Geoscience Reference
In-Depth Information
m
, the modal distance sensitivity
parameter for sector
m
;
g
m
, the ratio of monetary shipment value to weight used to
convert the output of sector
m
from dollars to tons ($/ton); and
X
i
m
, the total output
of sector
m
in zone
i
($).
The total cost to be minimized is represented by the sum of the link flows times
link distances by mode, the interzonal commodity shipments by mode, and the
freight and passenger flows by routes and links within modes. Minimization of the
objective function also results in the optimal use of land in zone
i
represented by
x
i
smw
, in three-dimensional variable of
s
. Not all of these commodity shipments and
flows, however, are uniquely determined by the solution.
At the same time, there exist caveats for entropy maximization models and
spatial models with gravity formulation (Nijkamp and Reggiani
1988
). Nijkamp
(
1975
) proved that “entropy results lead to an underestimation, and linear program-
ming results in an overestimation of intraregional flows. For irrterregional flows,
however, the entropy outcomes appear to overestimation: the actual flows, whereas
the linear programming out- comes appear to underestimate the actual flows. Both
methods appear to lead to considerable prediction errors, which can be mainly
explained from the heterogeneity of the commodities in question.”
The exogenous parameters and variables are:
ʱ
Lagrangian Analysis and Optimality Conditions
The Lagrangian function for deriving the optimality conditions is constructed using
objective function (6), constraints (1-5), and Lagrangian multipliers as follows:
2
0
1
#
Z
X
w
a
X
g
m
X
ij
x
ij
X
i
1
4
@
A
d
a
x
ij
ln
min
h;x;ʼ;ʸ;ʴ
Lh
ð
;
x
; ʼ; ʸ; ʴ
Þ ¼
ðÞ
d
ˉ þ
ð
Þ
0
aw
m
2
4
0
@
1
A
#
"
!
þ
#
X
m
g
m
X
ijws
x
smw
ij
x
ij
X
mj
ʼ
X
a
smn
X
k
X
1
x
smw
ij
m
j
x
jk
y
j
x
ij
þ
ln
þ
ʱ
m
sn
j
0
@
1
A
!
X
mij
ʸ
X
x
mw
ij
g
m
X
m
ij
x
ij
x
mw
ij
m
ij
h
mw
ijr
þ
þ ʴ
m
r
h
mw
s
:
t
:
ijr
0
ð
17
:
7
Þ
ʴ
ij
mw
) in vector form. The Karush-
Kuhn-Tucker optimality conditions are obtained by taking partial derivatives with
respect to the unknown variables as follows:
ʼ
j
m
), (
ʸ
ij
m
), (
The Lagrange multipliers are (
