ATwo-PhaseGradientProjectionAlgorithmforSolvingthe ...

Research ArticleA Two-Phase Gradient Projection Algorithm for Solving theCombined Modal Split and Traffic Assignment Problem withNested Logit Function

Seungkyu Ryu 1 Anthony Chen 2 and Songyot Kitthamkesorn3

1Korea Institute Science Technology Information 245 Daehak-ro Yuseong-gu Daejeon Republic of Korea2Department of Civil and Environmental Engineering e Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong3Department of Civil Engineering Chiang Mai University 239 Huay Kaew Road Chiang Mai 50200 ailand

Correspondence should be addressed to Seungkyu Ryu ryuseungkyugmailcom

Received 25 February 2020 Revised 2 March 2021 Accepted 3 April 2021 Published 19 April 2021

Academic Editor Ludovic Leclercq

Copyright copy 2021 Seungkyu Ryu et al(is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

(is study provides a gradient projection (GP) algorithm to solve the combined modal split and traffic assignment (CMSTA)problem (e nested logit (NL) model is used to consider the mode correlation under the user equilibrium (UE) route choicecondition Specifically a two-phase GP algorithm is developed to handle the hierarchical structure of the NLmodel in the CMSTAproblem (e Seoul transportation network in Korea is adopted to demonstrate an applicability in a large-scale multimodaltransportation network (e results show that the proposed GP solution algorithm outperforms the method of the successiveaverages (MSA) algorithm and the classical Evanrsquos algorithm

1 Introduction

Combined travel demandmodels (or combined models) havebeen used as a network equilibrium approach to resolve theinconsistency problem (ie flow) in the traditional four-steptravel demand model (see eg [1]) Several combined modelswere provided in the literature including the combined dis-tribution and assignment (CDA) model [2ndash6] combined tripdistribution modal split and traffic assignment model [7ndash9]combined modal split and traffic assignment (CMSTA)problem [8 10ndash13] and combined trip generation tripdistribution modal split and traffic assignment model[2 14 15]

Various formulations ranging from mathematical pro-gram (MP) nonlinear complementarity problem (NCP)variational inequality (VI) and fixed point (FP) have beenprovided to represent different combined models listed abovePlease refer to [6 12 16 17] for a review of the differentformulation approaches for modeling CDA CMSTA andcombined travel demand models (ese combined modelshave been adopted to represent different emerging techno-logical applications such asmixed gasoline and electric vehicles

with destination route and parking choices [16] electric ve-hicle charging stations using the CDA model [17] and ride-sharing as a new mode choice option using the CMSTAmodel[18] Most of the above studies adopted either the completelinearization method of the FrankndashWolfe algorithm [19] or thepartial linearization method of Evanrsquos algorithm [2] for solvingthese emerging technological applications Computationalresults conducted by LeBlanc and Farhangian [20] revealedEvanrsquos partial linearization method performed better thanFrankndashWolfersquos complete linearization method while Ryu et al[21] recently demonstrated the superiority of the gradientprojection (GP) algorithm over Evanrsquos algorithm Note thatvery few studies have focused on developing solution algo-rithms for solving large-scale problems with multiple modes ina multimodal transportation network [24ndash26]

For solving traffic assignment problems numerous algo-rithms were developed based on link-based path-based andbush-based algorithms (e link-based algorithms operate inthe space of link flows and do not require path storage but itrequires more computational efforts comparing to othergroups of algorithms On the other hand path-based algo-rithms require explicit path storage in order to directly

HindawiJournal of Advanced TransportationVolume 2021 Article ID 1986851 18 pageshttpsdoiorg10115520211986851

compute path flow equilibration and they have faster con-vergence comparing to link-based algorithms (e reason isthat the link-based algorithm exhibits a zigzagging trajectory asit approaches the optimal solution On the other hand thepath-based algorithm does not exhibit the zigzagging effect forthe same initial solution In the transportation literature[26 27] the computational performance of link- and path-based algorithms for the traffic assignment problem is com-pared in medium- to large-scale randomly generated gridnetworks and real-size networks Based on the conclusion apath-based algorithm is consistently faster in approaching theneighborhood of the optimal solution Bush-basedmethods aredeveloped to solve an acyclic network problem(ese methodshave improved the existing state-of-the-art of algorithms andare fast and memory efficient However using the algorithmsfor solving large-scale networks may remain impractical [28](e solution algorithms for the traffic assignmentmodel can besummarized below Interested readers may refer to [27 29] forthe detailed descriptions of the algorithm procedure

(i) Link-Based Algorithm (is includes search directionimprovement [30] restricted simplicial decomposition[31] and nonlinear simplicial decomposition [32]Mitradjieva and Lindberg [30] introduced conjugateand biconjugate algorithms (ese algorithms im-prove the search direction by using previous directioninformation Restricted simplicial decomposition andnonlinear simplicial decomposition algorithms arerepresented by a convex combination of extremepoints (ey first generate new extreme points andthen optimize the restricted master problem

(ii) Path-Based Algorithm (is includes disaggregatedsimplicial decomposition [32] gradient projection[21 33ndash35] projected gradient [36] improved so-cial pressure [37] greedy algorithm [38 39] andstep size improvement [40]

(e disaggregated simplicial decomposition algo-rithm proposed by Larsson and Patriksson [41] issimilar to the restricted simplicial decompositionalgorithm However the extreme points are gen-erated by path flows instead of link flows (egradient projection algorithm adjusts the O-Ddemand between nonshortest paths and theshortest path (e projected gradient algorithm isbased on the idea of adjusting flow between the setof path with a cost greater than the average pathcost and the set of path with a cost less than theaverage path cost Improved social pressure is alsousing the path flow equilibrium between costlierpaths and cheaper paths but the flow updatingstrategy is more complicated To solve the re-stricted subproblem a greedy algorithm is intro-duced Recently Du et al [40] introducedBarzilaindashBorwein step size to adjust path flows

(iii) Bush-Based Algorithm (is includes origin-basedalgorithm [42 43] modified origin-based algorithm[44] algorithm B [45] linear user cost equilibrium[46] and paired alternative segments [47]

(e bush-based algorithm sequentially solves a list ofnode-based subproblems defined on the bush rooted (ealgorithm B adjusts flows between the longest paths and theshortest ones in the bush rooted path tree A paired alter-native segment (PAS) is defined as two completely disjointpath segments and then the flows are adjusted in disjointpath segments

In this paper we are interested in developing solutionalgorithms for solving the MP formulation of the CMSTAproblem that can represent large-scale multimodal trans-portation networks with both private and public modes andcorrelation among multiple public modes using the nestedlogit (NL) model under the network user equilibrium (UE)framework Contrast to the simple structure of the multi-nomial logit (MNL) model the NL model has a hierarchicalstructure that decomposes the choice probability into twolevels represented by the marginal and conditional proba-bilities A two-phase GP algorithm is developed to solve theCMSTA problem with a hierarchical nested modal splitstructure and UE traffic assignment along with variousimprovement strategies to speed up the convergence of thepath-based GP algorithm

In the literature a path-based GP algorithm has beenadopted for solving various traffic assignment problems suchas UE assignment problem the nonadditive cost assignmentproblem side constrained problem stochastic user equi-librium assignment problem transit assignment problemsystem optimal assignment problem and freight traffic as-signment problem [21] Although the algorithm was de-veloped for solving various traffic assignment problems thealgorithm was rarely applied for solving CMSTA problem

Our goal is to demonstrate that the improved GP algo-rithm can solve large-scale multiclass or multimodal trafficassignment problems in CMSTA with NL function In ad-dition the proposed algorithm can apply to other logit modelshaving a hierarchical structure such as the generalized ex-treme value model by [48] including the cross-nested logit(CNL) paired combinatorial logit (PCL) and generalizednested logit (GNL) To show proof of concept a large-scalemultimodal transport network in Seoul Korea will be used todemonstrate the applicability of the two-phase GP algorithm

(is paper is organized as follows After the introduc-tion a relevant background of the NL model for the modalsplit problem and UE conditions for the traffic assignmentproblem is introduced in Section 2 Section 3 provides anequivalent MP formulation for the NL-UE CMSTA problemalong with the equivalence property Section 4 describes thetwo-phase GP algorithm for solving the NL-UE CMSTAproblem improvement strategies and its overall solutionprocedure Section 5 provides numerical results to dem-onstrate the applicability of the two-phase GP algorithm forsolving large-scale multimodal transportation networksSome concluding remarks are provided in Section 6

2 Mode Choice and Route Choice

In this section we provide some background of the nestedlogit (NL) model for the modal split problem and the UE

2 Journal of Advanced Transportation

model for the traffic assignment problem A list of variable isprovided first for convenience followed by the NL modeland multiclass UE model

21 List of Variable Consider a transportation networkG (N A) whereN is the set of nodes andA is the set of linkson a given network We define the list of variables in Table 1

22 Nested Logit Model for Mode Choice (e NLmodel is widely used to represent mode choice in theliterature (see eg [49ndash51]) It considers the modesimilarity through a tree structure as shown in Figure 1where each mode m isin Mrs

u may share an upper nestu isin Urs

(e NL mode choice probability can be expressed as

Prsm(NL)

exp θrsτrsu( 1113857 Ψrs

um minus μrsum( 1113857( 1113857 1113936misinMrs

uexp θrsτrs

u( 1113857 Ψrsum minus μrs

um( 1113857( 11138571113960 1113961τrs

u minus 1

1113936tisinUrs1113936sisinMrs

uexp θrsτrs

u( 1113857 Ψrsts minus μrs

ts( 1113857( 11138571113960 1113961 forallm isin Mrs

u rs isin RS (1)

With the tree structure the NL probability can bedecomposed as

Prsm P

rsm|uP

rsu forallm isin Mrs

u rs isin RS

Prsmu


um minus μrsum( 1113857( 1113857

1113936nisinMrsuexp θrsτrs

u( 1113857 Ψrsun minus μrs

un( 1113857( 1113857 forallm isin Mrs

u u isin Urs rs isin RS

Prsu

1113936misinMrsuexp θrsτrs


um( 1113857( 11138571113960 1113961τrs

u


uexp θrsτrs


ts( 1113857( 11138571113960 1113961τrs

u

forallu isin Urs rs isin RS

(2)

Note that if the parameter (τrsu ) is equal to 10 the NL

model reduces to the MNL model

23 Multiclass User Equilibrium for Route Choice For theroad-based transportation mode such as passenger car andbus we have the KarushndashKuhnndashTucker (KKT) conditions atequilibrium as follows

frskm c

rskm minus πrs

m( 1113857 0

forallm isin Mrsu rs isin RS k isin Km

rs

crskm minus πrs

m ge 0 forallm isin Mrsu rs isin RS k isin Km

rs

frskm ge 0 forallm isin Mrs

u rs isin RS k isin Kmrs

(3)

Equations (3) and (4) define a user equilibrium flowpattern in the route choice [52]

