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1d..XU0BxWZOAPOM Handout book

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									TABLE OF CONTENT
THE PRODUCTION/OPERATIONS FUNCTION IN BUSINESS ............................................................ 1
  A. TRUE/FALSE ....................................................................................................................... 1
  B. MULTIPLE CHOICES ............................................................................................................ 2
  C. FILL IN THE BLANKS AND CROSS-MATCH QUESTIONS....................................................... 9
  D. SHORT ANSWER ................................................................................................................ 10
  E. ESSAY TYPE QUESTIONS ................................................................................................... 10
PRODUCTIVITY .......................................................................................................................... 13
  A. MULTIPLE CHOICES .......................................................................................................... 13
  B. PROBLEMS......................................................................................................................... 14
FORECASTING ............................................................................................................................ 17
  A. MULTIPLE CHOICES ......................................................................................................... 17
  B. ESSAY QUESTIONS ........................................................................................................... 17
  C. PROBLEMS ........................................................................................................................ 18
DECISION MAKING .................................................................................................................... 39
  A. TRUE / FALSE .................................................................................................................... 39
  B. QUESTIONS........................................................................................................................ 39
  C. PROBLEMS ........................................................................................................................ 39
INVENTORY CONTROL ............................................................................................................... 65
LINEAR PROGRAMMING ............................................................................................................. 71
  A. SIMPLEX METHOD ............................................................................................................ 71
  B. ASSIGNMENT METHOD ..................................................................................................... 84
  C. TRANSPORTATION METHOD.............................................................................................. 91
BREAK-EVEN ANALYSIS............................................................................................................ 91
ANSWERS TO SELECTED QUESTIONS .................................................................................... 83
  INTRODUCTION TO PRODUCTION/OPERATIONS MANAGEMENT ............................................. 95
     A. TRUE OR FALSE ............................................................................................................ 95
     B. MULTIPLE CHOICES ...................................................................................................... 95
     C. FILL IN THE BLANKS AND CROSS-MATCH QUESTIONS.................................................... 95
     D. SHORT ANSWERS .......................................................................................................... 96
     E. ESSAY TYPE QUESTIONS ................................................................................................ 96
  PRODUCTIVITY ...................................................................................................................... 99
     A. MULTIPLE CHOICES ....................................................................................................... 99
     B. PROBLEMS..................................................................................................................... 99
  FORECASTING ...................................................................................................................... 101
     A. MULTIPLE CHOICE ...................................................................................................... 101
     B. ESSAY ........................................................................................................................ 101
     C. PROBLEMS................................................................................................................... 102
  INVENTORY CONTROL ........................................................................................................... 93
  LINEAR PROGRAMMING ....................................................................................................... 127
     A. SIMPLEX METHOD ...................................................................................................... 127
TABLES AND FORMULAS ................................................................................................... 161
                    Introduction to Production / Operations Management


THE PRODUCTION/OPERATIONS FUNCTION IN BUSINESS
A. TRUE/FALSE
1. Production/operations Management refers to creation of goods whereas production refers to the
    creation of services.
2. All organisations, including service organizations such as banks and educational institutions, have
    a production function.
3. Production is a creation of goods and services.
4. W. Edwards Deming is known as the Father of Scientific Management.
5. Lillian Gilbreth is credited for the early popularization of interchangeable parts.
6. The person most responsible for initiating use of interchangeable parts in manufacturing was
    Whitney Houston.
7. The origins of the scientific management movement are generally credited to James Taylor.
8. Operations Management is the set of the activities that create goods and services by transforming
    inputs into outputs.
9. Operations Management only applies to the creation of tangible goods.
10. An example of a “hidden‟ production function is money transfers at banks.
11. Operations management has benefited from advances in other fields of study.
12. In order to have a career in operations management, one must have a degree in statistics or
    quantitative methods.
13. The operations manager performs the management activities of planning, organizing, staffing,
    leading, and controlling of the POM function.
14. “Should we make or buy this item?” is within the Human Resources and Job Design critical
    decision area.
15. Marketing is one of the three functions critical to an organization‟s survival.
16. Students wanting to pursue a career in operations management will find multi-disciplinary
    knowledge beneficial.
17. The quality of a product is more difficult to measure than that of a service.
18. Consumer interaction is often high during the manufacturing process.
19. A company is considered excellent only if it is the best in its business.
20. The three primary functions in a business organization are operations/production, finance, and
    marketing.
21. Business functions are autonomous, thus each function can set objectives without much
    coordination.
22. In batch manufacturing, a few or several products share the same production resources.
23. Productivity and quality are easier to measure in manufacturing operations than in service
    operations.
24. Since customers are present in all service operations, service operations can provide only custom
    services.
25. Batch manufacturing must be capable of performing a wider variety of tasks as compared to job
    shop manufacturing.
26. A project for a service organization might be development of a computer software package.
27. There is a clear dividing line between manufacturing operations and service operations.
28. Specialization means each component of a product is fashioned to fit that particular item and
    should not fit any other item.
29. Because of the use of specialization, the industrial revolution brought about the need for a less
    formal procedure and a less sophisticated method of management.
30. Management Science (because of its use of mathematical theory) is the same as scientific
    management.
31. The primary difference between Taylor‟s study of management and Fayol‟s is that Fayol‟s was a
    top-down approach, with emphasis on overall administration, whereas Taylor‟s study was a
    bottom-up approach, with an emphasis on shop management.
32. The Industrial Revolution began in Japan.



Prof.Dr.Dr.M.Hulusi DEMIR                                                                           1
                     Introduction to Production / Operations Management


33. According to Adam Smith, specialization was more likely to lead to the development of
    mechanical devices to assist operations.
34. Sergio Farmerson‟s success is attributable to the use of specialization and interchangeable parts.
35. In using F. Taylor‟s scientific management, a duty of management was to select the best worker
    for a job, so that not much time or money need be spent on training.
36. Quality is easier to measure in a service organization.
37. An organization‟s mission statement is its broad statement of purpose.
38. Once an organization‟s mission has been decided upon, each functional area within the firm
    determines its own supporting mission.
39. Operations strategies are implemented in the same way in all types of organizations.
40. An organization‟s behaviour will be optimized if each of its departments optimizes their
    behaviours independently.
41. Top-level managers usually define the missions of each functional area, and then merge these
    missions to define the mission of the organization.
42. Strategies are mostly the same from one manufacturing company to another.
43. An organization‟s mission and its strategy are basically the same thing.
44. An organization‟s mission statement provides a plan of action.
45. An organization‟s strategy provides the purpose of the organization.
46. Differentiation, cost, and response are the three strategies for achieving competitive advantage.
47. An organization‟s ability to generate unique advantages over competitors is central to a successful
    strategy implementation.
48. Errors made within the location decision area may overwhelm efficiencies in other areas.
49. Decisions regarding quality are among the core decisions of POM.
50. Decisions regarding the location are among the core decisions of POM.
51. In order to maintain focus, an organization‟s strategy must not change during the product‟s life
    cycle.
52. Opportunities and threats are classified as internal factors of strategy development.
53. Strategies change because an organization‟s internal strengths and weaknesses may change.
54. The operations function is most likely to be successful when the operations strategy is integrated
    with other functional areas.
55. For the greatest chance of success, an organization‟s POM strategy must support the company‟s
    strategy.
56. Taylor‟s shop system was directed principally at improving the performance of top managers.
57. Time study, motion study and work sampling were all important techniques in scientific
    management.
58. Most of the techniques and approaches of scientific management eventually were developed into
    the modern field of industrial engineering.
59. New P/O Management computer applications today are in the areas of payrolls, billings, cost
    reports and inventory transactions.
60. Production functions are usually called manufacturing departments in manufacturing firms and
    operations departments in retailing and tracking firms.

B. MULTIPLE CHOICES
1. Which of the following is NOT a major activity of operations in supporting company
   success?
    a. provide products/services suited to the company‟s capabilities.
    b. produce product with consistent quality level.
    c. minimize cost.
    d. provide a product/service which has sufficient market.
2. Operations are concerned with ___________while marketing is concerned with____________.
    a. demand, quality
    b. efficiency, cost
    c. supply, demand
    d. demand, supply


2
                    Introduction to Production / Operations Management


3. Job shops are
     a. the same as batch.
     b. do not relate to service operations.
     c. often have large percentages of their inventory as work in process.
     d. are generally set up for repeat business.
4. Common characteristics of operations do NOT include
     a. fixed output capacity
     b. continuous improvement
     c. feedback from the pool of customers and potential customers
     d. the need to obtain inputs
5. The achievement of high quality is most closely related to ________________.
     a. repetitive operations.
     b. design specifications
     c. service operations
     d. customer needs.
6. The transformation of a set of inputs into a set of outputs is a characteristic of
     a. universities.
     b. prisons
     c. automobile assembly plants
     d. all of the above
 7. Services such as a chartering a bus or repairing an automobile are similar to the following
     a. project operations
     b. batch operations
     c. job shop operations
     d. productivity
8. All of the following are differences between manufacturing and service operations EXCEPT
     a. quality is more easily measured in service operations.
     b. productivity is easier to measure in manufacturing operations
     c. contact with customers is more prevalent with persons working in service operations.
     d. accumulation or decrease in inventory of finished products is more applicable to manufacturing
     operations.
9. According to Adam Smith, which of the following was NOT an advantage of specialisation of
     labour?
     a. rapid development of dexterity
     b. saving time in task shifts
     c. division of work between management and workers
     d. development of mechanical devices
10. Who of the following is NOT associated with scientific management
     a. Frederick W. Taylor
     b. Henry L. Gantt
     c. Elton Mayo
     d. Henry R. Towne
11. Lillian and Frank Gilbreth are responsible for principles of
     a. sociotechnical systems
     b. zero inventory
     c. motion study
     d. interchangeable parts
12. The principles of scientific Management included
     a. the rise of the service sector.
     b. increased motivation through additional employee fringe benefits
     c. the implementation of the 44 hrs. work week
     d. development of cooperation between management and production workers.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                            3
                     Introduction to Production / Operations Management


13. Which of the following is least related to the management science era
     a. efficiency experts
     b. operational research
     c. optimum solution
     d. statistical theory
14. POM is applicable
     a. mostly to the service sector
     b. mostly to the manufacturing sector
     c. to manufacturing and service sectors
     d. to services exclusively
     e. to the manufacturing sector exclusively
15. The person most responsible for popularizing interchangeable parts in manufacturing was
     a. Eli Whitney
     b. Whitney Houston
     c. Sergio Farmerson
     d. Lillian Gilbreth
     e. Frederick Winslow Taylor
16. The “Father of Scientific Management” is
     a. Frank Gilbreth
     b. Frederick W. Taylor
     c. W. Edwards Deming
     d. Walther Shewhart
     e. Just a figure of speech, not a reference to a person
17. Walter Shewhart is listed among the most important people of POM because of his
     contributions to
     a. assembly line production
     b. measuring productivity in the service sector
     c. statistical quality control
     d. Just-in-Time inventory methods
     e. Lean production and MRP I and MRP II
 18. Henry Ford is noted for his contributions to
     a. quality control
     b. assembly line operations
     c. scientific management
     d. standardization of parts
     e. time and motion studies
 19. Taylor and Deming would have both agreed that
     a. EMU is one of the best universities in the world
     b. Management must do more to improve the work environment and its processes so that quality
        can be improved
     c. Eli Whitney was an important contributor to statistical theory
     d. Productivity is more important than quality
     e. The era of POM will be succeeded by the era of scientific management
 20. Which one of the following statements is TRUE?
     a. The person most responsible for initiating use of interchangeable parts in manufacturing was Eli
        Whitney
     b. The person most responsible for initiating use of interchangeable parts in manufacturing was
        Whitney Houston
     c. The origins of management by exception are generally credited to Enrique Iglesias
     d. The origins of the scientific management are generally credited to James Taylor
     e. All of the above statements are TRUE
21. The field of POM is shaped by advances in which of the following fields?
     a. industrial engineering and management science               b. biology and anatomy
     c. information sciences                                        d. chemistry and physics
     e. ecology and zoology

4
                  Introduction to Production / Operations Management


22. The responsibilities of Production and operations manager include
    a. planning, organizing, staffing, procuring, and reviewing
    b. planning, organizing, staffing, leading and controlling
    c. forecasting, designing, accounting and financing
    d. marketing, selling, advising and auditing
    e. none of the above
23.Which of the following is not an element of the management process?
    a. staffing
    b. planning
    c. controlling
    d. leading
    e. pricing
24. Which of the following is TRUE about business strategies?
    a. All firms within an industry will adopt the same strategy.
    b. Well defined missions make strategic development much easier.
    c. Strategies are formulated independently of SWOT analysis.
    d. An organization should stick with its strategy for the life of the business.
    e. Organizational strategies depend on the knowledge given in EMU.
25. Which of the following statements about organizational missions is FALSE?
    a. They reflect a company‟s purpose.
    b. They indicate what a company intends to contribute to society.
    c. They define a company‟s reason for existence.
    d. They provide guidance for functional area missions.
    e. They are formulated after strategies are known.
26. Which of the following activities takes place once the mission has been developed?
    a. The firm develops alternative or back-up missions in case the original mission
       fails.
    b. The functional areas develop their functional area strategies.
    c. The functional areas develop their supporting missions.
    d. The ten POM decision areas are prioritized.
    e. None of the above.
27. The fundamental purpose for the existence of any organization is described by its
    a. Policies           b. Strategy              c. Bylaws
    d. Procedures         e. Mission
28. Which of the following is true? The impact of strategies on the general direction and
       basic character of a company is
    a. long range                  b. Short ranged         c. Minimal
    d. medium range                e. Temporal
29. Which of the following is true?
    a. corporate strategy is shaped by functional strategies
    b. corporate mission is shaped by corporate strategy
    c. functional strategies are shaped by corporate strategy
    d. external conditions are shaped by corporate mission
    e. corporate mission is shaped by functional strategies
30. The fundamental purpose of an organization‟s mission statement is to
    a. define the organization‟s purpose in the society
    b. define the operational structure of the organization
    c. generate good public relations for the organization
    d. define the functional areas required by the organization
    e. create a good human relations climate in the organization
31. Which of the following is not a key way in which business organizations compete with one
    another?
    a. production cost b. Product duplication                        c. Flexibility
    d. quality            e. Time to perform certain activities



Prof.Dr.Dr.M.Hulusi DEMIR                                                                 5
                    Introduction to Production / Operations Management


32. A strategy is
    a. a broad statement of purpose
    b. a simulation model used in TT classes
    c. a plan for cost reduction
    d. an action plan to achieve the mission
    e. to persuade parents for a new car.
33. Which of the following is not an operations strategy?
    a. Response                             b. Low cost
    c. Differentiation                      d. Technology
    e. all the above are operations strategies
34. Henry Ford is noted for his contributions to
    a. Prof.Demir‟s POM courses and TT‟s MIS presentations
    b. Quality control
    c. Assembly line operations
    d. Interchangeable parts
    e. Time and motion studies
35. Which one of the following is not typical question dealt with operations managers?
    a. how much capacity will be needed in months ahead?
    b. What is s satisfactory location for a new facility?
    c. Which products/services should be offered?
    d. How to motivate employees?
    e. All are typical of operations management.
36. Which one does not use operations management?
    a. a CPA firm                  b. a bank                 c. a hospital
    d. a supermarket               e. they all use.
37. Which one is not generally considered to be an advantage of using models for decision
      making?
    a. Providing a systematic approach to problem solving
    b. Emphasizing quantitative information
    c. Providing an exact representation of reality
    d. Enabling managers to answer “what if” questions
    e. Requiring users to be specific about objectives
38. Which came last in the development of manufacturing techniques?
    a. Lean production          b. Division of labour
    c. Mass production          d. Craft production           e. Interchangeable parts
39. If inputs decrease while output remains constant, what will happen to productivity?
    a. It will increase            b. It will decrease          c. It will remain the same
    d. It is impossible to tell    e. It depends on which inputs decreases
40. The foremost pioneers in scientific management are
    a. Ikujiro Monako, Hitotaka Takeuchi, Yotaro Kobayashi, Yuhua Cui
    b. M. Hulusi Demir, Tayfun Turgay, Serhan Ciftcioglu, Ilhan Dalci
    c. Chris Argyris, K. Imai, Elton Mayo, F.J. Roethlisberger, Herbert Simon
    d. Jay Heizer, Barry Render, Hamdy Taha, Richard Levin, Howard J. Weiss
    e. Frederick W. Taylor, Frank B. Gilbreth, Henry L. Gantt, Carl G. Barth, Henry Ford.
41. The scientific study of work
    a. applies the scientific method of the management of work
    b. has in some cases been misapplied by management.
    c. can be reconciled with a modern socio-technical approach.
    d. all of these.
42. The differences between the actual demand for a period and the demand forecast for that period is
    called:
    a. Forecast error                                b. weighted arithmetic mean
    c. Decision process.                             d. Mean square error
    e. Bias



6
                    Introduction to Production / Operations Management


43. All of the following decisions fall within the scope of operations management EXCEPT for
    a. Financial analysis
    b. Design of products and processes
    c. Location of facilities
    d. Quality management
    e. Facility Management
44. Which is not a discipline used by the production/operations function?
    a. Economics
    b. General management principles
    c. Quantitative analysis
    d. None of these
45. The industrial revolution:
    a. fostered the domination of manufacturing over service organizations
    b. substituted manpower for machine power
    c. came about through the efforts of F.W.Taylor
    d. has continued application in the service industries.
46. Harris‟ EOQ, Shewhart‟s quality control approach, and Dantzig‟s simplex method are examples
    of;
    a. mathematical decision making models
    b. linear programming
    c. computer systems
    d. accurate analysis
47. Computers serve Production/Operations Management by;
    a. eliminating clerical processing
    b. reducing need for the middle managers
    c. allowing use of sophisticated mathematical models
    d. all of these
    e. none of these
48. A productive systems approach;
    a. views production/operations as a separate organizational function
    b. must provide feedback information for control of process inputs and technology
    c. is of limited use in service organizations
    d. disregards human and social concerns
49. A service organization;
    a. is relieved of workforce decisions by marketing function
    b. falls at the extreme end of the goods-services continuum
    c. is faced with a highly perishable product that can‟t be stored in inventory
    d. all of these
50. Which of the following is not a characteristic of most service system?
    a. product is tangible
    b. quality of output can be highly variable
    c. production and consumption occur simultaneously
    d. no finished goods inventory is accumulated
    e. mark this answer if all the above are service system characteristics
51. The scientific management era spanned approximately what time period?
    a. 1945-present                b. 1640-1840                       c. 1875-1925
    d. 1776-1865                   e. none of the above
52. Frederick Winslow Taylor is called;
    a. father of operations research                b. father of scientific management
    c. father of industrial engineering             d. b and c                 e. none of the above.
53. P/O managers closed view of their external environments provide their organizations with
    a. adaptability                        b. growth                  c. efficiency
    d. all of the above                    e. none of the above




Prof.Dr.Dr.M.Hulusi DEMIR                                                                              7
                     Introduction to Production / Operations Management


54. P/O managers rely heavily on computers in their decision making because
    a. short planning horizon                 b. optimal goals        c. a and b
    d. open view of external environment                              e. all of the above.
55. Which phrase best describes the term “Production Management”?
    a. has evolved from terms like manufacturing management
    b. is concerned primarily with marketing and public relations
    c. is restricted to activities in profit making organizations
    d. does not extend to service activities.
56. Which of the following are not inputs into the production process?
    a. time        b. energy         c. labour        d. materials    e. finished goods
57. Which of the following are ways of classify services?
    a. labour intensity
    b. customer contact
    c. vendor relationship
    d. extent of customisation
    e. vertical integration.
58. Which of the following is not a way of organising a production process?
    a. continuous flow               b. job shop               c. repetitive flow
    d factory                        e. batch process
59. High-contact services:
    a. usually involve the customer in the execution of the process
    b. have limited uncertainty in customer arrival rates
    c. require extensive technical training for service personnel
    d. have high variability in customer requirements
    e. lend themselves to appointment system.
60. During the mass-production era of operations:
    a. standardisation of production was possible
    b. high-volume production was possible
    c. high-volume, standardised production was possible
    d. work was largely based on multi-skilled artisans
    e. intensive training was required.
61. Operations management is concerned with production and distribution of:
    a. products and services         b. products and goods            c. components and products
    d. goods and services            e. components and services       f. none of the above.
62. The person who developed the economic order quantity model was:
    a. Walter Shewhart               b. George Dantzig                c. Frederick W. Taylor
    d. Henry Gantt                   e. Ford Harris                   f. Henry Fayol
63. The founder of the scientific management movement was:
    a. Frank Gilbreth                b. Walter Shewhart               c. Frederick W.Taylor
    d. Ford Harris                   e. Henry Gantt                   f. Lillian Gilbreth
64. The Hawthorne Studies stimulated the development of:
    a. the scientific management movement
    b. the human relations movement
    c. the socio-technical movement
    d. the lean production movement.
65. Walter Shewhart developed:
    a. the economic order quantity model
    b. the human factors engineering field
    c. linear programming models
    d. statistical quality control techniques
    e. operations sequencing charts.
66. The moving assembly line was developed by:
    a. Elton Mayo                    b. Frederick W. Taylor           c. Clark Gable
    c. Eli Whitney                   d. Henry Ford                    e. Ray Charles



8
                   Introduction to Production / Operations Management


C. FILL IN THE BLANKS AND CROSS-MATCH QUESTIONS

1. _____________________is the set of activities that transforms inputs into goods and services
2. Operations is concerned with ______________ while marketing is concerned
    with______________________.
3. The achievement of high quality is most closely related to _____________ ____________.
4. Lillian and Frank Gilbreth are responsible for principles of _________ _________.
5. Adam Smith‟s idea to increase productivity a system of specialisation or a division of labour
   included:
    i.
    ii.
    iii.
6. Henry Ford‟s focus was largely on manufacturing efficiency.
    a.
    b.
    c.
7. Match this list of contributions with the originator
    a. father of scientific management             1. Henry Ford
    b. motion study principles                     2. Henry Gantt
    c. human relations movement                    3. Frank Gilbreth
    d. division of labour                          4. Adam Smith
    e. a few factors are important                 5. Elisabeth Taylor
    f. charts for planning and scheduling          6. Vilfred Pareto
    g. Total Quality Management                    7. Whitney Houston
8. Match each pioneer with appropriate description
    a. Henry Gannt                                 i. mass production and the moving assembly line
    b. F.W. Taylor                                 ii. interchangeable parts
    c. Frank Gilbreth                              iii. father of scientific management
    d. Henry Ford                                  iv. Motion study principles
    e. Eli Whitney                                 v. Charts used for scheduling
9. Match each pioneer with the appropriate description
    a. Richard Trevitchick                         i. Total Quality management
    b. Henry Gantt                                 ii. First train
    c. F.W. Taylor                                 iii. Mass production and moving assembly line
    d. Frank &Lillian Gilbreth                     iv. Motion study principles
    e. Henry Ford                                  v. Charts used for scheduling
    f. Sergio Bauersohn                            vi. First Quantitative Approach formulas
10. Match this list of contributions with the originator
    a. father of scientific management             1. Henry Ford
    b. motion study principles                     2. Henry Gantt
    c. human relations movement                    3. Frank Gilbreth
    d. division of labour                          4. Adam Smith
    e. a few factors are important                 5. Frederick W. Taylor
    f. charts for planning and scheduling          6. Vilfred Pareto
    g. Total Quality Management                    7. Yuhua Cui
                                                   8. Sergei Bauersohn




Prof.Dr.Dr.M.Hulusi DEMIR                                                                       9
                      Introduction to Production / Operations Management


D. SHORT ANSWER
1. List three primary functions of a business
2. State five reasons for the claim that service sector productivity is difficult to improve.
3. How do services differ from goods? List five ways.
4. List five elements of the management process.
5. According to the textbook, why should you study POM?

E. ESSAY TYPE QUESTIONS
1. What are the four major improvements in management history? Briefly describe the emphasis or
    concerns of each..
2. Discuss four conditions or changes that will continue to affect operations managers.
3. Discuss three major changes in organizations caused by the information age and reduced trade
    barriers.
4. Discuss the differences between manufacturing and service operations.
5. Identify the duties of management and indicate what management tries to do in performing these
    duties.
6. Briefly state the relative importance of technical competence and behavioural competence of
    managers.
7. Distinguish between repetitive production and batch production
8. Diagram the operations function or production system (transformation process.)
9. Explain the advantages of the division of labour, as noted by Adam Smith in “Wealth of Nations”.
10. According to Frederick Winslow Taylor, what are the four major duties of management?
11. Describe how an organization‟s mission and strategy have different purposes
12. What are the THREE conceptual ways to compete advantage proposed by the authors of your text-
    book Heizer and Render?
13. Classify the problems of management in the POM function.
14. Prepare a table showing the continuum of characteristics (differences between) services producer
    and goods producer.
15. Classify and explain briefly the types of production in two traditional ways? (If possible support
    your explanation with a diagram)
16. What examples of pure service can you identify? What is being transformed in each of these
    service processes?
17. What are the differences among Pure Service, Quasi Service and Manufacturing operations from a
    customer‟s point of view? From the operation‟s point of view?
18. Why was scientific management in the early 1900s aimed at the shop level?
19. Who were the foremost pioneers in scientific management, and what were their contributions?
20. In what ways is management of production/operations different from executive management?
21. Which event at about 1776 was especially significant in the development of industry?
22. Describe how the concept of division of labour applies to the following situations:
    a. university teaching                b. accounting
    c. the construction trades            d. a fast-food restaurant
23. Using the history of production management, what approaches have been used to improve
    productivity over the last century? Can these same approaches be used to improve productivity in
    today‟s world and in the future?
24. For the organizations listed below, describe the inputs, the transformation process, and outputs of
    the productive system.
    a. a high school/university library              b. hotel               c. a small manufacturing firm
25. Explain how production activities fit into the cultural pattern of a society, that is where they belong
    and what they accomplish.
26. Which aspect, or principle, of Taylor‟s philosophy of scientific management corresponds most
    closely with some firm‟s efforts to improve the quality of work life today?
27. Identify different approaches to management and then define what you mean by the term
    “management”.


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                    Introduction to Production / Operations Management


28. Briefly describe the following terms:
    a. Production/Operations           b. Production/Operations Management
    c. System                         d. Pareto Phenomenon
    e. Division of Labour
29. Identify the three major functional areas of business organizations and briefly describe how they
    interrelate.
30. List the important differences between goods production and service operations.
31. Briefly discuss each of these terms related to the historical evolution of POM.
    a. Industrial revolution          b. Scientific Management
    c. Interchangeable parts          d. Division of labour
32. Is McDonald‟s a service operation, a manufacturing organisation, or both?
33. Briefly describe the term “Production/Operations Management”. Describe also the
    production/operations function and the nature of production/operations manager‟s job?
34. List the key ways that organisations compete.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                         11
                    Introduction to Production / Operations Management


PRODUCTIVITY
A. MULTIPLE CHOICES
1. Ahmet Uslu produces cast bronze valves on an assembly line. If 1600 valves are produced in an 8-
     hour shift, the productivity of the line is
     a. 1600 valves/hr
     b. 200 valves/hr
     c. 80 valves/hr
     d. 40 valves/hour
     e. 2 valves/hr
2. The ABC plant produces 500 cypress packing boxes in two 10-hour shifts. Due to higher
     demand, they have decided to operate three 8-hour shifts instead. They are now able to produce
     600 boxes per day. What has happened to production?
     a. it has increased by 50 sets/shift
     b. it has increased by 37.5 sets/hour
     c. it has increased by 20%
     d. it has decreased by 8.3%
     e. it has decreased by 9.1%
3. Productivity measurement is complicated by
     a. the competition‟s output
     b. the fact that precise units of measure are often unavailable
     c. stable quality
     d. the workforce size
     e. the type of equipment used
4. ABC Co. produces cast bronze valves on an assembly line, currently producing 1600 valves
    each 8-hour shift. If the productivity is increased by 10%, it would then be
     a. 1760 valves/hr
     b. 880 valves/hr
     c. 220 valves/hr
     d. 200 valves/hr180 valves/hr
5. The ABC Box plant produces 500 cypress packing boxes in two 10-hour shifts. The use of
     new technology has enabled them to increase productivity by 30%. Productivity is now
     approximately
     a. 32.5 boxes/hr
     b. 60 boxes/hr
     c. 65 boxes/hr
     d. 150 boxes/hr
     e. 300 boxes/hr
6. Productivity can be improved by
     a. increasing inputs while holding outputs steady
     b. decreasing outputs while holding inputs steady
     c. increasing inputs and outputs in the same proportion
     d. decreasing inputs while holding outputs steady
     e. all of the above
 7. Three commonly used productivity variables are
     a. quality, external elements, and precise units of measure
     b. technology, raw materials, and labour
     c. education, diet, and social overhead
     d. labour, capital and management
     e. quality of the student, efficiency of the student to work and money




Prof.Dr.Dr.M.Hulusi DEMIR                                                                       13
                     Introduction to Production / Operations Management


B. PROBLEMS
1. Suzan has a part-time “cottage-industry” producing seasonal plywood yard ornaments for resale at
   local craft fairs and bazaars. She currently works a total of 4 hours per day to produce 10
   ornaments.
   a. What is her productivity?
   b. She thinks that by redesigning the ornaments and switching from use of a wood glue to a hot-
      glue gun she can increase her production to 20 ornaments per day. What is her new
      productivity?
   c. What is her percentage increase (or decrease) in productivity?

2. Ahmet grows domatoes in his 100 by 100 meters garden. He then sells the crop at the local
   farmer‟s market. Two summers ago, he was able to produce and sell 1200 kgs of tomatoes. Last
   summer, he tried a new fertilizer that promised a 20% increase in yield. He harvested 1350 kgs.
   Did the fertilizer live up to its promise?

3. A company has asked YOU to evaluate the firm‟s productivity by comparing this year‟s
   performance with last year‟s. The following data are available:

             ______________Last Year This Year
             OUTPUT          10 500 units 12 100 units
             Labour Hours    12 000         13 200
             Utilities        7 600 MU       8 250 MU
             Capital         83 000 MU      88 000 MU

     Has the company improved its PRODUCTIVITY during the past year?

4. A firm cleans chemical tank cars in the Bay Gazimagusa area. With standard equipment, the firm
   typically cleaned 60 chemical tank cars per month. They utilized 10 gallons of solvent, and two
   employees worked 20 days per month, 6 hours a day.
   The company decided to switch to a larger cleaning machine. Last February, they cleaned 60 tank
   cars in only 15 days. They utilized 12 gallons of solvent, and two employees worked 6 hours a
   day.
   a. What was their productivity with the standard equipment?
   b. What is their productivity with the larger machine?
   c. What is the change in productivity?

5. Serra‟s Ceramics spent 3 000 MU on a new kiln last year, in the belief that it would cut energy
   usage 25 % over the old kiln. This kiln is an oven that turns “greenware” into finished pottery.
   Serra is concerned that the new kiln requires extra labour hours for its operation. Serra wants to
   check the energy saving of the new oven, and also to look over other measures of their
   productivity to see if the change really was beneficial.

     Serra has the following data to work with:

                                                  Last Year This Year
                 Production (finished units)         4000         4000
                 Greenware (pounds)                  5000         5000
                 Labour (hrs)                          350          375
                 Capital (MU)                       15000        18000
                 Energy (kWh)                        3000          2600

     Were the modifications BENEFICIAL?




14
                     Introduction to Production / Operations Management


6. The Cool-Tech Co. produces various types of fans. In May, the company produced 1728 window
   fans at a standard price of 40 MU. The Co. has 12 direct labour employees whose compensation
   (including wages and fringe benefits) amounts to 21 MU/hour. During May, window fans were
   produced on 9 working days 9of 8 hours each), and other products were produced on other days.
   a. Determine the productivity of the window fans.
   b. In June, the Cool-Tech Co. produced 1 730 fans in 10 working days. What is the percentage in
       labour productivity of windows from May?

7. Mr. Ilhan DALCI makes billiard balls in his Beyarmudu plant. With a recent increase in taxes, his
   costs have gone up and he has a newfound interest in efficiency. Mr.Dalci is interested in
   determining the productivity of his organisation. He would like to know if his organisation is
   maintaining the manufacturing average of 3% increase in productivity. He has the following data
   representing a month from last year and an equivalent month this year.
               __________________Last year               Now

                Units produced         1 000              1 000
                Labour (hours)           300                275
                Resin (kg.s)              50                 45
                Capital invested (MU) 10 000             11 000
                Energy (BTU)           3 000              2 850

   Show the productivity change for each category and then determine the IMPROVEMENT for
    labour- hrs, the typical standard for comparison.

8. Ilhan‟s, a local bakery, is worried about increased costs – particularly energy. Last year‟s records
    can provide a fairly good estimate of the parameters for this year. Ilhan Balci, the owner, does not
    believe things have changed much, but he did invest an additional 3 000 MU for modifications to
    the bakery‟s ovens to make them more energy efficient. The modifications were supposed to make
    the ovens at least 15 % more efficient. I. Balci has asked you, as a brilliant graduate of EMU, to
    check the energy savings of the new ovens and also look over other measures of the bakery‟s
    productivity to see if the modifications were beneficial. You have the following data to work with:

                                                 Last Year               Now
                Production (dozen)                1 500                  1 500
                Labour (hours)                      350                    325
                Capital Investment (MU)          15 000                 18 000
                Energy (kw-hrs)                   3 000                  2 750

9. Haldun LOP, the production manager of LOP Chemicals, in Gazimagusa, TRNC, is preparing his
   quarterly report which is to include a productivity analysis for his department. One of the inputs is
   production data prepared by Meltem SERIN, his operation analyst. The report, which she gave him
   this morning, showed the following:
                                                          2005                    2006
                Production (units)                        4 500                   6 000
                Raw Material Used (barrels of
                Petroleum by-products)                      700                     900
                Labour Hours                     `       22 000                  28 000
                Capital Cost applied to the
                Department (MU)                         375 000                 620 000
   Haldun LOP wondered if his productivity had increased at all. He called Meltem into his office and
   conveyed the above information to her and asked her to proceed with preparing this part of the
   report. (Include your interpretations for each productivity figure)




Prof.Dr.Dr.M.Hulusi DEMIR                                                                            15
                     Introduction to Production / Operations Management


10. A Turkish manufacturing company operating a subsidiary in TRNC shows the following results:

                                                  TURKEY                   TRNC
                Sales (in units)                  100.000                  20.000
                Labour (hours)                     20.000                  15.000
                Raw materials (in MU)              20.000                   2.000
                Capital Equipment (hrs)            60.000                   5.000

     a. Calculate single factor productivity figures of labour and capital for the parent and subsidiary.
        Do the results seem misleading?
     b. Now compute multi-factor labour and capital productivity figures. Are the results better?
     c. Finally, calculate raw material productivity figures. Explain why these figures might be
        greater in TRNC.

11. Ahmet Uslu makes wooden boxes in which to ship motorcycles. Ahmet and his three employees
    invest 40 hours per day making the 120 boxes.
     a. What is their productivity?
     b. Ahmet and his employees have discussed redesigning the process to
        improve efficiency. If they can increase the rate to 125 per day, what
        would be their new productivity?
     c. What would be their increase in productivity?

12. Magusa Metal Works produces cast bronze valves on an assembly line. On a recent day, 160
    valves were produced during an 8-hour shift. Calculate the productivity of the line.

13. Kleen Karpet cleaned 65 rugs in April, consuming the following resources:
       Labour: 520 hours at 13 MU/hour
       Solvent: 110 litres at 5 MU/litre
       Machine Rental: 20 days at 50 MU/day
    a. What is the labour productivity?
    b. What is the multifactor productivity?

14. Ilhan Dalci is president of Ilhandir Manufacturing, a producer of Go-Kart Tires. Dalci makes 1000
    tires per tires per day with the following resources:
       Labour: 400 hours at 12.50 MU/hr
       Raw Material: 20 000 kgs/day at 1MU/kg
       Energy: 5 000 MU/day
       Capital: 10 000 MU/day
    a. What is the labour productivity for these tires at Ilhandir Manufacturing?
    b. What is the multifactor productivity for these tires at Ilhandir Manufacturing?
    c. What is the percent change in multifactor productivity if Ilhandir can reduce energy bill by
       1000 MU without cutting production or changing any other inputs?




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                    Introduction to Production / Operations Management


FORECASTING
A. MULTIPLE CHOICES
1. A large value of “alpha” ( α ) puts more weight on
     a. recent             b. oder
2. If the data being observed can be best thought of as being generated by random deviations about a
     stationary mean, a
     a. arge               b. small
3. The Delphi Method
     a. relies on the power of written arguments.
     b. requires resolution of differences via face-to-face debate.
     c. is mainly used as an alternative to exponential smoothing.
     d. none of the above.
4. Qualitative forecasting methods include
     a. delphi             b. Panel of experts       c. trend adjusted exponential smoothing
     d. (a) and (c)        e. (a) and (b)            f. (b) and (c)
5. The method that considers several variables that are related to the variable being predicted is
     a. Exponential smoothing                b. Causal forecasying
     c. Weighed moving average               d. All of the above             e. None of the above
6. Exponential smoothing is an example of causal model
     a. true               b. False
7. With regard to a regression based forecast, the standard error of the estimate gives a measure of
     a. the overall accuracy of the forecast.
     b. the time period for which the forecast is valid.
     c. the time required to derive the forecast.
     d. the maximum error of the forecast.
     e. none of the above.

B. ESSAY QUESTIONS
1. Is there a difference between forecasting demand and forecasting sales?
2. Define the terms “Qualitative Methods”, “Trend Analysis Method (Time Series Method), and
    “Causal Forecast”. Describe the uses of them.
3. The manager of a local firm says “the forecasting techniques are more trouble than they are worth.
    I don`t forecast at all, and I`m doing 25% more business than last year”. Comment.
4. What do you see as the main problem with qualitative (judgmental) forecasts? Are they ever better
    than “objective” methods?
5. A firm uses exponential smoothing with a very high value of alpha. What does this indicate with
    respect to the emphasis if placed on past data.
6. Regression and correlation are both termed “causal” methods of forecasting. Explain how they are
    similar in this respect and also how they are different.
7. Describe briefly the steps to develop a forecasting system.
8. Describe briefly the “Delphi Method”.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                         17
                       Introduction to Production / Operations Management


C. PROBLEMS
1. A manufacturing company has monthly demand for one of its products as follows:

                 MONTH         DEMAND
                 January        520       Develop a three-period average forecast and a three
                 February        490      period weighted moving average forecast
                 March           550      weights of 5, 3 and 2 for the most recent demand
                 April           580      values, in that order. Indicate which forecast would
                 May             600      seem to be most accurate
                  June           420      Make a forecast of september by using both approaches.
                  July           510
                 August          610

2. A computer software firm has experienced the following demand for its “Personal Finance”
    software package.

                  Period           Units
                  1                56
                  2                61    Develop an exponential smoothing forecast using
                  3                55    an alpha value of 0.40
                  4                70
                  5                66
                  6                65
                  7                72
                  8                75

3. The head of Business Department at EMU wants to forecast the number of students who will enroll
    in production/operations management next semester in order to determine how many sections to
    schedule. The department has accumulated the following enrollment data for the past 8
     semesters.

                  Semester         Students enrolled in POM
                           1                80
                           2                90
                           3                70
                           4                84
                           5               100
                           6               115
                           7                98
                           8               130

     a.   Compute a 3-semester moving average forecast for semester 4 through 8
     b.   Compute the exponentially smoothed forecast (alpha=0.20) for the enrollment data.
     c.   Compare two forecasts and indicate the most accurate.
     d.   Make a forecast for the next semester (semester 9) with the most accurate approach.




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                       Introduction to Production / Operations Management


4. ABC Hardware handles spare parts for lawn-mowers. The following data were collected for
   one week in April when replacement for lawn-mower blades were in high demand.

                         Day           Demand
                          10             15
                          12             16
                          13             18
                          15             22
                          17             21
                          20             23
                          21             24

   Simulate a forecast using simple smoothing for the week, starting with F = 15 and alpha=0.2. Find
   also the forecast for the 8th day.


5. Fill in the blank places.

                               Quarter        Quantity
                2007             I                26
                                 II               38
                                 III              54
                                 IV               34
                 __________________________              Moving Totals
                  2008           I                34            160
                                 II               50            172
                                 III              58            176
                                 IV               38            180
                  2009           I                ___           190

                                 II               ___           197.2

                                 III              ___           204.4

                                 IV               ___           211.6




Prof.Dr.Dr.M.Hulusi DEMIR                                                                         19
                      Introduction to Production / Operations Management


6. Using total moving average method to forecast the quarterly values of 2007.

                        Years           Quarters          Sales (million bottles)
                         2007                I                     18.2
                                             II                    29.2
                                            III                    22.2
                                            IV                     17.4
                         2008                I                     19.2
                                             II                    30.8
                                            III                    24.2
                                            IV                     18.2
                         2009                I                     21.6
                                             II                    33.2
                                            III                    26.2
                                            IV                     20.8

7. The general manager of a building materials production plant feels the demand for
   plasterboard shipments may be related to the number of construction permits issued in the
   municipality during the previous quarter. The manager has collected the data shown in the
   accompanying table.

                 Construction       Plasterboard
                   Permits            Shipments
                            15                 6
                             9                 4
                            40                16
                            20                 6
                            25                13
                            25                 9
                            15                10
                            35                16
     a. Find a regression forecasting equa
     b. Determine a point estimate for plasterboard shipments when the number of construction permits
        is 30.
     c. Given the data on permits and shipments, compute the standard deviation of regression.
     d. Find the prediction interval of 90%.(std.-t table)
     e. Find the prediction interval of 95.5% (normal) for the specific amount of shipments when the
        permits number 30. (for this part assume your regression equation has been derived from a
        sufficiently large sample that the prediction interval form equal to y+/-z.s may be used.)
     f. Determine r and coefficient of determination and interpret them.
     g. Test the correlation coefficient at 5% level of significance. Is the correlation
        coefficient significant at the level 5%?
     h. By using correlation coefficient analysis find the regression forecasting equation, and explain
        why this equation is different than the one you found in (a).




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                      Introduction to Production / Operations Management


8. ABC Hardware handles spare parts for lawn mowers. The following data were collected for
   one week in April when replacement lawn-mower-blades were in high demand. The firm also
   collected necessary data on the total sales dollars generated by the store. The manager of the
   store would like to know if the total sales are a good predictor of lawn-mower-blade
   sales.

                Day                 Demand for               Total sales
                                    Lawn-mowers           of the store(000MU)
                 1                     10                        10
                 2                     12                        13
                 3                     13                        14
                 4                     15                        16
                 5                     20                        19
                 6                     25                        20
                 7                     24                        20

  a. For the above data calculate the correlation coefficient between Demand for lawn-mower blade
      and Total sales of the store, and interpret the result.
  b. What percentage of variation in lawn-mower blade sales can be explained by total sales of the
      store?
  c. Test the correlation coefficient at 5% level of significance.
  d. Compute the forecast of 8th day total sales of the store.
  e. Using the forecast of total sales you found at (d), find the forecasted demand for lawn-mower
      blade sales for the same date with 90% probability.

9. Ali and Arzu are planning to set up an ice-cream stand in Laguna/Gazimagusa. After six months of
   operation, the observed sales of ice-cream (in MU) and the number of Laguna visitors are

            Month        Ice-cream sales (MU)       Laguna Visitors
                1               200                       800
                2               300                       900
                3               400                      1100
                4               600                      1400
                5               700                      1800
                6               800                      2000

    a. Determine a regression equation treating ice-cream sales as the dependent variable and Laguna
        visitors as the independent variable.
    b. If you expect the Laguna visitors to peak out at about 3000 persons next month, what would be
        the expected ice-cream sales?
    c. Express your forecast with 68.3% probability limits.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                           21
                      Introduction to Production / Operations Management


10. In a manufacturing process the assembly-line speed (meter/minute) was thought to affect the
    number of defective parts found during the inspection process. To test this theory, management
    devised a situation where the same batch (lot) of parts was inspected visually at a variety of line
    speeds. The following data were collected.

                  # of defective Line
                  parts found speed
                         21       20
                         19       20
                         15       40
                         16       30
                         14       60
                         17       40

     a. Develop the estimated Regression Equation that relates line speed to the number of defective
        parts found.
     b. Use the equation developed in part (a) to forecast the number of defective parts found for a line
        speed of 50 meters per minute.
     c. Express your forecast within 95.5% probability limits. (Assuming n is large)


11. Sergio‟s Restaurants collected the following data on the relationship between advertising
    and sales at a sample of five restaurants.


                   Advertising          Sales
                  Expenditures        (000 MU)
                   (000 MU)
                        1                 19
                        4                 44
                        6                 40
                       10                 52
                       14                 55

     a. Determine the strength of the causal relationship between advertising expenditures and sales of
        the restaurants and interpret the result.
     b. What is the coefficient of determination? What does it mean to you?
     c. Test the correlation coefficient you found in (a) at 5% level of significance. Is the correlation
        coefficient significant at this level?
     d. Using correlation coefficient find regression forecasting equation.




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                      Introduction to Production / Operations Management


12.
                  Year     Quarter        Demand (tons)
                  2007        I             105
                             II             150
                            III              93
                            IV              121
                  2008        I             140
                             II             170
                            III             105
                            IV              150
                  2009        I             150
                             II             170
                            III             110
                            IV              130

      Use Moving Totals to forecast the quarterly demand for the year 2010.

13. The data shown in the accompanying table include the number of lost-time accidents for the Izmir
    Lumber Company over the past 7 years. Some additional calculations are included to help you
    answer the following questions. Manager of the company uses the number of employees (in
    thousands) to predict the number of accidents.

                  YEAR        NO. OF   NO. OF
                            EMPLOYEES ACCIDENTS
                               (000)

                  2003            15              5                225         25      75
                  2004            12              20              144         400     240
                  2005            20              15              400          225    300
                  2006            26              18              676          324     468
                  2007            35              17              1225         289     595
                  2008             30             30              900          900    900
                  2009            37              35              1369        1225    1295
                  Totals          175             140               4939       3388    3873

      a. Use the normal equations to develop a linear regression equation for forecasting the
         number of accidents on the basis of the number of employees. State the equation. Use
         the equation to forecast the number of accidents when the number of employees is
         33(000).
      b. Assuming n is large, calculate the 95.5 percent confidence limits for the number of
         accidents when the number of employees is 33(000).
      c. What is the correlation coefficient between number of employees and the number of
         accidents? Interpret your result.
      d. What percentage of the variation in the number of accidents is explained by the
         employment level?
      e. Is the correlation significant at the 5% level?




Prof.Dr.Dr.M.Hulusi DEMIR                                                                         23
                      Introduction to Production / Operations Management


14. Kitchens of Tomorrow Inc. has collected the following data to learn if the number of building
    permits might be a useful predictor of their cabinet sales.

     BUILDING          CABINET
     PERMITS           SALES
      (00)             (000 MU)           a. Use the normal equations to derive a regression
                                              forecasting equation.
        2                3                b. Compute the standard deviation of regression
        5                5                c. Assume your regression has been derived from a
        1                5                   sufficiently large sample that the interval estimate
        2                6                   form equal to Y ±Z.Syx may be used.
        5                7                   Establish a 99.7% prediction interval estimate for
        4                6                   the specific amount of cabinet sales (000 MU)when
        3                5                   permits number 4.4(00).
        4                5                d. Compute the coefficient of correlation and explain
        1                3                   the meaning of it.
       27               45                e. Test the significance of r for 10% and n=9.
                                          f. Use the correlation coefficient formula to derive a
                                               regression forecasting equation.
                                           g. Is there any difference between the two equations
                                               you derived at a and f.

