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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 2 linear programming models graphical and computer methods

Chapter 2Linear 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
Steps in Developing a Linear Programming (LP) Model
  • Formulation
  • Solution
  • Interpretation and Sensitivity Analysis
properties of lp models
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
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
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
Flair Furniture Co. Data
  • Other Limitations:
    • Make no more than 450 chairs
    • Make at least 100 tables
slide7
Decision Variables:

T = Num. of tables to make

C = Num. of chairs to make

Objective Function: Maximize Profit

Maximize $7 T + $5 C

constraints
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)

slide9
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
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
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

add a new constraint
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

revised model for flair furniture
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
End of Chapter 2
  • No Graphical Solution will be discussed
graphical solution
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.
slide19

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

slide20

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

slide21

C

1000

600

450

0

Max Chair Line

C = 450

Min Table Line

T = 100

Feasible

Region

0 100 500 800 T

slide22

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

slide23

C

500

400

300

200

100

0

Additional Constraint

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
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
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 lp26
Special Situation in LP
  • Infeasibility – when no feasible solution exists (there is no feasible region)

Example: x < 10

x > 15

special situation in lp27
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 lp28
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