A Pareto-improving and revenue-neutral scheme to manage mass transit congestion with heterogeneous commuters

Available online at www.sciencedirect.com

ScienceDirect ScienceDirect Transportation Research Procedia 00 (2017) 000–000 Available online at www.sciencedirect.com

www.elsevier.com/locate/procedia

ScienceDirect

www.elsevier.com/locate/procedia

Transportation Research Procedia 00 (2017) 000–000

Transportation Research Procedia 38 (2019) 586–605 www.elsevier.com/locate/procedia

23rd International Symposium on Transportation and Traffic Theory, ISTTT 23, 24-26 July 2019, Lausanne, Switzerland 23rd International Symposium on Transportation and Traffic Theory, ISTTT 23, 24-26 July 2019, Lausanne, Switzerland

A Pareto-improving and revenue-neutral scheme to manage mass A Pareto-improving and revenue-neutral scheme to manage mass transit congestion with heterogeneous commuters transita, congestion with heterogeneous commuters b,c a b,c b,c

a a

Yili Tang *, Hai Yang , Bo Wang , Jianling Huang , Yunyun Bai a b,c b,c YiliUniversity Tangofa,*, HaiandYang , BoDepartment Wangb,c , Jianling HuangEngineering, , Yunyun BaiBay, The Hong Kong Science Technology, of Civil and Environmental Clear Water Hong Kong

b Beijing Transportation Information Center, ShouFa Building, Liuliqiao Nanli, Fengtai District, Beijing 100161, PR China c The Hong Kong University of Sciencefor andComprehensive Technology, Department of CivilMonitoring and Environmental Engineering, Clear Water Bay, Hong Kong Beijing Key Laboratory Traffic Operation and Service, Beijing 100161, PR China b Beijing Transportation Information Center, ShouFa Building, Liuliqiao Nanli, Fengtai District, Beijing 100161, PR China c Beijing Key Laboratory for Comprehensive Traffic Operation Monitoring and Service, Beijing 100161, PR China

Abstract Abstract A hybrid fare scheme (HFS) is proposed in this paper that combines a fare-reward scheme (H-FRS) and a non-rewarding uniform fare scheme (H-UFS) by considering the heterogeneity in transit commuters’ scheduling flexibility. It aims at reducing peak-hour A hybrid fare scheme proposed in alternative this paper that combines a fare-reward (H-FRS) a non-rewarding congestion in the urban(HFS) transitis system with options catering for various scheme commuters. In theand H-FRS, a commuteruniform will be fare scheme (H-UFS) by considering the heterogeneity in transit commuters’ scheduling flexibility. It aims at reducing peak-hour rewarded with a free ride during the periods preceding or following a given shoulder period after taking a certain number of paid congestion thecentral urban transit alternative options acatering for various In thebut H-FRS, a commuter will be rides duringinthe period system in peakwith hours. In the H-UFS, commuter needs tocommuters. pay a different uniform fare during peak rewarded with a free ride during the periods to preceding or of following a given shoulder period after taking a certain number of paid hours. Commuters will have the opportunity join either the sub-schemes according to their scheduling flexibility of departure rides choice. during The the central period in peak hours. Inthe thefree H-UFS, a commuter needs to pay a different but fares uniform faresub-schemes. during peak time hybrid fare scheme determines fare intervals, the rewarding ratio, and the new for the hours. Commuters will have the opportunity to join either of the sub-schemes according to their scheduling flexibility of Our results demonstrate that the proposed HFS is not only revenue-preserving but also Pareto-improving. Depending on thedeparture original time choice. The hybrid farehybrid scheme determines theachieve free farea reduction intervals, in thetotal rewarding ratio, the new sub-schemes. fare, an optimally designed fare scheme can time costs byand at least 25%fares with for thethe optimal free fare Our results demonstrate that the proposed HFS is not only revenue-preserving but also Pareto-improving. Depending on the original interval. fare, an optimally designed hybrid fare scheme can achieve a reduction in total time costs by at least 25% with the optimal free fare interval.

© 2017 Authors. Published by Elsevier B.V. © 2019 The The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 23rd International Symposium on Transportation and Peer-review responsibility of the scientific © 2017 The under Authors. Published by Elsevier B.V.committee of the 23rd International Symposium on Transportation Traffic Theory. and Traffic Theory. Peer-review under responsibility of the scientific committee of the 23rd International Symposium on Transportation Keywords: Departure time choice; Peak hours; Fare-reward scheme; Revenue-preserving; Pareto-improving and Traffic Theory. Keywords: Departure time choice; Peak hours; Fare-reward scheme; Revenue-preserving; Pareto-improving

* Corresponding author. Tel.: +852 23587175. E-mail address: [email protected] * Corresponding author. Tel.: +852 23587175. E-mail address: [email protected] 2352-1465 © 2017 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the scientific committee of the 23rd International Symposium on Transportation and Traffic 2352-1465 Theory. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 23rd International Symposium on Transportation and Traffic Theory.

2352-1465  2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 23rd International Symposium on Transportation and Traffic Theory. 10.1016/j.trpro.2019.05.031

2

1.

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

587

Introduction

Demand for public transportation is growing faster than transit capacity in many metropolises. Mass transit operators are thus facing the significant challenge of managing peak-hour demand and overcrowding. When passenger density is low and everyone can board the first arriving train, the expected usual waiting time at a station is approximately half the headway of the service. However, during peak hours, the high occupancy of trains with limited capacity results in on-platform queueing and passenger discomfort. Oldfield and Bly (1988) discussed the effects of bus occupancy and frequency on average waiting time if passengers failed to board the first bus. In Beijing’s subway system, 47% of daily trips are made in peak hours and some lines are overloaded and extremely crowded; the queueing time at the station platform is estimated to be up to 25 minutes. Crowd control measures are implemented at 96 entry stations, about 30% overall (China Intelligent Transportation System Association, 2018). The peak-hour congestion has negative effects on both transit operators and users which decreases the stability and reliability of transit services as well as user satisfaction. Policymakers are now giving increasing consideration to the peak-hour congestions and demand management strategies which take two main forms: increasing the service supply and reducing travel demand at peak times. Increasing the service supply is often done through adjusting the service frequency to accommodate variable passenger demand. These adjustments affect both service capacity and passenger waiting time (service quality). Notably, applying the bottleneck model (Vickrey, 1969; Arnott et al., 1990) to mass transit, Kraus and Yoshida (2002) determined the optimal fares and service frequency for minimizing long-term system costs. On the other hand, reducing peak demand is often done through fare differentials to spread demand, which may involve peak-fare charging, off-peak discounting and combinations of the above (for a recent review, see (Liu and Charles, 2013)). Halvorsen et al. (2016) investigated the Early Bird discount promotion on Hong Kong’s MTR system that prevented peak hour ridership from growing much. However, off-peak discounts or free fare strategies come at the expense of the government or transit operator. Due to their limited sources in funding, governments and transit operators usually prefer to charge passenger high fares than to lose revenue which can raise issues of equity and customer acceptability. Adding a surcharge to travel in peak periods is rarely seen as acceptable by users, particularly as those who work in lower-paid jobs (Faber Maunsell Ltd., 2007). With regard to the efficiency and public acceptability, it is therefore important to explore alternatives and more acceptable congestion management strategies. Such proposals have been explored for road traffic in theory and in practice. One direction is the tradable credit or permits schemes in the static equilibrium model as well as in the dynamic bottleneck model (Verhoef, 1997; Yang and Wang, 2011; Nie and Yin, 2013). Another idea focuses on the charges in combinations of the rewards, subsidies or refunding such as the theoretical analysis on the pricing and rationing (Daganzo, 1995), the exemptions from paying tolls (Daganzo and Garcia, 2000), pricing and revenuerefunding (Guo and Yang, 2010; Rouwendal et al., 2012), and the recent surcharges with rewards in ride-sourcing market (Yang et al., 2018). While combined strategies for demand management in road traffic is well researched, its investigations and applications for mass transit system are still emerging. One reason is that the behaviors of mass transit travelers are affected by service such as the fare costs and the timetables. Besides, the revenue and costs can be an important aspect to influence transit operator’s willingness to implement the congestion management strategies. Among the few existing studies on combined strategies in public transport system, Whelan and Johnson (2004) showed that a combined strategy of increased peak fares and reduced off-peak fares have a larger effect than just a single policy. Douglas et al. (2011) further simulated passenger assignment under different fares and found that the combination of discount and surcharge was the most effective one to reduce peak loads. Different from the anonymous fare strategies, Yang and Tang (2018) proposed an individual-based fare-reward scheme (FRS) with homogeneous commuters in which 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. The theoretical analysis demonstrated the effective reduction of the time costs depending on the original fares. Previous studies opened various avenues for further research on the combined strategies for demand management in public transport system. With the fast population and spatial growth in many mature cities, such demand management strategies become increasingly concerned and important. In this paper, we consider the continuous heterogeneity in commuters’ scheduling flexibility and propose an incentive-based hybrid fare scheme (HFS) with a revenue-neutral property. The HFS combines the fare-reward scheme

