Hani S. Mahmassani (Ed.): The 16th ISTTT, pp. 717-739, 2005

36

A TIME-DENPENDENT ACTIVITY AND TRAVEL CHOICE MODEL WITH MULTIPLE PARKING OPTIONS Hai-Jun Huang and Zhi-Chun Li School of Management, Beijing University of Aeronautics and Astronautics, China

William H. K. Lam Department of Civil and Structural Engineering, Hong Kong Polytechnic University, China

S. C. Wong Department of Civil Engineering, The University of Hong Kong, China

INTRODUCTION

In recent years, parking has become an increasingly serious problem in most densely populated cities due to the dramatic increase in the number of vehicles. Consequently, traffic conditions, particularly in peak periods, have tended to deteriorate in many urban areas, and the search time for parking has substantially increased in central business districts. Parking is closely tied to road usage (Glazer and Niskanen, 1992), and every car has to park at the end of a trip. Hence, parking policy offers a potentially strong instrument for influencing traffic flows (Verhoef et al., 1995). It has widely been recognized that parking policy plays an important role in the planning and management of transportation systems in densely developed urban areas (Young et al., 1991; Wong et al., 2000; Tong et al., 2004). It has been observed that drivers spend a significant percentage of their total trip time searching for a parking space (Axhausen and Polak, 1991). This can significantly affect the level of traffic congestion and environmental quality within the urban 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 even the socio-economic system (Bifulco, 1993). To make the urban transportation system work more

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efficiently, it is necessary to understand and predict the parking choice behaviour of travellers with regard to various parking policies, and also their travel choice behaviour in relation to changes in parking supply and parking charges. There is a body of literature on parking and travel choice models. Generally, these models can be classified into three categories: (1) the discrete choice approach (Polak et al., 1990; Hunt and Teply, 1993; Thompson and Richardson, 1998; Hensher and King, 2001; Hess and Polak, 2004); (2) the probability based approach (Wong et al., 2000; Dell’ Orco et al., 2003); and (3) the network based approach (Florian and Los, 1980; Bifulco, 1993; Lam et al., 2002). However, these studies mainly focus on time-stationary (static) equilibrium analysis, and little attention has been paid to time-dependent analysis. The static model cannot reveal the temporal and spatial interaction between road traffic and parking congestion, and hence may not be able to capture the congestion effects on travel choices and parking behaviour that can be carried over time during the day. All the aforementioned studies belong to the traditional trip-based approach, which treats the trip as an independent unit of analysis (Kitamura, 1984). The trip-based approach fails to recognize the existence of linkages among trips, and between trips and activity participation (Jones et al., 1990). In some instances, the forecasts of trip-based models have proved to be inaccurate due to their misrepresentation of travel behaviour (Recker et al., 1986). Therefore, the trip-based approach cannot predict many individual responses to various transport policies and leads to biased predictions of the effects of these transport policies (Pas, 1983). As a matter of fact, the travel demand that is generated within a city can be regarded as the result of the many decisions that people make about participating in activities in various locations, constrained in time and space. In other words, travel is usually undertaken not for its own sake, but rather for participation in something at the end of a trip (Damm and Lerman, 1981). Thus, travel demand forecasting must be based on a reasonable theory that takes account of human activities. In recent years, it has been widely recognized that travel is a derived demand that results from the needs and desires of individuals for participation in various activities (Mahmassani, 1988). Interaction between travel and activity appears to be a key to understanding travel-related choices (Supernak, 1992). Activity participation is the end of travel that such participation produces, whereas travel that connects the activities in the time-space continuum is the means of activity participation (Recker et al., 1986). Therefore, to better understand the travel behaviour, it is necessary to incorporate activity participation into the traditional travel demand analysis approach. In the past three decades, activity-based analyses have received widespread attention and have made substantial contributions to the better understanding of travel behaviour. This approach studies travel patterns in the context of the structure of activities, of the individual or household, with a framework that emphasizes the importance of time and space constraints (Goodwin, 1983).

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Damm and Lerman (1981) formulated a theory of activity scheduling for urban workers to investigate whether they participated in a non-work activity on the way from home to work or from work to home, and if so, for how long. Kitamura (1983) and Kitamura and Kermanshah (1984) employed the sequential modelling approach to analyze the time-, history-, or future-dependency of the activity and destination choices in a trip chain. Pas (1983) employed a methodology of the systematic identification of class to analyze daily travel-activity behaviour. Horowitz (1980) developed a structural model for multi-destination non-work travel based on the principle of utility maximization. Kitamura (1984) derived a model of daily time allocation from the concept of random utility maximization. Axhausen (1990) combined an activity chain simulation model with a traffic flow simulation model to achieve a simultaneous simulation of activity chains and traffic flows. Hamed and Mannering (1993) and Mannering et al. (1994) developed a methodology capable of modelling daily activity-travel patterns using discrete/continuous econometric models and hazard duration models with survival distributions. Bhat (1998a) examined travellers' activity behaviour after returning home at the end of the work tour using data from a Boston region household activity survey. Comprehensive reviews of the activity-based approach can be found in Kitamura (1988), Jones et al. (1990), and Bowman and Ben-Akiva (2000). Recently, Lam and Yin (2001) applied the activity-based approach to modelling time-dependent traffic assignment. Lam and Huang (2002, 2003) developed a combined activity/travel choice model for hunting for the dynamic user equilibrium solution on congested road network with queues. Huang et al. (2003) explored the activity and travel schedule behaviour under advanced traveller information system. Abdelghany and Mahmassani (2003) presented a stochastic temporal-spatial micro-assignment and activity sequencing model for activity/trip chains in a dynamic framework. However, the combination of the parking problem with activity-based analyses in a unified framework has not been explored. As a matter of fact, there is a close relation between activity-travel scheduling and parking choice. An activity-travel schedule may significantly interact with the choice of parking location. For instance, travellers who depart from home to perform a work or non-work activity during rush hours may spend a lot of time searching for an available parking space because of high parking occupancy. This may also give rise to an increase in cruising time on the road network, and hence create additional traffic congestion and pollution problems. In contrast, the characteristics of parking locations such as parking availability, the search time delay for parking, and parking charges, may plausibly be expected to vary according to the activity-travel patterns of travellers across the time of day. Therefore, travellers may change their choices of parking location and parking duration according to the characteristics of different parking locations at different times of the day. In this paper, we propose a time-dependent equilibrium model for studying the parking behaviour of travellers in the activity-based paradigm. To capture the temporal and spatial constraints of the activity-travel schedule, we introduce the concept of time windows for each activity in a traveller’s itinerary. The travellers who start to perform the activity outside the time windows will

