Dynamic traffic assignment with departure flow estimation based on preferred arrival time

KSCE Journal of Civil Engineering (2012) 16(4):633-643DOI 10.1007/s12205-012-1210-8

− 633 −

www.springer.com/12205

Transportation Engineering

Dynamic Traffic Assignment with Departure Flow Estimation based onPreferred Arrival Time

Hyunmyung Kim*, Yongtaek Lim**, and Sungmo Rhee***

Received May 8, 2010/Revised June 20, 2011/Accepted July 10, 2011

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Abstract

This study develops a new method for estimating the departure time of OD flows. Although departure time choice is very sensitiveto changes in the travel environment, it has not been included in transportation planning. There exist some theoretical methods, butmost have not been applied to practical studies. In addition, the behavior of travelers regarding departure time choice has not beenadequately addressed. To overcome these problems, the proposed model takes two behavioral aspects into account. The first is themaximization of in-home activity time before a departure, and the second is the dynamic user equilibrium in departure time choice.This study investigates the effect of incorporating these two aspects through numerical tests. The results indicate that the developedmethod has a reasonable capability to forecast the resulting departure pattern change after the modification of network attributes. Inaddition, the method is simpler than existing approaches such as the schedule delay function; thus, it can be helpful for studyingpractical problems.Keywords: departure flow estimation, preferred arrival flow pattern, dynamic traffic assignment, in-home activity time, dynamic userequilibrium

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1. Introduction

Developing a practical departure time choice model has beena popular topic among network researchers in last few decadesbecause departure time choice is one of the most common dailyproblems for travelers. When a travel condition such as thepreferred arrival time at a destination is changed, the most easilyadjustable travel choice is the departure time. In general, a travelmode and a route are the result of long-run experience. As aresult, mode or route choice is conducted habitually, and mostcommuters do not modify their choice for minor variation in thetravel environment. This is one of the reasons why the consider-ation of departure choice is important. In addition, traffic conges-tion is also closely related to the temporal departure flow pattern.As Kitamura et al. (1996) addressed, the main cause of trafficcongestion is not the excessive flow itself but the excessivespatial convergence of traffic flows during a specific time period.Therefore, the temporal profile of departure flows is a key factorin traffic congestion analysis. The motive behind DTA (dynamictraffic assignment) development is to make a tool for analyzingthe temporal pattern of traffic flows.

Many transportation researchers have proposed a number ofapproaches for modeling departure time choice behavior, but

little success has been achieved to date. Some theoretical ap-proaches such as the use of the schedule delay function havebeen proposed, but they have not shown sufficient practicalcapability. For evidence, few commercial DTA packages have adeparture time choice model. This implies that most existingtransportation planning methods are not appropriate for trafficcongestion analysis. In this regard, this paper proposes a newmethod for including departure time choice in DTA. The approachdoes not consider the microscopic behavioral aspects of travelersbut is appropriate for describing the aggregate choice pattern ofdeparture flows from a macroscopic point of view. The collectivedemand of travelers for their preferred arrival times is representedas an arrival time-based OD table. A new gradient-based optimi-zation method is developed, and its performance is verified witha numerical example. In the tests, the hypothesis regarding theshortcomings of the DTA model with fixed departure times isalso confirmed.

2. Literature Review

Chang and Mahmassani (1988) investigated the departure timeadjustment behavior of commuters with the data surveyed at theUniversity of Texas at Austin. The study identified two important

*Assistant Professor, Dept. of Transportation Engineering, College of Engineering, Myongji University, Yongin 449-728, Korea (Corresponding Author,E-mail: [email protected])

**Member, Associate Professor, Dept. of Transportation & Logistics, Chonnam National University, Yeosu 550-749, Korea (E-mail: [email protected])

***Member, Professor, Dept. of Civil and Environmental Engineering, Seoul National University, Seoul 151-742, Korea (E-mail: [email protected])

Hyunmyung Kim, Yongtaek Lim, and Sungmo Rhee

− 634 − KSCE Journal of Civil Engineering

points: the use of the preferred arrival time for the departure timeadjustment, and the consideration of the capability limit ofhuman beings in decision making. The indifference band andbounded rationality were combined in the departure timeadjustment process. These two aspects have been the core partsin departure time choice modeling to date, and thus, they areaccepted in this work.

Ran et al. (1996) used the schedule delay function for combin-ing a departure time choice problem with a route choice model.The schedule delay function used the preferred arrival time andthe indifference band, and a late arrival was penalized more thanan early arrival. Yang and Meng (1998) used a schedule delayfunction to assess the impact of congestion pricing on the spatialand temporal traffic patterns in the network. Jou (2001) studied asimilar topic and discussed the role of a priori information indeparture time and route choice. Data collected from a homeinterview survey of commuters in Taiwan were used for calibrat-ing a probit model. Lam and Huang (2002) addressed the activitytopic; they formulated a combined model based on the equili-brium concept for the location, route, and departure time choiceproblems. A similar model can be found in Huang and Lam(2002). In general, departure time choice models have mainlyaddressed the commuting travel problem, although there havebeen some exceptions such as Lam and Huang (2002), in whicha non-work travel purpose was taken into account. The proper-ties of non-mandatory travel were investigated in Bhat and Steed(2002). In their study, a hazard model was employed to describethe departure time choice behavior of shopping travelers. Thedeparture time choice for non-mandatory activities is substantiallydifferent from that for mandatory activities because there is nodesired arrival time in the choice problem.

