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

ADCA / MCA (II Year)
Term-End Examination

December, 2005

CS51 (S): Operations Research

Time: 3 hours
Maximum Marks: 75

1. (a) A manufacturer produces three models (I, II and III) of a certain product. She uses two types of raw materials (A and B) of which 4000 and 6000 units, respectively, are available. The raw material requirements per unit of the three models are as follows :

The labour time of each unit of Model I is twice that of Model II and three times that of Model III. The entire labour force of the factory can produce an equivalent of 2500 units ol Model I. A market survey indicates that the minimum demand of the three models is : 500, 500 and 375 units, respectively. However, the ratios of the number of units produced must be equal to 3 : 2 : 5. Assume that the profit per unit of Models I, II and III is Rs. 60, 40 and 100, respectively.
Formulate this problem as an LPP model to determine the number of units of each product which will maximize profit. (6)

(b) List the steps involved in the most general case of the simulation process. (4)

(c) Define the following dynamic programming terms : (6)
(i) Stage
(ii) State Variable
(iii) Decision variable
(iv) Immediate return
(v) Optimal return
(vi) State transformation function

(d) List the major limitations of the PERT model. (4)

(e) Determine the optimal strategies for both the players, and the value of the game, for the following payoff matrix : (6)

		  Player B

		2  -2  4   1

Player A     6   1  12  3

               -3   2  0   6

                2  -3  7   1

(f) A farmer buys a quantity of cabbage seeds from a company that claims that approximately 80% of the seeds will germinate if planted properly. lf 4 seeds are planted, what is the probability that
(i) exactly two will germinate ?
(ii) at least two will germinate ? (4)

2. (a) A company has factories at F₁, F₂ and F₃, which supply to warehouses at W₁, W₂ and W₃. Weekly factory capacities are 200, 160 and 90 units, respectively. Weekly warehouse requirements are 180, 120 and 150 units, respectively. Unit shipping costs (in rupees) are as follows :

Determine the optimal distribution for this company to minimize total shipping cost. (8)

(b) Each unit of an item costs a company Rs. 40. Annual holding costs are 18 percent of the unit cost for interest charges, 1 percent tor insurance, 2 percent allowances for obsolescence, Rs. 2 for building overheads, Rs. 1.50 for damage and loss, and Rs. 4 miscellaneous costs. The annual demand for the item is constant at 1000 units and each oder costs Rs. 100 to place.
(i) Calculate the EOQ and the total costs associated with stocking the item.
(ii) If the supplier of the item will only deliver batches of 250 units, how are the stock holding costs affected ? (7)

3. (a) The pattern of demand for a seasonal product is as follows :

The cost ot product is Rs. 80 per unit and selling price is Rs. 120. How many units should be purchased for the season so as to maximize expected profit ? Also, if the salvage price of the product is Rs. 20, then would there be any change in the purchase decision ? (7)

(b) In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day. Assume that the inter-arrival time follows an exponential distribution, and the service time (ie., the time taken to service a train) distribution is also exponential with an average of 36 minutes. Calculate the
(i) expected queue size (line length);
(ii) probability that the queue size exceeds 10.
If the input of trains increases to an average of 33 per day, what will the changes be in (i) and (ii) ? (8)

4. (a) A man is engaged in buying and selling identical items. He operates from a warehouse having a capacity of 500 items. Each month he can sell any quantity that he chooses up to the stock at the beginning of the month. Each month, he can buy as much as he wishes for delivery at the end of the month so long as his stock does not exceed 500 Items.
For the next four months he has the following error-free forecasts of cost and sales price :

If he currendy has a stock of 200 units, what quantities should he sell and buy in the next four months ? Find the solution using dynamic Programming. (10)

(b) Use the Kuhn - Tucker conditions to solve the following nonlinear programming problem : (5)
Max z = 10x₁-x₁²+10x₂-x₂²
s.t. x₁+x₂ ≤ 8
-x₁+x₂ ≤ 5
x₁, x₂ ≥ 0

5. Solve the following LP problem by using the two-phase method: (15)
Minimize z = x₁+x₂
s.t. 2x₁+4x₂ ≥ 4
x₁+7x₂ ≥ 7
and x₁, x₂ ≥ 0