34. A simplex tableau is shown below.
\begin{tabular}{cc|cccccc|c}
& & $\mathrm{x}_{1}$ & $\mathrm{x}_{2}$ & $\mathrm{x}_{3}$ & $\mathrm{~s}_{1}$ & $\mathrm{~s}_{2}$ & $\mathrm{~s}_{3}$ & \
Basis & $\mathrm{CB}_{3}$ & 3 & 5 & 8 & 0 & 0 & 0 & \
\hline $\mathrm{s}_{1}$ & 0 & 3 & 6 & 0 & 1 & 0 & -9 & 126 \
$\mathrm{~s}_{2}$ & 0 & $-5 / 2$ & $-1 / 2$ & 0 & 0 & 1 & $-9 / 2$ & 45 \
$\mathrm{x}_{3}$ & 8 & $1 / 2$ & $1 / 2$ & 1 & 0 & 0 & $1 / 2$ & 18 \
\hline & $\mathrm{Z}_{\mathrm{i}}$ & 4 & 4 & 8 & 0 & 0 & 4 & 144 \
& $\mathrm{c}_{2}-\mathrm{Z}_{i}$ & -1 & 1 & 0 & 0 & 0 & -4 &
\end{tabular}
a. Do one more iteration of the simplex procedure
b. What is the current complete solution?
c. Is this solution optimal? Why or why not?

Question

34. A simplex tableau is shown below.
\begin{tabular}{cc|cccccc|c} 
& & $\mathrm{x}_{1}$ & $\mathrm{x}_{2}$ & $\mathrm{x}_{3}$ & $\mathrm{~s}_{1}$ & $\mathrm{~s}_{2}$ & $\mathrm{~s}_{3}$ & \
Basis & $\mathrm{CB}_{3}$ & 3 & 5 & 8 & 0 & 0 & 0 & \
\hline $\mathrm{s}_{1}$ & 0 & 3 & 6 & 0 & 1 & 0 & -9 & 126 \
$\mathrm{~s}_{2}$ & 0 & $-5 / 2$ & $-1 / 2$ & 0 & 0 & 1 & $-9 / 2$ & 45 \
$\mathrm{x}_{3}$ & 8 & $1 / 2$ & $1 / 2$ & 1 & 0 & 0 & $1 / 2$ & 18 \
\hline & $\mathrm{Z}_{\mathrm{i}}$ & 4 & 4 & 8 & 0 & 0 & 4 & 144 \
& $\mathrm{c}_{2}-\mathrm{Z}_{i}$ & -1 & 1 & 0 & 0 & 0 & -4 &
\end{tabular}
a. Do one more iteration of the simplex procedure
b. What is the current complete solution?
c. Is this solution optimal? Why or why not?

Accepted Answer

∻Solution∻ ‖ a ⋮ To perform one more iteration of the simplex procedure, we need to identify the entering and leaving variables. The entering variable is the one with the most positive value in the $c_j - z_j$ row, which is $x_2$ with a value of 1. Next, we determine the leaving variable by calculating the minimum ratio of the right-hand side values to the corresponding positive coefficients in the $x_2$ column. The ratios are: $$ \frac{126}{6} = 21, \quad \frac{45}{-1/2} = \text{not applicable}, \quad \frac{18}{1/2} = 36 $$ The smallest ratio is 21, so $s_1$ is the leaving variable. We then perform the pivot operation on the element in the $s_1$ row and $x_2$ column. ‖ ‖ b ⋮ The current complete solution can be read from the tableau. The basic variables are $s_1$, $s_2$, and $x_3$. Their values are: $$ s_1 = 126, \quad s_2 = 45, \quad x_3 = 18 $$ The non-basic variables $x_1$, $x_2$, and $s_3$ are zero. ‖ ‖ c ⋮ This solution is not optimal because there is still a positive value (1) in the $c_j - z_j$ row, indicating that further improvement is possible. ‖ ∻Answer∻ ⚹ The current solution is not optimal. ⚹ ∻Key Concept∻ ⚹ Simplex Method Iteration ⚹ ∻Explanation∻ ⚹ The simplex method iterates by selecting entering and leaving variables to improve the objective function until no further improvement is possible. ⚹◊What is the current value of the objective function in the simplex tableau shown by the OCR model?⍭ Generate me a similar question◊

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