Practical Management Science
6th Edition
ISBN: 9781337406659
Author: WINSTON, Wayne L.
Publisher: Cengage,
expand_more
expand_more
format_list_bulleted
Related questions
Question
![### Using the Big M Method: Step-by-Step Simplex Method
**Problem Statement:**
Maximize \( Z = 4x_1 + 2x_2 + 3x_3 + 5x_4, \)
subject to
\[
\begin{align*}
2x_1 + 3x_2 + 4x_3 + 2x_4 & = 300, \\
8x_1 + x_2 + x_3 + 5x_4 & = 300,
\end{align*}
\]
and
\[
x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0, \quad x_4 \geq 0.
\]
**Explanation:**
This problem is a linear programming example with the objective to maximize a linear function \( Z \) of four variables \( x_1, x_2, x_3, \) and \( x_4 \). There are two linear equality constraints that these variables must satisfy, and all variables are constrained to be non-negative.
The Big M method is an approach used in linear programming to solve optimization problems with equality constraints, particularly where artificial variables are introduced to convert inequalities into equalities. This is crucial for applying the simplex method effectively.
In this setup, each constraint is already an equality, simplifying the need for artificial variables. The goal is to find the values of \( x_1, x_2, x_3, \) and \( x_4 \) that maximize \( Z \), while respecting the constraints. This can be solved using the simplex algorithm, which iteratively improves the value of \( Z \) while moving through feasible solutions.](https://content.bartleby.com/qna-images/question/e2f2c4bd-bf4c-4a3b-a0a2-6333c3306a45/87815b29-c2e2-42f2-9b0e-cd6329f452cf/lig009r_processed.png)
Transcribed Image Text:### Using the Big M Method: Step-by-Step Simplex Method
**Problem Statement:**
Maximize \( Z = 4x_1 + 2x_2 + 3x_3 + 5x_4, \)
subject to
\[
\begin{align*}
2x_1 + 3x_2 + 4x_3 + 2x_4 & = 300, \\
8x_1 + x_2 + x_3 + 5x_4 & = 300,
\end{align*}
\]
and
\[
x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0, \quad x_4 \geq 0.
\]
**Explanation:**
This problem is a linear programming example with the objective to maximize a linear function \( Z \) of four variables \( x_1, x_2, x_3, \) and \( x_4 \). There are two linear equality constraints that these variables must satisfy, and all variables are constrained to be non-negative.
The Big M method is an approach used in linear programming to solve optimization problems with equality constraints, particularly where artificial variables are introduced to convert inequalities into equalities. This is crucial for applying the simplex method effectively.
In this setup, each constraint is already an equality, simplifying the need for artificial variables. The goal is to find the values of \( x_1, x_2, x_3, \) and \( x_4 \) that maximize \( Z \), while respecting the constraints. This can be solved using the simplex algorithm, which iteratively improves the value of \( Z \) while moving through feasible solutions.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 2 steps with 1 images
Knowledge Booster
Similar questions
- Use the Big M method to solve the following LPs:arrow_forwardUse the Big M method to solve the following LPs:arrow_forwardConsider the following linear program, which maximizes profit for two products: regular (R) and super (S): MAX 5R + 75S s.t. 1.2 R+ 1.6 S ≤ 600 assembly (hours) 0.8 R +0.5 S ≤ 300 paint (hours) 0.16 R+ 0.4 S ≤ 100 inspection (hours) R,S >=0 See the sensitivity report provided below: Variable Cells Cell $C$28 Regular-R $C$29 Super-S Constraints Name Cell $F$6 assembly (hours) $F$7 paint (hours) $F$8 inspection (hours) Name Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 25 1E+30 62.5 0 250 -25 0 400 125 100 Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 5 75 1E+30 0 0 187.5 600 300 100 1E+30 1E+30 50 Regarding the resource "assembling (hours)". Please choose the option that best fit the empty space above. 200 175 100 O If the company has more hours of this process they will be willing to pay more for this resource. O If the company has only 450 hours of this process they will be willing to pay less for this…arrow_forward
- Answer the following questions based on the Tornado and Sensitivity graphs given below. Which input variables are the most and the least impactful variables? Please mention two for each case. How the objective function changes by changing the most impactful input variable?arrow_forwardEstimate a range for the optimal objective value for the following LPs: *(a) Minimize z = 5x₁ + 2x₂ subject to (b) Maximize z subject to x₁ + 5x₂ + 3x3 X₁ X₂ ≥ 3 2x₁ + 3x₂ = 5 X1, X₂ ≥ 0 x₁ + 2x₂ + x3 = 30 2x₁ - x₂ = 40 X1, X2, X3 ≥ 0arrow_forwardConsider the following problem using the simplex method and find all the solutionsarrow_forward
- Assume that $1,285,673 is to be invested in a project whose annual projected cash flows are as follows: Year 1 = $235,678, Year 2 = $267,854, Year 3 = $298,054, Year 4 = $314,567, and Year 5 = $325,678. Assume that the required rate of return is 5.25%. Compute the NPV by using Microsoft Excel's NPV function. Be sure to show your work. Should the project be pursued? Why or why not?arrow_forwardFind the values of x1 and x2 where the following two constraints intersect. (Negative values should be indicated by a minus sign. Round your answers to 3 decimal places.) (1) 12x1 + 9x22 73 (2) 6x1 + 2x2 12 x1 x2 Noxtarrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,
Operations ManagementOperations ManagementISBN:9781259667473Author:William J StevensonPublisher:McGraw-Hill Education
Operations and Supply Chain Management (Mcgraw-hi...Operations ManagementISBN:9781259666100Author:F. Robert Jacobs, Richard B ChasePublisher:McGraw-Hill Education 
Purchasing and Supply Chain ManagementOperations ManagementISBN:9781285869681Author:Robert M. Monczka, Robert B. Handfield, Larry C. Giunipero, James L. PattersonPublisher:Cengage Learning
Production and Operations Analysis, Seventh Editi...Operations ManagementISBN:9781478623069Author:Steven Nahmias, Tava Lennon OlsenPublisher:Waveland Press, Inc.
Practical Management Science
Operations Management
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:Cengage,
Operations Management
Operations Management
ISBN:9781259667473
Author:William J Stevenson
Publisher:McGraw-Hill Education
Operations and Supply Chain Management (Mcgraw-hi...
Operations Management
ISBN:9781259666100
Author:F. Robert Jacobs, Richard B Chase
Publisher:McGraw-Hill Education
Purchasing and Supply Chain Management
Operations Management
ISBN:9781285869681
Author:Robert M. Monczka, Robert B. Handfield, Larry C. Giunipero, James L. Patterson
Publisher:Cengage Learning
Production and Operations Analysis, Seventh Editi...
Operations Management
ISBN:9781478623069
Author:Steven Nahmias, Tava Lennon Olsen
Publisher:Waveland Press, Inc.