3 Mathematical Programming(MP) Formulation

Without loss of generality two types of transport modes (ieprivate vehicle and public transport) are considered in theupper nest while the lower nest can accommodate more thantwomodes (eg bus metro and bus-metro) Note that it canbe extended to include more than two types of transportmode in the upper nest Assuming that the number of publictransport modes operated on a link in the road network isgiven (ie fixed number of buses on a link) the in-vehicletime of public transport mode (ie bus) is affected by thenumber of private vehicles on a link In addition the roadcapacity on a link is reduced as the number of buses in-creases (ese assumptions enable to present the excessdemand function analytically Consider the following MPformulation

minZ Z1 + Z2 + Z3 + Z4 + Z5 1113944aisinA

1113946xa

0ta(ω)dω + 1113944

rsisinRS1113946

qrsT

0

1θrs ln

ωq

rsminus ω

1113888 11138891113888 1113889dω + 1113944rsisinRS

1113944misinMrs

u

τrsT

θrs qrsm ln q

rsm minus 1( 1113857

minus 1113944rsisinRS

τrsT

θrs 1113944misinMrs

u

qrsm

⎛⎝ ⎞⎠ ln 1113944misinMrs

u

qrsm

⎛⎝ ⎞⎠ minus 1⎛⎝ ⎞⎠ + 1113944rsisinRS

1113944misinMrs

u

qrsmgc

rsm

(4)

st

Journal of Advanced Transportation 3

Table 1 List of variables

Notation DefinitionSetR Set of origins R sub NS Set of destinations S sub NRS Set of O-D pairs rsKrs Set of routes connecting O-D pair rsUrs Set of upper nests connecting O-D pair rsMrs

u Set of lower nests connecting O-D pair rs in upper nest uMrs

T Set of transit modes connecting O-D pair rsParameters andinputsΨrs

um Given exogenous utility of mode m in nest u connecting O-D pair rsμrs

um Expected perceived utility of mode m in nest u connecting O-D pair rsθrs O-D specific parameterτrs

u Upper nest parameter of each O-D pair rs (ie 0lt τrsu lt θ

rs)Prs

(mu) Conditional probabilityPrs

u Marginal probabilityεinner Convergence error in the inner loopεouter Convergence error in the outer loopdB

a dMa Dwelling times on link a of bus and metro

ηBa ηM

a ηTa Equal to 1 for link a with a bus line metro line and bus-metro line and 0 otherwise

la Length on link aWs Walking speed (ie 4 kmh)Wa Weighting time for the transit mode (bus and metro) on link aFrs Transit fare connecting O-D pair rsVOT Inverse value of time for converting transit fare into equivalent time unitqrs Total travel demand connecting O-D pair rsCa Capacity on link aDecision variablesfrs

km Route flow on the mode m of route k connecting O-D pair rscrs

km Route cost on the mode m of route k connecting O-D pair rsπrs

m Minimum cost on the mode m connecting O-D pair rsfrs

k Route flow of the private transport mode on route k connecting O-D pair rscrs

k Route cost of the private transport mode route k connecting O-D pair rscrs

krs

Route cost of the private transport mode shortest route connecting O-D pair rssrs

k Positive-definite scaling factor between route k and the shortest route krs of O-D pair rsxa Link flow of private transport mode on link ata(middot) Travel time of private transport mode on link aδrs

ka Route-link indicator is 1 if link a is on route k between O-D pair rs and 0 otherwiseqrs

T Total demand of public transport mode connecting O-D pair rsqrs

m Demand from public transport mode m in the lower nest connecting O-D pair rsgcrs

m Generalized cost of public transport mode m connecting O-D pair rs

δrsBa δ

rsMa

Bus and metro route-link indicators 1 if link a is on bus route B or metro route M connecting O-D pair rs and 0otherwise

AbbreviationsGP Gradient projectionCMSTA Combined modal split and traffic assignmentNL Nested logitUE User equilibriumMSA Method of successive averagesCDA Combined distribution and assignmentMP Mathematical programNCP Nonlinear complementarity problemVI Variational inequalityFP Fixed pointMNL Multinomial logitCNL Cross-nested logitPCL Paired combinatorial logitKKT KarushndashKuhnndashTuckerMAE Mean absolute errorRMSE Root-mean-square errorATT Average travel timeIGP Improved gradient projection


1113944kisinKrs

frsk + q

rsT q

rs forall rs isin RS

1113944misinMrs

T

qrsm q

rsT forall rs isin RS

frsk ge 0 forall rs isin RS k isin Krs

qrsm ge 0 forall rs isin RS m isin Mrs

T

xa 1113944rsisinRS

1113944kisinKrs

frsk δ

rska forall a isin A

(5)

(e objective function consists of three terms Z1 is thewell-known Beckmannrsquos [53] transformation Z2minusZ5 cor-respond to the NL model Equation (5) shows the flowconservation constraints Note that gcrs

m consists of traveltime waiting time and other personal utilities (eg faretransfer time and inconvenience)

To set up the generalized travel costs (ie the lower nestmodes) we adopt a flow-dependent travel time for the automode and bus mode and a flow-independent travel time forthe metro mode as follows

auto crsk 1113944

aisinAta xa( 1113857δrs

k

bus(B) gcrsB 1113944

aisinAδrs

Ba ta xa( 1113857 + dBa1113872 1113873ηB

a +1 minus ηB

a1113872 1113873la

ws + Wa

⎛⎝ ⎞⎠ + VOT middot Frs

metro(M) gcrsM 1113944

aisinAδrs

Ma tMa + d

Ma1113872 1113873ηM

a + +1 minus ηM

a1113872 1113873la

ws + Wa


B minus M gcrsBM 1113944

aisinAδrs

BMa min ta xa( 1113857 + dBa1113872 1113873 t

Ma + d

Ma1113872 11138731113872 1113873ηT

a +1 minus ηT

a1113872 1113873la

ws + Wa


(6)

Travel times on each link are assumed to follow theBureau of Public Roads (BPR) function

ta xa( 1113857 t0a 1 + η

xa

Ca

1113888 1113889

β⎛⎝ ⎞⎠ foralla isin A (7)

where t0a is free-flow travel time of link a and η and β aregiven parameters

Proposition 1 e MP formulation in equation (4) throughequation (5) gives the mode choice solution of the NL model

(e Lagrangian of the equivalent minimization problemwith respect to the equality constraints can be formulated asfollows

L Z + 1113944rsisinRS

ϕrsq

rsminus 1113944

kisinKrs

frsk + 1113944

misinMrsu

qrsm

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (8)

where ϕrs the dual variableGiven that the Lagrangian has to be minimized with

respect to non-negative route flows and modal splits thefollowing conditions have to hold

Mode 1 Mode 2 Mode 3

MNL

(a)

Nest u

Nest m


NL

θrs

τurs

(b)

Figure 1 Tree structure of the MNL and NL models (a) multinomial logit (b) nested logit


zL

zfrsk

crsk minus ϕrs

0 forall rs isin RS k isin Krs

zL

zqrsm

1θrs ln

1113936misinMrsu

qrsm

qrs

minus 1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ + gcrsm minus ϕrs

0 forall rs isin RS m isin Mrsu

(9)

Note that equation (20) presents the UE conditionFrom equation (9) we have

1θrs ln q

rsT +

τrsu

θrs ln qrsm minus

τrsu

θrs ln qrsT + gc

rsm

1θrs ln q

rsA + c

rsk

(10)

where qrsA is auto mode demand qrs

A (qrs minus 1113936misinMrsu

qrsm)

Let

1θ rs ln q

rsT +

τrsu

θ rs ln qrsm minus

τrsu

θ rs ln qrsT + gc

rsm λ

1θ rs ln q

rsA + c

rsk λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)

Rearranging equation (11)

qrsm q

rsT( 1113857

1minus τrsT( )τrs

u( )( ) expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⟹ 1113944

misinMrsu

qrsm 1113944

misinMrsu

expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

τrsu

qrsA( 1113857

1τrsu( )( ) exp

θrs

τrsu

λ minus crsm( 11138571113888 1113889rArrq

rsA exp

θrs

τrsu

λ minus crsm( 11138571113888 11138891113888 1113889

τrsT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

From equation (12) we have the marginal probability forthe auto mode as follows

PrsA

qrsA

qrsA + 1113936misinMrs

uq

rsm

1113936kisinKrs f

rsk

qrs

exp minusθrsc

rsk( 1113857

exp minusθrsc

rsk( 1113857 + 1113936misinMrs

uexp θrsτrs

u( 1113857 minus gcrsm( 1113857( 11138571113872 1113873

τrsu

(13)

From qrsm(qrs

T )(((1minus τrsT

)τrsu )) exp((θrsτrs

u )(λ minus gcrsm)) in

equation (11) we have the conditional probability as follows

qrsm

qrsT

exp θrsτrs

u( 1113857 λ minus gcrsm( 1113857( 1113857

qrsT( 1113857

1τrsu( )( )

exp θrsτrs

u( 1113857 minusgcrsm( 1113857( 1113857


u( 1113857 minusgcrsm( 1113857( 1113857

(14)

(ese probabilities are consistent with the NL modelillustrated in the previous section for a case with two uppernests (is completes the proof

Proposition 2 e MP formulation in equation (4) throughequation (5) yields a unique solution

For the proof of Proposition 2 we need to prove thatobjective function in equation (5) is strictly convex withrespect to mode-route flow variables and the convexity of thefeasible region For the objective function taking the Hes-sian matrix in equation (4) with respect to the mode-routeflow variables gives

z2Z

zfrsk f

rsj

dta xa( 1113857

dxa

δrska ge 0 if j k

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)

which implies the positive-definite matrix when the link traveltime (ta(xa)) is a strictly monotonic increasing function


z2L

z qrsmq

rsn

1θrs

11113936misinMrs

uq

rsm

+1

qrs


qrsm

⎛⎝ ⎞⎠ minusτrs

T

θrs

1q

rsm

minus1

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ge 0 if m n

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

which also implies the positive-definite matrix (ereforethe MP formulation model has a unique solution (iscompletes the proof

From equation (9) the excess demand function of thelower nest modes in the NL modal split function can bedefined as follows (see Appendix for the detailed derivation)

wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsuq

rsm

qrs

minus 1113936misinMrsuq

rsm

⎛⎝ ⎞⎠ +τrs

u

θrs lnq

rsm

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ + gcrsm

(17)

Hence when these two costs are equilibrated (iecrs

k (frsk ) wrs

m(qrsm)) the flows between the private and

public transport modes achieve equilibrium conditionFigure 2 provides a graphical illustration of the equilibrated

mode choice probabilities in the two-level tree structure of theNL model (ree modes are considered including auto busand metro with their cost functions (e bus and metro sharethe same upper nest while the auto is an independent modeSince the auto and bus use the road space these two modeshave a flow-dependent travel time Meanwhile the metro isassumed to have a flow-independent travel time according adedicated right of way as presented in Figure 2(a) InFigure 2(b) the upper nest modal splits are determined andthen the estimated upper nest mode demands are used todetermine the modal splits for the lower nest modes inFigure 2(c) For more detail Figure 2(d) shows the change intravel cost and excess demand cost for the auto and two publictransportmodesWe can observe that the cost of auto increaseswhen the transit mode flows decrease However the excessdemand cost of the bus is not affected by the decreasing busmode flow(e reason is that the excess demand cost of the bus

consists of three terms (see equation (9)) As the transit modeflows decrease the first terms and the second terms are de-creased Meanwhile the third term is increased because theauto mode cost function is increased (is is according to theincorporation of the bus cost (ie gcB) On the other hand theexcess demand cost of the metro is decreased with decreasingthe metro flow (is is because gcM is not affected by automode cost In Figure 2(e) there are three different areas (ered area indicates that the travel cost of auto mode is higherthan the excess demand cost of transit modes (e red pointindicates the equilibrated flows (e travel cost of auto modeand the excess demand cost for transit modes are equal to 44In this case the travel cost of bus mode is 52 with 167 flowsand metro mode is 50 with 287 flows Figures 2(f) and 2(g)verify the validity of the probabilities of each mode (ie satisfyconservation) and equilibrium cost of each mode respectivelyand it also satisfies the NL probabilities (ie marginal andconditional probabilities)

4 Path-Based Gradient Projection Method

(e gradient projection (GP) algorithm is a well-knownpath-based algorithm for solving various traffic assignmentproblems Before considering the gradient projection (GP)algorithm for solving the CMSTAmodel (or NL-UEmodel)we provide a brief review on the ordinary GP algorithm forsolving the UE traffic assignment problem with a fixeddemand Equation (18) shows the flow update equations [35]for solving a fixed demand traffic assignment problem Ineach iteration the algorithm finds a descent search directionby solving a shortest path problem and updates the solutionby scaling it with the second derivative information

frsk (n + 1) max f

rsk (n) minus

αs

rsk (n)

crsk (n) minus c

rs

krs(n)1113874 11138751113890 1113891 01113896 1113897 forall rs isin RS k isin Krs kne krs

frs

krs(n + 1) qrs minus 1113944

kisinKrs

kne krs

frsk (n + 1) forall rs isin RS (18)

where n is the iteration number α is the step size maxf 0denotes the projection of the argument onto the non-neg-ative orthant of the decision variables (ie nonshortest routeflows) and frs

k (n + 1) and frs

k(n + 1)are the flows on route k

and shortest route krs O-D pair rs at iteration n + 1Equation (19) shows the route travel time difference

(crsk (n) minus crs

krs

(n)) and the scaling factor (srsk (n)) to find the

search direction

crsk minus c

rs

krs 1113944


ka minus 1113944aisinA

ta xa( 1113857δrs

krsa

srsk 1113944

aisinAtaprime xa( 1113857 δrs

ka minus δrs

krsa1113874 1113875

2

(19)

where ta(xa) and taprime(xa) are the travel time and its derivative

of the auto mode on link aFor a detailed description of the GP algorithm readers

should refer to [2635] (e modifications for solving the


CMSTA model with NL modal split function and the im-provement strategies for solving large-scale multimodaltransportation networks are described as follows

