J. Cent. South Univ. (2013) 20: 2897−2904 DOI: 10.1007/s11771­013­1811­5

Modified stochastic user­equilibrium assignment algorithm for urban rail transit under network operation

ZHU Wei(朱炜) 1, 2, HU Hao(胡昊) 1, XU Rui­hua(徐瑞华) 2, HONG Ling(洪玲) 2 1. Center of Transportation Research, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China; 2. School of Transportation Engineering, Tongji University, Shanghai 201804, China © Central South University Press and Springer­Verlag Berlin Heidelberg 2013

Abstract: Based on the framework of method of successive averages (MSA), a modified stochastic user­equilibrium assignment algorithm was proposed, which can be used to calculate the passenger flow distribution of urban rail transit (URT) under network operation. In order to describe the congestion’s impact to passengers’ route choices, a generalized cost function with in­vehicle congestion was set up. Building on the k­th shortest path algorithm, a method for generating choice set with time constraint was embedded, considering the characteristics of network operation. A simple but efficient route choice model, which was derived from travel surveys for URT passengers in China, was introduced to perform the stochastic network loading at each iteration in the algorithm. Initial tests on the URT network in Shanghai City show that the methodology, with rational calculation time, promises to compute more precisely the passenger flow distribution of URT under network operation, compared with those practical algorithms used in today’s China.

Key words: urban rail transit; stochastic user equilibrium; assignment algorithm; method of successive averages; network operation

too obvious to be neglected in influencing passengers’ 1 Introduction behaviors, and passengers may choose different routes for the same origin−destination (O−D) pair. Passenger flow is the foundation of making and 2) Under the condition of network operation, coordinating operation plans for urban rail transit (URT) whether a route is chosen by a URT passenger is systems. Models to solve passenger flow assignment subjected to not only the network’s topology and problems can be classified according to whether different travel costs but also the rail lines’ service time. Wardrop’s principle is followed. One model is the In other words, during some certain period of time, a non­equilibrium assignment, and the other is the route cannot be taken part in the passenger flow equilibrium assignment model which is sounder in theory assignment and thus will be excluded in the route choice and can be applied into a large­scale network with rapid set in case of some rail lines of it beyond their service development of computer technology. time. Currently, with the fast development of economy, Moreover, it is assumed that passengers’ choice the URT systems of big cities in China, such as Beijing, processes in reality have more or less random Shanghai and Guangzhou, have entered into the new characteristics because of imperfect knowledge of travel stage of network operation with technology for time, individual differences, measurement errors, and so one­ticket transfer, whose characteristics with regard to on [1−3]. Therefore, confronting with today’s URT passenger flow assignment are concluded as follows: systems in China, the result from passengers’ route 1) The dramatic expansion of those cities’ URT choices can be described more appropriately by the networks has enabled more flexible and complex travels stochastic user­equilibrium (SUE) with time constraint, within the URT systems because more and more transfer which is also proved by some simulation experiments stations have provided better connecting services. On the [4−5] and full­scale case tests [6]. other hand, most of the URT lines have been overloaded The SUE problem has been studied for a long time. since travel demand increases rapidly in these Thorough reviews were presented in part of the fast­expanding cities. Hereby, the factor of crowding is literature [7−10]. It can be solved either in the space of

Foundation item: Project(2007AA11Z236) supported by the National High Technology Research and Development Program of China; Project (2012M5209O1) supported by China Postdoctoral Science Foundation Received date: 2012−04−23; Accepted date: 2013−03−25 Corresponding author: ZHU Wei, PhD; Tel: +86−21−62933091; E­mail: [email protected] 2898 J. Cent. South Univ. (2013) 20: 2897−2904