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undergo a large penalty. Therefore, to some extent the time windows of the activities implicitly determine the order of the activities and even the activity patterns. We assume that the likelihood that a traveller selects a particular activity is based on utility maximization, where the traveller selects the activity that maximizes his/her perceived utility. We use the nested logit formulae to describe the choices on activity type, activity location, and activity duration, and employ the deterministic equilibrium conditions to govern the choices on parking location and travel route. All these choices follow a hierarchical structure and are finally integrated by a variational inequality problem. The proposed modelling approach can be used to assess the effects of parking policies on activity-travel scheduling behaviour and to make long-term strategic plans. The remainder of this paper is organized as follows. In the next section, we introduce the basic considerations about the proposed modelling approach. Then, a time-dependent equilibrium model is presented and a solution algorithm for solving this model is given. Furthermore, we provide a numerical example to test the proposed model and solution algorithm. Recommendations for future researches are given in the last section.

BASIC CONSIDERATIONS

Assumptions

To facilitate the presentation of the essential ideas of this paper, the following assumptions are made.

A1 The whole study horizon [0, T] is discretized into equally spaced intervals, sequentially numbered },,1,0{ TTt =∈ , and δ is the length of a period so that TT =δ .

A2 All commuters are homogenous concerning their values of time and utility perception; however, their activities are differentiated by the activity type, activity location and activity duration. In addition, the parking facilities are characterized by type (e.g., on-street and off-street) and time-varying parking charges.

A3 The decision-making on choices of activity, route and parking location is done before entering the network, i.e., any en route shifting is not allowed. This assumption is appropriate for assessing transportation policies at a strategic level.

A4 The trip-chain pattern of a person is not pre-specified before having a trip for an activity but caused by his or her choices on a series of decision-makings within a day. A person always makes decision on his or her next activity once he or she completes the current one.

A5 The decision-making process on activity, travel and parking choices follows a hierarchical structure, i.e., travellers first decide what to do (activity type choice) and where (activity location choice), then decide how long for it (activity duration choice), and finally where to park their cars

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(parking location choice) and the path by which to reach the chosen parking location (travel route choice).

A6 The travellers have full information about traffic and parking conditions, so that they can find parking locations with minimal disutility and travel routes with minimal travel times.

A7 Each activity is associated with a specific time window, outside which a traveller starting to perform the activity will suffer a large penalty.

A8 The parking charge depends on both the time that a traveller enters the parking location and the parking duration. Walking time is dependent on the walking speed and the distance between the parking site and the final destination (Bifulco, 1993).

A9 One or more activities may be performed in a zone. If two or more activities are performed in a zone, travellers first perform the primary activity and then other ones (Bowman and Ben-Akiva, 2000).

Travel and activity utilities

Consider a city network G = (N, A), where N is the set of all nodes, including origin nodes, intersection nodes, destination nodes, and parking locations, A is the set of all directed links, including centroid connectors from origin to road network, road links, parking access links from road network to parking location, and pedestrian or walk links from parking location to final destination. Let r denote an origin node, Nr∈ , and s be a destination node, . An origin node, which generates trips, may also be a destination node for an activity, and vice versa. Let denote the set of all activities j, , and the set of all feasible activity durations

Ns∈J

j J∈ jD jτ

associated with activity j, . Let be the set of all feasible parking locations serving the travellers between OD pair (r, s).

jj D∈τ rsI

In the activity-based travel demand analysis, activity utility, defined as the importance or the degree of need for an activity, plays an important role because the daily activity schedules of individuals directly influence travel demand (Supernak, 1992). When deciding whether or not to participate in an activity, one makes a trade-off between the utility of activity and the disutility of associated travel-parking. In this paper, we assume that every traveller attempts to make an activity-travel schedule that has the maximum overall utility. Define as the comprehensive utility gained by a traveller who departs from origin r ),( j

jrs tU τ

for destination s in period t and performs activity j for duration jτ in the destination zone s. This comprehensive utility can be expressed as

( , ) ( , ) [ ( , ) ]j j jrs j rs j j j s sU t C t V t V Vτ τ α τ= − + + + , JjDTtNsNr j ∈∈τ∈∈∈∀ ,,,, (1)

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where is the travel disutility from r to s departing in period t for performing activity j for duration , is the total utility of performing activity j by a traveller, who leaves origin r for s in period t and performs the activity j for duration

),( jj

rs tC τ

jτ ),( jj tV τ

jτ , is the systematic component of utility common to all elements with destination s (e.g., these caused by area, population and employment), is the systematic component of utility varying across both activity and destination (e.g., these caused by business conditions around the destination if j represents shopping activity). α is a conversion factor between travel disutility and activity utility.

