Geoscience Reference
In-Depth Information
24.4
Hardening Facilities: Protection Models
Interdiction models are a valuable tool for assessing facility criticality and worst-
case scenario losses in case of disruption. However, it can be easily demonstrated
that securing those facilities that are identified as the most critical in an optimal
interdiction solution does not necessarily result in the most effective protection
strategy (Church and Scaparra
2007b
). Interdiction is a function of what is protected
and this interdependency must be captured explicitly into a modeling framework
to guarantee that limited protective resources are allocated in an optimal way.
Most of the facility protection models existing in the literature incorporate an
interdiction model as a tool for evaluating worst-case losses in response to protection
plans. These models are expressed mathematically as bilevel optimization programs
(Dempe
2002
) which emulate a game played between a system
defender
(the leader)
and a system
attacker
or
interdictor
(the follower). In this bilevel structure, the upper
level problem involves decisions on which facilities to harden, whereas the lower
level problem identifies which unprotected facilities to attack to inflict maximum
damage.
In the following, we show how the model presented for the r-IMP in the previous
section can be embedded within a protection model to optimize security investments
in median systems (Scaparra and Church
2008a
).
24.4.1
The
r
-Interdiction Median Problem with Fortification
The bilevel formulation of the r-IMP with Fortification (r-IMPF) is as follows.
minimize H.
z
/
(24.12)
subject to
X
i2F
z
i
D
b
(24.13)
z
i
2f
0;1
g8
i
2
F;
(24.14)
where
H.
z
/
D
max
X
i2F
X
d
j
c
ij
x
ij
(24.15)
j2J
s.t. s
i
1
z
i
(24.16)
(
24.2
)-(
24.6
):
The leader objective (
24.12
) is to minimize the highest possible level of demand-
weighted service cost, H, following the disruption on r facilities by allocating b
protective resources (
24.13
).Theworst-casecostH is computed in the follower
