Geoscience Reference
In-Depth Information
15.2.2 Formulation of the Disaster Relief Supply Chain Network
Model
Let
f
a
denote the flow of the disaster relief product on link
a
, be it a procurement,
storage, transportation, processing, or distribution link. Let
c
a
(
f
a
) and
c
a
fðÞ
denote
the unit operational cost function and the total operational cost function on link
a
,
respectively. The link total cost functions are assumed to be convex and continu-
ously differentiable. We have:
^
c
a
fðÞ¼f
a
c
a
fð
,
8a
∈
L
:
ð
15
:
1
Þ
P
k
denotes the set of paths connecting the origin (node 1) to demand point
k
with
P
denoting the set of all paths joining the origin node to the destination nodes.
The total number of paths in the supply chain, i.e., the number of elements in
P
is
given by
n
p
.
In the model, we assume that the demand is uncertain due to the unpredictability
of the actual demand at the demand points. Similar examples of system-optimized
models with uncertain demand and associated shortage and surplus penalties can be
found in the literature (see, e.g., Dong et al.
2004
; Nagurney et al.
2011
,
2012a
;
Nagurney and Masoumi
2012
).
The probability distribution of demand is assumed to be available. It may
be derived using census data and/or information gleaned and obtained over the
course of the preparedness phase. If
d
k
denotes the actual (uncertain) demand at
destination point
k
, we have:
ð
D
k
0
F
k
ðÞdt
,
P
k
DðÞ¼P
k
d
k
D
k
ð
Þ ¼
k ¼
1,
...
,
n
R
,
ð
15
:
2
Þ
where
P
k
and
F
k
denote the probability distribution function, and the probability
density function of demand at point
k
, respectively.
Let
v
k
be the “projected demand” for the disaster relief item at point
k
;
k ¼
1,
,
n
R
. The amounts of shortage and surplus of the aid item at destination
node
k
are denoted by
...
Δ
k
and
Δ
+
, respectively, and are calculated as follows:
k
max 0,
d
k
v
k
f
g
,
k ¼
1,
...
,
n
R
,
ð
15
:
3a
Þ
Δ
Δ
k
max 0,
v
k
d
k
f
g
,
k ¼
1,
...
,
n
R
:
ð
15
:
3b
Þ
Hence, based on the probability distribution of the demand, the expected values
of shortage and surplus at each demand point are:
ð
1
Δ
k
¼
E
ð
t v
k
Þ
k
ðÞdt
,
k ¼
1,
...
,
n
R
,
ð
15
:
4a
Þ
v
k
ð
v
k
k
¼
E
ð
v
k
t
Þ
k
ðÞdt
,
k ¼
1,
...
,
n
R
:
ð
15
:
4b
Þ
Δ
0
