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Given a set of potential facility locations
N
=
{
1
,
2
,
· · ·
, n
}
and a set of customers
I
=
{
1
,
2
,
· · ·
, m
}
. A
facility placed at location
j
costs
c
j
for
j
∈
N
. The total cost of satisfying the demand of client
i
from a
facility at
j
is
h
ij
. The optimization problem is to choose a subset of locations at which to place facilities
and then to assign the clients to these facilities so as to minimize total cost.
Note:
The demand of a
Figure 1: Uncapacitated facility location problem
customer can be satisﬁed by more than one facilities.
(a)
Give a formulation of this optimization problem.
(b)
Now suppose facility place at
j
has capacity
u
j
for
j
∈
N
and customer
i
has demand
b
i
for
i
∈
I
.
Moreover, the
unit cost
of serving customer
i
by facility at
j
is
s
ij
.
Formulation this problem
again.
Solution:
(a) Deﬁne variables as follows:
x
j
:=
±
1
if a facility is placed at
j
0
otherwise
,
y
ij
:= the fraction of the demand of customer
i
that is satisﬁed from a facility at
j.
1
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 Spring '09
 Nedich

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