Research Article Optimal Design of the Feeder-Bus Network Based on the Transfer System

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1 Discrete Dynamics in Nature and Society Voume 2013, Artice D , 10 pages Research Artice Optima Design of the Feeder-Bus Network Based on the Transfer System Lianbo Deng, Wei Gao, Yanbing Fu, and Weniang Zhou Schoo of Traffic and Transportation Engineering, Centra South University, Changsha , China Correspondence shoud be addressed to Yanbing Fu; quanshuiqq@163.com Received 25 August 2013; Accepted 25 October 2013 Academic Editor: Huimin Niu Copyright 2013 Lianbo Deng et a. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. This paper studied the cassic feeder-bus network design probem (FBNDP), which can be described as foows: for the passenger trave demand between rai stations and bus stops on a given urban transit network, it designs the optima feeder bus routes and frequencies so as to minimize the passengers trave expense and the operator s cost. We extended the demand pattern of M-to-1 in most existing researches to M-to-M. We comprehensivey considered the passenger trave cost, which incudes the waiting and riding cost on the bus, riding cost on rai, and transfer cost between these two transportation modes, and presented a new genetic agorithm that determines the optima feeder-bus operating frequencies under strict constraint conditions. The numerica exampes under different demand patterns have been experienced and anaysed, which showed the robustness and efficiency of the presented agorithm. We aso found that the distribution pattern of the trave demand has a significant infuence on the feeder-bus network construction. 1. ntroduction As the two main transport modes in an urban transit system, the rai ine usuay pays the roe of the transport trunk, whie the feeder-bus network services act as a branch of and a suppement to the former. The integration and coordination of urban rai transit and the bus network can effectivey promote the service efficiency and simutaneousy improve the financia status of the system [1]. Stanger and Vuchic [2] pointed out that coordinative schedue optimization of the two modes coud ead to operating cost savings. Some cities, such as Atanta, Miami, and Washington, DC, gave top priority to the bus/rai coordination during the deveopment process of the transportation systems. Dunn Jr. [3] showed that the coordination and integration of transport services have been a precondition for improving pubic transportation. A good feeder-bus network significanty improves the pubic transport system s service eve, operation efficiency, and market competitiveness. The feeder-bus network transports transfer passengers who come from the urban perimeter zone and wi arrive at their fina business or work destination by rai. Each bus ine in the feeder-bus system usuay connects to a specia raiway station and serves a sequence of bus stops with a certain frequency. Thus, the feeder-bus network design probem (FBNDP) can be described as foows: for a given urban rai ine, the stop ocations and the passenger trave demand between bus stops and raiway stations, the optima feeder bus routes, and their frequencies are determined so as to minimize the passenger trave cost and the bus operation cost [4 6]. 2. Literature Review The existing research on the FBNDP mainy foows two approaches, that is, the anaytic approach and network programming (aso known as mathematica programming). Most eary research used anaytic approaches to deduce the optima route spacing, operating headway, and stop spacing basedonassumptionsregardingtheshapeofthestreetgeometry and the spatia distribution of the passenger demand. According to the assumption of the eary research, the demand is distributed in a rectanguar region in which an existing rai ine is serviced (accessed) by some parae bus routes perpendicuar to the rai ine. Byrne and Vuchic [4] studied the optima ocation and headway of parae bus ines

2 2 Discrete Dynamics in Nature and Society and presented a method to determine the optima number of bus ines. On the basis of [4], Byrne [7] determined the engths, positions, and headways of bus ines that coud minimize the user trave time and operating costs in response to a genera popuation density function with differing ine speeds. Hurde [8]studiedtheoptimaocationandschedue of parae feeder ines with variabe passenger density with different trip origins and times. Wirasinghe et a. [9] put forward optimization formuations for the optima raiway interstation spacing, feeder-bus zone boundary, and train headways mainy by the use of basic cacuus in conjunction with continuum approximations of certain discrete parameters. Wirasinghe [10] researched a feeder-bus system with a demand pattern of M-to-1 (i.e., mutipe bus stops and a singe station). An approximate anaytica mode and corresponding soution agorithm were successfuy appied to the Cagary (South Corridor) LRT system. uah and Per [5] optimized the route spacing, operating headway, and stop spacing simutaneousy, and anaysed the infuencing factors of