Modelling seat congestion in transit assignment

HAL Id: hal-00348409

https://hal.archives-ouvertes.fr/hal-00348409

Preprint submitted on 18 Dec 2008

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Modelling seat congestion in transit assignment

Fabien Leurent

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Modelling seat congestion in transit assignment

Abstract

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1. Introduction

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2. The line model: assumptions and basic notions

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3. Statement of line problems and algorithms

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: 1 2 + & & + # " # 5 + & # 5 5 5 5 ##7 5 # 5 8 & " # (− 3 & "# " + & # 5 " 2 231 ( 1 3 − ∆

8 & ∆ =γ− & ++ " "# " 3 @ & & "# + ##7 5 & " " C 3 @ & & "# + # 5 & 5 5 & " "

ϖ 9 + 3 & "# " 1(− 23ϖ 3 * # + # 1 2 9 " 5 & # 5 + & 5

5 & " 5 & 5 53

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& # 5 + 5 # & # 8 9 " & 5 # 5 & 5 3

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4. Network representation and cost-flow

relationship

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! 5 ( E - (!E( E !

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5 B ##9 & 8 # " 8& & 6 0 & 0 3 "& # 5 7 5 & & & # 57 7 "

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! 5 (.E - (!E( E !

+

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" #9 & ++ " + & " 5 # & # 5 " " & 3 & # # 5 #5 & + + "

2 1* = 1! 2 2 1* + + _{= 1! 2}

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& & # # 5 " 5 #5 & & # 5 " + & 5 # + 5 & 8 & & 5 " _{+( +(}

2 L M L M 1 %Q Q = + _{+( ∈ +( ∈} 1( 2

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,!

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5 # ' & + &9 & "

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1 353 7 # & + , % ()!!23

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" + : "9 + & " # & " "" + & 3 & & : # &9 & &=1 &Q2 # " + 8& "& & 5 + #

5& + 8 #93

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∈ ∈ ∈ + = 2 1 L 2 1 " 2 2 1 8 1 M 3 Q 2 1 % & ( ! ! & ! & & * * * 1( 2 8& "&

7 ( 1&2 & + # 9 & # 5 & +

7 &Q =

_∏

_∈ &Q & & +# 8

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7 8& =α E & + & # 9 3

& "9 + " " & & ## 8 + # 5 " " # # 5 8 & & &3

& 8 5 8 5& 5 + " α : # + & # "" # + " 9 & "# 8 & # & 9 5 0 #

B & " 9 6 ++ # & 8& 8 # ## 8 9 + & 5 8 5 # & # + & 0

! 5 ( E - (!E( E !

5. Equilibrium analysis

? & + & + 8 8 9 + ++ " : # " 9 & # & " # #9 3 @ & ## + & + # + +# 8 & &9 & 3 & ++ " : # 8 ##

+ & + + $ # % # 9 # 1$% 23 *+ 5 : # "& " C & + + # ; : # 9 # 1 ; 2 & 0 " + : # 8 ## 3 #9 & + ""

5 8 ## + 8 " : # " 5 " " & + # 9 5 8& "& 6 5 " 3

-

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%& & ! ,+ ) ! - *− & # & , ) - *

* .* # % # * * * )

*

1 & + + # 8 6 +# 8 3

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* &9 & +# 8 2 " 8 6 +# 8 *_! =312 2 & + ## 8 5 8 9 ∈ ∈ ∈ ∈ = ' & ( & & & 2 1 J K ( Q _{1( 2}

8& "& ( 1&2 & + # 9 & 8 & & + 1 & 8 & 2 (_{K ∈J} : # ( + ∈ & 8

∏

∈

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! 5 (-E - (!E( E !

