Available online at ScienceDirect. Procedia Engineering 137 (2016 ) GITSS2015

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1 Avalable onlne at ScenceDrect Proceda Engneerng 137 (2016 ) GITSS2015 Optmzaton of Urban Sngle-Lne Metro Tmetable for Total Passenger Travel Tme under Dynamc Passenger Demand Pan Shang a, Rumn L a, *, Lya Yang b a Insttute of Transportaton Engneerng, Tsnghua Unversty, Bejng , Chna b School of Publc Admnstraton and Polcy, Renmn Unversty of Chna, Bejng , Chna Abstract Ths paper studes the optmzaton of an urban sngle-lne metro tmetable for total passenger travel tme adapted to dynamc passenger demand, whch arses n an urban metro servce and s a common problem n major ctes. After analyzng the components of the total passenger travel tme, a model s presented wth the am of mnmzng total passenger travel tme. An S -pattern functon s proposed to represent the cumulatve demand functon for each par of orgn and destnaton n an urban sngle-lne metro. Furthermore, a spatal branch and bound algorthm that s applcable to the model s presented. The advantages of desgnng a tmetable that optmzes the total passenger travel tme adapted to dynamc passenger demand are depcted through extensve computatonal experments on several cases derved from a real urban sngle-lne metro. An extensve computatonal comparson of a regular tmetable, a tmetable optmzng average watng tme, and a tmetable optmzng total passenger travel tme tmetable are performed. Crown 2016 Copyrght The Authors Publshed by Elsever Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the Department of Transportaton Engneerng, Bejng Insttute of Technology. Peer-revew under responsblty of the Department of Transportaton Engneerng, Bejng Insttute of Technology Keywords: tran tmetable; total passenger travel tme; dynamc demand; regular tmetable; spatal branch and bound algorthm. 1. Introducton Urban metro system plannng s a sub-aspect of the complex publc transportaton system plannng and s usually decomposed nto several stages, ncludng networ desgn, lne desgn, tmetablng, rollng stoc, and staffng [8,10]. These problems have been tradtonally optmzed from the operator s perspectve. Ths paper provdes an approach that consders the operator s restrctons and mnmzes the passengers total travel tme, * Correspondng author. Tel.: ; fax: E-mal address: lrmn@tsnghua.edu.cn Crown Copyrght 2016 Publshed by Elsever Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the Department of Transportaton Engneerng, Bejng Insttute of Technology do: /j.proeng

