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As a sustainable transportation mode, high-speed railway (HSR) has been developing rapidly during the past decade in China. With the formation of dense HSR network, how to improve the utilization efficiency of train-sets (the carrying tools of HSR) has been a new research hotspot. Moreover, the emergence of railway transportation hubs has brought great challenges to the traditional train-sets’ utilization mode. Thus, in this paper, we address the issue of train-sets’ utilization problem with the consideration of railway transportation hubs, which consists of finding an optimal Train-set Circulation Plan (TCP) to complete trip tasks in a given Train Diagram (TD). An integer programming TCP model is established to optimize the train-set utilization scheme, aiming to obtain the one-to-one correspondence relationship among sets of train-sets, trip tasks, and maintenances. A genetic algorithm (GA) is designed to solve the model. A case study based on Nanjing and Shanghai HSR transportation hubs is made to demonstrate the practical significance of the proposed method. The results show that a more efficient TCP can be formulated by introducing train-sets being dispatched among different stations in the same hub.

During the past decade, great developments have been achieved in the high-speed railway (HSR) in China. Being acknowledged as high efficiency, high capacity, and low energy, HSR has been one of the most influential travel modes [

Thus, this paper addresses the issues of the TCP problem with the consideration of railway transportation hubs and maintenance requirements, which is one of the most significant aspects in the railway operation and management studies. Roughly speaking, it consists of finding the optimal assignment of train-sets so as to complete a set of trip tasks in the given TD with higher utilization efficiency, i.e., with less number of train-sets, and considering the maintenance requirements. The contribution of this study lies in the following aspects.

Firstly, an innovative train-set utilization mode is put forward to improve the train-sets’ utilization efficiency. In previous studies, train-sets can only undertake two adjacent trips when the departure station of the latter trip is the same with the arriving station of the former trip. In this case, when the time interval between the connected two trip tasks is long, train-sets must wait at the station for undertaking the next trip. Obviously, it will cause huge waste of the train-set capacity. In this paper, we propose an innovative train-set utilization mode that train-sets can be dispatched among different stations in the same railway transportation hub. Thus, a train-set can be dispatched to other stations to undertake trip tasks, instead of waiting a long time at its arriving station for the next trip task. Clearly, by relieving the constraint that the departure station of latter trip must be the same with the arriving station of the former trip, the flexibility and the efficiency of train-set utilization can be significantly enhanced. This utilization mode turns out to be more suitable to be applied in the dense HSR network.

Secondly, an integer programming TCP model is proposed to optimize the train-set utilization scheme with the consideration of maintenance requirements. Accumulated variables have been introduced to represent the running distance and running time of train-sets. The model aims to obtain an optimal TCP that determines the one-to-one correspondence relationship among sets of train-sets, trip tasks, and maintenances. Thus, the dispatcher can determine each trip should be undertaken by which train-set, the sequence of the trips undertaking by the same train-set, and when and where each train-set should be maintained. The objective of the model is set to simultaneously minimize the number of using train-sets and the total maintenances costs.

Thirdly, a genetic algorithm (GA) is designed to solve the TCP optimization model. We use an innovative representation method to formulate the relationship between maintenance arcs and normal connection arcs. Furthermore, effective crossover and mutation processes have been designed. The TCP problem has been proved to be a traditional NP-hard problem, which cannot be solved efficiently or directly by ready-made software, especially in large-scale cases. Thus, in the process of model solution, the GA is applied to search for a near-optimal TCP. The results show that the proposed GA for TCP problem has obvious advantages over ant colony algorithm (ACA) and simulated annealing (SA) in the solution quality, and good performance can be found in both computational efficiency and stability.

Finally, a case study based on Nanjing and Shanghai HSR transportation hubs is carried out to demonstrate the practical significance of the proposed method. The results show that, by introducing train-sets being dispatched among different stations in the same hub, a more efficient TCP can be formulated compared with traditional utilization mode. To complete the same trip tasks, both the number of needing train-sets and the maintenance times can be reduced. Thus, such a mode is a feasible method to utilize train-sets with lower costs and high efficiency.