A unique viewpoint regarding the departure choice problemwas proposed by Fujii and Kitamura (2004). Instead of an opti-mization framework, they adapted cognitive theory and modeledthe driver’s mental representation of travel time. Ettema et al.(2005) shared a similar idea. To consider travel uncertainty, alearning process based on cognitive theory was employed. Limand Heydecker (2005) developed a logit-based combined depar-ture time and dynamic stochastic user equilibrium (DDSUE)assignment. According to their definition, the UE (user equilib-rium) principle was extended to the departure time choice problem.In other words, all drivers for the same OD pair should have thesame perceived travel time under the DDSUE condition. In thesame year, Heydecker and Addison (2005) developed an equilib-rium-based departure time choice model, which considered routechoice and departure time choice simultaneously. They proposeda mathematical formulation for the problem, and the developedmodel was applied to a congestion toll problem. Bellei et al.(2006) considered the uncertainty of departure time choice withrandom utility theory. In their study, a nested logit structure isused in order to combine variable travel demand with departuretime choice problem. Zhang et al. (2008) proposed a departurechoice model for morning and evening commuters by consideringdelays at a parking lot. Contrasting with the use of user equilib-

rium principle, social optimum is used as an objective of depar-ture time choice problem in Chow (2009). The mathematicalformulation is also proposed as an optimal control problem. Thedeveloped model is able to calculate the departure flow pattern,which minimizes the total travel cost in the network.

In summary, studies of departure time choice can be categor-ized into two groups. The first group collects survey data, con-structs a behavioral model, and calibrates it with real data. In thistype of research, the heterogeneous behavioral aspects havedrawn close attention (Chang and Mahmassani, 1988; Bhat andSteed, 2002; Fujii and Kitamura, 2004). By contrast, the secondtype of research assumes that all drivers are homogeneous andfollow a common principle such as user equilibrium (Ran et al.,1996; Yang and Meng, 1998; Lam and Huang, 2002; Heydeckerand Addison, 2005; Bellei et al., 2006; Zhang et al., 2008; Chow,2009). The departure time choice based on utility maximizationtheory can also be categorized into the second type (Jou, 2001;Lim and Hydecker, 2005).

3. Departure Time Choice in Network Analysis

3.1 Departure Time as a Travel Choice AlternativeA general commuter has three alternatives for facing with the

change of travel environment; 1) travel mode, 2) travel route, and3) departure time. Among the three alternatives, departure time isthe most easily adjustable one. In Fig. 1, the first figure shows thecase of departure time change. A small adjustment in travelers’departure time does not cause a substantial difference in theirtravel condition. The traffic condition on the same path does notchange significantly within a short time.

It means a small change in departure time does not come witha big uncertainty. This is not the same in the case of routechange. Even in the overlapping path case, a route change givesrise to uncertainty among travelers because they have no experi-ence in the non-overlapping part of the new path. This implies

Fig. 1. Uncertainty Over Travel Attribute Change

Dynamic Traffic Assignment with Departure Flow Estimation based on Preferred Arrival Time

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that a departure time change is easier to accommodate than aroute change for commuters.

Mode and destination change causes bigger uncertainty sincethe choice alternatives does not share common properties. Forexample, the shopping mall that a customer usually visits is closedbut that the customer needs to buy some goods. In this case, eventhough the customer has sufficient experience shopping at thatspecific shopping mall, the experience does not provide adequateinformation on the properties of other shopping malls. Hence,the customer faces substantial uncertainty.

The above discussion illustrates why departure choice shouldbe included in transportation planning. Compared with other travelattributes, departure time adjustments entail less uncertainty. If aheavy traffic jam is announced in the morning, most commutersattempt to depart earlier to be at work on time. According toOrtuzar and Willumsen’s (1990), more than 87% of morningtrips were mandatory trips for work and education in the GreatSantiago. It implies a large portion of morning travelers preferredfixed arrival time, not fixed departure time. Therefore, withoutconsidering the fixed arrival flow pattern, a comprehensivedynamic network analysis would be impossible.

3.2 Problem from a Fixed Departure Time AssumptionIn transportation policy evaluation, the absence of departure

time choice may cause a serious problem. Kim and Jayakrishnan(2006) explained the problems of the DTA model with fixeddeparture flows in the network design problem. If travel environ-ment changes, as discussed above, travelers in reality try toadjust their departure times first. However, most existing trafficsimulation packages do not allow for departure time adjustments.

Evaluating transportation policy without considering departuretime choice may result in false results. It is acceptable when amajor portion of the OD flow is returning travelers from theiroffice or school in the evening. By contrast, for a commuting tripin the morning, the assumption makes little sense. Most workerstraveling in the morning have a preferred arrival time based onwhen their work starts. As a result, if OD travel time is changed,travelers tend to adjust their departure times.

In Fig. 2, it is assumed that OD travel time decreases becauseof network capacity expansion. With fixed departure flows, thereduction in travel time causes early arrival flows at destinations.It means that commuters arrive at their office much earlier thantheir preferred times, and have to wait for their work to start.From an activity perspective, this is not plausible, and travelerswant to avoid the schedule delay. In addition, if OD travel timedecreases in reality, commuters delay departure times accordingto their preferred arrival times. As a result, peak-hour congestionis also delayed. Fig. 2 illustrates that a DTA model cannot forecasta traffic pattern correctly after the network design is changed.

As shown above, assessing the impact of travel environmentchange on fixed departure flows is inappropriate. To avoid atime-reverse traffic loading process, Kim and Jayakrishnan (2006)proposed a bi-level framework for an arrival flow-based DTAmodel. However, they only developed mathematical formulationswith the VI (variational inequality), the GLS (general least square)method, and the FIFO (first-in-first-out) violation function. Nooperational algorithm and analytical result was given.

Kim (2009) developed a mathematical program for an arrivalflow-based DTA model with a GLS function. The preferred arrivalflow pattern was used directly for estimating departure flowswithout making use of a schedule delay function. Accordingly,the developed model does not require any calibration burdenrelated to a schedule delay function. However, the developedmodel may have multiple solutions. It means different departurepatterns generate the same arrival flow patterns. From a mathe-matical perspective, it is an underdetermined problem. To lessenthis problem, this study proposes a new heuristic objective for-mulation.

