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Managing rail transit peak-hour congestion with a fare-reward scheme
Transportation Research Part B 110 (2018) 122–136
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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb
Managing rail transit peak-hour congestion with a fare-reward scheme Hai Yang, Yili Tang∗ Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR China
a r t i c l e
i n f o
Article history: Received 23 January 2017 Revised 6 February 2018 Accepted 8 February 2018
Keywords: Rail transit bottleneck Queuing congestion Fare-reward scheme Departure time choice User equilibrium
a b s t r a c t This paper describes a new fare-reward scheme for managing a commuter’s departure time choice in a rail transit bottleneck, which aims to incentivize a shift in departure time to the shoulder periods of the peak hours to relieve queuing congestion at transit stations. A framework of the rail transit bottleneck is provided and the user equilibrium with a uniform-fare and the social optimum with service run-dependent fares are determined. A fare-reward scheme (FRS) is then introduced that rewards a commuter with one free trip during shoulder periods after a certain number of paid trips during the peak hours. For a given number of peak-hour commuters and ex-ante uniform fare, the FRS determines the free fare intervals and the reward ratio (the ratio of the free trips to the total number of trips, which is equivalent to the ratio of the number of rewarded commuters to the total number of commuters on each day during the peak hours). The new fare under the FRS is determined so that the transit operator’s revenue remains unchanged before and after introducing the FRS. Our study indicates that, depending on the original fare, FRS results in an optimal reward ratio up to 50% and yields a reduction of system total time costs and average equilibrium trip costs by at least 25% and 20%, respectively. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Since Vickrey’s bottleneck model (Vickrey, 1969), there has been a substantial stream of development of research in this area (Arnott et al., 1990; Laih, 2004; Lindsey et al., 2012), and various extensions and applications such as elastic demand and general queuing networks were made (Braid, 1989; Arnott et al., 1993; Yang and Meng, 1998). While research up to date has been overwhelmingly focused on the road traffic, a few studies were devoted to the peak-hour congestion in public transportation, where a bottleneck occurs when passenger demand exceeds transit service capacity. Existing studies on transit bottleneck and queuing congestion are generally concerned with transit capacity choice, scheduling and fare pricing. Unlike road traffic management, transit authorities usually adjust service frequency to accommodate variable passenger demand, which affects both service capacity and passenger waiting time (service quality). Notably, applying the bottleneck model in mass transit, Kraus and Yoshida (2002) considered optimal fare and service frequency to minimize long-term system costs. Various fare pricing schemes have been considered for transit services in practice such as time-based differential pricing that involves peak-fare charging, off-peak discounting and combinations. Pilot programs and temporal fare regulations are ∗
Corresponding author. E-mail address: [email protected] (Y. Tang).
https://doi.org/10.1016/j.trb.2018.02.005 0191-2615/© 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. The transit bottleneck in the peak period.
conducted worldwide. For instance, implementation of temporally differential fares in Taipei reveals that metro riders are indeed sensitive to fares, especially peak-hour surcharges (Lan et al., 2010). Metropolitan areas such as Washington DC and London introduced time-based fares. In the case of the metro rail service in Washington DC, a peak fare is combined with a distance-based fare. While a peak surcharge acts as a penalty for commuters during rush hours, rewarding offers an endogenous and incentive-compatible strategy for morning commute rationing. Note that, free fare is the largest discount for off-peak commuters. In the case of the ‘Early Bird’ scheme in Melbourne, commuters enjoy free travel to the central business district if they finish their trips before 7 a.m.; it was observed that 23% of the off-peak ridership was made by the commuters who shift from the rush-hour period (Currie, 2010). A similar peak avoidance experiment in The Netherlands was implemented by rewarding commuters for travelling off-peak to reduce peak-hour ridership (Peer et al., 2016). A major consideration of the above demand management is to offer monetary compensation or other incentives at the expense of the government or the transit operator. The fare-free adoption for off-peak hours in Mercer County, New Jersey led to a reported loss of around one-quarter of revenue (Perone, 2002) and as a result, the scheme was abolished. With limited sources of funding, governments or transit authorities would rather charge a high fare to passengers than lose revenue. In addition, the above fare-free strategy also results in excessive off-peak ridership with the majority coming from untargeted passengers who would otherwise travel by other modes such as walking or bicycling. To address the issues associated with the aforementioned demand management strategies for peak-hour transit operation; this paper introduces a controlled free fare reward scheme (FRS). Under the proposed FRS scheme, a commuter is rewarded with one free trip during pre-specified shoulder periods after taking a certain number of paid trips during the peak hours. Implementation of such a FRS is straightforward with the current smart card systems. The planner can determine the rewarding ratio and periods to minimize commuters’ trip costs and to control the number of free trip commuters, namely, a certain proportion of commuters will be induced to shift their departure times and thus passenger flow congestion at the transit bottleneck will be smoothed out over time. Furthermore, the FRS scheme creates a win-win situation for both commuters and the transit operator. Commuters enjoy reduced travel time without paying more on average. In the meantime, the operator’s revenue remains intact, which is in the same spirit of the ‘revenue-neutral’ traffic congestion management schemes in the literature, such as the Pareto-improving congestion pricing and revenue-refunding scheme (Guo and Yang, 2010) and the tradable credit scheme (Yang and Wang, 2011; Xiao et al., 2013; Nie and Yin, 2013). The paper is organized as follows. Section 2 introduces the framework of the urban rail transit bottleneck model with batch arrivals at stations and the problems of user equilibrium with a uniform fare and the social optimum with service rundependent fares. Section 3 develops the fare-reward scheme model within the framework of the transit bottleneck and seeks the optimal rewarding ratio and periods to minimize the total commuting costs. Section 4 assesses the system performance under the FRS in comparison with the original bottleneck situation. Sensitivity analysis of the results is conducted with respect to the initial system configuration and assumptions. Conclusions and further development are provided in Section 5. 2. The transit bottleneck model in the absence of FRS 2.1. Problem settings Table 1 summarizes the parameters, variables and abbreviations used in this paper to describe the transit bottleneck and fare-reward scheme. The symbols are also explained in the text. Consider a single origin and destination connected by an urban rail line where each train is running with capacity s and headway h. Suppose a bottleneck occurs in peak hours, as shown in Fig. 1. During the peak hours, a given total number of N commuters travel with M uniformly spaced service runs through the bottleneck. All trains passing through the bottleneck are indexed such that the service run 1 is the first train and M is the last train through the bottleneck. Departure time of each service run is denoted by tm , m = 1, 2, ..., M. The peak period lasts for L = (M − 1)h. Individual commuters are assumed to have an identical work starting time t∗ . The service run scheduled before or at ¯ , namely 0 ≤ t ∗ − tm¯ < h. Like the road bottleneck model and without loss of generality, work starting time is labeled as m commuters arrive at the transit station immediately upon their departing from home, and in-transit time is ignored so that commuter’s arrival time at work is the train’s departure time. In this case, a passenger who takes a service run labeled as m encounters a fare cost p(m), a queuing time q(m), a schedule delay e(m) if they arrive at work early or l(m) if they arrive late, all of which are functions of the service run m
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136 Table 1 Parameters, variables and abbreviations. h s N M L t∗ ¯ m p(m) q(m) e(m) l(m)
α β γ
p0 p∗0 p
λ λ∗ φ ϕ
FRS AEC TTC UFI FFI ETC
Headway of transit service Capacity of a train Total number of commuters Total number of service runs Length of the peak period Commuter’s work starting time Service run scheduled before or at work starting time Fare cost of taking service run m Queuing time of taking service run m Early arrival schedule delay of taking service run m Late arrival schedule delay of taking service run m Shadow value of travel time Shadow value of early arrival Shadow value of late arrival Uniform fare in the original transit bottleneck Reference original fare after fare-reward scheme Uniform fare after fare-reward scheme Reward ratio in fare-reward scheme Optimal reward ratio in fare-reward scheme System efficiency (relative reduction of system total time costs) Individual efficiency (relative reduction of average equilibrium trip costs) Fare-reward scheme Average equilibrium costs of individual commuters Total time costs in system Uniform fare interval Free fare interval Equilibrium time costs of individual commuters
in the bottleneck. With homogeneous passengers, the generalized trip costs, inclusive of time cost, fare cost given by (1), is the same for all passengers1
c ( m ) = α q ( m ) + β e ( m ) + γ l ( m ) + p( m )
(1)
The unit costs α , β , and γ are the shadow values of queuing time at the station, arriving early and arriving late respectively. As in the road bottleneck, it is assumed that β < α < γ (Arnott et al., 1990). Note that, unlike the road bottleneck model, commuters arrive at stations in batches and their schedule delay cost is a step function of discrete transit runs. 2.2. User equilibrium under a uniform fare With a fixed total number of commuters during the peak hours, a uniform fare in the urban rail transit has no effect on a commuter’s departure in relation to the fare-free user equilibrium. With long headways, commuters may follow the timetable and thus arrive at stations in batches. But for peak-hour urban transit operation with high frequency, commuters usually arrive at stations without considering the service schedule. In this case, the expected usual waiting time at the station (without overflow queuing) is approximately a half headway and is hence a constant for a given frequency. For these reasons and without loss of generality, we only focus on the overflow queuing time and ignore the usual unsaturated waiting time. In user equilibrium, commuters in each train have the same generalized costs based on the trade-off between schedule delay cost and queuing time cost. Commuters with the highest schedule delay cost do not encounter waiting time. Therefore, the equilibrium total time costs are either defined by the schedule delay cost of the first or the last service run, whichever is greater (hereinafter, ‘0’ refers to the original uniform fare case),
TTC0 = max {β (t ∗ − t1 )N, γ (tM − t ∗ )N}
(2)
The transit authority should set the optimal timetable to minimize the total time costs:
min {max {β (t ∗ − t1 )N, γ (tM − t ∗ )N}}
(3)
1 A few studies also take account of the in-vehicle-crowding costs in transit commuters’ costs (Huang et al., 2004; Tian et al., 2007; de Palma et al., 2015; de Palma et al., 2017), which is a linear function of the number of commuters taking the same train. The linear specification is also supported by empirical studies of crowding cost estimation (Whelan and Crockett, 2009; Wardman and Whelan, 2011; Haywood and Koning, 2015). In line with this, we assume the in-vehicle-crowding cost function, g(nm ), is a monotonically increasing linear function of the total number of passengers, nm , taking the same service run m . Realistic statistics show that the rail service runs are mostly fully occupied by commuters with 97.33% average loading of six main rail lines in Hong Kong (Legislative Council Panel on Transport, 2016). The very high in-vehicle congestion implies that the in-vehicle crowding cost is approximately the same for all commuters during peak period or g(nm ) = g(s) becomes a constant. Thus, we exclude the in-vehicle crowding cost in our model.
