LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 49

FOURTH SEMESTER – April 2009

ST 4807 – ADVANCED OPERATIONS RESEARCH

 

 

 

Date & Time: 23/04/2009 / 9:00 – 12:00  Dept. No.                                                    Max. : 100 Marks

 

 

SECTION -A                                                                                                                                 

Answer all the questions                                                                  10 x 2 = 20 marks

 

1. When a solution to an LPP is called infeasible?
2. How dual simplex method differs from other simplex methods ?
3. Define holding and penalty costs.
4. Write basic components of a queuing model.
5. Write the significance of integer programming problem.
6. Define Dynamic Programming Problems.
7. Differentiate goal programming from other programming problems.
8. Write a note on complementary slackness condition.
9. Provide any two  applications for parallel  and sequence service systems.
10. For a single item static model if D = 100 , h = $0.02 , K = $100 and lead

time is 10 days,find the economic order quantity and re order point.

 

                                                                SECTION -B                                                                                                                                 

Answer any five questions                                                                 5 x 8 = 40 marks

 

11. Use the graphical method to solve the following LPP:

Maximize Z = 2x₁ + 3x₂

Subject to the constraints:                                                                                                           x₁ + x₂ ≤ 30 ,  x₁ – x₂ ≥ 0 , x₂ ≥ 3 , 0≤ x₁ ≤ 20 and 0 ≤ x₂ ≤ 12.

12. Write big M method algorithm.
13. Use duality to solve the following LPP:

Maximize Z = 2x₁ + x₂

Subject to the constraints:

x₁ + 2x₂ ≤ 10  ,   x₁ + x₂ ≤ 6 ,  x₁ – x₂ ≤ 2 , x₁ – 2x­₂ ≤ 1 ; x₁,x₂ ≥ 0 .

14. Write briefly about inventory management.
15. Derive the steady state measures of (M/M/1) : (GD/∞/∞) queuing model.
16. Write Beale’s algorithm to solve Quadratic Programming Problem.
17. Obtain the set of necessary and sufficient conditions  for the following NLPP.

Minimize Z = 2x₁² – 24x₁ + 2x₂² – 8x₂ + 2x₃² – 12x₃ + 200

Subject to the constraints:

x₁ + x₂ + x₃ = 11 ,  x₁,x_2, x₃ ≥ 0 .

18. Solve the following NLPP using Kuhn- Tucker conditions :

Maximize Z =  –x₁² – x₂² – x₃² + 4x₁ + 6x₂

Subject to the constraints:

x₁ + x₂  ≤ 2  ,   2x₁ + 3x₂ ≤ 12 ; x₁, x₂ ≥ 0

 

 SECTION -C                                                                                                                              

Answer any two questions                                                                    2x 20 = 40 marks

 

19.(a) Use two-phase simplex method to

Maximize Z =  5x₁ + 8x₂

Subject to the constraints:

3x₁ + 2x₂ ≥ 3 , x₁ + 4x₂ ≥ 4 , x₁ + x₂ ≤ 5 ; x₁, x₂ ≥ 0

 

• Use dynamic programming to solve:

Minimize Z = x₁² + 2x₂² + 4x₃

Subject to the constraints:

x₁ + 2x₂ + x₃ ≥ 8 ;  x₁ ,x₂ , x₃ ≥ 0.

(12 + 8 )                                                                 20(a)  Derive probabilistic EOQ model.

 

(b)  Electro uses resin in its manufacturing process at the rate of 1000 gallons

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 optimum ordering policy for Electro.

(10 + 10)

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

Maximize Z = 6x₁ + 3x₂ – 4x₁x₂ – 2x₁² – 3x₂²

Subject to the constraints:

x₁ + x₂ ≤ 1  ,  2x₁ + 3x₂ ≤ 4 ; x₁, x₂ ≥ 0 .

 

22. Use cutting plane algorithm to solve the following LPP:

Maximize Z = 200x₁ + 400x₂ + 300x₃

Subject to the constraints:

30x₁ +  40x₂ + 20x₃ ≤  600

20x₁ + 10x₂ + 20x₃  ≤  400

10x₁ + 30x₂ + 20x₃  ≤  800

x₁, x₂, x₃ ≥ 0  and are integers.

 

 

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