588

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

3

(H-FRS) with a non-rewarding uniform fare scheme (H-UFS). In the FRS originally proposed by Yang and Tang (2018), commuters need to occasionally change their scheduling decisions (departure times or arrival times) to make use of their free rides. This may not be acceptable by all users, because some users tend to have less flexibility in their departure times. The HFS provides a more flexible and acceptable design to various users having different scheduling flexibility. The scheduling flexibility is defined as an arrival time flexibility interval (or departure time flexibility interval by the first-in first-out principle). The value of time for traveling outside one’s arrival time flexibility interval (ATFI) is greater than traveling inside the ATFI. Hence the length of the ATFI represents a commuter’s scheduling flexibility as a commuter prefers to travel inside the ATFI because of the lower schedule delay penalty. Commuters will join either scheme according to the design of the H-FRS and the H-UFS, and their ATFIs. The HFS will differentiate and determine the fares for paid rides under the H-FRS and the fares for the H-UFS to ensure equity and flexibility while preserving the operator’s revenue. In addition, the fare differentials will be optimized to incentivize commuters with flexible scheduling decisions to join the H-FRS. Commuters who voluntarily continue to use the H-UFS may need to pay a marginally higher fare each time (to maintain the operator’s revenue) than before, but they are still better off with the hybrid scheme due to the overall improvement (reduced queuing time costs and thus reduced individual trip costs with peak-hour transit services). This paper is organized as follows. Section 2 describes the mathematical formulation of the heterogeneity of commuter’s ATFI and introduces the framework of the urban rail transit bottleneck with batch arrivals at stations and the problem of departure time equilibrium with heterogeneous commuters under a uniform fare. Section 3 develops the optimization of the HFS model within the framework of the transit bottleneck. Section 4 assesses the system performance and conduct sensitivity analysis of the HFS in comparison with the original bottleneck. Conclusions and recommendations for further development and implementations are provided in Section 5. 2.

The transit bottleneck model with scheduling flexibility

2.1. Commuter’s scheduling flexibility and the arrival time flexibility interval Following investigations of departure times (Vickrey, 1969; Small, 1982), various empirical studies have been conducted on scheduling decision (departure times or arrival times) changes and the flexibility of commuters. It is found in the behavioral survey in Seattle that commuter’s departure time changes were affected by the length of the travel time and work schedule flexibility, where 62.6% of commuters have at least some flexibility in departure times in home-to-work trips, within which 20.9% of them reported a lot of flexibility (Mannering, 1989; Mannering et al., 1994). Meanwhile, the survey of commuter behavior conducted in Dallas, US demonstrated that the average preferred time for early arrival is up to 15 minutes and more than 50% of commuters accept late arrival in excess of 5 minutes at the workplace (Jou and Mahmassani, 1996). All of these suggest the heterogeneity in scheduling flexibility for commuters to respond differently to the incentive strategies such as the fare-reward scheme. A number of studies have incorporated commuter’s heterogeneity and scheduling flexibility in the traditional bottleneck model. For instance, suppose commuters have the different unit cost of travel time but the same unit costs of schedule delay early and schedule delay late. Then in equilibrium, commuters arrive in different time intervals such that the group with highest value of time is the first to arrive early (Arnott et al., 1989; Lindsey, 2004; Van den Berg and Verhoef, 2011). Moreover, for commuters with a continuously different work starting time, their equilibrium arrival time interval includes either early arrivals or late arrivals (Small, 1982; Peer and Verhoef, 2013). In the equilibrium under the above heterogeneity, a commuter’s departure time choice and scheduling flexibility are based on the trade-off between the schedule delay cost and the queuing time cost. Empirical studies indicate that the trade-off includes not only the above two time costs but also a flexible time interval, where there is a jump in the values of time and the schedule delay cost becomes higher beyond the flexible time interval (Abkowitz, 1980; Small, 1982; Ubbels et al., 2005). Some researchers assumed that commuters have a preferred arrival time interval and consider the extreme case where the schedule delay cost is zero within the preferred interval (Xiao et al., 2014). In this paper, we assume a representative commuter i has a preferred work starting time ti* and an arrival time flexibility interval (ATFI) tie , til  , where the length of the interval is denoted by Ti= t li −tie . The unit costs of travel

4

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

589

time, schedule delay early and schedule delay late inside the ATFI are denoted by  ,  and  respectively while beyond the interval are denoted by ,  and  , which is uniform among commuters for mathematical tractability. Besides, empirical studies also indicate the similarity of transit commuters’ socio-economic factors such as incomes and career categories in some metropolitans (Lehmann, 2018; Beijing Subway, 2017; Beijing Municipal Bureau of Statistics, 2018). Furthermore, statistical analysis demonstrated that the relative difference in the unit costs within and beyond the time interval can be as high as 5.8 (Small, 1982; Ubbels et al., 2005). Hence, we suppose the value of  is a sufficiently large parameter in our paper. Interpretation can be that firms adopt step penalties. It is also assumed      as in traditional bottleneck model. Since the ATFI represents a commuter’s flexibility to arrive early and late, the trade-off between early and late arrivals is supposed to be balanced such that the schedule delay cost is the same at the two ends of the ATFI, namely

 ( t * − tie ) =  ( til − t * )

(1)

For simplicity, we assume all commuters have the same work starting time t * but differ in their ATFI length. As shown in Fig. 1(a), the ATFI can be represented as the interval length Ti for commuter i , such that   tie = t* − Ti , til = t* + Ti + +

(2)

Suppose the length T of ATFI is distributed continuously across a large number of commuters over the domain  0,T  , where T is the longest ATFI of commuters during the peak period. Note that T can be greater than the length of the peak period L . Let F (T ) be the cumulative distribution function or the fraction of commuters whose ATFI is shorter than or equal to T , as shown in Fig. 1(b). N



ATFI = T j

Cumulative number of commuters

SDC



ATFI = Ti





early

t

*

F (T )  N

Ti

late Peak period

Tj

Length of the AFTI (minutes)

(a)

T

(b)

Fig. 1. (a) Individual’s arrival time flexibility interval and (b) cumulative distribution of ATFI during the peak period.

2.2. The transit bottleneck model with arrival time flexibility interval We explore the equilibrium with heterogeneous commuters by considering a transit bottleneck where a single origin and destination is connected by a rail line. The physical capacity of each service run is s and the uniform headway is h , as shown in Fig. 2. The service rate of the transit bottleneck can be regarded as s h . Suppose there are M service runs operated during the peak period with a total number of N commuters, the peak period lasts for = L The departure time of each service run is denoted by tm , m = 1, 2,..., M .

h

t1

t2

...

...

h

t3

tm t * tm+1

...

tm

Fig. 2. The transit bottleneck in the peak period

tM

( M −1) h .

