68
Chapter 6
Goal Programming
Chapter 6
Goal Programming
6.1 MAX Z:
4X
1
+ 3X
2
+ 5X
3
– 50V
1
 50V
2
Subject to:
X
1
+
X
2
+
X
3

V
1
+ S
1
= 10
2X
1
+
X
2
+ 2X
3
+
V
2
– S
2
= 50
(X
i
, S
i
, V
i
≥
0)
Solution:
X
1
= 0, X
2
= 0, X
3
= 35, V
1
= 15, S
1
= 0, V
2
= 0, S
2
= 0,
Z
MAX
= 125
(Note that the maximum value of Z is computed without
the violation
variables’ effect, since their weights are arbitrary.)
This solution
violates the first constraint by 15 units.
6.2
MAX Z:
2X
1
+ X
2
+ 4X
3
+ 5X
4
– 50V
1
– 50V
2
– 50V
3
– 50V’
3
S.T.:
3X
1
+ 2X
2
+ X
3
+ 4X
4
+ V
1
– S
1
= 100
X
1
+ 2X
2
+ 3X
3
+
X
4
– V
2
+ S
2
=
50
2X
1
+ 3X
2
+ 4X
3
+ 2X
4
+ V
3
– V’
3
= 40
(X
1
, X
2
, X
3
, X
4
, S
i
, V
i
≥
0)
Solution:
X
1
= 0, X
2
= 0, X
3
= 9.1, X
4
= 22.7, V
1
= 0, S
1
= 0, V
2
= 0, S
2
= 0, V
3
=
0, V’
3
= 41.8, Z
MAX
= 149.1.
(Note that we compute Z without
the effect of violation variables.)
This solution violates the equality constrain (#3) by 41.8 units on the
side away from the origin.
6.3 MAX Z:
100V
1
+ 100S
1
100V
2
+ 100S
2
Subject to:
4X
1
+
2X
2
+
8X
3
+ V
1
– S
1
= 100
X
1
+ 12X
2
+
X
3
+ V
2
– S
2
= 100
X
1
+
2X
2
+
3X
3
≥
18
5X
1
+
X
2
+
X
3
≤
48
(X
1
, X
2
, X
3
, S
i
, V
i
≥
0)
Solution:
X
1
= 0
X
2
= 48
X
3
= 0
V
1
= 4
S
1
= 0
Z
1
= 96
V
2
= 0
S
2
= 476
Z
2
= 576
6.4 MAX Z:
500V
1
+ 500S
1
– 100V
2
+ 100S
2
Subject to: 2X
1
+ 5X
2
+ 8X
3
+ V
1
– S
1
= 100
X
1
+ 2X
2
+ 5X
3
+ V
2
 S
2
= 100
3X
1
+ 5X
2
+
X
3
≤
60
X
1
+ 6X
2
+ 4X
3
≤
64
8X
1
+
X
2
+ 2X
3
≥
20
(X
1
, X
2
, X
3
, S
i
, V
i
≥
0)
Solution:
X
1
= 0
X
2
= 0
X
3
= 16
V
1
= 0
S
1
= 28
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Solutions Manual and Workbook
69
V
2
= 20
S
2
= 0
MAX Z:
100V
1
+ 100S
1
500V
2
+ 500S
2
Subject to:
2X
1
+ 5X
2
+ 8X
3
+ V
1
– S
1
= 100
X
1
+ 2X
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