Use the Two Phase method to determine if the LP problem is feasible (you can stop at a tableau showing it is feasible but show each step leading up to that one) Min z = s.t. 7x1 + 5x₂ ₁+3x₂ 3x₁ +4x2 3x₁+x₂ X1, X₂ ≥ 0 80 24 12
Use the Two Phase method to determine if the LP problem is feasible (you can stop at a tableau showing it is feasible but show each step leading up to that one) Min z = s.t. 7x1 + 5x₂ ₁+3x₂ 3x₁ +4x2 3x₁+x₂ X1, X₂ ≥ 0 80 24 12
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Using the Two-Phase Method to Determine Feasibility of an LP Problem**
In this tutorial, we will use the Two-Phase method to determine if a given Linear Programming (LP) problem is feasible. For clarity, we will show each step leading up to confirming the feasibility.
### Problem Statement:
We need to solve the following LP problem:
\[ \text{Minimize } z = 7x_1 + 5x_2 \]
subject to the constraints:
\[
\begin{aligned}
x_1 + 3x_2 & \leq 80 \\
3x_1 + 4x_2 & \geq 24 \\
3x_1 + x_2 & \geq 12 \\
x_1, x_2 & \geq 0
\end{aligned}
\]
### Steps:
1. **Formulate the Phase 1 Problem:**
- Convert inequalities into equalities by introducing slack variables and artificial variables where necessary.
2. **Set up the Phase 1 Objective Function:**
- The objective is to minimize the sum of the artificial variables.
3. **Solve the Phase 1 Problem:**
- Use the simplex method to solve the Phase 1 problem.
- If the minimum value of the objective function is zero, the original problem is feasible.
- If not, the LP problem is infeasible.
4. **Formulate the Phase 2 Problem:**
- Use the feasible tableau from Phase 1 to set up the Phase 2 problem.
- Remove the artificial variables and revert to the original objective function.
5. **Solve the Phase 2 Problem:**
- Use the simplex method to find the optimal solution to the original LP problem.
**Note:** For simplification, the actual tableau setup and simplex iterations are not shown here. This tutorial assumes familiarity with tableau methods used in linear programming.
**Conclusion:**
By following these steps in sequence, one can determine if the LP problem is feasible. The process involves turning the inequalities into equalities, introducing necessary variables, and systematically solving the simpler problems using the simplex method. If you arrive at a tableau showing the problem is feasible, you can stop and confirm feasibility.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c7c0620-7337-4c3c-adbf-2e0717dabbe8%2Fdec8754f-82b6-406b-bf18-f22b88d4f18f%2F5hjfvvb_processed.png&w=3840&q=75)
Transcribed Image Text:**Using the Two-Phase Method to Determine Feasibility of an LP Problem**
In this tutorial, we will use the Two-Phase method to determine if a given Linear Programming (LP) problem is feasible. For clarity, we will show each step leading up to confirming the feasibility.
### Problem Statement:
We need to solve the following LP problem:
\[ \text{Minimize } z = 7x_1 + 5x_2 \]
subject to the constraints:
\[
\begin{aligned}
x_1 + 3x_2 & \leq 80 \\
3x_1 + 4x_2 & \geq 24 \\
3x_1 + x_2 & \geq 12 \\
x_1, x_2 & \geq 0
\end{aligned}
\]
### Steps:
1. **Formulate the Phase 1 Problem:**
- Convert inequalities into equalities by introducing slack variables and artificial variables where necessary.
2. **Set up the Phase 1 Objective Function:**
- The objective is to minimize the sum of the artificial variables.
3. **Solve the Phase 1 Problem:**
- Use the simplex method to solve the Phase 1 problem.
- If the minimum value of the objective function is zero, the original problem is feasible.
- If not, the LP problem is infeasible.
4. **Formulate the Phase 2 Problem:**
- Use the feasible tableau from Phase 1 to set up the Phase 2 problem.
- Remove the artificial variables and revert to the original objective function.
5. **Solve the Phase 2 Problem:**
- Use the simplex method to find the optimal solution to the original LP problem.
**Note:** For simplification, the actual tableau setup and simplex iterations are not shown here. This tutorial assumes familiarity with tableau methods used in linear programming.
**Conclusion:**
By following these steps in sequence, one can determine if the LP problem is feasible. The process involves turning the inequalities into equalities, introducing necessary variables, and systematically solving the simpler problems using the simplex method. If you arrive at a tableau showing the problem is feasible, you can stop and confirm feasibility.
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