Chapter 7
Linear Programming
Models: Graphical
and Computer
Methods
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
7-1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to:
• Understand the basic
assumptions and properties of
linear programming (LP).
• Formulate small to moderatesized 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
7-2
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives continued
• 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
7-3
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Chapter 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
7-4
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Chapter Outline continued
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
7-5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Examples of Successful
LP 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 machinehours and labor-hours available while
maximizing the firm’s products
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
LP 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
7-7
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Requirements of a Linear
Programming 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
7-8
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Basic Assumptions of
Linear Programming
• Certainty
• Proportionality
• Additivity
• Divisibility
• Nonnegativity
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
7-9
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Flair Furniture Company
Data - Table 7.1
Hours Required to Produce One Unit
T
C
Tables Chairs
Department
Carpentry
Painting
&Varnishing
4
2
3
1
Profit Amount
$7
$5
Available
Hours This
Week
240
100
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
7-10
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Flair Furniture Company
Constraints
120
Number of Chairs
100
Painting/Varnishing
80
60
40
Carpentry
20
0
20
40
60
80
100
Number of Tables
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
Feasible Region
Number of Chairs
120
Painting/Varnishing
100
80
60
Carpentry
40
20
Feasible
Region
0
20
40
60
80
100
Number of Tables
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
Isoprofit Lines
120
Number of Chairs
100
Painting/Varnishing
80
7T + 5C = 210
60
7T + 5C = 420
40
Carpentry
20
0
20
40
60
80
100
Number of Tables
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
Optimal Solution
Isoprofit Lines
120
Number of Chairs
100
Painting/Varnishing
80
60
Solution
(T = 30, C = 40)
40
Carpentry
20
0
20
40
60
80
100
Number of Tables
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
Optimal Solution
120
Corner Points
2
Number of Chairs
100
Painting/Varnishing
80
60
Solution
(T = 30, C = 40)
40
Carpentry
3
20
1
0
20
4 60
80
Number of Tables
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
40
7-15
100
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Flair Furniture - QM
for Windows
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 - Excel
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
7-17
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Holiday Meal Turkey
Ranch
Minimize: 2 X1 + 3 X 2
Subject to : 5 X1 + 10 X 2  90
( A)
4 X1 + 3 X 2  48
(B)
11/2 (C)
½ X1
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
Problem
Corner Points
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
Problem
Isoprofit Lines
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
7-20
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Special Cases in LP
• Infeasibility
• Unbounded Solutions
• Redundancy
• Degeneracy
• More Than One Optimal
Solution
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
7-21
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
A Problem with No
Feasible Solution
X2
8
6
Region Satisfying
3rd Constraint
4
2
0
2
4
6
8
X1
Region Satisfying First 2 Constraints
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
7-22
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
A Solution Region That is
Unbounded to the Right
X2
15
X1 > 5
X2 < 10
10
Feasible Region
5
X1 + 2X2 > 10
0
5
10
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
15
7-23
X1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
A Problem with a
Redundant Constraint
X2
30
25
Redundant
Constraint
2X1 + X2 < 30
20
X1 < 25
15
10
X1 + X2 < 20
5
Feasible
Region
0
X1
5
10
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
15
20
7-24
25
30
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
An Example of Alternate
Optimal Solutions
8
Optimal Solution Consists of
All Combinations of X1 and
X2 Along the AB Segment
7
6
5
A
Isoprofit Line for $8
4
Isoprofit Line
for $12
Overlays Line
Segment
3
B
2
AB
1
0
1
2
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
3
4
7-25
5
6
7
8
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Sensitivity 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
7-26
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Changes in the Technological
Coefficients for High Note
Sound Co.
(a) Original Problem
X2
(b) Change in Circled
X2
Coefficient
60
2X1 + 1X2 < 60
3X1 + 1X2 < 60
Stereo Receivers
40
20
Optimal Solution
0 a
Still Optimal
a
2X1 + 4X2 < 80
b
2X1 + 4X2 < 80
d
c
20
e
40 X1
20
CD Players
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
30
40
CD Players
7-27
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
X1
Changes in the Technological
Coefficients for High Note
Sound Co.
(a) Original Problem
X2
(c) Change in Circled
X2
Coefficient
60
3X1 + 1X2 < 60
3X1 + 1X2 < 60
Optimal Solution
Optimal Solution
Stereo Receivers
40
20
0 a
2X1 + 4X2 < 80
b
2X1 + 5X2 < 80
f
g
c
20
c
40
X1
CD Players
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
20
40
CD Players
7-28
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
X1