sV

jsV

In general, the utility that is gained from an activity depends on the starting time of the activity and the duration that it lasts. When a traveller’s departure time from home and activity duration is determined, the beginning and ending times for his/her activity can be calculated. Therefore, the activity and travel schedule of a traveller is governed by departure time and activity duration. Let be the starting time of an activity for these travellers who depart from origin r in period

t, which is also the time arrival at the activity location (i.e. destination). Let be the in-vehicle travel time for those who depart from r in period t and park their cars at location i,

be the searching time delay for finding an available parking space for those who

depart from r in period t and arrive at location i at time t+ ) , and be the walking time from i to s. Then, is given by

)(tσ

)(tTri

))(( tTtd rii +

(tTri isw)(tσ

,))(()()( isriiri wtTtdtTtt ++++=σ Tt∈∀ , (2) The utility in Equation (1) is the sum of all utilities gained from performing activity j from time to time

),( jj tV τ

)(tσ jt τ+σ )( . It can be expressed as ( )

( )( , ) ( ), , ,

jk t

j j j jk t

V t v k t T D j Jσ τ

σ

τ= +

=

τ= ∀ ∈ ∈ ∈∑ (3)

where is the marginal activity utility in time interval t. )(tv j

Define as the travel disutility caused by leaving r for s during period t for performing activity j for duration and parking in location i, formulated as

),(, jj

irs tC τ

jτ

)),(()(),(, jrij

isrijj

irs tTtCtTtC τ++=τ , (4) JjDTtNsNrIi jrs ∈∈τ∈∈∈∈∀ ,,,,,

where is the travel disutility by traversing parking location i for destination s, and performing activity j for duration

)),(( jrij

is tTtC τ+

jτ there, expressed as

)]([)),(())(()),(( 321 twtTtztTtdtTtC jssisjriiriijri

jis σΘα+α+τ+α++α=τ+ ,

JjDTtNsNrIi jrs ∈∈τ∈∈∈∈∀ ,,,,, (5)

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where is the time-varying parking charge, is the schedule delay

penalty cost of early or late arrival at destination s for performing activity j, , are the reciprocal transformation coefficient between searching time, walk time and travel disutility, and

, are the conversion factors between parking charge, schedule delay cost and travel disutility.

)),(( jrii tTtz τ+ )]([ tjs σΘ

1α 3α

2α sα

The parking charge, which depends on the arrival time and desired duration for parking, can be formulated as (Tong et al., 2004)

)),(()),(()),(( tTtptTthtTtz riijriijrii +×τ+×θ=τ+ (6) JjDTtNsNrIi jrs ∈∈τ∈∈∈∈∀ ,,,,,where is the parking charge discount given by location i’s policy (in general, a smaller iθ iθ corresponds to a longer duration for parking), )),(( jri tTth τ+ is the parking duration related to

activity duration and walking time from parking site to activity location, and is the hourly parking fee which is related to the arrival time interval at parking location. The parking duration, or h for short, equals the activity duration plus the time needed for the round trip on foot between parking location and activity location, i.e.,

))(( tTtp rii +

)),(( jri tTth τ+

( ( ), ) 2 , , , , ,ri j j is rsh h t T t w i I r N s N t T j Jτ τ= + = + ∀ ∈ ∈ ∈ ∈ ∈ (7) The walking time , which depends only on the distance from parking location to destination, can be defined as

isw

( , ) , , ,is rsw i s i I r N s= Γ ω ∀ ∈ ∈ ∈N (8) where is the walking distance from location i to destination s, and ω is the walking speed (km/h).

),( siΓ

The schedule delay costs of early or late arrival at destination s, through parking location i for performing activity j, can be formulated as (Huang and Lam, 2002)

JjTtNstttt

tttt

t js

js

js

jsj

js

js

js

jsj

js ∈∈∈∀

⎪⎪⎩

⎪⎪⎨

⎧

σ<Δ+Δ−−σλ

σ>Δ−σ−Δ−ζ

=σΘ ,,,

otherwise,0

)(if],)([

)(if)],([

)]([ **

**

(9)

where is the desired time window for starting activity j at destination s

without any schedule delay penalty, is the middle point of the time window, and

],[ ** js

js

js

js tt Δ+Δ−

*jst jζ and jλ

are the unit costs of early and late arrivals, respectively. Given by Equation (2), is the start time of having an activity for travellers who depart from origin r in period t.

)(tσ

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As in Lam et al. (2002), the BPR (Bureau of Public Roads) type function for searching time is adopted in this study, i.e., ( )40( ( )) 0.15 ( ( )) , , , ,i ri i i ri i rsd t T t d D t T t K i I r N s N t T+ = + + ∀ ∈ ∈ ∈ ∈ (10)

where is the free-flow access time for parking a car in location i, is the parking

accumulation or occupancy in location i when vehicles arrive at time , and K

0id ))(( tTtD rii +

))(( tTt ri+ i is the capacity of location i for parking. The parking accumulation in location i can be computed from path inflows. For OD pair (r, s), the cumulative arrivals, , for performing activity j and parking their vehicles in location i by the period t (not including period t) can be represented by

)(tU jri

(11) ( )

1

,1

( ) ( , ), , , , ,ri j

k T kri

tj j

ri ri p j rsk p P

U t f k i I r N s N t T j Jξ

ξ τ

τ+ =

−

= ∈

= ∀ ∈ ∈ ∈∑ ∑ ∑∑ ∈ ∈

∈

∈ ∈

∈

where is the set of all paths between origin r and parking location i, is the flow

rate of travellers who depart from r to i in period k by path p and perform activity j for duration around destination s. Then, the total cumulative arrivals in any parking location i till period t

from all OD pairs for performing activity j, , can be computed by

riP ),(, jj

pri kf τ

jτ

)(tU ji

,( ) ( ), , ,j j

i ri rsr s

U t U t i I t T j J= ∀ ∈ ∈∑ (12)

