A Link-Node Nonlinear Complementarity Model for a Multiclass Simultaneous Transportation

Dynamic User Equilibria

Mohamad K. Hasan

Associate Professor Department of Quantitative Methods and Information Systems

College of Business Administration, Kuwait University P.O. Box 5486, Safat, 13055, Kuwait

Fax: (965) 2483-9406 Tel.: Work: + (965)2498-8453, Mobile: + (965)9782-2073

Email: mkamal@cba.edu.kw

And

Xuegang (Jeff) Ban Assistant Professor

Department of Civil and Environmental Engineering Rensselaer Polytechnic Institute

Jonsson Engineering Center, room: 4034 110 8th Street, Troy, New York 12180 Fax: 518-276-4833, Tel.: 518-276-8043

Email: banx@rpi.edu

Accepted for Publication in International Journal of Operations Research and Information Systems (IJORIS), IGI Global, USA, May 2011

Presentation Outlines

1. Introduction 2. The Nonlinear Complementarity Formulation for DUE 3. Multiclass Simultaneous Transportation Equilibrium Model

(MSTEM) 4. Discrete-Time Domain Finite-Dimensional Nonlinear

Complementarity DUE Multiclass Simultaneous Transportation Equilibrium Model (DMSTEM)

5. Solution Algorithm 6. Conclusions and Future Research

Transportation Network Equilibrium Models

Deterministic ModelsDynamic Models Stochastic Models

Sequential ModelsSimultaneous

Models

Single ClassMulticlass Variational Inequality

1- Trip Generation2- Trip Distribution3- Modal Split4- Traffic assignment

Equivalent Convex program

Variational Inequality

Distribution , Modal Split and Assignment (De Cea et. al 2003)ESTRAUS

Generation, Distribution, Modal split, Assignment, and Departure Time (Hasan and Dashti 2007)

MSTEM Distribution and

Assignment (Evans 1976)

Distribution , Modal Split and Assignment ( Florian and Nguyen 1978)

Generation, Distribution, Modal split, and Assignment (Safwat and Magnanti 1988)STEM

PGSTEM, GSTEM (Hasan 1991)

Traffic Assignment

(Jeff and et. al. 2008)

Traffic Assignment and Departure

Time (Jeff and et. al. 2008)

DSTEM (Hasan and Jeff )

MUE LU-MUELU-MUEMSTEM

(Hasan)Work in progress

Traffic (Passengers ) ModelsFreight Models (Hasan UN-ESCWA)

1. Introduction

Dynamic Traffic Assignment (DTA) models, which aim to predict future dynamic traffic states in a short-term fashion, have been extensively studied for decades. DTA studies accelerated in the last fifteen years with the advent of intelligent transportation systems (ITS).

Within DTA studies, the dynamic user equilibrium (DUE) problem, which tries to determine the distribution of time-dependent traffic flow of a traffic network assuming travelers follow rational route choices (such as to minimize travel time or other forms of cost), is one of the most challenging.

Two distinct approaches have dominated the methodologies applied to DUE research:

The simulation-based (microscopic/mesoscopic) approach.

The analytical (macroscopic) approach. Variational Inequality (VI) and Nonlinear Complementarity (NC) have been shown to be the most effective approaches.

The simulation-based (microscopic/mesoscopic) approach.

Multiclass DUE models have also been studied with these analytical approaches (Ran and Boyce, 1996; Belimer et al., 2003) or simulation-based approaches (Ben-Akiva et al, 2001; Mahmassani et al., 2001). The early studies, however, focused on different vehicle classes and a few social-economic factors of drivers, such as value of times (VOD) preferences (Lu et al., 2008). With this goal, Ramadurai and Ukkusuri (2010) integrate activity-based models for demand analysis into DUE. The model is solved using a super-network representation of activities and traffic networks.

By investigating further how socio-economic factors of drivers may affect trip generation, trip distribution, mode choice, and departure-times in a dynamic (i.e. time-dependent) fashion, and integrating the findings into DUE modeling, we expect to improve the practicality and accuracy of DUE models.

With this objective, we combine the NCP-based DUE model in Ban et al. (2008) and the Multiclass Simultaneous Transportation Equilibrium Model (MSTEM) in Hasan and Dashti (2007) to develop a multiclass combined Trip Generation (TG)-Trip Distribution (TD)-Nested Modal Split (NMS)-Modal Split (MS)-Dynamic Trip Assignment (DTA)-Departure Time (DT) Model (NCPDUE-TG-TD-NMS-MS-DTA-DT).

2. The Nonlinear Complementarity Formulation for DUE Ban et al. (2008) evenly divided the entire study period into 'K time intervals by introducing the length of each time interval such that '' TK . We use index k to denote the k-th time interval ),)1[( kk for '1 Kk . The notation for the discrete-time model is first listed as

follows (Ban et al., 2008):

ksid

ddk

ksikdd

xxk

ksaxx

vvk

kavv

uuk

kauu

xxk

ksakxx

ksav

ksau

vvk

ksakvv

uuk

ksakuu

kis

siksikis

iskis

kaka

A

Ss

kas

ka

kaka

A

Ss

kas

ka

kaka

A

Ss

kas

ks

ksakas

askas

kas

kas

ksakas

askas

ksakas

askas

interval timeduring n destinatio to node from generated demands the:

)( , interval entire theduringconstant be toassumed isich wh

, interval timeof beginning at the n destinatio to node from generated rate demand the:))1((

)( , interval entire theduringconstant be toassumed isich wh

, interval timeof beginning at the n destinatio towardslink of flows aggregated the:

)( , interval entire theduringconstant be toassumed isich wh

, interval timeof beginning at the link from rate flowexit aggregated the:

)( , interval entire theduringconstant be toassumed ishich w

, interval timeof beginning at the link torate inflow aggregated the:

)( , interval entire theduringconstant be toassumed ishich w

, interval timeof beginning at the n destinatio towardslink of flows the:))1((

interval timeduring n destinatio towardslink from flowsexit the:

interval timeduring n destinatio towardslink toinflows the:

)( , interval timeentire theduringconstant be toassumed also isch whi

, interval timeof beginning at the n destinatio towardslink from rate flowexit the:))1((

)( , interval timeentire theduringconstant be toassumed also isich wh

, interval timeof beginning at the n destinatio towardslink torate inflow the:))1((

,,,

,

,

,

,,

,,

,,

kaka

eeu

kauka

Ckuka

e

siksikis

ksikis

kis

kaka

CCukaka

Cuka

C

,)( , offunction a

, interval timeof beginning at the entering esfor vehicl exit time the:)(1)1()(

,,,)( , interval timeof end at the n destinatio to node from time travelminimum the:)(

,)( , offunction a , interval timeof end at the link of time travelthe:)()(

Ban et al. (2008) showed that the above discretization scheme leads to the following NCP formulation for DUE with fixed demand: find ),( u such that

0),(0 uu u

0),(0 u

where the two functions ),( uu and ),( u are defined as:

,

,,/)(11

1)](,,31[)(,,3)(),(

ksa

ks

al

luka

el

ls

ah

ulka

ls

ah

ulka

uka

Cuu

(1)

ksisiiBa uk

aekuk

aek

kas

uukka

uka

kas

uukka

uka

kis

diAa

kas

uu

,,,)( )(1')1()(':'

]1'))(,',21()(1',1')(,',2)(',1[ )(

),(

(2)

𝑤ℎ𝑒𝑟𝑒 𝑒𝑎𝑘ሺ𝒖ሻ= ሺ𝑘− 1ሻ∆+ 𝐶𝑎(𝑘−1)ሺ𝒖ሻ is the exit time of vehicles entering a at the

beginning of the k-th interval, which is a function of the inflow vector 𝒖 since 𝐶𝑎(𝑘−1) is so.

