Complexity

Volume 2017 (2017), Article ID 3717654, 14 pages

https://doi.org/10.1155/2017/3717654

## Application of Multiple-Population Genetic Algorithm in Optimizing the Train-Set Circulation Plan Problem

^{1}Department of Transportation Management Engineering, School of Traffic and Transportation, Beijing Jiaotong University, Beijing, China^{2}Ministry of Education (MOE) Key Laboratory for Urban Transportation Complex System Theory and Technology, School of Traffic and Transportation, Beijing Jiaotong University, Beijing, China^{3}Department of Civil, Environmental, and Infrastructure Engineering, Volgenau School of Engineering, George Mason University, Fairfax, VA, USA^{4}Center for Advanced Transportation System Simulation, Department of Civil Environment Construction Engineering, University of Central Florida, Orlando, FL, USA

Correspondence should be addressed to Leishan Zhou

Received 2 January 2017; Revised 5 April 2017; Accepted 28 May 2017; Published 2 July 2017

Academic Editor: Jose Egea

Copyright © 2017 Yu Zhou 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

The train-set circulation plan problem (TCPP) belongs to the rolling stock scheduling (RSS) problem and is similar to the aircraft routing problem (ARP) in airline operations and the vehicle routing problem (VRP) in the logistics field. However, TCPP involves additional complexity due to the maintenance constraint of train-sets: train-sets must conduct maintenance tasks after running for a certain time and distance. The TCPP is nondeterministic polynomial hard (NP-hard). There is no available algorithm that can obtain the optimal global solution, and many factors such as the utilization mode and the maintenance mode impact the solution of the TCPP. This paper proposes a train-set circulation optimization model to minimize the total connection time and maintenance costs and describes the design of an efficient multiple-population genetic algorithm (MPGA) to solve this model. A realistic high-speed railway (HSR) case is selected to verify our model and algorithm, and, then, a comparison of different algorithms is carried out. Furthermore, a new maintenance mode is proposed, and related implementation requirements are discussed.

#### 1. Introduction

During the last few decades, high-speed railway (HSR) has developed rapidly in China, with a total length of 19,000 kilometers [1], accounting for approximately 60% of all HSR in the world. Due to the characteristics of high-capacity, high efficiency, and low-energy, HSR has been a major transportation mode to satisfy passengers’ travel demands. In HSR, the train-set functions as the means to carry passengers, which is a collection of passenger cars. Generally, the passenger cars can be classified into two types: () self-propelled cars, which means that the engine is installed in the cars and can be treated as power units, and () cars without engines. In China, a train-set is usually comprised of 8 passenger cars, including either 4 or 6 self-propelled cars. Moreover, the length of a train can only be increased by coupling train-sets.

As the carrying tools of HSR, train-sets play a vital role in the HSR management, the utilization efficiency of which greatly influences the HSR operating cost. How to maximize the train-sets’ utilization efficiency has been a research hotspot for many years. On the one hand, due to the high cost of purchasing a train-set, it is crucial to reduce the number of required train-sets, meaning to fulfill the passengers’ travel demands by using as few train-sets as possible. On the other hand, due to the high train-sets’ maintenance expense, how to reduce the times of maintenance has been a great challenge for HSR operating company.

When a timetable is published, train-sets must be assigned to specific routes to satisfy passengers’ travel demands. The train-set circulation plan (TCP) is a technical scheme that guides the train-set assignments, which determines the connecting order and maintenance time of train-sets, as well as the corresponding relationships between train-sets and trip tasks in the timetable. Once the TCP has been formulated, the required number of train-sets and their maintenance times can be obtained. A good TCP can make full use of the available train-sets and fulfill the transportation tasks with fewer train-sets. Thus, it can be said that obtaining a high-quality TCP is also the key to improve the HSR operation efficiency.

However, due to the multiple kinds of train-sets as well as the large number of trip tasks needing to be covered, it is quite challenging from an optimization viewpoint to obtain a high-quality TCP. Besides, when taking maintenance constraints into consideration, the problem becomes much more complicated. Although many researchers have established mathematical programing models and corresponding algorithms to solve this problem, achieving an optimal solution for these models currently appears to be out of reach. This paper addresses the issues involved in formulating a TCP suiting for train-set utilization in China. The contribution of this study lies in the following aspects:(i)*A train-set utilization model is established to optimize the number of required train-sets and their maintenance times in an HSR system*. The goal of this study is to design a high-quality TCP. The objective of the model is to minimize the weighted sum of the number of required train-sets and the maintenance times. Thus, decision makers can determine the number of train-sets they should dispatch, when and where train-sets should be dispatched and be maintained.(ii)*A multiple-population genetic algorithm (MPGA) is designed to solve the train-set circulation plan problem (TCPP)*. MPGA evolves a number of subpopulations in parallel, each of which is connected by an immigration operator. The MPGA’s detection ability and computational efficiency are superior to the standard genetic algorithm (SGA). The connecting order of transportation and maintenance tasks is used to formulate the chromosomes. The reciprocal of the total connecting time is selected as the fitness function. After conducting crossover and mutation processes, children are generated from parents.(iii)*A realistic case study based on the Beijing-Shanghai HSR is carried out to test our model and algorithm. Furthermore, we compared MPGA with other algorithms*. The results show that such an approach is feasible for formulating a good-quality TCP and that the computation speed and solution quality of the MPGA are better than those of traditional algorithms.(iv)*We propose an alternative to the current maintenance practice in China*. While currently each train-set is assigned to a certain Train-set Utilization Base (TUB) and its maintenance must be carried out in a corresponding Depot of Inspection and Repair (DIR) near the TUB, we investigate how beneficial it would be to allow maintenance of train-sets in each DIR. The results show that the utilization efficiency of train-sets can be significantly improved.