2 152 Pan Shang et al. / Proceda Engneerng 137 ( 2016 ) whch s measured as the sum of passenger average watng tme (AWT) at statons and average rdng tme (ART) on trans. Ths study ams to solve the urban sngle-lne metro tmetable problem, whch conssts of determnng departure tme at each staton. Furthermore, the optmal speed between every segment of each tran servce to and from each staton along an urban sngle-lne metro over a plannng horzon, whch s determned by a dynamc passenger orgn-destnaton demand, s also addressed. The urban metro tmetable problem can be classfed nto two forms: perodc and non-perodc tmetables. [17] proposed a mathematcal model for the perodc schedulng problem, whch consdered the extenson to the perodc phenomena of ordnary schedulng wth precedence constrants. The advantage of perodc tmetables s that they can be easly memorzed by passengers and are able to deal wth large-scale ralway networs [15]. The alternatve, whch nvolves constructng a non-perodc tmetable, s approprate when the demand cannot be assumed constant over tme. Varous nteger lnear programmng models have been proposed for the non-perodc tmetable problem. Carey [6] presented a model, as well as algorthms and strateges to dspatch trans wth dfferent speeds and stoppng patterns for a double trac ral lne dedcated to trans n one drecton; they further extended t to nclude general and complex ral networs, wth choces of lnes and staton platforms. Caprara [5] proposed a model to determne a perodc tmetable for a set of trans that are unable to volate trac capactes and able to satsfy some operatonal constrants. They proposed a graph theory-based model for modellng the problem usng a drected mult-graph n whch nodes correspond to departures/arrvals at certan statons and gven tme nstants. Vansteenwegen and Oudheusden [19]dealt wth mprovng passenger servce on a small porton of the Belgan ralway networ by tang watng tmes and delays nto account. They establshed a two-phase model. Frst, deal buffer tmes are calculated based on delay dstrbutons of arrvng trans and the weghts of varous watng tmes to safeguard connectons when the arrvng tran s late; and second, standard lnear programmng s used to construct an mproved tmetable wth well-scheduled connectons and deal buffer tmes whenever possble. Cacchan and Toth [4] surveyed exstng studes that prmarly dealt wth the tran tmetable problem n ts nomnal and robust versons, whch satsfed trac capacty constrants. They amed to optmze an objectve functon wth varous meanngs that correspond to requests of the ralway company. Most of the papers mentoned optmzed an objectve functon relevant to the servce operator. From the nfrastructure operator s pont of vew, one common objectve s to mnmze devaton from a tmetable plan proposed by the operator, whch s frequently used n the perodc case. From the users perspectve, the objectve of mnmzng watng tme has been consdered under dynamc passenger demand [2]. However, these contrbutons have not been explctly consdered the total travel tme under dynamc passenger demand. The followng contrbutons focused on the dynamc structure of demand behavour. Hänseler [11] nvestgated ways of computng dynamc OD matrces for tran statons. Yano and Newman [20] proposed a dynamc programmng algorthm for the tmetable of trans used to transport contaners that arrve dynamcally at the orgn. Cordone and Redaell [7] too nto account the recprocal nfluence between tmetable qualty and transport demand amount captured by the ralway. They proposed a mxed-nteger nonlnear model wth a non-convex contnuous relaxaton. Nu and Zhou [16] focused on optmzng a passenger tran tmetable n a heavly congested urban ral corrdor under a dynamc demand scenaro. The researchers were nterested n system behavour under congeston once demand exceeds capacty. Ths paper focuses on constructng an urban sngle-lne metro tmetable that s adapted to dynamc passenger demand. The proposed model pursues two objectves: mnmzng passenger AWT at statons and ART on trans, respectvely, to represent the total passenger travel tme (TTT). One of the man features of ths model s the S- pattern functon that s mostly used n urban metro systems. Ths pattern s proposed to descrbe the cumulatve demand for each par of orgn and destnaton, as well as each spatal branch and bound algorthm to optmze the model. The soluton s an urban sngle-lne metro tmetable adapted to dynamc passenger demand over a fnte plannng horzon. The remander of ths paper s organzed as follows. Secton 2 formally states the problem and the objectves, and Secton 3 presents the proposed mathematcal model. Spatal branch and bound algorthm are further descrbed n Secton 4, and results of extensve computatonal experments and conclusons are presented n Sectons 5 and 6 respectvely. 2. Problem statement and analyss Ths study focuses on constructng an urban sngle-lne metro tmetable adapted to a dynamc passenger

3 Pan Shang et al. / Proceda Engneerng 137 ( 2016 ) demand usng the objectve AWT smlar to [1,2,16], and ART. Urban sngle-lne metro tmetable s normally represented n tme-space dagrams, as shown n Fgure 1[2]. The x-axs represents the plannng horzon, and the y- axs represents the statons n one lne, more concretely, the dstance from a certan staton to the frst one. Fgure 1(a) llustrates a regular tmetable, n whch the headway between consecutve trans and the speeds of dfferent trans between consecutve statons are constant. Fgure 1(b) llustrates a non-regular tmetable, n whch headways and speeds are not necessarly constant and tran frequency s normally hgh around pea hours. Fgure 1 Tme space dagrams of tran tmetables for a one lne metro. (a) Regular and (b) Non-regular tmetables. Source: Barrena et al., 2014b. Let S {1,..., n} be the ordered set of statons consdered n one lne as depcted by the y-axs n Fgure 1. Plannng horzon s dscretzed nto tme ntervals of length, whch can be set dscretonarly by the necessary precson. Thus, tme nstant t T {0,1,..., p} corresponds to t tme unts elapsed snce the begnnng of the plannng horzon as showed by the x-axs n Fgure 1. Let d j (t) be the cumulatve demand functon from statons to j before tme t,, j S, j. Let l 1, be the dstance between statons 1 and, h mn be the mnmum headway whch refers to the mnmum tme requred between the departure of two consecutve trans at each staton, w be the dwellng tme at the staton, s max and s mn be the nverse of the maxmum and mnmum travellng tran speeds. Speed nverse s used to avod problem non-lnearty n the constrants., 1 s the rdng tme between adjacent statons, and t j s the rdng tme between non-adjacent statons and j. The metro operator has a set M {1,..., m} of avalable trans. The decson varable s t x, whch refers to the departure tme of tran at staton. All the varables and functons used are lsted n Table 1. The model ams to determne tran departure tmes at statons and speeds on ral segments, respectvely, to mnmze the sum of passenger AWT at statons and ART between each staton. In the followng, the demand pattern and reasons for choosng the sum of passenger AWT at statons and ART between each staton are presented as objects Demand functon To determne an optmal tmetable under dynamc passenger demand, the cumulatve demand functon for each par of orgn and destnaton s consdered usng cumulatng functon d j (t). Several authors dealt wth varable demand and descrbed t through functons defned from dfferent perspectves. Hurdle [14] starts from a stepwse cumulatve demand functon and assumes that the arrval pattern of passengers can be reasonably approxmated through a dfferentable functon. Although not requred, the algebrac expresson of ths functon can be drawn. Barrena et al. [2] approxmated the cumulatve demand by usng a smooth functon as the sum of sgmodal functons as well. Gven the problem of dynamc passenger demand, the cumulatve demand functon for each par of orgn and

