Hindawi Publishing Corporation Advances in Mechanical Engineering Volume 2013, Article ID 848292, 12 pages http://dx.doi.org/10.1155/2013/848292

Research Article Coordination Optimization of the First and Last Trains’ Departure Time on Urban Rail Transit Network

Wenliang Zhou, Lianbo Deng, Meiquan Xie, and Xia Yang

School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China

Correspondence should be addressed to Meiquan Xie; [email protected]

Received 17 August 2013; Revised 11 October 2013; Accepted 6 November 2013

Academic Editor: Wuhong Wang

Copyright © 2013 Wenliang Zhou 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.

Coordinating the departure times of different line directions’ of first and the last trains contributes to passengers’ transferring. In this paper, a coordination optimization model (i.e., M1) referring to the first train’s departure time is constructed firstly to minimize passengers’ total originating waiting time and transfer waiting time for the first trains. Meanwhile, the other coordination optimization model (i.e., M2) of the last trains’ departure time is built to reduce passengers’ transfer waiting time for the last trains and inaccessible passenger volume of all origin-destination (OD) and improve passengers’ accessible reliability for the last trains. Secondly, two genetic algorithms, in which a fixed-length binary-encoding string is designed according to the time interval between the first train departure time and the earliest service time of each line direction or between the last train departure time and the latest service time of each line direction, are designed to solve M1 and M2, respectively. Finally, the validity and rationality of M1, M2, and their solving genetic algorithms are verified with numerical analysis, in which the effects of the parameters in M1 and M2 on coordination optimization result are analyzed.

1. Introduction linear programming model to optimize the train timetable for minimizing the total train travel time, subject to overtaking With the rapid development of urban rail transportation, and crossing headway constraints. Higgins and Kozan [2] urban rail transit networks have been formed in Beijing, described the development and use of a model designed to Shanghai, Guangzhou, and many other big cities in China. optimize train schedules on single line rail corridors. Zhou Under the network operation and management of urban and Zhong [3] proposed a generalized resource-constrained rail transportation, passengers’ alternative travel routes are project scheduling formulation to minimize the total train increased substantially because they can choose different sta- travel time of the train timetable on the single-track railway. tions to transfer, which greatly facilitates passengers’ traveling Li et al. [4] present a simulation method for solving the train but also increases the operation and organization difficulties timetabling problem to minimize the total travel time on of urban rail transportation. Coordinating trains of different the single-track railway. Zhou and Zhong [5]proposeda linesisoneofthemainproblemsofnetworkoperationof multimode resource-constrained project scheduling formu- urban rail transportation. The coordination of arrival and lation to consider acceleration and deceleration time losses departure time of trains from different line directions at in double-track train timetabling applications. Carey [6]and transfer time not only can effectively reduce the passenger Carey and Lockwood [7] solved the train timetabling and transfer waiting time, but also can make lines transport pathing problem in a rail network with one-way and two-way capacities match each other better to improve all trains’ tracks. Lee and Chen [8] presented an optimization heuristic operation efficiencies on urban rail transit network. that includes both train pathing and train timetabling. Carey Train schedule optimization was studied extensively, and and Carville [9] devoted to scheduling and platforming trains a series of excellent achievements has been made. The original at busy complex stations. researches aimed more at train schedule optimization of It is very difficult to solve the railway train schedule. Cai only one line. For example, Szpigel [1]firstdevelopeda et al. [10]andCapraraetal.[11] regarded the train scheduling

Downloaded from ade.sagepub.com by guest on June 23, 2016 2 Advances in Mechanical Engineering problemtobeNP-hard.Therearenoaccuratealgorithms train schedules of rail transit networks only with transfer time for solving the train schedule of large-scale and complex [20, 21], or both the transfer time and vehicle operating cost rail network within an acceptable time nowadays. In order [20]. Carey and Crawford [21] were devoted to finding and to obtain a satisfactory train schedule within an acceptable resolving the conflicts in draft train schedules on a network time, the heuristic algorithms are usually designed to solve of busy complex stations. Ghoseiri et al. [22]developinga this problem. Greenberg [12], JovanovicandHarker[´ 13], and multiobjective optimization model for the passenger train Higgins et al. [14] developed a branch-and-bound solution scheduling problem on a railroad network. Liu and Kozan framework to find feasible timetables. Kraft15 [ ]presented [15] presented a feasibility satisfaction procedure algorithm a branch-and-bound approach for solving the train conflict to solve the train scheduling problem which is regarded as to minimize a weighted sum of delay. Based on the branch- a blocking parallel-machine job shop scheduling problem. and-bound algorithm, Zhou and Zhong [5]incorporated Burdett and Kozan [19]proposedanovelhybridjobshop effective dominance rules into this algorithm to generate approach to create new train schedules. Based on the TAS pareto solution for train scheduling problem. Zhou and method proposed by Dorfman and Medanic [16], Li et al. [4] Zhong [3] presented to use a Lagrangian relaxation base lower proposed an algorithm based on the global information of the bound rule, an exact lower bound rule, and a tight upper train to obtain an effective travel advance strategy of the train. boundwhichareadaptedinittoreducethesolutionspace. The coordination of the first and last trains’ arrival and The priority rules, which determine the priority of each train departure time at transfer station is particularly prominent on in a conflict, depend on an estimate of the remaining crossing urban rail transit network. Most reviewed researches about and overtaking delay, as well as the current delay used in some the first and last trains only struggled to reduce passenger algorithm can effectively improve the optimal quality of train transfer waiting time; however, besides that, the first train timetable. Carey [6] and Carey and Lockwood [7]devel- departure time influences passengers origin waiting time oped an iterative decomposition approach which contains for the first train, while the last train departure time affects the several node branching, variable fixing, and bounding the inaccessible passenger volume and passengers’ accessible strategies to reduce the search space for solving the train reliability for last train. These passengers travel requirements timetabling and pathing problem. Dorfman and Medanic are worthy of concerns especially when passengers care more [16] incorporated some priority rules into a discrete event about their travel service levels today. After analyzing the simulation framework to solve large-scale real-world train organization requirements of the first and last trains on urban scheduling problem. S¸ahin [17], Higgins et al. [14], and Liu rail transit network this paper aims to coordinate the first and and Kozan [18] extended some priority rules to backtracking last trains’ departure time of each line direction so that first search, look-ahead search, and metaheuristic algorithms for and last trains’ arrival and departure time connect better at train scheduling, respectively. Burdett and Kozan [19]divided each of the transfer stations to meet as far as possible all travel the scheduling process into two levels: global scheduling service requirements of passenger. to establish an initial train diagram without considering The organization of this paper is as follows. Section 2 conflicts and local scheduling to repair conflicts. Dorfman analyzes the coordination optimization goals and constraints and Medanic [16] developed a local feedback based travel of the first and last trains’ departure time of all rail line advance strategy (TAS) by using a discrete event model of the directions. Section 3 describes passengers’ travel route choice train advance along the lines of railway. Li et al. [4], based on problem based on Logit model. Section 4 presents two the TAS method proposed an algorithm based on the global optimization models, respectively, for the first and last trains’ information of the train to obtain an effective travel advance departure time on urban rail transit network. Section 5 strategy of the train. In some literature, the train scheduling deals with the genetic algorithm development, and Section 6 problem is modeled as a blocking parallel-machine job shop reports on our computational experiments. Finally some scheduling problem solved by the alternative graph model conclusions are drawn in Section 7. in some literature. Liu and Kozan [18]presentafeasibility satisfaction procedure algorithm to solve the train scheduling problem which regarded as a blocking parallel-machine job 2. Analysis of the First and Last shop scheduling problem. Burdett and Kozan [19]proposeda Trains’ Departure Time Coordination novel hybrid job shop approach to create new train schedules. With network operation of urban rail transportation, In this section, some assumptions and symbol definitions more researchers became interested in train schedule opti- are given firstly for the coordination optimization of the mization of the urban rail transit network. Compared to a first and last trains’ departure time in Section 2.1.Thenthe single line, a new problem, that is, the arrival and departure coordination optimization goals of the first and last trains’ time coordination of trains with different line directions, departure time are analyzed, respectively, in Sections 2.2 and has arisen in the schedule optimization of urban rail transit 2.3, and the constraints of the first and last trains’ departure network. An optimal train schedule of urban rail transit timeareanalyzedinSection2.4. network is extremely difficult to obtain, even for a small urban railtransitnetworkbecauseofthelargenumberofvariables and constraints, the discrete nature of the variables, and the 2.1. Symbol Definitions and Assumptions. Coordination opti- nonlinearity of the objective function and the constraints. mization of the first and last trains’ departure time based In the past, attempts have been made to develop optimal on urban rail transit network and passenger travel demands

