Hindawi Journal of Advanced Transportation Volume 2017, Article ID 3097681, 17 pages https://doi.org/10.1155/2017/3097681

Research Article Optimized Skip-Stop Line Operation Using Smart Card Data

Peitong Zhang, Zhanbo Sun, and Xiaobo Liu

School of Transportation and Logistics, Southwest Jiaotong University, Chengdu,

Correspondence should be addressed to Xiaobo Liu; [email protected]

Received 4 June 2017; Accepted 14 August 2017; Published 13 November 2017

Academic Editor: Giulio E. Cantarella

Copyright © 2017 Peitong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Skip-stop operation is a low cost approach to improving the efficiency of metro operation and passenger travel experience. This paper proposes a novel method to optimize the skip-stop scheme for bidirectional metro lines so that the average passenger travel time can be minimized. Different from the conventional “A/B” scheme, the proposed Flexible Skip-Stop Scheme (FSSS) can better accommodate spatially and temporally varied passenger demand. A genetic algorithm (GA) based approach is then developed to efficiently search for the optimal solution. A case study is conducted based on a real world bidirectional metro line in , China, using the time-dependent passenger demand extracted from smart card data. It is found that the optimized skip-stop operation is able to reduce the average passenger travel time and transit agencies may benefit from this scheme due to energy and operational cost savings. Analyses are made to evaluate the effects of that fact that certain number of passengers fail to board the right train (due to skip operation). Results show that FSSS always outperforms the all-stop scheme even when most passengers of the skipped OD pairs are confused and cannot get on the right train.

1. Introduction a heuristic algorithm-based deadheading optimization strat- egy. Fu et al. [7] presented a new strategy to have all-stop and The time required for stops at stations is usually a major skip-stop trains operate in pairs. The interests of operators travel time component for urban transit systems. Cortes´ et and passengers were balanced in the model by using an al. [1] discussed different approaches and operation strategies exhaustive search procedure. Skip-stop operation is applied that can reduce total passenger travel time. With the goal of in practice by the Transmilenio system in Bogota and Metro reducing travel time and increasing overall transit operation Rapid system in Los Angeles for bus operation and has been efficiency, skip-stop operation is a scheme that can have fleets proved to be highly effective according to the US National not making all designated stops along a route. Skip-stop Research Council [8] and Leiva et al. [9]. operation has been widely applied in bus transit based on the For urban metro systems, the A/B mode-based skip-stop A/B mode. Under the A/B mode, operating fleets are labeled strategy was first developed for the metro systems in Chicago as either type A or type B; stations are also predetermined as city since 1948 [10, 11]. It was later implemented in Philadel- type A, type B, or type AB stations. Type A fleets can only phia and the City of New York [12]. However, due to budget stop at type A and type AB stations and type B fleets can only constraints and large train headways caused by operating stop at type B and type AB stations [2, 3]. Eberlein et al. [4] different types of trains, it switched back to the all-stop mode proposed a real-time deadheading strategy to determine the after several years. A/B mode has also been implemented dispatching time. Deadheading vehicles and skip operations in the metro system of Santiago, Chile, since 2007, which were optimized to minimize the total passenger cost. Sun and has demonstrated a significant benefit to passengers and Hickman [5] formulated the real-time skip-stop problem for transit operators [10]. Lee et al. [13] and Jamili and Aghaee bus operation and presented an enumeration method with [14] reported that, under skip-stop operation, passengers good computation speed. Yu et al. [6] studied the service experienced shorter travel time and operators benefited from reliability of a bus route in the city of Dalian and proposed the scheme by having reduced fleet size and operational costs. 2 Journal of Advanced Transportation

In addition, skip-stop operation has been commonly used as passenger arrival rates. Little work has been done using a recovery strategy in case of special events such as demand dynamic passenger demand from real operation data. With surges and major disruptions according to Cao et al. [15]. respect to the last challenge, although it is widely recognized Withrespecttotheuseofskip-stopmetrooperation, as a major drawback of skip-stop operation, the effect of “fail there are some challenges that have not been fully addressed to board the right train” has often been overlooked and little in the existing literature. The first is that since overtaking relevant analysis has been reported in literature. is prohibited on single-track metro lines, a larger departure In this paper, we develop a FSSS for bidirectional metro headway is needed for the case that the target train travels operation during off-peak periods. The problem is formu- faster (due to skip operation) than the previous train. This lated as a mixed integer nonlinear programming (MINLP) mayalsoreducethelinecapacityandcausesomedifficulty model that minimizes the average passenger travel time. The in timetable design. The second challenge is brought by proposed scheme is able to consider the train operation at the infrastructure constraints at the terminal stations, where the turnaround terminal with departure time optimization trains need to wait and/or be stored at the turnaround and storage capacity. The departure times at the two terminal terminal before serving the backward direction. Therefore the stations are constrained by three types of departure intervals, safe and efficient operation at the turnaround terminals needs that is, the minimum safety headway, headway that meets to be properly considered. The third challenge is the precise the passenger demand, and the turnaround headway. A GA- estimation of dynamic passenger demand pattern, which basedsolutionapproachisthendevelopedtoefficientlysolve significantly affects the design of skip-stop scheme. Last the problem. The proposed scheme is tested using a real metro but not least, skip-stop operation usually leads to passenger line in Shenzhen, China. The case study considers time- confusion. As a result, certain number of passengers may fail dependent demand information inferred from the smart card to board the right train. data of the metro system. Compared to the all-stop scheme, To better address the first challenge, recent research we find that the proposed FSSS can reduce the average interest has shifted from the conventional A/B skip-stop passenger travel time. Transit agencies may also benefit from mode to Flexible Skip-Stop Schemes (FSSS). A summary this scheme due to energy and operational cost savings. For and comparison of the existing skip-stop schemes for metro the cases that certain percentage of passengers fail to board operation is provided in Table 1. Sun and Hickman [5], Zheng the right train, results show that FSSS always outperforms the et al. [16], Freyss et al. [10], Lee et al. [13], and Cao et al. all-stopschemeevenwhenmostpassengersoftheskipped [15] used different approaches to optimize A/B skip-stop OD pairs are confused and cannot get on the right train. The scheme for a single direction urban transit line based on remainder of this paper is organized as follows. Section 2 assumed origin-destination (OD) information. Sogin et al. presents the FSSS model formulation. Section 3 proposes a [17] developed a mixed integer program that could be solved GA-based solution approach to efficiently solve the model. by commercial solvers for small scale problems. A genetic The experiment and numerical results are summarized in algorithm (GA) based solution approach was developed to Section4,followedbytheconcludingremarksinSection5. solve larger scale problems with more stations and trains considered. Niu et al. [19] optimized a skip-stop scheme based 2. FSSS Timetabling on predetermined train stop plans; a mixed integer nonlinear programming model was constructed and solved by a GAMS In this section, we first introduce the notations, followed by solver in their paper. Wang et al. [18] developed optimization a brief description of bidirectional train operation problem. models that offer real-time operations with Flexible Skip-Stop We then introduce the proposed FSSS model formulation. Schemes; that is, trains are allowed to adjust their skip-stop patterns in real-time to minimize passenger travel time and 2.1. Notations energy consumption. Jamili and Aghaee [14] built a model 𝑘: train index; to determine the “uncertain skip patterns” which means that trains can flexibly skip or stop at stations along the line so that 𝑖,: 𝑗 station indices; the train operation speed is optimized. 𝑔: time period index. Anotheroftenoverlookedchallengeistheoperation at turnaround terminals. The only work that considers Input Parameters turnaround operation is Wang et al. [18], which treated the turnaround terminal as an intermediate station. The authors 𝑁𝑠:numberofstationsofaone-wayline; believe it is more realistic to consider turnaround operation 𝑁 tr: number of trains; with departure time optimization and storage capacity. This 𝑙: length of each time period; enables more flexible bidirectional operation that can lead to 𝑡𝑖 𝑖 𝑖+1 more efficient line utilization. For the third challenge, recent tr: section travel time between stations and ,in research has shown that transit smart card data can be utilized seconds; to better interpret the temporal and spatial dynamics of 𝑡𝑖 𝑖 dw: dwell time at station ,inseconds; passenger demand [20, 21]. Niu et al. [19] used time-varying 𝑟𝑖,𝑗 𝑖 passenger demand obtained from the high-speed rail system. 𝑔 : time-dependent passenger’s OD from stations to 𝑗 𝑔 For metro operation, researchers (e.g., Sogin et al. [17]; Wang during period ,inpeoplepersecond; et al. [18]; Jamili and Aghaee [14]) usually assume uniform 𝛿: time loss during a stop, in seconds; Journal of Advanced Transportation 3

Table 1: Existing skip-stop schemes for metro operation.