bus stop spacing in three different cases. Supposing that the ocation of the rai ine was predetermined, Chien and Schonfed [11] cut the urban corridor into severa traffic zones with different engths but the same width and jointy optimized the rai ine ength, raiway station spacing, bus headways, bus stop spacing, and bus route spacing under the conditions that the passenger fow density in each traffic zone was the same and that ony one feeder-bus ine connected to the same raiway station. Chien and Yang [12] deveoped a mode for finding the optima bus route ocation and its operating headway in a heterogeneous service area whie considering intersection deays. n these modes, irreguar and discrete M-to-1 demand distributions were considered. A heuristic agorithm [12] and ater a genetic agorithm [13] were designed to sove the above mode. n recent decades, the network programming approach has been introduced to dea with the FBNDP. n this approach, the urban transport network is usuay represented by a graphic framework, in which nodes denote bus stops or raiway stations and inks denote route segments between the two successive nodes. For simpicity, it takes bus stops as the origin and the rai station as the destination of the trave demand. uah and Per [6] deveopedamathematicaprogramming mode for the FBNDP under the M-to-1 demand patternanddesignedaheuristicagorithmbasedonthe savings approach. The demand pattern of M-to-M (i.e., the demand pattern in which mutipe stations are the destinations) was transformed and generaized to M-to-1 by separating the bus stops into dummy chid nodes with the same number of raiway stations. Furthermore, the sensitivity ofthemodewasanaysedforchangesinthedesignobjective, passenger demand variabiity, vehice capacity, abour and fue costs, and rai ine. Martins and Pato [14] further presented two strategies to generate the initia soution (i.e., the continuous construction method and the two-phase method) and designed a oca search as we as tabu search heuristics with diversification and intensification strategies. Shrivastav and Dhingra [15] discussed the FBNDP for the operationa integration of the suburban raiway and bus transit system and deveoped a heuristic agorithm using different node seection and insertion strategies. uan et a. mainy focused on the appication of metaheuristic agorithms to the FBNDP, such as simuated anneaing and tabu search [16], genetic agorithm,and ant coony optimization [17] and anaysed and compared the optima resuts obtained by these agorithms. More recenty, Ciaffi et a. [18] deat with the FBNDP using a two-phase method. n the first phase, a heuristic agorithm was used to generate two different and compementary sets of feasibe routes, in order to provide a proper baance between the maximization of the service coverage area and the minimization of the overa trave time. n the second phase, the sets generated in the first phase were used as input data and a GA was designed to find a suboptima set of routes with the associated frequencies. Amost a the existing research assumed that the trave demand starts from mutipe bus stops but ends at one particuar raiway station near the centra business district (i.e., the demand pattern of M-to-1). n this paper, this drawback is modifiedtoadaptwetothereaisticpassengerdistribution. We consider that origin-destination (OD) pairs may exist between any bus stop and any raiway station (M-to-M). After computing the passenger trave cost from the view of the whoe feeder-bus network, we present a network optimization mode with the objective of minimizing the passenger trave cost and the bus operation cost. Furthermore, a new generation agorithm (GA) is deveoped and the optima resuts under different passenger patterns are anaysed and compared. 3. Probem Description The feeder-bus network mainy transports transfer passengersbetweenthebusandtheraiwaysystem.weregardthe nodes, incuding bus stops and rai stations, as traffic points of passenger coection and distribution. n the cassica FBNDP, a passengers are supposed to have a certain raiway station as their destination. We widen this assumption to the M-to-M pattern; that is, the passenger demand is distributed between any bus stop and any raiway station. Under this demand pattern, the feeder-bus routes obey the foowing assumptions in most previous studies. (1) Each bus stop is served by one feeder-bus route ony. (2) Each bus route does not cross its feeder raiway station butterminatesatthestation. (3) A bus routes have a uniform capacity and operating speed, and the voume of transport passengers shoud not exceed their capacity. (4) Each bus must hat at a the stops aong its route and the skip-stop running strategy is not considered. When the feeder-bus network obeys the above assumptions, there is an M-to-1 connection reationship between bus stops and raiway stations. However, the route structure and the feeder station are infuenced by the demand distribution of the M-to-M demand pattern, and the operating frequency canbeobtainedthroughtheoptimacostofthetransitoperator and transfer passengers.

