Geoscience Reference
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case of this problem was addressed by Albareda-Sambola et al. (
2010
), assuming
that each customer requires service only in a subset of periods. Additionally, it
is possible not to fulfil the request in one or several of those periods but in this
case, a penalty cost is paid. Several mathematical programming formulations were
proposed for the problem, which were compared computationally.
A multi-period discrete facility location problem was also investigated by
Gourdin and Klopfenstein (
2008
). The problem is motivated within the context of
telecommunications network design and consists of planning for the location of
modular equipment over a finite planning horizon. Operating capacity constraints
are considered for the nodes and for the links. The goal is to progressively expand
the capacity of the equipment as well as the capacity of its links to the demand
nodes. In that paper, the mathematical programming model initially proposed for
the problem was enhanced via polyhedral analysis.
11.6
The Value of the Multi-Period Solution
Multi-period modeling frameworks like those proposed in the previous sections,
involve one extra dimension in the decision space: the time. Models tend to be large
and thus more difficult to tackle, even for instances of moderate size. Accordingly,
one may ask whether it is worth considering this extra dimension. In other words,
let us consider a situation in which it is possible to make a static decision even with
costs, demands (and possibly other parameters) varying over time. Is it still worth
considering a multi-period modeling framework? An answer to this question can be
given by the
value of the multi-period solution
, which is a concept first introduced
by Alumur et al. (
2012
) in the context of a multi-period reverse logistics network
design problem.
The value of the multi-period solution compares the optimal value of the multi-
period problem and the value of a solution found by solving a static counterpart.
A static counterpart is a problem that takes into account the information available
for the planning horizon and looks for a static (time invariant) solution. Given the
optimal solution to a static counterpart, one can consider again the original multi-
period problem and set such solution for all periods of the planning horizon. If, by
doing so, we obtain a feasible solution to the multi-period problem, the difference
between its value and the optimal value of the multi-period problem gives the value
of the multi-period solution. In general, several static counterparts can be associated
with a multi-period problem. Depending on the one that is considered, a different
static solution may be obtained. Accordingly, the value of the multi-period solution
may not be unique.
In a multi-period facility location problem, costs, demands, and possibly other
parameters are assumed to change over the planning horizon. A static counterpart
is a problem that looks for a static location for the facilities, i.e., that can be
implemented at the beginning of period 1 and remain unchanged until the end of the
planning horizon. One possibility for building a static counterpart is to somehow