For a certain OD pair, the travellers with parking duration h, who arrive at parking location i before period (t-h) should leave this location till period t. So, the cumulative departures from location i by period t are

(13) ( )

1

,1

( ) ( , ), , , , ,ri j

k T kri

t hj j

ri ri p j rsk p P

V t f k i I r N s N t T j Jξ

ξ τ

τ+ =

− −

= ∈

= ∀ ∈ ∈ ∈∑ ∑ ∑∑

Considering all OD pairs, the total cumulative departures from any parking location i, are (14)

,( ) ( ), , ,j j

i ri rsr s

V t V t i I t T j J= ∀ ∈ ∈∑Therefore, the parking accumulation in location i during period t, for having activity j, can be computed by

( ) ( ) ( ), , , , ,j j ji i i rsD t U t V t i I r N s N t T j J= − ∀ ∈ ∈ ∈ ∈ ∈

∈

(15) The total parking accumulation in location i during period t is (16) ( ) ( ), ,j

i ij

D t D t i I t T= ∀ ∈∑where I denotes the set of all parking lots in the network.

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Behaviour modelling

According to A5 made previously, the activity and travel choices follow a hierarchical decision-making process. Specifically, before making a trip, a traveller first determines his or her activity type and activity location, and then selects the desirable activity duration. We now describe the hierarchical choice process using the nested logit (NL) model, which is not subject to the independence of irrelevant alternatives (IIA) property as in the multinomial logit (MNL) model. The NL model is often used to characterize the travel and activity choices in the literature (Horowitz, 1980; Kitamura and Kermanshah, 1984; Mannering et al., 1994). In addition, the NL model was used by Ben-Akiva et al. (1986) for studying the departure time and route choices, by Bhat (1998b) for investigating the travel mode and departure time choices, and by Huang and Lam (2003) for formulating the departure time and route choices under advanced traveller information systems. Let )1( −tNr be the number of potential travellers within the origin zone r at the end of period t-1, these people will make decisions on travel and activity schedules in the next time period t. Let

be the travel demand departing from r for s in period t and performing activity j for duration . Using the nested logit model, we have

),( jj

rs tq τ

jτ

,

exp[ ( , ) / ] exp[ ( ,*) / ]( , ) ( 1) ,exp[ ( , ) / ] exp[ ( ,*) / ]

j

j jrs jj rs

rs j r j jrs j rs

s j

U t U tq t N tU t U t

τ

τ β γτ = −

τ β γ∑ ∑ JjDTtNsNr jj ∈∈τ∈∈∈∀ ,,,, (17)

where , which reflects the accessibility of performing activity

j at destination s by the travellers originating from r at period t.

∑τ

βτβ=j

jj

rsj

rs tUtU ]/),(exp[ln,*)(

β and are the scaling parameters that measure the variability in choice behaviour among the travellers. To be consistent with random utility theory and reasonable signs of cross-elasticity, the mixed discrete/continuous nested logit model requires that .

γ

0>β≥γ Let be the marginal travel demand departing from r for s at period t for performing activity j. From Equation (17), one obtains

)(tq jrs

,

exp[ ( ,*) / ]( ) ( 1) , , , ,exp[ ( ,*) / ]

jj rs

rs r jrs

s j

U tq t N t r N s N t T j JU t

γγ

= − ∀ ∈ ∈ ∈ ∈∑

(18)

Combining Equation (17) and (18), we get

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,]/),(exp[

]/),(exp[)(),(∑τ

βτ

βτ=τ

j

jj

rs

jj

rsjrsj

jrs

tU

tUtqtq JjDTtNsNr j ∈∈τ∈∈∈∀ ,,,, (19)

Now we formulate the parking location choices for travellers departing from origin r at time slice t and performing activity j for duration jτ at destination s. Following A6, travellers park their cars in such location that has the minimum disutility from origin to final destination. This results in a deterministic parking location choice equilibrium: for each OD pair, travellers who depart in the same time interval and perform activity j for duration jτ , experience identical and minimal generalized travel cost; no traveller would be better off by unilaterally changing the parking location. In other words, the travel cost of any unused parking location for those who perform activity j for duration , is greater than or equal to the minimal travel cost. The parking location choice equilibrium can mathematically be expressed in a complementary form as

jτ

(20) , ,[ ( , ) ( , )] ( , ) 0, , , , , ,j j jrs i j rs j rs i j rs jC t C t q t i I r N s N t T D j Jτ τ τ τ− = ∀ ∈ ∈ ∈ ∈ ∈ ∈

∈

τ

(21) , ( , ) ( , ), , , , , ,j jrs i j rs j rs jC t C t i I r N s N t T D j Jτ τ τ≥ ∀ ∈ ∈ ∈ ∈ ∈

(22) , ( , ) 0, , , , , ,jrs i j rs jq t i I r N s N t T D j Jτ ≥ ∀ ∈ ∈ ∈ ∈ ∈ ∈

,( , ) min ( , ), , , , ,rs

j jrs j rs i j j

i IC t C t r N s N t T D j Jτ τ τ

∈= ∀ ∈ ∈ ∈ ∈ ∈ (23)

where is the travel demand departing from origin r for destination s via parking location i in period t and performing activity j for duration

