Journal of Advanced Transportation

Volume 2018, Article ID 7805168, 15 pages

https://doi.org/10.1155/2018/7805168

## Joint Operating Revenue and Passenger Travel Cost Optimization in Urban Rail Transit

^{1}School of Transportation & Logistics, Southwest Jiaotong University, Chengdu 610031, China^{2}National United Engineering Laboratory of Integrated and Intelligent Transportation, Southwest Jiaotong University, Chengdu 610031, China^{3}Department of Civil and Environmental Engineering, University of Waterloo, Canada N2L3G1

Correspondence should be addressed to Chao Wen; nc.utjws@oahcnew

Received 4 July 2018; Revised 16 September 2018; Accepted 25 October 2018; Published 16 December 2018

Academic Editor: Monica Menendez

Copyright © 2018 Wenxin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Urban rail transit (URT) scheduling requires designing efficient timetables that can meet passengers’ expectations about the lower travel cost while attaining revenue management objectives of the train operators. This paper presents a biobjective timetable optimization model that seeks maximizing the operating revenue of the railway company while lowering passengers’ average travel cost. We apply a fuzzy multiobjective optimization and a nondominated sorting genetic algorithm II to solve the optimization problem and characterize the trade-off between the conflicting objective functions under different types of distances. To illustrate the model and solution methodology, the proposed model and solution algorithms are validated against train operation record from a URT line of Chengdu metro in China. The results show that significant improvements can be achieved in terms of the travel cost and revenue return criteria when implementing the solutions obtained by the proposed model.

#### 1. Introduction

Being fast, reliable, safe, and convenient, URT has been able to provide satisfactory trip services and thus mitigate urban traffic congestion in Metropolitan cities, e.g., Tokyo [1], Beijing [2], and New York [3]. In Beijing, for example, seventeen URT lines are currently operating to serve over 10 million passengers each day [4]. Satisfying passengers’ expectation about service quality and operators’ expectations regarding economically viable return have been among the main operational challenges of managing this massive transit mode. These issues have been the focus of many research efforts and many sophisticated solutions, with a special focus on timetable optimization, have been proposed over decades. Since the price of URT tickets and thus the passenger fares do not change for short-term periods, the managerial decisions are restricted and there is more focus on reducing the operating costs. As studies on URT system operations show, train moving operations consumes more than 50% of total electrical energy during train operations [5]. However, reducing energy consumption alone may lead to a timetable with long travel times and thus diminish the service quality. Therefore, an operating timetable should consider both passengers’ point of view and operators’ objectives. In this regard, the New Haven line of the Metro-North Commuter Railroad can be mentioned as a real-world case where minimizing energy consumption in track alignment, speed limit, and schedule adherence objectives are considered to satisfy passengers and operators expectations simultaneously [6]. In another effort, to obtain energy-efficient train operations and distribute the total trip time among different sections, a numerical algorithm is proposed in [7].

Since the impact of train speed on energy consumption is significant, train energy consumption mainly depends on the driving strategies and the operational timetable. Also, any change in the train timetable can affect the operating costs, including passenger travel times (costs) and ticket fares, and thus the quality of transport services. The operating revenue and travel cost are sensitive to the duration of operations defined in the timetable. This necessitates an integrated model to plan train operations such that the operating revenue and travel cost objectives are optimized. In this paper, we present a biobjective timetable optimization model that maximizes the operating revenue and minimizes passengers’ average travel cost, to optimize the (average) operating speed of trains. To solve the optimization problem, we apply a fuzzy multiobjective optimization and a nondominated sorting genetic algorithm II. Next, using a case study from Chengdu metro in China, we perform a numerical analysis to characterize the behavior of the incompatible objectives at different scenarios, to figure out dominating operational strategies for improved and efficient train services.

The rest of this paper is organized as follows. The next section provides a brief review of the relevant literature on train speed and timetable optimization problems. In Section 3, we analyze the passenger boarding behavior and present our biobjective optimization model. In Section 4, we apply a fuzzy multiobjective optimization method and nondominated sorting genetic algorithm II to identify the relationship between different decision variables and objective functions. In Section 5, a case study from Chengdu metro in China is presented to verify the performance of the proposed model. Finally, Section 6 comes with conclusions and directions for future research.

#### 2. Literature Review

To meet the unpredictable and varying operational requirements, timetable rescheduling is the most common practice in URTs. This problem has been tackled from various theoretical and operational perspectives by practitioners. From saving energy perspective, Zhou and Xu proposed a multitrain coordinated operation optimization algorithm that considers both the buffer time and safety constraints [8]. Miyatake and Ko used three different methods, including dynamic programming, gradient method, and sequential quadratic programming, to solve the URT timetable rescheduling problem for URT operations by optimizing energy consumption [9]. Some studies have also investigated improving multiple factors such as train movement profile, passenger comfort, safety, and operation stability. For instance, Su et al. considered timetable optimization and speed profiles among successive stations for energy-efficient and optimized train operations [7]. A stochastic optimization model is proposed in [10] that redistributes the time supplements and buffer times in a given timetable, to improve the safety and operation stability of URT system. Assis and Milani analyzed the evolution of train headways and train passenger loads along URT lines and presented a methodology to optimize train timetables in URT lines [11].

