Geoscience Reference
In-Depth Information
The upper level objective (
24.17
) is to maximize the weighted sum of covered
demand before and after interdiction by locating p facilities (
24.18
). The demand
covered before interdiction is determined by constraints (
24.19
), whereas the worst-
case demand-weighted coverage after interdiction, H.y/, is computed in the lower
level problem (
24.22
)-(
24.26
). This is a simple modification of the r-ICP problem
(
24.7
)-(
24.11
), where constraints (
24.8
) are replaced by (
24.24
). These constraints
state that customer j must be covered after disruption (v
j
D
1/ unless all the open
facilities covering customer j are interdicted.
Bilevel location-interdiction problems such as the MCLIP are even more difficult
to solve than the protection-interdiction problems discussed in Sect.
24.4
and some
efficient approaches devised for protection models, such as the implicit enumeration
algorithm for r-IMPF, are not applicable to them. In O'Hanley and Church (
2011
),
the MCLIP is solved by a decomposition method using
supervalid inequalities
.
Another example of location/interdiction models can be found in Parvaresh
et al. (
2012
)forp-hub median problems. In this case, the bilevel model is solved
heuristically via simulated annealing and tabu search.
Note that design and protection decisions may be coupled within the same
modeling framework. Examples of risk-averse design models including the option
of hardening some of the facilities to be located can be found in Aksen et al. (
2011
),
Aksen and Aras (
2012
) and Shishebori and Jabalameli (
2013
).
24.5.2
Planning for a Risk-Neutral Designer
In this class of models, facilities are assumed to fail at random and the objectives
typically deal with expected costs or performances.
Although the first paper to consider unreliable facilities which fail with a given
probability appeared more than a couple of decades ago (Drezner
1987
), a renewed
interest in this type of problems has only emerged more recently with the reliability
problems investigated by Snyder and Daskin (
2005
): the Reliability p-Median
Problem (RPMP) and the Reliability Fixed-Charge Location Problem (RFLP). Both
problems aim at locating a set of facilities so as to minimize the costs incurred by
the system when all the facilities are operational and the expected transportation
costs after facilities failures.
In the RPMP model, each open facility may fail with the same fixed probability
, failures are independent and several facilities can fail simultaneously. If customer
j is not served by any facility, either because all open facilities fail or because it is
too costly to receive service by the closest operational facility, the system incurs
a lost-sale cost per unit of demand. To model this situation, the set I of potential
locations for the facilities is augmented with a dummy emergency facility. Let m
be the cardinality of the augmented set
j
I
j
and the index of the emergency facility.
The emergency facility m never fails and has unitary service cost c
mj
to customer
j. As facility m is forced to open, p
C
1 facilities must be located instead of p as in
standard p-median problems.
