In this paper, we develop a model of travel in a chain of trips joining several locations througha congested network. We develop a microscopic analysis of individual benefits obtained byspending time at each of the locations and costs incurred through travel between them. This iscombined with a macroscopic equilibrium model of travel during congested peak periods toshow how individuals? travel choices are influenced by the congestion that result fromcorresponding choices made by others. We show how different travellers can achieveidentical net utilities by making different combinations of choices within the equilibrium. Theresulting model can be used to investigate the effect on travel behaviour and individual utilityof various transport interventions, and we illustrate this by considering the effect of a peakperiodcharge that eliminates congestion

Heydecker, B

English

UCL Discovery

Submitted to the BiNs Workshop to be held on 22-23 JULY 2004 IN SEOUL, KOREA   EQUILIBRIUM ANALYSIS OF TRIP CHAINS IN CONGESTED NETWORKS  BG Heydecker  Centre for Transport Studies University College London London, England    ABSTRACT   In this paper, we develop a model of travel in a chain of trips joining several locations through a congested network. We develop a microscopic analysis of individual benefits obtained by spending time at each of the locations and costs incurred through travel between them. This is combined with a macroscopic equilibrium model of travel during congested peak periods to show  how  individuals’  travel  choices  are  influenced  by  the  congestion  that  result  from corresponding  choices  made  by  others.  We  show  how  different  travellers  can  achieve identical net utilities by making different combinations of choices within the equilibrium. The resulting model can be used to investigate the effect on travel behaviour and individual utility of various transport interventions, and we illustrate this by considering the effect of a peak-period charge that eliminates congestion.     2Submitted to the BiNs Workshop to be held on 22-23 JULY 2004 IN SEOUL, KOREA  EQUILIBRIUM ANALYSIS OF TRIP CHAINS IN CONGESTED NETWORKS  BG Heydecker Centre for Transport Studies University College London London, England   1.  Introduction  The decision entailed in undertaking a trip from one location to another is influenced by the benefits that would be gained by remaining at the origin, the conditions that are encountered in the making the trip, and the benefits gained through arriving at the destination. A traveller who undertakes a trip will generally do so because the benefits of reaching the destination outweigh the losses associated with leaving the origin and the costs of making the trip. In making a trip, the traveller will use a public access transport system of a kind that can become congested at peak times. Thus the collective behaviour of travellers influences the conditions experienced by each of them in travelling.   From the point of view of the individual travellers, the requirement to travel arises from the range of locations at which different activities can be undertaken. This view of travel leads to a  microscopic  analysis  of  trip-making  behaviour  by  individuals.  On  the  other  hand,  the collective  effect  of  travellers  can  be to cause congestion that will impact on the travellers themselves through increased journey times and decreased convenience of travel. This view leads to a macroscopic analysis of travel in congested networks.  In the present paper, we consider how these two distinct approaches to the analysis of travel behaviour can be combined into a self-consistent model. This brings a utility-based analysis of  trip  chains  that  describes  the  requirement  to  travel  together  with  an  equilibrium-based analysis of the choices that are available to travellers, and in particular those of departure-time and route.    3According to this model, benefits are obtained through time spent at the different locations, whilst costs are incurred through travel between them. This represents the role of travel as a means to the end of gaining access to facilities at a range of locations and hence emphasises its nature as a derived demand. In this model, we represent the benefit of attendance in the form of utility that depends on timing and duration of attendance. The resulting trip chaining analysis is based upon consideration of the individuals’ requirements for travel, and is framed in terms of the benefits that they gain through having travelled. We represent the cost of travel between locations, including congestion delays that are incurred as a consequence of travel during  peak  periods,  in  the  form  of  travel  time.  The  present  approach  combines  these elements  in  a  single  framework,  for  which  we  present  analytical  results.  We  apply  this analysis to a simple example and show how in equilibrium, different travellers can achieve identical  net  utility  through  different  combinations  of  utility  and  travel  cost  by  scheduling their travel at different times.    2.  