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1 should not only meet passenger originating wait time require- 𝑦𝑙𝑘: the first train’s arrival time of rail line direction 𝑙 ment on each rail line direction, but also make the first at station 𝑘; and last trains’ arrival and departure time of each rail line 𝑒 𝑥𝑙𝑘: the last train’s departure time of rail line direction direction effectively connect at transfer stations so as to 𝑙 at station 𝑘; better meet passenger transfer time requirement from one 𝑒 𝑦𝑙𝑘: the last train’s arrival time of rail line direction 𝑙 at rail line to another, especially to reduce the inaccessible 𝑘 passengers volume on urban rail transit network. In this station ; 𝑚 paper, the coordination optimization of the first and last 𝑥𝑙𝑘: departure time of train 𝑚 from station 𝑘 on rail trains’ departure time of each rail line direction is based on line direction 𝑙; the following assumptions. 𝑚 𝑦𝑙𝑘 : arrival time of train 𝑚 at station 𝑘 on rail line Assumption 1. The origin and terminal station of the first and direction 𝑙; 𝑠 last trains of one line direction are, respectively, line’s two 𝑡𝑙 : the earliest possible service time of rail line direc- terminal stations, if this line is noncircular, otherwise, they tion 𝑙; are the station connecting to the railcars servicing depot. 𝑒 𝑡𝑙 : the latest possible service time of rail line direction Assumption 2. Each train capacity meets all passengers’ 𝑙; boarding requirements on each line direction, which means 𝑘 𝑏 󸀠 : passenger minimum walk time while passengers that all passengers can get on the earliest arriving train after 𝑙𝑙 transfers from rail line direction 𝑙∈𝐿𝑘 to rail line they arrived at station. 󸀠 direction 𝑙 ∈𝐿𝑘 at transfer station 𝑘; 𝑠 Assumption 3.Passengers’travelroutechoiceobeystheLogit 𝐼𝑙 : train departure time interval of rail line direction 𝑙 assignment principle based on generalized travel cost while morning; there are multiple existing routes to be chosen for passengers. 𝐼𝑒 𝑙 The symbols related to coordination optimization include 𝑙 : train departure time interval of rail line direction the symbols for describing urban rail transit network, OD at night; 𝑘,𝑘+1 travel demands, and a series of train operation parameters. 𝑡𝑙 : Train section running time including starting All symbols used in this paper are defined as follows: and stopping additional time in secton (𝑘, 𝑘 + 1) on 𝑙 𝐿 𝑙∈𝐿 rail line direction ; : the set of rail line directions, ; 𝑘 𝜏𝑙 : train stop time at station 𝑘 on rail line direction 𝑙. 𝐾𝑙: the station sequence of rail line direction 𝑙∈𝐿, 𝑘∈𝐾𝑙; 𝐾 2.2. Coordination Optimization Goals of First Train’s Depar- ℎ:thesetoftransferstations; ture Time. The coordination optimization of first train’s 𝐿𝑘: the set of rail line directions linked to transfer departure time should not only reduce passengers’ originat- station 𝑘, 𝐿𝑘 ∈𝐿; ing wait time for first train, but also make all first trains’ 𝐸:thesetofODpairs,inwhich(𝑟, 𝑠) ∈𝐸 represents arrival and departure time effectively connect at transfer theODfromoriginstation𝑟 to terminal station 𝑠; station to minimize passengers transfer wait time from one rail line to another to the greatest extent. Therefore, the 𝑃𝑟𝑠(𝑡): the set of alternative travel routes for OD pair coordination optimization goals of first trains’ departure time (𝑟, 𝑠) passengers arriving at station at time 𝑡,inwhich on urban rail transit network are selected as follows in this each travel route 𝑝∈𝑃𝑟𝑠(𝑡) is composed one or more paper. ordered rail line directions; (1) Reduce Passengers’ Total Originating Wait Time for First 𝐿𝑝: the set of rail line directions constituting travel Train. Passenger’s originating wait time for first train is the route 𝑝; total time from the passenger’s arriving at the station to his or 𝑡s (𝑟, 𝑠) 𝑟𝑠: the earliest trip time of OD pair passenger; her getting on the first train. Obviously, originating wait time 𝑒 𝑤s (𝑡) (𝑟, 𝑠) 𝑟 𝑡𝑟𝑠: the latest trip time of OD pair (𝑟, 𝑠) passenger; 𝑟𝑠 of OD passengers, who arrive at origin station 𝑡(𝑡<𝑥1 ) 𝜆𝑟𝑠(𝑡):ODpair(𝑟, 𝑠) passenger’s arrival rate with time at time 𝑙𝑟 and then get on the first train of rail line s 𝑒 𝑙 𝑡 (𝑡𝑟𝑠 ≤𝑡≤𝑡𝑟𝑠); direction ,isasfollows: 𝑝 𝜑 (𝑡) 𝑝∈ s 1 𝑟𝑠 : selection probability of alternative route 𝑤𝑟𝑠 (𝑡) =𝑥𝑙𝑟 −𝑡. (1) 𝑃𝑟𝑠(𝑡) for passengers arriving at station at time 𝑡, which can be calculated according to passenger gen- The total originating wait time 𝑊𝑙 forthefirsttrainofrail eralized travel cost of each alternative travel route; line direction 𝑙 is calculated as follows: 𝑥1 𝑥𝑠 𝑙 : the first train’s departure time of rail line direction 𝑙𝑟 𝑝 𝑠 𝑙 𝑊𝑙 = ∑ ∑ ∫ 𝜆𝑟𝑠 (𝑡) 𝜑𝑟𝑠 (𝑡) 𝑤𝑟𝑠 (𝑡) d𝑡. ; 𝑡𝑠 𝑒 (𝑟,𝑠):(𝑟,𝑠)∈𝐸, 𝑟∈𝐾𝑙 𝑝:𝑝∈𝑃𝑟𝑠,𝑙∈𝐿𝑝 𝑟𝑠 𝑥𝑙 : the last train’s departure time of rail line direction 𝑙; (2) 1 𝑥𝑙𝑘: the first train’s departure time of rail line direction The first coordination optimization goal of first trains’ 𝑙 at station 𝑘; departure time is to reduce the total originating wait time

