Geoscience Reference
In-Depth Information
a
b
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4
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10
c
d
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Fig. 12.3
Access network with direct connections (
a
), multi-stops (
b
), tours (
c
), and complete
subgraphs (
d
)
reasons, the use of flow-independent costs may not only miscalculate the overall
flow cost of the hub network, but could also erroneously select the optimal set of
hub nodes and the assignment pattern of O/D nodes to hubs.
Several authors have pointed out these anomalies and different hub location
models able to capture the flow-dependency of discounted costs have been proposed.
The first hub location model that explicitly accounts for scale economies by
allowing discount factors on hub arcs to be a function of flows was introduced
in O'Kelly and Bryan (
1998
). This model, referred to as FLOWLOC, uses a non-
linear cost function, in which costs increase at a decreasing rate as flows increase, to
compute the flow cost in each hub arc. For any amount of flow, the cost is assumed
to be always less than the linear cost associated with a constant discount factor. This
function is approximated with a piecewise linear function to obtain a linear integer
programming formulation for the problem. Bryan (
1998
) provides some extensions
of the FLOWLOC model that relax the assumption of full interconnection between
hubs, by using a minimum threshold value to activate a hub arc, and that incorporate
a flow-dependent cost function for both the hub and access arcs. Klincewicz (
2002
)
shows that, once the location of the hubs is known, the FLOWLOC model can be
reduced to a classical UFLP. Horner and O'Kelly (
2001
) present a different non-
linear flow cost function based on link performance functions commonly used in