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

590

5

With a uniform fare in the transit bottleneck model, the equilibrium is the same as the fare-free user equilibrium. For peak-hour transit service with high frequency and the unsaturated condition that everyone can board the first arriving service run, the expected usual queuing time at the station is approximately a half headway which is a constant for a given frequency. Hence without loss of generality, we only focus on the overflow queuing time and the unsaturated queuing time is ignored. Moreover, we assume a commuter arrives at the rail station as soon as he or she leaves home and arrives at work upon departing from the rail station. The generalized costs for user i taking a train with departure time tm is the sum of the queuing time costs, schedule delay costs and a uniform fare cost p0 ,





  q ( tm ) + max  ( t * − tm ) ,  ( tm − t * ) + p0   e * e l l *   q ( tm ) + max  ( ti − tm ) +  ( t − ti )  ,  ( tm − ti ) +  ( ti − t ) + p0 





tm  tie , til  tm  tie , til 

(3)

The commuter’s schedule delay cost is a step function of discrete departure time of service runs. Since  is sufficiently large, arriving at a time outside the ATFI indicates a large schedule delay cost, which increases the transit system costs. Hence we first introduce the following assumption: 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 in which all service runs are at full capacity and the physical capacity of the service runs in the transit system satisfies T   0, T  , s  M (T )  F (T )  N , where M (T ) represents the total number of service runs within the interval T . Justification of assumption 1 is given in Appendix A. Assumption 1 is based on the sufficiently large value of schedule penalty of arriving outside the flexibility interval and the transit operator’s costs. The transit operator can adjust the service to accommodate the commuters and reduce the total system costs, for instance, expand the capacity of each service run. To some extent in practical situations, service adjustment has its limitations in the case that the operator could not incur the prohibitively huge cost when the infrastructure has to be expanded for the additive capacity, especially in heavily-congested transit lines. In such a situation, this assumption is identical to restrict the distributions of the ATFI based on the service run capacity and the headway. While the practical situations may vary with the infrastructures and operators, in this paper we suppose the operator is able to adjust the service and the distributions of the ATFI are exogenous. Assumption 1 demonstrates that under the long-term optimal transit service, all commuters will travel within their ATFI in equilibrium with the same unit costs which are  ,  and  , and work starting time t * . The equilibrium cost under assumption 1 is the same as if all commuters are identical but the departure time order is in general indeterminate. In the extreme case when ATFI is uniformly distributed over the peak hours, then each commuter will choose a certain service run. The equilibrium total time costs are determined by the schedule delay cost of the first service run or the last service run, whichever is greater





TTC0= max  ( t * − t1 ) ,  ( tM − t * ) N

(4)

The transit operator sets the optimal timetable (second best under the uniform fare) for minimizing the above total

 

 

time costs such that min max  ( t * − t1 ) ,  ( tM − t * ) N

(Yoshida, 2008; Yang and Tang, 2018). Hence the

commuters taking the first service run and the last service run incur the same schedule delay cost without the queuing time costs,

 ( t * − t1 ) =  ( tM − t * ) Since tM − t1 = AEC = 0

(5)

( M − 1) h , the individual average equilibrium costs (AEC) are given by  ( M − 1) h + p0 + 

The queuing time thus can be expressed as

(6)

6

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

  ( tm − t1 )  ( m − 1) h =     q ( tm ) =   t − t  M − m) h ( ) ( M m =    

3.

591

m  (1, m 

(7)

m   m + 1, M )

The hybrid fare scheme with a fare-reward scheme and a uniform fare scheme

3.1. Basic considerations and design criteria The hybrid fare scheme (HFS) consists of a fare-reward scheme (H-FRS) and a non-rewarding uniform fare scheme (H-UFS). Commuters have opportunity to join either scheme according to their arrival time flexibility interval (ATFI). In the H-FRS, a commuter will be rewarded with one free ride during a pre-specified shoulder peak interval after taking a certain number of paid trips at a new fare p f during the period central to peak hours. Implementation of the H-FRS changes the original uniform fare structure according to the commuters’ free ride entitlement. The peak period of interest is divided into two free fare intervals (FFI) and one uniform fare interval (UFI). UFI is the central period within the peak hours that spans an interval including the work starting time t * and FFI includes the two shoulder intervals before and after the UFI, as shown in Fig. 3(a). After a certain number of paid rides in UFI, a commuter is entitled to a free ride (free-ride commuters) only during the FFI; a commuter without such a free ride reward (nonfree ride commuters) pays a uniform fare p f during the whole peak period, as shown in Fig. 3(b). pf

FFI

t1

...

ti −1

ti

...

UFI

tm t * (a)

pf

...

tj

FFI ...

t j +1

tM

t1

t2

...

ti

...

tm t *

...

tj

...

tM −1

tM

(b)

Fig. 3. The fare-reward scheme for (a) free ride commuters and (b) non-free ride commuters

As mentioned above, the H-FRS requires commuters to occasionally change their scheduling decisions to make use of their rewards. However, commuters with a short arrival time flexibility interval (ATFI) may become worse off by joining the H-FRS if they have to travel outside their ATFI to make use of the free rides, which results in a high schedule delay cost. Therefore, they rather choose to use the H-UFS and may need to pay a marginally higher fare pu each time, but they are still no worse off due to the overall reduced individual trip costs (reduced queuing time cost) after the overall improvement to peak period transit services. For a given number of peak-period commuters, the ex-ante uniform fare and the uniform headway, the planner determines the free fare intervals, the reward ratio  , the new fare p f for paid trips in H-FRS and the new fare pu for H-UFS. The determination of the free fare intervals is represented by the FFI ratio  , which is the ratio of the number of service runs in FFI to the total number of service runs in the peak period. The reward ratio  is the ratio of the free trips to the total number of trips for H-FRS commuters, which is equivalent to the ratio of the number of freeride commuters to the total number of H-FRS commuters during the daily peak period, supposing the total number of commuters is large enough. To ensure the participation of both the transit operator and commuters in the proposed hybrid fare scheme, the hybrid fare scheme must meet the following design criteria: Design criterion 1: The total number of commuters, the rail transit headway and the operator’s fare revenue remain the same before and after the implementation of the hybrid fare scheme. Design criterion 1 means that the hybrid fare scheme is designed to influence a commuters’ departure time choice without sacrificing the operator’s benefits. For a commuter’s individual equilibrium trip costs, we also introduce the following design criteria: Design criterion 2: The hybrid fare scheme is designed so that both the fare-reward scheme and uniform fare scheme commuters are no worse off.

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

592

7

This design criterion ensures public acceptability that commuters will participate in the hybrid fare scheme by enjoying lower trip costs. Furthermore, the H-FRS under the HFS must meet the following design criteria: Design criterion 3: The H-FRS scheme 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. Design criterion 3 states that the H-FRS is effective in incentivizing commuters to shift their departure time by making use of the free-ride rewards. Additionally, from the perspective of the system total time costs, the H-FRS is designed with criterion 4, such that Design criterion 4: In H-FRS, the FFI ratio  is chosen such that M is an integer. Design criterion 4 states that free ride and non-free ride commuters can be segregated into the FFI and UFI respectively. The total queuing time costs will be reduced depending on the design of the hybrid fare scheme. Justification of design criterion 4 is provided in Appendix B. 3.2. The equilibrium arrival patterns with the hybrid fare scheme Under the design criteria, commuters’ equilibrium arrival patterns depend on the arrival time flexibility interval and the determination of the hybrid fare scheme, which includes FFI ratio  , reward ratio  and the new fares pu and p f . Let the two disjoint FFIs correspond to the time interval t1 , ti −1  and t j +1 , tM  , containing in total M service runs. The length of the time interval ( ti −1 , t j +1 ) between the two FFIs is t j +1 − ti −= 1

( (1 −  ) M + 1) h , as shown

in Fig. 3(a). The central UFI is the time interval ti , t j  , containing (1 −  ) M number of service runs from service run i to service run j . As mentioned above, the H-FRS requires commuters to occasionally change the departure time, thus we first introduce the following scheme choice: Theorem 1. Commuters whose ATFI is within the interval ( ti −1 , t j +1 ) , namely the length of ATFI is shorter than

( (1 −  ) M + 1) h , will choose the H-UFS. This is because these commuters will encounter a sufficiently high schedule delay cost as well as individual trip costs with the values of time , , and  if they travel in the FFI. Hence, they will choose to travel only in the UFI. Moreover, the fare of the paid trips of H-FRS in the central period will be set higher than the uniform fare of HUFS for individual rationality. Therefore, it is always better for them to use the H-UFS with a uniform fare pu . The total number of these commuters is the cumulative number of commuters whose ATFI is shorter than the time interval  N P T  ( (1 −  ) M + 1) h  N , which is a function of  for a given M , ( ti −1 , t j +1 ) , namely, F ( (1 −  ) M + 1) h =