Regarding train operation costs and total passenger travel time, Ghoseiri et al. and Chang et al. developed multiobjective optimization models for railways, to minimize fuel costs and total travel times [12, 13]. Li et al. proposed a multiobjective train scheduling model by minimizing the train energy and carbon emission costs as well as the total travel time of passengers [14]. They applied a fuzzy multiobjective optimization algorithm to solve the model. Corman et al. proposed an optimization model to generate timetables and to effectively manage the traffic in real-time, which illustrated the effects of changing trains’ speed profile in open corridors [15]. Chevrier et al. introduced the speed profiles found by the evolutionary algorithm produced by a set of solutions optimizing both the running time and energy consumption, which can be used to optimize the running of the trains [16].

The literature listed above mainly has focused on the long-distance railways. In the URT system, there are similar models to deal with the fuel costs reduction and travel-time saving. A biobjective integer programming model with headway time and dwell time control and a genetic algorithm with binary encoding to find the optimal solution were conducted in [17]. The idea of optimizing metro train speed profiles was also applied to reduce energy consumption [18]. A bilevel train scheduling optimization is proposed in [19] that takes into account stop-skipping strategies and the passenger travel time, and energy consumption gave the origin-destination-dependent passenger demand.

To improve the quality of metro service and reduce passenger costs, demand-sensitive orientation timetable models were presented in [20]. Moreover, a bilevel demand-oriented approach was applied to obtain a timetable for a suburban railway [21]. Considering the passenger demands, transfers, and passenger flow splitting, an event-driven train scheduling model for a URT network was proposed in [22] that concludes the nonfixed headway train schedules have a better performance.

Though some researchers have focused on the train timetable scheduling problem, few of them considered the timetable scheduling problem from an operational efficiency perspective and the concerns about the lower travel cost from passengers’ side. This paper tries to fill this gap, proposing a multiobjective optimization model considering operating revenue and travel costs simultaneously.

#### 3. Biobjective Optimization Model

##### 3.1. Train Energy Consumption and Speed Analysis

Accelerating, coasting, and braking are the main phases of a train movement when performing running activities between successive track sections [23]. One can ignore small variations in train speed as the preventive maintenance of infrastructures and facilities keeps the operational conditions at the required level. With this simplifying assumption, train motion formula can be described as shown in (1). In this equation, we consider the basic line resistance, the track gauge, the maximum speed, and the signal systems, for instance, but we do not consider the curve and tunnel resistances. where is the acceleration of train operation, is the speed of train operation, is the time of train operation, is the maximum unit traction of the train, is the coefficient of the traction output ratio, and is the unit basic resistance of the train; it indicates the basic resistance of trains under unit weight, N/kN.

Using this formula, when a train runs in a section, the relationship between energy consumption and the maximum speed can be derived as follows:where is the optimal maximum speed of the train in section ; it is not equal to the maximum speed limit of the train in section ; the maximum speed limit (the maximum speed limit of subway trains is equal to 80 km/h in China); is the weight of the train. Equation (3) is another expression proposed by Gu et al., to calculate the energy consumption of a train in a section from the perspective of traction work [24].More simplified formulations can be derived from (3) as follows: there is , because, in practice, is much smaller than and , when the train runs at the accelerating phase in a section [24]. Therefore, (2) and (4) can be reformulated into (5), given .

In (4) and (5), is the running distance of the train at the accelerating phase in section . We know that , where is the average running speed of the train in section ; is the quantity of the state that satisfies . It refers to the time period when the speed of the train increases from to . After some calculation and simplification, (5) can be further developed into

Owing to the theoretical derivation of the mathematical formula, we assume the correlation coefficient between and , to be

We approximate the value of using train movement data in track sections, including the optimal maximum speed, average operating speed, and the relationship between the optimal maximum speed and the average operating speed [23]. The instance is shown in Figure 1, in which represents the average operating speed while represents the maximum operating speed and the correlation value can be calculated by (8). Given this, (2) can be transformed into (9). Therefore, we get the relation between average speed and energy consumption, but this relation has the range of validity as follows.(1)The object of the study must be urban rail transit system; the maximum speed of trains should not exceed 80km/h. And considering the comfort of passengers, the maximum train acceleration is restricted to 1 m/s^{2,} and the maximum train deceleration is constrained as -1 m/s^{2}.(2)The distance between the two stations is very short (the distance between the two stations should be less than 3 kilometers in principle); the train running in the section only includes three phases: accelerating, coasting, and braking, without cruising phase.