Applied Soft Computing

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1 Appied Soft Computing 11 (2011) Contents ists avaiabe at ScienceDirect Appied Soft Computing j ourna ho mepage: Parae genetic agorithm in bus route headway optimization Bin Yu a,, Zhongzhen Yang a, Xueshan Sun a, Baozhen Yao b, Qingcheng Zeng a, Eri Jeppesen c a Transportation Management Coege, Daian Maritime University, Daian , PR China b Schoo of Civi Engineering & Architecture, Beijing Jiaotong University, Beijing , PR China c Centre of Maritime Research, University of Southern Denmar, DK-5230 Odense, Denmar a r t i c e i n f o Artice history: Received 7 June 2010 Received in revised form 26 March 2011 Accepted 23 May 2011 Avaiabe onine 12 June 2011 Keywords: Transportation Bus route headway Optimization Weight and PGA a b s t r a c t In this paper, a mode for optimizing bus route headway is presented in a given networ configuration and demand matrix, which aims to find an acceptabe baance between passenger costs and operator costs, namey the maximization of service quaity and the minimization of operationa costs. An integrated approach is aso proposed in the paper to determine the reative weights between passenger costs and operator costs. A parae genetic agorithm (PGA), in which a coarse-grained strategy and a oca search agorithm based on Tabu search are appied to improve the performance of genetic agorithm, is deveoped to sove the headway optimization mode. Data coected in Daian City, China, is used to verify the feasibiity of the mode and the agorithm. Resuts show that the reasonabe resource assessment can increase the benefits of transit system Esevier B.V. A rights reserved. 1. Introduction With the increase in concern on the environment poution and traffic congestion, authorities of most cities in China have formed many strategies on giving priority to the deveopment of urban pubic transportation system. During transit operation, there are some important tass incuding networ design, frequency design (headway design), setting timetabes, scheduing vehices to trips, and assignment of drivers [12]. Among these tass, timetabe (dispatching schedue) of bus vehices is one of the most important aspects in transit operation. The determination of dispatching time of each vehice is based on the pre-panned time interva between two adjacent vehices. In this study, a bus headway (i.e., schedued dispatching time interva of two successive buses) optimization mode is proposed to minimize the tota costs of passengers and operators in a given networ configuration and demand matrix. At the same time, an integrated approach is proposed to expore the trade-off between passenger costs and operator costs, in which the reative importance and the difference between the two conficting objectives are considered. Transit scheduing probems in the rea word are often inefficient to be soved by cassica optimization techniques because of the arge numbers of trips, bus routes and stations [13]. Recenty, the heuristics are considered as feasibe toos to sove combinato- Corresponding author. E-mai addresses: ybzhyb@163.com, minfish@yahoo.com.cn (B. Yu). ria optimization probems [7]. Genetic agorithm [14], which is a mutipurpose optimization too, has successfuy been appied in a wide range of optimization probems [5,10] incuding transportation fieds [3,4,23,25]. For this reason, genetic agorithm (GA) is used in this study to determine bus headways of routes. Since the proposed mode is to be appied in a rea transit system, a oca agorithm based on Tabu search and a coarse grain parae strategy are introduced into GA to improve the performance of the agorithm. This paper is organized as foowing: Section 2 is about the probems of basic notations and formuations; Section 3 contains the soution methods of determining bus route headway; Numerica anaysis is carried out in Section 4; and asty, the concusions are drawn in Section Optimization mode In this study, the maximized socia benefits are defined as minimizing the sum of passenger costs and operator costs [8]. In genera, it is reasonabe to provide enough capacity for a transit passengers on routes in panning stage. There are, however, the situations in which it is not feasibe to provide enough transit capacity to avoid congestion, especiay in the rea transit system. In this study, the probem of determining bus route headways can be formuated as a noninear program subject to the vehice feet constraint. In the optimization mode, the decision variabes are the headway of each direction of bus routes (h, denotes the headway in the direction of the bus route ). Firsty, passenger costs (in money) and operator costs (in money) are described separatey, and then the two sub-probems are integrated into one singe mode /$ see front matter 2011 Esevier B.V. A rights reserved. doi: /j.asoc

