INTRODUCTION TO
MANAGEMENT
SCIENCE, 13e
Anderson
Sweeney
Williams
Martin
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2008 Thomson South-Western. All Rights Reserved
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Slide 1
Chapter 2
An Introduction to Linear Programming
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Linear Programming Problem
Problem Formulation
A Simple Maximization Problem
Graphical Solution Procedure
Extreme Points and the Optimal Solution
Computer Solutions
A Simple Minimization Problem
Special Cases
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©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 2
Linear Programming
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Linear programming has nothing to do with computer
programming.
The use of the word “programming” here means
“choosing a course of action.” or planning
Linear programming involves choosing a course of
action when the mathematical model of the problem
contains only linear functions.
Maximize or Minimize
subject to some constraints
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©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 3
Linear Programming (LP) Problem
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The maximization or minimization of some quantity is
the objective in all linear programming problems.
All LP problems have constraints that limit the degree
to which the objective can be pursued.
A feasible solution satisfies all the problem's
constraints.
An optimal solution is a feasible solution that results in
the largest possible objective function value when
maximizing (or smallest when minimizing).
A graphical solution method can be used to solve a
linear program with two variables.
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2008 Thomson South-Western. All Rights Reserved
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Slide 4
Linear Programming (LP) Problem
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Linearity Assumption
• Objective function and constraints are all linear
What is Linearity?
• Proportionality
• Additivity
• Divisibility
Practical Assumption
• Linearity
• Deterministic : parameter values are fixed
not stochastic
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2008 Thomson South-Western. All Rights Reserved
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Slide 5
Problem Formulation
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2008 Thomson South-Western. All Rights Reserved
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Slide 6
Guidelines for Model Formulation
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Understand the problem thoroughly.
Describe the objective.
Describe each constraint.
Define the decision variables.
Write the objective in terms of the decision variables.
Write the constraints in terms of the decision variables.
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 7
Example 1: Graphical Solution
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First Constraint Graphed
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2008 Thomson South-Western. All Rights Reserved
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Slide 8
Example 1: Graphical Solution
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Other Constraints
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 9
Example 1: Graphical Solution
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Combined-Constraint Graph Showing Feasible Region
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 10
Example 1: Graphical Solution
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Objective Function Lines
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3
4
5
6
7
8
9
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
10
Slide 11
Example 1: Graphical Solution
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Optimal Solution
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5
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7
8
9
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©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
10
Slide 12
Summary of the Graphical Solution Procedure
for Maximization Problems
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Prepare a graph of the feasible solutions for each of
the constraints.
Determine the feasible region that satisfies all the
constraints simultaneously.
Draw an objective function line.
Move parallel objective function lines toward larger
objective function values without entirely leaving the
feasible region.
Any feasible solution on the objective function line
with the largest value is an optimal solution.
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 13
Slack Variables
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2008 Thomson South-Western. All Rights Reserved
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Slide 14
Slack Variables (for < constraints)
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Example 1 in Standard Form
s1 , s2 , s3, and s4
are slack variables
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 15
Slack and Surplus Variables
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A linear program in which all the variables are nonnegative and all the constraints are equalities is said to
be in standard form.
Standard form is attained by adding slack variables to
"less than or equal to" constraints, and by subtracting
surplus variables from "greater than or equal to"
constraints.
Slack and surplus variables represent the difference
between the left and right sides of the constraints.
Slack and surplus variables have objective function
coefficients equal to 0.
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©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 16
Extreme Points and the Optimal Solution
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If
Max
5S  9 D
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©
2008 Thomson South-Western. All Rights Reserved
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Slide 17
Extreme Points and the Optimal Solution
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The corners or vertices of the feasible region are
referred to as the extreme points.
An optimal solution to an LP problem can be found at
an extreme point of the feasible region.
When looking for the optimal solution, you do not
have to evaluate all feasible solution points.
You have to consider only the extreme points of the
feasible region.
But, if the number of decision variables is 50
and the number of constraints is 100
the number of extreme points is
150 
150!
40
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
2.01287

