Geoscience Reference
In-Depth Information
X
d
a
f
a
˕
L
∂
aw
ijr
d
mw
ij
ijr
¼
0
8
m, w, r, i, j
ð
17
:
8
Þ
h
mw
∂
a
!
ln
x
smw
mw
ij
g
m
ij
þ
ʴ
L
1
∂
ij
x
ij
þ
m
¼
1
:
0
ʸ
0
8
m
,
w
,
i
,
j
,
i
6¼
j
ð
17
:
9
Þ
x
smw
ij
ʱ
m
g
m
∂
X
n
ʼ
m
X
w
ln
x
mw
x
mw
ij
x
ij
L
1
g
m
1
∂
ij
X
i
þ
n
i
a
nm
j
ij
x
ij
¼
1
:
0
þ
ʼ
þ ʸ
0
8
m
,
i
,
j
m
ʱ
ʳ
∂
ð
17
:
10
Þ
The complementary slackness conditions are:
(
)
X
d
a
f
a
˕
L
ijr
∂
h
mw
h
mw
ijr
aw
d
mw
ij
ijr
¼
ijr
¼
0
8
mwrij
ð
17
:
12
Þ
h
mw
∂
a
(
!
)
x
mw
ij
x
ij
þ
d
mw
ij
g
m
∂
L
1
a
m
g
m
x
mw
ij
h
mw
ijr
m
ij
ijr
¼
ln
1
:
0
ʸ
þ
¼
0
8
mwij
,
i
6¼
j
ð
17
:
13
Þ
h
mw
∂
(
)
X
i
ʴ
i
X
s
a
m
g
m
X
w
ln
x
ij
x
mw
ij
x
ij
L
1
g
m
1
x
ij
∂
x
ij
a
snm
j
ij
x
ij
¼
X
i
þ
1
:
0
þ
ʴ
þ ʸ
∂
¼
0
8
mij
ð
17
:
14
Þ
Conditions representing route flows and distances may be derived from
Eqs. (
17.8
) and (
17.12
) as follows:
0,
then
X
a
f
a
φ
d
a
if h
mw
aw
mw
ij
1
:
ijr
>
ijr
¼ ʴ
;
0,
then
X
a
f
a
φ
ð
17
:
15
Þ
if h
mw
ijr
d
a
aw
ijr
mw
ij
2
:
>
ʴ
;
The interpretation of these conditions is as follows: if the route flow
h
ijr
mw
is
positive, then
X
a
f
a
φ
d
a
aw
ijr
, the cost of using route
r
is equal to the equilibrium
shipment cost
d
ij
mw
from
i
to
j
by mode
w
for sector
m
; if the route flow
h
ijr
mw
is
zero, then the cost of using route
r
is not less than the equilibrium shipment cost.
Hence, Lagrange multipliers
ʴ
ij
mw
have an interpretation of equilibrium shipment
cost. These optimality conditions are equivalent to the equilibrium conditions
proposed by Wardrop and Whitehead (
1952
) for user-optimal route choice of
drivers in road networks.
