# June 2001

## Question Paper of CS 51 – Operations Research of June 2001 from IGNOU

**Time**: 2 hours

**Max. Marks**: 75

**Note**: Question No 1 is compulsory. Attempt three more questions from questions numbered as 2 to 6.

1.

(a) Enumerate six applications of Operations Research (OR) � 3 marks

(b) Discuss three major limitations of OR. � 3 marks

(c) Product Mix Problem: Give the Linear Programming formulation of the following program: The Product A, B and C are produced in three machine centers X, Y and Z. Each product involves operation of each of the machine centers. The time required for each operation for unit amount of each product is given below. Time available at machine centers X, Y and Z are 100, 77 and 80 hours respectively. The profit per unit of A, B and C is Rs. 12, Rs. 3 and Rs. 1 respectively.

Products Machine Centers

X � Y � Z Profit Per Unit

A 10 � 7 � 2 12

B 2 � 3 � 4 3

C 1 � 2 � 1 1

- 3 marks

(d) Explain the following concepts in context of Linear Programming / OR: – 3 marks (i) Objective Function (ii) Convex Polygon (iii) Redundant Constraint

(e) Explain the following in context of Transportation Problem (not exceeding three sentences each) � 3 marks (i) Stepping Stone Method (ii) Degenerated Transportation Problem (iii) the Modified Distribution Method

(f) Explain the following in context of Assignment Problem (not exceeding three sentences for each): – 3 marks (i) Balanced Assignment Problem (ii) Hungarian Method (iii) An Infeasible Assignment

(g) Company XYZ produces two products. The Maximum sales potential for Product 1 and Product 2 are 30 units and 40 units respectively. Write the goal constraints for achieving the sales goal by incorporating the deviational variables. � 3 marks

(h) Explain the following concepts in context of Dynamic Programming (not exceeding three sentences for each) � 3 marks (i) Principle of Optimality (ii) State (iii) Stage

(j) Explain the following in context of Inventory Control � 3 marks (i) Decoupling (ii) VED Classification (iii) Delivery Lag (k) Explain the Minimax Criterion of Optimality in context of Game Theory � 3 marks

2. Solve the Product Mix Problem given above as Q. No. 1 (c), using either Graphical Method or Simplex Method of Linear Programming. � 15 marks 3. Using Stepping Stone Method, solve the following transportation problem for minimum cost of transportation � 15 marks

Factory Distributor

1 � 2 � 3 Inventory

1 2 � 1 � 5 10

2 7 � 3 � 4 25

3 6 � 5 � 3 20

Order 15 � 22 � 18 55

4. (a) Explain the various steps in developing a Goal Programming Model of an optimization problem. – 5 marks (b) Consider the problem of assigning three jobs to three men. Each man is capable of doing all the jobs, however, the time taken by the different men on each job is different and can be assumed to be known. The assignment has to be done so that each job is assigned only once, each man gets only one job and the total time taken by all jobs is minimized.

Formulate the above problem as an Integer Programming (IP) with the decision variables defined as: – 10 marks Xij = 1 if the ith man is assigned to job Xij = 0 otherwise

5. (a) An item is produced at the rate of 50 units per day and is consumed at the rate of 25 units per day. If the set up cost is Rs. 100 per production run and holding cost in stock is Rs. 365 per unit per year, find – 7 marks (i) economic lot size per run (ii) number of runs per year (iii) total related cost

(B) At a certain petrol pump, customers arrive according to a Poisson Process with an average time of 5 minutes between arrivals. The service time is exponentially distributed with mean time = 2 minutes. On the basis of this information, find out – 8 marks (i) What would be the average queue length? (ii) What would be the average number of customers in the queuing system?

6.

(a) Determine the solution of the game whose pay-off matrix is given by – 7 marks

I II III

I -4 -6 3

II -3 -3 6

III 2 -3 4

(b) Determine which course of action Player B will not use in the following game. Obtain the best strategy for each of the two players and value of the game. – 8 marks

I II III

I -3 -1 7

II 4 1 -2