Information Technology Reference
In-Depth Information
We first present the formulation of each level separately, and then describe the
linkages.
The first level addresses the decision of inventory pre-positioning, the loca-
tion of evacuation centres and assignment of the communities to the centres,
formulated as follows:
x,y,z
i∈I
(
L1
):
min
(
f
i
y
i
+
h
i
z
i
)+
RecoveryCost
(
x, y, z
;
δ
;
r, u, s, t
)
subject to
y
i
≤
E,
(1)
i∈I
x
ij
=1
∀
j
∈
J,
(2)
i∈I
d
ij
x
ij
≤
my
i
∀
i
∈
I,j
∈
J,
(3)
z
i
≤
C
i
y
i
∀
i
∈
I,
(4)
p
j
x
ij
≤
z
i
∀
i
∈
I,
(5)
j∈J
y
i
,x
ij
∈{
0
,
1
}∀
i
∈
I,j
∈
J,
(6)
z
i
≥
∀
i
∈
I.
0
(7)
The objective minimizes the sum of the costs of establishing the centres, of
maintaining the pre-positioned inventories and the recovery costs post-disaster.
The constraint (1) limits the maximum number of evacuation centres to be
established. Constraints (2) and (3) determine the assignment of communities
to their designated centres, which must be within a distance
m
. Constraints (4)
and (5) ensure that the level of inventory to be pre-positioned at the centres
established is within the capacity of the centre and sucient to support the
designated communities should a disaster occur. For modelling simplicity, we
aggregate all necessary supplies as one commodity and assume that one unit is
needed per person. (Considering commodities separately does not substantially
alter the structure of the model. The amount of different commodities needed are
generally proportional to the number of people served and likely to be sourced
from the same location.)
In the second level, the disaster is considered as an adversary that tries to
inflict maximum damage. The “decision” for this level is formulated as:
(
L2
):
max
δ
RecoveryCost
(
x, y, z
;
δ
;
r, u, s, t
)
subject to
δ
i
≤
D,
(8)
i∈I
δ
i
≤
y
i
∀
i
∈
I,
(9)
δ
i
∈{
0
,
1
}∀
i
∈
I.
(10)
