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In the above model, the non-anticipativity principle 2 is implicitly considered: each
first-stage decision variable has the same value for all scenarios.
So far, no capacities have been considered for the facilities. When they exist,
several adjustments are required. Denote by q i the capacity of a facility established
at i 2 I. A model for the capacitated stochastic facility location problem is obtained
if we replace ( 8.30 ) with
X
d j x ij q i y i ; i 2 I:
(8.36)
j2J
With the inclusion of these constraints, it may happen that for some first-stage
feasible solution, no feasible completion exists in the second stage for one or
several realizations of the random vector, i.e., the problem no longer has relatively
complete recourse. This feasibility issue adds an extra difficulty to this stochastic
programming problem. Infeasibility in the second stage is often an indication of an
undesirable first-stage solution. A natural way for addressing this issue is to penalize
the non-satisfied demand, which makes sense from a practical point of view. In fact,
such penalties correspond, for instance, to costs associated with opportunity losses.
Denote by j the demand of customer j 2 J which is not supplied and denote by
j the corresponding unitary penalty cost. Note that j is also a random variable as
it depends on the occurring realization of the random vector . We can still consider
the first stage problem ( 8.26 )-( 8.27 ). However, the second stage problem becomes
the following:
Minimize X
i2I
X
c ij d j x ij C X
j2J
j j
(8.37)
j2J
subject to
( 8.31 ), ( 8.36 )
d j X
i2I
x ij C j D d j ; j 2 J
(8.38)
j 0; j 2 J:
(8.39)
Again, if a finite set of scenarios exists, we can consider scenario-indexed recourse
variables and parameters, and we can write the deterministic equivalent in its
extensive form.
In the capacitated model just described, capacities are exogenous. Louveaux
( 1986 ) considers a stochastic facility location problem with endogenous capacities.
In particular, capacity decisions are ex ante decisions, i.e., the capacities of the
facilities must be decided in advance before uncertainty is disclosed. A unitary cost
g i is considered for the capacity to be installed at location i 2 I. Additionally,
2 A decision should depend only on the information available at the time it is made (see Rockafellar
and Wets 1991 ).
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