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Which of the following statements are true? Give a short proof or a counter example in
support of your answer.
(i) For any two square matrices A and B, AB = BA.
(ii) If the following table is obtained in the intermediate stage while solving an LPP by the Simplex method, then the LPP has an unbounded solution:
____________________________
. -1 -2 0 0 0
____________________________
1 x1 1 2 -1 0 1
0 x4 0 3 -1 1 2
____________________________
. 0 4 -1 0 1
____________________________
(iii) The number of basic variables in a feasible solution of a transportation problem with m sources and n destinations is mn.
iv) An optimal assignment of the assignment problem with cost matrix C is also an optimal assignment of the assignment problem with cost matrix Ct
(v) (1,2) is an optimal solution to the following LPP:
Max Z = 2x1 + 4x2 subject to
x1 + 2x2 ≤ 5
x1 + x2 ≤ 4
x1, x2 ≥ 0
Solve the (4x3) game with pay off matrix.
A=
8 5 8
8 6 5
7 4 5
6 5 6
At each stage, clearly explain the steps involved.
Write the dual of the following LPP after reducing it to canonical form.
Min Z = 3x1 + 4x2 + 3x3
Subject to
2x1+4x2 =12
5x1+3x3 ≥11
6x1+ x2 ≥ 8
x1,x2,x3≥0
Find the initial basic feasible solution of the following transportation problem using North- West corner method:
P1 P2 P3 P4 Requirement
M1 19 11 23 11 11
M2 15 16 12 21 13
M3 30 25 16 39 19
Availability 6 10 12 15 113
Also, find the optimal solution.
Using graphical method, solve the game whose pay-off matrix is given as:
Player B
I II III IV
Player A I 1 3 -3 7
II 2 5 4 -6
A manufacturer has two products P1 and P2 , both of which are produced in two steps by machines M1 and M2. The process time per hundred for the products on the machines
M1 M2 Profit(in
thousand Rs.
per 100 units)
P1 4 5 10
P2 5 2 5
Available 100 80
hours
The manufacturer can sell as much as he can produce of both products. Formulate the problem as LP model. Determine optimum solution, using simplex method.
Consider the system of equations
2x1+x2+4x3=11
3x1+x2+5x3=14
feasible solution is x1=2,x2=3,x3 =1.Reduce this feasible solution to a basic feasible solution.
Show that the set S ={(x y)|3x2 + 5y2 ≤15}is convex.
While solving problems, clearly indicate which part of which question is being solved.
For the following matrix game, write down the equivalent LPPs for solving the game.
B
A= -1 2
1 0
