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The righthand side of constraint (8) is the maximum number of centres destroyed
by the disaster, which can be considered as a parameter for the severity of the
damage, on which sensitivity analysis can be performed. Constraints (9) state
that a centre is not destroyed unless it has been established (in the first level).
The third level represents the response and recovery stage after the disaster,
where the decisions involve (re-)establishment of evacuation centres, re-supply
and/or transfer of goods to/between centres, and re-allocation of people to the
centres. It is formulated as follows.
r,u,s,t,w
RecoveryCost
(
x, y, z
;
δ
;
r, u, s, t
)=
i∈I
(
F
i
u
i
+
H
i
s
i
)+
i∈I
(
L3
): min
T
ik
t
ik
k∈I
−
(
y
i
−
δ
i
)
≥
u
i
,
∀
i
∈
I,
subject to
1
(11)
r
ij
∀
j
∈
J,
=1
(12)
i
∈
I
u
i
+(
y
i
−
δ
i
)
≥
r
ij
∀
i
∈
I,j
∈
J,
(13)
C
i
(
y
i
−
δ
i
)
≥
w
i
∀
i
∈
I,
(14)
w
i
≤
z
i
∀
i
∈
I,
(15)
t
ik
≤
w
i
∀
i
∈
I,
(16)
k∈J
w
i
+
s
i
+
k∈J
t
ki
−
t
ik
≥
p
j
r
ij
∀
i
∈
I,
(17)
k∈J
j∈J
u
i
∈{
0
,
1
}∀
i
∈
I,
(18)
r
ij
,t
ik
,s
i
,w
i
≥
0
∀
i,k
∈
I,j
∈
J.
(19)
The
RecoveryCost
includes cost of (re-)establishing centres and cost of supplying
and transhipping inventory to and among operational centres, in order to cover
the population accommodated at the centres. Constraint (11) states that a centre
canbeestablishedinalocationnotpreviouslyusedorinalocationwherean
established centre has been destroyed. Constraints (12) and (13) indicate in
which evacuation centres (that are operational post-disaster) the population for
each community should be accommodated. In this level, the population of each
community may be split among several centres. The variables
w
j
with constraints
(14) and (15) determine how much inventory is available post-disaster; we assume
all inventory is rendered unusable if a centre is destroyed. Finally, constraints
(17) ensure that the inventory at each centre (including existing, newly-supplied
and transhipped) is sucient to support the population accommodated at the
centre post-disaster.
The linkage between the three levels of the formulation is via the
Recov-
eryCost
post-disaster. The rationale for the formulation is that the decisions at
the preparedness stage (in the first level) should anticipate the recovery cost
post-disaster. Our formulation also takes an interdiction approach rather than
an expected-value or chance-constrained approach in modelling the disaster.
Probability-based formulations are much affected by the pool of scenarios con-
sidered and their associated probabilities, which may be dicult to ascertain for