The remainder of this paper is organized as follows. Section

The rolling stock planning (RSP) problem is one of the most significant aspects in the application and assignment problem of transportation vehicles. How to improve the utilization efficiency of transportation vehicles has always been the research hotspot in transportation management, including not only RSP in railway management, but also Aircraft Routing Problem (ARP) in the airline operation, Vehicle Routing Problem (VRP) in the logistics filed, and so on. Specifically, extensive research on RSP problem has been carried out worldwide, leading to the development of various rolling stock utilization models and techniques. Interested readers can refer to Cacchiani et al. [

Two main categories of RSP problem have been studied during different development stages of passenger railways. Before the emergence of HSR, the transportation vehicles used in the railways are carriages and locomotives. The former is a kind of vehicle without traction engines, which can be coupled individually and independently to a convoy. The latter is a necessary part of trains which can provide traction power and pull the convoy [

Then, with the rapid development of HSR, matched railway vehicles named train-sets have been introduced. Different from above traditional railway vehicles, the train-set is a kind of self-contained trains with an engine and passenger seats, which means self-propelled train units that are not required to be pulled by a locomotive. These units consist of a fixed number of carriages and have their own traction engines [

To ensure the operation safety of train-sets, maintenances should be carried out as long as a train-set has been utilized for certain time periods or accumulated running distances. Extensive research has focused on the rolling stock problem considering maintenance constraints. Maróti G and Kroon L are the pioneers in exploring the maintenance routing problem in railway management. A transition model was put forward [

Another important issue which should be addressed in the train-set utilization procedure is how to estimate the time for two consecutive trips in terminal stations. In the operational stage, possible delays or fluctuations, which can occur during operations, will influence the connection between two trips. So compensating possible delays with the planned timetable is of great importance. Many scholars put forward approaches to estimate time rates involved in the preparation phase between two successive trips. D’Acierno et al. [

Recently, more and more scholars have shifted their attentions to the combination optimization of train-set utilization problem with other problems related to railway management. They argued that the train-set utilization problem should be simultaneously optimized with service demand prediction problem, timetabling problem, crew scheduling problem, etc. Wang et al. [

Although a comprehensive body of literature on RSP is available, there still exist limitations and gaps in the aspect of TCP problem. Firstly, most of existing researches on TCP are suitable for the train-set utilization in Europe, which is quite different from that in China. In Europe, scholars exploring the TCP problem mainly focused on train-sets’ coupling and uncoupling so as to form trains to carry out trip tasks in a given TD. They sought to minimize the seat shortages as well as the empty train movements. But, in China, train-sets are fixed composition containing 8 train-sets or 16 train-sets, which are utilized as a whole. Thus, there is no need to consider the problem of train-sets’ coupling and uncoupling in China. Secondly, the approach proposed in previous studies cannot be applied directly in this paper while the train-set utilization modes are quite different. Most scenarios of existing research on TCP are set under the condition that train-sets’ departure station of the next trip should be the same with the arrive station of the former trip, while no studies are available for the utilization mode that train-sets can be dispatched among different stations in the same railway transportation hub. The former can obtain a feasible TCP only when the HSR network has not been formed and the distances between railway stations are a bit long. But, in China, a complex HSR network has been formed and the distribution density of railway stations is quite high, while there could be several railway stations in some metropolis. As thus, this study makes a new attempt about the issue on TCP.

Figure

The outline of the TCP problem.

As previously mentioned, the expanded of HSR network and the construction of HSR transportation hubs have caused enormous complexity of train-set application, leading to a challenge for developing the optimal solution of algorithms. Compared with the traditional utilization mode that train-sets can only departure from stations they arrive, the train-set utilization efficiency can be considerably improved by introducing dispatching train-sets among different stations in the same transportation hub. Figure

Topological structure of sample.

In the example, it is assumed that there are three trips needing to be undertaken, the basic information of which is as listed in Table

Basic information of the sample.

Departure Station | Arriving Station | Departure Time | Arriving Time | |
---|---|---|---|---|

Trip | Station A | Station D | 20:00 | 23:00 |

Trip | Station D | Station A | 12:00 | 15:00 |

Trip | Station D | Station B | 15:00 | 19:00 |

Several assumptions are made throughout the paper for simplicity of the model and are explained as follows:

The TD of HSR are drawn up in pairs, which means that the number of arrival trains is equal to the number of departure trains in each station.

