Discomfort in mass transit and its implication for scheduling and pricing

Transportation Research Part B 71 (2015) 1–18

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Discomfort in mass transit and its implication for scheduling and pricing André de Palma a,c, Moez Kilani b,⇑, Stef Proost c a

École Normale Supérieure de Cachan, 61 avenue du Président Wilson, Cachan 94230, France University of Lille, Domaine universitaire du Pont de Bois, Villeneuve d’Ascq 59653, France c KU Leuven, Naamsestraat 69, 3000 Leuven, Belgium b

a r t i c l e

i n f o

Article history: Received 11 March 2013 Received in revised form 18 July 2014 Accepted 4 October 2014

Keywords: Crowding and discomfort in public transport Congestion Timetable Schedule delay cost Seat capacity Seat allocation

a b s t r a c t This paper discusses the formulation of crowding in public transport and its implications for pricing, seating capacity and optimal scheduling. An analytical model is used to describe the user equilibrium and the optimal equilibrium for different stylized conditions. For the one OD pair case with identical desired arrival time, we derive the optimal dynamic pricing and optimal share of seats. For the uniformly distributed desired arrival times case, we derive the optimal time table and the optimal pricing. Next we generalize the results to the case of a small network with several stations, stochastic choice and allocation of seats. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The quality of mass transit and in particular crowding has become a severe problem in several metropolitan areas in developing countries, but also in European countries. In Paris, London, and several other metropolitan areas, it is hard to get into the metro or local commuter trains at peak times. As a consequence, passengers may change their departure time, their route or mode in order to avoid excessive congestion. Many metropolitan areas have adopted a second best, low price, policy to attract car users into public transportation. This policy can be justified (cf. Parry and Small, 2009; Proost and Van Dender, 2008), but the necessary capacity decisions calling upon public revenues have not always been adopted and this has resulted in severe congestion in mass transit. In this paper, we focus on the characterization of congestion in mass transit, which is a key ingredient of cost-benefit analysis in transit. Note that there are very few studies of congestion in mass transit while there is a very large number of studies of congestion in private transportation. This may be understood in the US where the use of public transport is typically low except in some cities like New York and Boston. However, this is more difficult to understand for European cities where the fraction of commuters using transit can exceed 50% in the peak period. To the best of our knowledge, public transportation congestion is not modeled in commercial software that describes public and private transportation. There is a long tradition in dealing with road congestion. The so-called BPR formula – Bureau of Public Roads – is now the standard in the literature. But the micro-economic theory of mass transit has not dealt frequently with crowding and riding ⇑ Corresponding author. E-mail addresses: [email protected] (A. de Palma), [email protected] (M. Kilani), [email protected] (S. Proost). http://dx.doi.org/10.1016/j.trb.2014.10.001 0191-2615/Ó 2014 Elsevier Ltd. All rights reserved.

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comfort. There are only a few exceptions in the literature. To study the implications of crowding in public transport, we need a simple formulation of crowding. As there is not yet a consensus on the best way to model crowding in public transit, we present in Section 3, three alternative functional forms that can be used to study optimal pricing and scheduling in public transport. These are, first the power function in the spirit of the BPR function, second, the step function with a discontinuous threshold when all seats are taken and the third function is the step function with continuous step. In Section 4, we derive the properties of user equilibrium and optimal time table and pricing in a dynamic mass transit model for the case of a uniform desired arrival time and one OD pair. In this section we also compare the first best and second best where the allocation of space to seats is optimized according to time of day. In Section 5, we deal with scheduling and pricing in the case of a uniformly distributed desired arrival time. An assignment model with multiple stations is used in Section 6. This section generalizes the results to the case of a small network with several stations. In Section 7, we conclude and discuss possible generalizations. Our paper deals mainly with the stylized case of a homogeneous population that wants to make a fixed number of trips between an origin and a destination. Another simplification we use is that the congestion in the transit vehicles does not slow down public transport. 2. Review of literature Vuchic (2005) provides a major contribution to the recent theory of transit but does not explicitly deal with crowding in urban transit. Jara-Díaz and Gschwender (2005) offers an informal discussion of crowding. Initial papers dealing with rail (and bus) transport were mainly concerned with the optimal service frequency and vehicle capacity (cf. Mohring, 1972; Jansson, 1980; Rietveld et al., 2001). An approach of congestion in public transportation based on multi-prize contests has been proposed recently by de Palma and Munshi (2013). Kraus (1991) is a pioneering contribution where a distinction is made between the value of time for standing and for seated passengers. Kraus and Yoshida (2002) focus on the congestion on the rail platform where passengers may have to wait for several trains before they can enter. Several papers have considered congestion in public transport in a multi-modal model, but rely on a very simplified congestion technology for public transportation. Rouwendal and Verhoef (2004) have crowding as an increasing function of occupancy ratio and Huang (2000) has comfort costs linearly increasing in the number of users. Cortés et al. (2013) consider stochastic boarding decisions and incorporate congestion in the bus stop and the increased in-vehicle travel time. The authors do not, however, consider seat allocation. Tian et al. (2007) discuss crowding and seat allocation. They comment on the difficulty in addressing the problem (seat allocation) on a general network (cf. page 626) and limit their analysis to a simplified situation. In the numerical example, they consider a crowding cost, per unit time, given by gðnÞ ¼ 5 lnð1  ðn=nÞ1=4 Þ, where n is the number of passengers in the vehicle and n is the maximum capacity of the vehicle. This formulation is simple and reflects an increase in crowding as the number of passengers increases. Still, it does not directly consider the number of available seats. A recent literature, that we briefly review in Section 6, explores equilibrium computation in transit assignment models and considers stochastic dynamic frameworks. These models propose algorithms to compute transit equilibrium on time-expanded graphs. Many empirical and experimental studies confirmed the importance of discomfort conditions in rail and public transport (cf. Wardman and Whelan, 2011; Li and Hensher, 2011). The value of time (VOT) in mass transit depends on riding conditions. There are mainly three different situations a passenger may face: (i) a seat is available, and in that case the value of time can be assumed to be independent of the number of passengers in the vehicle; (ii) the passenger has to stand but the vehicle is not crowded (in that case the VOT will be higher but again constant); and (iii) the passenger has to stand and there are too many passengers in the vehicle. The case that the VOT will be higher, or the riding conditions, would depend on the number of passengers. We do not consider here the case where the vehicles are so crowded that some passengers cannot enter and have to wait for the next available vehicle. Similar situations were considered in Lam et al. (1999), where they distinguish between three discrete situations in their empirical analysis. Prud’homme et al. (2012) and Koning and Haywood (2011) have conducted a survey to evaluate the generalized cost in the Paris subway. Their conclusion is that the crowding cost is about three times the fare, a value that confirms the importance of crowding. Their estimation was based on a linear model that does not explicitly considers the availability of seats. In the next section, we briefly compare three alternative analytical formulations of the discomfort function and show how their properties affect optimal scheduling and pricing. 3. Defining discomfort functions in mass transit Time cost in public transit depends on riding conditions. The most comfortable situation is when a passenger has a seat. Not having a seat is not enjoyable but acceptable when there is no crowding and the trip is not too long. Discomfort becomes particularly important when too many passengers have to stand and having a seat guaranties some comfort during the whole trip. Notation and assumed numerical values are provided in Appendix A. Let ns denote the number of seats in the vehicle and let nx denote the standing capacity. The standing capacity is sometimes defined by the manufacturer of the bus or by the regulating authority. But it is often exceeded at peak times. Vuchic (2005) distinguishes between five situations that range from ‘‘independent standing, easy circulation’’ when passenger density is less than one per m2, ‘‘crashes loads, possible

