Geoscience Reference
In-Depth Information
firms, the market area of each of the firms is a cell in a Voronoi diagram. Each
firm attempts to maximize the area of its Voronoi cell. Voronoi cells are in global
optimum with the hexagonal pattern. It is noted that results in one-and two-
dimensional spaces are markedly different: the pairing in one dimension does not
carry over to the two-dimensional plane. Another attempt in the two dimensional
plane was reported by Okabe and Suzuki (
1987
). The authors use the same concept
as in the previous paper, but locate finite numbers of facilities (32-256) in a bounded
market the shape of a square. Global optimization techniques are sequentially and
repeatedly applied. The result is a honeycomb-type pattern that, however, self-
destructs again and rebuilds. The instability is likely to be the result of “boundary
effects” that distort the results.
Aoyagi and Okabe (1993) consider a Hotelling model in the plane with totally
inelastic demand, identical facilities, and customers who purchase the good from the
closest facility. Customers are assumed to be located in a compact and convex subset
Z
of the two-dimensional Euclidean plane. The authors demonstrate that for
n
D
2,
an equilibrium exists if and only if the market is point-wise symmetric with respect
to some point in
Z
. The firms will then locate at that point. For three facilities, no
global equilibrium exists except maybe in the case of a equilateral triangle.
14.4.4
UD1a, Plane, von Stackelberg Solution
The first author to discuss competitive location problems in the plane given location
leaders and followers appears to have been Drezner (
1981
,
1982
). His contribution
first considers the simple case, in which each firm locates a single facility in the
presence of
n
demand points. The follower's best location is arbitrarily close to that
of the leader. Sorting of angles from the leader's point to the demand points yields
an O(
n
log
n
) algorithm for the follower's problem. The leader's problem (given he
locates one facility and expects the follower to do the same) is shown to be solvable
in O(
n
4
log
n
) time. In case a minimum separation of some prespecified distance
R
is required between leader and follower, the complexity of the two problems is still
O(
n
log
n
)andO(
n
5
log
n
), respectively. Other cases include the problem in which
the leader locates one, the follower
r
> 1 facilities. This problem is easy: the leader
is wedged in and his optimal strategy is to locate right on the point with the largest
demand, as that is all he will get. If the leader locates
p
> 1 facilities and the follower
locates one facility, then the follower's problem can be solved in O(
n
2
log
n
)
time.
Shigehiro et al. (
1995
) consider a duopoly with firms
A
and
B
in a bounded
subset of the two-dimensional plane. Given fixed and equal prices, both firms are
market share maximizers. Given demand at grid points and the one of
A
's two
facilities being already located, firm
B
locates a single facility, followed by firm
A
locating its second facility. It turns out that firm
A
will locate its second facility next
to it competitor's facility, thus re-establishing the pairing of facilities known from
one-dimensional markets. An algorithm for the centroid problem is also described.