3.3 Problem of the Schedule Delay FunctionThe schedule delay function has been widely used for model-

ing the departure time choice behavior of travelers (Chen, 1999).In a dynamic context, the arrival time of a traveler (tar) is equal tothe sum of his departure time (tdp) and OD travel time (cij(tdp)):

(1)

Assume the traveler has a preferred arrival time (tpref). Then theschedule delay function for the late or early arrival can bedefined as shown in Fig. 3, and the schedule delay function canbe written as a family of equations as shown in Eq. (2):

if (2a)

if (2b)

if (2c)

In Eq. (2), Dij(tar) denotes the schedule penalty for the OD pairij when a driver arrives at time tar. The interval tpref ±∆ representsthe indifference band for the preferred arrival time. In reality,human beings have a perception limit, so the indifference bandreflects reality. The condition |θ+| > |θ−| is added because a latearrival causes a more serious schedule penalty then an earlyarrival. In addition, θ+ should be non-negative, and θ− should be

tar tdp cij tdp( )+=

Dij tar( ) θ−– tpref ∆–( ) tar–{ }⋅= tar tpref ∆–≤

Dij tar( ) 0= tpref ∆– tar tpref ∆+≤ ≤

Dij tar( ) θ+ tar tpref ∆–( )–{ }⋅= tar tpref ∆+≥

Fig. 2. Impact of Travel Condition Change on Morning Commuters



non-positive.Although the schedule delay function has been widely used in

academic works (Ran et al., 1996; Yang and Meng, 1998; Chen,1999), few studies have been conducted in practice. The firstdifficulty regarding the schedule delay function is how to set theindifference band. The indifference band describes travelers’insensitivity to a small violation of the preferred arrival time. Thecalibration of parameters θ+ and θ− should depend on the socio-economic status of travelers. The slopes also might differ accordingto the OD pair, the time of day, and the type of subsequent activity.As a result, finding the appropriate values for the parameters fora whole network might be complicated.

Another difficulty in using a schedule delay function is that aDTA model with a schedule delay function might have multipleoptimal departure flow patterns. As Sheffi (1985) demonstratedwith a static case, there are multiple path flow solutions that cangenerate the same link flow pattern. A similar example can befound in the case of the dynamic OD estimation problem withlink traffic counts (Kim, 2008). There are multiple OD flows thatgenerate the same link arrival flows. Similarly, more than twodeparture flow patterns can generate the same arrival flow pattern.In this regard, the present study proposes a heuristic approach formodeling departure choice behavior. In the developed model, anOD table is defined in terms of the arrival time instead of thedeparture time.

4. Developing an Objective Function for an Ar-rival Time-based DTA Model

Developing a new objective function for an arrival time-basedDTA model starts from previous work. Kim and Jayakrishnan(2006) proposed three formulations: 1) VI (Variational Inequality)formulation, 2) the GLS (General Least Square) method, and 3)the FIFO (First-In-First-Out) violation function. However, theydid not propose any operational model. In Kim (2009), anoperational model was formulated as a GLS function as follows:

(3)

where is the simulated arrival flows and is thegiven arrival time-based OD flows. The rationale behind theobjective function in Eq. (3) is simple. It only minimizes the dis-crepancy between the simulated and preferred arrival OD flowpatterns. It is assumed that if the magnitude of the discrepancy

between the simulated arrival flows and the given flows isminimized, then the optimal departure flow can be estimated.Compared with the schedule delay approach, there is onedifference in Kim (2009)’s model. As shown in Eq. (1), ODtravel time (t ij(tdp)) is defined as a deterministic value in the DUEassignment model because the travel time of all used path atequilibrium is equal to the minimum OD travel time. Conversely,Kim (2009) assumed that OD travel time is a probabilistic distri-bution. Using a probabilistic distribution for molding OD traveltime is useful when there is vehicle dispersion in the network.The OD travel time distribution is denoted as Φ ij(tdp,tar). This is aportion of OD ij’s departure flows that depart during the timeinterval tdp and arrive during the time interval tar. Then the arrivalflows ( ) can be calculated by summing the OD departureflows (Tij(tdp)):

(4)

where Φ ij(tdp,tar) is defined as shown in Fig. 4. As explainedabove, there are numerous departure flow combinations that canreproduce the same arrival flow pattern. Accordingly, withminimizing the objective function, Eq. (3), with a constraint, Eq.(4), finds only a feasible solution among a number of multiplesolutions. To increase the probability for finding a more plausibleone, the present study takes some behavioral aspects into account.

The estimation of departure flow patterns in this study relies onthree assumptions. The first assumption is that the discrepancybetween simulated arrival pattern and the given preferred arrivaldistribution is minimal. The second is that travelers want to delaytheir departure times as much as possible. The third is that alltravelers in the same OD pair have the same travel time and thatit is equal to the minimum OD travel time. The third conditiontakes the DUE (Dynamic User Equilibrium) principle intoaccount. With the three assumptions, a heuristic GLS objectivefunction is formulated as Eq. (5).

(5)

The meaning of the first term in Eq. (5) is straightforward. The

Z Min T̂ij tar( ) Tij tar( )–{ }2

tar T∈∑

ij OD∈∑=

T̂ij tar( ) Tij tar( )

T̂ij tar( )

T̂ij tar( ) Tij tdp( ) Φij tdp tar,( )⋅tdp T∈∑=

Z Min T̂ij tar( ) Tij tar( )–{ }2

tar t∈∑

ij IJ∈∑ Tij tdp( ) tref tave

ij– tdp–( )⋅tdp t∈∑

ij IJ∈∑+=

Tij tdp( ) tij tdp( ) π ij–( )⋅tdp t∈∑

ij IJ∈∑+

Fig. 4. Definition of the Probability Distribution of OD Travel Time

Fig. 3. A Sample Schedule Delay Function


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discrepancy between simulated arrival flows and preferredarrival flows should be minimal. In the second term, Tij(tdp)represents the departure flows for OD ij at the time interval tdp,and tref is the reference time for calculating the in-home activityduration. Further, I is the average travel time from the origin ito the destination j.

