Geoscience Reference
In-Depth Information
Table 15.1
Sensitivity analysis of the optimal solution to the unit shortage penalty at the demand
point for Variant II
λ
R
1
x
p
3
x
p
4
z
p
3
z
p
4
ˉ
p
3
ˉ
p
4
Value of objective function (21)
2,500
0.50
5.56
5.09
7.66
35.66
122.58
5,081.96
5,000
0.33
6.26
8.54
14.09
59.77
225.49
8,440.02
7,500
0.20
6.79
11.18
19.02
78.25
304.39
11,021.81
10,000
0.09
7.22
13.26
22.91
92.80
366.49
13,035.31
12,500
0.01
7.56
14.94
26.05
104.57
416.72
14,655.25
Table
15.1
displays the optimal values of the path flows,
x
p
3
and
x
p
4
, the path time
deviations,
z
p
3
and
z
p
4
, and the Lagrange multipliers,
p
3
and
p
4
, as the unit shortage
ˉ
ˉ
R
1
, is increased from 2,500 to 12,500 in Variant II.
As seen in Table
15.1
, as the shortage penalty increases, the organization will
be fulfilling a higher projected demand,
x
p
3
þ x
p
4
, by assigning higher quantities
to the path that uses air transportation to the affected region. At
penalty,
λ
R
1
¼
12, 500, the
optimal path flow on the ground transportation path is almost zero, which means
that the organization relies on the air transport mode. Handling larger volumes
of goods increases the congestion on paths which, in turn, worsens the lateness
of deliveries to the region.
λ
15.3
The Algorithm
In this section, we recall the Euler method, which is induced by the general iterative
scheme of Dupuis and Nagurney (
1993
). Its realization for the solution of disaster
relief supply chain network problems governed by variational inequality (
15.24
)
(and (
15.27
)) yields subproblems that can be solved explicitly and in closed form.
Specifically, recall that at an iteration
˄
of the Euler method (see also Nagurney and
Zhang
1996
) one computes:
,
X
˄þ
1
¼ P
K
X
T
a
˄
FXðÞ
ð
15
:
29
Þ
where
P
K
is the projection on the feasible set
K
and
F
is the function that enters
the variational inequality problem: determine
X
∈K
such that
FXð
,
X X
h
i
0,
8X
∈K
,
ð
15
:
30
Þ
R
n
, and
F
(
X
)isan
n
-dimensional function from
K
to
R
n
, with
F
(
X
) being continuous (see
also (
15.28
)).
As shown in Dupuis and Nagurney (
1993
); see also Nagurney and Zhang (
1996
),
for convergence of the general iterative scheme, which induces the Euler method,
among other methods, the sequence {
a
˄
} must satisfy:
where
h
,
i
is the inner product in
n
-dimensional Euclidean space,
X
∈
∑
˄ ¼
1
a
˄
¼1
,
a
˄
>
0,