15. A company wants to develop a means to forecast its carpet sales. The store manager believes that
    the store‟s sales are directly related to the number of new housing starts in town. The manager has
    gathered data from Chamber of Commerce records of monthly house construction permits and
    from store records on monthly sales. These data as follows:

         Monthly Construction             Monthly Carpet
           Permits                        Sales (000 metres)
               42                                 20
               70                                 40
               20                                 16
               24                                 12
               32                                 32
               18                                  8
               82                                 48
               30                                 44
               36                                 36
               52                                 56

     a. Develop a linear Regression Model for these data and forecast carpet sales if 30
        construction permits for new homes are filed.
     b. Calculate the standard deviation of regression.
     c. State your forecast in the confidence limits of 90%.




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                      Introduction to Production / Operations Management


16. Demand for hockey skates at a local sports store for the past eight weeks has been

             Week               Demand
               1                  122
               2                  130
               3                   98
               4                  121
               5                   96
               6                  152
               7                  113
               8                  124

    Use a simple exponential smoothing model with alpha=0.6. Assume the forecast for Period 1 was
    120. Make a forecast for period 9.

17. A retail chain of eyewear specialist has been experimenting with sales price of contact lenses.
    The following data have been obtained.

         Average lenses        Price per
         per day_______        lens, MU
               200                24
               190                26
               188                27
               180                28
               170                29
               162                30
               170                32

    a. For the above data calculate the correlation coefficient between lens price and lens sales and
    interpret the result.
    b. What percentage of variation in lens sales can be explained by prices.
    c. Test the correlation coefficient at 5% level of significance.
    d. What is 95% confidence interval for demand at price 28 MU? (Hint: n=7)

18. Fill in the blank places

                         Year       Quarters     Demand(tons)
                         2007          I            105
                                       II           150
                                       III           95
                                       IV           120
                                                         Moving TOTALS
                         2008              I        150       515
                                           II       200       565
                                           III      125       595
                                           IV       175       650
                         2009              I       ____       690
                                           II      ____       733.5
                                           III     ____       777
                                           IV      ____       820.5




Prof.Dr.Dr.M.Hulusi DEMIR                                                                             25
                     Introduction to Production / Operations Management


19. Compute a forecast for the demand in each of the quarters of the following years, 2010.
      Year Quarter                     Demand
      2008       1                        92
                 2                        82
                 3                        84
                 4                        92
      2009       1                        90
                 2                        80
                 3                        82
                 4                        90

20. A company has collected the following data to learn if the number of building permits
    might be a useful predictor of their kitchen cabinet demand.
               Building permits           Cabinet Sales
               (00 MU) x                    (000 MU) y__
                        2                          6
                        5                         10
                        1                         10
                        2                         12
                        5                         14
                        4                         12
                        3                         10
                        4                         10
                        1                           6

     a. Use the normal equations to derive a regression forecasting equation.
     b. Compute the standard deviation of regression
     c. Assume our regression equation has been derived from a sufficiently large sample.
        Establish a 95.5% confidence limits estimate for the specific amount of cabinet sales (000 MU)
        when permits number is 4.4 (00).
     d. Find the prediction interval of 90%, when permits number is 4.4 (00).
     e. Determine r and interpret it.
     f. Determine coefficient of determination and interpret it.
     g. Test the correlation coefficient at 5% level of significance.
     h. By using correlation coefficient analysis find the regression forecasting equation, and explain
        why this equation is different than the one you found in (a).

21. A company wants to develop a means to forecast its carpet sales. The store manager believes
    that the store‟s sales are directly related to the number of new housing starts in town. The
    manager has gathered data from Chamber of Commerce records of monthly house construction
    permits and from store records on monthly sales.

        Monthly Construction        Monthly Carpet
            Permits                 Sales (000 metres)
              42                         10
              70                         20
              20                          8
              24                          6
              32                         16
              18                          4
              82                         24
              30                         22
              36                         18
              52                         28



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                        Introduction to Production / Operations Management


    a. Develop a linear Regression Model for this data and forecast carpet sales if 27 construction
        permits for new homes are filed.
    b. Calculate the standard deviation of regression.
    c. State your forecast in the confidence limits of 95%.
    d. Determine r and interpret it.
    e. Determine the strength of the causal relationship between monthly sales and new home
        construction using correlation.
    f. Test the correlation coefficient at 10% level of significance.

22. Using total moving average method to forecast the quarterly values of 2010.

        Years Quarters           Sales
                            (million bottles)
        2007       I                 91
                  II               146
                 III               111
                 IV                  87
        2008       I                 96
                  II               154
                 III               121
                 IV                  91
        2009        I              108
                   II              166
                  III              131

23. In a manufacturing process the assembly-line speed (meter/minute) was thought to affect the
    number of defective parts found during the inspection process. To test this theory, management
    devised a situation where the same batch (lot) of parts was inspected visually at a variety of line
    speeds. The following data were collected.

         # of defective Line
         parts found     speed
                 22      20
                 20      20
                 18      40
                 19      30
                 15      60
                 20      40
    a. Develop the estimated Regression Equation that relates line speed to the number of defective
       parts found.
    b. Use the equation developed in part (a) to forecast the number of defective parts found for a line
       speed of 50 meters per minute.
    c. Express your forecast within 95.5% probability limits.

24. Room registrations in the Magusa Plaza Hotel have been recorded for the past nine years.
    Management would like to determine the mathematical trend of guest registration in order to
    project future occupancy. This estimate would help the hotel management to determine whether a
    future expansion will be needed. Given the following time-series data, develop a trend equatin
    relating to registrations to time.
    Then,
    a. Forecast next year‟s registrations.
    b. Give your next year‟s forecast with 95% probability (i.e. assuming the level of significance is
       equal to 5%)
    c. Assuming n is large (i.e. n≥30), show your confidence limits for the next year with %95.5
       probability.

Prof.Dr.Dr.M.Hulusi DEMIR                                                                            27
                      Introduction to Production / Operations Management



                  Years    Registrants(000)
                  2001           17
                  2002           16
                  2003           16
                  2004           21
                  2005           20
                  2006           20
                  2007           23
                  2008           25
                  2009           24


25.             Time    1 2 3 4 5 6 7 8 9 10 11 12
                Demand 10 14 19 26 31 35 39 44 51 55 61 54

      a. Use a simple four-period moving average to forecast the demand for periods 5-13.
      b. Find the mean absolute devaiation (average error).
      c. Use a four-period moving average with weights 4,3,2 and 1 to forecast demand for time13.
      d. Assume F1 = 8 and α = 0.3 . Use an exponential smoothing factor to forecast demand in periods
          2-13.
      e. Find the mean absolute deviation of exponential smoothing.
      f. Compare the above methods. Which one you prefer? Why?
      g. Repeat the analysis using alpha = 0.5.
      h. If you were to use an exponential smoothing model to forecast this time series, would you
          prefer alpha = 0.3, a larger (α≥0.3), or smaller (α≤0.3) value of alpha? Why?

26.
                          Year    Quarter    Demand for
                                             fertilizer (tons)
                          2007       I            50
                                     II           73
                                    III           45
                                    IV            60
                          2008        I           71
                                     II          85
                                    III           50
                                    IV            61
                          2009        I           71
                                     II           80
                                    III           55
                                    IV            65
      a. Compute a three-quarter moving average forecast. Compute also the forecast error for each
          quarter.
      b. Compute the quarterly forecasted demand for the year 2010.




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                     Introduction to Production / Operations Management


27. The manager of Magusa Transport Co. wishes to forecast the number of miles driven by his trucks
    for the coming three years.
                 Years Thousands of
                          Miles driven
                  2004          22
                  2005          24
                  2006          34
                  2007          30
                  2008          40
                  2009          50

    a. Compute the forecast of miles driven for the next three years (2010, 2011 and 2012)
    b. Give your forecast for the year 2007 with %95 probability (i.e. assuming the level of
        significance is equal to %5)
    c. Assuming n is large (i.e. n≥30), show your confidence limits for the year 2008 with %68.3
        probability.

28. November             Demand
      10                  20
      11                  28
      12                  38
      13                  52
      14                  62
      15                  70

    a. Use a simple 3-period moving average to demand for 13 November-15 November.
    b. Find the average error for that period.
    c. Assume that F1=24 and α= 0.6. Use an exponential smoothing method to forecast demand in
        periods 11 November-15 November. Find the average error.
    d. Compare the methods and state which one you prefer and why?

29. The monthly sales for Telco Batteries Inc., were as follows:
               Month            Sales
               January          20
               February         21
               March            15
               April            14
               May              13
               June             16
               July             17
               August           18
               September        20
               October          20
               November         21
               December         23

    Forecast past sales using each of the following;
    a. A three-month moving average,
    b. a 6-month weighted average using 1,1,2,2,2, and 3 with the heaviest weights applied to the most
        recent months.
    c. Exponential smoothing using an α = 0.3 and a January forecast of 20.
    d. Which method you prefer and why?
    e. using the method you chose, forecast January sales of the coming year.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                          29
                      Introduction to Production / Operations Management


30. Dr. Alev Yakar, a Magusa psychologist, specializes in treating patients who are agoraphic (afraid
    to leave their homes). The following table indicates how many patients Dr. Yakar has seen each
    year for the past 10 years.

                          Year      No.of Patients
                          2000            36
                          2001            33
                          2002            40
                          2003            41
                          2004            40
                          2005            55
                          2006            60
                          2007            54
                          2008            58
                          2009            61

     a. Using trend analysis, predict the number of patients Dr. Yakar will see in years 2010 and 2011.
     b. What is the standard error of the forecasts?
     c. Forecast number of patients in 2007 at 5% level of significance.
     d. Assuming sample is large (i.e. n>30), state your forecast of 2007 within 95.5%confidence
         interval.

31. Data collected on the yearly demand for 50-kg bags of fertilizer at Ilhandir Garden Supply are
    shown in the table below.
                                 DEMAND FOR
                                 FERTILIZER
                     YEAR         (000 of BAGS)
                        1                 4
                        2                 6
                        3                 4
                        4                 5
                        5                10
                        6                 8
                        7                 7
                        8                 9
                        9                12
                        10               14
                        11               15

     a. Develop a three-year moving average to forecast sales.
     b. Develop a four-year moving average for demand for fertilizer.
     c. Estimate demand again with weighted three-year moving average in which sales in the most
        recent year are given a weight of 2 and sales in other two years are each given weight of 1.
     d. Three different forecasts were developed for the demand for fertilizer. These three
        forecasts are a three-year moving average, four-year moving average and a weighted moving
        average. Which one would you use and explain why?
     e. Use exponential smoothing with a smoothing constant of 0.3 to forecast the demand for
        fertilizer. Assume that last period‟s (year‟s) sales forecast for year 1 is 5 000 bags to begin the
        procedure.
     f. Would you prefer to use the exponential smoothing model or one of the above models. Explain
        your choice. And according to your choice forecast the year 12.




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                     Introduction to Production / Operations Management


32. Girne Manufacturing Company‟s demand for electrical generators over the period 2003 - 2009 is
    shown in table below.
                                 Electrical
                 Year            Generators
                                  Sold
                 2003               74
                 2004               79
                 2005               80
                 2006               90
                 2007              105
                 2008              142
                 2009              122

   a.   Develop a linear trend line by using the least squares method.
   b.   Estimate the demand in 2010 and 2011.
   c.   Calculate the standard error of the past record.
   d.   Give your forecast for the year 2011 at 5% level of significance.
   e.   Assume n is large (n>30), give your forecast for the year 2010 within 95.5%
        confidence interval.

33. The following gives the number of pints of type O (Rh+) blood used at Nalbantoglu Hospital       in
    the past 6 weeks:

                 Week of         Pints Used
                 August 4          360
                 August 11         389
                 August 18         410
                 August 25         381
                 September 1       368
                 September 8       374

   a. Forecast the demand for the week of September 15 using a 3-week moving average.
   b. Use a 3-week-weighted moving average, with weights of 1,3, and 6, using 6 for the most recent
      week. Forecast demand for the week September 15.
   c. Compute the forecast for the above data using exponential smoothing with a forecast for
      August 4 of 360 and α =0.2. Forecast the demand for the week of September15.
      (Show all your calculations and errors in tabular form.)

34. The manager of the Petroco Service Station wants to forecast the demand for unleaded gasoline
    next month so that the proper number of gallons can be ordered from the distributor. The owner
    has accumulated the following data on demand for unleaded gasoline from sales during the past
    10 months.
                        Gasoline
        MONTH           Demanded (gallons)
        November        800                   a. Compute an exponentially smoothed forecast
        December        725                       using α = 0.3 and F1 = 700.
        January         630                    b. Compute the error of each month and find the
        February        500                      average error for the past record.
        March           645                   c. Forecast the demand for the coming month
        April           690                       September.
        May             730
        June            810
        July            1200
        August          980


Prof.Dr.Dr.M.Hulusi DEMIR                                                                            31
                      Introduction to Production / Operations Management


35. Quarterly data for the failures of certain aircraft engines at a local military base during the last
     two years are

                     Quarters Engine failures
                         1                 200
                         2                 250
                         3                 175
                         4                 186
                         5                 225
                         6                 285
                         7                 305
                         8                 190

     a. Determine one-step-ahead forecasts for periods 4 and 8 using three-period moving averages
        method.
     b. Let us assume that the forecast for period 1 was 200. Also suppose that  = 0.1. Determine
        one-step-ahead forecasts for periods 2 and 8.
     c. Compare the above mentioned methods for the periods 4 and 8. Based on this comparison
        conclude which method is a superior method for the given series.

36. Bicycle sales at TT‟s Bikes are shown below.
                                        Actual
                         Week           Bicycle Sales
                         1                   8
                         2                  10
                         3                   9
                         4                  11
                         5                  10
                         6                  13

     a. Use 3-week moving average for forecasting week 4, week 5, week 6 and week 7.
     b. If
                                  Weights
                                  Applied       Period
                                        3       last week
                                        2       2 weeks ago
                                        1       3 weeks ago

        Forecast the weeks 4, 5, 6 and 7.
     c. Which method would you prefer and why?
     d. Use exponential smoothing to forecast bike sales. Assume that the forecast for
        Week 1 was 9 and α = 0.7.

37. The sales manager of a large apartment rental complex feels the demand for apartments may be
    related to the number of newspaper ads placed during the previous month. She has collected the
    data shown in the accompanying table.
                 Ads Purchased         Apartments leased
                        15                      6
                         9                      4
                        40                     16
                        20                      6
                        25                     13
                        25                      9
                        15                     10
                        35                     16

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                    Introduction to Production / Operations Management


    a. Find the mathematical equation by using the least squares regression approach.
    b. If the number of ads is 30, estimate the number of apartments leased.
    c. Given the data on ads and apartment rentals as above, compute the standard deviation of
        regression (Syx).
    d. Compute the correlation coefficient and interpret.
    e. Compute the determination coefficient and interpret.
    f. Test the hypothesis, i.e. r = 0, at 5% level of significance

38. Given below are 2 years of quarterly demand data for a particular model of personal computer
    from a local computer store.

                        Year    Quarter      Demand
                        2008       I           40
                                  II           46
                                 III           39
                                 IV            42
                        2009       I           44
                                  II           57
                                 III           43
                                 IV            45

    a. Deseasonalize the data with a moving total and compute a linear equation for the trend in
        demand.
    b. Using the trend you have developed, compute a forecast for the demand in each quarters of the
        following year.
39. Bus and subway ridership for the summer month in London, England, is believed to be tied heavily
    to the number of tourists visiting the city. During the past 12 years, the following data have been
    obtained.

                YEAR          NO. OF       RIDERSHIP
                           TOURISTS        (in millions)
                           (in millions)

                1998             7             1.5           49  2.25      10.5
                1999             2             1.0            4  1.00       2.0
                2000             6             1.3           36  1.69       7.8
                2001             4             1.5           16  2.25       6.0
                2002            14             2.5          196 6.25       35.0
                2003            15             2.7          225 7.29       40.5
                2004            16             2.4          256  5.76      38.4
                2005            12             2.0          144  4.00      24.0
                2006            14             2.7          196  7.29      37.8
                2007            20             4.4          400 19.36      88.0
                2008            15             3.4          225 11.56      51.0
                2009             7             1.7           49  2.89      11.9
                TOTALS         132            27.1         1796 71.59      352.9

    a. Use the normal equations to develop a linear regression equation for forecasting the number of
        ridership on the basis of the number of tourists. State the equation.
    b. Use the equation to forecast the number of ridership when the number of tourists visit London
        in a year is 10 million.
    c. Explain the predicted ridership if there are no tourists at all.
    d. Assuming n is large, calculate the 95.5 percent confidence limits for the number of ridership
        when the number of tourists is 10 million.



Prof.Dr.Dr.M.Hulusi DEMIR                                                                           33
                      Introduction to Production / Operations Management


      e. What is the correlation coefficient between number of ridership and the number of tourists?
          Interpret your result.
      f. What percentage of the variation in the number of ridership is explained by the tourist level?
      g. Is the correlation significant at the 5% level?

40. Sales of Volkswagen‟s Beetle have grown steadily at auto dealership in Istanbul during the past 5
    years (see the table below).
                 Year Sales
                 2005 450
                 2006 495
                 2007 518
                 2008 563
                 2009 584

      a. The sales manager had predicted in 2004 that 2005 sales (F1) would be 410 VWS. Using
         exponential smoothing with a weight of α = 0.30, develop forecast for 2006 through 2009.
      b. Use a 3-year moving average to forecast the sales of VW beetles in Istanbul through 2008.
      c. Which method you would use, exponential smoothing with α = 0.3 or a 3-year moving average.
         (Use average errors)
      d. According to the method you have chosen, forecast 2010 sales.

41.                       Year        Quarter Demand (Units)
                  2008            I              92
                                  II             82
                                  III            84
                                  IV             92
                 2009             I              90
                                  II             80
                                  III            82
                                  IV             94
      Compute a forecast for the demand in each of the quarters of the following year, 2010.

42. Following are the actual tabulated demands for an item for a nine-month period, from January
    through September. Your supervisor wants to test three forecasting methods to see which method
    was better over this period.
                         Month     Actual Demand
                         January       110
                         February      130
                         March         150
                         April         170
                         May           160
                         June          180
                         July          140
                         August        130
                         September     140

      a. Forecast April through September using a 3-month simple moving average.
      b. Using a weighted moving average with weights 6, 3, 1 from recent to oldest, forecast April
         through September.
      c. Use simple exponential smoothing to estimate April through September (α = 0.3) and assume
         that the forecast for March was 130.
      d. Use absolute errors to decide which method produced be better forecast over the six-month
         period.




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                      Introduction to Production / Operations Management


43. Dumlupinar Sports Club wants to develop its budget for the coming year using a forecast for
    football attendance. Football attendance accounts for the largest portion of its revenues, and the
    Vice Director Mr. T. Turgay believes attendance is directly related to the number of wins by the
    team. The Vice Director has accumulated total attendance figures for the last eight months.

               WINS      ATTENDANCE
                4             3 630
                6             4 010
                6             4 120
                8             5 300
                6             4 400
                7             4 560
                5             3 900
                7             4 750

    a. Develop a simple regression equation.
    b. Forecast attendance for at least 7 wins next year.
    c. If “ r = 0.948 “, what is the coefficient of determination. Interpret both.
    d. Test the correlation coefficient at 5 % level of significance. Is the correlation coefficient
        significant (meaningful) at this level?
    e. Using correlation coefficient find regression equation and explain the difference between two
        regression equations you have calculated.
    f. Calculate standard deviation of regression equation.

44. The Carpet City Store has kept records of its sales (in m2) each year, along with the number of
    permits that were issued for new houses in its area.. Carpet City`s operations manager believes
    that forecasting carpet sales is possible if the number of new housing permits is known for that
    year.

        Year    No. Of      Housing Sales
                Permits      (in 000m2)___
        2001     18            14
        2002     15            12
        2003     12            11
        2004     10             8
        2005     20            12
        2006     28            16
        2007     35            18
        2008     30            19
        2009     20            13

    c. Use linear relationship and find regression forecasting equation.
    d. Suppose that there are 25 new housing permits granted in 2010. What would be the 2010 sales?
    e. Find the correlation coefficient and interpret it.
    f. How much of the changes in the dependent variable are “explained” by the changes in the
       independent variable?
    g. Test the hypothesis r = 0 at 5% level of significance.
    h. Using correlation coefficient find regression forecasting equation.
    i. Forecast 2010 sales based on forecasted permits for that year.
    j. Compute the standard deviation of regression.
    k. Find confidence limits of 90% for the forecasted sales.
    l. Assuming “n” is large, find 95.5% confidence interval.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                          35
                      Introduction to Production / Operations Management


45.                                  Thousand of
        Month        Tires Used      Miles Driven
         1               100             1 500
         2               150             2 000
         3               120             1 700
         4                80             1 100
         5                90             1 200
         6               180             2 700

      The manager of Azim Trucking Co. Believes that Demand for Tires Used on his trucks is closely
      related to the number of miles driven. Accordingly, the above data covering the past 6 months
      have been collected.
      a. What percentage of variation in tire use can be explained by mileage driven?
      b. Test the correlation coefficient at 10% level of significance.
      c. Using correlation coefficient find regression forecasting equation.
      d. Compute 7th month tires used based on the forecasted thousands of miles driven for that month.
      e. Find confidence limits of 90 % for the 7th month forecast.

46. In the Magusa area, the number of daily calls for repair of Speedy Copy Machines has been
    recorded as follows:

          October 2007       Calls
                    1           92
                    2         127
                    3         103         a. Prepare a three-period weighted moving average
                    4         165            forecast using weights of w1 = 5, w2 = 3, w3 = 2.
                    5         132         b. Prepare exponentially smoothed forecast for
                    6         111               α = 0.3, F1 = 90.
                    7         174
                    8          97

47.       Year Quarters           Demand (units)
          2008 I                  350
               II                 460
               III                280
               IV                 360
          2009 I                  500
               II                 590
               III                450
               IV                 530

      a. Deseasonalise the data above bu computing 4-Quarter Moving Averages with a mean absolute
         deviations (errors) and also forecast Quarter I of 2010.
      b. Determine the trend line for the above data and forecast the next quarter.
      c. Determine exponentially smoothed forecast with α = 0.2 and F1 = 400 units. Determine the
         errors for this model. Forecast the following quarter.




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                   Introduction to Production / Operations Management


48. The sales manager of a local building material supply chain suspects that the sales of roofing
    materials are correlated with the amount of fraing lumber sold.

   Month Lumber Roofing
   ________Sales  Sales____
   1         90     50                4500       8100       2500
   2         115    52                5980     13225        2704
   3         120    60                7200     14400        3600
   4         125    64                8000     15625        4096
   5         145    72               10440     21025         5184
   6         145    74               10730     21025         5476
   7         150    74               11100     22500         5476
   8         140    84               11760     19600         7056
   9         135    82                11070     18225        6724
   10        120    72                 8640    14400         5184
   11        115    72                 8280    13225         5184
   12        100    60                 6000    10000         3600
             1500   816              103700    191350       56784
        Φ = 125    68

   a. Using the sales data above, develop a regression equation to express the number of units of
      roofing that you would expect to sell as a function of the number of units of lumber sold.
   b. Forecast the expected roofing sales for the next month in which 125 units of framing lumber is
      expected to be sold.
   c. Find correlation coefficient and determination coefficient. Interpret them.
   d. Test the correlation coefficient at 5 % level of significance. Is the correlation coefficient
      meaningful (significant) at this level?
   e. Using the correlation coefficient, find regression equation and explain the difference between
      two regression equations in (a) and (e).
   f. Calculate standard deviation of regression equation and express your forecast (found in (b))
      within 90% probability limits.
   g. Assuming n is large state your forecast within 95.5 %confidence intervals.

49. Mr. Salim Selim, sales manager for Magusa Gas Grills Ltd., needs a sales forecast for the next
    year. He has the following data from the last 2 years. (Sales are in 000 grills)
    Year Quarter        Sales           Year Quarter Sales
    2008         I       60             2009       I        105
                II       91                       II        130
               III       277                      III       522
               IV        34                       IV         73
    Compute quarterly sales forecasts for the coming year.

50. TT Construction Company renovates old homes in Magusa. Over time, the company has found
    that its MU volume of renovation work is dependent on the Magusa area payroll. The figures for
    TT‟s revenues and the amount of money earned by wage earners in Magusa for the past six years
    are presented in the table below.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                        37
                     Introduction to Production / Operations Management



     Years          Sales      Payroll
                 (100.000MU) (100.000.000MU)
         2004       2.0             1
         2005       3.0             3
         2006       2.5             4
         2007       2.0             2
         2008       2.0             1
         2009       3.5             7

     a. Using sales data above develop a regression equation.
     b. Find correlation coefficient and determination coefficient and interpret.
     c. Test the correlation coefficient at 5% level of significance. Is the correlation coefficient
        meaningful (significant) at this level?
     d. Using correlation coefficient, find regression equation and explain the difference between the
        two regression equations in (a) and (d).
     e. Calculate standard deviation of the regression equation and express your forecast within 90%
        probability limits, if the local chamber of commerce predicts the Magusa area payroll will be
        600 million MU next year.
     f. Find the forecast of Magusa Area Payroll for the year 2010.
     g. Find the regression equation using the forecast found in (f)
     h. Assuming sample is large (n>30) find the confidence intervals for 65.5% probability.

51. The sales manager of a local building material supply chain suspects that the sales of roofing
    materials are correlated with the amount of framing lumber sold.
       Years Lumber Sales                Roofing Sales
       2003                9                      5
       2004              10                       5
       2005              12                       6
       2006              14                       6
       2007              15                       8
       2008              18                       9
       2009              20                     10

     a. Find correlation coefficient and determination coefficient. Interpret them.
     b. Using correlation coefficient, find regression equation.
     c. Is the correlation significiant at 5% level.
     d. Forecast the expected roofing sales for the next year (2010) depending on the forecast of lumber
        sales for 2010.
     e. Calculate standard deviation of the regression equation and express your forecast found in (c)
        within 90% probability limits, i.e. 10% level of significance.
     f. Assuming n is large, state your forecast for 2010, within 95.5 confidence interval.




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                     Introduction to Production / Operations Management


DECISION MAKING
A. TRUE / FALSE
1. The decision maker has the option to choose the best state of nature available.
2. Decision-making under risk is issued when probability information about the states of nature is
    unavailable.
3. The consequence of each alternative needs to be known when using decision making under
    certainty.
4. Decision-making under risk requires the use of a payoff table.
5. The maximum criterion leads to a pessimistic alternative that is appropriate when the decision-
    making is seeking to avoid risk.
6. The equally likely criterion leads to an optimistic alternative that is appropriate when the decision-
    making is seeking to be exposed to risk.
7. The criterion of realism relies on a weighted average approach when choosing an alternative.
8. Minimax regret calculates the expected monetary value of each alternative thereby minimizing any
    regret.
9. To maximax criterion is part of decision making under uncertainty.
10. Calculate the probabilities for various states of nature are a step of Decision Theory process.

B. QUESTIONS
1. Describe what is involved in the decision process.
2. a decision table (excluding conditional values) to describe this situation. What is an alternative?
   What is a state of nature?
3. Discuss the differences among decision making under certainty, decision making under risk, and
   decision making under uncertainty.
4. Ayse Mutlu is trying to decide whether to invest in real estate, stocks, or certificates of deposit.
   How well she does depends on whether the economy enters a period of recession or inflation.
   Develop

C. PROBLEMS
1. You are planning your wedding day and need to decide this week whether the reception will be
   outdoors, outdoors with a tent or indoors. Your level of satisfaction will be affected by the weather
   on the day of reception. It will be sunny, cloudy or rainy. The table below summarizes your level
   of satisfaction for the various combinations on a sale 1 – 10 (10 = most satisfied)


                    Alternative               Sunny           Cloudy             Rainy

                     Outdoor                    10                6                1

                Outdoor with tent               9                 6                3

                      Indoor                    4                 5                7
    Which alternative would you choose by using the following criteria?
    a. Maximax
    b. Maximin
    c. Equally likely
    d. Realism (α = 0.7)
    e. Minimax regret




Prof.Dr.Dr.M.Hulusi DEMIR                                                                             39
                      Introduction to Production / Operations Management


2. Consider the following payoff table for three product decision (A, B and C) and the
   three future market conditions (payoffs = $ millions)

                                                            Market Conditions
                               Decision               1              2             3
                                  A                 $0.10           $2            $0.50
                                  B                  0.8            1.2            0.9
                                  C                  0.7            0.9            1.7

     Determine the best decision using the following decision criteria:
     a. Maximax
     b. Maximin

3. Demir Comp is a Turkey-based manufacturer of personal computers. It is planning to
   build new manufacturing and distribution facility in either W. Cyprus, Azerbaijan,
   Kazakhstan, Turkmenistan and Kirghizia. The cost of the facility will differ between
   Countries depending on the economic and political climate, including monetary
   Exchange rates. The Company has estimated the facility cost ( in $ millions) in each
   Country under three different future economic / political climates as follows

     D                                                  Economic / Political Climate
     e               Country                   Decline             Same              Improve
     t              N.Cyprus                     21.7               19.7                15.2
     e              Azerbaijan                    19                18.5                17.6
     r             Kazakhstan                    19.2               17.1                14.9
     m            Turkmenistan                   22.5               16.8                13.8
     i               Kirghizia                    25                21.2                12.5
     n
     e the best decision using the following decision criteria. (Note that since payoff is the cost,
     the maximax criteria becomes minimin and maximin becomes minimax)
     a.   Minimin
     b.   Minimax
     c.   Hurwicz (α = 0.40)
     d.   Equally likely

 4. Serin Cumbul has come into an inheritance from her grandparents. She is attempting to decide
    among several investment alternatives. The return after 1 year is dependent on the interest rate
    during the next year. The rate is currently 7% and she anticipates it will stay the same or go up or
    down by at most 2 points. The various investment alternatives plus their returns ($10000) given
    the interest rate changes are shown in the following table:

                                                            Interest Rate
               Investments                5%       6%           7%          8%            9%
            Money market fund               2       3.1          4          4.3             5
            Stock growth fund              -3       -2          2.5          4              6
                Bond fund                   6        5           3           3              2
            Government fund                 4       3.6         3.2          3             2.8
                Risk fund                  -9      -4.5         1.2         8.3           14.7
               Saving funds                 3        5          3.2         3.4            3.5



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                     Introduction to Production / Operations Management


   Determine the best investment using the following decision criteria.
    a. Maximax
    b. Maximin
    c. Equal likelihood
    d. Minimax regret
    e. Hurwicz (α = 0.40)

5. Sergio Bauersohn is the principal owner of Double T Oil Inc. After quitting his university teaching
   job, Sergio has been able to increase his annual salary by a factor of over 100. At the present time,
   Sergio is forced to consider purchasing some more equipment for Double T Oil because of
   competition. His alternatives the are shown in the following table:
                                                 STATES OF NATURE
         Equipment                   Favorable                    Unfavorable       Market
                                     Market(MU)                   (MU)
         Sub 100                     300.000                      -200.000
         Order MHD                   250.000                      -100.000
         Petrosan                    75.000                       -18.000

A. Sergio has always been a very optimistic decision maker
    a. What type of decision is Sergio facing?
    b. What decision criterion should he use?
    c. What alternative is best?

B. Although Sergio is the principal owner, his friend. N. Jayfer is credited with making the company a
    financial success. N. Jayfer is vice-president of finance. He attributes his success to his pessimistic
    attitude about business and the oil industry. He is likely to arrive a different decision than Sergio.
    What decision criterion should N. Jayfer use, and what alternative will he select?

6. Even though independent gasoline stations have been having a difficult time, Serin Cumbul has
   been thinking about starting her own independent gasoline station. Serin‟s problem is to decide
   how large her station should be. The annual returns will depend on both the size of her station and
   a number of marketing factors related to the oil industry and demand for gasoline. After a careful
   analysis, Serin developed the following table.

      Size of gas station    Good Market(MU)         Fair Market(MU)         Poor Market(MU)
      Small                  50.000                  20.000                  -10.000
      Medium                 80.000                  30.000                  -20.000
      Large                  100.000                 30.000                  -40.000
      Very Large             300.000                 25.000                  -160.000

    For example, if Serin constructs a small station and the market is good, she will realize a profit of
    50 000 MU.
    a. Develop a decision table for this decision
    b. What is the maximax decision?
    c. What is the maximin decision?
    d. What is the equally likely decision?
    e. What is the criterion of realism decision? Assume  = 0.80
    f. What is the minimax regret decision?




Prof.Dr.Dr.M.Hulusi DEMIR                                                                               41
                      Introduction to Production / Operations Management


7. Ilhan‟s Hardware does a brisk business in Girne during the year, but during Chrismas, Ilhan‟s
   Hardware sells Christmas trees for a substantial profit. Unfortunately, any trees not sold at the end
   of the season are totally worthless. Thus, the number of trees that are stocked for a given season is
   a very important decision. The following table reveals the demand for Christmas trees.

        Demand                                        Probability
        50                                            0.05
        75                                            0.10
        100                                           0.20
        125                                           0.30
        150                                           0.20
        175                                           0.10
        200                                           0.05

      Ilhan sells trees for 15 MU each, but his cost is only 6 MU.
     a. How many trees should Ilhan stock at his hardware store?
     b. If the cost increased to 12 MU per tree and Ilhan continues to sell trees for 17 MU each, how
        many trees should Ilhan stock?
     c. Ilhan is thinking about increasing the price to 18 MU per tree. Assume that the cost/tree is
        6MU. It is expected that the probability of selling 50, 75, 100, or 125 trees will be 0.25 each.
        Ilhan does not expect to sell more than 125 trees with this price increase. What do you
        recommend?

8. In addition to selling Christmas trees during the Christmas holidays, Ilhan‟s Hardware sells all the
   ordinary hardware items. One of the most popular items is Great Glue HD, glue that is made just
   for Ilhan‟s Hardware. The selling price is 2 MU per bottle, but unfortunately the glue gets hard
   and unusable after one month. The cost of the glue is 0.75 MU. During the past several months,
   the means sales of glue have been 60 units, and the standard deviation is 7. How many bottles of
   glue should Ilhan‟s Hardware stock? Assume that sales follow a normal distribution.

9. Demir Chemical, Inc, develops industrial chemicals that are used by other manufacturers to p
   roduce photographic chemicals, preservatives, and lubricants. One of their products, MHD-158, is
   used by several photographic companies to make a chemical that is used in the film developing
   process. To produce MHD-158 efficiently, Demir Chemical uses the batch approach, in which a
   certain number of gallons is produced at one time. This reduces set-up costs and allows Demir
   Chemical to produce MHD-158 at a competitive price. Unfortunately, MHD-158 has a very short
   shelf life of about one month.
   Demir Chemical produces MHD-158 in batches of 500 gallons, 1000 gallons, 1500 gallons, and
   2000 gallons. Using historical data, Mehmet Demir was able to determine that the probability of
   selling 500 gallons of MHD-158 is 0.2. The probabilities of selling 1000, 1500 and 2000 gallons
   are 0.3, 0.4, and 0.1 respectively. The question facing Mehmet is how many gallons to produce of
   MHD-158 in the next batch run. MHD-158 sells for 20 MU/gallon. Manufacturing cost is 12
   MU/gallon, and handling and warehousing costs are estimated to be 1 MU/gallon. In the past,
   Mehmet has allocated advertising costs to MHD-158 at 3 MU/gallon. If MHD-158 is not sold after
   the batch run, the chemical loses much of its important properties as a developer. It can, however,
   be sold at a salvage value of 13MU/gallon. Furthermore, Mehmet has guaranteed to his suppliers
   that there will always be an adequate supply of MHD-158. If Mehmet does run out, he has agreed
   to purchase a comparable chemical from a competitor at 25 MU/gallon. Mehmet sells the entire
   chemical at 20 MU/gallon. Mehmet sells the entire chemical at 20 MU/gallon, so his shortage
   means that Mehmet loses the 5 MU to buy more expensive chemical.
   a. Develop a decision tree of this problem.
   b. What is the best solution?
   c. Determine the EVPI



42
                     Introduction to Production / Operations Management


10. Serin Cumbul is not sure what she could do. She can build a quadplex (i.e. building with four
    apartments), build a duplex, gather additional information or simply do nothing. If she gathers
    additional information, the result could be either favorable or unfavorable, but it would cost her 3
    000 MU to gather the information. Serin believes that there is a 50-50 chance that the information
    will be favorable. If the rental market is favorable, Serin will earn 15 000 MU with the quadplex
    or 5 000 MU with the duplex. Serin does not have the financial resources to do both.
    With an unfavorable rental market, however, Serin could lose 20 000 MU with the quadflex or 10
    000 MU with the duplex. Without gathering additional information, Serin estimates that the
    obability of a favorable rental market is 0.7. A favorable report from the study would increase the
    probability of a favorable rental market to 0.9. Furthermore, an unfavorable report from the
    additional information would decrease the probability of a favorable rental market to 0.4. Of
    course, Serin could forget all of these numbers and do nothing.
    What is your advice to Serin?

11. The Steak and Chop Butcher Shop purchases from a local meatpacking house. The meat is
    purchased on Monday at 2.00 MU/kg and the shop sell the steak for 3.00 MU/kg. Any steak left
    over at the end of the week is sold to a local zoo for 0.50 MU/kg. The possible demands for steak
    and the probability for each are as follows:

                             Demand (kg)       Probability
                                 20               0.1
                                 21               0.2
                                 22               0.3
                                 23               0.3
                                 24               0.1

    The shop must decide how much steak to order in a week?

12. Place-Plus, a real estate development firm, is considering several alternative development
    projects. These include building and leasing an office park, purchasing a parcel of land and
    building an office building to rent, buying and leasing a warehouse, building a strip shopping
    center, and building and selling condominiums. The financial success of these projects depends on
    interest rate movement in the next 5 years. The various development projects and their 5 year
    financial return (MU millions) given that interest rates will decline, remain stable or increase are
    shown in the following payoff table:

                                                        Interest Rates
                        Projects            Decline        Stable      Increase
                Office Park                  0.5             1.7          4.5
                Office Building              1.5             1.9          2.4
                Warehouse                    1.7             1.4           1
                Shopping Center              0.7             2.4          3.6
                Condominiums                 3.2             1.5          0.6

    Determine the best investment using the following decision criteria:
    a. Maximax
    b. Maximin
    c. Minimax regret
    d. Equally Likely
    e. Hurwicz (α = 0.3)



Prof.Dr.Dr.M.Hulusi DEMIR                                                                            43
                    Introduction to Production / Operations Management


     13. The Magusa Livestock Company receives order for an average of 6000 dozen quail eggs a
         week. The standard deviation of weekly orders is 425 dozen. The eggs cost 7 MU/dozen and
         are resold for 10 MU/dozen. If the eggs are not shipped within a week, their fertility is
         impaired and Magusa`s can not sell them as first-quality; they can however be sold for 1
         MU/dozen. Calculate Magusa`s optimum weekly order of eggs.

     14. The manager must decide how many machines of certain type to buy. The machines will be
         used to manufacture a new gear for which there is increased demand. The manager has
         narrowed the decision to two alternatives: buy one machine or buy two. If only one
         machine is purchased and demand is more than it can handle, a second machine can be
         purchased at a later time. However, the cost per machine would be lower if the two
         machines were purchased at the same time.
         The estimated probability of low demand is 0.30, and the estimated probability of high
         demand is 0.70. The net present value associated with the purchase of two machines initially
         is 75 000 MU if demand is low, and 130 000 MU if demand is high. The net present value
         for one machine and low demand is 90 000 MU. If demand is high, there are three options:
         One option is to do nothing, which would have a net present value of 90 000 MU. A second
         option is to subcontract; that would have a net present value of 110 000 MU. The third
         option is to purchase a second machine. This option would have a net present value of 100
         000 MU.
         How many machines should the manager purchase initially? (Use a decision tree to
         analyse this problem.)

     15. A company is faced with the decision of how many units of product to prepare before the
         tourism season at the local market. Each unit of product costs 3 MU and sells for 12 MU per
         unit. Past records indicate that 3 500 units are enough to prevent any shortage, and this is the
         number prepared before tourism season in the past 10 years. Unsold product is disposed of
         at a total loss. The following data summarizes the sales history.

       DEMAND            FREQUENCY
       2 700                     8
       2 800                    12
       2 900                    20
       3 000                    25
       3 100                    15
       3 200                    10
       3 300                     5
       3 400                     5
       3 500                    10
         a. How many units of this type of product should be prepared prior to tourism sector each
         year?
         b. What is the long-run expected loss under the current policy?

     16. Seaman‟s Fish Market buys fresh Izmir Bluefish daily for 1.40 MU/kg and sells for 1.90
         MU/kg. At the end of each business day, any remaining blue fish is sold to a producer of a
         cat food for 0.80 MU/kg. Daily demand can be approximated by a normal distribution with a
         mean of 80 kg. and a standard deviation of 10 kg.
        What is the optimal stocking level?

     17. The owner of Double-T Pizza is considering a new oven in which to bake the firm‟s
       signature dish “Vegeterian Pizza”. Oven A type can handle 20 pizzas an hour. The fixed
       costs associated with oven A are 20 000 MU and the variable costs are 200 MU/pizza. Oven B
       is larger and can handle 40 pizzas an hour. The fixed costs associated with Oven B are 30 000
       MU and the variable costs are 1.25 MU/pizza. The pizzas sell for 14 MU each.
          a.    what is the break-even point for each oven?

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                       Introduction to Production / Operations Management


    b.    if the owner expects to sell 9 000 pizzas, which oven should the owner purchase?
    c.    if the owner expects to sell 12 000 pizzas, which oven should the owner purchase?
    d.    at what volume should the owner switch ovens?

18. A) A group of medical professional is considering the construction of a private cardiology clinic,
    Hospital of Cardiology (HOC). If the medical demand is high (i.e. there is a favorable market for
    the clinic), the physicians could realize a net profit of 100000 MU. If the market is not favorable,
    they could lose 40000 MU. Of course they do not have to proceed at all, in which case there is no
    cost. In the absence of any market data, the best the physicians guess is that there is a 50 – 50
    chance the clinic will be successful. Construct a decision tree to help analyzing this problem. What
    should the medical professionals do?

    B) The phsycians have been approached by a market research firm that offers to perform a study
       of the market at a fee of 5 000 MU. The market researchers claim their experience enables
       them to use Bayes` theorem to make the following statements of probability;
         Probability of a favorable market given a favorable study = 0.82
         Probability of a unfavorable market given a favorable study = 0.18
         Probability of a favorable market given an unfavorable study = 0.11
         Probability of an unfavorable market given an unfavorable study = 0.89
         Probability of a favorable research study = 0.55
         Probability of an unfavorable research study = 0.45
       a. Develop a new decision tree for the medical professionals to reflect the options
          now open with the market study
       b. Use the EV approach to recommend a strategy
       c. What is the expected value of sample information? How much might the
          physicians be willing to pay for a market study?

  19.
                                                                                 Excess
                       To                                                        Supply
                                W            X           Y            Z
              From

                   A
                                12           4            9            5           55
                   B
                                 8           1            6            6           45
                   C
                                 1          12            4            7           30
                Unfilled
                Demand          40          20           50           20

         Use Vogel‟s Approximation method to find an initial assignment of the excess supply




Prof.Dr.Dr.M.Hulusi DEMIR                                                                            45
                    Introduction to Production / Operations Management


20. The purchase agent of Magusa Plumbing Co. wishes to purchase 3 000 meters of pipe A, 2 000
    meters of pipe B and 3 000 meters of pipe C.
    Three manufacturers (X,Y, and Z) are willing to provide the needed pipe at the costs
    given below (in MU per 1 000 meter). Magusa Plumbing wants delivery within I month.
    Manufacturer X can provide 6 000 meters, Manufacturer Y can provide 5 000 meters and
    Manufacturer Z can provide 3 000 meters. Determine Magusa Plumbing Co‟s least–cost
    purchasing plan for the pipe should be? (Use VAM method)

21. During the Gulf War, Operation Desert Storm required large amounts of military material
    and supplies to be shipped daily from supply depots in the USA to bases in the Middle
    East. The critical factor in the movement of these supplies was speed. The following table
    shows the number of planeloads of supplies available each day from each of six supply
    depots and the number of daily loads demanded at each of five bases. (each planeload is
    approximately equal in tonnage). Also included are the transport hours per plane,
    including loading and fuelling, actual flight time, and unloading and refuelling.
     Determine the OPTIMAL DAILY FLIGHT SCHEDULE that will minimize total transport time.



                                           Types of Pipe
                                      (Cost MU/1000 Metres)

                          A                  B               C              Available

          X               580                600             520

          Y               620                560             580

          Z               600                580             580
          Amount
          Needed

       Supply                    Military Base
       Depot              A       B      C     D     E                     Supply
      _________________________________________________                  _________
       #1                  36     40     32    43    29                       14
        #2                 28     27     29    40    38                       20
       #3                  34     35     41    29    31                       16
       #4                  41     42     35    27    36                       16
       #5                  25     28     40    34    38                       18
       #6                  31     30     43    38    40                        6
      _________________________________________________

        Demand                  18      12         24   16         20



22. ABC Air Conditioners operates factories in four different cities. Each of these factories is
    responsible for maintaining warehouse supplies in 5 different warehouses. Because of varying
    distances, transportation charges from factory to warehouse are not uniform. Shipping charges per
    unit are summarized below:




46
                    Introduction to Production / Operations Management


     WAREHOUSE
       FACTORY         1          2   3  4          5_
         F 1________ 8            9 12  7          18
         F 2________ 6            8 13  9          21
         F3           20           7 10 11          8
         F4          12           7 14 15          22

   Factory output and warehouse supplies that must be maintained are as follows:

   Factory Units produced/day                   Warehouse        Daily Supply
      #1             35                               1                 15
      #2             25                               2                 12
      #3             40                               3                 22
      #4             32                               4                 30
                                                      5                 20
   Determine;
   a. The best possible factory-to-warehouse shipping program using Vogel‟s Approximation
      Method.
   b. What is the cost of this shipping program?

23. The YUHUA Disk Drive Co. Produces drives for personal computers. YUHUA produces drives in
    three plants (factories) located in IZMIR/TURKEY, MAGUSA/TRNC and BEIJING/CHINA.
    Periodically, shipments are made from these three production facilities to four distribution
    Warehouses located in Turkey, namely: ISTANBUL, ANKARA ADANA and BURDUR. Over
    the next month, it has been determined that these warehouses should receive the following
    proportions of the company‟s total production of the drives.

           Warehouse                % of Total Production
           Istanbul                    31
           Ankara                      30
           Adana                       18
           Burdur                      21

  The production quantities at the factories in the next month are expected to be (in thousand
  of units)

         Plant Anticipated Production(000 units)
          Izmir                      45
          Magusa                   120
          Beijing                   95

   The unit costs for shipping 1000 units from each plant to each warehouse is given in the
   table below. The goal is to minimize total transportation cost. (use VAM)
   (Hint: When finding total production at the three plants you may round the figures to the
   nearest unit)
   Shipping costs per 1000 units in MU:
                          Istanbul Ankara Adana Burdur
              Izmir          250     420        380       280
           Magusa           1280     990      1440      1520
            Beijing         1550    1420      1660      1730

24. ABC ship supplies from 4 principal manufacture to four regional stores. The manufactures
    are located at Izmir, Manisa, Aydin and Denizli. The regional stores are located in Isparta,
    Burdur, Antalya and Afyon. In order to reduce the cost of meeting demand for supplier,


Prof.Dr.Dr.M.Hulusi DEMIR                                                                        47
                       Introduction to Production / Operations Management


         ABC has decided to allocate its material according to the standard transportation
         model. An analysis of daily shipping records reveal that the following costs per unit
         are typical for the current shipping operations.

                       TO                               Antalya       Afyon        SHIP-
             FROM            Isparta      Burdur                                   MENT
             Izmir              44           22            30           20           70
             Manisa             34           28            26           15           50
             Aydin              25           30            34           40           90
             Denizli            32           40            22           25          100

             NEEDS              90            50           60           80

     a. Determine an initial shipping program
     b. Calculate the daily cost of this program.

