Competitive Location Models - Location Science - page 374

the market whose locations are known, The paper demonstrates that the follower's
objective function is submodular, and that, given appropriate redefining of the
problem's parameters, the problem can be formulated as an r -median model.

14.4.22 UP2, Plane, von Stackelberg Solution

Drezner et al. ( 2002 ) discuss a medianoid problem in the plane, in which customers'
choices are modeled probabilistic and are based on attraction functions. The
follower's objective is to minimize the probability that the new facility's revenue
falls short of a given threshold. The optimal locations tend to markedly differ from
those that are the result of the maximization of the expected market share, especially
in those cases, in which the probability of failure is relatively small.

14.4.23 UP2, Network, von Stackelberg Solution

The main contribution of the work by Serra and Colomé ( 2001 ) is the comparison
of various customer choice models. The basic setting includes fixed demand at the
nodes of a network, one homogeneous good, and two profit-maximizing firms with
identical cost structures. There are presently q facilities on the market. One new
firm enters the market and attempts to locate p new facilities. Customer behavior
is modeled as follows. Model 1 is the usual all-or-nothing assumption based on
the closest facility, while Model 2 is a multiplicative competitive interaction Model
(Nakashani and Cooper 1974 ), which assumes that the proportion of demand of
customer i captured by facility j equals 1/customer-facility distance raised to the
power of a parameter that indicates a customer's sensitivity with respect to distance,
divided by the sum of such expressions, taken over all facilities. Model 3 is the
standard proportional model, and Model 4 assumes partially binary preferences. It
turns out that the simple Model 1 appears to be most robust, meaning that it has
never more than an 8 % deviation from the solution that is based on the correct
customer behavior.

14.4.24 Summary, Extensions, and Outlook

This chapter has described the basic Hotelling model, outlined its major compo-
nents, described a three-stage procedure that models customer behavior, and has
surveyed the literature regarding results of different models. While many different
features have been included, most models, while they have some explanatory power,
lack many facets of customer decision-making.