41 Two-Phase Equilibration Unlike the UE traffic assign-ment problem the CMSTA problem considered in thispaper (ie UE for route choice and NL for mode choice)requires not only equilibrating route flows of the auto modefor route choice but also equilibrating modal splits amongmultiple available modes (ie both private and publictransport modes) following the NL model for mode choice

Based on the hierarchical choice structure given in Figure 1a two-phase equilibration method is developed to performthe equilibration for the two travel choices In the first phaseGP equilibrates between the private auto mode and publictransit mode using the minimum route travel time of theauto mode and the excess demand cost of the transit modeIn the second phase the lower nested probability function isapplied to determine the mode demands among the transitmodes From the lower nested probability function theexcess demand cost function for the transit mode in theupper nest (see Appendix for the detailed derivation) can bederived as follows

B B

M M

Walk time (1) Walk time (2)Bus travel time (05 middot cA)

qndash = 100

qA

qB

qM

Walk time (1)

Auto travel time (cA = 3 + (qA5)4)

θ = 05

τT = 02

gcM = 10 + 30 + 10

gcB = 10 + 05 middot cA + 20

Metro travel time (3) Walk time (1)Auto Bus Metro

(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Trav

el ti

me (

auto

)

Flow (auto)

Equilibrium point

Auto = 547

wT (qT)θ = 05τ = 02

cA (qA) = 444w (qT) = 444

cA (qA)

Transit = 453

(b)

Equilibrium point

2

3

4

5

6

7

00 05 10 15 20 25 30 35 40 45

Exes

s dem

and

cost

Flow (transit)

Metro = 287 Bus = 167

w (qB)

w (qB)

Excess demand cost 444

(c)

00204060

4040

80100120

Tim

e (c a

nd w

)

140

3535

Metro (M)

Bus (B) Auto (A)

160180

3030 2525 2020 1515Bus flow Metro flow1010 0505 0000

(d)

00

3025

20

1015

0505

1015

20

q M = 287 amp c M

= 500

qB = 167 amp c

B = 522

2530

00

35

40

wB gt wM cA

cA = wB = wM

cA = wB wM

wM gt wB cA

qA = 547 amp cA = 444

40

35

(e)

exp (ndash0502 middot 444)02

= 0547exp (ndash0502 middot 444)02 + (exp (ndash0502 middot 522) + exp (ndash0502 middot 500))02

PA =

exp (ndash0502 middot 522)= 0167

exp (ndash0502 middot 522) + exp (ndash0502 middot 500)PB = PT middot PBT = (1 ndash 0546)middot

exp (ndash0502 middot 500)

exp (ndash0502 middot 522) + exp (ndash0502 middot 500)PM = PT middot PMT = (1 ndash 0546)middot = 0287

(f )

cA (qA) = 444

167 + 287 167021ln+ + 522 = 444

05ln

10 ndash (167 + 287) 05 167 + 287wB (qB) =

1 167 + 287 28702lnln + + 500 = 444

05 10 ndash (167 + 287) 05 167 + 287wB (qM) =

(g)

Figure 2 Illustration of the mode choice equilibration under the nested logit modal split function (a) network demand and cost functions(b) upper level equilibrium (marginal prob) (c) lower level equilibrium (conditional prob) (d) travel time change by flows (e) equilibriumsolution (f ) probabilities verification (g) cost verification


wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs

minus 1113936misinMrsTq

rsm


T

θrs ln 1113944misinMrs

T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)

with the minimum route travel time of the auto mode (crs

krs

)

and the excess demand cost for transit mode in equation(20)

Figure 3 provides a graphical illustration of the firstphase equilibration in the two-phase GP algorithm whichinvolves the modal splits of the upper nest (ie private carand public transit) If wrs

T (qrsT )lt crs

krs

(see Figure 3(a)) routeflows from the auto mode should be decreased accordinglyto equilibrate themodal costs of both auto and public modesOn the other hand if wrs

T (qrsT )gt crs

krs

(see Figure 3(b)) routeflows from the automode should be increased accordingly tobalance the two auto and public modal costs

Once the upper nest is equilibrated in the first phase (iethe auto demand (1113936kisinKrs frs

k ) and the public transit demand(qrs

T ) are determined) the second phase is performed to splitthe public transit demand (qrs

T ) into multiple transit modes(qrs

m) in the lower nest to achieve equilibrium in the modechoice For the lower nest the modal split probabilityfunction can be directly calculated as follows

qrsm(n + 1) q

rsT (n + 1) middot

exp θrsτrsT( 1113857 minusgc

rsm(n + 1)( 1113857( 1113857

1113936misinMrsTexp θrsτrs

T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)

42 ColumnGeneration Scheme (e disaggregate simplicialdecomposition (DSD) algorithm proposed by Larsson andPatriksson [41] is one of the earlier path-based traffic as-signment algorithms (e algorithm consists of a linearizedsubproblem and a master problem to determine the nextsolution point From the subproblem extreme points (ieroute generation) are resulted and then the master problemis solved with extreme points generated in the subproblemwhich is equilibration In the algorithm the master problemis iterated until the route flows stabilized before generatingnew paths in the column generation procedure Korpelevich[54] introduced a double projection method to relax thestrongly monotone condition in the mapping in the vari-ational inequality problem From the predictor step thestrongly monotone condition is relaxed and the solution isadjusted to obtain a more accurate solution in the correctorstep Galligari and Sciandrone [55] recently proposed a fastpath equilibration algorithm with an adaptive columngeneration scheme for each O-D pair Lu et al [56] usedcolumn generation scheme under the dynamic user equi-librium traffic assignment problem Wang et al [57] in-troduced the tolerance-based strategies for the columngeneration scheme in the dynamic user equilibrium trafficassignment problem Ameli et al [22 58] recently proposedtrip-based dynamic traffic assignment and simulation-baseddynamic traffic assignment (ese two algorithms adopt thecolumn generation scheme for path generation In additionthey assessed the computational performance of the innerloop methods in the first paper and proposed algorithm for

solving large-scale dynamic traffic assignment in the secondpaper

In this study we apply a similar adaptive column gen-eration scheme in the gradient projection algorithm Toimprove the computational efficacy in the few iterations weadopt the following condition in the inner loop (masterproblem) as the termination step

εinner lt εouter middot c (22)

where 0lt cle 10 is the given parameterFirst the convergence error in the outer loop is com-

puted after the column generation step (en the inner loop(equilibration) step is performed without the column gen-eration step until it satisfies the termination condition inequation (22) (e overall flowchart is shown in Section 5

Another implementation improvement is the columndropping scheme If the path set Krs(n) for each O-D pair rscontains relatively few paths the path-based algorithms willperform satisfactorily without column dropping Howeverthe number of routes in Krs(n) can be very large for somesituations (eg problems with mostly simple paths con-necting each O-D pair heavily congested networks ormultimodal transportation networks) Under these situa-tions it may be necessary to remove the inactive (zero flowor near-zero flow) paths from the route set to reduce theamount of calculation and bookkeeping needed in eachiteration Chen and Jayakrishnan [59] have shown that GPwhich automatically achieves column dropping during theprojection step reduces computational time if the number ofactive (positive flow) paths is considerably less than thenumber of generated paths In this study we adopt thecolumn dropping step suggested by Chen and Jayakrishnan[59]

43 SolutionProcedure Figure 4 shows the overall flowchartof the solution procedure After the initialization stepcolumn (or route) generation is performed With a new pathset flow updated is active and then restrict equilibration isperformed without new path generation If convergenceerror computed in restrict equilibration satisfies the innerloop convergence criteria they go to column dropping step(e process is iterated until convergence error satisfies theouter loop convergence criteria

Figure 5 shows the detailed flow update procedurefollowing the two-phase gradient projection algorithm

As described above modified GP extends the equili-bration of private vehicles (ie traffic assignment) to alsoconsider equilibration of multiple transport modes (iemodal splits) by using a two-phase procedure which is theequilibration between public and private modes as the firstphase and equilibration of modal splits among the publictransport modes as the second phase

In the column generation step we adopt origin-basedcolumn generation to improve computation efficiency See[60] for more detailed results Based on the literatures (seeeg [26 35 61 62]) a unit stepsize (ie α 10) is adoptedfor all iteration n since the second derivative information foran automatic scaling and the one O-D at-a-time flow update


strategy is used in the equilibration procedure (is schemehas been found to be helpful in reducing computationalefforts In this paper we also used unit step size (ie α 10)with ldquoone-at-a-timerdquo flow update strategy However a linesearch step (eg self-adaptive strategies) can be used todetermine a suitable stepsize to help better convergenceespecially when highly accurate solutions are sought

5 Numerical Experiments

(e transportation network in the city of Seoul Korea isused to investigate the performance of the proposed two-stage GP algorithm in solving the NL-UE model (e

convergence characteristics are tested and then parametersensitivity analysis is examined

51 Description of Seoul Network and Experimental Set Up(e Seoul transportation network shown in Figure 6 consistsof 425 zones 7512 nodes 11154 links and 107434 O-Dpairs with a total of 2874441 trips Table 2 summarizes somedetails of the multimodal transportation network in SeoulKorea (e nested logit structure and modal split functionsfor modeling mode choice are shown in Figure 7

(e occupancy of auto mode and bus mode is set 13 perpassenger car and 1927 per bus respectively (MTA 2015)(e transit has the distance-based fare structure Without

n = 0 xn = 0 f n = 0 K = oslashInitialization

Column generation(route generation)

Flow update(Restricted) equilibration

Flows update(Initial equilibration)

Terminate

Outer iteration

Inner iteration

Columndropping

lowastE stopping criteria

n =

n +

1

Y

N

Compute convergenceerror (εouter)

Compute convergenceerror (εinner)

εouter lt E

εouter lt εouter middot γ

qAn = 05 middot qndash qT

n = 05 middot qndash

Figure 4 Gradient projection algorithm with restricted equilibration

w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)

(α (n)skrs (n)) (ck

rs (n) ndash w (qTrs (n)))

Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)

(α (n)sTrs (n)) (w (qT

rs (n)) ndash ckndashrs (n))

qndashrs ndash ΣkєKrs

fkrs (n + 1) ndash qT

rs (n + 1)


rs (n) ndash ckndashrs (n))

Cost

Flow

(b)

Figure 3 Demand adjustment of the first phase equilibration (ie private car and public transit) (a) Demand adjustment on the auto modeaccording to crs

k(n)gtwrs

T (qrsT (n)) (b) Demand adjustment on the public transport modes according to wrs

T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation

Update link cost tn = tnndash1 (xnndash1)

Search shortest route of transit mode m є MTrs

Destination s = 1


Compute path cost ckrs k є Krs

Search shortest route kndash

rs = arg min ckrs | k є Kn

rsColumn generation kndashrs forall s isin SUpdate path set Kn

rs = Krsnndash1 cup kndashrs forall s isin S

Compute transit modes cost cmrs m є MT

rs

Compute transit excess demand cost wTrs (qT

rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y

Upper nested equilibration (first phase)

Lower nested equilibration (second phase)


(fkrs)n+1 = max [(fk

rs)n ndash (αskrs)((ck

rs)n ndash (ckrs)n)] 0



rs)n ndash (wTrs)n)] 0

(fkndashrr

ss)n+1 = qndashrs ndash Σ (fk

rs)n+1 ndash (qTrs)n+1

(qTrs)n+1 = qndashrs ndash Σ (fk

rs)n+1

kєKrs

kєKrs

kneKndashrs

(qTrs)n+1 = max [(qT

rs)n ndash (αsTrs)((wT


ckndashrs gt wT

rs (qTrs)

Pm|T = exp ((θrsτrs) (ndashgcmrs))

(qmrs)n+1 = (qT

rs)n+1 middot Prsm|T

exp ((θrsτrs) (ndashgcnrs))Σ

nєMTrs

rs

Figure 5 Two-phase flow update strategy

Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued


loss of generality the metro mode is assumed to provideenough operation capacity such that there is no congestionin this mode One hundred maximum number of inner loop

is set and c value is set 01(e tolerance error of the relativegap (RG) shown in equation (23) is set 1E-8

RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)