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Trains on line l Decreasing passengers’ transfer wait time for first train is ··· regarded as the second coordination optimization goal of first trains’ departure time, namely, First train 2 3 Transfer ∑ ∑ 𝐻𝑚s. station k min 𝑙𝑙󸀠 Time 󸀠 󸀠 (6) 1 ··· 𝑘∈𝐾ℎ 𝑙,𝑙 ∈𝐿𝑘;𝑙=𝑙̸ First train 󳰀 Trains on line l 2.3. Coordination Optimization Goals of Last Trains’ Depar- Figure 1: Passengers’ transferring between two line directions ture Time. Passengers arriving at station after the last train linked to transfer station 𝑘. departing will not reach their destination by urban rail transportation because there are no trains to take them at Trains on line l that time, which is called inaccessibility. Compared with the first trains’ departure time, the last trains’ departure time ··· Last train influences passenger’s travel quality more extensively. It not 2 only determines passengers transfer wait time for the last Transfer train, but also affects network inaccessible passenger volume station k Time and passenger accessible reliability for last trains. In this ··· 1 Last train Trains on line paper, the coordination optimization goals of last trains’ departure time are summarized as follows. Figure 2: Passenger transfer to a line direction last train. (1) Minimize Passengers’ Transfer Wait Time for the Last Train. Passenger transfers from one line direction train to another 𝑊𝑙 of originating passengers on urban rail transit network, line direction last train can be shown in Figure 2. 𝑥𝑒 namely, Only when last train departure time 𝑙󸀠𝑘 of rail line 󸀠 𝑚 direction 𝑙 is more than train 𝑚 arrival time 𝑦𝑙𝑘 of rail line min ∑𝑊𝑙. 𝑘 (3) direction 𝑙 plus passenger minimum walk time 𝑏 󸀠 between 𝑙∈𝐿 𝑙𝑙 these two rail line directions at transfer station 𝑘 and any 󸀠 (2) Decrease Passengers’ Transfer Wait Time for First Train. other direction 𝑙 nonlast train’s departure time at transfer 𝑒−1 𝑚 𝑘 𝑒 Generally, there are four transferring cases as shown in station 𝑘 is less than that, namely, 𝑥𝑙󸀠𝑘 ≤𝑦𝑙𝑘 +𝑏𝑙𝑙󸀠 ≤𝑥𝑙󸀠𝑘 is Figure 1: (1) from one line direction first train to another line established, passengers will conveniently transfer from train 󸀠 direction first train, (2) from one line direction nonfirst train 𝑚 of direction 𝑙 to the last train of rail line direction 𝑙 ,and to another line direction first train, (3) from one line direction theirtransferwaittimeisasfollows: nonfirst train to another line direction nonfirst train, and (4) 𝑚𝑒 𝑒 𝑚 𝑘 from one line direction first train to another line direction ℎ𝑙𝑙󸀠 =𝑥𝑙󸀠𝑘 −𝑦𝑙𝑘 −𝑏𝑙𝑙󸀠 . (7) nonfirst train. In view that we only discuss the coordination 𝐻𝑚𝑒 𝑚 optimization of first train departure time, two transferring The total transfer wait time 𝑙𝑙󸀠 from train of rail line 󸀠 cases, which passenger transferred from one line direction direction 𝑙 to rail line direction 𝑙 last train at transfer station 𝑚𝑒 first train or nonfirst train to another line direction first train, 𝑘 is obtained as follows by summing up transfer wait time ℎ𝑙𝑙󸀠 are considered only in this paper. of each passenger: 1 Only when first train departure time 𝑥𝑙󸀠𝑘 of rail line 𝑚e 𝑚𝑒 𝑚𝑒 󸀠 𝑚 𝐻 󸀠 = ∑ ℎ 󸀠 𝑞 󸀠 , direction 𝑙 is not less than train 𝑚 arrival time 𝑦𝑙𝑘 of rail line 𝑙𝑙 𝑙𝑙 𝑘𝑙𝑙 𝑘 𝑒−1 𝑚 𝑘 𝑒 (8) 𝑚:𝑚≥1; 𝑥 󸀠 ≤𝑦 +𝑏 󸀠 ≤𝑥 󸀠 direction 𝑙 plus passenger minimum walk time 𝑏𝑙𝑙󸀠 ,namely, 𝑙 𝑘 𝑙𝑘 𝑙𝑙 𝑙 𝑘 𝑚 𝑘 1 𝑦𝑙𝑘 +𝑏𝑙𝑙󸀠 ≤𝑥𝑙󸀠𝑘 is established, passengers will conveniently 𝑚𝑒 󸀠 where 𝑞𝑘𝑙𝑙󸀠 is the passenger volume transferring from train transfer from train 𝑚 of direction 𝑙 to direction 𝑙 first train, 󸀠 𝑚 of rail line direction 𝑙 to rail line direction 𝑙 last train at and their transfer wait time is transfer station 𝑘. 𝑚𝑠 1 𝑚 𝑘 ℎ𝑙𝑙󸀠 =𝑥𝑙󸀠𝑘 −𝑦𝑙𝑘 −𝑏𝑙𝑙󸀠 . (4) Reducing passengers’ transfer wait time for last trains is regarded as the first coordination optimization goal of last 𝐻𝑚s The total transfer wait time 𝑙𝑙󸀠 of all passengers transfer- trains’ departure time, namely, ring from train 𝑚 of rail line direction 𝑙 to rail line direction 󸀠 𝑚𝑒 𝑙 first train at transfer station 𝑘 is obtained as follows by ∑ ∑ 𝐻 󸀠 . 𝑚𝑠 min 𝑙𝑙 ℎ 𝑘∈𝐾 󸀠 󸀠 (9) summing up transfer wait time 𝑙𝑙󸀠 of each passenger: ℎ 𝑙,𝑙 ∈𝐿𝑘;𝑙=𝑙̸

