significantly

Transportation Research Part B 40 (2006) 368–395www.elsevier.com/locate/trbModeling time-dependent travel choice problemsin road networks with multiple user classesand multiple parking facilitiesWilliam H.K. Lam a, *, Zhi-Chun Li b , Hai-Jun Huang b , S.C. Wong ca Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, Chinab School of Management, Beijing University of Aeronautics and Astronautics, Beijing 100083, Chinac Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, ChinaReceived 26 July 2004; received in revised form 4 May 2005; accepted 17 May 2005AbstractThis paper proposes a time-dependent network equilibrium model that simultaneously considers a travelerÕschoice of departure time, route, parking location and parking duration in road networks with multipleuser classes and multiple parking facilities. In the proposed model, travelers are differentiated by theirtrip purpose and parking duration, parking locations are characterized by facility type and parking charge,and the decision-making process of travelers on travel and parking choices is assumed to follow a hierarchicalchoice structure. The model is formulated as a variational inequality problem, and is solved by a heuristicsolution algorithm. Numerical results for two example networks are presented to show the solutionquality and investigate the solution sensitivities to some input data. It is found that parking behavior is significantlyaffected by travel demand, walking distance, parking capacity, and parking charge. The proposedmodel provides a useful tool for studying the complex temporal and spatial interaction between road trafficand parking congestion, and can be used to assess the effects of various parking policies and infrastructureimprovements at a strategic level.Ó 2005 Elsevier Ltd. All rights reserved.* Corresponding author. Tel.: +852 2766 6045; fax: +852 2334 6389.E-mail address: cehklam@polyu.edu.hk (W.H.K. Lam).0191-2615/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.trb.2005.05.003

W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 369Keywords: Time-dependent network equilibrium; Hierarchical choice structure; Multiple user classes; Multiple parkingfacilities; Parking location; Parking charge; Variational inequality1. IntroductionParking has become an increasingly serious problem in large cities around the world due tothe dramatic increase in the number of vehicles. Consequently, traffic conditions, particularlyin peak periods, have tended to deteriorate in many urban areas, and the searching time forparking has substantially increased in the central business districts. In recent years, researchershave advocated road-use pricing strategies to alleviate peak-period traffic congestion (Hau,2005a,b). Although the successes in Singapore and recently in London of such schemesprovide us with many ideas, it is generally believed that road pricing is difficult to widelyimplement in practice because of social and political impediments (Lewis, 1994; Button andVerhoef, 1998). The attention of researchers, therefore, has been mainly focused on alternativetransport policies that balance traffic demand and supply. Parking is closely tied to road usage(Glazer and Niskanen, 1992), and every car has to park at the end of a trip. Hence, parkingpolicy offers a potentially strong instrument for influencing road traffic flows (Verhoef et al.,1995).It is widely recognized that parking policy plays an important role in the planning and managementof transportation systems in densely developed urban areas (Young et al., 1991; Lam et al.,1998; Wong et al., 2000; Lau et al., 2005). It has been observed that drivers spend a significantpercentage of their total trip time searching for a parking space (Axhausen and Polak, 1991),which significantly affects the level of traffic congestion and environmental quality within anurban area (Feeney, 1989). Parking, as an integral component of urban transportation systems,influences not only the parking system itself, but also the whole transportation system and eventhe socio-economic system (Bifulco, 1993). To make urban transportation systems work more efficiently,it is necessary to understand and predict the parking choice behavior of travelers undervarious parking policies, and also their travel choice behavior in relation to changes in parkingsupply and parking charges.There is a substantial body of literature on parking and travel choice models. Generally, threeclasses of modeling approaches can be classified: the discrete choice approach (Van der Goot,1982; Polak et al., 1990; Hunt and Teply, 1993; Lambe, 1996; Thompson and Richardson,1998; Hensher and King, 2001; Hess and Polak, 2004), the probability-based approach (Wonget al., 2000; DellÕOrco et al., 2003), and the network-based approach (Dirickx and Jennergren,1975; Florian and Los, 1979, 1980; Nour Eldin et al., 1981; Gur and Beimborn, 1984; Goyaland Gomes, 1984; Spiess, 1993, 1996; Bifulco, 1993; Lam et al., 1999a, 2002; Sattayhatewa andSmith, 2003; Tong et al., 2004). However, these studies mainly focus on time-stationary (static)equilibrium analyses, and little attention has been paid to time-dependent analysis. Static modelsare also unable to reveal the temporal and spatial interaction between road traffic and parkingcongestion. In view of this, Bifulco (1993) developed a quasi-dynamic model in which the traveldemand in each time interval is in a steady-state equilibrium. It is assumed that travelers start and

370 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395complete their journeys within the same time interval, and that the connection between successivetime intervals is represented by parking occupancy that is carried over to the next interval. However,BifulcoÕs model cannot be used to examine the choice sensitivity of departure time, route,parking lot, and parking duration to the time-varying characteristics of road networks and parkingfacilities.On a strategic level, researchers have recently paid much attention to the development ofvarious time-dependent travel choice models (Bell et al., 1996; Lam et al., 1999b; Tong andWong, 2000; Lam and Yin, 2001; Huang et al., 2003). However, studies that combine the parkingproblem with a time-dependent travel choice in a unified framework have not yet been explored.There is actually a close relation between departure time choice and parking activity,and departure time choice may significantly interact with the choice of parking location.Specifically, parking spaces in an urban center, especially in the central business districts,are usually fully occupied during peak hours, but are often unoccupied during off-peak hours.Therefore, travelers who depart during rush hours may spend a lot of time searching for anavailable parking space because of the high parking occupancy at that time. This may also giverise to an increase in cruising time on the road network, which has an adverse impact on trafficcongestion and environmental pollution. Consequently, travelers may consider changing theirdeparture times on future trips. In contrast, the characteristics of parking locations, such asparking availability, searching time delay for parking, and parking charges, may plausiblybe expected to vary according to the time of day, which will also affect the decisions oftravelers on the start and end times of their activities. Therefore, travelers may change theirchoices of departure time and parking duration according to the characteristics of differentparking locations at different times of day. For instance, higher parking charges in rush hoursmay compel travelers to shorten their parking durations or depart from their origins earlier orlater.In this paper, we propose a network equilibrium model for studying the joint behavior of traveland parking in a time-dependent paradigm. We assume that the decision-making process of atraveler on travel and parking choice follows a hierarchical choice structure. The time-dependentnetwork equilibrium model is formulated as an equivalent variational inequality (VI) problem.The proposed model is a significant extension of the existing static models, because it simultaneouslyconsiders the choices of travelers on departure time, route, parking location, and parkingduration in road networks with multiple user classes and multiple parking facilities. We develop aheuristic algorithm to solve the equivalent VI problem, and compare the results that are generatedby our model with the static version through numerical experiments.It should be noted that the time-dependent model that is proposed in this paper belongs to theclass of quasi-dynamic models for strategic planning purposes, because the physical queues ofvehicles at road intersections are not considered explicitly. Moreover, the period that is used inour model is generally longer than which is used in dynamic models (Daganzo, 1994; Peetaand Mahmassani, 1995; Lo, 1999; Tong and Wong, 2000).The remainder of this paper is organized as follows. In the next section, some basic componentsof the proposed model are first described. In Section 3, a time-dependent network equilibriummodel is formulated. A heuristic solution method is presented in Section 4. Section 5 providestwo numerical examples to demonstrate the effectiveness of the proposed model and solution algorithm.Finally, the concluding remarks are given in Section 6.

