Loading...

Messages

Proposals

Stuck in your homework and missing deadline? Get urgent help in $10/Page with 24 hours deadline

Get Urgent Writing Help In Your Essays, Assignments, Homeworks, Dissertation, Thesis Or Coursework & Achieve A+ Grades.

Privacy Guaranteed - 100% Plagiarism Free Writing - Free Turnitin Report - Professional And Experienced Writers - 24/7 Online Support

What is an isoprofit line

10/12/2021 Client: muhammad11 Deadline: 2 Day

Linear Programming Models: Graphical and Computer Methods

7

To accompany Quantitative Analysis for Management, Twelfth Edition,

by Render, Stair, Hanna and Hale

Power Point slides created by Jeff Heyl

Copyright ©2015 Pearson Education, Inc.

After completing this chapter, students will be able to:

LEARNING OBJECTIVES

Copyright ©2015 Pearson Education, Inc.

7 – 2

Understand the basic assumptions and properties of linear programming (LP).

Graphically solve any LP problem that has only two variables by both the corner point and isoprofit line methods.

Understand special issues in LP such as infeasibility, unboundedness, redundancy, and alternative optimal solutions.

Understand the role of sensitivity analysis.

Use Excel spreadsheets to solve LP problems.

Copyright ©2015 Pearson Education, Inc.

7 – 3

7.1 Introduction

7.2 Requirements of a Linear Programming Problem

7.3 Formulating LP Problems

7.4 Graphical Solution to an LP Problem

7.5 Solving Flair Furniture’s LP Problem using QM for Windows, Excel 2013, and Excel QM

7.6 Solving Minimization Problems

7.7 Four Special Cases in LP

7.8 Sensitivity Analysis

CHAPTER OUTLINE

Introduction

Many management decisions involve making the most effective use of limited resources

Linear programming (LP)

Widely used mathematical modeling technique

Planning and decision making relative to resource allocation

Broader field of mathematical programming

Here programming refers to modeling and solving a problem mathematically

Copyright ©2015 Pearson Education, Inc.

7 – 4

Requirements of a Linear Programming Problem

Four properties in common

Seek to maximize or minimize some quantity (the objective function)

Restrictions or constraints are present

Alternative courses of action are available

Linear equations or inequalities

Copyright ©2015 Pearson Education, Inc.

7 – 5

LP Properties and Assumptions

PROPERTIES OF LINEAR PROGRAMS
1. One objective function
2. One or more constraints
3. Alternative courses of action
4. Objective function and constraints are linear – proportionality and divisibility
5. Certainty
6. Divisibility
7. Nonnegative variables
TABLE 7.1

Copyright ©2015 Pearson Education, Inc.

7 – 6

Formulating LP Problems

Developing a mathematical model to represent the managerial problem

Steps in formulating a LP problem

Completely understand the managerial problem being faced

Identify the objective and the constraints

Define the decision variables

Use the decision variables to write mathematical expressions for the objective function and the constraints

Copyright ©2015 Pearson Education, Inc.

7 – 7

Formulating LP Problems

Common LP application – product mix problem

Two or more products are produced using limited resources

Maximize profit based on the profit contribution per unit of each product

Determine how many units of each product to produce

Copyright ©2015 Pearson Education, Inc.

7 – 8

Flair Furniture Company

Flair Furniture produces inexpensive tables and chairs

Processes are similar, both require carpentry work and painting and varnishing

Each table takes 4 hours of carpentry and 2 hours of painting and varnishing

Each chair requires 3 of carpentry and 1 hour of painting and varnishing

There are 240 hours of carpentry time available and 100 hours of painting and varnishing

Each table yields a profit of $70 and each chair a profit of $50

Copyright ©2015 Pearson Education, Inc.

7 – 9

Flair Furniture Company

The company wants to determine the best combination of tables and chairs to produce to reach the maximum profit

HOURS REQUIRED TO PRODUCE 1 UNIT
DEPARTMENT (T) TABLES (C) CHAIRS AVAILABLE HOURS THIS WEEK
Carpentry 4 3 240
Painting and varnishing 2 1 100
Profit per unit $70 $50
TABLE 7.2

Copyright ©2015 Pearson Education, Inc.

7 – 10

Flair Furniture Company

The objective is

Maximize profit

The constraints are

The hours of carpentry time used cannot exceed 240 hours per week

The hours of painting and varnishing time used cannot exceed 100 hours per week

The decision variables are

T = number of tables to be produced per week

C = number of chairs to be produced per week

Copyright ©2015 Pearson Education, Inc.

7 – 11

Flair Furniture Company

Create objective function in terms of T and C

Maximize profit = $70T + $50C

Develop mathematical relationships for the two constraints

For carpentry, total time used is

(4 hours per table)(Number of tables produced) + (3 hours per chair)(Number of chairs produced)

First constraint is

Carpentry time used ≤ Carpentry time available

4T + 3C ≤ 240 (hours of carpentry time)

Copyright ©2015 Pearson Education, Inc.