2 5082 B. Yu et a. / Appied Soft Computing 11 (2011) Passenger costs Passenger costs are defined as the passenger trave time-costs, which incude waiting time-costs at stops, riding/dweing timecosts in vehices and the boarding/aighting time-costs on/from vehices Waiting time-costs Basic notations, which is a binary sign, denotes that the first direction or the return direction of route. Here, = 1 denotes the direction with more vehices and = 1 denotes the other directions. T w,, passenger waiting time-costs in the direction of the bus route. Cp w, coefficients of waiting time-costs. u,,, the number of passengers waiting for buses in the direction of the bus route at the stop. h,, headway in the direction in the bus route. It is the decision variabe in the mode. Since there are severa vehices but not infinity vehices for a route, it is reasonabe that the headway of a route has a ower imit. In addition, over many vehices wi induce a arger demand for paring and storing vehices. Here, the headway of a route is bigger than 60 s. t e,, expected waiting times for the direction of the bus route. Since passenger arrivas foow a uniform process, the expected waiting times of a passenger is the haf of the headway, i.e., t e, = h, /2. q,,, aighting proportion of the stop in the direction of the bus route. It means that the aighting passengers are divided by the aighting passengers in a stops between current stops and the stops to the destination. For exampe, there are three stops, 1, 2 (destinations), the aighting passengers are 2, 4, 4, respectivey, thus q,, = 20% (2/( ) = 20%), q,,1 = 50%, q,,2 = 100%. a,,, the number of passengers aighting from buses in the direction of the bus route at the stop. a,, = v,, 1 q,, (1) V max, maximum capacity of a standard vehice. n,,, the number of buses going through the stop in the direction of the bus route during the researched period. Assume that the first bus of a routes starts from its origin stop at the same time during the researched period and the times is set to zero. The number of buses going through the stop can be formuated as foow. H t,, 1 n,, = (2) h, where y is the foor integer symbo, which returns the argest integer ess than or equa to y. For exampe, if y = 5.2, y = 5. v,, +1, the number of passengers in buses in the direction of the bus route from the stops to + 1. v,, +1 = v,, 1 + b,, a,, (3) b,,, the number of passengers boarding on buses in the direction of the bus route at the stop. b,, = { u,, u,, n,, (V max v,, 1 + a,, ) n,, (V max v,, 1 + a,, ) otherwise introduced, which denotes the number of non-served passengers at the stop in the direction of the bus route. ı,, +1 = u,, b,, (5) ϕ, penaty coefficient for additiona waiting times of non-served passengers. Formuation The time-costs T w, of passengers waiting for buses in the direction of the bus route are formuated as foow: T w, = Cp w [u,, t e, + ϕ ı,, h, ] (6) If the bus capacity can satisfy the passenger demand at the segment + 1, the non-served passengers is 0, i.e., a the passengers can catch the first bus arriving at stop in the direction of the route after their arriva. Otherwise, the non-served passengers, besides the waiting times for the first bus, yet have to wait for the foowing buses. We assume that passengers are sequentiay (on a first come, first served basis). Thus, the additiona waiting times of non-served passengers equa to the headway in the direction of the route. Generay, the additiona waiting times of passengers are more costy. ϕ is a constant, which is used to refect the infuence of additiona waiting times of passengers. It can increase with the increment of the missed vehices or it may be a constant integer bigger than Riding/dweing time-costs Basic notations Cp r, coefficients of the riding/dweing time-costs. ω,, +1, comfortabe eve [11,25] of passengers from the stops to + 1 in the direction of the bus route. Here, the comfort index is approximated by crowded eve, which is used to tune with the weight of bus crowding. ω,, +1 = v,, +1 n,, V t,, +1, average running times from the stops to + 1 in the direction of the bus route. In most existing researches, average running times in the two directions of a route are set as the same. However, considering inappropriate ayout of iving area and woring area in most arge cities in China, tide traffic phenomenon usuay occurs in some main roads, especiay during the rush hours. Thus the traffic in two directions of the main roads is imbaanced. Therefore, average running times of two directions of one route are computed, respectivey. t d,,, bus dweing times at the stop in the direction of the bus route. t d 0, constant times for vehice acceeration, deceeration and door opening/cosing. H, researched period (e.g. a rush hour).,,, crowded coefficient for boarding and aighting at the stop in the direction of the bus route. Zofaghari et a. [28] pointed out dweing times of buses at stops are reated to the bus oad, besides the number of passengers boarding and aighting. This indicated that the boarding and aighting times of passengers woud increase in the crowded conditions. Here, (7) (4) ı,, +1, the number of non-served passengers at the stop in the direction of the bus route. Since the provision of capacity on one route cannot aways carry a passengers, ı n,, +1 is the crowded coefficient is used to refect the infuence of the bus oad to the boarding and aighting times of passengers.