4 154 Pan Shang et al. / Proceda Engneerng 137 ( 2016 ) destnaton generally exhbts a smooth growth durng non-pea hours. However, t wll acheve dramatc growth durng pea hours. To descrbe the phenomenon, an S -pattern functon s proposed as Equaton (1) n Fgure 2 (a), where K j, a j, and b j are parameters for each par of orgn and destnaton. Parameters can be acheved through fttng wth actual data. Table 1. Varables and functons assocated wth urban sngle-lne metro tmetable decsons Varable/functon j m M n S p 1, Descrpton Index of statons Index of statons Index of trans Number of avalable trans Set of avalable trans, M {1,..., m} Number of statons Set of statons, S {1,..., n} Unt of tme nterval Number of tme ntervals T Set of tme ntervals, T {0,1,..., p} 1, l Dstance between staton 1 and staton, S s Inverse of travelng speed of tran between staton 1 and staton, S, M s, s Inverse of the maxmum and mnmum travelng speed of a tran max, 1 t j t hmn mn Rdng tme between adjacent statons Rdng tme between staton and staton j Mnmum headway w Dwellng tme at staton, S x Departure tme of tran at staton, S, M dj (t) Cumulatve demand functon from staton to staton j before tme t,, j S, j bt j d (t) K (1 a e ) j j j (1) Fgure 2 (a) S-pattern cumulatve functon and (b) Explanaton for passenger watng and rdng tmes.

5 Pan Shang et al. / Proceda Engneerng 137 ( 2016 ) Illustraton of passenger watng and rdng tmes A servce s defned as a trp from an orgn to a fnal destnaton, where a tran refers to the servce t operates. The tran operator can determne the departure tme and the speed between statons. The trp can be dvded nto three stages for each passenger: (1) enterng the staton and watng for the tran; (2) rdng on the tran; (3) dsembarng from the tran and leavng the staton. Apart from the tme of the last stage, whch conssts of dsembarng from the tran and leavng the staton, whch are decded by the passengers, most of the tme spent durng the frst stage conssted of watng for the tran, whch s decded by the departure tme of the next tran. Rdng the tran, whch s another large proporton of the tme, s decded by the speed of the tran between statons. As shown n Fgure 2 (b), the arrow mars the moment when passengers arrve at the staton and begn to wat for the next tran. Untl the tran departure, the watng tme s mared by the red lne. Travelng tme between staton n 1 and staton n s mared by the blue lne n Fgure 2 (b), whch s the tme between the tran departure from staton n 1 and arrval at staton n. Rdng tme s decded by the speed of tran, represented by the green lne. Both the watng and rdng tmes for and on the tran, respectvely, are decded by the operator, whch means that the TTT s decded by the operator, whch s represented by the yellow lne n Fgure 2 (b). The model to mnmze the passenger TTT n urban sngle-lne metro s proposed n Secton Integer programmng model 3.1. Objectve functon 1 AWT ( ( d ( x ) d ( x ))( x x )) 2 (2) j j 1 1 S M ( m 1) j S, j j S, j ART ( (d (x ) d (x )) t ) S j S, j M j j j 1 (3) mn( AWT ART) (4) As demonstrated n Secton 2.2, the sum of AWT and ART should serve as the objectve functon. The total demand s constant and should not be consdered. The objectve functon (2) s referred to as passenger AWT and functon (3) as passenger ART, whch ams to mnmze the sum of passenger AWT and ART as a functon (4) Constrants x0 0, S (5) x p S (6) m 1, x h,, \{ } mn x 1 S M m (7), 1, 1, t l s 1, S, M (8) s s s S M (9) 1,,, mn max t t, 1 j, M (10) j j 1, j