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injuries, forced movements’’ when there are 6.7 passengers per meter square (cf. Vuchic, 2005, Table 1.2, page 12). Of course these standards vary when some passengers are carrying some luggage or strollers. We assume that the standing capacity still allows passengers to travel in comfortable conditions. What matters here is not how it is defined but that the discomfort of standing is increasing with the number of passengers. So, the total seating and comfortable standing capacity of the vehicle is ns þ nx . The most natural formulation of discomfort may correspond to a piecewise linear expression, where a constant cost is associated for the seating passengers, then an upwards jump reflects the cost increase for those who are standing. The cost is then constant for this group of passengers until the vehicle capacity is reached. At this stage the vertical section of the cost shift reflects the impossibility to put more passengers in the vehicle. This cost function is given in Fig. 1(a). If it is urgent to get a tractable expression, a quadratic formulation is convenient, since it preserves the continuity in the cost and its convexity leads to unique solutions in general. But, the quadratic formulation misses most important features of discomfort public transport. It is not difficult to describe situations where the quadratic formulation leads to odd conclusions. For example, consider two vehicles with a distinct number of seats. Suppose that the passengers have a slightly higher preference for the highest capacity bus because it arrives at a better time to the destination, but the difference is small. How should a given number of passengers be distributed between the two trains? With a quadratic formulation, since the difference is small the travel cost in the higher capacity train will soon be equal to the travel cost in the other train, and we obtain an equilibrium where the passengers are almost equally split between the two trains. In reality, however, since the travel cost is flat as long as some seats are available, most users will choose the higher capacity train until no seats are no longer available. For this reason we disregard the quadratic formulation in the rest of the paper. The main advantage of the piecewise linear formulation is its relevance to real situations, but its inconvenience is related to the discontinuity and lack of tractability. As a trade-off we propose a smooth function that approximates the piecewise linear function that preserves the advantages and removes the disadvantage of discontinuity and piecewise definition of the cost. The user cost of the nth passenger, is given by1

wðnÞ ¼ a0 þ

a1  a0 s x þ b ec ðnn n Þ 1 þ eaðns nÞ

ð1Þ

where a; b and c are positive parameters that reflect how crowding impacts user cost. The user cost for those who can sit is

a0 . The user cost for the passenger that has to stand is a1 when there is no crowding, i.e. when n 6 ns þ nx . Crowding impacts the user cost through the last term which becomes dominating for n > ns þ nx (b and c are positive). Beyond that threshold, it increases rapidly with occupancy. Function wðnÞ is illustrated in Fig. 1(b) next to the piecewise formulation, first for smaller values of parameters a; b and c (cf. curve ðiÞ), and then for a higher values of these parameters (cf. dashed curve ðiiÞ). In Appendix B we consider a discontinuous formulation of discomfort and show that wðnÞ appropriately approximates it. The total cost corresponding to (1) is

 TCðnÞ ¼

if n 6 ns

n wðnÞ s

s

s

n wðn Þ þ ðn  n ÞwðnÞ if n > ns :

ð2Þ

The average cost is ACðnÞ ¼ TCðnÞ=n, and the marginal social cost is obtained by differentiation of (2) with respect to n, i.e. SCðnÞ ¼ d TCðnÞ=d n. The user cost is illustrated on Fig. 1. From here on, we refer to the formulation of comfort (Eqs. (1) and (2)) as the MAS formulation.2 For n < ns þ nx and a positive value of parameter c the term with exponential is very small and can be neglected. As the number of passengers increases and becomes higher than the ‘‘capacity’’ of the vehicle (ns þ nx ), crowding increases and this is captured in the exponential term. If there are less than ns passengers then they all have a seat and the marginal passenger contributes to total travel cost by a0 . If the number of passengers is between ns and nx , the marginal social cost is almost constant and equal to a1 . For more than ns þ nx passengers the marginal social cost increases, reflecting crowding and the difficulty to get into the vehicle. Note that the user cost is not the same for all passengers: those who have a seat have lower travel cost. The value of parameters a; b and c can be obtained from empirical observations. For example, to make the cost function fit the empirical observation of Prud’homme et al. (2012), which states that crowding cost is three times the fare, one has to find parameter values that yield wðns þ nx Þ ¼ 3a0 , since ns þ nx is the threshold for crowding. When the modeler has multiple observations, then she can estimate the parameter values through a standard fitting procedure like least squares, and possibly imposing some constraints on the functional form. More empirical work is clearly required. 4. User equilibrium, scheduling and optimal pricing with identical desired arrival time In this section we explore optimal scheduling and pricing in a simple dynamic model where users want to travel via mass transit from one origin to a given destination. We study first the simpler case of identical desired arrival times. The objective is to illustrate how crowding in rail or bus system forms in peak hours. On the side of passengers, we assume a group of N identical individuals3 who have the same desired arrival time window ðt; tÞ, and who incur a schedule 1 This is a cost function per unit of time. If the travel time is t, then total cost for the nth passenger is given by t  CðnÞ. A more general formulation would set 0 total cost to hðtÞ  CðnÞ where h ðtÞ P 0. The increasing function hðtÞ reflects a higher marginal discomfort for a longer trip. 2 MAS is a rearrangement of the initial letters of the authors’ first names. 3 This is a strong assumption which is relaxed in the next section.

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Fig. 1. Modeling user cost in public transportation.

delay cost whenever they arrive too early (before time t) or too late (after time t) at their destination. Let t denote the actual arrival time of a given passenger. We consider the following penalty function:

SDCðtÞ ¼



ðt  tÞ b if t < t ðt  tÞ c

if t > t;

ð3Þ

where b and c are the schedule delay parameters, respectively, for early and late arrival. It is usually assumed that b < c, i.e. the penalty of an early arrival is lower than the penalty of a late arrival. We assume that in-vehicle travel time and ‘‘walking’’ time at the arrival station are both constant. So, the passengers have to choose the train with respect to its departure time and its loading and they are assumed to have perfect knowledge of the timetables (we ignore waiting time). Let ti denote the arrival time of a train i. Notice that this train departs at time t i  T, where T is the (fixed) travel time. As in Kraus and Yoshida (2002), we consider successive departures of trains that arrive at successive times4 t i ¼ ðt þ tÞ=2 þ d i, where i ¼ . . . ; 2; 1; 0; 1; 2; . . ., and d is the technical time interval between two departures. So, train arrivals are centered on the desired arrival interval ðt; tÞ. Denote by ni the number of passengers that select the train that arrives at time t i . We consider a simple configuration where all passengers board from the same (single) station and have the same destination. We discuss the user equilibrium, the system optimum, and the optimal pricing case. 4.1. User equilibrium At a user equilibrium, each passenger’s objective is to minimize his own travel cost plus schedule delay cost, referred to as the generalized user cost. If there is no crowding (unlimited capacity) then the best solution for all passengers is to board a train that arrives in time interval ðt; tÞ. With crowding, each passenger will trade off the schedule delay cost with discomfort cost to select the best departure time. At equilibrium, no passenger will have an incentive to change his departure time. Each passenger does not take into account the external costs he generates, and user equilibrium may not correspond to the system optimum. The total in-vehicle cost in train i is TCðni Þ, and the average cost is TCðni Þ=ni , where ni is the loading of train i. Taking into account the schedule delay cost, the generalized user cost in train i is wðni Þ þ SDCðti Þ. At equilibrium no passenger has an incentive to switch to another train and this means that generalized user costs are equalized among all trains (assuming all trains are used). Formally, we have

wðni Þ þ SDCðt i Þ ¼ ce ; where ce is the generalized user cost in all the trains.5

