Scheduling Combination Optimization Research for Bus Lane Line

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1 TELKOMNIKA Indonesian Journal of Electrical Enineerin Vol., No., January 04, pp. 809 ~ 87 DOI: Schedulin Combination Optimization Research for Bus Lane Line Hao Xiaoni*, Jin Wenzhou, Yan Yazao 3, School of Civil Enineerin and Transportation, South China University of Technoloy Wushan Road 38, Tianhe District, Guanzhou, China, Phone/Fax: Chonqin Key Laboratory of Traffic and Transportation, Chonqin Jiaoton University Xuefu Avenue 66, Nanan District, Chonqin, China, Phone/Fax: *Correspondin anthor, xnhao@scut.edu.cn, scutyyz@mail.com Abstract Different schedulin forms can be adopted in bus lane system to meet passeners travel demand well and improve operational efficiency. Therefore, this paper researched the optimal headway and bus schedulin combination of a bus lane line. Bus schedulin combination is composed of a sequence of fulllenth, express bus and short-turn. We established a model to minimize passeners waitin cost and vehicles operation cost and to optimize headway and bus schedulin combination, under the constraint of the headway restriction for each stop, the minimum number of vehicle trips in one hour and the proportion of short-turn and express bus trips in total trips. The model was solved by improved enetic alorithm, and the optimal solution was obtained by repeatin the operation of enetic alorithm L times. The results of numerical example show that the whole cost can be saved by.77% at most after optimization, which indicate the model and alorithm we presented are reasonable and practicable. Keywords: Schedulin Combination, Headway, Bus Lane Line, Genetic Alorithm Copyriht 04 Institute of Advanced Enineerin and Science. All rihts reserved.. Introduction Bus lane system is a bus form between ordinary buses and bus rapid transit, speed and priority hiher than the former, lower than the latter. Bus schedulin is the bus oranization forms in vehicles operation. As shown in Fiure, full-lenth bus is a basic schedulin form. Short-turn buses only travel in the lare passener flow section of bus lines, and express buses drive the entire route and only stop at certain stations with lare passener volume []. Fiure. Stops of three schedulin forms Some related literature is as follows. Jian first put forward the idea of combination schedulin and made a preliminary discussion on vehicle schedulin combination problem []. With the shortest bus operation turnover time for the taret and nown headways, Zhan et al. researched the vehicles combination schedulin order [3]. Zou proposed a reional schedulin model to optimize vehicle schedulin mode and dispatch schedule simultaneously, but she Received May 0, 03; Revised Auust 3, 03; Accepted Auust 30, 03

2 80 ISSN: didn t ive a specific alorithm for that model [4]. Xu et al. put forward the express bus schedulin model and applied it to practice [5]. Bai et al. optimized the combination frequency of BRT line and standard line by simulation of tabu simulated annealin alorithm [6]. Sun et al built the optimization model based on minimizin passener travel time costs and transit vehicles operation time costs, and studied and determined the headway and schedulin combination form of Bus Rapid Transit [7]; however, she didn t tae into account the number constraints of short-turn and express bus, and the restrictions on maximum waitin time of passener at the middle bus stop, which maes the obtained optimization results impracticable. The current bus schedulin researches are mostly concentrated in timetable and frequency optimization of traditional bus lines [8-9]; moreover, schedulin researches on BRT are not many. Furthermore, schedulin researches on bus lane lines is rare, and the literature studyin vehicle schedulin combinations and headway optimization simultaneously is also considerably less. Thus, this article intends to study the vehicle schedulin combination and headway optimization of bus lane lines.. Optimization Model of a Bus Lane Schedulin.. Problem Description and Assumption Bus lane line runs smoothly with special lanes, therefore, different schedulin forms can be adopted to meet bus passeners travel demand well in various sections and at the same time to increase operational revenue of bus companies. In summary, the ey to optimize the bus schedulin prorams and improve the service level of bus transit is to establish a rational and effective optimization model on