Geoscience Reference
In-Depth Information
the fixed points are close to center, otherwise they are close to the boundaries of
the market. The paper then introduces fixed and variable relocation costs, which are
subsequently used to force an equilibrium.
14.4.14
UD2a, Plane, von Stackelberg Solution
Drezner (
1994b
) locates a single new facility in the Euclidean plane with a winner-
take-all allocation rule. For each customer, the paper determines a circle around
the customer location, so that any facility located inside that circle will capture the
customer. Such circles are then constructed for all customer points. This is then used
to optimally locate a new facility with given attraction.
14.4.15
UD2a, Network, Nash Equilibria
Eiselt and Laporte (
1991
) investigate the existence of locational Nash equilibria on
a tree, given an attraction function of the type facility attraction divided by distance
to some power greater or equal to one. When the base attractions of the facilities
are equal, equilibria always exist with either both facilities at the median of the
tree (in case co-location is permitted) or with one facility at the median and the
other adjacent to it in the largest subtree spanned by the median. For unequal base
attractions, if co-location is permitted and the winner-take-all allocation rule applies,
then an equilibrium never exists; otherwise (i.e., with co-location permitted and an
allocation proportional to the attractions and in case location at the same vertex is
prohibited), equilibria may or may not exist.
14.4.16
UD2A, Network, von Stackelberg Solution
von Stackelberg problems in networks enjoy quite some popularity among oper-
ations researchers. The main reasons are their relative tractability (the problems
can, at least in their basic form, be formulated as integer linear programming
problems). This is very much in contrast to the leader's problem, which is a bilevel
integer programming problem. Suárez-Vega et al. (
2007
) employ an attraction
function, defined as facility weight divided by an increasing concave function of
the distance. Customers purchase proportionally from the facilities they are most
attracted to,
provided
they are attracted to them by a measure that exceeds a
minimally acceptable threshold. The authors describe a finite dominating set. They
deal with the case of a single new facility, but the results generalize to multiple
facilities (even though the computations will be more complex). Benati (
2003
)
does not fix the number of facilities the follower is going to locate. Customer
