Journal of Modern Transportation Volume 19, Number 4, December 2011, Page 233-239 Journal homepage: jmt.swjtu.edu.cn DOI: 10.1007/BF03325763 Optimization of train plan for urban rail transit in the multi-routing mode Lianbo DENG1*, Qiang ZENG1, Wei GAO1, Song BIN2 1. School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China 2. China Railway Siyuan Survey and Design Group Co. Ltd., Wuhan 430063, China Abstract: The train plan of urban rail transit under multi-routing mode can be divided into three parts: train formation, train operation periods and corresponding train counts of each routing in each period. Based on the analysis of passen- ger’s general travel expenses and operator’s benefits, the constraints and objective functions are defined and the multi- objective optimization model for the train plan of urban rail transit is presented. Factors considered in the multi- objective optimization model include transport capacity, the requirements of traffic organization, corporation benefits, passenger demands, and passenger choice behavior under multi-train-routing mode. According to the characteristics of this model and practical planning experience, a three-phase solution was designed to gradually optimize the train for- mation, train counts as well as operation periods. The instance of Changsha Metro Line 2 validates the feasibility and efficiency of this approach. Key words: urban rail transit; multi-train-routing; train plan; multi-objective model; three-phase solution method © 2011 JMT. All rights reserved. 1. Introduction gramming. Bussieck et al. [8] treated the train plan as a train schedule (i.e. a line plan), and determined the ver years, the trains on urban railways in China number of trains connecting two terminal stations of a O have to stop at every station along a single long serving line in a fixed time interval. Claessens et al. [9] route. The daily train plan depends on the cross-section built an integer nonlinear model to maximize the num- flow in each operation period [1]. ber of direct travelers for the Dutch railway system. Sun In order to provide service efficiently and to decrease et al. [10] investigated the multi-train-routings of urban cost effectively, many cities in China have to take the rail, and proposed a two-phase solution for the maximal approaches such as train formation adjustment [2] and seat occupancy rate and the minimal count of train op- multi-routing operation [3] to make their rail transit eration periods. Deng et al. [4] considered single train networks more competitive. The traditional train plan of routing based on the analysis of passenger’s general urban rail transit has expanded into more advanced one, travel expenses, and established a multi-objective model which consists of train formation, train operation period with respect to transport capacities, transport organiza- and train frequency for each routing [4]. Among these tions, economic benefits and traffic demands. Refs. [11- components, the most important determinant of trans- 12] studied passenger’s choice behavior and flow as- port capacity is train formation. In the multi-train- signment in the urban rail transit network. Sang et al. routing mode, the train plan needs to consider not only [11] presented a graph theoretic framework for the pas- the heterogeneous distribution of passenger volumes in senger assignment problems that simultaneously en- the time-space, but also the passengers’ choices [5-6]. compassed the departure time and the route choice. Tian Chang et al. [7] presented a multi-objective optimiza- et al. [12] analyzed the equilibrium properties with in- tion model to minimize operation cost and total travel vehicle crowding effect and schedule delay cost in a time of passengers. Train stops, service frequency and many-to-one transit system. train count were optimized by fuzzy mathematic pro- Although the train plans of urban rail transit are es- sentially network problems. Unlike national railway network, they are independent of each other to some ex- Received Aug. 29, 2011; revision accepted Nov. 17, 2011 *Corresponding author. E-mail: [email protected] (L.B. tent. In this paper, the train plan is targeted on one line DENG) with the consideration of the passenger choice. For sim- © 2011 JMT. All rights reserved plification, passengers transfer is ignored. doi: 10.3969/j.issn.2095-087X.2011.04.003 234 Lianbo DENG et al. / Optimization of train plan for urban rail transit in the multi-routing mode 2. Related concepts on passenger train plan tTb . Ht e The train plan of urban rail transit : determines Train routing set is expressed as Uusel ^ / ll, train formation and the number of trains for