Geoscience Reference
In-Depth Information
For each facility i we now solve the following univariate “capacity optimization”
model:
maximize r
i
VC
i
i
i
i
/
subject to
i
D
max
i
exp.
w
i
0:
Aboolian et al. (
2012
) show that the solution to this model is unique and can
be found through a simple univariate search. Note that the solution yields both,
the optimal capacity
i
and the corresponding demand level
i
. It is convenient
to represent these quantities as functions of the allocated maximum demand:
.
ma
i
/;.
ma
i
/. Substituting these quantities into the original model (
17.62
)-
(
17.64
), (
17.55
), (
17.56
) we obtain
maximize Z
D
r
X
i2I
.
ma
i
/
FC
X
i2I
y
i
VC
X
i2I
.
ma
i
/
subject to
(
17.55
), (
17.56
), (
17.65
),
where the only non-linearities occur in the objective function. By solving the
capacity optimization model repeatedly over a range of possible values of
max
i
,
we can construct a piecewise linear approximation of the functions .
max
i
/ and
.
ma
i
/ to any desired level of tolerance. Using these approximations in the model
above yields a linear MIP which can be solved using standard off-the-shelf software.
As noted earlier, the separation of capacity optimization and customer allocation
problems is a common feature of Explicit Customer-Response models and has been
used by a number of authors. However, an important driver of the exact approach
outlined above is that the model in Aboolian et al. (
2012
) is of DR type, i.e.,
directed assignment and single-sourcing are both assumed. The analysis presented
in Aboolian et al. (
2012
) suggests that neither of these assumptions is very restrictive
(echoing the results in Castillo et al.
2009
discussed earlier). It was observed that in
the vast majority of instances solved, customers were, in fact, assigned to facilities
that minimize their sum of waiting and travel times, i.e., the facilities they would
have selected under an FR model. Also, by splitting the original customer nodes into
k copies each containing 1=k of the original demand, and allowing each of these
new nodes to be assigned to a different facility, the impact of the single-sourcing
assumption was examined. Again, it turned out that for the instances solved, the
violation of this assumption was rare (all copies of the original node were assigned
to the same facility in the vast majority of the cases) and when split assignments
occurred, they did not have a large impact on the objective function. Intuitively, both
effects can be explained by the fact that in DR models the incentives of customers
and the decision-maker, while not identical, are well-aligned: by forcing customers
to use a less convenient facility, the realized demand (and the revenue) are reduced.
