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The Theory and Practice of a Dual Criteria Assignment
Model with a continuously distributed Value-of-Time
Fabien Leurent
To cite this version:
Fabien Leurent. The Theory and Practice of a Dual Criteria Assignment Model with a con-tinuously distributed Value-of-Time. J.B. Lesort. ISTTT, Jul 1996, Lyon, France. Pergamon,pp. 455-477, 1996, ISTTT Proceedings. <hal-00348537>
HAL Id: hal-00348537
https://hal.archives-ouvertes.fr/hal-00348537
Submitted on 19 Dec 2008
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1 A dual criteria traffic assignment model
F.M. Leurent (1996)
in Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon, Exeter, England.
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2 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
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3 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
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Grs
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4 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
" .(
Pm
Tm
cost
time
!
(
qrs ! &
qrs dHrs(v)v rs
m−1
v rsm
C
v rs
m : = +∞
v rs
0: = 0
)*, %$"($"-#(
$#
D!-
$ ta(f ) -
E zb(f ) ≤ 0 %
zb(f ) = 0 ! wb ≥ 0
wb∂zb
∂frsk '"
xa(f ) − Ca ≤ 0 %
')
wa∂ (xa−Ca )
∂frsk = waδrs
ak
'
Trsk
= δrsak
taa( )+ wb∂zb
∂frskb
>
2- t1 = 5+ 2x1
t2 = 4 + x2 % P2 P1 = 0
' q = 10 = x1 + x2
P2
1!(
8< 3: :3
(
5 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
" 1 !!
0
2
4
6
8
0 500 1000 1500
0
500
1000
1500
2000
2500
3000
0 500 1000 1500
P2 = 0 % P2
C P2
>0 P2
)*. ##/0$"# &(&
@
&- qrs Drs
) Srs # qrs = Drs(Srs )
(
)# Srs = min kGrs
k(v) dHrs(v)
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7 1!
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Toll P2
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4
6
8
10
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6 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
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& & R rs # kRrs l Prs
k= Prs
l-
'8@!
-'8@ . m rs - Mrs(k)
'*$+' ∆ rs
ki# = 1 i = Mrs(k) :
> P rs
m!!
T rs
m
> qrs
m:= ∆rs
kifrs
kk
!8@ $ Qrs
m:= qrs
ii≤m $
& ! (
qrs = qrs
ii = Qrs
m rs - qrs
0= Qrs
0: = 0
'
!' G rs
m(v):= T rs
m+ P rs
m/ v
7 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
- Hrs
Ωrs : = v ; Hrs(v) ∈]0 ; 1[ % Hrs
−1 Hrs(Ωrs ) Ωrs
BB dHrs(v) > 0
51@$ 5$8@!
! #
Ers
m: = v ∈Ωrs ; G rs
m(v) = min G rs
(v)
" ∀rs, Ωrs = m Ers
m Ers
m= ∅ !
Ers
m '
v0 ,v1 ∈Ers
m
v0 G rs
m(v0 ) = min n v0 G rs
n(v0 )
v1 G rs
m(v1) = minn v1 G rs
n(v1) ∀α ∈ [0; 1]
vα = αv0 + (1−α )v1 vα G rs
n(vα ) = αv0 G rs
n(v0 ) + (1− α )v1G rs
n(v1)
vα G rs
m(vα ) ≤ vα G rs
n(vα )
vα ∈Ers
m
Ers
mErs
n n ≠ m #
Ers
m
Ers
n
'!% ∀u ∈Ers
m,
G rs
m(u) ≤ G rs
n(u)
(P rs
m− P rs
n) / u ≤ T rs
n− T rs
m
∀v ∈Ers
n,
G rsm
(v) ≥ G rsn
(v)
(P rs
m− P rs
n) / v ≥ T rs
n− T rs
mF
∀u ∈Ers
m, ∀v ∈Ers
n, (P rs
m− P rs
n) / u ≤ T rs
n− T rs
m≤ (P rs
m− P rs
n) / v .!
( n > m P rs
n> P rs
m u ≤ v
supErs
m≤ inf Ers
n
557 8@
! v rs
m= sup Ers
m
Ers
m≠ ∅ :
Ers
m= ∅ (
Ers
m≠ ∅ !
??. 28@!
