msccst304

Roll No. ......................

Total No. of Questions : 13] [Total No. of Pages : 04

[2037]M.Sc. (CST) (Semester - 3rd)

OPERATIONS RESEARCH (M.Sc. (CST) - 304)

Time : 03 Hours Maximum Marks : 75

Instruction to Candidates:

1) Section - A is compulsory.

2) Attempt any Nine questions from Section - B.

J-3698[S-1554]

Section - A

Q1) (15 x 2 = 30)a) List down the main characteristics of operations research.

b) Explain the principles of modelling.

c) What is a redundant constraint?

d) What is meant by standard form of LPP?

e) What is a key column and how it is selected?

f) Define Artificial variable.

g) Define dual of LPP.

h) List the advantages of duality.

i) Explain how degeneracy in a transportation problem may be resolved.

j) State the major differences between an assignment and transportationproblem.

k) Define a pure integer programming problem.

l) Why not round off the optimum values instead of resorting to integerprogramming? Explain.

m) What do you understand by devision tree analysis?

n) Sketch the Branch and Bound method in integer programming.

o) Give some examples of integer programming problems.

P.T.O.

Section - B

(9 x 5 = 45)Q2) Solve the following all integer programming problem using branch and

bound method :

Maximize z = 6x1 + 8x

2

Subject to : 4x1 + 16x

2 ≤ 32

14x1 + 4x

2 ≤ 28

x1, x

2 ≥ 0 and are integers.

Q3) Use dynamic programming to solve the following LPP

Maximize z= 3x1 + 7x

2

Subject to : x1 + 4x

2 ≤ 8

x2 ≤ 2

x1, x

2 ≥ 0.

Q4) Consider a small plant which makes two types of automobile parts, say Aand B. It buys casting that are machined, bored and polished. The capacityof machining is 25 per hour for A and 40 per hour for B. Capacity ofboring is 28 per hour for A and 35 per hour for B and the capacity ofpolishing is 35 per hour for A and 25 per hour for B.Casting for part A costs Rs. 2 each and for part B cost Rs. 3 each. Theysell for Rs. 5 and Rs. 6 respectively. The three machines have runningcosts of Rs. 20, Rs. 14 and Rs. 17.50 per hour. Assuming that anycombination of parts A and B can be sold, what product mix maximisesprofit?

Q5) Solve the following LP problem using Simplex method.


2 + 20x

3


2 + 6x

3 ≤ 24

3x1 + 9x

2 + 6x

3 ≤ 30

x1, x

2 and x

3 ≥ 0.

Q6) Solve the following LP problem graphically.

Maximize z = 5x1 – 2x

2

Subject to : x1 ≤ 2

–x1 + 2x

2 ≥ 4.

x1, x

2 ≥ 0.

J-3698[S-1554] -2-

Q7) Solve the following LP problem using the result of its dual problem.

Minimize z1 = 24x

1 + 30x

2


2 ≥ 10

4x1 + 9x

2 ≥ 15

6x1 + 6x

2 ≥ 20

x1, x

2 ≥ 0.

Q8) For the LPP.


2 + 4x

3


2 ≤ 8

2x2 + 5x

3 ≤ 10

3x1 + 2x

2 + 4x

3 ≤ 15

x1, x

2, x

3 ≥ 0

find that range over which C3, C

4 and b

2 can be changed maintaining the

optimality of current solution.

Q9) A manufacturing company has three factories F1, F

2 and F

3 with monthly

manufacturing capacities of 7000, 4000 and 10,000 units of a product.The product is to be supplied to seven stores. The manufacturing costs inthese factories are slightly different but the important factor is the shippingcost from each factory to a particular store. The store requirements are1500, 2000, 4500, 4000, 2500, 3500 and 3000 units. The followingtable shown the unit cost (in Rs.) of shipping from each factory to eachstore. Find the optimal transportation plan so as to minimize thetransportation cost.

Stores

S1

S2

S3

S4

S5

S6

S7

F1

5 6 4 3 7 5 4

Factory F2

9 4 3 4 3 2 1

F3

8 4 2 5 4 8 3

J-3698[S-1554] -3-

Q10) Consider the problem of assigning four sales persons to four differentsales regions as shown in the table below such that the total sales ismaximized.

Region

1 2 3 41 10 22 12 14

Salesman 2 16 18 22 10

3 24 20 12 18

4 16 14 24 20

Q11) A company is evaluating four alternative single-period investmentopportunities whose returns are based on state of economy. The possiblestates of the economy and the associated probability distribution is asfollows :State : Fair Good GreatProbability : 0.2 0.5 0.3The returns of each investment opportunity and each state of economy areas follows :

State of Economy

Alternative Fair Good Great

W Rs. 1,000 Rs. 3,000 Rs. 6,000

X Rs. 500 Rs. 4,500 Rs. 6,800

Y Rs.0 Rs. 5,000 Rs. 8,000

Z Rs. 4,000 Rs. 6,000 Rs. 8,500Using decission-tree approach, which alternative investment proposalwould you recommend if EMV criterion is to be employed?

Q12) Find the optimum integer solution to the following LP problem.


2


2 ≤ 8

4x1 + x

2 ≤ 10

x1, x

2 ≥ 0 and integers.

Q13) Use revised simplex method to solve the following L.P.P :

Minimize z = x1 + x

2


2 ≥ 7

4x1 + x

2 ≥ 6

x1, x

2 ≥ 0.

J-3698[S-1554] -4-

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