13th COTA International Conference of Transportation Professionals (CICTP 2013)

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1 Avaiabe onine at ScienceDirect Procedia - Socia and Behaviora Scien ce s 96 ( 03 ) th COTA Internationa Conference of Transportation Professionas (CICTP 03) Optima design of feeder-bus networ reated to urban rai ine based on transfer system Lian-bo DENG a, Wei GAO b, Wen-iang ZHOU a *, Tian-zhen LAI a a Schoo of Traffic and Transportation Engineering, Centra South University, Changsha 40075, China b The Tthird Raiway survey and Design Institite Group Corporation, Tianjin 3004 Abstract Based on the transfer networ composed of urban rai ine and bus routes, this paper studies an optima design probem of feeder-bus networ reated to urban rai ine under M-to-M passenger demand pattern between bus stops and raiway stations, to minimize passenger trave cost and transit operating cost. More comprehensivey considered passenger trave cost incude bus waiting cost and accurate bus riding cost, transfer cost between buses and trains, rai riding cost. We present a new genetic agorithm which determines optima feeder-bus operating frequency under stricty constraints condition. The numerica exampes under passenger demand patterns of M-to-M are provided. It shows that the agorithm can provide reasonabe resuts with satisfactory efficiency and passenger demand distributions have significant infuence on feeder bus networ construction. 03 The Authors. Pubished by Esevier by Esevier Ltd. Open B.. access under CC BY-NC-ND icense. Seection and/or peer-review under under responsibiity responsibiity of Chinese of Overseas Chinese Overseas Transportation Transportation Association (COTA). Association (COTA). Keywords: urban rai transit; feeder-bus networ; transfer networ; genetic agorithm. Introduction As two main modes in urban pubic transport system, the integration of urban rai transit and bus networ can effectivey promote the service efficiency and simutaneousy improve financia status of the system(may, 99). Stanger and uchic (979) points out that coordinative schedue optimization of two modes coud ead to operating cost savings and the transportation system s deveoping in some cities, such as Atanta, Miami and Washington, D.C., shoud give top priority to bus/rai coordination. Dunn (980) thins that coordination and integration of transport services has been a pre-condition to improving pubic transportation. * Corresponding author. Te.: E-mai address: zw_063@63.com The Authors. Pubished by Esevier Ltd. Open access under CC BY-NC-ND icense. Seection and peer-review under responsibiity of Chinese Overseas Transportation Association (COTA). doi: 0.06/j.sbspro

2 384 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) Reasonabe design of feeder-bus networ reated to urban rai transit can promote service eve, efficiency and competitiveness of pubic transport. Feeder-bus networ is the set of bus ines which serves transfer passenger between buses and trains. Each bus ine connects to one feeder raiway station, serves some bus stops in a certain order, and operates in certain frequency. Thus the feeder-bus networ-design probem (FBNDP) is to determine feeder-bus routes composed of the feeder station, the route structure and the operating frequency (Byrne & uchic, 97; Kuah & Per, 988; Kuah & Per, 989).. Literature review Existing research on FBNDP incudes anaytic approach and networ programming (aso nown as mathematica programming). Eary research mainy uses anaytic approach, which derives the optima route spacing, operating headway and the optima stop spacing based on assumptions regarding the shape of the street geometry and the spatia distribution of demand, those assumptions have their imitations. Supposing that passenger demand distributes in rectanguar region composed of a rai ine and parae bus ines which are perpendicuar to the rai ine, Byrne and uchic (97) studied the optima ocation and headway of parae bus ines, and the method to determine the optima number of bus ines was given. On the basis of this wor, Byrne (976) determined the engths, positions and headways of bus ines which coud minimize user trave time and operating costs in response to a genera popuation density function and differing ine speeds. Hurde (973) studied how parae feeder ines shoud be ocated and how their schedues shoud respond to a passenger arriva pattern that varied with trave ocation and time. Wirasinghe, Hurde and Newe (977) put forward optimizations formuations for the optima raiway interstation spacings, feeder-bus zone boundary and train headways mainy by use of basic cacuus in conjunction with continuum approximations of certain discrete parameters. Wirasinghe (980) supposed that a rai pus feeder bus system served a pea-period demand type of M-to-, and presented an approximate anaytica mode and corresponding soution agorithm, the mode was appied to the Cagary (South Corridor) LRT system. Kuah and Per (988) optimized the route spacing, operating headway and the stop spacing simutaneousy, and anayzed infuencing factors of bus-stop spacing in three different cases. Supposing that the ocation of rai ine to be studied was predetermined, Chien and Schonfed (998) cut urban corridor into severa traffic zones with different ength and same width, and jointy optimized rai ine ength, raiway station spacing, bus headways, bus stop spacing and bus route spacing under conditions that passenger fow density in each traffic zone was same and that ony one feeder-bus ine connected to the same raiway station. Chien and Yang (000) deveoped a mode for finding the optima bus route ocation and its operating headway in a heterogeneous service area whie considering intersection deays. In the mode, irreguar and discrete M-to- demand distributions were considered, the near-optima agorithm was demonstrated. Networ programming approach coud better dea with feeder-bus networ-design probem. In the approach, urban traffic networ is composed of two types of nodes: rai nodes and bus nodes, which represent bus stops and raiway stations respectivey, bus sections represent feeder-bus route segments and the demand is assumed to be concentrated at bus nodes. Kuah and Per (989) deveoped a mathematica programming mode for FBNDP under M-to- demand pattern and designed heuristic agorithm based on savings approach. They generaized M- to-m FBNDP to M-to- FBNDP by separating bus stops into dummy chid nodes the number of which is same with raiway stations, and they anayzed sensitivity of the mode for change of the design objective, passenger demand variabiity, vehice capacity, abour and fue costs, the rai networ. On the above basis, Martins and Pato (998) buit up the initia soution through a sequentia savings or a two-phase method, and designed oca search as we as tabu search heuristics with diversification and intensification strategies. Shrivastav and Dhingra (00) discussed deveopment of feeder routes for operationa integration of suburban raiway stations and pubic buses, and deveoped a heuristic agorithm using different node seection and insertion strategies. Kuan and Ong mainy focused on the appication of meta-heuristic agorithms on FBNDP, such as simuated anneaing and tabu search

