Geoscience Reference
In-Depth Information
Now, a formulation for finding the best allocation of customers within the set of
facilities S is given by:
GAP
.S/ minimize
z
D
X
i2S
X
c
ij
x
ij
(3.13)
j2J
subject to
X
i2S
x
ij
D
1
j
2
J
(3.14)
X
d
j
x
ij
q
i
i
2
S
(3.15)
j2J
x
ij
2f
0;1
g
i
2
S; j
2
J:
(3.16)
So far we have presented FLPs as a minimization problems in which both
types of decisions incur costs. However, the type of objective function depends on
the decision maker. Minimization FLPs usually appear in the public sector when
locating facilities for essential services: public hospitals or schools, dumps for
garbage collection, etc. In the private sector, however, service to customers produces
a profit to companies so that the objective of companies facing location decisions
for their service centers is to maximize the net profit, defined as the difference
between the revenue derived from the serviced customers and the cost for the
location of the selected facilities. There is indeed an essential difference between
these two models: while minimization FLPs impose that all customers be served
(no demand point can be excluded from an essential service), in maximization FLPs
not all users necessarily have to be served. The company may not have enough
incentive for servicing all customers and only those generating a profit in an optimal
location setting will be served. However, as we will next see, from a mathematical
programming point of view the maximization and minimization versions of the FLP
are equivalent.
Consider a maximization FLP where b
ij
denotes the profit for servicing customer
j
2
J from facility i
2
I. As indicated in Cornuéjols et al. (
1990
), typically, b
ij
is
a function of the unit production costs at facility i.h
i
/, the unit transportation costs
from facility i to customer j.t
ij
/, and the service price for customer j.s
j
/.Thatis,
b
ij
D
d
j
.s
j
h
i
t
ij
/. Then, the objective function for a maximization FLP is
maximize
z
D
X
i2I
f
i
y
i
C
X
i2I
X
b
ij
x
ij
:
(3.17)
j2J
In principle, if not all customers have to be served, allocation constraints should
be stated as inequalities, i.e.
P
i2I
x
ij
1, j
2
J. However, such constraints are
easily transformed into equalities by simply defining a fictitious potential facility
0, representing the facility to which all unserved demand is allocated. To this end,
we assume a sufficiently large capacity for the fictitious facility, q
0
D
P
j2J
d
j
,