𝑚s 𝑚𝑠 𝑚𝑠 𝐻𝑙𝑙󸀠 = ∑ ℎ𝑙𝑙󸀠 𝑞𝑘𝑙𝑙󸀠 , (2) Decreasing Network Inaccessible Passengers Volume.A 𝑚:𝑚≥1; 𝑦𝑚+𝑏𝑘 ≤𝑥s (5) 𝑙𝑘 𝑙𝑙󸀠 𝑙󸀠𝑘 travel route is inaccessible when passenger arrives at the station after the last train departs from there in the process 𝑞𝑚𝑠 where 𝑘𝑙𝑙󸀠 is the passenger volume transferring from train of travel with this route. For OD (𝑟, 𝑠) passengers, if all 󸀠 𝑚 of rail line direction 𝑙 to rail line direction 𝑙 first train at their alternative travel routes are inaccessible, they will not transfer station 𝑘. reach their destination by urban rail transportation on that

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𝑑 day, which is called OD inaccessibility. Denote 𝑡𝑝 as the routes in which passengers have to transfer to one or more last latest accessible time of travel route 𝑝∈𝑃𝑟𝑠,whichcanbe trains. The third coordination optimization goal of last trains’ calculated according to the last train departure time on each departure time is to improve travel route accessible reliability rail line direction included in this travel route and passenger withaweightofpassengervolume,whichisequalto minimum walk time between two adjacent line directions. 𝑝 𝑝 𝑑 min ∑ ∑ (1 − 𝑅 )𝑞 , Then the latest accessible time 𝑡𝑟𝑠 of OD (𝑟, 𝑠) passenger is 𝑟𝑠 𝑟𝑠 (15) (𝑟,𝑠)∈𝐸 𝑝∈𝑃 calculated as follows: 𝑟𝑠 𝑝 𝑑 𝑑 where 𝑞𝑟𝑠 is the passenger volume of OD (𝑟, 𝑠) selecting route 𝑡𝑟𝑠 = max {𝑡𝑝 |𝑝∈𝑃𝑟𝑠}. (10) 𝑝∈𝑃𝑟𝑠 to travel. 𝑄𝑏 (𝑟, 𝑠) In fact, the third goal contradicts with the first goal in the Therefore, the inaccessible passenger volume 𝑟𝑠 of OD process of coordination optimization of last trains’ departure is identified as follows: time. Thus, these two goals should be balanced according to 𝑡𝑒 passengers’ actual requirements. So each goal is given weight 𝑏 𝑟𝑠 𝑄𝑟𝑠 = ∫ 𝜆𝑟𝑠 (𝑡) d𝑡. (11) and weighed by setting the weight value. 𝑑 𝑡𝑟𝑠

Minimizing inaccessible passenger volume of all ODs 2.4. Constraints of First and Last Trains’ Departure Time. is another coordination optimization goal of last trains’ Besides meeting passengers’ travel requirement, first and last departure time, namely, trains’ departure times should not only satisfy the earliest and latest operating time of each rail line but also fulfill ∑ 𝑄𝑏 . train minimum occupancy rate requirement to ensure train min 𝑟𝑠 (12) (𝑟,𝑠)∈𝐸 operation efficiency. The constraints of meeting earliest operating time and (3) Improve Passengers’ Accessible Reliability for Last Trains. minimum occupancy rate of each line direction first train can Passenger’s transfer time at transfer station changes randomly be expressed as follows: because of the influence of some random factors such as 1 𝑠 train arrival delay and walk interference transferring between 𝑥𝑙 ≥𝑡𝑙 , different two line directions. The random uncertainty of pas- (16) 𝜂𝑠 ≥ 𝜂̂𝑠, senger transfer time may lead to passenger fail in transferring. 𝑙 𝑙 If this fail is just for transferring to a nonlast train, it has 𝑠 𝑠 where 𝜂̂𝑙 ,𝜂𝑙 is the minimum required and actual effective no effect on passengers because they still have other trains 𝑙 to transfer. Otherwise, it affects passengers badly because occupancy rate when the first train of line direction departs: this transfer fail leads to their travel inaccessibility. Transfer 𝑡𝑝𝑙 ∑ ∑ 𝑑𝑝𝑙 ∫ 𝑟𝑠 𝜆𝑠 (𝑡) 𝜑𝑠𝑝 (𝑡) 𝑡 reliability for a last train is defined to describe the probability (𝑟,𝑠)∈𝐸 𝑝:𝑙∈𝑝∈𝑃𝑟𝑠 𝑟𝑠 𝑡𝑠 𝑟𝑠 𝑟𝑠 d 𝜂𝑠 = 𝑟𝑠 , (17) that passengers transfer smoothly to a last train with a given 𝑙 𝑑𝑙𝜀𝑙 longest transfer time. The transfer reliability for a nonlast ̂ ℎ 󸀠 𝑝𝑙 train is regarded as 1. Denote 𝑙𝑙 as the maximum transfer where 𝑑 is the travel mileage on line direction 𝑙 for OD (𝑟, 𝑠) 𝑙󸀠 𝑙 𝑟𝑠 time to line direction lasttrainfromlinedirection , passengers travel with route 𝑝, 𝑑𝑙 is the total mileage of line which is determined according to last train running time, and 𝑙 𝜀 󸀠 direction ,and 𝑙 is train’s fixed passenger number of line ℎ 󸀠 𝑙 𝑙 denote 𝑙𝑙 as the transfer time between direction and ;then direction 𝑙. last train transfer reliability is described as follows: The constraints of meeting latest operating time and ̂ minimum occupancy rate of each line direction last train can 𝑅 󸀠 =𝑝{ℎ 󸀠 ≤ ℎ 󸀠 }. 𝑙𝑙 𝑙𝑙 𝑙𝑙 (13) be described as follows: 𝑥𝑒 ≥𝑡𝑒, Passengers will reach their destinations with a travel 𝑙 𝑙 (18) route only when they successfully transfer between any two 𝜂𝑒 ≥ 𝜂̂𝑒, connecting line directions included in this route. Travel route 𝑙 𝑙 can be regarded as a system with one or more ordered transfer 𝑒 𝑒 where 𝜂̂ ,𝜂 is the minimum required and actual effective stations which are independent of each other. Thus, accessible 𝑙 𝑙 𝑝 occupancy rate when the last train of line direction 𝑙 departs, reliability 𝑅𝑟𝑠(𝑡) of OD (𝑟, 𝑠) passengers, who arrive at station 𝑒 of which 𝜂𝑙 is calculated according to formula (17)analo- at time 𝑡 and choose route 𝑝∈𝑃𝑟𝑠(𝑡) for traveling, is gously. represented as