25. A firm that plans to expand its product line must decide whether to build a large or a small plant to
    produce the new products. If it builds a large plant and demand is high, the estimated net present
    value is 80 000 MU. If demand turns out to be low, the net present value will be -1 000 MU. The
    probability that demand will be high is estimated to be 0.70. If a small plant is built and the
    demand is low, the net present value after deducting for building costs will be 40 000 MU. If
    the demand is high, the firm can either maintain the small plant or expand it. Expansion would
    have a net present value of 45 000 MU, and maintaining small plant would have a net present
    value of 5 000 MU. The probability of low demand is 0.40.
     a. analyze using a tree diagram.
     b. compute the EVPI. How would this information be used?

26. The Our-Bags-Don‟t-Break (OBDB) plastic bag company manufactures three plastic refuse bags
    for home use: a 5-kg garbage bag, a 10-kg garbage bag, and a 15-kg leaf-and-grass bag. Using
    purchased plastic material, three operations are required to produce each end product: cutting,
    sealing and packaging.The production time required to process each type of bag in every
    operation and the maximum production time available for each operation are shown
     (Note that the production time figures in this table are per box of each type of bag).

                                           TYPE OF BAG                               TIME
                         5-kg Bag            10-kg Bag    15-kg Bag                  AVAILABLE
        Cutting          2 Seconds/Box      3Seconds/Box 3 Seconds/Box               2 Hours
        Sealing          2 Sec./box         2 Sec./Box   3 Sec./Box                  3 Hours
       Packaging         3 Sec./Bag         4 Sec./Box    5 Sec./Box                 4 Hours

     If OBDB‟s profit contribution is 0.10MU for each box of 5-kg bags produced, 0.15MU for each
     bpx of 10-kg bags, and 0.20 MU for each box of 15-kg bags, what is the optimal product mix?

27. M&D Chemicals produces two products that are sold as raw materials to companies
    manufacturing bath soaps and laundry detergents. Based on an analysis of current
    inventory levels and potential demand for the coming month, M&D‟s management has
    specified that the combined production for products 1 and 2 must total at least 700 Kgs.
    Separately, a major customer‟s order for 250 kgs of product 1 must also be satisfied.
    Product 1 requires 2 hours of processing time per kg. While product 2 requires 1 hour of
    processing time per kg, and for the coming month, 1200 hrs of processing time are
    available. M&D‟s objective is to satisfy the above requirements at a minimum total
    production cost. Production costs are 2 MU/kg for product 1 and 3 MU/kg for product 2.
    Construct the GENERAL SIMPLEX MODEL properly. Place the figures of the model in

48
                     Introduction to Production / Operations Management


    an initial simplex tableau and find which variable is entering and which variable is leaving.

28. A national car rental service has a surplus of one car in each of cities 1,2,3,4,5,6, and a deficit of
    one car in each of cities 7,8,9,10,11,12. The distances between cities with a surplus and cities with
    a deficit are displayed in the matrix below. How should the car be dispatched so as to minimize
    the total mileage travelled?

                                                   To
                          7            8            9            10           11           12

                 1       41           72           39            52           25           51

                 2       22           29           49            65           81           50
         From 3          27           39           60            51           32           32
                 4       45           50           48            52           37           43
                 5       29           40           39            26           30           33
                 6       82           40           40            60           51           30



29. The Izmir Aerospace Company has just been awarded a rocket engine development contract. The
    contract terms require that at least five other smaller companies be awarded subcontracts for a
    portion of the total work. So Izmir requested bids from five small companies ( A, B, C, D, and E )
    to do subcontract work in five areas ( I, II, III, IV and V ). The bids are as follow:

    Cost information:

                                            Subcontract bids
                              I             II            III               IV             V
       Company
          A             45000MU        60000MU          75000MU        100000MU       30000MU
            B            50000             55000          40000          100000         45000
            C            60000             70000          80000          110000         40000
            D            30000             20000          60000           55000         25000
            E            60000             25000          65000          185000         35000



    a. Which bids should Izmir accept in order to fulfil the contract terms at the least cost?
    b. What is the total cost of subcontracts?




Prof.Dr.Dr.M.Hulusi DEMIR                                                                              49
                    Introduction to Production / Operations Management


30. Azim Kola has assets of 300 000 MU and wants to decide whether to market a new melon-
    flavoured soda, Melcola. Melcola has three alternatives:
    Alternative 1       Test market Melcola locally, then utilize the results of the market study to
                        determine wherher to market Melcola nationally.
    Alternative 2       Immediately (without test marketing) market Melcola nationally.
    Alternative 3       Immediately (without test marketing) decide not to market Melcola
                        nationally.
    In the absence of a market study Azim Kola believes that Melcola has a 55% chance of being a
    national success and a 45% chance of being a national failure. If Melcola is a national success,
    Azim Kola`s asset position will increase by 600 000 MU, and if Melcola is a national failure,
    Azim Kola`s asset position will decrease by 200 000 MU.

   If Azim Kola performs a market study (at a cost of 60 000 MU), there is a 60% chance that the
   study will yield favourable results (referred to as a local success) and a 40% chance that the study
   will yield unfavourable results (referred to as a local failure). If a local success is observed, there
   is an 85% chance that Melcola will be a national success. If a local failure is observed, there is
   only 10% chance that Azim Kola will be a national success. If Azim Kola is a risk-neutral (wants
   to maximise its expected final asset position), what strategy should the company follow?
                     Introduction to Production / Operations Management


INVENTORY CONTROL
1. The probability distribution of the demand for a product has been estimated to be

        Demand          Prob. of Demand
           0               0.05
           1               0.15
           2               0.30
           3               0.35
           4               0.10
           5               0.05
           6               0.00

    Each unit sells for 50 MU, and if the product is not sold, it is completely worthless. The purchase
    costs of a unit are 10 MU. Assuming no reordering is possible, how many units should be
    purchased?

2. Demand for a product is approximately normal with a mean 40 units and standard deviation 12
   units. The product costs 2 MU per unit and sells for 5 MU. Unsold units have no value. What is the
   optimal order size?

3. Sweet cider is delivered weekly to Sergio‟s Produce stand. Demand varies uniformly between 300
   litres and 500 litres per week. Sergio pays 0.20 MU/litre for the cider and charges 0.80 MU/liter for
   it. Unsold cider has no salvage value and cannot be carried into the next week due to spoilage.
   Find the optimal stocking level and the stock-out risk for that quantity.

4. A wholesaler of stationery is deciding how many desk calendars to stock for the coming year. It is
   impossible to reorder, and leftover units are worthless. The following table indicates the possible
   demand levels and the wholesaler‟s prior probabilities.


        Demand(in 000s)          Prob. Of Demand
             100                        0.10
             200                        0.15
             300                        0.50
             400                        0.25

    The calendars sell for 100 MU per thousand, and the incremental purchase cost is 70 MU. The
    incremental cost of selling (commissions) is 5 MU per thousand. Use marginal analysis to find how
    many calendars should be ordered.

5. A camera manufacturer makes most of its sales during the New Year selling season. For each
   camera sold, it makes a unit profit of 20 MU, if a camera is unsold after the major selling season, it
   must be sold at a reduced price, which is 5 MU less than the variable cost of manufacturing the
   camera. The manufacturer estimates that demand is normally distributed with a mean of 10 000
   units and a standard deviation of 1 000 units. What is the optimum number to order?




Prof.Dr.Dr.M.Hulusi DEMIR                                                                             65
                      Introduction to Production / Operations Management


6. Ahmet Koc owns and operates a large fresh fruit stand in Gazimagusa, TRNC. Fresh greens are his
   primary produce. Each case of greens sells for 15 MU. Ahmet‟s cost is 5 MU for each case. Cases
   that are not sold can be sold for 1 MU a case at the end of the day to a small grocery store. The
   probabilities of sales for cases of greens are as follows:

        Daily sales (cases)    Probability at this level
                5                        0.1
                6                        0.1
                7                        0.2
                8                        0.3
                9                        0.2
               10                        0.1__________

     Determine the best policy to stock each week?

7. A special style of sweater can be purchased by retail store for 18.25 MU on a one-time opportunity.
    The store plans to offer the sweater at a retail price of 34.95 MU during the season. Any sweaters
    left at the end of the season will be sold for 14.95 MU. It is estimated that the demand for this item
    at this location will have a normal probability with a mean of 80 and a standard deviation of 22.
    How many of these sweaters should the store stock?

8. A magazine shop owner orders a popular monthly magazine, the demand of which varies from
   1000 to 2400 copies. The magazines cost 250 MU/hundred and sell for 4.50 MU/each. When
   purchase in lots at this price, the publisher accepts no returns. What should be the ordering
   quantity for the next period?

9. Seaman‟s Fish Market buys fresh Izmir Bluefish daily for 1.40 MU/kg and sells for 1.90 MU/kg. At
    the end of each business day, any remaining blue fish is sold to a producer of a cat food for 0.80
    MU/kg. Daily demand can be approximated by a normal distribution with a mean of 80 kg. and a
    standard deviation of 10 kg. What is the optimal stocking level?

10. Ali Caliskan sells New Year trees, which he grows on his farm in Guzelyurt. Because bad
    weather and heavy rain is common in the month December. Ali has always harvested the trees
    he intends to sell in a given year by December 1. Ali has been selling trees for many years, and
    has kept detailed records of sales in previous years. From this data, he has determined that
    probability of selling various quantities of trees in a given year as follows:

         DEMAND                  PROBABILITY
         500                          0.15
         550                          0.20
         600                          0.25
         650                          0.30
         700                          0.10
         750 +                        0.00

     For the coming year, Ali will sell his trees for an average of 25 MU each. His cost to grow and cut
     each tree is estimated to be 10 MU. Any unsold trees at the end of the year can be sold for kindling
     wood at a price of 5 MU a piece. What is the optimal number of trees that Ali should harvest?




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                     Introduction to Production / Operations Management


11. The manager of a drugstore is wondering how many New Year Cards to order before December.
    Each card costs 1.30 MU, but retails for 2.20 MU if sold before New Year. After New Year the
    store reduces the price by 60%. On the basis of past records, the manager has developed the
    following table.

          Demand        Probability
           3 000           0.05
           3 500           0.15
           4 000           0.25
           4 500           0.25
           5 000           0.15
           5 500           0.15

    How many cards should be ordered?

12. A style item can be purchased for 65 MU/unit before the season, and no additional units can be
    ordered. The product will sell for 130 MU during the season and any units left at the end of the
    season will be for 50 MU. The probability distribution of demand during the season is estimated
    normally distributed with a mean of 200 units and a standard deviation of 50. Determine the
    amount to stock to order that will give the maximum expected profit?
13. Ahmet Caliskan experiences an annual demand of 220 000 MU for quality tennis balls at the
    Gazimagusa Tennis Supply Company. It costs Ahmet 30 MU to place an order and his carrying
    cost is 18%. How many orders per year should Ahmet place for the balls?

14. Ayse Guzel, owner of Computer Village, needs to determine an optimal ordering policy for Genius
    Computers. Annual demand for the computers is 28.000 MU and carrying cost is 23%. Ayse has
    estimated order costs to be 48 MU/order. What is optimal MU per order? (optimal quantity in
    monetary units)

15. A large bakery buys flour in 25-kg bags. The bakery uses an average of 4860 bags a year. Preparing
    an order and receiving a shipment of flour involves a cost of 4 MU per order. Annual carrying costs
    are 30 MU/bag.
        a. Determine the economic order quantity
        b. What is the average number of bags on hand?
        c. How many orders per year will there be?
        d. Compute the total cost of ordering and carrying flour
        e. If annual ordering cost were to increase by 1 MU per order. How much would that affect
            the minimum total annual cost?

16. A large law firm uses an average of 40 packages of copier paper a day. The firm operates 260
    days a year. Storage and handling costs for the paper are 3 MU a year per pack, and itcosts
    approximately 6 MU to order and receive a shipment of paper.
        a. What order size would minimize total ordering and carrying costs?
        b. Compute the total annual inventory cost using your order size from part a.
        c. Except for rounding, are annual ordering and carrying costs always equal at EOQ?
        d. The office manager is currently using an order size of 200 packages. The partners of the firm
           expect the office to be managed “in a cost-efficient manner”. Would you recommend that
           the office manager use the optimal order size instead of 200 packages? Justify your answer.

17. Garden Variety Flower Shop uses 750 clay pots a month. The pots are purchased at 2 MU each.
    Annual carrying costs are estimated to be 25 percent of cost, and ordering costs are 30 MU per
    order.
    a. Determine the economic order quantity and the total annual cost of carrying and ordering.

Prof.Dr.Dr.M.Hulusi DEMIR                                                                             67
                      Introduction to Production / Operations Management


     b. Suppose an analysis shows actual carrying costs are roughly double the current estimate. If the
        order size wasn‟t changed, how much extra cost would the firm incur?

18. A produce distributor uses 800 packing crates a month, which it purchases at a cost of 10 U/crate
    and carrying cost is 35% of the purchase price per crate. Ordering costs are 28 MU. Currently the
    manager orders once a month. How much could the firm save annually in ordering and carrying
    costs by using economic order quantity?

19. Demand for jelly doughnuts on Saturdays at Ilhan‟s Doughnut Shoppe is shown in the following
    table. Determine the optimal number of doughnuts, in dozens, to stock if labour, materials, and
    overhead are estimated to be 0.80 MU per dozen, doughnuts are sold for 1.20 MU per dozen, and
    leftover doughnuts at the end of each day are sold the next day at half price. What is the resulting
    service level?

         Demand(dozens) Relative Probability
              19            0.01
              20            0.05
              21            0.12
              22            0.18
              23            0.13
              24            0.14
              25            0.10
              26            0.11
              27            0.10
              28            0.04
              29            0.02

20. Burger Prince buys top-grade ground beef for 1.00MU/kg. A large sign over the entrance
     guarantees that the meats fresh daily. Any leftover meat is sold to the local high school
     cafeteria for 0.80/kg. Four hamburgers can be prepared from each kg. of meat. Burgers sell
     for 0.60 MU/each. Labour, overhead, meat, buns, and condiments costs0.50 MU/burger.
     Demand is normally distributed with a mean of 400 kgs per day and a standard deviation
     of 50 kgs a day. What daily order quantity is optimal?
      *(HINT: Shortage cost must be in MU/kg)

21. Ali Uslu sells bicycles. One particular model is highly popular with annual sales of 2000 units per
    year. The cost of one such bicycle is 800 MU. Annual holding costs are 25% of the item‟s cost and
    the ordering cost is 40 MU. The store is open 250 days a year.
        a. What is the economic order quantity?
        b. What is the optimal number of orders?
        c. What is the optimal number of days between orders?
        d. What are the annual total costs?
        e. What are total annual ordering costs and annual total holding costs? Verify your results.

22. The soft goods department of a large department store sells 150 units per month of a certain large
    bath towel. The unit cost of a towel to the store is 2.50 MU and the cost of placing an order has
    been estimated to be 12.00 MU. The store uses an inventory carrying charge of 27% per year.
    Determine the optimal order quantity, order frequency, and the annual cost of inventory
    management. If through automation of the purchasing process, the ordering cost can be cut to 4
    MU, what will be the new EOQ, order frequency and the annual inventory management cost?
    Explain these results.

23. EMU uses 96 000 MU annually of a particular toner cartridge for laser printers in the student
    computer labs. The purchasing director of the university estimates the ordering cost at 45MU and


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                     Introduction to Production / Operations Management


    thinks that the university can hold this type of inventory at an annual storage cost of 22% of the
    purchase price. How many months‟ supply should the purchasing director order at one time to
    minimize the total annual cost of purchasing and carrying?

24. Given the following data :
       C = 72 000 units/year; s = 120 MU/set-up, p = 4 MU/unit; Z = 25% /year
    Calculate EOQ and calculate annual costs following EOQ behaviour.

25. A local firm has traditionally ordered a supply item 60 units at a time. The firm estimates that
    carrying cost is 40% of the 10 MU unit cost, and that annual demand is about 240 units /year. The
    assumptions of the EOQ model are thought to apply. For what value of ordering cost would their
    action be optimal?

26. A firm that makes electronic circuits has been ordering a certain raw material 60 kgs at a time. The
    firm estimates that carrying cost is 30% per year, and that ordering cost is about 20 MU/order. The
    current price of the ingredient is 200 MU/kg. The assumptions of the EOQ model are thought to
    apply. For what value of annual demand is their action optimal?

27. The Rushton Trash Co. stocks, among many other products, a certain container, each of which
    occupies four square feet of warehouse space. The warehouse space currently available for storing
    this product is limited to 600 square feet. Demand for the product is 12000 units per year. Holding
    costs are 2 MU/container/year. Ordering costs are 5 MU/order.
        a. What is the cost-minimizing order quantity decision for Rushton?
        b. What is the total inventory-related cost of this decision?
        c. What is the total inventory-related cost of managing the inventory of this product, when the
       limited    amount of warehouse space is taken into consideration?
        d. What would the firm willing to pay for additional warehouse space?

28. A local distributor for a national tire company expects to sell approximately 9600 steel belted
    radial tires of a certain size and tread design next year. Annual carrying cost is 16 MU/tire and
    ordering cost is
    75 MU. The distributor operates 288 days a year.
      a. What is the EQO?
      b. How many times per year does the store reorder?
      c. What is the length of an order cycle?
      d. What is the total annual inventory costs if the EOQ is ordered?

29. TT Manufacturing Co. produces commercial refrigeration units in batches. The firm‟s estimated
    demand for the year is 10 000 units. It costs 100 MU to set up the manufacturing (production)
    process and the carrying cost is about 0.50 MU/unit-year. Once the production process is set up, 80
    refrigeration units can be manufactured daily. The demand during the production period has
    traditionally been 60 units each day.
           a. How many refrigeration units should TT Manufacturing produce in each batch?
           b. How long should the production part of the cycle?
           c. What is the maximum inventory level at this production rate?
           d. What is the minimum annual total inventory cost?

30. Demand during lead-time varies uniformly between 8.000 Units and 12.000 Units. Each unit costs
    3.00 MU, sells for 4.00 MU, and has a salvage value of 1.20 MU, if not sold.
    Use the single-period model to find the optimal level of inventory to stock.

31. A local supermarket sells a popular brand of Shampoo at a fairly steady state of 380 bottles per
    month. The cost of each bottle to the supermarket is 0.45 MU and the cost of placing an order has
    been estimated at 8.50 MU. Assume that holding costs are based on a 25% annual interest rate.

Prof.Dr.Dr.M.Hulusi DEMIR                                                                                69
                      Introduction to Production / Operations Management


         a. Determine the economic order quantity and the time between placements of orders for this
            product.
         b. If the procurement lead-time is two months, find the reorder point.
         c. If the shampoo sells for 1.00 MU, what is the total annual cost of the shampoo?
         d. What is the total annual holding cost? Verify your result.
         e. Determine the optimal number of orders.

32. Azim Co. manufactures Product A. Projected demand for Product A equals 200 000 units. Each
    production run requires an outlay of 160 MU/machine set-up, and each unit carried in inventory
    costs 100 MU. The estimated cost of a back-order is 600 MU. Each back-order is filled as soon as
    the production run is completed. Determine the following:
         a. The optimal size of each production run?
         b. The maximum level of inventory that the firm can expect to have on hand?
         c. The back-order quantity?
         d. The optimal number of productiın runs in a year?
         e. The time between runs (assume 250 days/year)?
         f. The total annual cost of the inventory policy?
         g. If annual demand is doubled at Azim Co. and a wage increase doubles the set-up cost,
             what effect does this have on Azim‟s original inventory policy?

33. Osman Sabit sells New Year trees, which he grows on his farm in Guzelyurt. Because bad weather
    and heavy rain is common in the month December. Osman has always harvested the trees he
    intends to sell in a given year by December 1. Osman has been selling trees for many years, and
    has kept detailed records of sales in previous years. From this data, he has determined that
    probability of selling various quantities of trees in a given year as follows:

       DEMAND                    PROBABILITY
        501                           0.10
        551                           0.25
        601                           0.25
        651                           0.35
        701                           0.05
        750 +                         0.00

      For the coming year, Osman will sell his trees for an average of 30 MU each. His cost to grow and
      cut each tree is estimated to be 15 MU. Any unsold trees at the end of the year can be sold for
      kindling wood at a price of 5 MU a piece. What is the optimal number of trees that Ali should
      harvest?

34. The probability distribution of the demand for a product has been estimated to be

       Demand            Prob. of Demand                         Demand           Prob. of Demand
              7               0.05                                    11             0.10
              8               0.15                                    12             0.05
              9               0.30                                    13             0.00
              10              0.35
     Each unit sells for 50 MU, and if the product is not sold, it is completely worthless. The purchase
     costs of a unit are 10 MU. Assuming no reordering is possible, how many units should purchased?

35. Demand for a product is approximately normal with a mean 40 units and standard deviation 12
    units. The product costs 2 MU per unit and sells for 5 MU. Unsold units have no value. What is the
    optimal order size?




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                     Introduction to Production / Operations Management


36. Sweet cider is delivered weekly to Sergio‟s Produce stand. Demand varies uniformly between 300
    litres and 500 litres per week. Sergio pays 0.20 MU/liter for the cider and charges 0.80 MU/liter for
    it.Unsold cider has no salvage value and cannot be carried into the next week due to spoilage. Find
    the optimal stocking level and the stockout risk for that quantity.

37. A wholesaler of stationery is deciding how many desk calendars to stock for the coming year. It is
    impossible to reorder, and leftover units are worthless. The following table indicates the possible
    demand levels and the wholesaler‟s prior probabilities.

        Demand(in 000s)              Prob. Of Demand
             101                        0.10
             201                        0.15
             301                        0.50
             401                        0.25

    The calendars sell for 100 MU per thousand, and the incremental purchase cost is 70 MU. The
    incremental cost of selling (commissions) is 5 MU per thousand. Use marginal analysis to find how
    many calendars should be ordered.
38. A camera manufacturer makes most of its sales during the New Year selling season. For each
    camera sold, it makes a unit profit of 20 MU, if a camera is unsold after the major selling season, it
    must be sold at a reduced price, which is 5 MU less than the variable cost of manufacturing the
    camera.The manufacturer estimates that demand is normally distributed with a mean of 10 000
    units and a standard deviation of 1 000 units. What is the optimum number to order?

39. Azim Manufacturing produces a product for which the annual demand is 10 000. Production
    averages 100 per day, while demand is 40 per day. Holding costs are 1.00 MU per unit per year;
    set-up costs 200.00 MU. If they wish to produce this product in economic batches,
        a. What size batch should be used?
        b. What is the maximum inventory level?
         c. How many order cycles are there per year?
         d. How much does management of this good in inventory cost the firm each year?

40. Lead-time for one of Azim Manufacturing‟s fastest moving product is 3 days. Demand during this
    period averages 100 units per day. What would be an appropriate re-order point?

41. The new office supply discounter, Paper Clips Etc. (PCE) sells a certain type of ergonomically
    correct office chair, which costs 300 MU. The annual holding cost rate is 40%, annual demand is
    600, and the order cost is 20 MU per order. The lead-time is 4 days. The store is open 300 days
    per year.
                a.What is the optimal order quantity?
                b. What is the reorder point?

42. A toy manufacturer makes its own wind-up motors, which are then put into toys. While the toy
    manufacturing process is continuous, the motors are intermittent flow. Data on the manufacture of
    the motors appears below.
        Annual Demand= 50 000 units                      Daily subassembly production rate = 1 000
        Set-up cost = 65 MU per batch                    Daily subassembly usage rate = 200
        Carrying cost = 0.10 MU per unit-per year
      a. To minimize cost, how large should each batch of subassemblies be?
      b. Approximately how many days are required to produce a batch?
      c. How long is a complete cycle?
      d. What is the total inventory cost (rounded to nearest MU) of the optimal behaviour in this
          problem?

Prof.Dr.Dr.M.Hulusi DEMIR                                                                              71
                     Introduction to Production / Operations Management


43. Jayfer‟s Sewing machines Co. expects next year‟s sales to be 360 000 units. Each production run
    requires an outlay of 100 MU for machine set-up, and each unit is carried in inventory 25% of the
    purchasing price 72 MU. It is estimated that the cost of permitting a back-order is 9 MU/unit/year.
    Each back-order is completed as soon as the production run is completed.
        a. Determine the complete size for each run,
        b. Determine the maximum level of inventory that the manufacturer can expect to have on
      hand.
        c. Find average inventory level.
        d. Calculate the number of runs in a year.
        e. Find how much such a policy will cost to the company.
        f. Determine the stock-out time.

44. Usage: 200 000 units/year               Set-up cost: 80 MU/set-up
    Carrying cost: 25 % of the price        Price: 200 MU/unit        Back-order      cost:     950
    MU/unit-year
      a. Optimal size of each production run?
      b. The maximum level of inventory/
      c. average inventory level?
      d. The back-order quantity?
      e. The optimal number of runs in a year?
      f. The time between runs in a year? (assume 311 days/year)
      g. The total annual inventory cost?
      h. The total annual cost?
      i. What effect does an increase of yearly usage to 400 000 units have on the firm‟s inventory
      policy?

45. One of the top-selling items in the container group at the museum‟s gift shop is a bird-feeder. Sales
    are 18 units per week and the supplier charges 60 MU/Unit.. The cost of placing an order with the
    vendor (supplier) is 45 MU. Annual holding cost is 25% of the feeder‟s value.The museum
    operates 52 weeks/year. Management chose a 390-unit lot size so that orders could be placed less
    frequently.
        a. What is the annual cost of using a 390-unit lot size?
        b. Would a lot size 468 be better?
        c. Find the optimal order size (EOQ).
        d. Find the total inventory cost of the optimal order policy.
        e. Find the minimum annual ordering cost. Show your verification.
        f. Find optimal order number.
        g. How long is the ordering period (in weeks)?
        h. If lead time is 1 week , find the reorder point.

47. A special style of sweater can be purchased by a retail store for 17.85 MU on a one-time
    opportunity. The store plans to offer the sweater at a retail price of 35.85 MU during the season.
    Any sweaters left at the end of the season will be sold for 13.85 MU. It is estimated that demand
    for this item at this location will have a normal probability distribution with a mean of 75 and a
    standard deviation of 21.
   How many of these sweaters should the store stock?

48. Sergio Manufacturing, Inc. makes and sells specialty hubcaps for the retail automobile after-
    market. Sergio‟s forecast for its wire-wheel hubcap is 1 000 units next year. However, the
    production process is most efficient at 8 units per day. Given the following values, solve for the
    optimum number of units per order.
        Set-up cost = 10 MU/run                  Holding cost = 0.50 MU/unit/year
    (Note: This plant schedules production of this hubcap only as needed, during the 250 days/year the
    shop operates.)



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                     Introduction to Production / Operations Management


49. A company is faced with the decision of how many units of product to prepare before the
    tourism season at the local market. Each unit of product costs 3 MU and sells for 12 MU per unit.
    Past records indicate that 3 500 units are enough to prevent any shortage, and this is the number
    prepared before tourism season in the past 10 years. Unsold product is disposed of at a total loss.
    The following data summarizes the sales history.

        DEMAND           FREQUENCY
        2 700                 8
        2 800                12
        2 900                20
        3 000                25
        3 100                15
        3 200                10
        3 300                 5
        3 400                 5
        3 500                10

    a. How many units of this type of product should be prepared prior to tourism sector each year?
    b. What is the long-run expected loss under the current policy?

50. Product X is produced at a rate of 100 units a day. The assembly line uses the product at a rate of 40
    units a day. The firm operates 250 days each year. Set-up costs total 50 MU and the average annual
    holding cost is 0.50 MU/unit-year. Each product X costs 7 MU and requires a lead-time of 7 days.
    Determine;
        a. Optimal Lot Size for each production run,
        b. The reorder point,
        c. The total annual cost of the OLS policy,
        d. The total annual cost
        e. The time between runs,
        f. The time between production runs.

51. A style can be purchased for 32.5 MU a unit before the season, and no additional units can be
    ordered. The product will sell for 64.95 MU during the season, and any units left at the end of the
    season will be sold or 24.95 MU. The probability distribution of demand during the season is
    estimated to be normally distributed with a mean of 160 units and a standard deviation of 45 units.
    Determine the amount of stock to order that will give the maximum expected profit.

52. Sergio Farmerson‟s machine shop uses 2 500 brackets during the course of a year, and this usage is
    relatively constant throughout the year. These brackets are purchased from a supplier 100 kms.
    Away for 15 MU each and the lead-time is 2 days. The holding cost per bracket per year is 10% of
    the unit cost and the ordering cost is 18.75 MU. There are 250 working days per year.
        a. What is the EOQ?
        b. Given the EOQ, what is the average inventory?
        c. What is the annual inventory holding cost?
        d. In minimizing cost, how many orders would be made each year?
        e. What would be the annual ordering cost?
        f. Given the EOQ, what is the total annual cost (including purchase cost)?
        g. What is the time between orders (days)?
        h. What is the reorder point level?

53. Sergio Farmerson (see Problem 52) wants to reconsider his decision of buying the brackets and
    is considering making the brackets in-house. He has determined that set-up costs would be 25 MU
    in machinist time and lost production time, and 50 brackets could be produced in a day once the
    machine has been set-up. Sergio estimates that the cost (including labour time and materials) of
    producing one bracket would be 14.80 MU. The holding costs would be 10% of this cost.
Prof.Dr.Dr.M.Hulusi DEMIR                                                                              73
                     Introduction to Production / Operations Management


        a. What is the daily demand rate?
        b. What is the optimal production quantity?
        c. How long will it take to produce the optimal quantity?
        d. How much inventory is sold during the production run time?
        e. If Sergio uses the optimal production quantity, what would be the maximum inventory level?
        f. What would be the average inventory level?
        g. What is the total annual inventory cost?
        h. What is the reorder point level, if the lead time is one-half day?

54. The annual demand for rackets is 5000 units per year. Machinery set-up costs to produce these
    rackets are 400MU. The annual holding cost is 25 % of the value of the racket. The racket is worth
    45 MU. The production rate is 30 rackets per day. Assume there are 250 working days in a year.
      a. What is the optimal lot size?
      b. The TOTAL annual set-up and inventory holding cost for this item.
      c. The time between runs, or cycle time for OLS?
      d. The production time per lot.
      e. The maximum inventory level and the number of runs in a year.

55. TT Company produces material for National Defence Ministry of Turkey. Projected demand for a
   secret material TT007, equals 200 000 units. Each production run requires an outlay of 80 MU for
   machine set-up. Each unit carried in the inventory costs 50 MU. The estimated cost of a back-order
   is 550 MU. Each back-order is filled as soon as the production run is completed. Determine the
   following:
        a. The optimal size for each production
        b. The maximum level of inventory that TT Co. can expect to have on hand?
        c. The time between runs (assume 250 working days/year)
        d. The annual cost of the optimal system?
        e. The back-order size (shortage quantity) and the optimal number of runs?

56. A style item can be purchased for 65 MU/unit before the season, and no additional units can be
    ordered. The product will sell for 130 MU during the season and any units left at the end of the
    season will be for 50 MU. The probability distribution of demand during the season is estimated
    normally distributed with a mean of 200 units and a standard deviation of 50. Determine the
    amount to stock to order that will give the maximum expected profit?

57. A chemical firm produces Sodium Bisulphate in 100 kg bags. Demand for this product is 20 tons
    per day. The capacity for producing the producing the product is 50 tons per day. Set-up costs
    100 MU, and storage and handling costs are 5 MU per ton per year. (Hint: 1 ton: 1 000 kg : 10
    bags)
         a. How many bags per run are optimal?
         b. Calculate maximum inventory level of this firm.
         c. What would the average inventory be for this lot size?
         d. Determine the approximate length of a production run, in days.
         e. About how many runs/year would there be?
         f. Calculate minimum total inventory cost.
         g. How much could the company save annually if the set-up cost could be reduced to
            25 MU/run?
58. Stitch-in-Time, a manufacturer of sewing machines, expects next year‟s sales to be 180 000 units.
    Each production run requires an outlay of 100 MU for machine set-up, and each unit carried in
    inventory costs 9 MU. It is estimated that the cost of permitting a back-order is 16 MU/unit/year.
    Each back-order is filled as soon as the production run is completed.
         a. Determine the optimal size (quantity) for each production run.
         b. Determine the maximum level of inventory that the manufacturer can expect to have on
            hand.


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                     Introduction to Production / Operations Management


        c. Determine the time between runs.
        d. Find how much such a policy will cost to the company.

59. Product X is a standard item in TT‟s inventory. One of the components/parts of Product X is
    produced within TT‟s facilities at a rate of 100 units/day. The assembly line uses the component at
    a rate of 40 units/day. The firm operates 250 working days/year. Set-up costs total 50 MU and the
    average annual holding cost is 0.50 MU/unit/year. The component costs 7 MU and requires a lead-
    time of 7 days. Using this data determine the following:
        a. Optimal Production Lot Size (OLS)
        b. The reorder point
        c. The annual cost of the optimal lot size policy
        d. What is the TOTAL ANNUAL COST OF PRODUCTION AND INVENTORY SYSTEM?
        e. What is the optimal number of runs per year?
        f. What is the time between runs (in days)?

60. Blast-Off Inc., manufactures Material X. Projected demand for Material X equals 100 000
    units.Each production run requires an outlay of 80 MU/machine setup, and each unit carried in
    inventory costs 25 MU. The estimated cost of a back-order is 600 MU. Each back-order is filled as
    soon as the production run is completed. Determine the following:
        a. The optimal size of each production run?
        b. The maximum level of inventory that the firm can expect to have on hand?
        c. The back-order quantity
        d. The optimal number of production runs in a year
        e. The time between runs (assume 250 days/year)
        f. The total annual cost of the optimal inventory policy
        g. If annual demand is doubled at Blast-Off and a wage increase doubles the set-up cost, what
           effect does this have on Blast-off‟s original inventory policy?

61. Cheap-Shot Sales Inc., uses a fixed-quantity model as the basis for its inventory policy. For the past
    five years, demand has been relatively constant. However, recent demand has become somewhat
    unstable,    and management has asked for an update on its reorder policy. At the present time, the
    reorder point is set at 150 units, a policy that incurs no stock-outs 68% of the time. The following
    data summarizes company records:

              Reorder period
              (Units)_________ Frequency of Use
                50                      15
                100                     21
                150                     32
                200                     16
                250                     10
                300                   _ 6_
                                      100
    Cheap-Shot currently places orders five-times/year and has estimated that the cost of running out of
    stock is 25 MU/unit and holding cost is 30 MU.Calculate the total expected annual cost of each
    Safety Stock options open to Cheap-Shot and choose the best option.

62. Product A is produced at a rate of 200 units a day. The assembly line uses the product at a rate of 80
    units a day. Set-up costs total 25 MU and the average holding cost is 0.50 MU/unit/year. Each
    product A costs 7 MU and requires a lead-time of 7 days. The firm operates 250 days each year.
    Determine;
        a. Optimal Lot Size for each production run,
        b. The reorder point,
        c. The total annual cost of the Optimal Lot Size policy,
        d. The annual cost,
Prof.Dr.Dr.M.Hulusi DEMIR                                                                              75
                     Introduction to Production / Operations Management


        e. The time between runs,
        f. The time between production runs,
        g. The number of runs per year.

63. A company is faced with the decision of how many units of product to prepare before the tourism
    season at the local market. Each unit of product costs 6 MU and sells for 24 MU per unit. Past
    record indicate that 7 000 units are enough to prevent any shortage, and this is the number prepared
    before tourism season in the past 10 years. Unsold product is disposed of at a total loss. The
    following data summarizes the sales history.

        DEMAND FREQUENCY
        5 400          8
        5 600         12                    a. How many units of this type of product should
        5 800        20                        be prepared prior to tourism sector each year?
        6 000         25
        6 200         15                    b. What is the long run expected loss under current
        6 400         10                       policy?
        6 600          5
        6 800          5
        7 000         10

64. Serin Cumbul experiences an annual demand of 220 000 MU for quality tennis balls at the Cyprus
    Tennis Supply Co. It costs Serin 30 MU to place an order and his carrying cost is 18%. How many
    orders per year should Serin place for the balls?

65. Demand = 200 000 units/year                          Set-up cost = 80 MU/set-up
    Holding cost = 50 MU/unit/year                       Back-order cost = 550 MU/unit/year
    Number of days/year = 250 days
        a. Optimal size for each production run?
        b. Maximum level of inventory
        c. Time between runs
        d. The total annual inventory cost
        e. Back-order quantity
        f. Number of runs per year

66. ABC Motor Co. has determined that the cost of being stocked out is 150 MU/unit. The EOQ
    analysis indicates that the company should reorder 10 times a year. Carrying costs are 25
    MU/motor. The company is considering dropping the reorder point from 255 to 220 units. Based
    on the information in the table below, what would you advise the company to do?

        USAGE DURING                    PROBABILITY OF
        REORDER PERIOD                   THIS USAGE
             200                             0.10
             220                             0.08
             240                             0.06
             260                             0.04
             280                             0.02

67. The manager of LEMAR is wondering how many New Year trees to order before December.
    Each tree costs13 MU but retails for 22 MU if sold before New Year. After New Year the trees will
    have no salvage value. On the basis of past records, the manager has developed the following
    table?




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             Demand         Probability
               300             0.05
               350             0.15
               400             0.25
               450             0.20
               500             0.20
               550             0.15

    How many trees should be ordered? (Add your interpretations to every step)

68. A) Sweet cider is delivered weekly to Sergio‟s Cider Bar. Demand varies uniformly between 300
       liters and 500 liters per week. Sergio pays 0.20 MU per liter for the cider and charges 0.80 MU
       per liter for it. Unsold cider has no salvage value and cannot be carried over into the next week
       due to spoilage.
       Find the optimal stocking level and its stock-out risk for that quantity.
    B) Sergio‟s Cider Bar also sells a blend of cherry juice and apple cider. Demand for the blend
      is approximately normal, with a mean of 200 liters per week and a standard deviation of 10
      liters per week. Cost=0.20 MU/liter, Price=0.80 MU/liter, and salvage value is 0 MU.
      Find the optimal stocking level for the apple-cherry blend

69. A large bakery buys flour in 25-kg bags. The bakery uses an average of 4860 bags a year. Preparing
    an order and receiving a shipment of flour involves a cost of 10 MU per order.
    Annual carrying cost is 7.5% of its price, 1000 MU per bag.
    a. Determine the economic order quantity.
    b. What is the average number of bags on hand?
    c. How many orders per year will there be?
    d. Compute the total cost of ordering and carrying flour.
    e. If ordering costs were to increase by 1 MU per order, how much would that effect the
       minimum total inventory cost?

70. As New Year promotion LEMAR is going to sell turkeys. Each turkey will cost LEMAR 8.50 MU
    and will sell them for 11.99 MU each. Since LEMAR is not in the turkey business, they will give
    all unsold turkeys to an orphanage.If demand for turkeys is estimated to be normally distributed,
    with a mean of 550 and a standard deviation of 40, how many turkeys should LEMAR ırder, if one
    order is allowed?

71. TT Distribution Company can purchase TV sets for 285 MU a set and sell these sets at 490 MU
    through regular channels. Any sets unsold at the end of the model year can be sold to another
    distributor, Bauersohn Co. For 215 MU.
    Calculate P(C)* and the distributor‟s recommended order quantity based on the probability
    distribution of demand for the TV sets and the assumption that the distributor can only order these
    new sets one time.

                  Demand          Probability
                  8 and fewer          0.00
                       9               0.27
                      10               0.34
                      11               0.19
                      12               0.12
                      13               0.08
                   14 or more          0.00

72. Gulum Iren, Inc., which sells children‟s art sets, has an ordering cost of 40 MU for the TT-1 set.
    The carrying cost for TT-1 set is 5 MU per set per year. In order to meet demand, Gulum orders
    large quantities of TT-1 seven times a year. The stock-out cost is estimated to be 50 MU per set.
Prof.Dr.Dr.M.Hulusi DEMIR                                                                            77
                      Introduction to Production / Operations Management


     Over the last several years, Gulum has observed the following demand for TT-1 during the lead
     time:

        Demand During Lead Time Probability
                   40                          0.1
                   50                         0.2
                   60                         0.2
                   70                         0.2
                   80                         0.2
                   90                          0.1
                                              1.0
     The reorder point for TT-1 is 60 units. What level of safety stock should be maintained for TT-1?

73. Assume Carpet Discount Store allows shortages and the shortage cost, d, is 2 MU/metre/year. All
    other costs are as follows:
        Annual Demand : 10 000 metres Annual Carrying cost : 0.75 MU/metre/year
        Ordering Cost : 150 MU/order         Total working days : 311 days/year
    Find;
       a) Xo             b) S            c) Imax          d) Ke            e) No       f) to

74. Azim Furniture Co. handles several lines of furniture, one of which is the popular Layback Model
    TT Chair. The manager, Mr. Sergio Farmerson, has decided to determine by use of the EOQ model
    the best quantity to obtain in each order. Mr. Farmerson has determined from past invoices that he
    has sold about 200 chairs during each of the past five years at a fairly uniform rate, and he expects
    to continue at that rate.
    He has estimated that preparation of an order and other variable costs associated with each order
    are about 10 MU, and it costs him about 1.5% per month to hold items in stock. His cost for the
    chair is
    87 MU.
        a. How many layback chairs should be ordered each time?
        b. How many orders would there be?
        c. Determine the approximate length of a supply order in days.
        d. Calculate the minimum total inventory cost.
        e. Show and verify that the total holding cost is equal to the annual ordering cost (due to
             rounding the figures may be approximately equal

75. Suppose that TT Beverage Co. has a soft-drink product that has a constant annual demand rateof 3
    600 cases. A case of the soft drink costs TT 3 MU. Ordering costs are 20 MU per order and holding
    costs are 25% of the value of the inventory. There are 250 working days per year and the lead-time
    is 5 days. Identify the following aspects of the inventory policy.
          a. Economic order quantity.
          b. Reorder point.
          c. Cycle time (in days).
          d. Total annual inventory cost.
          e. A general property of the EOQ inventory model is that total inventory holding and total
              ordering costs are equal at the optimal solution. Use data above to show that this result is
              true.
76. Azim Electronics supplies microcomputer circuitry to a company that incorporates
    microprocessors into refrigerators and other appliances. One of the components has an annual
    demand of 250 units, and this is constant throughout the year. Carrying cost is estimated to be 1
    MU/unit/year and the ordering cost is 20 MU/order.
         a.      To minimize cost, how many units should be ordered each time an order is placed?
         b.      How many orders per year are needed with the optimal policy?
         c.      What is the average inventory if costs are minimized?


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                     Introduction to Production / Operations Management


    d. Suppose the ordering cost is not 20 MU, and Azim has been ordering 150 units each
        time an order is placed. For this order policy to be optimal, what would the ordering
        cost have to be?
77. Azim Accessories produces paper slicers used in offices and art stores. The minislicer has been one
    of its most popular items: Annual demand is 6 750 units and is constant throughout the year.
    Minislicers are produced in batches. On average, the firm can manufacture 125 minislicers/day.
    Demand for these slicers during the production process is 30 minislicers/day. The set-up cost for
    the equipment necessary to produce the minislicers is 150 MU. Carrying costs are 1 MU/minislicer
    per year. How many minislicers should Azim manufacture in each batch?

78. Sergio Farmerson is the owner of a small company that produces electric scissors use to cut fabric.
    The annual demand is for 8 000 scissors, and Sergio produce 150 scissors per day, and during the
    production process, demand for scissors has been about 40 scissors per day. The cost to set-up the
    production process is 100 MU, and it costs Sergio 0.30 MU to carry one pair of scissors for one
    year.
    How many scissors should Sergio produce in each batch?

79. A. The Call-Us Plumbing Co. stocks thousands of plumbing items sold to regional plumbers,
       contractors, and retailers. The firm‟s general manager wonders how much money could be
       saved annually if EOQ were used instead of the firm‟s present rules of thumb. He instructs
       an inventory analyst to conduct an analysis of one material to see if significant savings might
       result from using the EOQ. Necessary information is as follows:
       C = 10 000 units/year           Xcurrent = present order quantity = 400 units/order
       E = 0.40 MU/unit/year           B = 5.50 MU/order

    B. The Co. has an adjacent production department that could produce the item. If the units were
       produced in-house in production lots, they would flow gradually into inventory at the main
       warehouse for use. The carrying cost, ordering or set-up cost and annual demand would
       remain about the same. Because the units actually flow into inventory rather than being
       received all at once as a batch. The firm‟s general manager wonders how this would effect the
       order quantity and annual stocking (inventory) cost.
       The estimates are;
               C = 10 000 units/year             E = 0.40 MU/unit/year          s = 5.50 MU/order
               R = 120 units/day                 1 year = 250 working days

    C. If the general manager to back-order some units and to fill each back-order as soon as the
       order cycle is completed. If the cost estimation indicates back-order cost as 5.60 MU/order.
       Find how this would effect the order quantity and annual inventory cost.

80. The manager of a bottling (bottle-filling)plant which bottles soft drinks needs to decide how long a
    “run” of each type of drink to ask the lines to process. Demand for each type of drink is reasonably
    constant at 80 000 per month (a month has 160 production hours).The bottling lines fill ata rate of 3
    000 bottles per hour but take an hour to change over between different drinks. The cost of each
    changeover (cost of labour and lost production capacity) has been calculated at 100 MU/hour.
    Stock holding costs are counted at 0.1 MU/bottle-month.
        a.       How many bottles the company produce on each run?
        b.       The staff who operate the lines have devised a method of reducing the changeover
                 time from 1 hour to 30 minutes. How would that change the Economic Lot Size?

81. Jantsan Co. makes and sells specialty hubcaps for the retail automobile aftermarket. Jantsan`s
    forecast for its hubcap is 1000 units next year, with an average daily demand of 4 units. However,
    the production process is most efficient at 8 units per day. (So the Co. produces 8 per day but uses
    only 4 per day.) Given the following values, solve for the optimum number of units per run.
    Annual demand = C = 1 000 units               Set-up cost = s = 10 MU
    Holding cost = E = 0.50 MU/unit/year                  Daily production rate = R = 8 units daily
Prof.Dr.Dr.M.Hulusi DEMIR                                                                             79
                   Introduction to Production / Operations Management



     82. As a part of a factory-wide JIT program to reduce set-up times so that production lot sizes
         can be smaller, a firm wants to determine what length of the set-up time of a manufacturing
         operation should be in order to accommodate an OLS of 10 units of a part. A production
         analyst has developed these data for the operation:
          C = 10 000 units/year          c = 250 units/day              R = 500 units/day
          OLS = 10 units/run             E = 5 MU/unit/year             s = ? (to be determined)
         If the labour rate for the operation is 10 MU/hour, what set-up time results in an economic
         production lot size of 10 units?

     83. Carpet Discount Store in Gazimagusa stocks carpet in its warehouse and sells it through an
         adjoining showroom. The store keeps several brands and styles of carpet in stock; however,
         its biggest seller is Super Shag Carpet. Given an estimated annual demand of 10 000 meters
         of carpet, an annual carrying cost of 0.75 MU/meter, and an ordering cost of 150 MU/order,
         the store wants to determine
                a.       the optimal order size
                b.       total inventory cost for this brand of carpet
                c.       total ordering cost and verify it is ½ of total inventory cost
                d.       the number of orders that will be made annually
                e.       the time between orders
                (The store is open 311 days annually.)

     84. Assume that the Carpet Discount Store has its own manufacturing facility in which it
         produces Super Shag Carpet. We will further assume that the ordering cost, B, is the cost of
         setting up the production process to make Super Shag carpet. Daily demand is 32 meters and
         daily production is 32 meters of the carpet. Determine and interpret the optimal lot size.