Ers
m≠ ∅
Ers
n≠ ∅
P rs
m< P rs
n
Ers
i= ∅ ∀i ∈]m;n[
T rs
m> T rs
n
v rs
m= (P rs
n− P rs
m) / (T rs
m− T rs
n) = inf Ers
n
55A8@5$
8@ q = [qrs
m]m
Ers
m
qrs
m≥ 0
qrs
mm = qrs
qrs
m= qrs dHrs(v)
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Ers
m
(P rs
m;T rs
m)m Ωrs 8@
5 8@
8@
)#
∀v ∈Ωrs ! G rs
m(v) ≤ minn G rs
n(v)
Ers
m
! ' !
qrs
m= qrs dHrs(v)
Ersm F
8 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
8@
# 8@
5
??1) q = [qrs
m]m
qrs
m≥ 0
qrs
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Ersm ∀n,
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Hrs !
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v rs
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v rs
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n) / (T rs
n− T rs
c(n)) 8@ '
e = e(m rs ) T rs
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n− P rs
c(n)) /
v rs
n= T rs
c(n)%
T rs
n+ (P rs
l− P rs
c(l)) /
v rs
ll efficient=ne(e −1) = T rs
e
(P rs
l− P rs
c(l)) /
v rs
l
T rs
c(l)− T rs
l &!
' T rs
n+ (P rs
l− P rs
l+1) /
v rs
ll =nm rs−1 = T rs
e + (P rs
e − P rs
m rs ) /v rs
e
(P rs
m− P rs
m+1) /
v rs
lm=lc(l) = (P rs
l− P rs
c(l)) /
v rs
l e(m) = l
v rs
m=v rs
l
(
' &
e F&
&8@ qrs
n> 0
5508@ 8@
I rsn
(f ;w ) = T rsn
+ (P rsl
− P rsl+1
) /v rs
ll =nm rs−1
? ?7 q = [qrs
m]m
qrs
m≥ 0
Ers
m
∀n, qrs
n> 0 I rs
n= min m I rs
m
?7 - &#
&!
& $ ./6/! *
9 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
+*+ %" "4$&'# ""5'#6"'(
553&& (f ;w)
$ frs
k≥ 0 / wb
zb(f ) ≤ 0 #
! " #
Trsk
(f ;w ) = δrsak
ta(f )a( )+ ( wb∂zb(f )
∂frskb )
wb ≥ 0 zb(f ) ≤ 0 wbzb(f ) = 0
!$' # qrs = Drs min kGrs
k(v) dHrs(v)( )
!) 5$8@
! ∀k ∈m, frs
k> 0 Trs
k= T rs
m 8@
[qrs
m]m $
qrs
m: = ∆ rs
kmfrs
kk
Ers
m
(P rs
m;T rs
m)m
T rs
m: = min k∈mTrs
k
" ?1 &
8@ )
556 >
Frs(x): = d t
Hrs−1
(t)0
x
' Irs
k(f ;w ) =
Trsk
+P
rsi
− P rsi+1
Hrs−1(Q
rsi /qrs )i=Mrs(k)
m rs−1
−−++−+
−
=−
+−.
. ,.
.. !!*
,!!"??!."?!5
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!
$$$
$$
$
$
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$$
!$$$
),
),
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,)
(f ;w) Trs
k
Q
rsi
5 Mrs(k) ! Irs
k= I rs
m+ Trs
k− T rs
m− Brs Brs
8@
' # Brs = ′ I rs + Drs
−1− ′ S rs
′ S rs = T rs
n qrsn
qrs+ P rs
n[Frs(
Qrsn
qrs) − Frs(
Qrsn−1
qrs)]n
= G rsn
(v)dHrs(v)v rs
n−1
v rsn
n &
Srs : = minnG rs
n(v) dHrs(v)
∀v ∈ [
v rs
n−1;v rs
n]
G rs
n(v) = min mG rs
m(v)
′ I rs = ( T rsn q
rsn
qrsn ) +(P
rsl − P
rsl+1)
v rs
l
Qrsl
qrsl=1m rs −1 &
Irs : = min nI rs
n=
qrsn
qrsI rsn
n
10 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
. ) & (f ;w)
&#
!
