Chapter 2 Linear Programming Models: Graphical and Computer Methods

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Chapter 2Linear Programming ModelsGraphical
and Computer Methods

• Jason C. H. Chen, Ph.D.
• Professor of MIS
• School of Business Administration
• Gonzaga University
• Spokane, WA 99223
• chen_at_jepson.gonzaga.edu

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Steps in Developing a Linear Programming (LP)
Model

• Formulation
• Solution
• Interpretation and Sensitivity Analysis

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Properties of LP Models

• Seek to minimize or maximize
• Include constraints or limitations
• There must be alternatives available
• All equations are linear

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Example LP Model FormulationThe Product Mix
Problem

• Decision How much to make of gt 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. Data

• 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 lt 2400 (hours)
• Have 1000 hours of painting time available
• 2 T 1 C lt 1000 (hours)

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• More Constraints
• Make no more than 450 chairs
• C lt 450 (num. chairs)
• Make at least 100 tables
• T gt 100 (num. tables)
• Nonnegativity
• Cannot make a negative number of chairs or tables
• T gt 0
• C gt 0

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Model Summary

• Max 7T 5C (profit)
• Subject to the constraints
• 3T 4C lt 2400 (carpentry hrs)
• 2T 1C lt 1000 (painting hrs)
• C lt 450 (max chairs)
• T gt 100 (min tables)
• T, C gt 0 (nonnegativity)

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Using Excels Solver for LP

• Recall the Flair Furniture Example
• Max 7T 5C (profit)
• Subject to the constraints
• 3T 4C lt 2400 (carpentry hrs)
• 2T 1C lt 1000 (painting hrs)
• C lt 450 (max chairs)
• T gt 100 (min tables)
• T, C gt 0 (nonnegativity)
• Go to file 2-1.xls

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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 gt 75
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Revised Model for Flair Furniture

• Max 7T 5C (profit)
• Subject to the constraints
• 3T 4C lt 2400 (carpentry hrs)
• 2T 1C lt 1000 (painting hrs)
• C lt 450 (max chairs)
• T gt 100 (min tables)
• - 1T 1C gt 75
• T, C gt 0 (nonnegativity)
• Go to file 2-2.xls

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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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C 600 0
Carpentry Constraint Line 3T 4C
2400 Intercepts (T 0, C 600) (T 800, C 0)
Infeasible gt 2400 hrs
3T 4C 2400
Feasible lt 2400 hrs
0 800 T
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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
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C 1000 600 450 0
Max Chair Line C 450 Min Table Line T 100
Feasible Region
0 100 500 800 T
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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
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C 500 400 300 200 100 0
Additional Constraint Need at least 75 more
chairs than tables C gt T 75 Or C T gt 75
New optimal point T 300, C 375
T 320 C 360 No longer feasible
0 100 200 300 400
500 T
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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

• Redundant Constraints - do not affect the
  feasible region
• Example x lt 10
• x lt 12
• The second constraint is redundant because it is
  less restrictive.

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Special Situation in LP

• Infeasibility when no feasible solution exists
  (there is no feasible region)
• Example x lt 10
• x gt 15

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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 lt 10 T lt
5 C lt 6 T, C gt 0
All points on Red segment are optimal
2T 2C 20
0 5 10 T
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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 gt 6 T,
C gt 0
0 1 2 3 T