Geoscience Reference
In-Depth Information
assumption leads to flow equilibrium-type conditions with the ensuing structural and
computational difficulties. An alternative approach is based on the assumption that
customers allocate their demand among many (possibly all) facilities in proportion
to the utility derived from these facilities. Essentially, each customer node j
2
J is
viewed as a “market” with facilities competing for the shares of this market. These
models, that are axiomatically rooted in the stochastic utility theory, have generated
a large body of literature, particularly in economics and marketing; in the latter they
are accepted as the dominant model for customer choice in the presence of many
substitutable alternatives (e.g., predicting market share of a particular brand when
many other brands are available).
In the competitive location literature these models have appeared under many
names, including “competitive interaction models”, “Huff-type models”, “gravity
models”, “multinomial logit models”, “market-share models”. While there are
minor specification differences between these, the basic structure remains the same;
we refer the reader to the recent review by Berman et al. (
2009a
).
Since SLCIS models of AR and FR type can be regarded as bi-level games played
between the decision-maker and the customers, proportional allocation mechanism
can be applied to the SLCIS context as well (in effect, it specifies the solution to the
non-cooperative game played between customers once the decision-maker's strategy
is specified). The specification is quite simple: for customers at j
2
J and (open)
facility at i
2
I, the demand allocation is given by
U.d.i;j/;W
i
/y
i
P
k2I
U.d.k;j/;W
k
/y
k
x
ij
D
;
(17.42)
where the numerator represents the utility derived from facility i and the denomina-
tor is the total utility derived by customers at j from all open facilities. Note that if
there are any pre-existing competitive facilities that may a
ttr
act customer demand,
they should be included as an extra sum
P
k2C
U.d.k;j/;W
k
/ in the denominator,
where C is the set of competitive facilities. To simplify the exposition, we will
assume no competitive facilities in the remainder of the current section.
This specification implies that the demand allocations are fractional, and the
demand rate from j attracted by facility i is (as before)
j
x
ij
,where
j
is elastic
for FR models and inelastic in AR case.
Note that from Eq. (
17.42
) it follows that market shares add up to 1,i.e.,
all available demand from j is served. This may be unrealistic if none of the
available facilities provide good service to j. The easy modification is
to
introduce a
“dummy” facility 0, representing “no service”, and letting U.d.0;j/;W
0
/
D
u
j0
—
a constant representing the utility value of not getting served (e.g., the customer
may choose to consume a different product). The popular Multinomial Logit (MNL)
specification (McFadden
1974
) employs exponential utilities, leading to
exp.
d
d.i;j/
w
W
i
/y
i
P
k2I
exp.
d
d.k;j/
w
W
k
/y
k
x
ij
D
;
(17.43)