     85. Assume now Carpet Discount Store allows shortages and the shortage cost, d, is 4
         MU/metre/year. All other costs are as follows;
         Annual demand: 10 000 meters              Annual Carrying Cost: 0.75 MU/meter/year
         Ordering Cost: 150 MU/order               Total working days: 311 days/year
         Find;
               a.     the optimal order size
               b.     the shortage level
               c.     the maximum inventory level
               d.     the total minimum inventory cost
               e.     the total number of orders per year
               f.     the time between orders
               g.     the time during which inventory is on hand
               h.     the time during which there is a shortage

     86. Bur-Al Auto Sales is offering a special car attachment at the unheard-of-price of 2000
         MU/unit. The attachment cost Bur-Al 1400 MU/unit. Unsold units can be salvaged for 600
         MU/unit. Management has projected the following weekly demand pattern.

          Weekly Demand        Probability
           (units)             of Demand
              70                 0.10
              71                 0.15
              72                 0.25
              73                 0.25
              74                 0.15
              75                 0.10
              76 +               0.00



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              a. Using marginal analysis, determine the optimal stock level.
              b. Suppose that restocking is a continual process. If a unit is not sold in one period, it
                 is held over to the next period. However, there is an additional cost of 300 MU for
                 handling and storage. What is the optimal stock level under these conditions?
                 (Use marginal analysis and assume that any unsold unit is held over for one period
                 only.)




Prof.Dr.Dr.M.Hulusi DEMIR                                                                             81
                       Introduction to Production / Operations Management


PERT/CPM
1. A planning consultant has collected the following estimates (days) for optimistic (x), most likely
   (m), and pessimistic (y) times for the activities associated with installation of a new computer
   centre.


            ACTIVITY          x     m       y
                12          4      6      14
                13          4      6      14
                14          2      4       8
                25          6      9      12
                35          3      4       5
                46          8     12      20
                56          1      3       5
                67          2      4       6

     a. Compute the estimated time (te) and the variance (δ2) of each activity. State which activity has the
        most precise time and which has the most uncertain time.
     b.Draw a PERT network of the installation plan in the space below and show “TE”
     c. Show “TE” and “TL” of each event on the network
     d.Find the critical path, duration of the project and mark also the critical path on the network with a
        heavy line.
     e. What is the probability the installation will be completed within a scheduled 5 weeks (25 working
        days)?

2. An advertising campaign uses a network as shown below:

                  Activity         x       m       y
                  12              4       5       6
                  13              3       4       8
                  24              1       2       5
                  25              5       6       9
                  34              2       3       4
                  35              2       3       6
                  46              4       5       6
                  56              3       4       8

     a. Draw a network and label each activity with its expected time and variance.
     b. Calculate the expected completion time and variance for the entire project.
     c. What is the probability that the project is completed in 18 days?
     d. What is the probability that the project be completed in 15 days?
     e. What are the PERT assumptions used to calculate the probability in part (c) realistic in this case?
        Why or why not?
     f. What is the effect of the large variance in activity 13?




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3. Project activities and their time estimates are given in the following table.

        Nodal             Time Estimates (days)
        Sequence          x      m         y___
        1 2              2       3      10
        1 3              5       6       7
        2 3              6     10       14
        2 4              3      6       15
        3 4              2      6       10
        3 5              3      7       11
        4 5              3      6        9
        4 6              1      4        7
        5 6              6     10       14
        5 7              5      7        9
        6 7              6      8       16
        7 8              1      3        5

   a. Draw a PERT network.
   b. Calculate te, TE, and σ2
   c. Find Project Duration Time and Project standard Deviation ( √Σσ2cp)
   d. Find the probability that the task can be completed in 56 days.
   e. Find the probability that the task can be completed in 45 days.

4. A complex NASA project has the following time estimates in weeks:

               Activity   Optimistic      Most likely       Pessimistic      te     σ2
               Time       Time            Time
                1 2       1              2                 4
                2 3       2              4                 6
                2 4       2              6                 10
                3 5       6              8                 10
                4 5       4              6                 8
                4 6       6              10                14               10     1.78
                57        8              10                12               10     0.44
                6 7      12              14                16               14     0.44
                7 8       4              8                 12               8      1.78
                7 9      10              12                16               12.3   1.00
                8 10      2              4                 6                4      0.44
                9 10      6              10                14               10     1.78

   a. Construct a network diagram
   b. Determine te for each activity. Write the answer next to the appropriate letter on the network.
   c. Calculate TE and TL for each node (event). Write your answer on the network above each node.
   d. What is the CRITICAL path? Give it‟s completion time and variance.
   e. What is the slack between the paths containing Event 3 and the critical path?
   f. What is the slack of event 3?
   g. Compute the probability that the project will be completed within 49 weeks?
   h. Compute the probability that the project will be completed within 60 weeks?




Prof.Dr.Dr.M.Hulusi DEMIR                                                                               83
                    Introduction to Production / Operations Management


5.       Activity        Predecessors
            A                C, F
            B                H, I
            C                D
            D                None (Start)
            E                B, J
            F                D
            G                C
            H                C
            I                G
            J                A

     Construct a CPM network for the project.




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Introduction to Production / Operations Management
                         Introduction to Production / Operations Management


LINEAR PROGRAMMING
A. SIMPLEX METHOD
1. Maximize Z = 6A + 3B (revenue)
   Subject to
   Material      20A + 6B  600 1bs
   Machinery 25A + 20B  1000 hrs
   Labour        20A + 30B  1200 hrs
                         A, B  0
   a. What are the optimal values of decision variables and Z?
   b. Do any constraints have (non zero) slack? If yes, which one(s) and how much slack does each have?

2. An appliance manufacturer produces two models of microwave ovens: H and W. Both models require
   fabrication and assembly work; each H uses four hours of fabrication and two hours of assembly; and each
   W uses two hours of fabrication and six hours of assembly. There are 600 fabrication hours available this
   week and 480 hours of assembly. Each H contributes $40 to profits, and each W contributes $30 to profits.
   What quantities of H and W will maximize profits?

3. A small candy shop is preparing for the holiday season. The owner must decide how many bags of deluxe
   mix and how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 pound
   raisins and 1/3 pound peanuts, and the standard mix has ½ pound raisins and ½ pound peanuts per bag. The
   shop has 90 pounds of raisins and 60 pounds of peanuts to work with.
   Peanuts cost $0.60 per pound and raisins cost $1.5 per pound. The deluxe mix will sell for $2.90 per pound,
   and the standard mix will sell for $2.55 per pound. The owner estimates that no more than 110 bags of one
   type can be sold.
   a. If the goal is to maximize profits, how many bags of each types should be prepared?
   b. What is the expected profit?

4. Solve each of these problems by computer and obtain the optimal values of the decision variables and the
   objective function.

   a. Maximize          4x1   +   2x2 + 5x3
      Subject to        1x1   +   2x2 + 1x3      ≤   25
                        1x1   +   4x2 + 2x3      ≤   40
                        3x1   +   3x2 + 1x3      ≤   30
                                    x1, x2, x3   ≥    0
   b. Maximize          10x1+     6x2 + 3x3
      Subject to        1x1 +     1x2 + 2x3      ≤   25
                        2x1 +     1x2 + 4x3      ≤   40
                        1x1 +     2x2 + 3x3      ≤   40
                                    x1, x2, x3   ≥    0

5. The Stevens Fertiliser Co. markets two types of fertiliser, which are manufactured in two departments.
   Type A contributes 3 MU/ton, and Type B contributes 4 MU/ton.

        Department                     Hours/ton           Max. Hours
                         Type A          Type B           worked per week

        I                     2              3                   40
        II                    3              3                   75




    Prof.Dr.Dr.M.Hulusi DEMIR                                                                               71
                           Introduction to Production / Operations Management


     Set up a linear programming problem to determine how much of the two fertilisers to make in order to
     maximise profits. Use simplex algorithm to solve your problem. (Levin, R. et.al. “Quantitative Approaches
     to management)
6. Gul‟s Craft Shoppe manufactures two products in two departments.
   Product X1 contributes 6 MU and takes 6 hours in Dept. 1 and 6 hours in Dept.2.
   Product X2 contributes 14 MU and takes 8 hours in Dept.1 and 12 hours in Dept.2.
   Dept.1 has a capacity of 38 hours and Dept. 2 has a capacity of 42 hours.
   Indicate the maximum production level in units and the maximum monetary units ($ or TL) contribution
   production level, and show the MU contribution between the two.

7. The following is tableau for maximisation problem:

     Cj      Product       Quantity        8       6        0       0        0____
               Mix                         X1      X2       S1      S2      S3
     _________________________________________________________
      8           X1        4 units/day    1       .75      2.5     0        0
      0           S2       4 hours/day     0       .05      -.5     1        0
      0           S3     1.4 houra/day     0       .175     -.75    0        1
                  Zj        32 MU/day      8        6       20      0        0
               Cj-Zj                       0        0       -20     0        0


     a. Is this an optimal solution?
     b. Is there more than one optimal solution to this problem? If so, find another one.
     c. What is the optimal objective value?

8. Solve the following problem using the simplex algorithm.
   Maximise!      D + 2F
   Subject to     D + 3F < 50
                 6D + 9F < 150
                 3D + 8F < 120
                     D, F > 0
   What conclusions can you reach about this problem?

9. Hurşit Manufacturing has contracted to build two products. A and B, for an out- of –state purchaser. The
   purchaser has indicated that all of the units that are manufactured will be bought. Hurşit plans to
   manufacture as many units as possible each operating day. However, capacity restrictions are such that
   Hurşit can produce at most 10 units of an at most 6 units of B per day.
   An analysis of current assembly operation revealed the following: Product A requires 5 man-hours per unit
   and Product B requires 6 man-hours per unit. Product B also requires twice as much inspection time as does
   Product A, which requires 1 man-hour per unit. Hurşit has a maximum of 60 man-hours per day for
   producing both products and at most 16 man-hours for inspection. Product A return a profit of $ 2 per
   unit.and product B returns a profit of $3 per unit. Use the simplex method to determine the most profitable
   daily combination.

10. Gramco Industries operates two assembly lines. Each line is used to produce three grades of quality metal-
    frame toy trailers, small, medium, and large. Daily outputs for each line-product combinations are fixed, as
    shown below.



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                         Introduction to Production / Operations Management


                Trailer frame            Line 1           Line 2
                Small                    300              100
                Medium                   100              100
                Large                    200              600

   On the basis of past records, the firm can expect to sell at least 2400 small metal-frame trailers, and at least
   1600 medium metal-frame trailers, and at least 4800 large metal-frame trailers. Daily production costs for
   the two lines average $600 for Line 1 and $400 for Line 2. Gramco wants to minimize total production cost
   and satisfy demand. Determine the number of days the two lines should run to meet these requirements.

11. Deep-Hole Mining has 1000 tons of B1 grade ore, 2000 tons of B2 grade ore, and 500 tons of B3 grade ore.
    Three products, X1, X2, and X3, can be made from these ores at one of Deep-Hole‟s subsidiaries.
    Management wishes to determine how much of each product to make from the available ores so as to
    maximize the profit from the overall operation. Ore requirements per unit of product produced are as
    follow. (1) Product X1, requires 5 tons of grade B1 ore, 10 tons of grade B2 ore, and 10 tons of grade B3
    ore. (2) Product X2 requires 5 tons of grade B1 ore, 8 tons of grade B2 ore, and 5 tons of grade B3 ore. (3)
    Product X3 requires 10 tons of grade B1 ore, 5 tons grade B2 ore, and none of grade B3 ore. Each unit of
    Product X1 contributes $100 to profit and each unit of Product X2 contributes $200 per unit to profit. Profit
    contribution from Product X3 is $50 per unit.

   a. Set up the appropriate linear program
   b. Determine the optimal mix of products X1, X2, and X3.
   c. Identify any existing unused resource.
   d. What is the optimal profit from Deep-Hole‟s operation at the subsidiary?

12. The Zingo Bakery produces three types of baked goods – bread, rolls, and doughnuts. Bread contributes $2
    per pan to profit. Rolls contribute $4 per pan to profit. Doughnuts contribute $3 per pan profit. Each pan of
    the baked goods passes through three baking centres, where the time in each centre per pan of baked goods
    is as follows.

                               Man-hours per pan
        Product         Centre 1     Centre 2             Centre 3
        Bread             3            2                    1
        Rolls             4            1                    3
        Doughnuts         2            2                    2

   Each one of the three baking centres has a limited amount of man-hours available for the daily operation of
   the bakery. These hours are as follows: Centre 1, 60 man-hours; Centre 2, 40 man-hours; and Centre 3, 80
   man-hours.
   a. Set up the appropriate linear program.
   b. Determine the optimum product mix for Zigo‟s daily operation.
   c. What is the maximum daily profit?




    Prof.Dr.Dr.M.Hulusi DEMIR                                                                                   73
                          Introduction to Production / Operations Management


13. Schurman Orchards has apple trees and cherry trees. The apples and cherries that are grown at Schurman
    Orchards are used to produce both apple cider and cherry cider. Weekly sales commitments by the owners
    of Schurman Orchards require at least 50 gallons of apple cider and at least 20 gallons of cherry cider.
    Schurman Orchards has the weekly capacity to produce at least 100 gallons of apple cider or at least 50
    gallons of cherry cider or any linear combination of apple cider and cherry cider. Each gallon of apple cider
    cost Schurman Orchards $4; each gallon of cherry cider cost $6.
    a. Set up the appropriate linear program
    b. Solve the result of (a) using the simplex algorithm.

14. Each weekend in his spare time, Ali Caliskan uses his wood lathe to produce either Cigar Boxes or
    Cigarette Boxes. He spends 20 hours each weekend in this pursuit. Each cigar box requires 30 minutes
    machine time while each Cigarette box requires 25 minutes of machine time. Next week, Ali has a firm
    commitment to deliver 25 cigar boxes. Otherwise, he can expect to sell as many as many boxes as he can
    produce. Cigar boxes contribute 9MU per box to profit, and Cigarette boxes yield a contribution of 8 MU
    per box. How many of each type of box should Ali make this weekend in order to maximize profit?

15. Bagwell Distributors packages and distributes industrial supplies. A standard shipment can be packaged in
    a class A container, a class K container, or a class T container. A single class A container yields a profit of
    $8; a class K container, a profit of $6; and a class T container, a profit of $14. Each shipment prepared
    requires a certain amount of packing material and a certain amount of time, as seen in the following table:
        Class of                         Packing Material                  Packing Time
        Container                           (Pounds)                          (Hours)____
            A                                     2                                 2
            K                                     1                                 6
            T                                     3                                 4
        Total amount of resource          120 pounds                           240 hours
        Available each week                                                                _

     Bill Bagwell, head of the firm, must decide the optimal number of each class of container to pack each
     week. He is bound by the previously mentioned resource restrictions, but he also decides that he must keep
     his six full-time packers employed all 240 hours (6 workers, 40 hours) each week. Formulate and solve this
     problem using the simplex method.

16. The Roniger Company produces two products: bed mattresses and box springs. A prior contract requires
    that the firm produce at least 30 mattresses or box springs, in any combination. In addition, union labor
    agreements demand that stitching machines be kept running at least 40 hours per week, which is one
    production period. Each box spring takes 2 hours of stitching time, while each mattress takes 1 hour on the
    machine. Each mattress produced costs $20, each box spring costs $24.
    a. Formulate this problem so as to minimize total production costs.
    b. Solve using the simplex method.

17. The Statewide Trucking Company needs to haul 20 tons of fertilizer from Masena to Pottsdam. They can
    use either or both of two types of trucks – model M or model P. Each model M truck is capable of hauling a
    load of 10 tons at a cost of $300 for the trip. Each model P truck can haul 5 tons at a cost of $100 for the
    trip. Because of prior commitments, only two model P trucks can be made available for the scheduled haul.
    Use the simplex method to determine how many of each type of truck should be scheduled to haul the 20
    tons at minimal cost.

18. Use the simplex algorithm to find the optimal solutions to the following linear programming problem.

     Minimize: 3X + 4Y
     Subject to: 3X – 2Y  30
                 X + 2Y  40
                 6X + 8Y  240
                  X,Y  0

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19. Objective function: Max! Z= 9x1 + 7x2
     Subject to: 2xx + x2 ≤ 40
                    x1 + 3x2 ≤ 30
                    x1 , x2 ≥ 0
       a) Solve the above LP problem and give the final solution.
       b) Find the shadow prices for the two constraints.

20. The Magusa Development Co. is building two apartment complexes. It must decide how many units to
       construct in each complex subject to labour and material constraints. The profits generated for each
       apartment in the first complex is estimated at 900 MU, for each apartment in the second complex
       1 500 MU. A partial initial simplex tableau for Magusa is given in the following table:

                   Prod.         900          1 500 0       0
          Cj        Mix Quantity x1             x2 s1       s2
                         3 360   14             4   1       0
                            9 600       10       12     0   1
                    Zj
                   Cj- Zj
          ___________                  ___________________


        a)     Complete the initial tableau.
        b)     Reconstruct the problem‟s original constraints (excluding slack variables).
        c)     Write the problem‟s original objective function.
        d)     What is the basis for the initial solution?
        e)     Which variable should enter the solution at the next iteration?
        f)     Which variable should leave the solution at the next iteration?
        g)     How many units of the variable entering the solution next will be in the basis in the second tableau?
        h)     How much will profit increase in the next solution?

21. Objective function: Maximize Earnings! Z = 0.8x1 + 0.4x2 + 1.2 x3 – 0.1 x4
     Subject to: x1 + 2x2 + x3 + 5x4 ≤ 150
                                 x2` - 4x3 + 8x4 = 70
                            6x1 + 7x2 + 2x3 – x4 ≥ 120
                                    x1, x2, x3 x4 ≥ 0
   a)   Convert these constraints to equalities by adding the appropriate slack, surplus, or artificial variables.
        Also add the new variables into the problem‟s objective function.
   b)   Set up the complete initial simplex tableau for this problem. Do not attempt to solve.
   c)   Give the values for all variables in this initial solution.

22. The management of Parks Resource National Forest is concerned with the influx of visitors to the general
    recreation area. In response to this concern, a recent study was conducted in which it was found that two
    basic categories of visitors used the general recreation area, A and B. The study has also revealed that
    Category B visitors required twice as many as man-hours per week from the park rangers as Category A
    visitors. In addition, the eating area could accommodate 10 of the Category B visitors to 3 of the Category
    A visitors.
    At no point in time were there more than 300 of the Category A visitors in the park.



    Prof.Dr.Dr.M.Hulusi DEMIR                                                                                    75
                         Introduction to Production / Operations Management


     Because of other duties, the park rangers cannot devote more than 400 man-hours/week to the visitors,
     regardless of the category. The eating area could accommodate at most 1 500 persons.
     If the park makes a profit of 2 MU from each Category A visitor and 1.5 MU from each Category B visitor,
     how many of each category should be admitted each week? What is the maximum profit?

23. Azim Co. markets two products: ABC and XYZ. Manufacturing time and monthly capacities are given
    below;
                                manufacturing time maximum hours
                            per unit in hours  available
                              ABC        XYZ        ___ ________
      Machining              4.0         2.0  1 600
      Fitting and Assembly   2.5         1.0  1 200
      Testing                4.5         1.5  1 600
     _________________________________________________________

       The ABC model costs 250 MU and sells for 400 MU. The XYZ model costs 375 MU and sells for 575
       MU. Market demand is such that Azim can sell either product. However, management is interested in
       optimizing its product mix.
        a) Set up the appropriate linear program.
        b) Solve this problem using the simplex algorithm and interpret the resulting solution.

24. Bauersohn Chemical Corporation must produce exactly 2000 kilos of a special mixture of phosphate and
    potassium for a customer. Phosphate costs 10 MU/kg and potassium costs 12 MU/kg. No more than 600
    kilos of phosphate can be used, and at least 300 kilos of potassium must be used. The problem is to
    determine the least-cost blend of two ingredients. (Please indicate the total cost, and quantities of each
    ingredient.)

25. Emre Uslu manufactures inexpensive set-it-up-yourself furniture for EMU students. He currently makes
    two products- bookcases and tables. Each bookcase contributes 6 MU to profit and each table, 5 MU. Each
    product passes through two manufacturing points, CUTTING and FINISHING. Bookcases take 4 hours in
    cutting and 4 hours in finishing. Tables require 3 hours a unit in cutting and 5 in finishing. There are
    currently 40 hours available in cutting and 30 in finishing.
      a. Use simplex algorithm to find the product mix that produces the maximum profit for Emre.
      b. Use whatever computer package is available to solve this problem. (You are not supposed to submit
          this to the instructor.)

  26. The initial matrix of a maximization linear programming problem with all ≤ constraints was found to be
      as follows:

        Cj                    187   45    95     0     0      0
              Product Quantity X1    X2    X3     S1    S2     S3
               Mix_______________________________________________________
        0        S1     600    200   180   80     1     0      0
        0        S2     500    500   0     90     0     1      0
        0        S3     120    40    40    0      0     0      1______
                 Z      0      0     0     0      0     0      0
                 Cj-Zj         187   45    95     0     0      0______

        a. What is the objective function and what are the constraints?
        b. Solve the problem manually.




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 27. Write the following linear program in tableau form and complete the initial tableau. State also which
     variable should enter the basis and which variable should leave the basis for the next iteration (second
     simplex tableau).
       Maximize :              Z = 3X1 + 4 X2
       Subject to :            6X1 – 4 X2  60
                               -2X1 + 4 X2  80
                               12X1 + 16 X2  480
                                       X1, X2  0

 28. A food supplement for livestock is to be mixed in such a way as to contain
       -- exactly 400 kgs of vitamin A,
       -- at least 240 kgs of vitamin B, and
       -- at least 640 kgs of vitamin C.
     The supplement is to be made from two commercial feeds, feed #1 and feed #2. Each bag of feed
     #1 contains 2kgs of A, 6kgs of B and 4kgs of C. A bag of feed #2 contains 4 kgs of A, 1 kg of B
     and 3kgs of C. Each bag of feed #1 costs 5 MU and a bag of feed #2 costs 3 MU.
       a). Formulate the objective function and constraints for a LP problem (i.e.General and standard
           form of LP model).
       b). Set up the initial simplex tableau and state which variable is leaving and which variable is entering
           the solution.

 29. ABC ceramics offers 2 of its best figurines for sale to the general public. Style 1 costs 2MU per unit,
     style 2 costs 1MU per unit. Both figurines are made in a common oven and require the use of a common
     type of clay. Style 1 uses 1.6 kilos of clay and 2 hours of oven time. Style 2 uses 0.8 kilos of clay per unit
     and only 1 hour of oven time. On a weekly bases ABC ceramic has available a minimum of (at least) 32
     kilos of clay, but only 65 hours of oven time. How many figurines of each style should the firm produce
     each week to optimise the operations?

 30. The Sweet Dreams Company produces two products: Bed Mattresses and Box Springs. A prior contract
     requires that the firm produce at least 30 mattresses or box springs, in any combination. In addition,
     union labour agreements demand the stitching machines be kept running at least 40 hour/week, which
     is one production period. Each box spring takes 2 hours of stitching time, while each mattress takes 1
     hour on the machine. Each mattress produce costs 20 MU and each box spring costs 24 MU.
       a. Formulate this problem so as to minimise total production costs.
       b. Solve using the simplex method.

31. Azim Specialties produces wall shelves, bookends, and shadow boxes. It is necessary to plan the
    production schedule for next week. The wall shelves, bookends and shadow boxes are made of oak. The
    company currently has 600 square meters of oak boards. A wall shelf requires 4 sq. meters; a bookend
    requires 2 sq. meters, and a shadow box requires 3 sq. meters.
    The Co. has a power saw for cutting the oak boards. A wall shelf requires 30 minutes, a bookend requires
    15 minutes, and a shadow box requires 15 minutes. The power saw is available for 32 hours next week.
    After cutting, the pieces are hand finished in the finishing department. There are 4 skilled labourers in
    the department, and each labourer is expected to operate for 80 hours next week. A wall shelf requires 30
    minutes of finishing, bookends require 60 minutes and a shadow box requires 90 minutes.
    The company has a commitment to produce 10 wall shelves for Business Department.
    The profit contribution for each wall shelf is 12 MU, for each bookend 7 MU and for each shadow box is 8
    MU.
    The firm normally operates to achieve maximum contribution.
    a. Solve this problem using simplex method.
    b. For maximum contribution, how much of each product should be produced?
    c. How much contribution selling the output will make?



   Prof.Dr.Dr.M.Hulusi DEMIR                                                                                    77
                         Introduction to Production / Operations Management


32. The Cyprus Foundry is developing a long-range strategic plan for buying scrap metal for its foundry
    operations. The foundry can buy scrap metal in unlimited quantities from two sources: IZMIR (IZ) and
    ISTANBUL (IST), and it receives the scrap daily in railroad cars.The scrap is melted down, and lead and
    copper are extracted for use in the foundry processes. Each railroad car of scrap from source IZ yields 1 ton
    of Copper and 1 ton of lead and costs 10 000 MU. Each railroad car of scrap from source IST yields 1 ton
    of copper and 2 tons of lead and costs 15 000 MU. If the foundry needs at least 5/2 tons of copper and at
    least 4 tons of lead per day foreseeable future. How many railroad cars of scrap should be purchased from
    source IZ and source IST to minimize the long-range scrap metal cost?

33. Write the following linear program in tableau form and complete the initial tableau. State also which
    variable should enter the basis and which variable should leave the basis for the next iteration (second
    simplex tableau).

         Objective Function:               Minimize !        Z = 3X1 + 4 X2
         Subject to:                                              6X1 – 4 X2  60
                                                                 -2X1 + 4 X2  80
                                                                12X1 + 16 X2  480
                                                                     X1, X2  0

34. The initial simplex tableau given below was developed by Ilhan Balci. Unfortunately Mr. Balci quit before
    completing this important LP application. Ms. Ayse Sumbul, the newly hired replacement, was
    immediately given the task of using LP to determine what different kinds of ingredients should be used to
    minimize costs. Her first need was to be certain that Balci correctly formulated the objective function and
    constraints. She could find no statement of the problem in the files, so she decided to reconstruct the
    problem from the initial simplex tableau.
      a. What is the correct formulation, using real decision variables (i.e. Xi „s) only?
      b. Which variable will enter this current solution mix in the second tableau? Which basic variable will
           leave? What are the new values of the entering variable?

     Solution         12 18 10 20 7 8                     M 0      0 M      0    M    0     0    M
Cj    Mix           Quantity X\ X2 X3 X4                X5 X6       A1 s2   s3   A3   s4    A4   s5   s6   A6
M      A1              100   1 0 -3 0                    0 0        1 0      0    0    0    0    0    0     0
0     s2              900    0 25   1 2                  8 0         0 1     0    0     0   0    0    0     0
M      A3             250    2 1    0 4                  0 1        0 0     -1    1     0    0   0    0     0
M      A4             150 18 -15 -2 -1                  15 0        0 0      0   0    -1    1    0    0     0
0      s5             300 0     0  0  0                  0 25       0 0     0    0     0    0    1    0     0
M      A6              70 0     0  0  0                  2 6       0 0      0    0    0     0    0    -1    1
      Zj            570 M      21M -14M    - 5M 5M    21M    M     M    0   -M   M -M       M 0       -M   M
     Cj – Zj               12-21M        10+5M       7-21M          0   0    M   0    M     0    0    M    0
                                    18+14M     20-5M         8-M


35. The Double-T Corporation manufactures two electrical products: air-conditioners and large fans. The
    assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air
    conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and
    1 hour of drilling. During the next production period 240 hours of wiring time are available and up to 140
    hours of drilling time may be used.
    Management decides that to ensure an adequate supply of air conditioners for a contract, at least 20 air
    conditioners should be produced. Because Double-T incurred an oversupply of fans in the preceding
    period; management also insists that no more than 80 fans be produced during this production period. Each
    air conditioner sold yields a profit of 25 MU. Each fan assembled may be sold for a 15 MU profit.
    Formulate and solve this LP production mix situation to find the best combination of air conditioners and
    fans that yields the highest profit.




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                           Introduction to Production / Operations Management


36. A commercial fertilizer manufacturer produces three grades X1, X2, and X3, which net the firm 40 MU, 50
    MU, and 60 MU in profits per ton respectively. The products require the labour and materials per batch that
    are shown in the accompanying table.

                                             X1       X2       X3      Total Available
        ---------------------------------------------------------------------------------------------
        Labour hrs                           4        4        5       80 hours
        Raw Material A (kg)                 200      300      300      6000 kg
        Raw Material B (kg)                 600      400      500      5000 kg
        ---------------------------------------------------------------------------------------------
    a) Set up the initial simplex tableau
    b) Use hand calculations (not computer program) to find the mix of products that would yield
       maximum profits.
    c) Indicate what variables are in the final solution and the optimal profit value.

37. A data processing manager wishes to formulate a LP model to help him decide how to use his
    personnel as programmers (X1) or system analysts (X2) in such a way as to maximise revenues (Z).
    Each programmer generates 40 MU/hr in income and system analysts bring in 50 MU/hr.
    Programming work during the coming week is limited to 50 hrs (maximum). The production
    scheduler has also specified that the total of programming time plus two times the system analysis
    time be limited to 80 hrs or less.
    a) State the objective function and constraints.
    b) Set up the initial simplex tableau.
    c) From optimal solution

       How many hrs of time should the manager schedule for systems analysis work?
       How many hrs of time (in total) should be scheduled?
       How much revenue can the firm expect to gain from the optimal scheduling plan?
       How much more revenue would be gained if there were one more hr. of programming work
        available?
       What is the shadow price associated with the 80 hrs total time constraint?
       How much could the systems analysis time be increased?
       What would be the effect upon profits of such a change (i.e. MU amount of increase or
        decrease)?

38. A company producing a standard line and a deluxe line of electric clothes dryers has the following
    time requirements (in minutes) in departments where either model can be processed:

        Activity                            Standard          Deluxe
        ----------------------------------------------------------------------------
        Metal Frame Stamping                    3                 6
        Electric Motor Installation 10                          10
        Wiring                                10                15
        ----------------------------------------------------------------------------

    The standard models contribute 30 MU each and the deluxe 50 MU each to profits. The motor installation
    production line has a full 60 minutes available each hour, but the stamping machine is available only 30
    minutes per hour. There are two lines for wiring, so the time availability is 120 minutes per hour.
        a) State the objective function and constraints.
        b) Use the simplex method to solve the problem manually.


    Prof.Dr.Dr.M.Hulusi DEMIR                                                                               79
                           Introduction to Production / Operations Management



39. The initial matrix of a maximisation LP problem with all ≤ constraints was found to be as       follows:

       Cij →                                187   45      95      0       0       0
       ↓    variable Quantity               X1    X2      X3      S1      S2      S3

        0      S1         600               200   180     80      1       0       0
        0      S2         500               500    0      90      0       1       0
        0      S3         120                40   40      0       0       0       1
              Zj           0                  0    0      0       0       0       0
             Cj – Zj                        187   45      95      0       0       0

        a) What is the objective function?
        b) What are the constraints?

40. ABC Company has contracted to produce a special mix for use in a high grade agriculture fertilizer. The
    contract specifies that ABC Company will provide exactly 1000 pounds of the mix. Three ingredients are
    used in this special mix: Z100, X23, and HC5. Z100 costs $5 per pound; X23 costs $6 per pound; and HC5
    costs $7 per pound. Because of EPA restrictions, no more than 300 pounds of Z100 can be used. However,
    the mix must contain at least 150 pounds of X23 and at least 200 pounds of HC5. What is the least-cost blend
    of two ingredients? (Please indicate the total cost, and quantities of each ingredient.)

41. A manufacturer makes 4 MU profit on each unit of X1 and 2 MU on X2. Each product requires different hours
    of time on each of two machines as shown.

                       X1 req’ts      X2 req’ts           Total Available
            Lathe         6              4                       12 hrs
            Mill          2              8                       16 hrs

        a) State the objective function and constraints
        b) Use the simplex algorithm to find the optimal values of X1 and X2 to maximise profits.

  42. Use the simplex method to maximise objective function
                    Max Z = 20 X1 + 40 X2
       Subject to the constraints
                    3 X1 + X 2 ≤ 9
                    2 X1 + 2 X2 ≤ 10
                              X2 ≤ 4
                         X1, X2 ≥ 0

  43. The initial matrix of a maximisation LP problem with all ≤ constraints was found to be as follows:

       Cij →                                187   45      95      0       0       0
       ↓ Variable Quantity                  X1    X2      X3      S1      S2      S3

        0       S1        600               200   180     80      1       0       0
        0       S2        500               500    0      90      0       1       0
        0       S3        120                40   40      0       0       0       1
              Zj           0                  0    0      0       0       0       0
             Cj – Zj                        187   45      95      0       0       0



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                          Introduction to Production / Operations Management


      a) What is the objective function?
      b) What are the constraints?
  44. The following partial initial simplex tableau is given
      a. Complete the initial tableau
      b. Write the problem in original linear program
      c. What is the value of the objective function at this initial solution
      d. For the next iteration (tableau), which variable should enter the basis, and which variable
         should leave the basis
      e. How many units of entering variable will be in the next solution? What do you think will be
         the value of the objective function after the second simplex tableau?
      f. Find the optimal solution using the simplex algorithm and interpret.

 Cij → Product           Quantity         5       20      25      0        0       0
  ↓    Mix                                X1      X2      X3      S1       S2      S
                      40        2     1     0     1     0     0
                      30        0     2     1     0     1     0
                      15        3     0     -1/2 0      0     1
 _______________________________________________________________
       Zj
       Cj -Z ____________________________________________________

 45. A small construction firm specializes in building and selling single-family homes. The firm offers two
     basic types of houses, MODEL A and MODEL B. Model A houses require 4000 labour hours, 2 tons of
     stone and 2000 board meters of lumber. Model B houses require 10000 labour hours, 3 tons of stone and
     2000 board meters of lumber. Due to long lead times for ordering supplies and scarcity of skilled and
     semi-skilled workers in the area, the firm will be forced to rely on its present resources for the upcoming
     building season. It has 400 000 hours of labour, 150 tons of stone, and 200 000 board meters of lumber.
     What mix of Model A and B houses should the firm construct if Model As yield a profit of 1 000 MU per
     unit and Model Bs yield 2 000 MU per unit? Assume that the firm will be able to sell all the units it builds.

 46. A retired couple supplement their income by making fruit pies, which they sell to a local grocery store.
     During the month of September, they produce apple and grape pies. The apple pies are sold for 1.50 MU to
     the grocer, and the grape pies are sold for 1.20 MU. The couple is able to sell all the pies they produce
     owing to their high quality. They use fresh ingredients. Flour and sugar are purchased once each month.
     For the month of September, they have 1200 cups of sugar and 2100 cups of flour. Each apple pie requires
     3/2 cups of sugar and 3 cups of flour. Each grape pie requires 2 cups of sugar and 3 cups of flour.
      a. Determine the number of grape and the number of aplle pies that will maximize revenues if the couple
         working together can make an apple pie in 6 minutes and grape pie in 3 minutes. They plan to work no
         more than 60 hours.
      b. Determine the amounts of sugar, flour and time that will be unused.

47.   A small firm makes three similar products, which will allow the same three-step process, consisting of
      milling, inspection and drilling. Product A requires 12 minutes of milling, 5 minutes for inspection, and 10
      minutes of drilling per unit; product B requires 10 minutes of milling, 4 minutes for inspection, and 8
      minutes of drilling per unit; and Product C requires 8 minutes of milling, 4 minutes for inspection, and 16
      minutes of drilling. The department has 20 hours available during the next period for milling, 15 hours for
      inspection, and 24 hours for drilling. Product A contributes 2.40 MU per unit to profit, B contributes 2.50
      MU per unit and C contributes 3.00 MU per unit. Determine the optimal mix of products in terms of
      maximizing contribution to profits for the period.




      Prof.Dr.Dr.M.Hulusi DEMIR                                                                                81
                         Introduction to Production / Operations Management



48.   Maximize 10X1+ 6X2+3X3
      Subject to:
             X1+X2+2X3 ≤ 25
            2X1+X2+4X3≤ 40
             X1+3X2+3x3≤ 40 X1, X2,X3≥0

49.   Each weekend in his spare time, Ali Caliskan uses his wood lathe to produce either Cigar Boxes or
      Cigarette Boxes. He spends 20 hours each weekend in this pursuit. Each cigar box requires 30 minutes
      machine time while each Cigarette box requires 25 minutes of machine time. Next week, Ali has a firm
      commitment to deliver 25 cigar boxes. Otherwise, he can expect to sell as many as many boxes as he can
      produce. Cigar boxes contribute 9MU per box to profit, and Cigarette boxes yield a contribution of 8 MU
      per box. How many of each type of box should Ali make this weekend in order to maximize profit?

50.   The Farmerson Company needs to produce 40 units of Product A tomorrow. They can produce on either
      machine X or machine Y or both. Each unit of Product A when processed on machine X takes 30 minutes
      of time, while a unit processed on machine Y takes 25 minutes. It costs the company 2 MU per minute
      and 3 MU per minute respectively to operate machines X and Y. Tomorrow, Machine X has only 10 hours
      available to produce Product A, while Machine Y can be operated as long as desired.
      Construct the model to be used to determine how many hours to schedule on each machine to minimize
      production costs. Use simplex algorithm to solve the model.

51.   Emre Uslu manufactures inexpensive set-it-up-yourself furniture for EMU students. He currently makes
      two products- bookcases and tables. Each bookcase contributes 6MU to profit and each table, 5 MU. Each
      product passes through two manufacturing points, CUTTING and FINISHING. Bookcases take 4 hours in
      cutting and 4 hours in finishing. Tables require 3 hours a unit in cutting and 5 in finishing. There are
      currently 40 hours available in cutting and 30 in finishing.,
       a.Use simplex algorithm to find the product mix that produces the maximum profit for Emre.
       b.Use whatever computer package is available to solve this problem. (You are not supposed to submit
         this to the instructor.)

 52. Alev Yakar assembles stereo equipment for resale in her shop. She offers two products, VCDs and DVDs.
     She makes a profit of 10 MU on each VCD and 6 MU on each DVD. Both must go through two steps in
     her shop –assembly and bench checking. A VCD takes 12 hours to assemble and 4 hours to bench check.
     A DVD takes 4 hours to assemble and 8 hours to bench check. Looking at this month‟s schedule. Alev
     sees that she has 60 assembly hours uncommitted and 40 hours of bench-checking time available. Use
     simplex algorith to find her best combination of these two products. What is the total profit on the
     combination you found?

 53. Solve the following
        Objective function:
                 Minimize!       Z = 2X1+ 7X2 – 3 X3
        Subject to:
                                 3X1 + 2 X3 = 9
                                  2X1 + 3X2 ≥ 4
                                   X1 + X2 ≥ 1
                                  X1, X2, X3 ≥ 0

 54. The Our-Bags-Don‟t-Break (OBDB) plastic bag company manufactures three plastic refuse bags for home
     use: a 5-kg garbage bag, a 10-kg garbage bag, and a 15-kg leaf-and-grass bag. Using purchased plastic
     material, three operations are required to produce each end product: cutting, sealing and packaging. The
     production time required to process each type of bag in every operation and the maximum production time




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    available for each operation are shown (Note that the production time figures in this table are per box of
    each type of bag).



                                        TYPE OF BAG                              TIME
                       5-kg Bag           10-kg Bag      15-kg Bag               AVAILABLE
       Cutting         2 Seconds/Box      3Seconds/Box 3 Seconds/Box             2 Hours
      Sealing          2 Sec./box        2 Sec./Box    3 Sec./Box                3 Hours
      Packaging        3 Sec./Box        4 Sec./Box    5 Sec./Box                4 Hours

    If OBDB‟s profit contribution is 0.10MU for each box of 5-kg bags produced, 0.15MU for each box of 10-
    kg bags, and 0.20 MU for each box of 15-kg bags, what is the optimal product mix?

55. M&D Chemicals produces two products that are sold as raw materials to companies manufacturing bath
    soaps and laundry detergents. Based on an analysis of current inventory levels and potential demand for
    the coming month, M&D‟s management has specified that the combined production for products 1 and 2
    must total at least 700 kgs. Separately, a major customer‟s order for 250 kgs of product 1 must also be
    satisfied. Product 1 requires 2 hours of processing time per kg. While product 2 requires 1 hour of
    processing time per kg, and for the coming month, 1200 hrs of processing time are available. M&D‟s
    objective is to satisfy the above requirements at a minimum total production cost. Production costs are 2
    MU/kg for product 1and 3 MU/kg for product 2.
    Construct the GENERAL SIMPLEX MODEL properly. Place the figures of the model in an initial
    simplex tableau and find which variable is entering and which variable is leaving.




   Prof.Dr.Dr.M.Hulusi DEMIR                                                                               83
                            Introduction to Production / Operations Management


B. ASSIGNMENT METHOD
1. Estimated project completion times (days) for the ABC assignment problem are as follows, make the
   optimal assignment and state the solution time.

        Client
          ↓        Project       1                  2               3
         Ahmet                   10                 15               9
         Hüseyin                  9                 18               5
         Mehmet                   6                 14               3

     (Hint: time=26 days)

2.. ABC Company is an accounting firm that has 3 new clients. Project leaders will be assigned to the three
    clients. Based on the different backgrounds and experiences of the leader, the various leader client
    assignments differ in terms of projected completion times. The possible assignment and the estimated
    completion time in days are:
                                           Client____________
        Project leader           1             2              3
        Ahmet                    10           16             32
        Hüseyin                  14           22             40
        Mustafa                  22           24             34

     What is the optimal assignment? What is the total time required?

3. Assume that in problem 2 and additional employee is available for possible assignment. The following
   table shows the assignment alternatives and the estimated completion time.


                                                      Client_____________
         Project leader           1                  2              3
         Ahmet                    10                16             32
         Hüseyin                  14                22             40
         Mustafa                  22                24             34
         Emine                    14                18             36

     a. What is the optimal assignment?
     b. How did the assignment change compared to the best assignment possible in Problem 2? Was there any
        savings associated with considering Emine as one of the possible project leaders?
     c. Which project leader remains unassigned?

4. A national car - rental service has a surplus of one car in each of cities 1, 2, 3, 4, 5, 6 and a deficit of one car
   in each of cities 7,8,9,10,11,12. The distances between cities with a surplus and cities with a deficit are
   displayed in the matrix below. How should the cars be dispatched so as to minimise the total mileage
   travelled?
                                                  To
                                 7        8       9        10      11        12

             1                    51       82       49      62       35       61
             2                    32       39       59      75       91       60
             3                    37       49       70      61       42       42
        From 4                    55       60       58      62       47       53
             5                    39       50       49      36       40       43
             6                    92       50       50      70       61       40


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5. Five customers must be assigned to five stockholders in a brokerage house estimated profits for the
   brokerage house for all possible assignments are show below:


                                                        BROKERS

                                  1                2                  3             4      5

                       A         $500            $525              $550            $600   $700

                       B         625             575                 700           550    800

                       C         825             650                 450           750    775

  CUSTOMERS            D         590             650                 525           690    750

                       E         450             750                 660           390    550



   a. Use the assignment method to assign the five customers to the five different brokers to maximize profits
      for the brokerage house.
   b. What are the profits from your assignment in part (a)?

6. Baseball umpiring crews are currently in four cities where three-game series are beginning. When these are
   finished, the crews are needed to work games in four different cities. The distances (km) from each of the
   cities where the crews are currently working to the cities where the new games will begin are shown in the
   table below.
                                                To
                From              Kansas Chicago Detroit Toronto

                Seattle           1500        1730          1940          2070_
                Arlington          460         810          1020          1270_
                Oakland           1500        1850          2080            X__
                Baltimore          960         610           400           330__

7. EMU is moving its Business and Economics Faculty into a new building, which has been designed to house
    six academic departments. The average time required for a student to get to and from classes in the building
    depends upon the location of the department in which he or she is taking the class. Based on the distribution
    of class loads, the dean estimated the following mean student trip times in minutes, given the departmental
    locations.

                                      L O    C     A    T   I    O    N
                             A        B        C            D               E        F
                1           13        18      12            20             13        13
                2           18        17      12            19             17        16
                3           16        14      12            17             15        19
                4           18        14      12            13             15        12
                5           19        20      16            19             20        19
                6           22        23      17            24             28        25




    Prof.Dr.Dr.M.Hulusi DEMIR                                                                                 85
                         Introduction to Production / Operations Management


8.      A national car rental service has a surplus of one car in each of cities 1,2,3,4,5,6, and a deficit of one
        car in each of cities 7,8,9,10,11,12. The distances between cities with a surplus and cities with a deficit
        are displayed in the matrix below.
        How should the car be dispatched so as to minimize the total mileage travelled?
                                                  To
                          7             8           9            10            11           12

                 1       41           72           39           52           25             51

                 2       22           29           49           65           81             50

         From 3          27           39           60           51           32             32
                 4       45           50           48           52           37             43
                 5       29           40           39           26           30             33
                 6       82           40           40           60           51             30


9.      Merkez Kooperatif Bank, headquartered in Lefkoşa, wants to assign 3 recently hired EMU graduates,
        Cemal, Beton and Halil to branch offices in Lefke, Girne and Güzelyurt.But the bank also has an
        opening in DAU Campus Branch and would send one of the three there if it were more economical than
        to move to Lefke, Girne or Güzelyurt.
        It will cost 1000 MU to relocate Cemal to DAU Campus Branch, 800 MU to relocate Beton there and
        1500 MU to move Halil.
        What is the optimal assignment of personnel to branches.

                                              Lefke             Girne          Güzelyurt
                       Branch

                       Hire
                           Cemal            800 MU            1100 MU             1200 MU

                              Beton         500 MU            1600 MU             1300 MU

                              Halil         500 MU            1000 MU             2300 MU


10.   An electroplating shop scheduler has four jobs to schedule through a plating operation. Some jobs can be
      done in any one of the five plating tanks, but some of the tanks are restricted to a specific use. The
      scheduling alternatives and variable costs of power, plating material, and labour are shown in the table.
      Which assignment of jobs to plating tanks will minimize the total cost?
                _________________________________________
                                __ Plating Tank Cost (MU)_____
                _Job            1        2        3       4      5_____
                  A             120     -        100     -       200
                  B             80      70       50      130     300
                  C             40      70       90      -       180
                  D             110     -        150     -       190____

11.     A market research firm has three clients who have each requested that the firm conduct a sample
        survey. Four available statisticians can be assigned to these three projects; however, all four
        statisticians are busy, and therefore each can handle only one of the clients. The following data show
        the number of hours required for each statistician to complete each job; the differences in time are
        based on experience and ability of the statisticians.


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                           Introduction to Production / Operations Management


                                    C   L   N T S
                                             I   E
                 Statistician       A       B     C
                 _____________________________________________________
                          I        300     420    540
                          II       340     460    440
                          III       360    460    450
                          IV        320    480    460

12.      The Izmir Aerospace Company has just been awarded a rocket engine development contract. The
         contract terms require that at least five other smaller companies be awarded subcontracts for a portion
         of the total work. So Izmir requested bids from five small companies ( A, B, C, D, and E ) to do
         subcontract work in five areas (I, II, III, IV and V ). The bids are as follow:
         Cost information:

                                            Subcontract bids
                             I               II            III               IV             V
         Company
            A           45000MU         60000MU          75000MU        100000MU       30000MU
             B            50000             55000          40000          100000         45000
             C            60000             70000          80000          110000         40000
             D            30000             20000          60000           55000         25000
             E            60000             25000          65000          185000         35000


         a). Which bids should Izmir accept in order to fulfil the contract terms at the least
             cost?
         b). What is the total cost of subcontracts?

