Stochastic Location Models with Congestion - Location Science - page 459

arrival rate at each facility. When the capacities i are decision variables, these
reformulations remain non-linear. However, if one makes a simplifying assumption
that all facilities have identical service rate (for multi-server facilities, this
implies assuming identical number of servers at all facilities), non-linearities
disappear. This is a common assumption in coverage-oriented (and other SLCIS)
models: Berman et al. ( 2006 ), Kakhki and Moghadas ( 2010 ), Marianov and Serra
( 1998 ) assume identical and pre-specified service rates at the facilities. Under this
assumption, ( 17.21 ) takes the form

i N ;

where the right-hand side is a constant which depends on the desired service level
and is computable in advance. This shows the equivalence of a cover-type SLCIS
model with fixed service rates to the capacitated location problems. Such connection
is discussed at length in Boffey et al. ( 2006 ).
The resulting linear MIP may, in principle, be solved exactly using off-the-shelf
software, such as CPLEX. However, as pointed out in Berman et al. ( 2006 ), the
formulation resulting from the addition of linearized service level constraints and
the “closest assignment” constraints tends to be large and not very tight, causing
computational difficulties for even moderately-sized instances. This has led Berman
et al. ( 2006 ) and other authors to develop heuristic approaches.
Finally, we note an important result from Baron et al. ( 2008 ), who studied a very
general version of the coverage-type SLCIS model, where both the number and the
capacities of facilities are decision variables and the facility-related costs are quite
general (in their version, all customer demand must be served and the objective is
to minimize fixed and variable location costs). They show that, under quite general
conditions, the optimal facility configuration is one that ensures that each facility
sees (approximately) the same demand, i.e., ideally, i D k should hold for all
open facilities i;k 2 I (identical demand may not be possible to achieve when
customer demand originates from discrete nodes and single-sourcing assumption is
made). Once the facility locations are determined, the optimal capacities i can be
determined through a separate optimization model.
This result provides an important insight for coverage-type models: when the
goal is to ensure “satisfactory” service experience, the optimal design should
equalize loads at the facilities. This leads to an “Equitable Location Problem”—
a deterministic problem where one seeks to locate a set of facilities so that the
attracted demand is distributed as evenly as possible. Such problem was addressed
in Baron et al. ( 2007 ), Berman et al. ( 2009b ), and Suzuki and Drezner ( 2009 ).

17.5.2 Service-Objective Models

Service-objective models seek to design a system that optimizes “customer service”
using limited resources. These models are denoted by “S” in the “Model Type”