Model and algorithm for the first-level maintenance operation optimization of EMU trains

Abstract

As a key operation for the daily maintenance of electric multiple units (EMU), the first-level maintenance operation directly affects the utilization efficiency of the EMU. The fixed operation sequence of EMU trains, the limitation of the track capacity and inconsistent arrival time of EMU trains give rise to such problems as extended waiting time, idle tracks and waste of maintenance capacity. To solve these problems and optimize the assignment of EMU-to-track, we propose a flexible job-shop sequence scheduling (Flexible-JSS) mode for the first-level maintenance of EMU trains, and a flexible sequence and tracks sharing (FSTS) model for the first-level maintenance at electric multiple units depot (EMUD) has also been proposed in this paper. The FSTS model is designed to shorten the latest completion time after taking into account the constraints such as the train length, track capacity, the operation sequence of all EMU trains, the operation process of a single EMU train, and the train-set scheduling plan. A modified genetic algorithm is used to solve the model. The feasibility and effectiveness of the model and algorithm are verified by a real case, and the comparison with the other two fixed job-shop sequence scheduling (Fixed-JSS) modes proves that the Flexible-JSS mode can improve the efficiency and ability of the first-level maintenance at EMUD impressively.

Keywords

First-level maintenance operation flexible job-shop sequence scheduling mode flexible sequence and tracks sharing (FSTS) model modified genetic algorithm the latest completion time

1 Introduction

By the end of 2020, the mileage of China’s High-Speed Railway (CHSR) has reached 37500 kilometers, ranking first in the world. With the continuous construction and opening of railway lines, higher service requirements and more EMU trains are needed to meet the growing passenger demand. Beyond that, as more and more EMU trains need to be maintained, the maintenance capability of EMUD has gradually become one of the main factors affecting the train-set scheduling plan. In China, there are two main types of EMU trains in China: short EMU train with 8 cars and long EMU train with 16 cars, as shown in Fig. 1. The maintenance of EMU trains is divided into five levels according to their running time and mileage. The first-level maintenance is a key link of daily maintenance of EMU trains, which mainly includes external washing and maintenance of basic equipment. And its efficiency is related to the effective utilization of transportation resources, which has a great impact on the use of EMU trains and the efficiency of transportation production. Thus, it has become an urgent scientific problem to design an efficient first-level maintenance plan for EMU trains.

Fig. 1

Two types of China’s EMU trains.

The EMU trains usually return to the EMUD at night for routine maintenance. Due to the different transportation tasks they perform, the time distribution of the EMU trains returning to the EMUD is uneven, and a large number of EMU trains may arrive intensively. The limited equipment resources of EMUD do not have the conditions to meet all the EMU trains to be operated immediately when they arrive in a concentrated manner. The above factors will lead to the phenomenon of uneven busyness and low utilization of equipment, and may even cause the EMU trains to fail to put into use on time the next day due to unreasonable operation arrangements.

The purpose of optimizing the first-level maintenance operation should be to improve the utilization rate and reduce the operation time on the basis of ensuring safety and quality. For the first-level maintenance operation of EMU trains, the sequence of the EMU trains, the selection of operating track, and the sequence of the operating processes of single EMU train are not fixed. Therefore, the optimization of the first-level maintenance operation is a combination optimization problem, and the purpose is to improve the efficiency of the first-level maintenance of EMUD by optimizing the arrangement of EMU-to-track.

The maintenance plan of EMU trains has an important impact on the efficiency of the EMUD and the utilization of EMU trains. Therefore, many scholars have conducted extensive research on the optimization of EMU maintenance schedule.

