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11/15/2011
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Chapter 12
Discrete Optimization Methods
12.1 Solving by
Total Enumeration
•
If model has only a few discrete decision variables, the
most effective method of analysis is often the most
direct: enumeration of all the possibilities. [12.1]
•
Total enumeration solves a discrete optimization by
trying all possible combinations of discrete variable
values, computing for each the best corresponding
choice of any continuous variables.
Among
combinations yielding a feasible solution, those with the
best objective function value are optimal. [12.2]

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Swedish Steel Model
with All-or-Nothing Constraints
min 16
(75)y
1
+10
(250)y
2
+8 x
3
+9 x
4
+48 x
5
+60 x
6
+53 x
7
s.t.
75y
1
+
250y
2
+ x
3
+ x
4
+ x
5
+ x
6
+ x
7
= 1000
0.0080
(75)y
1
+ 0.0070
(250)y
2
+0.0085x
3
+0.0040x
4
6.5
0.0080
(75)y
1
+ 0.0070
(250)y
2
+0.0085x
3
+0.0040x
4
7.5
0.180
(75)y
1
+ 0.032
(250)y
2
+ 1.0 x
5
30.0
0.180
(75)y
1
+ 0.032
(250)y
2
+ 1.0 x
5
30.5
0.120
(75)y
1
+ 0.011
(250)y
2
+ 1.0 x
6
10.0
0.120
(75)y
1
+ 0.011
(250)y
2
+ 1.0 x
6
12.0
0.001
(250)y
2
+ 1.0 x
7
11.0
0.001
(250)y
2
+ 1.0 x
7
13.0
x
3
…x
7
0
y
1
, y
2
= 0 or 1
(12.1)
Cost = 9967.06
y
1
* = 1, y
2
* = 0, x
3
* = 736.44, x
4
* = 160.06
x
5
* = 16.50, x
6
* = 1.00, x
7
* = 11.00
Swedish Steel Model
with All-or-Nothing Constraints
Discrete
Combination
Corresponding Continuous Solution
Objective
Value
y1
y2
x3
x4
x5
x6
x7
0
0
823.11
125.89
30.00
10.00
11.00
10340.89
0
1
646.67
63.33
22.00
7.25
10.75
10304.08
1
0
736.44
160.06
16.50
1.00
11.00
9967.06
1
1
561.56
94.19
8.50
0.00
10.75
10017.94

11/15/2011
3
Exponential Growth
of Cases to Enumerate
•
Exponential growth makes total enumeration impractical
with models having more than a handful of discrete
decision variables. [12.3]
12.2 Relaxation of Discrete
Optimization Models
Constraint Relaxations
•
Model (
°
) is a constraint relaxations of model (
P
) if
every feasible solution to (
) is also feasible in (
°
) and
both models have the same objective function. [12.4]
•
Relaxation should be significantly more tractable than
the models they relax, so that deeper analysis is
practical. [12.5]

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