Game Theory

The assignment cost of assigning any one operator to any one machine is given in the following Table:

Machines \ Operators	I	II	III	IV
A	10	5	13	15
B	3	9	18	3
C	10	7	3	2
D	5	11	9	7

Find the optimal assignment.

Find the values of $p$ and $q$ which will render the entry (2, 2) a saddle point for the game:

$$Player\ B$$
$$Player\ A \begin{bmatrix} 2 & 4 & 5 \ 10 & 7 & q \ 4 & p & 6 \end{bmatrix}$$

What is operations research? Why operations research is important in the field of Mechanical Engineering?

Alumco manufacturers Aluminum sheets and Aluminum bars. The maximum production capacity is estimated at either 800 sheets or 600 bars per day. The maximum daily demand is 550 sheets and 560 bars. The profit per ton is $40 per sheet and $35 per bar. Formulate the problem as a linear programming problem.

Use graphical method to solve the following problem

$$Maximize \quad z = 2x_1 + 3x_2$$

$$Subject \ to \quad x_1 + x_2 \le 1$$

$$3x_1 + x_2 \le 4$$

$$x_1, x_2 \ge 0$$

Solve the TSP as to minimize the cost per cycle.

From \ To	1	2	3	4	5
1	$\infty$	10	25	25	10
2	1	$\infty$	10	15	2
3	8	9	$\infty$	20	10
4	14	10	24	$\infty$	15
5	10	8	25	27	$\infty$

For the game with pay off matrix

Player A \ Player B	1	2	3
1	-1	2	-2
2	6	4	-6