Modeling and Optimization of Urban Rail Transit Scheduling with Adaptive Fruit Fly Optimization Algorithm
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Open Phys. 2019; 17:888–896 Research Article Jin Li, Guangyin Xu*, Zhengfeng Wang, and Zhanwu Wang Modeling and optimization of urban rail transit scheduling with adaptive fruit fly optimization algorithm https://doi.org/10.1515/phys-2019-0094 Received Oct 23, 2019; accepted Dec 02, 2019 1 Introduction Abstract: Despite the rapid development of urban rail tran- Along with the explosive growth of economy and the ris- sit in China, there are still some problems in train oper- ing level of urbanization in China, urban rail transit plays ation, such as low efficiency and poor punctuality. Tore- an increasingly important role in domestic public trans- alize a proper allocation of passenger flows and increase port system and it develops rapidly in major and medium train frequency, this paper has proposed an improved ur- cities. Its advantages include less pollution, higher pas- ban rail transit scheduling model and solved the model senger carrying capacity, good travel speed and punctu- with an adaptive fruit fly optimization algorithm (AFOA). ality. As one of the core tasks of urban rail transit, depar- For the benefits of both passengers and operators, the ture scheduling heavily affects the operating costs, service shortest average waiting time of passengers and the least quality and transport efficiency. Therefore, the optimiza- train frequency are chosen as the optimization objective, tion of urban rail transit scheduling has become a bench- and train headway is taken as the decision variable in mark problem in this field. the proposed model. To obtain higher computational effi- How to accurately and efficiently allocate reasonable ciency and accuracy, an adaptive dynamic step size is built train intervals (train headway) becomes one of the op- in the conventional FOA. Moreover, the data of urban rail timization tasks for urban rail transit scheduling. Since transit in Zhengzhou was simulated for case study. The scheduling optimization is a NP-complete problem, the comparison results reveal that the proposed AFOA exhibits traditional methods are ineffective to solve it. The nature- faster convergence speed and preferable accuracy than the inspired metaheuristic algorithm is proved to be an effi- conventional FOA, particle swarm optimization, and bacte- cient way of optimizing public transit scheduling. In [1– rial foraging optimization algorithms. Due to these superi- 3], a bus scheduling optimization model is built on the ba- orities, the proposed AFOA is feasible and effective for op- sis of actual passenger flows and solved by an improved timizing the scheduling of urban rail transit. genetic algorithm. Ref. [4] takes into account the number of trains, waiting time and deadheading time to establish Keywords: Urban rail transit; Scheduling model; Train a mixed integer programming model and proposes an im- headway; Fruit fly optimization algorithm proved fruit fly optimization algorithm for the optimiza- PACS: 89.40.Bb, 07.05.Tp tion of regional bus scheduling. In Ref. [5], a modified bac- terial foraging optimization algorithm is used to decide the headways of bus services. However, the factors consid- ered wherein are not comprehensive enough for all prac- tical purposes. Ref. [6] applies the integer programming theory to study electric bus timetables. It has established a network model to optimize the solution with column gen- eration algorithm and improved the efficiency of electric bus timetables. In ref. [7], a scheduling model is estab- *Corresponding Author: Guangyin Xu: College of Mechanical & lished based on the framework of intelligent bus schedul- Electrical Engineering, Henan Agricultural University, Zhengzhou ing system. It uses the hybrid of particle swarm optimiza- 450002, China; Email: [email protected] tion and ant colony optimization to prove the feasibility. Jin Li, Zhengfeng Wang, Zhanwu Wang: College of Mechanical & In ref. [8], genetic algorithm is used to solve the coordi- Electrical Engineering, Henan Agricultural University, Zhengzhou nation optimization model of departure times of differ- 450002, China Open Access. © 2019 J. Li et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License Modeling and optimization of urban rail transit scheduling Ë 889 ent lines. Ref. [9] has developed an application-oriented sales are recorded as company income. The price of model to estimate the waiting times so as to calculate subway ticket for one-way travel is fixed and pas- bus departure time intervals. Ref. [10] presents a mixed- sengers could go anywhere by transferring between integer-programming optimization model for solving the trains. problem of schedule synchronization and