As shown in Fig. 5, if the second term is minimized, a driverdelays his departure from the current location as long as possible.For morning commuters, this means that the duration of their in-home activity is maximized. In an activity perspective, the utilityof in-home activity is generally higher than that of waiting forsubsequent activity participation. Instead of waiting at their officewithout work, most workers delay their departure times. The thirdterm is introduced to consider the departure time equilibriumprinciple. According to the DUE principle, all departure timeswith flows should have the same OD travel time, and its valueshould be smaller than departure times without flows. In theequation, π ij is the minimum OD travel time for the OD pair ij.Note that the second and third terms always have a positivevalue; thus, squaring them is unnecessary.

5. Developing Solution Algorithms

5.1 A Gradient-based MethodSpiess (1990) proposed a gradient method for a static OD

estimation problem. In the study, the methodology was appliedto a departure flow estimation problem. The updating proceduremakes use of the gradient of the objective function with respectto the departure flow of a single OD pair. For the gradientcalculation, has to be substituted by OD departure flowsTij(tdp) with Eq. (4). From the proportions (Φ ij(tdp,tar)), the traveltime distribution for the OD pair ij can be identified. If the traveltime can be determined as a deterministic value, only one arrivaltime interval has a unit value for Φ ij(tdp,tar), and other arrival timeintervals should have zero value for it. In this case, all departingvehicles during the same time interval for the same OD pairreach the destination during the same time interval. However, ifvehicle dispersion is allowed, then multiple arrival time intervalshave arriving vehicles departing at the same departure timeinterval for the same OD pair.

From Eqs. (4) and (5), the gradient of Z with respect to thedeparture flow can be calculated as follows:

(6)

Denote the gradient of the objective function with respect tothe OD departure flow by . To updateOD departure flows, a recursive formula for OD departure flowsis given as follows:

(7)

By plugging Eqs. (4) and (7), Eq. (5) is reformulated as afunction of λ as follows:

(8)

To find the minimum of Z, the optimal λ value, whichminimizes the Z value, is found from differentiating Z by λ:

(9)

The optimal λ can be found by setting Eq. (9) to be equal tozero. If Eq. (9) is rearranged with respect to λ, then an equationfor calculating the optimal λ can be given as follows:

(10)

Where,

(i) ,and

(ii) .

5.2 A Convex-combination MethodThe gradient algorithm proposed by Spiess (1990) is based on

a simple mathematical idea. At the optimal solution of theobjective function, the first derivative of the function with respectto an independent variable is equal to zero. The gradient value ∇ij

taveij

T̂ij tar( )

∂Z∂Trs tdp'( )-------------------- 2 Tij tdp( )

tdp t∈∑ Φij tdp tar,( )⋅ Tij tar( )–

⎩ ⎭⎨ ⎬⎧ ⎫

Φrs tdp' tar,( )⋅tar t∈∑

ij IJ∈∑⋅=

tref tavers– tdp'–( ) trs tdp( ) π rs–( )+ +

2 T̂ij tar( ) Tij tar( )–{ } Φij tdp tar,( )⋅[ ]tar t∈∑

ij IJ∈∑⋅=

tref tavers– tdp'–( ) trs tdp( ) π rs–( )+ +

∂Z ∂Trs tdp'( )⁄ ∇ij tdp( )=

Tn 1+ij tdp( ) Tn

ij tdp( ) 1 λ ∇nij tdp( )⋅–{ }⋅=

min Z Tnij tdp( ) 1 λ ∇n

ij tdp( )⋅–{ } Φnij tdp tar,( )⋅ ⋅[ ]

tdp t∈∑ Tij tar( )–⎝ ⎠

⎛ ⎞2

tar t∈∑

ij IJ∈∑=

Tnij tdp( ) 1 λ ∇n

ij tdp( )⋅–{ } tref taveij– tdp–( )⋅ ⋅

tdp t∈∑

ij IJ∈∑+

Tnij tdp( ) 1 λ ∇n

ij tdp( )⋅–{ } tij tdp( ) π ij–( )⋅ ⋅[ ]tdp t∈∑

ij IJ∈∑+

∂Z∂λ------ 2 Tn

ij tar( ) Tij tar( )–{ }tar t∈∑

ij IJ∈∑ Tn

ij tdp( )tdp t∈∑ ∇n

ij tdp( ) Φij tdp tar,( )⋅⋅–⎩ ⎭⎨ ⎬⎧ ⎫

⋅ ⋅=

2λ Tnij tdp( ) ∇n

ij tdp( ) Φij tdp tar,( )⋅ ⋅tdp t∈∑

⎩ ⎭⎨ ⎬⎧ ⎫2

tar t∈∑

ij IJ∈∑+

∇nij tdp( ) Tn

ij tdp( ) tref taveij– tdp–( )⋅ ⋅

tdp t∈∑

ij IJ∈∑–

∇nij tdp( ) Tn

ij tdp( ) tcij tdp( ) π ij–( )⋅ ⋅tdp t∈∑

ij IJ∈∑+

λ i( ) ii( )+

2 Tnij tdp( )

tdp t∈∑ ∇n

ij tdp( ) Φij tdp tar,( )⋅⋅⎩ ⎭⎨ ⎬⎧ ⎫2

tar t∈∑

ij IJ∈∑⋅

---------------------------------------------------------------------------------------------------------=

