Lecture Introduction to Management Science with Spreadsheets: Chapter 3 - Stevenson, Ozgur

Chapter 3 Linear programming: basic concepts and graphical solution, after completing this chapter, you should be able to: Explain what is meant by the terms constrained optimization and linear programming, list the components and the assumptions of linear programming and briefly explain each, name and describe at least three successful applications of linear programming,...

Stevenson and Ozgur

First Edition

Introduction to Management Science

Part 2 Introduction to Management Science and Forecasting

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Learning Objectives

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

After completing this chapter, you should be able to:

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Learning Objectives (cont’d)

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

After completing this chapter, you should be able to:

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Decisions and Linear Programming

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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Example 3–1

Example 3–1

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Table 3–1 Successful Applications of Linear Programming Published

Table 3–1 Successful Applications of Linear Programming Published

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Formulating LP Models

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

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

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

Key Terms in Graphing

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Figure 3–1 A Graph Showing the Nonnegativity Constraints

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 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

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 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

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

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 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

Table 3–3 Extreme Point Solutions for the Server Problem

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The Objective Function (Iso-Profit Line) Approach

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 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

Figure 3–7 Finding the Optimal Solution to the Server Problem

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Graphing—Objective Function Approach

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 Figure 3–8 A Comparison of Maximization and Minimization Problems

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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 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

Example 3-4

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Table 3–3 Summary of Extreme Point Analysis for Example 3-4

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 Table 3–5 Computing the Amount of Slack for the Optimal Solution to

the Server Problem

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Some Special Issues

Some Special Issues

• No Feasible Solutions

– Occurs in problems where to satisfy one of the constraints,

another constraint must be violated.

• 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 Figure 3–10 Infeasible Solution: No Combination of x1 and x2, Can

Simultaneously Satisfy Both Constraints

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Figure 3–12 Examples of Redundant Constraints

Figure 3–12 Examples of Redundant Constraints

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Figure 3–13 Multiple Optimal Solutions

Figure 3–13 Multiple Optimal Solutions

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Figure 3–14 Constraints and Feasible Solution Space for

Solved Problem 2 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

Figure 3–15 A Graph for Solved Problem 3

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Figure 3–16 Graph for Solved Problem 4

Figure 3–16 Graph for Solved Problem 4