Geoscience Reference
In-Depth Information
of case, one might want to cast disruption as a budget constrained process (see
for example Losada et al.
2012b
). However using an interdiction budget requires
information that may be completely hidden from the system operator, including the
costs of striking and the available budget itself. The use of cardinality constraints
such as (
24.3
) can be seen as a surrogate to knowing exact budget values of the
interdictor.
The r-IMP can be cast as an integer linear programming model which can be
solved with general-purpose integer programming software. The above formulation
of the r-IMP can be streamlined by consolidating redundant assignment variables
under special proximity conditions. The resulting variable reduction of this consoli-
dation mechanism, which was initially proposed by Church (
2003
)forthep-median
problem, can be substantial. Scaparra and Church (
2008a
) report reductions of up to
80 % of the initial number of variables. The same authors also analyze and compare
different formulations of the closest assignment constraints (
24.4
) to identify the
most efficient formulation for the r-IMP. Although other approaches could be
devised to solve the r-IMP, including decomposition methods or heuristics, solving
the streamlined model by commercial software is usually quite effective, even for
problem instances of significant size.
Clearly, the r-IMP makes some simplifying assumptions which may limit its
practical applicability. For instance, it assumes that every strike or disruption is
successful and always results in a complete impairment of the affected facility.
In reality, the chances of losing a facility following a natural disaster or a man-
made attack are based upon some probability. Church and Scaparra (
2007a
)
introduced a probabilistic version of r-IMP where an attempted interdiction is
successful only with a given probability. The same authors also show how to
build a
reliability envelope
for identifying the range of possible impacts associated
with losing one or more facilities. Losada et al. (
2012b
) further extended this
probabilistic r-IMP by assuming that the probability of impairing a facility depends
on the intensity of the disruption or on the amount of offensive resources used
in the attack. In a further extension, Lei and Church (
2011
) address the issue
of interdiction when not all demands are served by their closest facility after a
disruption.
The r-IMP also assumes no restrictions on the facilities capacity, thus implying
that after a disruption, the unaffected facilities have enough combined capacity
to supply all the demand. This may not be a realistic assumption as most real
supply systems usually operate with capacity limits. The capacitated version of the
r-IMP can be found in Scaparra and Church (
2012
). Another interesting variation of
the r-IMP which considers capacity restrictions is the partial interdiction problem
introduced by Aksen et al. (
2012
). In this model, an interdicted facility may preserve
part of its capacity; the capacity loss due to interdiction is commensurate to the
intensity of the attack and the unmet demand after interdiction can be outsourced at
some cost.