4

These trains have departed at times t i  T. If there are k trains, i ¼ 1 . . . k, we have a system of k þ 1 nonlinear equations: k equations, one for each train stating wðni Þ þ SDCðti Þ ¼ ce , plus the condition that each user chooses only one train n1 þ    þ nk ¼ N. The unknowns are the train loadings ni and the generalized travel cost ce . To find a solution to this problem, it suffices to solve the first k equations by considering ce as a parameter, and then find the value of ce that yields the condition on the total number of users. Moreover, wMAS ðnÞ is monotone increasing if parameter a is not too large, and in that case uniqueness is guaranteed. Indeed, ni , the solution to P P wMAS ðni Þ þ SDCðti Þ ¼ ce for i ¼ 1 . . . k, is monotone increasing in ce , and so is i ni . It is not difficult to see that for ce slightly higher than a0 ; ki¼1 ni < N and for Pk e e large values of c ; i¼1 ni > N. By continuity and monotonicity, there exists a unique value of c that yields the equilibrium condition. 5

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4.2. System optimum P The optimum distribution of users over trains minimizes total transport cost, i.e. i ðTCðni Þ þ SDCðti Þ ni Þ, where ni represents the number of users of the train leaving at time t i . Travel cost considered here is based on the MAS cost function given in (1) that represents the discomfort costs as a function of arrival order. Total number of users is fixed and we have Pk are minimized. We can formulate this i¼1 ni ¼ N. The optimal occupancy rate in each train is determined so that total costs Pk1 problem as an unconstrained minimization program if we substitute for nk ¼ N  i¼1 ni . If the optimum solution is an interior solution where ni > ns , we can use first-order condition for a given train i (where i ¼ 1; . . . ; k  1), that is

wðni Þ þ SDCðt i Þ ¼ wðnk Þ þ SDCðt k Þ: This condition implies that, at the optimum, the marginal cost of a new passenger in-vehicle i is equal to the marginal social gain obtained from removing the same user from vehicle k. The optimality conditions form a set of n  1 non-linear equations that can be solved numerically. We illustrate the user equilibrium and the system optimum in Table 1 using the parameters of Appendix A. Recall that in-vehicle travel cost is wðnÞ  T, where T is the travel time, here equal to 15 min. In the same table we also illustrate the solution where the users are arbitrarily distributed uniformly over the trains (one third of the total population in each train). The difference between the user equilibrium and the system optimum is the suboptimal allocation of users over vehicles. The system optimum is to load the trains arriving too early or on time more or less like the trains arriving too late and the occupancy rate is not too different from uniform (compare first and third lines in Table 1). There are two forces at play: discomfort (which itself tends to spread demand uniformly) in the vehicle and schedule delay cost (which plays a concentration role). The optimal solution reaches the best trade off between these two forces. The reduction of schedule delay cost made possible by loading more passengers on the first trains is limited by the crowding it will induce. Since unit early delay cost b is smaller than unit late delay cost c in the example of Table 3 (first line) there are more passengers in the first train than in the third one. Consider now the user equilibrium. Users make efforts to improve their comfort and try to minimize the user cost also by picking the wrong train. They disregard the extra in-vehicle comfort costs they generate for the other users. This leads to trains with too many passengers, when they arrive early or on time. In the example of Table 1 (line 2) there is more crowding in the first and second train and much fewer passengers in the third one. This suboptimal distribution of passengers over the trains leads to a much higher total travel cost. Comparing the uniform distribution of passengers over the trains with the system optimum tells us that, in our example, the inefficient allocation over trains is less important in terms of efficiency than the efforts of the users to improve their in-vehicle comfort. In this case, the travel cost in equilibrium is higher by more than 35% than the cost obtained with the optimal loadings of the trains. The travel cost in the uniform distribution is higher only by less than 3%. Notice however that the uniform distribution yields a satisfactory total cost (only slightly higher than the optimum), because the number of passengers is higher than the available capacity of the trains and crowding matters. If the number of passenger were smaller with seats available, then the uniform distribution will not be an attractive alternative, since in that case the optimum solution requires most users to board on the second train and only few users, if at all, to board on the first and third trains. A similar point is discussed in Section 5.2 below.6 4.3. Decentralizing the optimum The source of inefficiency is the bad allocation of passengers over the different trains. In particular, too much passengers use trains 1 and 2 and too few passengers use train 3. Removing this inefficiency can be reached through a differentiation of the charges for the different trains so that the following system optimum condition is satisfied (for an interior optimum):

SCðni Þ þ SDCðti Þ ¼ SCðnk Þ þ SDCðt k Þ;

i ¼ 1 . . . ; k  1:

In order to internalize the crowding externalities, the trains with higher crowding must charge more or alternatively, those trains with the best arrival times must charge more. This is an illustration of the peak load pricing principle for the case of public transportation. Optimum fares are given in the first three columns of Table 2 (optimum in the sense they reach the optimum in Table 1). The values of the distinct fares reflect the objective to reduce the number of passengers in trains 1 and 2 and increase those who take train 3. The seating capacity has not been changed (ns ¼ 20). As an alternative (or complementary) to pricing, the operator can adjust the number of seats to manipulate the passengers choice. Increasing the number of seats allows more passengers to benefit from comfortable travel conditions but reduces the available space for standing passengers and the total capacity of the vehicle. By reducing the number of seats, the operator increases the capacity of the vehicle but most passengers do not find an available seat. We can evaluate the impacts of this alternative using the same framework. Let S denote the total number of standing spaces available in the vehicle and let m be the number of spaces that are required to add one seat. From the notation above, we have Si ¼ m nsi þ nxi , where the index i refers to the train (in the example above i ¼ 1; 2; 3). We assume that it is possible 6 The interested reader may check the Mathematica notebook accompanying this paper that we make available from http://perso.univ-lille3.fr/mkilani/ codes/.

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Table 1 Train loading and user costs (per day) under three alternatives (parameters from Appendix A). Abbreviations: ‘‘C’’ stands for crowding, and ‘‘SD’’ for schedule delay. Travel time is 15 min. Train loadings

System optimum User equilibrium Uniform

Average costs

1 30 mins

2 on time

3 þ30 mins

C ($/passenger)

SD

Total

53.5 61.5 50.0

55.1 63.3 50.0

41.4 25.3 50.0

2.08 3.75 1.99

1.70 1.36 1.89

3.78 5.11 3.88

Table 2 Cost reduction through fares setting and adjustment of the number of seats. Fares

Fares Seats 1 Seats 2 Seats 3 Full ⁄

Seats

Loadings

Average costs

p1

p2

p3

ns1

ns2

ns3

n1

n2

n3

C

SD

Total

2.12 0⁄ 0⁄ 0⁄ 0

3.65 0⁄ 0⁄ 0⁄ 1.64

0.0⁄ 0⁄ 0⁄ 0⁄ 0⁄

20⁄ 3.3 20⁄ 12.3 22

⁄

20⁄ 20⁄ 20⁄ 12.3 20⁄

53.5 89.7 45.3 66.6 41.1

55.1 60.3 104.7 83.4 108.9

41.4 3.92 0 0 0

2.08 2.64 3.3 2.99 2.24

1.70 1.00 0.5 0.74 0.45

3.78 3.64 3.8 3.73 2.70

20⁄ 5.2 12.3 0

Fixed exogenousely.

