Time of day choice modelling

Time of day choices of travellers are important for transport planning mainly because in dense urban areas they are central to the generation and dissipation of congestion waves (Mahmassani, 2000). Transport analysts have been working on understanding the determinants for these decisions for several decades in order to devise demand side strategies that serve to spread peaks in demand. This is vital for improvements in the operational reliability of transport networks in urban areas. Demand side strategies may in fact avoid resorting to capacity increases, especially in situations in which this may not be physically, economically or politically viable. Moreover, they allow other issues associated with congestion externalities to be tackled: particularly poor urban air quality and the health risks that this is associated with (Levy et al., 2010).

Most of the studies analysing the choice of the time of travel are based on the concept the individuals have a preferred time of travel and moving away from that causes disutility. In particular the first important contribution for this view is the model of Vickrey (1969), in which it is assumed that individuals choose their time of travel as result of a trade off between travel time and a measure of early arrival and late arrival to work (Vickrey‟s studies only commuting trips). The measures of early arrival and late arrival are schedule delay early ( ) and schedule delay late ( ) defined as follows

(3.1)

where is the departure time, is the departure time dependent travel time and is the preferred arrival time. The choice of travel time is treated in standard microeconomic perspective as a result of the maximisation of the following utility function

96

(3.2)

All the model parameters and are assumed to be negative since individuals derive disutility for longer travel times and for departure times that are shifted from the preferred one, whether earlier or later. This model assumes that travellers trade-off travel time and schedule delays, i.e. depending on the relative magnitudes of the marginal utilities, and that the optimal choice may result in accepting an earlier departure in order to reduce the travel time.

Vickrey‟s theoretical model was then reformulated and estimated empirically in a discrete choice framework using revealed preference data by Small (1982), who again considers only the trade-offs between travel time and schedule delays. Small‟s (systematic) utility function also considers an additional term to capture the jump in utility in the presence of a delay. While the models by Vikery‟s model and by Small‟s model consider only time of day choice, later studies have combined time of day with other choices. Mannering (1989), Arnott et al. (1990), Mahmassani et al. (1991), and Khattak et al. (1995) have developed models for jointly analysing travellers‟ time of day ad route choices. Hendrickson and Plank (1984); Bhat (1998a); Bhat (1998b); de Jong et al. (2003); Hess et al. (2007) and Lizana et al. (2013) have jointly studied the choice of travel timing and mode. In other cases, the choice of time of travel has been studied in conjunction with choice of activity timing, for example by Polak and Jones (1994); Wang (1996); Ettema et al. (2004) and Ettema et al. (2007).

Moreover, while the models by Vickrey and Small have considered only tradeoffs between travel time and schedule delays, later studies have considered also other sources of tradeoffs: namely travel reliability and travel costs. The idea that travellers trade between travel times, schedule delay and charging costs is at the base of road pricing schemes. A review of studies considering the effect of travel time reliability is provided by Bates et al. (2001). Of those explicitly including travel costs, the majority were developed to analyse the effect of time-of- use road pricing (see for example, Polak and Jones, 1994, de Jong et al., 2003, Arellana et al., 2013).

Amongst the studies analysing the effect of price-based traffic management policies, some departed for the original trip based approach of Vickrey and Small to consider a tour-based perspective. Polak and Jones (1994) developed a theoretical framework for the simultaneous choice of the timing of the outbound and inbound legs of home-based tours. The advantage of a tour-based approach to time of travel choice is that it allows the explicit consideration of “the linkage between timing decisions of journeys within an overall activity pattern” (Polak

97 and Jones 1994). The basic idea is that a traveller undertaking a daily commute maximises the utility that he derives from spending time on:

 home activities before the outbound journey,  travelling to their destination,

 activities (work) at destination,  travelling back home,

 and home activities after the inbound journey.

The utility of activities including travelling is time dependent, intrinsically and because of scheduling constraints. It also depends on the duration. In their paper Polak and Jones establish the link between outbound and inbound legs and the scheduling of activities at home and out of home. To represent the intrinsic preference for activity timing Polak and John use marginal utilities for activity participation in continuous time. Thus they represent the utility that an individual derives in taking part in an activity, in terms of its time dependent flow rate drawing from Winston‟s theory on timing of the economic activities (Winston, 1982):

(3.3)

where is the activity start time and is duration.

Polak and Jones derive then the utility of a (two leg) home based tour as sum of individual contributions: the utility attained by spending time at home before travelling (starting from midnight); the (dis-)utility from travel at destination D, the utility from spending time at destination, the utility from travelling back home and the utility from spending the rest of the available time , (e.g. 24 hours), time at home.

