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urban transportation planning. This function is used to model flow-dependent costs
in both hub and access arcs.
Racunica and Wynter (
2005
) study an extension of HLPs arising in the design of
intermodal transportation networks for freight rail. Their model uses another type
of non-linear concave function to model flow-dependent discounted costs only on
the transfer and distribution legs. Contrary to the FLOWLOC model, this function
is based on an efficiency threshold that considers that discounted flow costs should
be higher than the linear cost up to a threshold, and less costly thereafter.
Kimms (
2006
) introduces a different approach for modeling flow-dependent
discounted costs in all the arcs of the network, which is based on fixed-charge
cost functions commonly used in other network design problems. This function
consist of a fixed flow-independent set-up cost and of a variable flow-dependent
(or marginal) cost. This paper presents three different models: an uncapacitated
model, a capacitated model, and a multimodal model with different capacities for
each mode of transportation. Mirzaghafour (
2013
) consider a stepwise function to
model flow-dependent costs on both hub and access arcs. This type of functions are
commonly used to model the transportation cost in most vehicle routing problems
(see Laporte
2009
).
12.4.3
Capacitated Models
Similar to FLPs, an important extension to HLPs is the incorporation of capacity
considerations when designing hub networks. However, in the case of HLPs the
capacity constraints may arise not only at the hub facilities but also at the arcs of the
network. Moreover, when considering capacitated models with multiple assignment
patterns, commodities may be split over several paths and thus, splittable and non-
splittable commodity variants arise. In the former case, commodities are allowed to
be split over several paths between their origins and destinations. However, in the
latter case the commodities cannot be split, meaning that each commodity will be
routed through the network from its origin to its destination through a unique path.
Note that a multiple assignment pattern that allows splitting is highly desirable when
minimizing the total flow cost. However, splitting commodities may not be feasible
in some applications.
Capacitated versions of HLPs with multiple assignments are studied by Campbell
(
1994b
), Ebery et al. (
2000
), and Boland et al. (
2004
), with capacity constraints on
the incoming or outgoing flow at the hubs. Bryan (
1998
) introduces a model in
which capacities are associated with the hub arcs rather than with the hub nodes.
Marín (
2005b
) studies a capacitated model in which commodities are splittable.
Rodríguez-Martín and Salazar-González (
2008
) study another model where com-
modities can be split into several routes. Capacity constraints are imposed on the
incoming flow of each hub, whether it originated from non-hub nodes or from hub
nodes. In addition, an upper limit is imposed on the flow traversing any link of the
network.