rs link flows or in the space of path flows [11−12]. Among min Z ( X ) = − q rs S rs [ c ( X )] + x ∑ various solution algorithms, the well­known method of rs x successive averages (MSA) developed by SHEFFI and x t ( x ) − a t (ω ) d ω (6) ∑ a a a ∑∫ 0 a POWELL [13] is the first algorithm applied to solve the a a SUE problem and can be applied with any stochastic s.t. Eqs. (3)−(5). network loading method. Especially, the MSA is widely where ta(ω) is the unit generalized travel cost on link a, it considered as a kind of efficient algorithm for two is the function of the flow (ω) on link a. c rs( X) is the reasons [1]. First, the algorithm will converge if the vector of perceived generalized travel cost for the O−D search direction is a descent vector only on the average. rs pair (rs). Srs[c (X)] is the expected perceived generalized Second, the algorithm is based on a predetermined travel cost for the O−D pair (rs). sequence of move sizes along the descent direction so The above description is a SUE model and there are that the difficult calculation of objective function can be several features needed to be noticed when it is applied avoided. Even though the algorithm has been applied in into URT networks: the urban transit analyses extensively in the past, few 1) The generalized travel cost is affected studies have analyzed it with regards to a URT network considerably by in­vehicle congestion rather than [6, 14]. The objective of this work is to propose a vehicle­to­vehicle congestion; modified stochastic user­equilibrium assignment 2) The route­based assignment algorithm is algorithm based on the framework of MSA, which can easier to be implemented due to both of less complex solve passenger flow assignment problem more network compared to road network and time constraint appropriately and precisely for the URT systems under under network operation; network operation especially in today’s China. 3) The transferring time at the transfer station influences the route choices of rail passengers 2 SUE assignment model for URT network considerably.