13.      NG Marketing Research has four project leaders available for assignment to three clients. Find the
         assignment of project leaders to clients that will minimize the total time to complete all projects. The
         estimated project completion times in days are as follows:

                          Project              C l i e n t
                          Leader            1       2      3
                          Emre              10      15     9
                          Baran              9      18     5
                          Berkay            6       14     3
                          Sevki              8      16     6


14.      In a job shop operation, four jobs may be performed on any of four machines. The hours required for
         each job on each machine are presented in the following table. The plant supervisor would like to
         assign jobs so that total time is minimized. Use the assignment method to find the best solution.

              M A C      H I N ES
              W X           Y    Z_
J       A      10 14      16   13_
O       B     12 13      15   12_
B       C      9 12      12    12_
S       D     14 16      18   16_




      Prof.Dr.Dr.M.Hulusi DEMIR                                                                               87
                       Introduction to Production / Operations Management


15.   Use the assignment method to obtain a plan that will minimize the processing cost in the following
      table under these conditions:
      a.      The combination 2-D is undesirable.
      b.      The combinations 1-A and 2-D are undesirable.

                Machi ne
                 A  B         C        D       E__
          1    14   18        20       17      18_
          2    14   15        19       16      17_
Job       3    12   16        15       14      17_
         4    11    13        14       12      14_
        5     10    16        15       14      13_


16.   Human Care Laboratories has just been notified that it has received three government grants. The lab
      administrator must now assign research directors to each of these projects. There are four researchers
      available now who are free from other duties. The time required to complete the required research
      activities will be the function of experience and ability of the research director who is assigned to the
      project. The lab administer has estimated the project completion time (in weeks) for each director-grant
      combination. What assignments should be made to minimize the total time?

                              Grant
                      1       2   3__
               NG     80      90  54_
              TT      54      108 30_
              SC      46      104 48_
              SA      72      96  48_

17.   A shop has four machinists to be assigned to four machines. The hourly cost of having each machine
      operated by each machinist as follows.

                                       Machine
              Machinist            A    B    C       D
                    1             12     11     8      14
                    2            10     9   10         8
                    3            14     8     7      11
                    4            6     8     10      9

      However, because he does not have enough experience machinist 3 cannot operate Machine B.
      Determine the optimal assignment and compute total minimum cost.

18.   The Santapharma pharmaceutical firm has five salespersons, whom the firm wants to assign to five
      sales regions. Given their various contacts, the salespersons are able to cover the regions in different
      amounts of time. The amount of time (days) required by each salesperson to cover each city is shown in
      the following table. Which salesperson should be assigned to each region to minimize total time?
      Identify the optimal assignments and compute total minimum time.

                                               R E G I O N

              SALESPERSON              A       B         C     D        E
                   1                   17      10        15    16       20
                   2                   12       9        16     9       14
                   3                   11      16        14    15       12
                   4                   14      10        10    18       17
                   5                   13      12         9    15       11

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19.      The Bunker Manufacturing firm has five employees and six machines and wants to    assign     the
         employees to the machines to minimize cost. A cost table showing the cost incurred by each
         employee on each machine follows. Because of union rules regarding    departmenta      transfers,
         employee 3 cannot be assigned to Machine E and employee 4 cannot assign to Machine B. Solve this
         problem, indicate the optimal assignment and compute total minimum cost.

                                                           Machine
                 Salesperson      A       B       C        D       E         F
                        1         12      7       20       14      8         10
                        2         10      14      13       20      9         11
                        3           5      3       6        9      7         10
                        4          9      11       7       16      9         10
                        5         10       6      14         8     10        12

20.      The Business Administration Department head of EMU has five instructors to be assigned to four
         different courses. All of the instructors have taught the courses in the past and have been evaluated by
         the students. The rating for each instructor for each course is given the following table (a perfect score
         is 100). The department head wants to know the optimal assignment of instructors to courses that will
         maximize the overall average evaluation. The instructor who is not assigned to teach a course will be
         assigned to grade exams.

                        __________________________________________________
                                            Course             _      __
                 Instructor     A           B            C            D_
                        1      80           75           90           85_
                        2      95           90           90           97_
                        3      85           95           88           91_
                        4      93           91           80           84_
                        5      91           92           93           88_

21.      Sergio‟s Department Store has six employees available to assign to four departments in the      store-
         home furnishings, china, appliances, and jewelry. Most of the six employees have worked in each of
         the four departments on several occasions in the past and have demonstrated that they perform better in
         some departments than in others. The average daily sales for each of the employees in the each of the
         four departments are shown in the following table.

                                                  Department Sales (MU)
                 Employee         H.Furn.         China        Appl.                Jewelry
                       1          340             160          610                  290
                       2          560             370          520                  450
                       3          270             --           350                  420
                       4          360             220          630                  150
                       5          450             190          570                  310
                       6          280             320          490                  360

         Employee 3 has not worked in the china department before, so the manager does not want to assign this
         employee to china.
         Determine which employee to assign to each department and indicate the total expected daily sales.

22.      Cem Tanova, Chairman of EMU‟s Business Department, has decided to use decision modelling to
         assign professors to courses next semester. As a criterion for judging who should teach each
         course, Tanova reviews the past two years‟ teaching evaluations (which were filled by students). Since
         each of the four professors taught each of the four courses at one time or another during the two-year

      Prof.Dr.Dr.M.Hulusi DEMIR                                                                                 89
                        Introduction to Production / Operations Management


      period, Tanova is able to record a course rating for each instructor. These ratings are shown in the
      following table. Find the best assignment of professors to courses to maximize the overall teaching
      ratings.

                              Courses
                         QM MRKT MIS OR
      Instructors    TT  90  65     95 40
                     MHD 70  60     80 75
                     SF  85 40    80 60
                     NG  55  80     65 55

23.   A local college is sponsoring a community job fair that requires hiring 4 temporary employees to
      handle 4 separate tasks. The table below provides the number of approximate hours each employee
      would require to perform each task along with their hourly labour costs. Assuming each employee can
      be assigned only one task, assign employees to tasks in a manner that minimizes total labour costs.

              Employee       Mailings Phone Registration Set Up     Hrly.Labour
                                       Calls                        Costs (MU)
              Osman          15         11         8        6               10
              Kadriye        19        10          5        8               15
              Gokhan         14        13       7           5               14
              Hale          17           8        6         4               12




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C. TRANSPORTATION METHOD
1. You have begun a business of your own and have decided to produce one or more of products A, B, C, and
   D. You have approached four banks – W, X, Y, and Z – with your ideas on these projects in order to obtain
   the necessary financing. The following table reflects the level of financing required for each project, the
   interest rate each of the banks is willing to charge on loans for each of the projects, and the total line of
   credit each of the banks is willing to lend you.

                                                     PROJECT
                                                  ( Interest Rate )
                                                                                                MAX
                     BANK                   A              B           C          D            CREDIT
                      W                   16%             18%         19%       17%            $20,000
                       X                   15              17          20        16            10,000
                       Y                  17             16            18         18            20,000
                        Z                 18             19            19         18            30,000
             AMOUNT REQUIRED            $40,000        30,000        20,000     20,000


    As each project should be as attractive profitwise as any other, you have decided to undertake all or part of
    any number of projects you can at the lowest total interest cost. Which projects should you adopt and from
    which banks should you finance them?

2. A Company has 6 warehouse and 4 stores. The warehouses altogether have a surplus of 17 units of
   a given commodity, divided among the 4 stores. Costs of shipping one unit of the commodity from
   warehouse i to store j are displayed in the following matrix. Find feasible (not necessarily optimal)
   solutions, and the cost associated with each.

                 Warehouses
                                    A       B         C          D         E       F       Required
               Stores

                      I             5       3          7         3          8      5             3
                      II            5       6         12         5          7     11             4
                      III           2       8         3          4          8      2             2
                      IV            9       6         10         5         10      9             8
                  Available                                                                          17
                                    3       3         6          2          1      2
                                                                                           17

3. Solve the following transportation problem with Vogel‟s approximation method and show the calculations
   and find the minimum feasible solution.
                                  TO
                                          Manisa           Aydm       Muğla     Factory Totals
            FROM

                      Izmir                  31             21         42                400
                    Istanbul                 20             21         30                100
                    Ankara                   23             20         15                600

               Warehouse Totals             300            900         800


    Prof.Dr.Dr.M.Hulusi DEMIR                                                                                 91
                          Introduction to Production / Operations Management




4. ABC Air Conditioners operates factories in four different cities. Each of these factories is responsible for
     maintaining warehouse supplies in 5 different warehouses. Because of varying distances, transportation
     charges from factory to warehouse are not uniform. Shipping charges per unit are summarized below:

                                            WAREHOUSE
               FACTORY                  1   2  3 4    5_
                 F 1________            8   9 12 7 18
                 F 2________            6   8 13 9    21
                 F3                    20   7 10 11    8
                 F4                    12   7 14 15 22

     Factory output and warehouse supplies that must be maintained are as follows:

     Factory Units produced/day                   Warehouse        Daily Supply
        #1             35                               1                 15
        #2             25                               2                 12
        #3             40                               3                 22
        #4             32                               4                 30
                                                        5                 20
     Determine;
     c. The best possible factory-to-warehouse shipping program using Vogel‟s Approximation Method.
     d. What is the cost of this shipping program?

5. The YUHUA Disk Drive Co. Produces drives for personal computers. YUHUA produces drives in three
   plants (factories) located in IZMIR/TURKEY, MAGUSA/TRNC and BEIJING/CHINA. Periodically,
   shipments are made from these three production facilities to four distribution warehouses located in
   Turkey, namely: ISTANBUL, ANKARA ADANA and DIYARBAKIR. Over the next month, it has been
   determined that these warehouses should receive the following proportions of the company‟s total
   production of the drives.

         Warehouse                % of Total Production
            Istanbul                       30
            Ankara                         30
            Adana                          15
            Diyarbakir                     25
     The production quantities at the factories in the next month are expected to be (in thousand of units)

           Plant                  Anticipated Production (000 units)
            Izmir                         50
           Magusa                        140
           Beijing                       110

     The unit cost for shipping 1000 units from each plant to each warehouse is given in the table below. The
     goal is to minimize total transportation cost. (use VAM)
     (Hint: When finding total production at the three plants you may round the figures to the nearest unit)
     Shipping costs per 1000 units in MU:
                        Istanbul Ankara Adana Diyarbakir
             Izmir         250        420       380        280
             Magusa       1280         990     1440       1520
             Beijing      1550        1420      1660      1730




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6. ABC ship supplies from 4 principal manufacture to four regional stores. The manufactures are located at
   Izmir, Manisa, Aydin and Denizli. The regional stores are located in Isparta, Burdur, Antalya and Afyon. In
   order to reduce the cost of meeting demand for supplier, ABC has decided to allocate its material according
   to the standard transportation model. An analysis of daily shipping records reveals that the following costs
   per unit are typical for the current shipping operations.

                           To                                                         SHIPME
                                   Isparta          Burdur   Antalya        Afyon
                   From                                                                 NT
                   Izmir              44              22        30           20          70
                   Manisa             34              28        26           15          50
                   Aydin              25              30        34           40          90
                   Denizli            32              40        22           25         100
                     NEEDS            90              50        60           80
     c. Determine an initial shipping program
     d. Calculate the daily cost of this program.

7.
                            To                                                        Excess
                                      W              X         Y             Z
                  From                                                                Supply
                        A             12             4         9             5            55
                        B              8             1         6             6            45
                       C               1             12        4             7            30
                  Unfilled
                                      40             20        50           20
                  Demand

     Use Vogel‟s Approximation method to find an initial assignment of the excess supply.

8. The purchase agent of Magusa Plumbing Co. wishes to purchase 3 000 meters of pipe A, 2 000 meters of
   pipe B and 3 000 meters of pipe C. Three manufacturers (X,Y, and Z) are willing to provide the needed
   pipe at the costs given below (in MU per 1 000 meter). Magusa Plumbing wants delivery within I month.
   Manufacturer X can provide 6 000 meters, Manufacturer Y can provide 5 000 meters and Manufacturer Z
   can provide 3 000 meters. Determine Magusa Plumbing Co‟s least–cost purchasing plan for the pipe should
   be? (Use VAM method)

                                                     Types of Pipe
                                               (cost MU/ 1000 Meters)
                                            A             B            C          Available
                           X               580           600          520
                           Y               620           560          580
                           Z               600           580          580
                         Amount
                         Needed


9. The Demir Manufacturing Company has orders for three similar products:

                  PRODUCT Orders(Units)
                       A      2000
                       B       500
                       C     1200




     Prof.Dr.Dr.M.Hulusi DEMIR                                                                              93
                                Introduction to Production / Operations Management


           Three machines are available for the manufacturing operations. All three machines can produce all the
           products at the same rate. However, due to varying defect percentages of each product on each machine, the
           unit costs of the products vary depending on the machine used. Machine capacities for the next week, and
           the unit costs, are as follows:

                       MACHINE        Capacity (units)
                       1               1 500
                       2               1 500
                       3               1 000

                              Product
                          Machine     A                     B             C___
                              1     1.00 MU              1.20 MU        0.90 MU
                              2     1.30 MU              1.40 MU        1.20 MU
                              3     1.10 MU              1.00 MU        1.20 MU

           Use TRANSPORTATION MODEL to develop the minimum-cost production schedule for the products and
           machines.

10.         During the Gulf War, Operation Desert Storm required large amounts of military material and supplies to
            be shipped daily from supply depots in the USA to bases in the Middle East. The critical factor in the
            movement of these supplies was speed.
           The following table shows the number of planeloads of supplies available each day from each of six supply
           depots and the number of daily loads demanded at each of five bases. (each planeload is approximately
           equal in tonnage). Also included are the transport hours per plane, including loading and fuelling, actual
           flight time, and unloading and refuelling.
           Determine the OPTIMAL DAILY FLIGHT SCHEDULE that will minimize total transport time.

                Military
         Supply Base
         Depot                   A     B      C     D      E     Supply
         _________________________________________________ _________
             #1                  36    40     32    43     29    14
             #2                  28    27     29    40     38    20
             #3                  34    35     41    29     31    16
             #4                  41    42     35    27     36    16
             #5                  25    28     40    34     38    18
             #6                  31    30     43    38     40     6
      ____________________________________________________
             Demand              18    12     24    16     20

11.        An air conditioning manufacturer produces room air conditioners at plants in Houston,   Phoenix,    and
           Memphis. These are sent to regional distributors in Dallas, Atlanta and Denver. The shipping costs vary,
           and the company would like to find the least-cost way to meet the demands at each of the distribution
           centers.
           How many units should be shipped from each plant to each regional distribution center? What is the total
           cost for this?
                                                                            FACTORY
               FROM \ TO DALLAS               ATLANTA          DENVER CAPACITY_
               HOUSTON            8               12              10             850____
               PHOENIX           10               14               9             650____
               MEMPHIS           11                 8             12             300____
              WAREHOUSE 800                      600             200
               REQUIREMENTS_____________________________________________

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12.    A soft drink manufacturer has recently begun negotiations with brokers in areas  where it intends to
       distribute new products. Before making final agreements, however, the firm wants to determine shipping
       routes and costs. The firm has 3 plants with capacities as follows:

         Plant Capacity
               (cases/week)
         Metro 40 000
         Ridge 30 000
         Colby 25 000

         Estimated demands in each of the warehouse localities are:
         Warehouse       Demand
                       (cases/week)
         SR 1          24 000
         SR 2          22 000
         SR 3          23 000
         SR 4          16 000
         SR 5          20 000

         The estimated shipping cost/case for the various routes are:
                                         TO
         FROM            SR 1     SR2       SR3 SR 4 SR5
         Metro            0.80    0.75      0.60 0.70 0.90
         Ridge            0.75    0.80      0.85 0.70 0.85
         Colby            0.70    0.75      0.70 0.80 0.80

         Determine the feasible shipping plan that will minimize total shipping cost (using VAM).

13.    TRNC has three major power-generating companies (A,B, and C). During the months of peak demand,
       KIB-TEK authorizes these companies to pool their excess supply and to distribute it to smaller
       independent power companies that do not have generators large enough to handle demand. Excess supply
       is distributed on the basis of cost/kw-hr. transmitted.
       The following table shows the demand and supply in millions of kw-hrs. and costs per kw-hr of
       transmitting electric power to four small companies in cities of Girne, Guzelyurt, Lefkosa and
       Gazimagusa.

            To Girne Guzelyurt    Lefkosa Gazimagusa Excess
      From                                              Supply_
       A       12 MU    4MU       9 MU       5 MU         55
       B        8       1         6         6             45
       C        1      12         4         7             30___
      Unfilled
      Power     40     20        50        20
      Demand______________________________________________

       Use Vogel‟s Approximation Method to find an initial transmission assignment of the excess power
       supply.

14.    A concrete company transports concrete from three plants to three construction    sites. The supply
       capacities of the three plants, the demand requirements at the     three sites, and the transportation
       costs per ton as follows.




      Prof.Dr.Dr.M.Hulusi DEMIR                                                                           95
                          Introduction to Production / Operations Management


         _____________________________________________________________
             Site
                             A             B            C      Supply (tons)
         Plant                        _________________________________
           1                 8             5            6      120
         ______________________________________________________________
           2                15           10           12        80
         _______________________________________________________________
           3                 3             9          10        80
         _______________________________________________________________
         Demand (tons)     150            70         100 __________________

         Solve this problem using Vogel‟s approximation method.

15.   Oranges are grown, picked, and then stored in warehouses in Yesilyurt, Lefke and Girne. These warehouses
      supply oranges to markets in Lefkosa, Magusa, Iskele and Mersin. The following table shows the shipping
      costs per truckload (100 MU), supply and demand.
          __________________________________________________________
                   TO
          FROM             LEFKOSA MAGUSA ISKELE MERSIN SUPPLY
          YESILYURT            9           14        12        17        200___
          LEFKE               11          10         6        10        200____
          GIRNE              12            8        15        7         200____
          DEMAND            130          170       100      150 ____________
      Solve this problem using VAM.

16.   A manufacturing firm produces diesel engines in four cities – Bursa, Manisa, Kayseri and Trabzon. The
      company is able to produce the following numbers of engines per month.

         Plant           Production
         1. Bursa                5
         2. Manisa               25
         3. Kayseri              20
         4. Trabzon              25
         Three trucking firms purchase the following numbers of engines fot their plants in three cities.

         Firm            Demand
         A. Gaziantep       10
         B. Adana          20
         C. Konya         15

      The transportation costs per engine (100 MU) from sources to destinations are shown in the following
      table. Solve this problem by using VAM and find feasible total transportation cost.
          _____________________________________________
                    To
          From             A              B              C__
          1                7              8              5___
          2                6              10             6___
          3                10             4              5___
          4                3              9              11___

17.. A large manufacturing company is closing three of its existing plants and intends to transfer some of its
     more skilled employees to three plants that will remain open. The number of employees available for
     transfer from each closing plant is as follows.



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                       Introduction to Production / Operations Management


       Closing Plant           Transferable Employees
          1                                60
          2                               105
          3                                70
                               Total      235

       The following number of employees can be accommodated at the three plants remaining open.
       Open Plants           Employees Demanded
               A                             45
               B                             90
               C                             35
                             Total         170

       Each transferred employee will increase product output per day at each plant as shown in the following
       table. The company wants to transfer employees to ensure the maximum increase in product output.
               To
       From                    A               B               C
               1               5               8               6
               2               10              9               12
               3               7               6               8
       Solve this problem by using VAM.

 18.   A Company has 5 warehouse and 5 stores. The warehouses altogether have a surplus of 32 units of a
       given commodity, divided among the 5 stores. Costs of shipping one unit of the commodity from
       warehouse i to store j are displayed in the following matrix. Find feasible (not necessarily optimal)
       solutions, and the cost associated with each.


     MARKET            A          B           C          D          E          Available
           
WAREHOUSE
I                      73         40          9          79         20         8
II                     62         93          96         8          13         7
III                    96         65          80         50         65         9
VI                     57         58          29         12         87         3
V                      56         23          87         18         12         5
Required               6          8           10         4          4                  32
                                                                               32




  Prof.Dr.Dr.M.Hulusi DEMIR                                                                               97
     Introduction to Production / Operations Management




98
                     Introduction to Production / Operations Management


BREAK-EVEN ANALYSIS
1. The owner of Double-T Pizza is considering a new oven in which to bake the firm‟s signature dish
   “Vegetarian Pizza”.
   Oven A type can handle 20 pizzas an hour. The fixed costs associated with oven A are 20 000 MU
   and the variable costs are 200 MU/pizza. Oven B is larger and can handle 40 pizzas an hour. The
   fixed costs associated with Oven B are 30 000 MU and the variable costs are 1.25 MU/pizza. The
   pizzas sell for 14 MU each.
   a. what is the break-even point for each oven?
   b. if the owner expects to sell 9 000 pizzas, which oven should the owner purchase?
   c. if the owner expects to sell 12 000 pizzas, which oven should the owner purchase?
   d. at what volume should the owner switch ovens?

2. Mrs. Gulmez KARGUDER, owner of the buffet at the entrance of the faculty building, wanted to
   develop a break-even report for her food-service operations. She developed the following table
   showing the suggested selling prices, and her estimate of the variable costs, and the percent
   revenue by item. It also provides and an estimate of the percentage of the total revenues that would
   be expected for each of the items based on the historical sales data.

          Item                  Selling          Variable        Percent
                                Price/unit       cost/unit       revenue
        Soft Drink              1.50 MU/unit     0.75 MU/unit     25%
        Coffee                  2.00             0.50             25
        Hot Dogs                2.00             0.80             20
        Hamburgers              2.50             1.00             20
        Misc.Snacks             1.00             0.40             10

   Fixed Costs = 105 850 MU
   a. What is the break-even sales for the Buffet (in MU) ?
   b. What her unit sales would be at break-even for each item?
   c. What the expected profit would be, if the sales is 200 000 MU?
   d. What her unit sales would be at 100 000 MU profit for each item?

3. A company produces product A which is sold for 300 MU each. At volume of 100 units per month,
   their labor, materials, overhead and other costs total is 40,000 MU and a volume of 500 units per
   month the total is 100,000 MU.
   a. What is your best estimate of the variable cost per unit?
   b. Now the general manager is considering the addition of a new machine to its present assembly
      line which is expected to reduce variable cost by 10% per unit; however, it will add 20,000 MU
      to total fixed cost. Given that the current production volume is 500 units per month, and assume
      no other change, should the company purchase the new machine? Why or why not?

4. DEMIR Furniture Co. manufactures and sells bedroom suites. Each suite costs 500 MU and sells
   for 800 MU. Fixed costs at DEMIR Furniture total 150.000 MU.
   Determine the breakeven-point using
   a. Algebraic analysis
   b. The general formula approach.

5. TT Co. Ltd. Sells four basic products: Ovens, refrigerators, washing machines, and electric fans.
   During preceeding year, the total fixed cost associated with the four products was 420 000 MU.
   The respective sales volumes, unit prices and unit costs are summarized below.


Prof.Dr.Dr.M.Hulusi DEMIR



                                                                                                       91
                    Introduction to Production / Operations Management


       Product         Unit Sales        Unit variable
                       Volume Unit Price    Cost
       Oven             2000      500 MU      450 MU
       Refrigerator     1000      600         500
       Wash.Mach.       5000      320         280
       Elect.Fan        4000      200         160


6. Process A has fixed costs of 80 000 MU per year and variable cost of 18 MU/unit, whereas Process
   B has fixed costs of 32 000 MU per year and variable costs of 48 MU/unit.
   At what production quantity X0 are the total costs of A and B are equal?




Prof.Dr.Dr.M.Hulusi DEMIR


92
ANSWERS TO SELECTED QUESTIONS
                      Introduction to Production / Operations Management


INTRODUCTION TO PRODUCTION/OPERATIONS MANAGEMENT
A. TRUE OR FALSE

1.      F                16.       T              31.   T         46.      T
2.      T                17.       F              32.   F         47.      T
3.      T                18.       F              33.   T         48.      T
4.      F                19.       F              34.   T         49.      T
5.      F                20.       T              35.   F         50.      T
6.      F                21.       F              36.   F         51.      F
7.      F                22.       T              37.   T         52.      F
8.      T                23.       T              38.   T         53.      T
9.      F                24.       F              39.   F         54.      T
10.     T                25.       F              40.   F         55.      T
11.     T                26.       F              41.   F         56.      T
12.     F                27.       F              42.   F         57.      T
13.     T                28.       F              43.   F         58.      T
14.     F                29.       F              44.   F         59.      F
15.     T                30.       F              45.   F         60.      T


B. MULTIPLE CHOICES
1.      c                15.       a              29.   c         43.      a
2.      c                16.       b              30.   a         44.      d
3.      c.               17.       c              31.   b         45.      d
4.      a                18.       b              32.   d         46.      a
5.      d                19.       b              33.   d         47.      c
6.      d                20.       a              34.   c         48.      b
7.      c                21.       all true       35.   e         49.      c
8.      a                22.       b              36.   e         50.      a
9.      c                23.       e              37.   c         51.      c
10.     c                24.       b              38.   a         52.      d
11.     c                25.       c              39.   a         53.      c
12.     d                26.       c              40.   e         54.      c
13.     a                27.       e              41.   d         55.      a
14.     c                28.       a              42.   a         56.      a

                57.      b and d                        62.   e
                58.      d                              63.   c
                59.      c and d                        64.   b
                60.      c                              65.   d
                61.      d                              66.   d


C. FILL IN THE BLANKS AND CROSS-MATCH QUESTIONS
1. Production/Operations Management
2. supply, demand
3. customer needs
4. motion study
5. i. skill development on the part of the workers,
  ii. avoidance of lost time due to changing jobs,
 iii. the use of specialised machines.
6. a. by adopting fix work-stations,


Prof.Dr.Dr.M.Hulusi DEMIR                                                      95
                     Introduction to Production / Operations Management


  b. increasing task specialisation,
  c. moving work to the worker.

D. SHORT ANSWERS
1. i. Marketing,
  ii. Production/Operations
 iii. Finance/Accounting
2. i. Typically labour intensive
  ii. Frequently individually processed
 iii. Often an intellectual task performed by professionals
 iv. Often difficult to mechanise and automate
  v. Often difficult to evaluate for quantity.
3. Pick the following:
  * A service is tangible
  * It is often produced and consumed simultaneously
  * Often unique
  * It involves high customer interaction
  * Product definition is inconsistent
  * Often knowledge-based
  * Frequently dispersed.
4. Planning, organising, staffing, leading, and controlling.
5. a. POM is one of the three major functions of any organisations, and it is integrally related to all the
       other business functions. Therefore, we study how people organise themselves for productive
       enterprise.
    b. We want to know how goods and services are produced.
    c. We want to understand what production/operations managers do. This will help us explore the
       numerous and lucrative career opportunities in POM.
    d. It is such a costly part of an organisation. It provides a major opportunity for an organisation to
       improve its profitability and enhance its service to society.


E. ESSAY TYPE QUESTIONS
16. Examples of pure services include university lectures, many physical examinations, and legal
    opinions. Information is being transformed, primarily into usable knowledge. There may be some
    “consumption” of physical materials during the service process, but that is not transformation.
17. In pure services the customer is usually involved in the service operation; pure service might
    therefore be synonymous with the high contact service. The customer therefore has a direct say in
    the type and quality of the service, and the time required to perform the service. Quasi
    Manufacturing, or low contact, services do not involve the customer in the performance of the
    service itself, although the customer may be at a service desk close to the actual operation. In a
    manufacturing operation the customer is likely to be close to the operation at all. The further
    removed the customer, the less the ability to influence the performance of the operation at the time
    at which it is being performed and, therefore, the need to more clearly and comprehensively state
    needs and expectations before the operation starts. From the operator‟s point of view, the more
    influence the customer has in the process the greater the degree of uncertainty that needs to be
    accommodated. The closer to the customer, therefore, the greater the need for appropriate surplus
    resources, the more complicated the resource scheduling and the higher the likely cost to the
    customer.
18. Taylor and his associates concentrated on the problems of foremen, superintendents, and lower
    middle managers in factories because it was here that most of management‟s problems of the day
    were found. What was needed most was mass production and efficiency in the factories to respond
    to the great western markets.




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                     Introduction to Production / Operations Management


19. Frederick W. Taylor          Father of Scientific Management
    Frank B. Gilbreth            Motion Study, methods, therbligs
    Lillian M. Gilbreth          Fatique Studies, human factor in work
    Henry L. Gantt               Gantt Charts
    Henry Ford                   Inaugurated assembly-line mass production for
                                  autos
20. a. Production/operations managers are usually inseparately related to the productive system.
    b. P/O managers are usually strive toward optimal short run goals, their daily routine is relatively
       more predictable, their view of the external environment is relatively closed, and their decisions
       are principally based upon computations.
    c. Executives, on the other hand, strive toward sufficient long-run goals, have their daily routines
       that are unpredictable, their view of the external environment is relatively open and they deal
       principally with people and ideas in their daily jobs.
21. a. James Watt‟s steam engine
     b. Adam Smith‟s “Wealth of Nations”.
22. a.University teaching duties are divided according to academic specialisation: among
        faculties/schools within the university among departments within the faculties/schools and
        among instructors within a department.
    b. Accounting is divided into several disciplines for instructional purposes: financial, cost, tax
       accounting, and auditing. Certification of accountants is accomplished through separate
       examinations and licences for CPA`s and CMA`s. Accounting departments within organisations
       hire accountants with these distinct specialisations.
    c. In the construction industry, labour is divided among trades according to skills and materials
       required. Carpentry work, for example, is supplied by carpentry contractors.
    d. A fast-food restaurant produces services and facilitating goods by assigning food preparation,
       cooking, assembly and customer services tasks to specifically trained workers.
25. Production activities are a major part of technology and economics. Their purpose is to deliver
    goods and services that enhance the level of existence of society.
26. Taylor‟s principle (3) of striving for a spirit of cooperation between management and the workers
    was also aimed at fostering higher productivity.

30.      More Like A Goods Producer More Like A Serices Producer___
         * Physical, durable products            * Intangible, perishable products
         * Product can be resold                 * Reselling a service is unusual
         * Output can be inventoried             * Many outputs cannot be inventoried
         * Low customer contact                  * High customer contact
         * Long response time to demand          * Short response time to demand
         * Regional, national, or international  * Local markets
          markets
         * Large facilities with economies       * Small facilities (often difficult to automate
          of scale
         * Capital intensive                     * Labour intensive
         * Quality easily measured               * Quality not easily measured
         * Site of the facility is important     * Site of the facility is important for customer
           for cost                                contact
         * Selling is distinct from production   * Selling is often a part of the service
         * Product is transportable              * Provider, not product, is often transportable
31. a.The industrial revolution began in the 1770s in England, and spread to the rest of Europe and to
        the US in the late eighteenth century and the early nineteenth century. A number of inventions
        such as steam engine, the spinning Jenny, and the power loom helped to bring about to bring
        this change. There also were ample supplies of coal and iron ore to provide the necessary
        materials for generating the power to operate and build the machines which were much
        stronger and more durable than the simple wooden ones they replaced.
     b. Frederic W. Taylor, who is often referred as the father of scientific management, spearheaded
        the scientific management movement. The science of management was based on observation,

Prof.Dr.Dr.M.Hulusi DEMIR                                                                             97
                     Introduction to Production / Operations Management


       measurement, analysis, improvement of work methods and economic incentives. Management
       should be responsible for planning, carefully selecting and training workers, finding the best
       way to perform each job, achieving cooperation between management and workers, and
       separating management activities from work activities.
    c. Parts of a product made to such precision that each part would fit any of the identical items
       being produced. It meant that individual parts would not have to be custom made because they
       were standardised.
    d. Breaking up a production process into a series of tasks, each performed by a different worker.
       It enabled workers to learn jobs and become proficient at the more quickly; avoiding the delays
       of workers shifting from one activity to another.
32. McDonald‟s is either, or, or both, depending on the unit of analysis. At the counter McDonald‟s is
    a service; in the back of restaurant operations McDonald‟s is very much a manufacturing. This
    points out the need to carefully identify the aspect of the firm‟s operations that is being analysed.




98
                            Introduction to Production / Operations Management


PRODUCTIVITY
A. MULTIPLE CHOICES
1. b                 5. a
2. c                 6. d
3. b                 7. d
4. c


B. PROBLEMS
1. a. Productivity = (output)/(input)
      Plabour = (10 ornaments/day) / (4 hours/day) = 2.5 ornaments/hour
    b. Plabour = (20 ornaments/day) / (4 hours/day) = 5 ornaments/hour
    c. Change in productivity = 5 ornaments/hour – 2.5 ornaments/hour = 2.5 ornaments/hour
          Percent change = (2.5 ornaments/hour) / (2.5 ornaments/hour) x100 = 100%

2. Productivity = (1200 kgs)/ (100m x 100m) = (1200 kgs)/(10 000m2) = 0.12 kg/m2
   Productivity = (1350 kgs) /(10 000m2) = 0.135 kg/m2
   Change = 0.135 kg/m2 – 0.12 kg/m2 = 0.015 kg/m2
   Percent change = (0.015 kg/m2)/(0.12 kg/m2) x 100 = 12.5%
   No, the fertilizer didn‟t live up to its promise. The increase in productivity was 12.5% not 20%.
   The fertilizer was not good as advertised.

3. Resource            Last Year             This Year              Change             Percent Change
       Labour        10500 units/12000 hrs 12100 units/13200hrs 0.92 – 0.88      0.04 units/hr/0.88units/hr
                      = 0.88 units/hr       = 0.92 units/hr      = 0.04 units/hr = 0.048 = 4.8 %
       Utilities   10500units/7600MU 12100units/8250MU           1.47-1.38       0.09units/MU/ 1.38units/MU
                      = 1.38 units/MU       = 1.47 units/MU       = 0.09 units/MU = 0.06 = 6.2 %
       Capital     10500units/83000MU 12100units/88000MU        0.14 – 0.01      0.01units/MU-0.13units/MU
                      = 0.13 units/MU       = 0.14 units/MU     = 0.01 units/MU     = 0.078 = 7.8%
       Productivity improved in all three categories this year. Utilities showed medium, capital
       showed the greatest and labour the least.

4. Resource Standard                    Larger Machine            Percent Change
                     Equipment__________________________________________________
       Solvent       60 tanks/10 gallons      60 tanks/12 gallons     [(5-6) tanks/gallon]/6tanks/gallon
                     = 6 tanks/gallon         = 5 tanks/gallon        = - 0.1667 = - 16.67%
       Labour        60 tanks/240 hrs         60 tanks/180 hrs        [(0.33-0.25)tanks/hr]/0.25 tanks/hr
                     = 0.25 tanks/hr          = 0.33 tanks/hr         = 0.32 = 32%

5. Resource Last Year                      This Year           Change              % Change_________
       Labour   4000units/350hrs   1500units/325hrs (10.67-11.43) units/hr (-0.76units/hr)/11.43units/hr)
                  = 11.43 units/hr   = 10.67 units/hr = - 0.76 units/hr      = - 0.067 = -6.7%
       Capital 4000units/15000MU 1500units/18000MU (0.22-0.27)units/MU (-0.04units/MU)/ (0.27units/MU)
                  = 0.27 units/MU    = 0.22 units/MU = - 0.04 units/MU       = - 0.167 = - 16.7%
       Energy 4000units/3000kw 1500units/kw          (1.54-1.33)units/kw (0.21units/kw)/(1/33units/kw)
                  = 1.33 units/kw    = 1.54 units/kw    = 0.21 units/kw      = 0.154 = 15.4%
       The energy modifications did not generate the expected savings; labour and capital
       productivity decreased.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                                   99
                      Introduction to Production / Operations Management


FORECASTING
A. MULTIPLE CHOICE
1.   a.   recent
2.   b.   small
3.   a.   relies on the power of written arguments
4.   e.   (a) and (b)
5.   b.   causal forecasting
6.   b.   false
7.   a.   overall accuracy of the forecast


B. ESSAY
Demand is a measure of the amount desired by customers. Sales measures the amount actually
   purchased by customers. Sales will actually reflect demand if there have been no stock-outs.
   Demand can only be forecast from historical sales data if there have been no stock-outs, or if the
   data are adjusted for stock-outs.
2. Qualitative Method forecasts which rely on the judgment of individuals or groups. Qualitative
   forecasts are useful for long range time horizons and for such purposes as process design, capacity
   planning and facilities location. They are most useful when historic data exists or when existing
   data are not applicable.
   Time Series Method forecasts which assume that time is the only important independent variable.
   Time series forecasts are primarily useful in the short range for purposes such as materials
   management, purchasing, and scheduling.
   Causal Method forecasts which assume that the variable to be forecast is causally related to one or
   more intrinsic or extrinsic variables. Causal models are primarily useful in the medium range for
   aggregate planning and budgeting. They may be useful in the long range if applicable historic data
   exist and in the short range if the cost of the method is low relative to its benefits.
3. When a manager of a local firm says he‟s doing 25% more business than before, he is tacitly
   acknowledging that he has made a forecast. The forecast is subjective, and perhaps unconscious,
   but he is assuming that has happened in the past will persist in the future. It would undoubtedly be
   improved if it were performed in a methodical, systematic manner.
4. Qualitative (Judgmental) forecasts tend to be variable among individual forecasters, difficult to
   analyse, not precise, and lacking an objective basis for improvement.
   They do, however, have some advantages over objective forecasts in that they can incorporate
   intangible and subjective inputs along with objective ones. Thus they may, at times, be better than
   objective methods.
5. The smoothing constant dictates how much weight should be given to the past versus current
   demand. A high value of α emphasizes recent demand and causes the forecast to follow demand
   closely. A low value damps out fluctuations and yields a much smoothed forecast.
6. Regression and correlation methods are similar in that both describe the association among two or
   more variables. They may be simple or multiple, linear or nonlinear, depending upon the data.
   The regression equation states how the dependent variable changes as a result of changes in the
   independent variable. The regression curve expresses the nature of the relationship between two
   or more variables. Correlation is different in that it is a means of expressing the degree of
   relationship between two or more variables which are not considered dependent upon one another,
   but rather of equal status.
8. The Delphi Method involves a panel of (usually) anonymous people who fill in questionnaires and
   return them to the coordinator. The consolidated results are sent out again; the outliers are required
   to explain the reasons for their divergence from general consensus. This process is repeated for a
   set number of rounds or until consensus is reached, whichever is sooner. Studies show that the
   general public are likely as experts to produce reasonable forecasts.



Prof.Dr.Dr.M.Hulusi DEMIR                                                                           101
                      Introduction to Production / Operations Management



C. PROBLEMS
1. Month         Demand          MA3     ERROR            WMA3 ERROR
   January       520             -             -            -          -
   February      490             -             -            -          -
   March         550             -             -            -          -
   April         580             520.00        60           526.0     54
   May           600             540.00        60           553.0     47
   June          420             576.00     - 156           584    - 164
   July          510             533.33     - 23.33         506         4
   August        610             510.00        100          501       109
                                         __________              _______
                                 Totals        399.33                 378__
                                 Average         79.87                 75.6_

      Forecast for September:
                                MA3= 5 13.33 Units
                             WMA3 = 542 Units
      The weighted moving average is slightly better.

Period Units Last Period’                          Forecast of            Smoothed Forecast
                   Units             α    α(LPU) (1-α) Last Period (1-α)(FLP) for the period

          1   56         -           -     -        -       -            -       -
          2   61         56         0.4   22.4     0.6    56 (Guess)   33.6      56.0
          3   55         61         0.4   24.4     0.6    56           33.6      58.0
          4   70         55         0.4   22.0     0.6    58           34.8      56.8
          5   66         70         0.4   28.0     0.6    56.8         34.08     62.08
          6   65         66         0.4   26.4     0.6    62.08        37.25     63.65
          7   72         65         0.4   26.0     0.6     63.65        38.19    64.19
          8   75         72         0.4   28.0     0.6    64.19         38.51    67.31
      FORECAST:
          9               75        0.4    30.0     0.6    67.31       40.39      70.39

3. a.           Semester Students       MA3         Error
                    9          80         -           -
                    10         90         -           -
                    11         70         -           -
                    12         84 (80+90+70)/3 = 80   4.00
                    13         100        81.33       18.67
                    14         115        84.67       30.33
                    15         98         99.67        1.67
                    16         130       104.33       25.67
                                          Total       80.34
                                          Average     16.07




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  b.              Previous α (Previous         Smoothed Forecast (1-α). Smoothed
 Smtr. Enrollm.  Enrolm. α Enrolm.) 1-α of Previous Enrolm. (SFPE) Forecast ERROR
     1   80         -      -      -        -         -                -         -             -
     2   90        80    0.2 16           0.8       80 (Guess)       64        80            10
     3   70        90    0.2 18           0.8      80                64        82            12
     4   84        70    0.2 14           0.8      82                65.6      79.6            4.4
     5  100        84    0.2 16.8         0.8      79.6              63.9      80.5          19.5
     6  115      100      0.2 20          0.8       80.5             64.4      84.4          30.6
    7    98      115     0.2 23           0.8      84.4              67.5      90.5             8.5
     8  130        98     0.2 19.6        0.8       90.5             72.4      92             38__
                                                                             Total           123
                                                                       *(neglecting the sign)
                                                                          Average, Φ       17.58
 The average error of MA3 is 16.07 and the average error of exponential smoothing is 17.58. Three
 period moving average is preferred.
 MA3 forecast for the coming semester is (115+98+130)/3 = 114.33

7. Plasterboard shipments --- Dependent variable, y
   Construction permits ------ Independent variable, x
       x              y         x.y    x2     y2
       15             6         90    225    36
       9              4         36     81    16
       40            16       640    1600 256
       20             6       120     400    36
       25            13       325     625 169
       25             9       225     625    81
       15            10       150     225 100
       35            16       560 1 225 256
      184            80      2 146 5 006 950

   a.           ∑y = n.a + ∑x        (1)      80 = 8a + 184b              (1)
              ∑xy = ∑x + ∑x2        (2)       2146 = 184 a + 5006 b       (2)
        Multiplying (1) by (23) -1840 = -184 a + 4232 b                    (1)
                                     2146 = 184a + 5006 b                  (2)
        Therefore        b = 0.395
        Substituting b = 0.395 in Eq. (1)     80 = 8a + 184(0.395)  a = 0.91
        Trend forecasting equation is
                         Y = 0.91 + 0.395 X

   b. Y = 0.91 + 0.395X               Y = 0.91 + 0.395(30)             Y= 12.76
                                Y = 13 shipments

   c. Syx = √ [∑y2 - a∑y - b∑xy] / (n-2)
      Syx = √[950 – 0.91(80) – 0.395 (2146)] / (8-2) = 2.2 shipments

   d. Prediction interval (confidence limits) of 90% is
              Y +/- t.Syx              12.76 +/- 1.943 (2.2)
                       17.03   8.49 shipments
     If the number of permits is 30, the value of Y (the demand for plasterboard shipments) can be
     expected to lie with 90% probability within the interval of 17 shipments and approximately 8
     shipments.

   Prediction interval (confidence limits) of 95.5 % is,
                Y +/- 2Syx              12.76 +/- 2(2.2)
                        17.16   8.36 shipments

Prof.Dr.Dr.M.Hulusi DEMIR                                                                      103
                            Introduction to Production / Operations Management


           There is a 95.5% probability that the shipments for 30 permits will lie between 8 and 17
           shipments.

      f.    r = [n∑xy - ∑x∑y]/√ [n∑x2 – (∑x)2][n∑y2 – (∑y)2]
            r = [8(2146) – 184(80)]/[(8(5006)-(184)2][8(950) – (80)2]       = 0.90
           There is a very strong relation between the number of permits and the demand for plasterboard
           permits.

      g. r2 = (0.90)2 = 0.81
        The demand for plasterboard shipments and the change in demand depends 81% on
        construction permits and 19% on other factors.

   h. The significance of value of r = 0.90 can, however, be tested under a hypothesis that there is no
      correlation between the number of permits and demand for plasterboard shipments, that is, Hor
      = 0.
      The computed value of r (statistical-t value of r) is compared with a tabled value of r
      (theoretical-t value of r) for a given size (n = 8) and significance level of 5%.
      If the statistical-t value of r ( tc ) > theoretical-t value of r (tk), the hypothesis is rejected, the
      correlation is deemed significant at specified level.
                 tc = |r|√[(n-2)/(1-r2)]                    tc =| 0.9|√[(8-2)/(1-0.81)]
                 tc = 5.06
     Level of significance (α) = 0.05               Degree of freedom (n-2) = 6
     From student-t table  tk = 2.447
                 tc > tk                   5.06 > 2.447 The hypothesis is rejected.
                                                             The computed r, i.e. r = 0.90 is meaningful.

      i.     b = [n∑xy - ∑x∑y]/[n(∑x2 – (∑x)2

             b = [8(2146) – 184(80)]/[8(5006) – (184)2]
             b = 0.395

             a = y –b.x      a = 10 – 0.395(23)        a = 0.91

                               Y = 0.91 + 0.395X

             There is no difference between the both regression equations.

8. Day Demand for     Total sales
       Lawn-mowers, y of the store(000MU), x (x –x) (y – y) (x - x)2              (y –y)2 (x –x)(y –y)
   1      10               10                  -6     -7      36                    49         42
   2      12               13                  -3     -5       9                    25         15
   3      13               14                  -2     -4       4                   16           8
   4      15               16                   0    -2         0                    4          0
   5      20               19                   3       3      9                     9          9
   6      25               20                   4       8     16                   64          32
   7      24               20                   4       7     16                   49          28
         119              112                   0       0     90                  216         134
               y = 17               x = 16
  a.         r = [∑(x – x)(y – y)] / √[∑(x – x)2.∑(y – y)2]
             r = 134 / √(90)(216)            r = 0.96
           There is a very strong relation between the total sales of the store and lawn-mowers.
  b.         rr = (0.96)2            r2 = 0.92


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     Since determination coefficient is 0.92, we could say that 92% of the variation of the lawn
     mower blade sales is explained by total sales of the store. Only 8 % of the variation is explained
     by other factors.
   c. The significance of the value of r = 0.96, can however, be tested under a hypothesis that there is
      no correlation between total sales of the store and lawn mowers, that is Hor = 0.
     The computed statistical-t value of r is compared with a theoretical-t value of r for a given size
     (n = 7) and significance level of 5%.
                tc = | r |√[(n -2)/(1 – rr)]
       tc = 0.96√[(n -2)/(1 – 0.92)]                           tc = 7.589
       tk = 2.571
                          tc (7.589) > tk (2.571
       The hypothesis r = o is rejected. The computed r is meaningful.

   d. Day, x     Total Sales, y x.y      x2
      1    0           10                 0     0
      2    1           13                13     1
      3    2           14                28     4
      4    3           16                48     9
      5    4           19                76    16
      6    5           20               100    25
      7    6           20               120    36
         21         112                 385    91

       ∑y = n.a + b.∑x               (1)                112 = 7a + 21b
       ∑xy = a∑x + b∑x2              (2)                385 = 21a + 91b
                      a = 10.75      b = 1.75
                              Y = 10.75 + 1.75X

       Y8 = 10.75 + 1.75(7) = 23 (000)MU

   e. b = ∑(x – x) / ∑(x – x)2
      b = 134/90 = 1.489
      a = y – b.x                               a =17 – 1.48(16)         a = - 6.82
      Y = -6.82 + 1.48X                 Y = -6.82 + 1.48(23) Y = 27.22 MU

        x      y        x.y     y2 __
        10     10       100     100
        13     12       156     144
        14     13       182     169
        16     15       240     225
        19     20       380     400
        20     25       500     625
        20     24       480     576
       112    119      2038    2239

                Syx = √[(∑y2 - a∑y - b∑x.y) / (n – 2)]
       Syx = √[(2239 -6.68x119 – 1.48x2038(/(7 – 2)]           Syx = 1.897
                Y +/- t.Syx
               22.3 +/- (2.015)(1.897)          27.22 +/- 3.82
                31.04   23.4 MU

   Assuming total sales of 8th day be 23 MU, demand for lawn mower blade sales for the 90%
   probability fall between 31.04 MU and 23.4 MU.