headways and bus schedulin combinations. Considerin the passener flow uneven with the time, under reasonable assumptions, an optimization model will be built to research headway and schedulin form combination of bus lane lines in a certain period in this paper, whose obective is to minimize the sum of bus company operation costs and passeners waitin costs. In order to facilitate research, the followin assumptions are iven: (a)the article studies headway and schedulin combinations only in the sinle direction at a bus lane line; (b)passener demand is stable with the headway and bus schedulin combination chanin; (c)the runnin time between stations is certain. Specifically, tain the averae runnin time; (d)in the study period, the passeners are uniformly arrived; (e)the station dwell time of vehicles is fixed, obtained by continuous observation; (f)there are enouh vehicles in every fleet; ()The lane transit vehicles have a uniform departure interval at the oriin station... Symbol Definition In the paper, the variables involved are defined as follows: i - vehicle trip of a bus lane line at the oriin station, i,,,, i I s ; - stops on the bus line,,,, N ; Ka - stop set of full-lenth buses; Ke - stop set of express buses; These stops are selected from all stops accordin to passener flow volume, and they maybe discontinuous sites. To show the difference, with ' denotes them,,,..., n ; Ks - stop set of short-turn buses; K - intersection of set of full-lenth stops, express stops and short-turn stops, K K K K a e s ; - schedulin form, means full-lenth bus, is express bus, 3 means short-turn bus; s - the total number of bus trips, s 3 ; - the number of full-lenth bus trips; - the number of express bus trips; 3 - the number of short-turn bus trips; r - passener averae arrival rate at stop durin the study period (per/min); t 0 - departure time of the first bus trip at the oriin station in the iven period; a i, - the arrival time of bus trip i at stop ; a i, - the arrival time of bus trip i at stop, obtained by sortin a i, ascendin; d i, - the departure time of bus trip i at stop ; d i, - the departure time of bus trip i at stop, obtained by sortin d i, ascendin; h - headway between bus trips at the oriin station; - headway between bus trip i and i at stop, when stop belonin to the set K, and h i d i d,, i, ; h i, - headway between vehicle trip i and i at stop,when stop not belonin to the set K, and h i, TELKOMNIKA Vol., No., January 04:

3 TELKOMNIKA ISSN: h i d d, i, i,, i I, K, K a ; t, - vehicle runnin time between stop and ; b0 - vehicle dwellin time at stops; - outbound acceleration and inbound deceleration time, and the time will be saved when vehicles don t stop at some stations; S, S i, i, - remainin passeners at stop when bus i or i leavin stop. In subsequent calculations, the number of passener left at stops is enerated by a random number enerator in the above-mentioned ranes. xi - schedulin form of bus trip i ; Althouh the variable x i doesn t appear in the model, the value of x i, is determined by x. i x i, - 0- variable, for the schedulin form, when vehicle trip i stops at station, the value is, otherwise is 0; h, h min max - the iven minimum and maximum headway (min); v w - passener waitin cost (yuan/min); v o - vehicle operatin cost (yuan/min);, - in the obective function, the respective weihts of passener waitin costs and vehicles operatin costs; Qt - durin the tth hour, passener flow of the hihest section; Q0 - vehicle rated passener capacity; r 0 t - durin the tth hour, vehicle actual load rate quota; - the endurance time when passeners waitin for vehicles at station. Generally, it is 0 to 5 minutes for pea period, flat period to tae 5 to 0 minutes; - the proportion of short-turn and express bus trips in total trips; T - the study period (min);.3. Model Formulation This paper researches mornin pea hour passener flow in upstream direction of a certain bus lane line. The numbers of boardin passeners at each stop can be obtained by dealin with the data of readin bus cards times durin the iven period, and the numbers of alihtin passeners can be also acquired. When nowin the numbers of boardin and alihtin passeners at each stop, we can determine the stops for short-turn and express buses. This article assumes that all vehicles depart from the oriin station due to the land utilization restriction of middle stops. We build the optimization model to minimize passener s waitin fees and vehicles operatin costs in this paper. However, both