LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

AB 21

FIFTH SEMESTER – November 2008

MT 5504 – OPERATIONS RESEARCH

 

 

 

Date : 07-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer all the questions. (10 X 2 = 20)

 

1. Define feasible solution of a general Linear Programming Problem.
2. Give the symmetric form of a LPP.
3. State the necessary and sufficient condition for a Transportation problem to have a feasible solution.
4. How can you convert a maximization assignment problem to a minimization problem?
5. What is a two person zero sum game.
6. Define EOQ.
7. Define total float of an activity.
8. What is payoff matrix?
9. What is a project? List the 3 main phases of a project.
10. What are the different types of inventory?

 

SECTION B

Answer any five questions.(5 X 8 = 40)

 

11. Solve the following L.P.P by the Graphical method:

Max Z = 3x₁ + 2x₂ subject to

-2x₁ + x₂ ≤ 1,

x₁≤ 2,

x₁ + x₂ ≤ 3 and x₁, x₂ ≥ 0.

 

12. Find the initial basic feasible solution for the following transportation problem by least cost method.

To                          Supply

From          1          2          1          4          30

3          3          2          1          50

4          2          5          9          20

Demand          20        40        30        10

 

13. Solve the following 2 X 2 game:

B

A         5          1

4          0

 

14. Construct the network for the project whose activities are given below and compute the total, free and independent float of each activity and hence determine the critical path and the project duration:

 

Activity: 0-1    1-2       1-3       2-4       2-5       3-4       3-6       4-7       5-7       6-7

 

Duration:3       8          12        6          3          3          8          5          3          8

 

15. For an item, the production is instantaneous. The storage cost of one item is Re. one per month and the set up cost is 25 per run. If the demand is 200 units per month, find the optimum quantity to be produced per set-up and hence determine the total cost of storage and set-up per month.

 

16. A commodity is to be supplied at a constant rate of 200 units per day. Supplies for any amounts can be had at any required time, but each ordering costs Rs. 50. Cost of holding the commodity in inventory is Rs. 2.00 per unit per day while the delay in the supply of the items induces a penalty of Rs.10 per unit per delay of one day. Formulate the average cost function of this situation and find the optimal policy (q,t) where t is the reorder cycle period and q is the inventory level after reorder. What should be the best policy if the penalty cost becomes infinite?

 

17. The processing time in hours for the jobs when allocated to the different machines is indicated below. Assign the machines for the jobs so that the total processing time is minimum.

	Machines
Jobs		M1	M2	M3	M4	M5
	J1	9	22	58	11	19
	J2	43	78	72	50	63
	J3	41	28	91	37	45
	J4	74	42	27	49	39
	J5	36	11	57	22	25

 

 

18. Solve the following:

Maximize

Z=15x₁ + 6x₂+ 9x₃ + 2x₄

Subject to 2x₁ + x₂+ 5x₃+6x₄ ≤ 20

3x₁+x₂+3x₃+25x₄ ≤ 24

7x₁ + x₄ ≤70

x₁, x₂, x₃, x₄ ≥ 0.

SECTION C

Answer any two questions: (2 X 20 = 40)

19a) Use Penalty method to solve Z=2x₁ + x₂+ x₃

Subject to 4x₁ + 6x₂+ 3x₃ ≤ 8

3x₁– 6x₂– 4x₃ ≤ 1

2x₁ + 3x₂ – 5x₃  ≥ 4

x₁, x₂, x₃ ≥ 0.

 

19b) A person wants to decide the constituents of a diet which will fulfill his daily requirements of essential nutrition at the minimum cost. The choice is to be made from four different types of foods. The yields per unit of these foods are given in the following table:

Food type	Yield/unit			Cost / unit (Rs)
	Proteins	Fats	Carbohydrates
1	3	2	6	45
2	4	2	4	40
3	8	7	7	85
4	6	5	4	65
Minimum requirement	800	200	700

Formulate the linear programming model for the [problem.

 

20a) Solve the following Transportation problem to minimize the total cost of transportation:

 

	Destination
origin		D1	D2	D3	D4	supply
	O1	14	56	48	27	70
	O2	82	35	21	81	47
	O3	99	31	71	63	93
Demand		70	35	45	60	210

 

 

20b) Solve the following Travelling salesman problem:

		A	B	C	D
From	A	–	46	16	40
	B	41	–	50	40
	C	82	32	–	60
	D	40	40	36	–

 

21a) Three time estimates (in months) of all activities of a project are given below:

	Time in months
Activity	a	m	b
1-2	0.8	1.0	1.2
2-3	3.7	5.6	9.9
2-4	6.2	6.6	15.4
3-4	2.1	2.7	6.1
4-5	0.8	3.4	3.6
5-6	0.9	1.0	1.1

 

Find the expected duration and standard deviation of each activity,

Construct the project network,

Determine the critical path, expected project length and expected variance of the project length.

 

21b) For the pay-off matrix given below, decide optimum strategies for A and B

B

• 2

A         1          200      80

2          110      170

 

22a) Explain the purchase inventory model with n-price breaks.

 

22b) Find the optimal order quantity for a product for which the price break is as follows:

Quantity                                  Unit cost

0 ≤ Q₁ < 50                             Rs. 10

50 ≤Q₂<100                            Rs. 9

100 ≤Q₃                                   Rs. 8

The monthly demand for the product is 200 units, the cost of the storage is 25% of the unit cost and ordering cost is Rs. 20 per order.

 

 

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