LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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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
- When a solution to an LPP is called infeasible?
- How dual simplex method differs from other simplex methods ?
- Define holding and penalty costs.
- Write basic components of a queuing model.
- Write the significance of integer programming problem.
- Define Dynamic Programming Problems.
- Differentiate goal programming from other programming problems.
- Write a note on complementary slackness condition.
- Provide any two applications for parallel and sequence service systems.
- 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
- 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.
- Write big M method algorithm.
- 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 .
- Write briefly about inventory management.
- Derive the steady state measures of (M/M/1) : (GD/∞/∞) queuing model.
- Write Beale’s algorithm to solve Quadratic Programming Problem.
- 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,x2, x3 ≥ 0 .
- 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)
- 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 .
- 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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