3 B. Yu et a. / Appied Soft Computing 11 (2011) V, rated capacity of a standard vehice. { 1 v,, +1 (n,, V),, = (v,, +1 /(n,, V)) 2 otherwise b, ā, average boarding and aighting times of a passenger. t d,, = t d 0 + max(b,,,, b, a,,,, ā) (9) Formuation Riding/dweing time-costs of the passengers T r, in the direction of the bus route are defined beow: T r, = Cp r [(ω,, +1 t,, +1 v,, +1 ) (8) + ((v,, +1 b,, ) t d,, )] (10) Riding times of passengers on the segment + 1 are the products of the bus running times on the segment + 1 ( t,, +1 ) and the number of the passengers (v,, +1 ) that can be carried by the direction of the bus route. Athough some buses coud go through their current stops (e.g. stop ) and not arrive at their foowing stops (e.g. the stop + 1) at the end of the researched period, for simpification, we assume that these buses woud have reached their foowing stops and computed the riding time of the passengers in these buses. Dweing times of passengers are the waiting times of onboard passengers at the stop Boarding/aighting time-costs Basic notations T ba,, passenger boarding/aighting time-costs in the direction of the bus route. Cp ba, coefficients of boarding/aighting time-costs. Formuation Passenger boarding/aighting time-costs T ba, in the direction of the bus route are defined beow: T ba, = C ba p [ b,,,, b ] + a,,,, ā (11) T o, are consisting by fixed operationa costs and variabe operationa costs. T o, = C f H o + C v o n h,, t,, +1 (12), where y is the ceiing integer symbo, which returns the smaest integer more than or equa to y. For exampe, if y = 5.2, y = 6. The fixed costs C f o H/h, are consisting by capita discount costs, maintenance costs, saaries of the drivers, etc. The variabe costs mainy concern about fue consumptions. Lie cacuating passenger riding time-costs, we aso assume that a the buses can arrive at foowing stops at the end of the researched period Mode integration Bus route headway optimization shoud find the trade-off between passenger costs and operator costs. Generay, if headways are excessivey sma (too many buses are dispatched), operators have to suffer excessivey operationa costs. However, if headways are excessivey arge, some service criteria may not be met which resuted in unsatisfied passengers who may choose the aternative means of transportation. One of the most widey used methods for soving muti-objective optimization probems [16,17] is the weighted sum method, which can transform the mutipe objective optimization into a singe objective function by weight factors. For simpification, a convex combination of the two objective functions is used in this study and combined the mode of passenger costs (in money) and operator costs (in money) for tota costs (T) of passengers and operators can be expressed as: Basic notations g, approximate feet size of the bus route. Omitting dweing times of buses at termina, the needed feet size of the more one of two directions is viewed as approximate feet size of the route. ( ) g = max t,, +1 /h,1, t,, +1 /h,2 (13) G, the tota number of bus vehices. Formuation T = w passenger (T w, + T r, + T ba, ) + w operator T o, = w [C passenger p w (u,, t e, + ϕ ı,, h, ) + Cp r [(ω,, +1 t,, +1 v,, +1 ) + (v,, +1 b,, ) ] [ ] t d,, ] + Cp ba [b,,,, b + a,,,, ā] + w operator C f H o + Co v n h,, t,, +1, s.t. (14) 2.2. Operator costs Basic notations r,,, average arriva rate at the stop in the direction of the route. Assume passenger arrivas foow an uniform process during the researched period [6,27]. The average arriva rate equas to the boarding passengers at the stop during the period divided by H. Co, f Co v, coefficients of fixed operationa costs and variabe operationa costs, respectivey. T o,, operator time-costs in the direction of the bus route. Formuation Generay, operators and transit agencies a expect to provide the transit service in an economic efficienty way. Operator costs h, 60 (15) g G (16) ω,, +1 > 0 (17) w passenger, w operator 0 (18) w passenger + w operator = 1 (19) where w passenger and w operator are the factors controing the weights of passenger costs and operator costs. In practice, the proper seection of the weight factor for each objective is difficut to be determined, because the definition of weights is not precise, nor are the vaues given by a decision-maer [19]. In fact, w passenger and w operator can be determined by decision-maers, e.g.