Prof.Dr.Dr.M.Hulusi DEMIR                                                                          105
                       Introduction to Production / Operations Management


9. Month              Ice-cream sales    Laguna Visitors
                      (MU), y                    x          xy x2        y2_
           1               200                  800       160 000 640 000          40 000
           2               300                  900       270 000 810 000          90 000
           3               400                1100        440 000 1200 000        160 000
           4               600                1400        840 000 1960 000        360 000
           5               700                1800       1260 000 3240 000        490 000
           6               800                2000       1600 000 4000 000        640 000
               Totals     3000                8000       4570 000 11860 000      1780 000

      a.          ∑y = n.a + b.∑x              3000 = 6a + 8000b                        (1)
                ∑xy = a∑x + b∑x2                   4570000 = 8 000a + 11860 000b        (2)
                  (1) x 4000        12000 000 = 24000a + 32000 000b     (1)
                          (2) x 3          13710 000 = 24000a + 35580 000b               (2)
                  (2)-(1) 1710 000 = 3580 000b            b = 0.48
                  Substituting in Eq. (1)
                                   3000 = 6a + (8000)0.48
                                     a = -14

                                  Y = -140 + 0.48 X

      b. Y = - 140 + 0.48(3000)         Y = 1 300 ice-creams

      c. Syx = √{[∑ y2 - a∑y - b∑xy] / [n - 2]}
          Syx = √{[1780 000 – (-140)(3000) – 0.48(4570 000)] / [n – 2)}
          Syx = 45 ice-creams
                                  Y +/- Syx
                                  1 300 +/- 45          1 345   1255 ice-creams
      Ice-cream sales for 3 000 persons will fall with 68.3 probability within the range of 1 345 ice-
      creams and 1 255 ice-creams.

19.
          Year Quarter           Demand
      2005      I                   92
                II                  82
                III                 84
                IV                  92      Moving
      ___       ___          x_____350___   Totals, y         x.y      x2
      2006      I            0      90         348              0      0
                II           1      80         346            346      1
                III          2      82         344            688      4
                IV           3      90         342           1026      9
                             6                1380           2060 14
       ∑y = na + b∑x         (1)              1380 = 4a + 6b                            (1)
      ∑x.y = a∑x + b∑x (2)2
                                              2060 = 6a + 14b                           (2)
                                                      a = 348     b = -2
                                                              Y = 348 – 2X
       Y2007/I = 348 – 2(4) = 340
       Y2007/II = 348 – 2(5) = 338
       Y2007/III = 348 – 2(6) = 336
       Y2007/IV = 348 – 2(7) = 334
                                               2007 I         88 units
                                                      II      78 units
                                                      III     80 units
                                                      IV      88 units

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24.
      Years, x   Registrants(000), y     x.y     x2
      2000 0             17              0       0
      2001 1             16              16      1
      2002 2             16              32      4
      2003 3             21              63      9
      2004 4             20              80     16
      2005 5             20             100     25
      2006 6             23             138     36
      2007 7             25             175     49
      2008 8             24             192     64
           36           182            796     204

      ∑y = n.a + b.∑x  182 = 9a + 36b (1)
         ∑xy = a∑x + b∑x2  796 = 36a + 204b (2)
                  (1) . 4  728 = 36a + 144b (1)
                   (2). 1   796 = 36 + 204b (2)
         (2) – (1)        68 = 60 b        b = 1.13
         Substitute b=1.13 in Eq. (1)
                          182 = 9a + 36b  182 = 9a + 36(1.13)                 a = 15.69
                          Y = 15.69 + 1.13X

         Therefore trend forecasting equation is
                Y2009 = 15.69 + 1.13 (9)  Y2009 = 25.86  Y2009 = 25 860 Registrants

25. a. Time Demand MA3 Error
         1   10         -                -
         2   14         -                -
         3   19         -                -
         4   26         -                -
         5   31         17.25            13.75
         6   35         22.50            12.50
         7   39         27.75            11.25
         8   44         32.75            11.25
         9   51         37.25            13.75
         10  55         42.25            12.75
         11  61         47.25            13.75
         12  54         52.75              1.25
    b.                 Total            90.25
                      Average           11.28

      WMA4 forecast of period 13 = [4(54)+3(61)+2(55)+ 1(51)] / 10 = 56 Units




Prof.Dr.Dr.M.Hulusi DEMIR                                                                   107
                        Introduction to Production / Operations Management


      d.
                 Sales Last                      SF of P.          Smoothed Forecast
           Sales Period α         α(SLP)   (1-α) Period (1-α)(SFPP) for this period___ ERROR
            10     -         -     -        -      -        -          -                 -
            14     10       0.3    3.0      0.7    8       5.6         8.6               5.4
            19     14       0.3    4.2      0.7    8.6     6.02        10.22             8.78
            26     19       0.3    5.7      0.7    10.22   7.154       12.854           13.146
            31     26       0.3    7.8      0.7    12.85   8.998       16.798           14.202
            35     31       0.3    9.3      0.7    16.8    11.76       21.06            13.94
            39     35       0.3    10.5     0.7    21.06   14.74       25.24            13.76
            44     39       0.3    11.7     0.7    25.24   17.67       29.37            14.63
            51     44       0.3    13.2     0.7    29.37   20.56       33.66            17.34
            55     51       0.3    15.3     0.7    33.66   23.56       38.86            16.14
            61     55       0.3    16.5     0.7    38.86   27.20       43.70            17.30
            54       61     0.3    18.3     0.7    43.70   30.60       48.90              5.11

      e.                                                                 Total           139.214
                                                                         Average          12.66

   f. MA4 is preferable, since it has lower average error compared to smoothing forecast with α = 0.3.
   g.
 Sales       SF of P                        Smoothed Forecast
 Period      Period      α α(SLP) (1-α) for this period_ (1-α)(SFPP) SFTP ERROR_
  10            -        -       -       -        -                -         -         -
  14            10       0.5 5          0.5      8                4          9         5
  19            14      0.5 7           0.5      9                4.5       11.5       7.5
  26            19      0.5 9.5         0.5     11.5              5.75      15.25 10.75
  31            26      0.5 13          0.5     15.25             7.625     20.625 10.375
  35            31      0.5 15.5        0.5     20.625           10.3125 25.813 9.187
  39            35      0.5 17.5        0.5     25.813           12.906     30.406 8.594
  44            39      0.5 19.5        0.5     30.406             15.203 34.703 9.297
  51            44      0.5 22          0.5     34.703            17.352    39.352 11.648
  55            51      0.5 25.5        0.5     39.352           19.676     45.176      9.82
  61            55      0.5 27.5        0.5     45.176           22.588     50.088 10.912
  54            61      0.5 30.5        0.5     50.088           25.044     55.544 11.544
                                                                 Total               94. 631
                                                                 Average              8.603
   h. If you were to use an exponential smoothing factor larger than o.3 to forecast the given time-
       series, you will get smaller average error.




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39.
       YEAR NO. OF        RIDERSHIP
            TOURISTS (in millions)
            (in millions)
               x               y       x2  y2                  x.y ____________
      1996     7             1.5      49  2.25                10.5
      1997     2             1.0       4  1.00                 2.0
      1998     6             1.3      36  1.69                 7.8
      1999     4             1.5      16  2.25                 6.0
      2000    14             2.5     196 6.25                 35.0
      2001    15             2.7     225 7.29                 40.5
      2002    16             2.4     256 5.76                 38.4
      2003    12             2.0     144 4.00                 24.0
      2004    14             2.7     196 7.29                 37.8
      2005    20             4.4     400 19.36                88.0
      2006    15             3.4     225 11.56                51.0
      2007      7            1.7       49 2.89                11.9
      TOTALS 132            27.1    1796 71.59               352.9

      a. ∑y = n.a + b∑x         (1)                             27.1 = 12a + 132b            (1)
         ∑xy = a∑x + b∑x2       (2)                           352.9 = 132a + 1796 b          (2)
                        a = 0.51           b = 0.159
                                                           Y = 0.51 + 0.159X
      b. Y = 0.51 + 0.159 (10)                    Y = 2.1 = 2 100 000 persons

      c. If there are no tourists at all, the model predicts of 0.5 or 500 000 persons. One would not place
         much confidence in this forecast, however, because the number of tourists is outside the range
         of data used to develop the model.

      d. Syx = √{(∑y2 - a∑y - b∑x.y)/(n – 2)}
         Syx = √{[71.59 – 0.511(27.1) – 0.159(352.9)] / (12 -2)}
         Syx = 0.404 mil. Persons
                  Y +/- 2Syx                               2.1 +/- 2(0.404)
                                  2.9   1.3 mil. Persons
        There is 95.5% probability that the ridership will fall between 2 900 000 persons and 1 300
        000 persons, if the tourist population is 10 mil. People. There is only 4.5% risk that the
        ridership may fall outside these limits.

      e. r = [nΣxy - ΣxΣy] /√[nΣx2 – (Σx)2][nΣy2 – (Σy)2]
         r = [12(352.9) – 132(27.1)] / √[12(1796) – (132)2][12(71.59) – (27.1)2] = 0.917
         There is a very strong relationship between ridership and number of tourists.

      f.   r2 = (0.917)2  r2 = 0.84
           84% of variation in ridership depends on number of tourists, 16% depends on other factors.

      g. degree of freedom = 12 - 2 = 10              level of significance = 5%
         from normal distribution table tk = 2.228
         tc = | r |√[(n – 2) / (1 – r2)]
         tc = 0.917 √(10)/(1-0.84)             tc = 7.25
         tc (7.25) > tk (2.228)       Ho(r=0) Hypothesis is rejected. The computed r is meaningful.




Prof.Dr.Dr.M.Hulusi DEMIR                                                                             109
                         Introduction to Production / Operations Management


40. a. Year         Sales    P.Sales  α α(P.Sales) (1-α) SFPS (1-α)(SFPS) SFTP | Error |
       2003         450         -      -     -       -    -       -          -     -
       2004         495        450    0.3 135        0.7 410     287       422     73
       2005         518        495    0.3 148.5      0.7 422     295.4     443.9   74.1
       2006         563        518    0.3 155.4      0.7 443.9 310.73      466.13 96.87
       2007         584       563     0.3 168.9      0.7 466.13 326.29     495.19 88.81
                                                                       Total     332.78
                                                                       Average 83.20
      b. Year       Sales    MA3      | Error |
         2003       450        -         -
         2004       495        -         -
         2005       518        -         -
         2006       563       487.7      75.3
         2007       584       528.3      58.7
                                Total    134
                                Average 67

      c. Moving Average of 3-period is preferred, because it has less average error.
      d. MA3 for 2008 = *518 + 563 + 584)/3 = 555 units

41.         Year Quarter Demand (Units)
            2005 I              92
                   II           82
                   III          84
                   IV           92    Moving
            __________x___________350_ Totals          x.y  x2
            2006 I 0            90    348                0  0
                   II 1         80    346              346 1
                   III 2        82    344              344 4
                   IV 3         94    346             1038 9_____
                       6             1384             2072 14____

            ∑y = n.a + b∑b             (1)                        1384 = 4a + 6b         (1)
            ∑x.y = a∑x + b∑x2          (2)                        2072 = 6a + 14b        (2)
                                              a = - 0.8    b = 347.2
                                       Y = 347.2 – 0.8X

            Y2007/I = 347.2 – 0.8(4) = 344                 2007 I         88     units
            Y2007/II = 347.2 – 0.8(5) = 343.2                   II        79.2   un its
            Y2007/III= 347.2 – 0.8 (6) = 342.4                 III        81.2   units
            Y2007/IV = 347.2 – 0.8(7) = 341.6                  IV         93.2   units

      42.                       a.             b.
            Month Actual Demand MA3 | Error |             WMA3                | Error |
            January      110       -    -                  -                      -
            February     130       -    -                  -                      -
            March        150       -    -                  -                      -
            April        170     130   40     [(6x150)+3(130)+(110)]/ 10 = 140 30
            May          160     150   10                                160      -
            June         180     160   20                                162     18
            July         140     170   30                                173     23
            August       130     160   30                                154     24
            September    140     150   10                                138      2
                               Total  140                                       107
                              Average   23.33                                    17.83

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44.        Year  No. of Housing            Sales
                  Permits, x             (000m2), y        x.y      x2     y2___
           1999        18                    14            252      324    196
           2000        15                    12            180      225    144
           2001        12                    11            132      144    122
           2002        10                     8             80      100     64
           2003        20                    12            240      400    144
           2004        28                    16            448      784    256
           2005        35                    18            630     1225    324
           2006        30                    19            570      900    361
           2007        20                    13            260      400    169
           TOTALS     188                   123           2792     4502   1780

      a. Regression forecasting equation is found as follows;
                 Σy = n.a + b.Σx       (1)       123 = 9a + 188b                           (1)
               Σxy = Σx + bΣx2 (2)  2792 = 188 a + 4502b                                   (2)
                                                     b = 0.3757
         Substitute b=0.3757 in Equation (1), we get
                 123 = 9a + 188(0.3757)        a = 5.818
         Therefore the regression forecasting equation is    Y = 5.818 + 0.3757
      b. If we suppose that there are 25 new housing permits granted in 2008, the sales for 2008 will
         be
         Y2008 = 5.818 + 0.3757 (25)        Y2008 = 15.211 = 15 211 m2 of carpet
         (This assumes that the number of housing permits issued in a year is known at the
         beginning of the year.)

      c. The correlation coefficient is calculated as follows;
                 r = [nΣxy - ΣxΣy] /√[nΣx2 – (Σx)2][nΣy2 – (Σy)2]
                 r = [9(2792) – (188)(123)] / √[9(4502) – (188)2][9(1780) – (123)2]
                 r = 2004 / √(5174)(891)                   r= 0.93
         There is a very strong relationship between number of housing permits and carpet sales.

      d. Determination coefficient is therefore,
                  r2 = (0.93)2  r2 = 0.86
         86% of changes in carpet sales from year to year can be attributed to a change in the number
         of housing permits issued. Only 14 % of the changes in the carpet sales depend on other
         factors.

      e. Testing the hypothesis r = 0 at 5% level is significance is done as follows;
         Level of significance = 5%                 Degree of freedom = n – 2 = 9 – 2 = 7
                  tc = | r |√ [(n- 2)/(1 – r2)]
                  tc = | 0.93 | √[(9-2)/(1- 0.932)]         tc = 7.103
         From student-t table tk = 2.365
         tc (7.103) > tk ( 2.365) Hypothesis r =0 is rejected. The computed r is meaningful.

      f.   By using correlation coefficient formula, we can find
                  b = 2004/5174 = 0.387
                  a = y – b.x         mean value of x = 20.89        mean value of y = 13.67
                  a = 13.67 – 0.387(20.89) = 5.89
                  Y = 5.89 + 0.387 X

      g. Forecast 2008 sales based on forecasted permits for that year. First we have to forecast permits
         of 2008 by using trend analysis.


Prof.Dr.Dr.M.Hulusi DEMIR                                                                           111
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                   Year     x         Permits,y      x.y   x2
                   1999     0             18          0    0
                   2000     1             15         15    1
                   2001     2             12         24    4
                   2002     3             10         30    9
                   2003     4             20         80   16
                   2004     5             28       140    25
                   2005     6             35       210    36
                   2006     7             30       210    49
                   2007     8             20       160    64
                   Totals   36           188      869    204

          Σy = n.a + b.Σx         (1)             188 = 9a + 36 b
         Σx.y = aΣx + b.Σx2       (2)             869 = 36a + 204 b
                  a = 13.09                b = 1.95
         Therefore trend forecasting equation for permits is,
                  Y = 13.09 + 1.95X
         The forecasted permits for the year 2008 will be,
                  Y2008 = 13.09 + 1.95 (9) = 30.64 permits
         By using regression equation we may forecast 2008 expected sales,
                  Y2008 = 5.818 + 0.3757X
                  Y2008 = 5.818 + 0.3757 (30.64) = 17.329 = 17 329 m2 of carpet
      h. Syx = √{[Σy2 – aΣy - bΣxy] / (n-2)}
         Syx = √{[1780 – 5.818(123) – 0.3757(2792)]/(9-2)} = 1.485 (000) m2
         Confidence limits of 90% probability for the forecasted sales:
                  Y +/- Syx                       17.329 +/- (1.895)(1.485)
                          20.142   14.516 (000)m2
         Assuming permits of year 2008 be 30.64, with 90% probability carpet sales will fall between
         20 142 m2 and 14 516 m2. There is only 10% risk that forecasted sales may fall outside this
         range.
         Assuming n is large, the forecasted sales of 2008 with 95.5%probability will be
                  Y +/- 2Syx              17.329 +/- 2(1.485)
                          20.306   14.352 (000)m2
         Assuming permits of year 2008 be 30.64, carpet sales will fall with 95.5% probability within
         the limits of 20 306 m2 and 14 352 m2. There is still 4.5% risk that it may fall outside this
         interval.

45.                                     Thousand of
           Month      Tires Used, y     Miles Driven, x     x.y                  x2____
            1             100             1 500           150 000            2 250 000
            2             150             2 000           300 000            4 000 000
            3             120             1 700           204 000            2 890 000
            4              80             1 100            88 000            1 210 000
            5              90             1 200           108 000            1 440 000
            6             180             2 700           486 000            7 290 000
                   Totals 720            10 200         1 336 000           19 080 000

      a.   ∑y = n.a + b.∑x                        720 = 6a + 10 200 b                (1)
         ∑x.y = a∑x + b∑x2                  1 336 000 = 10 200a + 19 080 000         (2)
                          a = 11.2                 b = 0.064
                                  Y = 11.2 + 0.064X
      b. r = 0.987               r = 0.99
         There is a very strong relationship between tires used and miles driven.
         r2 = (0.987)2           r 2 = 0.974 = 97.4%
         97.4% of the variation in tires used is explained by the miles driven, which is a good fit.

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46. a. October 2007      Calls         WMA3                                    |Error|
             1             92
             2            127
             3            103
             4            165      [5(103) + 3(127 + 2(92)]/10 = 108 57
             5            132                                  = 138.8            6.8
             6           111                                   = 136.1           25.1
             7           174                                   = 128.1           45.9
             8            97                                   = 146.7           49.7
                                                                       Total    184.5
                                                                    Average      36.9
        Forecast for October 9 = 122.9 = 123 calls

   b.
       Calls Previous α α(P.Calls) (1-α) SFPC (1-α)(SFPC) S.Forecast Error
              Calls__________________________________________________________
        92       -      -       -         -       -         -      -         -
       127      92     0.3    27.6       0.7     90        63     90.6      36.4
       103 127         0.3    38.1       0.7     90.6      63.42 101.52      1.48
       165 103         0.3    30.9        0.7   101.52     71.06 101.96     63.04
       132 165         0.3    49.5       0.7    101.96     71.37 120.87      11.13
       111 132         0.3    39.6       0.7    120.87     84.61 124.21      13.21
       174 111         0.3    33.3       0.7    124.21     86.95 120.25      53.75
        97      174    0.3    52.2       0.7    120.25     84.18 136.38      39.38
                                                                      Total 218.39
                                                                     Average 31.2
   Forecast for October 9 = 0.3(97) + 0.7(136.38) = 124.57
                          = 125 calls

47. a. Year Quarters Demand (units)           MA4      | Error|
       2006 I            350
             II          460
             III         280
             IV          360
       2007 I            500                  362.      137.5
             II          590                  400       190
             III         450                  432.5      17.5
             IV          530                  475        55
                                              Total    400
                                                Φ      100
        2008   I                              517.5

   b. Year Quarters,x Demand (units),y x.y             x2
      2006 I       0     350             0             0
            II     1     460            460            1
            III    2     280            560            4
            IV     3     360           1080            9
      2007 I       4     500           2000           16
            II     5     590           2950           25
            III    6     450           2700           36
            IV     7     530           3710           49
                  28     3520         13460           140

   Σy = n.a + bΣx                (1)                  3520 = 8a + 28b                   (1)
   Σxy = aΣx + b Σx2             (2)                  13460 = 28a + 140b                (2)

Prof.Dr.Dr.M.Hulusi DEMIR                                                                      113
                     Introduction to Production / Operations Management


                         a = 345      b = 27.14
                              Y = 345 + 27.14X
      Y = 345 + 27.14X  Y = 345 + 27.14(8) = 562.14 units

c.
                         Previous
       Year     Demand   Demand α α (P.D.)     (1-α) (SFPD) (1-α)(SFPD) SFTP      | Error|
      2006/ I     350       -      -   -          -      -       -         -          -
           II     460     350     0.2 70        0.8    400     320       390          70
          III     280     460     0.2 92        0.8    390     312       404        124
          IV      360     280     0.2 56        0.8    404      323.2    379.2       19.2
      2007/I      500     360     0.2 72        0.8    379.2    303.36   375.36 124.64
          II      590     500     0.2 100       0.8    375.36   300.29   400.29 189.71
          III     450     590     0.2 118       0.8    400.29   320.23   438.23      11.77
           V      530     450     0.2 90        0.8    438.23   350.58    440.58     89.42
                                                                             Total 628.74
                                                                             Average 89.82
      2008/I               530     0.2 106       0.8 440.48    352.46    458.46

48. Month Lumber Roofing
    ________Sales,x Sales,y_           x.y        _x2__       y2
    1         90       50              4500      8100        2500
    2         115      52              5980     13225        2704
    3         120      60              7200     14400        3600
    4         125      64              8000     15625        4096
    5         145      72            10440      21025        5184
    6         145      74            10730      21025        5476
    7         150      74            11100      22500        5476
    8         140      84            11760      19600        7056
    9         135      82             11070     18225        6724
    10        120      72             8640      14400        5184
    11        115      72             8280      13225        5184
    12        100       60            6000      10000        3600
              1500    816           103700     191350       56784
          Φ = 125      68

      a. ∑y = n.a + b∑x          (1)            816 = 12a + 1500b                    (1)
         ∑xy = a∑x + b∑x2        (2)         103700 = 150a + 191350b                 (2)
                                 (1)x125      102000 = 1500a + 187500b                (1)
                                         1     03700 = 1500a + 191350b                (2)
                                 ___________________________________
                                              1700 = 3850b
                                                  b = 0/44155
         Substitute b=0.44 in Equation (1)
                                           816 = 12a + 1500(0.44)
                                           156 = 12a
                                             a = 13
         Y = 13 + 0.44X

      b. Y = 13 + 0.44(125)                    Y = 68 Units

      c. r = [n∑x.y - ∑x∑y] / √{(n∑x2 – (∑x)2}{n∑y2 – (∑y)2}
         r = [12(103700) – 1500(816)] / √[12(191350) – (1500)2][12(56784) – (816)2]
         r = 0.76
         There is a strong relationship between lumber sales and roofing sales.

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                       Introduction to Production / Operations Management


           r2 = (0.76)2 = 0.58
           58% of roofing sales depends on Lumber sales, and 42% depends on other factors.

      d. tc = | r |√ [ (n – 2) / (1 – r2)]
         tc = 0.76 √(12 – 2)/(1 – 0.58)                                tc = 3.71
         from normal distribution table; degree of freedom = 12 – 2 = 10,
                                               level of significance = 5%
                                       tk = 2.228
         tc (3.71) > tk (2.228)                Hypothesis r=0 is rejected. The computed r is meaningful.

      e. b = [12(103700) – 1500(816)] / [12(191350) – (1500)2] = 0.44
         a = y – bx                     a = 68 – 0.44(125)              a = 13
                          Y = 13 + 0.44X
         There is no difference between two regression equations. They are same.

      f.   Syx = √[∑y2 - a∑y - b∑xy]/(n – 2)]        Syx = √[56784 – 13(816) – 0.44(103700)/(12 – 2)
                                                      Syx = 7.4 units
           Y +/- t.Syx            68 +/- (1.812)(7.4)       81.4   54.59 Units
           There is 90% probability that lumber sales will fall within the limits of 55 units and 81
           units. There is only 10% risk that lumber sales may fall outside these limits.

      g. Y +/- 2 Syx             68 +/- 2(7.40)            82.8   53.2 Units
         There is 95.5 % probability that lumber sales will fall within the limits of 53 units and 83 units.
         There is only 4.5% probability that it may fall outside these limits.

49.      Year Quarter Sales
      2006       I     60
                II     91
               III    277
               IV      34      Moving
      ____________ x______462__ Totals, y             x.y            x2___
      2007        I 0 105           507                 0            0
                 II 1 130           546               546            1
                III 2 522           791              1582            4
                 IV 3 73            830              2490            9
                    6              2674              4618           14

      ∑y = n.a + b∑x               (1)                              2674 = 4a + 6b            (1)
      ∑x.y = a∑x + b∑x2            (2)                              4618 = 6a + 14b           (2)
                                            a = 486.4        b = 121.4

                  Y = 486.4 + 121.4X                Y2008/I = 486.1 + 121.4(4) = 972
                                                     Y2008/II = 486.1 + 121.4 (5) = 1093.4
      Year        Sales                              Y2008/III = 486.1 + 121.4(6) = 1214.8
      2008/I      247                                Y2008/IV = 486.1 + 121.4(7) = 1336.2
          II      251.4
         III      643.4
         IV       194.4




Prof.Dr.Dr.M.Hulusi DEMIR                                                                               115
                      Introduction to Production / Operations Management


50.
                Years     Sales           Payroll
                     (00.000MU),y (000.000.000MU),y           x.y      x2       y2
                2002        2.0              1                2.0       1       4
                2003        3.0              3                9.0      9        9
                2004        2.5              4               10.0     16        6.25
                2005        2.0              2                4.0      4        4
                2006        2.0              1                2.0      1         4
                2007        3.5              7               24.5     49       12.25
                           15.0            18                51.5     80       29.50
      y = 2.5           x=3
a.
      ∑y = n.a + b∑x                      (1)                        15 = 6a +18.b              (1)
      ∑x.y = a∑x + b∑x2                   (2)                      51.5 = 18a + 80b             (2)
      a = 1.75          b = 0.25                                     Y = 1.75 + 0.25X

b. r = [ n∑x.y - ∑x∑y]/√[(n∑x2 – (∑x)2)(n∑y2 – (∑y)2)]
   r = [6(51.5) – 18(15)] / √[6(80) – (18)2][6(39.5) – (15)2]           r = 0.901
   The r value of 0.901 appears to be a very strong correlation between sales and payroll.
   r2 = (0.901)2 = 0.81
   The determination coefficient indicates that 81% of the total variation is explained by the
   regression equation.

c. tc = | r |√[(n -2)/(1 – r2)]                            tc = |0.901| √[(6 – 2)/(1 – 0.81)]
                                                            tc = 4.129
      level of significance = 5%
      degree of freedom = n – 2 = 6 – 2 = 4         tk = 2.776
      tc (4.129) > tk (2.776)                       Ho(r=0) is rejected. The computed r is meaningful.

d. b = [n∑x.y - ∑x∑y] / {(n∑x2 – (∑x)2}
   b = [6(51.5) – (18)(15)]/ [6(80) – (18)2]                                b = 0.25
   a = y – b.x     a = 2.5 – 0.25(3)                                       a = 1.75
   Y = 1.75 + 0.25X
   There is no difference between the regression equations.

e. Syx = √[∑y2 - a∑y - b∑xy]/(n – 2)]
   Syx = √{[(39.5 – 1.75(15) – 0.25(51.5)] / (6 – 2)}                     Syx = 0.306 (000 000)MU
   Y = 1.75 + 0.25X                     Y = 1.75 + 0.25(6)               Y = 3.25 (000 000)MU
   Y +/- t.Syx           3.25 +/- (2.132)(0.306)                   3.902   2.598 (000 000) MU

      There is 90% probability that the sales will fall between 3 902 000MU and 2 598 000MU, if next
      year‟s payroll is 6 000 000 MU. There is still 10% risk that sales may fall outside these limits.

51. Years Lumber        Roofing
            Sales,x     Sales,y (x – x)       (y – y) (x – x)2   (y – y)2 (x – x)(y – y) ___
    2001      9          5         -5            -2      25         4           10
    2002     10          5         -4            -2      16         4            8
    2003     12          6         -2            -1       4         1            2
    2004     14          6          0           -1        0         1             0
    2005     15          8          1             1       1         1             1
    2006     18          9          4             2      16         4             8
    2007     20         10          6            3       36         9            18
             98         49          0            0       98        24            47
      x = 14    y=7

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                    Introduction to Production / Operations Management


   a. r = [∑(x – x)(y – y)] / √[∑(x – x)2.∑(y – y)2]                  r = 47/ √(98)(24)
                                                               r = 0.97
       There is a very strong relationship between Lumber sales and Roofing sales.

       r2 = (0.97)            r2 = 0.94
       94% of variation in lumber sales depends on roofing sales, only 6% depends on other
       factors.

   b. b = 47/98 = 0.48       a = y – b.x                  a = 7 – 0.48(14) = 0.28
                      Y = 0.28 + 0.48X

   c. tc = | r |√[(n – 2)/(1 – r)2]                     tc = 0.97 √(5/0.06) = 8.85
      tk = 2.05 at 10% level of significance and 5 as the degree of freedom.
      tc (8.85) > tk (2.015)      The computed r is meaningful.

   d. Years, x   Lumber
                 Sales, y  xy          x2
       2001    0      9     0          0
       2002    1    10     10          1
       2003    2    12     24          4
       2004    3    14     42          9
       2005    4    15     60         16
       2006    5    18     90         25
       2007    6    20    120         36
              21    98    346         91

       ∑y = n.a + ∑x            (1)                    98 = 7a + 21b            (1)
       ∑xy = a∑x + b∑x2         (2)                    346 = 21a + 91b          (2)
                                a = 8.43        b = 1.86
                        Y = 8.43 + 1.86X Trend Forecasting Equation
                        Y2008 = 8.43 + 1.87 (7)                Y2008 = 21.45 units
      Forecast of roofing sales of 2008;
                        Y2008 = 0.28 + 0.48 X                  Y2008 = 0.28 + 0.48(21.45)
                                                                Y2008 = 10.58 units
   e. left to the student
   f. left to the student




Prof.Dr.Dr.M.Hulusi DEMIR                                                                    117
INVENTORY CONTROL
6. Selling price = 15 MU/unit               Cost = 5 MU/unit         Salvage Value = 1 MU/unit
   If the store overstocks, the loss per case for every excess case at the end of the day will be;
        K0 = Cost/case – Salvage Value/case           Ko = 5 MU/case – 1 MU/case = 4 MU/case
   If the store/stand understocks, the opportunity cost for every case the stand could sell but did not
  stock will be;
          Ku = Price/case - Cost/case       Ku = 15 MU/case – 5 MU/case = 10 MU/case
  Therefore the critical ratio will be;
          P(C)* = Ko / (Ku + Ko)       P(C)* = 4/(4+10) = 0.29

  Daily Prob. at Cumulative
  Sales this level Probababity
   5      0.10        1.00
   6      0.10        0.90
   7      0.20        0.80
   8      0.30        0.60
   9      0.20        0.30
                            P(C)* = 0.29
   10     0.10        0.10

  The stand should order and sell 9 cases/day, i.e. 63 cases/week, because it has cumulative
  probability (0.30) > critical probability (0.29).

7. If the store overstocks, the loss per unit for every excess sweater at the end of the season will be;
          K0 = Cost/sweater – Salvage Value/sweater
         K0 = 18.25 MU/sweater – 14.95 MU/sweater = 3.3 MU/sweater
    If the store understocks, the opportunity cost for every sweater the store could sell but did not stock
    will be;
          Ku = Price/sweater – Cost/sweater
          Ku = 34.95 MU/sweater – 18.25 MU/sweater = 16.7 MU/sweater
    Thus the Service Level is;
          S.L. = Ku/(K0 + Ku)
          S.L. = 16.7/(3.3 + 16.7) = 0.835
    From normal distribution table Z = 0.97
          Iopt = µ + Z.σ
          Iopt = 80 + 0.97(22) =101.34 = 101 Sweaters
    The store should order and stock 101 sweaters.

8. ∆I = 2400 – 1000 = 1400 copies              Selling price/copy = 4.50 MU/copy
   Cost/copy = 2.50 MU/copy                    Salvage Value/copy = 0 MU/copy
   Ko = 2.50 MU/copy
   Ku = Price/unit – Cost/unit  Ku = 4.5 – 2.5 = 2 MU/copy
   P(C)* = Ko / (Ku + Ko)      P(C)* =2.5/(2.5 + 2) = 0.56
   Thus the service level is;
   S.L. = 1.00 – 0.56 = 0.44
   Iopt = Cmin + ∆I (S.L.)  Iopt = 1000 – 1400(0.44) = 1616 copies
   The magazine shop should order 1600 copies.
                      Introduction to Production / Operations Management


9. If the Fish Market overstocks, the loss per unit for every excess kg of blue fish at the end of the day
   will be;
         Ko = Cost/kg – Salvage Value/kg = 1.40 MU/kg – 0.80 MU/kg = 0.60 MU/kg
   If the Fish Market understocks, the opportuniy cost for every kg of blue fish the Market could sell
   but did not stock will be;
         Ku = Price/kg – Cost/kg = 1.90 MU/kg – 1.40 MU/kg = 0.50 MU/kg
   Thus the Service Level is;
         S.L. = Ku / (Ko + Ku)  S.L. = 0.50 / (060 + 0.50) = 0.45
   From normal distribution table Z = 0.13
         Iopt = µ + Z.σ
         Iopt = 80 – 0.13(10) = 78.7 kg
   The Fish Market should order and sell 79 kgs. of blue fish daily.


11. If the drugstore overstocks, the loss/unit for every excess unit at the end of the New Year will be;
          K0 = Cost/unit – Salvage Value/unit  K0 = 1.30 MU/unit – 0.88 MU/unit = 0.42 MU/unit
    If the drugstore understocks, the opportunity cost for every excess unit the store could sell but did
    not stock will be;
          Ku = Price/unit – Cost/unit  Ku = 2.20 MU/unit – 1.30 MU/unit = 0.90 MU/unit
    Thus the critical probability is:
          P(C)* = Ko/(K0 + Ku)        P(C)* = 0.42 / (0.42 + 0.90) = 0.32

               Demand       Probability    Cum. Prob.
                3 000            0.05            1.00
                3 500            0.15            0.95
                4 000            0.25            0.80
                4 500            0.25            0.55
                                                          CRITICAL PROBABILITY (0.32)
                 5 000            0.15             0.30
                 5 500            0.15             0.15

     We recommend Drugstore to order 4 500 cards, because it has cumulative probability (0.55)≥
     Critical probability (0.32).

13. Using equation
        No = √(CE/2B)          or      No = √ (Cp.Z/2B)
    we obtain
        No = √(220 000)(48))/2(30) = 25.69 orders/year

14. Using equation
        Xo (MU) = √(2CpB)/Z
    We obtain
        X0√(2(28 000)(48) / 0.23         Xo (MU) = 3 418.62 MU/order

15. a.   Xo = √(2CB)/E  Xo = √(2(4860)(4)/30 = 36 bags/order
    b.   Average number of bags on hand = X/2 = 36/2 = 18 bags/order
    c.   No = C/X = 4 860/36 = 135 orders/year
    d.   Ke = √(2CBE) = √(2(4860)(4)(30) = 1 080 MU/year
    e.   Ke = √(2(4 860)(5)(30) = 1207.48 MU/year
         Increase = 1 207.48 – 1 080 = 127.48 MU/year
     It will affect the total inventory cost to increase by 127.48 MU/year.

16. a. Usage = 40 packages/day x 260 days/year = 10400 packages/year
       Xo = √(2CB)/E        Xo = √(2(10 400)(6)) / 3 = 204 packages/order

94
       b. That is you have to use the formula
          Ke = √(2CBE)            Ke = √(2(10 400)(6)(3) = 611.88 MU/year
       c. Yes. Since we round the figures, the total annual ordering cost must be equal to the annual
          ordering cost at EOQ.
          Ke = (C/X)B + (X/2)E= (10 400/204)6 + (204/2)3 = 305.88 + 306 = 611.88 MU/year
       d. Ke = (C/X)B + (X/2)E = (10 400/200)6 + (200/2)3 = 312 + 300
          Ke = 612 MU/year
          No, I won‟t recommend. It will only save 0.12 MU/year, which is
          negligible.

 17.     Usage = 750 pots/month = 750 x 12 = 9 000 pots/order
         Price = 2 MU/pot        Carrying cost = 25% annually         Ordering cost = 30 MU/order
         a. Xo = √(2CB)/Zp  √{2(9 000)(30)/0/25(2) = ~ 1 039 pots/order
            Ke = √(2CBE)      √2 (9 000)(30)(0.25)(2) = 519.62 MU/year
         b. Ke = (C/X)B + (X/2)E  Ke = (9 000/1039)/30 + (1039/2)(2)(0.5) = 779.37 MU/year

 18.     Usage = 800 crates/month                          Purchase cost = 10 MU/crate
         Carrying cost = 35% of purchase cost annually     Ordering cost = 28 MU/order
         Ke according to EOQ:
         Ke = √(2CBE)  Ke = √{2(800x12)(28)(0.35)(10)} = 1 371.71 MU/year
       Ke according to current policy:
       Ke = (C/X)B + (X/2)E  Ke = 12(28) + (800/2)(0/35)(10) = 1 736 MU/year
       Saving due to using EOQ model;
       - Ke (EOQ) + Ke (current) = 1 736 - 1 371.71 = 364.29 MU

19. If İlhan‟s Doughnut Shoppe overstocks, the loss per dozen for every excess dozen at the end of the
    day will be;
         K0 = Cost/dozen- Salvage Value/dozen = 0.80 MU/dozen – 0.60 MU/dozen
         K0 = 0.20 MU/dozen
    If İlhan‟s Shoppe understocks, the opportunity cost for every dozen the Shoppe could sell but did
    not stock will be;
         Ku = Price/dozen – Cost/dozen =1.20 MU/dozen – 0.80 MU/dozen
         Ku = 0.40 MU/dozen
    Thus the critical probability is:
          P(C)* = K0/(K0 + Ku) = 0.20 /(0.20 + 0.40) = 0.33

           Demand(dozens)       Probability Cum.Prob.
                19                 0.01             1.00
                20                 0.05             0.99
                21                 0.12             0.94
                22                 0.18             0.82
                23                 0.13             0.64
                24                 0.14             0.51
                25                  0.10            0.37
                                                     CRITICAL PROBABILITY ( 0.33)
                  26               0.11             0.27
                  27               0.10             0.16
                  28               0.04             00.6
                  29               0.02             0.02
       The level of stock that will maximize expected profit is the highest level of stock that has a
       cumulative probability greater than or equal to 0.33 that will be sold. From the table you see that
       25 dozens of Doughnuts is the highest level wit a cumulative probability greater than 0.33.
                      Introduction to Production / Operations Management


21. Demand rate = 2 000 bikes/year                      Cost = 800 MU/bike
    Ordering cost = 40 MU/order                         Carrying cost = 25% item`s cost
    Store open 250days per year
    a. Xo = √(2CB)/Zp  X0 = √{2(2000)(40)/(0.25)(800)} = 28.28 = ~ 28 bikes/order
    b. No = C/X = 2000/28 = 71.43 = ~ 71 orders/year
    c. To = 1/No                       to = (1/71)250 = 3.5 days between orders
    d. Ke = √(2CBE)  Ke = √{2(2000)(40)(0.25)(800)} = 5 656.85 MU annually
    e. Annual ordering cost = N.B =70.72(40) = 2828.80 MU (due to rounding)
       Annual holding cost = (X/2)(Zp) = (28.28/2)(0.25x800) = 2828 MU

22.   Demand rate : 150 units/month = 1800 units/year              Cost/towel = 2.5 MU/towel
      Ordering cost = 12 MU/0rder                                  Carring cost = 27%/year
      Current process:
                   Xo = √(2CB)/Zp  Xo = √{2(1800)(12)/(0.27)(2.5)} = 252/98 = ~ 253 units/order
                  No = C/X = 1800/252.98 = 7.12 = ~ 7 orders/year
                 . Ke = √(2CBE)  Ke = √{2(1800)(12)(0.27)(2.5) = 170/76 MU/year
                  Annual ordering cost = 85.38 MU/year          Annual holding cost = 85.38 MU/year
      With automation:
          Cost of ordering = 4 MU/order
                  Xo = √(2CB)/Zp  Xo = √{2(1800)(4)/(0.27)(2.5)} = 146.06 units/order
                  No = C/X = 1800/146.06 = 12.32 orders/year
                        Ke = √(2CBE)  Ke = √{2(1800)(4)(0.27)(2.5)} = 98.59 MU/year
                        Annual ordering cost = 49.3 MU/year        Annual holding cost = 49.3 MU/year
      At order cost 12, EOQ „s 253 units/order and there are about 7 orders per year. Annual costs of
      inventory management are 170.76 MU. At order cost 4 MU, EOQ falls to 146 units/order, and
      order frequency rises to 12. Annual inventory costs fall to 98.59 MU/year. The lower order cost
      encourages smaller, more frequent orders.

23.   Demand rate = 96 000 MU annually                      Ordering costs = 45 MU/order
      Holding costs = 0.22 of purchase price/year
      First calculate EOQ from the data provided. In this problem the “units” are “MU”
       Xo (MU) = √(2CpB)/Z  Xo = √{2(96000)(45)/(0.22)} = 6 266.80 MU/order
       To = 1/No                 to = (6266.8/96000)12 = 0.78 month`s supply (x 4 = about 3 weeks
      usage)

24.   C = 72 000 units/year               s = 120 MU/st-up
      p = 4 MU/unit                       Z = 25%/year
      Qo = √(2Cs)/Zp  Qo = √{2(72000)(120)/(0.25)(4) = 4 156.92 units/order
      Ke = √(2CsZp)  Ke = √{2(72000)(120)(0.25)(4) = 4 156.92 MU/year
      (Annual set=up cost = 2078.46 MU    Annual holding cost = 2078.46 MU)

25.   Order quantity = 60 units/order     Carrying cost = 0.40 of units price
      Cost = 10 MU/unit                   Annual demand = 240 units/year
      Xo = √(2CB)/Zp  60 = √{2)240)(B)/(0.40)(10)  B = 30 MU/order

26.   Order quantity = 60 kgs/order       Carrying cost = 30% year
      Ordering cost = 20 MU/order         Price of the item = 200 MU/kg
      Xo = √(2CB)/Zp  60 = √{2)(C)(20)/(0/30)(200)  C = 5 400 kg/year

27.   A container occupies 4 ft2 of space
      Available space = 600 ft2
      Therefore the warehouse will hold 600/4 = 150 containers
      Demand = 12 000 units/year          Holding cost = 2 MU/unit-year
      Order cost = 5MU/order

 96
      a. Xo = √(2CB)/E  Xo = √{2(12000)(5)/2} = 244.95 = ~245 containers/order
      b.     Ke = √(2CsZp)  Ke = √{2(12000)(5)(2) = 489.90 MU/year
      c. Ke = (C/X)B + (X/2)E  Ke = (12000/150)(5) + (150/2)(2) = 550 MU/year
      d. Xo (EOQ) = 245 containers/order                    Ke (EOQ) = 489.90 MU/year
           Xo (current) = 150 containers/order              Ke (current) = 550 MU/year
           Extra        = 95 containers                                  = 61.1 MU/year
      Result:
      The warehouse will hold only 150 containers. The annual cost of inventory at Xo = 150 is 550
      MU. The economic order quantity is 245 containers, more than there is room to store. The total
      annual cost at 245 containers is 489.90 MU. This cost is 61.10 MU less than current cost which
      reflects the limited storage space. Rushton would consider paying up to 61.1 MU for a year`s
      rental of enough space to store 95 additional containers.

28.   C = 9 600 tires/yr          E = 16 MU/tire/yr
      B = 75 MU/order             days/yr = 288 days/yr
      a. Xo = √[(2CB/E]            Xo = √[2(9600)(75)/ 16]            Xo = 300 tires/order
      b. No = C/Xo                       No = 9600/300 = 32 orders/yr
      c. To = 1/No                       to = (1/32)288 = 9 days
      d. Ke = √[2CBE]                    Ke = √[2(9600)(16)(75)] = 4 800 MU/yr

29. C = 10 000 units/yr         s = 100 MU/set-up
    E = 0.50 MU/unit/yr         R = 80 units/day
    c = 60 units/day
    a. Qo = √[(2Cs)/(1 – c/R)]  Qo = √[(2)(10000)(100)] / [1- 60/80]  Qo = 4 000 units/run
    b. t1 = 4 000/80 = 50 days/run
    c. Imax = Qo(1 – c/R)                      Imax = 4 000(0.25) = 1 000 units/run
    d. Ke = √[2CBE(1 – c/R]
         Ke = √[2(10 000)(100)(0.50)(0.25)] = 500 MU/yr

31.    Sales = 380 bottles/month  Sales = 380x12 = 4 560 bottles/year
       Price = 0.45 MU/bottle             Order Cost = 8.50 MU/order
       Holding Cost = 25%
       a. No = √[CE/2B]
           No = √[4560(0.45)(0.25)/(2(8.50)]  No = 5.49 orders/year
       b. to = (1/N)240           to = (1/5.49)240 = 43.72 days = ~ 44 days
       c. Ro = c.tlt
           Ro = 380(2) = 760 bottles
       d. Xo = √[2CB/Zp]        Xo = √[2(4560x8.5)/(0.25x0.45)
           Xo = 830.10 units/order
      e. Ke = √[2CBZp]          Ke = √[2x4560x8.50x0.45x0.25]
           Ke = 93.39 MU/year

32.   Demand = 200 000 Units/year               Set-up cost = 160 MU/set-up
      Carrying cost = 100 MU/unit/year          Back-order cost = 600 MU/unit
      a. Qo = √{2Cs/E} .√{(E+d)/d}          Qo = √{2(200000)(160)/100}.√(100+600)/600
                                                 = 864.10 units/run
      b.   Imax = Qo (d/(E+d)  Imax = 864.10 (600/700) = 740.1 units/run
      c.   S = Qo – Imax  S = 864.1 – 740.1 = 124 units/run
      d.   No = C/Q          No= 200000/864.1 = 231.45 runs/year
      e.    to = ( X/C)250  t0 =(564.10/200000)250 = 1.08 days/run
      f.   Ke = S.d           Ke = 124(600) = 74 400 MU/year
                      Introduction to Production / Operations Management


      g. C = 400 000 units/year               s = 320 MU/set-up
         Qo = √{2Cs/E} .√{(E+d)/d}  Qo = √{2(400000)(320)/100}. √(600+100)/600
                                                Qo = 1728.20 units/run
          Imax = Qo (d/(E+d)     Imax = 1728.20 (600/700) = 1481.3 units/run

45.   a) We begin with computing the annual demand. C = 18 units/week x 52 weeks = 936 units/year
        The annual cost for the current policy is (ordering 390 units every time)
        Ke = (C/X)B + (X/2)Zp = (936/390)45 + (390/2)(0.25x60) = 3 033 MU/year
      b) The annual cost of 468 units-lot size is
         Ke = (C/X)B + (X/2)Zp = (936/468)45 + (468/2)(0.25x60) = 3600 MU
         Decision Point : A lot size of 468 units, which is a half year supply would be a more expensive
         option than the current policy.
      c) EOQ = X0=√[2CB/Zp] = √[2(936x45)/(0.25x60)] = 74.94 = ~75 units/order
      d) Ke = √[2CBZp]               Ke = √[2(936)(45)(0.25)(60) = ~1 124.10 MU/year
      e) Total Ordering Cost = Ke/2 =1124.10/2 = 562.05 MU/year
         (C/X)B = (936/75)(45) = 562 MU/year
      f) No = C/X = 936(75 = ~12.48 Orders/year
      g) to = X/C = 75/18 = 4.17 weeks/order
      h) Ro = c.to = 18 units/weekx1 week = 18 units

 46. Cost = 1.30 MU/unit            Price = 2.20 MU/unit Salvage Value = 2.20x0.40 = 0.88MU/unit
      If the store overstocks, the loss per unit for every excess unit at the end of the season will be;
        Ko = Cost/unit – Salvage Value/unit = 1.30 MU/unit – 0.88 MU/unit = 0.42 MU/unit
     If the store understocks, the opportunity cost for every unit the company would sell but did not
     stock      will be;
        Ku = Price/unit – Cost/unit = 2.20 MU/unit – 1.30 MU/unit = 0.90 MU/unit
        Thus the critical probability is:
         P(C)* = Ko/(K0 + Ku)       = 0.42/(0.42+ 0.90) = 0.32
      The level of stock that will maximize the expected profit is the highest level of stock that
      has a probability greater than or equal to 0.32. From the table you can see that 4 500 cards
      is the highest level with a probability greater than 0.32.

