The Theory and Practice of a Dual Criteria Assignment ... · The Theory and Practice of a Dual...

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

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https://hal.archives-ouvertes.fr/hal-00348537

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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<*< =

> 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)



∂

∂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

)



<* )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



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)

The Theory and Practice of a Dual Criteria Assignment ... · The Theory and Practice of a Dual...

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Transcript of The Theory and Practice of a Dual Criteria Assignment ... · The Theory and Practice of a Dual...