Skip-stop Input data Single/bidirectional Solution Publication Objective Model Results mode type/source line approach Per the specific scenario, the Maximize the skip-stop total time saving Train Calculate the operation results Suh et al. of a skip-stop operation total travel time Fixed A/B Static OD Single in less total [12] schedule in both simulation of five passenger travel saving in peak and model boarding types the peak period off-peak period compared to the off-peak period A 0-1 integer It is better to programming Minimize total assign a type AB Zheng et al. Static OD with model by Fixed A/B Single passenger travel Tabu search train between a [16] 13 stations analyzing the time type A train and passenger type B train traveling time Short lines with Minimize the fewer stations cost function are less Consider five that is favorable for passenger Assumed static comprised of skip-stop types under Continuous Freyss et al. OD from total passenger operation; larger Fixed A/B Single different approximation [10] Chilean metro transfer time, benefit can be combination of approach data waiting time, obtained in the AorBorAB and total lines with stations traction and smaller energy cost minimum headway Consider three Manipulated 17%–20% travel types (A, B, static OD based time saving for Minimize the and AB) of Lee et al. on boarding and Genetic passengers and Fixed A/B Single passenger travel passenger OD [13] alighting ratio algorithm reduced time and collision from the Seoul operational cost constraints in Metro line for the operator four scenarios A 0-1 mixed integer model Biobjective Smaller that considers Simulated static formulation that headway leads the constraints Cao et al. OD based on minimizes the to better Fixed A/B Single of two Tabu search [15] Beijing Metro passenger passenger successive data waiting time waiting time trains; train and trip time andtriptime operation simulation A mixed integer formulation GAMS and that considers CPLEX for Simulated static Skip-stop the tradeoff small scale OD using Minimize the operation can Sogin et al. between problems; Dynamic gravity Single total passenger reduce [17] passenger genetic model/Chicago travel time passenger travel travel time and algorithm for Metro lines time by 9.5% service large-scale frequency problems caused by skip-stop 4 Journal of Advanced Transportation

Table 1: Continued. Skip-stop Input data Single/bidirectional Solution Publication Objective Model Results mode type/source line approach Consider the impact of Bidirectional; boarding Static OD based Minimize the Bilevel mixed consider turnaround passenger on real data total passenger integer Time and Wang et al. terminal as an number on the Dynamic from the Beijing travel time and programming energy saving [18] intermediate station train dwell Metro energy model with up to 15% without storage time and train (Yizhuang) Line consumption rolling horizon capacity utilization at the departure terminal A unified The model can Time- quadratic be solved by dependent integer standard demand for nine programming Minimize the commercial Niu et al. Predetermined stations of the model with Single total passenger GAMS solver optimization [19] skip-stop Shanghai- linear waiting time packages with Hangzhou constraints; good High-Speed Rail consider safe computation Line in China operation and time train capacity Anonlinear Trip time is programming Decomposition- reduced from Static OD based Minimize the model that based algorithm 92.3 minutes to Jamili and Uncertain on an Iranian Single total passenger considers and simulated 86.6 minutes; Aghaee [14] skip-stop metro line travel time boarding annealing (SA) the average passengers and algorithm speed increases train capacity by 3.4 km/h Time- Three types of dependent Bidirectional; headways are arrival rate consider departure Minimize the considered; Our Genetic See numerical Flexible based on smart time optimization and average travel train operation approach algorithm section card data from storage capacity at the time simulation; the Shenzhen turnaround terminal turnaround Metro line operation

𝐼 min: minimum headway for safety operation, in Variables seconds; 𝑎 𝑘 𝑖 𝐼 𝑘,𝑖: Operation state of train at the station ;itequals 𝑑: headway that meets the passenger demand, in 1 if train 𝑘 stops at station 𝑖;otherwiseitiszero; seconds; 𝑑𝑘,𝑖: departure time of train 𝑘 at station 𝑖,inseconds; 𝐼𝑡: minimum turnaround headway, in seconds; 𝑐𝑘 𝑖 𝐼𝑝: maximum acceptable passenger waiting time, in 𝑖 : train stopped at station prior to the arrival of train 𝑘 seconds; ; 𝐶𝑘 𝑘 𝑐𝑘 𝑐: number of lines that allow trains to be stored at the 𝑖 :thesetoftrainsbetweentrains and 𝑖 that stop turnaround terminal; at station 𝑖 and skip station 𝑗; SN: set contains all the stations along a line; 𝑖𝑘:thenextstoppedstationoftrain𝑘 after departure 𝑛 from station 𝑖; ps: number of the solutions selected at the first step of the GA; 𝑘𝑒: the next available train after train 𝑘 that can 𝑖 𝑗 𝑟𝑐:crossoverrate; transport the confused passengers alighted at 𝑘 to ; 𝑟 𝑖,𝑗 𝑚: mutation rate; 𝑛𝑝𝑘 : number of passengers who board train 𝑘 at 𝑖 𝑗 𝑀: maximum iteration number; station , with a destination ; 𝑛 𝑖,𝑗 𝑐: number of initial chromosomes; tw𝑘 : total waiting time associated with passengers 𝑟 𝑖,𝑗 𝑓: confusion rate. 𝑛𝑝𝑘 ; Journal of Advanced Transportation 5

𝑖,𝑗 tv𝑘 : total in-vehicle time associated with passengers departure terminal; stations 𝑁𝑠 and 𝑁𝑠 +1represent the 𝑖,𝑗 turnaround terminal; {1, 2𝑁𝑠}, {2, 2𝑁𝑠 −1},...,{𝑁𝑠,𝑁𝑠 +1} 𝑛𝑝𝑘 ; are used to represent the same stations, which share the same 𝑛𝑘,𝑖 𝑘 pg: total number of onboard passengers for train at dwell time. The section travel times between two adjacent station 𝑖; stations are assumed to be the same for both directions. In 𝑡𝑘,𝑖 real practices, there are usually more than one turnaround wait: total waiting time for passengers who board train 𝑘 at station 𝑖; line at the turnaround terminal where trains are allowed to 𝑘,𝑖 be stored on the lines. As illustrated in Figure 1, train 𝑘−1up 𝑡 : total in-vehicle time for passengers who board veh to train 𝑘−𝑐−1are allowed to wait at turnaround terminal train 𝑘 at station 𝑖; before train 𝑘−𝑐departs from station 𝑁𝑠 +1.Aproofisgiven 𝑛 pg:numberofonboardpassengers; in Appendix to show that more turnaround lines (e.g., 𝑐≥1) 𝑡 can lead to better train utilization and therefore increase the wait: total passenger waiting time, in seconds; line capacity. The solution of the metro operation problem 𝑡 : total passenger in-vehicle time, in seconds; veh is to make skip/stop decisions and determine the departure SS: the candidate skipping station set that contains the time at each station. Thereby a complete operation timetable stations that can be potentially skipped; can be obtained. 𝑆𝑘: the set that records the skip-stop schemes of train The authors notice that there is a debate regarding 𝑘 for the forward direction; whether skip-stop is more suitable for off-peak or peak period 󸀠 operation. For example, Niu et al. [19] built mathematical 𝑆𝑘: the set that records the skip-stop schemes of train 𝑘 models to solve the rail optimization problem during peak for the backward direction; period. As reported in Suh et al. [12], the benefits received 𝑘 𝑚𝑠 : starting station of arc 𝑠 of train 𝑘; in off-peak period (due to skip-stop) are more significant 𝑘 compared to the peak period. The results are largely due to the 𝑛 : ending station of arc 𝑠 of train 𝑘; 𝑠 fact that there are larger passenger demands at most stations 𝑘 𝑘 𝑘 𝑚 arc𝑚𝑠 ,𝑛𝑠 : skip arc that connects two stations ( 𝑠 and during the peak period. The skip operation may lead to 𝑘 excessive waiting time at the skipped stations and it may also 𝑛𝑠 ), between which at least one station is skipped; 𝑘 cause station congestion. In this work, the authors consider Δ𝑡 𝑘 𝑘 𝑚 𝑚𝑠 ,𝑛𝑠 : maximum travel time difference between 𝑠 the off-peak operation. The train capacity is not considered 𝑘 and 𝑛𝑠 ; as a constraint since our early study [22] found out that trains are far from crowded during the off-peak period. Δ 𝑘: maximum travel time difference between train 𝑘− 1 and train 𝑘 for the forward direction; 󸀠 Δ 𝑘: maximum travel time difference between train 𝑘− 2.3. Assumptions. The key assumptions used in this paper are 1 and train 𝑘 for the backward direction. summarized and discussed as below. Assumption 1. The proposed method targets offline Outputs timetabling design and the passenger information (e.g., OD) pg is assumed to be known and can be accurately calibrated 𝑡 : average waiting time per passenger, in minutes; wait using smartcard data collected from their day-to-day travel 𝑡pg veh: average in-vehicle time per passenger, in min- activities. utes; 𝑡pg Assumption 2. The number of the supplied trains and the tv : average travel time per passenger which equals 𝑡pg +𝑡pg storage capacity at the departure terminal are unlimited. wait veh,inminutes; 𝑍:objectivevalue; Assumption 3. All stations can only accommodate one train 𝑛 ss:numberofskippedstations; at a time for one direction; overtaking is prohibited at any point of the line. This assumption is consistent with 𝑛 : total number of the passengers whose in-vehicle bpg the common infrastructure setup at most metro systems in time is reduced by skip-stop operation. China.