3 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) (004), genetic agorithm and ant coony optimization (006), anayzed and compared the optima resuts obtained by these agorithms. Latey, Ciaffi (0) deat with feeder-bus networ design probem in a soving procedure with phases. In the first phase, a heuristic agorithm was used to generate two different and compementary sets of feasibe routes, in order to provide a good baance between maximization of the service coverage area and minimization of the overa trave time. In the second phase, the sets generated in the first phase were used as input data and GA was designed to find a sub-optima set of routes with the associated frequencies. On this basis, they constructed a mode for ayout region of feeder buses. In this paper, in accord with reaistic passenger distribution characteristics, the existing assumption that passengers trave from mutipe origins to a singe destination (M-to-), is reaxed to more genera passenger demand pattern (M-to-M) that means passenger demand distributes in a stops and stations. Then, bus and rai transit are regarded as a whoe transfer service system and passenger trave cost is considered based on the transfer networ, feeder-bus networ-design probem is studied to minimize passenger trave cost and bus operation cost. Furthermore, a new generation agorithm (GA) is deveoped and optima resuts under different passenger patterns are anayzed and compared. 3. mode Construction The constraints of feeder-bus can be obtained according to the above assumptions and transit operating requirement. Compared with M-to- demand pattern, networ construction constraints under M-to-M pattern are competey identica.. Under M-to-M demand pattern, under considering passenger generaized trave cost, mutipe destinations of passengers at every stop wi effect on their feeder station seection, then feeder-bus route structure and feeder-bus networ construction wi be infuenced. 3.. Constraints anaysis To represent feeder-bus networ constraints, Y ij and nodes or nodes and routes. X ih are defined to denote reationship between,if bus node i is assigned to rai node j Yij, i,, I; j I,, I J otherwise,if node i precedes node h on bus route Xih ih=,,,i+ J;=,,K otherwise The foowing section wi anayze a constraints which feeder-bus networ need to satisfy. ) Feeder-bus networ connectedness constraint In the feeder-bus networ, any sub-set of bus stops must in to feeder stations directy or via other bus stops, i.e. the foowing connectedness constraint: K X,for a H () i H h H ih H is any proper subset of N containing the set of a rai nodes. ) Feeder-bus route integrity constraints Each bus route must in to a singe raiway station: I I J X =,,, K () i j I A route terminates in a certain feeder station d ij T where the route passengers are transported to, i.e.,