(

)

(

)

h and N . For the convenience of mathematical illustration, hereinafter, we define =  F

( ( (1 −  ) M + 1) h )

(8)

  is given as a function of  and its value is determined once  is chosen. The number of service runs during the

time interval ( ti −1 , t j +1 ) is (1 −  ) M . As discussed in assumption 1, the total capacity and the number of commuters

in the time interval ( ti −1 , t j +1 ) satisfies s  (1 −  ) M    N , where s  M = N as proved in Appendix A under long-

term optimal train services. Hence the relation between   and the FFI ratio is such that   1 −  . From design criterion 2, the rest of the (1 −  ) N commuters are incentivized to choose the H-FRS and undertake their free-rides in the FFI. Therefore, in the hybrid fare scheme, the H-UFS commuters and the H-FRS commuters for paid rides will travel together in the central period. We then demonstrate theorem 2 such that Theorem 2. For a given FFI ratio  , the optimal reward ratio is determined by

8

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

 = N ( N −  N )

593

(9)

Proof: The proof is given in Appendix C. Theorem 2 illustrates that the optimal reward ratio is set such that the supply in FFIs (UFI) can exactly accommodate the demand of the free-ride commuters (non-free ride commuters). The equilibrium arrival times (departure times) of commuters in FFI and UFI are segregated. This is because design criterion 4 decides that M is an integer hence FFIs and UFI are separated and free-ride commuters will voluntarily choose to travel in FFIs because of the less trip costs due to design criterion 3. Hence free-ride commuters travel only in FFIs and non-free ride commuters travel with the H-UFS commuters in UFI by paying p f and pu respectively, as shown in Fig. 4.

(p )

H-UFS commuters

f

Free ride commuters Non-free ride commuters ( pu ) Free ride commuters FFI FFI UFI ... ... ... ...

t1

ti −1

ti

tj

tm t *

t j +1

tM

Fig. 4. Equilibrium arrival pattern of H-UFS and H-FRS commuters

The above analysis show that an appropriate value of  is selected first according to design criterion 4 and then the corresponding values of design parameters   and   are determined. Since commuters in FFI and UFI travel separately, the equilibrium time costs in UFI and FFI are different. For free-ride commuters in FFI, the equilibrium time costs are determined by the schedule delay costs of the first or the  ( tM − t * ) . The system total time costs for FFI commuters at equilibrium are last service run, such that  ( t * − t1 ) =

TTCFFI =

 ( M − 1) h  N  ( + )

The queuing time for each service run of the two FFIs is given by   ( tm − t1 )  ( m − 1) h , m  1, i − 1 =    q ( tm ) =    ( tM − tm )  ( M − m ) h , m  j + 1, M =     

For H-UFS commuters and H-FRS commuters for paid rides in UFI, the uniform fare in UFI has no effect on equilibrium time costs in relation to the fare-free user equilibrium. From the equilibrium condition,  ( t * − ti ) =  ( t j − t * ) , the queuing time for each service run in UFI is given by

 ( tm − ti )  ( m − i ) h =     q ( tm ) =  t t  −  ( j m )  ( j − m) h =  

m  i, m  m   m + 1, j 

By design criterion 4 we have,

(1 − ) M =

j − i +1

(10)

Therefore, the system total time costs for commuters in the central period are

TTCUFI =

  ) ) N ( j − i ) h  (  + (1 −   )(1 −= ( +  )

 (1 −  ) M − 1 h  (1 −  ) N ( +  ) 

(11)

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

594

9

The average equilibrium trip costs of H-UFS commuters are = AECu

 ( (1 −  ) M − 1) h + pu + 

(12)

The average equilibrium trip costs of H-FRS commuters are

AEC f =

    ( M − 1) h    +  ( (1 −  ) M − 1) h + p f   (1 −   ) +   +   

(13)

3.3. The optimal hybrid fare scheme design Under the equilibrium arrival patterns, the hybrid fare scheme chooses the interval ratio  , the new fare p f for paid trips in FRS and the new fare pu for UFS to minimize the system total time costs. The minimization problem is given by min TTC =  ( M − 1) h  N +  (1 −  ) M − 1 h  (1 −  ) N

 , pu , p f

(14)

subject to

  Npu + (1 −   )(1 −  )  Np f = Np0

(15)

 (1 −  ) M − 1 h + pu   ( M − 1) h + p0

(16)

 (1 −  ) M − 1 h + pu   (1 −  ) M − 1 h + p f

(17)

(

)

(18)

 ( M − 1) h    +  (1 −  ) M − 1 h + p f  (1 −   )   ( M − 1) h + p0

(

)

(19)

 ( M − 1) h   (1 −  ) M − 1 h + p f

(20)

p0  0, p f  0, pu  0, 0    1

(21)

 ( M − 1) h    +  (1 −  ) M − 1 h + p f  (1 −   )   (1 −  ) M − 1 h + pu

where  =  ( +  ) . Constraint (15) is imposed to ensure the operator’s revenue is intact before and after the hybrid fare scheme; inequality (16) and (19) ensure that design criterion 2 is met for both H-UFS and H-FRS commuters respectively; while inequality (17) and (18) ensure that H-UFS and H-FRS commuters choose their respective subschemes according to ATFI and based on their generalized trip cost; and finally inequality (20) ensures design criterion 3 that the trip costs of free rides are no greater than those of non-free rides under the H-FRS. In the above minimization problem, constraint (19) is always satisfied because of the constraint (18) for H-FRS commuters and constraint (17) for H-UFS commuters. Moreover, under the revenue neutral constraint (15), p f can be mathematically represented by pu as p= f

( p0 −   pu ) (1 −   )(1 −   ) , and vice versa. Hereinafter we use

H-UFS fare pu to discuss the new fares and the performance of the hybrid fare scheme, thus the above constraints (15) - (21) can be simplified as

pu  pua

(22)

pu  pub

(23)

pu  puc

(24)

pu  pud

(25)

10

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

595

p0  0, pu  0, 0    1

(26)

pud  p0 − (1 −   )(1 −   )  Mh    . Inequality where pua = 2 Mh + p0 , pub = Mh + p0 ,= puc p0 (1 −  ) and = (22) comes from constraint (15) & (18) while inequalities (23), (24) and (25) come from constraints (16), (15) & (17) and (15) & (20), respectively. The above inequalities state that the value of the optimal new fares can be set within a certain range for a given value of the FFI ratio  . The optimal value of H-UFS fare pu (and thus p f ) is affected by the transit operational service M and h , the original fare p0 and the FFI ratio  . The same properties also hold for the new fare p f of H-FRS under the revenue neutrality constraint. One can analytically obtain the optimal solutions of the hybrid fare scheme. As shown in Table 1, the feasible range of FFI ratio  , the optimal value of  and the optimal range of new fare pu as well as p f depend on the ex-ante original fare. Fig. 5 shows the total time costs and the average equilibrium trip costs of H-FRS and H-UFS commuters before and after the hybrid fare scheme. Table 1. The optimal hybrid fare scheme performance (i) p0

Feasible 

*

(ii) Mh Mh  p0  4 2

(iii) Mh  p0 2

 1  1   0, −    + ,1  2  2 

( 0,1)

( 0,1)

1  2

12

12

Mh 0  p0  4

*

 *

 1 − *

1 2 − 20.5

1 2 − 20.5

TTC

 ( M − 1) h − p0  N

3    M − 1 h  N 4 

3    M − 1 h  N 4 

pf

p0 − *  pu

p0 − 0.5  pu

p0 − 0.5  pu

(1 −  )(1 −  )

(1 −  0.5 )(1 − 0.5 )

(1 −  0.5 )(1 − 0.5 )

a c p= p= p= pud u u u

pua  pu  puc

pua  pu  pub

*Mh

pua

AECu

 ( M − 1) h

AEC0.5

AEC f

 ( M − 1) h

AEC0.5

*

pu

Remarks:= 

*

puc 1    M − 1 h + 2 p0 2 

AECcf

pua

pub

AEC0.5

 ( M − 1) h + p0

AEC0.5

AECbf

 3 − 40.5  b 3   M − 1 h + p0 , 1 4 − p0 Mh , 0.5 =   M − 1 h + p0 , AEC f =  =0.5 , 0.5 = =0.5 , AEC0.5 = 4   4 − 40.5 