2. Basic components of the model2.1. AssumptionsTo facilitate the presentation of the essential ideas without loss of generality, the following basicassumptions are made in this paper:A1. The whole study period [0,T] is discretized into equal time intervals that are sequentiallynumbered t 2 T ¼f0; 1; ...; T g, and d is the length of an interval so that T d ¼ T . As the proposedmodel aims to assess the effects of various parking policies and infrastructure improvements forlong-term strategic planning, we use longer time intervals, such as 1 h (Lam et al., 1999b), 30 minor 15 min (Janson, 1991; Janson and Robles, 1995). It is assumed in this paper that the value of Tis sufficiently large to ensure that all travelers can complete their journeys within the study period[0,T].A2. All travelers in the network are differentiated by their trip purposes (e.g., work and nonwork)and parking duration. Parking facilities are characterized by type (e.g., on-street and offstreet)and parking charging scheme. Each class of travelers can choose a particular set of parkinglocations.A3. The choice of departure time is a pre-trip decision. This assumption has also been adoptedby many related studies, such as that of Ran et al. (1996) and Yin and Lam (2002), and is appropriatefor assessing transportation policies at a strategic level.A4. The decision-making process on travel and parking choices follows a hierarchical choicestructure, which means that travelers first make choices based on the time at which they will departfrom the origin and how long they will park their car at the destination. Specifically, the travelersfirst determine the start and end times of their activities, and then select a desirable parkinglocation that will minimize their travel costs from origin to destination. Finally, they choose theshortest route to reach the chosen parking location. This decision-making process is rational, becausewhen travelers choose a parking location, they always consider the following cost components:the travel time from origin to parking location, the searching time delay for parking,parking charge at the parking location, and the walking distance from the parking location tothe destination. Based on the trade-off between these components, travelers generally prefer to selecta parking location with a minimal total cost.A5. The parking charge depends on both the time at which a traveler enters the parking locationand the parking duration. Walking time depends on the distance between the parking locationand the destination, and also walking speed (Dirickx and Jennergren, 1975; Bifulco, 1993).2.2. Travel disutilityW.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 371Consider a transportation network G =(N,A), where N is the set of all nodes, including originnodes, intersection nodes, destination nodes, and parking locations, and A is the set of all directedlinks, including centroid connectors from the origin to the road network, road links, parking accesslinks from the road network to the parking location, and footways or walk links from theparking location to the final destination. Let R denote the set of origin nodes, r represent a singleorigin node r 2 R N, S denote the set of destination nodes, and s represent a destination nodes 2 S N. Let J denote the set of trip purposes of all travelers and j denote an element in J. Let L

372 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395denote the set of all possible parking durations (for instance, 1 h, 2 h, or 3 h) and l an element inL. Let M denote the set of all user classes and m a user class with a specific trip purpose j andparking duration l. Hence, M ={m:(j,l) 2 J · L}, where ‘‘·’’ denotes the Cartesian product.Within the context of a static demand–supply equilibrium with multiple user classes and multipletypes of parking facilities, Tong et al. (2004) adopted a set of admissible links to connect a setof demand locations and a set of parking locations in accordance with the admissibility of a vehicleclass to a particular type of facility. In the same vein, we introduce the concept of a feasibleparking location to capture the interrelation between parking supply and travel demand.Definition 1 (Feasible parking location). A parking location i is referred to as feasible for a certainorigin–destination (OD) pair (r,s) and for a certain user class if and only if the walking distancefrom the parking location to the destination is within an acceptable threshold.Define I m rsas the set of feasible parking locations by user class m for OD pair (r,s):I m rs ¼fi : Cði; sÞ 6 Dm rsg; 8m 2 M; r 2 R; s 2 S; ð1Þwhere C(i,s) is the physical distance of the walk link between parking location i and destination s,and D m rsis the threshold for the acceptable walking distance that depends on the OD pair and userclass. It is noted that other factors that are associated with parking location, such as parkingcharges and parking types, can also be incorporated into the criteria for the determination of feasibleparking locations in addition to the walking distance as in Definition 1. From this definition,it can easily be understood that a parking location is feasible for a certain specific user class andOD pair, but may be infeasible for other user classes and OD pairs.Define C m rs;iðtÞ as the disutility (measured in time unit) for user class m departing from origin r atinterval t and traveling to destination s via parking location i, which is formulated asC m rs;i ðtÞ ¼T m ri ðtÞþam 1 dm i ðt þ T m ri ðtÞÞ þ am 2 zm i ðt þ T m ri ðtÞÞ þ am 3 wm isþ a m s Hm s ðt þ T m ri ðtÞþdm i ðt þ T m ri ðtÞÞ þ wm is Þ; 8i 2 Im rs ; m 2 M; r 2 R; s 2 S; t 2 T ; ð2Þwhere T m riðtÞ is the in-vehicle travel time for user class m departing from origin r at interval t andtraveling to parking location i, d m i ðt þ T m riðtÞÞ is the searching time delay for an available parkingspace for user class m departing from origin r at interval t and arriving at location i at intervalt þ T m ri ðtÞ, zm i ðt þ T m ri ðtÞÞ is the parking charge for user class m at location i at interval t þ T m ri ðtÞ,w m isis the walking time for user class m from location i to destination s, and H m s ðt þ T m ri ðtÞþd m i ðt þ T m ri ðtÞÞ þ wm isÞ is the schedule delay cost of early or late arrival at destination s for user classm departing from origin r at interval t and arriving at destination s via parking location i at intervalt þ T m ri ðtÞþdm i ðt þ T m ri ðtÞÞ þ wm is. Some terms of Eq. (1) are multiplied with coefficients (a) forthe purpose of converting different quantities to the same unit that is defined in the disutilityfunction.Let C m is ðt þ T m riðtÞÞ beC m is ðt þ T m ri ðtÞÞ ¼ am 1 dm i ðt þ T m ri ðtÞÞ þ am 2 zm i ðt þ T m ri ðtÞÞ þ am 3 wm isþ a m s Hm s ðt þ T m ri ðtÞþdm i ðt þ T m ri ðtÞÞ þ wm is Þ;8i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T .ð3Þ