 52. a. EOQ = X0=√[2CB/Zp] = √[2(2500)(18.75)/(0.10)(15) = 250 units/order
     b. Average Inventory = Xo/2       Average Inventory = 250/2 = 125 units/order
     c. Annual inventory holding cost = (X/2)(Zp)
        An. Hold. Cost = (250/2)(0.10(15) = 187.50 MU/year
     d. No = C/X  No = 2500/250 = 10 orders/year
     e. Annual ordering costs = N.B       Annual ordering cost = 10(18.75) = 187.50 MU/year
     f. Ka = C.p + Annual ordering Cost + Annual carrying cost
        Ka = 2500(15) + 187.50 + 187.50 = 37 875 MU/year
     g. to = (X/C)no. of days     to = 1/10(250) = 25 days
     h. Ro = c.tlt     Ro = (2500/250)(2) = 20 units

 53. a. Daily Demand = C/250 = 10 units/day
     b. Q0=√[2Cs/E(1-c/R)]  Q0= √[2(2500)(25)/1.48(1-10/50)] =324.92 units/run
     c. t1 = Qo/R  t1 = 324.92/50 = 6.5 days/run
     d. Inventory sold = 10 units/day x 6.5 days/run = 65 units/run
     e. Imax = Qo(1 – c/R)                      Imax = 324.92(1-10/50) = 259.94 units/run
     f. Av. Inv. = Imax/2                       Imax = 129.97 units
     g. Ke = √[2CsE(1-c/R)]              Ke = √[2(2500)(25)(0.10)(14.80)(1-10/50)] =            384.71
     MU/year
     h. Ro = c.tlt     Ro = (2500/250)(0.5) = 5 units


 98
  59. Data Summary:
         R = 100 units/day                s = 50 MU/run                   c = 40 units/day
         E = 0.50 MU/unit-year            workdays = 250 days/year        p = 7 MU/unit
         Tlt = 7 days
      a. Q0=√[2Cs/E(1-c/R)]         Q0= √[2(40x250)(50) / (0.50)(1-40/100)]
                                         = 1825.74 = ~ 1826 units/run
      b.   Ro = c.tlt     Ro = 40 units/day(7 days) =280 units
      c.   Ke = √[2CsE(1-c/R)]          Ke = √[2(40x250)(50)(0.50)(1-40/100) = 547.72 MU/year
      d.   Ka = C.p +√[2CsE(1-c/R)]             Ke = (40x250)(7) + 547.72 = 70 547.72 MU/year
      e.   No = C/X  No = 10 000/1825/74 = 5.48 = ~ 6 runs/year
      f.   to = (X/C)no. of days  to = (1/5.48)(250) = 45.65 = ~ 46 days

  60. Data Summary:
          Demand = C = 100 000 units/year                    d = 600 MU/unit-year
          s = 80 MU/set-up                                   E = 25 MU/unit-year
       a. Q0=√[2Cs/E] √[(E+d)/d  Qo= √[2(100000)(80)/25]√[(25+600)/600]
                                              = 816.50 = ~ 817 units/run
       b. Imax = Q0 (d/(E+d)       Imax = 816.50 (600/625) = 783.84 = ~ 784 units/run
       c. So = Qo – Imax             So = 816.50 – 783.84 = 32.66 = ~ 33 units/run
       d. No = C/X  No = 100000/816.50 = ~ 122.47 runs/year
       e. to = (X/C)no. of days        to = (816.50/100000)250 = 2.04 = ~ 2 days
    f.    Ke = √[2CsE] √[(E+d)/d)]          Ke = √[2(100000)(80)(25)(625/600)]
                                                   = 19 595.96 MU/year
          or Ke = S.d                Ke = 32.66(600) = 19596 MU/year
          (slight difference is due to rounding the figures)
    g.    C = 200 000 units/year            s = 160 MU/set-up
            Q0=√[2Cs/E] √[(E+d)/d          Qo = √[2(200000)(160)/25]√[(25+600)/600]
                                                  = 1632.993 = ~ 1633 units/run
           Imax = Q0 (d/(E+d)       Imax = 1632.993(600/625) = 1567.67 = ~ 1568 units/run
           Ke = S.d               Ke = (1632.993 – 1567.6734)(600) = 39 192 MU/year
          All figures are doubled.

61. Ro = 150 units/order            No = 5 orders/yr Holding Cost = 30 MU/unit
    Safety Stock Levels = 0 units, 50 units, 100 units, 150 units
    Stock-out cost = 25 MU/unit/yr
  Ro SS Probability of                Annual                Total                           Total
           Total Safety Number Stock-out                     Stock-out       Carrying       Safety
           Being out        Short      Cost                 Cost              Cost          Stock Cost
  150 0 0.16 when 200 50            0.16(50)(5)(25)=1000
           0.10 when 250 100 0.20(100)(5)(25)=1250
           0.06 when 300 150 0.06(150)(5)(25)=1125 3375 MU/yr                   -             3375MU/yr
  150 50 0.10 when 250 50 0.10(50)(5)(25)= 625
           0.06 when 300 100 0.06(100)(5)(25)=750 1375 MU/yr 50(30)=1500MU/yr 2875MU/yr Min!
  150 100 0.06 when 300 50 0.06(50)(5)(25)= 375 375MU/yr 100(30)=300MU/yr 3375MU/yr
  150 150       -                 -           -          -                150(30)=4500MU/yr 4500MU/yr
        Safety stock level of 5o units is preferred. The new reorder level will be 200 units.

70. If LEMAR overstocks, the loss per turkey for every excess turkey at the end of the new year will be;
          Ko = Cost/turkey – SV/turkey = 8.50/turkey – 0 MU/turkey = 8.50 MU/turkey
    If LEMAR understocks, the opportunity cost for every turkey LEMAR could sell but did not stock
        will be;
          Ku= Price/turkey – Cost/turkey = 11.99 MU/turkey – 8.50 MU/turkey = 3.49 MU/turkey
                           Introduction to Production / Operations Management


      Thus service level is;
            SL = Ku/(Ko + Ku) = 3.49/(8.50 + 3.49) = 0.29
      The optimal inventory level will be
            Iopt = µ - Z.σ = 550 - 0.56 (40) = 527.6 = 528 turkeys
      LEMAR should order and stock 528 turkeys.

71. If TT overstocks, the loss per set for every excess set at the end of the model year will be;
          Ko = Cost/set – SV/set = 285 MU/set – 215 MU/set = 70 MU/set
    If TT understocks, the opportunity cost for every set TT could sell but did not stock will be;
          Ku = Price/set – Cost/set = 490 MU/set – 285 MU/set = 205 MU/set
    Thus the critical ratio will be;
          P(C)* = Ko/(Ko + Ku) = 70/(70 + 205) = 0.25
                      Demand            Probability Cum. Prob.
                      8 and fewer            0.00             1.00
                            9                0.27             1.00
                          10                 0.34             0.73
                          11                 0.19             0.39
                                                              CRITICAL PROBABILITY (0.25)
                          12                 0.12             0.20
                          13                 0.08             0.08
                          14 or more         0.00             0.00

        TT should order 11 TV sets, because it has cumulative probability which is greater than
           critical probability (0.25).

72.     Ordering cost, B = 40 MU/order              Carrying cost, E = 5 MU/unit/year
        Reorder point, Ro = 60 Units                Stock-out cost = 50 MU/unit
        Number of orders = 7 orders/year

 Ro SS Probability of        Annual               Total                                  Total
       Total Safety Number Stock-out              Stock-out Carrying                     Safety
        Being out     Short  Cost                 Cost      Cost                        Stock Cost
 60 0   0.2 when 70    10   0.2(10)(7)(50) = 700
        0.2 when 80    20   0.2(20)(7)(50) = 1400
        0.1 when 90    30   0.1(30)(7)(50) = 1050 3150MU/yr 0MU/yr                          3150MU/yr

 70 10       0.2 when 80       10   0.2(10)(7)(50) = 700
             0.1 when 90       20   0.1(20)(7)(50) = 700     1400MU/yr 10(5)=50MU/yr 1450MU/yr

 80 20       0.1 when 90       10   0.1(10)(7)(50) = 350     350MU/yr     20(5)=100MU/yr 150MU/yr

 90 30            -            -               -                     -    30(5)=150 MU/yr     150MU/yr Min!

 The optimal safety stock level is 30 units. The optimal reorder point is, therefore, 90 units.

73.     a.   X0=√[2CB/E] √[(E+d)/d         
                                           Xo = √[2(10000)(150)/0.75].√[(2+0.75)/2]
                                               = 2 345.2 metres/order
       b. So = Q0 (E/(E+d)       So = 2 345.2 (0.75/(2+0.75) = 639.6 metres/order
       c. Imax = Xo - So         Imax = 2345.2 – 639.6 = 1705.6 metres/order
       d. Ke =S.d                Ke = 639.6(2) = 1279.2 MU/year
       e. No = C/X               No = 10000/2345.2 = 4.26 orders/year
       f. to = (1/N)No. of days  to = (1/4.26)(311) = 73 days/order
        g.   t1 = Imax/c             t1 = (1705.6/10000)311 = 53.2 days/order
        h.   ts =S/c                 ts = (639.6/10000)311 = 19.9 days/order

 100
76.   a. The EOQ assumptions are met, so the optimal order quantity is
         EOQ = Xo = √(2CB/E)          Xo = √[2(250)(20)(1)] = 100 units/order
      b. No = C/X  No = 250/100 = 2.5 orders/year
      c. Average inventory = Xo/2         Average Inventory = 100/2 = 50 units/order
      d. Given an annual demand of 250, a carrying cost of 1 MU, and an order quantity of 150, the
         Co. must determine what the ordering cost would have to be for the order policy of 150 units
      to be optimal. To find the answer to this problem, we must solve the traditional EOQ equation
         for the ordering cost.As you can see in the calculations that follow an ordering cost of 45 MU
      is needed for the order quantity of 150 units to be optimal.
         EOQ = Xo = √(2CB/E)          (150)2 = 2(250)B/1  B = 22500/500 = 45 MU/order

77.       OLS = Qo = √(2Cs/E(1-c/R))  Qo = √[2(6750)(150)/1(1-30/125)] = 1632 minislicers/run

78.       OLS = Qo = √(2Cs/E(1-c/R))  Qo = √[2(8000)(100)/0/3(1-40/150)] = 2697 scissors/run

79.   A. aa. Let`s calculate the present total annual inventory cost:
             Ke = (C/X)B + (X/2)E          Ke = (10000/400)(5/5) + (400/2)(0.4) = 217.50 MU/year
         ab. EOQ is calculated as follows:
             EOQ = Xo = √(2CB/E)           Xo = √[2(10000)(5.5)/ 0.4] = 524.4 units/order
         ac. The total annual inventory cost if EOQ is employed calculated as follows:
             Ke = √[2CBE]                   Ke = √[2(10000)(5.5)(0.4)] = 209.76 MU/year
         ad. Estimated annual savings in inventort is calculated:
             Saving = Ke (current) – Ke (EOQ) = 217.50 – 209.76 = 7.74 MU/year
         ae. The inventory analyst concludes that if the annual savings on this one material were
             applied to the thousand of items in inventory, the savings from EOQ would be significant.

      B. ba. EOQ is calculated as follows:
             OLS = Qo = √(2Cs/E(1-c/R))  Qo = √[2(10000)(5.5)/(0.4)(1- (10000/250)/120)]
             Qo = 642.26 units/run
         bb. Maximum inventory in the stocks:
             Imax = Q0 (1 - c/R)    Imax = 624.26(1 – 40/120) = 428.17 units/run
         bc. The new total inventory cost is calculated:
             Ke = √(2CsE(1-c/R))  Ke = √[1(10000)(5.5)(0.4)(1-40/120)] = 171.26 MU/year
         bd. The EOQ and total annual inventory costs from A, when the units were delivered all at
              once, were 524.4 units/order and Ke =209.76 MU/year.
         be. The estimated savings are calculated:
             Savings = Ke (Model 1) – Ke (Model 2) = 209.76 -171.26 = 38.50 MU/year

      C. ca. The EOQ is:
              Q0=√[2Cs/E] √[(E+d)/d       Qo = √[2(10000)(5.5)/(0.4)]√[(5.6+0.4)/5.6]
                                              = 542.81 units/run
        cb. The max inventory level
            Imax = Q0 (d/(E+d)       Imax = 542.81 (5.6/6.0) = 506.62 units/run
        cc. The new total inventory cost is calculated:
            Ke = S.d      Ke = (542.81-506.62)(5.6) = 217.10 MU/year
        cd. The estimated savings are calculated:
             Savings = Ke (minimum from be) – Ke (Model 3) = 209.76 – 217.10 = - 7.34 MU/year
             The policy will result in loss, therefore this policy is not recommended.

80.   C = 80 000 bottles/month = 500 bottles/hour
      a. OLS = Qo = √(2Cs/E(1-c/R))  Qo = √[2(100)(80000)/0.1(1-500/3000)] = 13 856 bottles/run

      b. New s = 50 MU/set-up
                        Introduction to Production / Operations Management


              New Qo = √(2Cs/E(1-c/R))        Qo = √[2(50)(80000)/0.1(1-500/3000)] = 9 798
              bottles/run


      81.     Qo = √(2Cs/E(1-c/R))           Qo = √[2(1000)(10)/0.5(1-4/8)] = 282.8 hubcaps/run
                                              Qo= ~ 283 hubcaps/run

      82.     Qo = √(2Cs/E(1-c/R))     10 = √[2(10000)(s)/5(1-250/500)]
                                               100 = 20000s/(5/2)
                                                  s = 0.0125 MU/run
              Set-up time = s/ labour rate     set-up time = 0.125 MU/run/10 MU/hour
                                                            = 0.00125 hour/run
                                           or    set-up time = 0.075 minutes/run
                                           or                = 4.5 seconds/run

      83.     a. EOQ = Xo = √(2CB/E)       Xo = √[2(150)(10000)/0.75] = 2000 metres/order
              b. Ke = √[2CBE]             Ke = √[2(10000)(150)((0.75)] = 1500 MU/year
              c. Ke = 1500/2 = 750 MU/year Total Ordering Cost
                  (C/X)B = (10000/2000)150 = 750 MU/year
              d. No = C/X  No = 10000/2000 = 5 orders/year
               e. to =(X/C)(No. of days)    to = 1/5(311) = 62.2 days/order

      84.     c = 32 metres/day                    R = 32 metres/day
               Qo = √(2Cs/E(1-c/R))        Qo = √[2(10000)(150)/0.75(1-32/32)] = ∞
              Continuous production


      86. a. Cost = 1400 MU/unit           Price = 2000 MU/unit           Salvage Value = 600 MU/unit
             If the store overstocks, the loss per unit for every excess unit at the end of the season will
             be;
            Ko = Cost/unit – Salvage Value/unit = 1400 MU/unit – 600 MU/unit = 800 MU/unit
            If the store understocks, the opportunity cost for every unit the company would sell but did
            not stock will be;
            Ku = Price/unit – Cost/unit = 2000 MU/unit – 1400 MU/unit = 600 MU/unit
            Thus the critical probability is:
            P(C)* = Ko/(K0 + Ku)       = 800/(800 + 600) = ~ 0.57
              The level of stock that will maximize the expected profit is the highest level of stock
              that has a probability greater than or equal to 0.57. From the table you can see that
              72 units is the highest level with a probability greater than 0.57.

            b. Each unsold unit increases the cost of the unit by 300 MU.
               Ku = 2000 MU/unit – 1400 MU/unit = 600 MU/unit
               Ko = 1400 MU/unit + 300 MU/unit – 600 MU/unit = 1100 MU/unit
              P(C)* = Ko/(K0 + Ku)        = 1100/(1100 + 600) = ~ 0.6471
              The optimal level is again 72 units of attachment, because it`s cumulative probability (0.75)
              is greater than critical probability.




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LINEAR PROGRAMMING
A. SIMPLEX METHOD
      1. a) Objective Function: Z = 6A + 3B             Z = 6A + 3B + 0S1 + 0S2 + 0S3
      Subject to: 20A + 6B ≤ 600                  20A + 6B + S1         = 600
                  25A + 20B ≤ 1000                25A + 20B     + S2    = 1000
                  20A + 30B ≤ 1200                20A + 30B        + S3 = 1200
                        A,B ≥ 0                          All variables ≥ 0

       Initial Simplex Tableau:

               Product Quantity 6            3      0      0    0
       Cj      Mix            bi      A      B      S1     S2   S3       bi/aij_____
       0       S1         600         20     6      1      0    0       600/20 = 30Min! Leaving
       0       S2       1000          25     20     0      1    0       1000/25 = 40
       0       S3       1200          20     30     0      0    1       1200/20 = 60
               Zj           0         0      0      0      0    0
               Cj - Zj                6      3      0      0    0           ______
                                      Max! ENTERING VARIABLE
       A is entering while S1 is leaving.
       New A values: 600/3=30, 20/20=1, 6/20=0.3, 1/20=0.05, 0, 0

       New Values of S2                   New Values of S3
       1000 - 25(30) = 250                1200 – 20(30) = 600
         25 – 25(1) = 0                     20 – 20(1) = 0
         20 – 25(0.3) = 12.5                30 – 20(0.3) = 24
          0 – 25(0.05) = -1.25               0 – 20(0.05) = -1
          1 – 25(0) = 1                      0 – 20(0) = 0
          0 – 25(0) = 0                      1 – 20(0) = 1

       2nd Simplex Tableau:

               Product Quantity 6         3     0           0        0
       Cj      Mix         bi   A         B     S1          S2       S3 bi/aij_____
       6       A        30       1        0.3   0.05        0        0 30/0.3 = 100
       0       S2      250       0        12.5 -1.25        1        0 250/12.5=20Min! Leaving
       0       S3      600       0        24   -1           0        1 600/24= 25
               Zj      120       6        1.8   0.3         0        0
               Cj - Zj           0        1.2 - 0.3         0        0
                                           Max! Entering variable
       B is entering while S2 is leaving.
       New B values: 250/125=20, 0/12.5=0, 12.5/12.5=1, -1.25/12.5=-0.1, 1/12.5=0.08, 0

New Values of A                           New Values of S3
30 - 0.3(20) = 24                         600 – 24(20) = 120
 1 – 0.3(0) = 1                              0 – 24(0) = 0
0.3 – 0.3(1) = 0                            24 – 24(1) = 0
0.05 – 0.3(=0.1) = 0.08                    - 1 – 24(-0.1) = 1.4
  0 – 0.3(0.08) = - 0.024                    0 – 24(0.08) = - 1.92
  0 – 0.3(0) = 0                             1 – 24(0) = 1
   Prof.Dr.Dr.M.Hulusi DEMIR                                                                 127
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3rd Simplex Tableau:

               Product Quantity 6               3          0     0      0
       Cj      Mix         bi   A               B          S1    S2     S3
       6       A        24       1              0          0.08 -0.024 0
       3       B        20       0              1        - 0.1   0.08 0
       0       S3      120       0              0          1.4 - 1.92 1
               Zj      204       6              3          0.18 0.096 0
               Cj - Zj           0              0        - 0.18 - 0.096 0

There is no positive value in the row of “Cj – Zj”, therefore optimal solution is attained.
               A = 24 units
               B = 20 units
               Z = 204 MU

b)     S3 has non-zero slack.
       S3 has 120 hrs. of idle labour hour.

2.     Information from the question:

              Models           H (hrs./unit)              W (hrs./unit) Total hours
                                  X1                          X2       available
       Department
       Fabrication                  4                          2             600 hrs.
       Assembly                     2                          6             480 hrs.
       Profit/Unit                 40 MU                      30 MU

       Model building:
        Objective function: Max! Z = 40X1 + 30X2            Max! Z = 40X1 + 30X2 + 0S1 + 0S2
        Subject to:             4X1 + 2X2 ≤ 600             4X1 + 2X2 + S1      = 600
                                2X1 + 6X2 ≤ 480             2X1 + 6X2      + S2 = 480
                                    X1, X2 ≥ 0                     All variables ≥ 0

Initial Simplex Tableau:

               Product Quantity        40MU 30MU            0MU     0MU
       Cj      Mix       bi            X1    X2             S1       S2    bi/aij____
       0       S1       600            4     2              1        0    600/4=150 Min! Leaving
       0       S2       480            2     6              0        1   480/2=240
               Zj         0            0     0              0        0
               Cj - Zj                 40    30             0        0 ___________
                                        Max! Entering
       X1 is entering, while S1 is leaving.
       New X1 values: 600/4=150, 4/4=1,2/4= ½ , ¼, 0

       New values of S2
       480 – 2(150) = 180
         2 - 2(1) = 0
         6 – 2(1/2) = 5
         0 – 2(1/4) = - ½
         1 – 2(0) = 1

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Second Simplex Tableau:

               Product Quantity        40MU 30MU 0MU             0MU
       Cj      Mix       bi            X1     X2       S1         S2   bi/aij____
       40      X1       150            1      ½        ¼          0   150/0.5=300
       0       S2       180            0      5      -½          1    180/5 =36 Min! Leaving
               Zj      6000           40     20        10         0
               Cj - Zj                 0     10      - 10         0 ___________
                                               Max! Entering
       X2 is entering, while S2 is leaving.
       New X2 values: 180/5=36, 0,1,-0.1, 0.2

       New values of X1
       150 – ½ (36) = 132
          1 - ½ (0) = 1
        0.5 – ½ (1) = 0
         ¼ - ½ (-0.1) = 0.3
          0 – ½ (0.2) = -0.1

Third Simplex Tableau:

               Product Quantity       40MU 30MU 0MU 0MU
       Cj      Mix       bi           X1    X2     S1   S2 _
       40      X1       132           1     0    0.3  - 0.1
       30      X2        36           0     1  - 0.1    0.2
               Zj      6360           40   30     9     2
               Cj - Zj                0     0   -9    - 2___

There is no positive value in the row of “Cj – Zj”, therefore optimal solution is obtained.
We should produce 132 units of Model H,
                     36 units of Model W.
Maximum profit is 6360MU.

3.a)   Data summary from the question:

                      Deluxe Mix Standard Mix Total kgs.
                           (kgs.) X1      (kgs.) X2 Available
       Raisins                 2/3             ½         90 kgs.    (1/5MU/kg.)
       Peanuts                 1/3             ½         60 kgs.__ (0.60MU/kg.)
       Selling Price/kg.       2.9MU           2.55MU
       Cost/kg.     1.5(2/3)+0.6(1/3)=1.2 1.5(1/2)+0.60(1/2)=1.05
       Profit/kg.              1.7MU           1.5MU            ___

       Model construction:
             Objective function:      Max! Z = 1.7X1 + 1.5X2
             Subject to:              2/3X1 + ½ X2 ≤ 90
                                      1/3X1 + 1/2X2 ≤ 60
                                         X1         ≤ 110
                                                 X2 ≤ 110
                                             X1, X2 ≥ 0

   Prof.Dr.Dr.M.Hulusi DEMIR                                                                   129
                   Introduction to Production / Operations Management



       Changing the inequalities into equations, we have;
             Objective function: Max! Z = 1.7X1 + 1/2X2 + 0S1 + 0S2 + 0S3 + 0S4
             Subject to          : 2/3X1 + ½ X2 + S1                    = 90
                                     1/3X1 + 1/2X2      + S2            = 60
                                        X1                   + S3       =110
                                                 X2                + S4 = 110
                                                           All variables ≥ 0

Initial Simplex Tableau:

          Product Quantity 1.7MU 1.5MU                0MU 0MU 0MU 0MU
       Cj Mix        bi      X1    X2                 S1   S2 S3  S4___bi/aij
       0   S1       90      2/3    ½                  1    0  0    0 90/2/3=135
       0   S2       60      1/3    ½                  0    1  0    0 60/1/3=180
       0   S3      110      1      0                  0    0  1    0 110/1=110 Min! Leaving
       0   S4      110      0      1                  0    0  0    1   -
           Zj        0      0      0                  0    0  0    0
           Cj - Zj          1.7    1.5                0    0  0    0    ___
                                   Max! Entering


       X1 is entering, while S3 is leaving the tableau.
       New values of X1: 110/1=110, 1, 0, 0, 0, 1, 0

       New values of S1               New values of S2                  New values of S4
       90 – 2/3(110) = 50/3           60 – 1/3(110) = 70/3              110 - 0(110) = 110
       2/3 – 2/3(1) = 0               1/3 - 1/3(1) = 0                    0 – 0(1) = 0
       ½ - 2/3(0) = ½                 ½ - 1/3(0) = ½                      1 – 0(0) = 1
       1 - 2/3(0) = 1                 0 - 1/3(0) = 0                      0 – 0(0) = 0
       0 - 2/3(0) = 0                 1 - 1/3(0) = 1                      0 – 0(0) = 0
       0 - 2/3(1) = - 2/3             0 - 1/3(1) = -1/3                   0 – 0(1) = 0
       0 - 2/3(0) = 0                  0 - 1/3(0) = 0                     1 - 0(0) = 1

Second Simplex Tableau:

          Product    Quantity   1.7MU     1.5MU     0MU        0MU 0MU    0MU
       Cj Mix           bi      X1        X2        S1         S2   S3    S4___bi/aij
       0   S1        50/3       0         ½         1          0  -2/3    0 50/3/1/2= 100/3 Min! Leaving
       0   S2         70/3      0          ½        0          1  -1/3    0 70/3/1/2=140/3
       1.7 X1        110        1         0         0          0    1     0 110/0= -
       0   S4        110        0         1         0          0    0     1 110/1=110
           Zj        187        1.7       0         0          0    1.7   0
           Cj - Zj              0         1.5       0          0  - 1.7   0    ___
                                              Max! Entering

       X2 is entering the solution, while S1 is leaving the solution.
       New X2 values: 50/3/1/2=100/3, 0, 1, 2, 0,-2/3/1/2=-4/3, 0




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        New values of S2               New values of X1              New values of S4
        70/3 – ½(100/3) = 20/3         Since the key number          110 – 1(100/3) = 230/3
           0 - ½ (0)    =0             is zero, the row values         0 – 1(0)     =0
          ½ - ½ (1)     =0             remain same.                    1 – 1(1)     =0
           0 – ½ (2)    = -1                                            0 – 1(2)    = -2
           1 – ½ (0)    =1                                              0 – 1(0)    =0
        -1/3 – ½ (-4/3) = 1/3                                           0 – 1(-4/3) = 4/3
           0 – ½ (0)    =0                                              1 – 1(0)    =1

 Third Simplex Tableau:

           Product    Quantity   1.7MU    1.5MU   0MU        0MU 0MU    0MU
        Cj Mix            bi     X1       X2      S1         S2  S3     S4___bi/aij
        1.5 X2        100/3      0        1        2         0  -4/3    0     -
        0   S2          20/3     0        0       -1         1   1/3    0     20 Min! Leaving
        1.7 X1         110       1        0        0          0  1      0    110
        0   S4         230/3     0        0       -2          0  4/3     1    115/2
            Zj         237       1.7      1.5      3          0 -0.3    0
            Cj - Zj              0        0       -3          0  0.3    0    ___
                                                                  Max! Entering


        S3 is entering the solution, while S2 is leaving.
        New S3 values are: 20/3/1/3= 20, 0, 0, -3, 3, 1, 0

New values of X2                       New values of X1              New values of S4
100/3 – (-4/3)20 = 60                  110 – 1(20) = 90              230 - 4/3(20) = 50
    0 – (-4/3)0 = 0                      1 – 1(0) = 1                  0 – 4/3(0) = 0
    1 – (-4/3)0 = 1                      0 - 1(0) = 0                  0 – 4/3(0) = 0
    2 – (-4/3)(-3) = -2                  0 - 1(-3) = 3                -2 - 4/3(-3) = 2
    0 – (-4/3)3 = 4                       0 - 1(3) = -3                0 - 4/3(3) = -4
 -4/3 – (-4/3)1 = 0                      1 - 1(1) = 0                 4/3 – 4/3(1) = 0
    0 – (-4/3)0 = 0                       0 - 1(0) = 0                  1 - 4/3(0) = 1

 Fourth Simplex Tableau:

           Product Quantity 1.7MU 1.5MU 0MU 0MU                  0MU    0MU
        Cj Mix         bi    X1    X2    S1   S2                 S3     S4___
        1.5 X2      60       0     1     -2    4                 0      0
        0   S3       20      0     0     -3    3                 1      0
        1.7 X1       90      1     0      3   -3                 0      0
        0   S4       50      0     0      2   -4                 0      1
            Zj      243    1.7   1.5   2.1   0.9                 0      0
            Cj - Zj          0     0  -2.1  -0.9                 0      0   ___

        There is no positive value in the row of “Cj – Zj”, therefore we have obtained the
        optimal solution.

b)      We should prepare 90 bags of deluxe, 60 bags of standard. Expected maximum profit
        is 243 MU.


     Prof.Dr.Dr.M.Hulusi DEMIR                                                                  131
                      Introduction to Production / Operations Management


7. a)      An optimal tableau for a maximization problem must contain all zeroes or negative
           values in the “Cj – Zj” row. Therefore, the tableau is optimal.

     b)    We always find zero values in the “Cj – Zj” row beneath the coloumns associated with
           those variables in the product mix. In this case X1, S2 and S3 are in the product mix,
           and the variable coloumns X1, S2 and S3 all contain zeroes in the “Cj – Zj” row.
           However, variable X2 which is not in the product mix also has a zero “Cj – Zj” value.
           This means we can enter variable X2 in another iteration and still not change our
           optimal profit of 32 MU/day. In fact, whenever there exists another optimal solution,
           as in this case, there are an infinite number of optimal solutions. The most X2 we can
           expect is the least-positive quotient of the three:

                   Quantity     X2         Quotient (bi/aij)
                      4        0.75          5 1/3  Min!
                      4        0.05         80
                      ¼        0.175         8
           Therefore, we can introduce any amount of X2 in the continuous range of 0 to 5 1/3
           units per day giving rise to an infinite number of possible solutions.

     c)    The optimum value of Zj is 32MU/day.

9.        Data Summary:
                                 Product A      Product B         Available
                                     X1            X2             Capacity
           Man-hours             5 hours        6 hours           60 hours maximum
           Inspection Time       1 hour         2 hours           16 hours maximum
           Production: A         1                                10 units maximum
           Production: B                        1                  6 units maximum
           Profit Contribution   2MU/unit       3MU/unit

           Formulation of the problem:
              Objective function:
                 Max! Z = 2X1 + 3X2            Max! Z = 2X1 + 3X2 + 0S1 + 0S2 + 0S3 + 0S4
              Subject to:
                 5X1 + 6X2 ≤ 60                5X1 + 6X2 + S1                = 60
                  X1 + 2X2 ≤ 16                 X1 + 2X2      + S2           = 16
                  X1         ≤ 10               X1                 + S3      = 10
                          X2 ≤ 6                      X2                + S4 = 6
                      X1, X2 ≥ 0                                All variables ≥ 0

           Initial Simplex Tableau:

                  Product Quantity       2      3        0        0     0     0
           Cj     Mix       bi           X1     X2       S1       S2    S3    S4    bi/aij
           0      S1       60            5      6        1        0     0     0     60/6=10
           0      S2       16            1      2        0        1     0     0     16/2=8
           0      S3       10            1      0        0        0     1     0      -
           0      S4         6           0      1        0        0     0     1    6/1=6 Min! Leaving
                  Zj         0           0      0        0        0     0     0
                  Cj - Zj                2      3        0        0     0     0
                                                 Max! Entering


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        Variable X2 is entering and variable S4 is leaving the tableau.
        New X2 values are: 6/1=6, 0, 1/1=1, 0, 0, 0, 1
Old S1 Row – Key No.(New X2 Values) = New S1 Values    Old S2 Row – Key No.(New X2 Values) = New S2 Values
        60 – 6(6) = 24                                       16 – 2(6) = 4
         5 – 6(0) = 5                                         1 – 2(0) = 1
         6 – 6(1) = 0                                         2 – 2(1) = 0
         1 – 6(0) = 1                                         0 – 2(0) = 0
         0 – 6(0) = 0                                         1 – 2(0) = 1
         0 – 6(0) = 0                                         0 – 2(0) = 0
         0 – 6(1) = -6                                        0 – 2(1) = -2

Since the key number of S3 row is 0, therefore the values of S3 remain same.

Second Simplex Tableau:

                 Product Quantity          2          3      0       0        0        0
        Cj       Mix       bi              X1         X2     S1      S2       S3       S4    bi/aij
        0        S1       24               5          0      1       0        0      -6     24/5=4.8
        0        S2         4              1          0      0       1        0      -2     4/1=4 Min! Leaving
        0        S3       10               1          0      0       0        1        0     10/1=1
        3        X2         6              0          1      0       0        0        1    6/0=∞ -
                 Zj        18              0          0      0       0        0        0
                 Cj - Zj                   2          0      0       0        0       -3
                                            Max! Entering
        X1 is entering and S2 is leaving.
        New X1 values are as follows: 4,1, 0, 0, 1, 0, -2

Old S1 Row – Key No.(New X1 Values) = New S1 Values    Old S3 Row – Key No.(New X1 Values) = New S3 Values
        24 – 5(4) = 4                                                10 – 1(4) = 6
          5 – 5(1) = 0                                                1 – 1(1) = 0
          0 – 5(0) = 0                                                0 – 1(0) = 0
          1 – 5(0) = 1                                                0 – 1(0) = 0
          0 – 5(1) = -5                                               0 – 1(1) = -1
          0 – 5(0) = 0                                                1 – 1(0) = 1
         -6 – 5(-2) = 4                                               0 – 1(-2) = 2

        Since the key number of X2 is 0, therefore the values of X2 remain same.

Third Simplex Tableau:

                 Product Quantity          2          3      0       0        0        0
        Cj       Mix      bi               X1         X2     S1      S2       S3       S4    bi/aij
        0        S1        4               0          0      1      -5        0        4     4/4=1 Min! Leaving
        2        X1        4               1          0      0       1        0      -2       -
        0        S3        6               0          0      0      -1        1        2     6/2=3
        3        X2        6               0          1      0       0        0        1     6/1=6
                 Zj       26               2          3      0       2        0       -1
                 Cj - Zj                   0          0      0      -2        0        1
                                                                                        Max! Entering


    Prof.Dr.Dr.M.Hulusi DEMIR                                                                             133
                     Introduction to Production / Operations Management



                 S4 is entering and S1 is leaving.
                 New S4 values are: 1, 0, 0, ¼, -5/4, 0, 1

Old X1 Row – Key No.(New S4 Values) = New X1 Values    Old S3 Row – Key No.(New S4 Values) = New S3 Values
                 4 – (-2)1 = 6                                       6 – 2(1) = 4
                 1 – (-2)0 = 1                                       0 – 2(0) = 0
                 0 – (-2)0 = 0                                       0 – 2(0) = 0
                 0 – (-2)¼ = ½                                       0 – 2(1/4) = -½
                 1 – (-2)(-5/4) = -3/2                              -1 -2(-5/4) = 3/2
                 0 – (-2)0 = 0                                       1 – 2(0) = 1
                -2 – (-2)1 = 0                                       2 – 2(1) = 0

Old X2 Row – Key No.(New S4 Values) = New X2 Values
                 6 – 1(1) = 5
                 0 – 1(0) = 0
                 1 – 1(0) = 1
                 0 – 1(1/4) = -¼
                 0 – 1(-5/4) = 5/4
                 0 – 1(0) = 0
                 1 – 1(1) = 0

Fourth Simplex Tableau:

                 Product Quantity            2        3      0       0        0        0
        Cj       Mix      bi                 X1       X2     S1      S2       S3       S4
        0        S4        1                 0        0     ¼      -5/4       0        1
        2        X1        6                 1        0     ½       -3/2      0        0
        0        S3        4                 0        0    -½        3/2      1        0
        3        X2        5                 0        1    -¼        5/4      0        0
                 Zj       27                 2        3      ¼       3/4      0        0
                 Cj - Zj                     0        0     -¼      -3/4      0        0

        Inspection of 4th Tableau reveals that “Cj – Zj” ≤ 0 for all values of “Cj – Zj”, which
        means the optimal solution is attained.

        The solution is           X1 = 6
                                  X2 = 5
                                  S3 = 4
                                  S4 = 1 .

        In order to achieve maximum profit, it is necessary to produce 6 units of Product A, and
        5 units of Product B, This combination will result in a maximum of 27 MU. The S 3
        value indicates that Hurşit will have 4 units of unused capacity for producing Product A
        as given in the original formulation of the problem. The S4 value of 1 indicates that
        there will be 1 unit of unused capacity for producing Product B. Since S1 and S2 do not
        appear in the solution set, they both equal zero. Hence production an d inspection time
        will be totally consumed.




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11. a)
         Data Summary:

                                  Product             Product           Product    Capacity
                                    X1                  X2               X3     (Resource Limits)
         B1 Grade Ore                5                   5               10        1000
         B2 Grade Ore              10                    8                 5       2000
         B3 Grade Ore              10                    5                 -        500
         Profit Contribution      100 MU               200 MU             50 MU

         Formulation of the problem:
               Objective function:    Max! Z = 100X1 + 200X2 + 50X3
               Subject to:            5X1 + 5X2 + 10X3 ≤ 1000
                                     10X1 + 8X2 + 5X3 ≤ 2000
                                     10X1 + 5X2        ≤ 500
                                            X1, X2, X3 ≥ 0

         Changing the model into standard form, we‟ll have;
               Objective function: Max! Z = 100X1 + 200X2 + 50X3 + 0S1 + 0S2 + S3
               Subject to:           5X1 + 5X2 + 10X3 + S1            = 1000
                                    10X1 + 8X2 + 5X3         +S2      = 2000
                                    10X1 + 5X2                   + S3 = 500
                                                        All variables ≥ 0

Initial Simplex Tableau:

                 Product Quantity   100   200                  50       0       0       0
         Cj      Mix       bi       X1    X2                   X3       S1      S2      S3    bi/aij
         0       S1      1000       5     5                    10       1       0       0 1000/5=200
         0       S2      2000      10     8                     5       0       1       0 2000/8=250
         0       S3       500      10     5                     0       0       0       1 500/5=100 Min! Leaving
                 Zj        0        0     0                     0       0       0       0
                 Cj - Zj          100   200                    50       0       0       0_________
                                                       Max! Entering
         S3 is leaving and X2 is replacing.
         New values of X2:      500/5=100, 10/5=2, 5/5=1, 0, 0, 0, 1/5

Old S1 Row – Key No.(New X2 Values) = New S1 Values      Old S2 Row – Key No.(New X2 Values) = New S2 Values
         1000 – 5(100) = 500                                            2000 – 8(100) = 1200
            5 – 5(2) = - 5                                                10 – 8(2) = - 6
            5 – 5(1) = 0                                                   8 – 8(1) = 0
           10 – 5(0) = 10                                                  5 – 8(0) = 5
            1 – 5(0) = 1                                                   0 – 8(0) = 0
            0 – 5(0) = 0                                                   1 – 8(0) = 1
            0 – 5(1/5) = - 1                                               0 – 8(1/5) = - 8/5




    Prof.Dr.Dr.M.Hulusi DEMIR                                                                             135
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Second Simplex Tableau:

                 Product Quantity    100 200               50       0          0       0
        Cj       Mix        bi       X1  X2                X3       S1         S2      S3    bi/aij
        0        S1        500      -5   0                 10       1          0      -1 500/10=50 Min! Leaving
        0        S2       1200      -6   0                  5       0          1    -8/5 1200/5=240
        200      X2        100       2   1                  0       0          0     1/5 100/0= ∞ -
                 Zj      20000     400 200                  0       0          0       40
                 Cj - Zj          -300   0                 50       0          0      -40_________
                                                              Max! Entering


        S1 is leaving, X3 is replacing.
        New X3 values are: 500/10=50, -5/10=-1/2, 0, 1, 1/10, 0, -1/10

Old S2 Row – Key No.(New X3 Values) = New S2 Values   Old X2 Row – Key No.(New X3 Values) = New X2 Values
                 1200 – 5(50) = 950                        Since key number is zero, values of X2
                   - 6 – 5(-1/2) = -7/2                    row remain same.
                     0 – 5(0) = 0
                     5 – 5(1) = 0
                     0 – 5(1/10) = -1/2
                     1 – 5(0) = 1
                 - 8/5 – 5(-1/10) = -11/10

Third Simplex Tableau:

                 Product Quantity         100 200          50       0          0       0
        Cj       Mix        bi            X1  X2           X3       S1         S2      S3
        50       X3         50          -½    0             1     1/10         0    -1/10
        0        S2        950          -7/2  0             0     -½           1    -11/10
        200      X2        100            2   1             0       0          0     1/5
                 Zj      22500          375 200            50       5          0       35
                 Cj - Zj               -275   0             0      -5          0      -35_________

b)      The solution set (Quantity) in the 3rd simplex tableau is optimal. We have 0 and
        negative values for “Cj – Zj” row.
        Deep-Hole Mining should produce 50 units of X3,
                                         100 units of X2
                                          and no units of X1.

c)      The only unused resource is the one associated with the S2 variable. It is unused in
        the sense that it is not totally consumed. Since S2 = 950, this means that Deep-Hole
        Mining will consume all but 950 tons of B2 Grade Ore. All other grades, B1 and B3,
        will be totally consumed.

d)      The optimum profit for Deep-Hole Mining equals 22500 MU.




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12. a)
         Data Summary:

                         Bread Rolls Doughnuts                 Capacity
                           X1   X2     X3                            __
         Centre 1          3    4       2                        60
         Centre 2          2    1       2                        40
         Centre 3          1    3       2                        80___
         Profit/pan       2MU 4MU      3MU                           __

         Formulation of the problem:
           Objective function:
               Max! Z = 2X1 + 4X2 + 3X3                    Z = 2X1 + 4X2 + 3X3 + 0S1 + 0S2 + 0S3
           Subject to:
                       3X1 + 4X2 + 2X3 ≤ 60            3X1 + 4X2 + 2X3 + S1           = 60
                       2X1 + X2 + 2X3 ≤ 40             2X1 + X2 + 2X3        + S2     = 40
                        X1 + 3X2 + 2X3 ≤ 80             X1 + 3X2 + 2X3            +S3 = 80
                             X1, X2, X3 ≥ 0                              all variables ≥ 0
b)       Initial Simplex Tableau:

                 Product Quantity          2          4        3        0         0     0_
         Cj      Mix       bi              X1         X2       X3       S1        S2    S3       bi/aij
         0       S1       60               3          4        2        1         0     0    60/4=15 Min! Leaving
         0       S2       40               2          1        2        0         1     0    40/1=40
         0       S3       80               1          3        2        0         0     1    80/3
                 Zj         0              0          0        0        0         0     0
                 Cj – Zj                   2          4        3        0         0     0        ____
                                                       Max! Entering

         S1 is leaving, X2 is entering the solution.
         New X2 values are computed as follows: 15, ¾, 1, ½ , ¼, 0, 0

Old S2 Row – Key No.(New X2 Values) = New S2 Values      Old S3 Row – Key No.(New X2 Values) = New S3 Values
                 40 – 1(15) = 25                                        80 – 3(15) = 35
                  2 – 1(3/4) = 5/4                                       1 – 3(3/4) = -5/4
                  1 – 1(1) = 0                                           3 – 3(1) = 0
                  2 – 1(1/2) = 3/2                                       2 – 3(1/2) = ½
                  0 – 1(1/4) = -1/4                                      0 – 3(1/4) =-3/4
                  1 - 1(0) = 1                                           0 – (0) = 0
                  0 – 1(0) = 0                                           1 – 3(0) = 1

Second Simplex Tableau:

                 Product Quantity    2                4       3          0        0     0_
         Cj      Mix       bi        X1               X2      X3         S1       S2    S3      bi/aij
         4       X2       15       3/4                1       ½          ¼         0    0 15/1/2=30
         0       S2       25      5/4                 0      3/2        -¼        1     0 25/3/2=50/3 Min! Leaving
         0       S3       35      -5/4                0       ½         -¾        0     1 35/1/2 =70
                 Zj        60        3                4       2          1        0     0
                 Cj – Zj            -1                0       1         -1        0     0       ____
                                                                 Max! Entering

     Prof.Dr.Dr.M.Hulusi DEMIR                                                                            137
                      Introduction to Production / Operations Management


                 S2 is leaving, X3 is replacing.
                 New X3 values are: 50/3, 5/4/3/2=5/6, 0, 1, -1/6, 2/3, 0


Old S2 Row – Key No.(New X2 Values) = New S2 Values    Old S3 Row – Key No.(New X2 Values) = New S3 Values
                 15 – ½ (50/3) = 20/3                                35 – 1/2(50/3) = 80/3
                  ¾ - ½(5/6) = 1/3                                  -5/4 – ½(5/6) = -5/3
                  1 – ½(0) = 1                                         0 – ½(0) = 0
                  ½ - ½(1) = 0                                         ½ - ½(1) = 0
                  ¼ - ½(-1/6) = 1/3                                 -3/4 – ½(-1/6) = -2/3
                   0 – ½(2/3) = -1/3                                    0 – ½(2/3) = -1/3
                   0 – ½(0) = 0                                         1 – ½(0) = 1

Third Simplex Tableau:

                 Product Quantity    2                4     3     0           0       0_
        Cj       Mix       bi        X1               X2    X3    S1          S2      Sj
        4        X2       20/3      1/3               1     0    1/3         -1/3     0
        3        X3       50/3      5/6               0     1  -1/6          -2/3     0
        0        S3       80/3     -5/3               0     0  -2/3          -1/3     1
                 Zj       230/3    23/6               4     3    5/6           2/3    0
                 Cj – Zj          -11/6               0     0   -5/6          -2/3    0        ____
        The solution is optimal.
               X2 = 20/3 units
               X3 = 50/3 units.
               There will be 26 2/3 man-hours of unused capacity at Centre 3.

c)      The maximum daily profit for Zingo Bakery is 76.67MU.