3 Discrete Dynamics in Nature and Society 3 Considering an urban pubic transit network composed of bus stops and J raiway stations, we denote the set of bus stops by B={1,...,},thesetofraiwaystationsbyT={+ 1,...,}, and the set of network nodes by N=B T.We aso et the distance of a section between two adjacent nodes i, j be L ij, i,j N, the bus operating speed V B, and the train speed V T. Due to the time fuctuation of the urban passenger demand, we can determine the bus schedue for each panning period (such as one hour in the morning peak). n the given period, the demand between i and d can be expressed as P id,fori B, d T. Supposing the feeder-bus network Ω consists of feeder-bus routes, the path structure of bus route k(k = 1,...,)is denoted by ω k ={n k 1,nk 2...,nk P k 1,n P k },inwhich n Pk Tisthe feeder raiway station and n k 1,nk 2,...,nk P k 1 B arebusstopsaongtheroute.theoperatingfrequencyof route k is denoted by f k. 4. Mode Construction The constraints of the feeder-bus network can be obtained according to the above assumptions and the transit operating requirement. Compared with the demand pattern of M-to- 1, the network construction constraints under the M-to-M pattern are competey uniform. However, the generaized trave cost of transfer passengers under the demand pattern of M-to-M wi have a more compex infuence on their choice of feeder station in the raiway ine, thus affecting the feederbus route structure and further the network construction Constraints Anaysis. To represent the feeder-bus network constraints, Y ij and X ihk are defined to denote the reationship between nodes and routes: Y ij ={ 1, if bus node i is assigned to rai node j 0, otherwise, i=1,...,; j=+1,...,, X ihk ={ 1, if node i precedes node h on bus route k 0, otherwise, i,h=1,...,; k=1,...,. (1) A the constraints that need to be satisfied by the feederbusnetworkareasfoows. (1) Connectedness constraint of the feeder-bus network: in the feeder-bus network, any subset of bus stops must ink to feeder stations directy or via other bus stops, that is, the foowing connectedness constraint: X ihk 1, H, (2) i H h H k=1 where H is a set containing a the rai stations and somebusstops.obviousy,itisasoasubsetofn. (2) ntegrity constraints of the feeder-bus route: each bus route must ink to a singe raiway station: i=1 j=+1 X ijk =1, k=1,...,. (3) A route terminates at a certain feeder station d Tto which the route passengers are transported, i=+1 j=1 X ijk =0, k=1,...,. (4) n addition, a feeder-bus route shoud incude at east one stop and one feeder station, that is, the foowing nonempty constraint: X ijk 1, k=1,...,. (5) i=1 j=1 (3) Reationship constraints between routes and nodes: feeder-bus route k must hat at stop i ony once, i shoud be served ony by route k when i ω k,and route k shoud be acycic: h=1 X ihk =1, k=1 h=1 X pik 1, k=1 p=1 X ihk X pik 0, p=1 i=1,...,, i=1,...,, i=1,...,; k=1,...,. (6) n addition, reationship variabe Y ij between route k and nodes i, j satisfies the foowing constraint: h=1 X ihk + X pjk Y ij 1, p=1 i=1,...,; j=+1,...,; k=1,...,. (7) (4) Capacity constraints of the feeder-bus route and network: for route k, operatingfrequencyf k shoud meet the passenger transport capacity; that is, P id i=1 h=1 X ihk f k Cρ, k=1,...,, (8) where C is the bus capacity and ρ istheoadfactor.

4 4 Discrete Dynamics in Nature and Society Meanwhie, the route operating frequencies are restricted bythetotarunningmieageofbusvehicesinthepanning period; that is, f k k=1 i=1 h=1 L ih X ihk 1 2 NV B (T t), (9) where t is the average servicing and turn-around time of every bus vehice in period T and N isthenumberofvehices thatcanbeusedint Cost Anaysis. n order to ensure the good market competitiveness and operation effect, the feeder-bus network needs to consider the benefits both of passengers and of operators. For passengers, this refers to their generaized trave cost, which can be divided into three parts: the waiting and riding cost on the bus, the riding cost on the rai ine, and the transfer cost between these two modes. Compared with [6, 14], the transfer cost is taken into account in this paper andthecontentsofthepassengertravecostarecosertoa reaistic situation. n the passenger trave cost, the bus waiting and bus riding costs are λ w k=1 (1/2f k) i=1 h=1 P idx ihk and k=1 {L k P n k 1 d + (L k L n k 1 n k 2 ) P n k 2 d + + L n k Pk 1 n P k P np k d}λ k 1 r/v B, respectivey, where λ w and λ r are the corresponding monetary cost conversion coefficients, P k and L k are the tota passenger