We then haveC m rs;i ðtÞ ¼T m ri ðtÞþCm is ðt þ T m ri ðtÞÞ; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T . ð4ÞLet the searching time delay for an available parking space be computed by the following Bureauof Public Roads (BPR) function (Lam et al., 1999a):d m i ðt þ T m riðtÞÞ ¼ d0miþ 0.31 D iðt þ T m ri ðtÞÞ 4.03;C i8i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð5Þwhere d 0miis the free-flow parking access time at parking location i, D i ðt þ T m riðtÞÞ is the parkingaccumulation or occupancy at location i when travelers in user class m depart at intervalt from origin r and arrive at location i at interval t þ T m ri ðtÞ, and C i is the capacity of parking locationi.In the following, we derive the parking accumulation for location i that is used in Eq. (5). ForOD pair (r,s), the cumulative arrivals of user class m, U m riðtÞ, at parking location i by interval t (butexcluding interval t) is given by1U m riðtÞ ¼Xtn¼1X Xf mri;p ðkÞ; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð6Þk p2P rikþT m ri ðkÞ¼nwhere P ri is the set of all routes between origin r and parking location i, and fri;p m ðkÞ is the departureflow rate of user class m on route p between origin r and parking location i during interval k.Therefore, the total cumulative arrivals of user class m from all of the OD pairs, U m iðtÞ, at parkinglocation i by interval t can be computed byU m i ðtÞ ¼X U m ri ðtÞ; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T . ð7Þr;sFor a certain OD pair and user class m with parking duration l, travelers who arrive at parkinglot i before interval (t l) have already left parking location i till interval t, and thus the cumulativedepartures of user class m, V m riðtÞ, from parking location i by interval t is given bytV m ri ðtÞ ¼ X1l Xf mri;p ðkÞ; 8i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T . ð8Þp2P rin¼1W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 373XkkþT m ri ðkÞ¼nTherefore, the total cumulative departures of user class m for all OD pairs, V m iðtÞ, from parkinglocation i by interval t can be computed byV m i ðtÞ ¼X r;sV m ri ðtÞ; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T . ð9ÞThe parking accumulation at location i by user class m at interval t, D m iðtÞ, which equals to thecumulative arrivals U m i ðtÞ by the time t minus the cumulative departures V m iðtÞ by that time,can be represented asD m i ðtÞ ¼U m i ðtÞ V m i ðtÞ; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T . ð10Þ

374 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395Hence, the total parking accumulation at location i by all user classes at interval t, D i (t), isD i ðtÞ ¼ X D m iðtÞ; 8i 2 I; t 2 T ; ð11Þmwhere I denotes the set of all parking lots in the network.The parking charge, which is dependent on the desired parking duration and the arrival timeinterval at the parking location, can be defined as (Tong et al., 2004)z m i ðt þ T m ri ðtÞÞ ¼ h m l h i ðt þ T m ri ðtÞÞ; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð12Þwhere h m is the parking charge discount for user class m, and h i ðt þ T m riðtÞÞ is the hourly parkingfee at time interval t þ T m riðtÞ. In general, the longer the parking duration, the smaller the valueof h m .The walking time is computed byw m Cði; sÞis¼x ; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; ð13Þwhere C(i,s) is the walking distance from location i to destination s, and x is the average walkingspeed (km/h).The schedule delay costs of early or late arrival at the destination s can be defined as (Yang andMeng, 1998; Huang and Lam, 2002)8s m ½t msD m sðt þ T m ri ðtÞþdm i ðt þ T m ri ðtÞÞ þ wm is ÞŠ;>< if t mH m s ðt þ T m ri ðtÞþdm i ðt þ T m sD m s> ðt þ T m ri ðtÞþdm i ðt þ T m ri ðtÞÞ þ wm is Þri ðtÞÞ þ wm is Þ¼ k m ½ðt þ T m ri ðtÞþdm i ðt þ T m ri ðtÞÞ þ wm is Þ tm sD m s Š;if t msþ D m s< ðt þ T m ri ðtÞþdm i ðt þ T m ri ðtÞÞ þ wm is >:Þ0; otherwise;ð14Þwhere ½t msD m s ; tm sþ D m sŠ is the desired arrival time window for user class m at destination s withoutany schedule delay penalty, t msis the middle point of the time window, and s m (k m ) is the unitcost of arriving early (late) (i.e., schedule delay) at destination s for user class m.2.3. Travel and parking choicesAccording to A4 in Section 2.1, the travel and parking choices of travelers follow a hierarchicaldecision-making process. Specifically, before making a trip, a traveler first determines the startand end times of the activity, namely, the departure time and parking duration, and then selectsa desirable parking location that will minimize the travel disutility from origin to final destination.After choosing the parking location, the traveler will take the shortest route to travel to the parkinglocation. Similar treatments of the hierarchical choice structure can be found in Abrahamssonand Lundqvist (1999) and Boyce and Bar-Gera (2003).We now formulate the multinomial logit (MNL) model for the joint choice of departure timeand parking duration. The MNL model was adopted by Lerman (1979) for the investigation oftrip chaining behavior, by Ben-Akiva et al. (1986) in a study of the departure time and routechoice problem, by Bhat (1998) in an inquiry into the complication of travel mode and departure

time choices, and by Huang and Lam (2003) for the assessment of departure time and route choicebehavior in advanced traveler information systems.Let q j rsbe the given total demand of travelers with trip purpose j between OD pair (r,s), andq jlrsðtÞ be the portion of the demand that selects interval t for departure and parks their cars forl intervals. We then haveq jlrs ðtÞ expð b jl C jl¼qj rs ðtÞÞrsPt2TPl2L expð b ;jlC jlrsðtÞÞ 8j 2 J; l 2 L; r 2 R; s 2 S; t 2 T ; ð15Þor, in the form specified by user class,q m rs ðtÞ ¼qj rs PW.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 375t2Texpð b m C mðj;lÞrsðtÞÞPl2L expð b ; 8m 2 M; r 2 R; s 2 S; t 2 T ; ð16ÞmC mðj;lÞrsðtÞÞwhere C m rsðtÞ is the travel disutility of user class m departing at interval t between OD pair (r,s),and b m is a dispersion parameter that reflects the degree of familiarity of travelers with, or theirperception of, the variation of travel disutility.In the following, we examine the parking location choice of user class m departing at interval t.Following A4, travelers park their cars at a parking location that has the minimum travel disutilityfrom origin to destination. This leads to a parking location choice equilibrium, which canmathematically be expressed in a complementary form asðC m rs;i ðtÞ Cm rs ðtÞÞqm rs;i ðtÞ ¼0; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð17ÞC m rs;i ðtÞ P Cm rs ðtÞ; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð18Þq m rs;i ðtÞ P 0; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð19ÞwhereC m rsðtÞ ¼min C mi2I m rs;iðtÞ; 8m 2 M; r 2 R; s 2 S; t 2 T ; ð20Þrswhere q m rs;iðtÞ is the travel demand of user class m between OD pair (r,s) departing in interval t viaparking location i. Eqs. (17)–(19) state that for each OD pair and each user class that departs atthe same interval, the travel disutility in the selection of parking location is minimal, and no travelerwould be better off by unilaterally changing the choice of parking location. In other words,the travel disutility of any unused parking location for each OD pair and each user class is greaterthan or equal to the minimum.According to A4, the route choice of travelers follows a deterministic time-dependent multiclassequilibrium, which means that for each OD pair and each user class at each interval, the actualroute travel times that are experienced by travelers who depart at the same time are equal andminimal (Ran and Boyce, 1996). Let T m ri;pðtÞ be the route travel time for user class m departingfrom origin r at interval t to parking location i via route p. This time-dependent route choice equilibriumcan be written asðT m ri;p ðtÞ T m ri ðtÞÞf mri;p ðtÞ ¼0; 8p 2 P ri; i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð21ÞT m ri;p ðtÞ P T m ri ðtÞ; 8p 2 P ri; i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð22Þf mri;p ðtÞ P 0; 8p 2 P ri; i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð23Þ

