Geoscience Reference
In-Depth Information
by customer response type, as well as by the utility function used, if applicable.
The following three columns indicate the main underlying system assumptions: the
nature of the queuing system, and whether the number of facilities and the number of
servers are flexible or not. The next two columns identify the presence of coverage
and service level constrains. The following five columns indicate the presence of the
specific terms in the objective function. The last two columns briefly describe the
solution approach and any additional comments.
17.5.1
Coverage-Type Models
Coverage-type models aim to design the system that provides
adequate
service
to customers, where adequacy is usually defined through travel distance and
congestion delays, which are controlled through coverage and service level con-
straints, respectively. The defining feature of this model class is the presence of
coverage constraints (
17.51
). The coverage-type models are denoted by “C” in the
“Model Type” column of Table
17.2
; they include Baron et al. (
2008
), Berman et al.
(
2006
), Kakhki and Moghadas (
2010
), Marianov and Serra (
1998
). These models
were among the very first SLCIS models to be considered, dating back to Marianov
and Serra (
1998
), and stem directly from similar models for systems with mobile
servers (see Berman and Krass
2002
for an extensive discussion).
Coverage-type models usually assume that it may not be possible to provide
adequate service to all customers and thus demand losses may occur. The objective
is typically to maximize the “captured” demand, i.e., the total demand of customers
who get adequate service. The travel and congestion costs are not included in
the objective as these are controlled through the corresponding constraints. Earlier
models were of type NR (directed choice); later models tended to be of type AR,
but customer allocations were assumed to be only a function of travel distance,
i.e., the underlying utility is given by (
17.28
), avoiding all complications related to
equilibrium behaviors. It is interesting to note that even though demand is assumed
to be inelastic, the assumption of demand losses can be viewed as (a rather crude)
form of demand elasticity—corresponding to the implicit utility function which has
a stepwise function form, with customers using service provided by the facilities if
coverage and service level constraints are met, and not using it otherwise.
The typical formulation maximizes the objective consisting of (
17.44
) with r
D
1
(i.e., the captured demand), subject to constraints (
17.51
)-(
17.56
). For models of
type AR, one also adds constraints specifying the allocations. These enforce each
customer to travel to the closest available facility. These constraints can be specified
in various forms; see Berman et al. (
2006
) for a discussion.
It can be seen that this leads to a formulation which is a linear mixed-integer
program (MIP), except for the service level constraints. However, as discussed in
Sect.
17.2.3.2
, under some conditions, the latter can be linearized. Recall that a
general service level constraint can be recast as either (
17.20
), requiring adequate
service capacity at each facility, or (
17.21
), placing an upper limit on the allowed