He also define three types of indicator functions on 𝒖: 𝜆𝑎1,𝑘ሖሺ𝒖ሻ , 𝜆𝑎2,𝑘,𝑘ሖሺ𝒖ሻ and 𝜆𝑎3,𝑘,𝑞ሺ𝒖ሻ. First, flow propagation constraints can be represented by 𝜆𝑎1,𝑘ሖሺ𝒖ሻ, which is defined as: 𝜆𝑎1,𝑘ሖሺ𝒖ሻ= ∆𝐶𝑎𝑘ሖሺ𝒖ሻ−𝐶𝑎൫𝑘ሖ−1൯

ሺ𝒖ሻ+∆,∀𝑎,𝑘ሖ (3)

The relation between link inflow and exit flow rates can be represented by both 𝜆𝑎1,𝑘ሖሺ𝒖ሻ and 𝜆𝑎2,𝑘,𝑘ሖሺ𝒖ሻ with the latter defined as: 𝜆𝑎2,𝑘ሖሺ𝒖ሻ= 𝐶𝑎𝑘ሖሺ𝒖ሻ+൫𝑘ሖ+1−𝑘൯∆𝐶𝑎𝑘ሖሺ𝒖ሻ−𝐶𝑎൫𝑘ሖ−1൯

ሺ𝒖ሻ+∆ ∀𝑎,𝑘,𝑘ሖ, 𝑒𝑎𝑘ሖሺ𝒖ሻ≤ ሺ𝑘− 1ሻΔ < 𝑒𝑎൫𝑘ሖ+1൯ሺ𝒖ሻ (4)

Similarly, 𝜆𝑎3,𝑞ሺ𝒖ሻ is used to discretize (and interpolate) the minimum travel time 𝜋ℎ𝑎𝑠ሾ𝑡+ 𝐶𝑎ሺ𝑡ሻሿ in equation (1). It can be defined as: 𝜆𝑎3,𝑞ሺ𝒖ሻ= 𝑞− 𝑘− 𝐶𝑎𝑘ሺ𝒖ሻ∆ ,∀𝑞− 1 ≤ 𝑒𝑎ሺ𝑘+1ሻሺ𝒖ሻ∆ < 𝑞 (5)

The three indicator functions satisfy the following:

𝜆𝑎1,𝑘ሖሺ𝒖ሻ> 0,∀𝑎,𝑘ሖ 0 < 𝜆𝑎2,𝑘,𝑘ሖሺ𝒖ሻ≤ 1, ∀𝑎,𝑘,𝑘ሖ,𝑒𝑎𝑘ሖሺ𝒖ሻ≤ ሺ𝑘− 1ሻΔ < 𝑒𝑎൫𝑘ሖ+1൯

ሺ𝒖ሻ 0 < 𝜆𝑎3,𝑘,𝑞ሺ𝒖ሻ≤ 1,∀ 𝑞− 1 ≤ 𝑒𝑎ሺ𝑘+1ሻሺ𝒖ሻ∆ < 𝑞

Φ𝒖 is a vector function for any combination of link a, destination s, and time interval k. Each of its component represents, for the given a; s; k, the difference of the minimum travel times via two sets of paths from the starting node (or tail node) of link a to destination s at k. The first set of paths must traverse link a, which is a subset of the second set that includes all paths from the starting node of a to destination s. This difference should be zero if link a is selected (i.e. the inflow to link a at time k,𝑢𝑎𝑠𝑘 , is nonzero). Φ𝝅 , on the other hand, simply represents the flow conservation constraint at any node i, destination 𝑠≠ 𝑖 and time k. Detailed discussion on how Φ𝒖 and Φ𝝅 are derived can be found in Ban et al. (2008), which also showed (Theorem 3) that NCPDUE has a nonempty and compact solution set under certain conditions.

3. Multiclass Simultaneous Transportation Equilibrium Model (MSTEM)

A brief description of MSTEM model that developed by Hasan and Dashti (2007). The notation for MSTEM model is first listed as follows:

),( AN = A multimodal traffic network consisting of a set of N nodes and a set of A links

l = User class (e.g., income level, car availability, etc.) L = Set of all user classes o = Trip purpose (e.g., home-based-work, home-based-shopping, etc.) O = Set of all trip purpose

loI = Set of origin nodes for user class l and trip purpose o i = An origin node in the set loI for user class l with trip purpose o

loiD = Set of destination nodes that are accessible from a given origin i for user

class l with trip purpose o s = A destination node in the set lo

iD for user class l with trip purpose o loR = Set of origin-destination pairs is for user class l with trip purpose o , i.e., the

set of all origins loIi and destinations loiDs

m = Any transportation mode in the urban area n = Nest of transportation modes m that has a specific characteristics (e.g., pure modes including private and public or combined modes) that are available for user class l with trip purpose o travel between origin-destination pairs is

lois = Set of all nests of modes n that are available for user class l with trip purpose o

travel between origin-destination pairs is

lonM = Set of all transportation modes m in the nest n for user class l with trip purpose o

travel between origin-destination pairs is k = Departure time period for user class l with trip purpose o using mode m in the nest n to travel between origin-destination pairs is

lomK = Time horizon of the departure time periods k for users of class l with trip purpose

o using mode m between origin-destination pairs is a = A link in the set A in the multimodal network ),( AN 𝑪𝒂𝒍𝒐𝒏𝒎𝒌൫𝒖𝒍𝒐𝒏𝒎൯ = the travel time of vehicles entering link a at the end of the 𝒌th time interval

(=Departure time period k ) for user class l with trip purpose o using mode m in the nest n to travel between origin-destination pairs is

lowsA = the value of the thw socio-economic variable that influences trip attraction

at destination s for users of class l with trip purpose o

)(

1

lows

low

W

w

loiw

los AgA

=a composite measure of the effect that socio-economic variables, which,

are exogenous to the transport system, have on trip attraction at destination s for users of class l with trip purpose o .

)( lows

low Ag = a given function specifying how the thw socio-economic variable lo

wjA

influences trip attraction at destination s for users of class l with trip purpose

o , and

The quantities loi and lo

iw for Ww ..., ,2 ,1 are coefficients to be estimated, where 0loi .

