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aggregate the information available for all periods. For instance, consider time
varying demands. If facilities are uncapacitated, then several possibilities emerge
for aggregating this information: (1) the demands can be averaged over the planning
horizon, or (2) a reference value can be determined (e.g., the maximum value
observed throughout the planning horizon). If additional constraints exist (e.g.,
capacity constraints) then, choosing a reference value may render the resulting static
solution infeasible in some periods. In this case, one possibility for building a static
counterpart is to define the (time invariant) demand of each customer according
to the maximum value observed across all periods. In any case, the adequate
aggregation of multi-period data is very much problem-dependent.
In order to clarify the above explanation, we consider problem (
11.23
)-(
11.27
),
(
11.39
), and (
11.40
). A static counterpart can be obtained simply by considering the
UFLP with operation costs f
i
, i
2
I, equal to the average of the values f
it
, t
2
T
and distribution costs c
ij
, i
2
I, j
2
J, given by the average of the values c
ijt
, t
2
T .
When the value of the multi-period solution is obtained by aggregating the data
for all periods we refer to it as a
weak
value of the multi-period solution. On the other
hand, we obtain a
strong
value of the multi-period solution when no aggregation is
performed in the data. This is a possibility in some cases, namely when we can add a
set of constraints to the problem stating that some or all decisions are to be the same
in all periods of the planning horizon. In the case of a multi-period facility location
problem, a static counterpart must define a static location, i.e., a solution in which
the location of the facilities is the same for all periods of the planning horizon.
Consider, for instance, problem (
11.41
), (
11.42
), (
11.44
), (
11.45
), and (
11.51
). A
static counterpart yielding a strong value of the multi-period solution is obtained by
setting
z
it
D
0t
D
1;:::;
j
T
j
1; i
2
I
c
;
z
it
D
0t
D
2;:::;
j
T
j
;i
2
I
o
:
These conditions simply impose that the status of each location does not change
during the planning horizon. Therefore, the set of operating facilities will be the
same across all periods.
To the best of our knowledge, the only paper within the context of facility
location, in which the relevance of using a multi-period modeling framework is
measured is the one by Alumur et al. (
2012
).
11.7
Conclusions
In this chapter, we have presented and discussed several essential aspects related
with multi-period facility location problems. The existing literature reveals that the
topic has achieved a significant level of maturity. From a modeling point of view, it
is now clear how to capture several features of practical relevance and how to tackle
