Geoscience Reference
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a
b
Fig. 12.1
Solution network of a hub node location problem (
a
) and a hub arc location problem (
b
)
node and no hub arc, or two hub nodes and a single hub arc. Moreover, because of
Assumptions 2 and 4, each collection and distribution leg, if present, contains only
one access arc. O/D paths are thus of the form .i;k;m;j/,where.k;m/
2
N
N
is the ordered pair of hubs to which i and j are allocated, respectively. Note that
these paths contain one, two or at most three arcs, depending on the number of
visited hubs and on the function of origins and destinations (i.e. hub or non-hub
nodes). For each O/D pair, the flow cost of routing W
ij
along the path .i;k;m;j/
is then given by F
ijkm
D
W
ij
d
ik
C
Ǜd
km
C
ıd
mj
; where ; Ǜ,andı represent
the collection, transfer and distribution costs along the path. To reflect economies of
scale between hubs, we assume that <and <ı.
Figure
12.1
a shows an example of a solution network of a HNLP in which
different structures on O/D paths arise (squares represent hub nodes and circles
represent non-hub nodes). The path .1;2;9;10/ is a two-hub path formed by the
access arcs .1;2/, .9;10/ and the hub arc .2;9/. The path .2;2;9;6/is also a two-
hub path but containing only the access arc .9;6/ and the hub arc .2;9/. The path
.3;3;9;9/is yet another two-hub path formed only by the hub arc .3;9/. The path
.1;2;2;8/is a one-hub path containing only the access arcs .1;2/ and .2;8/.The
path .7;8;8;8/is also a one-hub path containing the single access arc .7;8/.
In HALPs, hubs are not necessarily fully interconnected due to the set up cost
on the hub arcs or because additional conditions on the network topology are
imposed. This causes O/D paths to become more involved, since they may use more
than three arcs and visit more than two hub nodes. Similar to HNLPs, because of
Assumptions 2 and 4, each collection and distribution leg, if present, employs either
one access arc or one bridge arc. However, the transfer leg can now use several
bridge and hub arcs, depending on the particular assumptions considered on the
structure of O/D paths.
To simplify the routing decisions in HALPs, an additional assumption stating
that O/D paths contain at most one hub arc can be considered. This limits paths
to have at most three arcs, being the first and last ones either access or bridge
arcs and the intermediate arc, if it exists, a hub arc. As mentioned in Campbell
et al. (
2005a
), this assumption is used to increase service level in classical HLPs