14.     Data Summary:
                             Cigar Boxes Cigarette Boxes                      Available Capacity
                                 X1           X2                                            ______
                 Machine Time    30 min.      25 min.                         20 hours (=1200 min.)
                 Order           1                                            25 boxes minimum___
                 Profit/box     9MU/box      8MU/box                                                _

        Objective function:
               Maximize weekend contribution!    Z = 9X1 + 8X2
        Subject to:
               Available time 30X1 + 25X2 ≤ 1200
               Commitment         X1        ≥ 25
               Non-negativity        X1, X2 ≥ 0
        After augmenting, we have:
               Objective function      Max!     Z = 9X1 + 8X2 + 0S1 + 0S2 – MA2
               Subject to                       30X1 + 25X2 + S1            = 1200
                                                   X1             - S2 + A2 = 25
                                                          X1, X2, S1, S2, A2 ≥ 0




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Initial Simplex Tableau:

   Product        Quantity         9       8          0       0         -M
Cj Mix               bi            X1      X2         S1      S2        A2__bi/aij)___
0   S1             1200            30      25         1       0         0 1200/30 =40
-M  A2                25             1       0        0      -1         1   25/1=25 Min! Leaving variable
    Zj            -25M            -M         0        0       M        -M
    Cj - Zj                      9+M         8        0      -M         M__________
                                   Max! Entering variable

        X1 enters the solution and A2 leaves the solution. (We eliminate A2 from the 2nd
        Simplex Tableau.)
        New X1 values are as follows: 25,1, 0, 0, -1

Old S1 Row – Key No.(New X1 Values) = New S1 Values
                 1200 – 30(25) = 450
                   30 – 30(1) = 0
                   25 – 30(0) = 25
                    1 - 30(0) = 1
                    0 – 30(-1) = 30

Second Simplex Tableau:

      Product     Quantity        9        8          0       0_
Cj    Mix           bi            X1       X2         S1      S2__bi/aij)___
0      S1          450              0      25         1      30 450/15 =15 Min! Leaving variable
9      X1            25             1        0        0      -1 25/-1= -
       Zj          225              9        0        0      -9
       Cj - Zj                      0        8        0       9      ______
                                                                Max! Entering variable
        S2 enters the solution and S1 leaves the solution.
        After necessary calculations we obtain:

Third Simplex Tableau:

      Product     Quantity        9       8        0         0_
Cj    Mix           bi            X1      X2       S1        S2__bi/aij)___
0      S2          15               0    25/30    1/30       1 15/25/30 =18 Min! Leaving variable
9      X1          40               1    25/30    1/30       0 40/25/30=48
       Zj          225              9    75/10    3/10       0
       Cj - Zj                      0      ½     -3/10       0      ______
                                             Max! Entering variable
        S2 leaves, X2 enters the solution.
        After necessary calculations we obtain the fourth simplex tableau.




     Prof.Dr.Dr.M.Hulusi DEMIR                                                                              139
                     Introduction to Production / Operations Management


Fourth Simplex Tableau:

      Product     Quantity       9        8        0       0_
Cj    Mix           bi           X1       X2       S1      S2
8      X2          18              0      1       1/25    6/5
9      X1          25              1      0        0      -1
       Zj          369             9      8       8/25    3/5
       Cj - Zj                     0      0      -8/25   -3/5

        Since all the variables in the “Cj - Zj” row are either o or negative, we obtained the
        optimum solution.
        Therefore; Cigar Boxes = 25 Boxes
                    Cigarette Boxes = 18 Boxes
        must be produced. Total profit will be 369 MU.

15.     Let Container A = X1
            Container B = X2
            Container C = X3
        Objective Function: Max! Z = 8X1 + 6X2 + 14X3
        Subject to           : 2X1 + X2 + 3X3 ≤ 120
                               2X1 + 6X2 + 4X3 ≤ 240
                                        X1, X2 ≥ 0
        After augmenting, we have:
                Objective Function : Max!      Z = 8X1 + 6X2 + 14X3 + 0S1 + 0S2
                Subject to :                       2X1 + X2 + 3X3 + S1          ≥0
                                                   2X1 + 6X2 + 4X3        + S2 ≥ 0
                                                                  All variables ≥ 0
        Initial Simplex Tableau:

        Product          Quantity         8        6          14   0    0__
Cj      Mix                bi             X1       X2         X3   S1   S2_   bi/aij
0       S1                120             2        1          3    1    0 120/3=40 Min! Leaving
0       S2                240             2        6          4    0    1 240/4=60
        Zj                  0             0        0          0    0    0
        Cj - Zj                           8        6          14   0    0_________
                                            Max! Entering
        X3 is entering the solution, while S1 is leaving.
        New values of X3 are as follows: 40, 2/3, 1/3, 1, 1/3, 0

        Old S2 Row – Key No.(New X1 Values) = New S2 Values
                 240 – 4 (40) = 80
                   2 – 4 (2/3) = -2/3
                   6 – 4 (1/3) = 14/3
                   4 – 4 (1) = 0
                   0 – 4 (1/3) = -4/3
                   1 – 4 (0) = 1




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Second Simplex Tableau:

         Product          Quantity         8       6       14      0      0__
Cj       Mix                bi             X1      X2      X3      S1     S2_   bi/aij
14       X3                 40             2/3     1/3      1      1/3    0 40/1/3=120
0        S2                 80           -2/3     14/3      0     -4/3    1 80/14/3=17.1   Min! Leaving
         Zj                560            28/3    14/3     14     14/3    0
         Cj - Zj                          -4/3     4/3      0    -14/3    0_________
                                                      Max! Entering

         X2 is entering the solution, while S2 is leaving the solution.
         New values of X2 : 120/7, -1/7, 1, 0, -2/7, 3/14

         Old X3 Row – Key No.(New X2 Values) = New X3 Values
                            40 – 1/3 (120/7) = 240/7
                          12/3 – 1/3 (-1/7) = 5/7
                           1/3 – 1/3 ( 1 )   =0
                             1 – 1/3 ( 0 ) = 1
                           1/3 – 1/3 (-2/7) = 3/7
                             0 – 1/3 (3/14) = -1/14

Third Simplex Tableau:

         Product          Quantity         8      6        14       0      0__
Cj       Mix                 bi            X1     X2       X3       S1     S2_
14       X3                 240/7          5/7     0        1       3/7    0
 6       X2                120/7         -1/7     1         0      -2/7 3/14
         Zj                 582.9        64/7      6       14      30/7 4/14
         Cj - Zj                          -8/7__ _0__       0     -30/7_ -4/14

         Since there is no positive value in the row of “Cj –Zj”, we have obtained yhe optimal
         solution.
                 Container A : No
                 Container K : 120/7 units = 17.14 units
                 Container T : 240/7 units = 34.29 units
                 Maximum profit : 582/9 MU

16.
         Let
                 X1 : Number of bed mattresses
                 X2 : Number of box springs
         Objective function:      Minimize!      Cost (Zj) = 20X1 + 24X2
         Subject to :                              X1 + X2 ≥ 30
                                                   X2 + 2X2 ≥ 40
                                                      X1, X2 ≥ 0
         After augmenting we have:
                Objective function:        Min! Zj = 20X1 + 24X2 + 0S1 + MA1 + 0S2 + MA2
                Subject to:                X1 + X2 – S1 + A1           = 30
                                           X1 + 2X2          - S2 + A2 = 40
                                                         All variables ≥ 0
      Prof.Dr.Dr.M.Hulusi DEMIR                                                                    141
                   Introduction to Production / Operations Management


Initial Simplex Tableau:

       Product Quantity 20     24                0          M        0     M
Cj     Mix        bi     X1    X2                S1         A1       S2    A2    bi/aij
M      A1         30     1     1                 -1         1        0     0     30
M      A2         40     1     2                  0         0        -1    1     20 Min! Leaving
       Zj        70M     2M    3M                -M         M        -M    M
       Cj - Zj         20-2M 24-3M                M         0        M     0_________
                                         Max (absolute)! Entering

       X2 is entering and A2 is leaving. Since A2 is leaving the solution, it must not appear in
       the tableau.
       New X2 values are : 20,1/2, 1, 0, 0, -1/2

       Old A1 Row – Key No.(New X2 Values) = New A1 Values
                       30 – 1 (20) = 10
                         1 – 1 (1/2) = ½
                         1 – 1 (1) = 0
                        -1 – 1 (0) = -1
                         0 – 1 (0) = 0
                         0 – 1 (-1/2) = 1/2

Second Simplex Tableau:

       Product Quantity 20    24                 0          M      0_
Cj     Mix         bi    X1   X2                 S1         A1     S2    bi/aij
M      A1          10    ½    0                  -1         1       ½    10/1/2 Min! Leaving
24     X2          20    ½    1                   0         0      -½    20/1/2= 40
       Zj      480+10M 12+M/2 24                 -M         M     M/2-12
       Cj - Zj          8-M/2 0                   M         0     12-M/2_____
                                Max (absolute)! Entering

       X1 is entering, while A1 is leaving. A1 must not appear in the next tableau.
       New X1 values are: 20,1, 0, -2, 1

              Old X2 Row – Key No.(New X1 Values) = New X2 Values
                       20 – ½ (20) = 10
                        ½ - ½ (1) = 0
                        1 – ½ (0) = 1
                        0 – ½ (-2) = 1
                       -½ - ½ (1) = - 1

Third Simplex Tableau:

       Product Quantity 20              24       0           0_
Cj     Mix        bi     X1             X2       S1          S2
20     X1         20     1              0        -2          1
24     X2         10     0              1         1         -1
       Zj        640    20             24       -16         -4
       Cj - Zj           0              0        16          4



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         Since there is no negative value in the row of “Cj – Zj”, we have obtained the optimal
         solution.
         We should produce 20 units of bed mattresses,
                             10 units of box springs.
         The minimum cost is 640 MU.

20.      a)     See the table below.
         b)     14X1 + 4X2 ≤ 3 360
                10X1 + 12X2 ≤ 9 600
                        X1, X2 ≥ 0
         c)     Maximisation profit = 900X1 + 1 500X2
         d)     Basis is S1 = 3 360
                         S2 = 9 600
         e)     X2 should enter basis next.
         f)     S2 will leave next.
         g)     800 units of X2 will be in the solution at the second tableau.
         h)     Profit will increase by “Cj – Zj” (units of variable entering the solution)
                                = (1500)(800) = 1 200 000 MU

         Tableau for the problem:

                Product Quantity       900  1500       0         0_
         Cj     Mix                    X1   X2         S1        S2     bi/aij
         0      S1       3 360         14   4          1         0    3360/4=840
         0      S2       9 600         10  12          0         1    9600/12=800 Min! Leaving
                Zj           0         0    0          0         0
                Cj - Zj               900 1500         0         0
                                                Max! Entering

21.
         a)     Objective function:
                       Max! Z = 0.8X1 + 0.4X2 +1.2X3 – 0.1X4 + 0S1 – MA2 + 0S3 – MA3
                       Subject to:   X1 + 2X2 + X3 + 5X4 + S1               = 150
                                    X2 – 4X3 + 8X4     + A2           = 70
                              6X1 + 7X2 + 2X3 – X4          - S3 + A3 = 120
                                                         All variables ≥ 0

         b)
                Product Quantity     0.8    0.4  1.2     -0.1           0          -M    0       -M
         Cj     Mix        bi        X1     X2   X3      X4             S1         A2    S3      A3
         0      S1        150        1      2    1       5              1          0     0       0
         -M     A2         70        0      1    -4      8              0          1     0       0
         -M     A3        120        6      7    2       -1             0          0     -1      1
                Zj       -190M -6M -8M           2M      -7M            0          -M    M       -M
                Cj - Zj          0.8+6M 0.4+8M 1.2-2M -0.1+7M           0          0     -M      0_

         c)     S1 = 150
                A2 = 70
                A3 = 120
                All other variables = 0

      Prof.Dr.Dr.M.Hulusi DEMIR                                                                  143
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22.
       Data Summary:

                                       A (X1)         B (X2)            Resource Limits
              Man-hours                1              2                 400 max.
              Eating Area              3              10                1500 max.
              Availability for Group A 1              -                 300 max.______
              Profit/visitor           2 MU           1.5 MU                         __

       Formulation of the problem:
         Objective function:
             Max! Z = 2X1 + 1.5X2                  Max! Z = 2X1 + 1.5X2 +0S1 + 0S2 + 0S3
         Subject to:
                      X1 + 2X2 ≤ 400                           X1 + 2X2 + S1           =0
                     3X1 + 10X2 ≤ 1500                        3X1 + 10X2     + S2      =0
                      X1        ≤ 300                          X1                 + S3 = 0
                         X1, X2 ≥ 0                                       All variables ≥ 0

Initial Simplex Tableau:

       Product Quantity       2      1.5       0      0         0
Cj     Mix         bi         X1     X2        S1     S2        S3    bi/aij
0      S1        400          1      2         1      0         0     400
0      S2       1500          3      10        0      1         0     500
0      S3         300         1      0         0      0         1     300 Min! Leaving
       Zj             0       0      0         0      0         0
       Cj - Zj                2.5    1.5       0      0         0_________
                                Max! Entering


       X1 enters the solution, while S3 leaves the solution.

Second Simplex Tableau:

       Product Quantity       2      1.5    0         0          0
Cj     Mix        bi          X1     X2     S1        S2         S3    bi/aij
0      S1       100           0      2      1         0        -1       50 Min! Leaving
0      S2       600           0     10      0         1        -3       60
20     X1        300          1      0      0         0          1       ∞
       Zj        600          2      0      0         0          2
       Cj - Zj                0      1.5    0         0         -2_________
                                       Max! Entering

       X2 enters the solution, while S1 leaves the solution.




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Third Simplex Tableau:

         Product Quantity         2        1.5    0          0     0
Cj       Mix       bi             X1       X2     S1         S2    S3
1.5      X2         50            0        1      ½          0   -½
0        S2       100             0        0     -5          1   -2
20       X1       300             1        0      0          0     1
         Zj       675             2       1.5 0.75           0  1.25
         Cj - Zj                  0        0   -0.75         0 -1.25_

From the third simplex tableau,    X2 = 50 visitors
                                   X1 = 300 visitors
                                   S2 = 100 spaces in the eating area.
                 Maximum weekly profit = 675 MU

23.
   a) ABC Model = X1                       Profit for X1 = 400 MU – 250 MU = 150 MU
      XYZ Model = X2                       Profit for X2 = 575 MU – 375 MU = 200 MU

         Objective function:
             Max! Z = 150X1 + 200X2                  Max! 150X1 + 200X2 + 0S1 + 0S2 + 0S3
         Subject to:
                  4X1 + 2X2 ≤ 1600                     4X1 + 2X2 + S1            = 1600
                2.5X1 + X2 ≤ 1200                    2.5X1 + X2       + S2       = 1200
             4.5X1 + 1.5 X2 ≤ 1600                   4.5X1 + 1.5X2          + S3 = 1600
                     X1, X2 ≥ 0                                    All variables ≥ 0

  b)     The basic initial simplex tableau is as follows:

                 Product    Quantity     150    200    0       0         0_
         Cj      Mix          bi           X1   X2     S1      S2        S3 bi/aij
         0       S1         1600           4.0  2.0    1       0         0 800 Min! Leaving variable
         0       S2         1200           2.5  1.0    0       1         1 1200
         0       S3         1600           4.5  1.5    0       0         1 3200/3
                 Zj              0         0    0      0       0         0
                 Cj - Zj                  150  200     0       0         0__
                                                 Max! Entering variable
         X2 will enter and S1 will leave the solution.
         New X2 values are: 800, 2, 1,1/2, 0, 0

Old S2 Row – Key No.(New X2 Values) = New S2 Values     Old S3 Row – Key No.(New X2 Values) = New S3 Values
                 1200 – 1 (800) = 400                                 1600 – 1.5(800) = 400
                   2.5 – 1 (2) = ½                                      4.5 – 1.5(2) = 3/2
                   1.0 – 1 (1) = 0                                      1.5 – 1.5(1) = 0
                     0 – 1 (1/2) = -1/2                                   0 – 1.5(1/2) = -3/4
                     1 – 1 (0) = 1                                        0 – 1.5(0) = 0
                     0 – 1 (0) = 0                                        1 – 1.5(0) = 1




      Prof.Dr.Dr.M.Hulusi DEMIR                                                                          145
                     Introduction to Production / Operations Management


Second Simplex Tableau:

        Product Quantity          150     200    0           0       0_
Cj      Mix         bi            X1      X2     S1          S2      S3
200     X2         800            2       1      ½           0       0
0       S2         400            ½       0     -½           1       0
0       S3         400            3/2     0   -3/4           0       1
        Zj      160000            400     200 100            0       0
        Cj - Zj                  -250     0    -100          0       0__

        Since every entry in the “Cj – Zj” row is less than or equal to zero, the solution set is
        optimal. Azim Co. should market only 800 units of ABC Models, and none of the
        XYZ Models. This will result in an optimal profit of 160 000 MU and the following
        surplus resources:
               S2 = 400, which means that fitting and assembly will have 400 unused hours.
               S3 = 400, which means that there will be 400 unused hours in testing.
        We know that X1 equals zero, because it is not present in the optimal solution set.

24.
        Objective Function:
           Min! Z = 10X1 + 12X2                      Min! Z = 10X1 + 12X2 + A1 + S2 – S3 + A3
        Subject to:
               X1 + X2 = 2000                        X1 + X2 + A1                = 2000
               X1       ≤ 600                        X1           + S2           = 600
                     X2 ≥ 300                             X2           - S3 + A3 = 300
                 X1, X2 ≥ 0                                        All variables ≥ 0
Initial Simplex Tableau:

        Product Quantity 10               12          M      0       0        M
Cj      Mix       bi      X1              X2          A1     S2      S3       A3 bi/aij
M       A1       2000     1               1           1      0       0        0 2000
0       S2        600     1               0           0      1       0        0  -
M       A3        300     0               1           0      0      -1        1 300 Min! Leaving
        Zj      2300M     M               2M          M      0     -M         M
        Cj - Zj          10-M            12-2M        0      0       M        0
                                           The MOST negative value. Entering value.

        X2 is entering, while A3 is leaving. A3 will not appear in the 2nd simplex tableau.
        New X2 values are: 300, 0, 1, 0, 0, -1, 1

Old A1 Row – Key No.(New X2 Values) = New A1 Values
                 2000 – 1(300) = 1700                        S2 remains same, since pivot (key)
                     1 – 1(0) = 1                            number is 0.
                     1 – 1(1) = 0
                     1 – 1(0) = 1
                     0 – 1(0) = 0
                     0 – 1(-1) = 1




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Second Simplex Tableau:

        Product Quantity 10                12         M      0        0
Cj      Mix        bi     X1               X2         A1     S2       S3 bi/aij
M       A1        1700    1                0          1      0        1  1700
0       S2         600    1                0          0      1        0   600 Min! Leaving
12      X2         300    0                1          0      0       -1    -
        Zj    1700M+3600 M                 12         M      0      M-12
        Cj - Zj          10-M              0          0      0      12-M
                                   The MOST negative value. Entering value.

        X1 is entering, while S1 is leaving.
        New X1 values are: 600, 1, 0, 0, 1, 0
Old A1 Row – Key No.(New X1 Values) = New A1 Values
                 1700 – 1(600) = 1100                        X2 remains same, since pivot (key)
                    1 – 1(1) = 0                             number is 0.
                    0 – 1(0) = 0
                    1 – 1(0) = 1
                    0 – 1(1) = -1
                    1 – 1(0) = 1

Third Simplex Tableau:

        Product Quantity          10       12         M    0    0
Cj      Mix        bi             X1       X2         A1   S2   S3 bi/aij
M       A1        1100            0        0          1   -1    1  1100            Min! Leaving
10      X1         600            1        0          0    1    0   -
12      X2         300            0        1          0    0   -1   -
        Zj    1100M+9600         10        12         M -M+10 M-12
        Cj - Zj                    0       0          0  M-10 12-M
                                                                      The MOST negative value. Entering variable

        S3 is entering and A1 is leaving. A1 will not appear in the 4th simplex tableau.
        New S3 values are: 1100, 0, 0, -1, 1
Old X2 Row – Key No.(New S3 Values) = New X2 Values
                 300 – (-1)(1100) = 1400                 Old X1 row will remain same, since 0 is
                    0 – (-1)(0) = 0                      the corresponding key number.
                    1 – (-1)(0) = 1
                    0 – (-1)(-1) = -1
                   -1 – (-1)(1) = 0
Fourth Simplex Tableau:

        Product Quantity         10       12        0        0
Cj      Mix       bi             X1       X2        S2       S3
0       S3       1100            0        0        -1        1
10      X1        600            1        0         0        1
12      X2       1400            0        1        -1        0
        Zj      22800            10       12       -2         0
        Cj - Zj                   0       0         2         0

     Prof.Dr.Dr.M.Hulusi DEMIR                                                                             147
                   Introduction to Production / Operations Management


       The optimal solution has been reached, because only positive and zero values appear in
       the “Cj-Zj” row.
       The Chemical Company‟s decision should be to blend 600 kgs. of phosphate (X1) with
       1400 kgs. of potassium (X2). This provides a surplus of (S3) of 1100 kgs. of potassium
       more than required by the constraint X2 ≥ 300 kgs. The cost of this solution is
       22800MU.

25.
       Data Summary:
                              Bookcases      Tables          Available
                                 X1            X2                   ___
              Cutting           4 hrs         3 hrs          40 hours
              Finishing         4 hrs         5 hrs          30 hours__
              Profit           6 MU          5 MU                   ___

       Formulation of the problem:
             Objective function:
                     Max! Z = 6X1 + 5X2              Z = 6X1 + 5X2 + 0S1 + 0S2
             Subject to:
                     4X1 + 3X2 ≤ 40                  4X1 + 3X2 + S1       = 40
                     4X1 + 5X2 ≤ 30                  4X1 + 5X2      + S2 = 30
                          X1 , X2 ≥ 0                       All variables ≥ 0

Initial Simplex Tableau:

             Product Quantity         6    5       0         0
       Cj    Mix       bi             X1   X2      S1        S2    bi/aij
       0     S1       40              4    3       1         0     10
       0     S2       30              4    5       0         1     15/2 Min! Leaving
             Zj         0             0    0       0         0
       _____Cj-Zj                     0    0       0         0__________
                                       Max! Entering

Second Simplex Tableau:

             Product Quantity         6       5       0      0
       Cj    Mix       bi             X1      X2      S1     S2
       0     S1       10              0      -2       1     -1
       6     X1       15/2            1      5/4      0      ¼
             Zj         4/5           6     15/2      0      3/2
       _____Cj-Zj                     0      -5/2     0      -3/2

       The solution is optimal. No positive values in the “Cj - Zj! row.
       Thus
              Bookcases (X1) = 15/2 units
              Available hours in cutting = 10 hours
              Total profit = 45 MU




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28. a)          Ingredient        Bag No. 1       Bag No. 2            Available
                                     X1              X2
                Vitamin A           2 kgs.          4 kgs.             exactly 400 kgs.
                Vitamin B           6 kgs.          1 kg.              at least 240 kgs.
                Vitamin C           4 kgs.          3 kgs.             at least 640 kgs.
                Cost/kg.            5 MU            3 MU                               _
         Objective function:
           Min! Z = 5X1 + 3X2        Min! Z = 5X1 + 3X2 + MA1 + 0S2 + MA2 + 0S3 + MA3
         Subject to:
           2X1 + 4X2 = 400          2X1 + 4X2 + A1                     = 400
           6X1 + X2 ≥ 240           6X1 + X2       - S2 + A2           = 240
           4X1 + 3X2 ≥ 640          4X1 + 3X2                - S3 + A3 = 640
              X1, X2 ≥ 0                                 All variables ≥ 0

 b)      Initial simplex tableau would be as shown below:

              Variables in Quantity 5     3                 M          0            M     0    M
         Cj   Solution         bi    X1   X2                A1         S2           A2    S3   A3 bi/aij
         M       A1          400     2    4                 1           0           0     0    0 400/2=200
         M       A2          240     6    1                 0          -1           1     0    0 240/6=40 Min! 
         M       A3          640     4    3                 0            0          0    -1    1 640/4+160
                 Zj        1280M 12M      8M                M         -M            M    -M    M
                 Cj - Zj           5-12M 3-8M               0           M           0     M    0
                                         Max! Entering (The most negative value)
         X1 enters the solution, while A2 leaves the solution.
         New X1 values are: 40,1, 1/6, 0, -1/6, 0, 0

         Notes for the student:
             Note that the “ =” constraint (Vitamin A requirements) requires one artificial
                variable (A1) to ensure its equality.
             The two “ ≥” constraints each require a slack variable and an artificial variable.
             The slack variables in “ ≥” constraints represent amounts that must be subtracted
                from the constraint values; hence they must have a negative sign.
             All artificial variables are assigned an extremely large cost “M” to ensure that they
                are driven out of solution by the simplex iterative procedure.
             The solution procedure is the same as in maximisation problems except that the
                variable with the most negative value in the bottom “Cj-Zj” row is always the one
                introduced.
             Problems such as this, or others that involve more than two or three variables or
                constraints, are most easily solved on a computer.

31.      Data Summary:
                               WS BE              SB        Available
                               X1  X2             X3        Capacity_
         Lumber                 4   2              3         600
         Saw                   30  15             15        1920 (=32x60)
         Finishing             30  60             90        19200 (=80x60x4)
         Commitment             1                              10 ≥ ___
         Contribution        12MU 7MU               8MU


      Prof.Dr.Dr.M.Hulusi DEMIR                                                                         149
                      Introduction to Production / Operations Management



        Formulation of the given problem as a mathematical model is as follows:
              Objective function:
                      Max! z = 12X1 + 7X2 + 8X3
              Subject to:
                      4X1 + 2X2 + 3X3 ≤ 600
                     30X1 + 15X2 + 15X3 ≤ 1920
                     30X1 + 60X2 + 90X3 ≤ 19200
                        X1                ≥    10
                                   X1, X2 ≥ 0

        After augmenting we have;
               Objective function:
                      Max! Z = 12X1 + 7X2 +8X3 + 0S1 + 0S2 + 0S3 + 0S4 – MA4
               Subject to:
                      4X1 + 2X2 + 3X3 + S1                      = 600
                    30X1 + 15X2 + 15X3     + S2                 = 1920
                    30X2 + 60X2 + 90X3          + S3            = 19200
                        X1                            - S4 + A4 = 10
                                                  All variables ≥ 0

Initial Simplex Tableau:

        Product Quantity   12              7          8     0        0        0        0       -M
Cj      Mix        bi      X1              X2         X3    S1       S2       S3       S4      A4 bi/aij
0       S1         600     4               2          3     1        0        0        0       0  150
0       S2       1920      30              15         15    0        1        0        0       0   64
0       S3      19200      30              60         90    0        0        1        0       0  640
-M      A4          10      1              0          0     0        0        0        -1      1  10 Min! Leaving
        Zj        -10M     -M              0          0     0        0        0        M       -M
        Cj - Cj          12+M              7          8     0        0        0       -M        0    __
                                   Max! Entering

        X1 enters in the place of A4. A4 will not appear in the following tableau.
        New X1 values are: 10, 1, 0, 0, 0, 0, 0, -1

Old S1 Row – Key No.(New X1 Values) = New S1 Values    Old S2 Row – Key No.(New X1 Values) = New S2 Values
                 600 – 4(10) = 560                                   1920 – 30(10) = 1620
                   4 - 4(1) = 0                                        30 – 30(1) = 0
                   2 – 4(0) = 2                                        15 – 30(0) = 15
                   3 – 4(0) = 3                                        15 – 30(0) = 15
                   1 – 4(0) = 1                                         0 – 30(0) = 0
                   0 – 4(0) = 0                                         1 – 30(0) = 1
                   0 – 4(0) = 0                                         0 – 30(0) = 0
                   0 – 4(-1) = 4                                        0 – 30(-1) = 30




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Old S3 Row – Key No.(New X1 Values) = New S3 Values
                 19200 - 30(10) = 1620
                    30 – 30(1) = 0
                    60 – 30(0) = 60
                    90 – 30(0) = 90
                     0 - 30(0) = 0
                     0 - 30(0) = 0
                     1 - 30(0) = 1
                     0 - 30(-1) = 30

Second Simplex Tableau:

        Product Quantity          12      7        8         0          0    0      0
Cj      Mix       bi              X1      X2       X3        S1         S2   S3     S4   bi/aij
0       S1        560              0      2        3         1          0    0      4    140
0       S2       1620              0     15       15         0          1    0     30      54 Min! Leaving
0       S3      18900              0     60       90         0          0    1     30      630
12      X1         10              1      0        0         0          0    0      -1    -
        Zj        120             12      0        0         0          0    0     -12
        Cj - Cj                     0     7        8         0          0    0      12        __
                                                                                     Max! Entering
        S4 enters in the place of S2.
        New S4 values are: 54, 0, ½, ½, 0, 1/3, 0, 1

Third Simplex Tableau:

        Product Quantity          12       7       8         0        0      0     0
Cj      Mix       bi              X1       X2      X3        S1       S2     S3    S4    bi/aij
0       S1        344              0       0       1         1     -2/15     0     0     344
0       S4         54              0      ½       ½          0       1/30    0     1     108 Min! Leaving
0       S3      17280              0     4/5      75         0        0      1     0     2304
12      X1         64              1       ½       ½         0       1/30    0     0      128
        Zj        768             12       6       6         0       2/15    0     0
        Cj - Cj                     0      1       2         0      -2/15    0     0          __
                                                       Max! Entering
        X3 enters in the place of S4.

Fourth Simplex Tableau:

        Product Quantity          12        7         8      0       0       0      0
Cj      Mix       bi              X1        X2        X3     S1      S2      S3     S4
0       S1        236              0      -1          0      1      1/15     0     -2
8       X3        108              0       1          1      0      1/15     0      2
0       S3       9180              0     -30          0      0       5       0    150
12      X1         10              1        0         0      0       0       0     -1
        Zj        984             12        8         8      0      8/15     0      4
        Cj - Cj                     0      -1         0      0     -8/15     0     -4_

        There is no positive value in the “Cj - Zj” row, thus optimal solution is obtained.



     Prof.Dr.Dr.M.Hulusi DEMIR                                                                        151
                     Introduction to Production / Operations Management


               Azim should produce
                              WS (X1) = 10 units
                              SB (X3) = 108 units
                       Maximum Profit = 984 MU
        There will be 236 m2 of oak boards (S1) and 9180 minutes free in the finishing department.

32.
        Data summary:

                                  Car Loads Scrap Purchased from
                                    Izmir, X1   Istanbul, X2     Available (tons)
                 Copper                  1            1             2½ ≥
                 Lead                    1            2             4 ≥        _
                 Cost/ton           10 000 MU 15 000 MU                        __

        Objective function:
               Min! Z = 10 000X1 + 15 000X2
        Subject to:
               X1 + X2 ≥ 2 ½
               X1 + 2X2 ≥ 4
                  X1, X2 ≥ 0
        After augmenting the model becomes;
               Objective function:
                       Min! Z = 10 000X1 + 15 000X2 + 0S1 + MA1 + 0S2 + MA2
               Subject to:
                       X1 + X2 - S1 + A1             =2½
                       X1 + 2X2            - S2 + A2 = 4
                                      All variables ≥ 0
        where X1 = carloads of scrap purchased from Izmir/day
                X2 = carloads of scrap purchased from Istanbul/day

Initial Simplex Tableau:

        Product Quantity 10000 15000 0                       M        0        M
Cj      Mix       bi        X1    X2     S1                  A1       S2       A2    bi/aij
M       A1       2½         1     1      -1                  1        0        0     2½
M       A2       4          1     2       0                  0        -1       1     2 --> Min! Leaving
        Zj       6½M        2M    3M     -M                  M        -M       M         (smallest + ve number)
        Cj - Zj        10 000-2M 15000-3M M                  0        M        0_________
                                            Max! Entering ( largest number among – ve signed figures)

        X2 enters the solution, while A2 leaves the solution. A2 will not appear in the next tableau.
        New X2 values are: 2, ½, 1, 0, 0, -1/2

Old A1 Row – Key No.(New X2 Values) = New A1 Values
                 2 ½ - 1 (2) = ½
                   1 – 1 (½) = ½
                   1 – 1 (0) = 0
                  -1 – 1 (0) = -1
                   1 – 1(0) = 1
                   0 – 1(-1/2) = ½

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Second Simplex Tableau:

      Product Quantity 10000 15000                  0          M        0
Cj    Mix          bi       X1     X2               S1         A1       S2       bi/aij
M     A1          ½         ½      0                -1         1        ½        1 Min! Leaving (smallest + ve number)
15000 X2          2         ½      1                 0         0       -½        4
      Zj      30000+M/2 7500+M/2 15000              -M         M      M/2-7500
      Cj - Zj           2500-M/2   0               M         0    7500-M/2 _____
                                   Max! Entering ( largest number among – ve signed figures)

         A1 leaves the solution and X1 enters the solution.
         New X1 values are : 1, 1, 0, -2, 1

         Old X2 Row – Key No.(New X1 Values) = New X2 Values
                         2 – ½ (1) = 1 ½
                         ½ - ½ (1) = 0
                         1 – ½ (0) = 1
                         0 – ½ (-2) = 1
                       - ½ - ½ (1) = -1

Third Simplex Tableau:

      Product Quantity 10000 15000 0                           0__
Cj    Mix        bi     X1    X2    S1                         S2__
10000 X1       1        1     0     -2                         1
15000 X2      1½        0     1     1                          -1
      Zj      32500    10000 15000 -5000                    -5000
      Cj - Zj           0     0     5000                       5000


         There is no “- ve” value in the “Cj – Zj” row. Thus optimum solution is attained in the
         minimisation problem.

                 Carloads of scrap from Izmir (X1) = 1 ton
                 Carloads of scrap from Istanbul (X2) = 1 ½ tons
                 Total Minimum cost = 32 500 MU

33.      Objective function:
                Min! Z = 3X1 + 4X2                 Min! Z = 3X1 + 4X2 + 0S1 + 0S2 +0S3 + MA3
         Subject to:
                6X1 – 4X2 ≤ 60                     6X1 – 4X2 + S1               = 60
               -2X1 + 4X2 ≤ 80                    -2X1 + 4X2      + S2          = 80
              12X1 + 16X2 ≥ 480                   12X1 + 16X2          -S3 + A3 = 480
                       X1 , X2 ≥ 0                 all variables ≥ 0




      Prof.Dr.Dr.M.Hulusi DEMIR                                                                              153
                   Introduction to Production / Operations Management


Initial Simplex Tableau:

       Product        Quantity        3     4      0         0          0          M
 Cj    Mix               bi           X1    X2     S1        S2         S3         A3     bi/aij
0      S1               60            6     -4     1         0          0          0        -
0      S2               80            -2      4    0         1          0          0    80/4 = 20 Min! Leaving
M      A3              480            12    16     0         0          -1         1    480/16=30
       Zj              480 M          12M 16M 0              0          -M         M
       Cj - Zj                      3-12M 4-16M 0            0          M          0_________
                                              Max! Entering
       X2 enters in the place of S2.
       New X2 values are = 20, - ½, 1,0, ¼, 0, 0

35.    Objective function:
         Max! Zj = 25X1 + 15X2                 Zj = 25X1 + 15X2 + 0S1 + 0S2 + 0S3 + MA3 + 0S4
       Subject to:
              3X1 + 2X2 ≤ 240                  3X1 + 2X2 + S1                    = 240
              2X1 + X2 ≤ 140                   2X1 + X2       + S2               = 140
                X1       ≥ 20                   X1                 - S3 + A3     = 20
                      X2 ≤ 80                         X2                     + S4 = 80
                   X1, X2 ≥ 0                                         All variables ≥ 0

Initial Simplex Tableau:

       Product    Quantity 25         15        0      0        0      -M          0
Cj     Mix            bi    X1        X2        S1     S2       S3     A3          S4    bi/aij
0      S1          240      3         2         1      0        0      0           0     240/3=80
0      S2          140      2         1         0      1        0      0           0     140/2=70
-M     A3            20     1         0         0      0        -1     1           0     20/1=20 Min! Leaving
0      S4           80      0         1         0      0        0      0           1     -
       Zj          - 20M -M           0         0      0        M     -M           0
       Cj - Zj            25+M        15        0      0       -M      0           0________
                               Max! Entering
       X1 enters the solution, A3 leaves the solution.
       New X1 values: 20, 1, 0, 0, 0, -1, 0

Second Simplex Tableau:

       Product    Quantity 25         15         0     0         0      0
Cj     Mix            bi   X1         X2         S1    S2        S3     S4         bi/aij
0      S1          180     0          2          1     0         3      0          60
0      S2          100     0          1          0     1         2      0          50 Min! Leaving
25     X1            20    1          0          0     0        -1      0          -
0      S4           80     0          1          0     0         0      1          -
       Zj          - 20M 25           0          0     0       -25      0
       Cj - Zj             0          15        15     0        25      0          ________
                                                                  Max! Entering
       S3 enters the solution, while S2 leaves the solution.




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Third Simplex Tableau:

        Product    Quantity 25         15     0        0        0    0
Cj      Mix            bi    X1        X2     S1       S2       S3   S4   bi/aij
0       S1            30     0         ½      1      -3/2       0    0    60 Min! Leaving
0       S3            50     0         ½      0        ½        1    0    100
25      X1            70     1         ½      0        ½        0    0    140
0       S4           80      0         1      0        0        0    1    80
        Zj          1750    25       25/2     0        50       0    0
        Cj - Zj              0        5/2     15      -50       0    0             ________
                                         Max! Entering

        X2 enters the solution, while S1 leaves the solution.

Fourth Simplex Tableau:

        Product    Quantity 25         15      0           0    0    0
Cj      Mix            bi    X1        X2      S1          S2   S3   S4
15      X2            60     0         1       2          -3    0    0
0       S3            20     0         0     -1            2    1    0
25      X1            40     1         0     -1            2    0    0
0       S4           20      0         0     -2            3    0    1
        Zj          1900    25        15       5            5    0   0
        Cj - Zj              0         0     - 5          -5    0    0_

        There is no positive value in the “Cj – Zj” row. Thus optimum solution is attained.
        Therefore
               Air conditioners (X1) = 40 units
               Air fans (X2)         = 60 units
               Total Profit          = 1900 MU
        Over the minimum air conditioner production (S3) = 20 units
        Unused air fan capacity      = 20 units

36. a) Objective function:
              Max! Zj = 40X1 + 50X2 + 60X3
       Subject to:
              Labour             4X1 + 4X2 + 5X3 ≤ 80
              Material A       200X1 + 300X2 + 300X3 ≤ 6000
              Material B       600X1 + 400X2 + 500X3 ≤ 5000
              Non-negativity               X1, X2, X3 ≥ 0

        After augmenting the model becomes:
               Objective function:
                      Max! Zj = 40X1 + 50X2 + 60X3 + 0S1 + 0S2 + 0S3

               Subject to:
                               4X1 + 4X2 + 5X3 + OS1             = 80
                             200X1 + 300X2 + 300X3   + 0S2       = 6000
                             600X1 + 400X2 + 500X3         + 0S3 = 5000
                                                   All variables ≥ 0

     Prof.Dr.Dr.M.Hulusi DEMIR                                                              155
                     Introduction to Production / Operations Management


        Initial Simplex Tableau:

                 Product Quantity         40       50      60    0         0      0
        Cj       Mix       bi             X1       X2      X3    S1        S2     S3        bi/aij
        0        S1         80               4        4       5  1         0      0         80/5=16
        0        S2      6000             200      300     300   0         1      0    6000/300 =20
        0        S3      5000             600      400     500   0         0      1    5000/500=10 Min! Leaving
                 Zj          0               0        0       0  0         0      0
                 Cj - Zj                    40      50      60   0         0      0
                                                             Max! Entering

  b)    Values for entering variable, X3 : 10, 6/5, 4/5, 1, 0,0, 1/500

Old S1 row – #. X3 = new S1 row   Old S2 row - #.X3 = new S2 row
        80 - 5(10) = 30             6000 – 300(10) = 3000
         4 – 5(6/5) = -2             200 – 300(6/5) = -160
         4 – 5(4/5) = 0              300 – 300(4/5) = 60
         5 – 5(1) = 0                300 – 300(1) = 0
         1 – 5(0) = 1                  0 – 300(0) = 0
         0 – 5(0) = 0                  1 – 300(0) = 1
         0 – 5(1/5) = -1/100           0 – 300(1/500) = -3/5

Second Simplex Tableau:

                 Product Quantity          40 50       60       0      0     0
        Cj       Mix       bi              X1 X2       X3       S1     S2    S3             bi/aij
        0        S1         30                -2 0        0      1     0  -1/100            ∞
        0        S2      3000             -160  60        0      0     1  - 3/5           3000/60 =50
        60       X3         10              6/54/5        1      0     0 1/500            10/4/5=25/2 Min! Leaving
                 Zj       600                72 48      60       0     0   6/50
                 Cj - Zj                    -32  2        0      0     0  -6/50
                                                  Max! Entering
        Values for entering variable, X2: 25/2, 3/2, 1, 5/4, 0, 0, 1/400

        Old S1 row – #. X2 = new S1 row   Old S2 row - #.X2 = new S2 row
      Remains same, because the 3000 -60(25/2) = 2250
      Key number is zero.       – 160 – 60(3/2) = -250
                                   60 – 60(1) = 0
                                    0 – 60(5/4) = -75
                                    0 – 60(0) = 0
                                    1 – 60(0) = 1
                                 -3/5 – 60(1/400) = - 9/20
Third Simplex Tableau:

                 Product Quantity          40      50      60       0      0       0
        Cj       Mix        bi             X1      X2      X3       S1     S2      S3__
        0        S1          30               -2      0       0      1     0    -1/100
        0        S2      2250             -250        0     -75      0     1    - 9/20
        50       X2       25/2              3/2       1      5/4     0     0     1/400
                 Zj        625               75     50     125/2     0     0      1/8
                 Cj - Zj                    -35       0      -5/2    0     0     -1/8__

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        Optimal solution is reached, because all values of “Cj – Zj” are either negative or zero.
        Only X2 is produced.
             X2 = 12.5 units
             S1 = 30 hours
             S2 = 2250 units

37. a) Objective function:
              Max! Z = 40X1 + 50X2
       Subject to:
              Programming constraint:         X1       ≤ 50
              Total time constraint:          X1 + 2X2 ≤ 80
                                                 X1, X2 ≥ 0
     b) Objective function:
               Max! Z = 40X1 + 50X2 + 0S1 + 0S2
        Subject to:
               X1        + S1      = 50
               X1 + 2X2       + S2 = 80
                     All variables ≥ 0

Initial Simplex Tableau:

        (Variables in solution) Quantity      40       50      0       0     Decision Variables
Cj      Product Mix                 bi        X1       X2      S1      S2     bi/aij
0               S1               50           1        0       1       0      ∞
0               S2               80           1        2       0       1   80/2=40 Min! Leaving
                Zj                0           0        0       0       0
                Cj - Zj                       40       50      0       0__
                                                        Max! Entering
        X2 enters the solution, while S2 leaves the solution. The variables in the second tableau
        will be S1 and X2.

 c)     Values of entering variable X2 are: 40, ½, 1, 0, ½

        Old S1 row – #. X2 = new S1 row
        Since the key number is zero, there will be no change in the values of S1 row.


Second Simplex Tableau:

        (Variables in solution) Quantity      40     50         0      0     Decision Variables
Cj      Product Mix                bi         X1     X2         S1     S2     bi/aij
0               S1               50           1      0          1      0   50/1=50 Min! Leaving
50              X2               40           ½      1          0      ½ 40/1/2=80
                Zj              2000        50/2    50          0     50/2
                Cj - Zj                       15     0          0    -50/2__
                                               Max! Entering
        X1 enters in the place of S1.
        Values of entering variable, X1 are: 50, 1,0,1,0


     Prof.Dr.Dr.M.Hulusi DEMIR                                                               157
                        Introduction to Production / Operations Management



           Old S1 row – #. X2 = new S1 row
                   40 – ½ (50) = 15
                   ½ - ½ (1) = 1
                   1 – ½ (0) = 1
                   0 – ½ (1) = - ½
                   ½ - ½ (0) = ½

Third Simplex Tableau:

           (Variables in solution) Quantity        40    50      0       0    Decision Variables
Cj         Product Mix                bi           X1    X2      S1      S2
40                 X1               50             1     0       1       0
50                 X2               15             0     1      -½       ½
                   Zj              2750           40    50      15      25
                   Cj - Zj                         0     0     -15     -25__

           Optimal solution is attained. X1 = 50 hrs.
                                         X2 = 15 hrs.
     ca)   The system analysts have to work 15 hours as shown for X2 under “Quantity”.
     cb)           X1 = 50 hrs.
                   X2 = 15 hrs
                Total = 65 hrs.
     cc)   The total revenue to be expected = 2750 MU
     cd)   15 MU (The shadow price under S1)
     ce)   25 MU (The shadow price under S2)
     cf)     ½ hr. (the -1/2 in the S1 column indicates that the variable in solution X2 could be
           increased by ½ hr@50MU = 25MU increase)
     cg)    -15 (results from the 40 MU loss of 1 hour of programming time
                               + 25 MU gain from ½ hour of system analysis time)

38. a) Objective function:
         Max!        Z = 30X1 + 50X2                    Max! Z = 30X1 + 50X2 + 0S1 +0S2 + 0S3
       Subject to:
                     3X1 + 6X2 ≤ 30                      3X1 + 6X2 + S1           = 30
                    10X1 + 10X2 ≤ 60                    10X1 + 10X2     + S2      = 60
                    10X1 + 15X2 ≤ 120                   10X1 + 15X2          + S3 = 120
                           X1, X2 ≥ 0                                All variables ≥ 0

  b)
Initial Simplex Tableau:

                   Product Quantity          30   50    0         0     0
           Cj      Mix        bi             X1   X2    S1        S2    S3    bi/aij
           0       S1        30               3    6    1         0     0     30/6=5 Min! Leaving
           0       S2        60              10   10    0         1     0     60/10=6
           0       S3       120              10   15    0         0     1     120/15=8
                   Zj          0              0    0    0         0     0
                   Cj - Zj                   30   50    0         0     0
                                                    Max! Entering


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        X2 enters in the solution in the place of leaving variable S1.
        New X2 values are: 5, 1/2, 1, 1/6, 0, 0

Old S2 row – #. X2 = new S2 row           Old S3 row - #.X2 = new S3 row
        60 – 10(5) = 10                           120 – 15(5) = 45
        10 – 10(1/2) = 5                           10 – 15(1/2) = 5/2
        10 – 10(1) = 0                             15 – 15(1) = 0
         0 – 10(1/6) = -5/3                         0 – 15(1/6) = -5/2
         1 – 10(0) = 1                              0 – 15(0) = 0
         0 – 10(0) = 0                              1 – 15(0) = 0

Second Simplex Tableau:

                Product Quantity          30    50        0       0        0
        Cj      Mix       bi              X1    X2        S1      S2       S3      bi/aij
        50      X2         5               ½      1     1/6       0        0       5/1/2=10
        0       S2       10                5      0 -5/3          1        0       10/5=2 Min! Leaving
        0       S3       45               5/2     0 -5/2          0        0       45/5/2=18
                Zj      250               25    50      25/3      0        0
                Cj - Zj                    5      0 -25/3         0        0
                                            Max! Entering

        X1 enters in the place of S2.
        New X1 values are: 2, 1, 0,-1/3, 1/5, 0

        Old X2 row – #. X1 = new X2 row           Old S3 row - #.X1 = new S3 row
                5 – ½ (2) = 4                             45 – 5/2(2) = 40
                ½ - ½ (1) = 0                             5/2 – 5/2(1) = 0
                1 – ½ (0) = 1                               0 – 5/2(0) = 0
              1/6 – ½ (-1/3) = 1/3                       -5/2 – 5/2(-1/3) = -5/3
                0 – ½ (1/5) = -1/10                         0 – 5/2(1/5) = -1/2
                0 – ½ (0) = 0                               0 – 5/2(0) = 1

Third Simplex Tableau:

                Product Quantity          30      50      0     0          0
        Cj      Mix       bi              X1      X2      S1    S2         S3
        50      X2         4               0       1     1/3 -1/10         0
        30      X1         2               1       0    -1/3  1/5          0
        0       S3       40                0       0    -5/3 -1/2          1
                Zj      260               30      50     20/3   1          0
                Cj - Zj                    0       0    -20/3 -1           0

        Optimal is arrived. Solution: X1 = 2 units/hr
                                      X2 = 4 units/hr
                                      S3 = 40 units/hr
                                      Z = 260 MU



    Prof.Dr.Dr.M.Hulusi DEMIR                                                                        159
                  Introduction to Production / Operations Management


39. a) Objective function:        Max! Z = 187X1 + 45X2 + 95X3
    b) Constraints are:           200X1 + 180X2 + 80X3≤ 600
                                  500X1          + 90X3≤ 500
                                   40X1 + 40X2         ≤ 120
                                           X1 , X2, X3 ≥ 0




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TABLES AND FORMULAS




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