4 5084 B. Yu et a. / Appied Soft Computing 11 (2011) w passenger = w operator, which refect the subjective judgment or intuition of decision-maers. However, anaysis resuts are based on the weights can be infuenced by the decision-maers due to their ac of nowedge or experience. Therefore, it is an essentia and chaenging tas to deveop an objective method to assess the reative weight of the aternatives. An exampe of bus headway optimization weight factor determination is described in the foowing section. 3. Parae genetic agorithm Genetic agorithm is a search agorithm based on the concepts of natura seection and genetic operations. Many researchers attempted to improve the performance of GA by some methods [22]. Recenty, parae genetic agorithms (PGAs) have become one of the most effective strategies. Actuay, PGA basicay consists of various GAs, each processing a part of the popuation or independent popuations, with or without communication between them. Therefore, PGA can increase the diversity of popuation and reduce computation time. Generay, it can be divided into three types [1], namey master-save type, coarse-grained type and fine-rained type. Here, coarse-grained PGA is used since it costs ess and can obtain a near-ine acceeration ratio. Moreover, coarse-grained paraeization schemes run severa subpopuations in parae. So it is especiay suitabe for the custer system with ower communication bandwidth Encoding In this research, decision variabes of the agorithm are the headways of two directions of each route. Here, an integer encoding scheme is seected to represent bus headways and then a typica chromosome is as foows: {e 1,1, e 1,2, e 2,1, e 2,2,..., e,1, e,2,..., e N,1, e N,2 } (20) The bus departure interva for a route is rarey ess than 1 min (for exampe, the shortest departure interva for Daian in China is 1 min), and thus we assume that each gene cannot be ess than 1 min (i.e., 60 s). The initia popuation of chromosomes are generated by a probabiistic methodoogy, Firsty, temporary headways (genes), e, (60 e, 3600), are obtained by producing some random numbers. Thus, some temporary chromosomes that consist of the temporary genes are constructed. Headways of routes are generay imited by the feet size of bus vehices. The temporary chromosomes need to be checed whether to satisfy the feet constraint. The approximate feet size g of each temporary chromosome is firsty cacuated. Then, a scaed coefficient is gained according as the approximate feet size and the tota feet size. The genes of the initia chromosomes are computed as formua (23). {e 1,1, e 1,2, e 2,1, e 2,2,..., e,1, e,2,..., e N,1, e N,2 } (21) = g G (22) { e 0, = e, e, > 60 (23) 60 otherwise If a gene e,1 is smaer than 0, the vaue of the gene is set as zero and the scaed coefficient is re-cacuated. For exampe, if e 0, < 60, then set e 0, = 60. An initia chromosome is as foows. {e (0) 1,1, e (0) 1,2, e (0) 2,1, e (0) 2,2,..., e (0),1, e (0),2,..., e (0) N,1, e (0) N,2 } (24) 3.2. Fitness function GA is an optima searching method to find the maximum fitness of the individua chromosome, so it is necessary to transform the minima objective of the probem to a maximum fitness function [10,14]. Here, a constant Q is introduced to transform the fitness function from the tota cost function. Generay, the genetic operations may vioate tota feet size constraint. There are two approaches to dea with this situation. The first one is to assign a very high penaty cost for such candidate soutions and accordingy reduce their probabiity of being seected in the forthcoming search. The second approach is to try to fix the resutant vioations by adjusting the headways. The advantage of the first approach over the second one is that it is more suitabe in according with natura seection and evoution, and it enabes GA to investigate further points in the search space. Therefore, the first approach is adopted to dea with the vioation situation. Then, the chromosomes are evauated as foows. F = () = Q T + () ( g G ) (25) { ˇ1ˇ2 if g > G 0 otherwise (26) where F is the fitness function. Q is a constant. Ф () is the penaty coefficient at the generation. ˇ1, ˇ2 are contro coefficients, which can determine the penaty extent for the invaid individuas. They can usuay be estimated through simuation Seection operation The basic part of the seection process is to seect from one generation to create the basis of the next generation stochasticay. The fittest individuas have a greater chance of surviva than weaer ones is required. Here, the Rouette whee seection method is used to seect the chromosomes. Besides that, the Eitism is aso used for the seection. Eitism is a seection method where the best chromosomes in the popuation are automaticay copied into the next generation. That is, if the eitism parameter was set to then the top chromosomes in the popuation are copied to the next generation Crossover operation The crossover operator is associated with a crossover rate p c. An arithmetic crossover [26] is designed. Here, a random mutistratum crossover method is adopted. For exampe, for a particuar crossover process, at the generation 1, the two seected parent chromosomes E 1 and E 2 are as Eq. (27). If the random ı > 0.5, that mean two genes wi be crossed, otherwise, the genes of parent chromosome woud remain to the new chromosome directy. The crossover of two parent chromosomes is as Eq. (28). E = {e () 1 1 1,1, e () 1 1,2, e () 1 2,1, e () 1 2,2,..., e () 1,1, e () 1,2,..., e () 1 N,1, e () 1 N,2} E = {e () 2 2 1,1, e () 2 1,2, e () 2 2,1, e () 2 2,2,..., e () 2,1, e () 2,2,..., e () 2 N,1, e () e () 1, = e () 2, = e () 1, = e ( 1) 1, e () 2, = e ( 1) 2, ı e( 1) 1, + (1 ı )e( 1) 2, ı e( 1) 2, + (1 ı )e( 1) 1, if ı where ı, ı are random numbers between (0, 1). 2 N,2} (27) > 0.5 otherwise (28)

