0 0 A Heuristic Method for Bus Rapid Transit Panning Based on the Maximum Trip Service Zhong Wang Associate professor, Schoo of Transportation & Logistics Daian University of Technoogy No., Linggong Road, Daian, Liaoning Province, China 0 TEL: +-000; Emai: zwang@dut.edu.cn Fengmin Lan Graduate Research Assistant, Schoo of Transportation and Logistics, Daian University of Technoogy No., Linggong Road, Daian, Liaoning Province, China 0 TEL: +-; Emai: anfm@mai.dut.edu.cn Lian Lian, Corresponding Author Schoo of Transportation and Logistics Daian University of Technoogy No., Linggong Road, Daian, Liaoning Province, China 0 TEL: +--0; Emai: ian.ian@dut.edu.cn Word count:, words text + tabes/figures x 0 words (each) =, words Submission Date: Juy 0 th, 0
Wang, Lan, Lian 0 Abstract Bus rapid transit (BRT) is characterized by higher speed, higher comfort and bigger capacity compared to conventiona pubic transportation service. Athough more and more cities choose BRT in recent years wordwide, there is an absence of scientific and quantitative approach for BRT network panning. The probem of BRT panning in a given network is very compex considering the constraints of road geometrics, reguations, right of way, trave demand, vehice operations and so on. This paper focuses on deveoping an optimization mode for BRT network panning, where the authors estabish a mathematica mode with the objective of maximizing the number of trips served by the network, subjected to a number of constraints incuding distance between stations, expense of construction, road geometrics, etc. In addition, the noninear coefficient of the BRT route is taken as a constraint, which is an important indicator but widey ignored in previous studies. A heuristic method is appied to generate optima soutions to the integer programming mode with respect to a constraints. A case study is conducted in Luoyang, China and the numerica resuts indicate that the method is effective therefore can be appied to improve BRT network panning. Keyword: Transportation Panning; BRT; Transit; Network Panning; OD; Heuristic Method
Wang, Lan, Lian 0 0 0. INTRODUCTION The rising private car ownership has caused remarkabe trave demand increase as we as traffic congestion, green gas emission, road accidents and energy consumption. To address these probems, deveoping efficient pubic transportation systems are extremey important. Various pubic transportation systems such as subway, ight rai, conventiona bus service, and BRT have been deveoped to compete with private cars. Among these pubic transportation systems, rai transit has the highest capacity and efficiency but needs arge capita investment and a ong impementation period. In addition, the rai transit ony aows rairoad car operations on fixed tracks and is ack of service fexibiity. BRT, on the other hand, provides an option of high capacity, comfort, fexibiity, quick impementation, and reativey ow cost. It is defined as a fexibe, rubber-tired form of rapid transit that combines stations, vehices, services, running ways and information technoogies into an integrated system with strong identity (). BRT possesses the advantages of both conventiona bus service and rai transit in capacity and fexibiity. It is considered one of those systems that can bridge the gap between the demand and suppy of transportation. In the past decade, it has been embraced by a number of major and medium-sized cities in the word (,,). To ensure the success of its impementation and to make fu use of the imited financia and road capacity resource, the BRT network and station ocations must be carefuy panned. The iterature reating to transit network design has continuousy grown over time. Magnanti et a. reviewed some of the appications of integer programming methods for transit network design, and introduced severa continuous and discrete choice modes and agorithms (). Guihaire and Hao presented a goba review of the design and scheduing of the transit network (). Laporte et a. reviewed some indices for the quaity of a rapid transit network, as we as mathematica modes and heuristics that can be used to sove the network design probem (). Farahani et a. presented a comprehensive review of the definitions, cassifications, objectives, constraints, network topoogy decision variabes, and soution approaches for the pubic transportation network design probems (). It has been widey recognized that operationa research methods can hep to determine the aignments and station sites of a transit network (). Gutiérrez-Jarpa et a. proposed a tractabe mode in