Compared with the maintenance plan of EMU trains, the research of aircraft hangar maintenance scheduling problems is earlier and more abundant. Regardless of model construction or algorithm design, the research on aircraft maintenance scheduling is relatively in-depth, and there are many valuable research results. In [1] and [2], an integrated aircraft maintenance scheduling and hangar layout planning problem was studied. In addition to the aircraft maintenance scheduling, the variation of parking capacity and blocking during aircraft movements was studied in-depth. In [3], a mixed-integer linear programming formulation was developed for aircraft scheduling and parking problems with the goal of reducing the total maintenance time. The maintenance plan of EMU trains has similarities with aircraft maintenance, but also has some unique points. The research of the EMU maintenance plan focuses on the optimization of the EMU-to-track arrangement. Hansmann et al. [4] first proposed the sorting of rolling stock problem at EMUD, and classified the problems by setting several parameters such as track design, track capacity, yard type, decomposition timing, and flexible or order. Most other scholars conducted in-depth research by adjusting parameters on the basis of this research. Wang Z et al. [5] first studied the integrative scheduling of EMU utilization plan and maintenance plan. On this basis, Wang Z et al. [6] further studied the optimization of the maintenance plan at the EMUD, and transformed the problem into a job-shop scheduling problem (JSP) with additional space and time constrains. Sun P et al. [7] built a Multi-project RCPSP model for EMU Overhaul Problem. A refined particle swarm optimization (PSO) is proposed through the design of the non-linear descending inertial weighting factor. Wang J et al. [8, 9] established an integer programming model on the basis of Wang Z with the goal of minimizing the number of shunting movements, and solved it with particle swarm optimization (PSO)-based optimization algorithm. However, the track capacity and EMU length attributes have not been considered in Wang J’s research, as the route conflicts and operation sequence take into the main considerations. Guo X et al. [10] established an optimized model for the shunting operation plan with the aim of minimizing the total delay time of EMU, while they did not consider the length of EMU and the section of track. Li H et al. [11, 12] focused on the impact of different types of depots on the EMU-to-track assignment. Li H et al. [11] optimized the shunting operations for two types of EMUDs. And in [12], the length of the EMU and the section attribute of the track were considered simultaneously to optimize the EMU shunting operations. Most of the studies on the EMU maintenance plan did not consider parameters together, such as the length of EMU trains, the section of the track, the operation sequence, and the train-set scheduling plan. In Table 1, some comparisons between the studies on the maintenance plan of EMU trains are listed.

Table 1

Comparison between the studies on the maintenance plan of EMU trains

Study	Type of EMUD yard	Length of EMU trains	operation sequence	Track sections	Train-set scheduling plan
Wang Z2013	Through	single	Fixed	one	Considered
Sun P2013	Through	single	fixed	one	Not Considered
Wang J2016	Through	single	flexible	one	Considered
Guo X2016	Through	single	flexible	one	Considered
Li H2019	Through &Stub-end	Variable	flexible	two	Not Considered
Tong J2016	Through	Variable	fixed	two	Considered
This study	Through	Variable	flexible	two	Considered

Wang Z et al. [6] referred to the JSP when they first studied the problem of EMU shunting operations. The shunting operation of the EMU is similar to the arrangement of the partial flexible job-shop scheduling problem, both of which determine the order of operations and the machines to be selected. Tong J [13, 14] studied the shunting operation problem of EMUD with fixed operation flow. They turned the problem into a hybrid flow-shop scheduling problem (HFJSP) with special process constraints. Haahr et al. [15] compared different models and solution approaches to solve the problem of EMU shunting and parking problem. The heuristic algorithm proved to be able to solve the current case in a very short time. And heuristic algorithms such as genetic algorithms are often used to solve JSP. Turkyilmaz A et al. [16] presented the forms of scrutiny of multi-objective FJSPs, and various heuristic techniques were used to solve problems in the last decade. Pezzella F et al. [17] proposed a genetic algorithm that integrates more strategies into the genetic framework in order to solve the flexible job-shop scheduling problem (FJSP). And the effectiveness of the genetic algorithm for solving FJSP is proved by comparing with the famous algorithm based on tabu search. Aghajani A et al. [18] designed a non-dominated sorting genetic algorithm (NSGAII) to find its Pareto optimal solution for the JSP considering the dual objective of minimum total completion time and the reliability of system. Fan K et al. [19] addressed the multiprocessor job-shop scheduling problem (MTJSP), constructed the optimization model with disjunctive formulation, and developed a Scatter Search (SS) based on the solution representation to solve the problem. There are also some differences between the EMU shunting operation and the FJSP. For example, each job in the FJSP has a fixed process, and a machine can only perform one job at a time. In the EMU shunting operation, the process of washing and maintenance is flexible, and one track can perform maintenance operations for two short EMU trains at the same time. That is, one machine can perform two operations at the same time. Therefore, the problem of EMU shunting operation cannot be directly transformed into a problem of flexible workshop, and some special treatment and design are required.

As shown in Table 1, there are few studies on the first-level maintenance of EMU trains that consider the two sections of tracks, the flexible operation sequence of single EMU train and the EMU train-set scheduling plan. Therefore, the contributions of this paper to the first-level maintenance problem are as follows.