6 156 Pan Shang et al. / Proceda Engneerng 137 ( 2016 ) j j 1 j r 1, r q r 1 q 1 (11) t t w, 1 j, M x x t w, S, M (12) 1 1, x N, S, M x [1,..., p], S, M (13) Constrant (5) defnes the frst dummy tran departure tme whose role s to ensure that the objectve functon taes all watng tmes and demands nto account. Constrant (6) defnes the last dummy tran departure tme whose role s the same as that of constrant (5). Constrant (7) ensures the departure of two trans from the same staton beyond the mnmum headway h mn. Constrant (8) defnes the relatonshp of rdng tme and tran speed between contguous statons 1 and. Constrant (9) ensures that the tran speed s under the control of the operator. Constrant (10) and (11) defnes the passenger s rdng tme between statons and j. Ths tme s the sum of the tran s run and dwell tmes between statons and j, whch denote the tme perod from the tran starts runnng at staton to passengers gettng off the tran at staton j, respectvely. Ths rdng tme s an mportant part n the computaton of the objectve functon. Constrant (12) defnes the departure tme between contguous statons. The departure tme for tran at staton s decded by the departure tme for tran at staton 1 and the dwell tme at staton, of whch t s the sum. Gven that we dscretzed the plannng horzon nto tme ntervals of length, whch can be set dscretonarly by the necessary precson, constrant (13) ensures x should be an nteger and can only be a value between 1 and p. 4. Spatal branch and bound algorthm The spatal branch and bound algorthm s an algorthm for solvng non-convex NLPs and mxed-nteger nonlnear programs; several varants le Dallwg et al. [8], Belotta et al. [3], Smth and Panteldes [18] exst. The spatal branch and bound algorthm used here s of the general form proposed by Horst and Tuy [13]. Most spatal branch and bound algorthms for global optmzaton conform to a general framewor n the followng form: Step 1: Intalzaton A lst of regons s ntalzed to a sngle regon that comprses the entre set of varable ranges. The convergence tolerance s set as 0, the best objectve functon value determned from the current step as U :, and the correspondng soluton pont s x * : (,..., ). Step 2: Choce of Regon If the lst of regons s empty, the algorthm s termnated wth soluton x * and objectve functon value U. Otherwse, a regon R s selected from the lst. R s deleted from the lst. Feasblty-based bounds tghtenng s performed on R. Step 3: Lower Bound Convex relaxaton of the orgnal problem s generated n the selected regon R and solved to obtan an underestmaton l of the objectve functon wth correspondng soluton x. If l U or the relaxed problem s nfeasble, then Step 2 s performed agan. Step 4: Upper Bound An attempt s made to solve the orgnal problem n the selected regon to obtan a (locally optmal) soluton x wth objectve functon value u. If t fals, then u: U and 0 x X s set. Step 5: Prunng * If u U, then x x and U : u are set. All regons wth lower bounds bgger than U n the lst are deleted as they cannot possbly contan the global mnmum.

