Chapter.2 Linear programming Models
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 + $5 C
Constraints:
• Have 2400 hours of carpentry time
available
3 T + 4 C < 2400 (hours)
• Have 1000 hours of painting time available
2 T + 1 C < 1000 (hours)
More Constraints:
• Make no more than 450 chairs
C < 450
(num. chairs)
• Make at least 100 tables
T > 100
(num. tables)
Nonnegativity:
Cannot make a negative number of chairs or tables
T>0
C>0
Model Summary
Max 7T + 5C
(profit)
Subject to the constraints:
3T + 4C < 2400
(carpentry hrs)
2T + 1C < 1000
(painting hrs)
T
C < 450
(max # chairs)
> 100
(min # tables)
T, C > 0
(nonnegativity)
Graphical Solution
• Graphing an LP model helps provide
insight into LP models and their solutions.
• While this can only be done in two
dimensions, the same properties apply to
all LP models and solutions.
Carpentry
Constraint Line
C
3T + 4C = 2400
Infeasible
> 2400 hrs
600
3T
Intercepts
(T = 0, C = 600)
(T = 800, C = 0)
+
4C
=
Feasible
< 2400 hrs
24
00
0
0
800 T
C
1000
1C
2T + 1C = 1000
+
2T
Painting
Constraint Line
000
=1
600
Intercepts
(T = 0, C = 1000)
(T = 500, C = 0)
0
0
500
800 T
Max Chair Line
C
1000
C = 450
Min Table Line
600
450
T = 100
Feasible
0
Region
0 100
500
800 T
+
7T
C
40
4,0
=$
7T + 5C = Profit
5C
Objective
Function Line
500
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
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 + $5 C
Constraints:
• Have 2400 hours of carpentry time
available
3 T + 4 C < 2400 (hours)
• Have 1000 hours of painting time available
2 T + 1 C < 1000 (hours)
More Constraints:
• Make no more than 450 chairs
C < 450
(num. chairs)
• Make at least 100 tables
T > 100
(num. tables)
Nonnegativity:
Cannot make a negative number of chairs or tables
T>0
C>0
Model Summary
Max 7T + 5C
(profit)
Subject to the constraints:
3T + 4C < 2400
(carpentry hrs)
2T + 1C < 1000
(painting hrs)
T
C < 450
(max # chairs)
> 100
(min # tables)
T, C > 0
(nonnegativity)
Graphical Solution
• Graphing an LP model helps provide
insight into LP models and their solutions.
• While this can only be done in two
dimensions, the same properties apply to
all LP models and solutions.
Carpentry
Constraint Line
C
3T + 4C = 2400
Infeasible
> 2400 hrs
600
3T
Intercepts
(T = 0, C = 600)
(T = 800, C = 0)
+
4C
=
Feasible
< 2400 hrs
24
00
0
0
800 T
C
1000
1C
2T + 1C = 1000
+
2T
Painting
Constraint Line
000
=1
600
Intercepts
(T = 0, C = 1000)
(T = 500, C = 0)
0
0
500
800 T
Max Chair Line
C
1000
C = 450
Min Table Line
600
450
T = 100
Feasible
0
Region
0 100
500
800 T
+
7T
C
40
4,0
=$
7T + 5C = Profit
5C
Objective
Function Line
500
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