```
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?
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Answer from Sia

Posted about 1 month ago

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.

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