Geoscience Reference
In-Depth Information
24.5.2.1
Scenario-Indexed Models
When the uncertainty associated with disruptions can be captured by a finite
set of scenarios, we can resort to scenario-indexed models. Within the context
discussed in this chapter, such models are an alternative way for writing two-
stage stochastic mixed integer programs. The non-anticipative first-stage decisions
concern the location of the facilities and are made in the presence of uncertainty
about the realization of future disruption scenarios. The second-stage (recourse)
decisions, which are conditional to the first-stage decisions, involve the assignment
of customers to facilities in response to specific disruption scenarios.
Below we show a scenario-indexed model for the p-median problem, where the
objective is to minimize the expected service cost over all failure scenarios. Let ǝ
be the set of disruption scenarios such that a
i!
D
1 if facility i fails in scenario !.
The probability that scenario ! occurs is denoted by
!
. The assignment decision
variables are defined for each scenario as follows:
x
ij
!
D
1 if customer j is assigned to facility i in scenario !
0 otherwise
The scenario-indexed model is then:
minimize
X
!2ǝ
!
X
i2I
X
d
j
c
ij
x
ij
!
(24.35)
j2J
subject to
X
j2J
x
ij
!
.1
a
i!
/y
i
8
i
2
I;!
2
ǝ
(24.36)
X
x
ij
!
D
1
8
j
2
J;!
2
ǝ
(24.37)
i2I
X
y
i
D
P
(24.38)
i2I
y
i
2f
0;1
g8
i
2
I
(24.39)
x
ij
!
2f
0;1
g8
i
2
I;j
2
J;!
2
ǝ
(24.40)
The objective function (
24.35
) minimizes the demand-weighted expected cost
across all scenarios. Constraints (
24.36
) prevent the assignment of customer j to
facility i in scenario ! if either i is not open or if it is open but not available in
scenario !. Constraints (
24.37
) guarantee that each customer is assigned to some
facility in every scenario. The remaining constraints are standard cardinality and
integrality constraints.
The expected performance criterion used in problem (
24.35
)-(
24.40
) yields
solutions that may perform poorly in certain scenarios. Solutions which are effective
no matter what scenario is realized can be obtained by incorporating robustness
measures into the model. An example is the LJ-robustness measure introduced by