4 386 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) I J I J i I j X 0,,, K (3) ij In addition, a feeder-bus route structure shoud incude at east one stop and one feeder station, i.e. the foowing nonempty constraint: I I J i j X,,, K (4) ij 3) Reationship constraints between routes and nodes Feeder-bus route must hat at stop i ony one time and i shoud be served ony by route when i shoud be a acycic route,: K I J h K I p, and X, i,, I (5) ih X, i,, I (6) pi I J I X X 0, i,, I;,, K (7) ih h p pi In addition, reationship variabey ij between route and node i, j satisfies the foowing constraint: I J I X X Y, i,, I; j I,, I J;,, K (8) ih pj ij h p 4) Feeder-bus route and networ capacity constraints For route, operating frequency f shoud meet the passenger transport capacity, i.e., where C is bus capacity, I I J I J P X f C, K (9) id ih i d I h is oad factor. Meanwhie, route operating frequencies is restricted by the tota running mieage of bus vehices in the given period: K I I J f LihXih NvB( T t ) (0) i h where t is average servicing and reentrant time of every bus vehice in period T, N is the number of vehices which can be used in T. 3.. Cost anaysis In order to ensure mareting competitiveness and operation effect, feeder-bus networ pan needs to consider interests both of passengers and operators. To passengers, they care for their generaized trave cost. Under M-to-M demand pattern, feeder-bus networ and rai operating ine constitute passengers transfer networ. On transfer networ, passenger trave cost can be divided into three parts: bus waiting cost and bus riding cost, transfer cost between buses and trains, rai riding cost. Compared with Kuah & Per (989) and Matins & Pato (998), transfer cost is incuded in passenger trave cost, so that integrated feeder system cost can be cacuated more overa and cose to reaistic operations.

5 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) Among passenger trave cost, bus waiting cost and bus riding cost are K I J I J I J nd nn nd nd n P n r B P d I d I d I K I J I I J P X w id ih f d I i h L P ( L L )( P P ) L P v respectivey, where and are w r corresponding monetary cost conversion coefficients, P and L stand for tota passenger demand and ength of route ; transfer cost at feeder station is cosey reated to transfer faciities and train operating density, so transfer cost for every passenger at station j( j I,, I J ) can be expressed as j ; the tota rai riding cost is v T T I J I J I d I j I i PYL id ij jd, where T is corresponding monetary cost conversion coefficient. To bus operator, operating cost of feeder bus routes mainy has reation to operating mieages and frequencies of feeder routes, which is denoted as per mie Optimization mode K I J I b ih ih h i and f L X (round trip), b is the cost per carriage Considering interest both of passengers and operators to minimize passenger generaized trave cost and operator s cost, the objective function of feeder-bus networ programming is I J I J I K I J I I J T min Z PidYijLjd w Pid Xih v f T d I j I i d I i h K I J I J I J L P ( L L )( P P ) L P r v nd nn nd B nd n P n P d I d I d I I J I J I K I J I j PY id ij Pij b f Lih Xih j I d I i h i Objective function () and constraints ()-(0) constitute feeder-bus networ-design probem based on transfer networ (FBNDP-TN). The optimization mode has beow main characteristics: () Transfer cost is incuded in passenger trave cost, so the trave cost on transfer networ can be cacuated more overa; The optima objective is to minimize passenger trave cost and transit operating cost, interests of users and operators in the pubic transport transfer system are both considered; The passenger demand do not be imited to a singe destination (M-to-), it is extended to the distribution between arbitrary origin and arbitrary destination (M-to-M), this we accords with reaistic demand distribution. 4. Mode soution The optimization mode with arge of independent variabes and constraints, is a NP hard probem (Matins & Pato, 998). Networ design probem is essentiay route optimization in the fied of operations research, meta-heuristic agorithms have better effects. Thus, genetic agorithm is used to sove the mode in this paper. In generation of initia soution individuas and new popuation individuas, constructing soution strategy which first generates routes, then optimizes route structure and determines operating frequency is used. In the process of generating feeder bus routes, feeder reations between stops and station and subordinations between routes and stops are determined according to measure rout cost. ()