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Therefore, the schedule delay cost of the first and the last service run is the same with the optimal timetable:
β (t∗ − t1 ) = γ (tM − t∗ ). In equilibrium, commuters taking the first service run and the last service run do not incur queuing time and have identical trip costs consisting of only schedule delay cost and a uniform fare cost. Namely,
β (t ∗ − t1 ) + p0 = γ (tM − t ∗ ) + p0 Since tM
− t∗
TTC0 =
+ t∗
(4)
− t1 = (M − 1)h, the system total time costs (TTC) are given by
βγ (M − 1 )hN (β + γ )
(5)
The individual average equilibrium trip costs (AEC) are given by
AEC0 =
βγ (M − 1 )h + p0 (β + γ )
(6)
Following the first-in-first-out principle, commuters who take the intermediate mth service run encounter the following queuing time at the station
q (m ) =
β (m−1)h
α γ (M−m )h α
¯] m ∈ ( 1, m ¯ +1, M ) m ∈ [m
(7)
2.3. Social optimum with run-dependent step fares At the social optimum under the above optimal timetable, queuing time at the station is eliminated and commuters only encounter the schedule delay cost. This could be achieved by implementing the following run-dependent step fares:
⎧ ⎪ ⎨ p¯
p˜ + β (m − 1 )h p( m ) = ⎪ ⎩ p˜ + γ (M − m )h p¯
m < 1, ¯ ], m ∈ [1 , m ¯ + 1 , M ], m ∈ [m m>M
where p¯ and p˜ are the basic fares for service runs outside the peak period and within the peak period respectively. In a road bottleneck, a time-dependent fine toll is required to eliminate queue for social optimum, which is difficult to implement in practice. Similarly, the above run-dependent transit fare pricing scheme represents an ideal situation, which, without sufficiently accurate information, is also difficult to implement in practice. 3. The transit bottleneck model in the presence of FRS 3.1. Basic considerations and assumptions It is accepted that, instead of fully realizing a social optimum in a first-best environment, various second-best toll pricing strategies can be employed to relieve congestion, such as the coarse tolls (Arnott et al., 1990; Laih, 2004; Xiao et al., 2012) and step tolls (Laih, 1994; Laih, 2004; Li et al., 2017) for managing congestion in a road bottleneck. The fare-reward scheme proposed here is not intended to achieve the above-mentioned first-best result under a run-dependent transit fare, rather, it attempts to strike a balance among efficiency, simplicity, and fairness in an incentive-compatible manner. The following assumptions are first introduced: Assumption 1. With the first-in-first-out principle, the peak-hour operation of the rail transit system is under a long-term optimal configuration where all service runs are at full capacity. Assumption 1 implies a predetermined optimal and stable urban rail transit service for the peak-hour commute. Kraus and Yoshida (2002) considered the optimal long-run peak-hour transit services without late arrival and showed that all service runs are at full capacity at the user equilibrium and social optimum. Yoshida (2008) extended the model with late arrivals and explored the optimal transit service under the first-in-first-out principle and random-access principle. Furthermore, according to the report of the Mass Transit Railway in Hong Kong, peak-period service runs are approximately at full capacity where the average loading of the six main rail lines in 2015 was 97.33% (Legislative Council Panel on Transport, 2016). Since commuters generally follow the first-in-first-out discipline so as to respect queuing rules in rail transit and in particular during the peak hours, with full capacity utilization, the number of service runs, the train service capacity and the total number of commuters during the whole peak period also satisfy the following relation in our model: M = N/s. Justification of the full capacity utilization under the optimal long-term transit service is provided in Appendix 1. Assumption 2. The total number of commuters and the rail transit headway remain the same and the operator’s fare revenue stays unchanged before and after the implementation of the proposed FRS. Assumption 2 means that the fare-reward scheme is designed to influence commuters’ departure time choice only in reducing individual and system commuting costs, without sacrificing the operator’s benefits.
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Fig. 2. Urban transit bottleneck with original fare.
Fig. 3. Fare-reward scheme for (a) free ride commuters and (b) non-free ride commuters.
Fig. 4. Fare intervals under FRS with design criteria.
Consider an original uniform fare p0 throughout the whole period in an urban rail transit bottleneck, as shown in Fig. 2. Implementation of the FRS changes the original uniform fare structure according to the commuters’ free ride entitlement. The peak hours of interest are divided into two free fare intervals (FFI) and one uniform fare interval (UFI). UFI is the central period within the peak period spanning the time interval [ti ,tj ] that contains service runs from i to j and includes the work starting time t∗ , as shown in Fig. 3(a). FFI includes the two shoulder intervals before and after the UFI. The uniform fare changes from p0 to p through the whole peak period. After a certain number of paid rides, a commuter is entitled to a free ride only during the FFI; a commuter without such an entitlement can choose either the UFI or FFI at a uniform fare p, as shown in Fig. 3(b). To ensure that a commuter will prefer to take the free ride during FFI, the FRS must meet the following design criterion: Design criterion 1: The FRS is designed so that the equilibrium trip costs in FFI with a reward (a free ride) are less than or equal to those in UFI without a reward. This design requirement ensures that the FRS is effective in motivating commuters to shift their departure time when receiving a free-ride reward. From the perspective of the system total time costs savings, we also introduce the following second design criterion: Design criterion 2: In FRS, a reward ratio λ is chosen such that λM is an integer. Design criterion 2 states that the proposed FRS will segregate the free ride and non-free ride commuters into the service runs respectively during FFI and UFI, as shown in Fig. 4. Therefore, the free ride and non-free ride commuters also follow the batch departures and the optimal service run under FRS is at full capacity too. Equivalently, the whole peak period under the FRS is divided into three fare intervals: the two shoulder intervals used only by free ride commuters and the central interval used only by the non-free ride commuters. Justification of design criterion 2 is given in Appendix 2.
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Table 2 Summary of the optimal FRS. p0
Feasible λ
(a)
p0 ≤ θ Mh 4
( 0,
(b)
p0 > θ Mh 4
(0, 1)
Remark: ϑ =
1 2
− ϑ] ∪
λ∗ [ 12
+ ϑ , 1)
1 2 1 2
±ϑ
TTC at λ = λ∗
Average fare cost
AEC at λ = λ∗
[θ h(M − 1) − p0 ]N
p0
θ h(M − 1)
θ h( 34 M − 1)N
p0
θ h( 34 M − 1) + p0
1/4 − p0 /θ hM.
3.2. User equilibrium and the optimal FRS design Here we consider how to determine the start and end time of the UFI illustrated in Fig. 4. Under Assumption 2 of the ex-ante and ex-post fixed commute demand and transit headway, the choice of UFI is equivalent to the determination of the reward ratio λ of the FFI (ratio of the free rides to the total number of rides), which is also equal to the ratio of the number of free ride commuters to the total number of commuters during daily peak hours, supposing that the total number of commuters is large enough. Let the central UFI correspond to the time interval [ti , tj ] containing the service runs from i to j, as depicted in Fig. 4. From the equilibrium condition,2 β (t∗ − ti ) + p = γ (tj − t∗ ) + p, the queuing time for each service run of UFI is given by
q (m ) =
β (m−i)h
α γ ( j−m )h α
¯] m ∈ [i, m ¯ + 1, j ] m ∈ [m
Considering design criterion 2, we have
( 1 − λ )M = j − i + 1
(8)
The total time costs for UFI commuters at equilibrium are
TTCUFI =
βγ βγ [(1 − λ )M − 1]h · (1 − λ )N ( j − i )h · (1 − λ )N = (β + γ ) (β + γ )
(9)
where the second equality comes from Eq. (8). Similarly, the queuing time for each service run of the two FFIs is given by
q (m ) =
β (m−1)h
, m ∈ [1, i − 1] α γ (M−m )h , m ∈ [ j + 1, M ] α
The total time costs for FFI commuters at equilibrium are
TTCFFI =
βγ (M − 1 )h · λN (β + γ )
For the given M and p0 , the minimization problem of the system total time costs is given by
min TTC(λ ) = θ (M − 1 )h · λN + θ (M − λM − 1 )h · (1 − λ )N
0<λ<1
(10)
subject to
N p0 = (1 − λ )N p
(11)
θ [(1 − λ )M − 1]h + p ≥ θ (M − 1 )h
(12)
where θ =βγ /(β + γ ) is a constant. Constraint (11) is imposed to ensure the operator’s revenue is unchanged before and after the FRS; inequality (12) is a mathematical statement of the aforementioned design criterion of the FRS. Appendix 3 analyses the effects of relaxing the revenue neutrality assumption. It is interesting to note that the FRS design problem reduces to a single variable optimization problem in terms of λ under a commute equilibrium constraint. In what follows, for convenience in the mathematical analysis, the reward ratio λ is treated as a continuous variable for the optimization problem, λ ∈ (0, 1). One can analytically obtain the optimal solution of λ, as summarized in Table 2, where the valid range and the corresponding optimal value of λ, together with the FRS performance, depend on the ex-ante uniform fare p0 . Case (a) is when p0 ≤ θ Mh/4. In this case, constraint (12) is always binding at the optimum (commuter’s equilibrium trip costs in FFI and UFI are identical), and the feasible value of λ consists of two disjoint intervals belonging to (0, 1). For a given p0 , two solutions of λ exist in the system of simultaneous nonlinear Eqs. (11) and (12). This can be explained by the fact that the ex-ante constant revenue can be achieved with a larger (smaller) number of non-free ride commuters at a 2 Appendix 1 states that when the headway is sufficiently short and the number of service runs is sufficiently large, the optimal timetable with FRS approaches the optimal timetable with a uniform fare.
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136 Table 3 Fare-reward scheme performances on urban rail transit bottleneck. p0
λ∗
(a)
0 ≤ p0 ≤ θ Mh 4
1 2
(b)
p0 > θ Mh 4
1 2
±
1 4
−
p0 θ hM
TTC
φ
AEC
ϕ
Np0
p0 θ h(M−1 )
p0
θ h(M−1 )+ p0 θ Mh/4 θ h(M−1 )+ p0
θ MhN 4
θ Mh
M 4(M−1 )
4
p0
lower (higher) fare p under the FRS. Moreover, at the optimal solution λ∗ , the trip costs of all commuters during the peak hours are equal to the right-hand side constant of constraint (12), which are also the time costs for free ride commuters in the two FFIs. This is always true, regardless of the value of p0 , whenever p0 ≤ θ Mh/4. The system total time costs, TTC, always decrease with p0 . Case (b) is when p0 > θ Mh/4. In this case, constraint (12) is always nonbinding, and any λ ∈ (0, 1) is feasible. The optimal λ∗ is uniquely determined to be 1/2, implying ‘pay one get one free’. Differing from Case (a), the minimum system total time costs, TTC, is a constant, independent of the original fare p0 . The average equilibrium commute costs (averaged over free and non-free rides), however, increase with p0 . Finally, we point out that the average fare cost (for the same commuter over days or for all commuters on the same day) is always p0 simply because the total number of commuters and the total revenue remains unchanged before and after the FRS. 4. Assessment of system performance To assess the system performance under the proposed FRS, we first introduce a set of system performance measures, analyze the effects of the FRS, and conduct the sensitivity analysis of system performance with respect to the initial system settings. 4.1. System performance measure The maximum reduction in system total time costs (TTC) by the FRS is defined by
TTC = TTC0 − TTC(λ∗ )
(13)
where TTC0 is the user equilibrium TTC given in Eq. (5). The percentage reduction of the system TTC (or system efficiency) is given by
φ=
TTC
(14)
TTC0
The reduction in individual average equilibrium costs (AEC) achieved by the FRS at the optimal reward ratio is given by
AEC = AEC0 − AEC(λ∗ )
(15)
where AEC0 is the user equilibrium AEC provided in Eq. (6). Finally, the percentage change in the individual AEC (or individual efficiency) is given by
ϕ=
AEC
(16)
AEC0
The above four system performance measures serve different purposes. The absolute reduction of the system TTC and individual AEC indicate the magnitude of the cost savings. The percentage reduction of the TTC and AEC show the efficiency and public acceptability of the FRS scheme respectively. 4.2. System performance versus original fare p0 The above four measures and reward ratio λ together reflect the performance of the FRS. As shown in Table 3, in the extreme case when the original fare p0 = 0, the FRS becomes inapplicable and all performance indexes become zero. As shown in Table 3 and Figs. 5 and 6, from the perspective of both system performance and individual commute costs, the FRS performs the best at a reference original fare p∗0 = θ Mh/4, with an optimal reward ratio λ∗ = 1/2. The optimal reward ratio incentivizes 50% of commuters in traveling free in the shoulder periods. This can be explained intuitively that when λ is small, increasing λ will increase the number of free ride commuters to relieve congestion. However, when λ is large, the excessive number of free ride commuters induces congestion in the shoulder periods, which results in higher system total time costs. At the reference original fare p∗0 , the absolute and percentage reduction in both system total time costs (TTC) and individual average equilibrium trip costs (AEC) reach maximum:
TTC =
θ MhN 4
,φ =
M , AEC = 4M − 4
θ Mh 4
,ϕ =
M , at p∗0 = 5M − 4
θ Mh 4
(17)
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Fig. 5. The reduction in TTC and AEC curves with respect to original fare p0 .
Fig. 6. Reward ratio, system efficiency and individual efficiency curves with respect to p0 .
