Max 1x₁ + 1x2 s.t. 5x₁ + 6x₂ ≤ 41 1x1 + 5x₂ ≤ 20 2x₁ + 1x₂ ≤ 15 X₁, X₂20 and integer
Max 1x₁ + 1x2 s.t. 5x₁ + 6x₂ ≤ 41 1x1 + 5x₂ ≤ 20 2x₁ + 1x₂ ≤ 15 X₁, X₂20 and integer
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
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I need help with part b and c. Please explain the steps on how to solve.
![### All-Integer Linear Program
Consider the following all-integer linear program:
**Objective:**
Maximize \( 1x_1 + 1x_2 \)
**Subject to Constraints:**
\[
\begin{align*}
5x_1 + 6x_2 & \leq 41 \\
1x_1 + 5x_2 & \leq 20 \\
2x_1 + 1x_2 & \leq 15 \\
\end{align*}
\]
**Variable Conditions:**
- \( x_1, x_2 \geq 0 \) and integer
#### Explanation:
This setup describes an optimization problem where the objective is to maximize the linear expression \( 1x_1 + 1x_2 \) given a set of linear inequality constraints. Each constraint limits the allowable combinations of \( x_1 \) and \( x_2 \), which must be non-negative integers. The goal is to find the values of \( x_1 \) and \( x_2 \) that maximize the objective function while satisfying all constraints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2fdea76-24b5-4e92-ab93-0d275ab88ec3%2Fd672c7fe-fbaa-4de5-be23-7d523e05d3b7%2Fmjmrrwj_processed.png&w=3840&q=75)
Transcribed Image Text:### All-Integer Linear Program
Consider the following all-integer linear program:
**Objective:**
Maximize \( 1x_1 + 1x_2 \)
**Subject to Constraints:**
\[
\begin{align*}
5x_1 + 6x_2 & \leq 41 \\
1x_1 + 5x_2 & \leq 20 \\
2x_1 + 1x_2 & \leq 15 \\
\end{align*}
\]
**Variable Conditions:**
- \( x_1, x_2 \geq 0 \) and integer
#### Explanation:
This setup describes an optimization problem where the objective is to maximize the linear expression \( 1x_1 + 1x_2 \) given a set of linear inequality constraints. Each constraint limits the allowable combinations of \( x_1 \) and \( x_2 \), which must be non-negative integers. The goal is to find the values of \( x_1 \) and \( x_2 \) that maximize the objective function while satisfying all constraints.
![(b) Solve the LP Relaxation of this problem.
\[ \boxed{\phantom{\text{answer}}} \text{ at } (x_1, x_2) = \left(\boxed{\phantom{\text{answer}}}\right) \]
(c) Find the optimal integer solution.
\[ \boxed{\phantom{\text{answer}}} \text{ at } (x_1, x_2) = \left(\boxed{\phantom{\text{answer}}}\right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2fdea76-24b5-4e92-ab93-0d275ab88ec3%2Fd672c7fe-fbaa-4de5-be23-7d523e05d3b7%2F8qyc01q_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Solve the LP Relaxation of this problem.
\[ \boxed{\phantom{\text{answer}}} \text{ at } (x_1, x_2) = \left(\boxed{\phantom{\text{answer}}}\right) \]
(c) Find the optimal integer solution.
\[ \boxed{\phantom{\text{answer}}} \text{ at } (x_1, x_2) = \left(\boxed{\phantom{\text{answer}}}\right) \]
Expert Solution
Step 1 Introduction;-
Linear programming is a mathematical technique that is also used in operations management departments. This technique is commonly used to find the best outcome with the maximum profit. Linear programming has various methods; each method is chosen by the companies as per their requirements or based on different factors.
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