6 388 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) Determination of the optima operating frequency for each route In feeder-bus networ, route structure and cost don t correate for different feeder-bus routes, therefore operating frequency of each route can be independenty set. For feeder-bus route, according to objective function (), its cost is np I J np inp v T i n d I i n = T id n P d w n P b C P L P f P P f L I J I J I J L P ( L L )( P P ) L P v r nd nn nd nd n B P n P d I d I d I In order to minimize cost of, first-order optimaity conditions of f is soved by ()to obtain the optima operating frequency without any constraint, which is f P L Considering constraint (9), the optima operating frequency of Operating by f *, minimum route cost of Thus, the tota operating cost of * w b * max w b, shoud be f P L P C is np I J np * T * * = id n P d w n P in P b v T i n d I i n C P L P f P P f L I J I J I J L P ( L L )( P P ) L P v r n d n n n d nd n P P B n d I d I d I is Z( )= Particuary, for direct-in route ij, which is directy constituted by stop i B and station j T, the optima operating frequency f * ij is K C * I J I J max P L, P C, the tota cost DC is ij w id b ij id d I d I () I J I J I J I J T * * ij = id jd w id ij r ij id B j id ij b ij ij vt d I d I d I d I DC P L P f L P v P P f L (3) 4.. Assisted optimization strategy for route structure In GA, any route of one popuation individua need to optimize its route structure when nodes on the route change or the generation update. When stops served by and the end of (feeder station) are determined, route structure optimization of can be come down to a open vehice routing probem with one depot (that is feeder station).thus, -opt strategy is enforced p times to optimize one route structure, p is the number of nodes on the rout. For that exceent gene segments of parent can be inherited by offspring with high probabiity, finite optimization of route structure for every offspring individua can reaize popuation improvement one generation by one generation.

7 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) Coding scheme of GA Feeder-bus networ is composed of feeder routes ining bus stops and raiway stations, so natura number coding is used to mae gene segment and in reation of nodes consistent. Each feeder-bus route is ined in order to form coding scheme of feeder-bus networ, and each substring ending at one station stands for one route. The ength of coding scheme of feeder-bus networ depends on the number of routes and concrete constitute of each route, so it is not fixed. When B=,,3,4,5,6,7,8,9,0 and T =,,3, one feeder-bus networ can be expressed as , substrings 3, 4 6, 5 7, stand for one route respectivey. Based on objective function of the optimization mode, the feasibiity of constraint (0) is considering simutaneousy, fitness function of individua is constructed as foow K I I J F( ) NvB ( T t) f LihX ih Z( ) i h is penaty factor Generation of the initia feeder-bus networ The initia feeder-bus networ is generated one route by one route. Due to that a routes end with raiway stations, so feeder-bus station j T can be seected first, then bus stop i Bis seected to insert the route which is connected to j or directy ined to j to form new route. In this way, networ is generated one rout by one route. In order to improve individua quaity of the initia popuation, connection evauation function between stops and stations is constructed to seect candidate stops to insert the existing routes with rouette seection method. * Let DC max DC + min DC, connection evauation function between stop i and station j is ij ij ij i B i B * Fij DCij DC ij For station j, the probabiity that stop i is seected is F / F. ij I i Beow is the concrete agorithm that generates the initia feeder-bus networ: Agorithm B stands for the set of stops which coud be seected to construct the current feeder-bus route, and B B. is the current feeder-bus networ. is number of route being constructed, and. If B =, the agorithm terminates. Otherwise, station j is randomy seected fromt with equa probabiity. Random number R radiom(0,) is created according to uniform distribution. Stop i is taen out from B, and satisfies i I i I F / F R F / F. ij ij ij ij i i i i i is directy ined to j to buid a new route ending with j. If there are no routes whose end is j in, set =, B B \ i,, turn to. Otherwise, the number of routes ending with j is supposed to be M in, and i is inserted in mm (,, M) route ending with j to form corresponding networ m, and = M ; Let arg min Z( ) m,, M, if, set B B \ i, M +, turn to. m ij Otherwise, B B \ i, turn to. n is the size of initia popuation, which coud be reasonaby determined according to I and J. The initia popuation is generated by caing agorithm n times.