The absolute and percentage reductions in TTC increase linearly with p0 up to p∗0 = θ Mh/4 (case (a) in Table 3), reach the optimum and then remain at the optimum (case (b) in Table 3). On the other hand, the absolute and percentage reductions in AEC increase with p0 up to p∗0 = θ Mh/4 but in a linear and nonlinear manner respectively. After reaching the optima, the former one remains at the optimum but the latter one decreases nonlinearly. Finally, we mention in Fig. 5 that the absolute reduction in AEC is exactly equal to the original fare p0 in case (a). The corresponding absolute reduction in TTC is then Np0 , which is exactly equal to the operator’s revenue. This is because the commuters’ average fare cost is unchanged, due to the constant revenue constraint, and thus the total reduction in TTC is equal to the reduction in AEC multiplied by the total number of commuters N.
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Fig. 7. Changes of efficiencies with M at the reference original fare.
4.3. Sensitivity analysis of system performance at p∗0 The FRS analyses so far are carried out for a given supply (transit headway h and number of service runs M) and demand (number of commuters N and their shadow values of time β and γ ). Here we conduct sensitivity analysis on the obtained results with respect to these exogenous inputs. For convenience, we fix the original fare at p0 = p∗0 , which is termed as the reference original fare hereafter. Fig. 7 shown the changes in system efficiency φ and individual efficiency ϕ with the total number of service runs M, M ∈ [2, ∞). Both system and individual efficiencies decrease with M. This is because the FRS aims at reducing the queuing time costs at train stations, and the total schedule delay costs do not change because all train services are already operating at full capacity. A larger M means a higher commuter demand and a longer peak period for a given headway and train capacity. The schedule delay costs increase with M more than queuing time costs under the implementation of FRS. Thus, the relative reduction of the queuing time costs (efficiency) decreases. It is also observed that the system efficiency is always greater than the individual efficiency. In fact, one can readily see from Table 3 that this observation is true for any original fare p0 . This can be accounted for by the fact that individual efficiency is measured by the full trip costs, inclusive of time costs and fare cost, while the system efficiency is measured by the time costs only, and the fare cost (or operator’s revenue) is a constant independent of M. In an extreme case with two service runs, system and individual efficiencies reach maximum values of 50.0% and 33.3%. When the number of service runs is sufficiently large, they approach their minimum values of 25.0% and 20.0%, respectively. The positive minimum value of efficiency ensures the effectiveness of FRS in a busy rail transit line. The system efficiency of the FRS scheme can be compared with that of the tolling strategies in the traditional road bottleneck model. In Arnott et al. (1990), the coarse toll can save at least 25% of the total system time costs, and the efficiency of the proposed fare-reward scheme would also save approximately 25% when the demand is large enough. This is not surprising because the batch commuter departures and arrivals approximate to the continuous departure and arrival rates in the road bottleneck case when the demand is high and the headway is short. Further mentioning in passing, Xiao et al. (2012) found that, at most, a 25% saving of the system costs could be achieved by the coarse toll under the assumption of “tactical waiting”. The existence of “tactical waiting” behavior reduces the capability of the coarse toll in a road bottleneck. In contrast, in our fare-reward scheme, the free fare in a transit bottleneck is only for eligible commuters who completed a certain number of trips and hence commuter “tactical waiting” is not applicable.3 Now we look at the impacts of the two exogenous parameters β and γ on system performance. Here, we fix the shadow value of early arrival β , and look at the difference between the shadow value of late arrival and early arrival in terms of the ratio, γ /β . We point out that the absolute reductions in TTC and AEC are affected by γ /β , but the percentage changes are independent of γ /β at the reference original fare p∗0 , from Table 3 and Eq. (17). Without loss of generality, we only
3 The discontinuities in the optimal coarse toll studied in Arnott et al. (1990) also induce a shift in the departure period. In order for all commuters to have the same equilibrium trip cost, no departures occur for an interval before the toll is applied. When the toll is lifted, a mass of commuters immediately departs. By contrast, in our model commuters who are entitled to a free ride incur a different equilibrium trip cost than commuters with no entitlement. No mass departures or changes in the aggregate timing of departures are required to assure equilibrium for both groups.
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Fig. 8. Change of AEC with h, N and γ /β at the reference original fare. Table 4 Three metro rail lines in Hong Kong. MTR Rail line
Distance
Fare for entire line (HKD)
Ma On Shan Line (O) Island line (I) West Rail line (W)
11.4 km 16 km 35.4 km
7.5 11.0 23.0
conduct sensitivity analysis for the absolute value of reduction in average equilibrium costs ( AEC), given β = 50 HKD/h and capacity s = 30 0 0. It is easy to check that AEC is positively related to the demand and headway, as shown in Fig. 8. For a given combination (N, h) of demand and headway, AEC is increasing with γ /β over the range γ /β ∈ (1, ∞). This is due to the fact that the schedule delay cost and system total time costs increase with γ . Particularly, AEC reaches a minimum value β Mh/8 as γ /β → 1 (θ → β /2) and a maximum value β Mh/4 as γ /β → ∞ (θ → β ) (late arrival is prohibited). These results illustrate that the FRS is more effective as the late arrival penalty increases. 4.4. Examples of realistic fares in the Hong Kong mass transit railway This section investigates whether the reference original fare p∗0 determined above is realistic under real transit settings, and then determines the system performance of the FRS. We consider three typical metro rail lines in Hong Kong Mass Transit Railway (MTR) shown in Table 4. We consider the following three different but realistic operational scenarios for the above three metro lines: (1) h = 2.5 min, N = 750 0 0, s = 30 0 0, M = 25, β = 50 HKD/h, γ = 100 HKD/h (2) h = 2.5 min, N = 450 0 0, s = 30 0 0, M = 15, β = 50 HKD/h, γ = 100 HKD/h (3) h = 2.5 min, N = 750 0 0, s = 30 0 0, M = 25, β = 50 HKD/h, γ = ∞ HKD/h The headways and passenger demands are assumed with reference to the Annual Report of Hong Kong MTR Cooperation (MTR Corporation Limited, 2015). The shadow values of early and late arrival are chosen with reference to Small et al. (2005) and Ubbels et al. (2005).4 Scenario 2 represents the case with a reduced demand and scenario 3 represents a case with prohibited late arrival. The results for the above scenarios are presented in Table 5. The system performance indices under the implementation of the FRS at the reference original fare are consistent with the analytical results obtained so far. For example, under scenario (1), the reference original fare is p∗0 = 8.7, and the actual fare for Line O is less than p∗0 = 8.7, thus two optimal reward ratios appeared, resulting in sub-optimal system performance. These results correspond to the curves at p0 < p∗0 = θ Mh/4 4 Early studies (Small et al., 2005; Ubbels et al., 2005) are used for reference in the determination of the relative magnitude of β and γ . In the Hong Kong study context, we have the value of time data (Hong Kong Transport Department, 2014) but do not have precise data of schedule delay early and late. Values of parameters for early and late arrival are chosen with due considerations of the relative β and γ values in the above early studies and Hong Kong value of time.
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136 Table 5 Three scenarios with realistic fares. Scenarios
Rail line
p0 (HKD)
λ∗
φ (%)
ϕ (%)
AEC (HKD)
TTC (× 105 HKD)
(1)
Optimum O I W
8.7 7.5 11.0 23.0
0.50 0.32 or 0.68 0.50 0.50
26.0 22.5 26.0 26.0
20.7 18.4 19.6 15.4
8.7 7.5 8.7 8.7
6.510 5.625 6.510 6.510
(2)
Optimum O I W
5.2 7.5 11.0 23.0
0.50 0.50 0.50 0.50
26.8 26.8 26.8 26.8
21.1 19.3 17.1 12.3
5.2 5.2 5.2 5.2
2.344 2.344 2.344 2.344
(3)
Optimum O I W
13.0 7.5 11.0 23.0
0.50 0.17 or 0.83 0.30 or 0.70 0.50
26.0 15.0 22.0 26.0
20.7 13.0 18.0 17.8
13.0 7.5 11.0 13.0
9.766 5.625 8.250 9.766
in Figs. 5 and 6. The actual fare for Lines I & W are both higher than p∗0 = 8.7, and thus only one optimal reward ratio is obtained. These results correspond to the curves at p0 > p∗0 = θ Mh/4 in Figs. 5 and 6. The results in Table 5 also clearly show that the system performances for the actual fares are different under the three scenarios with different values of demand N and shadow value of late arrival γ . When N decreases from 75,0 0 0 to 45,0 0 0 (M decreases from 25 to 15 correspondingly), rail line O changes status to the system optimum while rail lines I & W remain at the system optimum. Similarly, when late arrival is prohibited (γ → ∞), rail line I fails to reach the system optimum, resulting in a decrease in system efficiency from 26.0% to 22.0% in comparison with scenario (1).
5. Conclusions This paper proposes a novel fare-reward scheme (FRS) for managing peak-hour congestion of urban rail transit bottlenecks. It shifts commuters’ departure time to reduce commuter queuing at stations in an incentive-compatible manner while keeping the transit operator’s revenue intact. Commuter equilibrium choice of departure time is based on the trade-off between the schedule delay and queuing time and the eligibility for a free ride. A free ride during the shoulder periods is granted after a certain number of paid trips during the peak hours. We found that the performance of the FRS depends on the original fare. The best performance of the FRS is achieved at a reference original fare p∗0 = θ Mh/4, at which 50% of commuters travel free during the shoulder periods, giving rise to at least a 25.0% and 20.0% reduction in total time costs and average equilibrium trip costs, respectively. The FRS is more effective in system total time costs terms. Implementation of the proposed FRS is straightforward with the current smart card systems which are able to record all the commuter travel information. Usage rules of the free trip, like expiration dates and exclusions, can be properly set up to manage the distribution of free and non-free ride commuters and to avoid crowding during off-peak hours on particular days.5 Our preliminary work has opened potential avenues for further research. One avenue is to consider a transit line with multiple origin-destination pairs (Sumi et al., 1990; Tian et al., 2007). Another avenue is to explore optimal long-term urban rail transit services, such as headways, under the proposed FRS schemes. One may also take into account commuter heterogeneity in terms of shadow values of time (commute and early/late arrival).
Acknowledgements The authors wish to express their thanks to Associate Editor, Prof. Robin Lindsey and three anonymous reviewers, whose useful comments have improved the exposition of the paper. The authors are also thankful to Prof. Kenneth Small for his insightful comments in the 2016 ITEA Conference and Prof. Yasuo Asakura in Tokyo Institute of Technology for providing useful suggestions. This research was partly supported by a grant from the Hong Kong’s Research Grants Council (HKUST16205715).
5 The Netherlands are a good example where many of the travel products make the distinction between weekdays and weekends such as season tickets, Keuzedagen (chosen free travel days), etc. Commuters might tend to use the rewards on particular days of the week (e.g. Fridays or Mondays) because weekdays may differ in terms of the required workplace presence in a consistent way across the workforce, which may be driven by meeting or school schedules. As a consequence, it may be useful to adjust the usage limitations accordingly, for instance in such a way that rewards cannot be obtained (or only to a limited extent) on specific weekdays.
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Fig. 9. Optimal timetable with a uniform fare and with the FRS.