376 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395whereT m riðtÞ ¼min T mp2Pri;p ðtÞ;ri 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T . ð24ÞThe route travel time of user class m in Eqs. (21) and (22) can be represented by the sum of all ofthe link travel times along this route (Chen and Hsueh, 1998), i.e.,T m ri;p ðtÞ ¼X Xc m a ðkÞdrim apt ðkÞ; 8p 2 P ri; i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T ;a kPtð25Þwhere the indicator variable, d rimaptðkÞ, equals 1 if the flow on route p departing from origin r atinterval t to parking location i arrives at link a at interval k, and 0 otherwise. The travel timeof user class m on link a during interval k, c m aðkÞ, can be expressed as a function of the inflowsof all of the user classes that enter that link by interval k, i.e.,c m a ðkÞ ¼f ðu1 a ð1Þ; u1 a ð2Þ; ...; u1 a ðkÞ;...;um a ð1Þ; um a ð2Þ; ...; um a ðkÞÞ; 8a 2 A; m 2 M; k 2 T ; ð26Þor, in a concise form,c m a ðkÞ ¼f ðu að1Þ; u a ð2Þ; ...; u a ðkÞÞ; 8a 2 A; m 2 M; ð27Þwhere u a ðkÞ ¼fu 1 a ðkÞ; ...; um aðkÞg is a vector of the inflows of all of the user classes that enter link aduring interval k, andu m a ðkÞ ¼X X Xu rimaptðkÞ;ri p t8a 2 A; m 2 M; k 2 T ; ð28Þwhere u rimaptðkÞ is the inflow to link a during interval k that departs from origin r over route p towardparking location i during time interval t. This link inflow can be represented by the route inflowsthrough the use of the indicator variable d rimaptðkÞ, i.e.,u rim maptðkÞ ¼fri;p apt ðkÞ; 8a 2 A; p 2 P ri; i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T ; k 2 T . ð29ÞSo far, we have defined and formulated all of the travel choices that are to be investigated in thispaper. The parking location and route choices follow a time-dependent deterministic user equilibrium.The combination of departure time and parking duration choices is governed by a multinomiallogit formulation. We now formally define the time-dependent network equilibrium withmultiple user classes and multiple parking facilities as follows.Definition 2. A flow pattern ðfri;p m ðtÞ; qmrs;iðtÞ; qmrs ðtÞÞ is a time-dependent network equilibrium forthe joint choice problem of departure time, parking duration, parking location, and travel route ina network with multiple user classes and multiple parking facilities if it satisfies conditions (15)–(24) simultaneously.3. Model formulationIn this section, we present an equivalent variational inequality (VI) formulation of the problemthat is studied in this paper. First, we list all of the constraints that are associated with the jointtravel choice problem as follows.

W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 3773.1. ConstraintsLet X denote the set of route flows and demand variables that satisfy the following constraints:Xf mri;p ðtÞ ¼qm rs;i ðtÞ;p2P ri 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð30ÞXq m rs;i ðtÞ ¼qm rsðtÞ; 8m 2 M; r 2 R; s 2 S; t 2 T ; ð31ÞiXl;tq mðj;lÞrsðtÞ ¼q j rs; 8j 2 J; r 2 R; s 2 S; ð32Þf mri;p ðtÞ P 0; 8p 2 P ri; i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð33Þq m rs;i ðtÞ P 0; 8i 2 Im rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð34Þq m rsðtÞ P 0; 8m 2 M; r 2 R; s 2 S; t 2 T . ð35ÞEq. (30) states that the sum of all of the route flows of user class m between origin r and parkinglocation i should equal the travel demand between OD pair (r,s) when parking location i is selected.Eq. (31) is the parking demand conservation constraint. Eq. (32) states that the sum of demandover all of the intervals and parking durations for each OD pair should equal the total ODdemand with trip purpose j between that OD pair. Eqs. (33)–(35) are the usual non-negativity constraintson route flows and demand. In addition, the following definitional constraints should beincluded:T m ri;p ðtÞ ¼X au m a ðkÞ ¼X riXkPtX Xpc m a ðkÞdrim apt ðkÞ; 8p 2 P ri; i 2 I m rs; m 2 M; r 2 R; s 2 S; t 2 T ; ð36Þtf mri;p ðtÞdrim ðkÞ; 8a 2 A; m 2 M; k 2 T . ð37Þapt3.2. VI formulationThe equivalent VI formulation of the time-dependent network equilibrium conditions (15)–(24)is given below.Theorem 1. A time-dependent flow pattern in road networks with multiple user classes and multipleparking facilities reaches the user equilibrium state if and only if it satisfies the following VIcondition:X X X Xrimþ X rsptT mri;p ðtÞðf mri;p ðtÞX X 1ln q mrsb ðtÞðqm rs ðtÞmmsubject to ðfri;p m ðtÞ; qm rs;i ðtÞ; qm rsðtÞÞ 2 X.tf mri;p ðtÞÞ þ X rsX X Xmqm rsðtÞÞ P 0;ð38ÞitC mrs;i ðtÞðqm rs;i ðtÞqm rs;i ðtÞÞ

378 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395Proof. The VI (38) is in fact the integration of the following three VI sub-problems. For a givenq j rs, the logit-based equilibrium (16) for departure time and parking duration choice is equivalenttoX X X 1ðVI sub-problem 1Þln q mrsb ðtÞðqm rs ðtÞ qm rsðtÞÞ P 0;ð39Þmrsmtsubject to (32) and (35).For a given q m rsðtÞ, which is determined by VI sub-problem 1, the equilibrium (17)–(19) forparking location choice is equivalent toX X X XðVI sub-problem 2ÞC mrs;i ðtÞðqm rs;i ðtÞ qm rs;iðtÞÞ P 0;ð40Þrsmitsubject to (31) and (34).For a given q m rs;iðtÞ, which is given by VI sub-problem 2, the equilibrium (21)–(23) for routechoice is equivalent toX X X XðVI sub-problem 3ÞT mri;p ðtÞðf mri;p ðtÞ f mri;pðtÞÞ P 0;ð41Þrimptsubject to (30) and (33).For the proof of VI sub-problems 2 and 3, we refer the reader to Wie et al. (1995), Chen (1998),and Ran and Boyce (1996). Hence, we only need to prove the equivalence between equilibrium(16) and VI sub-problem 1. It is known that VI sub-problem 1 is equivalent to the mathematicalprogram that is given in the following (Facchinei and Pang, 2003):min X rsX X 1q m rsb ðtÞðln qm rsðtÞ 1Þ; ð42Þmmtsubject to (32) and (35). It is easy to show that the first-order optimality conditions of this mathematicalprogram can induce a logit-based equilibrium (16). As program (42) is strictly convex,the solution is unique, which ensures the equivalence between equilibrium (16) and program(42). This completes the proof of Theorem 1. hNote that VI sub-problem 3 is route-based, and therefore explicit route enumeration in both themodel formulation and algorithm design are required. Some effective methods have been developedto generate a set of feasible or efficient paths that are commonly used in traffic assignmentmodels (Friesz et al., 1992; Lim and Heydecker, 2005). In fact, this route-based VI can be convertedto a link-based equivalent problem. Substituting the definitional constraints (36) and(37) into (41) yieldsX X X XT mri;p ðtÞðf mri;p ðtÞrimptf mri;p ðtÞÞ ¼ X mX Xc m a ðkÞðum a ðkÞakPtum aðkÞÞ P 0. ð43ÞVI sub-problem 3 then becomes a link-based problem. Therefore, VI problem (38) can be rewrittenas

