VTU Mechanical Engineering (Semester 7)
Operations Research
May 2016
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Define the following with reference to linear programming model.
i) Unbounded solution
ii) Feasible solution
iii) Slack variable
iv) Surplus variable
v) Optimal Solution.
10 M
1(b) The whit window company is a company with only 3 employees which makes two different kinds of handcrafted windows a wood framed and an aluminum framed window. They earn $60 profit for each wood framed window and $30 profit for each aluminum framed window. Doug makes the wood frames and can make 6 per day. Linda makes the aluminium frames and can make 4 per day. Bob forms and cutsthe glass and can make 48 square feet of glass per day. Each wood framed window uses 6 square foot of glass and each aluminum framed windows used 8 square feet of glass. The company wishes to determine how many windows of each type to produce per day to it maximize total profit. Formulate it as LPP and solve Graphically.
10 M

2(a) Find all the basic solutions to the following systems of equations indentifying in each case the basic and non basic variable and finally the optimal solution.
Maximize Z = 5x1 + 3x2 + 4x3
Subject to
2x1 + x2 + x3 ≤ 20
3x1 + x2 + 2x3 ≤ 30
x1, x2, x3 ≥ 0.
10 M
2(b) Use the simplex method to solve the following problem.
Maximize Z = x1 + 2x2 + 4x3
Subject to
3x1 + x2 + 5x3 ≤ 10
x1 + 4x2 + x3 ≤ 8
2x1 + 2x3 ≤ 7
x1, x2, x3 ≥ 0.
10 M

3(a) Solve the following LPP using two phase method
Minimize Z = 2x1 + 3x2 + x3
x1 + 4x2 + 2x3 ≥ 8
3x1 + 2x2 ≥ 6
x1, x2, x3 ≥ 0.
10 M
3(b) Use Big M method to solve the problem
Minimize Z = 3x1 + 2x2 + 4x3
Subject to
2x1 + x2 + 3x3 = 60
3x1 + 3x2 + 5x3 ≥ 120
x1, x2, x3 ≥ 0.
10 M

4(a) Solve by revised simplex method
Maximize Z = 6x1 - 2x2 + 3x3
Subject to
2x1 - x2 + 2x3 ≤ 2
x1 + 4x3 ≤ 4 and x1, x2, x3 ≥ 0.
10 M
4(b) Use duality to solve ;
Minimize Zx =3x1 + x2
Subject to
x1 + x2 ≥ 1
2x1 + 3x2 ≥ 2, x1, x2, x3 ≥ 0.
10 M

5(a) Solve the following problem by dual simplex method.
Minimize Z = 2x1 + x2
Subject to
3x1 + x2 ≥ 3
4x1 + 3x2 ≥ 6
x1 + 2x2 ≥ 3
x1, x2 ≥ 0.
10 M
5(b) Solve the following problem by using lower bound technique.
Maximize Z = 10x1 + 15x2 + 8x3
Subject to
x1 + 2x2 + 2x3 ≤ 200
2x1 + x2 + x3 ≤ 220
3x1 + x2 + 2x3 ≤ 180
x1 ≥ 10, x2 ≥ 20, x3 ≥ 30.
10 M

6(a) Hindustan construction company needs 3, 3, 4 and 5 million cubic feet of fill at four earthern dams-sites in Punjab. It can transfer the fill from three mounds A, B and C where 2, 6 and 7 million cubic feet of fill is available, cost of transporting one million cubic feet of fill from mounds to the four sites in lakhs are given in the table.
Find IBFs by using any method and check for optimality.
10 M
6(b) Five men are available to do five different jobs. From past records the time (in hrs) that each man takes to do each job is known and given in the following table:
10 M

7(a) Define the following with reference to game theory with an example :
i) Pure strategy
ii) Mixed strategy
iii) Saddle point
iv) Pay off matrix
v) 2 person zero sum games.
10 M
7(b) In a game of matching coins with two players, suppose one player wins Rs 2 when there are two heads and wins nothing when there are two tails and loses Rs 1 when there are one head and one teil. Determine the payoff matrix, the best strategies for each player and the value of the game.
10 M

Explain briefly the following:
8(a)(i) Tabu search
5 M
8(a)(ii) Genetic Algorithm
5 M
8(a)(iii) Simulated annealing technique
5 M
8(a)(iii) Meta heuristics.
5 M

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