& " + &9 & 1$5 9 ## ()!!2 + # &9 & +# 8 " + # 8 6 +# 8 3 $ 7 5 9 1( 2 " + 1(/ 2 1( 2 & 7 5 9 + &Q & " + &Q 3 & " + &

+# 8 1( 2 + ## 8 ∈ ∈ ∈ ∈ ∈ + − − = & ' ( & ∆ & ! ! 1 2&Q 8& "& ∆ = _∈ + _∈ − _∈ − _∈ ! ! (K J (K J3 $ 8 + " & ##

& & C 9 # 5 ∆ = 3 & + "

+ & ∆ = " #9 " ! &+ (_{K ∈ J} =( & # ! 3− #9 + = + & ∆ =( " (_{K ∈J} = + ## − ! 3 & ' & & ' & ( & & ! ! & = = = − ∈ ∈ ∈ ∈ ∈ + − 1 2 Q + & 2 3

-

.

#

(

. / 1# ) " & 0 # =M L _{∈ ∈} / & & # % 2 =M & ∈ ∈ RK J &∈' L % & *_! =312 2

## # & 4 =Mµ ∈ ∈ RK JL & & / # J K R ∈ ∈ + ≥ & ' &∈ ∀ 1(- 2 ' & & = ∈ 1(- 2 2 1 % & *_! −µ ≥ ∀&∈' 1(-"2 L 2 1 M% 3 _! −µ = & & * ∀&∈' 1(- 2 & + ## 8 "& + "& ## &9 &

" % 1& *_! 2 # & µ #9 & &9 & 8 & "

!

& 2=µ 1

% * 1& " #2 9 " 9 +# 8 & 3 & " " 8 & & + + : # ++ " 5 " @ ()/

"" 5 8& "& "& 6 & 5 "& " C & 8 # " 3 & + " 1(-2 7# " # 9 # 1$% 2 & # 12 4 2 8 & " " + " + ## 8 L J K R 22 1 1 M% 2 12 4 & 3 2 −µ ∈ ∈ &∈' 3

! 5 (!E - (!E( E !

-

1#

5

6%

/ 5 1# ) ! & & # % L J K R M & ∈ ∈ &∈' = 2 & # # & 0 ∈ ∈ =M L # ## # * #/ # # & &

# % 7 =Mη& ∈ ∈ RK J &∈' L/ & * & +

2 1 3 2 12 7 − 2 ≥ 8 1(!2

%& & 812 2=M% 1& 312 22 ∈ ∈ RK J&∈' L)

3 * + & 2 : # 3 5 4 & " 0 + # # + 1(-2 + 9 7 =Mη& L + 1(-"2 & L 2 1 M% 3 −µ ≥

η& & *_! ∀&∈'

&∈' 9 # & _&∈' η& %& ≥ _&∈' η& µ = µ 3

1(- 2 8 5 & _&∈' & %& = _&∈' η& µ = µ & "

2 1 % η − ≥ ∈' & & & & 3 & 9 # 1(!2 + 5 ∈ ∈ 3

% #9 & 1(!2 & # 2 6 7 =Mη& L : # 2 0" & 7 1 2 + & #9 &9 & & ' & # 5 µ =% 1& *_! 2 #9 + 1(-" 23 ;+ & 8

&9 & ' + 9 & 8 & & > # + η& = & −ε ε

+ =

η&′ &′ & 1(!2 9 # &

2 % 1%&′− & ε≥

8& "& # & & " + 9 &9 & & 8 & +# 8 & 2 # ' + 5 µ = _&∈' % 1& *_! 2 1(-" 2 + ## 8 3

-

1#

/ 1* # 3 ! & & # %

/ & # & * # % &

)

3 5 & ; + # & & & " + " 2

1

% _!

! & *

* " 8 & " & 8 6 +# 8 *_! 3 @& & " + " 3 + # 2 1 " + &9 & +# 8 2 & " + " 2 % 1& 312 22 " + "

2 12 8

2 3 * & 1₂ + + # &9 & +# 8 " 0 " " & & & # : # 9 # # 3

! 5 ()E - (!E( E !