=2· Tnij tar( ) Tij tar( )–{ }

tar t∈∑

ij IJ∈∑ · Tn

ij tdp( )tdp t∈∑ Φij tdp tar, ·∇n

ij tdp( )( )⋅⎩ ⎭⎨ ⎬⎧ ⎫

∇nij tdp( ) Tn

ij tdp( ) tref taveij– tdp–( ) tij tdp( ) π ij–( )+{ }⋅⋅

tdp t∈∑

ij IJ∈∑=

Fig. 5. Maximization of the In-home Activity Duration in the Morn-ing



(tdp) (=∂Z/∂Trs(tdp)) indicates the reduction in Z by a unit changein Tij(tdp). Consequently, the gradient ∇ij(tdp) is an approximateddescent direction of Z. In terms of the current Tij(tdp), the descentdirection is relatively accurate, but the accuracy of the approxi-mation becomes worse as it moves away from the current Tij(tdp).As a result, the quality of gradient information is guaranteedwithin a limited boundary of the current Tij(tdp), and moving toofar along the approximated descent direction is not desirable. Inthis context, a heuristic updating method for OD departure flowsis developed. The method is formulated by using a convex-com-bination method. A recursive formula for updating OD departureflows at the n iteration is formulated as follows:

(11)

where is an auxiliary variable for OD departure flows atthe n iteration. The role of the auxiliary variable in the convexcombination method is to define the maximum boundary forchanges in OD departure flows. The values arecalculated by the equilibrium of the gradient. As Kim (2008)explained, all departure times should have the same ∇ij(tdp)values at the optimum. Assume that all departure flows have thesame gradient value. Note that a whole departure flow shouldbe conserved during the updating process. Accordingly, theincrease in the flow at a departure time interval decreases theflow at the other departure time. If ∇ij(tdp) perfectly reflects thesensitivity of the objective function value with respect to thedeparture flow change, then there is no decrease in theobjective function through the change in departure flows in thecondition below. If:

(12)

then no descent direction exists. This means that a small flowtransfer from one interval to another interval cannot decrease theZ value. For instance, if we move a unit flow at interval 1 tointerval 2, then the expected change in the objective functionvalue is:

(13)

As a result, if the gradients of all departure time intervals areequal, then no additional updates are possible in the algorithm,and the updating process stops.

The gradient equilibrium can be found in an iterative way. Asshown above, ∇ij(tdp') shows the change in the objective functionvalue Z corresponding to the change in the departure flow.Accordingly, the Z value can be decreased by moving some de-parture flows from high-gradient departure time intervals (∇ij(tdp)>∇ij

ave) to low-gradient departure time intervals (∇ij(tdp)≤ ∇ijave ). In

other words, if ∇ij(tdp)≤ ∇ijave, then the departure time tdp should

have more departure flows. Otherwise, the departure time losesdeparture flows.

The magnitude of the departure flow update at a specificinterval is assumed to be proportional to the discrepancy between

the average gradient and the gradient of the departure timeinterval. A heuristic equation for calculating the auxiliary variableDn

ij(tdp) is formulated as follows:

if (14a)

otherwise (14b)

In Eq. (14a), Fnij represents the total changeable flows that can

be calculated as follows:

(15)

where tdp* denotes a specific departure time.

Figure 6 illustrates an example of the calculation for Dnij(tdp). If

∇ij(tdp)>∇ijave, then departure flows at the interval (tdp) will de-

crease. In the example in Fig. 6, all OD departure flows are 70vehicles over the entire departure period. The departure intervalsfrom 11th to 14th have the maximum gradient, so the ODdeparture flows for the intervals should decrease as much aspossible. As a result, the auxiliary variables for the intervals areset to zero. In the lower figure, the sum of white bars is equal toFn

ij. After calculating the Fnij value, it should be distributed to

low-gradient departure intervals (∇ij(tdp)≤ ∇ijave). In the figure, the

intervals from the 28th departure time to the 57th departure timeget additional departure flows according to the size of the gapbetween ∇ij(tdp) and ∇ij

ave. In the figure, the total Dnij(tdp) value is

equal to the sum of the colored and hatched parts.If we plug Eqs. (4) and (11) into Eq. (5), then we have Eq. (16)

Tn 1+ij tdp( ) Tn

ij tdp( ) λ Dnij tdp( ) Tn

ij tdp( )–{ }⋅+=

Dnij tdp( )

Dnij tdp( )

∂Z∂Tij 1( )---------------- ∂Z

∂Tij 2( )---------------- … ∂Z

∂Tij N( )-----------------= = =

Z∆ ∂Z∂Tij 1( )----------------= 1–( )⋅ ∂Z

∂Tij 2( )----------------+ +1( )⋅ 0=

Dnij tdp( ) Tn

ij tdp( ) Fnij+

∇aveij ∇ij tdp( )–{ }

∇aveij ∇ij tdp'( )–{ }

td ' t∈∑

------------------------------------------------⋅=

∇ij tdp( ) ∇aveij≤

Dnij tdp( ) Tn

ij tdp( ) Tnij tdp( )–

∇ij tdp( ) ∇aveij–( )

Max ∇ij tdp'( ) ∇aveij–{ }

-------------------------------------------------⋅=tdp' t∈

Fnij tdp( ) Tn

ij tdp*( )

tdp* t∈∑

∇ij tdp* ( ) ∇ave

ij–( )Max ∇ij tdp'( ) ∇ave

ij–{ }-------------------------------------------------⋅=tdp' t∈

∇ij tdp( ) ∇aveij>

Fig. 6. Calculation of the Auxiliary Variable based on a GradientValue


Vol. 16, No. 4 / May 2012 − 639 −

as follows:

(16)

Finally, the only variable in Eq. (16) is λ, and it can be foundby a conventional interval reduction method such as the goldensection method or the bi-section method.

The present study proposes two solution algorithms for theobjective function, Eq. (5). Between the two, the convex-combi-nation approach has a better capability for finding the optimalsolution. The results of the numerical tests show that the convex-combination method is superior to the gradient-based method.The next chapter details the test. This study used DYNASMART-P (Mahmassani et al., 2004) to execute a dynamic trafficassignment with the UE option.

