This document provides an overview of linear programming models, including how to formulate an LP model, solve it, and interpret the results. It discusses key properties of LP models, such as seeking to maximize or minimize an objective function subject to constraints. The document then presents an example LP model for a product mix problem at Flair Furniture Company, showing how to set up the decision variables, objective function, and constraints to maximize profit. It also provides a graphical representation of the Flair Furniture LP model and describes the optimal solution.
2. Steps in Developing a Linear
Programming (LP) Model
1) Formulation
2) Solution
3) Interpretation and Sensitivity Analysis
3. 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
4. Example LP Model Formulation:
The Product Mix Problem
Decision: How much to make of > 2 products?
Objective: Maximize profit
Constraints: Limited resources
5. Example: Flair Furniture Co.
Two products: Chairs and Tables
Decision: How many of each to make this
month?
Objective: Maximize profit
6. Flair Furniture Co. Data
Tables Chairs
(per table) (per chair)
Profit Hours
$7 $5
Contribution Available
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
7. Decision Variables:
T = Num. of tables to make
C = Num. of chairs to make
Objective Function: Maximize Profit
Maximize $7 T + $5 C
8. Constraints:
• Have 2400 hours of carpentry time
available
3 T + 4 C < 2400 (hours)
• Have 1000 hours of painting time available
2 T + 1 C < 1000 (hours)
9. More Constraints:
• Make no more than 450 chairs
C < 450 (num. chairs)
• Make at least 100 tables
T > 100 (num. tables)
Nonnegativity:
Cannot make a negative number of chairs or tables
T>0
C>0
10. Model Summary
Max 7T + 5C (profit)
Subject to the constraints:
3T + 4C < 2400 (carpentry hrs)
2T + 1C < 1000 (painting hrs)
C < 450 (max # chairs)
T > 100 (min # tables)
T, C > 0 (nonnegativity)
11. Graphical Solution
• Graphing an LP model helps provide
insight into LP models and their solutions.
• While this can only be done in two
dimensions, the same properties apply to
all LP models and solutions.
12. C
Carpentry
Constraint Line
3T + 4C = 2400 Infeasible
600
> 2400 hrs
3T
+
Intercepts 4C
=
24
00
(T = 0, C = 600) Feasible
< 2400 hrs
(T = 800, C = 0)
0
0 800 T
13. C
1000
Painting
Constraint Line
2T
+
2T + 1C = 1000
1C
600
=1
000
Intercepts
(T = 0, C = 1000)
(T = 500, C = 0) 0
0 500 800 T
14. C
1000
Max Chair Line
C = 450
600
Min Table Line
450
T = 100
Feasible
Region
0
0 100 500 800 T
15. C
7T
Objective
+ 5C
Function Line
=$
4,0
500
7T + 5C = Profit
40
Optimal Point
(T = 320, C = 360)
7T
400
+5
C
=$
7T
300
2 ,8
+5
00
C
=$
200
2 ,1
100 00
0
0 100 200 300 400 500 T
16. C
Additional Constraint New optimal point
500 T = 300, C = 375
Need at least 75
more chairs than
tables 400 T = 320
C > T + 75 C = 360
No longer
300 feasible
Or
C – T > 75
200
100
0
0 100 200 300 400 500 T
17. 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
18. 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.
19. Special Situation in LP
2. Infeasibility – when no feasible solution
exists (there is no feasible region)
Example: x < 10
x > 15
20. Special Situation in LP
3. Alternate Optimal Solutions – when
there is more than one optimal solution
C
Max 2T + 2C 10
2T
Subject to:
+
All points on
2C
T + C < 10
=
Red segment
20
6
T < 5 are optimal
C< 6
T, C > 0
0
0 5 10 T
21. Special Situation in LP
4. Unbounded Solutions – when nothing
prevents the solution from becoming
infinitely large
n
C
ctio on
Max 2T + 2C re luti
Di so
Subject to: 2 of
2T + 3C > 6
T, C > 0 1
0
0 1 2 3 T
22. 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
