Geoscience Reference
In-Depth Information
and set to zero, both the fixed-charge cost of the fictitious facility (f
0
D
0)and
the allocation profits of all customers (b
0j
D
0, j
2
J). Thus, without loss of
generality we can assume that in the maximization FLP allocation constraints must
also be satisfied as equality.
Taking into account the expression of the coefficients b
ij
and because of the
equality allocation constraints, the second term in (
3.17
) can be rewritten as
X
X
b
ij
x
ij
D
X
i2I
X
d
j
.s
j
h
i
t
ij
/x
ij
i2I
j2J
j2J
D
X
i2I
X
d
j
s
j
x
ij
X
i2I
X
d
j
.h
i
C
t
ij
/x
ij
j2J
j2J
D
X
j2J
d
j
s
j
X
i2I
X
c
ij
x
ij
:
j2J
Hence objective (
3.17
) reduces to
2
4
X
i2I
3
X
f
i
y
i
C
X
i2I
X
5
:
c
ij
x
ij
d
j
s
j
min
(3.18)
j2J
j2J
Since the first term in (
3.18
) is a constant, the maximization FLP is equivalent to a
minimization FLP.
3.2.1
Set Partitioning Formulation of FLPs
Below we present alternative formulations for FLPs which use decision variables
to model the overall customers demand allocated to open facilities. Consider for
the moment the single allocation case and note that feasible assignments to a
given facility i
2
I are associated with subsets of customers T
J such that
P
j2T
d
j
q
i
. We will use the notation K
i
to denote the index set of feasible
assignment subsets for facility i
2
I, T
k
J the index set of the customers served
in feasible assignment k
2
K
i
,andp
ki
for the fixed-charge cost of facility i plus the
cost for assigning to i all the customers indexed in T
k
,i.e.p
ki
D
f
i
C
P
j2T
k
c
ij
.
Also, for i
2
I, k
2
K
i
, j
2
J,leta
ijk
D
1 if j
2
T
k
and 0 otherwise. Consider
now the following decision variables:
z
ki
D
1 if the subset of customers T
k
is assigned to facility i
0 otherwise.
