Geoscience Reference
In-Depth Information
T
kp
¼ T
k
h
p
,
8p
∈P
k
;
k ¼
1,
...
,
n
R
:
ð
15
:
15
Þ
We assume that the values of the
T
kp
's are all nonnegative; otherwise, we
remove path
p
from the network due to infeasibility. We further assume that there
is at least one path
p
0; otherwise, the organization
will have to relax the corresponding time target
T
k
(make it less restrictive).
Thus, inequality (
15.14
) can be re-written as:
X
∈P
k
for each
k
such that
T
kp
>
g
a
f
a
ʴ
ap
T
kp
,
8p
∈P
k
;
k ¼
1,
...
,
n
R
:
ð
15
:
16
Þ
a∈L
Note that the goal constraints introduced here are not hard constraints meaning
that, under certain circumstances, the organization might be forced to deviate from
the goal. However, this deviation will be minimized, as we shall see, along with
the minimization of total cost throughout the network subject to the uncertain
demand. For example, depending on the actual completion time, the sequence
of activities on path
p
leading to the delivery of the relief products to demand
point
k
will be completed either at, before, or after the determined time goal
target. Let
z
p
denote the amount of deviation with respect to target time
T
kp
corresponding to the “late” delivery of product to point
k
on path
p
, which was
assumed to be nonnegative. Using (
15.16
), we now construct
the following
constraints:
X
g
a
f
a
ʴ
ap
z
p
T
kp
,
8p
∈P
k
;
k ¼
1,
...
,
n
R
:
ð
15
:
17
Þ
a∈L
The path time deviations must be nonnegative for all paths in the network;
that is,
z
p
0,
8p
∈P
k
;
k ¼
1,
...
,
n
R
:
ð
15
:
18
Þ
Using (
15.8
), we can replace the link flows with the path flows in (
15.17
), so that
X
X
g
a
x
q
ʴ
aq
ʴ
ap
z
p
T
kp
,
8p
∈P
k
;
k ¼
1,
...
,
n
R
:
ð
15
:
19
Þ
q∈P
a∈L
ʳ
k
(
z
) denote the tardiness penalty function corresponding to demand point
k
which is a function of the time deviations on paths leading to that point. These
functions are assumed to be convex and continuously differentiable.
Interestingly, Nagurney et al. (
1996
) utilized goal targets in the case of spatial
economic markets, whereas Nagurney and Ramanujam (
1996
) considered penalties
associated with transportation targets with associated penalty functions that could
be nonlinear (as in the case above).
C
p
ðÞ
denotes the total operational cost function on path
p
and is constructed as:
Let