demand and ength of route k; the transfer cost at the feeder station is cosey reated to the transfer faciities and train operating density, so the transfer cost for every passenger at station j(j = + 1,...,)can be expressed as λ j,andthetotarairiding cost is (λ T /V T )( j=+1 i=1 P idy ij L jd ),whereλ T is the corresponding monetary cost conversion coefficient. For the bus operator, the operating cost of feeder-bus routes mainy reates to operating mieages and frequencies, which are denoted as 2λ b k=1 f k h=1 i=1 L ihx ihk,whereλ b is the cost per carriage per mie Optimization Mode. Considering the benefits of both the passengers and the operator, the objective function of feeder-bus network programming is to minimize the passengers generaized trave cost and the operator s cost: min Z = λ T ( V T +λ w k=1 +( j=+1 k=1 1 2f k i=1 {L k P id Y ij L jd ) P id X ihk i=1 h=1 P n k 1 d +(L k L n k 1 n k 2 ) P n k 2 d + j=+1 +2λ b λ j ( + +L n k Pk 1 n P k f k k=1 h=1 P id Y ij P ij ) i=1 L ih X ihk. i=1 P n k d}λ Pk 1 r) (V B ) 1 (10) Objective function (10)andconstraints(2) (9)constitute the optimization mode of the FBNDP. The optimization mode has the foowing main characteristics. (1)Thetransfercostisincudedinthepassengertrave cost, so the trave cost on the transfer network can be cacuated overa. (2) The optima objective is to minimize the passenger trave cost and transit operating cost, so benefits to users and operators in the pubic transit system are both considered. (3) The demands are not imited to a singe destination (M-to-1) and are extended to the distribution between every origin and destination (M-to-M), so the mode accordswewithareaisticdemanddistribution. 5. Mode Soution The optimization mode of the feeder-bus network, with many 0-1 variabes and many constraints, is NP hard [14]. t is essentiay a route optimization probem in the fied of operations research, which is appropriate for soving by some inteigent heuristic agorithms. Thus, in this paper, we present a genetic agorithm for the mode. n the generation process of initia soution individuas and new popuation individuas, we use the foowing strategy for constructing routes: it firsty generates the connection reations of stops and stations, then finay optimizes the routing path structure and determines the operating frequency. n the process of generating feeder-bus routes, feeder reations and the path of each bus route are determined according to the cost of candidate routes Optima Frequency for Each Route. n feeder-bus network Ω, the path structure and cost of a feeder-bus route are not correated with the other routes; therefore, the operating frequency of each route can be set independenty. For feederbus route ω k, according to the objective function (10), its cost is C k = λ n k P k 1 T ( P V id L npk d)+ λ wp k T 2f k i=n k 1

5 Discrete Dynamics in Nature and Society 5 +λ npk (P k n k P k 1 i=n k 1 P inpk )+2λ b f k L k +(λ r {L k P n k 1 d +(L k L n k 1 n ) 2 k P n k 2 d + +L n k P k 1 n P k P n k Pk 1 d}) (V B) 1. (11) n order to minimize C k, the optima operating frequency without any constraint can be obtained by the first-order optimaity condition of C k with respect to f k in (11) as foows: f k = 1 2 λ wp k λ b L k. (12) Then, the optima operating frequency of ω k under constraint (8)is f k = max {1 2 λ wp k P, k }. (13) λ b L k (Cρ) Under the optima frequency f k,theminimumroutecostof ω k is C k = λ n k P k 1 T ( P V id L npk d)+ λ wp k T 2f k i=n k 1 +λ npk (P k n k P k 1 i=n k 1 P inpk )+2λ b f k L k +(λ r {L k P n k 1 d +(L k L n k 1 n ) 2 k P n k 2 d + +L n k P k 1 n P k Thus, the tota operating cost of Ω is Z (Ω) = k=1 P n k Pk 1 d}) (V B) 1. (14) C k. (15) Particuary, the optima operating frequency f ij of directink route ω ij, which directy inks stop i B and station j T,ismax{(1/2) λ w P id/λ b L ij, P id/(cρ)}. Then, accordingy, the tota operating cost DC ij of ω ij is DC ij = λ T ( V T + λ rl ij P id P id L jd )+ λ w P id V B +λ j ( 2f ij P id P ij )+2λ b f ij L ij. (16) 5.2. Optimization Strategy for the Route Structure. n the process of the GA, a routes ω k in the popuation individuas need to optimize the path structures when the nodes in the routes are changed or the generation is updated. When the stops served by route ω k and the end (feeder station) of ω k are determined, the path structure optimization of ω k can come down to an open vehice routing probem with one depot (i.e., the feeder station). Thus, we adopt 2p k iterations of the 2-opt strategy to optimize the route