),(, jj

irs tq τ

jτ . According to A6, the route choice of all travellers follows a deterministic time-dependent equilibrium: for each OD pair, the actual travel times experienced by travellers departing in the same time period are equal and minimal (Ran and Boyce, 1996). Let be the path travel

time for travellers departing from origin r to parking location i in period t via path p, where and is the set of all paths between origin r and parking location i. This time-

dependent route choice equilibrium can be mathematically expressed as

)(, tT pri

riPp∈ riP

(24) , ,[ ( ) ( )] ( , ) 0, , , , , , ,jri p ri ri p j ri rs jT t T t f t p P i I r N s N t T D j Jτ τ− = ∀ ∈ ∈ ∈ ∈ ∈ ∈ ∈

∈ (25) , ( ) ( ), , , , ,ri p ri ri rsT t T t p P i I r N s N t T≥ ∀ ∈ ∈ ∈ ∈

, ( , ) 0, , , , , , ,jri p ri rs jf t p P i I r N s N t T D jτ > ∀ ∈ ∈ ∈ ∈ ∈ ∈ ∈ Jτ

∈

∈ ∈

(26)

(27) ,( ) min ( ), , , ,ri

ri ri p rsp PT t T t i I r N s N t T

∈= ∀ ∈ ∈ ∈

The path travel time in Equations (24)-(25) can be represented by the sum of all link travel times along the path (Chen and Hsueh, 1998) (28) , ( ) ( ) ( ), , , , ,ri

ri p a apt ri rsa k t

T t c k k p P i I r N s N t Tδ≥

= ∀ ∈ ∈ ∈∑∑

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727

∈ ∈

∈

∈ ∈

where , an indicator variable, is equal to 1 if the flow on path p departing in period t from

origin r to parking location i arrives at link a in period k, and 0 otherwise, is the link travel time when entering link a in period k. Similar to the work by Chen and Hsueh (1998), the link travel time can be formulated as a function of the inflows entering that link before period k, i.e.,

)(kriaptδ

)(kca

(29) ( ) ( (1), (2), , ( )), ,a a a ac k f u u u k a A k T= ∀

where is the inflow entering link a in period k, and )(kua

(30) ,

( ) ( ), ,ria apt

r i p tu k u k a A k T= ∀ ∈∑∑∑

where is the inflow to link a in period k for these travellers who departs from origin r in

period t and select path p toward parking location i. The link inflow, , is composed of path inflows as follows:

)(ku riapt

)(ku riapt

. (31) ,( ) ( , ) ( ), , , , , , ,j

ri j riapt ri p j apt ri rs

j

u k f t k a A p P i I r N s N t T k Tτ

τ δ= ∀ ∈ ∈ ∈ ∈ ∈∑∑So far, we have defined and formulated all behaviour choices to be investigated in this paper. In summary, both the path choice and parking location choice follow the time-dependent deterministic user equilibrium, whereas the combined choices of destinations, activities and activity durations are governed by a nested logit model. We now define the time-dependent network equilibrium with simultaneous choices on activity type, activity destination, activity duration, parking location and travel route as follows. Definition 1 A flow pattern ( ))(),,(),,(),,( ***

,*, tqtqtqtf j

rsjj

rsjj

irsjj

pri τττ is a time-dependent

equilibrium of the joint choices on activity type, activity location, activity duration, parking location and travel route, if it satisfies the conditions (18)-(31) simultaneously.

EQUIVALENT VARIATIONAL INEQUALITY FORMULATION

In this section, we present an equivalent variational inequality (VI) formulation for the problem given by Definition 1. First, we list all constraints for the problem.

Constraints

Let Ω denote the set of path flows and OD demands that satisfy the following constraints: , ,( , ) ( , ), , , , , ,j j

ri p j rs i j rs jp

f t q t i I r N s N t T D jτ τ τ= ∀ ∈ ∈ ∈ ∈ ∈∑ J∈

∈

(32)

, ( , ) ( , ), , , , ,j jrs i j rs j j

i

q t q t r N s N t T D j Jτ τ τ= ∀ ∈ ∈ ∈ ∈∑ (33)

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( , ) ( ), , , ,j

j jrs j rsq t q t r N s N t T j J

τ

τ = ∀ ∈ ∈ ∈∑ ∈ (34)

,( ) ( 1), ,j

rs rj s

q t N t r N t T= − ∀ ∈ ∈∑ (35)

, ( , ) 0, , , , , , ,jri p j ri rs jf t p P i I r N s N t T D j Jτ ≥ ∀ ∈ ∈ ∈ ∈ ∈ ∈ ∈τ (36)

, ( , ), ( , ), ( ) 0, , , , , ,j j jrs i j rs j rs rs jq t q t q t i I r N s N t T D j Jτ τ τ≥ ∀ ∈ ∈ ∈ ∈ ∈ ∈ (37)

where Equation (32) states the conservation of path flows. Equation (33) is the parking demand conservation. Equation (34) is the demand conservation for activity duration. Equation (35) is the conservation for activity type and location combination. Equations (36) and (37) are the usual non-negativity constraints on path flows and demands, respectively. In addition, the conservation of flows at origin node r should be considered

( ) ( )( ) ( 1) ( 1 ( )) ( 1), ,r r a a a

a A r a B rN t N t u t c t u t r N t T

∈ ∈

= − + − − − − ∀ ∈ ∈∑ ∑ (38)

where is the set of links with head node r and is the set of links with tail node r. And, the following definitional constraints should also be included

)(rA )(rB

(39) , ( ) ( ) ( ), , , , ,riri p a apt ri rs

a k t

T t c k k p P i I r N s N t Tδ≥

= ∀ ∈ ∈ ∈∑∑ ∈ ∈

,, ,

( ) ( , ) ( ), ,j

j ria ri p j apt

r i p j tu k f t k a A k T

τ

= τ δ ∀ ∈ ∈∑∑∑∑ . (40)

VI formulation

The VI formulation equivalent to conditions (18)-(31) can be stated as follows: Theorem 1 A time-dependent flow pattern in the context of daily activity-travel schedule reaches an equilibrium state if and only if it satisfies the following VI requirement:

