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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