 3 − 20.5  (1 − 20.5 ) p0 AECcf =  M − 1 h + 4 − 4  1 − 0.5 0.5  

In case (i) when 0  p0  Mh 4 , the feasible region of the FFI ratio includes two disjoint intervals. Two solutions of the optimal FFI ratios exist in the system. At the optimal value of the FFI ratio, the new fare pu is uniquely * a c determined such that p= p= p= pud , which indicate the average equilibrium trip costs of all commuters are the u u u same, inclusive of commuters in H-FRS, free ride and non-free ride commuters in H-FRS. In the meantime, the new uniform fare for H-UFS commuters pu is equal to the new fare of the paid rides of H-FRS commuters p f . In this

case the hybrid fare scheme performance always achieves a ‘win-win-win situation’ for the H-UFS commuters, the H-FRS commuters and the transit operator. All commuters enjoy lower trip costs after the hybrid fare scheme under the revenue neutrality, as shown in Fig. 5. In case (ii), when Mh 4  p0  Mh 2 , the feasible region of the FFI ratio is ( 0,1) with an optimal value of 1 2

. The optimal value of the reward ratio is 1 ( 2 − 20.5 ) , which is higher than 1 2 , implying that a commuter should

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

596

11

‘pay for one get more than one free’. The ‘win-win-win’ situation is always satisfied. Different from case (i), the optimal value of pu is an interval  pua , puc  at the optimal  =1 2 . FRS commuters always have a lower trip cost for their free-ride than that for their non-free rides in this case, because constraint (20) is non-binding when pua  pu  pub . Furthermore, the average equilibrium trip costs of the two sub-scheme commuters are the same if pu = pua . In case (iii) when p0  Mh 2 , the feasible region and the optimal value of the FFI ratio is the same as in case (ii). But the optimal range of pu is in the interval  pua , pub  with a different upper boundary. At the upper boundary when pu = pub , the average equilibrium trip costs of the H-UFS commuters are the same as the original equilibrium trip costs. AEC

pu = pub

 ( M − 1) hN

AEC of H-UFS commuters

pu = pua

3    M − 1 hN 4 

AEC of H-FRS commuters AEC of original case

pu = pub

 ( M − 1) h + p0

pu = puc  ( M − 1) h

m=2

m=

pu = puc Mh 4

TTC of hybrid fare scheme

m =1

1 − 20.5 1 − 0.5

Mh 2

p0

Fig. 5. The average equilibrium trip costs and system total time costs before and after the hybrid fare scheme.

4.

Assessment of hybrid fare scheme performance

4.1. Performance of the hybrid fare scheme The hybrid fare scheme affects the system total time costs and the individual equilibrium trip costs. To assess the performance of the hybrid fare scheme, we introduce the following set of performance measures, and conduct sensitivity analysis of the performance with respect to the exogenous system settings. The absolute reduction of the system total time costs (TTC) at the optimal FFI ratio is defined as

TTC = TTC ( * ) − TTC 0

(27)

The absolute reduction of the individual average equilibrium costs (AEC) at the optimal FFI ratio is defined as

AECu = AEC0 − AECu ( * ) for H-UFS commtuers

(28)

AEC f = AEC0 − AEC f ( * ) for H-FRS commuters

(29)

The corresponding relative reduction of the system total time costs (system efficiency) is given by

=

TTC TTC0

(30)

And the relative reduction of the individual equilibrium trip costs (individual efficiency) is given by

AECu u = for H-UFS commuters AEC0

(31)

12

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

AEC f f = for H-FRS commuters AEC0

597

(32)

The absolute cost reduction indicates the magnitude of cost savings. The system efficiency and the individual efficiency reflect the quality and public acceptability of the hybrid fare scheme. Table 2 and Fig. 6 summarize the hybrid fare scheme performance under the optimal reward ratio with respect to the original fare and the new fares. From the perspective of the total time costs in the system, the absolute and percentage reductions in TTC increase linearly with p0 up to p0 = Mh 4 , reach the optimum and then remain at the optimum, as shown in Fig. 6(a). The minimum system total time cost is achieved with an optimal FFI ratio  =1 2 , which can be explained that when  is small, increasing  will increase the number of free ride commuters shifting from the central period to shoulder periods, and thus relieve the congestion. However, when  is large, the excessive number of free ride commuters in the shoulder periods induces higher system total time costs. The corresponding optimal reward ratio is 1 ( 2 − 20.5 ) , and incentivizes more than 50% commuters in H-FRS to travel free in the shoulder periods. The individual efficiency gain (absolute reduction and relative reduction of the AEC) of H-FRS commuters and HUFS commuters varies with the new fare pu . We first mention in Table 2 that under the revenue neutrality constraint and the unique value of the system total time costs reduction, the average absolute reduction of the AEC over the two sub-schemes is the average reduction of individual time cost in the system as in (33). Thus an increase of AEC f indicates a decrease of AECu given the values of 0.5 and TTC .

AECu 0.5 + AEC f  (1 − 0.5 ) = TTC N

(33)

When p0  Mh 4 , the new fare pu is uniquely determined as pu = Mh . The absolute and relative reductions of the equilibrium average trip costs in the two sub-schemes are exactly the same, and increase linearly and nonlinearly with the original fares, as shown in Fig. 6(b). When p0  Mh 4 , the minimum system total time costs can be achieved by a range of the new fare pu and hence

p f as well. The individual efficiency gain of the H-FRS commuters is equal to or greater than that of the H-UFS commuters. The hybrid fare scheme performs best for H-UFS commuters at the critical original fare p10 = Mh 4 and pu = Mh 2 where

AEC = f

Mh M Mh * M Mh * f = u p10 pu 2 p10 ,= , AEC ,= , at= ,= u 4 5M − 4 4 5M − 4 4

(34)

The best performance of the hybrid fare scheme for H-FRS commuters depends on the relations of the cumulative distribution of ATFI and the number of service runs, as shown in Fig. 6(b). At pu = puc , the H-FRS individual

efficiency gain continues to increase if 0.5  M ( 6M − 4) . This can be explained that, at pu = puc , the free ride trip

costs are less than the non-free ride trip costs as mentioned in case (ii), a higher value of 0.5 means a higher value of the reward ratio which results in a higher reduction of average trip costs with the H-FRS and thus dominates the increase from the original fare. The corresponding individual efficiency gain increases. In the extreme case when 0.5 = 0 , the H-UFS is inapplicable hence the hybrid fare scheme is the same as the fare-reward scheme in Yang and Tang (2018), because all commuters are flexible in changing their departure times within the whole peak period. The optimal performance of the H-FRS can be summarized as Mh * M Mh Mh AEC*f =AEC*u = if 0.5  M ( 6 M − 4 ) ,  f =*u = , at p10 = , pu = 4 5M − 4 4 2 M (1 − 0.5 ) Mh Mh , *f = , AECu = f =0, at p02 = , pu =Mh if 0.5  M ( 6 M − 4 ) AEC*f = 4 − 40.5 2 ( 6M − 4 )

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

598

13

Table 2. The hybrid fare scheme performance with respect to the original fare Mh Mh  p0  4 2

Mh  p0 2

1  2

12

12

 *

* 1 − *

1 2 − 20.5

1 2 − 20.5

TTC

Np0

MhN 4

MhN 4



p0  ( M − 1) h

M 4M − 4

M 4M − 4

p0

0  p0 

*

Mh 4

p0 − *  pu

pf

p0 − 0.5  pu

(1 −  )(1 − 0.5 )

(1 −  )(1 − 0.5 )

a p= p= pud u u

pua  pu  pud

pua  pu  pub

(  Mh)

(p )

(p )

(p )

(p )

AECu

p0

Mh 4

Mh − p0 2

Mh 4

0

AEC f

p0

Mh 4

AECcf

Mh 4

u

p0 AEC0

Mh 4 AEC0

uc

Mh 4 AEC0

0

f

p0 AEC0

Mh 4 AEC0

cf

Mh 4 AEC0

bf

pu

a u

*

c Remarks: AEC0 =  ( M − 1) h + p0 , AEC f

uc =

p0 − 0.5  pu

(1 −  ) (1 − * )