to add or remove seats from the vehicle, and we assume that this operation is costless to the operator (these assumptions are not crucial but adopted to simplify the exposition). So, the operator takes S as given and can freely choose the value of ns . The output of the computations for a number of scenarios is reported in lines 2, 3 and 4 of Table 2. Each line corresponds to a scenario. Decision variables extend on the six first columns (p1 to ns3 ), and starred values indicates that these were fixed exogenously. In the first scenario (Fares) total travel cost is minimized with respect to the fares of the first and second trains taking the fares of the third train and the number of seats in each train as given. The outcome corresponds to the optimum output in Table 1. In the next three scenarios (Seats 1 to Seats 3) we consider unpriced crowding but allow now to adjust the number of seats to reduce total travel cost. For example in scenario ‘‘Seats 2’’ we take ns1 and ns3 as given and minimize costs over ns2 alone. On the last line (Full) we minimize over p1 ; p2 ; ns1 and ns2 . From the last column it appears that all scenarios reach comparable levels of efficiency, except the last one which reaches a much lower cost. In this scenario fares are imposed only on train 2 (the on-time train) in parallel to:  a slight increase in the number of seats in train 1 (the early train), and  removing all the seats in train 2 to make the maximum space available for passengers arriving early. In this solution, no passenger boards on train 3 (the late train). With respect to the high levels of crowding observed in public transportation networks in large metropolitan areas during peak hours, this simplified model favors reforms that create more capacity and introduce peak load pricing at the same time. The reduction of the number of seats to free more space for standing passengers would be definitely more convenient in the case of short trips where standing does not last for a long time. For example, in Paris region, that would concern the metro that travels in Paris but it will be less appropriate for the RER that travels on longer distances. Peak load pricing of public transport has been shown to be welfare improving by Kraus (2012) in a theoretical framework and Kilani et al. (2014) for the case of Paris region. In these two models, however, mode choice between private and public transportation was considered and was a key feature to generate the result, but in this framework only public transport is considered. The values in Table 2 are obtained for each scenario with a fixed number of passengers (here, N ¼ 150). In order to evaluate the impact of crowding on the solution, it is useful to start with a small number of passengers so that the crowding level is small, minimize the total travel cost, then progressively increase the number of the passengers to reach high levels of crowding. Doing so, for the scenarios considered earlier, leads to the curves in Fig. 2. These curves show how average transport costs increase as the number of passengers, and thus crowding, increases under each of the scenarios considered above. We can observe that most of the regulation instruments yield comparable impacts when the number of passengers is small or moderate (below 180 in this example). When crowding is stressful (high number of passengers), imposing fares has almost no impacts on the choice of the passengers. This is because crowding cost becomes so high by comparison to schedule delay costs that both equilibrium and optimum requires equal distribution of the passengers among the trains. In this case only a reallocation of the available space in the trains between seats and standing spaces can lead to a reduction in the average travel cost. In this example, scenario ‘‘Seats 3’’ where the adjustment of the number of the seats is the same for the three trains appear as an efficient alternative. The unregulated situation is an upper envelope of all the curves while the ‘‘Full’’ scenario is a lower envelope. This follows directly from the definition of the considered scenarios. So, for all levels of crowding it is better to minimize the cost through simultaneous peak load pricing and capacity adjustment. When this is

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Fig. 2. Crowding impact on travel cost.

not possible, a simple adjustment of the number of seats can reduce cost at a satisfactory level. At least partly, the allocation of space is already in use since some vehicles are equipped with fold-up seats that can be used or not depending on the loading levels of the vehicle. 5. User equilibrium, scheduling and optimal pricing with randomly distributed desired arrival times We now turn to the more general case where passengers’ desired arrival times are no longer identical. We consider a uniform distribution of desired arrival times over normalized time interval (0,1), as it allows us to derive analytical expressions. Travel times are assumed fixed and we refer to the trains by their arrival times. We start by studying the passenger choice facing two arrival times t A and tB , both located in time interval (0,1) and t A < t B .7 Next, we optimize the departure times for multiple trains and discuss pricing alternatives. Since the travel time T is fixed, we can equivalently optimize over arrival times t A and t B . 5.1. Train loadings with uniformly distributed desired arrival times Consider the choice between the two trains A and B. Train A arrives at t A 2 ð0; 1Þ, while train B arrives at tB where t A < t B < 1. Transit timetables are given and known to the passengers and we assume that the desired arrival times are continuously distributed over (0,1), with density q. We first compute the equilibrium average user costs for any pair ðt A ; tB Þ where tA < tB . For an equilibrium it is necessary that the last entrant is indifferent between the two trains. Denote the desired arrival time of the user indifferent between the two trains A and B by ^t. The number of users of train A is q^t. The generalized cost of user ^t is:

C GA ¼ wðq^tÞ þ bð^t  t A Þ: Similarly, the generalized cost for using train B, is

     C GB ¼ w q 1  ^t þ c t B  ^t : The indifference condition for a user equilibrium reads:

     ðb þ cÞ^t ¼ ctB þ btA þ w q 1  ^t  w q^t :

ð4Þ

The form of wðnÞ precludes a general analytical solution but we can inspect its properties numerically, as illustrated in Fig. 3(a). The case of a small enough q(when all passengers have a seat and incur the same travel cost) can be computed explicitly from Eq. (1), since then wðqt  Þ ! a0 and wðqð1  t ÞÞ ! a0 . Therefore, in this case, Eq. (4) reduces to bð^t  tA Þ ¼ cðtB  ^tÞ and

^t ¼ ctB þ btA : ðb þ cÞ

7

So, train A departs at t A  T and trains B departs at time t B  T.

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Fig. 3. Comparison of optimum and equilibrium when tA ¼ 0:2 and tB ¼ 0:8 and other parameters from Appendix A.

For c > b, a usual assumption which is empirically sound, we have ^t < 1=2. The reason is that transit users prefer to be one minute too early rather than one minute too late. This is the solution when congestion (crowding) does not matter (q ! 0). As the density parameter q increases there will be more and more passengers in train B. We check from Eq. (4) that

dnA < 0; dq

if

ctB þ btA > 0:5; ðb þ c Þ

and at the limit we have limq!1^t ¼ 0:5. As expected, the effect of crowding is to equalize the number of users in each train. If the total number of passenger is high enough, only congestion matters and the number of passengers in both trains equalizes. In Fig. 3(a) the two curves correspond to the number of users in train A under equilibrium and optimal regimes. When the number of users is small (by comparison to the vehicle capacity), external costs are small and the equilibrium outcome is optimal. As the number of users increases, comfort decreases, still, train A remains overused since each passenger considers his own cost, not the external crowding cost imposed on the other passengers. As the number of users continues to increase, crowding becomes the main concern and passengers are almost equally split between the two trains, both at equilibrium and at the optimum. This figure, however, hides a particular detail. From the fact that equilibrium loadings converge to optimum, one may conclude that pricing is no longer required in this case or that optimum fare of train A, denoted pA , satisfies pA ! 0, as q gets larger. This is a false conclusion. Indeed, optimum fare is increasing in q as shown in Fig. 3(b). The reason is that as the number of passengers increases, crowding increases strongly. Even if the difference in the two quantities is small, it still induces an increasing differential in cost (the number of passengers is treated as a continuous variable). In the next section we study optimal departure times and show that with optimized departure times and uniformly distributed arrival times, price differentiation between the two trains is not needed. 5.2. Optimal departure (or arrival) times with randomly distributed desired arrival times For an early study of optimal timetables see de Palma and Lindsey (2001). We start with the case of one train. The total number of users is fixed and so is the crowding discomfort and it plays no role in the optimization of the best departure time Rt R1 tA  T or its best arrival time t A . The total schedule delay cost is: 0A cðtA  tÞgðtÞdt þ tA bðt  tA ÞgðtÞdt, where gðtÞ denotes the users distribution of desired arrival times. The minimum cost is obtained at arrival time t A that satisfies

b  Nearly ¼ c  Nlate ; early

R1

ð5Þ late

R tA

where N ¼ tA gðtÞdt denotes the number of passengers that arrive before their desired arrival time, and N ¼ 0 gðtÞdt denotes the number of passengers that arrive after their desired arrival time. When gðtÞ is uniform over (0,1), we get a simple solution: c tA ¼ bð1  t A Þ. Solving this equation for tA we find that the train should arrive at time b=ðb þ cÞ. With k trains, the same optimal solution applies in each subinterval. The first-order condition for a maximum gives the optimal arrival times t i (for i ¼ 1 . . . k), that is

tunif ¼ i

  1 b þ ði  1Þ : k bþc

ð6Þ

The main result is that when users only differ in their preferred arrival time and the distribution of preferred arrival times is uniform, crowding does not modify the optimal departure times. This result is discussed graphically on Fig. 4. The top figure illustrates Eq. (5) for the uniform case. The total cost is the area of the two triangles above N early and N late , respectively. This area is minimized when the opposite sides of the two triangles have equal length, i.e. when b  N early ¼ c  N late . With one train, crowding is clearly independent of the arrival time since the train loading is constant. Consider then the case of two trains

