13 dta_departurechoice (ewgt2002)

8/14/2019 13 DTA_DepartureChoice (EWGT2002)

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A DEMAND MODEL WITH DEPARTURE TIME CHOICEFOR WITHIN-DAY DYNAMIC TRAFFIC ASSIGNMENT

Giuseppe Bellei, Guido Gentile, Lorenzo Meschini, Natale PapolaUniversit degli studi di Roma La Sapienza

E-mail: [email protected]

1 INTRODUCTION

The dynamic analysis of transportation networks gathered increasing attention through the past years. The static models, traditionally used in this field, are in fact unable to representrelevant phenomena, such as demand variations over time and temporary over saturation of

network elements. On the other hand, dynamic models are more complex than static ones: onthe supply side, they require to ensure temporal consistency, besides spatial consistency,among system variables (for instance, see Cascetta, 2001); on the demand side, the departuretime choice has to be modelled explicitly in order to achieve a correct representation of userstravel behaviour (Mahmassani and Chang, 1986; Van Vuren, Daly and Hyman, 1998;Mahmassani and Liu, 1999).Different approaches to the latter problem can be found in the literature. One of them is toregard departure time choice as a discrete choice among temporal intervals and use staticassignment to characterize the utility of each interval (Daly et al, 1990); models based on thisapproach yield a rough representation of travel demand during day time as a sequence of static equilibria. Another approach is based on the hypothesis that departure time and pathchoices are made jointly, so that, for each origin-destination pair, a discrete choice set isdefined whos finite number of alternatives is equal to the number of departure intervalsmultiplied by the number of paths (Arnott, De Palma and Lindsey, 1990; Cascetta, Nuzzoloand Biggierio, 1992); models based on this approach require explicit path enumeration and tointroduce a diachronic graph (Van der Zijpp and Lindveld, 2000), so they can be hardlyapplied to congested urban networks.

In this paper a Nested Logit demand model with five choice levels is presented. The model isconceived to extend to the case of elastic demand the multimodal within-day Dynamic TrafficAssignment (DTA) model proposed in Gentile, Meschini and Papola (2001). The abovemodel doesnt require to introduce a diachronic graph and allows implicit path enumeration.With reference to departure time choice, the proposed demand model adopts a continuousapproach, thus not requiring to enumerate explicitly the desired departure time intervals. Theresulting within-day DTA model is then capable of representing both supply and demanddynamic phenomena concerning congested multimodal urban networks, and leads to a fixed

point formulation that can be solved by an efficient MSA algorithm applicable to real

networks.


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2 EQUILIBRIUM MODEL

In analogy with the static case, the within-day DTA, regarded as a dynamic user equilibrium,can be consistently formalized through a fixed point problem expressed in terms of arc flow

temporal profiles, by combining the arc performance function with the Network Loading Map(NLM) and thus avoiding to introduce the Dynamic Network Loading (DNL) problem(Bellei, Gentile and Papola, 2000). To this purpose, the NLM is here extended to the case of elastic demand with departure time choice, as depicted in Figure 1.

Figure 1

3 CHOICE AND DEMAND MODEL

In modelling travel demand we follow the behavioural approach based on random utilitytheory, where it is assumed that each user is a rational decision-maker who, when making histravel choice: a) considers a positive, finite number of mutually exclusive travel alternatives constituting his choice set J ; b) associates to each travel alternative j of his choice set a

perceived utility, not known with certainty, and thus regarded by the analyst as a randomvariable U j; and c) selects the travel alternative that maximises his utility. With thesehypotheses the probability of alternative j is formally expressed as:

network loading map

network flow propagation model

actual demandmodel

arc performance model

implicit pathchoice model

arc flows arc performances

node satisfactions

arc conditional probabilities

actual demandflows

potential demand

departure timechoice model

arc performance function

departure time probabilities

trip probabilities

departure timesatisfactions

desired demandflows

trip choices model desired demandmodel


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P j = Prob[ k J k V j -V k + j ] , (1)where V j and j are respectively the systematic utility and the random residual of the genericalternative j. The expected value of the maximum perceived utility is called satisfaction :

W = E[max j J {V j + j }] (2)

As usual, we assume that it is possible to divide the choice process into a hierarchic sequenceof decisions; at each level the user has a specific choice set, dependent on the choices made at

previous levels. The utility associated to each alternative available at a given level is the sumof a specific term and of the satisfaction that takes into account the alternatives available atthe successive levels.The proposed demand model is a Nested Logit with five choice levels, namely: generation ,distribution , modal split , departure time choice , and path choice .