5 B. Yu et a. / Appied Soft Computing 11 (2011) Mutation operation Lie the crossover operator, the mutation operator is aso associated with a mutation rate (p m ) to determine whether the mutation operator is to be appied to the chromosome or not. Since there are two directions in each bus route, both the headways (genes) of two directions of each route need to be mutated in each mutation. If E 1 denotes a parent chromosome in the generation 1 and the genes e ( 1), e ( 1) are seected for the mutation, the resut of the,1,2 mutation of E 1 and the mutated chromosome in the generation are shown in (29). e (),1 = e ( 1) (1 + ı ),1 e (),2 = e ( 1) (1 ı ),2 if ı > 0.5 otherwise where ı, ı are random numbers between (0, 1) Loca search agorithm based on TABU search (29) GA is a suitabe method for goba optimization probems. To improve the oca optimization performance of the GA, a oca search agorithm based on TABU search [9] is introduced to find the oca optimum in a we-defined oca region. Since, the frequent oca searches can increase the computation time, the method is impemented with parae strategy of GA (Section 3.6) in this study. Tabu search is an iterative procedure that proceeds by transforming one soution into another by maing moves. It has successfuy been appied in soving the optimization probems in transportation fieds [2,15,18,20,21]. The heuristics requires an initia soution and a neighborhood structure. In this study, the initia soution in the oca search is to use the current optima soution in the GA proposed in the previous subsection. Then, the neighbors of the initia soution are examined and the best non-forbidden move is seected. The neighborhood structure in the oca search can be described as foows. For exampe, the chromosomes E denotes the initia soution (Eq. (30)). Firsty, randomy seect two genes, e.g. e (),1, e (),2 and e (),1, e (),2. A neighbor of the initia soution is as Eq. (31). E = {e () 1,1, e () 1,2, e () 2,1, e () 2,2,..., e (),1, e (),2,..., e (),1, e (),2,..., e () N,1, e () N,2} (30) { e (), = ı e (), + (1 ı )e (), e (), = ı e (), + (1 ı )e (), (31) where ı is the random number between (0, 1). In the oca search, a Tabu ist is used to prevent generating the degradation soution that has aready tested in previous iterations. The size of the Tabu ist can infuence the search quaity, and in our oca search the arge and fixed Tabu ist is used, i.e., the size of the Tabu ist is set to 20. The oca search agorithm continues unti the maximum tota number of the iterations or the maximum number of the iterations without improvement of the best soution Coarse-grained strategy The coarse-grain strategy runs severa subpopuations in parae. The information exchange among these subsets is done at certain the intervas (epoch) of iterations. By exchanging the outstanding chromosomes between subsets, the search spaces of the subsets are diversified to effectivey prevent the premature convergence. Let and P represent the amount of subsets and their scae, respectivey, thus the tota popuation P size = P. The common migrating strategy [1,26] is adopted, in which one best individua to migrate, and then to repace the worse individua in a subset with the migrated ones from nearby subset. Here a ring topoogy is used, which means that subsets x exchange individuay with subset x + 1 during migration Stopping criterion When the average of the fitness vaues of a the individuas is greater than 90% of the fitness vaues of the best individua or when the agorithm repeats the prepared maximum number of generations, the PGA is considered to have converged and therefore is stopped. The process of the PGA proposed in this study is described as Fig Numerica test The mode and the agorithm are tested with the data of Daian City in China. Daian s popuation is about 2 miion, the buid-up area is about 180 m 2, and the road networ consists of 3200 ins and 2300 nodes. There are totay 89 bus ines (Fig. 2) and 3004 bus stops, which extend 1130 m, and with 4130 vehices in it. Passenger origin-destination (OD) stop matrix is obtained from our former research [24] Weight identification Before optimizing the headways of routes, the weights of passenger costs and operator costs shoud be determined. It is often difficut for decision-maers to determine the weights because the definition of weights itsef is not precise. This paper proposed an integration approach to determine the weights which considers both the reative importance and the difference of operator costs and passenger costs. The sampe data consisted of the passenger costs and operator costs of the 89 bus routes (I = 89) of transit system in Daian City. The passenger costs and operator costs between the 89 routes are different due to the different numbers of passengers and vehices. Thus, the costs of the 89 routes need to be normaized. Firsty, the maximum passenger costs and operator costs among the 89 bus routes were used to scae the sampe data. Here, the two attributes were scaed to the range between 0 and 1. ˆT passenger ˆT operator { T passenger,max T operator,max { passenger,min T T operator,min = T passenger,max T passenger T passenger,max = T operator,max T operator,max = max = max = min = min T passenger,min T operator T operator,min (T passenger ) (T operator ) (T passenger ) (T operator ) (32) (33) (34) where T passenger and T operator denote the origin passenger costs and operator costs of the existing routes, respectivey. T passenger,max and T passenger,min indicate the maximum and minimum passenger costs. Simiary, T operator,max and T operator,min indicate the maximum and minimum operator costs. ˆT passenger and ˆT operator indicate the scaed passenger costs and operator costs of the existing routes.