each routing and operation period. For simplicity, we only consider 1, ,Hull ;se , S` where sl and el are the two termi- the train plan in a sequence of 1, 2, ,H s . The number nals of routing ul . The train routing modes include: ad- of trains on routing u during T is d l , and train set jacent routing, nested routing, and the mixed mode l k k l (Fig. 1). For smooth train operation and passenger is Ddk ^`ktu 1, 2, , Hl ; 1, 2, , H . According to boarding, H can not be excessive. Among all the rout- u above definitions, : ^`bT,, D can denote the train ings, the ones running in all operation periods are basic plan of urban rail transit. routings whereas others running in peak periods are non-basic ones. Adjacent routings are basic. The long routing of the nested routing mode is a basic one while 3. Passenger’s general travel expenses the short is not. The train plan for the adjacent routing mode can be optimized on a single routing basis. Com- Passenger flow of urban rail transit fluctuates in paratively, the train plan of the nested routing mode [,]TTs e . It is relatively steady in certain duration, which should be optimized according to passenger’s choice. is called passenger travel period. [,]TTs e can be divided Further, the train plan of the mixed mode is a combina- into H 0 passenger travel periods. The passenger flow tion of both plans over two routing modes. The train t 0 plan optimization of the nested routing mode is the fo- from station i to station j in period Tk is f (,ijk , ), for cus in this paper. 0 ij,1,2,,, Hs kH 1, 2, ,t . The train frequency is unified in one passenger travel period, and one train operation period may include one or more passenger (a) Adjacent routing mode travel periods. If the passenger travel period T 0 is in- k1 cluded in train operation period T , the relationship k2 0 0 (b) Nested routing mode can be expressed as TT , for kH 1, 2, , , and kk12 1 t kH2 1, 2, ,t . The passenger’s general travel expense usually in- (c) Mixed mode cludes ticket fare, travel time and congestion cost. Ticket fare is a monetary expenditure, which can be Fig. 1 Routing modes expressed as person-kilometers fare rate p . Individual ticket fare for passenger flow f (,ijk , ) is Let [,]TTs e denote the daily operation period of urban rail transit. The train formation is usually Pijk,, pwij , . unchanged. For the train plan : , b is the train formation length (i.e. number of carriages in a train), Travel time includes riding time and waiting time. and V is the average carrying capacity of carriages. Riding time is equal to the train running time between Considering the factors such as station capacity and the stations, which can be calculated by dividing the dis- track length, the feasible formation length is in a range tance wij , with the train speed J . The riding time of ªºbb, , where b and b are the shortest and the ¬¼ for passenger flow f (,ijk , ) is longest formation lengths respectively. According to the fluctuation of passenger flow in one wij , G ijk,, . day and the variation of train frequency in [,]TTs e , we J can define the train operation period Ti . Set Waiting time refers to the time period from a passen- ab TTtti ^`iii ,1,2,, Ht to be train operation ger’s arrival to departure. The waiting time is related to a b train frequency. When TT0 , the average waiting periods, where ti and ti denote the start and the end kk12 time of Ti respectively; Ht is the count of train time for the flow fijk(, ,1 ) is calculated by a operation periods under the conditions of tT1 s and Journal of Modern Transportation 2011 19(4): 233-239 235 The length restriction of the train formation is V Tk Z(,ijk , ) 2 , 1 d l ¦ k2 bbbdd, and b is an integer. (2) uUueijll:(,) The train operation periods should satisfy where T is the period length of T ; V is a parameter k2 k2 ba related with the passenger flow distribution, and V is ttkk 1,1,2,,1; k Ht a equal to 0.5 for uniform passenger flow. tTks , k 1; (3) f (,ijkl , , ) is the volume of passengers which select b tTke , kH t. train routing u in f (,ijk , ). Within the passenger l The train frequency should be restricted by train 0 travel period Tk , the passenger volume on routing ul in headway W [13]. The upper limit of train frequency is the section (,ii 1) E is Tk tW ,1,,,iHkH St1 1,2,,. el i l '' ¦ dk (4) g(,ikl , ) ¦¦ f ( i , j , kl , ). lU '' eii(, 1) u ji 1 is l l We introduce the congestion cost function The lowest train frequency for each routing varies. y gikl(, , ) to describe the passenger discomfort for For basic routings, such as long routings, its train fre- quency should be no less than the minimal operational routing ul and passenger flow period Tk .