Stochastic Location Models with Congestion - Location Science - page 441

17.3 Customer Response: Demand Levels and Allocations

In this section we discuss the two remaining key issues in SLCIS models: the
mechanism determining the allocation of customer demand to facilities, represented
by x ij variables, and the amount of demand j generated by customers at j 2 J.
In location modeling two approaches for allocating customer demand to facilities
are generally considered: directed choice , where the same decision-maker determin-
ing the number and locations of the facilities also has the power to assign customers
to the facilities in a way that will optimize the model objective, and user choice
where customers self-assign to facilities based on maximization of their own utility
functions, which may not be aligned with the overall model objective. For example,
a common customer utility function is the travel distance. Thus, in a user choice
environment, each customer will select the closest facility, while in the directed
choice case a customer may be assigned to a further facility even when a closer one
is open (if such assignment reduces the overall facility cost).
The same framework can be applied to the SLCIS models. However it may be
more useful to also classify the models in terms of the assumed customer reaction.
We differentiate four classes of models:

Type NR: Models with no customer reaction: customers do not control the
demand allocations and the demand rates are fixed (directed choice with inelastic
demand)
Type AR: Models with allocation-only reaction: customers select utility-
maximizing facilities, but the demand rates are fixed (user choice with inelastic
demand)
Type DR: Models with demand rate-only reaction: customer do not control the
demand allocations but do determine the demand rates (directed choice with
elastic demand)
Type FR: Models with full customer reaction: customers control both, the alloca-
tion of demand (by selecting the utility-maximizing facilities) and the demand
rates (user choice with elastic demand).

This classification is summarized in Table 17.1 .
The NR models correspond to the standard directed choice assumptions in the
literature: the values of the assignment variables x ij are entirely controlled by the
decision-maker and must only satisfy the basic constraints ( 17.3 )-( 17.5 ). One may
also interpret such models as describing a “social optimum” (also known as “first
best solution” in economics)—the customers will accept whatever assignments are
needed to optimize the overall system objective, even if that means that some of

Table 17.1 Model
classification by customer
response

Demand allocation
Decision-maker Customer

Inelastic demand NR AR

Elastic demand DR FR