A. de Palma et al. / Transportation Research Part B 71 (2015) 1–18

9

Fig. 4. Illustration for the uniform case.

and let us ignore for instance crowding costs. The solution that minimizes total schedule delay cost is obtained by splitting the time period in two intervals of equal length, and then by noticing that we can apply the same argument, as above, within each interval. When users’ desired arrival times are uniformly distributed, N early and N late are proportional to distances on the x-axis. So, the top graphical construction is replicated twice as indicated on the bottom figure and the loadings of the two trains are equal. Notice that the user indifferent between train A and train B has a preferred arrival time at 1/2. Then, we can check whether taking into account crowding can lead to lower total cost (we already know that it increases schedule delay cost). It is easy to see that if we pick a passenger from train A and load him into train B the passenger himself will have higher crowding cost (there is one more passenger). Crowding in train A decreases but this positive effect is dominated by the increase of crowding cost in train B because crowding costs are convex for n > ns þ nx . Thus, introducing crowding does not change the solution obtained from the minimization of schedule delay cost. From Fig. 4 it is clear why this is not a general result. Indeed, for more general distributions of desired arrival times, the scheduled delay costs no longer correspond to the identified triangles and we cannot geometrically identify the two groups of users N early and N late . For the sake of comparison, let us consider an alternative nonuniform distribution of desired arrival times given by

8 if 0 6 t  1=2; > < 4qt hðtÞ ¼ 4qð1  tÞ if 1=2 < t 6 1; > : 0 elsewhere:

ð7Þ

R1 Notice that 0 gðtÞdt ¼ q, so we have the same total number of users as in the case of a uniform distribution discussed above. With distribution hðtÞ, most users have a desired arrival time near 1/2. With only one train, Eq. (5) still applies, since there will be q passengers in the train independently of its arrival time. From the first-order condition we obtain the arrival time

th1 ¼

b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b þ bð2c  bÞ

and we check that this is an increasing function of b and decreasing function c, and that t h1 2 ð0; 1=2Þ for b < c. With two trains and more, we can no longer use condition (5). Instead, total user cost should be minimized with respect to arrival times of the two trains. The fares, pA and pB for trains A and B, respectively, are equal. In the next section we consider the case of differential fares. Let TSDðtA ; tB Þ denote total schedule delay cost for all users when the arrival times of the two trains are at tA and t B , respectively. The optimal arrival times minimize total cost TCðnA Þ þ TCðnB Þ þ TSDðtA ; tB Þ, where nA and nB denote the number of passengers in the first and second train, respectively. For any given tA and tB , one has to find the user who is indifferent between the two trains in order to compute nA and nB . A solution to this problem cannot be derived analytically, given the nonlinear expression of wðnÞ in Eq. (1), but may be solved numerically. For a numerical illustration,8 computing the solution with distribution hðtÞ defined above and parameter values in Appendix A, we find the solution values 8

The computational details are given in the Mathematica notebook. See footnote 6.

10

A. de Palma et al. / Transportation Research Part B 71 (2015) 1–18 Table 3 Optimizing over departure times and fares. Optimizing over tA and tB with

Average cost tA tB ^t % arriving late pA

Same fares (pA ¼ pB )

Optimized pA and pB ¼ 0

8.68 0.276 0.587 0.499 31.38 –

7.61 0.256 0.558 0.472 29 1.439

given in the second column (where pA ¼ pB ) in Table 3. Users are almost split equally between the two trains (^t ¼ :499) and we check, as expected, that t unif < t hA < t hB < t unif A B . 5.3. Optimal pricing We know that optimal pricing depends in principle on the arrival times as they determine the levels of congestion and the external congestion costs. In a few cases, differential fares are not necessary. This is clearly the case when there is no congestion. When there is only one train, congestion level is fixed and differential fares are not needed as long as total demand is fixed. More generally, for any set of arrival times, optimal prices are always equal to the marginal external congestion costs. When total demand is fixed only the differences in marginal external congestion costs of the different trains matters. Also, with a uniform distribution of desired arrival times there is no need for differential fares to decentralize the optimum. Indeed, in this case, arrival times are given by Eq. (6) and there are q=n passengers in each train. One can check that users with desired arrival time at i=n, for i ¼ 1; . . . ; n  1, are indifferent between train i and train i þ 1. So, the private decision leads to the optimal choice. To discuss the case of nonuniform distribution of arrival times, we consider the case of two trains. Desired departure times distribution is hðtÞ over (0,1), and the fares9 in the two trains are pA and pB , respectively. The passenger who is indifferent between train A and train B has a desired arrival time that depends on t A and t B and the fares pA and pB . The objective is to minimize the total user cost for all users. A numerical illustration, for hðtÞ given by Eq. (7), is shown on the last column of Table 3. Average user cost is (of course) smaller when fares are optimized. It is interesting to notice the difference between the case where differential fares are allowed and the case where they are not. When differential fares are possible, arrival times are chosen to minimize schedule delay cost and the values of the fares are chosen to eliminate external costs induced by crowding. Both instruments coordinate to reach the final objective to reduce total travel cost. When pA ¼ pB , only arrival times could be chosen so that both crowding costs and schedule delay costs are minimized. Consequently, arrival times are postponed because too many passengers want to board train A. With differential fares, it is the positive value of pA that keeps train A moderately loaded.

6. The multi-station case In this section we explore the case with multiple stations and origin–destination pairs. This analysis illustrates how our formulation can be applied to schedule based models of public transport. The development of a full assignment model of public transit is beyond the scope of this paper. We plan to develop such a model in future research. There is a large literature on the modeling of transit assignment models. Most of the papers consider dynamic allocation of seats, and appreciate comfort as the possibility of having a seat (cf. Hamdouch et al., 2011; Cortés et al., 2013; Hamdouch and Lawphongpanich, 2008; Sumalee et al., 2009). We consider a series of network examples where the impact of discomfort is evaluated. The first example below considers multiple origins and a single destination, and the second example envisages multiple destinations. In both cases, the equilibrium is derived by equating the generalized travel cost between the available alternatives (trains in this case). The third example explores stochastic choice models and illustrates a more general methodology. We limit this analysis to simplified networks. The last example considers the most complex network used in Hamdouch et al. (2011). This is a multiple lines and multiple origin–destination case. 6.1. Example 1: deterministic choice, single destination The network of the first example is illustrated in Fig. 5. Passengers board from the first three stations and are all alighting at the fourth one. Assume first that there are two trains serving this line, i.e. trains A and B. We consider interior solutions 9 When demand is not elastic, only one train need to be priced, with a value that may be positive or negative. So in this discussion one may assume that pA ¼ 0 and only consider fares on train B.