During the time period required to achieve the short-run equilibrium predicted in the model, socio-economic activities in the system will remain essentially unchanged; the composite effect

lojA for users of class l with trip purpose o of these activities is assumed to be a fixed constant. lo

iE = the value of the th socio-economic variable that influences the number of

trips generated from origin i for users of class l with trip purpose o , lo

iEq = a given function specifying how the th socio-economic variable, loiE ,

influences the number of trips generated from origin i for users of class l with trip purpose o , and

)(1

loi

loloi EqE

= a composite measure of the effect the socio-economic variables,

which are exogenous to the transport system, have on the number of trips generated from origin i for users of class l with trip purpose o

The quantities ..., ,2 ,1for and lo

lo are coefficients to be estimate OoLl , .

Similar to losA , lo

iE is assumed to be a fixed constant during the time period required to

achieve short-run equilibrium. 𝑢𝑎𝑠𝑙𝑜𝑛𝑚𝑘 = the inflow rate to link a towards destination s at the beginning of the 𝑘-th time

interval for user class l with trip purpose o using mode m in the nest n ,which

is assume to be constant during the entire 𝑘th time interval

𝜋𝑖𝑠𝑙𝑜𝑛𝑚𝑘 = the minimum travel time from node i to s destination at the end of time

interval k for user class l with trip purpose o using mode m in the nest n

loiS = the accessibility of origin loIi as perceived from user of class l with trip purpose

o traveling from that origin.

loiG = the number of trips generated from origin i for users of class l with trip purpose o .

lonmkisd = the number of trips of users of class l with trip purpose o traveling from the origin

node loIi to the destination node loiDs and whose already chose the mode of

transport lonMm from the nest of modes lo

isn and start their trip at the time

interval lomKk .

lonmisd = the number of trips of users of class l with trip purpose o traveling from the origin

node loIi to the destination node loiDj and whose already chose the mode of

transport lonMm from the nest of modes lo

isn

lonisd = the number of trips of users of class l with trip purpose o traveling from the origin

node loIi to the destination node loiDs and whose already chose the nest of

modes loisn .

loisd = the number of trips of users of class l with trip purpose o traveling from the origin

node loIi to the destination node loiDs .

MSTEM:

OoLlIi )Aθ(S lolos

lonmkis

Kk

loi

MmnDs

loi

lom

lon

lois

loi

,, expln,0 max

,, OoLlIiESG loloi

loi

loloi

OoLlRis )Aθ(

)Aθ(

Gd lo

los

lonmkis

Kk

loi

MmnDs

los

lonmkis

Kk

loi

Mmnloi

lois

lom

lon

lois

loi

lom

lon

lois

,,

exp

exp

OoLlRisn )Aθ(

)Aθ(

dd loloislo

slonmkis

Kk

loi

Mmn

los

lonmkis

Kk

loi

Mmlois

lonis

lom

lon

lois

lom

lon

,,, ,

exp

exp

OoLlRisnMm )Aθ(

)Aθ(

dd lolois

lonlo

slonmkis

Kk

loi

Mm

los

lonmkis

Kk

loi

lonis

lonmis

lom

lon

lom

,,,, ,

exp

exp

OoLlRisnMmKk )Aθ(

Add lolo

islon

lomlo

slonmkis

Kk

loi

los

lonmkis

loilonm

islonmkis

lom

,,,,,, exp

)exp(

lonmkisd can also be given by the following combined trip generation, trip distribution, modal split,

and departure time demand model (function):

OoLlRisnMmKk

)Aθ(

AESd

lolois

lon

lom

los

lonmkis

Kt

loi

MmnDj

los

lonmkis

loilo

iloi

lolonmkis

lom

lon

lois

loi

,,,,,

exp

)exp()(

4. Discrete-Time Domain Finite-Dimensional Nonlinear Complementarity DUE Multiclass Simultaneous Transportation Equilibrium Model (DMSTEM)

Following the same thought of Ban et al. (2008), the dynamic user equilibrium traffic assignment (NCPDUE-DTA) can be represented by the following two NCPs:

0 ≤ 𝒖𝑙𝑜𝑛𝑚 ⊥Φ 𝒖𝑙𝑜𝑛𝑚 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜,𝑛,𝑚 (6)

0 ≤ 𝜋𝑙𝑜𝑛𝑚 ⊥Φ 𝜋𝑙𝑜𝑛𝑚 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜,𝑛,𝑚 (7)

To extend the DTA model above to model simultaneous route and departure time, we assume the early/late arrival penalty function for each

user class l with trip purpose o using mode m in the nest n to travel between origin-destination pairs is departing at time 𝑘 (denoted as 𝐹𝑖𝑠𝑙𝑜𝑛𝑚 ) is:

𝐹𝑖𝑠𝑙𝑜𝑛𝑚𝑘 = 0.5൫𝑘∇+ 𝜋𝑖𝑠𝑙𝑜𝑛𝑚𝑘 − 𝑇𝑖𝑠𝑙𝑜𝑛𝑚𝑘൯2,∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠 𝑎𝑛𝑑 ∀ 𝑙,𝑜,𝑛,𝑚

where 𝑇𝑖𝑠𝑙𝑜𝑛𝑚𝑘 ,∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠 is the desired arrival time for user class l with trip purpose o using mode m in the nest n to travel between

origin-destination pairs is departing at time 𝑘. Therefore, ൫𝑘∇+ 𝜋𝑖𝑠𝑙𝑜𝑛𝑚𝑘 − 𝑇𝑖𝑠𝑙𝑜𝑛𝑚𝑘൯ represents early or late arrival for travelers of class l

with trip purpose o using mode m in the nest n departing 𝑖 to 𝑠 at time 𝑘. In other words, the cost (disutility) for such a traveler at time 𝑘 is

Ψ 𝑖𝑠𝑙𝑜𝑛𝑚𝑘 = 𝜋𝑖𝑠𝑙𝑜𝑛𝑚𝑘 + 𝐹𝑖𝑠𝑙𝑜𝑛𝑚𝑘 = 𝜋𝑖𝑠𝑙𝑜𝑛𝑚𝑘 0.5(𝑘Δ + 𝜋𝑖𝑠𝑙𝑜𝑛𝑚𝑘 − 𝑇𝑖𝑠𝑙𝑜𝑛𝑚𝑘 )2 ,∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠,∀𝑙,𝑜,𝑛,𝑚 Further, we denote the minimum disutility between 𝑖 and 𝑠 for user class l with trip purpose o using mode m in the nest n (for all time intervals) as 𝜇𝑖𝑠𝑙𝑜𝑛𝑚 = min𝑘 Ψ 𝑖𝑠𝑙𝑜𝑛𝑚𝑘 ,∀𝑖,𝑠,𝑖 ≠ 𝑠,∀𝑙,𝑜,𝑛,𝑚 Therefore, the departure time choice condition can then be expressed as: 𝑑𝑖𝑠𝑙𝑜𝑛𝑚𝑘 ≥ 0,𝑖𝑓 Ψ 𝑖𝑠𝑙𝑜𝑛𝑚𝑘 = 𝜇𝑖𝑠𝑙𝑜𝑛𝑚 ,∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠,∀𝑙,𝑜,𝑛,𝑚 𝑑𝑖𝑠𝑙𝑜𝑛𝑚𝑘 = 0,𝑖𝑓 Ψ 𝑖𝑠𝑙𝑜𝑛𝑚𝑘 > 𝜇𝑖𝑠𝑙𝑜𝑛𝑚 ,∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠,∀𝑙,𝑜,𝑛,𝑚 This condition simply states that travelers are trying to minimize their disutility when choosing departure times.