Appendix 1. Configuration of the optimal long-term transit service In this appendix, we prove that with the optimal timetable (derivation of Eq. (4)), the optimal capacity for long-run transit service is such that every service run is at full capacity. Then we prove the congruence between optimal timetables before and after the implementation of the FRS. 1.1. Full capacity utilization For long-term transit service, the total system costs include the commuters’ total time costs and transit authority’s costs. As illustrated in Eq. (5), the commuters’ total time costs are derived as
TTC0 = β (t ∗ − t1 )N =
βγ (M − 1 )hN β +γ
(18)
Different from Kraus and Yoshida (2002), our model does not consider the time for a roundtrip by a train set and does not distinguish between the number of service runs and the number of physical train sets. The transit authority’s costs are simplified to
TA = (v0 + v1 s )M + v2 s + v3
(19)
where v0 ,v1 ,v2 and v3 are all non-negative parameters. The first term (v0 + v1 s)M is the operating costs and fleet costs of service runs. The second term v2 s is the costs related to the train capacity, which is interpreted as capital costs of terminals since a terminal is proportional to the capacity of trains in Kraus and Yoshida (2002). The last term v3 is the fixed construction costs. The total system costs are obtained
TC =
βγ (M − 1 )hN + (v0 + v1 s )M + v2 s + v3 β +γ
(20)
Consider two capacity choices sa and sb , sa < sb , where Ma = N/sa is an integer but Mb = N/sb is not, and [Mb ] + =Ma , where [·] + is rounded up to the next integer. At s = sa , there are Ma service runs operated in the transit system. At s = sb , the number of service runs and commuters’ total time costs keep unchanged, but transit authority’s costs are higher as capacity increases. Therefore, it is clear to see that any level of capacity sb is dominated by sa . The optimal capacity should be set to achieve full capacity utilization for each service run. 1.2. Congruence between the optimal timetables before and after the FRS After the implementation of the FRS, free ride and non-free ride commuters travel separately with different equilibrium costs and (1 − λ)M = j − i + 1 (Design criterion 2). For free ride commuters, the total time costs are minimized when the schedule delay cost of the first and the last service run is the same as per Eq. (4). Therefore, the optimal timetable with the FRS is the same as the optimal timetable with a uniform fare. For non-free ride commuters, the total time costs are determined by the schedule delay cost of service run i or j as shown in Fig. 3(a). The optimal timetable with FRS for non-free ride commuters indicates that the schedule delay cost of service run i or j is the same, but it may differ from the optimal timetable with a uniform fare, because the departure time of service run i for the two timetables may be different. Namely, tiFRS for the optimal timetable with the FRS and tio for the optimal timetable with a uniform fare can be different. As shown in Fig. 9, hereinafter, ‘o’ refers to the optimal timetable with a uniform fare case and ‘FRS’ refers to the optimal timetable with the FRS case. We prove that when the headway is sufficiently short and the number of service runs M is sufficiently large, the optimal timetable with the FRS for non-free ride commuters approaches the optimal timetable with a uniform fare. We first derive the departure time of service run i, tiFRS , for the optimal timetable with FRS when the schedule delay cost of the service run i or j is the same, namely,
γ ( j − i )h β t ∗ − tiFRS = γ t FRS − t ∗ → tiFRS = t ∗ − j (β + γ )
(21)
Second, we follow the optimal timetable with a uniform fare (derivation of Eq. (4)) and therefore the total time costs of non-free ride commuters are either determined by the schedule delay cost of service run i or service run j. Without loss of
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136
Fig. 10. Fare-reward scheme (a) without mixed service run and (b) with mixed service run.
generality, suppose the total time costs of non-free ride commuters are minimized and determined by the schedule delay cost of service run i, the departure time tio satisfies the following conditions:
γ ( j − i )h γ ( j − i + 1)h γ t oj − t ∗ ≤ β t ∗ − tio ≤ γ t oj+1 − t ∗ → ≤ t ∗ − tio ≤ β +γ β +γ
(22)
The left-hand inequality ensures that the total time costs are determined by the service run i, the right-hand inequality ensures that the total time costs are the minimum value. The difference of tiFRS and tio works out
0 ≤ tiFRS − tio ≤
γh (β + γ )
(23)
Moreover, for a given λ, as above-mentioned, TTCoFFI = TTCFRS FFI and (1 − λ)M = j − i + 1, the relative difference of the total time costs with the two timetables is
0≤
FRS TTCoFFI +TTCoUFI − TTCFRS FFI + TTCUFI
FRS TTCFRS FFI + TTCUFI
≤
1
λ(M − 1 ) + (1 − λ )[(1 − λ )M − 1]
(24)
Therefore, when the number of service runs is sufficiently large, the total time costs with the two timetables are approximately the same. The optimal timetable with FRS approaches the optimal timetable with a uniform fare. This is true in the peak period as a mass transit railway system uses short headway with a large number of service runs to accommodate the high demand. For instance, the headway of the Island Line in Hong Kong is as short as 112 seconds in the morning peak hours (MTR Corporation Limited, 2015). Appendix 2. Justification of design criterion 2 Justification: Under the FRS design criterion 1 and with reference to Fig. 3, all free ride commuters will voluntarily choose their departure times within the shoulder intervals. Consider two reward ratio strategies λa and λb , λa > λb , where λa M is an integer but λb M is not, and [λb M] + =λa M, where [·] + is rounded up to the next integer. We begin the justification by clarifying the commuter departure time choice first. At λ = λa , free ride commuters choose to travel in FFIs containing λa M service runs over intervals [t1 ,ti − 1 ] and [tj + 1 , tM ]which can exactly satisfy the demand λa Ms. Non-free ride commuters travel separately in UFI which contains (1 − λa )M service runs over interval [ti ,tj ] respectively, as shown in Fig. 10(a). At λ = λb , to accommodate the number of free ride commuters λb Ms, FFIs should still contain [λb M] + =λa M service runs which carry more than the demand of free ride commuters. Therefore, at least one mixed train with both free and non-free commuters exists under the FRS. Without loss of generality, let the (i-1)th service run (train) be the mixed train, as shown in Fig. 10(b). The two reward ratios generate two different patterns of commuter departure time choices, which result in different system total time costs. At λ = λa , since λa M is an integer, free ride and non-free ride commuters travel separately. As discussed in Section 3.2, the non-free riders choose the (j − i + 1) service runs during the central period UFI, and their individual equilibrium time cost (ETC) is
ETCUFI = θ ( j − i )h = θ (M − λa M − 1 )h λ=λa
(25)
H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136
135
Table 6 Summary of the optimal solution without revenue-neutrality assumption.
(a)
p
Feasible λ
λ∗
0 < p < θ Mh 2
( 0, ( 0,
p θ hM 1 2
p ≥ θ Mh 2
(b)
p θ hM ] p θ hM ]
TTC at λ = λ∗ 2 [ θphM − p + h 34 M − 1
θ (
AEC at λ = λ∗
Average fare cost
θ h(M − 1 )]N )N
p− p 2
θ h(M − 1) θ h( 34 M − 1) +
p2 θ hM
p 2
Table 7 Fare-reward scheme performance without revenue neutrality assumption. p0
λ∗
(a)
0 < p < θ Mh 2
(b)
p ≥ θ Mh 2
p θ hM 1 2
φ
TTC (p − θ hMN
p2 θ hM
)N
4
1 θ h(M−1 )
(p −
p2 θ hM
M 4(M−1 )
)
AEC
ϕ
p0
θ h(M−1 )+ p0 +4 p0 −2 p 1 · θθMh 4 h (M−1 )+ p0
θ hM + p − p 0 4 2
p0
where θ = βγ /(β + γ ). For the free ride commuters in FFIs, the individual equilibrium time cost always equals the time cost of commuters taking the first train, which is
ETCFFI = θ (M − 1 )h
(26)
λ=λa
The system total time costs work out as
TTC(λa ) = θ (M − 1 )hλa N + θ (M − λa M − 1 )h(1 − λa )N = θ
λ2a − λa + 1 M − 1 hN
(27)
At λ = λb , the individual equilibrium time cost of free ride commuters can be seen to be unchanged
ETCFFI = θ (M − 1 )h
(28)
λ=λb
In the mixed train, both free ride commuters and non-free ride commuters incur the same equilibrium time cost, and the non-free ride commuters incur the equilibrium time cost shown in Eq. (28), where
ETCUFI = ETCmixed = ETCmixed = ETCFFI = θ (M − 1 )h UFI FFI λ=λb
λ=λb
λ=λb
λ=λb
(29)
Therefore, the system total time costs are
TTC(λb ) = θ (M − 1 )h · N
(30)
which is always greater than TTC(λa ) in Eq. (27), because (λ2a − λa + 1 ) < 1 given λa ∈ (0, 1). The above analysis shows that, with a reward ratio λb and mixed trains in UFI, the system total time costs are higher and all non-free commuters incur a higher time cost than the case with λ = λa . Put differently, any reward ratio strategy λb with mixed trains is strictly dominated by the strategy λa with non-mixed trains. Therefore, such inefficient reward ratios λb are not considered in the FRS design. Appendix 3. Effects of relaxing revenue-neutrality assumption Suppose revenue-neutrality in Assumption 2 is relaxed, the minimization problem (10)-(12) becomes
min TTC(λ ) = θ (M − 1 )h · λN + θ (M − λM − 1 )h · (1 − λ )N
0<λ<1
(31)
subject to
θ [(1 − λ )M − 1]h + p ≥ θ (M − 1 )h
(32)
where p is the uniform fare under the fare-reward scheme, called the FRS fare. Constraint (32) indicates λ ≤ p/θ hM. The analytical optimal solution of reward ratio λ is summarized in Table 6, where the valid range and the corresponding optimal value of λ, together with the FRS performance, depend on the FRS fare p. Following the aforementioned system measures in Section 4.1, the system performance is listed in Table 7. The optimal performance is reached at a fare p = θ Mh/2, which is termed as the reference FRS fare hereafter. Since the ex-ante uniform fare p0 and FRS fare p are independent of each other without the revenue-neutrality assumption, the revenue of the transit operator may change under the optimal FRS, depending on the ex-ante uniform fare p0 , as summarized in Table 8. Under the optimal FRS, the reference FRS fare p∗ = θ Mh/2 is a constant, independent of the ex-ante uniform fare p0 . In case (i), when p0 < θ Mh/4, the optimal FRS results in a revenue gain to the transit operator; in case (ii) with p0 = θ Mh/4 and p∗ = θ Mh/2, the optimal FRS conforms to the optimal FRS under the revenue neutrality assumption; in case (iii) when p0 > θ Mh/4, the optimal FRS results in a revenue loss.