A. de Palma et al. / Transportation Research Part B 71 (2015) 1–18

11

Fig. 5. A simple line with a single destination.

where both trains are used. If the passengers are using both trains then, at equilibrium, their generalized costs are equal. For passengers boarding at station 1:

d1 C A ðnA1 Þ þ d2 C A ðnA1 þ nA2 Þ þ d3 C A ðnA1 þ nA2 þ nA3 Þ þ SDCA1 ¼ d1 C B ðnB1 Þ þ d2 C B ðnB1 þ nB2 Þ þ d3 C B ðnB1 þ nB2 þ nB3 Þ þ SDCB1 ;

ð8Þ

where SDCA1 and SDCB1 are the schedule delay costs for trains A and B, respectively and where d1 ; d2 ; d3 represent the time between stations 1 and 2, 2 and 3, 3 and 4, respectively. C i ðnÞ is discomfort, per unit of time, in train i (for i ¼ A; B).10 The indexes, with respect to the trains, show that we can consider situations where the trains have distinct capacities (either with respect to seats ns or standing spaces nx ). Only interior solutions are considered. In this case, passengers boarding at station 1 incur the same generalized cost when using either train A or train B. For passengers boarding at station 2, the equilibrium condition is

d2 C A ðnA1 þ nA2 Þ þ d3 C A ðnA1 þ nA2 þ nA3 Þ þ SDCA2 ¼ d2 C B ðnB1 þ nB2 Þ þ d3 C B ðnB1 þ nB2 þ nB3 Þ þ SDCB2 ;

ð9Þ

and finally for passengers boarding at station 3, we have

d3 C A ðnA1 þ nA2 þ nA3 Þ þ SDCA3 ¼ d3 C B ðnB1 þ nB2 þ nB3 Þ þ SDCB3 :

ð10Þ

This condition states that the generalized cost (congestion plus schedule delay cost) is the same for trains A and B. This implies that a passenger arriving at node 3 will always stay in the same train and that the travel cost incurred before arriving at node 3 should be the same for trains A and B. This holds because waiting time at a node (here node 3) is more costly than staying at home for the same period of time. If the schedule delay cost depends only on the arrival time of the train, and not on the origin, then SDCA1 ¼ SDCA2 ¼ SDCA3 . A similar condition holds for train B. Using these conditions and (8) and (9) we get

C A ðnA1 Þ ¼ C B ðnB1 Þ:

ð11Þ

That is, the travel costs are equalized on the first leg of the trip. As we will see below, they will also be equalized on the second leg of the trip from node 2 to node 3. Note that this condition implies that the number of passengers is the same if trains A and B are identical (we assume that N 1 > 2ns ). Eq. (11) results from the assumption that schedule delay cost of user 2 is the same for train A and B. So comfort conditions for passengers boarding in station 1 also have to be the same. Eq. (11) has one unknown, nA1 , since nB1 ¼ N 1;4  nA1 . Using again the condition in the schedule delay cost and Eqs. (9) and (10), we obtain

C A ðnA1 þ nA2 Þ ¼ C B ðnB1 þ nB2 Þ:

ð12Þ

This condition corresponds to the equalization of travel time on the second leg of the trip from node 1 to node 4. As discussed above, this condition has to be logically valid for passengers going from node 2 to node 4. By recurrence, equality of travel cost also holds upstream (from node 1 to node 2: see (11). Thus, passengers boarding at station 3 are the only ones who trade-off discomfort and schedule delay cost. By equating the generalized cost on the last link the other passengers boarding at station 1 focus on discomfort in the first link (before station 2) and disregard discomfort. Similarly, passengers boarding at station 2 focus on discomfort between station 2 and station 3 (and disregard discomfort, too). The solution procedure starts by solving nA1 in Eq. (11). Using the substitution rule nB1 ¼ N 1;4  nA1 , we get an equation of a single unknown. In a second step we substitute the solution in Eq. (12) and solve it for nA2 . Finally we substitute the solutions in Eq. (10) and solve for nA3 . As we have assumed above, this procedure works when all passengers board on both trains. So, if the two trains have the same capacities, then from Eq. (11) we have nA1 ¼ nB1 ¼ N 1 =2. Substituting in Eq. (12) we get nA2 ¼ nB2 ¼ N 2 =2. If for example, N 1 ¼ 20; N 2 ¼ 30 and N 3 ¼ 40, then nA1 ¼ nB1 ¼ 10; nA2 ¼ nB2 ¼ 15, and for passengers boarding at station 3, we solve d3 C A ðN 1 =2 þ N 2 =2 þ nA3 Þ þ SDCA ¼ d3 C B ðN 1 =2 þ N 2 =2 þ N 3  nA3 Þ þ SDCB . With parameter values for the crowding cost function considered earlier (cf. Appendix A), and for SDCA ¼ 0 and SDCB ¼ 5, we find that most passengers at station 3 board on train A, with nA3 ¼ 34. So, in total, about 65% of the passengers board on train A, and the majority of these boarded at station 3. The solution generalizes easily for any number of stations (while keeping the same spirit), and for 10

For the numerical implementation we use the MAS formulation given in Eq. (1).

12

A. de Palma et al. / Transportation Research Part B 71 (2015) 1–18

Fig. 6. A simple line with multiple destinations.

asymmetric congestion costs. Notice that if crowding does not matter (and there is no capacity limitation) all passengers will board on the train with the smallest schedule delay cost. 6.2. Example 2: deterministic choice, multiple destinations The network of the second example is illustrated in Fig. 6. With respect to Example 1, the main difference is that we have two destination nodes. The equilibrium concept we use to describe the loading of each train is the same as before. So, each group of passengers using the two trains equates the generalized travel cost related to both. Since we have two possible destinations, we keep the two indexes in the number of passengers, e.g. nAo;d is the number of passengers boarding in train A at station o and alighting at station d. For passengers 1 ! 2 (boarding at station 1 and alighting at station 2), the equilibrium condition is

d1 C A ðnA1;2 þ nA1;3 Þ þ SDCA1;2 ¼ d1 C B ðnB1;2 þ nB1;3 Þ þ SDCB1;2 :

ð13Þ

For passengers 2 ! 3, the equilibrium condition is

d2 C A ðnA1;3 þ nA2;3 Þ þ SDCA2;3 ¼ d2 C B ðnB1;3 þ nB2;3 Þ þ SDCB2;3 :

ð14Þ

For passengers 1 ! 3, the equilibrium condition is

d1 C A ðnA1;2 þ nA1;3 Þ þ d2 C A ðnA1;3 þ nA2;3 Þ þ SDCA1;3 ¼ d1 C B ðnB1;2 þ nB1;3 Þ þ d2 C B ðnB1;3 þ nB2;3 Þ þ SDCB1;3 :

ð15Þ

If SDCA1;3 ¼ SDCA2;3 and SDCB1;3 ¼ SDCB2;3 then Eqs. (14) and (15) lead to C A ðnA1;2 þ nA1;3 Þ ¼ C B ðnB1;2 þ nB1;3 Þ. From (13) this yields the logical condition SDCA1;2 ¼ SDCB1;2 .11 So these conditions hold only in the case where both trains have the same schedule delay cost. If this condition is not met then a corner solution is obtained. For example, if SDCA1;2 < SDCB1;2 , then all passengers 1 ! 2 use train A. 6.3. Example 3: stochastic choice The two former examples are static and do not consider the dynamic allocation of seats. Also, passengers’ decisions are deterministic. The results shed some light on theoretical issues but the model is still too simple for most empirical applications. We wish here to go one step ahead toward more realism. Stochastic choice models are much more flexible and convenient for empirical applications. We consider again the network of Example 1 above but assume a discrete choice process. The passengers compare the generalized travel costs, each one being a sum of a crowding cost and a schedule delay cost, and decide on which train to board. We make here an explicit distinction between passengers who get a seat (and keep it for the whole trip) and those who stand. Ignore for instance the train index. When n1 passengers board at station 1, n1 get a seat and ^ 1 remain standing. In this case we have n1 ¼ minðns ; n1 Þ and n ^ 1 ¼ n1  n1 . For passengers boarding at station 2, those who n ^ ¼ n2  n2 . Finally, for passengers boarding at station 3, we find a seat are n2 ¼ minðns  n1 ; n2 Þ and those who stand are n ^ 3 ¼ n3  n3 who stand. By definition, we have have n3 ¼ minðns  n1  n2 ; n3 Þ passengers who find a seat and n ^ i ; i ¼ 1; 2; 3. So, the expected generalized travel cost for passengers 1 ! 4 is given by ni ¼ ni þ n

K 1 ðn1 ; n2 ; n3 ; SDCÞ ¼

1 ^1 þ n ^ 2 Þ þ d3 wðns þ n ^1 þ n ^2 þ n ^ 3 ÞÞn ^ 1  þ SDC: ½ðd1 þ d2 þ d3 Þwðns Þn1 þ ðd1 wðns þ n^1 Þ þ d2 wðns þ n n1 ð16Þ

The first term is the cost for the passengers who have a seat, the second and third terms are the cost for the standing passengers and the schedule delay cost. A similar evaluation for passengers boarding at station 2 yields,

K 2 ðn1 ; n2 ; n3 ; SDCÞ ¼

11

1 ^1 þ n ^2 Þ þ d3 wðns þ n ^1 þ n ^2 þ n ^ 3 ÞÞn ^ 2  þ SDC; ½ðd2 þ d3 Þwðns Þn2 þ ðd2 wðns þ n n2

This result is robust, and is a logical consequence of the equilibrium conditions. It does not depend on the specification used.