which trave cost was minimized and traffic capture maximized (). The branch-and-cut method was used to sove the probem in the CPLEX framework. Besides, heuristic methods were often used to sove simiar probems. Bruno et a. presented a mathematica mode to maximize the tota popuation covered by the rapid transit aignment, and a two-phase heuristic was used to generate a rapid transit aignment in an urban setting (). However, the objective function in this study was ony subjected to interstation spacing constraint, which was rather simpistic to refect the reaity. Nikoić and Teodorović deveoped a mode to optimize the number of satisfied passengers, the tota number of transfers and the tota trave time of a served passengers (). The probem was soved using the Bee Coony Optimization (BCO) meta-heuristics. For the same probem in Nikoic and Teodorović s study, Nayeem et a. deveoped a Genetic Agorithm (GA) to sove it (). Beside the studies focusing on the mode and the agorithm, some researchers have proposed methods to anayze and compare transit networks in forms of star, cartwhee,
Wang, Lan, Lian 0 0 triange, grid and so on. Laporte et a. suggested that in grid cities, the modified grid and haf-grid configurations were the best in terms of passenger-network effectiveness but inferior to grid configurations with respect to passenger-pane effectiveness (). Hosapujari and Verma proposed an approach to deveop a hub and spoke mode for bus transit network services (). Due to its reativey short history and imited impementations in the deveoped countries, BRT has attracted itte attention and the iterature in BRT network design is imited. In practice, the panning of BRT route or network often depends on panners experience instead of a scientific approach with soid quantitative anaysis. Taking the aforementioned studies in transit network design as a foundation, the authors aim to putting the BRT panning probem in a mathematica framework and sove it. The remainder of this paper is structured as foows: In Section, the BRT panning probem is introduced and assumptions described; In Section, the mathematica mode is estabished; In Section, a heuristic method is given to sove the probem step by step; Section introduces an appication to rea case scenarios; The summary and future studies are presented in Section.. PROBLEM DEFINATION. The Right of Way for BRT BRT differs from conventiona pubic transportation modes in many aspects. It often requires dedicated right of way and specia stations. Thus, it is a unique urban surface pubic transportation system. Some countries have panning guideines and design standards for bus-ony or BRT right of way settings. For exampe, according to The setting for bus anes (00) () of China, bus-ony anes, incuding BRT anes, shoud be set up when the street inks meet a of the foowing conditions: a) The number of motor anes in one direction shoud not be ess than, or the tota width of a motor anes in one direction shoud not be ess than meters; b) The number of bus passengers in one direction shoud not be ess than,000 during peak hours, or the bus voume shoud not be ess than 0 per hour per direction during peak hours; c) The average traffic voume shoud be more than 00 vehices per ane during peak hours. BRT can be impemented on the roads that meet the conditions above. When deveoping the optimization mode, these design standards or panning guideines shoud be incorporated into the constraint set to refect the reguation requirements, or we can first make a scan of the roads in the panning area and then search for feasibe BRT routes on the candidate inks. The scan criteria can be expressed mathematicay as foows: 0 LN Q Q B,, ave 0 veh / h 00 veh / h Where i and j denote the seria number of intersections in the panning area and i j; () LN is
Wang, Lan, Lian the number of anes from intersection i to intersection j; QB, is the bus voume on the ink from intersection i to intersection j during peak hours; Q is the average traffic voume, ave on the ane from intersection i to intersection j during peak hours. Eq. () is the first constraint for the mode, which is considered as geometric and reguation constraints. Ony two-way roads are considered in this study, and the BRT route is assumed to be on a ink in both directions.. Trips Served by the BRT Network In genera, the OD matrices are estabished based on traffic anaysis zone (TAZ) data such as and use, popuation, empoyment, etc. In this study, it is assumed the ODs in the panning area are known by modes of transportation and are presented in OD matrices. Since