Location and GIS - Location Science - page 498

Otherwise demand may be assumed to be concentrated on discrete points. In any
case, it is always possible to transform continuous into discrete demand and vice
versa through appropriate procedures. However, during these operations particular
attention should be paid to approximations and errors introduced in the model
(Current and Schilling 1990 ; Francis et al. 2002 ).
When facilities provide different types of services, demand should concern
several kinds of services and the corresponding FLPs are referred to as multi-
commodity problems. Depending on each particular application, demand can be
deterministic or stochastic. In both these cases, it can be estimated either by
combining current data and/or attributes or by using appropriate forecasting tools.
Interactions Between Elements of a Problem In a FLP mainly two kinds of
interactions have to be taken into account: customer-facility interactions and
facility-facility interactions. In some applications customer-facility interactions
concern how customers patronize their own facilities or how they are “allocated”
to facilities. In some cases customers are free to decide on the basis of a utility
function which, in general, combines attributes of facilities and distances between
customers and facilities while, in other cases, customers are obliged to patronize
certain facilities according to given rules. Facility-customer interactions may also
concern the determination of the intensity of the effects produced by facilities to
the customers. This is typical, for instance, in problems where risks and/or damage
generated by obnoxious activities have to be evaluated on the population living in
the area around the facility position.
Facility-facility interactions take into account how facilities interact with each
other to capture the available demand. In some cases there is competition in order
to capture as much of the demand as possible (i.e. commercial stores of different
companies). This aspect is also known as cannibalization effect. On the other hand,
in some applications facilities are located in such a way that they cooperate in order
to assure a certain level of accessibility to the users (i.e. bank offices, public service
sites, franchising stores).
Objective Function(s) Location decisions can be made according to different
criteria or objective functions whose choice mainly depends on the nature of
facilities (desirable or undesirable). In the case of desirable facilities, efficiency is
the most commonly used criterion. Efficiency is typically associated to costs, and
distance is the most common proxy for costs. For this reason, objective functions are
in most cases expressed as functions of distances between customers and facilities,
possibly weighted by the demand associated with each customer.
Denoting with p the number of facilities to be located, problems differ according
to whether p is pre-defined or a decision variable. In the first case, the minimization
of the sum of the weighted distances between demand points and facilities to be
located (minisum objective) is the typical objective of the well known class of
p-median problems (Cooper 1963 ;Hakimi 1964 ; ReVelle and Swain 1970 ). When
p is a decision variable, the objective to be adopted is usually the minimization