(3.4)

where: the is the utility flow rate from spending time on home activities; _{is the} utility flow rate (assumed constant) from spending time travelling; is the utility flow

98 rate from spending time at the destination, is the departure time of the outbound leg, which is also equal to the time spent at home, from midnight, before departure, is the time spent at the destination; and are the respective travel times. is the utility from consumption of the generalised good G (assumed to be independent of time).

Travellers maximise by choosing and subject to the budget constraint (the time constraint is already implied by the integral limits):

(3.5)

where and are travel costs, is the unit price of G, Y is an unearned income and w is the wage rate (which is multiplied by since Polak and Jones consider specifically a home based tour to work).

They then derive an expression for the indirect utility by:

 Linearising the expressions of the utilities from activity participation, by using first order Taylor expansions around reference timings (e.g. those form a tour observed in a traveller‟s travel diary);

 Substituting first order Taylor expansions of the utility attained in home and destination activities into the Lagragian of the optimisation problem above.

Finally, they obtain an expression of the following form for the indirect utility of a tour:

(3.6)

where is the departure time of the outbound leg, is the time spent at destination; and are the respective travel times and and are the inbound and out bound travel costs respectively. The starred quantities are the respective quantities in an observed tour. In the expression above we can identify in the first term a schedule adjustment, in the second term an adjustment in activity participation at destination, often named participation time penalty Hess et al. (2007). The meaning of such an expression is that travellers “trade-off schedule delay against participation time, when adjusting to changes in travel times and costs”. Polak and Jones approach was then followed in other empirical applications by de Jong et al. (2003) and Hess et al. (2007).

Polak and Jones‟s model was further expanded by Ettema et al. (2007) in order to disentangle intrinsic time of day preference from the effect of scheduling constraints and to take into

99 account satiation effects in the utility attained from activity participation. Ettema et al. (2007), in order to express the intrinsic preference to take part in a specific activity at a specific time of day use a nonlinear (bell-shaped) functional form for the time of day dependent marginal utility of activities. This bell-shaped marginal utility means that there is a specific time of day in which the marginal utility for taking part in that activity is at a maximum. The functional form they choose for the marginal utility (that of a Cauchy distribution) has a closed form integral, so that they do not need linearization to express the utility. To express satiation effects they use the logarithm of the activity participation time, so that the longer the time spent in an activity, the lower the marginal utility the individual attains from it. Finally, in order to capture the effect of scheduling constraints, Ettema et al. (2007) use schedule delay terms as in Small‟s approach. These three contributions to the utility attained from activity participation are assumed to be additive.

They thus obtain the expression of the utility for activity participation as:

(3.7)

where is the marginal utility for participating in an additional instant to activity A, is the duration of the activity, the activity start time, _{is the early schedule delay} with respect to a preferred start time _{of the activity, is the late schedule delay} with respect to a preferred start time _.

The expression above is then used in a formulation that expresses the utility attained in an activity-travel as a sum of contributions from activity participation and contributions from the time spent travelling.

It should be pointed out that all the tour-based models mentioned have been estimated using SP data, whereas, there are examples of trip-based models in which the estimation was also based on revealed preference data. Apart from the first RP study by Small (1982), more recent work by Lizana et al. (2013) estimates a trip timing model jointly using SP and RP data from a recent survey carried out in Santiago, Chile (Arellana et al., 2013). Furthermore, often, instead of calculating the schedule delays with respect to the preferred arrival time (or departure time), since this needs to be explicitly asked of survey respondents, observed arrival (departure) times are used as a reference. This approach has been questioned by Bates (2008b) because it is inconsistent with the Small-Vickery method. When observed timings are

100 used as a reference, the utility of several scheduling alternatives is relative to the status quo. This may not be representative of the preferred condition; therefore a schedule delay may not necessarily cause disutility. The status quo is likely to be the result of a series of seamless adaptations in travellers‟ interlinked activities, however, and therefore a change is very likely to cause a disutility, resulting in disruption to current behaviour. This is demonstrated by the consistently negative estimates in schedule delay parameters obtained in studies using observed travel timings as reference points in the definition of the schedule delay terms. In the next section for the formulation of the model for joint EV use scheduling and charging choices, we adopt delay/participation time penalty formulation, as the novelty of the present work is intended to be the joint analysis of charging and activity timing choices. The nuances introduced by Ettema et al. are for the sake of simplicity, avoided in the present treatment.