2.2. Bidirectional Metro Operation Problem. Consider a bidi- Assumption 4. The travel time between two adjacent stations 𝑁 +1 rectional metro line. The system is comprised of tr and the dwell time at each station varies along the line but 𝑁 𝑁 𝑁 trains and 𝑠 stations, indexed from 0 to tr and 1 to 𝑠, theyareassumedtobefixedforalltrains.Thetimeloss respectively. Train 0 is an all-stop train that departs at time associated with a stop (due to braking and acceleration) is 0; this all-stop train is operated to ensure the system is in assumedtobeaconstantforalltrainsandallstops. operation at the beginning of the optimization period. Station 1 is the departure terminal and station 𝑁𝑠 is the turnaround Assumption 5. It is assumed that if train 𝑘 stops at station 𝑖 terminal. The line has only one track in each direction as and is about to skip station 𝑗, certain percentage of passengers shown in Figure 1. To avoid confusion, stations are denoted as from OD pairs 𝑖 to 𝑗 will mistakenly board the train (this 1,2,...,𝑁𝑠,𝑁𝑠 +1,...,2𝑁𝑠,inwhichstations1(2𝑁𝑠)indicate percentage is referred to as the confusion rate). It is assumed 6 Journal of Advanced Transportation

Train k−c

Ns +2 Ns +1 2Ns 2Ns −1 Turnaround terminal Train k−c−1 Departure terminal 2 N 1 Ns −1 s Train k−1

Train k

Figure 1: Illustration of a bidirectional metro system.

𝑛𝑘,𝑖 𝑡𝑘,𝑖 𝑡𝑘,𝑖 that these passengers shall notice their mistake (e.g., via train uniform. The resulting pg, wait,and veh can be calculated as radio broadcast) once they get on the train, and they will 𝑑 −𝑑𝑘 in (3)–(5). Here 𝑘,𝑖 𝑐𝑖 ,𝑖 represents the departure interval alight at the next stopping station. These passengers will 𝑘 𝑖,𝑗 𝑘 𝑐 𝑖 𝑟 (𝑑 −𝑑𝑘 ) between trains and 𝑖 at station ; therefore 𝑔 𝑘,𝑖 𝑐𝑖 ,𝑖 then wait at the station until being transported by the next 𝑖 available (correct) train to their destination 𝑗. represents the amount of passengers who arrive at station andplantotraveltostation𝑗 during this time interval. 𝑑𝑘,𝑗 − 𝑑 −𝑡𝑗 2.4. Objective Function. The objective of FSSS is to determine 𝑘,𝑖 dw refers to the in-vehicle travel time for passengers 𝑘 the operation scheme and departure times to minimize the travel from 𝑖 to 𝑗.Thesecondcaseiswhentrains𝑘 and 𝑐𝑖 average passenger travel time including the in-vehicle travel 𝑑 𝑘 ∈ [(𝑔 − 1)𝜏, 𝑔𝜏) ∧ depart in different time periods (i.e., 𝑐𝑖 ,𝑗 time and waiting time. The operation for train 𝑘 at station 𝑖 is 𝑘,𝑖 𝑘,𝑖 𝑑𝑘,𝑖 ∈ [𝑔𝜏, (𝑔 )+ 𝑠)𝜏 ∀𝑠≥1). The corresponding 𝑛 , 𝑡 , defined as a binary variable 𝑎𝑘,𝑖. pg wait 𝑡𝑘,𝑖 and veh canbecalculatedasin(6)–(8).Theobjectivefunction {1, if train 𝑘 stops at the station 𝑖 is then formulated to minimize the average passenger travel 𝑎𝑘,𝑖 = { (1) time; see formulation (9). 0, . { otherwise 2𝑁𝑠 𝑘,𝑖 𝑖,𝑗 𝑛 𝑛 = ∑ 𝑎𝑘,𝑖𝑎𝑘,𝑗𝑟𝑔 (𝑑𝑘,𝑖 −𝑑𝑐𝑘,𝑖), (3) The total number of onboard passengers pg,thetotalpassen- pg 𝑖 𝑡 𝑗=𝑖+1 ger waiting time wait,andthetotalpassengerin-vehicletime 𝑘,𝑖 𝑘,𝑖 𝑘,𝑖 𝑡 𝑛 𝑡 𝑡 1 2 veh can be calculated using (2), in which pg, wait,and veh 𝑘,𝑖 𝑘,𝑖 𝑡 = 𝑛 (𝑑𝑘,𝑖 −𝑑𝑐𝑘,𝑖) , (4) are the number of onboard passengers, the passenger waiting wait 2 pg 𝑖 time, and the passenger in-vehicle time for train 𝑘 at station 2𝑁𝑠 𝑖,respectively. 𝑘,𝑖 𝑖,𝑗 𝑗 𝑡 = ∑ 𝑎𝑘,𝑖𝑎𝑘,𝑗𝑟 (𝑑𝑘,𝑖 −𝑑𝑘 )(𝑑𝑘,𝑗 −𝑑𝑘,𝑖 −𝑡 ), veh 𝑔 𝑐𝑖 ,𝑖 dw (5) 𝑁 2𝑁 𝑗=𝑖+1 tr 𝑠 𝑛 = ∑ ∑𝑛𝑘,𝑖, pg pg 2𝑁𝑠 𝑔+𝑠−1 𝑘=1 𝑖=1 𝑘,𝑖 𝑖,𝑗 𝑖,𝑗 𝑛 = ∑ 𝑎 𝑎 [𝑟 (𝑔𝜏 −𝑑 𝑘 )+ ∑ 𝜏𝑟 pg 𝑘,𝑖 𝑘,𝑗 𝑔 𝑐𝑖 ,𝑖 𝑡 𝑁 2𝑁 tr 𝑠 𝑗=𝑖+1 𝑡=𝑔+1 𝑡 = ∑ ∑𝑡𝑘,𝑖 , (6) wait wait (2) 𝑘=1 𝑖=1 𝑖,𝑗 +𝑟𝑔+𝑠 (𝑑𝑘,𝑖 −(𝑔+𝑠−1)𝜏)], 𝑁 2𝑁 tr 𝑠 𝑡 = ∑ ∑𝑡𝑘,𝑖 . veh veh 𝑘=1 𝑖=1 𝑘,𝑖 1 𝑘,𝑖 2 𝑡 = 𝑛 (𝑑𝑘,𝑖 −𝑑𝑐𝑘,𝑖) , (7) wait 2 pg 𝑖 We use 𝑑𝑘,𝑖 to denote the departure time of train 𝑘 at 2𝑁 𝑔+𝑠−1 𝑖 𝑖 𝑡𝑖 𝑠 station .Thedwelltimeatstation is dw, which is determined 𝑘,𝑖 𝑖,𝑗 𝑖,𝑗 𝑡 = ∑ 𝑎𝑘,𝑖𝑎𝑘,𝑗 [𝑟𝑔 (𝑔𝜏𝑐 −𝑑 𝑘,𝑖)+ ∑ 𝜏𝑟𝑡 based on the real metro operation data. The arrival time at veh 𝑖 𝑖 𝑡𝑖 𝑑 𝑗=𝑖+1 𝑡=𝑔+1 station can be easily acquired by subtracting dw from 𝑘,𝑖. (8) 𝑐𝑘 𝑖 Variable 𝑖 isusedtodenotethetrainthatstoppedatstation 𝑗 𝑘 +𝑟𝑖,𝑗 (𝑑 −(𝑔+𝑠−1)𝜏)](𝑑 −𝑑 −𝑡 ), prior to the arrival of train 𝑘, which can be referred to as 𝑐𝑖 = 𝑔+𝑠 𝑘,𝑖 𝑘,𝑗 𝑘,𝑖 dw 󸀠 󸀠 max{𝑘 |𝑘 <𝑘,𝑎𝑘,𝑖 =𝑎𝑘󸀠,𝑖 =1}according to Niu et al. [19]. The time-dependent passenger OD rate between stations (𝑡 +𝑡 ) 𝑖 𝑗 𝑔 𝑟𝑖,𝑗 𝑍= wait veh . and during time period is denoted as 𝑔 and the length of min 𝑛 (9) 𝜏 𝑛𝑘,𝑖 𝑡𝑘,𝑖 𝑡𝑘,𝑖 pg each period is .Tocalculatethevaluesof pg, wait,and veh, two cases are considered; see Figure 2. The first case is when 𝑘 2.5. Model Constraints. The model is subject to a few safety, 𝑘 𝑐 𝑑 ,𝑑 𝑘 ∈ trains and 𝑖 depart in the same time period (i.e., 𝑘,𝑖 𝑐𝑖 ,𝑖 operational, and infrastructure constraints. Train headway [(𝑔 − 1)𝜏,). 𝑔𝜏) In this case, the passenger arrival rate is is an important concept that is directly related to operation Journal of Advanced Transportation 7