the centroids of TAZs, where trips start and end, are usuay not station sites, the OD matrix based on TAZ centroids needs to be further converted into OD matrix based on potentia stations. To simpify the probem, the induced trave demand is not considered when a BRT network is constructed. In the form of piecewise function, an attraction function () is introduced as β ki, that depends on the Eucidean metric distance d ki between the passenger ocation k and the potentia station i. The passenger ocations are concentrated at the centroids of TAZs. The attraction function is then assumed as foows: 0 0, ( dki 0) 0. (0 dki 00), ki 0., (00 dki 00) () 0., (00 dki 0) 0, ( dki 0) Before cacuating t, which is the OD from i to j, it must be known whetherβ ki and β kj equa to 0. If both β ki andβ kj are not 0, then (β ki +β kj )*(BOD kk +BOD kk )/ composes of the OD from i to j, where BOD kk is BRT OD from zone k to zone k. If mutipe stations are seected to serve the passengers from a TAZ, then the passengers wi be aocated to these stations proportionay based on their distances to the centroid. In this study, it is assumed the amount of ODs served by the BRT network contains two portions: direct trips and trips with a transfer. To simpify the probem, it is assumed that trips with more than one transfers are at a ow eve. Aso, there are transfers between BRT and the conventiona pubic transportation. To obtain this portion of transfer trips, the conventiona pubic transportation network in the panning area must be known we. A significant amount of work needs to be done to address these issues, which wi be considered in future studies. Figure -a gives a simpe exampe of a BRT route and the gray ces in Figure -b represent the direct trips served by the route. Figure -a shows two individua BRT routes: a-c-e-g-h-m-p and d-f--m-o-q-t. Passengers make transfers at station m when necessary. Figure -b presents the trips with a transfer served by the network.
Wang, Lan, Lian FIGURE Direct trips served by BRT. 0 FIGURE Trips with a transfer served by BRT. Then the serving trips of a BRT network can be cacuated through Equation (): ' '' ( t ) ( ) t ji v tim tmj vim vmj () i; i, m; im Where t is the amount of OD from i to j; v is a 0- variabe indicating whether i and j are on the same route and the trips from i to j are directy served trips, so are ' im v and. The Potentia Stations and Connections To simpify the probem, it is assumed the BRT stations can be set in the intersection areas. A intersection areas that are on the quaified road inks can be potentia BRT station sites. The stations wi then be automaticay searched and seected for the BRT route according to the mode estabished in the foowing sections. Using geographic information data, the distance ω between each pair of adjacent intersections can be obtained. For those intersections not adjacent to each other, their distances are set to be infinite. Based on ω, the Foyd Agorithm is empoyed to cacuate the distances between any two potentia stations with the assumption of intersection-potentia '' mj v.
Wang, Lan, Lian 0 station overap. The distance matrix and the shortest paths between each pair of potentia stations are then generated. The procedure of the Foyd Agorithm is described as foows (): Step 0: Let k=0, and every potentia stations are given a seria number u, u, u n. Create a matrix D 0 0, i j, in which the eements is d and a matrix P 0, whose eements 0, i j p 0 is i. Step : k = k+, derive matrix D k from matrix D k-, and derive matrix P k from matrix P k- k k k k k k k,if k. For a u i and u j, d min( d, dik dkj ) and p d d p. k k k k pkj,if d dik dkj Step : If k = n, stop; ese, go to Step.. THE MATHEMATICAL MODEL After a scan of quaified roads in the panning area according to eq. (), we wi further find BRT routes and stations to estabish the BRT network. It is defined that x is whether the ink from i to j is chosen by BRT route : if x =, then the ink from i to j is chosen; if x =0, then the ink from i to j is not chosen. y i is whether the potentia station i is chosen by BRT route : if y i =, then a station is set at intersection i; if y i = 0, then a station is not set at intersection i. The objective of the mode is to pursue the maximum number of trips served by the BRT network. As shown in Eq. (), the number of trips served is cacuated as: ' '' ji im mi im mj () T( i, j) ( t t ) v ( t t ) v v i; i, m; im Thus, the mathematica mode for this probem can be formuated as: max T( i, j ) () subject to: x =,when dmin d dmax () E x ei yi Emax () i i x yi ; If x =, then yi, y j ( i, j) D in s ( d x ) d C od N () () v y ; i v y ; j v im y ; i v im yi () Where e i is the construction expenditure of station i; E is the construction expenditure to impement BRT right of way on the ink from i to j; d min and d max are the minimum and the maximum distances between two adjacent stations, respectivey; C N is a constant, i.e. the