The remainder of this study is organized as follows. Section 2 reviews relevant studies in the literature. Section 3 develops a modeling framework for the TCPP, including a problem statement and optimization model construction. Section 4 introduces the MPGA to solve this model. In Section 5, a case study based on realistic HSR is performed, and the comparison with other algorithms is carried out. In Section 6, a new maintenance mode is put forward, and relevant preparation work is discussed. Finally, Section 7 provides conclusions and suggestions for future research directions.

#### 2. Literature Review

Many scholars and researchers have studied the ARP and the VRP with maintenance constraints. Compared with the VRP, the ARP with maintenance is much more similar to the TCPP. The ARP with maintenance is to determine which aircraft should fly which segment and when and where each aircraft should undergo different levels of maintenance checks required by the Federal Aviation Administration [2]. Related studies usually consider type-A checks, and the maintenance standard is approximately 3-4 days. It is assumed that the maintenance can be done only at night [3–5]. The maintenance standard of the TCPP is scheduled by the running time and running distance. However, due to the high utilization efficiency, the running distance usually first reaches the maintenance standard, and the maintenance can be performed during the entire day, in addition to the evening. These factors make the TCPP more flexible, and, therefore, the approaches to solve the ARP with maintenance cannot be directly implemented in the TCPP. However, the approaches to solve the ARP with maintenance have inspired scholars and researchers to solve the TCPP. The solution methodologies for the ARP with maintenance were categorized into three approaches by Liang et al. [6]. The most common approach is to model a sequence of aircraft rotations as connecting flight strings and find the optimal routing by solving a set partitioning problem [7–9]. The second approach is to model the ARP as Euler tour problem or asymmetric traveling salesman problem with side constraints [10–12]. The last approach is to convert the problem into a network flow problem [13, 14]. In related studies, the flight leg can be carried out by any type of aircraft, but, in the TCPP, different trips need different type of train-sets. For example, there are two HSR lines between Beijing and Shanghai. One is the new-built HSR (design speed: 300 km/h) and the other is the rebuilt HSR from the conventional railway (design speed: 200 km/h). If one trip runs on the new HSR, only the CRH3 series train-sets or other higher level train-sets can be used to carry out the trip. If the trip runs in the rebuilt HSR, most of the train-set types can be used. The differences in infrastructures (signal, power supply, train control systems, etc.) between lines make the TCPP slightly different from the ARP with maintenance. However, the idea of modeling and the solution approaches of the ARP have inspired corresponding studies of the TCPP. Zhao et al. [15] modeled the TCPP as the TSP with side constraints similar to the second approach mentioned above. In brief, the VRP and the ARP with maintenance have inspired researchers to study the TCPP, but these approaches cannot be directly used for the TCPP due to the different situations between aircraft utilization and train-set utilization.

The train-set utilization problem has been a hot research topic for several decades and belongs to the field of railway routing and scheduling. Extensive studies have been conducted on this problem worldwide. HSR first developed rapidly in Europe; consequently, many European scholars and researchers have focused on this problem. Schrijver [16] first studied the train-set scheduling problem and proposed a basic model based on the minimum cost flow theory. Then, the model was solved using CPLEX software. His model and solution approaches inspired later researchers. Abbink et al. [17] studied the marshaling of train-sets during morning rush hours; the goal of the model was to minimize shortages of train seats. This study contributed to passenger service during rush hours. Peeters and Kroon [18] developed a model that addressed train-set marshaling and rolling stock utilization. Their work was unique because they applied D-W decomposition and used branch and bound techniques to solve their model, which provided ideas for later studies. Arianna et al. [19] addressed the train-set utilization problem on a single train line and for a single day using an integer programing model to obtain the rolling stock circulation while considering the order of the train units in the compositions. Fioole et al. [20] proposed a mixed integer programing model based on widely adopted previous research achievements and applied an improved branch and bound algorithm to obtain an optimized solution. Together, these studies exploring the train-set utilization problem represent great contributions to this research field. Most of the studies refer to previous locomotive and car assignment problems [21–24]. However, these studies have rarely considered maintenance constraints. In practical utilization, maintenance must be conducted on train-sets after running for a certain time or distance. As train-sets are the main carrying mechanism, carrying out maintenance is one of the most important aspects to ensure HSR operation safety. Moreover, proper maintenance can keep the train-sets in good operation status, which will effectively reduce the possibility of perturbations and disruptions and, thus, contribute to a high level of punctuality and high service availability. Therefore, these previous studies have some limitations that make them difficult to apply in practice.

