ISE 536–Fall03: Linear Programming and Extensions
November 26, 2003
Lecture 23: Integer Programming, Modeling
Lecturer: Fernando Ord´
o˜nez
1
Mixed Integer Programming Problem
Here we study linear programming problems with some variables constrained to be integer.
For example:
min
c
t
x
+
d
t
y
s
.
t
.
Ax
+
By
=
b
x, y
≥
0
x
integer
•
Mixed integer programming problem:
•
Integer programming problem:
•
Binary programming problem:
Example
(zeroone knapsack problem).
We are given
n
items, each has weight
w
j
and
value
c
j
. Given a bound
K
on the weight that we can carry on the knapsack, we would like
to select the items that maximize the total value. Formulate this as an integer programming
problem.
1
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Things that can be represented with integer variables:
•
x
≤
y
and
x, y
binary:
•
x
1
+
x
2
+
x
3
≤
1,
x
i
binary:
•
x
1
+
x
2
+
x
3
= 1,
x
i
binary:
•
The vectors
a
i
≥
0,
i
= 1
, . . . , m
and
a
t
i
x
≥
b
i
y
i
i
= 1
, . . . , m
m
i
=1
y
i
≥
k
y
i
∈
{
0
,
1
}
i
= 1
, . . . , m
•
x
=
∑
m
j
=1
a
j
y
j
,
∑
m
j
=1
y
j
= 1, and
y
j
binary:
•
How can we use integer variables to represent a piecewise linear function? (consider
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 Spring '05
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 Operations Research, Linear Programming, Optimization, LP, George Dantzig, integer programming problem

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