Wang, Lan, Lian threshod of noninear coefficient. Constraint () is the station spacing constraint that ensures the distance between any two adjacent stations fas into a certain range. Constraint () is the cost constraint that ensures the expenditure of the BRT network is within the budget. Constraint () prevents a route from passing a station more than once and ensures the route is consecutive. Constraint () is the noninear coefficient constraint that ensures the overa ayout of the route maintain a certain eve of straightness. The noninear coefficient is the ratio between the Manhattan and the Eucidean distance from the start point to the end point of a route. In some countries reguations require that the noninear coefficient of a transit route shoud be under a threshod such as.(). Constraint () is to match the OD pairs with the chosen stations.. THE PROCESS OF HEURISTIC METHOD Based on the mode proposed above, there wi be two stages to sove the probem: ) obtain singe route soutions, and ) compose the routes into a network. The procedure of this heuristic can be found in Figure. Determine panning area, obtain road network and trave demand Geometric and regeation constraints (eq.) Construction expenditure of every potentia station Potentia stations sites Candidate BRT inks Construction expenditure matrix for inks from i to j Station spacing and consecution constraints (eq.&eq.) Feasibe soution set I Expenditure constraint (eq.) Feasibe soution set II Noninear coefficient constraint (eq.) Feasibe soution set III Cacuate served trips Top-ranked feasibe soutions Identify key stations FIGURE Soving process of the mode. Determine the BRT network configuration and identify the optima soution
Wang, Lan, Lian 0 0 Step 0: As described in section., the distance matrix and the paths between each pair of potentia stations are cacuated using the Foyd Agorithm. Step : If a path goes from i to j via M inks, then the E can be cacuated according to Eq. (). The expenditure matrix wi then be estabished. The vaue of c m, which represents the cost per unit ength of right of way construction, can be determined according to ane number, ength, width, even breadth of the road. M E d c () m m m Step : A new matrix with vaues in or 0 is introduced, which indicates whether two stations can be seected consecutivey on a route. The vaues of the matrix are determined by whether the distance d of any two potentia stations meets the station spacing constraint Eq (). Step : Starting from each potentia station, a feasibe soution set I is obtained, which contains the routes whose stations can be derived through the 0- matrix estabished in step. If a station has aready been seected by a route, it wi not be seected or passed through again. This ensures that BRT routes meet the route consecution constraint Eq (). Step : Examine the feasibe routes generated from Step with respect to cost constraint Eq (). The feasibe soution set II is then obtained, which contains those routes meet station spacing, consecution, and cost constraints. Step : Cacuate each route s ength and the Eucidean distance between the start station and the end station, and then cacuate each route s noninear coefficient. Eiminate the routes whose noninear coefficients exceed the threshod, as shown in Eq (), and obtain the feasibe soution set III meeting a constraints. Step : Cacuate and rank the quantity of direct trips for each feasibe route. Step : Identify the key stations that frequenty appear on the top-ranked feasibe routes. Step : On the basis of key stations, compose the BRT network configuration by making combinations of the top-ranked routes. Cacuate the trips with a transfer, then cacuate T(i,j), and finay find the best combination as the optima soution to the probem.. THE CASE STUDY AND NUMERICAL RESULTS To vaidate the mode and agorithm proposed in this study, Luobei District in the City of Luoyang, China (seeing Figure ) was chosen for a case study. The road network and trave demand data were obtained from oca panning agencies. The data of popuation, traffic anaysis zones and transport network used in this study came from an estabished trave demand forecast mode, which was ast updated in 0.