10


50

 50!100!
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2008 Thomson South-Western. All Rights Reserved
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Slide 18
Computer Solutions
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Management Scientist
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2008 Thomson South-Western. All Rights Reserved
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Slide 19
Computer Solutions
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LP problems involving 1000s of variables and 1000s of
constraints are now routinely solved with computer
packages.
Linear programming solvers are now part of many
spreadsheet packages, such as Microsoft Excel.
Leading commercial packages include CPLEX, LINGO,
MOSEK, Xpress-MP, and Premium Solver for Excel.
Using Excel
Using LINGO
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2008 Thomson South-Western. All Rights Reserved
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Slide 20
Reduced Cost
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The reduced cost for a decision variable whose value is
0 in the optimal solution is:
the amount the variable's objective function
coefficient would have to improve (increase for
maximization problems, decrease for minimization
problems) before this variable could assume a
positive value.
The reduced cost for a decision variable whose value is
> 0 in the optimal solution is 0. (see. p.115)
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2008 Thomson South-Western. All Rights Reserved
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Slide 21
Example 2: A Simple Minimization Problem
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LP Formulation
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 22
Example 2: Graphical Solution
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Constraints Graphed
1
2
3
4
5
6
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 23
Example 2: Graphical Solution
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Optimal Solution
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 24
Summary of the Graphical Solution Procedure
for Minimization Problems
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Prepare a graph of the feasible solutions for each of
the constraints.
Determine the feasible region that satisfies all the
constraints simultaneously.
Draw an objective function line.
Move parallel objective function lines toward smaller
objective function values without entirely leaving the
feasible region.
Any feasible solution on the objective function line
with the smallest value is an optimal solution.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a
©
2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 25
Surplus Variables
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Example 2 in Standard Form
s1 and s2 are
surplus variables
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2008 Thomson South-Western. All Rights Reserved
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Slide 26
Example 2: Spreadsheet Solution
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Interpretation of Computer Output
 Excel output
 LINGO output
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2008 Thomson South-Western. All Rights Reserved
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Slide 27
Feasible Region
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The feasible region for a two-variable LP problem can
be nonexistent, a single point, a line, a polygon, or an
unbounded area.
Any linear program falls in one of four categories:
• is infeasible
• has a unique optimal solution
• has alternative optimal solutions
• has an objective function that can be increased
without bound
A feasible region may be unbounded and yet there may
be optimal solutions. This is common in minimization
problems and is possible in maximization problems.
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 28
Special Cases
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Alternative Optimal Solutions
In the graphical method, if the objective function line
is parallel to a boundary constraint in the direction of
optimization, there are alternate optimal solutions,
with all points on this line segment being optimal.
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 29
Special Cases
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Infeasibility
• No solution to the LP problem satisfies all the
constraints, including the non-negativity
conditions.
• Graphically, this means a feasible region does not
exist.
• Causes include:
• A formulation error has been made.
• Management’s expectations are too high.
• Too many restrictions have been placed on the
problem (i.e. the problem is over-constrained).
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 30
Example: Infeasible Problem
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Consider the following LP problem.
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Slide 31
Example: Infeasible Problem
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Slide 32
Example: Unbounded Solution
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Max
s.t.
Consider the following LP problem.
20X + 10Y
X
> 2
Y > 5
X, Y > 0
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2008 Thomson South-Western. All Rights Reserved
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Slide 33
Special Cases
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Unbounded
• The solution to a maximization LP problem is
unbounded if the value of the solution may be
made indefinitely large without violating any of
the constraints.
• For real problems, this is the result of improper
formulation. (Quite likely, a constraint has been
inadvertently omitted.)
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 34
End of Chapter 2
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2008 Thomson South-Western. All Rights Reserved
publicly accessible website, in whole or in part.
Slide 35