IMT-25: OPERATION RESEARCH

PART - A

Q1. 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.

Q2. 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.

Q3. 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.

Q4. Solve the following Linear Programming Problem?

Maximize P = 10q + 9r

5q + 4r < 14

4q + 5r = 9

7q - 9r = 11

Q5. 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.

PART - B

Q1. 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?

Q2. What is a trans-shipment problem? What are the main characteristics of a transshipment problem? Explain how a trans-shipment problem can be solved as a transportation problem.

Q3. Determine an initial basic feasible solution to the following transportation problem by using (a) the least cost method, and (b) Vogel's approximate method..

Q4. A salesman has to visit five cities A, B, C, D and E. The distances (in hundred kilometers) between the five cities are as follows :

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?

Q5. How would you deal with the assignment problems, where (a) the objective function is to be maximized? (b) Some assignments are prohibited?

PART - C

Q1. 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?

Q2. In a toy manufacturing company suppose the product acceptance probabilities are not known but the

Following data is known:

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?

Q3. 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?

Q4. The following maintenance job has to be performed periodically on the heat exchangers in a refinery:

(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.

Q5. (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?

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 cheek-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:

Q1: Determine the average amount of time that a guest spends checking in. How would this change under each of the stated options?

Q2: 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.