Supplement F - Linear Programming

1
Supplement F -Linear Programming
2
Basic Concepts

• Objective function
• Decision variables
• Constraints
• Feasible region
• Parameters
• Linearity
• Nonnegativity

3
Stratton Company
Pipes R Us
4
(No Transcript)
5
Linear Programming
Step 1Define the decision variables

• x1 amount of type 1 pipe produced and sold next
  week, 100-foot increments
• x2 amount of type 2 pipe produced and sold next
  week, 100-foot increments

Example F.1
6
Linear Programming
Step 2Define the objective function
Example F.1
7
Linear Programming
Step 2Define the objective function
Max Z
Objective
Example F.1
8
Linear Programming
Step 2Define the objective function
Max Z
Example F.1
9
Linear Programming
Step 2Define the objective function
Max Z x1 x2
Decision variables
Example F.1
10
Linear Programming
Step 2Define the objective function
Max Z x1 x2
Example F.1
11
Linear Programming
Step 2Define the objective function
Max Z 34 x1 40 x2
Coefficients
Example F.1
12
Linear Programming
Step 2Define the objective function
Max Z 34 x1 40 x2
Example F.1
13
(No Transcript)
14
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
Example F.1
15
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
48 (extrusion)
Right-hand side value
Example F.1
16
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
48 (extrusion)
RHS value
Example F.1
17
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
48 (extrusion)
Example F.1
18
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
? 48 (extrusion)
Type of limit
Example F.1
19
Linear Programming
Step 3Formulate the constraints
Example F.1
20
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
? 48 (extrusion)
Example F.1
21
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
x1 x2 ? 48 (extrusion)
Decision variables
Example F.1
22
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
x1 x2 ? 48 (extrusion)
Example F.1
23
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
4 x1 6 x2 ? 48 (extrusion)
Coefficients
Example F.1
24
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
4 x1 6 x2 ? 48 (extrusion)
Example F.1
25
Linear Programming
Step 3Formulate the constraints
Max Z 34 x1 40 x2
4 x1 6 x2 ? 48 (extrusion) 2 x1 2 x2 ? 18
(packaging) 2 x1 x2 ? 16 (additive mix)
Example F.1
26
Linear Programming
Graphical solution
27
Linear Programming
Figure F.1
28
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
(0, 8)
4x1 6x2 ? 48 (extrusion)
(12, 0)
2 4 6 8 10 12 14 16 18
x1
Figure F.1
29
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
(0, 8)
4x1 6x2 ? 48 (extrusion)
(12, 0)
2 4 6 8 10 12 14 16 18
x1
Figure F.1
30
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
(0, 9)
2x1 2x2 ? 18 (packaging)
4x1 6x2 ? 48 (extrusion)
(9, 0)
2 4 6 8 10 12 14 16 18
x1
Example F.2
31
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
(0, 9)
2x1 2x2 ? 18 (packaging)
4x1 6x2 ? 48 (extrusion)
(9, 0)
2 4 6 8 10 12 14 16 18
x1
Example F.2
32
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
(0, 16)
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (packaging)
4x1 6x2 ? 48 (extrusion)
(8, 0)
2 4 6 8 10 12 14 16 18
x1
Example F.2
33
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
(0, 16)
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (packaging)
4x1 6x2 ? 48 (extrusion)
(8, 0)
2 4 6 8 10 12 14 16 18
x1
Example F.2
34
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (packaging)
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
Figure F.2
35
Linear Programming
Figure F.3
36
Linear Programming
Feasible region
Figure F.3
37
Linear Programming
x2
Feasible region
2x1 x2 ? 10
2x1 3x2 ? 18
x1
Figure F.3
38
Linear Programming
Figure F.3
39
Linear Programming
Figure F.3
40
Linear Programming
Figure F.3
41
Linear Programming
x2
Feasible region
2x1 x2 ? 10
6x1 5x2 ? 5
x1 ? 7
x2 ? 5
Feasible region
2x1 3x2 ? 18
x1
Figure F.3
42
Linear Programming
Test point
Figure F.3
43
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
4x1 6x2 ? 48 (extrusion)
Feasible region
D
E
2 4 6 8 10 12 14 16 18
x1
A
Figure F.4
44
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
34x1 40x2 272
C
(0,6.8)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
A
Figure F.5
E (8,0)
45
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
(0,6.8)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
A
E (8,0)
46
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
(0,6.8)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
A
E (8,0)
47
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Graphical solution
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
Optimal solution (3,6)
C
(0,6.8)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
A
Figure F.6
E (8,0)
48
Linear Programming
49
Linear Programming
Optimal corner point
4x1 6x2 48 2x1 2x2 18
50
Linear Programming
Optimal corner point
4x1 6x2 48 (4x1 4x2 36)
51
Linear Programming
Optimal corner point
4x1 6x2 48 (4x1 4x2 36) 2x2 12