Combining equations (6) and (7) with the following two NCPs we can combined the dynamic traffic assignment with the departure time ((NCPDUE-DTA-DT) :

0 ≤ 𝒅𝑙𝑜𝑛𝑚 ⊥Φ 𝒅𝑙𝑜𝑛𝑚 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜,𝑛,𝑚 (8)

0 ≤ 𝝁𝑙𝑜𝑛𝑚 ⊥Φ 𝝁𝑙𝑜𝑛𝑚 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜,𝑛,𝑚 (9)

Similarly, combining equations (6)-(9) with the following two NCPs we can combined the Dynamic Traffic Assignment, Departure Time and Modal Split ((NCPDUE-MS-DTA-DT) :

0 ≤ 𝒅𝑙𝑜𝑛 ⊥Φ 𝒅𝑙𝑜𝑛 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜,𝑛 (10)

0 ≤ 𝝁𝑙𝑜𝑛 ⊥Φ 𝝁𝑙𝑜𝑛 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜,𝑛 (11)

Combining equations (6)-(11) with the following two NCPs we can combined the Dynamic Traffic Assignment, Departure Time, Modal Split, and Nested Modal Split (NCPDUE-NMS-MS-DTA-DT):

0 ≤ 𝒅𝑙𝑜 ⊥Φ 𝒅𝑙𝑜(𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜 (12)

0 ≤ 𝝁𝑙𝑜 ⊥Φ 𝝁𝑙𝑜(𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜 (13)

Combining equations (6)-(13) with the following two NCPs we can combined the Dynamic Traffic Assignment, Departure Time, Modal Split, Nested Modal Split, and Trip Distribution (NCPDUE-TD-NMS-MS-DTA-DT):

0 ≤ 𝒅𝑅𝑙𝑜𝑙𝑜 ⊥Φ 𝒅𝑅𝑙𝑜𝑙𝑜 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜 (14)

0 ≤ 𝝁𝐼𝑙𝑜 ⊥Φ 𝝁𝐼𝑙𝑜(𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜 (15)

Combining equations (6)-(15) with the following two NCPs we can combined the Dynamic Traffic Assignment, Departure Time, Modal Split, Nested Modal Split, Trip Distribution, and Trip Generation (NCPDUE-TG-TD-NMS-MS-DTA-DT):

0 ≤ 𝑺𝐼𝑙𝑜𝑙𝑜 ⊥Φ 𝑺𝐼𝑙𝑜𝑙𝑜 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜 (16)

0 ≤ 𝝁⊥Φ 𝝁(𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) ≥ 0,∀𝑙,𝑜 (17)

where the decision variables are defined as follows:

𝒖𝑙𝑜𝑛𝑚 = ሺ𝑢𝑎𝑠𝑙𝑜𝑛𝑚𝑘 ,∀𝑎,𝑠,𝑘ሻ,∀𝑙,𝑜,𝑛,𝑚 , 𝝅𝑙𝑜𝑛𝑚 =൫𝜋𝑖𝑠𝑙𝑜𝑛𝑚𝑘 ,∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠൯,∀𝑙,𝑜,𝑛,𝑚

𝒅𝑙𝑜𝑛𝑚 = ൫𝑑𝑖𝑠𝑙𝑜𝑛𝑚𝑘 ,∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠൯,∀𝑙,𝑜,𝑛,𝑚 , 𝝁𝑙𝑜𝑛𝑚 =൫𝜇𝑖𝑠𝑙𝑜𝑛𝑚 ,∀𝑖,𝑠,𝑖 ≠ 𝑠൯,∀𝑙,𝑜,𝑛,𝑚

𝒅𝑙𝑜𝑛 = ൫𝑑𝑖𝑠𝑙𝑜𝑛𝑚 ,∀𝑖,𝑠,𝑚,𝑖 ≠ 𝑠൯,∀𝑙,𝑜,𝑛, 𝝁𝑙𝑜𝑛 =൫𝜇𝑖𝑠𝑙𝑜𝑛 ,∀𝑖,𝑠,𝑖 ≠ 𝑠൯,∀𝑙,𝑜,𝑛

𝒅𝑙𝑜 =൫𝑑𝑖𝑠𝑙𝑜𝑛 ,∀𝑖,𝑠,𝑛,𝑖 ≠ 𝑠൯, ∀𝑙,𝑜 , 𝝁𝑙𝑜 =൫𝜇𝑖𝑠𝑙𝑜,∀𝑖,𝑠,𝑖 ≠ 𝑠൯,∀𝑙,𝑜

𝒅𝑅𝑙𝑜𝑙𝑜 = (𝑑𝑖𝑠𝑙𝑜,∀𝑖,𝑠,𝑖 ≠ 𝑠), ∀𝑙,𝑜, 𝝁𝐼𝑙𝑜𝑙𝑜 = (𝜇𝑖𝑙𝑜,∀𝑖), ∀𝑙,𝑜

𝑺𝐼𝑙𝑜𝑙𝑜 = (𝑆𝑖𝑙𝑜,∀𝑖), ∀𝑙,𝑜, 𝝁= (𝜇𝐼𝑙𝑜𝑙𝑜 ,∀𝑙,𝑜)

𝜇𝑖𝑠𝑙𝑜𝑛 = min𝑚 μ 𝑖𝑠𝑙𝑜𝑛𝑚 ,∀𝑖,𝑠,𝑖 ≠ 𝑠,∀𝑙,𝑜,𝑛, 𝜇𝑖𝑠𝑙𝑜 = min𝑛 μ 𝑖𝑠𝑙𝑜𝑛 ,∀𝑖,𝑠,𝑖 ≠ 𝑠,∀𝑙,𝑜

𝜇𝑖𝑙𝑜 = min𝑠∈𝐷𝑖𝑙𝑜 μ 𝑖𝑠𝑙𝑜 ,∀𝑖,𝑖 ≠ 𝑠,∀𝑙,𝑜, 𝜇𝐼𝑙𝑜𝑙𝑜 = min𝑖∈𝐼𝑙𝑜 μ 𝑖𝑙𝑜 ,∀𝑙,𝑜 .