The SUE model is a kind of assignment model 3 Framework of modified algorithm based based on the principle of stochastic user equilibrium, on MSA which is described as: in a stochastic user equilibrium network no user believes he/she can improve his/her The basic algorithm step of the SUE problem above travel cost by unilaterally changing routes. is to search for the descent direction and move size, both The conditions of the SUE are formulated as of which are difficult to be found. However, MSA can be f rs (C rs − C rs ) = 0 (1) used to solve the SUE problem despite the difficulties k k above. The descent direction and move size of the MSA rs rs C k − C ≥ 0 (2) can be written as follows: f rs ≥ 0 (3) 1 k λ n = (7) rs n where C k is the perceived generalized travel cost on rs n n n route k for the O−D pair (rs) in equilibrium. C k can be d =y −x (8) expressed as C rs = c rs + ε rs , where c rs is the actual k k k k y n = q ⋅ P rsn ⋅ δ rs (9) travel cost and ε rs is the random component. C rs is the a ∑∑ rs k a , k k rs k perceived generalized travel cost for the O­D pair(rs) in n equilibrium and is the expectation of { C rs }. f rs is the where n is the number of iterations; y is the auxiliary k k n number of trips on route k for the O−D pair (rs). link flows vector at the n­th iteration; y a is the auxiliary rsn Moreover, there are two important relationships can be flow on link a at the n­th iteration; Pk is the choice respectively shown as probability of route k for the O−D pair (rs) at the n­th iteration; δ rs =1 if link a is on route k, otherwise f rs = q (4) a, k ∑ k rs δ rs =0; xn is the link flows vector at the n­th iteration; λn k a, k is the move size; and dn is a descent direction vector rs rs n x a = ∑∑∑ f k ⋅ δ a , k (5) computed at x . r s k Based on the conventional MSA, the modified where qrs is the number of trips for the O−D pair (rs). xa algorithm’s framework is summarized by Algorithm 1. rs is the flow volume on link a. δ a ,k =1, if link a is on rs route k for the O−D pair (rs); otherwise, δ a ,k =0. Algorithm 1: Modified algorithm for SUE model based The equations above can be solved by indirect on MSA solution where a minimization program is expressed as 1 Input: URT network, rail­use O−D matrix, and all J. Cent. South Univ. (2013) 20: 2897−2904 2899 parameters expressed as 2 Output: link flows vector x n in stochastic user­ T rs = α ⋅(t rs + t rs ) (11) equilibrium 2,k,m 1,k,m, pq 2,k, q n rs 3 Initialize n←0, t a ←0 where t is the walking time for transferring from n 1,k,m, pq 4 Generate the choice set {k } line p to line q at station m of the k­th route for the O−D 5 Load O−D matrix to network using a route choice pair (rs), and it can be induced from the walking distance model and obtain x 1, let n=1 for transferring and the passenger’s walk speed; t rs 6 Update the set of travel cost on links { t n } with the 2,k, q a is the waiting time at station m, which can be half of the set of current flows x n train’s headway; α is the amplification factor for 7 Load O−D matrix to network again with { t n }, and a transferring time, which is greater than 1 since the obtain a set of auxiliary flows { y n } a perceived transferring time is greater than the rail­ride 8 Calculate the current flows as: xn+1 = x n + 1/ a a travel time in terms of the same actual value [18]. n( yn − xn ) , ∀a ∈ A  a a  The in­vehicle congestion of the k­th route for the 9 Calculate the difference between two consecutive O−D pair (rs) is defined as n+1 n 2 n iterations as ∑ (xa − xa ) / ∑ xa < ε rs 2 x a a H rs = (β rs ) ⋅T rs = ( k, a ) 2 ⋅ T rs (12) k ∑ k,a 1,k,a ∑ rs 1,k, a n+1 n 2 n a a Dk , a 10 If ∑ (xa − xa ) / ∑ xa < ε (ε is a given value) a a rs where β k, a is the in­vehicle congestion rate of rail link 11 let n=n+1, and go to Line 6 rs a of the k­th route for the O−D pair (rs); xk , a is the 12 Else rs 13 break hourly link flow of link a; Dk , a is the hourly traffic 14 End if flow capacity of link a. n n 15 Return x ={ x a }. 4.2 Choice set generation Although the inaccuracies in passenger flow Due to network operation, there may be several forecasts for URT systems are due to various factors of alternative rail routes for an O−D pair (rs) and trip distribution, deliberately slanted forecasting, trip passengers, in practice, will choose not only the shortest generation and so on [15], in terms of assignment route but also the second, the third, …, the k­th shortest algorithms, there are three keys, which have important path due to the factor of congestion. Therefore, firstly, an influence on the forecasts’ accuracy, need to be improved deletion algorithm (DA) based on depth­first highlighted: travel cost function, choice set and route traversal (DFT) is introduced to find the k­th shortest choice model [16−17]. route and the initial choice set for a given O−D pair (rs) is obtained. 4 Modifications for URT assignment However, there is no benefit to enumerating routes algorithm under network operation that no passenger would consider. It is observed from travel surveys for URT passengers in China that if the 4.1 Generalized cost function travel cost difference between an alternative route and Based on the insight from travel surveys for URT the shortest route exceeds a threshold, there is little passengers in China, the generalized cost function of a possibility for the alternative route to be chosen by route, in which the factor of crowding is considered, is passengers. The threshold can be calculated as specified as rs rs rs rs rs rs ∆Gmax = min{δ ⋅Gmin , ∆G } (13) Gk = ∑T1,k,a + ∑ T2,k, m + H k (10) a m rs where ∆G ma x is the threshold of travel cost difference rs where G k is the generalized cost for the O−D pair (rs); between an alternative route and the shortest route for the rs rs T1 ,k, a is the rail­ride travel time of the k­th route for the O−D pair (rs); Gmi n is the travel cost of the shortest rs O−D pair (rs); T2 ,k, m is the passenger’s stay time at the route for the O−D pair (rs); δ and ΔG are a proportion rs station m of the k­th route for the O−D pair (rs); H k is coefficient and a constant with the same unit to travel the in­vehicle congestion of the k­th route for the O−D cost, respectively, both of which can be decided by the pair (rs). result of travel survey for URT passengers. rs There are two different formulas to calculate T2 ,k, m . Moreover, due to network operation, it is also rs If the station m is a pass station, T2 ,k, m is the train’s stay necessary to consider the constraint of rail line’s service rs time. If the station m is a transfer station, T2 ,k, m can be time. During some certain period of time, a route is 2900 J. Cent. South Univ. (2013) 20: 2897−2904 unable to be chosen by passengers and thus will be Grs − G rs ∆C rs = k min (16) excluded from the choice set in case of some circuits of it k rs min{δ ⋅Gmin , ∆G } beyond their service time. rs Based on the above descriptions, a method, based where ∆C k is the generalized travel cost difference on the k­th shortest route algorithm, for generating between the shortest route and route k; σ is the standard choice set with time constraint can be described as deviation of normal distribution, which is a constant to follows: all the O−D pairs and can be analyzed and drafted Step 1: Obtain the initial choice set using the k­th through the results of travel surveys. Other variables are shortest route algorithm based on topology of the URT the same as described before. network. Step 2: Judge the rationality of alternative routes 5 Test on URT network in Shanghai based on the threshold of travel cost difference between an alternative route and the shortest route. 5.1 Network and data used in test Step 3: Judge the rationality of alternative route In order to test the modified assignment algorithm again based on the constraint of rail lines’ service time, proposed in this work, a full­scale test network building and generate the final choice set. on the 2010 URT network in Shanghai (see Fig. 2) is constructed firstly. The test network includes the links 4.3 Route choice model for the rail lines, transfers at rail stations, waits at rail In the modified assignment algorithm, a route stations as well as rail stations. choice model which is derived from travel surveys for URT passengers in China is introduced to perform the stochastic network loading at each iteration: U rs p rs = k (14) k rs ∑ U k k rs where p k is the choice probability of the route k for a rs given O−D pair (rs); U k is the utility of the route k for the O−D pair (rs), assuming the utility of the shortest route is the maximum and equals to 1. How an alternative route’s generalized travel cost Fig. 2 Urban rail transit network in Shanghai, China (2010) oversteps the shortest route’s? Based on travel survey of rs URT passengers in China, the utility U k can be related The data used in the test to analyze the new rs to the degree ∆C k , and the relationship between them algorithm’s performance is from the 2010 Travel Survey is illustrated in Fig. 1, and could be expressed as of URT passengers in Shanghai conducted by the rs 2 Institute of Shanghai Urban Transportation Planning rs −(∆C k ) U k = exp( ) (15) Research. The passenger travel data include the records 2σ 2 of rail journeys with an origin and destination, travel rail lines, transfer stations and the beginning and the end time.