Rate

i,j Period “g” rg

Time

Station i g−1  g

d  d c ,i k,i (a)

Rate Period Period “g+s” Period i,j “g” rg “g+1”

··· Time

Station i g−1  g g+1  g+s−1  g+s 

d  dk,i c ,i (b)

𝑘 𝑘 Figure 2: The departure of two consecutive trains; (a) trains 𝑘 and 𝑐𝑖 depart in the same time period; (b) trains 𝑘 and 𝑐𝑖 depart in different time periods. safety and the capacity of a metro line. Three types of the departure time (of train 𝑘) at the departure terminal; see train headways are considered in this study, which are the the trajectory of train 2. 𝐼 Δ 𝑆 𝑆󸀠 minimum safety headway min, the headway that meets To calculate the value of 𝑘,set 𝑘 and set 𝑘 are defined the passenger demand 𝐼𝑑, and the minimum turnaround as the skip-stop plans of train 𝑘 for the forward direction and headway 𝐼𝑡. Different from the work in Wang et al. [18] who backward direction, respectively. Each set is comprised of a took the turnaround terminal as a normal station, we specif- series of the “skip arcs”; each “skip arc” connects two stations ically consider the departure time optimization and storage where train 𝑘 stops, between which at least one station has 𝑘 𝑘 capacity at the turnaround station. In this subsection, the been skipped. For example, as shown in Figure 4, arc𝑚1,𝑛1 headway-related constraints are discussed below, followed by 𝑘 denotes the fact that train 𝑘 stops at station 𝑚1 =1and stops the operational constraints. 𝑘 again at station 𝑛1 =4, between which station 2 and station 3 𝑚𝑘 𝑠 𝑛𝑘 2.5.1. Minimum Safety Headway. Equations (10) and (11) areskipped.Here 𝑠 refers to the starting station of arc ; 𝑠 represent that train 0 departures at time 0; after arriving at refers to the ending station of arc 𝑠.Thelast“skiparc”inset 𝑆 𝑘 𝑘 the turnaround terminal, the new departure time of train 𝑘 is denoted as arc𝑚𝑞,𝑛𝑞 . 0 is calculated by adding the section travel time, station dwell time, and the turnaround time. For safe operation, the 𝑆𝑘 ={arc𝑚𝑘,𝑛𝑘 ,...,arc𝑚𝑘,𝑛𝑘 ,...,arc𝑚𝑘,𝑛𝑘 }. (13) departure interval of two consecutive trains should satisfy 1 1 𝑠 𝑠 𝑞 𝑞 formulation (12) at all stations. 𝑛𝑘 ∑ 𝑠 (1−𝑎 ) 𝑑0,1 =0, (10) Under this definition, 𝑘 𝑘,𝑖 canbeusedtorecord 𝑖=𝑚𝑠 𝑘 𝑁 the total number of skip operations of train from stations 𝑠 𝑘 𝑘 𝑖 𝑖 𝑚𝑠 to 𝑛𝑠 with respected to arc𝑚𝑘,𝑛𝑘 . The travel time saving of 𝑑0,𝑁 +1 = ∑ (𝑡 +𝑡 )+𝐼𝑡, (11) 𝑠 𝑠 𝑠 tr dw 𝑘 𝑟=1 𝑘 𝑘 𝑛𝑠 train 𝑘 from stations 𝑚𝑠 to 𝑛𝑠 can be expressed as ∑ 𝑘 (1 − 𝑖=𝑚𝑠 𝑖 𝑖 𝑑𝑘,𝑖 −𝑑𝑘−1,𝑖 ≥𝐼 ∀1 ≤ 𝑖 ≤ 𝑠2𝑁 , 1≤𝑘≤𝑁 . (12) 𝑎 )(𝑡 +𝛿) 𝑡 𝑖 𝛿 min tr 𝑘,𝑖 dw ,where dw is the dwell time at station and is the time loss due to the acceleration and deceleration associated Note that skip operation leads to shorter travel time and with a stop at a station. Therefore, the maximum travel time it may cause some safety concerns. An illustrative example 𝑘 difference between train 𝑘−1and train 𝑘 from stations 𝑚𝑠 is provided in Figure 3. Suppose train 1 stops at all stations 𝑘 𝑛 Δ𝑡 𝑘 𝑘 and train 2 skips station 2 and station 3. If train 2 only keeps and 𝑠 , denoted as 𝑚𝑠 ,𝑛𝑠 , can be represented as 𝐼 the minimum safety headway min at the departure terminal, 𝑛𝑘 the minimum safety headway will not suffice at station 3. This 𝑠 𝑖 Δ𝑡 𝑘 𝑘 = ∑ (1−𝑎 )(𝑡 +𝛿) situation poses safety issues and requires further intervention 𝑚𝑠 ,𝑛𝑠 𝑘,𝑖 dw Δ 𝑘 control. To avoid this, an extra amount of time 𝑘 is added to 𝑖=𝑚𝑠 8 Journal of Advanced Transportation

Terminal form of constraints (17a) and (17b). Here 𝐼𝑑 is the headway station 4 Imin that meets the passenger demand. 𝑑 ≤𝑘𝐼, ∀1≤𝑘≤𝑁, 𝑘,1 𝑑 tr (17a) Station 3 𝑑 ≤𝑑 +𝑘𝐼 , ∀1≤𝑘≤𝑁. 𝑘,𝑁𝑠+1 0,𝑁𝑠+1 𝑑 tr (17b) This constraint, however, is too strict for skip-stop oper- ation. In the cases that the safety operation headway is larger Station 2 than the demand headway (due to skip operation), we cannot guarantee that the demand headway is satisfied at all times. Terminal Therefore we consider constraints (18a) and (18b) as a com- min station 1 I Δ 2 Time promise to constraints (17a) and (17b). Train 1 Train 2∗ Train 2 𝑑 ≤ {𝑘𝐼 ,𝐼 +Δ +𝑑 }, 𝑘,1 max 𝑑 min 𝑘 𝑘−1,1 Figure 3: The illustration of the method to obtain the train safety (18a) headway. ∀1≤𝑘≤𝑁, tr 𝑑 ≤ {𝑑 +𝑘𝐼 ,𝐼 +Δ +𝑑 }, 𝑘,𝑁𝑠+1 max 0,𝑁𝑠+1 𝑑 min 𝑘 𝑘−1,𝑁𝑠+1 𝑛𝑘 (18b) 𝑠 ∀1≤𝑘≤𝑁. − ∑ (1 − 𝑎 )(𝑡𝑖 +𝛿) tr 𝑘−1,𝑖 dw 𝑘 𝑖=𝑚𝑠 2.5.3. Turnaround Headway. Inourmodel,wespecifically 𝑘 𝑛𝑠 consider 𝑐 (𝑐≥1) turnaround lines at the turnaround ter- = ∑ (𝑎 −𝑎 )(𝑡𝑖 +𝛿). 𝑘−𝑐 𝑘−1,𝑖 𝑘,𝑖 dw minal. Under the infrastructure geometry, train should 𝑘 𝑁 +1 𝑘 𝑖=𝑚𝑠 depart from station 𝑠 before train departs from station 𝑁𝑠.Thatis,formulations(18a)and(18b)shouldbesatisfied. (14) Here 𝐼𝑡 refers to the minimum turnaround headway at the turnaround station. The formulation can better utilize the The next step is to calculate the maximum travel storage capacity at the turnaround station. time difference between train 𝑘−1and train 𝑘 at any 𝑑 +𝐼 −𝑑 ≥0, ∀𝑐≤𝑘≤𝑁. point along the metro line, which can be expressed as 𝑘,𝑁𝑠 𝑡 𝑘−𝑐,𝑁𝑠+1 tr (19) 𝑘 𝑠 𝑛𝑢 𝑖 max{∑𝑢=1 ∑ 𝑘 (𝑎𝑘−1,𝑖 −𝑎𝑘,𝑖)(𝑡 + 𝛿) | 𝑠 = 1,...,𝑞}.In 𝑖=𝑚𝑢 dw the cases that this maximum travel time difference is larger 2.5.4. Operational Constraint. To avoid intolerable waiting than zero (i.e., train 𝑘 travels faster than train 𝑘−1), an time at the trip origins, we assume that each viable OD pair should be served in an acceptable time period which is extra amount of time is added to the departure time of train 𝐼 𝑘 toavoidcollision.Inthecasethatthemaximumtravel marked as 𝑝; that is, constraint (20) needs to be satisfied. 󸀠 󸀠 𝑖,𝑗 𝑖,𝑗 time difference is smaller than zero, no adjustment is needed; Here 𝑥=max{𝑘 |𝑘 <𝑘,𝑎𝑥,𝑖𝑎𝑥,𝑗𝑛𝑝𝑥 >0∧𝑎𝑘,𝑖𝑎𝑘,𝑗𝑛𝑝𝑘 >0}, Δ󸀠 ∀1≤𝑘≤𝑁 0≤𝑖≤𝑁 𝑖<𝑗≤𝑁 see formulation (15). Similarly, we use 𝑘 to represent the tr, 𝑠, 𝑠. maximum travel time difference for the backward direction. 𝑑𝑘,𝑖 −𝑑𝑥,𝑖 ≤𝐼𝑝. (20) 𝑛𝑘 { { 𝑠 𝑢 } Δ = ∑ ∑ (𝑎 −𝑎 )(𝑡𝑖 +𝛿)|𝑠=1,...,𝑞 , In practice, some stations such as terminal stations and 𝑘 max {max { 𝑘−1,𝑖 𝑘,𝑖 dw } 𝑢=1 𝑘 { { 𝑖=𝑚𝑢 } transfer stations should not be skipped. For this, we further (15) define a set SS that contains the stations that can be skipped. } 0 . For the stations that do not belong to this set, they should not } ∪ = } be skipped; see constraint (21). Here SS SS SN, where SN is the set that contains all metro stations. To ensure minimum safety headway between two con- 𝑎𝑘,𝑖 =0, ∀𝑖∈SS. (21) secutive trains at any point of the line, the departure time of train 𝑘 is constrained at terminal stations 1 and 𝑁𝑠 +1by formulation (16a) and formulation (16b), respectively. 3. Solution Approach