minimizing the interchange waiting times of all passengers. Ref. [11] em- ploys genetic algorithm to solve the optimization model of 2.2 Scheduling model train operation scheme during rush hour. Ref. [12] adopts an improved Lagrangian algorithm to solve the schedule The scheduling of urban rail transit is a comprehensive optimization model of train. Ref. [13] uses particle swarm work. The goal of determining the train headway is to effi- optimization algorithm to optimize the operation adjust- ciently use the transport capacity so that lower operating ment model in urban rail transit. Ref. [14] combines linear costs and better passenger experience could be achieved weighting method and genetic algorithm to optimize the at the same time. timetable of urban rail transit. Passengers’ satisfaction depends mainly upon com- At present, intelligent optimization algorithms have muting time and waiting time. As it is assumed that trains been widely used in bus transit scheduling, but few new al- travel at a constant speed and travel time is not considered, gorithms are applied to optimize urban rail transit schedul- waiting time becomes the key factor to ensure passengers’ ing. Based on the real condition of urban rail transit in satisfaction. The less the waiting time, the more passenger Zhengzhou, this paper has proposed an improved schedul- will get satisfied. Since passengers arrive at different times, ing model and solved it with an adaptive fruit fly optimiza- it is difficult to decompose the waiting time of individual tion algorithm (AFOA). The comparison results validate passengers. Therefore, average waiting time is used in this the feasibility and effectiveness of the proposed AFOA. paper. As the passenger flows are evenly distributed, pas- sengers’ waiting time equals to half of the headway time. 1. The satisfaction model of passengers is as follows: 2 Model of urban rail transit K P mk k=1 scheduling fp = Ts (1) (hmax+hmin)/2 2.1 Model hypotheses where fp denotes passengers’ satisfaction; k is the kth period in a day; K is the set for time segments and Urban rail transit system is much different from the bus th k = f1, 2, ...Kg; mk is the departure times in the k transit system. It requires less factors to optimize the train period; Ts is the train service duration in a day; hmax schedule than the bus transit system. Road smoothness, and h are the maximum and minimal departure traffic jam and overtaking station are not considered inthe min interval, respectively. analysis of the train headway for urban rail transit system. 2. Model of enterprise costs Therefore, the results are easier and more accurate. How- The operation costs of an enterprise mainly include ever, to make the model more universal, some external fac- fixed personnel and equipment inputs. The number tors need to be appropriately simplified. The following hy- of trains is the most influential factor to the costs of potheses are proposed: an enterprise. Since there is enough travel capacity 1. The stopover time is fixed; the passenger flow in to carry passengers, the costs of an enterprise will each period is uniformly distributed; the time when be reduced as the number of trains declines. There- passengers get stranded at the station is not consid- fore, the model of enterprise costs is to minimize the ered; and the boarding time and the time from open- number of trains in use. ing to closing the doors are not taken into account; K J P P 2. Trains travel at a constant speed and no accident oc- (mk ρkj ∆tk)∆tk 2 k=1 j=1 curs; fv η = K J (2) 3. Since urban rail transit system is a public trans- P P ukj(hmax + hmin) 2 port infrastructure, its construction costs, revenue k=1 j=1 and expenditure are neglected, and its operating ex- th where fv denotes the enterprise cost, j is the j sta- pense only include fuel and staff wages. Only ticket tion, J is the station set and j = f1, 2, ...Jg, ρkj is 890 Ë J. Li et al. the passenger flow density at the jth station during will decrease, and η equals to the actual passenger th th the k period, ∆tk is the train headway at k period, rate of f1. ukj is the number of passenger arrivals at the station ( th 1, f ≤ 1 j during the k period. η = 1 (5) f f Therefore, the optimization objective is to minimize 1, 1 < 1 ≤ 1.2 the passengers’ waiting time and departure times: 3. Income from ticket sales is set to be higher than op- K erating costs. P mk K J k=1 F αfp βfv α P P min = + = Ts (3) ukj k=1 j=1 (hmax+hmin)/2 f2 = G1 − G2L f2 ≥ 0 (6) K J K P P P m (mk ρkj ∆tk)∆tk 2 k k k=1 j=1 =1 + βη K J where f denotes the revenue of the urban rail com- P P u h h 2 kj( max + min) 2 L k=1 j=1 panies, is the total mileage of lines in operation, G1 is average ticket price and G2 is the average oper- α where is the weighting coefficient of operation ating cost per kilometer.