(e improved GP algorithm is coded in Intel VisualFORTRAN XE and runs on a 360GHz processor and1600GB of RAM

52 Convergence Characteristics Figure 8 shows the con-vergence characteristics using the method of successive aver-ages (MSA) Evanrsquos algorithm ordinary GP algorithm and

improved GP (IGP) algorithm (e method of successiveaverages (MSA) is the most widely used method in the pre-determined line search scheme [52] It predetermines a di-minishing step size sequence such as 1 12 13 1n Onthe other hand Evanrsquos algorithm [2] is a classical algorithm forsolving the combined distribution and assignment (CDA)problem as well as the combined modal split and traffic

(c)

Figure 6 Multimodal transportation network in Seoul Korea (a) auto network (b) bus network and (c) metro network

Table 2 Summary of the multimodal transportation network in Seoul Korea

Auto Bus Metro Bus-metroTotal length (km) 2790 1202 467 1669Number of stations mdash 3635 354 3989Number of lines mdash 1765 14 1779

Auto Bus Metro Bus-metro

exp ((θrsτurs)(ndashπrs))τu

rs

lowastπrs is O-D cost of auto

= qArsqndashrs

rArr

NL

τurs

θrs exp ((θrsτurs)(ndashgcm

rs))τurs + exp ((θrsτu

rs)(ndashπrs))τurs

ΣmєMu

rs

exp ((θrsτurs)(ndashgcm

rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=

Figure 7 Nested mode choice structure and its modal splits


assignment problem (see eg [20])(e results reveal the slowconvergence of the MSA and Evanrsquos algorithm (e GP al-gorithm presents a slower convergence than the IGP algorithmWith the GP algorithm the solution cannot reach the highlyaccurate level of a RG of 1E-8 while the solution with IGP isreached to RG of 1E-8 in 3100 seconds Note that the toleranceerror of 1E-8 is much stricter than the typical one (ie 1E-4)required in practice [64]

In the following analyses we conduct the sensitivity analysiswith respect to the inner loop parameters c (e parameteradjusts the stability of path flows before generating new paths inthe column generation step Figure 9(a) shows the convergencewith different values (01 05 08) and Figure 9(b) provides

active number of inner iterations Note that the maximumnumber of inner iteration is 100 As c value increases thecomputational effort is significantly increasedWhen a higher c

value is used the required number of inner iteration is relativelysmaller but it requires more outer iterations On the otherhand using a smallc value requires more inner iterations withthe stabilized route flows but it requires less the number ofouter iteration In Figure 9(b) we can observe that a smaller c

requires more inner iterations (e algorithm is converged inthe smaller outer iteration After a few outer iterations theinner iteration reaches the maximum inner loop On the otherhand using higher c value is converged well in a few inneriterations but it requires more outer iteration

1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan

Figure 8 Convergence characteristics

1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)

Figure 9 Computational efforts under different parameters (a) computational efficacy with varying c (b) number of inner loop withvarying c


633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()

Figure 10 Effect of logit parameters on modal splits

ATT (min)case 1 2576case 2 2559

Flow differenceRMSE 13688MAE 9710

CASE1 - CASE2ndash10000 to ndash300ndash300 to ndash150ndash150 to 00 to 150150 to 300300 to 10000

ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)

Figure 11 Effect of logit parameter on flow allocation (a) auto flow difference (b) bus flow difference (c) metro flow difference (d) bus-metro flow difference


53 Application of the CMSTA Problem (is section ex-amines the sensitivity of the IGP algorithm with respect to thenested logit parameters (ie θrs and τrs

T ) Figures 10 and 11present the effects of these two parameters on modal splitsand average modal travel costs respectively Figure 10 showsthat θrs is sensitive to the modal splits between private andpublic modes (ie a 3 increase in the private car mode whenθrs increased from 01 to 0133) while τrs

T is fairly insensitive tothe modal splits When the transit demand transfers to automode the bus mode is affected more than metro and bus-metro mode (is is because the travel time of the bus andauto is dependent (e result also shows that the modal splitsare different albeit the ratio (θrsτrs

T 20) is the sameFigure 11 shows the link flow differences between case 1

(ie θrs 01 and τrsT 005) and case 2 (ie θrs 0133 and

τrsT 0067)(e green color shows link flows using case 1 arelarger than those of case 2 while the red color shows thereverse

Specifically the figure shows the mean absolute error(MAE) and the root-mean-square error (RMSE) forassessing the link flow difference and average travel time(ATT) (ie total travel timemode demand) Recall that themodal splits for case 1 are 7219 and 2781 for auto andtransit mode respectively and the modal splits for case 2 are7530 and 2470 for auto and transit mode respectivelyBecause the modal splits are different using different pa-rameters the link flow pattern is also a different pattern asshown in the GIS map (e highest RMSE value is 16416 inthe metro mode and the highest MAE is 9951 in the busmode Albeit the link flow patterns are significantly differentthe ATT values are similar values (is is because of thescaling effect in the large network Although the ATT dif-ference is less than 10minutes between two cases the totaltrip difference is 2874441 trips

6 Concluding Remarks

In this paper we presented (1) the CMSTA model with thenested logit (NL) model for the mode choice and the userequilibrium (UE) model for the route choice and (2) amodified-and-improved gradient projection (IGP) algo-rithm to solve the proposed NL-UE model (e proposedNL-UE model considered the mode similarity through theNL model under the congested network obtained by the UEcondition Specifically an equivalent MP formulation for theNL-UEmodel was provided using a modified access demandto incorporate the two-level nested tree structure On theother hand the IGP algorithm was developed in such a wayto perform a two-level equilibration In the first phase the

upper nest of the NL model is solved (en the lower nest ofthe NL model is applied with updated flows from the uppernest in the second phase For the solution algorithm theimproved GP algorithm with inner loop equilibrationprocedure was introduced and shown the effectiveness insolving the combined NL modal split and UE traffic as-signment (NL-UE) problem (is is because the excessdemand cost function consists of two logarithm terms(eseterms are sensitive to a small traffic flow change A morestabilized path flow is required before generating new paths

A real-size transportation network in Seoul Korea wasused to demonstrate the applicability of the IGP algorithmfor solving the proposed NL-UE model (e algorithm canconduct 1E-8 convergence criteria (e convergence time ofthe IGP algorithm seemed to be better than that of theordinary GP algorithm Further the NL parameters have asignificant impact on the convergence time and the modeshare results

For future research the improved GP algorithm shouldbe extended to consider mode (or vehicle) interactions withasymmetric cost functions In addition the route choicemodel for transit modes should be considered with transitinformation systems (eg arrival information) It would beinteresting to see how the improved GP algorithm performswhen more realistic features are incorporated into theCMSTA problem In addition a time-dependent model canbe considered with travel activity patterns

Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)

Equation (A1) is arranged as follows

11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ middot expθrs

τrsT

gcrsm1113888 1113889

minus ln 1113944misinMrs

T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)

Hence equation (A3) is expressed with equation (A2)as follows

wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs

minus 1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability

(e basic data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

Seungkyu Ryu collected and analyzed the data interpretedthe results and prepared the manuscript draft All authorscontributed to study conception and design reviewed theresults and approved the submitted version of themanuscript

Acknowledgments

(is research was supported by the Basic Science ResearchProgram through the National Research Foundation (NRF)of Korea by the Ministry of Science (grant no NRF-2016R1C1B2016254) the Ministry of Science ICT Republicof Korea (project no K-21-L01-C05-S01) the ResearchGrants Council of the Hong Kong Special AdministrativeRegion (project no 15212217) the Research Committee ofthe Hong Kong Polytechnic University (project no 1-ZVJV)and the Chiang Mai University

References

[1] N Oppenheim Urban Travel Demand Modeling John Wileyand Sons Inc New York NY USA 1995

[2] S P Evans ldquoDerivation and analysis of some models forcombining trip distribution and assignmentrdquo TransportationResearch vol 10 no 1 pp 37ndash57 1976

[3] W H K Lam and H-J Huang ldquoA combined trip distributionand assignment model for multiple user classesrdquo Trans-portation Research Part B Methodological vol 26 no 4pp 275ndash287 1992

[4] N Oppenheim ldquoEquilibrium trip distributionassignmentwith variable destination costsrdquo Transportation Research PartB Methodological vol 27 no 3 pp 207ndash217 1993

[5] J Yao A Chen S Ryu and F Shi ldquoA general unconstrainedoptimization formulation for the combined distribution andassignment problemrdquo Transportation Research Part BMethodological vol 59 pp 137ndash160 2014

[6] M Florian S Nguyen and J Ferland ldquoOn the combineddistribution-assignment of trafficrdquo Transportation Sciencevol 9 no 1 pp 43ndash53 1975

[7] D E Boyce and M S Daskin ldquoUrban transportationrdquo inDesign and Operation of Civil and Environmental EngineeringSystems C ReVelle and A McGarity Eds Wiley New YorkNY USA 1997

[8] M Florian ldquoA traffic equilibrium model of travel by car andpublic transit modesrdquo Transportation Science vol 11 no 2pp 166ndash179 1977

[9] K I Wong S CWong J HWu H Yang andW H K LamldquoA combined distribution hierarchical mode choice andassignment network model with multiple user and modeclassesrdquo inUrban and Regional Transportation Modelingpp 25ndash42 2004

[10] M Abdulaal and L J LeBlanc ldquoMethods for combiningmodal split and equilibrium assignment modelsrdquo Trans-portation Science vol 13 no 4 pp 292ndash314 1979

[11] R Garcıa and A Marın ldquoNetwork equilibrium models withcombined modes models and solution algorithmsrdquo Trans-portation Research Part B vol 39 no 3 pp 223ndash254 2005

[12] S Kitthamkesorn and A Chen ldquoAlternate weibit-basedmodelfor assessing green transport systems with combined modeand route travel choicesrdquo Transportation Research Part BMethodological vol 103 pp 291ndash310 2017

[13] Z X Wu and W H K Lam ldquoCombined modal split andstochastic assignment model for congested networks withmotorized and nonmotorized transport modesrdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 1831 no 1 pp 57ndash64 2003

[14] K N Ali Safwat and T L Magnanti ldquoA combined tripgeneration trip distribution modal split and trip assignmentmodelrdquo Transportation Science vol 22 no 1 pp 14ndash30 1988

[15] C Yang A Chen and X Xu ldquoImproved partial linearizationalgorithm for solving the combined travel-destination-mode-route choice problemrdquo Journal of Urban Planning and De-velopment vol 139 no 1 pp 22ndash32 2013

[16] S Kitthamkesorn A Chen X Xu and S Ryu ldquoModelingmode and route similarities in network equilibrium problemwith go-green modesrdquo Networks and Spatial Economicsvol 16 no 1 pp 33ndash60 2016

[17] D Boyce and H Bar-Gera ldquoMulticlass combined models forurban travel forecastingrdquo Networks and Spatial Economicsvol 4 no 1 pp 115ndash124 2004

[18] N Jiang C Xie J C Duthie and S T Waller ldquoA networkequilibrium analysis on destination route and parkingchoices with mixed gasoline and electric vehicular flowsrdquoEURO Journal on Transportation and Logistics vol 3 no 1pp 55ndash92 2013

[19] F He Y Yin and S Lawphongpanich ldquoNetwork equilibriummodels with battery electric vehiclesrdquo Transportation Re-search Part B Methodological vol 67 pp 306ndash319 2014

[20] O Bahat and S Bekhor ldquoIncorporating ridesharing in thestatic traffic assignment modelrdquo Networks and Spatial Eco-nomics vol 16 no 4 pp 1125ndash1149 2016

[21] L J LeBlanc E K Morlok and W P Pierskalla ldquoAn efficientapproach to solving the road network equilibrium trafficassignment problemrdquo Transportation Research vol 9 no 5pp 309ndash318 1975

[22] L J LeBlanc and K Farhangian ldquoEfficient algorithms forsolving elastic demand traffic assignment problems and modesplit-assignment problemsrdquo Transportation Science vol 15no 4 pp 306ndash317 1981

[23] S Ryu A Chen and K Choi ldquoSolving the combined modalsplit and traffic assignment problem with two types of transitimpedance functionrdquo European Journal of Operational Re-search vol 257 no 3 pp 870ndash880 2017

[24] S Ryu A Chen and K Choi ldquoSolving the stochastic multi-class traffic assignment problem with asymmetric interac-tions route overlapping and vehicle restrictionsrdquo Journal ofAdvanced Transportation vol 50 no 2 pp 255ndash270 2016

