Minimize $Z = 10X_1 + 15X_2 + 20X_3$

Subject to $2X_1 + 4X_2 + 6X_3 ≥ 24 \\ 3X_1 + 9X_2 + 6X_3≥ 30 \\ X_1, X_2, X_3≥ 0$

Converting the minimization function to a maximization function by multiplying by ‘-1’: Maximize $Z = - 10X_1 - 15X_2 - 20X_3$

Introducing surplus and artificial variables into the constraints:

$2X_1 + 4X_2 + 6X_3 ≥ 24 → 2X_1 + 4X_2 + 6X_3 – S_1 + A_1 = 24 \\ 3X_1 + 9X_2 + 6X_3 ≥ 30 → 3X_1 + 9X_2 + 6X_3 – S_2 + A_2 = 30$

By big M method, the maximization function becomes:

Z = - 10X_1 - 15X_2 - 20X_3 + 0S_1+ 0S_2–MA_1– MA_2

Iteration ends here. Max. Z = -82. Reconverting back:

$Min. Z = 82; X_1 = 0; X_2= \dfrac65 ; X3 = \dfrac{16}{5}$