LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 52

FOURTH SEMESTER – APRIL 2008

ST 4807 – ADVANCED OPERATIONS RESEARCH

 

 

 

Date : 23/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION- A

Answer all the questions.                        10 x 2 = 20 Marks

 

1. When an LPP is said to have an unbounded solution?
2. Write the significance of goal programming.
3. Write a note on holding and shortage costs in an inventory system.
4. What are the behaviours of customers in queuing analysis?
5. Provide the Khun -Tucker conditions for a maximization problem.
6. How Beal’s method differs from Wolfe’s method?
7. Give an account of methods used for solving an integer programming problem.
8. Define dynamic programming problem.
9. How to find an optimal inventory policy for multiple item static model?

10.When an LPP is called stochastic?

 

                                                                        SECTION- B

Answer any five questions.                5 x 8 =  40 Marks

11. Use two-phase simplex method to

Max. Z = 5x₁ + 3x₂

Subject to

2x₁ + x₂    1

x₁  + 4x₂  6

x₁  0 and x₂  0 .

12. An item sells for $25 a unit ,but a 10 % discount is offered for lots of 150 units or more. A company uses this item at the rate of 20 units per day. The setup cost for ordering a lot is $50 and the holding cost per unit per day is $0.3. Should the company take advantage of the discount?
13. Explain multi-item EOQ model with storage limitation.
14. Derive the steady-state measures of performance for (M / M / 1):(FIFO/ / ).
15. Explain generalized Poisson queuing model.
16. Provide branch and bound algorithm for solving I P P.
17. Use dynamic programming to

Min. Z = x₁² + x₂² + x₃²

Subject to

x₁ + x₂ + x₃  15

x₁ 0 , x₂  0 and  x₃  0.

18. Explain the two- stage programming technique used in stochastic programming.

 

    …2

 

 

-2-

 

 

SECTION – C

Answer any two questions.                2 x 20 =  40 Marks

 

19.(a)  Solve the following LPP graphically:

Max. Z = 2x₁ + 5x₂

Subject to

x₁ + x₂  1  ,     x₁ – 5x₂  0  ,   5x₁ – x₂ 0  ,  x₁ – x₂ -1 ,   x₁ + x₂  6  ,

x₂  3   ,   x₁  0 and x₂  0 .

 

• Use dynamic programming to solve the following LPP:

Max.Z = 3x₁ + 5x₂

Subject to

x₁  4 ,  x₂  6 ,  3x₁ + 2x₂   18

x₁  0 and x₂  0 .

 

20. (a) Derive the probabilistic EOQ model.

(b)  Electro uses resin in its manufacturing process at the rate of 100 gallons per

month. It cost Electro $100 to place an order for a new shipment. The holding

cost per gallon per month is $2 and the shortage cost per gallon is $10.Historical

data show that the demand during lead time is uniform over the range (0, 100)

gallons. Determine  the  optimal ordering policy for Electro.

 

21. Use Wolfe’s method to solve the following QPP:

Max. Z = 2x₁ + x₂ – x₁²

Subject to

2x₁ + 3x₂   6

2x₁ + x₂     4

x₁  0 and x₂  0 .

 

22. Solve the following mixed-integer programming problem using Gomory’s cutting plane algorithm:

Max. Z = 3x₁ + x₂+ 3x₃

Subject to

-x₁  + 2x₂ + x₃    4

4x₂ – 3x₃            2

x₁  – 3x₂ + 2x₃  3

x_i   0     (i = 1,2,3) where x₁ and x₃ are integers.

 

 

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