Chapter 2Linear Programming Models:Graphical and Computer Methods

Jason C. H. Chen, Ph.D.Professor of MIS

School of Business AdministrationGonzaga UniversitySpokane, WA 99223

chen@jepson.gonzaga.edu

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Steps in Developing a Linear Programming (LP) Model

1) Formulation

2) Solution

3) Interpretation and Sensitivity Analysis

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Properties of LP Models

1) Seek to minimize or maximize

2) Include “constraints” or limitations

3) There must be alternatives available

4) All equations are linear

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Example LP Model Formulation:The Product Mix Problem

Decision: How much to make of > 2 products?

Objective: Maximize profit

Constraints: Limited resources

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Example: Flair Furniture Co.

Two products: Chairs and Tables

Decision: How many of each to make this month?

Objective: Maximize profit

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Flair Furniture Co. DataTables

(per table)

Chairs(per chair)

Hours Available

Profit Contribution

$7 $5

Carpentry 3 hrs 4 hrs 2400

Painting 2 hrs 1 hr 1000

Other Limitations:• Make no more than 450 chairs• Make at least 100 tables

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Decision Variables:

T = Num. of tables to make

C = Num. of chairs to make

Objective Function: Maximize Profit

Maximize $7 T + $5 C

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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)

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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 > 0C > 0

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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)

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Using Excel’s Solver for LPRecall 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

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Flair Furniture

T C

  Tables Chairs

Number of units      

Profit 7 5 =SUMPRODUCT(B6:C6,$B$5:$C$5)

Constraints:      

Carpentry hours 3 4 =SUMPRODUCT(B8:C8,$B$5:$C$5) <= 2400

Painting hours 2 1 =SUMPRODUCT(B9:C9,$B$5:$C$5) <= 1000

Maximum chairs   1 =SUMPRODUCT(B10:C10,$B$5:$C$5) <= 450

Minimum tables 1   =SUMPRODUCT(B11:C11,$B$5:$C$5) >= 100

      LHS Sign RHS

Flair Furniture

T C

  Tables Chairs

Number of units 320.0 360.0  

Profit $7 $5 $4,040.00

Constraints:      

Carpentry hours 3 4 2400.0 <= 2400

Painting hours 2 1 1000.0 <= 1000

Maximum chairs   1 360.0 <= 450

Minimum tables 1   320.0 >= 100

      LHS Sign RHS

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Microsoft Excel 10.0 Answer Report

Worksheet: [2-1.xls]Flair Furniture

   

Target Cell (Max)  

  Cell Name Original Value Final Value  

  $D$6 Profit $0.00 $4,040.00  

   

Adjustable Cells  

  Cell Name Original Value Final Value  

  $B$5 Number of units Tables 0.0 320.0  

  $C$5 Number of units Chairs 0.0 360.0  

   

Constraints  

  Cell Name Cell Value Formula Status Slack

  $D$8 Carpentry hours 2400.0 $D$8<=$F$8 Binding 0.0

  $D$9 Painting hours 1000.0 $D$9<=$F$9 Binding 0.0

  $D$10 Maximum chairs 360.0 $D$10<=$F$10 Not Binding 90.0

  $D$11 Minimum tables 320.0 $D$11>=$F$11 Not Binding 220.0

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Add a new constraint

• 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

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Revised Model for Flair FurnitureMax 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

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Flair Furniture - Modified Problem

T C

  Tables Chairs

Number of units 300.0 375.0  

Profit $7 $5 $3,975.00

Constraints:      

Carpentry hours 3 4 2400.0 <= 2400

Painting hours 2 1 975.0 <= 1000

Maximum chairs 0 1 375.0 <= 450

Minimum tables 1 0 300.0 >= 100

Tables vs Chairs -1 1 75.0 >= 75

      LHS Sign RHSFlair Furniture

T C

  Tables Chairs

Number of units 320.0 360.0  

Profit $7 $5 $4,040.00

Constraints:      

Carpentry hours 3 4 2400.0 <= 2400

Painting hours 2 1 1000.0 <= 1000

Maximum chairs   1 360.0 <= 450

Minimum tables 1   320.0 >= 100

      LHS Sign RHS

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End of Chapter 2

• No Graphical Solution will be discussed

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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.

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Carpentry

Constraint Line

3T + 4C = 2400

Intercepts

(T = 0, C = 600)

(T = 800, C = 0)

0 800 T

C

600

0

Feasible

< 2400 hrs

Infeasible

> 2400 hrs

3T + 4C = 2400

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Painting

Constraint Line

2T + 1C = 1000

Intercepts

(T = 0, C = 1000)

(T = 500, C = 0)

0 500 800 T

C1000

600

0

2T + 1C = 1000

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0 100 500 800 T

C1000

600

450

0

Max Chair Line

C = 450

Min Table Line

T = 100

Feasible

Region

Dr. Chen, Decision Support Systems 220 100 200 300 400 500 T

C

500

400

300

200

100

0

Objective Function Line

7T + 5C = Profit

7T + 5C = $2,100

7T + 5C = $4,040

Optimal Point(T = 320, C = 360)7T + 5C

= $2,800

Dr. Chen, Decision Support Systems 230 100 200 300 400 500 T

C

500

400

300

200

100

0

Additional Constraint

Need at least 75 more chairs than tables

C > T + 75

Or

C – T > 75

T = 320C = 360

No longer feasible

New optimal pointT = 300, C = 375

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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

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Special Situation in LP

1. Redundant Constraints - do not affect the feasible region

Example: x < 10

x < 12

The second constraint is redundant because it is less restrictive.

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Special Situation in LP

2. Infeasibility – when no feasible solution exists (there is no feasible region)

Example: x < 10

x > 15

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Special Situation in LP

3. Alternate Optimal Solutions – when there is more than one optimal solution

Max 2T + 2CSubject to:

T + C < 10T < 5 C < 6

T, C > 0

0 5 10 T

C

10

6

0

2T + 2C = 20All points on Red segment are optimal

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Special Situation in LP

4. Unbounded Solutions – when nothing prevents the solution from becoming infinitely large

Max 2T + 2CSubject to: 2T + 3C > 6

T, C > 0

0 1 2 3 T

C

2

1

0

Directi

on

of solutio

n