6 5086 B. Yu et a. / Appied Soft Computing 11 (2011) Fig. 1. The fowchart of the PGA. Then, the passenger weight and operator weight can be computed. w passenger = 1 2 (wpassenger,g + w passenger,c ) w operator = 1 2 (woperator,g + w operator,c ) (35) x passenger = 1 L x operator = 1 L L =1 L =1 ˆT passenger ˆT operator (37) w passenger,g x passenger = x passenger + x operator (36) w operator,g = 1 w passenger,g w passenger,c s operator = s passenger + s operator (38) w operator,c = 1 w operator,c

7 B. Yu et a. / Appied Soft Computing 11 (2011) Tabe 1 Parameters in headway optimization mode. Parameter H V V max b, ā ϕ t d 0 Cp w Cp r Cp ba C f o C v o Vaue 3600s s 2 3 s 2.7 RMB a /h 2.0 RMB/h 1.0 RMB/h 8.75 RMB/vehice 3 RMB/min a RMB (Reminbi). (s passenger ) 2 = 1 L (s operator ) 2 = 1 L L =1 L =1 (ˆT passenger (ˆT operator x passenger ) 2 x operator ) 2 (39) 4.2. Resuts The ists of parameters used in the headway optimization mode and PGA are shown in Tabes 1 and 2. The agorithm is impemented in C++, using message passing interface (MPI) ibrary, on 8-computer custer architecture: windows XP patform environment. where w passenger and w operator denote the weights for passenger costs and operator costs of the routes. L denotes the number of the sampe data, here L = 89. w passenger,g and w operator,g indicate the weights considering the reative importance between mutipe objectives (the passenger costs and operator costs), i.e., the average proportion of passenger/operator costs of each route in its tota costs. Simiary, w passenger,c and w operator,c indicate the weights considering the difference between mutipe objectives (the passenger costs and operator costs), i.e., the difference between passenger/operator costs of routes. x passenger and x operator denote the mean passenger costs and operator costs of the sampe data. s passenger and s operator denote the variance for passenger costs and operator costs of the sampe data. To describe the computation process of the weight identification, we proposed a simpe exampe. There are three routes. Assume that the passenger costs and operator costs of the three routes are (passenger costs = 0.8, operator costs = 0.76), (passenger costs = 0.9, operator costs = 0.8) and (passenger costs = 0.7, operator costs = 0.78), respectivey. Then, the functionaity weights and proportionaity weights of two indexes are computed, w passenger,g = 0.49, w operator,g = 0.51, w passenger,c = 0.55 and w operator,c = Thus, the weights of the passenger costs and the operator costs are determined, w passenger = 0.52, w operator = From caibration resuts, there is amost no significant difference between the coefficients of the passenger costs and the operator costs. However, the coefficients are caibrated by practice data of bus system in Daian City. This indicates that passenger costs and operator costs are simiar in the city Performance of the proposed agorithm In order to show the basic behavior of our parae genetic agorithm, experimenta resuts are given here for various conditions in which there were two sequentia GAs with 60 and 80 individuas and eight PGAs: P size = 240 or 320 and = 4, 6, 8 and 16 nodes. As the Tabe 2 shown, a the parameters are same except P size which change from 240 to 320. Fig. 3 shows the experimenta resuts. The sequentia GAs (SGA) are the agorithms with = 1. Since the proposed mode is a compicated probem, the SGA tends to step into premature convergence. It is obvious that the PGAs observe a better quantity than the SGAs. This comportment can be expained principay because when > 1, the migration operation between sub-popuations can diversify each subset, widen