Minimize $Z = X_1 + X_2 + 3X_3$

Subject to $3X_1 + 2X_2 + X_3 ≤ 3 \\ 2X_1 + X_2 + 2X_3 ≥ 3 \\ X_1, X_2, X_3 ≥ 0$

Introducing slack, surplus & artificial variables in the constraints:

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

Converting the objective function from a minimization problem to a maximization one by multiplying by ‘-1’, and introducing ‘M’ (Big M method):

Maximize $Z = -X_1- X_2- 3X_3 -0S_1 + 0S_2- MA_1$

So $X_1= \dfrac34 , X_3 = \dfrac34 , Max \ Z = -3$

Reconverting (multiplying by ‘-1’): $X_1= \dfrac34 , X_3 = \dfrac34 , Max Z = 3$