structure, where p k is the number of nodes on the route. Because of the high probabiity that some eite gene segments of the offspring can be inherited from the parents, the fixed iterations of the route structure optimization for every individua of each generation can improve the popuation quaity Genetic Coding. n this paper, we use an intuitiona stye to code the feeder-bus pan. Thus, every node in the network is expressed by a natura number; we aso identify the bus stops or raiway stations with different number sets. Then, a feeder-bus route woud be a number substring ending at a raiway station, and the whoe coding scheme of the network woud be the sequentia connection of these route substrings. Note that the coding ength of the network pan depends on thetotaroutenumberdesignedinthetransitsystemandit is not fixed. For exampe, when B={1,2,3,4,5,6,7,8,9,10}and T= {11, 12, 13}, a sampe feeder-bus network can be expressed as ; substrings12311, 4612, 57 12, and stand for 4 feeder-bus routes, respectivey, in which the bodface numbers stand for the feeder stations. Based on objective function (10) and considering the feasibiity of constraint (9) simutaneousy, the fitness function of individua Ω is constructed as foows: 1 F (Ω) = λz (Ω) +λ[1 2 NV B (T t) where λ is a penaty factor. f k k=1 i=1 h=1 L ih X ihk ], (17) 5.4. nitia Popuation. Each feeder-bus network in the initia popuation is generated one route by one route. Due to that, a the routes end at raiway stations, so a feeder-bus station j Tcanbeseectedfirst;thenwechooseabusstopi B with a choice probabiity, insert this stop into an existing route that terminates at node j, or ink it directy with node j to

6 6 Discrete Dynamics in Nature and Society generate a new route. n this way, the generation of a network is finished when a the bus stops have been seected. n order to improve the individua quaity of the initia popuation, we construct a function to evauate the connecting reationship between a bus stop and a raiway station so that the rouette seection method is utiized to compute the seection probabiity. Let DC ij = max i B DC ij + min i B DC ij ; then, the evauation function between stop i and station j is F ij =DC ij DC ij. (18) For station j, the seection probabiity of stop i is F ij / i=1 F ij. The construction agorithm of a feeder-bus network of the initia popuation is as foows. Agorithm 1. (1) Let B =B,whereB stands for the set of stops that coud be seected to construct the current feederbus route. Ω=0is a feeder-bus network and k=0is the number of routes that have been generated. (2) f B =φ, the agorithm terminates. Otherwise, randomy seect a station j from T with equa probabiity. (3) ArandomnumberR (0, 1) is created according to uniform distribution. Take a stop i from B,which satisfies i 1 i=1 F ij/ i=1 F ij R< i i=1 F ij/ i=1 F ij. (4) Let ω ij be the direct route from i to j and M the number of routes ending at j in Ω. fm=0,setω=ω {ω ij }, B =B \i, k = k+1;goto(2). Otherwise,form= 1,...,M,insertiinto the mth route that ends at j in Ω, form the corresponding network Ω m and et Ω M+1 = Ω {ω ij };etω = arg min{z(ω m ) m = 1,...,M+1};ifΩ = Ω M+1, k=k+1; Ω=Ω, B =B \i;goto(2). Let n be the size of the initia popuation, which coud be reasonaby determined according to the vaues of and J. The initia popuation is constructed by caing Agorithm 1 n times Genetic Operators (1) Seection and Repication Operator. To strengthen the searching abiity of the GA, competition and intrusion mechanisms are introduced to construct the parent popuation. The formeris that the popuationwith n individuas is repicated to form a new popuation with 2n individuas firsty; then these 2n individuas are divided into n pairs arbitrariy and the better individuas are preserved by comparing the fitness ofeach pair. Theatter means that αn new individuas by Agorithm 1 are introduced to repace the αn worst ones of the current popuation, where α is an intrusion ratio. The vaue of α isdynamicaycontroedintherangeof[α, α]. Set α=min(2 α, α) when the best soution has not been improved in T α times generations; set α=αwhen the best soution is improved. (2) Crossover Operator. Here, we take two parents to generate two offspring with crossover probabiity P c. n order to ensure that offspring individuas can inherit the eite gene from the parents, gene segments of routes of which the average cost per passenger is ower are chosen and inserted into the offspring individuas. n individua Ω,etthenode of ocus i be V i,etthenumberofroutesn, and et the average