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136 Table 8 Summary of operator’s revenue changes. Reference FRS fare p = θ Mh 2 ∗
Ex-ante uniform fare p0 (i) (ii) (iii)
p0 < θ Mh 4 p0 = θ Mh 4
p0 > θ Mh 4
Revenue change
(
p∗ 2
Revenue outcome
− p0 )N
Revenue gain
( p2 − p0 )N
Revenue loss
0
Revenue neutrality ∗
Therefore, when the revenue-neutrality assumption is relaxed, the transit operator may gain or lose, depending on the level of the ex-ante uniform fare p0 . Gaining more revenue with the FRS means more monetary transfer from the commuters to the transit operator, thereby reducing its public acceptability. Losing revenue with the FRS may require governmental compensation to the transit operator in order to sustain the service. Therefore, a revenue neutral FRS is desirable. References Arnott, R., De Palma, A., Lindsey, R., 1993. A structural model of peak-period congestion: a traffic bottleneck with elastic demand. Am. Econ. Rev. 83, 161–179. Arnott, R., De Palma, A., Lindsey, R., 1990. Economics of a bottleneck. Journal of Urban Economics 27 (1), 111–130. Braid, R.M., 1989. Uniform versus peak-load pricing of a bottleneck with elastic demand. J. Urban Econ. 26 (3), 320–327. Currie, G., 2010. Quick and effective solution to rail overcrowding: free early bird ticket experience in Melbourne, Australia. Transp. Res. Rec. 2146, 35–42. de Palma, A., Kilani, M., Proost, S., 2015. Discomfort in mass transit and its implication for scheduling and pricing. Transp. Res. Part B 71, 1–18. de Palma, A., Lindsey, R., Monchambert, G., 2017. The economics of crowding in rail transit. J. Urban Econ. 101, 106–122. Guo, X., Yang, H., 2010. Pareto-improving congestion pricing and revenue refunding with multiple user classes. Transp. Res. Part B 44 (8–9), 972–982. Haywood, L., Koning, M., 2015. The distribution of crowding costs in public transport: new evidence from Paris. Transp. Res. Part A 77, 182–201. Hong Kong Transport Department, 2014. Travel Characteristics Survey 2011 Final Report. (Government publication). Hong Kong: Transport Department. Huang, H., Tian, Q., Yang, H., Gao, Z., 2004. Modeling the equilibrium bus riding behavior in morning rush hour. In: Proceedings of the Ninth Annual Conference of the Hong Kong Society of Transportation Studies. Hong Kong, pp. 434–442. Kraus, M., Yoshida, Y., 2002. The commuter’s time-of-use decision and optimal pricing and service in urban mass transit. J. Urban Econ. 51 (1), 170–195. Laih, C., 2004. Effects of the optimal step toll scheme on equilibrium commuter behaviour. Appl. Econ. 36 (1), 59–81. Laih, C., 1994. Queueing at a bottleneck with single- and multi-step tolls. Transp. Res. Part A 28 (3), 197–208. Lan, L.W., Lee, H., Wen, C., 2010. Effects of temporally differential fares on Taipei metro riders’ mode and time-of-day choices. Int. J. Transport Econ. 37 (1), 97–118. Legislative Council Panel on Transport, 2016. Capacity and Loading of Trains in the MTR Network. Legislative Council Panel on Transport, Hong Kong (No. CB(4)854/15-16(07)). Li, Z., Lam, W.H.K., Wong, S.C., 2017. Step tolling in an activity-based bottleneck model. Transp. Res. Part B 101, 306–334. Lindsey, R., van den Berg, V.A.C., Verhoef, E.T., 2012. Step tolling with bottleneck queuing congestion. J. Urban Econ. 72 (1), 46–59. MTR Corporation Limited. Annual report. Hong Kong. Retrieved from http://www.mtr.com.hk/en/corporate/investor/2015frpt.html. Nie, Y.M., Yin, Y., 2013. Managing rush hour travel choices with tradable credit scheme. Transp. Res. Part B 50, 1–19. Peer, S., Knockaert, J., Verhoef, E.T., 2016. Train commuters’ scheduling preferences: Evidence from a large-scale peak avoidance experiment. Transp. Res. Part B 83, 314–333. Perone, J.S., 2002. Advantages and Disadvantages of Fare-Free Transit Policy. National Center for Transit Research, Center for Urban Transportation Research, University of South Florida (No. NCTR-473-133). Small, K.A., Winston, C., Yan, J., 2005. Uncovering the distribution of motorists’ preferences for travel time and reliability. Econometrica 73 (4), 1367–1382. Sumi, T., Matsumoto, Y., Miyaki, Y., 1990. Departure time and route choice of commuters on mass transit systems. Transp. Res. Part B 24 (4), 247–262. Tian, Q., Huang, H., Yang, H., 2007. Equilibrium properties of the morning peak-period commuting in a many-to-one mass transit system. Transp. Res. Part B 41 (6), 616–631. Ubbels, B., Tseng, Y., Verhoef, E.T., 2005. Value of time, schedule delay and reliability-estimates based on choice behaviour of Dutch commuters facing congestion. ERSA Conference Papers (No. ersa05p202). Vickrey, W.S., 1969. Congestion theory and transport investment. Am. Econ. Rev. 59, 251–260. Wardman, M., Whelan, G., 2011. Twenty years of rail crowding valuation studies: evidence and lessons from British experience. Transport Rev. 31 (3), 379–398. Whelan, G.A., Crockett, J., 2009. An investigation of the willingness to pay to reduce rail overcrowding. International Choice Modelling Conference 2009. Xiao, F., Qian, Z., Zhang, H.M., 2013. Managing bottleneck congestion with tradable credits. Transp. Res. 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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb
Managing rail transit peak-hour congestion with a fare-reward scheme Hai Yang, Yili Tang∗ Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR China
a r t i c l e
i n f o
Article history: Received 23 January 2017 Revised 6 February 2018 Accepted 8 February 2018
Keywords: Rail transit bottleneck Queuing congestion Fare-reward scheme Departure time choice User equilibrium
a b s t r a c t This paper describes a new fare-reward scheme for managing a commuter’s departure time choice in a rail transit bottleneck, which aims to incentivize a shift in departure time to the shoulder periods of the peak hours to relieve queuing congestion at transit stations. A framework of the rail transit bottleneck is provided and the user equilibrium with a uniform-fare and the social optimum with service run-dependent fares are determined. A fare-reward scheme (FRS) is then introduced that rewards a commuter with one free trip during shoulder periods after a certain number of paid trips during the peak hours. For a given number of peak-hour commuters and ex-ante uniform fare, the FRS determines the free fare intervals and the reward ratio (the ratio of the free trips to the total number of trips, which is equivalent to the ratio of the number of rewarded commuters to the total number of commuters on each day during the peak hours). The new fare under the FRS is determined so that the transit operator’s revenue remains unchanged before and after introducing the FRS. Our study indicates that, depending on the original fare, FRS results in an optimal reward ratio up to 50% and yields a reduction of system total time costs and average equilibrium trip costs by at least 25% and 20%, respectively. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Since Vickrey’s bottleneck model (Vickrey, 1969), there has been a substantial stream of development of research in this area (Arnott et al., 1990; Laih, 2004; Lindsey et al., 2012), and various extensions and applications such as elastic demand and general queuing networks were made (Braid, 1989; Arnott et al., 1993; Yang and Meng, 1998). While research up to date has been overwhelmingly focused on the road traffic, a few studies were devoted to the peak-hour congestion in public transportation, where a bottleneck occurs when passenger demand exceeds transit service capacity. Existing studies on transit bottleneck and queuing congestion are generally concerned with transit capacity choice, scheduling and fare pricing. Unlike road traffic management, transit authorities usually adjust service frequency to accommodate variable passenger demand, which affects both service capacity and passenger waiting time (service quality). Notably, applying the bottleneck model in mass transit, Kraus and Yoshida (2002) considered optimal fare and service frequency to minimize long-term system costs. Various fare pricing schemes have been considered for transit services in practice such as time-based differential pricing that involves peak-fare charging, off-peak discounting and combinations. Pilot programs and temporal fare regulations are ∗
Corresponding author. E-mail address: [email protected] (Y. Tang).
https://doi.org/10.1016/j.trb.2018.02.005 0191-2615/© 2018 Elsevier Ltd. All rights reserved.
H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136
123
Fig. 1. The transit bottleneck in the peak period.
conducted worldwide. For instance, implementation of temporally differential fares in Taipei reveals that metro riders are indeed sensitive to fares, especially peak-hour surcharges (Lan et al., 2010). Metropolitan areas such as Washington DC and London introduced time-based fares. In the case of the metro rail service in Washington DC, a peak fare is combined with a distance-based fare. While a peak surcharge acts as a penalty for commuters during rush hours, rewarding offers an endogenous and incentive-compatible strategy for morning commute rationing. Note that, free fare is the largest discount for off-peak commuters. In the case of the ‘Early Bird’ scheme in Melbourne, commuters enjoy free travel to the central business district if they finish their trips before 7 a.m.; it was observed that 23% of the off-peak ridership was made by the commuters who shift from the rush-hour period (Currie, 2010). A similar peak avoidance experiment in The Netherlands was implemented by rewarding commuters for travelling off-peak to reduce peak-hour ridership (Peer et al., 2016). A major consideration of the above demand management is to offer monetary compensation or other incentives at the expense of the government or the transit operator. The fare-free adoption for off-peak hours in Mercer County, New Jersey led to a reported loss of around one-quarter of revenue (Perone, 2002) and as a result, the scheme was abolished. With limited sources of funding, governments or transit authorities would rather charge a high fare to passengers than lose revenue. In addition, the above fare-free strategy also results in excessive off-peak ridership with the majority coming from untargeted passengers who would otherwise travel by other modes such as walking or bicycling. To address the issues associated with the aforementioned demand management strategies for peak-hour transit operation; this paper introduces a controlled free fare reward scheme (FRS). Under the proposed FRS scheme, a commuter is rewarded with one free trip during pre-specified shoulder periods after taking a certain number of paid trips during the peak hours. Implementation of such a FRS is straightforward with the current smart card systems. The planner can determine the rewarding ratio and periods to minimize commuters’ trip costs and to control the number of free trip commuters, namely, a certain proportion of commuters will be induced to shift their departure times and thus passenger flow congestion at the transit bottleneck will be smoothed out over time. Furthermore, the FRS scheme creates a win-win situation for both commuters and the transit operator. Commuters enjoy reduced travel time without paying more on average. In the meantime, the operator’s revenue remains intact, which is in the same spirit of the ‘revenue-neutral’ traffic congestion management schemes in the literature, such as the Pareto-improving congestion pricing and revenue-refunding scheme (Guo and Yang, 2010) and the tradable credit scheme (Yang and Wang, 2011; Xiao et al., 2013; Nie and Yin, 2013). The paper is organized as follows. Section 2 introduces the framework of the urban rail transit bottleneck model with batch arrivals at stations and the problems of user equilibrium with a uniform fare and the social optimum with service rundependent fares. Section 3 develops the fare-reward scheme model within the framework of the transit bottleneck and seeks the optimal rewarding ratio and periods to minimize the total commuting costs. Section 4 assesses the system performance under the FRS in comparison with the original bottleneck situation. Sensitivity analysis of the results is conducted with respect to the initial system configuration and assumptions. Conclusions and further development are provided in Section 5. 2. The transit bottleneck model in the absence of FRS 2.1. Problem settings Table 1 summarizes the parameters, variables and abbreviations used in this paper to describe the transit bottleneck and fare-reward scheme. The symbols are also explained in the text. Consider a single origin and destination connected by an urban rail line where each train is running with capacity s and headway h. Suppose a bottleneck occurs in peak hours, as shown in Fig. 1. During the peak hours, a given total number of N commuters travel with M uniformly spaced service runs through the bottleneck. All trains passing through the bottleneck are indexed such that the service run 1 is the first train and M is the last train through the bottleneck. Departure time of each service run is denoted by tm , m = 1, 2, ..., M. The peak period lasts for L = (M − 1)h. Individual commuters are assumed to have an identical work starting time t∗ . The service run scheduled before or at ¯ , namely 0 ≤ t ∗ − tm¯ < h. Like the road bottleneck model and without loss of generality, work starting time is labeled as m commuters arrive at the transit station immediately upon their departing from home, and in-transit time is ignored so that commuter’s arrival time at work is the train’s departure time. In this case, a passenger who takes a service run labeled as m encounters a fare cost p(m), a queuing time q(m), a schedule delay e(m) if they arrive at work early or l(m) if they arrive late, all of which are functions of the service run m
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136 Table 1 Parameters, variables and abbreviations. h s N M L t∗ ¯ m p(m) q(m) e(m) l(m)
α β γ
p0 p∗0 p
λ λ∗ φ ϕ
FRS AEC TTC UFI FFI ETC
Headway of transit service Capacity of a train Total number of commuters Total number of service runs Length of the peak period Commuter’s work starting time Service run scheduled before or at work starting time Fare cost of taking service run m Queuing time of taking service run m Early arrival schedule delay of taking service run m Late arrival schedule delay of taking service run m Shadow value of travel time Shadow value of early arrival Shadow value of late arrival Uniform fare in the original transit bottleneck Reference original fare after fare-reward scheme Uniform fare after fare-reward scheme Reward ratio in fare-reward scheme Optimal reward ratio in fare-reward scheme System efficiency (relative reduction of system total time costs) Individual efficiency (relative reduction of average equilibrium trip costs) Fare-reward scheme Average equilibrium costs of individual commuters Total time costs in system Uniform fare interval Free fare interval Equilibrium time costs of individual commuters
in the bottleneck. With homogeneous passengers, the generalized trip costs, inclusive of time cost, fare cost given by (1), is the same for all passengers1
c ( m ) = α q ( m ) + β e ( m ) + γ l ( m ) + p( m )
(1)
The unit costs α , β , and γ are the shadow values of queuing time at the station, arriving early and arriving late respectively. As in the road bottleneck, it is assumed that β < α < γ (Arnott et al., 1990). Note that, unlike the road bottleneck model, commuters arrive at stations in batches and their schedule delay cost is a step function of discrete transit runs. 2.2. User equilibrium under a uniform fare With a fixed total number of commuters during the peak hours, a uniform fare in the urban rail transit has no effect on a commuter’s departure in relation to the fare-free user equilibrium. With long headways, commuters may follow the timetable and thus arrive at stations in batches. But for peak-hour urban transit operation with high frequency, commuters usually arrive at stations without considering the service schedule. In this case, the expected usual waiting time at the station (without overflow queuing) is approximately a half headway and is hence a constant for a given frequency. For these reasons and without loss of generality, we only focus on the overflow queuing time and ignore the usual unsaturated waiting time. In user equilibrium, commuters in each train have the same generalized costs based on the trade-off between schedule delay cost and queuing time cost. Commuters with the highest schedule delay cost do not encounter waiting time. Therefore, the equilibrium total time costs are either defined by the schedule delay cost of the first or the last service run, whichever is greater (hereinafter, ‘0’ refers to the original uniform fare case),
TTC0 = max {β (t ∗ − t1 )N, γ (tM − t ∗ )N}
(2)
The transit authority should set the optimal timetable to minimize the total time costs:
min {max {β (t ∗ − t1 )N, γ (tM − t ∗ )N}}
(3)
1 A few studies also take account of the in-vehicle-crowding costs in transit commuters’ costs (Huang et al., 2004; Tian et al., 2007; de Palma et al., 2015; de Palma et al., 2017), which is a linear function of the number of commuters taking the same train. The linear specification is also supported by empirical studies of crowding cost estimation (Whelan and Crockett, 2009; Wardman and Whelan, 2011; Haywood and Koning, 2015). In line with this, we assume the in-vehicle-crowding cost function, g(nm ), is a monotonically increasing linear function of the total number of passengers, nm , taking the same service run m . Realistic statistics show that the rail service runs are mostly fully occupied by commuters with 97.33% average loading of six main rail lines in Hong Kong (Legislative Council Panel on Transport, 2016). The very high in-vehicle congestion implies that the in-vehicle crowding cost is approximately the same for all commuters during peak period or g(nm ) = g(s) becomes a constant. Thus, we exclude the in-vehicle crowding cost in our model.
