Stochastic Location Models with Congestion - Location Science - page 443

represents the negative of the total expected time spent by the customer in the system
(until the end of service).
There are two other common specifications of U.d; w /. The simpler one is

U D .d; w / D d d;

(17.28)

i.e., customer's utility is simply proportional to the traveling distance (representing
the travel cost) and is independent of the waiting time. This is a very popular
specification form appearing (often implicitly) in numerous SLCIS models. While
the lack of dependence on w may seem counterintuitive, it is usually justified
by assuming that customers do not have advance knowledge of waiting times at
the facilities and thus must make their decisions based on travel times only. This
justification is not entirely convincing sine in a steady-state system some learning
about expected waiting times should, presumably, occur. Alternative justification is
that the waiting costs are dominated by the travel costs. Perhaps more importantly,
as will be seen below, specification ( 17.28 ) avoids many technical complications
that occur when a more general utility structure is used and can thus be treated as
an approximation.
Another natural specification is the log-linear form

U E .d; w / D exp . d d w w /;

(17.29)

which is quite similar to ( 17.27 ) with the advantage of the utility being non-negative,
convex and bounded by 1. Note that U E .d; w / D 1 when d D w D 0,i.e.,when
the customer incurs neither travel nor waiting cost, and U E .d; w / ! 0 as d; w !
1 . This makes it convenient to interpret U E .d; w / as the proportion of maximum
available demand realized from customer j if this customer is faced with travel
distance d and expected wait w . This interpretation will be useful when describing
elastic demand models below.
Finally, we note that a utility function can be defined in terms of service measures
other than the expected waiting time W — one can use the probability of waiting
P.W q >0/, or any other performance measure of the queuing system operated at
the facilities. The specifications of the utility can also be generalized to incorporate
other decision variables, such as the price charged by the facility operator for service
(see Berman et al. 2014 for an example).

17.3.2 SLCIS Models with Customer Reaction

Once a utility function is specified, it should be possible to specify the customer
reaction as well. At a first glance, this seems fairly straightforward: a SLCIS model
with customer reaction can be viewed as a bi-level game, where the decision-maker
first specifies the number, locations and capacities of the facilities (i.e., values of m,
y i and i for i 2 I) and then each customer selects a utility-maximizing strategy.
Unfortunately, as we will see shortly, complications quickly arise. This has to do,