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vehicle fleet, time horizon, etc.) give rise to a large variety of LRPs. A comprehen-
sive recent classification, following the ideas already presented in Laporte ( 1988 )
can be found in Borges Lopes et al. ( 2013 ).
The main difficulty when modeling LRPs through mathematical programming
formulations is to ensure that each vehicle tour is connected to exactly one
facility; that is, there are no closed tours visiting only customers, and there are
no paths connecting two different facilities. Therefore, incorporating the design of
vehicle routes within facility location problems entails a relevant additional level
of difficulty. Furthermore, as some authors argue, facility location is most often a
strategic decision, while vehicle routing is operational. These facts have discouraged
many researchers from considering combined LRPs. However, although routing
decisions can be readjusted relatively often once the facilities are established, the
possible configurations of the routes are strongly conditioned by these locations.
Therefore, if locations are chosen without taking into account the routing component
of the final system, initial savings in the facilities set up costs may not compensate
for large losses in distribution in the long run. Consider, for instance, the extreme
situation depicted in Fig. 15.2 . In this example, assume that the capacity of any
of the two candidate facilities (black squares) is sufficient to serve all customers
(white circles), and there is only one vehicle available at each location, also with
a large enough capacity. If one single location is to be chosen and routing costs
are ignored (i.e. if an uncapacitated facility location problem is considered in this
setting) obviously, the facility will be located at 2. However, if a tour needs to
be defined to serve all the customers once this facility is set, its cost will be
2M C .11M/=6 ' 7:76M. On the other hand, if the facility is set at node 1,
a better route, with cost 2M ' 6:28M can be defined. Since distribution is most
often a repetitive activity, this extra routing cost for having chosen facility location
2 will be incurred regularly and, after some time, these accumulated extra costs can
be larger than the initial possible savings in set up costs.
π M
6
M
1
2
Fig. 15.2 Influence of
facility location on the
routing costs
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