( )* * *, , , , ,

, , ,( ) [ ( , ) ( , )] ( , ) ( , ) ( , )

j j

j j j j jri p ri p j ri p j is j rs i j rs i j

r i p t j r s i j tT t f t f t C t q t q t

τ τ

⎛ ⎞⎜ ⎟τ − τ + τ τ − τ⎜ ⎟⎝ ⎠

∑∑∑ ∑∑ ∑∑∑∑ *

( )*

**

, ,

( , )ln ( ( , ) ) ( , ) ( , )

( )j

jrs j j j j

j j s s rs j rs jjr s j t rs

q tV t V V q t q t

q tτ

⎛ ⎞τ+ β −α τ + + τ −⎜ ⎟

⎜ ⎟⎝ ⎠

∑∑∑ τ

( )*

*

,

( )ln ( ) ( ) 0( 1)

jj jrs

rs rsrr s j t

q t q t q tN t

− γ −−∑∑∑ ≥ (

subject to ( ) Ω∈τττ )(),,(),,(),,( ,, tqtqtqtf jrsj

jrsj

jirsj

jpri , where Ω is given by (32)-(40). The

variables with asterisks in the above VI are the optimal solutions. The proof of Theorem 1 is similar to that in Lam et al. (2004). It is omitted here due to the paper length constraint, but will be provided if readers ask for. It should be pointed out that the VI (41)

Time-dependent activity & travel choice model with parking options

729

is route based, hence route enumeration is required in both model formulation and solution algorithm. It should also be noted that the indicator variable in (28) depends on link travel times which in turn depend on link inflows. Consequently, the path travel times are essentially non-linear and non-convex. It can be proven that all functions (given or endogenous) are continuous, hence there exists at least one solution (Huang and Lam, 2002).

)(kriaptδ

SOLUTION ALGORITHM

In this section, a route-based heuristic algorithm is developed to solve the model proposed in this paper. The basic logic of the algorithm follows the method of successive averages (MSA), but in each of the steps a time-dependent version of the Gauss-Seidel decomposition approach is used. This decomposition approach is one of the partial linearization methods (Patriksson, 1993). The step-by-step procedure of the algorithm is given below. Step 0 Let the iteration count n = 1. For each OD pair and time period, on the base of )0(rN ,

set the initial demand pattern { })()( tq njrs , { }),()(

jnj

rs tq τ and { }),()(, jnjirs tq τ , and

then compute the time-dependent path flow pattern { }),()(, jnjpri tf τ .

Step 1 Compute the link inflows by (30) and (31), and then the link travel time

by (29).

)()( ku na

)()( kc na

Step 2 Calculate the path travel time by (28), let and )()(, tT npri )(min)( )(

,)( tTtT n

priPpn

riri∈

=

{ }( ) ( ) ( ),: ( ) ( ),n n n

ri ri p ri riP p T t T t p P= = ∈ .

Step 3 Compute the generalized travel cost by (4)-(16), let ),()(, jnjirs tC τ ( ) ( , )j n

rs jC t τ =

and ( ),min ( , )

rs

j nrs i j

i IC t

∈τ { }( ) ( ) ( )

,: ( , ) ( , ),n j n j nrs rs i j rs j rsI i C t C t i I= τ = τ ∈ .

Step 4 Compute the comprehensive utility by (1)-(3), update ),( jj

rs tU τ )1( −tNr by (38), and

then get the travel demand and by (18) and (19), respectively. )()( tg njrs ),()(

jnj

rs tg τ

Step 5 Assign to the parking sizes in ),()(j

njrs tg τ ( )n

rsI defined in Step 3, get the parking

demand . ),()(, jnjirs tg τ

Step 6 Assign onto the paths in defined in Step 2, get the auxiliary path

inflows .

),()(, jnjirs tg τ ( )n

riP

),()(, jnjpri th τ

Step 7 Update the path departure flows by MSA,

)),(),((1),(),( )(,

)(,

)(,

)1(, j

njprij

njprij

njprij

njpri tfth

ntftf τ−τ+τ=τ+ .

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

730

Step 8 If , , ε<τ−τ+ |),(),(| )(,

)1(, j

njprij

njpri tftf ε<τ−τ+ |),(),(| )(

,)1(

, jnjirsj

njirs tqtq ( 1)| ( , )j n

rs jq t+ τ −

and , stop; set n = n+1 and go to Step 1, otherwise. The parameter is a pre-specified precision, taking 0.01 in this paper.

( ) ( , ) |j nrs jq t τ < ε ε<−+ |)()(| )()1( tqtq nj

rsnj

rs

ε In Step 0, the initial time-dependent path flows can be determined by assigning the initial demand patterns onto an empty network. The operations in Steps 1 to 6 directly follow the equations presented in previous sections. In Step 7, the MSA is used to update the path departure flows with predetermined step size.

NUMERICAL EXAMPLE

In this section, a numerical example is given to illustrate the model and algorithm in this paper. We will investigate the effects of various parking schemes on both activity and travel schedules.

