Geoscience Reference
In-Depth Information
The original RPMP model presented in Snyder and Daskin (
2005
) is slightly
more general than model (
24.27
)-(
24.34
) in two aspects: (1) some of the facilities
may be considered completely reliable and (2) the objective is to minimize the
weighted sum of normal costs and expected failure costs. The authors show that
by varying the weights of the resulting bi-objective model, one can generate a
trade-off curve for identifying good compromise solutions. This type of analysis
demonstrates that large reductions in failure costs can often be attained with only
minor increases in operation costs.
The Reliability Fixed-Charge Location Problem, which we do not report for the
sake of brevity, can be formulated in a similar way to RPMP. Both problems can be
tackled by Lagrangian relaxation (Snyder and Daskin
2005
). Efficient metaheuristic
approaches have also been devised for RPMP by Alcaraz et al. (
2012
), which report
very good results for large scale instances.
One of the major limitations of this structure for reliability models is that it
relies on the assumption that all facilities fail with the same probability. Without
this assumption, calculating expected transportation costs becomes significantly
more complicated due to the need of expressing probability products using high-
degree polynomials. Site-dependent probabilities were considered for the first time
by Berman et al. (
2007
) but the resulting model is highly non-linear and is only
solved heuristically. Several attempts at modelling heterogeneous facility failure
probabilities using a linear mixed integer program have appeared in recent years
(see for example Cui et al.
2010
and Lei and Tong
2013
). Particularly noteworthy
is the
probability chains
linearization technique proposed by O'Hanley et al. (
2013
)
for solving the RPMP with site-dependent probabilities. The technique, which is
general and can be extended to other model classes as well, is based on the idea
of using a specialized network flow structure for evaluating compound probability
terms. Empirical experiments indicate that this technique is quite effective in solving
reliability models of significant size.
Other important issues in modeling location problems with unreliable facilities
are correlation and informational uncertainty. Correlation concerns the extent to
which the failure of one facility affects the operational status of other facilities.
In many real situations neighboring facilities may be exposed to similar hazards
and, therefore, fail simultaneously. Examples of models with correlated disruptions
can be found in Li and Ouyang (
2010
) and Berman et al. (
2013
). Informational
uncertainty relates to the information available to customers about the operational
state of the facilities. It is clear that optimal location patterns and optimal service
costs may differ if customers do not have prior information about the state of the
facilities and must travel to different facilities before they can receive service. The
role of information in reliable facility design is analyzed in Berman et al. (
2009
)and
Berman et al. (
2013
).
Finally, as for the bilevel design models discussed in the previous section,
location and hardening decisions can be combined into a probabilistic design model
for identifying reliable and cost-efficient configurations of hardened and unhardened
facilities (see, for example, Lim et al.
2010
and Li et al.
2013
).
