Geoscience Reference
In-Depth Information
One example of a functional form of h that satisfies the required assumptions is the
exponential utility U
E
given by (
17.29
), leading to the popular “exponential decay”
demand specification:
j
.d.i.j/;j/;W
i.j/
/
D
ma
j
exp.
d
d.i.j/;j/
w
W
i.j/
/; j
2
J: (17.41)
While this expression is assumed in several published DR models, most of the
results below apply to more general functional forms as well. Observe that (
17.40
)
implicitly defi
ne
s an equilibrium condition: the left-hand side depends on the
waiting time W
i.j/
at facility i.j/, which is a function of demand
i.j/
D
P
j2J
j
x
i.j/;j
seen by this facility. Thus, (
17.40
) should be seen as a system of
j
J
j
equations that must be solved to yield the actual demand rates; this system
decouples into subsystems consisting of all customers j
2
J with i.j/
D
i for
each facility i with y
i
D
1. Thus, even though the allocation variables x
ij
are fixed
(or, rather, set by the decision-maker) for DR models, the issues related to existence
and uniqueness of equilibria must be dealt with. The following result is based on
Berman et al. (
2014
), where it is established for the case where price r is also a
decision variable.
Theorem 17.3
For any given facility location, capacity, and demand alloca-
tions
y
i
;
i
;x
ij
fo
r
i
2
I;j
2
J
, the
re
exist unique equilibrium arrival rates
j
.d.i.j/;j/;W
i.j/
/
and waiting times
W
i
.
Note that, unlike the case for AR models, this result holds with binary demand
allocations x
ij
(it obviously extends to the fractional allocations as well). As
illustrated in Aboolian et al. (
2012
), as well as in Berman and Kaplan (
1987
),
computation of the equilibrium demand is relatively simple in this case, based on
the fixed-point iteration approach.
An interesting feature of the DR model is that it is self-regulating: as waiting
times become longer at the facilities, customer demand is automatically reduced.
Thus, the system stability is assured by (
17.40
) without the need for explicit
constraints (
17.7
). Moreover, even though customer assignments are “dictated” by
the decision-maker through the specification of x
ij
, assigning customer j to a more
distant or more congested facility leads to lower demand
j
, with the resulting
loss of revenue. Thus, the model assures that customer assignments must take
customer utilities into consideration, while avoiding the complexities of full traffic
equilibrium treatment. In fact, Aboolian et al. (
2012
) report (based on computational
experiments) that optimal solutions where some customers are
not
assigned to their
utility-maximizing facility are quite rare, though they do occur.
The behavior of DR model involves an interesting feedback loop: as the service
offered by the facilities is improved (by locating the facilities closer to customer
nodes, or allocating more capacities to the facilities), the customers respond by
generating more demand (positive feedback), which leads to increased congestion
at the facilities, leading to reduced demand (negative feedback). Thus one could
legitimately ask whether models with elastic demand may lead to counter-intuitive
