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Write the dual of (P). Solve the dual model using the graphical solution method. Solve (P) by using the complementary slackness relations from the optimal dual solution you obtained in Part 6.

Write the dual of (P). Solve the dual model using the graphical solution method. Solve (P) by using the complementary slackness relations from the optimal dual solution you obtained in Part 6.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 23EQ: 23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes...
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5 through 7 please

Consider the following linear optimization model: (P) s.t. min 2x₁ +4x2+ 10x3 + 15x4 −X1 + X2 + X3 + 3x4 ≥ 1 X1 - - x2 + 2x3 + x4 ≥ 1 X1, X2, X3, X4 ≥ 0. 1. Write a Phase-I model for (P). Add the slack/excess variables first (call them x5, 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 compose of the two artificial variables. When selecting variables to enter the basis, always select the eligible variable with smallest index. When selecting variables to leave the basis, always select the eligible variable with 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 (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 (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 (P). 6. Solve the dual model using the graphical solution method. 7. Solve (P) by using the complementary slackness relations from the optimal dual solution you obtained in Part 6.
Transcribed Image Text:Consider the following linear optimization model: (P) s.t. min 2x₁ +4x2+ 10x3 + 15x4 −X1 + X2 + X3 + 3x4 ≥ 1 X1 - - x2 + 2x3 + x4 ≥ 1 X1, X2, X3, X4 ≥ 0. 1. Write a Phase-I model for (P). Add the slack/excess variables first (call them x5, 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 compose of the two artificial variables. When selecting variables to enter the basis, always select the eligible variable with smallest index. When selecting variables to leave the basis, always select the eligible variable with 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 (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 (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 (P). 6. Solve the dual model using the graphical solution method. 7. Solve (P) by using the complementary slackness relations from the optimal dual solution you obtained in Part 6.
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