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Chapter 2Linear Programming Models:Graphical and Computer Methods
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. 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
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 > 0C > 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)
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.
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
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
0 100 500 800 T
C1000
600
450
0
Max Chair Line
C = 450
Min Table Line
T = 100
Feasible
Region
0 100 200 300 400 500 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
0 100 200 300 400 500 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
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
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
Special Situation in LP
• 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
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