Trsk
(f ;w ) = δrsak
ta(f )a( )+ ( wb∂zb(f )
∂frskb ) wb ≥ 0 zb(f ) ≤ 0 wbzb(f ) = 0
! ∀r − s − k, frs
k> 0 Irs
k= min l Irs
l
! ∀r − s, qrs > 0 qrs = Drs min kGrs
k(v) dHrs(v)( )
?# ( 53!& .!53!
' 8@! Trs
k= T rs
m
qrs
m> 0
I rsm
= minn (I rsn
) #
Irs
k= I rs
m− Brs
= minn (I rsn
) − Brs = minn (I rsn
− Brs ) ≤ min lIrs
l .
!
?#>! qrs
m> 0 !'
frs
k> 0
I rsm
− Brs = Irsk
= minl (Trsl
− T rsMrs(l) + I rs
Mrs(l) − Brs ) I rsm
≤ min n I rsn
" Irs
k= minl Irs
l≤ minl∈m Irs
l
Trs
k= T rs
m
> # (ℜ+
)N
→ ℜN C $
/! (f ;w) V (f ;w) = ([Irs
k(f ;w )]rsk ;[− zb(f )]b)
1&
(f ;w) &B" (f ;w) ≥ 0
V(f ;w) ≥ 0 V(f ;w) ⋅ (f ;w) = 0B
# ! Irs
k≥ 0 ! zb ≤ 0 !
frs
kIrsk
= 0 !
wbzb = 0
. !⇔ 1!! wb ≥ 0 53 1 .!
Brs = Irs + Drs
−1− Srs
min k Irsk
= 0 .
!F 1!! frs
k> 0
Irs
k= 0 ≤ min l Irs
l .
!!
7&& (˜ f ; ˜ w ) ≥ 0
& & (?!#
B ∀(f ;w ) ≥ 0, V( ˜ f ; ˜ w ) ⋅ (f − ˜ f ;w − ˜ w ) ≥ 0 B
(? (ℜ+
)N 7& 1
>
Jbic(f ):= qrs [ P rsi
(Frs(Q
rsi
qrs) − Frs(
Qrsi−1
qrs))
i=1
m rs
]rs # > .//7!
∂
∂frsk Jbic(f ) = Irs
k− Trs
k+ Drs
−1(qrs )
11 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
A) &( ta(f ) J t(f )
f ≥ 0 & )
min f≥0 J(f ) = J t(f ) + Jbic(f ) − Drs
−1(u) du
0
qrsrs E
zb(f ) ≤ 0 ( ta(f ) = ta(xa ) J t(f ) = ta(u) du
0
xaa
? J(f ) = £(f ;0) > G £(f ;w) = J(f ) + zb .wbb #
(? 7&
+*, &'5'"'#
0'( ta zb
Hrs
−1
Drs Drs
−1 f0 ≥ 0
zb(f0 ) ≤ 0 ∀b '&
? H
#(? 7
3 & H A 0
ta Drs
−1&
(˜ f ; ˜ w ) #
! ta xa(˜ f ) &
! Drs
−1 5 qrs(˜ f ) &
!8@ qrs
m/ qrs &
?H 3 9 ' ( ta 9
' xa ( Drs
−1 9'
qrs 3!' Jbic >
.//7!
2
#&8@
&
,*
- &
7 &!
7.! >
71!
) " $ ./41! ' 5
./4:!#
12 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
,*) %2((" 7' "&
>.//7!8$ 8$!&
8$
>.//7!"-8
I .//A! ' > .//0!
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?@ # 5
&)E
&
!&5
-8$?@ %?@
8$ "-
?F.1:8*)?
?
1:::
.A. )5
q = q0 (S / S0 )−0.6
8<60"",:3
J8$&@#
KJ8$@# Rel(step n): = 1
a (xa
(n)− x a
(n))2
a a
xa
(n)
x a
(n)= 1
nxa
(k)k=0n−1
K J8$@# Abs(step n): = 1
a (xa
(n)− xa
∞)2
a xa
∞
&
" A J ! !
"-8$?@ ?!
?@ %
! ?@ -
?@
8$"-
13 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
" A
!@J !
0,1
100
1 101 201 301
iteration n
MSA
PET
F −W
!@ !
0,1
100
1 101 201 301
iteration n
PET
MSA
F− W
" 0J!!
8$ 8$
?@ "! ?@
J8$@ ! $
J 8$
8$
" 0
!@J !