[25] I O Verbas H S Mahmassani M F Hyland and H HalatldquoIntegrated mode choice and dynamic traveler assignment inmultimodal transit networks mathematical formulationsolution procedure and large-scale applicationrdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2564 no 1 pp 78ndash88 2016

[26] G Wang A Chen S Kitthamkesorn et al ldquoA multi-modalnetwork equilibrium model with captive mode choice and


path size logit route choicerdquo Transportation Research Part APolicy and Practice vol 136 pp 293ndash317 2020

[27] A Chen D-H Lee and R Jayakrishnan ldquoComputationalstudy of state-of-the-art path-based traffic assignment algo-rithmsrdquo Mathematics and Computers in Simulation vol 59no 6 pp 509ndash518 2002

[28] O Perederieieva M Ehrgott A Raith and J Y T Wang ldquoAframework for and empirical study of algorithms for trafficassignmentrdquo Computers amp Operations Research vol 54pp 90ndash107 2015

[29] E Jafari V Pandey and S D Boyles ldquoA decompositionapproach to the static traffic assignment problemrdquo Trans-portation Research Part B Methodological vol 105 pp 270ndash296 2017

[30] J Xie and C Xie ldquoOrigin-based algorithms for traffic as-signmentrdquo Transportation Research Record Journal of theTransportation Research Board vol 2498 no 1 pp 46ndash552015

[31] M Mitradjieva and P O Lindberg ldquo(e stiff is moving-conjugate direction frank-wolfe methods with applications totraffic assignmentlowastrdquo Transportation Science vol 47 no 2pp 280ndash293 2012

[32] D W Hearn S Lawphongpanich and J A Ventura ldquoFi-niteness in restricted simplicial decompositionrdquo OperationsResearch Letters vol 4 no 3 pp 125ndash130 1985

[33] T Larsson M Patriksson and C Rydergren ldquoApplications ofsimplicial decomposition with nonlinear column generationto nonlinear network flowsrdquo in Network Optimizationpp 346ndash373 Springer Berlin Germany 1997

[34] A Chen Z Zhou and X Xu ldquoA self-adaptive gradientprojection algorithm for the nonadditive traffic equilibriumproblemrdquo Computers amp Operations Research vol 39 no 2pp 127ndash138 2012

[35] G Gentile ldquoSolving a dynamic user equilibrium model basedon splitting rates with gradient projection algorithmsrdquoTransportation Research Part B Methodological vol 92pp 120ndash147 2016

[36] R Jayakrishnan W T Tsai J N Prashker andS Rajadhyaksha ldquoA faster path-based algorithm for trafficassignmentrdquo Transportation Research Record vol 1443pp 75ndash83 1994

[37] M Florian I Constantin and D Florian ldquoA new look atprojected gradient method for equilibrium assignmentrdquoTransportation Research Record Journal of the TransportationResearch Board vol 2090 no 1 pp 10ndash16 2009

[38] A Kumar and S Peeta ldquoAn improved social pressure algo-rithm for static deterministic user equilibrium traffic as-signment problemrdquo in Proceeding 90th Annual Meeting of theTransportation Research Board Washington DC USAJanuary 2011

[39] L Feng J Xie Y Nie and X Liu ldquoEfficient algorithm for thetraffic assignment problem with side constraintsrdquo Trans-portation Research Record Journal of the Transportation Re-search Board vol 2674 no 4 pp 129ndash139 2020

[40] J Xie Y Nie and X Liu ldquoA greedy path-based algorithm fortraffic assignmentrdquo Transportation Research Record Journalof the Transportation Research Board vol 2672 no 48pp 36ndash44 2018

[41] M Du H Tan and A Chen ldquoA faster path-based algorithmwith Barzilai-Borwein step size for solving stochastic trafficequilibrium modelsrdquo European Journal of Operational Re-search Inpress vol 290 no 3 pp 982ndash999 2021

[42] T Larsson andM Patriksson ldquoSimplicial decomposition withdisaggregated representation for the traffic assignment

problemrdquo Transportation Science vol 26 no 1 pp 4ndash171992

[43] H Bar-Gera ldquoOrigin-based algorithm for the traffic assign-ment problemrdquo Transportation Science vol 36 no 4pp 398ndash417 2002

[44] D-H Lee Q Meng and W Deng ldquoOrigin-based partiallinearizationmethod for the stochastic user equilibrium trafficassignment problemrdquo Journal of Transportation Engineeringvol 136 no 1 pp 52ndash60 2010

[45] Y Nie ldquoA note on Bar-Gerarsquos algorithm for the origin-basedtraffic assignment problemrdquo Transportation Science vol 46no 1 pp 27ndash38 2011

[46] R B Dial ldquoA path-based user-equilibrium traffic assignmentalgorithm that obviates path storage and enumerationrdquoTransportation Research Part B Methodological vol 40no 10 pp 917ndash936 2006

[47] G Gentile ldquoLocal User Cost Equilibrium a bush-based al-gorithm for traffic assignmentrdquo Transportmetrica A Trans-port Science vol 10 no 1 pp 15ndash54 2014

[48] H Bar-Gera ldquoTraffic assignment by paired alternative seg-mentsrdquo Transportation Research Part B Methodologicalvol 44 no 8-9 pp 1022ndash1046 2010

[49] D McFadden ldquoModelling the choice of residential locationrdquoin Spatial Interaction eory and Residential LocationA Karlquist Ed pp 75ndash96 NorthHolland AmsterdamNetherlands 1978

[50] S Bekhor T Toledo and L Reznikova ldquoA path-based al-gorithm for the cross-nested logit stochastic user equilibriumtraffic assignmentrdquo Computer-Aided Civil and InfrastructureEngineering vol 24 no 1 pp 15ndash25 2009

[51] M Ben-Akiva and S Lerman Discrete Choice Analysis eoryand Application to Travel Demand MIT press CambridgeMA USA 1985

[52] E Cascetta and A Papola ldquoRandom utility models withimplicit availabilityperception of choice alternatives for thesimulation of travel demandrdquo Transportation Research Part CEmerging Technologies vol 9 no 4 pp 249ndash263 2001

[53] Y Sheffi Urban Transportation Network Equilibrium Anal-ysis with Mathematical Programming Methods Prentice HallEnglewood Cliffs NJ USA 1985

[54] M J Beckmann C B McGuire and C B Winsten Studies inthe Economics of Transportation Yale University press NewHaven Connecticut 1956

[55] G M Korpelevich ldquo(e extragradient method for findingsaddle points and other problemsrdquo Matecon vol 12pp 747ndash756 1976

[56] A Galligari and M Sciandrone ldquoA convergent and fast pathequilibration algorithm for the traffic assignment problemrdquoOptimization Methods and Software vol 33 no 2 pp 354ndash371 2018

[57] C-C Lu H S Mahmassani and X Zhou ldquoEquivalent gapfunction-based reformulation and solution algorithm for thedynamic user equilibrium problemrdquo Transportation ResearchPart B Methodological vol 43 no 3 pp 345ndash364 2009

[58] D Wang F Liao Z Gao and H Timmermans ldquoTolerance-based strategies for extending the column generation algo-rithm to the bounded rational dynamic user equilibriumproblemrdquo Transportation Research Part B Methodologicalvol 119 pp 102ndash121 2019

[59] M Ameli J P Lebacque and L Leclercq ldquoSimulation-baseddynamic traffic assignment meta-heuristic solution methodswith parallel computingrdquo Computer-Aided Civil and Infra-structure Engineering vol 35 no 10 pp 1047ndash1062 2020


[60] A Chen and R Jayakrishnan A Path-Based Gradient Pro-jection Algorithm Effects of Equilibration with a RestrictedPath Set under Two Flow Update Policies Institute ofTransportation Studies University of California Irvine CAUSA 1998

[61] A Chen ldquoEffects of flow update strategies on implementationof the Frank-Wolfe algorithm for the traffic assignmentproblemrdquo Transportation Research Record Journal of theTransportation Research Board vol 1771 no 1 pp 132ndash1392001

[62] D Bertsekas E Gafni and R Gallager ldquoSecond derivativealgorithms for minimum delay distributed routing in net-worksrdquo IEEE Transactions on Communications vol 32 no 8pp 911ndash919 1984

[63] C Sun R Jayakrishnan and W K Tsai ldquoComputationalstudy of a path-based algorithm and its variants for statictraffic assignmentrdquo Transportation Research Record Journalof the Transportation Research Board vol 1537 no 1pp 106ndash115 1996

[64] D Boyce B Ralevic-Dekic and H Bar-Gera ldquoConvergence oftraffic assignments how much is enoughrdquo Journal ofTransportation Engineering vol 130 no 1 pp 49ndash55 2004


compute path flow equilibration and they have faster con-vergence comparing to link-based algorithms (e reason isthat the link-based algorithm exhibits a zigzagging trajectory asit approaches the optimal solution On the other hand thepath-based algorithm does not exhibit the zigzagging effect forthe same initial solution In the transportation literature[26 27] the computational performance of link- and path-based algorithms for the traffic assignment problem is com-pared in medium- to large-scale randomly generated gridnetworks and real-size networks Based on the conclusion apath-based algorithm is consistently faster in approaching theneighborhood of the optimal solution Bush-basedmethods aredeveloped to solve an acyclic network problem(ese methodshave improved the existing state-of-the-art of algorithms andare fast and memory efficient However using the algorithmsfor solving large-scale networks may remain impractical [28](e solution algorithms for the traffic assignmentmodel can besummarized below Interested readers may refer to [27 29] forthe detailed descriptions of the algorithm procedure

(i) Link-Based Algorithm (is includes search directionimprovement [30] restricted simplicial decomposition[31] and nonlinear simplicial decomposition [32]Mitradjieva and Lindberg [30] introduced conjugateand biconjugate algorithms (ese algorithms im-prove the search direction by using previous directioninformation Restricted simplicial decomposition andnonlinear simplicial decomposition algorithms arerepresented by a convex combination of extremepoints (ey first generate new extreme points andthen optimize the restricted master problem

(ii) Path-Based Algorithm (is includes disaggregatedsimplicial decomposition [32] gradient projection[21 33ndash35] projected gradient [36] improved so-cial pressure [37] greedy algorithm [38 39] andstep size improvement [40]

(e disaggregated simplicial decomposition algo-rithm proposed by Larsson and Patriksson [41] issimilar to the restricted simplicial decompositionalgorithm However the extreme points are gen-erated by path flows instead of link flows (egradient projection algorithm adjusts the O-Ddemand between nonshortest paths and theshortest path (e projected gradient algorithm isbased on the idea of adjusting flow between the setof path with a cost greater than the average pathcost and the set of path with a cost less than theaverage path cost Improved social pressure is alsousing the path flow equilibrium between costlierpaths and cheaper paths but the flow updatingstrategy is more complicated To solve the re-stricted subproblem a greedy algorithm is intro-duced Recently Du et al [40] introducedBarzilaindashBorwein step size to adjust path flows

(iii) Bush-Based Algorithm (is includes origin-basedalgorithm [42 43] modified origin-based algorithm[44] algorithm B [45] linear user cost equilibrium[46] and paired alternative segments [47]

(e bush-based algorithm sequentially solves a list ofnode-based subproblems defined on the bush rooted (ealgorithm B adjusts flows between the longest paths and theshortest ones in the bush rooted path tree A paired alter-native segment (PAS) is defined as two completely disjointpath segments and then the flows are adjusted in disjointpath segments

In this paper we are interested in developing solutionalgorithms for solving the MP formulation of the CMSTAproblem that can represent large-scale multimodal trans-portation networks with both private and public modes andcorrelation among multiple public modes using the nestedlogit (NL) model under the network user equilibrium (UE)framework Contrast to the simple structure of the multi-nomial logit (MNL) model the NL model has a hierarchicalstructure that decomposes the choice probability into twolevels represented by the marginal and conditional proba-bilities A two-phase GP algorithm is developed to solve theCMSTA problem with a hierarchical nested modal splitstructure and UE traffic assignment along with variousimprovement strategies to speed up the convergence of thepath-based GP algorithm

In the literature a path-based GP algorithm has beenadopted for solving various traffic assignment problems suchas UE assignment problem the nonadditive cost assignmentproblem side constrained problem stochastic user equi-librium assignment problem transit assignment problemsystem optimal assignment problem and freight traffic as-signment problem [21] Although the algorithm was de-veloped for solving various traffic assignment problems thealgorithm was rarely applied for solving CMSTA problem