In the case of path choice, the utility of the alternatives, beside varying over time, are relatedto values of arc performances taken at different instants. Then, in order to adopt an implicit

path enumeration approach, we need to assume that the distribution of random residuals isconstant over time; this hypothesis is essential in order to define arc conditional probabilitiesconsistent with the path choice model (see Bellei, Gentile and Papola, 2000).

In the following sections the models related to each choice level will be shortly examined.

3.1 Implicit path choice model and network flow propagation modelThe multimodal network is defined on a graph G( N , A), where N is the node set and A is the

arc set. At any instant , each mode m is characterized by specific arc flows and performances:

f am( ) entering flow of arc a ,

cam( ) generalized costs of arc a ,

t am( ) exit time of arc a .Referring to users travelling towards destination d on mode m, the implicit path choice andthe network flow propagation models require to define the following variables for every

instant :

w xmd ( ) satisfaction of node x,

pamd ( ) conditional probability of arc a ,

d omd ( ) actual demand flow from origin o,

f amd ( ) flow on arc a .With reference to the Logit case, we have (Bellei, Gentile and Papola, 2000):

w xmd ( ) = R ln(a FSE md ( x) exp((- cam( ) +w HD (a)md (t am( ))) / R)) (3) pamd ( ) = exp((- cam( ) +w HD (a)md (t am( )) -wTL(a)md ( )) / R) (4)

f amd ( ) = pamd ( ) [d TL(a)md ( ) + b BSE md (TL(a)) [ f bmd (t bm -1( )) t bm -1( ) / ]] (5)

f am

( ) = d N f amd

( ) (6)


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where FSE md ( x) and BSE md ( x) are respectively the efficient forward and backward stars of node x, while TL(a) and HD (a) are respectively the initial and final node of arc a .

3.2 Departure time choice and actual demand models

The role of the actual demand model is to transform the desired demand temporal profile of every o-d pair and mode m into the corresponding actual demand temporal profile, on the

basis of the departure time probabilities produced by the departure time choice model.With reference to the latter model, we assume that the choice set is an appropriateneighbourhood of the desired departure instant. Note that, by considering a continuous choiceset, we release one of the main hypotheses of discrete choice models (here we have an infinitenumber of alternatives). Moreover, we assume that the random residuals associated to eachinfinitesimal alternative are independently and identically distributed (i.i.d.) Gumblevariables; so that the resulting choice model belongs to the Logit family. In the following werefer to users travelling from origin o to destination d by mode m, then the correspondingindices will be dropped from the notation in order to improve the readability. With reference

to a desired departure instant , lets define:

q( ) desired demand flow

p( )( ) probability density function of leaving at time

V( )( ) systematic utility of leaving at time

T ( ) choice set

By thinking the continuous choice set T ( ) as dicretized into an infinite number of infinitesimal departure intervals, the Logit formula can be still applied to calculate the

probability of choosing the generic departure interval [ -d /2, +d /2] as follows:

( ) ( )

( ) ( )

( )( )( )

Vexp

p d dV

exp d

DT

DT T

x x

=

, (7)

where, clearly, the summation is replaced by an integration. In analogy to the discrete case,

the denominator in equation (7) is directly related to the departure time satisfaction:

( ) ( )( )( )

Vln exp d DT

DT T

xW x

= (8)

We assume that the continuous choice set is the following interval:

T ( ) = [ - ADV , + DEL ] , (9)

where ADV and DEL are respectively the maximum advance and delay accepted. Moreover,we assume that the systematic utility is given by the difference between the satisfaction of

leaving at time (expressed through the satisfaction of the origin yielded by the path choicemodel) and a disutility term proportional to the departure advance or delay:


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V( )( ) = w( ) -max{ b ADV ( - ), b DEL ( - )} (10)

With these hypotheses we have:

( ) ( ) ( ) ( ) ( )exp exp d exp d

DEL ADV DEL

DT DT DT x ADV x

W w x b x w x b x x

+

= =

= +

(11)We now address the problem of transforming the desired demand temporal profile into theactual demand temporal profile. The number of trips started within the infinitesimal departure

interval [ -d /2, +d /2] is given by the integral, for each desired departure time such that T ( ), of the number of desired trips within the infinitesimal interval [ -d /2, +d /2],

multiplied by the probability that the departure occurs within the interval [ -d /2, +d /2]:

( ) ( ) ( )( ) +

=

= ADV

DEL

qd

d pdd (12)

On the basis of (7), (8), (9) and (10), equation (12) becomes:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )exp d exp d ADV

DEL ADV

DT DT DEL

w b W w b W d q q

+

= =

= + (13)

As in general the profiles q( ), W ( ) and w( ) can take any form, the integrals in equations(11) and (13) cannot be solved in closed form; then they are to be calculated numerically. An

ad-hoc procedure based on the assumption that the profile q( ) is piece-wise constant, while

the profiles W ( ) and w( ) are piece-wise linear, is presented in section 4.1 .

3.3 Modal split modelReferring to users travelling from origin o toward destination d and to a desired departure

instant , lets define:

qod ( ) desired demand flow

V mod ( ) specific utility of modal alternative m

P mod ( ) choice probability of modal alternative m

W od ( ) modal split satisfaction

In the Logit case, we have:

( ) ( )ln expod

mod M

m M M

V W

= , ( ) ( ) ( )exp

od od mod

mM

V W P

= (14)

As usual, we assume:

V mod ( ) = M T X mod +W odm( ) , (15)

where X mod is a vector of attributes and M is the corresponding vector of coefficients, while

W odm( ) is the departure time satisfaction given by (8).

Thus, the desired demand flow on mode m at time is:


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q odm( ) = q od ( ) P mod ( ) (16)

3.4 Distribution modelReferring to users departing from origin o and to a desired departure instant , lets define:

q o( ) desired demand flowV d

o( ) specific utility of destination alternative d

P d o( ) choice probability of destination alternative d

W o( ) distribution satisfactionIn the Logit case, we have:

( ) ( )ln expo

d o D

d N D

V W

= , ( ) ( ) ( )exp

o od o

d D

V W P

= (17)

We assume:V d o( ) = DT X d o +W od ( ) , (18)

where X d o is a vector of attributes and D is the corresponding vector of coefficients, while

W od ( ) is the modal split satisfaction given by (14).

Thus, the desired demand flow travelling toward destination d at time is:

q od ( ) = q o( ) P d o( ) (19)

3.5 Emission modelReferring to users potentially departing from origin o and to a desired departure instant ,lets define:

N o( ) flow of users potentially travelling (assumed to be known)

V o( ) specific utility of travelling

P o( ) choice probability of travelling

S o( ) total satisfactionIn the Logit case, we have:

( ) ( )ln 1 expo

o E

E

V S = + , ( ) ( ) ( )exp

o o

o

E

V S P = (20)

We assume:

V o( ) = E T X

o +W o( ) , (21)

where X o is a vector of attributes and E is the corresponding vector of coefficients, while

W o( ) is the distribution satisfaction given by (17). Thus, the desired demand flow at instant is:

qo

( ) = N o

( ) P o

( ) (22)


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4 ALGORITHM

In order to implement the proposed DTA model with departure time choice and elasticdemand, it is necessary to divide the period of analysis into a sequence of temporal intervals

( i-1, i], i = 1, , I . Then, each temporal profile is approximated through a function either piece-wise constant, or piece-wise linear, consistent with the above intervals, so that it is

represented numerically by means of a ( I +1 1) vector. Specifically, the temporal profiles of the flows are assumed to be piece-wise constant, while the temporal profiles of performances,satisfactions and choice probabilities are assumed to be piece-wise linear. The resulting fixed

point problem formalizing the DTA is solved through the Method of Successive Averages(MSA) as outlined in the following:

0) k = 0 , f k +1 = {0 | f iniz} initialization1) k = k +1 update the iteration counter 2) t k = t( f k ) , c k = c( f k , t k ) calculate the arc performances3) w k = w( c k , t k ) , p k = p( w k , c k , t k ) implicit path choice model4) W DT k = W DT (w k ) departure time choice satisfactions5) W M k = W M (W DT k ) , P M k = P M (W M k , W DT k ) modal split model6) W D k = W D(W M k ) , P D k = P D(W D k , W M k ) distribution model7) S k = S( W D k ) , P G k = P G ( S k , W D k ) emission model8) q k = q( N , P G k , P D k , P M k ) calculate the desired demand flows9) d k = d( q k , W DT k , w k ) calculate the actual demand flows