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Therefore, the schedule delay cost of the first and the last service run is the same with the optimal timetable:
β (t∗ − t1 ) = γ (tM − t∗ ). In equilibrium, commuters taking the first service run and the last service run do not incur queuing time and have identical trip costs consisting of only schedule delay cost and a uniform fare cost. Namely,
β (t ∗ − t1 ) + p0 = γ (tM − t ∗ ) + p0 Since tM
− t∗
TTC0 =
+ t∗
(4)
− t1 = (M − 1)h, the system total time costs (TTC) are given by
βγ (M − 1 )hN (β + γ )
(5)
The individual average equilibrium trip costs (AEC) are given by
AEC0 =
βγ (M − 1 )h + p0 (β + γ )
(6)
Following the first-in-first-out principle, commuters who take the intermediate mth service run encounter the following queuing time at the station
q (m ) =
β (m−1)h
α γ (M−m )h α
¯] m ∈ ( 1, m ¯ +1, M ) m ∈ [m
(7)
2.3. Social optimum with run-dependent step fares At the social optimum under the above optimal timetable, queuing time at the station is eliminated and commuters only encounter the schedule delay cost. This could be achieved by implementing the following run-dependent step fares:
⎧ ⎪ ⎨ p¯
p˜ + β (m − 1 )h p( m ) = ⎪ ⎩ p˜ + γ (M − m )h p¯
m < 1, ¯ ], m ∈ [1 , m ¯ + 1 , M ], m ∈ [m m>M
where p¯ and p˜ are the basic fares for service runs outside the peak period and within the peak period respectively. In a road bottleneck, a time-dependent fine toll is required to eliminate queue for social optimum, which is difficult to implement in practice. Similarly, the above run-dependent transit fare pricing scheme represents an ideal situation, which, without sufficiently accurate information, is also difficult to implement in practice. 3. The transit bottleneck model in the presence of FRS 3.1. Basic considerations and assumptions It is accepted that, instead of fully realizing a social optimum in a first-best environment, various second-best toll pricing strategies can be employed to relieve congestion, such as the coarse tolls (Arnott et al., 1990; Laih, 2004; Xiao et al., 2012) and step tolls (Laih, 1994; Laih, 2004; Li et al., 2017) for managing congestion in a road bottleneck. The fare-reward scheme proposed here is not intended to achieve the above-mentioned first-best result under a run-dependent transit fare, rather, it attempts to strike a balance among efficiency, simplicity, and fairness in an incentive-compatible manner. The following assumptions are first introduced: Assumption 1. With the first-in-first-out principle, the peak-hour operation of the rail transit system is under a long-term optimal configuration where all service runs are at full capacity. Assumption 1 implies a predetermined optimal and stable urban rail transit service for the peak-hour commute. Kraus and Yoshida (2002) considered the optimal long-run peak-hour transit services without late arrival and showed that all service runs are at full capacity at the user equilibrium and social optimum. Yoshida (2008) extended the model with late arrivals and explored the optimal transit service under the first-in-first-out principle and random-access principle. Furthermore, according to the report of the Mass Transit Railway in Hong Kong, peak-period service runs are approximately at full capacity where the average loading of the six main rail lines in 2015 was 97.33% (Legislative Council Panel on Transport, 2016). Since commuters generally follow the first-in-first-out discipline so as to respect queuing rules in rail transit and in particular during the peak hours, with full capacity utilization, the number of service runs, the train service capacity and the total number of commuters during the whole peak period also satisfy the following relation in our model: M = N/s. Justification of the full capacity utilization under the optimal long-term transit service is provided in Appendix 1. Assumption 2. The total number of commuters and the rail transit headway remain the same and the operator’s fare revenue stays unchanged before and after the implementation of the proposed FRS. Assumption 2 means that the fare-reward scheme is designed to influence commuters’ departure time choice only in reducing individual and system commuting costs, without sacrificing the operator’s benefits.
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Fig. 2. Urban transit bottleneck with original fare.
Fig. 3. Fare-reward scheme for (a) free ride commuters and (b) non-free ride commuters.
Fig. 4. Fare intervals under FRS with design criteria.
Consider an original uniform fare p0 throughout the whole period in an urban rail transit bottleneck, as shown in Fig. 2. Implementation of the FRS changes the original uniform fare structure according to the commuters’ free ride entitlement. The peak hours of interest are divided into two free fare intervals (FFI) and one uniform fare interval (UFI). UFI is the central period within the peak period spanning the time interval [ti ,tj ] that contains service runs from i to j and includes the work starting time t∗ , as shown in Fig. 3(a). FFI includes the two shoulder intervals before and after the UFI. The uniform fare changes from p0 to p through the whole peak period. After a certain number of paid rides, a commuter is entitled to a free ride only during the FFI; a commuter without such an entitlement can choose either the UFI or FFI at a uniform fare p, as shown in Fig. 3(b). To ensure that a commuter will prefer to take the free ride during FFI, the FRS must meet the following design criterion: Design criterion 1: The FRS is designed so that the equilibrium trip costs in FFI with a reward (a free ride) are less than or equal to those in UFI without a reward. This design requirement ensures that the FRS is effective in motivating commuters to shift their departure time when receiving a free-ride reward. From the perspective of the system total time costs savings, we also introduce the following second design criterion: Design criterion 2: In FRS, a reward ratio λ is chosen such that λM is an integer. Design criterion 2 states that the proposed FRS will segregate the free ride and non-free ride commuters into the service runs respectively during FFI and UFI, as shown in Fig. 4. Therefore, the free ride and non-free ride commuters also follow the batch departures and the optimal service run under FRS is at full capacity too. Equivalently, the whole peak period under the FRS is divided into three fare intervals: the two shoulder intervals used only by free ride commuters and the central interval used only by the non-free ride commuters. Justification of design criterion 2 is given in Appendix 2.
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Table 2 Summary of the optimal FRS. p0
Feasible λ
(a)
p0 ≤ θ Mh 4
( 0,
(b)
p0 > θ Mh 4
(0, 1)
Remark: ϑ =
1 2
− ϑ] ∪
λ∗ [ 12
+ ϑ , 1)
1 2 1 2
±ϑ
TTC at λ = λ∗
Average fare cost
AEC at λ = λ∗
[θ h(M − 1) − p0 ]N
p0
θ h(M − 1)
θ h( 34 M − 1)N
p0
θ h( 34 M − 1) + p0
1/4 − p0 /θ hM.
3.2. User equilibrium and the optimal FRS design Here we consider how to determine the start and end time of the UFI illustrated in Fig. 4. Under Assumption 2 of the ex-ante and ex-post fixed commute demand and transit headway, the choice of UFI is equivalent to the determination of the reward ratio λ of the FFI (ratio of the free rides to the total number of rides), which is also equal to the ratio of the number of free ride commuters to the total number of commuters during daily peak hours, supposing that the total number of commuters is large enough. Let the central UFI correspond to the time interval [ti , tj ] containing the service runs from i to j, as depicted in Fig. 4. From the equilibrium condition,2 β (t∗ − ti ) + p = γ (tj − t∗ ) + p, the queuing time for each service run of UFI is given by
q (m ) =
β (m−i)h
α γ ( j−m )h α
¯] m ∈ [i, m ¯ + 1, j ] m ∈ [m
Considering design criterion 2, we have
( 1 − λ )M = j − i + 1
(8)
The total time costs for UFI commuters at equilibrium are
TTCUFI =
βγ βγ [(1 − λ )M − 1]h · (1 − λ )N ( j − i )h · (1 − λ )N = (β + γ ) (β + γ )
(9)
where the second equality comes from Eq. (8). Similarly, the queuing time for each service run of the two FFIs is given by
q (m ) =
β (m−1)h
, m ∈ [1, i − 1] α γ (M−m )h , m ∈ [ j + 1, M ] α
The total time costs for FFI commuters at equilibrium are
TTCFFI =
βγ (M − 1 )h · λN (β + γ )
For the given M and p0 , the minimization problem of the system total time costs is given by
min TTC(λ ) = θ (M − 1 )h · λN + θ (M − λM − 1 )h · (1 − λ )N
0<λ<1
(10)
subject to
N p0 = (1 − λ )N p
(11)
θ [(1 − λ )M − 1]h + p ≥ θ (M − 1 )h
(12)
where θ =βγ /(β + γ ) is a constant. Constraint (11) is imposed to ensure the operator’s revenue is unchanged before and after the FRS; inequality (12) is a mathematical statement of the aforementioned design criterion of the FRS. Appendix 3 analyses the effects of relaxing the revenue neutrality assumption. It is interesting to note that the FRS design problem reduces to a single variable optimization problem in terms of λ under a commute equilibrium constraint. In what follows, for convenience in the mathematical analysis, the reward ratio λ is treated as a continuous variable for the optimization problem, λ ∈ (0, 1). One can analytically obtain the optimal solution of λ, as summarized in Table 2, where the valid range and the corresponding optimal value of λ, together with the FRS performance, depend on the ex-ante uniform fare p0 . Case (a) is when p0 ≤ θ Mh/4. In this case, constraint (12) is always binding at the optimum (commuter’s equilibrium trip costs in FFI and UFI are identical), and the feasible value of λ consists of two disjoint intervals belonging to (0, 1). For a given p0 , two solutions of λ exist in the system of simultaneous nonlinear Eqs. (11) and (12). This can be explained by the fact that the ex-ante constant revenue can be achieved with a larger (smaller) number of non-free ride commuters at a 2 Appendix 1 states that when the headway is sufficiently short and the number of service runs is sufficiently large, the optimal timetable with FRS approaches the optimal timetable with a uniform fare.
