Geoscience Reference
In-Depth Information
a set of potential locations for the facilities (see Chap.
5
). A customer is said to
be covered if a facility is established within a maximum distance or travel time
specified in advance. Accordingly, for each customer, we can find the subset of
potential locations for the facilities which cover the customer. The goal is to cover all
the demand minimizing the number of facilities installed. The “classical” covering
constraints are
X
y
i
1; j
2
J;
(8.77)
i2I
j
where I
j
denotes the set of locations covering customer j
2
J. The probabilistic
version of these constraints is the following:
P
ŒAt least one location is available for serving customer j
Ǜ; j
2
J: (8.78)
These constraints have as a deterministic equivalent,
X
y
i
LJ;
(8.79)
i2I
j
with LJ
Dd
ln.1
Ǜ/=ln
e
. In fact, the probability that no location among those
covering customer j
2
J is available to serve the customer immediately is given by
P
i2I
j
y
i
. Thus, the probability that at least one location covering customer j
2
J
can serve it immediately is given by 1
P
i2I
j
y
i
which, together with (
8.78
) leads
to the deterministic equivalent just presented.
8.6
Challenges and Further Reading
Despite all the existing work on facility location problems under uncertainty, many
challenges still exist. In this section, we provide the reader with some notes on
relevant issues not discussed in the previous pages, and we give suggestions for
further reading.
8.6.1
Multi-Stage Stochastic Programming Models
In all the stochastic facility location problems discussed above, it was assumed that
there is a single moment for uncertainty to be revealed. However, in many situations,
this is not the case. Instead, we may observe uncertainty being progressively
revealed in more than one occasion. When this happens, the two-stage stochastic
programming modeling framework discussed in Sect.
8.4
is no longer appropriate,