6. Results of the Model Test

6.1 Test Problem 1To investigate the properties of the developed model, two

numerical tests were designed. The first network is a toy networkwith a single OD pair. It was designed for an in-depth modelbehavior analysis. Fig. 8 shows Test Network 1, in which thelength and capacity of links are denoted. The total number of

vehicles traveling from Node 1 to Node 2 is 4,200 (vehs).With the given initial departure flows, the error in the simulated

arrival flows was 43.33% with respect to the preferred arrivalflows, but it decreased to 3.08% after 38 iterations. Fig. 9 showsthe history of arrival flow updates. In the figure, the length of asingle arrival interval is 10 (min), so the 7th arrival interval rangesfrom the 60th minute to the 70th minute from the start of the simu-lation. In the initial simulation (i.e., iteration 0), the arrival flowpattern was flat, but it kept converging to the given arrival demandpattern over iterations. The trivial discrepancy between the simu-lated and given arrivals at the 38th iteration supported the capabilityof the developed algorithm. Note that the optimal arrival patternmay be coincident with the given. If there is a capacity constraintat the link exit, the preferred arrival flow during a specific timeinterval is larger than the capacity. In that case, the given arrivalvehicle numbers cannot be achieved exactly.

Figure 10 shows the history of departure flow updates. In theinitial simulation (i.e., at iteration 0), a uniform departure patternfrom the 11th minute to the 70th minute was assumed. The size ofthe uniform departure flow rate was 70 (vehs/min). With iterations,a peak became apparent, and the maximum flow rate was higherthan 400 (vehs/min). After the 50th interval, another departurepeak appeared, which represented the late-departure vehicles.However, the magnitude was not large. Fig. 11 shows the influ-ence of the initial departure pattern on the final departure flowsolution. The base case used a uniform departure pattern fromthe 11th minute to the 70th minute with 70 (vehs/min). Case 1 alsoassumed a uniform flow rate from the 6th minute to the 75th

minute with 60 (vehs/min), and Case 2 had a biased peak patternranging from the 11th minute to the 70th minute. The final depar-ture patterns nearly overlapped one another, although all cases

Z Min Tnij tdp( ) λ Dn

ij tdp( ) Tnij tdp( )–{ }⋅+[ ] Φij tdp tar,( )⋅

tdp t∈∑ Tij tar( )–⎝ ⎠

⎛ ⎞2

tar t∈∑

ij IJ∈∑=

Tnij tdp( ) λ Dn

ij tdp( ) Tnij tdp( )–{ }⋅+[ ] Φij tdp tar,( ) tref tave

ij– tdp–( )⋅⋅tdp t∈∑

ij IJ∈∑+

Tnij tdp( ) λ Dn

ij tdp( ) Tnij tdp( )–{ }⋅+[ ] Φij tdp tar,( ) tij tdp( ) π ij–( )⋅ ⋅

tdp t∈∑

ij IJ∈∑+

Fig. 9. Simulated Arrival Flows over Iterations

Fig. 7. Process of Departure Flow Estimation

Fig. 8. Test Network 1 and the Total OD Volume Fig. 10. History of Departure Flow Updates with Iterations



started from substantially different initial patterns.As explained, the developed model has three terms as shown

in Eq. (5). However, Kim (2009) used the single-term modelshown in Eq. (3). The single-term model takes only the gapbetween a preferred arrival flow pattern and a simulated one intoaccount. Accordingly, the first-term model stops the departureflow adjustment when the gap becomes trivial.

As shown in Fig. 11, the departure pattern resulting from thesingle-term model had a lower peak and late departure flows.The initial departure pattern of the single-term model was thesame as that of the base case of the three-term model. However,more people departed late in the three-term model case. Theresults of the three-term models are consistent with the rationaleof Eq. (3) because people delayed their departure times as muchas possible to maximize their in-home activity time.

6.2 Test Problem 2The topology of Test Network 2 shown in Fig. 12 follows the

famous Nguyen-Dupuis network, but the network attributes arearbitrarily assumed.

As shown in Fig. 13, the decreases in the objective function bythe convex-combination method and the gradient method werecompared. During the first several iterations, the gradient-basedmethod reduced the value of the objective function more than theconvex-combination method. However, the superiority of the

gradient-based method disappeared after the 22nd iteration. Afterthe 22nd iteration, the convex-combination method gave a lowerobjective value. Another important difference between the twomethods was the stability of the updating process. As shown inFig. 13, the convergence trajectory of the gradient method hadgreater fluctuations. Accordingly, the convex-combination methodshowed better efficiency. Therefore, the convex-combinationmethod was used in the following numerical analysis.

Figure 14 shows the estimated departure flow patterns of fourOD pairs. In all OD pairs (except the OD pair 4→3), driversbegan departing from the 31st minute; drivers in the OD pair4→3 started to depart from the 38th minute. In addition, the ODpairs 1→2 and 4→3 had smoother departure patterns with lowerpeaks in comparison with the OD pairs 1→3 and 4→2. Theseresults might have been due to the OD travel time. The averageOD travel times for 1→2 and 1→3 were 28.27 (min) and 30.96(min), respectively. The average OD travel time for the OD pair4→2 was 33.24 (min), but it was only 23.77 (min) for the ODpair 4→3. As a result, some drivers in the OD pair 4→3 departeduntil the 76th minute. By contrast, no driver in other OD pairsdeparted after the 71st minute.

Figure 15 shows the change in the simulated arrival flows overiterations. At iteration zero, a large number of vehicles arrivedearlier than the given preferred times. For example, no driver

Fig. 14. The Resultant Departure Flows for Four OD Pairs

Fig. 11. Departure Flows Resulting from Initial Solutions and a Sin-gle-term Model

Fig. 12. Test Network 2 and the Total OD Volume

Fig. 13. The Convex-Combination Method and the Gradient Method


Vol. 16, No. 4 / May 2012 − 641 −

wanted to arrive before the 7th interval in the OD pair 1→2, butapproximately 500 vehicles reached Node 2 before the timeinterval. The early arrival made the early arrival travelers delaytheir departure times.