8 390 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) Genetic operators ) Seection and repication operator To strength the searching abiity of GA, competition and intrusion mechanisms are introduced to construct parent popuation. The former is that the popuation with n individuas is repicated to form new popuation with n individuas, then, n individuas are divided into n pairs. The better individuas are preserved by comparing fitness of each pair. The atter means that n individuas are introduced to repace n worst ones of the current popuation by caing agorithm, is intrusion ratio. aue of is dynamic controed, the range of is,. When the soution has not improved int (intrusion ratio contro generation), set min *, ; when improved, set. ) Crossover operator Two parents are used to generate two offspring with P c, Pc is crossover probabiity. In order to ensure that offspring individuas can inherit exceent gene from parent ones, gene segments of routes whose average cost per i passenger is east are chose to form offspring individuas. In, node of gene ocation i is, the number of routes is n, average cost per passenger of is AC. of Choose parents and, and generate offspring based on is seected to form two gene segments with nodes at its next gene ocation in. First, node at first gene ocation and. Compare average cost per passenger of the two gene segments, and reject from and. The better one wi become one gene segment of. Corresponding node at ast gene ocation of is used to continue seecting gene segment from parents, and construction of the current route competes unti that the node is raiway station. Then, for parents, routes which can not satisfy constraint (4) wi be deeted or two shortest routes wi be combined. With the above method, gene segments of parents are seected to join offspring constanty, and routes of offspring are constructed one by one unti offspring feeder-bus networ are formed. The concrete agorithm is as foows Agorithm and are parents, offspring to be generated is. Set =,, i. i i If n 0, turn to, otherwise, seect gene ocation i from, i and route is. Set. j i Find gene ocation j from, satisfying, and the corresponding route is j i Find gene ocation i from, satisfying, and the corresponding route is i i j j If AC AC, set, \, \, i i, and if i, turn to., turn to. is not raiway station, i j i j turn to, otherwise n n, turn to ; otherwise, set, = \, \, i i, if is not raiway station, turn to, otherwise n n, turn to. i Chec every route of and to judge whether they satisfy constraint (4), and deete those do not satisfy. If a routes satisfy constraint (4) in or, combine two shortest routes in the corresponding parent, turn to. Optimize route structure of n routes in respectivey. Based on, another offspring can be generated in the same way. Construct the foowing parents, crossover operator that generates offspring based on Parent is shown as foows:

9 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) Parent : Parent : First, node at the first gene ocation in parent is used as node of Offspring at gene ocation. Then, gene segment - in Parent and -9 in Parent are compared, - is supposed to join Offspring. Subsequenty, -3 and - are compared,. The foowing is Offspring : Offspring : 3) Mutation operator Mutation operator is executed to strength agorithm capacity of jumping out oca optimum with probabiity P c. According to types of genes randomy seected from one individua, exchange or insertion mutation is done: if the seected gene is stops, it wi be randomy inserted in other gene ocation (insertion mutation); if the seected gene is stations, it wi be randomy repaced by another station. The probabiity of mutation soution is: Z 0 p ( Z/ Z0 ) e Z 0 Z0 is the fitness vaue of best so-far individua, Z is objective difference of soution mutation pre and post. Eite preservation strategy and agorithm termination rues To preserve the exceent individuas of parent popuations, 4% offspring individuas with the owest fitness are repaced by 4% parent ones with the highest fitness. Agorithm termination rues mae use of maximum evoutionary generationt max or maximum generation T 0 in which function vaue eeps invariant. 5. Anaysis of exampes The benchmar probem is taen from Kuah and Per (989), the networ incudes 55 bus stops and 4 raiway stations, serving *.5 square mies, the demand density of each stop in per period (one hour) is 00 passengers. The vaues of mode parameters are shown in Tabe. GA is designed based on C# anguage. In GA, n =0, P =0.8, P =0.08, T =00, max T =00 0, =0.05, =0.30, T =0. c m Tabe Mode parameters Parameters Units aue Parameters Units aue C seat 50 T. j t hour 0.6 b $/passenger-hour 4.5 $/passenger 0.03 $/vehice-mie 3 N vehice 0 B mie/hour 0 w $/passenger-hour 8 T mie/hour 30 r $/passenger-hour To study change aws of the optima networ under different demand distributions, passenger demand is distributed between 4 raiway stations by arithmetic progression whose initia vaue is a, equa difference is q. For exampe, when a 0, q 0, passengers form each stop to station 56, 57, 58, 59 is 0, 40, 60, 80 respectivey. Obviousy, imbaance of demand distribution increases when a decreases from 50 to 0. Tabe

10 39 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) shows the cacuation resuts, figure is the optima feeder-bus networ when demand uniformy distributes between 4 raiway stations ( a 50 ). Fig. Optima feeder-bus networ when a 50 Tabe Indicators of optima soutions under various demand distributions Station s demand from Average Average trave Diverted System Station s Transfer Passengers Numbers of Average each stop route time routing cost($) route route ength frequency (bus : train) factor : : : : : : 0.4. From Tabe, some aws can be obtained: () For one raiway station, with the increase of passengers whose destination is this station, the number of passengers who choose to feeder at this station increase correspondingy, as Figure shows: Fig. Reation between demand density and feeder passengers at stations

