Chapter 2 Linear Programming Models: Graphical and Computer Methods

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# Chapter 2 Linear Programming Models: Graphical and Computer Methods - PowerPoint PPT Presentation

Chapter 2 Linear Programming Models: Graphical and Computer Methods. Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA 99223 chen@jepson.gonzaga.edu. Steps in Developing a Linear Programming (LP) Model. Formulation Solution

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## Chapter 2 Linear Programming Models: Graphical and Computer Methods

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### Chapter 2Linear Programming Models:Graphical and Computer Methods

Jason C. H. Chen, Ph.D.

Professor of MIS

Gonzaga University

Spokane, WA 99223

chen@jepson.gonzaga.edu

Steps in Developing a Linear Programming (LP) Model
• Formulation
• Solution
• Interpretation and Sensitivity Analysis
Properties of LP Models
• Seek to minimize or maximize
• Include “constraints” or limitations
• There must be alternatives available
• All equations are linear
Example LP Model Formulation:The Product Mix Problem

Decision: How much to make of > 2 products?

Objective: Maximize profit

Constraints: Limited resources

Example: Flair Furniture Co.

Two products: Chairs and Tables

Decision: How many of each to make this month?

Objective: Maximize profit

Flair Furniture Co. Data
• Other Limitations:
• Make no more than 450 chairs
• Make at least 100 tables
Decision Variables:

T = Num. of tables to make

C = Num. of chairs to make

Objective Function: Maximize Profit

Maximize \$7 T + \$5 C

Constraints:
• Have 2400 hours of carpentry time available

3 T + 4 C < 2400 (hours)

• Have 1000 hours of painting time available

2 T + 1 C < 1000(hours)

More Constraints:
• Make no more than 450 chairs

C < 450 (num. chairs)

• Make at least 100 tables

T > 100 (num. tables)

Nonnegativity:

Cannot make a negative number of chairs or tables

T > 0

C > 0

Model Summary

Max 7T + 5C (profit)

Subject to the constraints:

3T + 4C < 2400 (carpentry hrs)

2T + 1C < 1000 (painting hrs)

C < 450 (max # chairs)

T > 100 (min # tables)

T, C > 0 (nonnegativity)

Using Excel’s Solver for LP

Recall the Flair Furniture Example:

Max 7T + 5C (profit)

Subject to the constraints:

3T + 4C < 2400 (carpentry hrs)

2T + 1C < 1000 (painting hrs)

C < 450 (max # chairs)

T > 100 (min # tables)

T, C > 0 (nonnegativity)

Go to file 2-1.xls

• A new constraint specified by the marketing department.
• Specifically, they needed to ensure theat the number of chairs made this month is at least 75 more than the number of tables made. The constraint is expressed as:

C - T > 75

Revised Model for Flair Furniture

Max 7T + 5C (profit)

Subject to the constraints:

3T + 4C < 2400 (carpentry hrs)

2T + 1C < 1000 (painting hrs)

C < 450 (max # chairs)

T > 100 (min # tables)

- 1T + 1C > 75

T, C > 0 (nonnegativity)

Go to file 2-2.xls

End of Chapter 2
• No Graphical Solution will be discussed
Graphical Solution
• Graphing an LP model helps provide insight into LP models and their solutions.
• While this can only be done in two dimensions, the same properties apply to all LP models and solutions.

C

600

0

Carpentry

Constraint Line

3T + 4C = 2400

Intercepts

(T = 0, C = 600)

(T = 800, C = 0)

Infeasible

> 2400 hrs

3T + 4C = 2400

Feasible

< 2400 hrs

0 800 T

C

1000

600

0

Painting

Constraint Line

2T + 1C = 1000

Intercepts

(T = 0, C = 1000)

(T = 500, C = 0)

2T + 1C = 1000

0 500 800 T

C

1000

600

450

0

Max Chair Line

C = 450

Min Table Line

T = 100

Feasible

Region

0 100 500 800 T

C

500

400

300

200

100

0

Objective Function Line

7T + 5C = Profit

7T + 5C = \$4,040

Optimal Point

(T = 320, C = 360)

7T + 5C = \$2,800

7T + 5C = \$2,100

0 100 200 300 400 500 T

C

500

400

300

200

100

0

Need at least 75 more chairs than tables

C > T + 75

Or

C – T > 75

New optimal point

T = 300, C = 375

T = 320

C = 360

No longer feasible

0 100 200 300 400 500 T

LP Characteristics
• Feasible Region: The set of points that satisfies all constraints
• Corner Point Property: An optimal solution must lie at one or more corner points
• Optimal Solution: The corner point with the best objective function value is optimal
Special Situation in LP
• Redundant Constraints - do not affect the feasible region

Example: x < 10

x < 12

The second constraint is redundant because it is less restrictive.

Special Situation in LP
• Infeasibility – when no feasible solution exists (there is no feasible region)

Example: x < 10

x > 15

Special Situation in LP
• Alternate Optimal Solutions – when there is more than one optimal solution

C

10

6

0

Max 2T + 2C

Subject to:

T + C < 10

T < 5

C < 6

T, C > 0

All points on Red segment are optimal

2T + 2C = 20

0 5 10 T

Special Situation in LP
• Unbounded Solutions – when nothing prevents the solution from becoming infinitely large

C

2

1

0

Direction of solution

Max 2T + 2C

Subject to:

2T + 3C > 6

T, C > 0

0 1 2 3 T