4. Reconsider the model in Problem 3 with objective function Maximize Z = C₁x₁ + ₂x₂, where C₁ ≥ 0, C₂ ≥ 0. Use the sensitivity analysis on the objective coefficients to answer the following questions. (a) Find the range of such that the optimal solution remains as (x₁, x₂) = (3,4). C₁ (b) Find the range of such that the optimal solution remains as (x₁,x₂) = (4,2). C₂ (c) Find the range of such that the optimal solution remains as (x₁, x₂) = (0,5). (d) If c₁ and c₂ are allowed to be negative, find the conditions on c₁ and C₂, and the range of such that the optimal solution remains as (x₁, x₂) = (4,0). C₂ Question 3 is given as follows: 3. Consider the following problem. Maximize Z = 3x₁ + 2x₂, subject to X1 ≤4 x₁ + 3x₂ ≤ 15 2x₁ + x₂ ≤ 10 (resource 1) (resource 2) (resource 3) and x₁ ≥ 0, x₂ ≥ 0, x3 ≥ 0. The optimal solution is (x₁, x₂) = (3,4) with Z* = 17. (a) Is any of these three constraints binding constraint? (b) Use graphical analysis to determine the shadow prices for the respective resources. 3
4. Reconsider the model in Problem 3 with objective function Maximize Z = C₁x₁ + ₂x₂, where C₁ ≥ 0, C₂ ≥ 0. Use the sensitivity analysis on the objective coefficients to answer the following questions. (a) Find the range of such that the optimal solution remains as (x₁, x₂) = (3,4). C₁ (b) Find the range of such that the optimal solution remains as (x₁,x₂) = (4,2). C₂ (c) Find the range of such that the optimal solution remains as (x₁, x₂) = (0,5). (d) If c₁ and c₂ are allowed to be negative, find the conditions on c₁ and C₂, and the range of such that the optimal solution remains as (x₁, x₂) = (4,0). C₂ Question 3 is given as follows: 3. Consider the following problem. Maximize Z = 3x₁ + 2x₂, subject to X1 ≤4 x₁ + 3x₂ ≤ 15 2x₁ + x₂ ≤ 10 (resource 1) (resource 2) (resource 3) and x₁ ≥ 0, x₂ ≥ 0, x3 ≥ 0. The optimal solution is (x₁, x₂) = (3,4) with Z* = 17. (a) Is any of these three constraints binding constraint? (b) Use graphical analysis to determine the shadow prices for the respective resources. 3
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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![### Sensitivity Analysis and Optimization Problem
#### Problem 4
Reconsider the model from Problem 3 with the objective function:
\[
\text{Maximize } Z = c_1 x_1 + c_2 x_2
\]
where \( c_1 \geq 0, c_2 \geq 0 \). Use sensitivity analysis on the objective coefficients to answer the following questions:
(a) Find the range of \(\frac{c_1}{c_2}\) such that the optimal solution remains as \((x_1^*, x_2^*) = (3,4)\).
(b) Find the range of \(\frac{c_1}{c_2}\) such that the optimal solution remains as \((x_1^*, x_2^*) = (4,2)\).
(c) Find the range of \(\frac{c_1}{c_2}\) such that the optimal solution remains as \((x_1^*, x_2^*) = (0,5)\).
(d) If \( c_1 \) and \( c_2 \) are allowed to be negative, find the conditions on \( c_1 \) and \( c_2 \), and the range of \(\frac{c_1}{c_2}\) such that the optimal solution remains as \((x_1^*, x_2^*) = (4,0)\).
#### Question 3
Given as follows:
Consider the following problem:
\[
\text{Maximize } Z = 3x_1 + 2x_2,
\]
subject to:
- \( x_1 \leq 4 \) (resource 1)
- \( x_1 + 3x_2 \leq 15 \) (resource 2)
- \( 2x_1 + x_2 \leq 10 \) (resource 3)
- \( x_1 \geq 0, \, x_2 \geq 0, \, x_3 \geq 0 \).
The optimal solution is \((x_1^*, x_2^*) = (3,4)\) with \( Z^* = 17 \).
(a) Is any of these three constraints a binding constraint?
(b) Use graphical analysis to determine the shadow prices for the respective resources](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f2c4bd-bf4c-4a3b-a0a2-6333c3306a45%2F66dbfbe6-9fb0-42e4-a012-61502897af75%2Frxhfdx9a_processed.png&w=3840&q=75)
Transcribed Image Text:### Sensitivity Analysis and Optimization Problem
#### Problem 4
Reconsider the model from Problem 3 with the objective function:
\[
\text{Maximize } Z = c_1 x_1 + c_2 x_2
\]
where \( c_1 \geq 0, c_2 \geq 0 \). Use sensitivity analysis on the objective coefficients to answer the following questions:
(a) Find the range of \(\frac{c_1}{c_2}\) such that the optimal solution remains as \((x_1^*, x_2^*) = (3,4)\).
(b) Find the range of \(\frac{c_1}{c_2}\) such that the optimal solution remains as \((x_1^*, x_2^*) = (4,2)\).
(c) Find the range of \(\frac{c_1}{c_2}\) such that the optimal solution remains as \((x_1^*, x_2^*) = (0,5)\).
(d) If \( c_1 \) and \( c_2 \) are allowed to be negative, find the conditions on \( c_1 \) and \( c_2 \), and the range of \(\frac{c_1}{c_2}\) such that the optimal solution remains as \((x_1^*, x_2^*) = (4,0)\).
#### Question 3
Given as follows:
Consider the following problem:
\[
\text{Maximize } Z = 3x_1 + 2x_2,
\]
subject to:
- \( x_1 \leq 4 \) (resource 1)
- \( x_1 + 3x_2 \leq 15 \) (resource 2)
- \( 2x_1 + x_2 \leq 10 \) (resource 3)
- \( x_1 \geq 0, \, x_2 \geq 0, \, x_3 \geq 0 \).
The optimal solution is \((x_1^*, x_2^*) = (3,4)\) with \( Z^* = 17 \).
(a) Is any of these three constraints a binding constraint?
(b) Use graphical analysis to determine the shadow prices for the respective resources
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