5.2 Parameter estimation Some parameters in the new algorithm should also be estimated in advance. The rail­ride travel time and trains’ stay time at rail stations are induced from train rs schedules. The walking time for transferring ( t1 ,k,m, pq ) is decided by the layout of the rail station. The waiting time rs ( t2 ,k, q ) is calculated according to the train’s headway. rs And the hourly flow capacity ( Dk , a ) which can be used to calculate the in­vehicle congestion rate of rail link a rs ( β k, a ), is decided by the rail line’s frequency and Fig. 1 Route’s utility value distribution loading capacity. The information above of train J. Cent. South Univ. (2013) 20: 2897−2904 2901 schedules, stations’ layouts, and trains’ loading capacities And the third one is the new algorithm proposed in this are provided by Operation Center of Shanghai Metro. work. In addition, based on the results of the travel survey, Shanghai’s Algorithm is an all­or­nothing (AON) δ, ΔG and α are 60%, 10 min and 1.5, respectively. The assignment algorithm based on mileages, while Beijing/ service time of each route is induced from rail lines’ Guangzhou’s Algorithm is a multi­route probabilistic departure time of the first and last trains. assignment algorithm based on time. Although the route choice of SUE is difficult to be specified, the route 5.3 Calculation and results choice from the last iteration of the new algorithm is The rail­use O−D matrixes on the day when the used to be the estimated route choice. travel survey was carried are obtained from the Three algorithms are test under the same Automatic Fare Collection (AFC) data and prepared to computation condition: 2.50 GHz Intel Core i5­2520 M, be the input of the assignment. The experiential data Windows 7 operating aystem, and the comparisons of from the Operation Center of Shanghai Metro are used their computation performance are given in Table 3. for the initial values of in­vehicle congestion rates in the Figure 3 shows a comparison of the shortest route rail links for the assignment. The convergence criteria (ε) choice probabilities estimated by the different algorithms of the assignment algorithm can be set to 0.05. Table 1 with the observed shortest route choice probabilities. and Table 2 give the results of assignment using the new Although it will be perfect if comparisons are conducted algorithm proposed in this work. from both aspects of route choices and link flows, the algorithms are not examined by the comparison of the 5.4 Analysis and validation estimated link flows and the observed link flows in our To examine the algorithm’s fitness, three algorithms analysis. This is because there are not enough data for are tested and the estimated route choices are compared obtaining observed link flows as the corresponding with the observed route choices from the travel survey survey of passenger flows was not conducted before. for each algorithm. The first one, which can be named Figures 4 and 5 show a case analysis in which the Shanghai’s Algorithm, is the present method for flow route choices estimated by the three algorithms are analysis in practice by URT operators in Shanghai. The compared for a specific O−D pair (from Guilin Rd. second one, which can be named Beijing/Guangzhou’s station to People’s Square station). There are two routes Algorithm, is the present method for flow analysis in that connect the origin station Guilin Rd. and the practice by URT operators in Beijing and Guangzhou. destination station People’s Square. The first route is the

Table 1 Estimated maximal link flows for each line in peak hour Line (up) Link Flow Line (down) Link Flow North Zhongshan Rd. to Line 1 Caobao Rd. to Shanghai Indoor Stadium 53 026 Line 1 41 734 Shanghai Railway Station