0,1
100
1 101 201 301
iteration n
PET
( )MSAs indiscernible
!@ !
0,1
100
1 101 201 301
PET
( )
iteration n
MSAs indiscernible
.
& -
14 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
?@
.J
$ 5
J (! J (!
"- :0 8$
14
8$ :7 8$ :A
?@ 71 ?@ A.
)'
"-
,*+ '0(&0"0(%&&#/%%2((" 7 "&
>
> A! >
) wb! !%
!'
( xa(f ) − Ca ≤ 0 >
LA (f ;w ; τ ): = J(f ) + 1
2τ (max0;wa + τ (xa(f ) − Ca )2
− wa2
)a τ > 0 5
ta[n]
(u): = ta(u) + max 0;wa[n]
+ τ (u − Ca )
min f J[n]
(f ) E f ≥ 0
J[n]
(f ): = LA(f ;w[n]
; τ) = ta
[n](u) du
0
xaa( )+ Jbic(f ) − Drs−1
(u) du0
qrsrs
2 f[n]
wa
[n+1]: = max0 ; wa
[n]+ρ (xa(f
[n]) − Ca ) ρ > 0
C(n) = w
[n+1]− w
[n]≤ ε
- 7. τ = ρ =. 05 ε = 5 ,
F 10
( w[0]
= 0 F.!<706%F0!<.74%F.:!</1%F1:!<04@!
?@ #
' ?@ 0
15 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
.*
-
) -
'
' A.!
'
!
A1! '
A7!
.*) (('(#8#%&("$%
'
'
>.//0!
) '
Gk(i) = ri,b .Xb,kb + vi Tk = Pk+ vi Tk
' Xb,k '
'!
ri,b vi Tk '
b + d #
b ri,b
d
)µσ d = 2 !
- 5'
fk '
L(θ ) = ln πki
(θ )i∏ = fk ln(πk(θ))k
θ = θn n
= rb b∪ µ ; σ πk(θ )
' ki
(
πk(θ ) = Hµ ,σ(Uk ) − Hµ,σ(Lk )
Uk Lk ''=.'
.'
8' L(θ ) #> θn
πk θn'!"*>
'
16 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
1J?#5
" .//7 Tk Pk
ˆ µ <A14A σ <:7A4
ˆ σ ˆ µ < :::A0
ˆ σ ˆ σ < ::. Cov( ˆ µ ; ˆ σ ) / ˆ µ ˆ σ = −7e − 6
66."",
18 C5
Pk""! Tk! ,
,
C " C17 30A :/36 .0:: .011
.. 436 :3.6 70:: 7A66
C " C17 10A :A: 13:: 131.
.. 77/ :7: .7:: .16/
" C17 A:. :036 0:: A03
.. 0:4 :A.6 A0: A/A
.*+ " !#$"""!"!0
L '
ε X M ( '
L < "M! " ε X
εY = (∇X F)ε X εY
ε X
( F(X,Y) = 0 ∇Y F
εY = [∇Y F]−1
[−∇X F]ε X ./43!("
'9 F = ∇X J
εY = [∇Y∇Y J]−1
[−∇X∇Y J]ε X
∇Y J 9 L ∇Y∇Y J *9
L ∇X∇Y J ' ∇Y J M
(M Drs
Hrs ! ta
zb
.*, /#8(!#
- A1"
'>.//A!
> #
"!
!>#
K TF TT
N(T F ; σTF
) N(T T ; σTT
)
17 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
K q N(q ; σq )
K M σ
N(M ; σM ) N(σ ;σσ )
K p
∆T = TF − TT > 0
v =
p
∆T
fT = q 1− H
v ( )( )
R = pq 1− H
v ( )( )
- X = ∆T, q, M, σ
'# ''
'
> ∆T = 0.2145h p = 15FF q = 3000 veh/h
M = 60FF/h σ = 0.6
'
σq
q =
σσ
σ =
σM
M = 10%
σ∆T
∆T = 15% '
0N
!
7!