Mh − 2 p0 1  2  ( M − 1) h + p0

c u

Mh 1 − 20.5 p0  0.5 =  + 4 1 − 0.5 1 − 0.5

a u

, cf =

b u

Mh 4 − 40.5

Mh (1 − 20.5 ) + 4 p0  0.5

(  ( M − 1) h + p ) ( 4 − 4 ) 0

,

0.5

Mh

, bf =

(  ( M − 1) h + p ) ( 4 − 4 ) 0

Time cost Reduction

0.5

AEC f

TTC

N Mh 4

pu = pub

Mh 4 − 40.5

pu = p

M 100 4M − 4 M 100 5M − 4

a u

pu = p

1

M

pu = puc

( 6M − 4 )(1 − 0.5 ) pu = pub

0.5  M = 0.5 M

pu = pua

pu = puc Mh 4

Individual efficiency u System efficiency FFI ratio

50

c u

Mh 4

Individual efficiency  f

(%) 75

AECu

pu = pub Mh 2

pu = pub

pu = puc

p0

Mh 4

Mh 2

0.5  M

( 6M − 4 ) ( 6M − 4 ) ( 6M − 4 ) p0

Fig. 6. (a) Absolute reduction of costs under the optimal FFI ratio; (b) relative reduction of costs (efficiencies) under the optimal FFI ratio

4.2. Sensitivity analysis We conduct sensitivity analysis on the obtained results with respect to the exogenous inputs inclusive of supply

14

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

599

(transit headway h and number of service runs M ) and demand (number of commuters, the cumulative distribution 0.5 and the shadow values of time  and  ). As mentioned above, the individual efficiency of H-UFS and H-FRS achieve the optimum at the two different critical original fares. We thus conduct sensitivity analysis at two critical original fares, as shown in Table 3. Table 3. The hybrid fare scheme performance at the two critical original fares

p0

pu

AECu

AEC f

TTC

u

f



(1) p10 =

Mh 4

Mh 2

Mh 4

Mh 4

MhN 4

M 5M − 4

M 5M − 4

M 4M − 4

(2) p02 =

Mh 2

3Mh 4

Mh 4

Mh 4

MhN 4

M 6M − 4

M 6M − 4

M 4M − 4

Mh 4 − 40.5

MhN 4

0

M 2 ( 3M − 2 )(1 − 0.5 )

M 4M − 4

0

Mh

At p0 = Mh 4 and pu = Mh 2 , the individual efficiency of H-UFS and the system efficiency are at the optimum. The individual efficiency of the two sub-schemes are the same. The absolute reductions of TTC and AEC increase with M , h and  ( +  ) . Given a fixed unit cost of early arrival  , the absolute reductions of AEC and TTC increase with the ratio of unit costs of late arrival and early arrival,   , which indicates that the hybrid fare scheme is more effective as the late arrival penalty increases. The two sub-schemes’ individual efficiencies and the system efficiency decrease with the number of service runs, as shown in Fig. 7. This is because the hybrid fare scheme reduces the queuing time costs at the stations M but does not change the schedule delay costs because all service runs are at full capacity. A higher value of M indicates a higher commuter demand and a longer peak period given a fixed headway and train capacity. The schedule delay costs increase with M more than the queuing time costs thus the relative reduction of the queuing time costs (efficiencies) decreases. Moreover, the system efficiency is always higher than the individual efficiency in this case. This is due to the fact that the average absolute reduction of AEC over the two sub-schemes is the average reduction of individual time costs in the system, as mentioned in Section 4.1. Since the absolute reductions of the AEC in two sub-schemes are the same in this case, the average reduction of the individual time costs in the system is the same as the absolute reduction of AEC. Therefore, the individual efficiency is lower than the system efficiency by the fact that the system efficiency is measured by the time costs only, while the individual efficiency is measured by the full trip costs, including the time costs and the fare cost. System efficiency

(%)

u and  f at p0 = Mh 4 and pu = Mh 2

50.0

 f at p0 = Mh 2

u at p0 = Mh 2 33.3 a 0.5  b0.5

25.0

a 0.5 Mh ) ( pu =

20.0

0

b0.5 ( pu = Mh ) ( pu = 3Mh 4 )

2

30

M

Fig. 7. Change of efficiencies with M and 0.5 at the two reference original fares.

At p0 = Mh 2 , the optimal range of new fare pu is the interval 3Mh 4, Mh . At the lower boundary

pu = 3Mh 4 , the AEC of the two sub-schemes commuters are the same, hence the performance of the hybrid fare

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

600

15

scheme has the similar properties as discussed above. Our interest here is at the upper boundary pu = Mh , where the individual efficiency of H-UFS is zero and the individual efficiency of H-FRS is related to the cumulative distribution of the ATFI. It is clear that the H-FRS individual efficiency decreases with the number of service runs and increases with the cumulative distribution of ATFI at  =1 2 , namely 0.5 . When the cumulative distribution 0.5 is high and the number of service runs are large, the individual efficiency of H-FRS may be higher than the system efficiency, as shown in Fig. 7. This is because the absolute reduction of the AEC in H-FRS is higher than the reduction of the individual time costs with a relation AEC f  (1 − 0.5 ) = TTC f N at pu = Mh . As 0.5 and M increase, the reduction of the AEC in H-FRS increases more than the reduction of the individual time costs. Thus, the H-FRS individual efficiency becomes higher than the system efficiency. 4.3. Examples of realistic transit service in Hong Kong Transit Railway This section investigates whether the proposed hybrid fare scheme is realistic under the real transit service and peak period commuter demand. We consider three typical commuting origin and destination trips in Hong Kong Mass Transit Railway as shown in Table 4. Table 4. Three commuting origin and destination trips in Hong Kong MTR Origin

Destination

Distance

Fare

Dimond hill station

Kwun Tong Station

Tsing Yi Station

Central Station

(D-K)

5km

5.3 HKD

(T-C)

16 km

Tin Shui Wai Station

14.2 HKD

Kowloon Tong Station (T-K)

30 km

19.0 HKD

The above three OD trips have similar transit service and patronage:

= h 2= min, N 15500, = s 500, M= =31,  40 HKD/h = ,  120 HKD/h, L = 60 minutes The headway and commuter demand are assumed with reference to the annual report of Hong Kong MTR Cooperation (MTR Corporation Limited, 2016). The capacity is regarded as the average available vacancies of each service runs during the peak period. The shadow value of early arrival and late arrival are chosen with due consideration of the Hong Kong average salary (Hong Kong Census and Statistics Department, 2017) and the relative  and  values in earlier studies (Small et al., 2005; Ubbels et al., 2005). We then assume that the distribution of commuter’s arrival time flexibility interval follows a lognormal distribution with  =3.6 and  =0.5 , where the mode and the mean of the ATFI is approximately 30 minutes and 40 minutes respectively. The cumulative distribution of ATFI is shown in Fig. 8.

Fig. 8. Cumulative distribution of ATFI during peak period

The performance of the HFS under the above transit service and fare scenarios are summarized in Table 5. The two

16

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

601

critical original fares are p10 = 7.75 HKD and p02 = 15.5 HKD . At p02 , the optimal individual efficiency of H-FRS is higher than the system efficiency, which is consistent with the analytical results obtained so far. Table 5. The hybrid fare scheme performance with realistic fares and transit service p0

*

*

(HKD) H-UFS optimum H-FRS optimum D-K trip

7.75 15.5 5.3

0.5 0.5 0.78 or 0.22

0.76 0.76 0.80 or 0.22

T-C trip

14.2

0.5

0.76

( pua )

19

0.5

0.69

AEC f

u

(HKD)

(%)

7.75 0.00 5.30

7.75 11.84 5.3

f

TTC



(%)

( 104 HKD)

(%)

20.53 0.00 14.60

20.53 26.02 14.60

12.013 12.013 8.215

25.83 25.83 17.67

12.013

25.83

7.75

7.75

17.53

17.53

c u

1.30

11.15

2.94

25.24

a u

(p )

7.75

7.75

15.82

15.82

( pub )

0.00

11.84

0.00

24.17

(p ) T-K trip

AECu (HKD)

As discussed above, the actual fare of the D-K trip is less than the critical fare p10 = 7.75 HKD , thus there exists two optimal FFI ratios and reward ratios, resulting in the sub-optimum in system perfromance. These results correpond to the curves at p0  Mh 4 in Fig. 6. The actual fare for T-C trip is between the two critical original fares which corresponds to the curves at Mh 4  p0  Mh 2 in Fig. 6. The system efficiency achieves the optimum and the individual efficiencies of H-UFS and H-FRS are both in a certain range. Moreover, the actual fare for the T-K trip corresponds to the curves at p0  Mh 2 , where the H-UFS individual efficiency varies from 0 to 15.82% and the HFRS individual efficiency varies from 15.82% to 24.17%. 5.