7 – 12

Flair Furniture Company

Similarly

Painting and varnishing time used ≤ Painting and varnishing time available

2 T + 1C ≤ 100 (hours of painting and varnishing time)

This means that each table produced requires two hours of painting and varnishing time

Both of these constraints restrict production capacity and affect total profit

Copyright ©2015 Pearson Education, Inc.

7 – 13

Flair Furniture Company

The values for T and C must be nonnegative

T ≥ 0 (number of tables produced is greater than or equal to 0)

C ≥ 0 (number of chairs produced is greater than or equal to 0)

The complete problem stated mathematically

Maximize profit = $70T + $50C

subject to

4T + 3C ≤ 240 (carpentry constraint)

2T + 1C ≤ 100 (painting and varnishing constraint)

T, C ≥ 0 (nonnegativity constraint)

Copyright ©2015 Pearson Education, Inc.

7 – 14

Graphical Solution to an LP Problem

Easiest way to solve a small LP problems is graphically

Only works when there are just two decision variables

Not possible to plot a solution for more than two variables

Provides valuable insight into how other approaches work

Nonnegativity constraints mean that we are always working in the first (or northeast) quadrant of a graph

Copyright ©2015 Pearson Education, Inc.

7 – 15

Graphical Representation of Constraints

100 –

80 –

60 –

40 –

20 –

C

| | | | | | | | | | | |

0 20 40 60 80 100

T

Number of Chairs

Number of Tables

This Axis Represents the Constraint T ≥ 0

This Axis Represents the Constraint C ≥ 0

FIGURE 7.1 – Quadrant Containing All Positive Values

Copyright ©2015 Pearson Education, Inc.

7 – 16

Graphical Representation of Constraints

The first step is to identify a set or region of feasible solutions

Plot each constraint equation on a graph

Graph the equality portion of the constraint equations

4T + 3C = 240

Solve for the axis intercepts and draw the line

Copyright ©2015 Pearson Education, Inc.

7 – 17

Graphical Representation of Constraints

When Flair produces no tables, the carpentry constraint is:

4(0) + 3C = 240

3C = 240

C = 80

Similarly for no chairs:

4T + 3(0) = 240

4T = 240

T = 60

This line is shown on the following graph

Copyright ©2015 Pearson Education, Inc.

7 – 18

Graphical Representation of Constraints

100 –

80 –

60 –

40 –

20 –

C

| | | | | | | | | | | |

0 20 40 60 80 100

T

Number of Chairs

Number of Tables

(T = 0, C = 80)

FIGURE 7.2 – Graph of Carpentry Constraint Equation

(T = 60, C = 0)

Copyright ©2015 Pearson Education, Inc.

7 – 19

FIGURE 7.3 – Region that Satisfies the Carpentry Constraint

Graphical Representation of Constraints

100 –

80 –

60 –

40 –

20 –

C

| | | | | | | | | | | |

0 20 40 60 80 100

T

Number of Chairs

Number of Tables

Any point on or below the constraint plot will not violate the restriction

Any point above the plot will violate the restriction

(30, 40)

(30, 20)

(70, 40)

Copyright ©2015 Pearson Education, Inc.

7 – 20

Graphical Representation of Constraints

The point (30, 40) lies on the line and exactly satisfies the constraint

4(30) + 3(40) = 240

The point (30, 20) lies below the line and satisfies the constraint

4(30) + 3(20) = 180

The point (70, 40) lies above the line and does not satisfy the constraint

4(70) + 3(40) = 400

Copyright ©2015 Pearson Education, Inc.

7 – 21

Graphical Representation of Constraints

100 –

80 –

60 –

40 –

20 –

C

| | | | | | | | | | | |

0 20 40 60 80 100

T

Number of Chairs

Number of Tables

(T = 0, C = 100)

FIGURE 7.4 – Region that Satisfies the Painting and Varnishing Constraint

(T = 50, C = 0)

Copyright ©2015 Pearson Education, Inc.

7 – 22

Graphical Representation of Constraints

To produce tables and chairs, both departments must be used

Find a solution that satisfies both constraints simultaneously

A new graph shows both constraint plots

The feasible region is where all constraints are satisfied

Any point inside this region is a feasible solution

Any point outside the region is an infeasible solution

Homework is Completed By:

Writer Writer Name Amount Client Comments & Rating
Instant Homework Helper

ONLINE

Instant Homework Helper

$36

She helped me in last minute in a very reasonable price. She is a lifesaver, I got A+ grade in my homework, I will surely hire her again for my next assignments, Thumbs Up!

Order & Get This Solution Within 3 Hours in $25/Page

Custom Original Solution And Get A+ Grades

  • 100% Plagiarism Free
  • Proper APA/MLA/Harvard Referencing
  • Delivery in 3 Hours After Placing Order
  • Free Turnitin Report
  • Unlimited Revisions
  • Privacy Guaranteed

Order & Get This Solution Within 6 Hours in $20/Page

Custom Original Solution And Get A+ Grades

  • 100% Plagiarism Free
  • Proper APA/MLA/Harvard Referencing
  • Delivery in 6 Hours After Placing Order
  • Free Turnitin Report
  • Unlimited Revisions
  • Privacy Guaranteed

Order & Get This Solution Within 12 Hours in $15/Page

Custom Original Solution And Get A+ Grades

  • 100% Plagiarism Free
  • Proper APA/MLA/Harvard Referencing
  • Delivery in 12 Hours After Placing Order
  • Free Turnitin Report
  • Unlimited Revisions
  • Privacy Guaranteed

6 writers have sent their proposals to do this homework:

Math Guru
Engineering Help
Math Exam Success
Writing Factory
Financial Assignments
Top Class Results
Writer Writer Name Offer Chat
Math Guru

ONLINE

Math Guru

After reading your project details, I feel myself as the best option for you to fulfill this project with 100 percent perfection.