ð17Þ

13

A. de Palma et al. / Transportation Research Part B 71 (2015) 1–18 Table 4 Equilibrium with stochastic choice. Passengers

Train

Sitting

Standing

n

wðnÞ

n

wðnÞ

1!4

A B

12.70 7.30

5.62 5.62

0.00 0.00

– –

2!4

A B

7.30 10.97

3.75 3.75

11.72 0.00

8.22 –

3!4

A B

0.00 1.72

– 1.88

31.29 6.98

5.97 2.25

Fig. 7. A multi-lines network.

Table 5 The OD matrix. OD

y

r

q o

15 10

10 15

and finally for those who board at station 3 we have

K 3 ðn1 ; n2 ; n3 ; SDCÞ ¼

1 ^1 þ n ^2 þ n ^ 3 ÞÞn ^ 2  þ SDC: ½d3 wðns Þn3 þ d3 wðns þ n n3

ð18Þ

Note that we assume that the available seats are shared proportionally between the passengers, but this can be made more sophisticated by including the strategic behavior that emerges from the competition to get a seat. Then, we distinguish between the trains by inserting exponents A and B. The number of passengers in train A boarding from station i; nAi for i = 1, 2 and 3, is given by a binary logit model (cf. Anderson et al., 1992)

nAi ¼

A A A A A N i eK i ðn1 ;n2 ;n3 ;SDC Þ=li A B A A A A B A A A eK i ðn1 ;n2 ;n3 ;SDC Þ=li þ eK i ðN1 n1 ;N2 n2 ;N3 n3 ;SDC Þ=li

;

ð19Þ

where li is the parameter reflecting the preference for varieties (cf. Anderson et al., 1992). Keeping parameter li distinct for each group of users makes the model more flexible. The fixed point problem given by Eq. (19) can be solved once the parameter values are known. We conjecture that the solution is unique. As a solution, we have for each origin, the number of passengers boarding on each train, the number of those having a seat, and those who don’t. Conversely, one can observe empirically the number of passengers in each train, how many get a seat and how many stand, and then adjust the parameters (those of the MAS formula and parameters li ) so that the model provides a satisfactory estimation. Definitely, a modeler can analyze fare changes (through the generalized travel cost) or forecast transit assignments. The procedure is illustrated below for a simple network, but we plan to consider a more general network in future research. Let us illustrate the model with a numerical example. We consider a situation with N 1 ¼ 20; N 2 ¼ 30 and N 3 ¼ 40, and SDCA ¼ 0 and SDCB ¼ 5 (same values as in Example 1 above). All other parameter values are as in the former examples. The solution for li ¼ 1; i ¼ 1; 2; 3, is given in Table 4. Most passengers from station 1 board on train A and they all get a seat (in both trains). Passengers from station 2 use all the remaining seats in train A. Some also prefer those seats available in train B and all those who do not find a seat prefer to be standing in train A, because of the schedule delay advantage. Only some seats remain available in train B for passengers from station 3 (there are no more seats in train A). Most passengers prefer to be standing, but crowding costs are so high that some passengers prefer to be standing in train B.

14

A. de Palma et al. / Transportation Research Part B 71 (2015) 1–18

Fig. 8. A time-expanded graph for the network in Fig. 7.

6.4. Example 4: a more complex network All the examples considered above are limited to a single line. We consider now a situation where users have to choose between two or more lines to make their trip. For this purpose, we use the network given in Fig. 7. There are two origins o and q and two destinations r and y. The other vertices, a to d, are stations where passengers can board or alight. The numbers on the edges denote the travel times, which are fixed. The departure and arrival times of the distinct trains are also exogenously fixed. Edges starting from o and q or r and y correspond to walking and the other edges correspond to public transit. There are three transit lines: Line (i), a ! b ! c ! d; Line (ii), a ! c; Line (iii), b ! d. So, for example, a passenger traveling from q to y can use line (i) or line (ii). The combination of origins and destinations yields four groups of users, with an origin– destination matrix given in Table 5. The passengers have a desired arrival times to their destination, and this may create crowding on some trains. As a result, some passengers would prefer to switch to another train on the same line but with a later arrival time or switch to another line if they expect that crowding is lower on that line. In general, the analysis of this problem is complex. The usual methodology relies on the so called time-expanded network (cf. Hamdouch and Lawphongpanich, 2008). Starting from the network in Fig. 7, a graph that explicitly reflects time is constructed. The major benefit of this step is that distinct trains serving the same line can be distinguished as it can be seen on Fig. 8. Each vertex has a name and an index. The name makes the correspondence with the vertex on the initial graph, and the index indicates the time period. For example, d4 refers to node d at time period 4. Thus, the departures of the trains on the same line can be easily located. Links, like b2 ! b3 , going from the same node but incremented indices correspond to waiting for the next train. From Fig. 8 we see that line (i) has departures from vertex ðaÞ at times 1, 3, 5 and 7, respectively. The description here is very short, for a more detailed description the reader is refereed to Hamdouch and Lawphongpanich (2008). The schedule delay penalties will depend on the arrival terminals. Passengers who want to arrive early will try to reach the arrival vertex with the smaller index. The generalized travel cost is the sum of waiting times, travel times (over all edges),

15

A. de Palma et al. / Transportation Research Part B 71 (2015) 1–18 Table 6 Example of an equilibrium for the multi-line case. OD

Paths

o!r

o1 o1 o3 o3

o!y

o1 ! b2 ! c3 ! y4 o3 ! b4 ! c5 ! y6

q!r

q0 q2 q0 q0 q2

! a1 ! a3 ! a1 ! a1 ! a3

q!y

q3 q0 q0 q2

! a4 ! a1 ! a3 ! a3

! b2 ! b2 ! b4 ! b4

Number of users

User cost

5.18 4.19 4.08 1.54

15.75

6.07 3.93

7.24

! b2 ! d4 ! r 5 ! b4 ! d6 ! r 7 ! c 3 ! d4 ! r 5 ! b2 ! c3 ! d4 ! r 5 ! b4 ! c5 ! d6 ! r 7