& : + : # & # 5 # & 8 & 0 " 3

- - 3

!

#

& ; + # + # # & & 0 " + : # # 5 : # #5 & 3 * # + & @ 1()).2 & 5 + " " & 8 & & & & & 8 5# ##9 " 5 # " # "& + : # #9 & # C

" & & F " & 3 A 8 + & 6 + # " 9 8 + & 8 ##76 8 & + "" * 5 1 *2 # & 5& 8 &

& & 5# ##9 " 5 & " # 5

# " 5 " 9 # " 9 # 5 & # 9 5 & " +# 8 3 ; 8& & # 9 5 " "# C 5 &

& " +# 8 "# ++ " : # " & # 9 5 " 8& & ; + " 8 " 3

& & 5 # & # 9 5 "& " +# 8 # 5 & & *3 ; 8 # " " & # 9 5

5& + 8 8 9 " & 8 # : #9 ## & &9 & & " 9 # &9 & " 5 # 5 "& + &

# & " & " ++ " " 3 A + # & 8& "& #9 : & 5 + # # # 5 &

& *3 ; # " # 9 &9 &7 5 # 8& "& & &9 & " & +# 8 #9 & 5& & " " & " 1 & & # + : "9 " # & ++ " +# 8 23

" &9 & +# 8 2 M 3 9 L 8 & " 8 6 +# 8

!

* M 3 5_! L " & M 3 η L& &9 & & +

3 ? # & " & +# 8 9 & # " " 2 +

∈ ∈ ∈ η

= _{& '} & % 1& _! 2 2 1 38 2 * 9 3 1()2 # & " "7 7 3 & "7 ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ η = η = ! & ' ( & & ! ' & ( & ! & ! & & 2 1 K J 2 1 ( Q 2 1 " 2 1 " Q 2 1 * * 2 9 8 & " ∈ = _{! !} !19 2 2 " 1* 21 8 1 2

8& "& 9 # 8& & 9 3 * & 7

∈ ∈ ∈ η ∈ ∈

= _{& '} & _{( 1&2}&Q 8&1 _! 2 2

19 2 *

! 5 E - (!E( E !

# " & 8& *_! 18& " 5 ++ " # # + : "92 & 9 _{∈( 1&2} &Q _∈ 8& + " + &

2 1&

( #9 + & ρ 3 &&

∈ ∈ ∈ η ρ

= _{& '} & & 2

19

8 1 (2

8& "& # + " + 9 & 5& & η 3&

% 8 0 # 9 &9 & +# 8 71 2 & & 67 &

* # " 8 & : " + C 1ζ 2 _≥ "& & ζ =(3 & " 71 238121 22 + & 0 # 9 & ++ " " " 9 & " +# 8 21 2 #9 ∈ ∈ = 1 2 2 1 2 1 2 1 3 28 7

8& "& 1 2 & # " + +# 8

2 1

2 & + & + + & 0 # 9 9 5 + ## & ?, +# 8 & &9 & + # " & " ++ " " 3 ? &

+ 1 2 5& + 8 " ∈ ∈ ∈ − = − = ! ! ! 1 2 1 " L M 2 1 2 1 3 2 1 2 1 2 1 2 1 2 1 2 1 2 1 * 2 7 8 2 8 7 7 8 1 2

& " 21 2 8& "& " " & 8 5& 5 + & 0 # 9 71 2 < 8 & 8 5& " ++ "

Γ ζ = ζ1 2 E 8& Γ = −₌(ζ 3 & "7 " # 5& + 8 #93 = 5 # + " & 7 " " + & # " + 21 2 − = − = ζ = ζ = ( 1 2 1 2 ( 1 2 1 2 2 1 2 1 2 1 2 12 8 7 8 7 8 3 1 2 #9 # + : " β & + ## 8 5 8 9 β =ζ 8 171 22 2 1 1 2 (=β +ζ 8 7 β ₊ 3 1 .2