cost per passenger of ω k be AC k. Choose the two parents Ω 1 and Ω 2 and generate offspring Ω based on Ω 1. Firsty, compare the average trave cost per passenger of the two gene segments that start the node V 1,thefirstocusofΩ 1,inΩ 1 and Ω 2.Thebetter option is to choose Ω and deete V 1 from the two parents. Then, make the comparison of the gene segments that start the ast node in Ω in the two parents unti Ω terminates at a raiway station and a route in Ω is generated. Then, deete the routes that cannot satisfy constraint (5) orcombinethe two shortest routes in each parent. With the above method, genesegmentsoftheparentsareseectedtojointheoffspring constanty, and the routes of the offspring are constructed one by one unti the whoe offspring feeder-bus network is formed. The detaied agorithm is described as foows. Agorithm 2. (1) Take the parents Ω 1 and Ω 2,andthegenerated offspring Ω.SetΩ =Ω 1, Ω 2 =Ω 2,andi=1. (2) f n =0,goto(7); otherwise,seecttheocusi = 1 and the corresponding route is ω k1 in Ω.SetV i =V i. (3) Find ocus j from Ω 2,satisfyingV j 2 corresponding route is ω k2 ;goto(5). (4) Find ocus i from Ω,satisfyingV j 2 corresponding route is ω k1 ;goto(5). =V i,andthe =V i,andthe (5) When AC k1 AC k2,set V i+1 =V i +1, Ω 2 =Ω 2 \V j, 2 Ω 2 =Ω 2 \V j, i=i+1,andifv i+1 2 is not a raiway station, go to (3);otherwise,n =n +1;goto(6). When AC k1 >AC k2,setv i+1 =V j +1, Ω 2 =Ω \V i, Ω 2 =Ω 2 \V j, i=i+1,andifv i+1 2 is not a raiway station, go to (4);otherwise,n =n +1;goto(6). (6) Deete those routes in Ω 1 and Ω 2 that do not satisfy constraint (5). fatheroutessatisfyconstraint(5) in Ω or Ω 2, combine the two shortest routes in the corresponding parent; go to (2). (7) Optimize the path structure of n routes in Ω,respectivey. Based on Ω 2, another offspring can be generated in the same way. Taking the foowing parents, the crossover operator that generates offspring 1 based on parent 1 is shown as foows: Parent1: Parent2: First, node 1 at the first ocus in parent 1 is used as the node of offspring 1 at ocus 1. Then, gene segments 1-2 in parent 1 and 1 9 in parent 2 are compared; 1-2 are supposed to join offspring 1. Subsequenty, 2-3 and 2 11 are compared. Offspring1isasbeow: Offspring 1 :

7 Discrete Dynamics in Nature and Society 7 (3) Mutation Operator. A mutation operator with mutation probabiity P c is used to strengthen the goba optimization abiity of the GA. According to the types of genes randomy seected from one individua, exchange or insertion mutationsaremade:iftheseectedgeneisabusstop,itwibe randomy inserted into another ocus (insertion mutation); if the seected gene is a station, it wi be randomy repaced by another station (exchange mutation). To guarantee the quaity of the mutation, the acceptance probabiity of a mutation soution is 1, ΔZ < 0, p={ e ( ΔZ/Z0) (19), ΔZ 0, where Z 0 is the fitness of the best individua so far and ΔZ is the objective difference of the soution mutation before and after. (4) Eite Preservation Strategy and Agorithm Termination Rues.Topreservetheeiteindividuasofparentpopuations, the worst 4% of offspring individuas are repaced by the same proportion of the best ones. The termination rues of thegamakeuseofthemaximumgenerationst max or the maximum generations T 0 without improving the best soution so far. 6. Numerica Exampes The benchmark probem is taken from [6]. The network incudes 55 bus stops and 4 raiway stations, serving square mies. The demand density of each stop per period (one hour) is 200 passengers. The vaues of the mode parameters are shown in Tabe 1.The GA is designed based on the C# anguage. n the GA, we take n = 120, P c = 0.8, P m = 0.08, T max = 1200, T 0 = 100, α = 0.05, α = 0.30, andt α = M-to-1 Demand Pattern. Raiway station 56 is regarded as the centra business district of the service area and the destination of a the passengers, so the demand is a distribution pattern between mutipe stops and one station. The best feeder-bus network is shown in Figure 1 and Tabe 2. Figure 2 shows a change in the objective function when the number of generations increases in the soving process. t iustrates that the GA presented in this paper performs a fast convergence speed Comparison of Best Soutions. To compare the optima soutions with other approaches, we negect the transfer cost (i.e., λ j =0) and make the tota cost of this paper accord with other methods, incuding saving heuristics [6], dispacement heuristics, basic TS [14], and TS with intensification [16]. However, the bus riding-time cost is roughy