the searching space, and improve the optimization quaity. Furthermore, it can be observed that the better performance among PGAs appears at = 6 and = 8. Compared with computation time of severa agorithms, the convergence speeds of the PGAs with = 8 and = 16 are faster than the ones of other PGAs. Weighting the optimization quaity and computation time, we seect the PGA with = 8 and P size = 320. To examine the efficacy of the PGA, we continue experimenting 10 times, and Fig. 4 shows the convergence of the cacuation. It can be observed that the fitness increases fast before the 1100th generation, and then it changes smoothy. The best fitness appears at about the 1500th generation. Furthermore, the fitness among ten experiments hardy changes again. This means our agorithm has a good converge and we can concude that after about 1500 generations of evoution the optima soutions can be found. Then, we test the coarse-grained strategy and the oca search agorithm. The performances of severa agorithms are compared, which incude SGA, SGA with the oca search agorithm (denoted by SGA-L), GA with coarse-grained strategy (denoted by CGA) and the proposed agorithm (PGA). The severa agorithms continue experimenting 10 times and the best soution, the worst soution and the average soution of the 10 resuts are shown in Fig. 5. We can find that the performances of SGA-L, CGA and PGA are better than the one of SGA. This is just as expected as the more efforts an agorithm expends, the better performance it certainy gains. Compared with SGA-L, CGA generay provides better soution. This can be attributed that the coarse-grain strategy diversifies the popuation and prevents the agorithm from trapping in oca optimization. Furthermore, the introduction of the oca search agorithm into CGA (PGA) can adequatey search the oca region and improve the soutions. This indicates that the incorporation Tabe 2 Parameters in PGA. Parameter Q p c p m max ˇ1 ˇ2 epoch Fig. 2. Bus networ in Daian City. Vaue 10,

8 5088 B. Yu et a. / Appied Soft Computing 11 (2011) Fig. 3. Experimenta resut of severa parameter combinations. Fig. 4. The resut of each cacuation. of the coarse-grained strategy and the oca search agorithm can greaty improve the performance of the agorithm. In order to further examine the performance of the agorithm, here, a muti-objective genetic agorithm (MOGA) is introduced. There are two objective functions in the MOGA: one is the minimum tota passenger cost, and the other is the minimum operation costs. The coding of the MOGA is consistent to the agorithm proposed in this paper. Fig. 5. The comparison of severa agorithms.

9 B. Yu et a. / Appied Soft Computing 11 (2011) Fig. 6. The computationa resuts of two agorithms. First, each chromosome is sorted according to two objectives, respectivey. After sorting each object, the objective function of the overa performance can be got. E i (X j ) = { (N R i (X j )) 2 R i (X j ) > 1 N 2 R i (X j ) = 1 (40) E(x j ) = E i (X j ) (41) i where n is the number of the object; N is the tota number of individuas; X j is the individua j in popuation; R i is the number for sorting a individua quaity in the popuation; E i (X j ) is the fitness of X j on the target i; K is the constant between (1, 2), which is used to increase the fitness when the individua function vaue performs optima. Individua choice is adopted by the rouette whee way. Here, K is set to 1.5. E(X j ) is the fina fitness vaue of the chromosomes j. For exampe, there are three chromosomes in a popuation. Assume that the order of three chromosomes according to two Fig. 7. Comparison of three headways in the direction with more passengers.