X X Xc m a ðkÞðum a ðkÞmaþ X rskPtmW.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 379tum a ðkÞÞ þ X rsX X 1ln q mrsb ðtÞðqm rs ðtÞmX X XC mrs;i ðtÞðqm rs;i ðtÞmqm rsðtÞÞ P 0.ð44Þitqm rs;i ðtÞÞIt can be shown that all of the functions that appear in VI model (38) or (44), expressed explicitlyor implicitly, are continuous, and thus a solution to the model exists. However, the indicator variablein Eq. (36) depends on the link travel times, which in turn depend on link inflows. Consequently,the path travel times are essentially non-linear and non-convex (Huang and Lam,2002). Thus, VI model (38) or (44) is non-convex. This implies that multiple local solutionsmay exist in the proposed model (Chen and Hsueh, 1998).4. Solution algorithmIn this section, a heuristic solution algorithm is developed to solve the model that is proposed inthis paper. With reference to the proof of Theorem 1, the original VI problem is decomposed intothree sequential sub-problems, the proofs of which are also sequential. The proof provides us withan avenue for the development of a solution algorithm. The proposed algorithm consists of threesequential phases, each of which corresponds to one sub-problem. It is basically a time-dependentgeneralization of the Gauss–Seidel decomposition approach in a static environment (Patriksson,1993). The flowchart of the algorithm is given in Fig. 1. Note that the proposed algorithm is anintuitive implementation of the proof, and that its convergence has not been rigorously investigated.Nonetheless, numerical experiments show that this algorithm is able to obtain a satisfactorysolution. The step-by-step procedure is described in the following:Step 0. Initialization.Set iteration j = 1. Choose the initial OD travel pattern fqrsmðjÞ ðtÞg.Step 1. First loop operation.Set n = 1. For each OD pair and each time interval, choose the initial parking demandpattern fq mðnÞrs;i ðtÞg based on the current OD travel pattern.Step 2. Second loop operation.Step 2.1. Assign fq mðnÞrs;i ðtÞg to the set of routes between origin r and parking location i by solvingVI sub-problem 3, and obtain the in-vehicle travel time fT mðnÞri ðtÞg and the route flowff mðnÞri;p ðtÞg.Step 2.2. In accordance with the solution of VI sub-problem 3, compute the searching time delayfor parking, the parking charge, the walking time and the schedule delay cost for eachpair of parking location i and destination s by Eqs. (5)–(14). Determine the travel disutilityC mðnÞrs;i ðtÞ by Eqs. (2)–(4) and the minimal travel disutility C mðnÞrsðtÞ.Step 2.3. Use the all-or-nothing method to assign fqrsmðjÞ ðtÞg to each parking location i with minimaltravel disutility and obtain the auxiliary parking demand pattern fg mðnÞrs;i ðtÞg.Step 2.4. Update the parking demand using the method of successive averages (MSA)q mðnþ1Þrs;i ðtÞ ¼q mðnÞrs;i ðtÞþ 1 n ðgmðnÞ rs;i ðtÞq mðnÞrs;i ðtÞÞ.

380 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395Fig. 1. Flowchart of the proposed algorithm.Step 3. Convergence check for the second loop operation.If the parking location choice equilibrium conditions are satisfied, then go to Step 4;otherwise, set n = n + 1 and go to Step 2.1.Step 4. According to the minimal travel disutility C mðnÞrsðtÞ, determine the auxiliary OD travelpattern fg mðjÞrsðtÞg by Eq. (16).

Step 5. Update the OD travel demand using the MSAq mðjþ1ÞrsW.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 381ðtÞ ¼q mðjÞrsðtÞþ 1 j ðgmðjÞ rsðtÞ q mðjÞrsðtÞÞ.Step 6. Convergence check for the first loop operation.If the equilibrium condition (16) for the joint choice of departure time and parkingduration is reached, then terminate and output the solution; otherwise, set j = j +1and go to Step 1.In Step 2.1, the route-based or link-based VI sub-problem 3 can be solved by many existingsolution methods, such as the disaggregate simplicial decomposition method, the gradient projectionmethod or the Frank–Wolfe method. In our numerical studies, the gradient projectionmethod is adopted because its computational efficiency is superior to the other two methods(Chen et al., 1999). It should be pointed out that the CPU time that is required for the proposedalgorithm is mainly dominated by Step 2.1, because in this step a full time-dependent route choiceequilibrium problem has to be solved.5. Numerical studiesIn this section, two test scenarios are used to illustrate the model and solution algorithm thatare proposed in this paper. The first scenario is intended to demonstrate the solution quality. Thenumerical results show that the proposed model produces a solution that satisfies the time-dependentnetwork equilibrium conditions (15)–(24). The second scenario is used to show the convergenceof the solution and to investigate the sensitivity of the solution to travel demand, parkingcapacity and parking charge. We also compare the outcomes that are generated by our timedependentmodel and those that are generated by a static model. The static model is constructedby eliminating the time dimension from our time-dependent model. As a result, the departure timechoice does not exist in the static model. However, the structure of the parking location and routechoices is similar to that in the time-dependent model.5.1. Scenario 1The network for Scenario 1 is shown in Fig. 2, and consists of one OD pair, four roadnodes, five road links and three parking locations (off-street parking locations A and B, andon-street parking location C). It is assumed that the walking distances from locations A, B,and C to the destination are 1.0 km, 0.75 km and 0.5 km, respectively. The study period runsfrom 06:00 to 20:00 and is divided into 14 hourly intervals. All of the travelers are grouped intotwo user classes, namely, non-commuters and commuters. The non-commuters are further categorizedinto three sub-classes according to whether their parking durations last for 1, 2 or 3 h.The commuters are divided into two sub-classes according to whether their parking durationslast for 4 or 8 h. The number of user classes in this example is therefore five, i.e., jMj = 5. Hereinafter,we employ the superscripts ‘‘c’’ to denote commuters and ‘‘nc’’ to denote noncommuters.

