# Question of Operation Research

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```					Subject Code: IMT-25
Subject Name : OPERATION                                       RESEARCH
Notes:
a.    Write answers in your own words as far as possible and refrain from copying from the text books/handouts.
b.    Answers of Ist Set (Part-A), IInd Set (Part-B), IIIrd Set (Part – C) and Set-IV (Case Study) must be sent
together.
c.    Mail the answer sheets alongwith the copy of assignments for evaluation & return.
d.    Only hand written assignments shall be accepted.
A. First Set of Assignments:                          5 Questions, each question carries 1 marks.
B. Second Set of Assignments:                         5 Questions, each question carries 1 marks.
C. Third Set of Assignments:                          5 Questions, each question carries 1 marks. Confine your answers to 150
to 200 Words.
D.    Forth Set of Assignments:                       Two Case Studies : 5 Marks. Each case study carries 2.5 marks.

ASSIGNMENTS
PART– A
1. Executives of all levels in business and industry come across the problems of making decisions at every
stage in their day-to-day activities. Operations Research provides them with various quantitative
techniques for decision making and enhances their ability to make long range plans and solve everyday
problems of running a business and industry with greater efficiency, competence and confidence.’
Elaborate the statement with examples.
2. Consider a small plant which makes two types of automobile parts, say A and B. It buys castings that
are machines, bored and polished. The capacity of machining is 25 per hour for A and 24 per hour for
B, capacity of boring is 28 per hour for A and 35 per hour for B, and the capacity of polishing is 35 per
hour for A and 25 per hour for B. Castings for part A cost Rs 2 and sell for Rs. 5 each and those for
part B cost Rs.3 and sell for Rs. 6 each. The three machines have running costs of Rs. 20, Rs. 14 and
Rs.17.50 per hour. Assuming that any combination of parts A and B can be sold, formulate this
problem as an LP model to determine the product mix which would maximizes profit.
3. What is integer linear programming? Explain the merits and demands of ‘ rounding-off a continuous
optimal solution to an LP problem in order to obtain an integer solution.
4. Solve the following Linear Programming Problem?
Maximize P = 10q + 9r
5q + 4r < 14
4q + 5r ≤ 9
7q – 9r ≤ 11

5. Solve the following integer programming problem using Gomory’s cutting plane algorithm.
Maximize Z = x1 + x2
Subject to the constraints
(i)     3x1 + 2x2 ≤ 5, (ii) x2< 2
And x1,x2 ≥ 0 and are integers.

Operation Research .................................................. Page 1 of 4 ............................................................................... IMT-25
PART– B
1. Given a mathematical formulation of the transportation problem and the simplex methods, what are the
differences in the nature of problems that can be solved by using these methods?
2. What is a trans-shipment problem? What are the main characteristics of a transhipment problem?
Explain how a trans-shipment problem can be solved as a transportation problem.
3. Determine an initial basic feasible solution to the following transportation problem by using (a) the least
cost method, and (b) Vogel’s approximate method..

Destination
D1          D2                        D3                 D4               Supply
S1                  1                  2                  1                 4                   30
Source                   S2                  3                  3                  2                 1                   50
S3                  4                  2                  5                 9                   20
Demand               20                  40                30                10

4. A salesman has to visit five cities A, B, C, D and E. The distances (in hundred kilometres) between the
five cities are as follows :
To City
A            B           C             D           E
A          -            1           6             8           4
From City             B          7             -          8             5           6
C          6            8           -             9           7
D          8            5           9             -           8
E          4            6           7             8           -

If the salesman starts from city A and has to come back to city A, which route should he select so that the
total distance travelled is minimum?

5. How would you deal with the assignment problems, where (a) the objective function is to be
maximized? (b) Some assignments are prohibited?

PART – C
1. Discuss the difference between decision-making under certainty, under uncertainty and under risk. What
techniques are used to solve decision-making problems under uncertainty? Which technique results in
an optimistic decision? Which technique results in a pessimistic decision?
2. In a toy manufacturing company suppose the product acceptance probabilities are not known but the
following data is known:

Anticipated First Year Profit (000 Rs)
Product Line
_____________________________
Product                               Full             Partial          Minimal
Acceptance
Good                                  8                        70                      50
Fair                                 50                        45                      40
Poor                               - 25                       -10                       0

Determine the optimal decision under each of the following decision criteria and show how you arrived at
it: (a) Maximax, (b) Maximin,(c) Equal likelihood and (d) Minimax regret?