Line 2 People’s Square to East Nanjing Rd. 44 961 Line 2 Dongchang Rd. to Lujiazui 37 686

Line 3 Yishan Rd. to Hongqiao Rd. 13 167 Line 3 Chifeng Rd. to Hongkou Stadium 24 970

Shanghai Indoor Stadium to Line 4 Yangshupu Rd. to Pudong Avenue 19 504 Line 4 14 946 Shanghai Stadium

Line 5 Chunshen Rd. to Yindu Rd. 7 092 Line 5 Chunshen Rd. to Xinzhuang 12 102

Line 6 Yuanshen Stadium to Century Avenue 15 521 Line 6 Century Avenue to Yuanshen Stadium 8 414

Line 7 Langao Rd. to Zhenping Rd. 24 178 Line 7 Changshu Rd. to Jing’an Temple 15 117

Line 8 Zhongxing Rd. to Qufu Rd. 23 133 Line 8 Lujiabang Rd. to Laoximen 21 275

Line 9 Hechuan Rd. to Caohejing Hi­Tech Park 22 234 Line 9 Yishan Rd. to Guilin Rd. 17 750

Line 10 Laoximen to Yuyuan Garden 10 338 Line 10 Hailun Rd. to North Sichuan Rd. 17 347

Line 11 Fengqiao Rd. to Caoyang Rd. 14 042 Line 11 Liziyuan to Qilianshan Rd. 7 914 2902 J. Cent. South Univ. (2013) 20: 2897−2904 Table 2 Estimated transfer flows for transfer stations in peak hour Transfer station Daily Peak hour People’s Square 389 338 55 141 Century Park 260 976 36 393 Jing’an Temple 96 399 10 750 Xinzhuang 90 062 13 159 Jiangsu Rd. 66 307 6 634 Shanghai Indoor Stadium 108 588 15 387 Zhongshan Park 62 957 10 024 Changshu Rd. 59 915 7 454 Zhenping Rd. 42 141 5 508 Zhaojiabang Rd. 61 503 7 189 Caoyang Rd. 75 899 10 111 Baoshan Rd. 47 023 8 228 South Xizang Rd. 57 955 7 326 Shanghai Railway Station 26 868 4 502 Laoximen 68 030 7 948 Xujiahui 68 735 8 334

Table 3 Comparison of computation performance among three algorithms Number of Computation Root mean Algorithm R 2 iteration times time/min square error Shanghai’s 1 3.26 — 0.055 6 Algorithm Beijing/ Guangzhou’s 1 8.17 0.976 0.008 7 Algorithm Algorithm proposed in 16 130.64 0.994 0.003 9 this work path through stations Yishan Rd., Hengshan Rd., Changshu Rd., South Shanxi Rd., and South Huangpi Rd. with the transfer station of Xujiahui, while the second route is the path through stations Yishan Rd., Xujiahui, Zhaojiabang Rd., Jiashan Rd., Dapuqiao, Madang Rd., Fig. 3 Comparisons of observed route choice probabilities Laoximen, Dashijie with the transfer station of Lujiabang versus computed route choice probabilities among three Rd. The travel time of the first route is about 29 min in algorithms: (a) Shanghai’s Algorithm; (b) Beijing/Guangzhou’s peak hours including about 9 min for transferring while Algorithm; (c) Proposed algorithm the travel time of the second route is 34 min in peak hours including about 6 min for transferring, and both second route. It is because that the factor of crowding is transfer times have been amplified. As we can see, the considered in the new algorithm proposed in this work choice probabilities of the first route computed by and the iteration strategy is carried out. Shanghai’s Algorithm and Beijing/Guangzhou’s In summary, according to the tables and figures Algorithm are much greater than the observed data, above, the proposed new algorithm delivered the most while the result from the new algorithm is the closest one accurate solution. Although it involves multi­routes to the observed data. The problem is the same to the search and iteration strategies which result in higher J. Cent. South Univ. (2013) 20: 2897−2904 2903

Fig. 4 Rail network connecting Guilin Rd. and People’s Square: (a) Guilin Rd. to Lujiabang Rd. to People’s Square is the first route connecting Guilin Rd. and People’s Square; (b) Guilin Rd.to Xujiahui to People’s Square is the second route connecting Guilin Rd. and People’s Square

practical methods used in today’s China. 3) The proposed algorithm can also be applied into those URT systems of similar giant cities in highly­populated areas, especially East Asian cities such as Tokyo, Seoul, Singapore, etc.