𝑑 −𝑑 ≥𝐼 +Δ , ∀1≤𝑘≤𝑁, In the previous section, the FSSS model is formulated as a 𝑘,1 𝑘−1,1 min 𝑘 tr (16a) mixed integer nonlinear programming (MINLP) problem, 󸀠 which is comprised of the objective function (9), constraint 𝑑𝑘,𝑁 +1 −𝑑𝑘−1,𝑁 +1 ≥𝐼 +Δ , ∀1≤𝑘≤𝑁. (16b) 𝑠 𝑠 min 𝑘 tr (1), constraint (10)-constraint (11), constraints (16a) and (16b), and constraints (18a) and (18b)–constraint (21). Note that the 2.5.2. Headway That Meets the Passenger Demand. Astrict model requires departure time optimization at each station, constraintthatsatisfiesthepassengerdemandshouldtakethe which imposes high computational complexity. To simplify Journal of Advanced Transportation 9

arc   arcm,n arc   m ,n 1 1 m ,n  

k k k k k k m1 n1 m2 n2 mq nq

1 234 Ns −2 Ns −1 Ns Skip the station Stop at the station

Figure 4: Definition of 𝑆𝑘 and “skip arcs.” theproblem,weapplyarule-basedsimulation((22)–(24)) allowed value that satisfies the safety and passenger demand that generates departure times to satisfy constraints (16a) constraints; therefore it should result in the smallest required and (16b) and constraints (18a) and (18b)-constraint (19). A number of trains. Similarly, the departure time at the similar rule-based computation of train departure times was turnaround terminal should take (23). Equation (24) is then implemented by Salzborn [23]. used to update the departure times at the intermediate First, we define the terminal departure time as (22); the stations. departure time at the terminal departure takes the maximum

𝑑 =𝑑 + {𝑘𝐼 −𝑑 ,𝐼 +Δ }+ {𝑑 −𝐼 −𝑑 ,0}, ∀𝑐≤𝑘≤𝑁, 𝑘,1 𝑘−1,1 max 𝑑 𝑘−1,1 min 𝑘 max 𝑘−𝑐,𝑁𝑠+1 𝑡 𝑘,𝑁𝑠 tr (22) 𝑑 = {𝑑 +𝐼,𝑑 +𝐼 +Δ󸀠 }, ∀1≤𝑘≤𝑁, 𝑘,𝑁𝑠+1 max 𝑘,𝑁𝑠 𝑡 𝑘−1,𝑁𝑠+1 min 𝑘 tr (23)

𝑖 V {𝑑 + ∑ (𝑡𝑖 +𝑎 𝑡𝑖 −(V − ∑𝑎 )𝛿), 𝑖≤𝑁, ∀1≤𝑘≤𝑁 { 𝑘,1 tr 𝑘,𝑖 dw 𝑘,𝑖 𝑠 tr { V=0 0 𝑑𝑘,𝑖 = { 𝑖 V (24) { {𝑑 + ∑ (𝑡𝑖 +𝑎 𝑡𝑖 −(V − ∑𝑎 )𝛿), 𝑖>𝑁, ∀1≤𝑘≤𝑁. 𝑘,𝑁𝑠+1 tr 𝑘,𝑖 dw 𝑘,𝑖 𝑠 tr { V=𝑁𝑠+1 0

It is particularly important to note that if the turnaround GA is motivated by the principles of natural selection and terminal is considered as an intermediate station (as in most survival-of-the-fittest individuals [24]. It is proved to be existing literature), an extra amount of the maximum travel efficient in solving NP-hard optimization problems in the 󸀠 time difference, which takes the form of max{Δ 𝑘,Δ𝑘}, needs transportation field [25–28]. This method was also used by 󸀠 to be added to the departure terminal. In the cases that Δ 𝑘 is Lee et al. [13] to find a solution to the A/B mode-based skip- larger than Δ 𝑘, the departure time at the departure terminal stop operation problem. In their work, gene is defined as the can be expressed by (25). And this leads to delayed departure stationtypeanditcantakethechoicesofA,B,orAB.The 𝑁 time compared to (22), which results in line capacity loss. time complexity of their problem is therefore 3 𝑠 .Compared 𝑑̃ =𝑑 + {𝑘𝐼 −𝑑 ,𝐼 +Δ󸀠 } to Lee et al. [13], the proposed FSSS model offers flexible 𝑘,1 𝑘−1,1 max 𝑑 𝑘−1,1 min 𝑘 schemes and it can further consider the operation at the turnaround terminal. Realistic operational and infrastructure + max {𝑑𝑘−𝑐,𝑁 +1 −𝐼𝑡 −𝑑𝑘,𝑁 ,0}, (25) 𝑠 𝑠 constraints are also considered. Therefore the proposed ∀𝑐≤𝑘≤𝑁 method is different from the one in Lee et al. [13]. The solu- tr tion procedure of the developed GA algorithm is shown in Furthermore, it is widely known that NP-hard problems Figure 5. suffer from the “Curse of Dimensionality”; separating the In our solution approach, a gene is defined as the opera- forward and backward operation scheme can significantly tion choice at the specific station (0 or 1 which denotes skip reduce the dimension of the problem; that is, the computa- or stop) and therefore a chromosome is defined as a binary 2𝑁𝑠𝑁 𝑁𝑠𝑁 +1 tional complexity reduces from 𝑂(2 tr ) to 𝑂(2 tr ). string that embodies {𝑎𝑘,𝑖 |𝑖∈SS}.Andthisensuresthat Using the simulation scheme, the departure times can be constraint (21) is satisfied throughout the searching process. determined, and the problem is thus reduced to the optimiza- In the initialization step, a number of 𝑛𝑐 chromosomes are tion of the binary variables 𝑎𝑘,𝑖. The problem is NP-hard randomly generated. Next, we check the feasibility of the and an exhaustive search of the skip-stop schemes has an generated chromosomes, using constraint (20). An arbitrarily 𝑁 𝑁 exponential time complexity 𝑂(2 𝑠 tr ). Toavoid enumerating large objective (fitness) value is assigned to the infeasible all possible outcomes, we develop a genetic algorithm (GA) chromosomes to penalize the infeasible predecessors; for the approach to guide the searching process. GA is a metaheuris- feasible predecessors, their fitness values need to be evaluated ticmethodthatimitatestheprocessofnaturalevolution. using the objective function. We then apply the GA operators 10 Journal of Advanced Transportation

Operation parameters O-D

Initialization

Yes Feasiblility No

Evaluate the objective value via Equ. (14)–(16a) and (16b) Penalize

No Chromosome list

Genetic algorithm

Generate the next generation

Stop criteria

Yes

Model output

Figure 5: A GA-based solution approach.