𝑝 𝑅 (𝑡) = ∏ 𝑅 󸀠 . 3. Passenger Travel Route Choice Based on 𝑟𝑠 𝑙𝑙 (14) 𝑙,𝑙󸀠∈𝑝;+ 𝑙 =𝑙󸀠 Logit Model In general, the more passengers transfer to a last train in a The choice of travel route mainly depends on originating wait travel route, the accessible reliability of this route gives more time, travel time, and ticket price of each alternative travel weight. This paper only considers the accessible reliability of route.ItisworthnotingthatODpassengersalternativetravel

Downloaded from ade.sagepub.com by guest on June 23, 2016 6 Advances in Mechanical Engineering routesarenotfixedandsomeroutesbecomeinaccessibleafter are decided by absolute cost differences between two routes’ the last train departs, so these routes will no longer be selected travel cost, which will lead to some unreasonable results in by passengers. the allocation process. Therefore, route choice probabilities For OD (𝑟, 𝑠) passengers arriving at time 𝑡,towhomtravel based on relative cost differences are calculated as follows: route 𝑝∈𝑃𝑟𝑠(𝑡) satisfies their inaccessibility requirement, their considered costs of travelling with travel route 𝑝 are (−𝜃𝑐𝑝 /𝑐𝑚) calculated as follows. 𝜉𝑝 (𝑡) = exp 𝑟𝑠 𝑟𝑠 , 𝑟𝑠 ∑ (−𝜃𝑐𝑝 /𝑐𝑚) (1) Originating Wait Time. Originating wait time is the time 𝑝 exp 𝑟𝑠 𝑟𝑠 (23) from the passengers’ arriving at station to their getting on 𝑝 ∑ 𝜉𝑝 =1, train. Denoting 𝑥𝑟𝑠(𝑡) as the departure time of the train, 𝑟𝑠 𝑝∈𝑃 which arrived at station earliest after time 𝑡, the originating 𝑟𝑠 wait time at time 𝑡 is calculated as follows: 𝑝 𝑝 𝜉𝑝 (𝑡) 𝑝∈𝑃 𝑤𝑟𝑠 (𝑡) =𝑥𝑟𝑠 (𝑡) −𝑡. (19) where 𝑟𝑠 is the probability of choosing route 𝑟𝑠 for passengers of OD (𝑟, 𝑠) arriving at time 𝑡, 𝜃 is the parameter 𝑚 (2)TravelTime. Passengers travel time contains all the time of Logit model, and 𝑐𝑟𝑠 is the minimum generalized travel cost taken from the origin to the destination, including train of all alternative travel routes, which is calculated as follows: running time in sections, train stop time at stations, transfer walking time, and transfer wait time. If all routes’ travel time is close, the less transfer times in a travel route, the greater 𝑚 𝑝󸀠 󸀠 𝑐𝑟𝑠 = min {𝑐𝑟𝑠 |𝑝 ∈𝑃𝑟𝑠}. (24) therateofbeingselected.Therefore,thesametransfertime outweighs the same travel time on train, and the transfer time should be appropriately enlarged with a punish coefficient. 𝑝 Finally, the passenger volume of each alternative travel Denote 𝜇 (𝑡) as the travel time of OD (𝑟, 𝑠) passengers 𝑟𝑠 route is obtained according to each route’s choiced probabil- arriving at time 𝑡 and selecting route 𝑝∈𝑃𝑟𝑠(𝑡) for travel; ity,andthenoriginatingpassengersandtransferpassengers then it is calculated as follows: of the first and last trains in each line can be calculated based 𝑝 𝑝 𝑝 on the route passenger volume. 𝜇𝑟𝑠 (𝑡) =𝑎𝑟𝑠 +𝛿𝑏𝑟𝑠, (20)

