Geoscience Reference
In-Depth Information
Snyder and Daskin (
2006
). Let
z
!
be the optimal cost for scenario !. By adding the
following constraint
X
X
d
j
c
ij
x
ij
!
.1
C
LJ/
z
!
8
!
2
ǝ;
(24.41)
i2I
j2J
it is possible to generate least-cost solutions whose relative regret in each scenario
is no more than LJ,foragivenLJ
0.
The LJ-robustness measure has been used in Peng et al. (
2011
)todesign
reliable multi-echelon supply chain networks. Other risk measures to generate
robust solutions in scenario planning models include the Ǜ-reliable minimax regret
(Daskin et al.
1997
)andtheǛ-reliable mean-excess regret (Chen et al.
2006
). In
Ǜ-reliable minimax models, the maximum regret is computed only over a subset
of scenarios, called the
reliability set
, whose total probability is at least Ǜ.TheǛ-
reliable mean-excess regret, which is closely related to the conditional value-at-risk
(CVaR) objective of portfolio optimization (Rockafellar and Uryasev
2000
), further
extends the Ǜ-reliable concept by ensuring that solutions perform reasonably well
even in the scenarios which are not included in the reliability set. Typically, the
objective function of these models minimizes a weighted sum of the maximum
regret over the reliability set and the conditional expectation of the regret over the
scenarios excluded from the reliability set. Although these measures have not been
explicitly used in facility location problems with disruptions, their application is
quite straightforward and certainly deserves future investigation.
When uncertainty can be captured by a finite set of scenarios and a scenario-
indexed model can be considered, it is easy to modify the model in a way that the
models discussed in Sect.
24.5.2
cannot. As an example, capacity restrictions can be
easily modeled by replacing constraints (
24.36
) with
X
d
j
x
ij
!
.1
a
i!
/q
i
y
i
8
i
2
I;!
2
ǝ;
(24.42)
j2J
where q
i
is the capacity of facility i.
Partial disruptions can also be captured by simply redefining a
i!
as the pro-
portion of facility i capacity which is lost in scenario ! to model the case where
disruptions only reduce the capacity but do not completely disable a facility.
One major drawback of scenario-indexed models is that they can become very
large if there are many scenarios (consider for example all the possible ways in
which subsets of facilities can fail). To obviate this difficulty, the scenario space
can be approximated using sampling techniques such as Sample Average Approx-
imation (Kleywegt et al.
2002
). Another alternative is to construct the scenario set
empirically by using historical data or expert judgement. As an example, Rawls and
Turnquist (
2010
) use a scenario planning approach to optimize facility locations and
emergency resource stockings in the face of natural disasters. In their case study, the
scenarios of concern are constructed by using historical records from a sample of
15 hurricanes.