As drivers delayed their departure times, their arrivals approachedtheir preferred times. At the 57th iteration, no substantial differ-ence between the simulated arrivals and the given arrivals wasfound. In general, late arrivals can be further adjusted by addi-tional iterations, but occasionally, the late arrivals can arise becauseof the random process in traffic simulation packages such asDYNASMART-P. This means that a different arrival pattern canoccur even without any departure flow change. Thus, to minimizethe random effect, multiple DTA runs are required when updat-ing departure flows. However, the present study adopted a single-execution approach to decrease the calculation burden.

7. Applications to Transportation Network Prob-lems

7.1 Network Capacity Increase (Change in the supply side)To investigate the role of departure time choice in the trans-

portation network analysis, two scenarios were designed. In thefirst scenario, the supply side of the network was expanded. As

shown in Fig. 2, the incorporation of departure time may result indifferent levels and patterns of traffic congestion. This test com-pared a fixed departure flow case with a departure time choicecase. The network improvement scenario is shown in Fig. 16, inwhich the two corridors’ capacities were increased. In addition,the free-flow speed of the links in the network was also increasedto 160 (km/hr).

Figure 17 shows the changes in departure and arrival flowsafter the network improvement. On the OD pair 1→2, the depar-ture flow profile of the fixed arrival case moved to the right. Itmeans that travelers delayed departure time after the networkimprovement. By contrast, the early arrival flows increased underthe fixed departure time assumption. Because of the shortenedOD travel time, drivers reached their destinations earlier thanbefore. In Fig. 17(b), late departures were observed after thenetwork improvement. The latest departure was delayed by morethan 10 minutes. By contrast, when the departure times of driverswere fixed, the drivers arrived more than 10 minute earlier. Thesimulation results are consistent with the expectation shown inFig. 2. When we used the OD table with fixed departure times,the early arrivals at destinations increased; this is inconsistentwith the results of prior activity-related research. The resultsshown in Fig. 17 support the hypothesis; in addition, they suggestthat the developed model can correctly forecast a temporalnetwork flow pattern.

The resultant number of running vehicles over the networkshown in Fig. 18 indicates why the consideration of departuretime is important in the transportation network analysis. Whenthe network was improved, two different traffic peaks appearedby two different departure time assumptions. With a fixed depar-ture time assumption, the peak of running vehicles disappearedearlier than before. By contrast, with a fixed arrival time assump-tion, the peak appeared later than before and disappeared at

Fig. 15. Arrival Flow Patterns of Four OD Pairs Over Iterations

Fig. 16. The Network Improvement ScenarioFig. 17. The Change in Departure/arrival Flows with Network Cap-

acity Increase: (a) OD Pair 1→2, (b) OD Pair 4→2



nearly the same time. Trips requiring fixed arrival times representthe bulk of morning commuters, and these commuters do notwant to spend their free time at the office waiting for the work tostart. Based on this rationale, the delayed peak generated by fixedarrival flows provides a more plausible temporal traffic pattern.

7.2 Travel Demand Increase (Change in the demand side)In this scenario, the impact of increases in travel demand under

the fixed departure time and fixed arrival time assumptions wasinvestigated. In the case of fixed departure time, departure flowsincreased proportionally. The flows in the OD pairs 1→2 and4→3 increased 1.8 times, and those in the other OD pairsincreased 1.2 times. Fig. 19(a) shows the change in departureflow patterns. The increase in travel demand increased OD traveltime. Consequently, two important changes were observed. First,the first and last departures occurred earlier than before after theincrease in travel demand. This means that travelers had todepart earlier. As shown in the right side of Fig. 19(a), the fixeddeparture time resulted in a tardy arrival pattern. Fig. 19(b)illustrates a similar tendency. Note that there were some delayedarrival vehicles even with the fixed arrival time assumption. Thissuggests that in the capacitated network, delayed arrivals may be

observed; thus, heavy penalties can be introduced to minimizelate arrivals. In this test, the number of delayed vehicles wastrivial; thus, a penalty for late arrivals was ignored in the analysis.

Figure 20 illustrates changes in running vehicles with increasesin travel demand. Under the fixed departure time assumption,peak-hour traffic congestion shifted to a later time. In general, adelayed peak is not plausible in the morning peak. If peopleexpect increased travel time, they can depart earlier than usual,which would induce an earlier peak. The fixed arrival time casegiven in Fig. 20 shows a result that is consistent with reality. Asdiscussed, the consideration of departure time choice is crucial intransportation planning. Without including the choice problem,forecasting a correct temporal traffic pattern after a change in thetravel condition or travel demand would be nearly impossible.

8. Conclusions

This paper presents a new estimation method for OD departureflows. The original idea was proposed by Kim and Jayakrishnan(2006), but their study did not propose an operational formula-tion. An executable model was developed in Kim (2009), but theabsence of behavioral consideration and the existence of multi-ple solutions were the shortcomings of the study. To overcomethese problems, the present study proposes a new mathematicalformulation and a solution algorithm. The developed model waseffective in estimating departure flow patterns, even in the multi-ple OD pair case. The present study developed a gradient-based aconvex-combination method as a solution algorithm. Behavioralaspects regarding the maximization of in-home activity time andthe satisfaction of the DUE condition for departure time choicewere also taken into account.

The developed model was applied to two test scenarios. In thefirst scenario, the network capacity and the free flow speed in-creased concurrently, and travel demand was augmented in thesecond scenario. The results indicated that if travelers were tohave preferred arrival times, the OD table with fixed departuretimes would give false forecasts. Thus, to foresee the changes inthe supply or demand side of the transportation system, the con-sideration of departure time choice would be necessary.