approximated by estimating the tota passenger-mies in these studies and there are some differences in constraints (8)and(9)inthemodes of [6, 16]. Tabe 3 gives the best soutions of these studies. The resuts show that the tota cost of the GA saves 8.0%, 1.3%, 1.2%, and 1.4% compared to the other approaches [6, 14, 16], respectivey. Because the ength of each route is Tabe 1: Mode parameters. Parameter Unit Vaue C Seat 50 ρ 1.2 t Hour 0.16 N Vehice 110 λ w $/passenger-hour 8 λ r $/passenger-hour 4 λ T $/passenger-hour 4.5 λ i $/passenger 0.03 λ b $/vehice-mie 3 V B Mie/hour 20 V T Mie/hour 30 λ Figure 1: Optima network under M-to-1. Tabe 2: Soution indicators under M-to-1. Tota cost ($) 6474 Number of routes 15 Average ength per route (mies) 1.02 Average frequency per route (trips/hour) not arbitrariy imited in our paper compared with the other studies, a better soution is obtained and the number of routes decreases. n the best soution of the GA, the number of routes and average route ength are simiar to those in the saving heuristics, whie the tota cost is ower M-to-M Demand Patterns. To study the change in the optima network under various demand distributions, the

8 8 Discrete Dynamics in Nature and Society Soution approaches Tota cost ($) Route no. Tabe 3: Comparison of the best soutions. Average ength per route (mies) Average frequency per route (trips/hour) Average computationa time (seconds) Saving heuristics [6] Dispacement heuristics [14] Basic TS [14] TS with intensification [16] GA (in this paper) a System cost ($) Tabe 4: ndicators of optima soutions under various demand distributions. Station s transfer passengers Number of routes Average route ength Average route frequency Average trave time (bus : train) Noninear coefficient : : : : : : Objective function ($) Generation Figure 2: Convergence efficiency of GA under M-to-1. patterns of passenger demand between each stop and 4 raiwaystationsaregeneratedbyanarithmeticprogression,the first term of which is a and the common difference is q. For exampe, when a = 20 and q = 20,thenumberofpassengers from each stop to stations is 20, 40, 60 and 80, respectivey. Obviousy, the imbaance of the demand distribution increases when a decreases from 50 to 0. The resuts under different vaues of a are shown in Tabe 4 and Figure 3 shows the optima feeder-bus network under the uniform distribution demand between the 4 raiway stations (q =0). Figure 4 shows a change in the objective function under the demand pattern of M-to-M and q=0. t shows that the convergence speed of GA under M-to-M is satisfactory as we as M-to-1. From Tabe 4, thefoowingobservations regarding the demand distributions effect on the optima feeder-busnetworkscanbefound. (1) For a given raiway station, with the increase in passengers who terminate at a station, the number of passengers who choose to feed into this station increases grossy, as seen in Figure Figure 3: Optima feeder-bus network under q= (2) The demand distribution has an obvious effect on the average riding time by bus and train, as shown in Figure 6. With the increase in the demand imbaance between the stations, the difference in the average trave time in the two traffic modes becomes graduay more significant and the tota trave time on the integrated transport network decreases simutaneousy becausethefeederstationandroutestructureare infuenced by major passengers

9 Discrete Dynamics in Nature and Society 9 Objective function ($) Feeder passengers Generation Figure 4: Convergence efficiency of GA under M-to-M Demands from each stop Figure 5: Effect of termination passengers on feeder passengers at stations. (3) As Figure 7 shows, with the increase in the demand imbaance between stations, namey, the concentration of demand destinations, passengers noninear coefficient fas and the system tota cost aso decreases remarkaby. The reason is that the concentration of demand destinations makes most passengers obtain a better service, which causes a decine in the system s tota cost. According to the above anaysis and the difference between Figures 1 and 3, wecandrawtheconcusionthat the demand distribution has a great effect on the voume of passengers choosing given feeder-bus stations and the path structure of feeder routes, which wi further infuence the tota cost of the whoe feeder system. 