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136 Table 3 Fare-reward scheme performances on urban rail transit bottleneck. p0
λ∗
(a)
0 ≤ p0 ≤ θ Mh 4
1 2
(b)
p0 > θ Mh 4
1 2
±
1 4
−
p0 θ hM
TTC
φ
AEC
ϕ
Np0
p0 θ h(M−1 )
p0
θ h(M−1 )+ p0 θ Mh/4 θ h(M−1 )+ p0
θ MhN 4
θ Mh
M 4(M−1 )
4
p0
lower (higher) fare p under the FRS. Moreover, at the optimal solution λ∗ , the trip costs of all commuters during the peak hours are equal to the right-hand side constant of constraint (12), which are also the time costs for free ride commuters in the two FFIs. This is always true, regardless of the value of p0 , whenever p0 ≤ θ Mh/4. The system total time costs, TTC, always decrease with p0 . Case (b) is when p0 > θ Mh/4. In this case, constraint (12) is always nonbinding, and any λ ∈ (0, 1) is feasible. The optimal λ∗ is uniquely determined to be 1/2, implying ‘pay one get one free’. Differing from Case (a), the minimum system total time costs, TTC, is a constant, independent of the original fare p0 . The average equilibrium commute costs (averaged over free and non-free rides), however, increase with p0 . Finally, we point out that the average fare cost (for the same commuter over days or for all commuters on the same day) is always p0 simply because the total number of commuters and the total revenue remains unchanged before and after the FRS. 4. Assessment of system performance To assess the system performance under the proposed FRS, we first introduce a set of system performance measures, analyze the effects of the FRS, and conduct the sensitivity analysis of system performance with respect to the initial system settings. 4.1. System performance measure The maximum reduction in system total time costs (TTC) by the FRS is defined by
TTC = TTC0 − TTC(λ∗ )
(13)
where TTC0 is the user equilibrium TTC given in Eq. (5). The percentage reduction of the system TTC (or system efficiency) is given by
φ=
TTC
(14)
TTC0
The reduction in individual average equilibrium costs (AEC) achieved by the FRS at the optimal reward ratio is given by
AEC = AEC0 − AEC(λ∗ )
(15)
where AEC0 is the user equilibrium AEC provided in Eq. (6). Finally, the percentage change in the individual AEC (or individual efficiency) is given by
ϕ=
AEC
(16)
AEC0
The above four system performance measures serve different purposes. The absolute reduction of the system TTC and individual AEC indicate the magnitude of the cost savings. The percentage reduction of the TTC and AEC show the efficiency and public acceptability of the FRS scheme respectively. 4.2. System performance versus original fare p0 The above four measures and reward ratio λ together reflect the performance of the FRS. As shown in Table 3, in the extreme case when the original fare p0 = 0, the FRS becomes inapplicable and all performance indexes become zero. As shown in Table 3 and Figs. 5 and 6, from the perspective of both system performance and individual commute costs, the FRS performs the best at a reference original fare p∗0 = θ Mh/4, with an optimal reward ratio λ∗ = 1/2. The optimal reward ratio incentivizes 50% of commuters in traveling free in the shoulder periods. This can be explained intuitively that when λ is small, increasing λ will increase the number of free ride commuters to relieve congestion. However, when λ is large, the excessive number of free ride commuters induces congestion in the shoulder periods, which results in higher system total time costs. At the reference original fare p∗0 , the absolute and percentage reduction in both system total time costs (TTC) and individual average equilibrium trip costs (AEC) reach maximum:
TTC =
θ MhN 4
,φ =
M , AEC = 4M − 4
θ Mh 4
,ϕ =
M , at p∗0 = 5M − 4
θ Mh 4
(17)
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Fig. 5. The reduction in TTC and AEC curves with respect to original fare p0 .
Fig. 6. Reward ratio, system efficiency and individual efficiency curves with respect to p0 .
The absolute and percentage reductions in TTC increase linearly with p0 up to p∗0 = θ Mh/4 (case (a) in Table 3), reach the optimum and then remain at the optimum (case (b) in Table 3). On the other hand, the absolute and percentage reductions in AEC increase with p0 up to p∗0 = θ Mh/4 but in a linear and nonlinear manner respectively. After reaching the optima, the former one remains at the optimum but the latter one decreases nonlinearly. Finally, we mention in Fig. 5 that the absolute reduction in AEC is exactly equal to the original fare p0 in case (a). The corresponding absolute reduction in TTC is then Np0 , which is exactly equal to the operator’s revenue. This is because the commuters’ average fare cost is unchanged, due to the constant revenue constraint, and thus the total reduction in TTC is equal to the reduction in AEC multiplied by the total number of commuters N.
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Fig. 7. Changes of efficiencies with M at the reference original fare.
4.3. Sensitivity analysis of system performance at p∗0 The FRS analyses so far are carried out for a given supply (transit headway h and number of service runs M) and demand (number of commuters N and their shadow values of time β and γ ). Here we conduct sensitivity analysis on the obtained results with respect to these exogenous inputs. For convenience, we fix the original fare at p0 = p∗0 , which is termed as the reference original fare hereafter. Fig. 7 shown the changes in system efficiency φ and individual efficiency ϕ with the total number of service runs M, M ∈ [2, ∞). Both system and individual efficiencies decrease with M. This is because the FRS aims at reducing the queuing time costs at train stations, and the total schedule delay costs do not change because all train services are already operating at full capacity. A larger M means a higher commuter demand and a longer peak period for a given headway and train capacity. The schedule delay costs increase with M more than queuing time costs under the implementation of FRS. Thus, the relative reduction of the queuing time costs (efficiency) decreases. It is also observed that the system efficiency is always greater than the individual efficiency. In fact, one can readily see from Table 3 that this observation is true for any original fare p0 . This can be accounted for by the fact that individual efficiency is measured by the full trip costs, inclusive of time costs and fare cost, while the system efficiency is measured by the time costs only, and the fare cost (or operator’s revenue) is a constant independent of M. In an extreme case with two service runs, system and individual efficiencies reach maximum values of 50.0% and 33.3%. When the number of service runs is sufficiently large, they approach their minimum values of 25.0% and 20.0%, respectively. The positive minimum value of efficiency ensures the effectiveness of FRS in a busy rail transit line. The system efficiency of the FRS scheme can be compared with that of the tolling strategies in the traditional road bottleneck model. In Arnott et al. (1990), the coarse toll can save at least 25% of the total system time costs, and the efficiency of the proposed fare-reward scheme would also save approximately 25% when the demand is large enough. This is not surprising because the batch commuter departures and arrivals approximate to the continuous departure and arrival rates in the road bottleneck case when the demand is high and the headway is short. Further mentioning in passing, Xiao et al. (2012) found that, at most, a 25% saving of the system costs could be achieved by the coarse toll under the assumption of “tactical waiting”. The existence of “tactical waiting” behavior reduces the capability of the coarse toll in a road bottleneck. In contrast, in our fare-reward scheme, the free fare in a transit bottleneck is only for eligible commuters who completed a certain number of trips and hence commuter “tactical waiting” is not applicable.3 Now we look at the impacts of the two exogenous parameters β and γ on system performance. Here, we fix the shadow value of early arrival β , and look at the difference between the shadow value of late arrival and early arrival in terms of the ratio, γ /β . We point out that the absolute reductions in TTC and AEC are affected by γ /β , but the percentage changes are independent of γ /β at the reference original fare p∗0 , from Table 3 and Eq. (17). Without loss of generality, we only
3 The discontinuities in the optimal coarse toll studied in Arnott et al. (1990) also induce a shift in the departure period. In order for all commuters to have the same equilibrium trip cost, no departures occur for an interval before the toll is applied. When the toll is lifted, a mass of commuters immediately departs. By contrast, in our model commuters who are entitled to a free ride incur a different equilibrium trip cost than commuters with no entitlement. No mass departures or changes in the aggregate timing of departures are required to assure equilibrium for both groups.
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Fig. 8. Change of AEC with h, N and γ /β at the reference original fare. Table 4 Three metro rail lines in Hong Kong. MTR Rail line
Distance
Fare for entire line (HKD)
Ma On Shan Line (O) Island line (I) West Rail line (W)
11.4 km 16 km 35.4 km
7.5 11.0 23.0
conduct sensitivity analysis for the absolute value of reduction in average equilibrium costs ( AEC), given β = 50 HKD/h and capacity s = 30 0 0. It is easy to check that AEC is positively related to the demand and headway, as shown in Fig. 8. For a given combination (N, h) of demand and headway, AEC is increasing with γ /β over the range γ /β ∈ (1, ∞). This is due to the fact that the schedule delay cost and system total time costs increase with γ . Particularly, AEC reaches a minimum value β Mh/8 as γ /β → 1 (θ → β /2) and a maximum value β Mh/4 as γ /β → ∞ (θ → β ) (late arrival is prohibited). These results illustrate that the FRS is more effective as the late arrival penalty increases. 4.4. Examples of realistic fares in the Hong Kong mass transit railway This section investigates whether the reference original fare p∗0 determined above is realistic under real transit settings, and then determines the system performance of the FRS. We consider three typical metro rail lines in Hong Kong Mass Transit Railway (MTR) shown in Table 4. We consider the following three different but realistic operational scenarios for the above three metro lines: (1) h = 2.5 min, N = 750 0 0, s = 30 0 0, M = 25, β = 50 HKD/h, γ = 100 HKD/h (2) h = 2.5 min, N = 450 0 0, s = 30 0 0, M = 15, β = 50 HKD/h, γ = 100 HKD/h (3) h = 2.5 min, N = 750 0 0, s = 30 0 0, M = 25, β = 50 HKD/h, γ = ∞ HKD/h The headways and passenger demands are assumed with reference to the Annual Report of Hong Kong MTR Cooperation (MTR Corporation Limited, 2015). The shadow values of early and late arrival are chosen with reference to Small et al. (2005) and Ubbels et al. (2005).4 Scenario 2 represents the case with a reduced demand and scenario 3 represents a case with prohibited late arrival. The results for the above scenarios are presented in Table 5. The system performance indices under the implementation of the FRS at the reference original fare are consistent with the analytical results obtained so far. For example, under scenario (1), the reference original fare is p∗0 = 8.7, and the actual fare for Line O is less than p∗0 = 8.7, thus two optimal reward ratios appeared, resulting in sub-optimal system performance. These results correspond to the curves at p0 < p∗0 = θ Mh/4 4 Early studies (Small et al., 2005; Ubbels et al., 2005) are used for reference in the determination of the relative magnitude of β and γ . In the Hong Kong study context, we have the value of time data (Hong Kong Transport Department, 2014) but do not have precise data of schedule delay early and late. Values of parameters for early and late arrival are chosen with due considerations of the relative β and γ values in the above early studies and Hong Kong value of time.
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136 Table 5 Three scenarios with realistic fares. Scenarios
Rail line
p0 (HKD)
λ∗
φ (%)
ϕ (%)
AEC (HKD)
TTC (× 105 HKD)
(1)
Optimum O I W
8.7 7.5 11.0 23.0
0.50 0.32 or 0.68 0.50 0.50
26.0 22.5 26.0 26.0
20.7 18.4 19.6 15.4
8.7 7.5 8.7 8.7
6.510 5.625 6.510 6.510
(2)
Optimum O I W
5.2 7.5 11.0 23.0
0.50 0.50 0.50 0.50
26.8 26.8 26.8 26.8
21.1 19.3 17.1 12.3
5.2 5.2 5.2 5.2
2.344 2.344 2.344 2.344
(3)
Optimum O I W
13.0 7.5 11.0 23.0
0.50 0.17 or 0.83 0.30 or 0.70 0.50
26.0 15.0 22.0 26.0
20.7 13.0 18.0 17.8
13.0 7.5 11.0 13.0
9.766 5.625 8.250 9.766
in Figs. 5 and 6. The actual fare for Lines I & W are both higher than p∗0 = 8.7, and thus only one optimal reward ratio is obtained. These results correspond to the curves at p0 > p∗0 = θ Mh/4 in Figs. 5 and 6. The results in Table 5 also clearly show that the system performances for the actual fares are different under the three scenarios with different values of demand N and shadow value of late arrival γ . When N decreases from 75,0 0 0 to 45,0 0 0 (M decreases from 25 to 15 correspondingly), rail line O changes status to the system optimum while rail lines I & W remain at the system optimum. Similarly, when late arrival is prohibited (γ → ∞), rail line I fails to reach the system optimum, resulting in a decrease in system efficiency from 26.0% to 22.0% in comparison with scenario (1).
5. Conclusions This paper proposes a novel fare-reward scheme (FRS) for managing peak-hour congestion of urban rail transit bottlenecks. It shifts commuters’ departure time to reduce commuter queuing at stations in an incentive-compatible manner while keeping the transit operator’s revenue intact. Commuter equilibrium choice of departure time is based on the trade-off between the schedule delay and queuing time and the eligibility for a free ride. A free ride during the shoulder periods is granted after a certain number of paid trips during the peak hours. We found that the performance of the FRS depends on the original fare. The best performance of the FRS is achieved at a reference original fare p∗0 = θ Mh/4, at which 50% of commuters travel free during the shoulder periods, giving rise to at least a 25.0% and 20.0% reduction in total time costs and average equilibrium trip costs, respectively. The FRS is more effective in system total time costs terms. Implementation of the proposed FRS is straightforward with the current smart card systems which are able to record all the commuter travel information. Usage rules of the free trip, like expiration dates and exclusions, can be properly set up to manage the distribution of free and non-free ride commuters and to avoid crowding during off-peak hours on particular days.5 Our preliminary work has opened potential avenues for further research. One avenue is to consider a transit line with multiple origin-destination pairs (Sumi et al., 1990; Tian et al., 2007). Another avenue is to explore optimal long-term urban rail transit services, such as headways, under the proposed FRS schemes. One may also take into account commuter heterogeneity in terms of shadow values of time (commute and early/late arrival).