52
Linear Programming
Optimal corner point
4x1 6x2 48 (4x1 4x2 36) 2x2 12
x2 6
53
Linear Programming
Optimal corner point
4x1 6(6) 48 (4x1 4x2 36) 2x2 12
x2 6
54
Linear Programming
Optimal corner point
4x1 6(6) 48 4x1 12 x1 3
55
Linear Programming
Slack variables
Slack and Surplus Variables
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (packaging)
Optimal solution (3,6)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
Example F.4
E (8,0)
56
Linear Programming
Slack variables
2x1 x2 16
Slack and Surplus Variables
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (packaging)
Optimal solution (3,6)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
Example F.4
E (8,0)
57
Linear Programming
Slack variables
2x1 x2 16 2x1 x2 s1 16
Slack and Surplus Variables
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (packaging)
Optimal solution (3,6)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
Example F.4
E (8,0)
58
Linear Programming
Slack variables
2x1 x2 16 2(3) 6 s1 16
Slack and Surplus Variables
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (packaging)
Optimal solution (3,6)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
Example F.4
E (8,0)
59
Linear Programming
Slack variables
2x1 x2 16 s1 4
Slack and Surplus Variables
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (packaging)
Optimal solution (3,6)
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
Example F.4
E (8,0)
60
Linear Programming
Objective function coefficients
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
61
Linear Programming
Objective function coefficients
34x1 40x2 Z
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
62
Linear Programming
Objective function coefficients
34x1 40x2 Z 40x2 34x1 Z
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
63
Linear Programming
Objective function coefficients
34x1 40x2 Z 40x2 34x1 Z x2

18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
34x1 Z 40 40
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
64
Linear Programming
Objective function coefficients
34x1 40x2 Z 40x2 34x1 Z x2

18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
c1x1 Z c2 c2
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
65
Linear Programming
Objective function coefficients
34x1 40x2 Z 40x2 34x1 Z x2

18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
c1x1 Z c2 c2
If c1 increases
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
66
Linear Programming
Objective function coefficients
34x1 40x2 Z 40x2 34x1 Z x2

18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
c1x1 Z c2 c2
If c1 increases
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
67
Linear Programming
Objective function coefficients
34x1 40x2 Z 40x2 34x1 Z x2

18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
c1x1 Z c2 c2
If c1 decreases
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
68
Linear Programming
Objective function coefficients
34x1 40x2 Z 40x2 34x1 Z x2

18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
c1x1 Z c2 c2
If c1 decreases
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
New Optimal Point
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
69
Linear Programming
Range of optimality
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Figure F.7
E
70
Linear Programming
Range of optimality
34 2 c2 3
1 ? ?
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.5
E
71
Linear Programming
Range of optimality
34 2 c2 3
1 ? ? 1 ?
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
34 c2
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.5
E
72
Linear Programming
Range of optimality
34 2 c2 3
1 ? ? c2 ? 34
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.5
E
73
Linear Programming
Range of optimality
34 2 c2 3
1 ? ? ?
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
34 2 c2 3
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.5
E
74
Linear Programming
Range of optimality
34 2 c2 3
1 ? ? 34 ?
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2c2 3
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.5
E
75
Linear Programming
Range of optimality
34 2 c2 3
1 ? ? 51 ? c2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.5
E
76
Linear Programming
Range of optimality
34 2 c2 3
1 ? ? 34 ? c2 ? 51
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.5
E
77
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
D
4x1 6x2 ? 48 (extrusion)
2 4 6 8 10 12 14 16 18
x1
A
E
78
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
Optimal solution before rotation
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
Optimal solution after rotation
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Figure F.8
E
79
Linear Programming
Coefficient sensitivity
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
Optimal solution before rotation
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
Optimal solution after rotation
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.6
E
80
Linear Programming
Coefficient sensitivity
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
Optimal solution before rotation
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
Optimal solution after rotation
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.6
E
81
Linear Programming
Coefficient Sensitivity

18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
Optimal solution before rotation
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
Optimal solution after rotation
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.6
E
82
Linear Programming
Coefficient Sensitivity