Wang, Lan, Lian FIGURE Case study area of Luoyang, China. First, the geometric and reguation constraints are empoyed to identify the candidate inks where BRT right of way can be impemented. Intersections on the candidate inks are then numbered as the potentia station sites, seeing Figure. As shown in the figure, the road network contains inks and intersections. FIGURE Candidate inks and station sites. In the case study, it is assumed min =0m, max =,00m, and C N =.. E max are assumed to be a certain budget vaue according to the number of stations and the tota ength of a BRT route. The average construction expenditure of a station is assumed to be miion CNY. Foowing the steps described in Section, more than 0,000 feasibe soutions are found ti Step. In Step, station,, are identified as the key stations because they
Wang, Lan, Lian 0 are the most frequenty seected ones among the top-ranked feasibe soution set. As pointed out by Laporte et a. (), there are severa basic configurations for a transit network incuding star, cartwhee, triange, grid and modified configurations. Based on the recommendations and distribution of the key stations, the modified grid configuration is seected for the case study area in order to achieve the best in terms of passenger-network effectiveness. Tabe and depict the detaied information of the network soution and routes incuded. Four network options, from a through d, are presented, each having three BRT routes. As shown in Tabe, each route wi serve,-, trips directy with ength varying from,0 to,0 meters. The transfer trips are added and shown in Tabe. The expenditure of each route is provided in Tabe whie the tota expenditure of each network option is given in Tabe, which indicate the budget constraint E max is effective. The noninear coefficients are under the constraint of.. However, from Figure it can be seen that the aignment of the seected route zigzags in the panning area with the vaue ranging from. to.. To improve this, the threshod vaue of the noninear coefficient constraint can be set ower according to the size of the panning area and the number of stations on the BRT route. TABLE Option a b c d TABLE Option Summary of the Soutions and Seected Routes Route No. Route aignment Served OD Length (m) Expenditure ( CNY) Noninear coefficient -------,,0,. -----0--,,0,0. -------,,,. -------,0,0,0. -----0--,,0,0. -------,,,. -------,,0,. 0-------,0,,. -----0--,,0,0. -------,0,0,0. 0-------,0,,. -----0--,,0,0. Summary and Comparison of the Soutions Directy-served OD OD with Transfer Objective Function Vaue Tota Length (m) Expenditure ( CNY) a 0,,,,0 0,0 b,,,0, 0, c,,,,0, d,,0,,,
Wang, Lan, Lian 0 FIGURE Layout of the network soutions. From Tabe it can be seen Option c is better than others in a aspects incuding the objective function vaue, the directy-served trips, served trips with transfers, tota ength and network expenditures. It is then considered the best soution for the case study area with respect to a the constraints. This BRT network option, containing routes, wi serve a tota of, direct and one-transfer BRT trips per day in the case study area. It proves that the proposed method is capabe of finding the BRT network that wi serve the maximum number of OD trips whie satisfying the constraints of budget, noninear coefficient, minimum and maximum stop distances, and road geometrics.. CONCLUSIONS AND OUTLOOK Considering the factors incuding distance between stations, expense of construction, road geometrics and so on, the authors proposed a mathematica mode for the BRT network panning probem, foowed by a heuristic agorithm and a case study in Luoyang, China. The objective function of the mode is to maximize the tota trips served by a BRT network, subjected to a number of geometric, reguation, station spacing, expenditure, and consecution constraints. Aso, noninear coefficient is introduced in the mathematica mode to measure the straightness of the BRT route aignment. A heuristic method is appied to sove the probem and to generate a BRT network soution. Athough the agorithm has reativey ow efficiency, it is effective to find the optima soution of BRT network incuding routes and stations, and it is easy to understand and impement. This study makes a contribution to improve the scientific and quantitative anaysis in BRT panning. Since the panning practice is often infuenced by poitics and the subjectivity of decision makers, the quantitative resuts from appying this method can be presented as decision support materias. In the future it is possibe to improve the agorithm
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