to the list of chromosomes to generate a list of new solutions. application of smart card data in public transit field and cate- The GA operators include (i) a selection operator, which gorizedthedatausageintothreelevels:thestrategiclevelfor 𝑛 is used to select ps fittersolutionsbasedontheRoulette long-term planning, the tactical level for service adjustment, Wheel Selection Algorithm [29]; (ii) a crossover operator, and the operational level for performance measurement. whichisusedtogeneratethechildsolutionsfromtheselected Munizaga and Palma [32] proposed a method to estimate solutions through a recombination process (with crossover metro passenger OD using GPS data and smart card data in rate 𝑟𝑐; see Back [30]); (iii) a mutation operator, which uses a Santiago. Hong et al. [33] developed an approach that can small probability (i.e., 𝑟𝑚) to model the genetic mutation pro- match passenger trips to trains. Zhang et al. [22] formulated cess. We further interpret the binary strings as Gray-coded a model to optimize the urban rail operation based on the integers, which is a proven practice that can be used to avoid detailed passenger transaction data during weekdays. It is thehammingcliffproblem(seeBack[30]).Wethencheck revealed by the literature that smart card data can better the generated new solutions to see if they satisfy the stopping reflect the passenger OD pattern. criteria; that is, the number of iterations exceeds a predefined The proposed FSSS model was tested using the smart card value (𝑀) or convergence is reached. If so, the final solu- data and operation parameters from a real world bidirectional tion is recorded; otherwise, return to check the feasibility. metro line in Shenzhen, China. The metro system is shown Readers can refer to Jong [24] for more detailed information in Figure 6. We selected (with 28 intermediate stations of GA. and two terminals; highlighted in green) to test the proposed FSSS model. 4. Case Study and Numerical Results ThedynamicpassengerODrateswereobtainedfromthe smart card data using the following steps: (i) a transfer rule 4.1. Data Processing and Experiment Setup. Smart card data was defined based on the least number of stations; that is, for collected from urban transit systems have been widely used a trip that starts and ends at different lines, the trip will use in transportation applications. Pelletier et al. [31] reviewed the the transfer station that yields the least number of stations Journal of Advanced Transportation 11

Danzhutou Shangtang Dafen Turnaround Airport terminal Hongshan East Mumianwan Honglang Liuxian University Shang Xiashui Change Xili Tanglang Bantian North Xingdong dong Town Changlingpi Minzhi Wuhe Yangmei shuijing jing long hourui Buji

Shenzhen Caopu Baishilong North Shuibel Minle Baigelong Xixiang Lingzhi Tianbei Shangmeilin Cuizhu Buxin Pingzhou Shenzhen Lianhua Fanshen North Shaibu Lianhuac Hongling Laojie Tai’an Bao’an un Stadium Children’s Palace Bao’an Yijing Center Futian Qiaocheng North Baohua Xin’an Shen Autuo Qiao Xiang Xiangmei Jing Lianhua Gangxia Huaqiang Yannan Grand Hubei Huangb Xinxiu Shenzhen Hi-Tech kang Hill xiang mi North tian West North North Theater eiling Liyumen Daxin Taoyua University Park guomao Zhuzilin huaqiang Science Linhai Qianhiwan Hongshu Window OCT Oiaocheng Chegong Xiangmihu Gangxia Rd Museum wan of the East miao Shopping Convention & Keyuan World Park Exhibition Center

Shixia Houhai Fumin Departure Luohu terminal Denglian

Figure 6: map. duringthetrip;(ii)thetransferrulewasappliedtotherecords were set according to the real world operating conditions. that either start or terminate at some station of Line 1; (iii) Parameter 𝑐 wassettobe2whichmeansthatonetrainis the origins and destinations of all the Line 1 trips were then allowedtobestoredattheturnaroundline. identified; (iv) by aggregating the OD data for the weekday The all-stop operation scheme was simulated by setting off-peak period (13:30–17:00) during one month, the dynamic parameter 𝑎𝑘,𝑖 to 1 for all the trains and stations. We then use 𝑖,𝑗 passenger OD rates (𝑟𝑔 )werecalculatedforeachtimeperiod formulation (1)–formulation (11) and formulation (22)–for- 𝑔, which was set to be 15 minutes in the experiment. mulation (24) to calculate the performance of the system This metro line currently operates under an all-stop under the all-stop scheme. It is found that the average waiting 𝑁 time 𝑡pg and the average in-vehicle time 𝑡pg are 3.0 minutes scheme. The round trip travel time is about 138.8 minutes. tr wait tv was set to be 10. The minimum safety headway, the demand and 13.5 minutes, respectively. The average travel time under headway, and the minimum turnaround headway were set to the all-stop scheme is 16.5 minutes. The total number of 𝑛 be 2 minutes, 6 minutes, and 3 minutes, respectively. Accord- onboard passengers pg is 26146. ing to the real world practice of Chicago Metro [11], the maximum acceptable waiting time 𝐼𝑝 was 12 minutes during 4.2. Skip-Stop Optimization Results. The proposed FSSS thepeakhours.Asourstudytargetsoff-peakoperation,𝐼𝑝 model is then solved using the GA-based solution approach. was set to be 15 minutes. The time loss 𝛿 was calculated based There are five major parameters used in GA, including the size 𝑛 𝑛 on the reachable maximum speed 𝑉 using formulation of the chromosome list ( 𝑐), population’s size ( ps), maximum max 𝑀 𝑟 (26). Here 𝛼 is the maximum acceleration rate and 𝛽 is number of iterations ( ), crossover rate ( 𝑐), and mutation 𝑉 𝛼 𝛽 rate (𝑟𝑚). The first three parameters were set to be 200, 500, the maximum deceleration rate. Parameters max, ,and 2 2 weresettobe80km/h,0.8m/s ,and1m/s ,respectively;the and2000,respectively.Thecrossoverrateandthemutation calculated 𝛿 was 25 seconds, using (26). Further discussions rate are particularly important and are often found to affect of this formulation can be found in Zheng et al. [16]. the modeling performance (Back [30]). In the experiment, these parameters were set to be 0.8 and 0.005. The selection (𝛼 + 𝛽) 𝑉 of parameters is based on a standard sensitivity analysis 𝛿= max . 432𝛼𝛽 (26) approach; details are not presented here. For the FSSS, we manually set SS to include 20 low The off-peak boarding and alighting counts at each demand stations, that is, SS = {10, 13, 14, 18, 21, 22, 23, 27, stationareshowninFigure7(a).Itisfoundthatthenumberof 28, 29, 32, 33, 34, 38, 39, 40, 43, 47, 48, 51}. The corresponding passengers is relatively low at station 21, station 22, and station chromosome length is 200 (20 binary variables multiple 10 23. An example of the time-dependent passenger demand trains). The problem is solved on a computer with 2.0 GB from station 8 to station 15 is illustrated in Figure 7(b). The RAM on a 2.0 GHz Intel Core i7 processor. The GA runtime is section travel time between two consecutive stations and the around 2 hours. However, as the FSSS model is proposed for dwelltimeateachstationareshowninTable2.Thedwell offline operations, the computation time is not a big concern. time at transfer stations {3, 4, 8, 9, 15, 24} is 45 seconds, which By solving the MINLP,the optimized FSSS is listed in Table 3, is higher compared to the 35-second dwell time at other only for the stations that belong to set SS. For the stations intermediate stations. The minimum turnaround time was set that belong to SS, they cannot be skipped. It is found that the to be 120 seconds. The section travel times and dwell times stations that have the lower passenger demand (e.g., station 12 Journal of Advanced Transportation

14 13 7000 12 6000 11

g 10 5000 9 8 4000 7 3000 6

Time period Time 5 2000 4 3 Demand (passengers) 1000 2 1 0

1 2 3 4 5 6 7 8 9 0.50 1 1.5 2 2.5 3 3.5 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Station Arriving rate (passenger per minutes)

Station 1−Ns Station (Ns + 1) −2Ns (a) Passenger demand at each station (b) Time-dependent passenger OD rates from station 8 to station 15

Figure 7

Table 2: Train operation parameters.

Station 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 𝑡𝑖 tr (seconds)0 95688510377145956213297931228485 𝑡𝑖 dw (seconds)353545453535354545353535353545 Station 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 𝑡𝑖 tr (seconds)6910190171839078100817210484204210156 𝑡𝑖 dw (seconds)353535353535353545353535353535

10, station 14, station 21–station 23, station 38–station 40, found that, compared with all-stop scheme, passengers who and station 51) are frequently skipped under the optimized boardtheskip-stoptrainshaveabetterchancetofindempty scheme.Atotalnumberof35stationsareskipped,which seats, meaning FSSS is more favorable to the passengers as it can lead to energy and operational cost savings for operating enhances their level of service. By analyzing the maximum agencies. The performance of FSSS is then compared to the section passenger load, we find that the highest passenger all-stop operation, as shown in Table 3. load under FSSS is 558, which is far below the train capacity The results present in Table 4 show that the objective value (1860). Therefore FSSS does not cause congestion on the of FSSS decreases by 5%, from 990 seconds to 940 seconds, trains. which indicates a travel time saving of about 50 seconds per passenger. It is found that the average in-vehicle time is 4.3. Analysis of the Passenger Confusion Rate. It is widely reduced by 6.5% with slightly increased (less than 4 seconds) recognized that the major problem in applying skip-stop 𝑛 passenger waiting time. As indicated by the value of bpg, operation is to keep the passengers informed regarding the about26.9%oftotalpassengers(6604outof24576)benefited skip-stop plans. The solution to it relies on information from FSSS. Compared to the all-stop operation, the number dissemination such as radio broadcast in stations/trains. of onboard passengers under FSSS slightly decreases by 6%. Nonetheless, the authors believe the effect of certain percent- Thisismainlybecausethelasttrainskipssomestationsand age of passengers being confused by the skip operation and some passengers are not served by this train. These passengers failing to find the right train is worth studying. The percent- areassumedtobepickedupbythesubsequenttrains[18]. age is referred to as the passenger confusion rate hereafter in The timetables of the two operation schemes are illus- the paper. As noted in Assumption 5, we assume that these trated in Figure 8. The light line denotes the all-stop opera- confused passengers shall notice their mistake (e.g., via train tion, while the dark line represents the FSSS operation. It is radio broadcast) once they board the train, and they will get found that FSSS has a slightly higher travel speed (the dark off at the next stopping station. They will then wait at the lines travel faster than the light lines). In specific, the average station until being transported by the next available (correct) round trip travel time under FSSS is 135.7 minutes, which train to their destination. Variable 𝑟𝑓 is introduced here to is more than 3 minutes shorter compared to the all-stop denote the confusion rate, and a sensitivity analysis was con- operation. The average departure interval is about 6 minutes ducted to see the effects of 𝑟𝑓 on the modeling performance. which is similar to the all-stop operation. The variable 𝑟𝑓 was tested by taking values from 0 to 100%, Figure 9 indicates the seat occupancy and maximum sec- with an increment of 20%. tion passenger load under the two schemes. Seat occupancy The calculation of the boarded passengers (on train 𝑘 is defined as the passenger load over the seat numbers. It is which stops at station 𝑖, with a destination 𝑗) can be divided Journal of Advanced Transportation 13