7C
@ fT
'
'
'
ε∆T ./6 :0/ :37
ε q .:: :/A :63
ε M ./6 :0/ :37
εσ ::76 ::.3 :::7
σ fT
f T=
σR
R 76N .1N .:N
- '
'
'
(
18 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
1*
(
# ) !
! "
1*) "0"#"&$$$'
$ )
Grs
k(v) = Trs
k+ Prs
k/ v
Grs
k(v) = G(Trs
k;Prs
k;v)
'
Grs
k(v) = Trs
k+ Prs
k τ rs(v)
τ rs )
Hrs(v)
.#
K Grs
k(v)
Trs
k+ Prs
k τrs(v)
Srs = min k Grs
k(v)dHrs(v)
K v rs
m
τ rs
m
τ rs (
v rs
m) τ rs
K I rsm
Irs
k &
1/Hrs
−1
τ rs Hrs
−1
K Frs(u)
(τrs Hrs−1
)(t)dt0
u τ rs τ rs
− (τrs Hrs
−1)(1− t)dt
0
u ( τ rs F5"τ
F5"
1*+ % 6%-"
Grs
k(v) = Trs
k+ Prs
k τ rs(v) + εrsk
(ω )
εrs
k(ω )
E vrs εrs
k
εrs
kk
19 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
vrs εrs
k8C
vrs εrs
kk ;ln vrs 8C
*
% 8$
1*, % $/$#/&!& ""
(
'
8
% &
#
K'
K
K I rsm
Irs
k'
K(?%
0 I rsm
Irs
k
'&
2'
1*. $%"&$''""" %
' > .//0!
. A>.//0!
8$
8 &# &
!
#
"
)
$
%
20 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
'
>.//0!
F./3.!>OO-../$
$ ./!$?9
F952./67!>P'/0 1
?
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>".//0!;7< =8 $33./$$%% $$ )$/%
(CJ@ $JJ./4(CJ@ $"
21 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
>".//0!
>/3$3 !?/4$%!
..6.14? JF>
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(CJ@ $"
8J./67!@ % !2 5$?
$ @$
8?5I.//A!
?3 $% 2>/13!14/.4/6F9
8*9.//:!!/: .4 . ! /
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J>./43!$ 55$$%@ ) $9
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-9;./01!$ J J?$ 1
4$?((710764
<*< =
> θm
πk #
∂L∂θm
= qPk
πk
∂πk∂θmk
∂2 L∂θm ∂θn
= qPk
πk
∂2π k
∂θm ∂θn−
1
πk
∂πk
∂θm
∂πk
∂θn
k
2 πm( ra ; µ ; σ ) µσ
Hµ ,σ ra Um Lm
v m
m+ n: =
Pm+n − Pm
Tm − Tm+n
2
∂
∂µ πm = ∂∂µHµ,σ(Um) − ∂
∂µHµ ,σ(Lm)
∀D ∈ ∂
∂µ ; ∂∂σ ; ∂2
∂µ∂σ ; ∂2
∂µ 2; ∂2
∂σ2
D πm = DHµ,σ(Um) − DHµ,σ( Lm)
ra
' Xa,k = Xa,m(k)
22 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
∂
∂ra
v m
m+n=
Xa,m+n − Xa,m
Tm − Tm+ n
∂2
∂ra∂rb
v m
m+n= 0
*
∂πm
∂ra=
∂ Hµ ,σ (Um)
∂x
∂Um
∂ra−
∂ Hµ ,σ (Lm)
∂x
∂Lm
∂ra
∂2πm
∂ra∂rb=
∂2 Hµ ,σ (Um)
∂x2
∂Um
∂ra
∂Um
∂rb−
∂2 Hµ,σ (Lm)
∂x2
∂Lm
∂ra
∂Lm
∂rb
∂2πm
∂ra∂µ=
∂2 Hµ ,σ (Um)
∂µ ∂x
∂Um
∂ra−
∂2 Hµ ,σ (Lm )
∂µ ∂x
∂Lm
∂ra
∂2πm
∂ra∂σ=
∂2 Hµ ,σ (Um)
∂σ ∂x
∂Um
∂ra−
∂2 Hµ ,σ ( Lm)
∂σ ∂x
∂Lm
∂ra
<*
&%
µσ H(x) = Φ(
ln(x)−µσ ) Φ
F5":.>φ
φ (t) = exp(−t2
/ 2) / 2π 5
ln(x) − µ
σ tx
#
H
−1(y) = exp µ + σ Φ
−1( y)( )
F(x) =
1
H−1(u)du
0
x
= exp(
σ2
2− µ )Φ Φ
−1(x) − σ( )
∂
/ ∂x
/ ∂µ
/ ∂σ
H(x) =
φ (tx )
σ
1 / x
−1
− tx
∂2
/ ∂x2
/ ∂x ∂µ / ∂x ∂σ
× /∂µ 2/ ∂µ ∂σ
× × / ∂σ2
H(x) =
φ (tx )
σ2
− (σ + tx ) /x2
tx / x − (1− tx2
) / x
× −tx 1− tx2
× × tx .(2 − tx2
)
23 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
<* )7+7,
? ?.# " .! ∃u ∈Ers
m
(P rs
n− P rs
m) / u > 0
0 < T rs
m− T rs
n >
v0 : = (P rs
n− P rs
m) / (T rs
m− T rs
n) # .!
