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

ADCA / MCA (II Year)
Term-End Examination

December, 2005

CS51: Operations Research

Time: 3 hours
Maximum Marks: 75

Note : Question number 1 is compulsory. Attempt any three more questions from questions numbered 2 to 5.

1. (a) (a) Define and give one example, of each of the following : (10)
(i) Economic order quantity
(ii) Variance of a distribution
(iii) Feasible region
(iv) Simulation process
(v) Degeneracy

(b) Consider a small plant which makes two types of automobile parts, say A and B. lt buys castings that are machined, bored and polished. The capacity of machining is 25 per hour for A and 24 per hour for B, the 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. Assume that any combination of parts A and B can be sold. Formulate this problem as an LP model to determine the product mix which maximizes profit. (6)

Player A	Player B
	B1	B2	B3
A1	2	4	5
A2	10	7	q
A3	4	p	6

(d) If S₁ and S₂ are two converse sets, check whether: (4)
(i) S₁ ∩ S₂ will always be convex.
(ii) S₁ ∪ S₂ will always be convex.

(e) List the steps involved in the application of CPM. (2)

(f) The number of cheques processed by a bank every day is normally distributed with a mean of 30,100 and standard deviation of 2450. Find the probability that the bank processes more than 32,000 cheques in a day. (3)
[Note that P[0 to 0.22] = 0.0871,
P[0 to 0.78] = 0.2823, and
P[0 to 1.78] = 0.4625

2. Use the Big-M method to solve the following LP problem. (15)
Maximize z=x₁+2x₂+3x₃-x₄
s.t.
x₁+2x₂+3x₃ = 15
2x₁+x₂+5x₃ = 20
x₁+2x₂+x₃+x₄ = 10
and x₁,x₂,x₃ ≥ 0

3. (a) A city corporation has decided to carry out road repairs on the four main roads of the city. The government has agreed to make a special grant of Rs. 50 lakhs towards the cost with the condition that the repairs be done at the lowest cost and quickest time. lf the conditions warrant, a supplementary token grant will also be considered favourably. The coporation has floted tenders and five contractors have sent in their bids. In order to expedite work, one road wiil be awarded to only one contractor.

Cost of Repairs (Rs. lakh)

			  Road

		     R₁   R₂   R₃   R₄

	         C₁ 9   14   19   15

                 C₂ 7   17   20   19

Contractors C₃ 9   18   21   18

		C₄ 10   12   18   19

		C₅ 10   15   21   16

(i) Find the best way of assigning the repair work to the contractors and the cost.
(ii) If it is necessary to seek supplementary grants, what amount should be asked for ? (7)

(b) A road transport company has one reservation clerk on duty at a time. She handles information of bus schedules and makes reservations. Customers arrive at a rate of 8 per hour and the clerk can service 12 customers on an average per hour, Alter stating your assumptions, answer the following:
(i) What is the average number of customers waiting for the service of the clerk ?
(ii) What is the average time a customer has to wait before getting service ?
(iii) The management is thinking of installing a computer system to handle the information and reservations. This is expected to reduce the service time from 5 to 3 minutes. The additional cost of having the new system works out to Rs. 50 per day. lf the cost of goodwill of having to wait is estimated to be 12 paise per minute spent waiting before being served, should the company install the computer system ? Assume an 8-hour working day. (8)

4. (a) Suppose there are n machines which can perform two jobs. If x of them do the first job, then they produce goods worth g(x) = 3x, and if y of them perform the second job, then they produce goods worth h(y) = 2.5y. Machines are subject to depreciation. So, after performing the first job only a(x) = x/3 machines remain unavailable and after performing the second job b(y) = 2y/3 machines remain available in the beginning of the second year. The process is repeated with the remaining machines. Obtain the maximum total return after three years, and also find the optimal policy in each year. (9)

(b) Solve the game whose payoff matrix is given below : (6)

       B1   B2   B3

 A1  30   40   -80

 A2  0    15   -20

 A3  90   20   50

5. (a) The production department for a company requires 3600 kg of raw material for matufacturing a particular item per year. It has been estimated that the cost of placing an order is Rs.36 and the cost of inventory carrying is 25 percent of the investment in the inventories. The price is Rs. 10 per kg. Determine an ordering policy for the raw material. (8)

(b) Derive the Kuhn - Tucker condition for the following problem, and find the value of x₁ and x₂ for which these conditions are satisfied.
Max z = 10x₁-x₁²+10x₂-x₂²
s.t. x₁+x₂ ≤ 14
-x₁+x₂ ≤ 6
and x₁, x₂ ≥ 0 (7)