Geoscience Reference
In-Depth Information
where
z
ih
is a binary decision variable taking the value of 1 if
i
D
h
and 0
otherwise, with the constraints
P
hD1
z
ih
1;i
2
I added to the model. Now
consider the function f.L/
D
L
1CL
. It is concave, and can thus be represented as
the minimum of tangent lines, yielding a linear form. This can be used to represent
the expression (
17.61
) as an infinite set of linear constraints (note that the objective
is already linear, in terms of the new variable L
i
). The resulting MIP can be solved
through a column generation approach. The reader should refer to Elhedhli (
2006
)
for details.
In summary, the simpler structure of balanced-objective models allows for
effective exact approaches to be developed. Another interesting observation is that
the “location-allocation” and “capacity determination” sub-problems often separate.
As noted earlier, these models, being of type NR, may assign individual customers
to rather distant facilities. However, since the travel cost is in the objective function,
these “undesirable” assignments can be controlled by increasing the corresponding
cost coefficients. The computational results in Castillo et al. (
2009
) suggest that
when travel costs are “reasonably” high, the overwhelming majority of customers
(over 99 % in the instances solved) are assigned to the closest open facility in the
optimal solution.
17.5.4
Explicit Customer Response Models
The final class we consider consists of SLCIS models where “explicit” customer
response mechanism is specified, i.e., they are of types AR, DR, or FR. These
models are listed in Table
17.4
. The demand in these models is generally elastic,
though in a few cases elasticity is specified implicitly through demand losses due
to blockages. The objective always includes the revenue term (
17.44
), and may also
include the facility cost terms (
17.47
), unless the number of facilities and servers is
given.
While this class of models has received much recent attention, the earliest
publications date back to the very beginning of the SLCIS modeling: see Berman
and Kaplan (
1987
). Some of the seminal early work is described in Brandeau et al.
(
1995
).
Many of the technical issues related to this class of models have been covered in
Sect.
17.3.2
. The problem of determining the optimal location for a single facility
(Berman and Drezner
2006
; Berman and Kaplan
1987
; Tong
2011
;Bermanetal.
2014
) can be solved exactly. However, the treatment of the multi-facility case
is generally quite difficult since, as noted earlier, in addition to the non-linear
objective function the underlying models include the feedback loop between the
customer demand and congestion and/or the equilibrium conditions for facility-
client allocations, or both. Thus, heuristic approaches are almost always employed
for multi-facility models. These heuristics are usually two-level: at the lower
level they incorporate subroutines for computing the equilibrium solutions (using