Our goal is to demonstrate that the improved GP algo-rithm can solve large-scale multiclass or multimodal trafficassignment problems in CMSTA with NL function In ad-dition the proposed algorithm can apply to other logit modelshaving a hierarchical structure such as the generalized ex-treme value model by [48] including the cross-nested logit(CNL) paired combinatorial logit (PCL) and generalizednested logit (GNL) To show proof of concept a large-scalemultimodal transport network in Seoul Korea will be used todemonstrate the applicability of the two-phase GP algorithm

(is paper is organized as follows After the introduc-tion a relevant background of the NL model for the modalsplit problem and UE conditions for the traffic assignmentproblem is introduced in Section 2 Section 3 provides anequivalent MP formulation for the NL-UE CMSTA problemalong with the equivalence property Section 4 describes thetwo-phase GP algorithm for solving the NL-UE CMSTAproblem improvement strategies and its overall solutionprocedure Section 5 provides numerical results to dem-onstrate the applicability of the two-phase GP algorithm forsolving large-scale multimodal transportation networksSome concluding remarks are provided in Section 6

2 Mode Choice and Route Choice

In this section we provide some background of the nestedlogit (NL) model for the modal split problem and the UE







Prsm(NL)



uexp θrsτrs


um( 1113857( 11138571113960 1113961τrs

u minus 1


uexp θrsτrs



u rs isin RS (1)


Prsm P

rsm|uP


u rs isin RS

Prsmu


um minus μrsum( 1113857( 1113857





Prsu



um( 1113857( 11138571113960 1113961τrs

u


uexp θrsτrs


ts( 1113857( 11138571113960 1113961τrs

u


(2)




frskm c

rskm minus πrs

m( 1113857 0


rs

crskm minus πrs


rs



(3)





1113946xa

0ta(ω)dω + 1113944

rsisinRS1113946

qrsT

0

1θrs ln

ωq

rsminus ω

1113888 11138891113888 1113889dω + 1113944rsisinRS

1113944misinMrs

u

τrsT

θrs qrsm ln q



τrsT


u

qrsm


u

qrsm

⎛⎝ ⎞⎠ minus 1⎛⎝ ⎞⎠ + 1113944rsisinRS

1113944misinMrs

u

qrsmgc

rsm

(4)

st









rs)Prs




ηBa ηM








krs







δrsBa δ

rsMa




1113944kisinKrs

frsk + q

rsT q


1113944misinMrs

T

qrsm q




T

xa 1113944rsisinRS

1113944kisinKrs

frsk δ


(5)




auto crsk 1113944


k


aisinAδrs

Ba ta xa( 1113857 + dBa1113872 1113873ηB

a +1 minus ηB

a1113872 1113873la

ws + Wa



aisinAδrs

Ma tMa + d

Ma1113872 1113873ηM

a + +1 minus ηM

a1113872 1113873la

ws + Wa



aisinAδrs

BMa min ta xa( 1113857 + dBa1113872 1113873 t

Ma + d

Ma1113872 11138731113872 1113873ηT

a +1 minus ηT

a1113872 1113873la

ws + Wa


(6)


ta xa( 1113857 t0a 1 + η

xa

Ca

1113888 1113889






ϕrsq

rsminus 1113944

kisinKrs

frsk + 1113944

misinMrsu

qrsm

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (8)




MNL

(a)

Nest u

Nest m


NL

θrs

τurs

(b)



zL

zfrsk

crsk minus ϕrs


zL

zqrsm

1θrs ln

1113936misinMrsu

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsu

qrsm



(9)


1θrs ln q

rsT +

τrsu

θrs ln qrsm minus

τrsu

θrs ln qrsT + gc

rsm

1θrs ln q

rsA + c

rsk

(10)



qrsm)

Let

1θ rs ln q

rsT +

τrsu

θ rs ln qrsm minus

τrsu

θ rs ln qrsT + gc

rsm λ

1θ rs ln q

rsA + c

rsk λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)


qrsm q

rsT( 1113857

1minus τrsT( )τrs

u( )( ) expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⟹ 1113944

misinMrsu

qrsm 1113944

misinMrsu

expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

τrsu

qrsA( 1113857

1τrsu( )( ) exp

θrs

τrsu


rsA exp

θrs

τrsu

λ minus crsm( 11138571113888 11138891113888 1113889

τrsT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)


PrsA

qrsA


uq

rsm

1113936kisinKrs f

rsk

qrs

exp minusθrsc

rsk( 1113857

exp minusθrsc

rsk( 1113857 + 1113936misinMrs

uexp θrsτrs

u( 1113857 minus gcrsm( 1113857( 11138571113872 1113873

τrsu

(13)

From qrsm(qrs

T )(((1minus τrsT




qrsm

qrsT

exp θrsτrs

u( 1113857 λ minus gcrsm( 1113857( 1113857

qrsT( 1113857

1τrsu( )( )

exp θrsτrs

u( 1113857 minusgcrsm( 1113857( 1113857


u( 1113857 minusgcrsm( 1113857( 1113857

(14)




z2Z

zfrsk f

rsj

dta xa( 1113857

dxa

δrska ge 0 if j k

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)



z2L

z qrsmq

rsn

1θrs

11113936misinMrs

uq

rsm

+1

qrs


qrsm


T

θrs

1q

rsm

minus1

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ge 0 if m n

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)



wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsuq

rsm

qrs


rsm

⎛⎝ ⎞⎠ +τrs

u

θrs lnq

rsm

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ + gcrsm

(17)


k (frsk ) wrs







frsk (n + 1) max f

rsk (n) minus

αs

rsk (n)

crsk (n) minus c

rs


frs


kisinKrs

kne krs



k (n + 1) and frs



(crsk (n) minus crs

krs


search direction

crsk minus c

rs

krs 1113944



ta xa( 1113857δrs

krsa

srsk 1113944


ka minus δrs

krsa1113874 1113875

2

(19)








B B

M M


qndash = 100

qA

qB

qM

Walk time (1)


θ = 05

τT = 02

gcM = 10 + 30 + 10

gcB = 10 + 05 middot cA + 20


(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Trav

el ti

me (

auto

)

Flow (auto)

Equilibrium point

Auto = 547

wT (qT)θ = 05τ = 02

cA (qA) = 444w (qT) = 444

cA (qA)

Transit = 453

(b)

Equilibrium point

2

3

4

5

6

7

00 05 10 15 20 25 30 35 40 45

Exes

s dem

and

cost

Flow (transit)

Metro = 287 Bus = 167

w (qB)

w (qB)


(c)

00204060

4040

80100120

Tim

e (c a

nd w

)

140

3535

Metro (M)

Bus (B) Auto (A)

160180


(d)

00

3025

20

1015

0505

1015

20

q M = 287 amp c M

= 500

qB = 167 amp c

B = 522

2530

00

35

40

wB gt wM cA

cA = wB = wM

cA = wB wM

wM gt wB cA

qA = 547 amp cA = 444

40

35

(e)



PA =





(f )

cA (qA) = 444

167 + 287 167021ln+ + 522 = 444

05ln

10 ndash (167 + 287) 05 167 + 287wB (qB) =

1 167 + 287 28702lnln + + 500 = 444

05 10 ndash (167 + 287) 05 167 + 287wB (qM) =

(g)



wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


rsm


T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)


krs

)



T (qrsT )lt crs

krs


T (qrsT )gt crs

krs







qrsm(n + 1) q

rsT (n + 1) middot


rsm(n + 1)( 1113857( 1113857


T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)























Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References











































































Prsm(NL)



uexp θrsτrs


um( 1113857( 11138571113960 1113961τrs

u minus 1


uexp θrsτrs



u rs isin RS (1)


Prsm P

rsm|uP


u rs isin RS

Prsmu


um minus μrsum( 1113857( 1113857





Prsu



um( 1113857( 11138571113960 1113961τrs

u


uexp θrsτrs


ts( 1113857( 11138571113960 1113961τrs

u


(2)




frskm c

rskm minus πrs

m( 1113857 0


rs

crskm minus πrs


rs



(3)





1113946xa

0ta(ω)dω + 1113944

rsisinRS1113946

qrsT

0

1θrs ln

ωq

rsminus ω

1113888 11138891113888 1113889dω + 1113944rsisinRS

1113944misinMrs

u

τrsT

θrs qrsm ln q



τrsT


u

qrsm


u

qrsm

⎛⎝ ⎞⎠ minus 1⎛⎝ ⎞⎠ + 1113944rsisinRS

1113944misinMrs

u

qrsmgc

rsm

(4)

st









rs)Prs




ηBa ηM








krs







δrsBa δ

rsMa




1113944kisinKrs

frsk + q

rsT q


1113944misinMrs

T

qrsm q




T

xa 1113944rsisinRS

1113944kisinKrs

frsk δ


(5)




auto crsk 1113944


k


aisinAδrs

Ba ta xa( 1113857 + dBa1113872 1113873ηB

a +1 minus ηB

a1113872 1113873la

ws + Wa



aisinAδrs

Ma tMa + d

Ma1113872 1113873ηM

a + +1 minus ηM

a1113872 1113873la

ws + Wa



aisinAδrs

BMa min ta xa( 1113857 + dBa1113872 1113873 t

Ma + d

Ma1113872 11138731113872 1113873ηT

a +1 minus ηT

a1113872 1113873la

ws + Wa


(6)


ta xa( 1113857 t0a 1 + η

xa

Ca

1113888 1113889






ϕrsq

rsminus 1113944

kisinKrs

frsk + 1113944

misinMrsu

qrsm

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (8)




MNL

(a)

Nest u

Nest m


NL

θrs

τurs

(b)



zL

zfrsk

crsk minus ϕrs


zL

zqrsm

1θrs ln

1113936misinMrsu

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsu

qrsm



(9)


1θrs ln q

rsT +

τrsu

θrs ln qrsm minus

τrsu

θrs ln qrsT + gc

rsm

1θrs ln q

rsA + c

rsk

(10)



qrsm)

Let

1θ rs ln q

rsT +

τrsu

θ rs ln qrsm minus

τrsu

θ rs ln qrsT + gc

rsm λ

1θ rs ln q

rsA + c

rsk λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)


qrsm q

rsT( 1113857

1minus τrsT( )τrs

u( )( ) expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⟹ 1113944

misinMrsu

qrsm 1113944

misinMrsu

expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

τrsu

qrsA( 1113857

1τrsu( )( ) exp

θrs

τrsu


rsA exp

θrs

τrsu

λ minus crsm( 11138571113888 11138891113888 1113889

τrsT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)


PrsA

qrsA


uq

rsm

1113936kisinKrs f

rsk

qrs

exp minusθrsc

rsk( 1113857

exp minusθrsc

rsk( 1113857 + 1113936misinMrs

uexp θrsτrs

u( 1113857 minus gcrsm( 1113857( 11138571113872 1113873

τrsu

(13)

From qrsm(qrs

T )(((1minus τrsT




qrsm

qrsT

exp θrsτrs

u( 1113857 λ minus gcrsm( 1113857( 1113857

qrsT( 1113857

1τrsu( )( )

exp θrsτrs

u( 1113857 minusgcrsm( 1113857( 1113857


u( 1113857 minusgcrsm( 1113857( 1113857

(14)




z2Z

zfrsk f

rsj

dta xa( 1113857

dxa

δrska ge 0 if j k

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)



z2L

z qrsmq

rsn

1θrs

11113936misinMrs

uq

rsm

+1

qrs


qrsm


T

θrs

1q

rsm

minus1

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ge 0 if m n

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)



wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsuq

rsm

qrs


rsm

⎛⎝ ⎞⎠ +τrs

u

θrs lnq

rsm

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ + gcrsm

(17)


k (frsk ) wrs







frsk (n + 1) max f

rsk (n) minus

αs

rsk (n)

crsk (n) minus c

rs


frs


kisinKrs

kne krs



k (n + 1) and frs



(crsk (n) minus crs

krs


search direction

crsk minus c

rs

krs 1113944



ta xa( 1113857δrs

krsa

srsk 1113944


ka minus δrs

krsa1113874 1113875

2

(19)








B B

M M


qndash = 100

qA

qB

qM

Walk time (1)