Similarly to the formulation of Ban et al. (2008), all Φ s functions can be defined as follows:

Φ 𝒖𝑙𝑜𝑛𝑚 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

(𝐶𝑎𝑙𝑜𝑛𝑚𝑘ሺ𝒖𝑙𝑜𝑛𝑚ሻ+ {𝜆𝑎3,𝑙𝑜𝑛𝑚𝑘 ,𝑞𝑞−1≤𝑒𝑎𝑙𝑜𝑛𝑚 ሺ𝑘+1ሻ൫𝑢𝑙𝑜𝑛𝑚 ൯∆ <𝑞

ሺ𝒖𝑙𝑜𝑛𝑚ሻ𝜋ℎ𝑎𝑠𝑙𝑜𝑛𝑚𝑞

+[1− 𝜆𝑎3,𝑙𝑜𝑛𝑚𝑘 ,𝑞ሺ𝒖𝑙𝑜𝑛𝑚ሻ]𝜋ℎ𝑎𝑠𝑙𝑜𝑛𝑚 ,𝑞+1} − 𝜋ℎ𝑎𝑠𝑙𝑜𝑛𝑚𝑘 ),∀𝑎,𝑠,𝑘),∀𝑙,𝑜,𝑛,𝑚 (18)

Φ 𝝅𝑙𝑜𝑛𝑚 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

( 𝑢𝑎𝑠𝑙𝑜𝑛𝑚𝑘𝑎𝜖𝐴ሺ𝑖ሻ – 𝑑𝑖𝑠𝑙𝑜𝑛𝑚𝑘

− {𝑎𝜖𝐵ሺ𝑖ሻ [𝜆𝑎1,𝑙𝑜𝑛𝑚𝑘ሖሺ𝒖𝑙𝑜𝑛𝑚ሻ

𝑘:ሖ𝑒𝑎𝑙𝑜𝑛𝑚 𝑘ሖ൫𝒖𝑙𝑜𝑛𝑚 ൯≤ሺ𝑘−1ሻΔ <𝑒𝑎𝑙𝑜𝑛𝑚 ൫𝑘ሖ+1൯൫𝒖𝑙𝑜𝑛𝑚 ൯

𝜆𝑎2,𝑙𝑜𝑛𝑚𝑘 ,𝑘ሖሺ𝒖𝑙𝑜𝑛𝑚ሻ𝑢𝑎𝑠𝑙𝑜𝑛𝑚𝑘ሖ

+ 𝜆𝑎1,𝑙𝑜𝑛𝑚൫𝑘ሖ+1൯ሺ𝒖𝑙𝑜𝑛𝑚ሻ൬1− 𝜆𝑎2,𝑙𝑜𝑛𝑚𝑘 ,𝑘ሖሺ𝒖𝑙𝑜𝑛𝑚ሻ൰𝑢𝑎𝑠𝑙𝑜𝑛𝑚൫𝑘ሖ+1൯]},∀𝑎,𝑠,𝑘),∀𝑙,𝑜,𝑛,𝑚 (19)

where

Φ 𝒅𝑙𝑜𝑛𝑚 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

൫ൣ�Ψ 𝑖𝑠𝑙𝑜𝑛𝑚𝑘 − 𝜇𝑖𝑠𝑙𝑜𝑛𝑚 ൧,∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠൯ ∀𝑙,𝑜,𝑛,𝑚 (20)

Φ 𝝁𝑙𝑜𝑛𝑚 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

ۉ

ۇ

ۏێێ𝑑𝑖𝑠𝑙𝑜𝑛𝑚𝑘ۍ − 0

exp

)exp()(

)Aθ(

AES

los

lonmkis

Kk

loi

MmnDs

los

lonmkis

loilo

iloi

lo

lom

lon

lois

loi

ےۑۑ𝑖,𝑠,𝑖∀,ې ≠ 𝑠

ی

𝑙,𝑜,𝑛,𝑚,𝑘∀ ۊ

(21)

Equations (21) ensure that if the demand 𝑑𝑖𝑠𝑙𝑜𝑛𝑚𝑘 and a period time ( departure time ) 𝑘 is positive it should be given by the MSTEM combined trip generation, trip distribution, modal split, and the departure time demand model (function) mentioned before.

Following the same concept we can define all others Φ 𝒅 ‘s and Φ 𝝁 ‘s functions where all Φ 𝝁 ‘s given by

MSTEM model equations.

Φ 𝒅𝑙𝑜𝑛 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

൫ൣ�μ 𝑖𝑠𝑙𝑜𝑛𝑚 − 𝜇𝑖𝑠𝑙𝑜𝑛 ൧,∀𝑖,𝑠,𝑚,𝑖 ≠ 𝑠൯ ∀𝑙,𝑜,𝑛 (22)

Φ 𝝁𝑙𝑜𝑛 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

ۉ

ۇ

ۏێێۍ

0exp

exp

)Aθ(

)Aθ(

ddlos

lonmkis

Kk

loi

Mm

los

lonmkis

Kk

loi

lonis

lonmis

lom

lon

lom

ےۑۑ𝑙,𝑜,𝑛,𝑚 ,∀𝑖,𝑠,𝑖∀ ې ≠ 𝑠

ی

(23) ۊ

Φ 𝒅𝑙𝑜(𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

൫ൣ�μ 𝑖𝑠𝑙𝑜𝑛 − 𝜇𝑖𝑠𝑙𝑜 ൧,∀𝑖,𝑠,𝑛,𝑖 ≠ 𝑠൯ ∀𝑙,𝑜 (24)

Φ 𝝁𝑙𝑜൫𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁൯=

ۉ

ۈۈۇ

ۏێێۍ

OoLlRisn )Aθ(

)Aθ(

dd loloislo

slonmkis

Kk

loi

Mmn

los

lonmkis

Kk

loi

Mmlois

lonis

lom

lon

lois

lom

lon

,,, , 0

exp

exp

ےۑۑې

,∀𝑖,𝑠,𝑖 ≠ 𝑠 (25) ی

ۋۋ ۊ

Φ 𝒅𝑅𝑙𝑜𝑙𝑜 ൫𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁൯=

൫ൣ�μ 𝑖𝑠𝑙𝑜 − 𝜇𝑖𝑙𝑜 ൧,∀𝑖,𝑠,𝑖 ≠ 𝑠൯ ∀𝑙,𝑜 (26)

Φ 𝝁𝐼𝑙𝑜(𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

ۉ

ۈۇ

ۏێێۍ

OoLlRis )Aθ(

)Aθ(

ESd lolos

lonmkis

Kk

loi

MmnDs

los

lonmkis

Kk

loi

Mmnloi

loi

lolois

lom

lon

lois

loi

lom

lon

lois

,, 0

exp

exp

)(

ےۑۑې

,∀𝑖,𝑖 ≠ 𝑠 (27) ی

ۋ ۊ

Φ 𝑺𝐼𝑙𝑜𝑙𝑜 (𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁) =

൫ൣ�𝜇𝑖𝑙𝑜 − 𝜇𝐼𝑙𝑜𝑙𝑜 ൧,∀𝑖൯ ∀𝑙,𝑜 (28)

Φ 𝝁൫𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁൯=

(29) ,, 0]expln,0 max([ OoLlIi )Aθ(S lolos

lonmkis

Kk

loi

MmnDs

loi

lom

lon

lois

loi

Theorem: The NCPDUE-TG-TD-NMS-MS-DTA-DT model (6) - (17) has a nonempty and compact solution set if the following four conditions are satisfied.