Acknowledgement The authors would like to appreciate Shanghai Shen Tong Metro Co., Ltd. and Institute of Shanghai Urban Transportation Planning Research for providing valuable information. Special thanks are given to the anonymous reviewers for giving very helpful comments. Fig. 5 Comparison of estimated route choices between Guilin Rd. and People’s Square among different algorithms References consumption of computation time compared with the other two methods, the new algorithm is still acceptable. [1] BEN­AKIVA M, LERMAN S R. Discrete choice analysis: Theory As the hardware improves, we believe that the algorithm and application to travel demand [M]. Cambridge, Massachusetts: derived from this research will become more viable and MIT Press, 1985: 31−57. [2] SMITH T E, HSU C C, HSU Y L. Stochastic user equilibrium model practicable. with implicit travel time budget constraint [J]. Transportation Research Record: Journal of the Transportation Research Board, 6 Conclusions 2008(2085): 95−103. [3] LIU Y L, BUNKER J, FERREIRA L. Transit users’ route­choice 1) A modified stochastic user­equilibrium modeling in transit assignment: A review [J]. Transport Reviews, assignment algorithm based on the frame of MSA 2010, 30(6): 753−769. algorithm is developed to solve the SUE problem for [4] LIU Tian­liang. A study on game equilibria and economic behaviors URT network. in transportation network with information provision [D]. Beijing: 2) Initial test shows that the computation of the School of Economics and Management, Beijing Jiaotong University, 2008: 14−27. (in Chinese) algorithm has a good performance and the results [5] ZHU Wei. Research on the model and algorithm of mass passenger indicate that it provides better accuracy than those flow distribution in network for urban rail transit [D]. Shanghai: 2904 J. Cent. South Univ. (2013) 20: 2897−2904 School of Transportation Engineering, Tongji University, 2011: [13] SHEFFI Y, POWELL W B. An algorithm for the equilibrium 15−24. (in Chinese) assignment problem with random link times [J]. Networks, 1982, [6] KATO H, KANEKO Y, INOUE M. Comparative analysis of transit 12(2): 191−207. assignment: Evidence from urban railway system in the Tokyo [14] LAM W H K, GAO Z Y, CHAN K S, YANG H. A stochastic user Metropolitan Area [J]. Transportation, 2010, 37(7): 775−799. equilibrium assignment model for congested transit networks [J]. [7] SHELFFI Y. Urban transportation networks: equilibrium analysis Transportation Research Part B, 1999, 33(6): 351−368. with mathematical programming methods [M]. Englewood Cliffs, [15] FLYVBJERG B, HOLM M K S, BUHL S L. Inaccuracy in traffic Prentice­Hall, Inc, 1985: 312−341. forecasts [J]. Transport Reviews, 2007, 26(1): 1−24. [8] THOMAS R. Traffic assignment techniques [M]. Aldershot: The [16] NIELSEN O A. A stochastic transit assignment model considering Academic Publishing Group, 1991: 32−46. differences in passengers utility functions [J]. Transportation [9] BELL M G H, IIDA Y. Transportation network analysis [M]. Research Part B, 2000, 34(5): 337−402. Chichester, West Sussex: John Wiley & Sons, 1997: 101−120. [17] BEKHOR S, BEN­AKIVA M, RAMMING S. Evaluation of choice [10] CASCETTA E. Transportation systems analysis: models and set generation algorithm for route choice models [J]. Annals of applications [M]. New York, NY: Springer Science & Business Operation Research, 2006, 144(1): 235−247. Media, 2009: 43−51. [18] XU Rui­hua, LUO Qin, GAO Peng. Research on the clearing method [11] BEKHOR S, TOLEDO T. Investigating path­based solution of beijing ACC on urban mass transit [R]. Shanghai: Tongji algorithm to the stochastic user equilibrium problem [J]. University, 2007. (in Chinese) Transportation Research Part B, 2005, 39(4): 279−295. (Edited by DENG Lü­xiang) [12] ZHANG T R, YANG C, CHEN D D. Modified origin­based algorithm for traffic equilibrium assignment problems [J]. Journal of Central South University, 2011, 18(5): 1765−1772.