$50 Chat With Writer
Engineering Help

ONLINE

Engineering Help

I have read your project description carefully and you will get plagiarism free writing according to your requirements. Thank You

$62 Chat With Writer
Math Exam Success

ONLINE

Math Exam Success

I can assist you in plagiarism free writing as I have already done several related projects of writing. I have a master qualification with 5 years’ experience in; Essay Writing, Case Study Writing, Report Writing.

$60 Chat With Writer
Writing Factory

ONLINE

Writing Factory

I have written research reports, assignments, thesis, research proposals, and dissertations for different level students and on different subjects.

$80 Chat With Writer
Financial Assignments

ONLINE

Financial Assignments

I have read your project details and I can provide you QUALITY WORK within your given timeline and budget.

$55 Chat With Writer
Top Class Results

ONLINE

Top Class Results

This project is my strength and I can fulfill your requirements properly within your given deadline. I always give plagiarism-free work to my clients at very competitive prices.

$75 Chat With Writer

Let our expert academic writers to help you in achieving a+ grades in your homework, assignment, quiz or exam.

Similar Homework Questions

Find the greedy looters remains to the north - Rams mortgage corporation limited - Research and evidence in practice la trobe - Ap macroeconomics loanable funds market - Projet 1 - Life in america's small cities by gs thomas - Chapeltown swimming baths sheffield - 700- to 1,050 word HR report - Expressed arguments in the media - Beneath the surface gary crew online book - Which market structure is characterized by a few interdependent firms - 7/9 as a recurring decimal - Unit 4/28 greg chappell drive burleigh heads 4220 - A disk with an initial angular velocity - Lack of cohesiveness - Set forth the distinctions between the valid, void and voidable contract. - Configure ip acls to mitigate attacks - Assignment - Developing your theoretical orientation in counseling - Human r - Rpm formula for turning - Ramona the brave chapter 5 - Colin jensen brisbane city council - The grass is always greener on the other side essay - Marquis de sade philosophy in the bedroom pdf - A pizza chain is considering opening a new store - Relationship between gold and oil - The norton introduction to literature 13th edition table of contents - Structural steel inspection forms - Week 7 Discussion Attachment - International financial management questions and answers pdf - Discussions(SPORT AND SOCIETY) - 8discu1 - What is a stance essay - Why are fats soluble in petroleum ether - Discussion 1.2 - 177 waterfall gully road - Value line publishing home depot lowes - Gcu learning lounge - 4 elements of malpractice claim nursing - Formulating your brief data visualization - Head loss due to pipe fittings experiment - Blair atholl caravan park - Pharmacology week 1 discussion - Read Exercise 17-1 Preparing for an OSHA Inspection on pages 551 and 552. - Case-1 - Direct and partial variation - Watch and review video - 06 hilux horn relay location - Discussion 1 WK 3 SOCW 6443: The Benefits and Limitations of Antidepressant Medications - Introduction to sociology seagull 11th edition pdf - Rmit connect to wifi - According to renaissance philosophy commoners often represent - Organic chemistry 9th edition - Discussion / Answer 2 questions and comment on 2 students / 200~300 words for answers / need in 4 hours - Watch the execution of wanda jean - Property valuation report pdf - If i only had a brain phoenix university - That funny feeling bo burnham lyrics meaning - Is diane ademu john nigerian - Uow final exam timetable - Family functioning style scale - Macbeth by peter moffat - Length of oleic acid molecule - Vigo vacations has $200 million - Apply for deferred exam griffith - Wearable computers change how we work - Chi square table ap bio - Political Science Week 7 Assignment - Nursing Project - Continuing education for pastors - Connect 4 3 player - Park f&b oe pub las vegas nv - Tas e travel system - Assignment - Individual butter portions morrisons - How does the water move - Five little firefighters fingerplay - Earthing rio in slabs - Shadow health neurological assessment - Missionary stories about prayer - Psychological First Aid test - Analogy of the cave meaning - Supply chain diagram of kfc - What do archaeologists use - Lum head primary school - Blood test whiston hospital - Is seaweed living or nonliving - Sir robert peel 9 principles of policing - Week 5 death - Ubc 1997 building code volume 2 - Sea doo rxt 260 rs specifications - Unit 7 Discussion 2 - How to make an animated gif in photoshop cs6 - Chapter 12 - Avion av dr 1401 - Kiwi jeans grasshopper soldier - Staff performance evaluation checklist - Anne boykin and savina schoenhofer nursing as caring - WEEK 8 ASSIGNMENT 1 SUBMISSION