7 Pan Shang et al. / Proceda Engneerng 137 ( 2016 ) Step 6: Chec Regon If u l, then u s accepted as the global mnmum for ths regon and Step 2 s performed agan. Otherwse, the next step s performed because the regon global mnmum may not have been located. Step 7: Branchng A branchng rule s appled to the current regon to splt t nto sub-regons. The sub-regons are added to the lst of regons, and they are assgned an (ntal) lower bound of l. Step 2 s then performed agan. 5. Computatonal experments A seres of mplementaton specfcatons s provded through cases used, and results of extensve computatonal experments are presented n Secton 5.2. Barrena et al. [2] proposed an optmzng tmetable model that uses AWT as an objectve. Ths AWT objectve s adopted to run computatonal experments usng our dataset wth our model to compare each model s advantages and dsadvantages. The regular tmetable s AWT and ART are computed for comparson purposes as well Set of cases The set of cases s generated accordng to the followng parameters: Number of statons n : 3, 4, 5; Horzon p : 120, 240, 480 mn; Dscretzaton constant : 1, 2 mn; Numbers of trans: 3, 4, 5; Fxed dwellng tme: 2 mn; Maxmum nverse speed of the trans s max : 3 mn/m; Mnmum nverse speed of the trans s mn :1 mn/m; Mnmum headway h mn : 2mn. The dstance parameters between each staton and dwell tme at each staton, as well as the other data necessary n the problem are obtaned from the Changpng lne of the Bejng metro. Passenger demand of the urban snglelne metro used n the computatonal experments ndcates a pattern smlar to that n Fgure 2 between each par of orgn and destnaton. These cases wll be referred to as TT-n-p-d-m as Barrena et al.[2], e.g., TT , correspondng to a tran tmetable case wth three statons: a plannng horzon of 120 mn, a dscretzaton constant of 2 mn, and a maxmum of 5 trans Computatonal results The summary of computatonal results for each mprovement between the regular and the optmzed AWT+ART and AWT tmetables on all cases are presented n Table 2, 8, and 6, respectvely. As Table 2, 3, and 4 shows, for cases across three statons, the proposed model for optmzng TTT reduced AWT, ART, and AWT+ART by 78.19%, 36.19%, and 68.05%, respectvely, compared wth the regular tmetable. Furthermore, t can reduce 4.50% ART and 1.57% AWT+ART although t ncreases 1.81% AWT compared wth the optmzed AWT tmetable. For four statons scenaro, the proposed model for optmzng the TTT can reduce AWT, ART, and AWT+ART by 72.89%, 38.11%, and 62.86%, respectvely, compared wth the regular tmetable. Furthermore, t can reduce 12.70% ART and 4.04% AWT+ART compared wth the optmzed AWT tmetable although t ncreases 8.61% AWT. For fve statons scenaro, the proposed model can reduce AWT, ART, and AWT+ART by 62.07%, 43.74%, and 56.30%, respectvely, compared wth the regular tmetable. Although t ncreases 16.43% AWT, t can reduce 27.13% ART and 8.58% AWT+ART compared wth the optmzed AWT tmetable.

8 158 Pan Shang et al. / Proceda Engneerng 137 ( 2016 ) Table 2. Improvement between the regular, optmzed AWT, and optmzed AWT+ART tmetables on cases across three statons (Unt: %) Case AWT+ART VS Regular Improvement AWT+ART VS AWT Improvement AWT ART AWT+ART AWT ART AWT+ART TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT Average Table 3. Improvement between the regular, optmzed AWT, and optmzed AWT+ART tmetables on cases across four statons (Unt: %) Case AWT+ART VS Regular Improvement AWT+ART VS AWT Improvement AWT ART AWT+ART AWT ART AWT+ART TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT Average

9 Pan Shang et al. / Proceda Engneerng 137 ( 2016 ) Table 4 Improvement between the regular, optmzed AWT, and optmzed AWT+ART tmetables on cases across fve statons (Unt: %) Case AWT+ART VS Regular Improvement AWT+ART VS AWT Improvement AWT ART AWT+ART AWT ART AWT+ART TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT Average Conclusons A model on an urban sngle-lne metro tmetable problem for optmzng total passenger travel tme TTT under dynamc passenger demand was proposed. Gven that the model s an nteger nonlnear program, spatal branch and bound algorthm are appled to the model. Extensve computatonal experments on the cases show that the model mproved AWT by around 70%, ART by around 40%, and TTT by around 60% compared wth the regular tmetable. Compared wth the optmzed AWT tmetable, our model mproved ART by around 15% and TTT by around 5%. The model that ams to optmze total passenger travel tme TTT that was measured by the sum of passenger AWT and ART s reasonable and equtable for all passengers. Thus, ths proposal can be adopted by operators to ncrease the performance of an urban metro system. Acnowledgement Ths research was partally supported by the Hang Lung Center for Real Estate, Tsnghua Unversty. Reference [1]Barrena, E., Canca, D., Coelho, L.C., Laporte, G., 2014a. Exact formulatons and algorthm for the tran tmetablng problem wth dynamc demand. Computers & Operatons Research 44, [2]Barrena, E., Canca, D., Coelho, L.C., Laporte, G., 2014b. Sngle-lne ral rapd transt tmetablng under dynamc passenger demand. Transportaton Research Part B: Methodologcal 70, [3]Belotta, P., Lee, J., Lbert, L., Margot, F., Wächter, A., Branchng and bounds tghtenngtechnques for non-convex MINLP. Optmzaton Methods and Software 24, [4]Cacchan, V., Toth, P., Nomnal and robust tran tmetablng problems. European Journal of Operatonal Research 219, [5]Caprara, A., Fschett, M., Toth, P., Modelng and solvng the tran tmetablng problem. Operatons Research 50, [6]Carey, M., A model and strategy for tran pathng wth choce of lnes, platforms, and routes. Transportaton Research Part B: Methodologcal 28,

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