Stochastic Location Models with Congestion - Location Science - page 434

to be d 0 .i;j/ D TC .d.i;j// for all j 2 J; i 2 I and use this new measure in
place of the original one. Thus, after suitably redefining distances and without loss
of generality, we can write

STC D X
j2J

X

LJd.i;j/ j x ij ; (17.10)

i2I

where LJ>0is a parameter relating the travel cost to other terms in the objective
function (defined below). We will use this linear form in place of ( 17.9 ) from this
point on.
Of course, a possible concern with the previous expression is that the short travel
cost of one customer will be added to the long travel cost of another, resulting in the
total quantity that may look reasonable, but will still provide poor service to some
customers. To assure that no customer faces an unreasonably long travel distance,
one can impose coverage constraints :
X

d.i;j/x ij R for all j 2 J; (17.11)

i2I

where R>0is the “coverage radius”, i.e., the maximum allowed travel distance
for a customer to be “covered” by a facility (this constraint should be interpreted
as referring to the “adjusted ” distance measure that incorporates the travel cost,
as discussed above). We note that most SLCIS models will include either ( 17.10 )
or ( 17.11 ); while, in principle, both can be used in the same model, such usage is
rare.

17.2.3.2 Congestion Costs and Service Level Constraints

While travel-related costs are present in all classes of location models covered in
the current volume, the congestion-related costs and constraints are, of course, a
defining feature of the stochastic location models with congestion. As discussed
earlier, the two common performance measures in a queueing system operated by
each open facility i 2 I are the system waiting time W i (recall that this includes the
service time; a closely related measure is W q i which only covers the waiting time in
queue) and the number of customers in the system L i , which are random variables
with certain steady-state distributions. The most co mm on way to define congestion
costs is in terms of expectations of these quantities, W i and L i , respectively. Since
the two are related by Little's Law, we will focus on the former (which is also more
commonly us ed) . For an M=G=1 queue, the expression for the mean waiting time
in the system W can be found in any standard reference on queuing (see, e.g., Gross
and Harris 1985 , p. 255):

D 1 C 2

W D W q C 1

1

1

C 1

(17.12)

2