𝑝 𝑝 where 𝑎𝑟𝑠 and 𝑏𝑟𝑠 are the travel time on train and the transfer time at transfer stations from passengers’ original station to 4. Coordination Optimization Model of their destination station, respectively, and 𝛿 is the punish First and Last Trains’ Departure Time coefficient for enlarging the transfer time. 4.1. Coordination Optimization Model of First Trains’ Depar- (3) Ticket Price. For convenience, ticket price is the product ture Time. The coordination optimization model M1, whose 1 of fare rate per mileage per passenger and passenger travel decision variables are first trains’ departure time 𝑥𝑙 (𝑙∈ mileages, which means that the longer the travel route, the 𝐿) of each line direction, is constructed to minimize the 𝜑𝑝 (𝑡) higher the ticket price. Denote 𝑟𝑠 as the ticket price paid total of originating wait time and transfer wait time for first (𝑟, 𝑠) 𝑡 by passengers of OD arriving at time and selecting trains subject to constraints of train’s minimum effective 𝑝∈𝑃 (𝑡) route 𝑟𝑠 for travel; then it is calculated as follows: occupancy rate requirement and earliest service time of rail line direction: 𝜑𝑝 (𝑡) =𝛾∑ 𝑑𝑝, 𝑟𝑠 𝑙 (21) 𝑙∈𝐿𝑝 𝑧 = ∑𝑊 + ∑ ∑ 𝐻𝑚s 𝑝 min 𝑠 𝑙 𝑙𝑙󸀠 𝛾 𝑑 󸀠 󸀠 where is the product of fare rate and 𝑙 is the travel mileage 𝑙∈𝐿 𝑘∈𝐾ℎ 𝑙,𝑙 ∈𝐿𝑘;𝑙 =𝑙̸ on line direction 𝑙 of passengers traveling with route 𝑝. 𝛽 1 𝑠 (25) By introducing passenger travel value of time to s.t.𝑥𝑙 ≥𝑡𝑙 ,𝑙∈𝐿, unify the unit of each travel cost, generalized travel cost of 𝜂𝑠 ≥ 𝜂̂𝑠,𝑙∈𝐿. passengers of OD (𝑟, 𝑠) arriving at time 𝑡 and traveling with 𝑙 𝑙 route 𝑝∈𝑃𝑟𝑠(𝑡) is described as follows: 𝑝 𝑝 𝑝 𝑝 𝑞 𝑐𝑟𝑠 (𝑡) =𝛽(𝑤𝑟𝑠 (𝑡) +𝜇𝑟𝑠 (𝑡))+𝜑𝑟𝑠 (𝑡) +𝜀𝑟 , (22) 4.2. Coordination Optimization Model of Last Trains’ Depar- ture Time. The coordination optimization model M2 is built 𝑞 𝑒 where 𝜀𝑟 is the random error term of generalized travel cost. to optimize last trains’ departure time 𝑥𝑙 (𝑙∈𝐿)ofall Generally, the less route generalized travel cost, the rail line direction on urban rail transit network, which greater route choice probability. Assuming that the random aims to reduce passengers’ transfer wait time for last trains 𝑞 error terms 𝜀𝑟 of each route travel cost are independent and and inaccessible passenger volume of all ODs, and improve obeythesameGumbeldistribution,routeselectedproba- passengers’ accessible reliability for last trains subject to bility is obtained by Logit stochastic route choice model. the constraints of trains’ minimum effective occupancy rate InthetraditionalLogitmodel,allroutechoiceprobabilities requirement and latest service time of rail line direction.

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The goal of this model is to minimize the generalized costs Encode for the problem of considering three objectives simultaneously as follows: coordination optimization of the first and last trains’departure time 𝑚e min 𝑧𝑒 =𝛼∑ ∑ 𝐻𝑙𝑙󸀠 󸀠 󸀠 𝑘∈𝐾ℎ 𝑙,𝑙 ∈𝐿𝑘;𝑙=𝑙̸ Initialize population + ∑ 𝛽 (𝑄𝑏 + ∑ (1 − 𝑅𝑝 )𝑞𝑝 ) 𝑟𝑠 𝑟𝑠 𝑟𝑠 𝑟𝑠 (26) (𝑟,𝑠)∈𝐸 𝑝∈𝑃𝑟𝑠 Evaluate population 𝑒 𝑒 s.t.𝑥𝑙 ≥𝑡𝑙 ,𝑙∈𝐿, (1) Individual decode (2) Calculate objective function value 𝑒 𝑒 𝜂𝑙 ≥ 𝜂̂𝑙 ,𝑙∈𝐿, (3) Calculate individual fitness where 𝛼 is passenger’s value of time and 𝛽𝑟𝑠 is the average increased cost when OD (𝑟, 𝑠) passengers travel with other Yes Terminate the Meet the termination condition? alternative traffic modes instead of urban railway transport. algorithm

5. Genetic Algorithm Design No GA’s operators Due to the complexities of train schedule, genetic algorithm (1) Selection (GA) is commonly designed to solve this train schedule (2) Crossover coordination problem. GA is a self-adaptive global opti- (3) Mutation mization probability search algorithm through simulating the genetic and evolutionary process of organisms in the natural environment. GA starts to search from a set of initial Generate new population solutions generated randomly, namely, population, in which each encoded individual is corresponding to a solution. Figure 3: The process of genetic algorithm. A certain number of individuals are selected as the next generation according to the fitness of each individual. GA can rapidly search good solutions in large solution space because it can treat multiple individuals at the same time and it is with invisible parallelism. departure time can be obtained by subtracting a smaller In the following, two GAs of coordination optimization of time length on the basis of the latest operating time. For first and last trains’ departure times are designed, respectively, example, the decreased time length is 45 minutes when the according to these optimization problems’ characteristics. latest operating time is 23:00 and the last train departure time The process of genetic algorithm is shown in Figure 3,which is 22:15. The solution code of last trains’ departure time is mainly involves encoding and decoding method, individual obtained by a string of fixed-length binary number which fitness calculation, selection operator, crossover operator, expresses the decreased time length. A fixed-length binary codeshowninFigure4 is 001111000100110010⋅⋅⋅011100 when mutation operator, algorithm running parameters setting, ... and so on. last trains’ departure times are 22:45, 22:56, 22:10, ,and (1) Encoding. Given that a first train departure time should 22:32, respectively. not be more than rail line earliest operating time, the first (2) Decoding and Fitness Calculation.Basedontheabove train departure time can be obtained by adding a smaller time encoding method, an incremental or decreased time length length on the basis of earliest operating time. For example, is obtained according to a solution code. Then the first train the incremental time length is 35 minutes when the earliest departure time is obtained by adding the earliest operating operating time is 6:00 AM and the first train departure time time to the increase time length, and the last train departure is 6:35 AM of one line direction. The solution code of first time is also obtained by subtracting the decrease time length trains’ departure time is obtained by two steps: the first is from the latest operating time. Based on this, trains’ arrival to express the incremental time length with a fixed-length and departure time at each passing stations can be calculated binary number which is usually only six lengths expressing according to train running time at each section and stop time the maximum time length of 63 minutes and the second is at each station: to order binary numbers. Then a fixed-length binary code shown in Figure 4 is 101101111000011100⋅⋅⋅100000 when first 𝑘−1 𝑘−1 trains’ departure times are 6:45, 6:56, 6:28, ...,and6:32, 󸀠 󸀠 󸀠 𝑦1 =𝑥1 + ∑ 𝑡𝑘 ,𝑘 +1 + ∑ 𝜏𝑘 , 𝑘=2,3,...,𝐾, respectively. This coding method not only can ensure the 𝑙𝑘 𝑙 𝑙 𝑙 𝑙 effectiveness of the code, but at the same time can reduce the 𝑘󸀠=1 𝑘󸀠=2 coding length and give play to the advantages of binary code. 𝑘−1 𝑘 1 1 𝑘󸀠,𝑘󸀠+1 𝑘󸀠 Similarly, given that last train departure time should not 𝑥𝑙𝑘 =𝑥𝑙 + ∑ 𝑡𝑙 + ∑ 𝜏𝑙 , 𝑘=1,2,...,𝐾𝑙 −1, be less than rail line latest operating time, the last train 𝑘󸀠=1 𝑘󸀠=2