Input data

To illustrate the application of the proposed model and algorithm clearly, we apply them to a simple network consisting of 3 zones, as shown in Figure 1. Four types of activities, i.e. staying at home (H), working (W), having dinner (E), and joining social activity (S) are considered. Each link is two-way link with identical free-flow travel time and capacity in two directions, as given in the Table 1. There exist two categories of parking locations around work zone and social activity zone, i.e. the on-street parking locations (c, d, e, g, i) and the off-street parking locations (b, f, h). There is only one off-street parking location (garage) a around residential zone. Both the W and E activities can take place in work zone, but W is the primary one. The marginal temporal utility profiles of these four activities are shown in Figure 2. The study horizon is from 00:00 to 24:00 and is discretized into 96 time periods each with 15 minutes. Table 2 gives the unit costs of the schedule delay penalty, the desired start time windows and the feasible durations for these four activities. It is assumed that there are 3000 seed travellers in the residential zone, i.e. . The parking access (and egress) times for all on-street parking sites and all off-street parking locations are assumed to be 0.05 hour and 0.1 hour, respectively. The charges over the time of day for on-street and off-street parking follow a step function, as shown in Figure 3. However, the off-street parking (garage) in home zone and the on-street parking around restaurant are free. Moreover, it is assumed that the parking for working activity is subsidized by the employer’s company. The distance from each off-street parking point to any nearby activity location is 0.5 km, while the distance from each on- street parking points to any nearby activity location is 0.25 km. The walking velocity is ω = 5.0 km/h. The discount coefficients for all parking locations are assumed to be 1.0. The capacities of nine parking locations are: K

3000)0( =N

iθ

a=3000, Kb=2600, Kc=Kd=200, Ke=300, Kf=Kh=1000, Kg=Ki=300, all in (veh/h).

Time-dependent activity & travel choice model with parking options

731

Other parameters are: 025.0=α , , , , , , . The case with the above inputs is called “base case”.

4.11 =α 1.02 =α 8.13 =α 3.0=α s 1.0=β 3.0=γ

Figure 1 The example network

Table 1 Input data associated with link travel time functions

Link 0ac (h) aC (veh/h) Link 0

ac (h) aC (veh/h)

(1, c) 0.45 1500 (g, d) 0.15 1000 (c, 2) 0.05 1000 (d, 2) 0.05 500 (1, i) 0.35 1500 (W, e) 0.15 1000 (i, 3) 0.05 1000 (e, E) 0.03 500 (3, g) 0.05 500 — — —

Table 2 Input parameters associated with activities

Activity type

Arrival early penalty jζ (HK$/h)

Arrival late penalty jλ (HK$/h)

Time window (o’clock)

Activity duration (h)

H 0.0 0.0 [00:00, 24:00] [8.0, 12.0] W 8.0 25.0 [08:45, 09:15] [5.0, 12.0] E 5.0 20.0 [12:30, 13:00] [0.5, 02.0] S 3.0 15.0 [18:00, 21:00] [1.0, 03.0]

Social activity zone

c Work zone Residential zone

a b

hOff-street parking location

On-street parking location

f

i

d

g

1 2 W

H e

E

3

S

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

732

Utility of working

Utility of eating

Utility of social activity

Utility of in-home

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00

Time of day (o'clock)

Mar

gina

l util

ity (H

K$)

24:00

Figure 2 Temporal utility profiles for the four activities

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0:00-1:00

2:00-3:00

4:00-5:00

6:00-7:00

8:00-9:00

10:00-11:00

12:00-13:00

14:00-15:00

16:00-17:00

18:00-19:00

20:00-21:00

22:00-23:00

Time period (hour)

Park

ing

char

ge (H

K$)

Off-street parking location

On-street parking location

Figure 3 Step charges designed for on-street and off-street parking locations

Numerical results

We firstly examine the convergence of the iterative algorithm. Figure 4 plots the value of the equilibrium criterion against iteration number. This equilibrium criterion is defined as

Time-dependent activity & travel choice model with parking options

733

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

, , , ,, , , , , ,

( , ) ( ) ( ) ( , ) ( , ) ( , )j j

j n n n j n j n j nri p j ri p ri rs i j rs i j rs j

r i j p t r s j i t

G f t T t T t q t C t C tτ τ

= τ − + τ τ −∑∑∑ ∑∑∑ τ

( )( ) ( ) ( )

( ), ,

exp( ( , ) / )( , )[ ( , ) ( ) ]

exp( ( , ) / )j

j

j nrs jj n j n j n

rs j rs j rs j nr s j t rs j

U tq t q t q t

U tττ

τ β+ τ τ −

τ β∑∑∑ ∑

( )( ) ( ) ( )

( ),

,

exp( ( ,*) / )( )[ ( ) ( 1) ]exp( ( ,*) / )

j nj n j n n rs

rs rs r j nr s j t rs

s j

U tq t q t N tU t

γ+ − −

γ∑∑∑ ∑ .

This is a gap function which can measure how closely the outputs generated at n-th iteration satisfy the time-dependent user equilibrium condition (19)-(32). Figure 4 shows that the algorithm converges well, although some oscillations exist in the first 150 iterations.

Figure 4 Convergence of the solution algorithm

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 50 100 150 200 250 300 350 400 450 500

Number of iterations

Val

ue o

f gap

func

tion

G

Figure 5 displays the parking occupancies in nine locations for parking. It can be seen that before 6:00 all people still stay at their homes, thus location a around H are full. After 6:00, some people start to leave their homes for working; at about 9:15, almost all people arrive at their offices due to the rigid work start time windows. Starting from this time, location a is idle and the parking locations around W become busy. We can see that the off-street parking location b is first occupied, due to its limited capacity then, the overflows go to the on-street parking locations c and d. After 11:00, some people leave their offices for having lunch; they first occupy the on-street parking location e around restaurants as it is free for parking; at about 12:30, location e is

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

734

fully occupied and some people turn to the off-street location f; at 13:30, the occupancy rate of location f reaches 82.3%. At about 14:00, most people return to offices for working. After 16:00, some people leave their offices for performing social activities. They first park their cars at the on-street parking location g, then at the off-street parking location h when g is congested. After about 21:45, all people end their daily activities and go back home. Nobody uses the parking location i because the chain “home-social activity-work” hardly occurs under the requirement of rigid work start time windows.

