LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2011

ST 4811/4807 – ADVANCED OPERATIONS RESEARCH

 

 

Date : 09-04-2011             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Section A

 

Answer ALL questions.                                                                                  (10 x 2 =20 marks)

1. Define General Linear Programming Problem.
2. Define Pure Integer Programming Problem.
3. What is the need for inventory control?
4. What is the behaviour of customers in a queue?
5. Define dynamic Programming Problem.
6. What do you mean by Non Linear Programming Problem?
7. Define a chance constrained model.
8. Show that Q = 2 x₁² + 2 x₂² + 3 x₃² + 2 x₁ x₂ + 2 x₂ x₃ is positive definite.
9. Write the significance of Goal Programming.
10. State the use of simulation analysis.

 

SECTION B

 

Answer any FIVE questions.                                                                    (5 x 8 =40 marks)

 

1. Apply the principle of duality to solve the following: Min Z = 2 x₁ + 2 x₂ , subject to the constraints,     2 x₁ +  4 x₂ ≥ 1,   x₁ + 2 x₂ ≥ 1,  2 x₁ + x₂ ≥ 1,      x₁ , x₂ ≥ 0.
2. Explain Generalized Poisson queuing model.
3. Explain the classical EOQ model.
4. Derive Gomory’s constraint for solving a Mixed Integer Programming Problem.
5. Use Dynamic Programming Problem to solve the following LPP; Max Z =  3 x₁ + 5 x₂ subject to the constraints,     x₁  ≤ 4, x₂  ≤ 6,    3 x₁ + 2 x₂ ≤ 18,    x₁ , x₂ ≥ 0.
6. Derive the KTNC for solving a GNLPP with one inequality constraint.

 

 

 

 

1. Find the deterministic equivalent of the following problem: Min Z =  3 x₁ + 4 x₂ subject to the constraints,     P[ 3 x₁ – 2x₂  ≤ b₁] ≥ ¾,  P[  x₁/7 + 2x₂  ≥ b_2;  x₁ + x₂ /9 ≥ b₃] = 1/4 , x₁ , x₂ ≥ 0, where b₁, b₂, and b₃ are independent random variables uniformly distributed in the intervals     (-2, 2), (0, 2), (0, 4) respectively.
2. An electronic device consists of 4 components, each of which must function for the system to function. The system reliability can be improved by installing parallel units in one or more of the components. The reliability R of a component with 1, 2 or 3 parallel units and the corresponding cost C ( in 000’s) are given in the following table. The maximum amount available for this device is  Rs. 1,00,000. Use DPP to maximize the reliability of the entire system.

	j = 1		j = 2		j = 3		j = 4
Uj	R1	C1	R2	C2	R3	C3	R4	C4
1	.7	10	.5	20	.7	10	.6	20
2	.8	20	.7	40	.9	30	.7	30
3	.9	30	.8	50	.95	40	.9	40

 

 

 

 

 

 

 

SECTION C

 

Answer any TWO questions.                                                                          (2 x 20 =40 marks)

 

1. Explain Branch and Bound algorithm for solving MIPP and hence solve the following problem:

Max z = 3 x₁+  x₂  + 3 x₃   subject to the following constraints, – x₁+  2 x₂  +  x₃ ≤ 4,

4 x₂ – 3 x₂   ≤ 2,  x₁ –  3 x₂  +  2 x₃ ≤ 3,   x₁ , x₂ , x₃  ≥ 0, x₁ , x₃  are integers.

 

20. Solve the following GNLPP using KTNC, Max Z = 2 x₁ – x₁² + x₂  subject to the constraints,         2 x₁ + 3 x₂ ≤ 6, 2 x₁ + x₂ ≤ 4,    x₁,  x₂ ≥ 0.

 

21. Max Z = 6 x₁ + 3 x₂ – 4 x₁ x₂ – 2 x₁² – 3 x₂² subject to the constraints,     x₁ +  x₂ ≤ 1,

2 x₁ + 3 x₂ ≤ 4,    x₁ , x₂ ≥ 0. Show that z is strictly concave and then solve the problem by Wolfe’s algorithm.

22. (i) Derive steady state measures of performance for (M│M│1) : (GD│∞│∞) queue system.

      (ii) Explain multi-item EOQ model with storage limitation.                                                                                          

 

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