We assume that only one type of train-set is taken into consideration.

In practical operation, five levels of train-set maintenance standard are set to ensure the transportation safety according to train-sets’ accumulated travelling time and distances. But, in this paper, only maintenance of Level One is taken into consideration. It is because of that, the TCP is usually drawn up by treating one day or several days as a cycle, which is much shorter than the travelling time standard of maintenance of Level Two (a week) to Level Five (several years). Moreover, when train-sets’ accumulated travelling time and distances reach to the standard of maintenance of Level Two or above, a specific train-set maintenance plan will be formulated.

We only consider a daily operation timetable; it means that means that the train timetable is the same every day.

The time for passengers alighting and boarding at stations is determined.

The following symbols are used in this paper:

The optimization model is based on a weighted directed graph

When a TD is given, lots of different TCPs can be formulated to complete all the trip tasks. To obtain an optimal TCP, an objective needs to be proposed to improve the train-set utilization efficiency as much as possible on the premise of satisfying constraints. For a TCP, the number of needing train-sets is a crucial indicator to measure the quality of the plan. Planners always seek to complete the same number of trip tasks in the given TD with less train-sets. Moreover, the number of maintenance is also an important factor influencing the train-set utilization efficiency. When train-sets are overhauled, lots of time is spent on train-sets running between railway stations and inspection and repair depots, which will reduce the effective working time of train-sets. Thus, in this paper, both the number of using train-sets and the total number of maintenances are considered into the objective function. To deal with different situations in practical operation and expand the model’s scope of application, variables _{1} and _{2} are introduced as weight values to the number of using train-sets and the number of maintenances, respectively. Moreover, the sum of _{1} and _{2} is equal to one.

The TCP problem is a traditional NP-hard problem. Due to the advantages over obtaining a feasible solution within a reasonable amount of time, intelligent algorithms are always used to solve this problem. Genetic algorithm (GA) has been proved that it can efficiently and effectively solve a variety of real-world issues, e.g., train timetabling problems, timetable rescheduling problems, and network design optimization problems. Thus, in this paper, a GA is designed to search for an optimal TCP. In general, the procedure of GA can be described as follows.

Coding and solution expression of GA chromosomes.

Generally, the initial population is generated randomly, allowing the entire range of possible solutions.

In this paper, the crossover process can be demonstrated as Figure

Crossover procedure.

Mutation procedure.

The processes of crossover and mutation are two important operations in GA, which decide the quality of solutions.

To evaluate the proposed model and algorithm, we used the HSR between Shanghai transportation hub and Nanjing transportation hub as a case study. Figure

Trips between different stations.

Nanjing Station | Nanjing South Station | |
---|---|---|

Shanghai Station | 35 pairs | 3 pairs |

Hongqiao Station | 14 pairs | 4 pairs |

Topological structure of case study.

The parameters related to the TCP are set as follows:

Minimum time for preparation work in station

Travelling distance standard of Level One maintenance:

Travelling time standard of Level One maintenance:

Fluctuation coefficient:

Running time from station

Basic coefficients for GA: the population size is set as 80; the probability of crossover is set as 0.6; the probability of mutation is set as 0.01; and the maximum iteration times is set as 50.

Running time between different stations in the same hub.

Departure Station | Arriving Station | Running Time |
---|---|---|

Shanghai Station | Hongqiao Station | 17 minutes |

Hongqiao Station | Shanghai Station | 18 minutes |

Nanjing Station | Nanjing South Station | 23 minutes |

Nanjing South Station | Nanjing Station | 23 minutes |

Based on the above information, we tested our proposed model and algorithm in the PC (WIN7, Intel Core i7-4779, 3.40GHz, and 16GRAM). Comparison experiments were conducted to evaluate whether introducing dispatching train-sets between different stations in the same hub can improve the train-set utilization efficiency. We tested 20 times for both conditions, and the best computational results are listed in Table

Computational results between two different utilization modes.