Operation Research .................................................. Page 2 of 4 ............................................................................... IMT-25
3. Explain the following terms in PERT/CPM.

(i) Earliest time, (ii) Latest time (iii) Total activity time (iv) Event Slake, and (v) Critical Path. What is
meant by the term critical activities , and why is it necessary to know about them?

4. The following maintenance job has to be performed periodically on the heat exchangers in a refinery:
Task                       Description                Immediate        Time (Days)
Predecessors
A          Dismantle pipe Connections                      -              14
B          Dismantle header, closure, and floating        A               22
C          Remove tube Bundle                             B               10
D          Clean Bolts                                    B               16
E          Clean header and floating head front           B               12
F          Clean tube bundle                              C               10
G          Clean Shell                                    C                6
H          Replace the bundle                            F,G               8
I         Prepare Shell Pressure Test                 D,E,H              24
J          Prepare the pressure test and make the          I              16
final reassembly
(a) Draw a network diagram of activities for the project.
(b) Identify the critical path. What is its length?
(c) Find the total float and free float for each task.

5. (a) What are the advantages and limitations of simulations models?
(b) What are the advantages and disadvantages of simulation over the use of analytical models? Is the
use of computers in simulation absolutely essential?

Operation Research .................................................. Page 3 of 4 ............................................................................... IMT-25
CASE STUDY-1

Winter Park Hotel
Donna Shader, manager of the Winter Park Hotel, is considering how to restructure the front desk to reach an
optimum level of staff efficiency and guest service. At Present, the hotel has five clerks on duty, each with a
separate waiting line, during the peak check-in time of 3.00 P.M. to 5.00 P.M. Observation of arrivals during this
time show that an average of 90 guests arrive each hour (although there is no upward limit on the number that
could arrive at any given time). It takes an average of 3 minutes for the front-desk clerk to register each guest.

Donna is considering three plans for improving guest service by reducing the length of time guests spend
waiting in line. The first proposal would designate one employee as a quick-service clerk for guests registering
under corporate accounts, a market segment that fills about 30% of all occupied rooms. Because corporate
guests are preregistered, their registration takes just 2 minutes. With these guests separated from the rest of
the clientele, the average time for registering a typical guest would climb to 3-4 minutes. Under plan I, non-
corporate guests would choose any of the remaining four lines.

The second plan is to implement a single-line system. All guests could form a single waiting line to be served
by whichever of five clerks became available. This option would require sufficient lobby space for what could be
a substantial queue.

The third proposal involves using an automatic teller machine (ATM) for chek-ins. This ATM would provide
approximately the same service rate as a clerk would. Given that initial use of this technology might be minimal,
Shader estimated that 20% of customers, primarily frequent guests, would be willing to use the machines. (This
might be a conservative estimate if the guests [perceive direct benefits from using the ATM, as bank customers
do. Citibank reports that some 95% of its Manhattan customers use its ATMs). Donna would set up single
queue for customers who prefer human check-in clerks. This would be served by the five clerks, although
Donna is hopeful that the machine will allow a reduction to four.

Question.
1. Determine the average amount of time that a guest spends checking in. How would this change under
each of the stated options?
2. Which option do you recommend?

CASE STUDY-2

In an airline hanger, jet-engines arrive at a rate of 2 per week. A jet engine costs nearly Rs. 1 crore.
There are 30 aircraft, each with 4 engines. When an engine breaks down, it is sent for reconditioning,
and the engine is immediately replaced by one from the spare engines’ bank of the reconditioned
engines. If mean rate of reconditioning is 3 engines per week, what should be the optimum number of
spare engines to be kept? Note that the cost of unavailability of a plane, if it is grounded, is Rs. 25 lakh
per week.

Operation Research .................................................. Page 4 of 4 ............................................................................... IMT-25

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