The research focus later shifted to the train-set utilization problem under maintenance constraints. There are two standards for maintenance, namely, running distances and running time. After a train-set has traveled a certain distance or has been operated for a specific length of time, maintenance must be undertaken to keep the train-set in good condition. When these maintenance constraints are taken into consideration, the train-set utilization problem becomes NP-hard, which has been proven in studies such as Cacchiani et al. [25]. There are no polynomial-time algorithms that can solve NP-hard problems [26]; therefore, the development of efficient algorithms for solving such problems has gained increasing attention. Heuristic algorithms are both suitable and efficient for NP-hard problems; their solutions are usually near-optimal, and their computational time is acceptable. These characteristics have popularized the use of heuristic algorithms for solving the train-set utilization problem. Maróti and Kroon [27] designed an interchange strategy with a shortest-path heuristic algorithm to solve the Netherland Passenger Rail rolling stock problem. This solution was put into practice on the Nederlandse Spoorwegen (NS) lines and turned out to be efficient. Cadarso et al. [28, 29] divided the train-set utilization problem into two subproblems, namely, the train-set assignment problem and the train-set routing problem, and then proposed a heuristic algorithm based on Bender’s decomposition. The RENFE (the main Spanish operator of suburban passenger trains) in Madrid, Spain, was used as a case study to verify the proposed approach. The results of this model, which can be solved in approximately 1 minute, were received positively by RENFE planners. Similar studies have been carried out around the world. For example, Hong et al. [30] formulated a two-stage heuristic algorithm for the Korea Train Express (KTX) case, and Thorlacius et al. [31] proposed a hill climbing heuristic to improve the existing rolling stock plans of the DSB S-tog in Copenhagen, Denmark. It is worth mentioning that the passenger demand constraint of these studies has usually been satisfied by considering maximization of passenger service and minimizing seat shortages. The inputs are passenger demand and train-sets, and the train-sets can be reconnected and decomposed. The papers in European aim to provide services as much as possible to reduce the shortage of seats. Thus, the goals and the constraints of the train-set utilization problem are different between China and Europe.

In contrast, in China’s HSR situation, train-set compositions are usually fixed, containing 8 or 16 passenger cars; consequently, the capacity for each trip is also fixed. The passenger demand is dealt with when working out the train diagram, which provides the train service frequency. In the process of formulating the train-set utilization plan, the inputs are all the trips in the train diagram and the train-sets. Over the past two decades, many scholars and researchers have studied the TCPP in China's HSR situation. Table 1 provides a systematic comparison of the key model components and solution methods in the existing studies in China. Most of these studies took maintenance constraints into consideration, and their objectives focus primarily on minimizing total connection time, minimizing maintenance costs, and balancing the utilization of train-sets. A variety of self-organization heuristic approaches have been designed to solve this problem. Zhao et al. [15] took the TSP as an example and introduced the train-set utilization network. Then, they used a TSP-based heuristic algorithm to solve the model, which turned out to be an efficient solution. Many scholars later tried to optimize the train-set utilization problem based on the approach of Zhao et al. Various types of bionic heuristic algorithms including the GA (Genetic Algorithm) [32], the ACA (Ant Colony Algorithm) [33], the PSO (Particle Swarm Optimization) [34], and an improved ACA [35] have been proposed. Although these algorithms have made great contributions to the problem, they are difficult to apply in practice due to their poor local optimization ability and slow computational speeds. As the scale of the problem increases, these algorithms become time-consuming and easily become stuck in local optimal solutions. In addition, the previous studies only considered one type of train-set, but there are many types of train-sets in practice. Only train-sets of the same type can connect with each other. After reviewing the shortcomings of these algorithms, we realized that the performance could be further improved by adopting more intelligent search mechanisms. The MPGA is based on partitioning a population into several semi-isolated subpopulations. Each subpopulation is associated with an independent GA and explores different promising regions of the search space. Therefore, we can propose a train-set utilization model that considers different types of train-sets. We design an efficient version of MPGA to solve the TCPP, and, then, we compare the MPGA with other algorithms. Finally, a realistic case study was carried out to verify our model and algorithm. The results showed that MPGA is efficient and obtains better solutions than previous approaches.