18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
Optimal solution before rotation
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
(Slope 1)
C
Optimal solution after rotation
D
4x1 6x2 ? 48 (extrusion)
(Slope 2/3)
2 4 6 8 10 12 14 16 18
x1
A
Example F.6
E
83
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (original packaging constraint)
B
C
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
A
E
84
Linear Programming
85
Linear Programming
Optimal corner point
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (original packaging constraint)
B
2x1 2x2 ? 19 (relaxed packaging constraint)
C
Increase in feasible region
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Figure F.9
E
86
Linear Programming
Optimal corner point
4x1 6x2 48 2x1 2x2 19
x2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (original packaging constraint)
B
2x1 2x2 ? 19 (relaxed packaging constraint)
C
Increase in feasible region
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Figure F.9
E
87
Linear Programming
Optimal corner point
4x1 6x2 48 2x1 2x2 19 x1 4.5 x2
5
x2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (original packaging constraint)
B
2x1 2x2 ? 19 (relaxed packaging constraint)
C
Increase in feasible region
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Figure F.9
E
88
Linear Programming
Optimal corner point
4x1 6x2 48 2x1 2x2 19 x1 4.5 x2
5
x2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
Z 34(4.5) 40(5) 353
2x1 x2 ? 16 (additive mix)
2x1 2x2 ? 18 (original packaging constraint)
B
2x1 2x2 ? 19 (relaxed packaging constraint)
C
Increase in feasible region
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Figure F.9
E
89
Linear Programming
Optimal corner point
4x1 6x2 48 2x1 2x2 19 x1 4.5 x2
5
x2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
Z 34(4.5) 40(5) 353
2x1 x2 ? 16 (additive mix)
353 - 342 11
2x1 2x2 ? 18 (original packaging constraint)
B
2x1 2x2 ? 19 (relaxed packaging constraint)
C
Increase in feasible region
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Figure F.9
E
90
Linear Programming
Optimal corner point
4x1 6x2 48 2x1 2x2 19 x1 4.5 x2
5
x2
18 16 14 12 10 8 6 4 2 0
Shadow price for packaging constraint
Sensitivity analysis
Z 34(4.5) 40(5) 353
2x1 x2 ? 16 (additive mix)
353 - 342 11
2x1 2x2 ? 18 (original packaging constraint)
B
2x1 2x2 ? 19 (relaxed packaging constraint)
C
Increase in feasible region
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Figure F.9
E
91
Linear Programming
x2
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
C
4x1 6x2 ? 48 (extrusion)
D
2 4 6 8 10 12 14 16 18
x1
A
E
92
Linear Programming
93
Linear Programming
Lower limit of shadow price
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
Packaging constraint for upper bound
Packaging constraint for lower bound
C
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Example F.7
E
94
Linear Programming
Lower limit of shadow price
B is the lowest feasible limit
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
Packaging constraint for upper bound
Packaging constraint for lower bound
C
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Example F.7
E
95
Linear Programming
Lower limit of shadow price
B is the lowest feasible limit For packaging, B,
x1 0 and x2 8 2(0) 2(8) 16 hours
18 16 14 12 10 8 6 4 2 0
Sensitivity analysis
2x1 x2 ? 16 (additive mix)
B
2x1 2x2 ? 18 (packaging)
Packaging constraint for upper bound
Packaging constraint for lower bound
C
4x1 6x2 ? 48 (extrusion)
C?
D
2 4 6 8 10 12 14 16 18
x1
A
Example F.7
E
96
Linear Programming
97
Linear Programming
Figure F.11
98
Linear Programming
Figure F.12
99
Linear Programming
Figure F.12
100
Linear Programming
Example F.8
101
Linear Programming
Example F.8
102
Linear Programming
Example F.8
103
Linear Programming
No, too expensive. 8 gt 3
Example F.8
104
Linear Programming
Example F.8
105
Linear Programming
Yes, increased revenue. 6 lt 11
Example F.8
106
Linear Programming
Example F.8
107
Linear Programming
Aggregate planning Production, Staffing,
Blends Distribution Shipping Inventory Stock
control, Supplier selection Location Plants or
warehouses Process management Stock
cutting Scheduling Shifts, Vehicles, Routing
108
Solved Problem 1
Figure F.13
109
Solved Problem 1
Figure F.14
110
Solved Problem 1
Figure F.15
111
Solved Problem 2
750 500 250 0
x2
Feasible region has no upper bound.
A
x1 x2 750 (minimum number)
0.30 x1 0.20 x2 150.00 (isocost line)
0.02 x1 0.04 x2 20 (minimum order)
B
C
5 10 15
x1
Figure F.16