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} The first train departure time { 6:45, 6:56, 6:28, ···, 6:32

··· Increase time length 45 56 28 32

Code 101101 111000 011100··· 100000

(a) The encoding method of the first train departure time

The last train departure time { 22:45, 22:56, 22:10, ···, 22:32 }

Decrease time length 15 4 50 ··· 28

··· Code 001111 000100 110010 011100 (b) The encoding method of the last train departure time

Figure 4: The encoding method of the first and last trains’ departure time.

Airport S

Jiahewanggang

Line 2 Line 3

Guangzhou east Guangzhou rail station Tianhe coach rail station terminal Line 5

Yangji Huangcun

Gongyuanqian Tiyu Xilu Jiaokou Fenghuang Zhujiang xincheng Chebeinan Wenchong Xincun Line 1 Line 8

Changgang Kecun Wangshengwei

Xilang Line 4

Line GF Panyu square

Guangzhou south Jinzhou Kuiqi Lu rail station

Figure 5: A sketch network based on Guangzhou Metro network.

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×104 ×104 22 3.5

20 3

18 2.5

16 2 Target value Target Target value Target 14 1.5

12 1

10 0.5 0 20 40 60 80 100 0 20 40 60 80 100 Iteration Iteration

Figure 6: GA convergence for first trains’ departure time. Figure 7: GA convergence for last trains’ departure time.

𝑘−1 𝑘−1 𝑒 𝑒 𝑘󸀠,𝑘󸀠+1 𝑘󸀠 this paper, the value of population size 𝑀 is 40, the value 𝑦𝑙,𝑘 =𝑥𝑙 + ∑ 𝑡𝑙 + ∑ 𝜏𝑙 , 𝑘=2,3,...,𝐾𝑙, of crossover probability 𝑝𝑐 is 0.5, and the value of mutation 𝑘󸀠=1 𝑘󸀠=2 probability 𝑝𝑚 is 0.01. The termination conditions are that 𝑘−1 𝑘 thebestsolutionkeepsamefor20iterationsorthenumber 𝑒 𝑒 𝑘󸀠,𝑘󸀠+1 𝑘󸀠 𝑥𝑙,𝑘 =𝑥𝑙 + ∑ 𝑡𝑙 + ∑ 𝜏𝑙 , 𝑘=1,2,...,𝐾𝑙 −1. of iterations reaches 100 iterations. 𝑘󸀠=1 𝑘󸀠=2 To sum up, genetic algorithm steps of coordination (27) optimization of first trains’ departure time are designed as follows. Each individual’s objective function value can be calcu- lated according to the first or last trains running time and the Logit model of passenger travel route choice. Then the fitness Begin. of each individual is calculated by the following formula: Let 𝑖=1, population size 𝑀=40,population𝑆=0, iteration step 𝑛=1, maximum iterations 𝑁 = 100, 𝑓𝑖 = max {𝑧𝑗 |𝑠𝑗 ∈𝑆}−𝑧𝑖, (28) and 𝐾=20. When 𝑖≤𝑀, do loops. where 𝑓𝑖 is the fitness of individual 𝑠𝑖, 𝑧𝑖 is the objective function value of individual 𝑠𝑖,and𝑆 is the population Begin 1. composed of all individuals. Generate a binary number with fixed length for each (3) Selection, Crossover, and Mutation Operator.Selection rail line direction randomly, and then arrange them as 𝑠 𝑆←𝑆∪{𝑠} 𝑖←𝑖+1 operator is the combination of roulette wheel selection one solution chromosome 𝑖, 𝑖 , . methodandoptimalpreservationstrategy;namely,thebest Return to begin 1. individual in current generation replaces the former best For 𝑠𝑖 ∈𝑆, calculate its object function value 𝑧𝑖 and its individual if it is better than the former one. Therefore, fitness function 𝑓𝑖 accordingly. the best individual will not be destroyed with the optimal preservation strategy in the process of evolutionary iteration. When 𝑛<𝑁or 𝑗<𝐾, do loops. The single-point crossover is selected as the crossover Begin 2. operator here because multipoint crossover can destroy some Calculatechoiceprobability𝑝𝑖 =𝑓𝑖/ ∑𝑠 ∈𝑆 𝑓𝑗, 𝑠𝑖 ∈𝑆, good individual pattern easily. The combination of uniform 𝑗 𝑠∗ mutation and basic bit mutation is adopted as the mutation and find the chromosome with maximal fitness 𝑓 operator, and each gene value is mutated randomly in a function 𝑖. smaller probability in the early stage, and a selection gene Selection operator:Firstly,select𝑀 individuals from 𝑆 value is mutated randomly in a smaller probability in the late in probability 𝑝𝑖 with roulette wheel selection method 󸀠 stage. to compose a new population 𝑆 , and then find out the ∗ 󸀠 ∗ best individual 𝑠𝐿 from 𝑆 .Ifindividual𝑠𝐿 is better (4) Genetic Algorithm Running Parameters. The parameters 𝑠∗ 𝑠∗ =𝑠∗ 𝑗=1 𝑗++ whose values need to be determined in the process of GA than individual ,then 𝐿, ,otherwise . design are mainly population size 𝑀,crossoverprobability Crossover operator: Pair individuals of population 󸀠 𝑝𝑐,mutationprobability𝑝𝑚, and termination condition. In 𝑆 , and then do crossover operator using single-point

