LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – STATISTICS

FourTh SEMESTER  – APRIL 2003

ST  4951/ S  1052   advanced operations  reSEaRCH

26.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

 

 

SECTION – A                      (10 ´ 2 = 20 Marks)

Answer ALL the questions.

1. State the Bellman’s principle of optimality.
2. Define a general non-linear programming problem.
3. What is posynomial and where it is used?
4. Define the mathematical formulation of an quadratic programming problem.
5. Define separable programming problem with an example.
6. Convert the chance constraint into equivalent deterministic constraint for the following problem.

Min  Z = 3x₁+ 4x₂

St   Pr (3x₁-2x₂ £ b₁) ³

 

x₁, x₂ ³ 0, where b₁, b₂, b₃ are independent random variables uniformly distributed

in the intervals (-2,2), (0,2) and (0,4) respectively.

7. Explain the concept of integer programming problem.
8. Explain the mathematical model of a stochastic linear program.
9. Explain a scenario where the goal programming concepts are applied.
10. State the Kuhn-Tucker conditions to solve an NLPP program.

 

SECTION – B                                   (5 ´ 8 = 40 Marks)

Answer any FIVE questions

11. Explain Wolfe’s Algorithm in solving a non linear programming problem.
12. Solve the following geometric programming problem

 

13. Explain clearly the piece wise linear approximation.
14. Solve the following Integer programming problem

Max Z = x₁+ x₂

s.t     3x₁+ 2x₂ £ 5

x₂ £ 2

x₁, x₂ ³ 0 and integers.

15. Derive the Gomary’s constraint in solving an integer programming problem
16. Solve the non linear programming problem using Lagrangian multipliers.

Maximize

s.t       x₁+ x₂ + x₃ =15

2x₁ -x₂ +2x₃ = 20

x₁,  x₂, x₃ ³ 0

 

 

 

17. Explain the concepts in solving the stochastic programming problem for

the following scenario  assuming the usual notations

(i) Aircraft Allocation problem    (ii) Two stage programming

18. Solve the following cargo loading problem assuming the usual notations when

there is no volume restriction  (i.e., Q = ¥) , W = 5 and N = 3 with the

numerical  data given below

 

item (n)	Weight (Wn)	Value (Vn)
1	2	7
2	3	10
3	1	3

 

SECTION – C                                  (2 ´ 20 = 40 Marks)

Answer any TWO questions.

19. Explain Beale’s Algorithm in solving a non linear programming problem.
20. Solve the following non linear programming problem using Beale’s Algorithm

 

s.t   x₁ +2x₂ £ 10

x₁ + x₂ £ 9

x₁ ,x₂ ³ 0

20. a) Explain Branch and Bound Technique in solving an Integer programming

problem

1. Derive the geometric -arithmetic mean inequality in solving a geometric

programming problem

21  a)   Explain the dynamic programming problem concepts in solving the Cargo

loading problem, assuming the usual notations.

1. Solve the following linear programming problem through Dynamic programming problem

Max  Z = 3x₁ + 4x₂

s.t         2x₁ + x₂ £ 40

2x₁ + 5x₂ £ 180

x₁, x₂ ³ 0

22. a) Approximate the following NLPP to LPP using separable convex

programming and piece wise linear combination concepts

 

Max f(x) = 3x₁ + 2x₂

                                                               

1. Explain the scenario of n component system in series in Reliability and

provide a solution to solve using a Dynamic programming problem.

 

 

 

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