Geoscience Reference
In-Depth Information
FXð
,
X X
∗
h
i
0,
8X
∈K
,
ð
15
:
27
Þ
where
h
,
i
denotes the inner product in
n
-dimensional Euclidean space.
If the feasible set is defined as
KK
, and the column vectors
X
(
x
,
z
,
ˉ
) and
F
(
X
)
(
F
1
(
X
),
F
2
(
X
),
F
3
(
X
)), where:
2
!
λ
!
!
X
F
1
ðÞ¼
∂
C
p
x
ðÞ
∂
k
P
k
X
q∈P
k
1
P
k
X
q∈P
k
X
a∈L
ˉ
q
g
a
ʴ
aq
ʴ
ap
,
4
k
x
p
þλ
x
q
x
q
q∈P
#
,
p
∈P
k
;
k¼
1,
...
,
n
R
2
3
F
2
ðÞ¼
∂ʳ
k
ðÞ
∂
4
5
,
z
p
ˉ
p
,
p
∈P
k
;
k¼
1,
...
,
n
R
and
"
#
,
F
3
ðÞ¼ T
kp
þ z
p
X
q∈P
X
g
a
x
q
ʴ
aq
ʴ
ap
,
p
∈P
k
;
k ¼
1,
...
,
n
R
,
ð
15
:
28
Þ
a
∈
L
then variational inequality (
15.24
) can be re-expressed as standard form (
15.27
).
We utilize variational inequality (
15.24
) for our computations to obtain the
optimal path flows and the optimal path time deviations. Then, we use (
15.8
)to
calculate the optimal link flows of disaster relief items in the supply chain network.
15.2.3 An Illustrative Example and Two Variants with Sensitivity
Analysis
We now present an illustrative numerical example as well as two variants along
with their solutions, accompanied by some sensitivity analysis, before proceeding
to the solution algorithm, which can be applied to solve large-scale disaster relief
supply chain networks in practice.
15.2.3.1 Illustrative Example
Consider the simple disaster relief supply chain network topology in Fig.
15.2
.
The organization is assumed to possess a single procurement facility and a
single storage facility, and aims to deliver the relief goods to one demand point
through one arrival portal and one processing facility in the affected region.
We allow two modes of transportation from the storage facility to the portal
of the affected region. The links are labeled as in Fig.
15.2
, i.e.,
a
,
b
,
c
,
d
,
e
,
f
,
and
g
, where links
d
and
e
represent ground and air transportation, respectively.
The total operational cost functions on the links are:
