Geoscience Reference
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information associated to them, a stochastic program is formulated, capturing the
uncertainty associated with the other set of parameters (those for which probabilistic
information exists). Then, a minmax regret formulation is proposed for the overall
problem. The reader should refer to the paper for further details.
As in the preceding section, when using a stochastic programming approach, it
is important to evaluate its relevance compared to a more simplified deterministic
approach. Although no robust measure exists for asserting such relevance, two
measures are often used to give an indication of such relevance: the EVPI and
the value of the stochastic solution (VSS). The EVPI is computed as described in
Sect.
8.3
. To obtain it, we have to solve the distributional problem (i.e., to find the
optimal value for each scenario). In many cases this is cumbersome, namely when
the number of scenarios is high or even infinity. The VSS emerges as an alternative
and can be obtained in two steps: (i) the expected value problem is solved. This is
the deterministic problem obtained when the random variables are replaced by their
expectation; (ii) the stochastic problem is considered and the difference between
its optimal value and the value of the solution obtained in (i) is computed. This
difference gives the VSS (the reader should refer to Birge and Louveaux
2011
for
further details).
8.5
Chance-Constrained Facility Location Problems
One important class of optimization problems under uncertainty includes chance-
constrained problems. The idea is that one or several constraints of the problem are
not required to always hold. Instead, the decision maker is satisfied if they hold with
some given probability. This type of constraints may be of relevance when dealing
with reliability issues.
In the particular case of a facility location problem, if demand is uncertain but
still the decision maker wants to plan for satisfying all the demand whatever it
may turn out to be, the resulting solution may call for an operational capacity
much above the demand level that turns out being observed. In such situation, one
alternative is to plan for assuring a certain service level, i.e., assuring that with some
pre-specified probability, the overall demand does not exceed the capacity of the
operating facilities.
In order to exemplify these modeling capability, we consider the classical single-
source capacitated facility location problem. Assume that fixed costs are associated
with the location of the facilities and also with the allocation of customers to the
facilities. Additionally, assume that facility i
2
I has capacity q
i
, and that demands
d
j
(j
2
J) are stochastic. We can formulate a capacitated facility location problem
with minimum service level as follows:
Minimize
X
i2I
f
i
y
i
C
X
i2I
X
c
ij
x
ij
(8.72)
j2J