10 5090 B. Yu et a. / Appied Soft Computing 11 (2011) Fig. 8. Decreasing or increasing costs by increasing an operation vehice. objectives are (1, 3), (2, 1) and (3, 2), respectivey. Then, N = 3, R 1 (x 1 ) = 1, R 2 (x 2 ) = 3, R 1 (x 2 ) = 2, R 2 (x 2 ) = 1, R 1 (x 3 ) = 3 and R 2 (x 3 ) = 2. Thus, the fitness vaues of three chromosomes are set to E(x 1 ) = 13.5, E(x 2 ) = 14.5 and E(x 3 ) = 1. In the MOGA, the same crossover and mutation operations with the proposed agorithm are used. To be fair, the MOGA aso uses the parae strategy. Then, in the same condition, we cacuated continuousy 10 times two agorithms. Fig. 6 shows the computationa resuts. Obviousy, both agorithms have good stabiity, for exampe, the difference between the best and worst soutions is ess than 5%. In addition, the soution optimized by the MOGA, passenger tota cost is ower whie there is a ower operator cost in the soution from our agorithm. This is because the MOGA is based on the raning of chromosomes to seect the target chromosome, rather than the objective vaues. Therefore, when the quaity of chromosomes has a arger difference, the MOGA has difficuty in distinguishing chromosomes according to evoution. In addition, from the computationa time, one can found the convergence speeds of the two agorithms are simiar. On the whoe, the optimization quaities of the two agorithms are simiar Headway optimization mode To vaidate the proposed mode, the optimized headways of routes are compared with the existing headways. The tota costs of the current transit system are thousand RMB (the operator costs is thousand RMB and the passenger costs is thousand RMB). Compared with current situation, the operator costs (272.2 thousand RMB) and the passenger costs (243.7 thousand RMB) of the transit system with optimized headways are decreased by about 6% and 14%, respectivey. This can attribute to the unreasonabe resource aocation in current situation and can aso indicate that the integrating resources, the service eve of system and the efficiency of resources can be improved. Furthermore, from optimization resuts, it is found that the tota feet size of the transit networ with optimized headways just equas to the existing one (the maximum feet size constraint). This impies that the feet size of the transit system in Daian City can perhaps not be enough for demands. Therefore, we compute the desired headways of routes using the same data sets as the proposed mode through reeasing maximum feet size constraint. Thus, if the costs to purchase vehices are not being considered, the tota costs of the transit system with the desired headways can be greaty decreased by about 18.3%. This can be seen as the one proof for the crowded condition of rush hours in Daian City. The detais of the comparison of three headways in the direction with more passengers of each route are shown in Fig. 7. It is obvious that the headways of the three situations are different. As a whoe, desired headways of most routes are ower or equa to the existing headways or optimized headways. It is aso observed that the optimized headways of some routes are much ower than the existing headways of these routes, e.g. route 1, 23, 31, 35, 36, 38, 39, 45, 74, 78, 79 and 86, etc. In fact, these routes are indeed heaviy crowed in the rush hours. It is necessary to increase the vehices to operation to improve service eve of these routes. Contrariy, the optimized headways of some routes are higher than the existing headways of these routes, e.g. route 15, 26, 50, 59, 63,

11 B. Yu et a. / Appied Soft Computing 11 (2011) , 67, 88 and 89, etc. Haf of these routes are comprised of an affiiated company, which incudes route The routes of the affiiated company mainy serve in suburb. From optimized resuts after resource integration, the comfortabe eve of passengers in crowed routes can be improved. From Fig. 7, the existing feet size is insufficient for the demand in rush hours. However, it is difficut and impossibe to purchase enough vehices to satisfy a routes simutaneousy. Therefore, it is necessary to anayze the sensitivity to increase operation vehices into a route. If the ower imit of headways of routes is set as 60 s, the sensitivity anaysis to the routes satisfying the ower imit constrain of headways is shown in Fig. 8. It can be observed that in the transit system with existing headways increasing an operation vehice into the route 36 can decrease the most costs of among a the routes, whie increasing an operation vehice into the route 8 can gain the most benefit in the transit system with the optimized headways. In addition, we can find that some routes in the existing situation or in the optimized situation bring more costs after increasing an operation vehice. This indicates that these routes are not crowed. Increasing more operation vehices to these routes wi aggravate the imbaance or brea the baance between passenger costs and operator costs. The sensitivity anaysis can provide a reference when the transit system or bus companies increase operation vehices. 5. Concusions Headway design is a necessary product for transit system, and it is aso true that a transit agency wi often evauate and determine headways of routes. This paper presents a headway optimization mode based on a given networ configuration and demand matrix. This mode syntheticay considers the passenger costs and operator costs. Aso, an objective approach integrating the functionaity and proportionaity to weight determination is proposed to find an acceptabe baance between the operator costs and the passenger costs. Parae genetic agorithm is used to sove the headway optimization mode and parameters in the agorithm are aso tested. Data of transit system in Daian City, China, is coected to test the mode and the agorithm. The existing passenger costs and operator costs of the bus routes of transit system in Daian City are used to determine the weight between two costs. Resuts show that PGA is a powerfu too for bus route headway optimization in this paper. Furthermore, resuts aso suggest that resource integration can improve the service eve of transit system. 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