v0 ∈[supErs
m; inf Ers
n] (
supErs
m= inf Ers
n
v ∈]sup Ers
m; inf Ers
n[
Ers
i
BB Ers
m
Ers
n$
v rs
m= sup Ers
m= inf Ers
n= (P rs
n− P rs
m) / (T rs
m− T rs
n)
? ?1# - 8@
Qrs
n= qrs
mmefficient ≤ n
= qrs dHrs(v)
Ersmmefficient ≤ n
= qrs Hrs(
v rs
n)
e(n) = n (
Qrs
n= Qrs
e(n)
Qrs
n= qrs Hrs(
v rs
e(n))
!
? # qrs
m≥ 0 Hrs
& n
v rs
e(n) %
qrs
mm = qrs
Qrs
m rs = qrs Hrs(v rs
e(m rs ))
= qrs Hrs(Hrs
−1(1)) = qrs ( e(n) = e(n − 1)
Qrs
n−1= Qrs
n
qrs
n= Qrs
n− Qrs
n−1= 0 (
qrs dHrs(v)
Ersm =
qrs Hrs(
v rs
n) − qrs Hrs(
v rs
e(n−1))
= Qrs
n− Qrs
n−1
= qrs
n
? ?7# ?1
&>
)&#
FA#
!! Qj = Qm
v rs
m=v rs
m
v rs
m∈Ers
m
G rs
m(v rs
m) = minn G rs
n(v rs
m)
≤ G rs
j(v rs
m) = T rs
j+ P rs
j/v rs
j
Qj = Qm
I rsm
≤ I rsj
(P rs
i− P rs
i+ 1) /
v rs
i
m < i < j P rs
m< P rs
j j < i < m
P rs
m> P rs
j
? # -
[qrs
m]m Ωrs > v ∈Ωrs #
Ωrs ⊂ m[
v rs
m−1;v rs
m] ' 8@ !
v ∈[
v rs
m−1;v rs
m]
v rs
m−1<v rs
m > ! )
n ≤ m I rsn
− I rsm
= T rsn
− T rsm
+ (P rsi
− P rsi+1
) /v rs
ii=nm−1 % n ≤ i < m
v rs
i≤v rs
m−1≤ v
(P rs
i− P rs
i+1) /
v rs
i≤ (P rs
i− P rs
i+1) / v 2
(P rsi
− P rsi+1
) /v rs
ii=nm−1 ≤ (P rs
i− P rs
i+1) / vi= n
m−1 = (P rs
n− P rs
m) / v
qrs
m> 0 0 ≤ I rs
n− I rs
m
0 ≤ T rs
n− T rs
m+ (P rs
n− P rs
m) / v
G rs
m(v) ≤ G rs
n(v) ( n ≥ m
v rs
i≥v rs
m−1≥ v
24 A dual criteria traffic assignment model
F.M. Leurent (1996) In Transportation and Traffic Theory, Lesort J.B. (ed), 455-477. Pergamon.
n ≥ i ≥ m − (P rs
i− P rs
i+1) /
v rs
i≤ −(P rs
i− P rs
i+1) / v!.
− (P rs
i− P rs
i+1) /
v rs
ii= nm−1 ≤ − (P rs
i− P rs
i+1) / vi=n
m−1 = (P rs
n− P rs
m) / v "
I rsn
− I rsm
= T rsn
− T rsm
− (P rsi
− P rsi+1
) /v rs
ii=mn−1
0 ≤ I rs
n− I rs
m
G rs
m(v) ≤ G rs
n(v)
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