A model of trip making 2.1  Introduction  We consider a population of travellers that is homogeneous in respect of their travel needs and their trip making decisions. Suppose that each of these travellers undertakes a chain of  trips that starts and ends at the same location (home) and visits a series of locations that we take in the first instance to be predetermined. The timing of these trips depends on the timeliness of attendance at each of the origin and the destination locations in respect of the benefits that accrue to the individuals. The duration of each trip depends on the traffic conditions that are encountered, and can be estimated by use of a traffic model. We suppose that travel between locations  is undertaken when the benefit of attendance at the destination surpasses that of remaining at the origin when allowance is made for the time and cost of travel. The timing of the trip is then a resolution of the tension between the benefits for the individual of being at the  origin  and  at  the  destination.  This  is  balanced  against  the  cost  of  travel  through  the network, which varies according to the congestion caused collectively by travellers. Hence the departure rates and consequent levels of congestion in the network are endogenous to the present analysis.     42.2  Analysis of trip-making  Consider a single trip  j  (1 < j < J)  made by a traveller that forms part of the day-long chain of  J  trips. Suppose that the traveller departs from location  j - 1  on trip  j  at time  sj  and consequently arrives at location  j  at time  tj(sj) so that the duration of this trip is  tj(sj) -  sj .  The arrival time for the trip is determined from the departure time by use of a traffic model according to the conditions that are encountered. The total travel time during a trip chain of this kind is then  Sj [tj(sj) - sj] .  We suppose that the time from  t  to  s  spent at location  j  confers a benefit to an individual, which  we  represent  as    fj(t, s) .  For  convenience,  we  express  this  in  terms  equivalent  to savings in travel time. This benefit can depend separately and jointly on each of the start time  t ,  the duration of attendance  s - t , and the end time  s . In a trip chain, the start time  t  at location  j  is given by the arrival time  tj(sj)  of journey  j , and the end time  s  is given by the departure time  sj+1  of journey  j+1 . Thus the benefit derived from attendance at location  j  is  fj[tj(sj), sj+1]  and the benefit accumulated during a trip chain of this kind is  Sj fj[tj(sj), sj+1] .  The net benefit to an individual of undertaking a trip chain of this kind with departure times  s  is then     ( ) [ ] ( ) [ ] ( ) [ ] ∑=+ - t - t + =Jjj j j j j j j s s s s f s f V11 1 0 , , 0 s           (1) where by convention we set  sJ+1 = 24 h  so that the final part of the day is spent at location  J .  Variations in departure time  sj  will affect the benefit that is obtained at the origin location  j – 1 , the duration of the journey, and the arrival time at the destination location and hence the benefit obtained there.   2.3  Analysis of network equilibrium   Suppose that the departure rate of individuals on trip j (1 < j < J)  at time  s  is  ej(s) . In equilibrium, the value V(s) achieved by each individual is identical – otherwise, some would have an incentive to change their departure times. Following Heydecker and Addison (2004),   5we note that while the departure rate is non-zero, the value of net utility  V  is invariant with respect to time so that   ( ) ( ) J jsVs ejj j £ £ =¶¶⇒ > 1 0 0             (2) Let the partial derivatives of the functions  f  be  ( )1 , j j f t s t f = ¶ ¶  and   ( )2 , j j f s s t f = ¶ ¶ . We note that whenever marginal increases in attendance at location  j  confer benefits,  01 £ j f  (for greater benefit with earlier arrival) and  02 ³ j f  (for greater benefit with later departure). Then we can express the equilibrium condition as ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) J j s s s f s f s e j j j j j j j j j j j £ £ = + t - t t + ⇒ > - 1 0 1 01 21 & & .    (3) Rearranging this gives   ( ) ( ) ( )( ) [ ] ( ) J js fs fs s ej j jj jj j j j £ £t -+= t ⇒ >- 1110 121 & .         (4) Now  flow  propagation  (see,  for  example,  Heydecker  and  Addison,  1996)  on  trip  j (1 < j < J) means that the arrival rate  gj(t) at location  j  satisfies   ( ) ( ) [ ] ( ) j j j j j j j s s g s e t t = &  .                (5) Thus the equilibrium departure profile  ej(s)  from location  j–1 (1 < j < J) is generated by the arrival profile  gj(t) at location  j  using the flow propagation relationship (5) together with the invariance relationship (4) as:   ( ) ( )( ) [ ] ( ) [ ] ( ) J j s gs fs fs e j j jj j jj jj j £ £ t t -+=- 111121  .          (6) The arrival rate profile  gj(t) at location  j  can be found from a suitable traffic model together with knowledge of departure profiles  ej(s)  from location  j - 1  at times for which  t(s) < t : Mun (2001) has investigated the suitability of models for this purpose.  