these costs are in opposite relation. Namely, bus companies always want to provide as lare departure interval and reduce the bus trips to decrease operation costs; however, passeners hope to et hih frequency bus services to reduce the costs of waitin and transferrin. Thus, there is a benefit inconsistency between passeners and bus companies. The bus schedulin combination optimization is a typical multi-obective optimization problem, and this problem can use weiht coefficient transformation method; that is to say, we can transform the multi-obective problem into a sinle-obective one by assinin different weiht coefficients to each sub-oal and main a linear weihted summation of all sub-oals. The mathematical model is formulated as follows: Z s N [ ( r hi, Si, hi, ) ( r hi, Si, hi i, ii K K, K )]} () vw{, N K e K s a Z v { ( t x b ) [ t ( ) ] ( )},, 0, x x,, x, b t x b () o i i i i 0 3, i, 0 min Z Z Z (3) s.t. h h (4) min h max h d d, i (5) h, i, i, i, d d, i (6), i, i, i, Schedulin Combination Optimization Research for Bus Lane Line (Hao Xiaoni)

4 8 ISSN: Q (7) 3 Q t 0 r 0 t ( ) 3 (8) a 3 i, t0 h ( b0 i ) [ t, ( x, x, ) x, ] i (9) i i d i, ai, xi, b0 (0) s h T () x i,, {0, },,,3, i () Equations () and () indicate passener waitin costs and vehicle operatin costs respectively. Equation () consist of two parts, the first part denotes passeners waitin cost when stop belonin to the set of K, the second part denotes passeners waitin cost of other stops. Passener waitin costs include the passener averae waitin cost and the extra waitin cost of passeners left by previous trip. From equation (), the vehicles operation cost includes the operation cost of full-lenth, express and short-turn buses. Constraint (4) ensures headway of the oriin station in a reasonable rane, namely, reater than or equal to the h min, and less than or equal to the h max. Constraints (5)-(6) assure that headway of intermediate station should be no more than passener waitin endurance time. Constraint (7) assures that the total number of bus trips should be reater than or equal to the minimum number of confiured vehicles in iven period to meet the requirement of actual load rate. Constraint (8) uarantees the number of short-turn and express vehicle trips can t exceed a certain proportion of the total. Constraints (9)-(0) express how to respectively calculate vehicles arrival time and departure time. Constraint () indicates that the studied period is divided into s intervals by uniform headway. Constraint () shows that variable x i, is a 0- variable. 3. Improved Genetic Alorithm for Bus Schedulin Combination Problem The enetic alorithm which simulates the evolution laws of mechanism of bein survival of the fittest in bioloical world is a robust search and optimization methods. Practice has proved that the enetic alorithm can quicly solve lare-scale combination optimization problems which the traditional methods can t solve. In this paper, the bus schedulin combination problem is a typical NP combination optimization problem, therefore, this paper uses the enetic alorithm to solve this problem. However, in the application of basic enetic alorithm, some unsatisfactory issues also appear, such as its premature phenomenon and poor local search ability. Thus, this paper maes some improvement in codin and enetic operators of basic enetic alorithm. The core of an improved enetic alorithm for solvin the bus schedulin combination problem is to determine the codin mode, initial population, fitness function and the desin of enetic operators [0-4]. 3.. Determination of Codin Mode The model contains two decision variables: headway and bus schedulin form combinations, and these variables are interrelated. Therefore, considerin two variables toether, chromosome lenth is variable. Namely, accordin to the number of bus trips, encodin the schedulin forms. If chromosome lenth is variable, and there will produce many invalid and infeasible solutions, which will reatly lower the efficiency of alorithm. Therefore, this paper aims at headways which divide study period into inteer trips, and respectively adopts enetic alorithm in solvin the optimal schedulin combination. Finally, comparison and TELKOMNIKA Vol., No., January 04:

5 TELKOMNIKA ISSN: analyses of these optimal vehicle schedulin form combinations are presented. Within [ h min, hmax ] interval and the number of bus trips bein inteer, valid headway will be finite number L. Therefore, we need to repeat the operation of enetic alorithm L times. For three different vehicle schedulin forms, 00 is represented as full-lenth bus, 0 as express bus, and 0 as short-turn bus. For example, when the headway is in the interval [3,5], to tae the headway 0min, the number of bus trips is six, and the possible codin is [ ], which ives one of the bus schedulin form combination. 3.. Determination of Decodin Method and Initial Population Besides combinations described above, 0 and can also produce another combination. For, when decodin, and it is desined for each randomly assined in this paper. Namely, when appears, there will produce a random number in [0,], and if , is decoded into full-lenth bus; if , is decoded into express bus; if 0.75, is decoded into short-turn bus. The initial population should be more widely representative and have enouh chromosomes in order to avoid fallin into local optimum and premature phenomenon. For bus schedulin optimization problem, the initial population consists of part of chromosomes by random and part of the feasible solution chromosomes Construction of Fitness Function In enetic alorithm, we use fitness function to evaluate the extent of the pros and cons of each individual; if individual fitness is reater, the individual is inherited to the next eneration with hiher probability. The oal of this bus schedulin combination model is to et minimum value, therefore, we can determine its fitness function by finitude construction method. The fitness function is expressed as: c f ( x) f ( x) c max max F ( x) (3) 0 f ( x) cmax Where c max represents the current eneration s maximum obective function value f (x) and it will vary with enerations Genetic Operators Realization For selectin operation, this paper adopts the roulette wheel selection method. The specific operation is as follows: first, calculate the sum of the fitness of all individuals in the roup; second, calculate the relative fitness value of each individual, and it is each individual s probability inherited to the next eneration populations; finally, simulated amblin dis operation to determine the times of each individual selected. In addition, this article also uses elitist stratey to ensure that the best individual obtained so far is not been destroyed by crossover and mutation operators. Crossover operation is a maor operation to enerate new individuals in the enetic alorithm, and it is an exchane of two individual s part of chromosomes with certain probability. This paper adopts one-point crossover, and the specific operation is as follows: first, pair individuals of roups randomly; secondly, in the individual encoded strin, select randomly a crossover point location accordin to the probability P c ; finally, exchane parts of the chromosomes of the two paired individuals. Mutation operation also can produce new individuals. In this paper we use simple mutation, namely, for ene value in one or a few locus assined randomly in the individual encoded strin doin mutation operation with mutation probability P m. For binary coded individuals, mutation operation means that the ene values of the variation points are inverted, with 0 replacin, or replacin Alorithm Steps Step : parameter settins, includin population size M, crossover probability P c, mutation probability P m, cycle numbers, maximum and minimum value of bus trips f ; Schedulin Combination Optimization Research for Bus Lane Line (Hao Xiaoni)

6 84 ISSN: Step : initialization, tae a valid inteer from bus trips and encode the schedulin forms with binary system. By random eneration and the supplementary part of the feasible solution, we can produce an initial population made up of M chromosomes; Step 3: for each individual of the population decodin, we can et the specific bus schedulin forms sequence, and then calculate the obective function value and fitness value of each individual, combined with the iven data and statistical data; Step 4: selection operation, eneratin new population with roulette selection; First, calculate the selected probability P i and cumulative probability q. Next, produce a uniformly distributed random number r in [0, ]. If r q, then select the first chromosome to replicate; If q r, then select the th chromosome to replicate. Finally, circulate step for M times to q produce the next population; Step 5: crossover operation, for chromosomes of new population main one-point crossover with the crossover probability P c ; Step 6: mutation operation, for the chromosomes of new population enerated by crossover operator doin mutation operation with mutation probability P m ; Step 7: repeat 3~6 steps until it meets preset evolution enerations, namely, complete the optimization of bus schedulin form combination, under a specific number of bus trips; Step 8: select new valid value of bus trips, and repeat ~6 steps, and obtain the optimal bus schedulin form combination under each valid bus trip. Finally, contrast these optimal values obtained by each cycle, and et the final optimal solution of the present model. 