θ = 05

τT = 02

gcM = 10 + 30 + 10

gcB = 10 + 05 middot cA + 20


(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Trav

el ti

me (

auto

)

Flow (auto)

Equilibrium point

Auto = 547

wT (qT)θ = 05τ = 02

cA (qA) = 444w (qT) = 444

cA (qA)

Transit = 453

(b)

Equilibrium point

2

3

4

5

6

7

00 05 10 15 20 25 30 35 40 45

Exes

s dem

and

cost

Flow (transit)

Metro = 287 Bus = 167

w (qB)

w (qB)


(c)

00204060

4040

80100120

Tim

e (c a

nd w

)

140

3535

Metro (M)

Bus (B) Auto (A)

160180


(d)

00

3025

20

1015

0505

1015

20

q M = 287 amp c M

= 500

qB = 167 amp c

B = 522

2530

00

35

40

wB gt wM cA

cA = wB = wM

cA = wB wM

wM gt wB cA

qA = 547 amp cA = 444

40

35

(e)



PA =





(f )

cA (qA) = 444

167 + 287 167021ln+ + 522 = 444

05ln

10 ndash (167 + 287) 05 167 + 287wB (qB) =

1 167 + 287 28702lnln + + 500 = 444

05 10 ndash (167 + 287) 05 167 + 287wB (qM) =

(g)



wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


rsm


T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)


krs

)



T (qrsT )lt crs

krs


T (qrsT )gt crs

krs







qrsm(n + 1) q

rsT (n + 1) middot


rsm(n + 1)( 1113857( 1113857


T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)























Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References













































































rs)Prs




ηBa ηM








krs







δrsBa δ

rsMa




1113944kisinKrs

frsk + q

rsT q


1113944misinMrs

T

qrsm q




T

xa 1113944rsisinRS

1113944kisinKrs

frsk δ


(5)




auto crsk 1113944


k


aisinAδrs

Ba ta xa( 1113857 + dBa1113872 1113873ηB

a +1 minus ηB

a1113872 1113873la

ws + Wa



aisinAδrs

Ma tMa + d

Ma1113872 1113873ηM

a + +1 minus ηM

a1113872 1113873la

ws + Wa



aisinAδrs

BMa min ta xa( 1113857 + dBa1113872 1113873 t

Ma + d

Ma1113872 11138731113872 1113873ηT

a +1 minus ηT

a1113872 1113873la

ws + Wa


(6)


ta xa( 1113857 t0a 1 + η

xa

Ca

1113888 1113889






ϕrsq

rsminus 1113944

kisinKrs

frsk + 1113944

misinMrsu

qrsm

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (8)




MNL

(a)

Nest u

Nest m


NL

θrs

τurs

(b)



zL

zfrsk

crsk minus ϕrs


zL

zqrsm

1θrs ln

1113936misinMrsu

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsu

qrsm



(9)


1θrs ln q

rsT +

τrsu

θrs ln qrsm minus

τrsu

θrs ln qrsT + gc

rsm

1θrs ln q

rsA + c

rsk

(10)



qrsm)

Let

1θ rs ln q

rsT +

τrsu

θ rs ln qrsm minus

τrsu

θ rs ln qrsT + gc

rsm λ

1θ rs ln q

rsA + c

rsk λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)


qrsm q

rsT( 1113857

1minus τrsT( )τrs

u( )( ) expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⟹ 1113944

misinMrsu

qrsm 1113944

misinMrsu

expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

τrsu

qrsA( 1113857

1τrsu( )( ) exp

θrs

τrsu


rsA exp

θrs

τrsu

λ minus crsm( 11138571113888 11138891113888 1113889

τrsT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)


PrsA

qrsA


uq

rsm

1113936kisinKrs f

rsk

qrs

exp minusθrsc

rsk( 1113857

exp minusθrsc

rsk( 1113857 + 1113936misinMrs

uexp θrsτrs

u( 1113857 minus gcrsm( 1113857( 11138571113872 1113873

τrsu

(13)

From qrsm(qrs

T )(((1minus τrsT




qrsm

qrsT

exp θrsτrs

u( 1113857 λ minus gcrsm( 1113857( 1113857

qrsT( 1113857

1τrsu( )( )

exp θrsτrs

u( 1113857 minusgcrsm( 1113857( 1113857


u( 1113857 minusgcrsm( 1113857( 1113857

(14)




z2Z

zfrsk f

rsj

dta xa( 1113857

dxa

δrska ge 0 if j k

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)



z2L

z qrsmq

rsn

1θrs

11113936misinMrs

uq

rsm

+1

qrs


qrsm


T

θrs

1q

rsm

minus1

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ge 0 if m n

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)



wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsuq

rsm

qrs


rsm

⎛⎝ ⎞⎠ +τrs

u

θrs lnq

rsm

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ + gcrsm

(17)


k (frsk ) wrs







frsk (n + 1) max f

rsk (n) minus

αs

rsk (n)

crsk (n) minus c

rs


frs


kisinKrs

kne krs



k (n + 1) and frs



(crsk (n) minus crs

krs


search direction

crsk minus c

rs

krs 1113944



ta xa( 1113857δrs

krsa

srsk 1113944


ka minus δrs

krsa1113874 1113875

2

(19)








B B

M M


qndash = 100

qA

qB

qM

Walk time (1)


θ = 05

τT = 02

gcM = 10 + 30 + 10

gcB = 10 + 05 middot cA + 20


(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Trav

el ti

me (

auto

)

Flow (auto)

Equilibrium point

Auto = 547

wT (qT)θ = 05τ = 02

cA (qA) = 444w (qT) = 444

cA (qA)

Transit = 453

(b)

Equilibrium point

2

3

4

5

6

7

00 05 10 15 20 25 30 35 40 45

Exes

s dem

and

cost

Flow (transit)

Metro = 287 Bus = 167

w (qB)

w (qB)


(c)

00204060

4040

80100120

Tim

e (c a

nd w

)

140

3535

Metro (M)

Bus (B) Auto (A)

160180


(d)

00

3025

20

1015

0505

1015

20

q M = 287 amp c M

= 500

qB = 167 amp c

B = 522

2530

00

35

40

wB gt wM cA

cA = wB = wM

cA = wB wM

wM gt wB cA

qA = 547 amp cA = 444

40

35

(e)



PA =





(f )

cA (qA) = 444

167 + 287 167021ln+ + 522 = 444

05ln

10 ndash (167 + 287) 05 167 + 287wB (qB) =

1 167 + 287 28702lnln + + 500 = 444

05 10 ndash (167 + 287) 05 167 + 287wB (qM) =

(g)



wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


rsm


T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)


krs

)



T (qrsT )lt crs

krs


T (qrsT )gt crs

krs







qrsm(n + 1) q

rsT (n + 1) middot


rsm(n + 1)( 1113857( 1113857


T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)























Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References






































































1113944kisinKrs

frsk + q

rsT q


1113944misinMrs

T

qrsm q




T

xa 1113944rsisinRS

1113944kisinKrs

frsk δ


(5)




auto crsk 1113944


k


aisinAδrs

Ba ta xa( 1113857 + dBa1113872 1113873ηB

a +1 minus ηB

a1113872 1113873la

ws + Wa



aisinAδrs

Ma tMa + d

Ma1113872 1113873ηM

a + +1 minus ηM

a1113872 1113873la

ws + Wa



aisinAδrs

BMa min ta xa( 1113857 + dBa1113872 1113873 t

Ma + d

Ma1113872 11138731113872 1113873ηT

a +1 minus ηT

a1113872 1113873la

ws + Wa


(6)


ta xa( 1113857 t0a 1 + η

xa

Ca

1113888 1113889






ϕrsq

rsminus 1113944

kisinKrs

frsk + 1113944

misinMrsu

qrsm

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ (8)




MNL

(a)

Nest u

Nest m


NL

θrs

τurs

(b)



zL

zfrsk

crsk minus ϕrs


zL

zqrsm

1θrs ln

1113936misinMrsu

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsu

qrsm



(9)


1θrs ln q

rsT +

τrsu

θrs ln qrsm minus

τrsu

θrs ln qrsT + gc

rsm

1θrs ln q

rsA + c

rsk

(10)



qrsm)

Let

1θ rs ln q

rsT +

τrsu

θ rs ln qrsm minus

τrsu

θ rs ln qrsT + gc

rsm λ

1θ rs ln q

rsA + c

rsk λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)


qrsm q

rsT( 1113857

1minus τrsT( )τrs

u( )( ) expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⟹ 1113944

misinMrsu

qrsm 1113944

misinMrsu

expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

τrsu

qrsA( 1113857

1τrsu( )( ) exp

θrs

τrsu


rsA exp

θrs

τrsu

λ minus crsm( 11138571113888 11138891113888 1113889

τrsT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)


PrsA

qrsA


uq

rsm

1113936kisinKrs f

rsk

qrs

exp minusθrsc

rsk( 1113857

exp minusθrsc

rsk( 1113857 + 1113936misinMrs

uexp θrsτrs

u( 1113857 minus gcrsm( 1113857( 11138571113872 1113873

τrsu

(13)

From qrsm(qrs

T )(((1minus τrsT




qrsm

qrsT

exp θrsτrs

u( 1113857 λ minus gcrsm( 1113857( 1113857

qrsT( 1113857

1τrsu( )( )

exp θrsτrs

u( 1113857 minusgcrsm( 1113857( 1113857


u( 1113857 minusgcrsm( 1113857( 1113857

(14)




z2Z

zfrsk f

rsj

dta xa( 1113857

dxa

δrska ge 0 if j k

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)



z2L

z qrsmq

rsn

1θrs

11113936misinMrs

uq

rsm

+1

qrs


qrsm


T

θrs

1q

rsm

minus1

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ge 0 if m n

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)



wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsuq

rsm

qrs


rsm

⎛⎝ ⎞⎠ +τrs

u

θrs lnq

rsm

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ + gcrsm

(17)


k (frsk ) wrs







frsk (n + 1) max f

rsk (n) minus

αs

rsk (n)

crsk (n) minus c

rs


frs


kisinKrs

kne krs



k (n + 1) and frs



(crsk (n) minus crs

krs


search direction

crsk minus c

rs

krs 1113944



ta xa( 1113857δrs

krsa

srsk 1113944


ka minus δrs

krsa1113874 1113875

2

(19)








B B

M M


qndash = 100

qA

qB

qM

Walk time (1)


θ = 05

τT = 02

gcM = 10 + 30 + 10

gcB = 10 + 05 middot cA + 20


(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Trav

el ti

me (

auto

)

Flow (auto)

Equilibrium point

Auto = 547

wT (qT)θ = 05τ = 02

cA (qA) = 444w (qT) = 444

cA (qA)

Transit = 453

(b)

Equilibrium point

2

3

4

5

6

7

00 05 10 15 20 25 30 35 40 45

Exes

s dem

and

cost

Flow (transit)

Metro = 287 Bus = 167

w (qB)

w (qB)


(c)

00204060

4040

80100120

Tim

e (c a

nd w

)

140

3535

Metro (M)

Bus (B) Auto (A)

160180


(d)

00

3025

20

1015

0505

1015

20

q M = 287 amp c M

= 500

qB = 167 amp c

B = 522

2530

00

35

40

wB gt wM cA

cA = wB = wM

cA = wB wM

wM gt wB cA

qA = 547 amp cA = 444

40

35

(e)



PA =





(f )

cA (qA) = 444

167 + 287 167021ln+ + 522 = 444

05ln

10 ndash (167 + 287) 05 167 + 287wB (qB) =

1 167 + 287 28702lnln + + 500 = 444

05 10 ndash (167 + 287) 05 167 + 287wB (qM) =

(g)



wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


rsm


T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)


krs

)



T (qrsT )lt crs

krs


T (qrsT )gt crs

krs







qrsm(n + 1) q

rsT (n + 1) middot


rsm(n + 1)( 1113857( 1113857


T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)























Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References






































































zL

zfrsk

crsk minus ϕrs


zL

zqrsm

1θrs ln

1113936misinMrsu

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsu

qrsm



(9)


1θrs ln q

rsT +

τrsu

θrs ln qrsm minus

τrsu

θrs ln qrsT + gc

rsm

1θrs ln q

rsA + c

rsk

(10)



qrsm)

