LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B. Ss. DEGREE EXAMINATION – STATISTICS

SIXTH SEMESTER – April 2011

ST6604 / ST6601 – OPERATIONS RESEARCH

Time: 3 hrs Max.:100 Marks

PART A

Answer ALL questions: (10 x 2 = 20)

Define Basic feasible solution.
What do you mean by Slack variable?
When do we go for artificial variables?
What are the two rules for selecting the entering and leaving variables?
Write the Standard form for the given Linear programming problem.

What do you understand by transportation problem?
What is a traveling salesman problem?
Define a Two- person zero sum game.
Distinguish between PERT and CPM.
What do you mean by Minimax criterion?

PART B

Answer any FIVE questions: (5 x 8 = 40)

Let us assume that you have inherited Rs. 1,00, 000 from your father – in – law that can be inversted in a combination of only two stock portfolios, with the maximum investment allowed in either portfolio set at Rs. 75, 000. The first portfolio has an average of 10%, whereas the second has 20%. In terms of risk factors associated with these portfolios, the first has a risk rating of 4 (on a scale of 0 to 10) and the second has a risk rating of 9. since you wish to maximize yours returns, you will not accept an average rate of return below 12% or a risk factor above 6. Hence, you face the important question. How much should you invest in each portfolio?

Formulate this as a Linear Programming Problem.

Explain the Two Phase method of solving an LPP.

Find the initial basic feasible solution for the following transportation problem using Vogel’s Approximation method.

From	To
	9	12	9	6	9	10	5
	7	3	7	7	5	5	6
	6	5	9	11	3	11	2
	6	8	11	2	2	10	9
	4	4	6	2	4	2

Solve the following linear programming problem by graphical method:

Maximize z = x₁ + 2x₂

Subject to

x₁ + 2x₂ £ 15

2x₁ + x₂ £ 20

x₁ + x₂ ³ 1

x₁, x₂ ³ 0

Consider a firm having two factories. The firm is to ship its products from the factories to three retail stores. The number of units available at factories X and Y are 200 and 300 respectively, while those demanded at retail stores A, B and C are 100, 150 and 250 respectively. Rather than shipping directly from factories to retail stores, it is added to investigate the possibility of transhipment. The transportation cost (in rupees) per unit is given below:

		Factory		Retail Store
		X	Y	A	B	C
Factory	X	0	8	7	8	9
Factory	Y	6	0	5	4	3
Retail Store	A	7	2	0	5	1
	B	1	5	1	0	4
	C	8	9	7	8	0

Find the optimal shipping schedule.

Solve the following game graphically:

Player B

Player A

Determine the total, free and independent floats and identify the critical path for the given project network.

Differentiate between Simplex method and Dual Simplex method.

PART C

Answer any TWO questions: (2 x 20 = 40)

a) Give the Dual for the following Primal.

Maximize z = 5x₁ + 2x₂

Subject to

6x₁ + x₂ ≥ 6

4x₁ + 3x₂ ≥ 12

x₁ + 2x₂ ≥ 4

x₁, x₂ ≥ 0

19 b) Solve the given problem using Big M method.

Minimize z = x₁ + 2x₂ + 3x₃– x₄

Subject to

x₁ + 2x₂ + 3x₃ = 15

2x₁ + x₂ + 5x₃ = 10

x₁ + 2x₂ + x₃ + x₄= 10

x₁, x₂, x₃, x₄≥ 0 (5 + 15)

a) Reduce a m x n game to a linear programming problem.:

b) Solve the following assignment problem to find the minimum total expected cost.

		Men
		I	II	III	IV
Job	A	42	35	28	21
	B	30	25	20	15
	C	30	25	20	15
	D	24	20	16	12

21. a) State the rules for drawing the network diagram.

21. b) A project consists of eight activities with the following relevant information:

Activity	Immediate Predecessor	Estimated duration (days)
Activity	Immediate Predecessor	Optimistic	Most likely	Pessimistic
A	–	1	1	7
B	–	1	4	7
C	–	2	2	8
D	A	1	1	1
E	B	2	5	14
F	C	2	5	8
G	D, E	3	6	15
H	F, G	1	2	3

(i) Draw the PERT network and find out the expected project completion time and variance

(ii) What duration will have 95% confidence for project completion?

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