Conclusions

The hybrid fare scheme (HFS) developed in this paper is incentive compatible and balances the competing claims of efficiency, simplicity and fairness. It captures the heterogeneity in commuter’s scheduling flexibility and provides diverse options for commuters with a fare-reward scheme (H-FRS) and a non-rewarding uniform fare scheme (HUFS). Commuters have the opportunity to join either scheme according to their flexibility in scheduling decisions. The hybrid fare scheme is an individual-based charge and reward strategy rather than an anonymous fare-differential pricing policy. First, the scheme will maintain the transit operator’s revenue, offering no monetary compensation or other incentives that incur costs for the government or transit authority. Second, rewarding eligible commuters with free trips in prescribed shoulder periods will not only smooth out peak-hour demand but also prevent excessive ridership in off-peak hours by untargeted customers who would otherwise use other modes of travel. Therefore, the HFS will create a revenue-preserving win-win-win situation for the transit operator, the flexible commuters and the non-flexible commuters. We found that the performance of the hybrid fare scheme varies with the original fare and the new fares after the HFS. For the system total time costs, the HFS performs best at p0  Mh 4 with at least a 25% reduction of the total time costs. For the H-UFS commuters, the HFS performs best at p0 = Mh 4 and pu = Mh 2 with at least 20% reduction of the individual equilibrium trip costs. And for the H-FRS commuters, the HFS can achieve at least max 20%,1 ( 6 − 60.5 ) individual efficiency depending on the relations between the cumulative distribution of the ATFI and the number of service runs. The optimal free fare intervals include half of the service runs during the peak period, which indicates 50% of commuters can travel free after the HFS. The hybrid scheme can be introduced without much complexity. Multiple travel products are usually available for most urban transit systems. On the Hong Kong MTR, for example, five types of Monthly Pass Extras are available with an Octopus card. Furthermore, the proposed HFS can be implemented gradually, starting with a wide time window and a small reward ratio. Appropriate conditions governing free rides, such as expiration dates and exclusions, can be established to manage the distribution of free and non-free ride commuters and to avoid crowding during

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

602

17

shoulder free ride intervals on particular days. The effects of the strategy on commuter’s scheduling decision changes can be easily monitored with the current smart card systems. Although the hybrid fare scheme in this paper focuses on the rail transit congestion utilizing the bottleneck model, the results obtained here are relevant for a wider range of congested areas where crowding management is important, in particular those used regularly by the same group (e.g. in tunnels, parking, recreation facilities). Appendix A. Justification of Assumption 1 In the long-term transit service, the total system costs include the commuters’ total time costs and transit operator’s costs (de Palma et al., 2017; Kraus and Yoshida, 2002). Our model does not distinguish between the number of service runs and the number of physical train sets, and does not consider the roundtrip time by a train set. Hence, the transit operator’s costs in our model are simplified to

TA = ( v0 + v1s ) M + v2 s + v3

(35)

where v0 , v1 , v2 and v3 are all non-negative parameters. The first term ( v0 + v1s ) M is the operating costs and fleet costs of the service runs. The second term v2 s refers to the costs related to the train capacity, which is interpreted as capital costs of the 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. From the above function, it is clear that the transit operator’s costs increase with physical capacity. We then prove that the physical capacity for the long-term transit system should be set so that T   0, T  , s  M (T )  F (T )  N

always holds where M (T ) is the total number of service runs within the interval T .

For a given arrival time flexibility interval T , the cumulative number of commuters whose ATFI is shorter than T is F (T )  N (demand). The total number of service runs within the T is defined as M (T ) which is a step function known by a fixed timetable and headway. The total number of vacancies (supply) within T is therefore s  M (T ) . Fig. 9 shows the cumulative ATFI distribution with two capacity choices, s1 and s2 , and the M (T ) function is approximated to a continuous function for mathematical analysis. The two capacity choices demonstrate

T   0, T  , s1  M (T )  F (T )  N

(36)

T  0, T  , s.t. s2  M (T )  F (T )  N

(37)

and

The total time costs of the commuters are different under the two capacity choices. At s = s1 , the cumulative number of commuters are always less than the total vacancies within the ATFI, hence all commuters are able to travel within their ATFI and have the same unit costs  ,  and  , and work starting time t * in equilibrium. The equilibrium trip costs are the same as if all commuters are identical, which are determined by the schedule delay cost of the first train or the last train, whichever is greater. Without loss of generality, we assume the equilibrium time costs are determined by the schedule delay costs of the first service run which are

 ( t * − t1 )

(38)

Therefore, the total system costs at s = s1 are

TC s = s1 =  ( t * − t1 ) N + ( v0 + v1 s1 ) M + v2 s1 + v3

(39)

However, at s = s2 , the demand exceeds the supply and some commuters have to travel beyond their ATFI. For instance, suppose an individual i has an arrival time flexibility interval length Ti , as shown in Fig. 9. The equilibrium

18

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

603

condition indicates that the time costs of traveling outside Ti for individual i is less or equal to the time costs of traveling inside Ti , otherwise commuters with ATFI shorter than Ti can always get less time costs by traveling inside

Cumulative number of commuters

Ti and all those commuters will choose to travel inside the interval Ti , which contradicts the insufficient supply condition at s = s2 .

F (T )  N

s1  M (T )

s2  M ( T )

Ti

T

Length of the ATFI

Fig. 9. cumulative distribution of ATFI and transit capacity choices

Suppose commuter i takes service run m outside the Ti , the equilibrium condition is therefore

 ( tie − tm ) +  ( t * − tie ) +q ( tm )   ( t * − tm ) + q ( tm )

(40)

where the left-hand side of the inequality (40) is the time costs of traveling outside the Ti by taking the service run m during the peak period and the right-hand side of the inequality is the time costs of traveling inside Ti by taking the service run m . The above equilibrium condition may not hold given a large value of  because of the finite value of the queuing time 0  q ( tm )  ( M − 1) h . Moreover, the total time costs in this case increase tremendously and dominate the reduced transit operator’s costs by s1 . We thus have

TC

TC

=s s2=s s1

(41)

Therefore, from the perspective of the total system costs, the long-term optimal transit service should be set so that all commuters are able to travel within their arrival time flexibility interval. With this capacity setting, commuters’ equilibrium total time costs are the same as the original transit bottleneck. We further demonstrate that the optimal capacity of the long-term transit system is such that all service runs are at full capacity as proven in Yang and Tang (2018) and is omitted here. Moreover, the full capacity is by the definition of the peak period of interest. Realistic statistics show that the service runs are mainly fully occupied by commuters in the peak period, for instance, the average loading of the six main rail lines in the Mass Transit Railway in Hong Kong was 97.33% in 2015 (Legislative Council Panel on Transport, 2016). Appendix B: Justification of design criterion 4 Justification is that the H-FRS aims at reducing commuters’ queuing time costs. In equilibrium, commuters taking the same service runs have the same queuing time costs. If M is not an integer, free-ride and non-free ride commuters will travel in the same service run which results in the same equilibrium arrival pattern and the same total queuing time costs as the original transit bottleneck. The exact proof can be accomplished in a similar manner as in Yang and Tang (2018). Note that this design criterion is derived from our theoretical analysis but it does not need to be met rigorously in practice in order to make the hybrid fare scheme more robust. Appendix C: Proof of Theorem 2 Proof: From Theorem 1, the number of H-UFS commuters is   N .The number of H-FRS commuters are

