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V Semester B.E. (CSE/ISE) Degree Examination, June/July 2015 (2K11 Scheme) CI 56 : OPERATION RESEARCH Time : 3 Hours

Max. Marks : 100

Instruction : Answer any five full questions selecting at least two from each Part. PART – A 1. a) Find a geometrical interpretation and solution as well for the following LP problem : Maximize Z = 3x1 + 5x2 Subject to restrictions : x1 + 2x2 ≤ 2000 x1 + x2 ≤ 1500 x2 ≤ 600 and x1 ≥ 0, x2 ≥ 0.

10

b) Solve the following LP problem by simplex method : Maximize Z = 5x1 + 7x2 Subject to x1 + x2 ≤ 4 3x1 – 8x2 ≤ 24 10x1 + 7x2 ≤ 35 and x1, x2 ≥ 0.

10

2. a) Solve the following LP problem by two phase method : Maximize Z = 3x1 – x2 Subject to the constraints 2x1 + x2 ≥ 2 x1 + 3x2 ≤ 2 x2 ≤ 4 and x1, x2 ≥ 0.

10 P.T.O.

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b) Solve the following LP problem by simplex method adding artificial variables : Maximize Z = 2x1 – 3x2 + 6x3 Subject to 3x1 – 4x2 – 6x3 ≤ 2 2x1 + x2 + 2x3 ≥ 11 x1 + 3x2 – 2x3 ≤ 5 x1, x2, x3 ≥ 0. 10 3. a) Use revised simplex method to solve the following linear programming problem : Maximize Z = x1 + x2 Subject to the constraints : 3x1 + 3x2 ≤ 6 x1 + 4x2 ≤ 4 x1, x2 > 0. 10 b) Obtain the dual of the following LP problem : Maximize Z = x1 – x2 + 3x3, Subject to the constraints x1 + x2 + x3 ≤ 10 2x1 – x3 ≤ 2 2x1 – 2x2 + 3x3 ≤ 6 x1, x2, x3 ≥ 0.

10

4. Consider the following problem : Maximize Z = 6x1 + 8x2, Subject to 5x1 + 2x2 ≤ 20 x1 + 2x2 ≤ 10 and x1 ≥ 0, x2 ≥ 0. i) Construct the dual problem for this primal problem. ii) Solve both the primal problem and the dual problem graphically. Identify the CPF solutions and corner-point infeasible solutions for both problems. Calculate the objective function values for all these solutions. iii) Use the information obtained in part (ii) to construct a table listing the complementary basic solutions for these problems. Use the column headings shown below. No.

Primal Problem Basic Solution Feasible

Z=W

Dual Problem Feasible Basic Solution

20

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PART – B 5. a) Consider the transportation problem having the following parameter table : Destination

Source

Supply

1

2

3

1

6

3

5

4

2

4

M

7

3

3

3

4

3

2

4

2

3

Demand

i) Use Vogel’s approximation method manually to select the first basic variable for an initial BF solution. ii) Use the northwest corner rule manually to construct a complete initial BF solution. 10 b) Determine the optimal solution to the following degenerate transportation problem : D1

D2

D3

D4

D5

ai

O1

4

7

3

8

2

4

O2

1

4

7

3

8

7

O3

7

2

4

7

7

9

O4

4

8

2

4

7

2

10

bj 8 3 7 2 2 6. a) A car hire company has one car at each of five depots a, b, c, d and e. A customer requires a car in each town, namely A, B, C, D and E. Distance (in kms) between depots (origins) and towns (destinations) are given in the following distance matrix : a b c d e A

160

130

175

190

200

B

135

120

130

160

175

C

140

110

155

170

185

D

50

50

80

80

110

E 55 35 70 80 105 How should cars be assigned to customers. So as to minimize the distance travelled ?

10

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b) Five operators have to be assigned to five machines. The assignment costs are given in the table below : Machine I

II

III

IV

V

A

5

5

–

2

6

B

7

4

2

3

4

Operator C

9

3

5

–

3

D

7

2

6

7

2

E 6 5 7 9 1 Operator a cannot operate machine III and operator C cannot operate machine IV. Find the optimal assignment schedule.

10

7. a) Find the optimal strategy for the game for which the pay-off matrix is given below : P2 I II I ⎡ 1 3⎤ P1 ⎢ ⎥ ⎢ ⎥⎦ 4 2 II ⎣ Also, fine the value of the game. 10 b) Define the following three criteria used in decision analysis. i) Maximum payoff criterion. ii) Maximum likelihood criterion. iii) Bayes’ Decision Rules.

10

8. Write short notes on the following : a) Metaheuristics. b) Tabu search. c) Simulated annealing. d) Genetic Algorithms.

(5×4=20) _______________________