Facility Location Under Uncertainty - Location Science - page 185

X

d j x ij q i ; i 2 I (8.55)

j2J

x ij 0; i 2 I; j 2 J: (8.56)

In the above model, b ij is a fixed cost for allocating customer j 2 J to facility
i 2 I. The other notation was already introduced before. Note that in this problem,
facilities are capacitated. Laporte et al. ( 1994 ) consider a finite set of scenarios and
solve the extensive form of the deterministic equivalent using the integer L-shaped
method previously proposed by Laporte and Louveaux ( 1993 ).
In line with the idea of allocating the customers before uncertainty is disclosed,
Albareda-Sambola et al. ( 2011 ) consider Bernoulli demands, which represent a
possible request for some service. This is an example of a problem in which the
presence or absence of customers is itself a source of uncertainty. The problem,
which we revisit below, is important to show that finding a deterministic equivalent
is not always straightforward (or even possible) as the models above could indicate.
In the problem studied by Albareda-Sambola et al. ( 2011 ), there is a limited
capacity for the facilities in terms of the number of customers that can be served.
In particular, for each facility i 2 I, there is a maximum number of customers, q i ,
that can be served from the facility. Due to the uncertainty in the demand, it makes
sense to allocate (a priori) to some facility more customers than the service capacity.
In the end, it may turn out that a facility has a number of requests for service
above its capacity. In this case, outsourcing is considered and the corresponding
costs incurred. An important assumption in many logistics systems that the authors
also consider is that, for each facility i 2 I, there should be a minimum number
of customers ` i allocated to it to justify its establishment. The problem can be
conceptually formulated as follows.
Minimize X
i2I

f i y i C E ŒService cost C Outsourcing cost (8.57)

subject to X
i2I

x ij D 1; j 2 J

(8.58)

x ij y i ; i 2 I; j 2 J (8.59)

` i y i X
j2J

x ij ; i 2 I (8.60)

y i ;x ij 2f 0;1 g ; i 2 I; j 2 J: (8.61)

Denote by j the demand of customer j 2 J that is assumed to follow a Bernoulli
distribution with parameter p j . For each first-stage solution, denote by z i the

number of customers assigned to facility i 2 I (i.e., z i D P j2J x ij ) and denote by
i the random variable representing the number of customers that request the service
(referred to as demand customers) among those assigned to facility i 2 I (i.e.,