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location. This yields a non-linear fractional model that the authors tackle by existing
methods and by ad-hoc procedures they propose in the paper.
One final aspect worth mentioning in this section regards the relevance of using
a model like the ones described above, instead of a “simplified” deterministic
model. When probabilities can be associated with the scenarios, we can measure
this relevance by using the expected value of perfect information (EVPI). The EVPI
indicates how much the decision maker would be willing to pay for getting perfect
information. Suppose we have an expected cost minimization problem. In this case,
the EVPI is obtained by computing the difference between the weighted sum of
the optimal values for all scenarios (using the probabilities as weights) and the
minimum expected cost. The reader should refer to Kouvelis and Yu (
1997
)for
further details.
8.4
Stochastic Facility Location Problems
A facility location problem under uncertainty, can often be casted within a stochastic
programming modeling framework if uncertainty can be described by some proba-
bility distribution. In this case, we say that we are dealing with a stochastic facility
location problem.
We start by considering the UFLP (
8.15
)-(
8.19
). In practice, several parameters
in this model may be uncertain. This is the case of the distribution costs and of
the demands. Let us assume that uncertainty can be measured probabilistically. In
particular, denote by the random vector containing all the random parameters
(e.g.,
D
.c
ij
/
i2I;j2J
;.d
j
/
j2J
). Furthermore, suppose that we know the joint
probability distribution of . Assuming
ex ante
location decisions, the model to
be adopted will depend on the
ex post
decisions, namely on the moment in time
where allocation or distribution decisions are to be implemented. If we have
ex
post
allocation decisions, the following stochastic uncapacitated facility location
problem with recourse can be considered:
Minimize
X
i2I
f
i
y
i
C
Q.y/
(8.26)
subject to y
i
2f
0;1
g
; i
2
I;
(8.27)
with Q.y/
D
E
ŒQ.y;/,andQ.y;/ denoting the optimal value of the following
problem:
Minimize
X
i2I
X
c
ij
d
j
x
ij
(8.28)
j2J
