Geoscience Reference
In-Depth Information
a
˄
!
0, as
˄!1
. Specific conditions for convergence of this scheme can be found
for a variety of networkbased problems, similar to those constructed here, in
Nagurney and Zhang (
1996
) and the references therein.
15.3.1 Explicit Formulae for the Euler Method Applied
to the Disaster Relief Supply Chain Network Variational
Inequality (
15.24
)
The elegance of this procedure for the computation of solutions to the disaster relief
supply chain network problem modeled in Sect.
15.2
can be seen in the following
explicit formulae. Indeed (
15.29
) for the supply chain network problem governed
by variational inequality problem (
15.24
) yields the following closed form
expressions for the product path flows, the time deviations, and the Lagrange
multipliers, respectively:
1
P
k
X
q∈P
k
x
p
¼
max
0,
x
p
þ a
˄
x
˄þ
1
p
k
λ
X
!
x
p
∂
C
p
x
ðÞ
∂
X
X
a∈L
ˉ
q
g
a
ʴ
aq
ʴ
ap
,
k
P
k
λ
x
p
q
∈P
k
q
∈P
8p
∈P
k
;
k ¼
1,
...
,
n
R
,
ð
15
:
31
Þ
,
¼
max 0,
z
p
þ a
˄
ˉ
p
∂ʳ
k
z
ðÞ
z
˄þ
1
p
8p
∈P
k
;
k ¼
1,
...
,
n
R
, and
∂
z
p
ð
15
:
32
Þ
(
)
,
ˉ
p
; þa
˄
;
X
q∈P
X
ˉ
˄þ
1
p
g
a
x
p
ʴ
aq
ʴ
ap
T
kp
z
p
¼
max 0,
a∈L
8p
∈P
k
;
k ¼
1,
...
,
n
R
:
ð
15
:
33
Þ
In the next section, we solve additional disaster relief supply chain network
problems using the above algorithmic scheme.
15.4 A Larger Numerical Example and Variant
The scenario for the first numerical example is built on the possibility of another
earthquake striking Haiti. We then construct a variant.
Figure
15.5
displays the disaster relief supply chain network topology
corresponding to the case of a Haiti earthquake. Node 1 is assumed to represent
the American Red Cross. We assumed that the Red Cross could utilize two of its
disaster aid zones in the US, one in Maryland—representing the Northeast and the
