Geoscience Reference
In-Depth Information
an aggregate, typically of a census tract or some other group of customers), some
members among the group may have different rankings and prefer what, on average,
is a higher-ranking facility.
This heterogeneity of customer tastes can be dealt with in different ways. One
such possibility is to use a
AD2: proportional allocation.
This allocation rule will allocate a customer's demand according to the relative
utility a customer has for a facility. For instance, the proportion of customer
i
's demand to facility
j
according to Hakimi's (
1990
) “proportional” rule equals
u
ij
ı
X
k
u
ik
. As an example, if a customer faces a duopoly, for whose facilities he has
computed utilities of 3 and 7, respectively, he will satisfy 30 and 70 % of his total
demand at the two respective facilities. Hakimi (
1990
) also designed a hybrid rule
based on AD1 and AD2. He refers to it as a “partially binary” allocation. According
to this rule, customers consider only the closest facility or branch of each of the
competing firms, and they then distribute their demand proportionally among those
branches. Suárez-Vega et al. (
2004
) investigated AD1, AD2, and the aforementioned
hybrid in detail.
Consider now probabilistic allocation functions. A natural extension of Reilly's
(
1931
) argument of attraction functions was Huff's (
1964
) allocation function,
which allocates a proportion of a customer's demand to a firm based on the firm's
attractiveness and its distance to the customer,
AP1a:p
ij
D
A
j
=d
ij
X
.
A
k
=d
ik
k
Huff suggested the selection of a location from a pre-specified set of locations,
whereas Drezner (
1994a
,
1995
) proposed a model for finding the best location any-
where in the plane. A multidimensional generalization of this idea was proposed by
Nakashani and Cooper (
1974
), the so-called multiplicative competitive interaction
model, or
MCI
for short. Assuming that
u
ijk
denotes the utility customer
i
has for
feature
k
of store
j
,let
p
ij
denote the probability that a customer at site
i
makes a
purchase at store
j
. The parameter ' reflects how sensitive is
p
ij
to feature
k
.The
MCI
model then asserts that
Y
u
Ǜ
k
ijk
X
k
Y
AP1:p
ij
D
.
u
Ǜ
`
ij
`
j
`
Following the arguments of McFadden (
1974
), the use of the probabilistic utility
function UP1 leads to the demand allocation rule
AP2:p
ij
D
e
.
R
i
p
j
td
ij
/
=
X
e
.
R
i
p
k
td
ik
/
=
,
