Stochastic Location Models with Congestion - Location Science - page 437

assume that WC . w / is non-negative and non-decreasing, noting that many models
make the simplifying assumption that the waiting cost is proportional to w . The total
expected waiting cost in the system can now be expressed as
SWC D X
j2J

X

WC.W i /x ij : (17.16)

i2I

In view of non-linear dependence of the expected waiting time W i on the decision
variables, SWC is a non-linear function even when the waiting cost is assumed to be
linear.
We note that since the waiting cost is only incurred by customers who are
assigned to some facility, we should also add a penalty term for customers that
are not assigned to any facility (i.e., not served)—otherwise the model may have
an incentive to not assign customers even if service capacity is available. The
“intentionally lost demand” customers may be represented in the revenue term
described later (i.e., they are treated as an opportunity cost of lost revenue).

Alternatively they can be represented by a term p P j2J 1 P i2I x ij which may
be added to the SWC expression above, where p represents the penalty for choosing
to not service a customer.
There are two potential issues with using ( 17.16 )asthe sole measure of service
quality (in terms of waiting times) at the facilities. First, as with the system travel
cost, a small value of SWC does not necessarily ensure that all customers are
receiving adequate service—a small expected waiting time at one facility may
“hide” a large expected waiting time at another. Thus, one may want to add the
constraints (these are traditionally stated in terms of waiting time, rather than system
time; we follow this tradition):

W q

i EW; i 2 I;

(17.17)

where EW represents the acceptable maximum waiting time at any facility.
Second, the expected waiting time may not be sufficient to express the desired
service quality; we may wish to ensure that most customers experience no waiting
at all or that the probability of “long” waits is sufficiently low. For this we need to
consider a constraint of the form

P.W q i >T/ Ǜ T ;i 2 I; (17.18)

where P. / is the steady-state distribution of W i , T>0is the specified threshold
for the waiting times, and Ǜ T 2 .0;1/ is the maximum acceptable probability
of waits longer than T at any facility. For example, Ǜ 0 represents the maximum
acceptable proportion of customers that must wait for service at any facility.
Both ( 17.17 )and( 17.18 ) above are examples of Service level Constraints (SCs)
that are quite common in SLCIS models. Since ( 17.17 ) refers to the expected
behavior of the system, while ( 17.18 ) refers to the probability of occurrence of