7. Concusions This paper studies the optima design probem of a feeder-bus networkunderthedemandpatternofm-to-m.thedrawback in most existing reated research, that ony a singe destination exists (M-to-1 demand pattern), is modified to the M- to-m pattern for better accordance with the reaistic demand distribution. n order to minimize the passenger trave cost and transit operating cost, an integrated pubic transport system of a feeder-bus network and raiway is regarded as a Riding time (hours) n bus n rai Tota Figure6:Effectofdemanddistributiononpassengers travetime. System cost ($) α System cost Noninear coefficient α Figure 7: Reation between demand distribution and system cost or noninear coefficient. whoe to cacuate the passenger trave cost overa. The resuts show that passenger demand distributions have a significant infuence on feeder-bus network construction, especiay on the feeder stations, the paths and frequencies, of feeder bus routes. Therefore, demand distributions shoud be considered when designing a feeder-bus network. Usuay pubic transportation network panning has symmetry, though differences in two directions are not considered in this paper. f the demand on a feeder-bus network has an obvious tida phenomenon with time distribution, and the operating frequencies in different directions differ greaty, a directed feeder-bus network shoud be designed according to the directiona demand. One prerequisite of the FBNDP in this paper is that the station ayout has to be determined. n our work, the optima frequencies of feeder bus routes are determined according to the feeder passenger between rai and bus transit system. f one feeder-bus route ony services few passengers, it coud be removed from the feeder-bus network. Acknowedgments ThisresearchissupportedbytheNationaNaturaScience Foundation of China ( , ), the Science and Noninear coefficient

10 10 Discrete Dynamics in Nature and Society Technoogy Research Deveopment Program of China Raiway Corporation (Major Program, 2013X004-A), and the Research Fund for Fok Ying Tung Education Foundation of Hong ong (Project no ). References [1] A. D. May, ntegrated transport strategies: a new approach to urban transport poicy formuation in the U, Transport Reviews,vo.11,no.3,pp ,1991. [2] R. M. Stanger and V. R. Vuchic, The design of bus-rai transit faciities, Transit Journa,vo.5,no.4, pp.61 72,1979. [3] J. A. Dunn Jr., Coordination of urban transit services: the german mode, Transportation,vo.9,no.1,pp.33 43,1980. [4] B. F. Byrne and V. Vuchic, Pubic transportation ine positions and headways for minimum cost, Traffic Fow and Transportation,pp ,1972. [5] G.. uah and J. Per, Optimization of feeder bus routes and bus-stop spacing, Transportation Engineering, vo. 114, no. 3, pp , [6] G.. uah and J. Per, The feeder-bus network-design probem, the Operationa Research Society, vo. 40, pp , [7] B. F. Byrne, Cost minimizing positions, engths and headways for parae pubic transit ines having different speeds, Transportation Research, vo. 10, no. 3, pp , [8] V. F. Hurde, Minimum cost ocations for parae pubic transit ines, Transport Science, vo. 7, no. 4, pp , [9] S. C. Wirasinghe, V. F. Hurde, and G. Newe, Optima parameters for a coordinated rai and bus transit system, Transportation Science, vo. 11, no. 4, pp , [10] S. C. Wirasinghe, Neary optima parameters for a rai/feederbussystemonarectanguargrid, Transportation Research A, vo.14,no.1,pp.33 40,1980. [11] S. Chien and P. Schonfed, Joint optimization of a rai transit ine and its feeder bus system, Advanced Transportation, vo. 32, no. 3, pp , [12] S.ChienandZ.Yang, Optimafeederbusroutesonirreguar street networks, Advanced Transportation, vo. 34, no. 2, pp , [13] S. Chien, Z. Yang, and E. Hou, Genetic agorithm approach for transit route panning and design, Transportation Engineering,vo.127,no.3,pp ,2001. [14] C. L. Martins and M. V. Pato, Search strategies for the feeder bus network design probem, EuropeanJournaofOperationa Research, vo. 106, no. 2-3, pp , [15] P. Shrivastav and S. L. Dhingra, Deveopment of feeder routes for suburban raiway stations using heuristic approach, Journa of Transportation Engineering,vo.127,no.4,pp ,2001. [16] S.N.uan,H.L.Ong,and.M.Ng, Appyingmetaheuristics to feeder bus network design probem, Asia-Pacific Operationa Research,vo.21,no.4,pp ,2004. [17] S.N.uan,H.L.Ong,and.M.Ng, Sovingthefeederbus network design probem by genetic agorithms and ant coony optimization, Advances in Engineering Software, vo. 37, no. 6, pp , [18] F. Ciaffi, E. Cipriani, and M. Petrei, Feeder bus network design probem: a new metaheuristic procedure and rea size appications, Procedia-Socia and Behaviora Sciences, vo. 54, pp , 2012.

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