382 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395Fig. 2. Example network for Scenario 1.For each time interval, the following BPR-type link travel time function is adopted:c a ðtÞ ¼c 0 a ð1.0 þ 0.15ðu aðtÞ=C a Þ 4 Þ;ð45Þwhere c 0 a is the free-flow travel time and C a the capacity of link a, and u a ðtÞ ¼ P m um aðtÞ. Theparameters of all of the link travel time functions are given in Table 1. For locations A and B,the free-flow parking access times and parking capacities that are used in the parking searchingtime functions (Eq. (5)) are 0.1 h and 2000 vehicles, respectively, and those for location C are0.05 h and 350 vehicles, respectively. The walking speed in Eq. (13) is assumed to be given andfixed, i.e. x = 5.0 km/h. The discount coefficients h m in Eq. (12) are assumed to be 1.0 for all userclasses. The uniform hourly parking fees are HK$16.0 at locations A and B, and HK$8.0 at locationC. The other parameters are a c 1 ¼ 1.4, ac 2 ¼ 0.05, ac 3 ¼ 1.8, ac s ¼ 1.0, ¼ 9.0, D c = 0.25,tcs c = 0.6 and k c = 5.6 for commuters; and a nc1¼ 1.4, anc2¼ 0.1, anc3¼ 2.0, ancs¼ 3.0, t nc ¼ 14.0,D nc = 1.0, s nc = 0.5 and k nc = 1.5 for non-commuters. In addition, b c = 0.3 for commuters andb nc = 0.2 for non-commuters. These values imply that commuters have a smaller variation in theirperception of travel disutility than non-commuters. This is reasonable because commuters generallypredict travel conditions more accurately than non-commuters due to their day-to-day learningand adaptation (Arentze and Timmermans, 2005). The total demand of non-commuters is1500 vehicles and that of commuters is 4000 vehicles during the study period.Table 2 shows the route inflows and the corresponding route travel times from origin to thethree parking locations, respectively. It can be seen that for each departure time interval, theroutes that are actually used have the minimum travel time, whereas the unused routes have equalor longer travel times. Hence, the deterministic equilibrium condition for route choice has beenreached. Table 3 shows the departure flow pattern of non-commuters heading for different parkinglocations and the corresponding disutilities. It can be verified that the parking locations thatare used at each interval have the minimum disutility, whereas the unused locations have equal orgreater disutility. The same finding can be observed in Table 4 for commuters. Therefore, thedeterministic equilibrium for parking location choice has indeed been reached.Table 1Parameters of the link travel time functions in Scenario 1Link c 0 a (h) C a (veh/h) Link c 0 a (h) C a(veh/h)(1,2) 0.30 800 (2,4) 0.50 800(1,3) 0.45 800 (3,C) 0.20 800(2,3) 0.15 500 (C, 4) 0.10 700

W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 383Table 2Route inflows and travel times (hour) in Scenario 1Departure time Route 1 Route 2 Route 3 Route 4 Route 5Flow Time Flow Time Flow Time Flow Time Flow Time06:00–07:00 367.7 0.82 224.4 0.82 538.8 0.82 18.6 0.70 24.0 0.7007:00–08:00 532.9 0.85 170.1 0.85 626.8 0.85 24.4 0.72 33.2 0.7208:00–09:00 453.8 0.83 187.1 0.83 572.9 0.83 34.5 0.71 43.2 0.7109:00–10:00 0.0 0.80 101.2 0.75 131.4 0.75 45.3 0.65 58.9 0.6510:00–11:00 0.0 0.80 32.4 0.75 42.2 0.75 61.2 0.65 79.6 0.6511:00–12:00 0.0 0.80 6.4 0.75 8.3 0.75 82.9 0.65 107.7 0.6512:00–13:00 0.0 0.80 1.2 0.75 1.6 0.75 103.0 0.65 133.9 0.6513:00–14:00 0.0 0.80 4.8 0.75 6.1 0.75 94.8 0.65 123.2 0.6514:00–15:00 0.0 0.80 38.2 0.75 49.6 0.75 61.2 0.65 79.6 0.6515:00–16:00 0.0 0.80 0.0 0.75 0.0 0.75 46.0 0.65 59.8 0.6516:00–17:00 0.0 0.80 0.0 0.75 0.0 0.75 22.8 0.65 29.7 0.6517:00–18:00 0.0 0.80 0.0 0.75 0.0 0.75 9.5 0.65 12.4 0.6518:00–19:00 0.0 0.80 0.0 0.75 0.0 0.75 3.9 0.65 5.0 0.6519:00–20:00 0.0 0.80 0.0 0.75 0.0 0.75 1.6 0.65 2.1 0.65Table 3Departure flow pattern of non-commuters heading for different parking locations and the corresponding disutilities(Disu.) in Scenario 1Departure time Parking for 1 h Parking for 2 h Parking for 3 hB C B C B CFlow Disu. Flow Disu. Flow Disu. Flow Disu. Flow Disu. Flow Disu.06:00–07:00 0.0 11.55 16.5 10.99 0.0 12.95 14.1 11.79 0.0 14.35 12.0 12.6007:00–08:00 0.0 10.03 22.3 9.48 0.0 11.43 19.0 10.28 0.0 12.83 16.2 11.0808:00–09:00 0.0 8.52 30.1 7.98 0.0 9.92 25.7 8.79 0.0 11.32 21.9 9.5909:00–10:00 0.0 7.06 40.4 6.52 0.0 8.46 34.5 7.32 0.0 9.86 29.4 8.1210:00–11:00 0.0 5.58 54.6 5.01 0.0 6.98 46.5 5.81 0.0 8.38 39.7 6.6111:00–12:00 0.0 4.09 73.9 3.50 0.0 5.49 63.0 4.30 0.0 6.89 53.7 5.1012:00–13:00 0.0 2.60 91.9 2.41 0.0 3.99 78.3 3.21 0.0 5.39 66.7 4.0113:00–14:00 10.4 2.60 78.2 2.60 0.0 4.00 75.5 3.40 0.0 5.40 64.3 4.2014:00–15:00 87.8 2.59 0.9 2.59 0.0 4.01 75.5 3.39 0.0 5.39 64.4 4.1915:00–16:00 0.0 7.09 41.1 6.44 0.0 8.49 35.0 7.24 0.0 9.89 29.8 8.0416:00–17:00 0.0 11.59 20.4 9.95 0.0 12.98 17.3 10.75 0.0 14.39 14.8 11.5517:00–18:00 0.0 16.10 8.5 14.33 0.0 17.50 7.2 15.13 0.0 18.89 6.2 15.9318:00–19:00 0.0 20.58 3.5 18.82 0.0 21.99 2.9 19.62 0.0 23.39 2.5 20.4219:00–20:00 0.0 25.09 1.4 23.32 0.0 26.48 1.2 24.12 0.0 27.89 1.0 24.92Note that in each iteration of the solution algorithm the combined demand of departure timeand parking duration is given by a logit model formulation. Hence, the logit-based equilibrium fordeparture time and parking duration choice is always satisfied. Therefore, it can be concluded thatthe proposed algorithm produces a solution that is truly consistent with the time-dependent networkequilibrium that is described in Definition 2.

384 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395Table 4Departure flow pattern of commuters heading for different parking locations and the corresponding disutilities (Disu.)in Scenario 1Departure time Parking for 4 h Parking for 8 hA B A BFlow Disu. Flow Disu. Flow Disu. Flow Disu.06:00–07:00 0.0 5.10 789.9 5.06 0.0 7.90 341.0 7.8607:00–08:00 684.1 4.52 244.8 4.52 295.3 7.32 105.7 7.3208:00–09:00 561.0 4.83 286.8 4.83 242.2 7.63 123.8 7.6309:00–10:00 87.7 10.33 74.8 10.33 37.9 13.14 32.3 13.1410:00–11:00 0.0 15.93 52.1 14.12 0.0 18.73 22.5 16.9311:00–12:00 0.0 20.03 10.3 19.53 0.0 22.83 4.5 22.3312:00–13:00 0.0 25.36 2.0 25.01 0.0 28.16 0.9 27.8113:00–14:00 0.0 30.95 0.4 30.59 0.0 33.75 0.2 33.3914:00–15:00 0.0 36.55 0.1 36.16 0.0 39.35 0.0 38.9615:00–16:00 0.0 42.13 0.0 41.76 0.0 44.93 0.0 44.5616:00–17:00 0.0 47.73 0.0 47.36 0.0 50.53 0.0 50.1617:00–18:00 0.0 53.33 0.0 52.96 0.0 56.12 0.0 55.7618:00–19:00 0.0 58.92 0.0 58.56 0.0 61.73 0.0 61.3619:00–20:00 0.0 64.53 0.0 54.17 0.0 67.33 0.0 66.95Table 4 also shows that the commuters first choose location B to park their cars at 07:00 andafter 1 h start to park at location A. This is because the walking distance from location A to thedestination is 0.25 km farther than that from location B to the destination. The non-commutersnever choose location A, and prefer to park their cars at off-street parking location C regardless ofthe congestion level at this location, as shown in Table 3. This can also be easily seen from Fig. 3,which depicts the parking occupancy distribution for different parking locations during the day.120Parking occupancy (%)'110100908070605040302010Location ALocation BLocation C06:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00Time of day (o' clock)Fig. 3. Parking occupancy at different locations during the day (Scenario 1).

