Test Papers / Previous Question Papers of IGNOU CS51 Operations Research June 2005

ADCA / MCA (II Year)
Term-End Examination

June, 2005

CS51: Operations Research

Time: 3 hours
Maximum Marks: 75

1. (a) A company produces two products. using two processes in sequence, Process 1 is operated for 10 hours and Process 2 can be operated for only 9 hours on any day. The company earns a profit of Rs. 20 per unit on Product 1 and Rs.25 per unit on Product 2. The following table summarises the other relevant information.

Formulate the linear programming problem for determining a product mix that maximizes the profit. (4)

(b) Patients arrive at a doctor's clinic at random and the average rate of arrival is 5 per hour. Determine the probabiity that during a period of one hour, there is no arrival. Also find the probability that during one hour there are. more than 2 arrivals. You may assume that the number of arrivals during an hour follows a Poisson distribution. (5)

(c) Describe the ABC and VED classificaton of items in an inventory and its use in inventory management. (8)

(d) List any three limitations of an OR-approach to solving practical problems. (3)

(e) Suppose A is an n x n integer matrix whose entries are 0, 1 and - 1. State the rule to determine if A is a unimodular matrix. Also explain the algorithmic use of the unimodularity property in integer programming. (5)

(f) In a job-shop, there are three operators and there are three tasks to be performed. The manager estimates the time in mintues of the three operators to do the three tasks as follows :

The manager wants to minimize the total time taken by the three operators, as this will minimize the electricity consumption. How should the manager assign the tasks to the operators ? (5)

2. (a) Solve the linear programming problem formulated in Q.1 (a), using the simplex method. (7)

(b) A mechanic who attends to flat tyres of the vehicles and replaces them, can attend to one vehicle at a time. Vehicles arrive according to a Poisson distribution with mean 2 per hour, for this service. The service time distribution is negative exponential with mean = 15 minutes.

(i) Calculate the steady state probabilities of finding K vehicles in the system.
(ii) Find the average time spent by a customer in the system
(iii) What is the proportion of time, the mechanic is free? (8)

3. (a) Write the dual of the problem in Q.2 (a). Find an optimal solution to the dual using the solution of Q.2 (a). (5)

(b) Give an interpretation of the dual variables in the problem above. If you want to increase either the operating hours of Process 1 or of Process 2, in order to increase the profit further, which one will you recommend for increment, and why ? (3)

(c) A big housing colony replaces the common neon lights at the rate of 16 per day. It costs Rs. 100/- to place an order. A neon light kept in storage costs Rs. 2/- a day.Assume that there is no lead time and that shortages are not allowed. What is the EOQ ? How frequently should the orders be placed ? What is the optimal cost (variable costs only) ? (7)

4. (a) Derive the Kuhn - Tucker conditions for the following problem : (7)
Maximize -(2x₁-5)² - (2x₂-1)²
subject to x₁+2x₂ ≤ 2
x₁ ≥ 0, x₂ ≥ 0.

(b) Using the concept of dominance, reduce the following 3 x 3 matrix to a two player zero sum matrix game, and solve for the optimal strategies of the players and value of the game. (8)

		Player II

Player I    1    -2    3

	     -2   -3     1

	      4     3     2

5. (a) Solve the following problem using dynamic programmmg :
Maximize 2x₁+5x₂+x₃
subject to x₁+2x₂+3x₃ ≤ 7
x_i ≥ 0, x_i integer,
i=1,2,3. (7)

(b) Define the following terms and give one example of each: (8)
(i) Extreme point
(ii) Mixed strategy
(iii) Deviational variable
(iv) Expectation of a random variable