Revised Simplex 15E36B

1. **State the problem:** We want to maximize the objective function $$Z = x_1 + 9x_2 + x_3$$ subject to the constraints: $$\begin{cases} x_1 + 2x_2 + 3x_3 \leq 9 \\ 3x_1 + 2x_2 + 2x_3 \leq 15 \\ 2x_1 + 3x_2 + x_3 \leq 14 \\ x_1, x_2 \geq 0 \end{cases}$$ 2. **Convert inequalities to equalities by adding slack variables:** Let $$s_1, s_2, s_3 \geq 0$$ be slack variables for each constraint: $$\begin{cases} x_1 + 2x_2 + 3x_3 + s_1 = 9 \\ 3x_1 + 2x_2 + 2x_3 + s_2 = 15 \\ 2x_1 + 3x_2 + x_3 + s_3 = 14 \end{cases}$$ 3. **Set up initial basis and tableau:** Initial basic variables: $$s_1, s_2, s_3$$ Initial basic feasible solution: $$x_1 = 0, x_2 = 0, x_3 = 0, s_1 = 9, s_2 = 15, s_3 = 14$$ 4. **Revised Simplex Method steps:** - Define matrices: $$A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 2 \\ 2 & 3 & 1 \end{bmatrix}, \quad b = \begin{bmatrix} 9 \\ 15 \\ 14 \end{bmatrix}$$ - Initial basis matrix $$B = I_3$$ (identity matrix for slack variables), so $$B^{-1} = I_3$$ - Cost coefficients for basic variables $$c_B = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$ - Cost coefficients for non-basic variables $$c_N = \begin{bmatrix} 1 \\ 9 \\ 1 \end{bmatrix}$$ 5. **Calculate reduced costs:** $$c_N - c_B^T B^{-1} N = c_N - 0 = c_N = \begin{bmatrix} 1 \\ 9 \\ 1 \end{bmatrix}$$ Since the largest positive reduced cost is 9 for $$x_2$$, enter $$x_2$$ into the basis. 6. **Compute direction vector:** $$d = B^{-1} A_2 = A_2 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix}$$ 7. **Compute step length:** $$\theta = \min \left\{ \frac{b_i}{d_i} : d_i > 0 \right\} = \min \left\{ \frac{9}{2}, \frac{15}{2}, \frac{14}{3} \right\} = \min \left\{4.5, 7.5, 4.666... \right\} = 4.5$$ 8. **Update basic variables:** The leaving variable corresponds to $$s_1$$ (first constraint). 9. **Update basis:** Replace $$s_1$$ with $$x_2$$ in the basis. 10. **Repeat the process:** - New basis matrix $$B = \begin{bmatrix} 2 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 0 & 1 \end{bmatrix}$$ (columns for $$x_2, s_2, s_3$$) - Calculate $$B^{-1}$$ and continue iterations similarly until no positive reduced costs remain. 11. **Final solution after iterations:** $$x_1 = 0, x_2 = 4.5, x_3 = 0, s_1 = 0, s_2 = 6, s_3 = 0$$ 12. **Calculate maximum value:** $$Z = x_1 + 9x_2 + x_3 = 0 + 9 \times 4.5 + 0 = 40.5$$ **Answer:** The maximum value of $$Z$$ is $$40.5$$ at $$x_1 = 0, x_2 = 4.5, x_3 = 0$$.