Sensitivity and error analysis of the dual criteria traffic assignment model

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https://hal.archives-ouvertes.fr/hal-00348420

Submitted on 18 Dec 2008

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assignment model

Fabien Leurent

To cite this version:

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/$ 8$ % ' 0 ' = ₌∂' ∂ $ = = ₊ = _$₊∂' ∂& $ = & =₊∂' ∂µ $ = µ =₊∂' ∂σ $ = σ =_{+ =}∂' ∂µ ∂' ∂σ E εµ)εσ$ .$ $+ # *' ''$, #* B & !""' $

9 = 9 = _&9 & @!DI _µ9µ = _σ9σ= 8I E ε_µ)ε_σ$ 9µ σ= !K

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* η ′ α$= !9 :−! α$ FD)!G ℜ :−! α$= %

{

) : $<α

}

: :−! GD)!F : $ ( D ! % ηN :−! D$> D η ′ D$ :−! D$= D η ′ D$ OP) #$ :−! D$= D ( ! :−! !$< +∞ η ′ !$ D B λ ' + ' = & * −λ = * −λ * = * 0 :−! ' 9 &$= 2 9 ) $ − )$$ "$ !$ :−! ' 9 &$= 0 , ' 9 &= : $ 3 L , 4 0 =$ L ! ' , - * *'-, , α ≡ ' 9 & ∈GD)!F & ?≡ F ' ) ' )λG #$

& 2≡ F* −λ ) * −λ ) ' + ' − &G = F − 2 ′ η α$−λ) −λ) ' + ' − &G B

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∇_?2 = ∂ ∂' − 2 & !−α$ = ∂ ′ η ∂α − 6 −!9 ∂ ∂' + 2 &α !−α$ ∂ ′ η ∂α − 6 −!9 ∂ ∂' + 2 &α !−α$ ∂ ′ η ∂α − 6 −!9 ∂ ∂' − 2 &α =∂ ′ η ∂α − 6 −!9 ∇_?2$−!= ! 6 ∂ ∂' − 2 &α =∂ ′ η ∂α − 6 −!9 ₋∂ ∂' − 2 &α !−α$ ∂ ′ η ∂α + 6 −!9 −∂ ∂' − 2 &α !−α$ ∂ ′ η ∂α + 6 −!9 ∂ ∂' − 2 & !−α$ =∂ ′ η ∂α − 6 −!9 !#$ 6 = ∇_?2 = ∂ ∂' ∂ ∂' − ∂ ∂' ∂ ∂' − 6 −!9 ∂∆ ∂' − ∂∆ ∂' − 2 & ∂ ′ η ∂α −2 & ∂ ′ η ∂α α =∂ ∂' + !−α$ =∂ ∂' +α !−α$ ∂ ∂' + ∂ ∂' $ % ∆ = − , - * *'-, , 3 >= F ) ) &_D)θ G &_D 6 6−!$ 3 0 ∇_>2= ∂2 ∂ ∂2 ∂ ∂2 ∂&_D ∂2 ∂θ = ! D −∂6−! ∂&_D 2F ∂η !$ ∂θ − ∂η α$ ∂θ − ! −α$ ∂ ′ η α$ ∂θ G D ! −∂6 −! ∂&_D 2F ∂η !$ ∂θ − ∂η α$ ∂θ +α ∂ ′ η α$ ∂θ G !"$ ∂' ∂ ∂' ∂ ∂' ∂&_D ∂' ∂θ = ! 6 ∂ ∂' + 2 &α !−α$ ∂ ′ η ∂α − 6 −!9 −∂ ∂' + 2 & !−α$ =∂ ′ η ∂α + 6 −!9 ∂∆ ∂' − !−α$ 2 & ∂ ′ η ∂α ∂6 −! ∂&_D − 2 ∂η !$ ∂θ − ∂η α$ ∂θ $ ∂∆ ∂' − 2 & !−α$ ∂ ′ η ∂α$+ ∂ ′ η ∂θ ∂ ∂' +α ∂∆ ∂' − 6 −!9_$ =D =D =D =D !A$ 3 '$ !/$ 4 =D$ & =D$ !/$ % &= 6 $ = &_D 9 _D$ D D→ D− % (

6−! &$= _D & 9 &_D$!9 D ₆−!9_&$₌ ₉

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6

6−!9 =D $ =D $ =D $

!/ $ !/ $ !/ $ =D $ !/ $ & 9 &_D

∂6−!9∂&_D = − &6−!99 &_D0 _D→ D− & 9 &_D ≈ ! R

∂ 9∂' ≥ ∂ 9∂' ∂ 9∂' ≥ ∂ 9 ∂' 6 0

6−!9 6 &= &_D &_D

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$ =

(

δ_$

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+

(

+.+∂ +9∂'$

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/&/ ) * " !*#* # 6* ) * "$*' # # * 7$ ' # $+ , : , % / , ! // / ' 0 1/ 5 6 L 0 $ L $′ 2_$ = 2_$_′ 3 3 % ! ) 3 * $ $ % 2 3$ & 2₎ $ ) ₎ & & )$≡ _{$ ) * $$=} '_$ ′ &₎ ≡ _≤)&

) & = & = ′ &₎ 3 &_D = ′ &_D≡ D

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$ ) ) ($≡ $+ 2 − 2 +!$η ′ &_&′ =* $ $ )−! − 6−! & $+ 2₎ η !$− 2_!η D$+ 2 − 2 _+!$ η &′ & − ′ & & η ′ ′ & & $ =! )−! =!$ (f ; w) $ &′ 5 5 6 =!$ + 2 η !$−η D$$− 6−! &$ B N , = ! 9 η !$−η D$$ =!$ - - ! / & ℜ+$T ℜT T ) ($ )) ($ ))( $= F F _$ ))( $G_$) F− ₊ ) $G₊G & !""' $ )U)(U$≥ 0 ==$0

M )U)(U$≥ 0 ∀ ))( $ ≥ 0 )U)(U$⋅ ) − )U)( − (U$ ≥ DM ==$

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1$# , ! * % %

0 1000

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12 Toll ($)

R

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$

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mean revenue of the

dual criteria model mean revenue of the

single criterion model

Value of Time Travel times Demand volume = * % 0 1000 2000 3000 4000 5000 6000 7000 8000 0 2 4 6 8 10 12 Toll ($) R e v e n u e ( $ /h

) mean revenue of the dual criteria model mean revenue of the

single criterion model Value of Time Travel times

Demand volume

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/ * 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1 2 3 4 5 6 7 8 9 10 11 Toll ($) R e v e n u e ( $ /h )

mean revenue of the dual criteria model mean revenue of the

single criterion model