(a) The link travel time is positive and finite for any finite 𝒖;

(b) 𝜆1 is bounded from above;

(c) 𝑇𝑖𝑠𝑙𝑜𝑛𝑚𝑘 ∀𝑖,𝑠,𝑘,𝑖 ≠ 𝑠 𝑎𝑛𝑑 ∀ 𝑙,𝑜,𝑛,𝑚 is positive and bounded from above; and

(d) 𝛷𝒖𝑙𝑜𝑛𝑚 ,𝛷𝜋𝑙𝑜𝑛𝑚 ,𝛷𝒅𝑙𝑜𝑛𝑚 ,𝛷𝝁𝑙𝑜𝑛𝑚 ,𝛷𝒅𝑙𝑜𝑛 ,𝛷𝝁𝑙𝑜𝑛 ,𝛷𝒅𝑙𝑜,𝛷𝝁𝑙𝑜,𝛷𝒅𝑅𝑙𝑜𝑙𝑜 ,𝛷𝝁𝐼𝑙𝑜,𝛷𝑺𝐼𝑙𝑜𝑙𝑜 , 𝑎𝑛𝑑 𝛷𝝁 are

continuous with respect to ൫𝒖𝑙𝑜𝑛𝑚 ,𝝅𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛𝑚 ,𝝁𝑙𝑜𝑛𝑚 ,𝒅𝑙𝑜𝑛 ,𝝁𝑙𝑜𝑛 ,𝒅𝑙𝑜,𝝁𝑙𝑜,𝒅𝑅𝑙𝑜𝑙𝑜 ,𝝁𝐼𝑙𝑜, 𝑺𝐼𝑙𝑜𝑙𝑜 ,𝝁൯.

5. Solution Algorithm

we presented a heuristic approach to solve the link-node based simultaneous Trip Generation (TG) - Trip Distribution (TD) - Nested Modal Split (NMS) - Modal Split (MS) - Dynamic Trip Assignment (DTA) - and Departure Time (DT) Model (NCPDUE-TG-TD-NMS-MS-DTA-DT). The solution convergence condition of the algorithm cannot be established mainly due to the fact, as shown in the definitions of functions 𝜱𝒖𝒍𝒐𝒏𝒎,𝜱𝝅𝒍𝒐𝒏𝒎,𝜱𝒅𝒍𝒐𝒏𝒎,𝜱𝝁𝒍𝒐𝒏𝒎,𝜱𝒅𝒍𝒐𝒏,𝜱𝝁𝒍𝒐𝒏,𝜱𝒅𝒍𝒐,𝜱𝝁𝒍𝒐,𝜱𝒅𝑹𝒍𝒐𝒍𝒐 ,𝜱𝝁𝑰𝒍𝒐,𝜱𝑺𝑰𝒍𝒐𝒍𝒐 ,𝒂𝒏𝒅 𝜱𝝁 in Equations (18)-(19), that the model is not close-formed. However, for a

given demands vectors 𝒅ഥ𝒍𝒐𝒏𝒎,𝒅ഥ𝒍𝒐𝒏,𝒅ഥ𝒍𝒐,𝒅ഥ𝑹𝒍𝒐𝒍𝒐 ,𝑺ഥ𝑰𝒍𝒐𝒍𝒐 (called base demand) and its

resulting inflow vector 𝒖ഥ𝒍𝒐𝒏𝒎 (called base inflow), all the indicator functions and the exit times (e) can be calculated and fixed via a so-called dynamic network loading procedure (see Ban et al. 2008). This leads to the so-called relaxed sub-problems that are explained in details in the appendix.

DSTEM Algorithm: Step 1: Initialization. Assign a base demands vectors 𝒅ഥ𝒍𝒐𝒏𝒎 (𝟎),𝒅ഥ𝒍𝒐𝒏 (𝟎),𝒅ഥ𝒍𝒐 (𝟎),𝒅ഥ𝑹𝒍𝒐𝒍𝒐 (𝟎),𝑺ഥ𝑰𝒍𝒐𝒍𝒐 (𝟎)

and a base inflow vector 𝒖ഥ𝒍𝒐𝒏𝒎 (𝟎)

Step 2: Major Iteration. Set 𝝊= 𝒐.

Step 2.1: Construct and solve RNCPDUE-DTA (A1) - (A2). Repeat this 𝝎𝟏 times. Each time using 𝒅ഥ𝒍𝒐𝒏𝒎 (𝝂)as the fixed demand. For the first

time, use 𝒖ഥ𝒍𝒐𝒏𝒎 (𝝂)as the base inflow. For other times, use inflow from

the previous solve as the base inflow. Denote the final inflow (after 𝝎𝟏 the solves) as 𝒖ෝ��𝒍𝒐𝒏𝒎 (𝝂).

Step 2.2: Construct and solve RNCPDUE-DTA-DT (A5) - (A8). Repeat this 𝝎𝟐 times. For the first time, use 𝒖ෝ��𝒍𝒐𝒏𝒎 (𝝂)

as the base inflow and use 𝒅ഥ𝒍𝒐𝒏𝒎 (𝝂) as the initial demand. For other times, use inflow and demand

obtained from the previous solve as the base inflow and initial demand.

Denote the final inflow and demand (after the 𝝎𝟐 solves) as 𝒖 𝒍𝒐𝒏𝒎 ሺ𝝂ሻ and 𝒅෩𝒍𝒐𝒏𝒎 ሺ𝝂ሻrespectively.

Step 2.3: Construct and solve RNCPDUE-DTA-DT-MS (A10) - (A15). Repeat

this 𝝎𝟑 times. For the first time, use 𝒖 𝒍𝒐𝒏𝒎 (𝝂) and 𝒅෩𝒍𝒐𝒏𝒎 (𝝂)

as the

base inflow and base demand respectively and use 𝒅ഥ𝒍𝒐𝒏 (𝝂) as the initial

demand. For other times, use inflow and demands obtained from the previous solve as the base inflow and initial demand. Denote the final

inflow and demands (after the 𝝎𝟑 solves) as 𝒖ሶ𝒍𝒐𝒏𝒎 (𝝂), 𝒅ሶ𝒍𝒐𝒏𝒎 (𝝂) , and 𝒅ሶ𝒍𝒐𝒏 (𝝂)

respectively.

Step 2.4: Construct and solve RNCPDUE-DTA-DT-MS-NMS (A17) - (A24).