In order for travel on journey  j  to be confined to a bounded interval of time, either the marginal value of attendance at the origin should decrease over time, or the marginal value of attendance  at  the  destination  should  increase  over  time,  or  both  of  these.  Equilibrium  is achieved  through  variations  in  travel  time  between  origin  and  destination  that  are complementary to these variations in utility.     62.4  Volume of travel  The volume of travel that takes place on each trip j is determined as the time integral of the departure rate. Travel starts when the net utility  V  of a trip made in uncongested conditions rises to the equilibrium value. Travel then continues until the net utility of further trips falls below the equilibrium value, even when the network is uncongested. Thus the volume Ej of travel of journey  j  is given by  ( ) ( ) J j ds s e Esj j £ £ =∫ 1 , where the integrand has non-zero value  only  within  a  certain  departure  time  interval  for  which  travel  is  possible  at  the equilibrium cost. Thus the start time of the interval during which travel takes place determines the  end  time,  and  together  with  the  inflow  profile  they  determine  the  volume.  For conservation of flow along the trip chain, we require that the volume  Ej  be equal for all trips  j (1 < j < J) .   The present model thus provides a relationship between the volume of trips made and the net utility  V  achieved through making them, as given by (1). We expect that as the volume of travel increases, so congestion will increase travel costs and also reduce the time available to gain  benefit  from  attendance  at  the  locations,  so  that  utility  decreases  on  each  of  these grounds.  This  relationship  can  be  used  in  conjunction  with  a  demand  function  D(V)  to establish a demand-performance equilibrium in which  ( ) ( ) J j V D E j £ £ = 1                  (7) where  V  is given by (1).   3.  Example calculations  By way of example, we consider a two-tip tour from home to work and back. We suppose that the marginal value of time at home is high early on during the day, and then falls to a constant value near the start of the morning peak period, representing a reluctance to depart too early: this is illustrated in Figure 1. Similarly, we suppose that there is a premium on attendance at work during core hours, and that some reward is given for flexible working beyond those: a marginal value of time at work that achieves this is illustrated in Figure 2.   7Utility of time at home050010001500400 500 600 700Departure time (minutes)Utility (minutes)          Utility of time at work0123400 800 1200Time (minutes)Marginal utility  Figure 1:  Utility of time at home       Figure 4: Marginal wage rate at work  Let  the marginal value of time at home, expressed in units of travel time saved, be 2.0 before 08:00  and  1.5  after  that.  Thus  the  utility    f0(s)    of  remaining  home  until  time  s    is  f0(s) = 1.5 s + 0.5 Min(s,  s0) ,  where    s0 = 8 h .  Suppose  that  the  value  of  time  at  work  is determined  by  three  elements:  the  time  of  arrival,  the  duration  of  stay,  and  the  time  of departure. Let the wage rate, expressed in units of equivalent travel time saved, be 1.0 outside the core hours of 09:00 to 17:00, and  2.0 within these core hours. Thus the benefit  f1(t, s)  of working from time  t  to time  s  is given by  f1(t, s) = s - t + Min(s, s1) - Max(t, t1) , where  s1 = 9 h  and  t1 = 17 h . Because time at home is valued at 1.5 throughout the day after 08:00, the return journey from work to home is influenced by that.   According to this specification, we have the following relationships for the marginal utilities that affect the two journeys: ( )><=00 20 5 . 10 . 2s ss ss f  , ( )> -< -=11 11 0 . 20 . 1t tt tt f  and  ( )><=11 21 0 . 10 . 2s ss ss f  , and ( ) 5 . 112 - = t f .                   (8) We suppose that the free-flow travel time for each journey is 30 minutes, and that the capacity of the network in each direction is 1800 vehicles/h .  When the start time of departures from home to work is 07:40 (460 minutes), the end time of departures is 09:40 (580 minutes) and the total volume of travel is 3,600 trips. In order to achieve the same volume for the return journey, departures from work to home start at time  16:00  (960  minutes),  and  end  at  time  18:00 (1,080  minutes).    The  departure  profiles  and   8consequent travel times that achieve equilibrium are shown in Figures 3 and 4: the initial peak in the profile of departures from home is a consequence of the higher marginal value of time at home before 08:00 (480 minutes).  Home to work0204060400 500 600 700Start time (minutes)Departure rate (per minute)0204060Travel time (minutes)        Work to home0204060900 1000 1100 1200Start time (minutes)Departure rate (per minute)0204060Travel time (minute) Figure 3: Journey from home to work     Figure 4: Journey from work to home  The travel time for departure from home at 07:40 is 30 minutes, and the utility of having remained home until that time is equivalent to having saved  920 minutes of travel time. An individual who departs at this time will then arrive at work at 08:10: if they remain only until the time of the first departure from work to home, at 16:00, they will gain utility equivalent to a saving of 890 minutes of travel time. Departing for home at that time gives a travel time of 30  minutes,  resulting  in  arrival  at  home  at  time  16:30  where  they  will  gain a further 675 minutes of utility during the remainder of the day. The net utility for this individual is then the sum of the two home utilities plus the work utility minus the two travel costs, giving 2,425 minutes, which will be identical for all travellers in this homogeneous group. These results are summarised in Table 1.  