Table 3: Optimized chromosome.

Train S10 S13 S14 S18 S21 S22 S23 S27 S28 S29 S32 S33 S34 S38 S39 S40 S43 S47 S48 S51 0 11111111111111111111 111110101111111001110 2 11111011111111111111 3 11111111111111011110 401111011111110101111 5 11110101111111111111 611111111111111001010 7 01010011111111111111 811111111111110001011 9 01011111001111111111 10 01001100001111111111 Skipped times 4 0 3 1 3 3 3 1 2 2 0 0 0 2 4 4 0 0 0 3

Table 4: Performance of all-stop operation and FSSS. 𝑡pg 𝑡pg 𝑡pg 𝑛 𝑍𝑛𝑛 Operation scheme wait (min) veh (min) tv (min) pg bpg ss All-stop 2.98 13.50 16.48 26,146 990 0 0 FSSS 3.04 12.62 15.66 24,576 940 6604 21 +0.06 (2%) −0.88 (6.5%) −0.82 (5%) −1570 (6%) −50 (5%) — —

󸀠 󸀠 into two cases. Case 1 is that train 𝑘 will stop at station 𝑗; these passengers shall wait until 𝑘𝑒 = min{𝑘 |𝑘 > 𝑖 𝑗 𝑘, 𝑎 󸀠 𝑎 󸀠 =1} 𝑖 >𝑗 therefore passengers (from OD to ) board the right train. 𝑘 ,𝑖𝑘 𝑘 ,𝑗 arrives. If 𝑘 , these passengers need to 𝑘 𝑗 󸀠 In Case 2, train will skip station ; people who board the 𝑘𝑒 = {𝑘 | 𝑘 take a train in the opposite direction, noted as min train are the confused passengers. Further define set 𝐶𝑖 to 𝑑𝑘󸀠,(2𝑁 +1−𝑖 ) >𝑑𝑘,𝑖 ∧𝑎𝑘󸀠,(2𝑁 +1−𝑖 )𝑎𝑘󸀠,(2𝑁 +1−𝑗) =1},where𝑁𝑠 𝑘 𝑠 𝑘 𝑘 𝑠 𝑘 𝑠 represent the set of trains between trains 𝑐𝑖 and 𝑘 that stop at is the total number of stations. The associated waiting time 𝑘 station 𝑖 and skip station 𝑗;thatis,𝐶𝑖 ={𝑥|𝑎𝑥,𝑖 =1∧𝑎𝑥,𝑗 =0, canbeobtainedusing(31),andthein-vehicletimecanbe 𝑘 ℎ calculated using (32) for 𝑖𝑘 <𝑗and (33) for 𝑖𝑘 >𝑗. 𝑐𝑖 ≤𝑥<𝑘}; the cardinality of the set is denoted as 𝑛; 𝑐 is used 𝑘 to denote the value of the ℎ-th element in set 𝐶𝑖 . 𝑖,𝑗 𝑖,𝑗 np =𝑟 (𝑑𝑘,𝑖 −𝑑𝑐𝑛,𝑖)∗𝑟𝑓, (30) For Case 1, the number of onboard passengers from 𝑘 𝑔 station 𝑖 to station 𝑗 via train 𝑘 can be calculated using (27). 𝑖,𝑗 𝑖,𝑗 tw𝑘 The corresponding passenger waiting time𝑘 tw and passen- 𝑖,𝑗 (31) 1 𝑖,𝑗 𝑖,𝑗 ger in-vehicle time tv𝑘 can be calculated using (28) and (29), = (𝑑𝑘,𝑖 −𝑑𝑐𝑛,𝑖) np𝑘 +(𝑑𝑘 ,𝑖 −𝑑𝑘,𝑖 + tw𝑖 ) np𝑘 , respectively. 2 𝑒 𝑘 𝑘 𝑘 𝑖,𝑗 𝑛−1 𝑖,𝑗 𝑖,𝑗 𝑖,𝑗 tv𝑘 np = ∑ (1 − 𝑟𝑓)∗𝑟 (𝑑𝑐ℎ+1,𝑖 −𝑑𝑐ℎ,𝑖)+𝑟 (𝑑𝑘,𝑖 𝑘 𝑔 𝑔 𝑖,𝑗 ℎ=1 (27) =(𝑑 −𝑑 − ) 𝑘,𝑖𝑘 𝑘,𝑖 tw𝑖𝑘 np𝑘 (32)

−𝑑𝑛 ), 𝑐 ,𝑖 +(𝑑 −𝑑 − ) 𝑖,𝑗, 𝑘𝑒,𝑗 𝑘𝑒,𝑖𝑘 tw𝑗 np𝑘 𝑛−2 1 𝑖,𝑗 = ∑ (1 − 𝑟 )[ 𝑟𝑖,𝑗 (𝑑 −𝑑 )2 𝑖,𝑗 tw𝑘 𝑓 2 𝑔 𝑐ℎ+1,𝑖 𝑐ℎ,𝑖 tv𝑘 ℎ=1 𝑖,𝑗 =(𝑑𝑘,𝑖 −𝑑𝑘,𝑖 − tw𝑖 ) np (33) 𝑖,𝑗 𝑖,𝑗 (28) 𝑘 𝑘 𝑘 +𝑟𝑔 (𝑑𝑐ℎ+1,𝑖 −𝑑𝑐ℎ,𝑖)(𝑑𝑘,𝑖 −𝑑𝑐ℎ+1,𝑖)] + 𝑟𝑔 (𝑑𝑘,𝑖 +(𝑑 −𝑑 − ) 𝑖,𝑗. 𝑘𝑒,(2𝑁𝑠+1−𝑗) 𝑘𝑒,(2𝑁𝑠+1−𝑖𝑘) tw(2𝑁𝑠+1−𝑗) np𝑘 2 −𝑑𝑐𝑛,𝑖) , By taking the summation of the above variables for all OD 𝑖,𝑗 = 𝑖,𝑗 ∗(𝑑 −𝑑 −𝑡𝑗 ). pairs of all trains, the total boarded passenger number, total tv𝑘 np𝑘 𝑘,𝑗 𝑘,𝑖 dw (29) waiting time, and in-vehicle time could be calculated, and the For Case 2, the number of confused passengers can be objectivevaluecouldthenbeobtainedusing(9).Theresults calculated using (30). They will get off at the nearest stop areprovidedinTable5. 󸀠 󸀠 denoted as 𝑖𝑘 = min{𝑖 |𝑖 >𝑖, 𝑎𝑘,𝑖󸀠 =1}after departure It is observed that as the confusion rate increases (e.g., from station 𝑖.If𝑖𝑘 <𝑗(destination has not been passed), poorer information dissemination), the number of confused 14 Journal of Advanced Transportation

Table 5: Sensitive analysis of the passenger confusion rate.