4. Numerical Example 4.. Parameter Value In this paper, parameter values are identified as follow: the passener waitin value is determined to be 0.5yuan/min, and the bus operation cost value is defined to be.5yuan/min. With the survey in stops, the vehicle dwellin time is min, the averae value of the acceleration and deceleration time is 30s. The runnin speed is 5m/h durin mornin pea. There are 0 stops and the studied period is hour. Passener waitin costs share a weiht of 0.5, and the vehicle operation costs account for 0.5. Passener waitin endurance time is 5mins. The proportion of short-turn and express vehicle trips in total trips is 0.4. Accordin to other relevant researches [-6], the crossover probability P c is 0.65, the mutation probability P m is 0.005, the population size is 0, and cycle number is 00. Stops of express buses include site,3,5,8,0,,,5,7,0, and the short-turn buses run between - sites. Passeners volumes and the route parameters are shown in Fiure and Tables. Passener Volume (Person/h) boardin passeners' number alihtin passeners'number Stop Fiure. Numbers of boardin and alihtin passener at stops TELKOMNIKA Vol., No., January 04:

7 TELKOMNIKA ISSN: Table. Parameters of bus lane line Stops Distances(m) Runnin time (min) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Results Analysis The enetic alorithm parameters, passener flow data and parameters of the bus lane line were set into the proram, and the calculation results of the numerical example are shown in Table. In the optimal bus schedulin form combination, represents full-lenth bus, represents express bus, and 3 represents short-turn bus. From the results, with the headway decreasin, the obective function values un-optimized decreased radually because of the reduced waitin time of passeners. For the optimized bus schedulin form combinations, with the number of bus trips, express and short-turn bus increasin, the passener waitin time at the main stops and overall vehicle operation time decreased, thus the whole system costs have been reduced in different deree. When the headway was 5min and specific bus schedulin form combination was 3 3 3, the optimal solution of the model is obtained and it was 83.3, which was save by 4.8% after optimization. When the headway is 6min, after optimization, the optimal solution was saved by.77% at most. Therefore, this model and alorithm to optimize the headway and bus schedulin form combinations of a bus lane line are reasonable and feasible, and they will save the whole system costs reatly. Headway (min) Table. Optimal schedulin form combination under each headway Saved cost(%) Vehicle trips Bus schedulin form Obective values Obective values combination (not optimized) Sensitivity Analysis When the proportion of passener waitin cost is 0.3~0.7, the chanes of optimal values are showed in Fiure 3. For the same headway, with the proportion of passener waitin cost increasin, we can see that the obective values increase, except the headway of 4 minutes, and headway is bier, the obective value increase quicly. More important, for the same proportion of passener waitin cost, the optimal obective value corresponds to the different headway. Therefore, we should carefully analysis and determine reasonable proportions of passener waitin cost and bus operation cost. Schedulin Combination Optimization Research for Bus Lane Line (Hao Xiaoni)