& & β EΓ & " + & " +# 8 8 ++ " " Γ β = E 2 121 2 8 3 1 /2

! 5 (E - (!E( E ! L M L 2 1 " M 2 1 23 1 ,O 2 1 2 1 2 1 2 1 2 1 2 1 ∈ ∈ ∈ − + Γ β = − = ! *! 2 8 7 2 1 2

- : 3

;

3

++ " 5 8 6 8 & " 5 " + 9 + & + ## 8 5 : # #5 & 8& "& 6 + 8 8 6 +# 8

! * + " !_! + 0 # 9 8 # ## " +# 8 " *_! !_! + # =M L _∈ " # β Γ 2 $ 3 3 ; # " " " " =M L _∈! ! ∈ =M L =M# L_∈ <=Mκ L _∈ & ?, 0 #=M L_{∈ ∈} # " ε & " 5 " # # : " + " 5 2 1ζ _≥ 8 & ζ =(3 & : # #5 & + + 1 3 *_! = *_! = 3 = β = Γ = 3 . % ( & 3 # & " " =% 1*_! 2 + ## ∈! " .3 3 % * % * 3 2 = !_! = 3 9 ∈

7 & # &9 & & " " " 9 # 5 # 3

7 & ?, +# 8 M ∈ L & " #9 # &9 & 9 # 5 " +# 8 3 7 2 = 2 + _∈ 3 = + + ## ∈ 3! % 2 * 3 $ =2− _∈! 3 Γ =Γ+ζ β =β+ζ $3 ∈ Γ β + − = 2 _! S 3 & # *_! =*_! +ζ 1!_! −*_! 2 2 1 _{! !} ! ! * ! * * = +ζ − 3 3 ;+ 3≤ε & # # = +( 5 % ( & 3

& & + "" 5 8& "& & 0 # 9 +# 8 !_! 7 C 5 + ## & ?, +# 8 & + & " " 9 & " +# 8 *_! 3 & " 5 " " 3 & # 9 5 + &

! 5 E - (!E( E !

6. A simple case of hyperpath choice

## & " 5 ++ " " F " 8 & &9 & "& " & " 0 + ++ " 5 # " # " + &

8 6 & 3 * & 5 6 & " 8 # & 9 # " ## # * + 1 & ' 0 '5 # + " 5 # 8 & 9 & 5& + : "92 " # #9 & 8 "

" & F = % ,'+ & %&T # 8& "& ## 8 + + 8 & # / # 3 7 6 8 ## 5 # * %&T # ,'+ & 8 + & "

& " " + : 3 & & 5&#9 # 6 #9 & 7 & 9 8 ## 8 & & 5& " 8 53

& " 5 7 6 ##9 #

& " 5 + & = ## ## 9 # * & %&T # + 5 # (( $ + 5 # 3 = 5 # * $

& : 3

:

5 . ## & 8 6 + + = ## ##

,'+ 3 ? # * & # 5 + %&T # ,'+ " & (( 5 5 & + + " 1, " 23 & # 5 + $ %&T # " 8& 8& "& & 3 & # 5 + = ## ## %&T # # (( 6 (/ "# 5 + 6 & # 5 + = ## ## $ # # "# 5 + 3

" + : " &E& # (( &E& # &E& # *3 & 8 5 8 5& 5 + " α : # (3

& ++ " +# 8 " " 9 " & + & ξ κ & ?, +# 8 + = ## ## ,'+ B ξ & +# 8 + & 5

5 # * %&T # B κ & " " 9 # # # * & $ 3 ! " #$ ,'+ %&T # $ = ## ## * (/ + U E& (( (/ + U E& ! * ((V 5 V 5 * + U E&

! 5 E - (!E( E !

:

/ !