10) y k

= ( p k

, t k

, d k

) network flow propagation model11) f k +1 = f k +1/ k ( y k - f k ) updates the arc flows12) if max a A i I | ya i k - f ai k | > and k < k max then goto 2 stop criterion

4.1 Departure time choiceIn this subsection, the integrals (11) and (13) are solved in closed form. Let ADV1(i) and DEL1 (i) be the first periodization instant respectively after i- ADV and i+ DEL , we have:

( )( )( )

( ) ( )( ) ( ) ( )( )( )( ) ( ) ( )( )

( )

( ) ( ) ( )( )( )

( )( )( )

1

1

11

1

expexp exp exp

expexp exp

expexp exp

expexp

A DV ii

ADV ADV i ADV i ADV ii ADV ADV A DV i

DT ADV

ji ADV j j j j

ADV ADV j j ADV i ADV

j DE L i DEL j j j j

DEL DEL j j i DEL

DE L i DEL

DE L i DEL

W ADV

= +

= +

= +

+ +

+ +

+

( ) ( )( ) ( ) ( )( )( )1exp DEL i DEL i DEL ii DEL DEL DEL +

(23)

where:1 1

1 1 1 11 1

1 1

1 1

,

,

j j j j j j i j j i

ADV DEL j j j j j j ADV DEL

DT DT

j j j j


DT DT

w w w ww b w b

w w w wb b

+ = =

+ = =


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Let ADV2(i) and DEL2 (i) be the first periodization instant respectively after i+ ADV and i- DEL , we have:

( )( )( )

( )( ) ( )( ) ( ) ( )( )( )

( ) ( ) ( )( )( )

( ) ( ) ( )( )( )

( )

2

2 2 2

2

1

2 1

2 11

1

2 1

expexp -exp

exp exp exp

expexp exp

exp

DEL i DEL DEL i DEL i DEL i DEL ii i

DEL DEL DEL i DEL

ji DEL j j j j j

DEL DEL j j DEL i DEL

j ADV i ADV j j j j j

ADV ADV j j i ADV

ADV ADV i

d q DEL

q

q

q

= +

= +

= +

+ +

+ +

+

( )( )

( )( ) ( )( ) ( ) ( )( )( )

2

2 1 2 2

2 exp exp

ADV i

ADV i ADV i ADV ii ADV ADV ADV i

ADV

ADV +

where: (24)

REFERENCES

Arnott R., De Palma A., Lindsey R. (1990). Departure time and route choice for the morning

commute. Transportation Research , vol. 24A, pp. 209-228

Cascetta E., Nuzzolo A., Biggiero L. (1992). Analysis and modelling of commuters departure timeand route choices in urban networks. Proceeding of the 2 nd International Seminar on Urban Traffic

Network , Capri, Italy.

Cascetta E. (2001). Transportation systems engineering: theory and methods , Kluwer AcademicPublisher

Daly A.J., Gunn H.F., Hungerink G., Kroes E.P., Mijjer P. (1990). Presented to PTRC Summer Annual

Meeting

Bellei G. , Gentile G. , Papola N. (2000). Un Modello Logit di Assegnazione Dinamica Intraperiodalealle Reti Stradali Urbane. In Metodi e Tecnologie dellIngegneria dei Trasporti , ed.s G. Cantarella, F.Russo, Franco Angeli s.r.l. , Milano, Italia

Gentile G., Meschini L., Papola N. (2001). Un modello Logit di assegnazione dinamica intraperiodalealle reti multi modali di trasporto urbano. Presented to IV seminario Metodi e tecnologiedellingegneria dei trasporti , Reggio Calabria

Mahmassani H. S., Chang G. L. (1986). Experiments with departure time choice dynamics of urbancommuters. Transportation Research , vol. 20B, pp. 297-320

Mahmassani H.S., Liu Y.H. (1999). Dynamic of commuting decision behaviour under advancedtraveller information systems. Transportation Research , vol. C, pp. 465-484

Van Vuren T., Daly A., Hyman G. (1998). Modelling departure time choice. ColloquiumVervoersplanologisch Speurwerk , Amsterdam

Van der Zijpp N.J., Lindveld C.D.R. (2000). Estimation of O-D demand for dynamic assignment with

simultaneous route and departure time choice. Presented at European Transport Conference ,University of Cambridge

1 1

1 1

1 1

,

,

i j i i j i j j ADV DEL ADV DEL

DT DT

j j j j


DT DT

w W b w W b

W W W W b b

+ = = = =

13 dta_departurechoice (ewgt2002)

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