Indira Gandhi National Open University (IGNOU) 2005 Diploma Computer Applications CS51 Operations Research - - Question Paper

ADCA / MCA (II Year)
Term-End exam

June, 2005

CS51: Operations Research

Time: three hours
Maximum Marks: 75

Note : ques. number one is compulsory. Attempt any 3 more ques. from ques. numbered two to 5.
1. (a) A company produces 2 products. using 2 processes in sequence, Process one is operated for 10 hours and Process two can be operated for only nine hours on any day. The company earns a profit of Rs. 20 per unit on Product one and Rs.25 per unit on Product 2. The subsequent table summarises the other relevant info.

Product Minutes per unit
Process one Process two
1 10 six
2 five 15

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

(b) Patients arrive at a doctor's clinic at random and the avg. rate of arrival is five per hour. Determine the probabiity that during a period of 1 hour, there is no arrival. Also obtain the probability that during 1 hour there are. more than two arrivals. You may presume that the number of arrivals during an hour follows a Poisson distribution. (5)

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

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

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

(f) In a job-shop, there are 3 operators and there are 3 tasks to be performed. The manager estimates the time in mintues of the 3 operators to do the 3 tasks as follows :

Operators Tasks
one two three
I 15 10 nine
II nine 15 10
III eight 12 10

The manager wants to minimize the total time taken by the 3 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 issue 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 1 vehicle at a time. Vehicles arrive according to a Poisson distribution with mean two per hour, for this service. The service time distribution is negative exponential with mean = 15 minutes.

(i) compute the steady state probabilities of finding K vehicles in the system.
(ii) obtain the avg. 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 issue in Q.2 (a). obtain an optimal solution to the dual using the solution of Q.2 (a). (5)

(b) provide an interpretation of the dual variables in the issue above. If you want to increase either the operating hours of Process one or of Process 2, in order to increase the profit further, which 1 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 subsequent issue : (7)
Maximize -(2x1-5)2 - (2x2-1)2
subject to x1+2x2 = 2
x1 = 0, x2 = 0.

(b) Using the concept of dominance, decrease the subsequent three x three matrix to a 2 player zero sum matrix game, and solve for the optimal strategies of the players and value of the game. (8)

Player IIPlayer I one -2 3 -2 -3 1 four three 25. (a) Solve the subsequent issue using dynamic programmmg :
Maximize 2x1+5x2+x3
subject to x1+2x2+3x3 = 7
xi = 0, xi integer,
i=1,2,3. (7)

(b) describe the subsequent terms and provide 1 example of each: (8)
(i) Extreme point
(ii) Mixed strategy
(iii) Deviational variable
(iv) Expectation of a random variable

Earning:   Approval pending.