Acknowledgements The authors wish to express their thanks to Associate Editor, Prof. Robin Lindsey and three anonymous reviewers, whose useful comments have improved the exposition of the paper. The authors are also thankful to Prof. Kenneth Small for his insightful comments in the 2016 ITEA Conference and Prof. Yasuo Asakura in Tokyo Institute of Technology for providing useful suggestions. This research was partly supported by a grant from the Hong Kong’s Research Grants Council (HKUST16205715).
5 The Netherlands are a good example where many of the travel products make the distinction between weekdays and weekends such as season tickets, Keuzedagen (chosen free travel days), etc. Commuters might tend to use the rewards on particular days of the week (e.g. Fridays or Mondays) because weekdays may differ in terms of the required workplace presence in a consistent way across the workforce, which may be driven by meeting or school schedules. As a consequence, it may be useful to adjust the usage limitations accordingly, for instance in such a way that rewards cannot be obtained (or only to a limited extent) on specific weekdays.
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Fig. 9. Optimal timetable with a uniform fare and with the FRS.
Appendix 1. Configuration of the optimal long-term transit service In this appendix, we prove that with the optimal timetable (derivation of Eq. (4)), the optimal capacity for long-run transit service is such that every service run is at full capacity. Then we prove the congruence between optimal timetables before and after the implementation of the FRS. 1.1. Full capacity utilization For long-term transit service, the total system costs include the commuters’ total time costs and transit authority’s costs. As illustrated in Eq. (5), the commuters’ total time costs are derived as
TTC0 = β (t ∗ − t1 )N =
βγ (M − 1 )hN β +γ
(18)
Different from Kraus and Yoshida (2002), our model does not consider the time for a roundtrip by a train set and does not distinguish between the number of service runs and the number of physical train sets. The transit authority’s costs are simplified to
TA = (v0 + v1 s )M + v2 s + v3
(19)
where v0 ,v1 ,v2 and v3 are all non-negative parameters. The first term (v0 + v1 s)M is the operating costs and fleet costs of service runs. The second term v2 s is the costs related to the train capacity, which is interpreted as capital costs of terminals since a terminal is proportional to the capacity of trains in Kraus and Yoshida (2002). The last term v3 is the fixed construction costs. The total system costs are obtained
TC =
βγ (M − 1 )hN + (v0 + v1 s )M + v2 s + v3 β +γ
(20)
Consider two capacity choices sa and sb , sa < sb , where Ma = N/sa is an integer but Mb = N/sb is not, and [Mb ] + =Ma , where [·] + is rounded up to the next integer. At s = sa , there are Ma service runs operated in the transit system. At s = sb , the number of service runs and commuters’ total time costs keep unchanged, but transit authority’s costs are higher as capacity increases. Therefore, it is clear to see that any level of capacity sb is dominated by sa . The optimal capacity should be set to achieve full capacity utilization for each service run. 1.2. Congruence between the optimal timetables before and after the FRS After the implementation of the FRS, free ride and non-free ride commuters travel separately with different equilibrium costs and (1 − λ)M = j − i + 1 (Design criterion 2). For free ride commuters, the total time costs are minimized when the schedule delay cost of the first and the last service run is the same as per Eq. (4). Therefore, the optimal timetable with the FRS is the same as the optimal timetable with a uniform fare. For non-free ride commuters, the total time costs are determined by the schedule delay cost of service run i or j as shown in Fig. 3(a). The optimal timetable with FRS for non-free ride commuters indicates that the schedule delay cost of service run i or j is the same, but it may differ from the optimal timetable with a uniform fare, because the departure time of service run i for the two timetables may be different. Namely, tiFRS for the optimal timetable with the FRS and tio for the optimal timetable with a uniform fare can be different. As shown in Fig. 9, hereinafter, ‘o’ refers to the optimal timetable with a uniform fare case and ‘FRS’ refers to the optimal timetable with the FRS case. We prove that when the headway is sufficiently short and the number of service runs M is sufficiently large, the optimal timetable with the FRS for non-free ride commuters approaches the optimal timetable with a uniform fare. We first derive the departure time of service run i, tiFRS , for the optimal timetable with FRS when the schedule delay cost of the service run i or j is the same, namely,
γ ( j − i )h β t ∗ − tiFRS = γ t FRS − t ∗ → tiFRS = t ∗ − j (β + γ )
(21)
Second, we follow the optimal timetable with a uniform fare (derivation of Eq. (4)) and therefore the total time costs of non-free ride commuters are either determined by the schedule delay cost of service run i or service run j. Without loss of
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Fig. 10. Fare-reward scheme (a) without mixed service run and (b) with mixed service run.
generality, suppose the total time costs of non-free ride commuters are minimized and determined by the schedule delay cost of service run i, the departure time tio satisfies the following conditions:
γ ( j − i )h γ ( j − i + 1)h γ t oj − t ∗ ≤ β t ∗ − tio ≤ γ t oj+1 − t ∗ → ≤ t ∗ − tio ≤ β +γ β +γ
(22)
The left-hand inequality ensures that the total time costs are determined by the service run i, the right-hand inequality ensures that the total time costs are the minimum value. The difference of tiFRS and tio works out
0 ≤ tiFRS − tio ≤
γh (β + γ )
(23)
Moreover, for a given λ, as above-mentioned, TTCoFFI = TTCFRS FFI and (1 − λ)M = j − i + 1, the relative difference of the total time costs with the two timetables is
0≤
FRS TTCoFFI +TTCoUFI − TTCFRS FFI + TTCUFI
FRS TTCFRS FFI + TTCUFI
≤
1
λ(M − 1 ) + (1 − λ )[(1 − λ )M − 1]
(24)
Therefore, when the number of service runs is sufficiently large, the total time costs with the two timetables are approximately the same. The optimal timetable with FRS approaches the optimal timetable with a uniform fare. This is true in the peak period as a mass transit railway system uses short headway with a large number of service runs to accommodate the high demand. For instance, the headway of the Island Line in Hong Kong is as short as 112 seconds in the morning peak hours (MTR Corporation Limited, 2015). Appendix 2. Justification of design criterion 2 Justification: Under the FRS design criterion 1 and with reference to Fig. 3, all free ride commuters will voluntarily choose their departure times within the shoulder intervals. Consider two reward ratio strategies λa and λb , λa > λb , where λa M is an integer but λb M is not, and [λb M] + =λa M, where [·] + is rounded up to the next integer. We begin the justification by clarifying the commuter departure time choice first. At λ = λa , free ride commuters choose to travel in FFIs containing λa M service runs over intervals [t1 ,ti − 1 ] and [tj + 1 , tM ]which can exactly satisfy the demand λa Ms. Non-free ride commuters travel separately in UFI which contains (1 − λa )M service runs over interval [ti ,tj ] respectively, as shown in Fig. 10(a). At λ = λb , to accommodate the number of free ride commuters λb Ms, FFIs should still contain [λb M] + =λa M service runs which carry more than the demand of free ride commuters. Therefore, at least one mixed train with both free and non-free commuters exists under the FRS. Without loss of generality, let the (i-1)th service run (train) be the mixed train, as shown in Fig. 10(b). The two reward ratios generate two different patterns of commuter departure time choices, which result in different system total time costs. At λ = λa , since λa M is an integer, free ride and non-free ride commuters travel separately. As discussed in Section 3.2, the non-free riders choose the (j − i + 1) service runs during the central period UFI, and their individual equilibrium time cost (ETC) is
ETCUFI = θ ( j − i )h = θ (M − λa M − 1 )h λ=λa
(25)
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Table 6 Summary of the optimal solution without revenue-neutrality assumption.
(a)
p
Feasible λ
λ∗
0 < p < θ Mh 2
( 0, ( 0,
p θ hM 1 2
p ≥ θ Mh 2
(b)
p θ hM ] p θ hM ]
TTC at λ = λ∗ 2 [ θphM − p + h 34 M − 1
θ (
AEC at λ = λ∗
Average fare cost
θ h(M − 1 )]N )N
p− p 2
θ h(M − 1) θ h( 34 M − 1) +
p2 θ hM
p 2
Table 7 Fare-reward scheme performance without revenue neutrality assumption. p0
λ∗
(a)
0 < p < θ Mh 2
(b)
p ≥ θ Mh 2
p θ hM 1 2
φ
TTC (p − θ hMN
p2 θ hM
)N
4
1 θ h(M−1 )
(p −
p2 θ hM
M 4(M−1 )
)
AEC
ϕ
p0
θ h(M−1 )+ p0 +4 p0 −2 p 1 · θθMh 4 h (M−1 )+ p0
θ hM + p − p 0 4 2
p0
where θ = βγ /(β + γ ). For the free ride commuters in FFIs, the individual equilibrium time cost always equals the time cost of commuters taking the first train, which is
ETCFFI = θ (M − 1 )h
(26)
λ=λa
The system total time costs work out as
TTC(λa ) = θ (M − 1 )hλa N + θ (M − λa M − 1 )h(1 − λa )N = θ
λ2a − λa + 1 M − 1 hN
(27)
At λ = λb , the individual equilibrium time cost of free ride commuters can be seen to be unchanged
ETCFFI = θ (M − 1 )h
(28)
λ=λb
In the mixed train, both free ride commuters and non-free ride commuters incur the same equilibrium time cost, and the non-free ride commuters incur the equilibrium time cost shown in Eq. (28), where
ETCUFI = ETCmixed = ETCmixed = ETCFFI = θ (M − 1 )h UFI FFI λ=λb
λ=λb
λ=λb
λ=λb
(29)
Therefore, the system total time costs are
TTC(λb ) = θ (M − 1 )h · N
(30)
which is always greater than TTC(λa ) in Eq. (27), because (λ2a − λa + 1 ) < 1 given λa ∈ (0, 1). The above analysis shows that, with a reward ratio λb and mixed trains in UFI, the system total time costs are higher and all non-free commuters incur a higher time cost than the case with λ = λa . Put differently, any reward ratio strategy λb with mixed trains is strictly dominated by the strategy λa with non-mixed trains. Therefore, such inefficient reward ratios λb are not considered in the FRS design. Appendix 3. Effects of relaxing revenue-neutrality assumption Suppose revenue-neutrality in Assumption 2 is relaxed, the minimization problem (10)-(12) becomes
min TTC(λ ) = θ (M − 1 )h · λN + θ (M − λM − 1 )h · (1 − λ )N
0<λ<1
(31)
subject to
θ [(1 − λ )M − 1]h + p ≥ θ (M − 1 )h
(32)
where p is the uniform fare under the fare-reward scheme, called the FRS fare. Constraint (32) indicates λ ≤ p/θ hM. The analytical optimal solution of reward ratio λ is summarized in Table 6, where the valid range and the corresponding optimal value of λ, together with the FRS performance, depend on the FRS fare p. Following the aforementioned system measures in Section 4.1, the system performance is listed in Table 7. The optimal performance is reached at a fare p = θ Mh/2, which is termed as the reference FRS fare hereafter. Since the ex-ante uniform fare p0 and FRS fare p are independent of each other without the revenue-neutrality assumption, the revenue of the transit operator may change under the optimal FRS, depending on the ex-ante uniform fare p0 , as summarized in Table 8. Under the optimal FRS, the reference FRS fare p∗ = θ Mh/2 is a constant, independent of the ex-ante uniform fare p0 . In case (i), when p0 < θ Mh/4, the optimal FRS results in a revenue gain to the transit operator; in case (ii) with p0 = θ Mh/4 and p∗ = θ Mh/2, the optimal FRS conforms to the optimal FRS under the revenue neutrality assumption; in case (iii) when p0 > θ Mh/4, the optimal FRS results in a revenue loss.
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H. Yang, Y. Tang / Transportation Research Part B 110 (2018) 122–136 Table 8 Summary of operator’s revenue changes. Reference FRS fare p = θ Mh 2 ∗
Ex-ante uniform fare p0 (i) (ii) (iii)
p0 < θ Mh 4 p0 = θ Mh 4
p0 > θ Mh 4
Revenue change
(
p∗ 2
Revenue outcome
− p0 )N
Revenue gain
( p2 − p0 )N
Revenue loss
0
Revenue neutrality ∗
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