To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Chapter 7Chapter 7

Linear Programming Linear Programming Models: Graphical Models: Graphical

and Computer and Computer MethodsMethods

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Learning ObjectivesLearning Objectives

Students will be able to:• Understand the basic

assumptions and properties of linear programming (LP).

• Formulate small to moderate-sized LP problems.

• Graphically solve any LP problem with two variables by both the corner point and isoline methods.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Learning Objectives - Learning Objectives - continuedcontinued

• Understand special issues in LP

- infeasibility, unboundedness,

redundancy, and alternative

optima.

• Understand the role of

sensitivity analysis.

• Use Excel spreadsheets to solve

LP problems.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Chapter OutlineChapter Outline

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 and

Excel

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Chapter Outline - Chapter Outline - continuedcontinued

7.6 Solving Minimization Problems

7.7 Four Special Cases

7.8 Sensitivity Analysis in LP

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Examples of Successful Examples of Successful LP ApplicationsLP Applications

1. Development of a production schedule

that will satisfy future demands for a

firm’s production and at the same time

minimize total production and

inventory costs

2. Selection of the product mix in a

factory to make best use of machine-

hours and labor-hours available while

maximizing the firm’s products

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Examples of Successful Examples of Successful LP ApplicationsLP Applications

3. Determination of grades of petroleum

products to yield the maximum profit

4. Selection of different blends of raw

materials to feed mills to produce finished

feed combinations at minimum cost

5. Determination of a distribution system

that will minimize total shipping cost

from several warehouses to various

market locations

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Requirements of a Linear Requirements of a Linear Programming ProblemProgramming Problem

• All problems seek to maximize or minimize some quantity (the objective function).

• The presence of restrictions or constraints, limits the degree to which we can pursue our objective.

• There must be alternative courses of action to choose from.

• The objective and constraints in linear programming problems must be expressed in terms of linear equations or inequalities.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Basic Assumptions of Basic Assumptions of Linear ProgrammingLinear Programming

• Certainty

• Proportionality

• Additivity

• Divisibility

• Nonnegativity

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Flair Furniture Company Data - Data - Table 7.1Table 7.1

Hours Required to Produce One Unit

DepartmentT

TablesC

Chairs

AvailableHours This

Week

CarpentryPainting &Varnishing

42

31

240100

Profit Amount $7 $5

Constraints: 4T + 3C 240 (Carpentry)

2T + 1C 100 (Paint & Varnishing)

Objective: Max: 7T + 5C

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Flair Furniture Company ConstraintsConstraints

Number of Tables

120

100

80

60

40

20

0

Num

ber

of C

hair

s

20 40 60 80 100

Painting/Varnishing

Carpentry

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Flair Furniture Company Feasible RegionFeasible Region

120

100

80

60

40

20

0

Num

ber

of C

hair

s

20 40 60 80 100

Number of Tables

Painting/Varnishing

Carpentry

FeasibleRegion

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Flair Furniture Company Isoprofit LinesIsoprofit Lines

Number of Tables

Num

ber

of C

hair

s

120

100

80

60

40

20

020 40 60 80 100

Painting/Varnishing

Carpentry

7T + 5C = 210

7T + 5C = 420

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Flair Furniture Company Optimal SolutionOptimal Solution

Num

ber

of C

hair

s

120

100

80

60

40

20

020 40 60 80 100

Number of Tables

Painting/Varnishing

Carpentry

Solution(T = 30, C = 40)

Isoprofit LinesIsoprofit Lines

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture Company Flair Furniture Company Optimal SolutionOptimal Solution

Num

ber

of C

hair

s

120

100

80

60

40

20

020 40 60 80 100

Number of Tables

Painting/Varnishing

Carpentry

Solution(T = 30, C = 40)

Corner PointsCorner Points

1

2

3

4

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture - QM Flair Furniture - QM for Windowsfor Windows

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Flair Furniture - ExcelFlair Furniture - Excel

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Holiday Meal Turkey Holiday Meal Turkey RanchRanch

(C)

(B)

toSubject

:Minimize

½ X

XX

A)(XX:

XX

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Holiday Meal Turkey Holiday Meal Turkey Problem Problem

Corner PointsCorner Points

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-20 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Holiday Meal Turkey Holiday Meal Turkey Problem Problem

Isoprofit LinesIsoprofit Lines

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Special Cases in LPSpecial Cases in LP

• Infeasibility

• Unbounded Solutions

• Redundancy

• Degeneracy

• More Than One Optimal

Solution

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

A Problem with No A Problem with No Feasible SolutionFeasible Solution

X2

X1

8

6

4

2

0

2 4 6 8

Region Satisfying3rd Constraint

Region Satisfying First 2 Constraints

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

A Solution Region That is A Solution Region That is Unbounded to the RightUnbounded to the Right

X2

X1

15

10

5

0

5 10 15

Feasible Region

X1 > 5X2 < 10

X1 + 2X2 > 10

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-24 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

A Problem with a A Problem with a Redundant ConstraintRedundant ConstraintX2

X1

30

25

20

15

10

5

0

5 10 15 20 25 30

Feasible Region

2X1 + X2 < 30

X1 < 25

X1 + X2 < 20

RedundantConstraint

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-25 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

An Example of Alternate An Example of Alternate Optimal SolutionsOptimal Solutions

8

7

6

5

4

3

2

1

01 2 3 4 5 6 7 8

Optimal Solution Consists of All Combinations of X1 and X2 Along the AB Segment

Isoprofit Line for $12Overlays Line Segment

Isoprofit Line for $8A

B

AB

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-26 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Sensitivity AnalysisSensitivity Analysis

• Changes in the Objective

Function Coefficient

• Changes in Resources (RHS)

• Changes in Technological

Coefficients

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-27 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Changes in the Technological Changes in the Technological Coefficients for High Note Coefficients for High Note

Sound Co.Sound Co.

Ste

reo

Rec

eive

rs

X1

60

40

20

0

CD Players

20 40

X2

(a) Original Problem

3X1 + 1X2 < 60

Optimal Solution

a2X1 + 4X2 < 80

b

c

X2

(b) Change in CircledCoefficient

Still Optimal

a2X1 + 4X2 < 80

d

e

2X1 + 1X2 < 60

20 40 X130

CD Players

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

7-28 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Changes in the Technological Changes in the Technological Coefficients for High Note Coefficients for High Note

Sound Co.Sound Co.

X1

Ste

reo

Rec

eive

rs

60

40

20

0

CD Players

20 40

X2

(a) Original Problem

3X1 + 1X2 < 60

Optimal Solution

a2X1 + 4X2 < 80

b

c

20 40

X2

X1

(c) Change in CircledCoefficient

3X1 + 1X2 < 60

Optimal Solution

f

2X1 + 5X2 < 80

g

c

CD Players