Repeat this 𝝎𝟒 times. For the first time, use 𝒖ሶ𝒍𝒐𝒏𝒎 (𝝂), 𝒅ሶ𝒍𝒐𝒏𝒎 (𝝂), and 𝒅ሶ𝒍𝒐𝒏 (𝝂)

as the base inflow and base demands respectively and use 𝒅ഥ𝒍𝒐 (𝝂)

as the initial demand. For other times, use inflow and demand obtained from the previous solve as the base inflow and initial demand. Denote

the final inflow and demands (after the 𝝎𝟒 solves) as 𝒖ሷ𝒍𝒐𝒏𝒎 (𝝂), 𝒅ሷ𝒍𝒐𝒏𝒎 (𝝂) ,𝒅ሷ𝒍𝒐𝒏 (𝝂)

, and 𝒅ሷ𝒍𝒐 (𝝂)respectively.

Step 2.5: Construct and solve RNCPDUE-DTA-DT-MS-NMS-TD (A26) -

(A35). Repeat this 𝝎𝟓 times. For the first time, use 𝒖ሷ𝒍𝒐𝒏𝒎 (𝝂), 𝒅ሷ𝒍𝒐𝒏𝒎 (𝝂) ,𝒅ሷ𝒍𝒐𝒏 (𝝂)

, and 𝒅ሷ𝒍𝒐 (𝝂) as the base inflow and base demands

respectively and use 𝒅ഥ𝑹𝒍𝒐𝒍𝒐 (𝝂) as the initial demand. For other times, use

inflow and demand obtained from the previous solve as the base inflow and initial demand. Denote the final inflow and demands (after the 𝝎𝟓

solves) as 𝒖ഺ𝒍𝒐𝒏𝒎 (𝝂), 𝒅ሸ𝒍𝒐𝒏𝒎 (𝝂) ,𝒅ሸ𝒍𝒐𝒏 (𝝂)

, 𝒅ሸ𝒍𝒐 (𝝂) , and 𝒅ሸ𝑹𝒍𝒐𝒍𝒐 (𝝂)

respectively .

Step 2.6: Construct and solve RNCPDUE-DTA-DT-MS-NMS-TD-TG (A37) - (A48).

Repeat this for 𝝎𝟔 times. For the first time, use 𝒖ഺ𝒍𝒐𝒏𝒎 (𝝂), 𝒅ሸ𝒍𝒐𝒏𝒎 (𝝂),𝒅ሸ𝒍𝒐𝒏 (𝝂)

,𝒅ሸ𝒍𝒐 (𝝂),

and 𝒅ሸ𝑹𝒍𝒐𝒍𝒐 (𝝂) as the base inflow and base demands respectively and use 𝑮ഥ𝑰𝒍𝒐𝒍𝒐 (𝝂)

as the initial demand. For other times, use inflow and demand obtained from the previous solve as the base inflow and initial demand. Denote the final inflow and

demands (after the 𝝎𝟔 solves) as 𝒖 𝒍𝒐𝒏𝒎 (𝝂),𝒅ෝ�𝒍𝒐𝒏𝒎 (𝝂) ,𝒅ෝ�𝒍𝒐𝒏 (𝝂)

, 𝒅ෝ�𝒍𝒐 (𝝂) , 𝒅ෝ�𝑹𝒍𝒐𝒍𝒐 (𝝂)

,

and 𝑺ෝ�𝑰𝒍𝒐𝒍𝒐 (𝝂) respectively . Call them the candidate solution

Step 2.7: Convergence Test. If certain convergence criterion is satisfied at the

candidate solution, go to Step 3; otherwise, go to Step 2.8.

Step 2.8: Update and Move. Set

𝒖ഥ𝒍𝒐𝒏𝒎 (𝝂+𝟏) = 𝒖ഥ𝒍𝒐𝒏𝒎 (𝝂) + 𝝈(𝒖 𝒍𝒐𝒏𝒎 ሺ𝝂ሻ − 𝒖ഥ𝒍𝒐𝒏𝒎 ሺ𝝂ሻ) 𝒅ഥ𝒍𝒐𝒏𝒎 (𝝂+𝟏) = 𝒅ഥ𝒍𝒐𝒏𝒎 (𝝂) + 𝝈(𝒅ෝ�𝒍𝒐𝒏𝒎 ሺ𝝂ሻ− 𝒅ഥ𝒍𝒐𝒏𝒎 ሺ𝝂ሻ) 𝒅ഥ𝒍𝒐𝒏 (𝝂+𝟏) = 𝒅ഥ𝒍𝒐𝒏 (𝝂) + 𝝈(𝒅ෝ�𝒍𝒐𝒏 ሺ𝝂ሻ − 𝒅ഥ𝒍𝒐𝒏 ሺ𝝂ሻ) 𝒅ഥ𝒍𝒐 (𝝂+𝟏) = 𝒅ഥ𝒍𝒐 (𝝂) + 𝝈(𝒅ෝ�𝒍𝒐 ሺ𝝂ሻ − 𝒅ഥ𝒍𝒐 ሺ𝝂ሻ) 𝒅ഥ𝑹𝒍𝒐𝒍𝒐 (𝝂+𝟏) = 𝒅ഥ𝑹𝒍𝒐𝒍𝒐 (𝝂) + 𝝈(𝒅ෝ�𝑹𝒍𝒐𝒍𝒐 (𝝂) − 𝒅ഥ𝑹𝒍𝒐𝒍𝒐 (𝝂)) 𝑺ഥ𝑰𝒍𝒐𝒍𝒐 (𝝂+𝟏) = 𝑺ഥ𝑰𝒍𝒐𝒍𝒐 (𝝂) + 𝝈(𝑺ෝ�𝑰𝒍𝒐𝒍𝒐 (𝝂) − 𝑺ഥ𝑰𝒍𝒐𝒍𝒐 (𝝂))

Set n = n + 1 and go to Step 2.1.

Step 3 Find an optimal solution 𝒖�� 𝒍𝒐𝒏𝒎 (𝝂),𝒅ෝ�𝒍𝒐𝒏𝒎 (𝝂) ,𝒅ෝ�𝒍𝒐𝒏 (𝝂)

, 𝒅ෝ�𝒍𝒐 (𝝂) , 𝒅ෝ�𝑹𝒍𝒐𝒍𝒐 (𝝂)

, and 𝑺ෝ�𝑰𝒍𝒐𝒍𝒐 (𝝂)

In Step 2.1 and 2.2 of DSTEM Algorithm, the calculation of the indicator functions and exit times are done via dynamic network loading, as detailed in Nie and Zhang (2005) and Ban et al. (2008). The relaxed sub-problems RNCPDUE-DTA and RNCPDUE-DTA-DT, RNCPDUE-DTA-DT-MS, RNCPDUE-DTA-DT-MS-NMS, RNCPDUE-DTA-DT-MS-NMS-TD, and RNCPDUE-DTA-DT-MS-NMS-TD-TG are solved directly by the PATH solver in GAMS (Ferris and Munson 1998). For detailed discussions, one can refer to Ban et al. (2008)