A flexible sequence and tracks sharing (FSTS) model was proposed to solve the first-level maintenance schedule optimization problem. Different from the previous studies, the sequence of operation can be scheduled flexibly, in which the first-level maintenance can be completed earlier. Meanwhile, the sharing maintenance of long tracks is considered, in which two short EMU trains can occupy the long track at the same time.

To construct the FSTS model, a parameter representing the length of EMU trains is proposed to build the relationship between the EMU trains and the tracks, in which the track sharing operation can be expressed by the mathematical formulations. Meanwhile, the earliest maintenance completion time is considered as the objective of FSTS model, and the constraints of the maintenance and washing duration and sequence, the train-set scheduling plan, and so on are mentioned. What’s more, a modified generical algorithm is proposed to solve the FSTS model.

A real-world case study of the EMUD in Tai Yuan China is introduced to verify the feasibility of the proposed model. Meanwhile, two other fixed-JSS modes are tested and compared with the proposed flexible-JSS mode. The results indicate that the flexible-JSS mode outperforms the other modes on both aspects of time-saving and track utilization.

The remainder of this paper is organized as follows. Section 2 gives a formal problem description for the first-level maintenance operation with flexible-JSS mode. The FSTS model is proposed to mathematically describe the first-level maintenance operation with flexible-JSS mode in Section 3. Section 4 develops a solution based on the modified genetic algorithm. In Section 5, a case of CHSR is selected to verify the proposed method, and two Fixed-JSS modes have also been compared and analyzed in this section. Finally, the conclusions are drawn in Section 6.

2 Problem statement

2.1 Problem description

The first-level maintenance mainly includes two operations, washing and maintenance, carried out on the washing yard and the maintenance yard, respectively. After these two operations, the EMU trains are sent to the storage yard waiting for the service online. This problem can be described as a process in which some EMU trains enter the EMUD to perform washing operations in several washing tracks and perform maintenance operations in several maintenance tracks. The EMU trains enter the EMUD at night after completing the transportation tasks during the day, and will be put into use next day after completing the first-level maintenance operation. The time and sequence of EMU trains entering the EMUD are fixed, but the sequence of the first-level maintenance operations is not fixed. Each yard has several operation tracks, and each EMU can complete the corresponding washing or maintenance operation on any selectable track. The number of EMU trains that can be operated on each track at the same time is determined by the length of the EMU and the capacity of track. The goal of the first-level maintenance operation plan is to determine the best EMU-to-track solution and the start time of each operation, so that the entire first-level maintenance operation takes the shortest time. For any EMU train, washing and maintenance are actually independent of each other, and the sequence of the two operations is not mandatory. Therefore, on the premise of considering the section attributes of the track and the length attributes of EMU trains, this paper proposes a flexible-JSS mode for the first-level maintenance operation. After the train enters the depot, the train no longer follows the “washing-maintenance” fixed job-shop sequence, but can choose washing or maintenance operation. After completing one operation, it will go to another yard to perform the corresponding operation. With all the operations done, it will wait in the storage yard to be put into use, as shown in Fig. 2.

Fig. 2

The flow chart of the first-level maintenance with flexible-JSS mode.

Therefore, the first-level maintenance operation optimization can be described as follows. Under the premise of knowing the attribute of tracks, the number of EMU trains, the start and end time requirements of the operations, and the duration of each operation, the first-level maintenance of all EMU trains can be completed in the shortest time by reasonably arranging the EMU-to-track and the operation sequence. In fact, it is difficult for a dispatcher to foresee the implications of a decision, and it is almost impossible to get the best schedule in a short time. Depending on the dispatcher, the same maintenance task of EMU trains may receive different schedules. Typically, manually produced schedules will have some time and track capacity waste. In this paper, we aim to determine the optimal arrangement of the first-level maintenance operation based on the FSTS model.

2.2 Assumptions

In order to facilitate the construction of the mathematical model, the following assumptions are made.

Assumption 1. As the EMU trains need to be powered off during washing, and the washing operation takes a short time. One washing track can be used to wash one train (regardless of the length of EMU trains) at the same time, as shown in Fig. 3. As the EMU trains do not interfere with each other during the maintenance operation, and the maintenance operation takes a long time. One maintenance track can accommodate one long EMU train or two short EMU trains at the same time (Shi et al. (2019)), as shown in Fig. 4.

Fig. 3

Washing yard.

Fig. 4

Maintenance yard.

Assumption 2. Each yard is a through-type yard, both ends of the throat area can be in and out of EMU trains without cross interference, as shown in Fig. 5.

