Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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5 through 7 please
![Consider the following linear optimization model:
\[
\begin{array}{rl}
\text{minimize} & 2x_1 + 4x_2 + 10x_3 + 15x_4 \\
(\mathcal{P}) \quad \text{subject to} & -x_1 + x_2 + x_3 + 3x_4 \geq 1 \\
& x_1 - x_2 + 2x_3 + x_4 \geq 1 \\
& x_1, x_2, x_3, x_4 \geq 0.
\end{array}
\]
1. **Write a Phase-I model for \((\mathcal{P})\).** Add the slack/excess variables first (call them \(x_5, x_6\)) and then add two artificial variables.
2. **Solve the Phase-I model using simplex.** The starting basis of your Phase-I model should be composed of the two artificial variables. When selecting variables to enter the basis, always select the eligible variable with the smallest index. When selecting variables to leave the basis, always select the eligible variable with the smallest index. (When providing an answer to this problem, report (at least) the simplex dictionary obtained at each iteration and state what variables are entering/leaving the basis.) At the conclusion of the algorithm, present the optimal solution you found.
3. **Write the simplex dictionary of \((\mathcal{P})\)** associated with the optimal basis you identified in Part 2.
4. **Starting from the simplex dictionary in Part 3**, find an optimal solution to \((\mathcal{P})\) using simplex. Use the same pivoting rules and present your work in the same format as in Part 2.
5. **Write the dual of \((\mathcal{P})\).**
6. **Solve the dual model using the graphical solution method.**
7. **Solve \((\mathcal{P})\)** by using the complementary slackness relations from the optimal dual solution you obtained in Part 6.](https://content.bartleby.com/qna-images/question/44cf4428-d50c-4dc1-a095-73926c558c67/0445d2c8-9f81-44bc-96f8-fd33f762c7c8/lfvaqwh_processed.png)
Transcribed Image Text:Consider the following linear optimization model:
\[
\begin{array}{rl}
\text{minimize} & 2x_1 + 4x_2 + 10x_3 + 15x_4 \\
(\mathcal{P}) \quad \text{subject to} & -x_1 + x_2 + x_3 + 3x_4 \geq 1 \\
& x_1 - x_2 + 2x_3 + x_4 \geq 1 \\
& x_1, x_2, x_3, x_4 \geq 0.
\end{array}
\]
1. **Write a Phase-I model for \((\mathcal{P})\).** Add the slack/excess variables first (call them \(x_5, x_6\)) and then add two artificial variables.
2. **Solve the Phase-I model using simplex.** The starting basis of your Phase-I model should be composed of the two artificial variables. When selecting variables to enter the basis, always select the eligible variable with the smallest index. When selecting variables to leave the basis, always select the eligible variable with the smallest index. (When providing an answer to this problem, report (at least) the simplex dictionary obtained at each iteration and state what variables are entering/leaving the basis.) At the conclusion of the algorithm, present the optimal solution you found.
3. **Write the simplex dictionary of \((\mathcal{P})\)** associated with the optimal basis you identified in Part 2.
4. **Starting from the simplex dictionary in Part 3**, find an optimal solution to \((\mathcal{P})\) using simplex. Use the same pivoting rules and present your work in the same format as in Part 2.
5. **Write the dual of \((\mathcal{P})\).**
6. **Solve the dual model using the graphical solution method.**
7. **Solve \((\mathcal{P})\)** by using the complementary slackness relations from the optimal dual solution you obtained in Part 6.
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