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are available. We emphasize that this tractability rests in the static nature of
customer-to-facility allocations (the demand allocations are determined once and
then remain in force for all states of the system). Thus, SLCIS models where
customers decide which facility to visit based on the current state of the system
(e.g., based on posted information about current waiting times), or where other
dynamic customer allocation mechanisms may be present, are likely to be closer
(in terms of tractability and solution approaches) to models with mobile servers.
On the other hand, models with mobile servers where static and non-intersecting
service regions are assumed for all facilities (effectively assuming away dynamic
customer reallocation) are quite similar to SLCIS models; many of the mobile server
models reviewed in Berman and Krass (
2002
) fall into this group. Thus, instead
of differentiating stochastic location models with mobile vs. immobile servers, it
would be more accurate to differentiate models with dynamic vs. static customer
assignments.
17.2.3
Costs, Revenues, and Constraints
To complete the description of the system it remains to specify two components: (1)
the mechanisms by which customers are “allocated” to the facilities, expressed by
the variables x
ij
(which would also determine the actual demand rates
j
;j
2
J),
and (2) the overall system costs and constraints assuring acceptable service levels.
We will postpone the discussion of (1) until Sect.
17.3
, focusing on the costs and
constraints in the current section and treating values of the key location, allocation,
capacity assignment and demand level decisions
f
y
i
;x
ij
;
i
;
i
g
;i
2
I;j
2
J as
fixed.
17.2.3.1
Travel Cost and Coverage Constraints
We assume that for each customer j
2
J and potential facility location i
2
I a
distance metric d.i;j/ is defined, satisfying the regular properties of distance. The
travel cost function
TC
.d/ for d
0, representing the cost of traveling distance d
is assumed to be non-decreasing and non-negative. This yields the System Travel
Cost of
STC
D
X
j2J
X
TC
.d.i;j//
j
x
ij
;
(17.9)
i2I
where we assume that constraint (
17.4
) ensures that customers are only assigned to
open facilities. This expression merely states that the system travel cost is the sum
of travel costs of all customers to their assigned facilities. We note that a frequent
assumption is that the travel cost is a linear function of distance. More generally,
since both J and I are discrete, one could simply redefine the distance measure