Fig. 5

Through-type yard.

Assumption 3. Due to the power of the EMU train itself, there is no need for shunting locomotive, so the cost of the shunting operation is not considered.

2.3 List of notations

The notations used in the model are introduced in Tables 2 and 3.

Table 2
Definition of sets, indices and parameters

Notation Description

N Set of EMU trains, N = {i|i = 1, 2, . . . n}

W Set of washing tracks, W = {j|j = 1, 2, . . . w}

M Set of maintenance tracks, M = {k|k = 1, 2, . . . m}

T ^wash The time needed for washing operation

T ^maint The time needed for maintenance operation

T ^turn The time it takes for the EMU train to switch yard

R A real number big enough

$T_{i}^{enter}$ The moment when the ith train enters the depot

$T_{i}^{out}$ The moment when the ith train is put into use the next day

y _i The length attributes of the train

Notation	Description
N	Set of EMU trains, N = {i\|i = 1, 2, . . . n}
W	Set of washing tracks, W = {j\|j = 1, 2, . . . w}
M	Set of maintenance tracks, M = {k\|k = 1, 2, . . . m}
T ^wash	The time needed for washing operation
T ^maint	The time needed for maintenance operation
T ^turn	The time it takes for the EMU train to switch yard
R	A real number big enough
$T_{i}^{enter}$	The moment when the ith train enters the depot
$T_{i}^{out}$	The moment when the ith train is put into use the next day
y _i	The length attributes of the train

Table 3

List of variables

Variables	Description
$x_{i, j}^{wash}$	Binary decision variable indicating Whether the ith train completes the washing operation on the jth washing track.
$x_{i, k}^{maint}$	Binary decision variable indicating Whether the ith train completes the maintenance operation on the kth maintenance track.
${tw}_{i, j}^{start}$	Non-negative decision variable stating the moment when the ith train starts washing at the jth washing track.
${tm}_{i, k}^{start}$	Non-negative decision variable stating the moment when the ith train starts maintenance at the kth maintenance track.
p _i	Binary decision variable, When the ith train completes the washing operation first, p_i = 1; otherwise, p_i = 0.
$t_{i}^{finish}$	Intermediate variable stating the moment when the ith train completes all operations.
T _total	Intermediate variable stating the moment when all EMU trains complete the first-level maintenance.

2.4 Illustration example

We conclude this section with an example that visually illustrates the problem of the first-level maintenance with flexible-JSS mode. In the example, there is one washing track and two maintenance tracks in the EMUD, and seven EMU trains that need to complete the first-level maintenance operation, the arrival time and attributes of trains are shown in Table 4. The washing and maintenance operations take 30 minutes and 180 minutes, respectively. And switching the yard takes 30 minutes.

Table 4
Information and attributes of the trains

N y _i $T_{i}^{enter}$ N y _i $T_{i}^{enter}$

1 1 7 : 30 5 1 9 : 00

2 1 8 : 00 6 1 9 : 30

3 1 8 : 10 7 1 9 : 40

4 2 8 : 30

N	y _i	$T_{i}^{enter}$	N	y _i	$T_{i}^{enter}$
1	1	7 : 30	5	1	9 : 00
2	1	8 : 00	6	1	9 : 30
3	1	8 : 10	7	1	9 : 40
4	2	8 : 30

Tables 5 and 6 provide two feasible first-level maintenance solutions for this illustrative EMUD. Figures 6 and 7 are Gantt charts of their corresponding EMU-to-track arrangements, and different colors represent different EMU trains.

Table 5

Solution 1 of the small case

N	Duration of washing	W	Duration of maintenance	M
1	8 : 00—8 : 30	1	9 : 00—12 : 00	1
2	8 : 30—9 : 00	1	9 : 30—12 : 30	1
3	12 : 10—12 : 40	1	8 : 40—11 : 40	2
4	9 : 00—9 : 30	1	12 : 30—3 : 30	1
5	9 : 30—10 : 00	1	10 : 30—1 : 30	2
6	10 : 00—10 : 30	1	11 : 40—2 : 40	2
7	10 : 30—11 : 00	1	1 : 30—4 : 30	2

Table 6

Solution 2 of the small case

N	Duration of washing	W	Duration of maintenance	M
1	11 : 30—12 : 00	1	8 : 00—11 : 00	1
2	12 : 00—12 : 30	1	8 : 30—11 : 30	1
3	8 : 40—9 : 10	1	11 : 00—2 : 00	1
4	12 : 30—1 : 00	1	9 : 00—12 : 00	2
5	9 : 30—10 : 00	1	11 : 30—2 : 30	1
6	10 : 00—10 : 30	1	12 : 00—3 : 00	2
7	10 : 30—11 : 00	1	12 : 00—3 : 00	2

Fig. 6

Gantt chart of solution 1.