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crossover method in probability 𝑃𝑐 for each individual transfer wait time and originating wait time per passenger pair. are obtained as shown in Table 2 while the train minimum required occupancy rate is, respectively, 50% and 60%. In Mutation operator:If𝑛≤𝑁/2, mutate each gene 󸀠 Table 2,TWTisthetransferwaittime,andOWTrepresents value of individual in population 𝑆 in probability the originating wait time. 𝑃𝑚; otherwise, select one gene randomly and mutate When the minimum occupancy rate is 50%, the transfer its value in probability 𝑃𝑚 for each individual in 󸀠 wait time per passenger is 3.8 min which is the smallest population 𝑆 . 󸀠 ∗ valueinthreesituations,andtheoriginatingwaittimeper Let 𝑆←𝑆 ∪{𝑠 }, 𝑛←𝑛+1. passenger is 8.8 min which is the smallest value in three Return to begin 2. situations. But the sum of transfer wait time and originating 1 wait time is larger than the minimum value 15.7 min attained First trains’ departure times set {𝑥𝑙 |𝑙∈𝐿}and its ∗ by model M1. The same conclusion can be obtained while the corresponding object function value 𝑧 are obtained minimum occupancy rate is 60%. by decoding the optimal individual. Secondly, model M2 with different optimal object is Return. solved by the designed genetic algorithm of coordinate optimization of the last train departure time in this paper. The genetic algorithm steps of coordination optimiza- Thus, transfer wait time per passenger, inaccessible passenger tion of last trains’ departure time on urban rail transit volume, and passenger accessibility reliability of last train network can be designed similarly. are obtained as shown in Table 3 when the train minimum required occupancy rates are, respectively, 50% and 60%. In 6. Numerical Analysis Table 3, TWT is the transfer wait time, IPV is the inacces- sible passenger volume, and ARL represents the accessibility Guangzhou Metro is China’s third largest metro network, reliability for last train. composed of 7 lines which are line 1, line 2, line 3, line 4, Comparing transfer wait time per passenger, inaccessible line 5, line 8, and GF line. The total operation mileage of this passenger volume, and accessibility reliability for last trains network is 236 km at present, and the long-term planning in Table 3, the following conclusions are drawn. length is 600 km. Metro has become one of the main travel (1) When the minimum occupancy rate is 50%, if transfer modes in Guangzhou, and the average daily passenger flow waiting time is not considered in the optimization is about 4.8 million. A ketch network shown in Figure 5, goal, it leads transfer waiting time per passenger reach is obtained by retaining all transfer, original and terminal 8.8 min; if minimizing network inaccessible passen- stations of Guangzhou Metro. ger volume is not considered, then the calculated Train departure time interval, earliest and latest operation network inaccessible passengers reach 1218 people. In time, and the earliest and latest travel time of each line are thesameway,ifaccessibilityreliabilityforthelast showninTable1. It is worth mentioning that the earliest and train is not considered, the accessibility reliability for latest operation time is just the earliest and latest time when the last train would reach 0.68. If all of them are trains are allowed to depart, and the actual earliest and latest considered, relatively ideal values for each objective service time depends on the first and the last trains, departure can be obtained simultaneously. The same situation time. also appears when the minimum occupancy rate is Passenger minimum walk time transferring from one 60%. linetoanotherateachtransferstationis2minutes,and trainstoptimeateachofthepassingstationsis30seconds. (2) With the increase of last train minimum required More train running data such as train running time between occupancy rate increased from 50% to 60%, the two adjacent stations on the rail network can be found at network inaccessible passenger volume increases, but http://www.gzmtr.com. In addition, passenger arriving rate the change of the transferring wait time per passenger of each OD pair in the morning and evening are generated and the accessibility reliability for last trains depends randomly. on passenger arriving rate. When the generation time is 100, the population size is 100, the cross probability is 0.6, and the variation probability 7. Conclusion is 0.005, the convergence of designed genetic algorithm of coordinate optimization of first and last trains’ departure time In this paper, we firstly analyzed passenger travel cost and the is shown in Figures 6 and 7,respectively. travel route choice behavior of passengers with Logit model. Figures 6 and 7 show that the designed genetic algorithm Then a coordination optimization model M1 was constructed of coordinate optimization of first and last trains’ departure to minimize the total passengers originating waiting time time can converge with 60 generation times, which demon- and transfer waiting time for first trains, subjecting to strates that two algorithms can converge efficiently and can the constraints of the minimum effective occupancy rate solve the first and last trains’ departure time effectively. and earliest service time. Similarly, a second coordination Firstly, M1, whose optimal goal is minimizing sum of pas- optimization model M2 was also built to reduce passengers’ sengers’ transfer wait time and originating wait time, is solved transfer waiting time for last trains, minimize the inaccessible using the designed genetic algorithm. Correspondingly the passenger volume of all OD pairs, and improve passengers’

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Table 1: Trains departing time interval, operating time range, and passenger travel timescope of each line.

Line Train departing time interval Line earliest Line latest Passenger Passenger Origin station Terminal station name Up Down service time service time earliest travel latest travel bound/min bound/min time time Guangzhou East Rail Line 1 Xilang 7.5 8 6:00 11:30 6:00 11:30 Station Guangzhou Line 2 Jiahewanggang South Rail 6.5 7 6:00 11:30 6:00 11:30 Station Tianhe Coach Line 3 Panyu Square 7 7.5 6:00 11:00 6:00 11:00 Terminal Line 3+ Airport S Tiyu Xilu 6.5 7 6:00 11:00 6:00 11:00 Line 4 Huangcun Jinzhou 7 7 7:00 10:30 7:00 10:30 Line 5 Jiaokou Wenchong 7.5 7 7:00 10:30 7:00 10:30 Line 8 Fenghuang Xincun Wangshengwei 8 7.5 7:00 10:00 7:00 10:30 Line GF Xilang Kuiqi road 8 8.5 7:30 9:40 7:30 9:40

Table 2: Coordination optimization result of first trains’ departure time with different optimal object.

Minimum occupancy rate: 50% Minimum occupancy rate: 60% OWT per TWT per OWT and TWT TWT per OWT per OWT and TWT Optimal object passenger/ passenger/ per passenger/ passenger/ passenger/ per passenger/ min min min min min min TWT 13.2 3.8 17 15 3.8 18.8 OWT 8.8 11.4 20.2 9.6 11.2 20.8 TWT & OWT 9.5 6.2 15.7 10.5 5.9 16.4

Table 3: Coordination optimization result of last trains’ departure time with different optimal object.

Minimum occupancy rate: 50% Minimum occupancy rate: 60% Optimal objects TWT per TWT per IPV ARL IPV ARL passenger/min passenger/min IPV & ARL 8.8 853 0.95 8.9 1106 0.97 TWT & IPV 5.6 954 0.68 5.4 1035 0.66 TWT & ARL 6.4 1218 0.82 6.4 1371 0.83 IPV & TWT & ARL 6.7 1023 0.84 6.8 1241 0.85

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