Let

1θ rs ln q

rsT +

τrsu

θ rs ln qrsm minus

τrsu

θ rs ln qrsT + gc

rsm λ

1θ rs ln q

rsA + c

rsk λ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)


qrsm q

rsT( 1113857

1minus τrsT( )τrs

u( )( ) expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⟹ 1113944

misinMrsu

qrsm 1113944

misinMrsu

expθrs

τrsu

λ minus gcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

τrsu

qrsA( 1113857

1τrsu( )( ) exp

θrs

τrsu


rsA exp

θrs

τrsu

λ minus crsm( 11138571113888 11138891113888 1113889

τrsT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)


PrsA

qrsA


uq

rsm

1113936kisinKrs f

rsk

qrs

exp minusθrsc

rsk( 1113857

exp minusθrsc

rsk( 1113857 + 1113936misinMrs

uexp θrsτrs

u( 1113857 minus gcrsm( 1113857( 11138571113872 1113873

τrsu

(13)

From qrsm(qrs

T )(((1minus τrsT




qrsm

qrsT

exp θrsτrs

u( 1113857 λ minus gcrsm( 1113857( 1113857

qrsT( 1113857

1τrsu( )( )

exp θrsτrs

u( 1113857 minusgcrsm( 1113857( 1113857


u( 1113857 minusgcrsm( 1113857( 1113857

(14)




z2Z

zfrsk f

rsj

dta xa( 1113857

dxa

δrska ge 0 if j k

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)



z2L

z qrsmq

rsn

1θrs

11113936misinMrs

uq

rsm

+1

qrs


qrsm


T

θrs

1q

rsm

minus1

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ge 0 if m n

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)



wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsuq

rsm

qrs


rsm

⎛⎝ ⎞⎠ +τrs

u

θrs lnq

rsm

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ + gcrsm

(17)


k (frsk ) wrs







frsk (n + 1) max f

rsk (n) minus

αs

rsk (n)

crsk (n) minus c

rs


frs


kisinKrs

kne krs



k (n + 1) and frs



(crsk (n) minus crs

krs


search direction

crsk minus c

rs

krs 1113944



ta xa( 1113857δrs

krsa

srsk 1113944


ka minus δrs

krsa1113874 1113875

2

(19)








B B

M M


qndash = 100

qA

qB

qM

Walk time (1)


θ = 05

τT = 02

gcM = 10 + 30 + 10

gcB = 10 + 05 middot cA + 20


(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Trav

el ti

me (

auto

)

Flow (auto)

Equilibrium point

Auto = 547

wT (qT)θ = 05τ = 02

cA (qA) = 444w (qT) = 444

cA (qA)

Transit = 453

(b)

Equilibrium point

2

3

4

5

6

7

00 05 10 15 20 25 30 35 40 45

Exes

s dem

and

cost

Flow (transit)

Metro = 287 Bus = 167

w (qB)

w (qB)


(c)

00204060

4040

80100120

Tim

e (c a

nd w

)

140

3535

Metro (M)

Bus (B) Auto (A)

160180


(d)

00

3025

20

1015

0505

1015

20

q M = 287 amp c M

= 500

qB = 167 amp c

B = 522

2530

00

35

40

wB gt wM cA

cA = wB = wM

cA = wB wM

wM gt wB cA

qA = 547 amp cA = 444

40

35

(e)



PA =





(f )

cA (qA) = 444

167 + 287 167021ln+ + 522 = 444

05ln

10 ndash (167 + 287) 05 167 + 287wB (qB) =

1 167 + 287 28702lnln + + 500 = 444

05 10 ndash (167 + 287) 05 167 + 287wB (qM) =

(g)



wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


rsm


T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)


krs

)



T (qrsT )lt crs

krs


T (qrsT )gt crs

krs







qrsm(n + 1) q

rsT (n + 1) middot


rsm(n + 1)( 1113857( 1113857


T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)























Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References






































































z2L

z qrsmq

rsn

1θrs

11113936misinMrs

uq

rsm

+1

qrs


qrsm


T

θrs

1q

rsm

minus1

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ge 0 if m n

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)



wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsuq

rsm

qrs


rsm

⎛⎝ ⎞⎠ +τrs

u

θrs lnq

rsm

1113936misinMrsu

qrsm

⎛⎝ ⎞⎠ + gcrsm

(17)


k (frsk ) wrs







frsk (n + 1) max f

rsk (n) minus

αs

rsk (n)

crsk (n) minus c

rs


frs


kisinKrs

kne krs



k (n + 1) and frs



(crsk (n) minus crs

krs


search direction

crsk minus c

rs

krs 1113944



ta xa( 1113857δrs

krsa

srsk 1113944


ka minus δrs

krsa1113874 1113875

2

(19)








B B

M M


qndash = 100

qA

qB

qM

Walk time (1)


θ = 05

τT = 02

gcM = 10 + 30 + 10

gcB = 10 + 05 middot cA + 20


(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Trav

el ti

me (

auto

)

Flow (auto)

Equilibrium point

Auto = 547

wT (qT)θ = 05τ = 02

cA (qA) = 444w (qT) = 444

cA (qA)

Transit = 453

(b)

Equilibrium point

2

3

4

5

6

7

00 05 10 15 20 25 30 35 40 45

Exes

s dem

and

cost

Flow (transit)

Metro = 287 Bus = 167

w (qB)

w (qB)


(c)

00204060

4040

80100120

Tim

e (c a

nd w

)

140

3535

Metro (M)

Bus (B) Auto (A)

160180


(d)

00

3025

20

1015

0505

1015

20

q M = 287 amp c M

= 500

qB = 167 amp c

B = 522

2530

00

35

40

wB gt wM cA

cA = wB = wM

cA = wB wM

wM gt wB cA

qA = 547 amp cA = 444

40

35

(e)



PA =





(f )

cA (qA) = 444

167 + 287 167021ln+ + 522 = 444

05ln

10 ndash (167 + 287) 05 167 + 287wB (qB) =

1 167 + 287 28702lnln + + 500 = 444

05 10 ndash (167 + 287) 05 167 + 287wB (qM) =

(g)



wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


rsm


T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)


krs

)



T (qrsT )lt crs

krs


T (qrsT )gt crs

krs







qrsm(n + 1) q

rsT (n + 1) middot


rsm(n + 1)( 1113857( 1113857


T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)























Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References









































































B B

M M


qndash = 100

qA

qB

qM

Walk time (1)


θ = 05

τT = 02

gcM = 10 + 30 + 10

gcB = 10 + 05 middot cA + 20


(a)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Trav

el ti

me (

auto

)

Flow (auto)

Equilibrium point

Auto = 547

wT (qT)θ = 05τ = 02

cA (qA) = 444w (qT) = 444

cA (qA)

Transit = 453

(b)

Equilibrium point

2

3

4

5

6

7

00 05 10 15 20 25 30 35 40 45

Exes

s dem

and

cost

Flow (transit)

Metro = 287 Bus = 167

w (qB)

w (qB)


(c)

00204060

4040

80100120

Tim

e (c a

nd w

)

140

3535

Metro (M)

Bus (B) Auto (A)

160180


(d)

00

3025

20

1015

0505

1015

20

q M = 287 amp c M

= 500

qB = 167 amp c

B = 522

2530

00

35

40

wB gt wM cA

cA = wB = wM

cA = wB wM

wM gt wB cA

qA = 547 amp cA = 444

40

35

(e)



PA =





(f )

cA (qA) = 444

167 + 287 167021ln+ + 522 = 444

05ln

10 ndash (167 + 287) 05 167 + 287wB (qB) =

1 167 + 287 28702lnln + + 500 = 444

05 10 ndash (167 + 287) 05 167 + 287wB (qM) =

(g)



wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


rsm


T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)


krs

)



T (qrsT )lt crs

krs


T (qrsT )gt crs

krs







qrsm(n + 1) q

rsT (n + 1) middot


rsm(n + 1)( 1113857( 1113857


T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)























Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References






































































wrsT q

rsT( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


rsm


T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(20)


krs

)



T (qrsT )lt crs

krs


T (qrsT )gt crs

krs







qrsm(n + 1) q

rsT (n + 1) middot


rsm(n + 1)( 1113857( 1113857


T( 1113857 minusgcrsm(n + 1)( 1113857( 1113857

(21)























Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References
















































































Terminate

Outer iteration

Inner iteration

Columndropping


n =

n +

1

Y

N



εouter lt E





w (qTrs (n))

w (qTrs (n + 1))

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckrs (fk

rs (n+1))

fkrs (n)

w (qTrs (n + 1))

qTrs (n + 1)

qTrs (n)

fkndashrs (n + 1)

qTrs (n) ndash Σ

kєKrs fk

rs (n + 1)



Cost

Flow

(a)

w (qTrs (n))

w (qTrs (n + 1))

qTrs (n)

fkndashrs (n)

fkrs (n + 1)

ckrs (fk

rs (n))

ckrs (fk

rs (n + 1))

ckndashrs (fkndash

rs (n))

ckndashrs (fkndash

rs (n + 1))

fkrs (n)

qTrs (n + 1)fkndash

rs (n + 1)





rs (n + 1)



Cost

Flow

(b)


k(n)gtwrs


T (qrsT (n))gt crs

k(n)


Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References






































































Origin r = 1

Column generation



Destination s = 1








rs


rs)

s le Ss = s + 1

r le Rr = r + 1

Terminate

N

YN

Y










(fkndashrr




rs)n+1

kєKrs

kєKrs

kneKndashrs




ckndashrs gt wT

rs (qTrs)


(qmrs)n+1 = (qT



nєMTrs

rs


Zone0 3

Kilometers6 9

(a) (b)

Figure 6 Continued




RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References








































































RG 1113944kisinKrs

frsk (n) middot c

rsk (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875 + q

rsT (n) w

rsT (n) minus min c

rs

krs(n + 1) w

rsT (n + 1)1113882 11138831113874 1113875

frsk (n) middot c

rsk (n) + w

rsT (n) middot q

rsT (n)

(23)




(c)






rs


= qArsqndashrs

rArr

NL

τurs




ΣmєMu

rs


rs))rArr

qmrs

qmrsΣ

mєMurs


rs))ΣmєMu

rs

=








1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References











































































1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 500 1000 1500 2000 2500 3000 3500 4000

Relat

ive g

ap (l

og10

)

CPT time (sec)

IGP

OGP

MSA

Evan


1E ndash 8

1E ndash 7

1E ndash 6

1E ndash 5

1E ndash 4

1E ndash 3

1E ndash 2

1E ndash 1

1E + 0

0 1000 2000 3000 4000 5000

Rela

tive

gap

(log1

0)

CPT time (sec)

γ = 01

γ = 05

γ = 08

(a)

0

20

40

60

80

100

120

1 11 21 31 41 51 61

of

inne

r ite

ratio

ns

Outer iteration

γ = 01

γ = 05

γ = 08

(b)



633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References






































































633

θrs = 01

τTrs = 0067

θrsτTrs = 15

θrs = 01

τTrs = 005

θrsτTrs = 20

θrs = 0133

τTrs = 0067

θrsτTrs = 20

633 399

1551 1553 1517

598 595 554

7218 7219 7530

000

2000

4000

6000

8000

10000

Bus

Metro

Bus-metro

Auto

()





ABS FLOW DIFF

1500 750 375

(a)




ABS FLOW DIFF

1500 750 375

(b)




ABS FLOW DIFF

1500 750 375

(c)




ABS FLOW DIFF

1500 750 375

(d)















Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References


















































































Appendix


rsm( 1113857( 1113857


T( 1113857 minusgcrsm( 1113857( 1113857

q

rsm

1113936misinMrsT

qrsm

(A1)


11113936misinMrs

Texp θrsτrs

T( 1113857 minusgcrsm( 1113857( 1113857

exp lnq

rsm

1113936misinMrsT

qrsm


τrsT

gcrsm1113888 1113889


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ +θrs

τrsT

gcrsm

(A2)


wrsm q

rsm( 1113857

1θrs ln

1113936misinMrsT

qrsm

qrs


qrsm

⎛⎝ ⎞⎠ +τrs

T

θrs lnq

rsm

1113936misinMrsT

qrsm

⎛⎝ ⎞⎠ + gcrsm⟹

wrsT q

rsT( 1113857

1θrs ln

qrsT

qrs

minus qrsT

1113888 1113889 minusτrs

T


T

expθrs

τrsT

minusgcrsm( 11138571113888 1113889⎛⎝ ⎞⎠

(A3)


Data Availability






Acknowledgments


References






































































Data Availability






Acknowledgments


References






































































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