Ave. waiting time Ave. in-vehicle Ave. waiting time Number of Scheduling of con- time of con- of con- pg pg pg 𝑟𝑓 confused 𝑡 (min) 𝑡 (min) 𝑡 (min) schemes fused/unconfused fused/unconfused fused/unconfused wait veh tv passengers passengers passengers passengers All-stop 0 0 NA NA NA 2.98 13.50 16.48 0 0 NA NA NA 3.04 12.62 15.66 0.2 79 3.05 12.63 15.68 FSSS 0.4 158 3.06 12.67 15.73 0.6 234 5.75/4.42 8.74/8.19 14.49/12.61 3.08 12.70 15.78 0.8 312 3.09 12.74 15.83 1.0 390 3.10 12.77 15.87

29 27 25 23 21 19 17 15 Station 13 11 9 7 5 3 1 00:06 00:12 00:18 00:24 00:30 00:36 00:42 00:48 00:54 01:00 01:06 01:12 01:18 01:24 01:30 01:36 01:42 01:48 01:54 02:00 02:06 02:12 02:18 02:24 02:30 02:36 02:42 02:48 02:54 02:00 03:06 03:12 03:18 03:24

All-stop scheme FSSS Figure 8: Timetables of all-stop scheme and FSSS. passengers increases proportionally. However, this number accommodate the spatially and temporally varied passenger only represents a small proportion of the some 25,000 demand. The problem was formulated as a mixed integer passengers in total. This is because only the OD pairs with nonlinear programming (MINLP) model that minimizes the skipped destinations are affected, and the skipped stations average passenger travel time. The proposed scheme is able have comparably low volume. For the confused passengers, it to optimize the skip-stop scheme and it also models the train is found that their in-vehicle time and especially their waiting departure time at the departure and turnaround terminals. time increase. The increased in-vehicle time is because some The departure intervals are constrained by three types of passengers missed their destination and have to take the headways, that is, the minimum safety headway, headway that trains to the inverse direction (as in Case 2, 𝑖𝑘 >𝑗). And meets the passenger demand, and the minimum turnaround the increased waiting time is mainly because the correct train headway. A GA-based solution approach was then developed has skipped station 𝑖𝑘 and the confused passengers are not to efficiently solve the problem. The proposed scheme was able to get on the correct train at that station. Per the specific tested using the smart card data collected at a real metro line experiment, we found that FSSS always outperforms the skip- in Shenzhen, China. Time-dependent passenger demands stop scheme even when all the passengers of the skipped were considered in the experiment and the passenger con- OD pairs get confused (i.e., 𝑟𝑓 =1). Therefore the system fusionratewasalsoanalyzedinthecasestudypart. performance is stable under FSSS. The model performance was assessed with respect to the passenger benefits and the operational benefits. Overall, the 5. Conclusion FSSS is effective in timetable design. It was found that the FSSS is able to reduce the average passenger travel time by In this study, we developed a FSSS approach for bidirec- 0.82 minutes, which corresponds to 5% travel time saving. tional metro line operation during off-peak periods to better About 26.9% of total passengers benefit from the skip-stop Journal of Advanced Transportation 15 scheme. It was also observed that the average round trip travel 1.5 1000 time under the skip-stop scheme is 135.7 minutes, which 900 is more than 3 minutes shorter compared to the all-stop 800 operation. The average train departure interval is 6 minutes 1 700 600 which is the same compared to the all-stop scheme. From the 500 timetable, it was found that 35 stations are skipped. Transit 400 agencies may benefit from this scheme due to the savings of 0.5 300

energy and operational costs. Through the sensitivity analysis Seat occupancy rate 200 of passenger confusion rate, we found that the proposed 100 section load Maximum scheme always outperforms the all-stop scheme even when 0 0 most passengers of the skipped OD pairs are confused and

fail to get on the right train. 1 Train 2 Train 3 Train 4 Train 5 Train 6 Train 7 Train 8 Train 9 Train Train 10 Train This paper mainly considers the skip-stop operation Max section load under FSSS during off-peak scenarios in which the demands for some sta- Max section load under all-stop scheme tionsarequitelow.Asshownbytheresults,theskipoperation Seat occup under FSSS does not lead to excessive waiting time at these stations. For Seat occup under all-stop scheme peak-hour operations, since the passenger demands at most stations are relatively large, the skip operation may lead to Figure 9: Seat occupancy and onboard passengers per train. longer waiting time at the skipped stations. In extreme cases, the number of waiting passengers may exceed the design capacity of the station. In future work, the tradeoff between By summing up the formulations in (A.1) and (A.2) above, it in-vehicle travel time and passenger waiting time needs to be can be shown that properly leveraged. The FSSS model currently requires a rule- 𝑑 −𝑑 ≥2𝑥𝐼. based simulation to calculate the train departure time. Under 𝑘+𝑥,𝑁𝑠+1 𝑘,𝑁𝑠+1 𝑡 (A.3) the rule-based simulation, the terminal departure times need to be adjusted by the maximum travel time difference Based on constraint (12), (A.4) can be derived: between two consecutive trains. This adjustment, however, may correspond to suboptimal solutions since departure 𝑑 −𝑑 ≥𝑥𝐼 . 𝑘+𝑥,𝑁𝑠+1 𝑘,𝑁𝑠+1 min (A.4) choice can be made more wisely at each station including the intermediate stations. The GA-based solution approach Therefore, given the turnaround departure time of train 𝑘,the requires a predetermined set (SS) that specifies the stations turnaround departure time of train 𝑘+𝑥should satisfy that can be skipped. This set needs to be determined using a more rigorous approach. The authors are also interested in 𝑑 −𝑑 ≥ {2𝑥𝐼 ,𝑥𝐼 } 𝑘+𝑥,𝑁𝑠+1 𝑘,𝑁𝑠+1 max 𝑡 min extending the current methodology to optimize the skip-stop (A.5) operation for a network-wide metro system. More results and ∀𝑐=1∧𝑐≤𝑥≤𝑁 −𝑘. findings will be provided in our subsequent studies. tr For the cases with more than one turnaround line, similar Appendix derivations are made as follows. When 𝑐>1∧𝑐≤𝑥≤𝑁 −𝑘,wehave We consider 𝑐 turnaround lines that can be utilized to store tr trains at the turnaround terminal. Below we prove that, under 𝑑𝑘+𝑥,𝑁 +1 −𝑑𝑘,𝑁 +1 ≥2𝐼𝑡. (A.6) certain circumstances, more turnaround lines (i.e., 𝑐>1)can 𝑠 𝑠 lead to reduced departure time at the turnaround station. Since (A.4) also holds for this case, the turnaround departure Letusfirstconsidersingleturnaroundline(i.e.,𝑐=1). time of train 𝑘+𝑥should satisfy (A.6). According to constraint (19), the following relationships hold. 𝑐=1∧𝑐≤𝑥≤𝑁 −𝑘 When tr , 𝑑 −𝑑 ≥ {2𝐼 ,𝑥𝐼 } 𝑘+𝑥,𝑁𝑠+1 𝑘,𝑁𝑠+1 max 𝑡 min (A.7) 𝑑𝑘+1,𝑁 −𝑑𝑘,𝑁 +1 ≥𝐼𝑡 ∀𝑐>1∧𝑐≤𝑥≤𝑁 −𝑘. 𝑠 𝑠 tr

𝑑𝑘+2,𝑁 −𝑑𝑘+1,𝑁 +1 ≥𝐼𝑡 𝑠 𝑠 𝑥𝐼 ≥2𝐼 ≥𝐼 Since min 𝑡 min isusuallytrueinrealpractices,the . (A.1) following relationships hold: .

𝑑𝑘+𝑥,𝑁 +1 −𝑑𝑘,𝑁 +1 ≥2𝑥𝐼𝑡 𝑑 −𝑑 ≥𝐼. 𝑠 𝑠 𝑘+𝑥,𝑁𝑠 𝑘+𝑥−1,𝑁𝑠+1 𝑡 ∀𝑐=1∧𝑐≤𝑥≤𝑁 −𝑘, tr For train 𝑘+𝑥, formulation (A.2) also holds: (A.8) 𝑑 −𝑑 ≥𝑥𝐼 𝑘+𝑥,𝑁𝑠+1 𝑘,𝑁𝑠+1 min 𝑑 −𝑑 ≥𝐼. 𝑘+𝑥,𝑁𝑠+1 𝑘+𝑥,𝑁𝑠 𝑡 (A.2) ∀𝑐>1∧𝑐≤𝑥≤𝑁 −𝑘. tr 16 Journal of Advanced Transportation

2𝑥𝐼 ≥𝑥𝐼 Since 𝑡 min,itcanbeconcludedthattheturnaround Transportation Research Part C: Emerging Technologies,vol.61, departure time can be reduced if they are more than one pp.63–74,2015. 2𝐼 ≥𝐼 turnaround line and 𝑡 min. [15]Z.Cao,Z.Yuan,andD.Li,“Estimationmethodforaskip-stop operation strategy for urban rail transit in China,” Journal of Conflicts of Interest Modern Transportation,vol.22,no.3,pp.174–182,2014. [16] L. Zheng, R. Song, S. W. He, and H. D. Li, “Optimization The authors declare that they have no conflicts of interest. model and algorithm of skip stop strategy for urban rail transit,” Journal of the Society,vol.6,pp.135–164,2009. Acknowledgments [17] S. L. Sogin, B. M. Caughron, and S. G. Chadwick, “Optimizing skip stop service in passenger rail transportation,” in Proceed- This research was financially supported by the National Nat- ings of the 2012 Joint Rail Conference, JRC 2012,pp.501–512,April ural Science Foundation of China under Grant no. 71671147. 2012. 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