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ISE 536–Fall03: Linear Programming and Extensions
November 5, 2003
Lecture 18: Large scale LP: Column Generation
Lecturer: Fernando Ord´o˜nez
1
Example
Consider the problem of operating a supermarket chain. There are a large number of
variables that apply only to each store: such as shelf space, shifts scheduling, inventory,
cleaning. And there are a number of variables that connect all the stores in the chain:
procurement of goods, budget,
employee medical beneﬁt negotiation
.
In the case of two stores this operation can be represented by a linear programming problem
such as:
(
P
)
z
= max
c
t
1
x
1
+
c
t
2
x
2
s
.
t
. B
1
x
1
+
B
2
x
2
=
b
A
1
x
1
=
b
1
A
2
x
2
=
b
2
x
1
,
x
2
≥
0
The column generation method that we discuss performs better if
m << m
1
,m
2
.
2
DantzigWolfe Decomposition
Deﬁne
S
i
:=
{
x

A
i
x
=
b
i
, x
≥
0
}
, for
i
= 1
,
2. The problem above is equivalent to
(
P
)
z
= max
c
t
1
x
1
+
c
t
2
x
2
s
.
t
. B
1
x
1
+
B
2
x
2
=
b
x
1
∈
S
1
x
2
∈
S
2
Since
S
i
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