Table 1: Utility obtained by an individual departing at the start of the morning peak   Home  Travel  Work  Travel  Home   Duration (minutes) Depart  Duration (minutes) Arrive  Duration (minutes) Depart  Duration (minutes) Arrive  Duration (minutes) Time  460  07:40  30  08:10  470  16:00  30  16:30  450 Utility  920    -30    890    -30    675  Consider now another individual who departs from home at the most congested time of 08:16 (496 minutes) and will arrive at work at 09.00 after a travel time of 44 minutes. Because in   9this example each of the marginal utilities (8) of attendance does not depend on other start and end times, the journey from home to work can be equilibrated separately from that from work to home. Compared with the earliest departing individual, this one will gain an equivalent of 64 minutes of additional utility at home, but will lose 14 minutes through increased travel time and a further 50 equivalent minutes of wages through later arrival. The results of this variation  in  departure  time  from  home  are  shown  in  Table  2.  This  illustrates  that  in equilibrium,  different  individuals  achieve  identical  net  utilities  through  different  interrelated combinations of costs and benefits by timing their journeys and activities differently.  Table 2: Utility obtained by an individual departing at the height of the morning peak   Home  Travel  Work  Travel  Home   Duration (minutes) Depart  Duration (minutes) Arrive  Duration (minutes) Depart  Duration (minutes) Arrive  Duration (minutes) Time  496  08:16  44  09:00  420  16:00  30  16:30  450 Utility  984    -44    840    -30    675  Finally, suppose that a time-varying charge is levied for travel during the morning peak, and that this is calculated to achieve the same arrival profile but without any congestion. In order to do this, the travel time should be identical for all travellers and the total utility maintained at the same constant value through variations in the charge. For a traveller who departs home at 08:16, the travel time of 30 minutes will lead to arrival at 08:46. Compared with the case shown in Table 2, this traveller will then save 14 minutes travel cost and consequently gain 14 minutes utility as payment for employment: the charge required to render this in equilibrium is then equivalent to 28 minutes of travel time rather than the 14 minutes that are saved. The results of these calculations are shown in Table 3. In this case, the individual has a shorter journey  and  also  earns  more  at  work,  but  passes  both  of  these  benefits  on  through  the congestion charge.   Table 3:  Utility obtained by an individual under congestion-eliminating charging   Home  Travel  Work  Travel  Home   Duration (minutes) Depart  Duration (minutes) Arrive  Duration (minutes) Depart  Duration (minutes) Arrive  Duration (minutes) Time  496  08:16  30  08:46  434  16:00  30  16:30  450 Utility  984    -30    854    -30    675   10  4.  Summary  The  present  model  provides  a  representation  of  the  way  in  which  travellers  choose  their departure times in trip chains according to the benefits that they gain through attendance at different locations and the travel conditions that they encounter between them. Choices made by individuals can, in equilibrium, include a range of possible balanced combinations of costs of travel and benefits obtained in attending the locations. We have shown how this can be used to investigate the effects of changes that might be made to travel provision (for example, to the free-flow travel time, capacity or monetary charges made for use of the network) and to the marginal utilities of attendance at the locations. The influence of changes of these kinds on trip making behaviour can then be explored by using the present model in conjunction with a demand relationship as part of a demand-performance equilibrium model.    References  Heydecker, BG and Addison, JD (1998) Analysis of traffic models for dynamic equilibrium traffic assignment. In: Transportation Networks: Recent Methodological Advances (ed MGH Bell). Oxford, Pergamon, ISBN: 0-08-043052-X, 35-49.  Heydecker, BG and Addison, JD (2004) Analysis of dynamic traffic equilibrium with departure time choice. Transportation Science (in press).  Mun, J-s (2001) A divided linear travel time model for dynamic traffic assignment. Proceedings of the 9th World Conference on Transport Research, Seoul, D1-05, 4189. 

Equilibrium analysis of trip chains in congested networks

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