The incorporation of departure time choice in the dynamictraffic assignment has been a long-cherished desire of transpor-

Fig. 20. The Change in Running Vehicles in the Network withDemand Increase

Fig. 18. The Number of Running Vehicles on the Network withCapacity Increase

Fig. 19. Changes in Departure and Arrival Flows with TravelDemand Increase: (a) OD Pair 1→3, (b) OD Pair 4→3


Vol. 16, No. 4 / May 2012 − 643 −

tation planners. However, there have been few practical methodsfor this purpose. No commercial DTA model has a departuretime choice module and assumes that dynamic OD flows havefixed departure times. This can be a critical error source in thedynamic transportation network analysis. In this regard, thepresent study proposes a practical remedy. An optimal departureflow pattern can be estimated more easily if we survey a sampledeparture time pattern. Moreover, the preferred arrival patternwould be similar in the morning because the mandatory activitystarting time would not be substantially different from one locationto another. This implies that the preferred arrival pattern exhibitsspatial transferability.

Acknowledgements

This work was supported by the National Research Foundationof Korea grant funded by the Korea government (NRF-2010-0029445) and also supported by Integrated Research Institute ofConstruction and Environmental Engineering Seoul NationalUniversity research Program funded by Ministry of Education &Human Resources Development.

References

Bellei, G., Gentile, G., Meschini, L., and Papola, N. (2006). “A demandmodel with departure time choice for within-day dynamic trafficassignment.” European Journal of Operational Research, Vol. 175,No. 3, pp. 1557-1576

Bhat, C. R. and Steed, J. L. (2002). “A continuous-time model ofdeparture time choice for urban shopping trips.” TransportationResearch Part B, Vol. 36, No. 3, pp. 207-234.

Chang, G.-L. and Mahmassani, H. S. (1988). “Travel time predictionand departure time adjustment behavior dynamics in a congestedtraffic system.” Transportation Research Part B, Vol. 22, No. 3, pp.217-232.

Chen, H. K. (1999). Dynamic travel choice models, Springer.Chow, A. H. F. (2009). “Properties of system optimal traffic assignment

with departure time choice and its solution method.” TransportationResearch Part B, Vol. 43, No. 3, pp. 325-344.

Ettema, D., Tammingga, G., Timmermans, H., and Arentze, T. (2005).“A micro-simulation model system of departure time using a per-ception updating model under travel time uncertainty.” Transpor-tation Research Part A, Vol. 39, No. 4, pp. 325-344.

Fujii, S. and Kitamura, R. (2004). “Drivers’ mental representation oftravel time and departure time choice in uncertain traffic networkconditions.” Networks and Spatial Economics, Vol. 4, No. 3, pp.243-256.

Heydecker, B. G. and Addison, J. D. (2005). “Analysis of dynamictraffic equilibrium with departure time choice.” TransportationScience, Vol. 39, No. 1, pp. 39-57.

Huang, H.-J. and Lam, W. H. K. (2002). “Modeling and solving thedynamic user equilibrium route and departure time choice problemin network with queues.” Transportation Research Part B, Vol. 36,No. 3, pp. 253-273.

Jou, R.-C. (2001). “Modeling the impact of pre-trip information oncommuter departure time and route choice.” Transportation Re-search Part B, Vol. 35, No. 10, pp. 887-902.

Kim, H. (2008). New dynamic travel demand modeling methods inadvanced data collecting environments, PhD Dissertation, Universityof California, Irvine.

Kim, H. (2009). “Dynamic traffic assignment based on arrival time-based OD flows.” Journal of Korean Society of Transportation, Vol.27, No. 1, pp. 143-155

Kim, H. and Jayakrishnan, R. (2006). “Dynamic traffic assignmentbased on arrival time-based OD demand.” The 85th Annual Meetingof the Transportation Research Board, January 22-26, Washington,D.C

Kitamura, R., Pas, E. I., Lula, C. V., Lawton, T. K., and Benson, P. E.(1996). “The Sequenced Activity Mobility Simulator (SAMS) : Anintegrated approach to modeling transportation, land use and airquality.” Transportation, Vol. 23, No. 3, pp. 267-297.

Lam, W. H. K. and Huang, H.-J. (2002). “A combined activity/travelchoice model for congested road network with queues.” Transporta-tion, Vol. 29, No. 1, pp. 5-29.

Lim, Y. and Heydecker, B. (2005). “Dynamic departure time andstochastic user equilibrium assignment.” Transportation ResearchPart B, Vol. 39, No. 2, pp. 97-118.

Mahmassani, H. S., Sbayti, H., and Zhou, X. (2004). DYNASMART-P:Intelligent transportation network planning tool, version 1.9 user’sguide, Maryland Transportation Initiative, University of Maryland.

Ortuzar, J. D. and Willumsen, L. G.. (1990). Modelling transport, JohnWiley & Sons Ltd.

Ran, B., Hall, R. W., and Boyce, D. E. (1996). “A link-based variationalinequality model for dynamic departure time/route choice.” Trans-portation Research Part B, Vol. 30, No. 1, pp. 31-46.

Sheffi, Y. (1985). Urban transportation networks, MIT press, NY.Spiess, H. (1990). A gradient approach for the O-D matrix adjustment

problem, Transportation Research Paper #692, University ofMontreal.

Yang, H. and Meng, Q. (1998). “Departure time, route choice and con-gestion toll in a queuing network with elastic demand.” TransportationResearch Part B, Vol. 32, No. 4, pp. 247-260.

Zhang, X., Huang, H.-J., and Zhang, M. (2008). “Integrate dailycommuting patterns and optimal road tolls and parking fees in alinear city.” Transportation Research Part B, Vol. 42, No. 1, pp. 38-56.

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