Loyola College M.Sc. Statistics April 2004 Advanced Operations Research Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

  FOURTH SEMESTER – APRIL 2004

ST 4951/S 1052 – ADVANCED OPERATIONS RESEARCH

12.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

  1. What is the need for an integer programming problem?
  2. Define a covex function.
  3. Is the following quadratic form negative definite?

j (x1, x2) = –

  1. Is the function f(x) = x1 separable? x = (x1, x2).
  2. Define a quadratic programming problem.
  3. Explain the Markovian property of dynamic programming.
  4. When do you say the Khun-Tucker necessary conditions are also sufficient for a maximization problem?
  5. Explain the need for Goal programming.
  6. What is zero-one programming?
  7. When do we need Geometric programming problem?

 

SECTION – B

 

Answer any FIVE questions                                                                        (5 ´ 8 = 40 marks)

 

  1. Solve the following LPP by dynamic programming.

max z = 3x1 + 4x2

Subject to

2x1+x2 £ 40

2x1+5x2 £ 180

x1+x2 ≥ 0

  1. State and prove the necessary condition for a function of n variables to have a minimum. Also prove the sufficient condition.
  2. Derive the Gomery’s constraint for a mixed algorithm.
  3. Solve by Beale’s method

max Z = 2x1 +3x2

Subject to

x1 + 2x2 £ 4, x1, x2 ≥ 0.

  1. Solve by using Khun – Tucker conditions

max Z = 10x1 + 4x2 – 2

Subject to: 2x1 + x2 £ 5, x1, x2 ≥ 0

 

 

 

 

  1. Reduce the following separable programming problem to an approximate linear programming problem.

f(x1, x2) = 2x1 + 3

Subject to 4x1 + 2£ 16, x1, x2 ≥ 0

  1. Consider the chance constrained problem

max Z = 5x1 + 6x2 + 3x3

Subject to

Pr [a11 x1 + a12 x2 + a13 x3 £ 8] ≥ .95

Pr [5x1 + x2 + 6x3 £ b2] ≥ .1, xj ≥ 0 “j = 1,2,3

~ N

b2 ~ N (7,9).  Reduce this problem to a deterministic model.

  1. Solve

minimize f(x1, x2) = 3 + 2

Subject to

x1 + x2 = 7

x1, x2≥ 0

 

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

  1. a) Solve the following all Integer programming problem

max  Z = x1 + 2x2

Subject to

x1 + x2 £ 7

2x1 £ 11

2x2 £ 7     x1, x2 ≥ 0, x1, x2 integers.

  1. b) Explain branch and bound method with an example.                                            (12+8)
  2. Solve by Wolfe’s method

max Z =  2x1 + 3

Subject to

x1 + 4 x2  £  4

x1 + x2 £ 2

1, x2 ≥ 0

  1. a) A student has to take examination in 3 courses A,B,C. He has 3 days available for the study.  He feels it would be best to devote a whole day to the study of the same course, so that he may study a course for one day, two days or three days or not at all.  His estimates of the grades he may get by study are as follows:-

Course  A      B    C

Days

0                  0      1     0

1                  1      1     1

2                  1      3     3

3                  3      4     3

How should he study so that he maximizes the sum of his grades?  Solve by Dynamic

Progrmming.

  1. b) Solve the following using dynamic programming

min Z =

Subject to

u1+u2+u3 ≥ 10,  u1, u2, u3 ≥ 0                                                       (15+5)

  1. a) Solve the following Geometric programming problem

f(X) = .

  1. b) Explain how will you solve if there is a constraint.        (15+5)

 

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