11 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) () Demand distribution has an obvious effect on average bus riding-time and average train riding-time, just as Figure 3 shows. With increase of imbaance of demand distribution at stations, feeder station and structure of routes are infuenced by advantage passengers, difference of average trave time at two traffic modes increases graduay. Simutaneousy, tota trave time on integrated transport networ decreases. Fig.3 Reation between demand distribution and trave time (3) As Figure 4 shows, with increase of imbaance of demand distribution at stations, namey concentration of demand destinations, circumambuate ratio fas and system tota cost aso decreases maredy. Reason is that concentration of demand destinations maes advantage passengers share better service and brings decine of system tota cost. Fig.4 The effect of termination passengers on feeder passengers at stations 6. Concusion We can see that passengers wi choose different feeder-bus stations and structure of trave routes wi differ under different demand distributions, a these wi infuence tota cost of the whoe feeder system. This paper studies optima design probem of feeder-bus networ reated to urban rai transit under M-to-M passenger demand pattern between bus stops and raiway stations, the passenger demand do not be imited to a singe destination (M-to-), which more accords with reaistic demand distribution aw. In order to minimize passenger trave cost and transit operating cost, integrated pubic transport system of feeder-bus networ and raiway is regarded as a whoe to overa cacuate passenger trave cost. Resuts show that passenger demand distributions have significant infuence on feeder bus networ construction. Therefore, demand distributions shoud be considered when designing feeder-bus networ reated to urban rai transit

12 394 Lian-bo Deng et a. / Procedia - Socia and Behaviora Sciences 96 ( 03 ) Usuay pubic transportation networ panning has symmetry, though differences at two directions are not considered in designing feeder-bus networ. If demand on feeder-bus networ has obvious tida phenomenon with time distribution, and operating frequencies at different directs differ greater, directed feeder-bus networ is designed according to directiona demand. Acnowedgements The wor described in this paper was supported by grants from the Nationa Natura Science Foundation of China (Project no ), the Research Fund for the Doctora Program of Higher Education of China (Project no ), and the Research Fund for Fo Ying Tong Education Foundation of Hong Kong (Project no. 307). References May A.D. (99). Integrated transport strategies: a new approach to urban transport poicy formuation in the UK. Transport Reviews, (3), Stanger R.M., uchic.r. (979). The design of bus-rai transit faciities. Transit Journa, 5, 6-7. Dunn J.A. (980). Coordination of urban transit services: the German mode. Transportation, 9, Byrne B.F., uchic. (97). Pubic transportation ine positions and headways for minimum cost. Traffic fow and transportation, Byrne B.F. (976). Cost minimizing positions, engths and headways for parae pubic transit ines having different speeds. Transport Research, 0(3), Hurde.F. (973). Minimum cost ocations for parae pubic transit ines. Transport Science, 7, Wirasinghe S.C., Hurde.F., Newe G.F. (977). Optima parameters for a coordinated rai and bus transit system. Transport Science, (4), Wirasinghe S.C. (980). Neary optima parameters for a rai/feeder-bus system on a rectanguar grid. Transport Research Part A, 4(), Kuah G.K., Per J. (988). Optimization of feeder bus routes and bus-stop spacing. Transport Engineering, 4(3), Chien S., Schonfed P. (998). Joint optimization of a rai transit ine and its feeder bus system. Journa of Advanced Transportation, 3(3), Chien S., Yang Z. (000). Optima feeder bus routes on irreguar street networs. Journa of Advanced Transportation, 34(), Kuah GK, Per J. (989). The feeder-bus networ-design probem. Journa of the Operationa Research Society, 40, Matins C.L., Pato M.. (998). Search strategies for the feeder bus networ design probem. Europe Journa Operationa Research, 06, Shrivastav P., Dhingra S.L. (00). Deveopment of feeder routes for suburban raiway station using heuristic approach. Journa of Transportation Engineering, 7(4), Kuan S.N., Ong H.L., Ng K.M. (004). Appying metaheuristics to feeder bus networ design probem. Asia-Pacific Journa of Operationa Research, (4), Kuan S.N., Ong H.L., Ng K.M. (006). Soving the feeder bus networ design probem by genetic agorithms and ant coony optimization. Advances in Engineering Software, 37, Ciaffi F, Cipriani E, Petrei M. (0). Feeder bus networ design probem: a new metaheuristic procedure and rea size appications. 5th meeting of the EURO Woring Group on Transportation.