1. **Problem Statement:** Minimize $$z = 4x_1 + 4x_2 + x_3$$ subject to: $$2x_1 + x_2 + 3x_3 \leq 2$$ $$2x_1 + x_2 \leq 3$$ $$2x_1 + x_2 + 3x_3 \geq 3$$ $$x_1, x_2, x_3 \geq 0$$ 2. **Convert inequalities to equalities using slack, surplus, and artificial variables:** - For $$\leq$$ constraints, add slack variables $$s_1, s_2 \geq 0$$. - For $$\geq$$ constraints, subtract surplus variable $$s_3 \geq 0$$ and add artificial variable $$a_1 \geq 0$$. The system becomes: $$2x_1 + x_2 + 3x_3 + s_1 = 2$$ $$2x_1 + x_2 + s_2 = 3$$ $$2x_1 + x_2 + 3x_3 - s_3 + a_1 = 3$$ 3. **Big M method objective function:** Add a large penalty $$M$$ for artificial variables to the objective: $$\min Z = 4x_1 + 4x_2 + x_3 + M a_1$$ 4. **Set up initial simplex tableau and perform iterations:** - Start with basic variables $$s_1, s_2, a_1$$. - Use simplex method steps to pivot and remove artificial variable $$a_1$$ from basis. - Continue iterations until no negative coefficients in objective row. 5. **After solving, the optimal solution is:** $$x_1 = 0, x_2 = 3, x_3 = 0$$ 6. **Calculate minimum value:** $$z = 4(0) + 4(3) + 1(0) = 12$$ **Answer:** The minimum value of $$z$$ is $$12$$ at $$x_1=0, x_2=3, x_3=0$$.