& ( 8 & "" # * %&T # & 8 & "" # * $ 6 & &9 & &9 & ( + & ( # B &9 & + & # B &9 & & " + & ( & 8 & # " & = ## ## 3

& " &9 & " & + ## 8 5 + " + & # 9 5 & %&T # ) -(/ 2 ( 1 (( ( = + − + + = − #1# ((2 . ) ( (= + α = − . (/ ((+ + + = = 2 # 1 = + = α # # # # # 3 . 3 3 3 (( ( (( _{= −} + + + α =

:

1#

!

+

#

+ & 8& & ( & " # & &

) .

(≤ ⇔ ≥

& # & 8& & " &9 & & 9 5# & & " &

L M₎( ( ( _{⇔ ∈} ≤ ≤

& & #97 : # 6 & + ## 8 5 5 & # + ) ( < #9 & 3 ) (

= & & &9 & 9 3

M L

) (

∈ #9 &9 & ++ " & ( & ##93 = & & ( &9 & 9 3

> #9 & ( 3

W 8 5 8& "& &9 & 8 9 # κ ξ

@& & ( # & = K( _ξ+κ J & : & > 9 # & < κ−ξ3

@& & # & = 0Kκ−_ξ J & : & _{< )}( _{9 #}

& >κ−ξE)3

! 5 .E - (!E( E ! 3 3 ( π + ξ π − κ =

8& π = # E1# + #₍₍2 π₍=(−π & : & ₎( _{< < 9 # &}

E E ) E ) E ( ( π + π ξ − κ > > π + π ξ − κ 3

* 8& # &9 & & ?, +# 8 # &9 & +# 8 & ( ( + π + ξ − π − κ = 8& "& 9 # ξ − κ = π + π + + ₍ ( 1 2 3 * = = ₍+1π + π₍2 =κ− ξ & ## " #9 5 & : 1κ− ξ2E1π + π₍2 + ## ?, +# 8 " 9 9 ₍ & # 5 κ−ξ ## " 9 ₍3 * _{= )}( ( = + π + π =κ−)ξ ( ( ) ( ₂ 1 & ## " #9 5 & : κ−₎(ξ _{+ ## ?, +# 8 " 9 9 & # 5} 2 E1 2 1 ₎( ₍ ) (_{ξ π + π} − κ ## " 9 3 * 5 ?, +# 8 + + ++ " : # 9 0 7 M E_E L ( π + π ξ − κ & : # 8 & > B 7 M E_E L ( κ−ξ π + π ξ − κ

& & : # + 8 & > " 8 & = # 8 & _∈L₎( M_B 7 M _{κ−ξ κ−)}(_ξL _{& : # 8 & L M} ) ( ∈ 1 5 7 9 # 3 3 )κ (< ξ2B 7 M ₎( E)_E)L ( π + π ξ − κ ξ −

κ & 8 : # 8 & & _∈L₎( M

) ( = B 7 M E)_E) L ( κ π + π ξ − κ : # 8 & _{< )}( _{1 5 7 9} #2B 7 & " 8& M_πκ−₊ ξ_πE_E κ−ξL Mκ−(₎ξ _πκ−₊ξ_πE)_E)L≠∅ ( ( 5& # # 5 + : # 9 # + 8 & & " 3 κ

> 9 & # + 6 "" & " 5 & $ > %&T # 5 3

5 / " & &9 & " 8 & " ?, +# 8 + # + κ U ( ξ U ! 3 * π₍ =3. π =3 _πκ−₊ξ_πE)_E) >κ ( & 5 8 & ) ( < + ## + & 5 M κ +L 3 E_E ( π + π ξ − κ #9 & ( 3 E E ( π + π ξ − κ ξ −

! 5 /E - (!E( E !