8 86 ISSN: Obective Value(Yuan) h=min h=0min h=7.5min h=6min h=5min h=4min Proportion of Passener Waitin Cost Fiure 3. Sensitivity of obective function to proportion of passener waitin cost Obective Value(yuan) min 5min 6min 7.5min 0min min 0.5,. 0.5,.5 0.5,.8 0.5,. 0.5,.5 0.5, ,. 0.35, ,.8 Passener Waitin Unit Cost and Vehicle Operatin Unit Cost Fiure 4. Sensitivity of obective function to passener waitin and vehicle operatin unit cost From the Fiure 4, it can be seen that when the passener waitin unit cost and bus operation unit cost tae nine different combinations, the optimal obective value of the same headway is a broen line, and for the same combination, the optimal obective value corresponds to different headway. Therefore, we should mae sufficient investiations on passener s time cost and bus operation cost, and then determine their reasonable values. 5. Conclusion and Discussion Considerin various practical constraints in bus operatin process, this paper built the model which aimed at minimizin the linear weihted sum of the passener waitin cost and vehicle operation cost in order to study the optimal headway and bus schedulin combination for a bus lane line in iven period, and solve the model by improve enetic alorithm. The optimal results obtained by numerical example can been saved by 4.8%, compared with the un-optimized value. Therefore, the model and the solvin method are practical and effective. Due to limited space and ability, this study has some defects inevitably, and the followin contents need to be deeply explored in further research. First, accordin to the actual arrival rate of passeners, without equal headway restriction, there can obtain more optimal results. Second, the value of proportion of short-turn and express bus trips in total trips also affects the optimal value of the model, and consequently, there needs careful study to determine the reasonable value in order to reduce passener waitin time at the main stops and not cause passener too lon waitin time at the rest stops. TELKOMNIKA Vol., No., January 04:

9 TELKOMNIKA ISSN: Acnowledement This wor was supported by the Fundamental Research Funds for the Central Universities under contract No. 0ZM009 to Hao Xiaoni, and National Science Foundation of China under contract No to Jin Wenzhou. Besides, this wor was supported in part by research rants from Science and technoloy research proect of Chonqin education commission (Proect No: KJ047) and Chonqin ey laboratory of traffic and transportation 00-0 open funds (Proect No: 0CQJY00). References [] Viayarahavan TAS, Anantharamaiah KM. Fleet assinment strateies in urban transportation usin express and partial services. Transportation Research Part A: Policy and Practice. 995; 9(): [] Jian GZ, He XC. Research on schedulin combination model for public transport routes. Journal of System Enineerin. 986; 3(): [3] Zhan FZ, Yan L, Fan YZ, Sun XF. Research on dispatchin methods of public traffic vehicles in intellient transport system. China Journal of Hihway and Transport. 003; 6(): [4] Zou Y. Study on bus reional schedulin travel plan oranizin method. Journal of Transportation Systems Enineerin and Information Technoloy. 007; 7(3): [5] Xu DW, Pei YL. Express bus schedulin model and application. Journal of Harbin Institute of Technoloy. 008; 40(4): [6] Bai ZJ, Son R, He GG, Lin JX. Simulation of tabu simulated annealin alorithm for optimizin BRT line combination frequency. Application Research of Computers. 008; 5(): [7] Sun CJ, Zhou W, Wan YQ. Schedulin combination and headway optimization of bus rapid transit. Journal of Transportation Systems Enineerin and Information Technoloy. 008; 8(5): [8] Kim Byun-ln, Kim Seonbae, Par Junhyu. A school bus schedulin problem. European Journal of Operation Research. 0; 8(): [9] Hao XN, Jin WZ, Wei M. Max-min ant system for bus transit multi-depot vehicle schedulin problem with route time constraints. 0th World Conress on Intellient Control and Automation. Beiin, China. 0; [0] Ni J, Yan NN. Genetic alorithm and its application in schedulin system. TELKOMNIKA Indonesian Journal of Electrical Enineerin. 03; (4): [] Yan YZ, Si HL, Hao XN, Luo JS. Traffic impact simulation for road construction proect. TELKOMNIKA Indonesian Journal of Electrical Enineerin. 0; 0(8): [] Kumar VS, Mohan MR. A enetic alorithm solution to the optimal short-term hydrothermal schedulin. International Journal of Electrical Power and Enery System. 0; 33(4): [3] Nin T, Guo C, Wan LJ. The shortest path optimization method usin hybrid enetic alorithm. International Journal of Advancements in Computin Technoloy. 0; 3(6): [4] Zou G, Li T, Qin Y. Automated uide vehicle dynamic schedulin based on annealin enetic alorithm. TELKOMNIKA Indonesian Journal of Electrical Enineerin. 03; (5): [5] Fuenschuh, Armin. A set partitionin reformulation of a school bus schedulin problem. Journal of Schedulin. 0; 4(4): [6] Jian J, Men LD. The stratey of improvin converence of enetic alorithm. Telomnia. 0; 0(8): Schedulin Combination Optimization Research for Bus Lane Line (Hao Xiaoni)