Geoscience Reference
In-Depth Information
where weights
d
;
w
can be estimated from the available consumer demand
allocation data using the MNL methodology.
The advantage of the proportional allocation approach is that the values of
x
ij
are directly computable from (
17.42
)or(
17.43
) without having to solve the
cumbersome flow equilibrium equations. Nevertheless, it is important to recognize
that an equilibrium condition is implicit in the definition above, even in case of
model
s w
ith inelastic demand: the expressions for x
ij
above are functions of waiting
times W
i
, which, in turn, are functions of x
ij
. Thus, (
17.42
) together with waiting
time specification (
17.15
) and facility-level demand specification (
17.6
)forma
system of non-linear equations. A solution to this system represents an equilibrium
demand allocations and waiting times. In case of FR models, one also has to add
the elastic demand specification (
17.40
) and the equilibrium solution includes the
demand rates at each customer node. Thus, the issues of existence and uniqueness
of the equilibrium must be addressed. These were examined in some detail by Lee
andCohen(
1985
). The existence follows directly from standard fixed-point results
and the continuity of x
ij
in (
17.42
) and is based on Theorem 1 in Lee and Cohen
(
1985
):
Theorem 17.5
There exists an equilibrium solution
.x
ij
;W
i
;
j
/;i
2
I;j
2
J
to
the proportional allocation model.
Lee and Cohen (
1985
) also examine uniqueness and stability of equilibria, where
stability refers to whether a system where customers start with some arbitrary
demand allocations, evaluate their utilities and then re-allocate according to (
17.42
)
will naturally reach an equilibrium. They derive sufficient conditions for both
uniqueness and stability. In the context of our AR and FR models, their results imply
the following:
Theorem 17.6
1. For AR models with proportional allocation the equilibrium is unique and stable
2. For FR models with proportional allocation the equilibrium is unique and stable
if
@
j
@
u
j
;
for all
j
2
J
where u
j
D
P
i2I
U.d.i;j/;W
i
/y
i
is the utility derived by customers at
j
from
all open facilities.
1
u
j
j
The condition in part (2) above states that the elasticity of demand from node j with
respect to the utility provided by all facilities must not exceed 1. As shown in Lee
and Cohen (
1985
) this holds automatically when the demands are given by (
17.41
),
as well as by many other common specifications of demand (we note that weaker,
but harder to verify, sufficient conditions are also provided in Lee and Cohen
1985
).
We close this section by noting that the analysis in Lee and Cohen (
1985
)
assumes that all location and capacity allocation decisions have already been made.