2.94 2.53 2.53 1.57 0.43

19.99

! c6 ! y7 ! c3 ! y4 ! b2 ! c3 ! y4 ! b4 ! c5 ! y6

7.22 3.82 2.00 1.96

11.22

! d4 ! r 5 ! c 3 ! d4 ! r 5 ! d6 ! r 7 ! c 5 ! d6 ! r 7

crowding cost (over all edges with crowding), switching costs (between distinct links) and schedule delay costs. In equilibrium, users of the same group (the groups are distinguished with respect to origin–destination nodes) have the same generalized travel cost. Initially, the users consider all possible paths from their origin to their destination. In practice we limit the length of the paths to a reasonable value, so the number of explored alternatives remains manageable. Therefore, the paths theoretically possible, but unrealistic, are eliminated. This makes sense also because some paths, which are theoretically possible, make no sense with respect to real situations. The five-step procedure used is summarized below: 1. define parameter values, OD matrix 2. for each group of users generate the set of all possible paths 3. compute the generalized travel cost on each link and for each path; Generalized travel costs for each group of users can then be computed 4. for each group of users, those who perform badly with respect to the whole group change their path and select another one with a better performance 5. at the end of each iteration compute the difference in the distribution of passengers among the available paths and stop when the change reaches a small threshold The implementation of this procedure has been used to explore some examples. At each iteration the change in passengers distribution is computed, and convergence is assumed when this change falls below a fixed small threshold. This adaptive procedure requires hundreds of iterations to converge. We have tried to keep close to parameter values used in Hamdouch et al. (2011) to be able to make comparative studies, but since our modeling of discomfort is not the same as theirs, the output of the two models cannot be directly compared. Table 6 shows an equilibrium for the case we have considered. For each group of users there are three, four or five paths used in equilibrium. The results show that all paths are used, but that only few paths from the time-expanded network survive the choice of the passengers. As expected, travel costs are equal for passengers of the same group, but are different for distinct groups. All users arrive at time periods 5, 6 and 7 because late arrivals induce a penalty. There are many different configurations that we can obtain by choosing other parameter values. By increasing crowding cost (parameter a1 ) less users will choose paths arriving later than time period 7. Also, for smaller wait time penalties, some users will wait in transfer station for less crowded trains.12 The precise choice of the passengers will depend on the magnitude of these parameter values. 7. Conclusion This paper has developed an analytical expression for the discomfort in mass transit. Our discomfort cost function distinguishes between passengers with a seat and those who have to stand. For those who have to stand, the discomfort depends on the number of standing passengers compared to the capacity of the vehicle. This formulation helps to derive optimal timetables and optimal user charges. We have also considered the case of multiple stations and computed a transit equilibrium with both deterministic and stochastic decisions processes. This step shows how the model can be used to simulate realistic situations. The model presented in this paper is very simple and many improvements can be envisaged. 12 We do not have enough space here to experiment with all these possibilities, but we provide Mathematica notebook that can be used for this purpose. We also intend to write a paper fully deserved to analysis of complex networks with more sophisticated equilibrium concepts.

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In particular, we have omitted the waiting time and the fact that when a train arrives, and is heavily congested, only a fraction of passengers are able to enter. The remaining users have thus to wait for the next train, incurring an extra waiting time. The parameters should be estimated in order to derive a congestion function for public transportation, comparable to the BPR function widely used for private transportation. Passengers make decisions on the basis of a given anticipated trip time and congestion level. An alternative approach consists in closed loop strategies where the passengers may reoptimize trip decisions at each node. This has been analyzed by Hamdouch et al. (2011) and some other papers, on the basis of a very simplified discomfort formulation, but remains outside the scope of our analysis. We hope this paper provides one additional step in this direction. Acknowledgments We acknowledge support of PREDIT (France, contract 09MT/CV14) research program and the EIBurs program (EIBurs-100113). For many useful comments, we are grateful to participants of Kuhmo-Nectar conference and in particular to Robin Lindsey, Ken Small and Guillaume Monchambert. The revised version has largely benefited from comments of Hai Yang and three anonymous referees. The usual disclaimer applies. Appendix A. The notation The main part of the notation used in the text as well the basic parameter values are given in the following table. Parameter

Comment

Illustrative value

a0 a1 ns nx ns þ nx b; c a ti T

VOT with seat VOT without seat Early arrival penalty Late arrival penalty Number of seats Standing capacity (legal) Vehicle capacity (legal) Discomfort parameters Discomfort parameters in wðnÞ Departure time of train i Travel time

6 ($/h) 9 ($/h) 5 ($/h) 12 ($/h) 20 (seat) 30 (passenger) 50 (passenger) (0.3, 0.3) 1.0 – 1/4 (h)

CðnÞ SC TC AC SDC TSD wðnÞ

User cost for the nth passenger (cf. Appendix B) Marginal social cost Total cost in the vehicle Average cost in the vehicle Schedule delay cost Total schedule delay cost MAS formulation of user cost (cf. Eq. (1))

b

c

Appendix B. Approximation of the MAS formula The user cost in the general case is given by

CðnÞ ¼



a0 if n 6 ns cðnns nx Þ a1 þ b e if n > ns :

ðB:1Þ

A drawback of the this expression is that it is not continuous at point ns . This may lead to solution existence problem in realistic problems. Hamdouch and Lawphongpanich (2008) assumes semi-continuity of the cost function to prove solution existence. Also, the conditional statement in definition (B.1) may lead to a complications in the practical implementation of the MAS formula. Whenever possible, a smooth function is preferred. A continuous alternative may be obtained by replacing the original function by a good approximation. We discuss here how one can construct a function w that approximates the user cost given in Eq. (B.1), and show that the MAS formulation, Eq. (1), used in this paper is a good candidate. There are several approximation procedures and techniques available. These are generally simple to apply to unidimensional functions. We use two standard techniques, a simple Chebyshev interpolating polynomials and a sophisticated implementation in Mathematica.13 Both of these solutions are illustrated on Fig. B.9. 13

The latter uses divided differences to construct Lagrange or Hermite interpolating polynomials.

A. de Palma et al. / Transportation Research Part B 71 (2015) 1–18

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Fig. B.9. Approximation of user cost function.

Fig. C.10. Comparison with initial formulation (bold curves) and max formulation (dashed curves).

For Chebyshev polynomials, we have used the standard procedure as given in Algorithm 6.2 in Judd (1998). Let us denote this approximation function by wc . The result shown in Fig. B.9(a) with m ¼ 150 (number of interpolation points) and n ¼ 50 (polynomial degrees). The approximation quality remains poor, due mainly to the discontinuity at ns . Increasing further the number of interpolation points does not improve the quality of the approximation. Increasing the degree of the interpolating algorithm leads to an instability in the output function and makes the computation much more complicate. The quality of this approximation is not good and there are two problems. For some values of n, particularly around ns , the values it genR 60 erates are not close to those of CðnÞ. This can be confirmed by measuring the error approximation 0 jCðnÞ  wc ðnÞjdn. The c second problem is that an equation of the form w ðnÞ ¼ A, where A is positive number may have more than one solution (depending on the values of A). This occurs because the approximation here does not preserve the monotonicity of CðnÞ. Definitely wc is not a good choice for the approximation of CðnÞ. The sophisticated approximation is denoted wl . Fig. B.9(b) shows both C and wl . In this case we obtain a better approximation. In particular, The error between the approximation and the original function is small. It is clear that ws ðnÞ fits the original function CðnÞ much better than wc . This observation may be confirmed by computing the error measure R 60 jCðnÞ  ws ðnÞjdn. 0 The explicit formula is actually a quiet long expression that can be handled by computers but not really useful for direct analytical usage. We can, however, provide a relatively simple formula that is comparable with respect to the error generated to the second approximation provided above. Function wðnÞ and introduced in Section 1 is a good approximation for the user cost as defined by Eq. (B.1). It is comparable to the function ws ðnÞ defined above but has the merit of being very simple and avoids all conditional expressions. Apart, the problem of multiple solutions (that appears only with large values of a) it could be used for practical purposes. It is interesting to notice the shape of wMAS ðnÞ for a smaller value of the constant a. Fig. 1 shows that, around the number of seats ns , the original cost function is averaged. This is particularly useful for empirical applications. Indeed, even if passengers care about the possibility of having a seat, they also consider alternative possibilities of giving it up for an older passenger. Sometimes a passenger avoids taking a seat next to another passengers she does not appreciate. With all the possibilities that can occur in real situations, it is suggested that an average value (and smooth) function is preferable to the discontinuous function.

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