W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 3855.2. Scenario 2The test network for Scenario 2, as shown in Fig. 4, consists of one OD pair, nine nodes, twelvelinks and four parking locations (i.e., off-street parking locations A and B, and on-street parkinglocations C and D). The parameters of all of the link travel time functions are given in Table 5.For locations A and B, the free-flow parking access times and parking capacities are 0.1 h and1200 vehicles, respectively, and those for locations C and D are 0.05 h and 250 vehicles, respectively.The total demand during the study period of non-commuters is 1500 vehicles and the totaldemand of commuters is 2500 vehicles. Let s c = 2.2, k c = 4.0, s nc = 0.9, and k nc = 1.0. The otherparameters are the same as those that are used in Scenario 1. The uniform hourly parking fees areHK$16.0 at locations A and B and HK$8.0 at locations C and D (see the parking charge Scheme 1in Table 6).Origin12345678C9DADestinationBFig. 4. Example network for Scenario 2.Table 5Parameters of the link travel time functions in Scenario 2Link c 0 a (h) C a (veh/h) Link c 0 a (h) C a(veh/h)(1,2) 0.20 800 (7,8) 0.30 500(1,4) 0.20 800 (5,6) 0.20 600(2,3) 0.20 600 (5,8) 0.20 600(4,7) 0.20 600 (6,D) 0.20 600(2,5) 0.30 500 (8,C) 0.20 600(4,5) 0.30 500 (D,9) 0.10 500(3,6) 0.30 500 (C,9) 0.10 500

386 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395Table 6Three parking charge schemes for Scenario 2 (charge in HK$)Arrival timeinterval atparking location06:00–07:00 07:00–08:00 08:00–09:00 09:00–10:00 10:00–11:00 11:00–12:00 12:00–13:00Scheme 1A, B 16 16 16 16 16 16 16C, D 8 8 8 8 8 8 8Scheme 2A, B 16 16 32 32 32 16 16C, D 8 8 8 8 8 8 8Scheme 3A, B 16 16 38 42 36 16 16C, D 8 8 8 8 8 8 8Arrival timeinterval atparking location13:00–14:00 14:00–15:00 15:00–16:00 16:00–17:00 17:00–18:00 18:00–19:00 19:00–20:00Scheme 1A, B 16 16 16 16 16 16 16C, D 8 8 8 8 8 8 8Scheme 2A, B 16 16 16 16 16 16 16C, D 16 16 16 8 8 8 8Scheme 3A, B 16 16 16 16 16 16 16C, D 16 18 14 8 8 8 8G-Value1.21.00.80.60.4Log (G -Value)0.5Log (No. of iterations)00.30 3.18 3.48 3.65 3.78 3.88 3.95 4.02 4.08 4.13 4.18 4.22 4.26 4.29-0.5-1-1.5-2-2.50.2-30.00 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000Number of iterationsFig. 5. Convergence of the proposed solution algorithm.

W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 387Fig. 5 plots the values of a comprehensive index, called the G-value, which measures how closelythe outputs that are generated at iteration n approach the time-dependent user equilibriumcondition (15)–(24). The G-value is defined asDeparture distribution (%)'0.20.180.160.140.120.10.080.060.04Parking for 1 hourParking for 2 hoursParking for 3 hoursTotalParking for 1 hourParking for 2 hoursParking for 3 hoursTotal0.02a006:00-07:0007:00-08:0008:00-09:0009:00-10:0010:00-11:0011:00-12:0012:00- 13:00- 14:00-13:00 14:00 15:00Interval (hour)15:00-16:0016:00-17:0017:00-18:0018:00-19:0019:00-20:000.450.40.35Departure distribution (%)'0.30.250.20.150.10.05Parking for 4 hoursParking for 8 hoursTotalParking for 4 hoursParking for 8 hoursTotalb006:00-07:0007:00- 08:00- 09:00- 10:00- 11:00- 12:00- 13:00- 14:00- 15:00- 16:00- 17:00- 18:00- 19:00-08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00Interval (hour)Fig. 6. (a) Departure distribution of non-commuters (solid lines denote demand and q c = 2500, and q nc =1500, anddotted lines denote demand q c = 1250, and q nc =750). (b) Departure distribution of commuters (solid lines denotedemand and q c = 2500, and q nc = 1500, and dotted lines denote demand q c = 1250, and q nc = 750).

388 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395G ¼ X riþ X rsþ X rsX X Xf mðnÞri;p ðtÞðT mðnÞri;p ðtÞmmpitX X Xq mðnÞrs;i ðtÞðC mðnÞX Xmttq mðnÞrsrs;i ðtÞðtÞ q mðnÞrsðtÞ q j rs P.X X X XT mðnÞri ðtÞÞri m p t.X X X XC mðnÞrsðtÞÞt2Tf mðnÞri;pðtÞT mðnÞriðtÞq mðnÞrs;i ðtÞC mðnÞrsrs m i t,expð b m C mðnÞrsðtÞÞXPl2L expð b mC mðnÞrsðtÞÞrsmðtÞX Xq mðnÞrsðtÞ. ð46Þt1300Cumulative arrivals and departures (veh)'12001100100090080070060050040030020010001188246Cumulative arrivals at ACumulative departures from ACumulative arrivals at CCumulative departures from C6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00Time of day (o'clock)Fig. 7. Cumulative arrivals at and departures from off-street parking location A and on-street parking location C(q c = 2500, q nc = 1500).Parking accumulation (veh) '1200110010009008007006005004003002001000Parking location AParking location AParking location CParking location C6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00Time of day (o'clock)Fig. 8. Parking accumulation at locations A and C (solid lines denote demand q c = 2500, and q nc = 1500, and dottedlines denote demand q c = 1250, and q nc = 750).