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

604

(1 −  )  N

19

on an average day. The total number of service runs in FFI is M and in UFI is (1 −  ) M . From design

criterion 2, free-ride commuters voluntarily travel in FFI because of the less trip costs. If  is set to be greater than  (1 −  ) , the number of service runs in FFI cannot accommodate the demand of free-ride commuters every day, hence some commuters cannot use up the rewards, and the transit operator will gain the revenue. On the contrary, if  is set to be less than  (1 −  ) , the available vacancies in UFI cannot accommodate the demand of the non-free ride commuters hence a proportion of commuters have to pay the fare p f and travel in the FFI interval. Their time

costs are  ( M − 1) h + p f , which are higher than the original equilibrium trip costs by the revenue neutrality

constraint. Therefore, from the perspective of both transit operator and the individual commuters, the optimal reward ratio is set such that the supply in FFI can exactly accommodate the demand of the free-ride commuters. Note that the optimal reward ratio is based on the number of commuters in FFI and UFI on an average day. A possible extreme counter-example may happen that the cash-out for the free rides in a month suddenly come up within a single day (e.g. Fridays or Mondays), and the system simply cannot accommodate such high demand in the shifted times, and therefore the equilibrium does not exist in this case. This may be driven by holidays events or work (school) schedules because departure times may differ in terms of the workforce. As a consequence, the usage limitations can be adjusted accordingly, for instance the reward cannot be used or obtained on specific days. Acknowledgements This research was supported by a grant from the Hong Kong’s Research Grants Council (HKUST16211218). The first author also wishes to acknowledge the support of the Hong Kong PhD Fellowship Scheme by Hong Kong’s Research Grants Council. References Abkowitz, M. D. (1980). The impact of service reliability on work travel behavior. Doctoral dissertation, Massachusetts Institute of Technology. Arnott, R., de Palma, A., and Lindsey, R. (1989). Schedule delay and departure time decisions with heterogeneous commuters. Transportation Research Record, 1197, 56-67. Arnott, R., De Palma, A., and Lindsey, R. (1990). Economics of a bottleneck. Journal of Urban Economics, 27(1), 111-130. Beijing Municipal Bureau of Statistics. (2018). Report of annual employee’s earnings for 2017. Beijing. Beijing Subway. (2017). Report of passenger satisfaction of Beijing subway survey for 2017. Beijing: Beijing Subway. China Intelligent Transportation System Association. (2018). Annual report for 2018: Twenty years of intelligent transportation system studies. China Daganzo, C. F. (1995). A pareto optimum congestion reduction scheme. Transportation Research Part B: Methodological, 29(2), 139-154. Daganzo, C. F., and Garcia, R. C. (2000). A pareto improving strategy for the time-dependent morning commute problem. Transportation Science, 34(3), 303-311. de Palma, A., Lindsey, R., and Monchambert, G. (2017). The economics of crowding in rail transit. Journal of Urban Economics, 101, 106-122. Douglas, N. J., Henn, L., & Sloan, K. (2011). Modelling the ability of fare to spread AM peak passenger loads using rooftops. Australasian Transport Research Forum. Faber Maunsell Ltd. (2007). Demand management techniques–peak spreading. Report for Department for Transport, Transport for London and Network Rail, London, United Kingdom. Guo, X., and Yang, H. (2010). Pareto-improving congestion pricing and revenue refunding with multiple user classes. Transportation Research Part B: Methodological, 44(8–9), 972-982. Halvorsen, A., Koutsopoulos, H. N., Lau, S., Au, T., and Zhao, J. (2016). Reducing subway crowding: Analysis of an off-peak discount experiment in Hong Kong. Transportation Research Record: Journal of the Transportation Research Board, (2544), 38-46. Hong Kong Census and Statistics Department. (2017). Report on annual earnings and hours survey for 2017. (Government publication). Hong Kong: Census and Statistics Department. Jou, R., and Mahmassani, H. (1996). Comparability and transferability of commuter behavior characteristics between cities: Departure time and route-switching decisions. Transportation Research Record: Journal of the Transportation Research Board, (1556), 119-130. Kraus, M., and Yoshida, Y. (2002). The commuter's time-of-use decision and optimal pricing and service in urban mass transit. Journal of Urban Economics, 51(1), 170-195.

20

Yili Tang et al. / Transportation Research Procedia 38 (2019) 586–605 Author name / Transportation Research Procedia 00 (2017) 000–000

605

Legislative Council Panel on Transport. (2016). Capacity and loading of trains in the MTR network. (No. CB(4)854/15-16(07)). Hong Kong: Legislative Council Panel on Transport. Lehmann, K. R. (2018). Understanding the effects of demographic and socio-economic factors on public transit ridership trends. Master's Dissertation, University of South Florida, Lindsey, R. (2004). Existence, uniqueness, and trip cost function properties of user equilibrium in the bottleneck model with multiple user classes. Transportation Science, 38(3), 293-314. Liu, Y., & Charles, P. (2013). Spreading peak demand for urban rail transit through differential fare policy: A review of empirical evidence. Australasian Transport Research Forum 2013 Proceedings, Mannering, F. L. (1989). Poisson analysis of commuter flexibility in changing routes and departure times. Transportation Research Part B: Methodological, 23(1), 53-60. Mannering, F., Kim, S., Barfield, W., and Ng, L. (1994). Statistical analysis of commuters' route, mode, and departure time flexibility. Transportation Research Part C: Emerging Technologies, 2(1), 35-47. MTR Corporation Limited. (2016). Annual report for 2015. Hong Kong: Retrieved from http://www.mtr.com.hk/en/corporate/investor/2015frpt.html; Nie, Y. M., and Yin, Y. (2013). Managing rush hour travel choices with tradable credit scheme. Transportation Research Part B: Methodological, 50(4), 1-19. Oldfield, R., and Bly, P. (1988). An analytic investigation of optimal bus size. Transportation Research Part B: Methodological, 22(5), 319-337. Peer, S., and Verhoef, E. T. (2013). Equilibrium at a bottleneck when long-run and short-run scheduling preferences diverge. Transportation Research Part B: Methodological, 57, 12-27. Rouwendal, J., Verhoef, E. T., and Knockaert, J. (2012). Give or take? rewards versus charges for a congested bottleneck. Regional Science and Urban Economics, 42(1-2), 166-176. Small, K. A. (1982). The scheduling of consumer activities: Work trips. The American Economic Review, 72(3), 467-479. Small, K. A., Winston, C., and Yan, J. (2005). Uncovering the distribution of motorists' preferences for travel time and reliability. Econometrica, 73(4), 1367-1382. 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. Van den Berg, V., and Verhoef, E. T. (2011). Winning or losing from dynamic bottleneck congestion pricing?: The distributional effects of road pricing with heterogeneity in values of time and schedule delay. Journal of Public Economics, 95(7-8), 983992. Verhoef, E. T. (1997). Tradeable permits: Their potential in the regulation of road transport externalities. Environment and Planning B: Planning and Design, 24(4), 527-548. Vickrey, W. S. (1969). Congestion theory and transport investment. The American Economic Review, 59, 251-260. Whelan, G., and Johnson, D. (2004). Modelling the impact of alternative fare structures on train overcrowding. International Journal of Transport Management, 2(1), 51-58. Xiao, L., Liu, R., and Huang, H. (2014). Congestion behavior under uncertainty on morning commute with preferred arrival time interval. Discrete Dynamics in Nature and Society, 2014. Yang, H., Shao, C., Wang, H., and Ye, J. (2018). Integrated reward scheme and surge pricing in a ride-sourcing market. Available at SSRN: https://ssrn.com/abstract=3198081. Yang, H., and Tang, Y. (2018). Managing rail transit peak-hour congestion with a fare-reward scheme. Transportation Research Part B: Methodological, 110, 122-136. Yang, H., and Wang, X. (2011). Managing network mobility with tradable credits. Transportation Research Part B: Methodological, 45(3), 580-594. Yoshida, Y. (2008). Commuter arrivals and optimal service in mass transit: Does queuing behavior at transit stops matter? Regional Science and Urban Economics, 38(3), 228-251.