Minimize $Z = 7.5X_1 - 3X_2$

Subject to $3X_1 - X_2 - X_3 \geq 3 \\ X_1 - X_2 + X_3\geq 2 \\ X_1, X_2, X_3 \geq 0$

Introducing surplus and artificial variables into the constraints:

$3X_1 - X_2 - X_3 ≥ 3 → 3X_1 - X_2 - X_3– S_1 + A_1 = 3 \\ X_1 - X_2 + X_3 ≥ 2 → X_1 - X_2 + X_3– S_2 + A_2 = 2$

Converting the minimization function to a maximization function by multiplying by ‘-1’:

Maximization $Z = -7.5X_1 + 3X_2 + 0X_3$

PHASE 1:

Maximization function becomes: $Z = 0X_1 + 0X_2 + 0S_1 + 0S_1 – A_1– A_2$

Iteration ends here, since all Cj–Zj are less than or equal to zero.

PHASE 2:

Iteration ends here.

$Max Z = -\dfrac32 ; reconverting back, Min. Z = \dfrac32 , X_1 = \dfrac15$