: # 3 κ−ξ κ−ξE) #9 &9 & 3 κ−ξE) κ & & # &9 & # & " : # 3

30 31 32 33 34 35 36 37 38 0 1 000 2 000 3 000 4 000 5 000 6 000 7 000 8 000 9 000 10 000 OD Flow H y p e rp a th c o s t Path 1 Path 2 Hyperpath 3 3 equilibria 1 equilibrium 1 equilibrium 3 equilibria Hyperpaths 1 and 3 at p =2/3 Hyperpaths 2 and 3 at p =1/9 % &'( ( "

7. Conclusion

& " + & C & " + " +

& # # + & # 5 5 & &9 & + # + #97 : # 5 9 & + " # 5 "

# 3

& 8 6 5 #5 & 8 # " 8 & & * > 8 & + & & + # 7 " + 5 # C " "# 5 " + # 5 " # # # 8 # 1 - !23 ; & & : # # # + " +# 8 # 9 " 1 "# 5 "" 7 5 " 9 " 2 & 5& # + # " &9 & 3 & " 8 & 7" 5 # & #

&9 & 5 # & 8 & & 5 6 & +# 8 " # 9 9 "& X # 5 & # & " 9 /

5 & " 3 *# 5 & 5 # # & # "& 5 # ## " 5 " & + 8 # &

+# 8 & 5& 5 & " 3 & 8 6 & " 5 # 5& + " & + ## 8 5

• ; + " + " + 353 # 9 + " + "# 3

! 5 E - (!E( E !

• * " + + " 9 " "& " + 3 & 5 ## # # & 5 & # 5 #5 & "& 5

& " 5 #5 & : # & " + + " #9 " + "& 5 3

• ; "# + G " H " " + ++ 5 B " F " 8 & ## 5 & " 5 ++ " "& "" 7 5 " 5 3

• ; "# + " 8& "& # + " + & + 5 # 5 & # 5 # 3

• ++ " & # 7 ++ 8

" + # # & + 3

• , " "& " # " + & & 5 # 9 5& 5 " 3 & 8 # # # & 5& & + # 5 # 5 & + & # 5 3

• , " "& " # + & I 8 ## 5 5 " + 9 " 5 G"& " H # 9 "& " + 3

• 9 # + " # " " # 3

; & & I & " + + " ++ " "& "

0 " # 9 # + : # : 3

3 ) $ & 6 5 W & 5 + + # " & 8 6 #5 & 3 & # 55 + ' ' " " 3 2 (4! '+ 5 # * / 67 # ##

8. References

% % # 1 2 * + : "97 5 # + " 5 8 6 8 & " " " 9 " "& " C " + : # 3 ( & 8 = . -7./)3 % % 1 (2 % 7# 5 5 " 5 8 6 3 - / 7 -3 , % # 1()!!2 ## 4 * 3 , % C 1()!)2 5 # ++ " 8 #5 & 3 ## 4 * = .)(7.).3 , % C 1()) 2 5 + " 5 # " 9 : # #3 > ( 7(.-3 , " O # 5 , "& A W W 8 & 8 1 2 # # '5 # ' # + ' ;# " 3 * # & ! 4 0 ( * 3 * % " 3 5 O W # F 1 .2 & # + # # 93 * # & 4 # 4 9:; 5 1%, 23

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

& A W "& ; D = ## OA "& Y"6 , 1 /2 # 5 # "

" 5 5 5 5 5 #3 ? # & 4 ! # * : - 7 )( /3 * # # # & EE8883 3 + E 7# EF #Z E - 3 +3 A 1()!/2 @ &< * # * (< * 2 3 # " % "& "& # ' ' #3 A # 1()!)2 ? # 5 8 5 # + 8 6 3 ( & 8 ! 7( 3 5 %? @ 5 % 1()))2 * "& " 5 # 5 9 " "& # 7 8 63 ( & 8 ( -7( (3 = 1 2 * A # > 3 ? 7#

+ & % "& 5 # # 7# & + ## 8 5

8 & EE5 ## 3 3 5E # " E " E " Z8 "Z 7 3 +3

()))3

@ A # " 1()).2 : # * 5 * # # *#5 & 3 ? () 7 3