In Step 2.7, there are several commonly used gap functions that can be used in the convergence test. The first one is based on the difference of the candidate solution and the base solution. In particular, we can check whether the following condition holds: 𝑮𝒂𝒑𝟏 = ቀቛ 𝒖�� 𝒍𝒐𝒏𝒎 ሺ𝝂ሻ − 𝒖തതത𝒍𝒐𝒏𝒎 ሺ𝝂ሻቛ𝟐 +ቛ 𝒅ෝ��𝒍𝒐𝒏𝒎 ሺ𝝂ሻ − 𝒅തതത𝒍𝒐𝒏𝒎 ሺ𝝂ሻቛ𝟐 +ቛ 𝒅ෝ��𝒍𝒐𝒏 ሺ𝝂ሻ − 𝒅തതത𝒍𝒐𝒏 ሺ𝝂ሻቛ𝟐 +ቛ 𝒅ෝ��𝒍𝒐 ሺ𝝂ሻ − 𝒅തതത𝒍𝒐 ሺ𝝂ሻቛ𝟐 +ቛ𝒅ෝ�𝑹𝒍𝒐𝒍𝒐 ሺ𝝂ሻ − 𝒅ഥ𝑹𝒍𝒐𝒍𝒐 ሺ𝝂ሻቛ𝟐 +ቛ𝑺ෝ�𝑰𝒍𝒐𝒍𝒐 ሺ𝝂ሻ − 𝑺ഥ𝑰𝒍𝒐𝒍𝒐 ሺ𝝂ሻ ቛ𝟐ቁ< 𝜺𝟏∀𝒍,𝒐 (30)

Especially if 𝒖�� 𝒍𝒐𝒏𝒎 ሺ𝝂ሻ = 𝒖തതത𝒍𝒐𝒏𝒎 ሺ𝝂ሻ, 𝒅ෝ��𝒍𝒐𝒏𝒎 ሺ𝝂ሻ = 𝒅തതത𝒍𝒐𝒏𝒎 ሺ𝝂ሻ , 𝒅ෝ��𝒍𝒐𝒏 ሺ𝝂ሻ − 𝒅തതത𝒍𝒐𝒏 ሺ𝝂ሻ, 𝒅ෝ��𝒍𝒐 ሺ𝝂ሻ = 𝒅തതത𝒍𝒐 ሺ𝝂ሻ , 𝒅ෝ�𝑹𝒍𝒐𝒍𝒐 ሺ𝝂ሻ = 𝒅ഥ𝑹𝒍𝒐𝒍𝒐 ሺ𝝂ሻ , and 𝑺ෝ�𝑰𝒍𝒐𝒍𝒐 ሺ𝝂ሻ = 𝑺ഥ𝑰𝒍𝒐𝒍𝒐 ሺ𝝂ሻ we obtain a fix point solution to the original problem NCPDUE-DTA-DT-MS-NMS-TD-TG.

We can also check directly the conditions of route choice, departure time choice, nest mode choice, mode choice from the nest, trip distribution choice, and trip generation choice by defining the second gap function: 𝑮𝒂𝒑𝟐 = ൬ 𝒖�� 𝒍𝒐𝒏𝒎 ሺ𝝂ሻ𝚽𝒖𝒍𝒐𝒏𝒎 + 𝒅ෝ��𝒍𝒐𝒏𝒎 ሺ𝝂ሻ𝚽𝒅𝒍𝒐𝒏𝒎 + 𝒅ෝ��𝒍𝒐𝒏 ሺ𝝂ሻ𝚽𝒅𝒍𝒐𝒏 + 𝒅ෝ��𝒍𝒐 ሺ𝝂ሻ𝚽𝒅𝒍𝒐+ 𝒅ෝ�𝑹𝒍𝒐𝒍𝒐 ሺ𝝂ሻ𝚽𝒅𝑹𝒍𝒐𝒍𝒐 + 𝑺ෝ�𝑰𝒍𝒐𝒍𝒐 ሺ𝝂ሻ𝚽𝑺𝑰𝒍𝒐𝒍𝒐 ൰< 𝜺𝟐∀𝒍,𝒐 (𝟑𝟏)

Similarly, if 𝑮𝒂𝒑𝟐 = 𝟎, both conditions are satisfied, the obtained candidate solution is indeed an optimal solution. In (30) and (31), 𝜺𝟏 and 𝜺𝟐 are both small scalars representing user-defined convergence accuracy.

6. Conclusions and Future Research In this paper we presented a link-node based discrete-time NCP model for

DUE that combined multiclass trip generation, trip distribution, nested modal split, specific modal split, dynamic trip assignment and departure time (DMSTEM) in a unified formulation.

This model combined the dynamic link-node based discrete-time NCP model that developed by Ban et al. (2008) and the static Multiclass Simultaneous Transportation Equilibrium Model (MSTEM) that developed by Hasan and Dashti (2007).

We also developed an iterative solution algorithm for the proposed model by solving several relaxed NCP in each iteration.

Most of the recent DUE models deal only with the traffic assignment and fixed demand flows between each O-D pair at each time period of the time horizon or an elastic dynamic demand (input to DTA models) that is usually modeled as departure-time choice for single user (traveler) class.

To apply DTA with departure-time choice (DTA-DT) in practice, we require fixed demand flows between each O-D pair during the whole time horizon and the DTA-DT to be embedded in more comprehensive transportation planning framework that results in an inconsistence prediction process.

The DMSTEM model will generate a better solution than this inconsistent prediction process within a comprehensive planning and operations framework.

We would apply the DMSTEM and it solution algorithm to a prototype example to get more insight and implication of its predictions and validity.

This prototype solution will encourage applying the model to real

world transportation network. A comparative study between DMSTEM as a dynamic model and MSTEM as static model in term of their prediction accuracy and solution complexity is very desirable in the transportation planning in general and in particular, the Intelligence Transportation System (ITS).

Transportation network analysis tools can be broadly categorized as

static analysis tools and dynamic analysis tools. The former focuses on long-term, steady traffic states, often referred as the four-step demand analysis models (Sheffi, 1985), while the latter focuses on short-term, dynamic traffic states, often called dynamic network analysis or dynamic traffic assignment (DTA, see Peeta and Ziliaskopoulos (2001)).

Traditionally, these two types of models have been applied in quite distinct scenarios: static tools for transportation planning purposes and dynamic tools for traffic operation purposes.

It would be ideal to combine the planning and operation tools to address transportation problems in a more comprehensive manner. In practice, such combination has been recently experimented and tested. Examples include the corridor management plan demonstration project (CCIT, 2006) and the state-wide corridor system management plan that is currently underway in California, which aims to integrate long-term planning tools (such as the static four-step demand analysis models) with operational tools such as microscopic traffic simulation and dynamic traffic assignment. The ICM program by FHWA also focuses on combining different transportation analysis tools for integrated corridor planning and operations.

Combining planning and DTA models in theory is not trivial because the two types of models are based on quite distinct assumptions. In this paper we present a combined model to simultaneously consider the origin, destination, mode, departure, and route choices of multiclass travelers.