Question

Use simplex method to solve the following LPP Maximize z=20x_{1}+20x_{2} subject to the constraints: 3x_{1}+3x_{2} ≤36, 5x_{1}+2x_{2} ≤50, 2x_{1}+6x_{2} ≤60, x_{1}≥ 0 and x_{2}≥0.

Answer 1

Given the constraints:

\[3x_1 + 3x_2 \leq 36\]

\[5x_1 + 2x_2 \leq 50\]

\[2x_1 + 6x_2 \leq 60\]

\[x_1 + x_2 = 12\]

Solving the system:

\[3x_1 + 3x_2 = 36\]

\[5x_1 + 2x_2 = 50\]

By substitution:

\[x_1 = \frac{26}{3} \approx 8.66\]

\[x_2 = \frac{10}{3} \approx 3.33\]

Considering the constraint:

\[2x_1 + 6x_2 = 60\]

And the equation:

\[x_1 + x_2 = 12\]

The solution is:

\[x_1 = 3\]

\[x_2 = 9\]The feasible region satisfying all the given conditions is OABCD.

  The coordinates of the corner points are:

\[O(0, 0), A(10, 0), B\left(\frac{26}{3}, \frac{10}{3}\right), C(3, 9), \text{ and } D(0, 10).\]

The maximum value of Z occurs at C(3, 9). Therefore, the optimal solution is:

\[x_1 = 3, x_2 = 9, \text{ and } Z_{\text{max}} = 330.\]