Introduction to
Management Science
with Spreadsheets
Stevenson and Ozgur
First Edition
Part 2 Introduction to Management Science and Forecasting
Chapter 3
Linear Programming:
Basic Concepts and
Graphical Solution
McGraw-Hill/Irwin
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Learning Objectives
After completing this chapter, you should be able to:
1. Explain what is meant by the terms constrained
optimization and linear programming.
2. List the components and the assumptions of linear
programming and briefly explain each.
3. Name and describe at least three successful
applications of linear programming.
4. Identify the type of problems that can be solved
using linear programming.
5. Formulate simple linear programming models.
6. Identify LP problems that are amenable to graphical
solutions.
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Learning Objectives (cont’d)
After completing this chapter, you should be able to:
7. Explain these terms: optimal solution, feasible
solution space, corner point, redundant constraint
slack, and surplus.
8. Solve two-variable LP problems graphically and
interpret your answers.
9. Identify problems that have multiple solutions,
problems that have no feasible solutions,
unbounded problems, and problems with redundant
constraints.
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Decisions and Linear Programming
• Constrained optimization
–Finding the optimal solution to a problem given that
certain constraints must be satisfied by the solution.
–A form of decision making that involves situations in
which the set of acceptable solutions is somehow
restricted.
–Recognizes scarcity—the limitations on the availability
of physical and human resources.
–Seeks solutions that are both efficient and feasible in
the allocation of resources.
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Linear Programming
• Linear Programming (LP)
–A family of mathematical techniques (algorithms) that
can be used for constrained optimization problems
with linear relationships.
• Graphical method
• Simplex method
• Karmakar’s method
–The problems must involve a single objective, a linear
objective function, and linear constraints and have
known and constant numerical values.
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Example 3–1
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Table 3–1
Successful Applications of Linear Programming Published
in Interfaces
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Table 3–2
Characteristics of LP Models
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Formulating LP Models
• Formulating linear programming models
involves the following steps:
1. Define the decision variables.
2. Determine the objective function.
3. Identify the constraints.
4. Determine appropriate values for parameters and
determine whether an upper limit, lower limit, or
equality is called for.
5. Use this information to build a model.
6. Validate the model.
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Example 3–2
x1 = quantity of server model 1 to produce
x2 = quantity of server model 2 to produce
maximize Z = 60x1+50x2
Subject to:
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Graphing the Model
• This method can be used only to solve problems
that involve two decision variables.
• The graphical approach:
1. Plot each of the constraints.
2. Determine the region or area that contains all of the
points that satisfy the entire set of constraints.
3. Determine the optimal solution.
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Key Terms in Graphing
• Optimal solution
• Feasible solution space
• Corner point
• Redundant constraint
• Slack
• Surplus
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Figure 3–1
A Graph Showing the Nonnegativity Constraints
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Figure 3–2
Feasible Region Based on a Plot of the First Constraint
(assembly time) and the Nonnegativity Constraint
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Figure 3–3
A Completed Graph of the Server Problem Showing the
Assembly and Inspection Constraints and the Feasible
Solution Space
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Figure 3–4
Completed Graph of the Server Problem Showing All of the
Constraints and the Feasible Solution Space
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Finding the Optimal Solution
• The extreme point approach
–Involves finding the coordinates of each corner point
that borders the feasible solution space and then
determining which corner point provides the best
value of the objective function.
–The extreme point theorem
–If a problem has an optimal solution at least one
optimal solution will occur at a corner point of the
feasible solution space.
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The Extreme Point Approach
1. Graph the problem and identify the feasible solution
space.
2. Determine the values of the decision variables at each
corner point of the feasible solution space.
3. Substitute the values of the decision variables at each
corner point into the objective function to obtain its
value at each corner point.
4. After all corner points have been evaluated in a similar
fashion, select the one with the highest value of the
objective function (for a maximization problem) or
lowest value (for a minimization problem) as the
optimal solution.
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Figure 3–5
Graph of Server Problem with Extreme Points of the Feasible
Solution Space Indicated
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Table 3–3
Extreme Point Solutions for the Server Problem
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The Objective Function
(Iso-Profit Line) Approach
• This approach directly identifies the optimal
corner point, so only the coordinates of the
optimal point need to be determined.
–Accomplishes this by adding the objective function to
the graph and then using it to determine which point is
optimal.
–Avoids the need to determine the coordinates of all of
the corner points of the feasible solution space.
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Figure 3–6
The Server Problem with Profit Lines of $300, $600, and $900
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Figure 3–7
Finding the Optimal Solution to the Server Problem
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Graphing—Objective Function Approach
1. Graph the constraints.
2. Identify the feasible solution space.
3. Set the objective function equal to some amount that is
divisible by each of the objective function coefficients.
4. After identifying the optimal point, determine which two
constraints intersect there.
5. Substitute the values obtained in the previous step into
the objective function to determine the value of the
objective function at the optimum.
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Figure 3–8
A Comparison of Maximization and Minimization Problems
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Example 3-3
Minimization
Determine the values of decision variables x1 and x2 that will
yield the minimum cost in the following problem. Solve using
the objective function approach.
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Figure 3–9
Graphing the Feasible Region and Using the Objective
Function to Find the Optimum for Example 3-3
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Example 3-4
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Table 3–3
Summary of Extreme Point Analysis for Example 3-4
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Table 3–5
Computing the Amount of Slack for the Optimal Solution to
the Server Problem
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Some Special Issues
• No Feasible Solutions
– Occurs in problems where to satisfy one of the constraints,
another constraint must be violated.
• Unbounded Problems
– Exists when the value of the objective function can be increased
without limit.
• Redundant Constraints
– A constraint that does not form a unique boundary of the feasible
solution space; its removal would not alter the feasible solution
space.
• Multiple Optimal Solutions
– Problems in which different combinations of values of the
decision variables yield the same optimal value.
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Figure 3–10
Infeasible Solution: No Combination of x1 and x2, Can
Simultaneously Satisfy Both Constraints
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Figure 3–11
An Unbounded Solution Space
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Figure 3–12
Examples of Redundant Constraints
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Figure 3–13
Multiple Optimal Solutions
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Figure 3–14
Constraints and Feasible Solution Space for
Solved Problem 2
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Figure 3–15
A Graph for Solved Problem 3
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Figure 3–16
Graph for Solved Problem 4
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