Managerial decision modeling with spreadsheets by stair render chapter 02

Chapter 2
Linear Programming Models:
Graphical and Computer Methods

© 2007 Pearson Education


Steps in Developing a Linear
Programming (LP) Model
1) Formulation
2) Solution
3) Interpretation and Sensitivity Analysis


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


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. Data
Tables

Chairs

(per table)

(per chair)

Profit
Contribution

$7

$5

Hours
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


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>0
C>0


Model Summary
Max 7T + 5C

(profit)

Subject to the constraints:

3T + 4C < 2400

(carpentry hrs)

2T + 1C < 1000

(painting hrs)

T

C < 450

(max # chairs)

> 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

C

3T + 4C = 2400

Infeasible

> 2400 hrs

600
3T

Intercepts
(T = 0, C = 600)
(T = 800, C = 0)

+

4C

=

Feasible
< 2400 hrs

24
00

0
0

800 T


C
1000

1C

2T + 1C = 1000

+
2T

Painting
Constraint Line

=1

600

000

Intercepts
(T = 0, C = 1000)
(T = 500, C = 0)

0
0

500

800 T


Max Chair Line

C
1000

C = 450

Min Table Line

600
450

T = 100
Feasible
0

Region
0 100

500

800 T


+
7T

C

=$
4
4,0

500

0

7T + 5C = Profit

5C

Objective
Function Line

7T

Optimal Point
(T = 320, C = 360)

400

C
+5
C
+5

00
2,8
=$

7T

300

=$
00
2,1

200

100

0
0

100

200

300

400

500 T


C

Additional Constraint
Need at least 75
more chairs than
tables

New optimal point
T = 300, C = 375

500

400

T = 320
C = 360
No longer
feasible

C > T + 75
Or
C – T > 75

300

200

100

0
0

100

200

300

400

500 T


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
1. 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
2. Infeasibility – when no feasible solution

exists (there is no feasible region)
Example:

x < 10
x > 15


Special Situation in LP
3. Alternate Optimal Solutions – when
there is more than one optimal solution
C
10

2T

Max 2T + 2C

All points on
Red segment
are optimal

2C
=
20

T + C < 10
T
< 5
C< 6
T, C > 0

+

Subject to:
6

0
0

5

10

T


Special Situation in LP
4. Unbounded Solutions – when nothing
prevents the solution from becoming
infinitely large
Max 2T + 2C
Subject to:

2T + 3C > 6
T, C > 0

on n
i
t
c

ri e lutio
D so
of

C

2

1

0
0

1

2

3

T


Using Excel’s Solver for LP
Recall the Flair Furniture Example:
Max 7T + 5C

(profit)

Subject to the constraints:

3T + 4C < 2400
2T + 1C < 1000
C < 450
T
> 100
T, C > 0

(carpentry hrs)
(painting hrs)
(max # chairs)
(min # tables)
(nonnegativity)

Go to file 2-1.xls