Using simplex method solve the following L.P.P

Max. Z =15x1 +6x2 +9x3 +2x4, Subject to 2x1 +x2 +5x3 +6x4 ≤20 ;

3x1 +x2 +3x3 +25x4 ≤24 ;7x1 +x4 ≤70 & x1 ,x2 ,x3 ,x4 ≥0

Solution:

Maximize subject to Z =

$$15 x_{1}+6 x_{2}+9 x_{3}+2 x_{4}+0 s_{1}+0 s_{2}+0 s_{3}$$

$$2 x_{1}+x_{2}+5 x_{3}+6 x_{4}+s_{1}+0 s_{2}+0 s_{3}=20$$

$$3 x_{1}+x_{2}+3 x_{3}+25 x_{4}+0 s_{1}+s_{2}+0 s_{3}=24$$

$$7 x_{1}+0 x_{2}+0 x_{3}+x_{4}+0 s_{1}+0 s_{2}+s_{3}=70$$

$x_{1}, x_{2}, x_{3}, x_{4}, s_{1}, s_{2}, s_{3} \geq 0$

so, The intial basic feasible solution is,

$s_1 = 20,s_2 = 24,s_3= 70 (x_1 = x_2 = x_3 = x_4 = o, non - basic)$

The initial simplex table is given by,

Initial iteration:

First iteration:

$(Z_2 - C_2) = -1 \lt0$ , the current basic feasible solution is not optimal.

now, Max Z = 132

$X_1 = 4, X_2 = 12, X_3 = 0, X_4 = 0$