Geoscience Reference
In-Depth Information
non-linear optimization techniques) for a given location set. At the upper level they
try improvement strategies to the determine a good set of open facilities, often using
meta-heuristics. As in the case of balanced-objective models, the determination
of the optimal capacity at a facility can often be done through a separate exact
optimization procedure, for a given location and customer-allocation scheme.
We illustrate the foregoing discussion with the approach loosely based on
Aboolian et al. (
2012
), who proposed one of the few exact approaches available
for Explicit Customer Response models (in fact, the approach outlined below is an
improvement on the original methodology). The model is of DR type, i.e., customers
accept directed assignments to facilities, responding by reducing their demand
when travel and congestion costs increase. Both M=M=K and M=M=1 queueing
systems can be considered; we will focus on the latter for si
mp
licity. The primary
queuing performance measure is the expected waiting time W
i
at each facility i.
While a general concave utility function may be used, we employ the exponential
utility (
17.29
) for transparency, with the elastic demand given by (
17.41
). The fixed
and variable costs are assumed to be uniform, i.e., identical for all locations.
We start by observing that if customers at node j
2
J are assigned to facility i,
the maximum demand is given by
max
ij
D
ma
j
exp.
d
d.i;j//;
quantities that can be pre-computed. The resulting model can be formulated as
follows:
maximize Z
D
r
X
i2I
i
FC
X
i2I
y
i
VC
X
i2I
i
(17.62)
y
i
i
i
subject to
W
i
D
i
2
I
(17.63)
i
D
X
j2J
max
ij
exp.
w
W
i
/x
ij
i
2
I
(17.64)
(
17.55
), (
17.56
)
This reflects the typical structure of DR models: explicit specification of the waiting
time and demand, in addition to regular constraints for location models. Note that
system stability constraints (
17.57
) are omitted, as the demand automatically adjusts
to the offered capacities.
The next observation is that once customer allocation variables x
ij
are specified,
both the optimal capacities at the facilities and the actual realized customer demands
are easy to determine. In fact, the latter only depend on x
ij
through the total
maximal
demand allocated to each facility:
D
X
j2J
max
i
max
ij
x
ij
:
(17.65)