0:001:152:303:455:006:157:308:45

10:0

011

:15

12:3

013

:45

15:0

016

:15

17:3

018

:45

20:0

021

:15

22:3

023

:45

abcdefghi

0102030405060708090100

Parking occupancy(%)

Time (o'clock) Parking location

Figure 5 Parking occupancies at different sites over the time of day

Figure 6 gives the population against day time in the three zones. It indicates that most people stay at home before 7:45 and after 21:15, work and have lunch from about 8:45 to 17:45, and join social activities from about 18:00 and 21:00. Figure 7 depicts the departure rates for working against day time and duration length. It is found that the highest departure rate for working reaches 200.4 vehicles per hour at around 8:15, these people work for 8.25 hours long. For examining the model’s sensitivities subject to different parking policies, we design the following four scenarios for comparison, i.e., Scenario I-base case, Scenario II-the capacity of location b is increased to Kb=3000, Scenario III-the charge of location h is decreased by 2 HK$, and Scenario IV-the combination of II and III. Table 3 summarizes the results generated in these scenarios. It can be seen that in Scenario II the average time for work increases by 0.23 hour and the average time for staying at home increases by 0.08 hour over the Scenario I, but the average time for social activity decreases by 0.12 hour. This is because the capacity increasing of location b reduces the difficulty of parking cars around W, so that people can postpone their departure

Time-dependent activity & travel choice model with parking options

735

times from home and work longer in offices. Some people leave their offices late and have to cancel their social activities. The alleviation of parking congestion improves relevant things in other aspects, e.g., the average travel time and parking delay time drop 0.1 hour and 0.09 hour, respectively.

0:001:152:303:455:006:157:308:45

10:0

011

:15

12:3

013

:45

15:0

016

:15

17:3

018

:45

20:0

021

:15

22:3

023

:45

HW

S

0

500

1000

1500

2000

2500

3000Population

Time (o'clock)Activity zone

Figure 6 Population in the three zones over the time of day

5:00-5:15

6:00-6:157:00-7:15

8:00-8:159:00-9:15

10:00-10:1511:00-11:15

11:45-12:00

5.06.07.08.09.010.011.012.00.0

30.0

60.0

90.0

120.0

150.0

180.0

210.0

Departure flow rate (veh/h)

Tim

e per

iod

(hou

r)

Work duration (hours)

Figure 7 Departure rates for working against time and duration length

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

736

In Scenario III, the charge reduction in parking location h attracts more people to participate in social activities after work, but not to go home directly. More people leave home for office ahead of time so as to win a slacker time budget for performing social activity, and thus the traffic congestion from home to workplace in the morning rush hour is alleviated. As a result, the average time for social activity gets an increase of 0.21 hour, the average time for staying at home drops 0.13 hour, and the average travel time decreases by 0.05 hour.

Table 3 Effects of different parking policies

Average time spent for (hour) Travel Parking Walking H W E S Scenario I 1.38 0.61 0.44 11.19 8.17 0.64 1.57 Scenario II 1.28 0.52 0.41 11.27 8.40 0.67 1.45 Difference (II-I) -0.10 -0.09 -0.03 0.08 0.23 0.03 -0.12Scenario III 1.33 0.65 0.48 11.06 8.08 0.62 1.78 Difference (III-I) -0.05 0.04 0.04 -0.13 -0.09 -0.02 0.21 Scenario IV 1.30 0.56 0.46 11.13 8.26 0.65 1.64 Difference (IV-I) -0.08 -0.05 0.02 -0.06 0.09 0.01 0.07

It should be noted that the results generated in Scenario IV are not simply the sum of those in Scenarios II and III. This implies that the systematic changes caused by parking capacity improvement and parking fee adjustment are much more complicated than expected. This numerical example shows that the proposed modelling approach provides a powerful tool for better understanding and assessment of the complex activity/parking/travel-related behaviours in various scenarios. By introducing the unified framework, we offer a new avenue of comprehensively evaluating parking schemes from strategic viewpoint.

CONCLUSIONS

In this paper, we employed the concept of the temporal utility profile of activities to model the activity choice and parking behaviours. We developed a time-dependent equilibrium model in the framework of the activity-based travel analysis. The model simultaneously considers traveller’s choices on activity type, activity location, activity duration, parking location, and traffic route over the time of day. The decision-making process on these choices follows a hierarchical structure. This model is formulated as an equivalent variational inequality problem and is solved by a heuristic solution algorithm. The numerical results show that the proposed model can truly reveal people’s choices on the travel and activity behaviour over the time of day, and the induced occupancies in various parking locations. The findings also show that parking schemes such as increasing parking capacity, decreasing parking charges and the combination of both, will greatly influence people’s travel, parking and activity patterns, and thus the associated time allocations. Our model can be considered as a coherent activity-choice parking demand analysis tool for long

Time-dependent activity & travel choice model with parking options

737

term strategy planning. By introducing the activity-based approach, we offer a new avenue of parking demand analysis for transportation planning and policy evaluation. Future research work may focus on the extension of the proposed model to multiple behaviourally heterogeneous groups and more explicit consideration of activity duration. The extension of incorporating modal choices into trip chains may also be conducted. The combined activity/parking choice models will also undoubtedly benefit from the attempt to empirically measure the utility functions of various activities. Although the numerical results that are presented in this paper are self-consistent, we recognize that various experimentations on larger networks are necessary for further validation of the proposed model.

ACKNOWLEDGEMENTS

The work described in this paper was supported by grants from the National Natural Science Foundation of China (70429001), and the Research Grant Council of the Hong Kong Special Administrative Region (PolyU 5040/01E, PolyU 5143/03E and HKU 7134/03E).

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