W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 389Fig. 5 shows that the G-value decreases as the iteration number increases. After 20,000 iterations,the G-value becomes less than 0.001. Hence, a satisfactory equilibrium solution is achieved.The use of logarithm scale shows a similar convergence pattern.Fig. 6a and b show the departure distributions of the non-commuters and commuters, respectively,during the day. It can be seen that the demand with a short parking duration is always largerthan the demand with a long parking duration. When the total demands of non-commutersand commuters are doubled, travelers depart from their origin earlier to reduce the late-arrivalDeparture distribution (%)0.20.180.160.140.120.10.080.060.040.02Parking for 1 hourParking for 2 hoursParking for 3 hoursTotalParking for 1 hourParking for 2 hoursParking for 3 hoursTotala006:00-07:0007:00-08:0008:00-09:0009:00-10:0010:00-11:0011:00-12:0012:00-13:0013:00-14:00Interval (hour)14:00-15:0015:00-16:0016:00-17:0017:00-18:0018:00-19:0019:00-20:000.4Departure distribution (%)0.350.30.250.20.150.10.05Parking for 4 hoursParking for 8 hoursTotalParking for 4 hoursParking for 8 hoursTotalb006:00-07:0007:00-08:0008:00- 09:00- 10:00- 11:00- 12:00- 13:00- 14:00- 15:00- 16:00- 17:00- 18:00- 19:00-09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00Interval (hour)Fig. 9. (a) Departure distribution of non-commuters before (solid lines) and after (dotted lines) parking capacityexpansion. (b) Departure distribution of commuters before (solid lines) and after (dotted lines) parking capacityexpansion.

390 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395penalty. Fig. 6a shows that the departure peak of non-commuters changes from 12:00–15:00 to12:00–14:00. For commuters, this peak moves from 8:00–9:00 to 7:00–8:00, as shown in Fig. 6b.Fig. 7 depicts the curves of cumulative arrivals at and cumulative departures from the off-streetand on-street parking locations. Only off-street location A and on-street location C are chosen forillustration as the example network for Scenario 2 is symmetric. The vertical distance between thecumulative arrival and cumulative departure curves represents the parking accumulation at thattime, which is displayed in Fig. 8. For instance, it is indicated in Fig. 7 that the maximum parking120011001000Parking accumulation (veh)900800700600500400300200100Scheme 1Scheme 2Scheme 3a06:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00Time of day (o'clock)Parking accumulation (veh)b250230210Scheme 1190Scheme 2170Scheme 31501301109070503010-106:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00Time of day (o'clock)Fig. 10. (a) Parking accumulation at off-street location A with different charging schemes. (b) Parking accumulation aton-street location C with different charging schemes.

W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395 391accumulations at the off-street parking location is 1188 vehicles, and at the on-street parking locationsis 246 vehicles. Fig. 8 shows that as the total demands are doubled, the peak for parking aton-street location C moves forwards about 1 h. It is also found that the peak for parking at offstreetlocation A appears at around 10:00, which is about 1 h later than the work start time of thecommuters.Suppose that the total demands are fixed as q c = 2500 and q nc = 1500. We examine the changesin departure distribution when the capacity of each parking location is increased by 100 vehicles.The results are shown in Fig. 9a for non-commuters and Fig. 9b for commuters. These two figuresshow that after parking conditions are improved, the departure distributions are slightly flattenedtowards the right. Hence, the parking capacity expansion positively affects the departure behaviorand relieves traffic congestion on the roads.In addition to looking at the effects of an increase in parking capacity, we can also make use ofthe proposed model to investigate the effects of different parking charge schemes on parking accumulation.In addition to the uniform charging in Scheme 1, Table 6 lists the other two parkingcharge schemes. Scheme 2 is a step-charging pattern and Scheme 3 offers a time-varying parkingcharge pattern. Fig. 10a shows the results for off-street parking location A and Fig. 10b those foron-street parking location C. It can be seen that the parking charging scheme significantly influencesthe parking accumulation. For instances, to reduce the parking fees, travelers change theirdeparture times in response to Schemes 2 and 3. In Fig. 10b, a concave part of the curve that iscaused by the time-varying parking charge can be observed at around 14:00. This is because thecharge at the on-street parking locations at around 14:00 is HK$2.0 higher than the charge atthe off-street parking locations. Consequently, some non-commuters move to park their cars atthe off-street parking locations.Finally, we use the numerical results of Scenario 2 to distinguish the time-dependent modelfrom its static version by comparing the average parking duration, average turnover, averagehourly occupancy at the off-street and on-street parking locations, and the parking revenues thatare generated. It is noted in Table 7 that for the off-street parking locations, the static modelunderestimates the average parking duration and overestimates the average turnover, averagehourly occupancy of parking location, and parking revenue compared to the time-dependentmodel, but that the opposite is the case for the on-street parking locations. This implies thatTable 7Main results generated by the time-dependent model and the static model for Scenario 2Model Parking type Average parkingduration (h)AverageturnoverAverage hourlyoccupancy ofparking location (%)Parking revenue(HK$)Time-dependent Off-street 5.11 1.04 38.00 102,150On-street 1.89 3.00 40.58 11,363Static Off-street 4.29 1.30 39.84 107,103On-street 2.46 1.76 30.96 8669Note: The average parking duration equals the total parking vehicle-hours divided by the total number of parked cars.The average turnover is the ratio of the total number of parked cars to parking capacity. The average hourly occupancyof a parking location equals the total parking vehicle-hours divided by the product of parking capacity and the numberof periods studied. Parking revenue is the sum of charges that are collected from all parked cars.

392 W.H.K. Lam et al. / Transportation Research Part B 40 (2006) 368–395the static and time-dependent models may give different results in the assessment of the usage ofon-street and off-street parking facilities.6. ConclusionsIn this paper, a time-dependent network equilibrium model is developed for the investigation ofthe joint choices on travel and parking in a road network with multiple user classes and multipleparking facilities. The model aims to assess the effects of various parking policies for strategicplanning purposes. The decision-making process of travelers on travel and parking choices is assumedto follow a hierarchical choice structure. The proposed model is formulated as a VI problem,and is solved by a heuristic solution algorithm. Two numerical examples are presented toillustrate the proposed model and algorithm. The first example verifies that the proposed modelcan indeed generate a time-dependent network equilibrium solution, and the second exampleshows that parking choice behavior is greatly influenced by travel demand, walking distance,parking capacity, and parking charge. In the second example, we also distinguish the time-dependentmodel from its static counterpart by comparing the performance of on-street and off-streetparking facilities. The proposed modeling approach can be used to reveal and study the complextemporal and spatial interactions between road traffic and parking congestion.It should be pointed out that although the numerical results that are presented in this paper areconsistent, case studies on large and real networks are necessary to further validate the proposedtime-dependent model. We are currently working toward this. In addition, some other relevantissues that we are investigating include the determination of the optimal parking capacities andcharging schemes for parking facilities, the assessment of the impacts of parking information systemswith different levels of market penetration on travel behavior, and the development of efficientsolution algorithms for large-scale networks.AcknowledgementsThe work described in this paper was supported by grants from the Research Grants Council ofthe Hong Kong Special Administrative Region (Project No. PolyU 5040/01E, PolyU 5143/03Eand HKU 7134/03E) and a grant from the National Natural Science Foundation of China.The authors would like to thank two anonymous referees for their helpful comments and suggestionson an earlier draft of the paper.ReferencesAbrahamsson, T., Lundqvist, L., 1999. Formulation and estimation of combined network equilibrium models withapplications to Stockholm. Transportation Science 33, 80–100.Arentze, T., Timmermans, H., 2005. Modelling learning and adaptation in transportation contexts. Transportmetrica 1,13–22.Axhausen, K.W., Polak, J., 1991. Choice of parking: stated preference approach. Transportation 18, 59–81.Bell, M.G.H., Lam, W.H.K., Iida, Y., 1996. A time-dependent multi-class path flow estimator. In: Lesort, J.B. (Ed.),Transportation and Traffic Theory. Elsevier, Oxford, pp. 173–194.

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