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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”. 10 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? 16 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. 18 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). 20 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. 22 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%. 24 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 26 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. 28 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. 30 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 32 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. 34 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. 36 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. 38 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 40 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? 44 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? 66 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 68 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? 70 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.) 72 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. 74 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? 76 Introduction to Production / Operations Management 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? 78 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 80 Introduction to Production / Operations Management 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 12 4 6 14 13 4 6 14 14 2 4 8 25 6 9 12 35 3 4 5 46 8 12 20 56 1 3 5 67 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 12 4 5 6 13 3 4 8 24 1 2 5 25 5 6 9 34 2 3 4 35 2 3 6 46 4 5 6 56 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 13? 82 Introduction to Production / Operations Management 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 57 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. 84 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. 72 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 74 Introduction to Production / Operations Management 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. 76 Introduction to Production / Operations Management 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. 78 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 80 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 82 Introduction to Production / Operations Management 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 84 Introduction to Production / Operations Management 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. 86 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 88 Introduction to Production / Operations Management 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 90 Introduction to Production / Operations Management 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 92 Introduction to Production / Operations Management 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_____________________________________________ 94 Introduction to Production / Operations Management 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. 96 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. 96 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 16.07 102 Introduction to Production / Operations Management 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 = 0.92 104 Introduction to Production / Operations Management 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 106 Introduction to Production / Operations Management 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. 108 Introduction to Production / Operations Management 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 17.83 110 Introduction to Production / Operations Management 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 Introduction to Production / Operations Management 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. 112 Introduction to Production / Operations Management 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. 114 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=7 116 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. 102 Introduction to Production / Operations Management 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 Introduction to Production / Operations Management 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) = 1 128 Introduction to Production / Operations Management 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, 0 130 Introduction to Production / Operations Management 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 132 Introduction to Production / Operations Management 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. 134 Introduction to Production / Operations Management 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 Introduction to Production / Operations Management 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. 136 Introduction to Production / Operations Management 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 ≥ 0 138 Introduction to Production / Operations Management 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) = 1 140 Introduction to Production / Operations Management 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 4 142 Introduction to Production / Operations Management 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 Introduction to Production / Operations Management 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. 144 Introduction to Production / Operations Management 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) = 1 146 Introduction to Production / Operations Management 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 148 Introduction to Production / Operations Management 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) = 30 150 Introduction to Production / Operations Management 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 152 Introduction to Production / Operations Management 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. 154 Introduction to Production / Operations Management 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__ 156 Introduction to Production / Operations Management 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 158 Introduction to Production / Operations Management 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 ≥ 0 160 TABLES AND FORMULAS 161