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1138491540's Question
Microeconomics
Posted about 1 month ago

34. A simplex tableau is shown below.
\begin{tabular}{cc|cccccc|c} 
& & x1\mathrm{x}_{1} & x2\mathrm{x}_{2} & x3\mathrm{x}_{3} &  s1\mathrm{~s}_{1} &  s2\mathrm{~s}_{2} &  s3\mathrm{~s}_{3} & \\
Basis & CB3\mathrm{CB}_{3} & 3 & 5 & 8 & 0 & 0 & 0 & \\
\hline s1\mathrm{s}_{1} & 0 & 3 & 6 & 0 & 1 & 0 & -9 & 126 \\
 s2\mathrm{~s}_{2} & 0 & 5/2-5 / 2 & 1/2-1 / 2 & 0 & 0 & 1 & 9/2-9 / 2 & 45 \\
x3\mathrm{x}_{3} & 8 & 1/21 / 2 & 1/21 / 2 & 1 & 0 & 0 & 1/21 / 2 & 18 \\
\hline & Zi\mathrm{Z}_{\mathrm{i}} & 4 & 4 & 8 & 0 & 0 & 4 & 144 \\
& c2Zi\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 cjzjc_j - z_j row, which is x2x_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 x2x_2 column. The ratios are: 1266=21,451/2=not applicable,181/2=36 \frac{126}{6} = 21, \quad \frac{45}{-1/2} = \text{not applicable}, \quad \frac{18}{1/2} = 36 The smallest ratio is 21, so s1s_1 is the leaving variable. We then perform the pivot operation on the element in the s1s_1 row and x2x_2 column
b
The current complete solution can be read from the tableau. The basic variables are s1s_1, s2s_2, and x3x_3. Their values are: s1=126,s2=45,x3=18 s_1 = 126, \quad s_2 = 45, \quad x_3 = 18 The non-basic variables x1x_1, x2x_2, and s3s_3 are zero
c
This solution is not optimal because there is still a positive value (1) in the cjzjc_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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