Scenario | Train-set Amount | Maintenance Amount | Train-set Utilization Efficiency |
---|---|---|---|

Dispatching train-sets between different stations | 24 (25.3) | 10 (11.5) | 51.9% (45.1%) |

Not dispatching train-sets between different stations | 26 (28.4) | 10 (12.1) | 47.9% (39.6%) |

From Table

To further illustrate the efficiency of the proposed approach, a convergence test of objective values is conducted. To reach the best found solution for the objective function, 30 iterations were performed. As shown in Figure

Convergence curve.

In order to highlight the performance of this paper in improving the train-set utilization efficiency, we compared our algorithm with ACA [

GA can obtain a better TCP than ACA and SA. After 30 times computations, 25.3 train-sets are needed on average to finish the tasks, which are less than the results of ACA (28.8) and SA (30.1). The average train-set utilization efficiency by GA is much higher than that of ACA and SA, which means that the TCP obtained by GA can save more costs and serve more passengers with the same amount train-sets.

The computational time of GA is about 8 seconds, which is similar to SA (8 seconds) but longer than ACA (4 seconds). The reason is that the operations of crossover and mutation are time-consuming.

Figure

Results of different algorithms.

Average Value | GA | ACA | SA |
---|---|---|---|

Train-sets Amount | 25.3 | 28.8 | 30.1 |

Average Maintenance Amount | 11.5 | 13.2 | 14.5 |

Efficiency | 45.1% | 39.4% | 35.2% |

Computational time (millisecond) | 8321.5 | 4522.2 | 8421.5 |

Comparison of convergence curves.

Based on the above analyses, it can be found that the GA for TCP has obvious advantages in the solution quality, and its computational efficiency and stability also have good performances. Thus, it can be said that the method proposed in this paper is a new way to utilizing train-sets with lower costs and high efficiency.

With the rapid development of HSR in China, the TCP problem, as a fundamental and vital part of railway management, has been receiving increasing attention. How to efficiently utilize the train-sets in the complex HSR network becomes a hotspot. With the emergence of the transportation hub in China, the higher challenge is faced by all corresponding scholars and operators. To cope with this new development trend, this paper aims to put forward a novel utilization mode to enhance the utilization efficiency of the train-set. We modified the traditional train-sets’ utilization mode that train-sets can only undertake trips that the departure stations are the same with the arriving stations of the trips the train-sets have just completed. Instead, an innovative train-set utilization mode is put forward that train-sets can be dispatched among railway stations in the same hub. We formulate an integer programming TCP model with the objective of simultaneously minimizing the number of using train-sets and the total maintenances times. In order to deal with the complicated maintenance constraints, accumulated variables have been introduced to represent the running distance and running time of the train-set. To obtain the optimal TCP, a genetic algorithm (GA) is designed. What makes the GA more unique is that our expression of the solution which considers the maintenance arc and connection arc and the crossover process makes good use of characteristics of maintenance constraints. In order to deal with the complicated maintenance constraints, accumulated variables have been introduced to represent the running distance and running time of the train-set. To obtain the optimal TCP, a genetic algorithm (GA) is designed. What makes the GA unique is our expression of the solution. The proposed GA considers the maintenance arc and connection arc, and the crossover process makes good use of characteristics of maintenance constraints. In order to verify the efficiency of the proposed utilization mode, numerical experiments and contrast experiments have been carried out based on the real data of Nanjing and Shanghai HSR transportation hubs. The results show that a high-quality train-set utilization scheme can be obtained. It can guide dispatchers to determine each trip should be undertaken by which train-set, the sequence of the trips undertaking by the same train-set, and when and where each train-set should be maintained. Also, the flexibility and the efficiency of train-set utilization can be significantly enhanced, even though it may cause empty running. Additionally, the designed GA has obvious advantages over ACA and SA in both the solution quality and computational efficiency. Further research work is recommended in twofold: one is to take the fluctuation and perturbation in the real world into consideration when formulating the TCP problem and study the rescheduling problem of train-set utilization. The other is to consider the impacts of passengers flow on the TCP problem, which is a vital factor influencing the dwell time and travel time in the TCP problem.

The data used to support the findings of this study are available from the corresponding author upon request.

Authors declare that they have no conflicts of interest regarding the publication of this paper.

This work is financially supported by the National Natural Science Foundation of China (91746201; 71621001).