Geoscience Reference
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results where service improvements result in a net loss of demand. Fortunately, this
is not the case as shown in the following result from Berman et al. (
2014
):
Theorem 17.4
Fo r
j
2
J
,let
j
.d
j
;
w
j
/
be the equilibrium demand rate when the
travel time is
d
j
and the expected waiting time is w
j
.Then
j
is non-increasing in
d
j
and w
j
(strictly decreasing when the utility function is strictly decreasing in the
corresponding parameter).
Thus, with a reasonably behaved utility function, when the service offered to
customers at j
2
J is improved in terms of either travel distance or waiting time,
or both, the demand rate increases, leading to higher revenue for the decision-
maker (for this customer node). Since nodes that are currently not served (i.e., with
P
i
x
ij
D
0) can be treated as having the travel distance that is so high that the
demand rate is negligibly close to 0, the decision to serve these nodes by assigning
them to
any
open facility can be treated as reducing the travel distance. This leads
to the following result:
Corollary 17.1
In the elastic demand case, there exists an optimal solution to
SLCIS where every demand node is served.
17.3.2.3
FR: Full Response Models
In this model class, the customer response to facility location and capacity allocation
decisions includes both the level and the allocation of demand. Thus, the equilibrium
values of x
ij
and
j
are described by a system that includes flow equilibrium
conditions (
17.36
)-(
17.38
), as well as the elastic demand equilibrium (
17.40
). The
existence and uniqueness of equilibria are assured by the following corollary:
Corollary 17.2
The equilibrium existence and uniqueness results of Theorems
17.1
and
17.2
extend to the FR model class.
The reader can refer to Brandeau et al. (
1995
) for further details; note that the
uniqueness result has the same limitations as for the AR models (i.e., uniqueness
can only be guaranteed with respect to the values of the objective, provided the
objective function is suitably defined). Also, just as in AR models, this corollary
requires fractional allocation vectors x
ij
.
The computation of equilibrium solutions presents even more challenges than
for AR models. This has lead to an alternative specification of demand allocation
vectors described in the following section.
17.3.2.4
FR and AR Models with Proportional Allocations: Market Share
Models
Our development of AR and FR models was based on the assumption that customers
allocate their demand in a utility-maximizing fashion. As we have seen, this