Fig. 7

Gantt chart of solution 2.

Both solutions have completed the first-level maintenance for all EMU trains, but their operation sequences and EMU-to-track arrangements are different. The trains in the first schedule preferentially chose to perform the washing operation first, and the completion time of all trains is 4 : 30. The trains in the second schedule have optimized the operation sequences, and the completion time of all trains is 3 : 00. Solution 2 completes all operations one and a half hours ahead of solution 1, which has higher track utilization and less waiting time of EMU trains. The scope of this example is small, but it is a good illustration of many feasible schedules. Deciding the optimal EMU-to-track arrangement of the first-level maintenance is not easy, and the length of the trains and the section of tracks make the situation more complicated.

3 The FSTS model formulation

In this section, we present a non-linear integer programming model for solving the first-level maintenance operation with flexible-JSS mode.

3.1 Objective function

The optimization goal of this paper is how to adjust the operation sequence and the EMU-to-track arrangement, so as to minimize the completion time of all trains, as shown in Equation (1). $min T_{total}$ (1)

3.2 Constraints

The proposed model is subject to a number of constraints. Now we introduce each of them in turn.

3.2.1 Constraints for the completion time

The completion time of each EMU train is shown in Equation (2). As shown in Equation (3), the completion time of all EMU trains should be no earlier than the completion time of each EMU train. $\begin{matrix} t_{i}^{finish} = (1 - p_{i}) (\sum_{j = 1}^{w} {tw}_{i, j}^{start} \cdot x_{i, j}^{wash} + T^{wash}) \\ + p_{i} (\sum_{k = 1}^{m} {tm}_{i, k}^{start} \cdot x_{i, k}^{maint} + T^{maint}) + T^{turn} \\ \forall i = 1, 2, . . . n \end{matrix}$ (2) $t_{i}^{finish} ⩽ T_{total} \forall i = 1, 2, . . . n$ (3)

3.2.2 Constraints for the number and location of operations

Each EMU train can only complete the corresponding operation on one track, and all EMU trains should complete the corresponding operations. $\sum_{j = 1}^{w} x_{i, j}^{wash} = 1 \forall i = 1, 2, . . . n$ (4) $\sum_{k = 1}^{m} x_{i, k}^{maint} = 1 \forall i = 1, 2, . . . n$ (5) $\sum_{i = 1}^{n} \sum_{j = 1}^{w} x_{i, j}^{wash} = n$ (6) $\sum_{i = 1}^{n} \sum_{k = 1}^{m} x_{i, k}^{maint} = n$ (7)

Equation (4) indicates that one train can only complete washing operation on one washing track; Equation (5) indicates that one train can only complete maintenance operation on one maintenance track. Equation (6) and Equation (7) indicate that there are n trains in total that need to finish washing and maintenance operations.

3.2.3 Constraints for duration and sequence of operation

The EMU trains can only proceed to the next operation after completing one operation and switching yard, and the completion of each operation must meet its required specified duration (Guo et al. (2016)). $\begin{matrix} \sum_{k = 1}^{m} {tm}_{i, k}^{start} \cdot x_{i, k}^{maint} + R (1 - p_{i}) ⩾ \\ \sum_{j = 1}^{w} {tw}_{i, j}^{start} \cdot x_{i, j}^{wash} + T^{wash} + T^{turn} \\ \forall i = 1, 2, . . . n \end{matrix}$ (8) $\begin{matrix} \sum_{j = 1}^{w} {tw}_{i, j}^{start} \cdot x_{i, j}^{wash} + {Rp}_{i} ⩾ \\ \sum_{k = 1}^{m} {tm}_{i, k}^{start} \cdot x_{i, k}^{maint} + T^{maint} + T^{turn} \\ \forall i = 1, 2, . . . n \end{matrix}$ (9)

Equation (8) is the constraint on washing operation, Equation (9) is the constraint on maintenance operation.

3.2.4 Constraints for the EMU train-set scheduling plan on the start and end time of each operation

$\begin{matrix} T_{i}^{out} - T^{wash} - T^{turn} ⩾ \sum_{j = 1}^{w} {tw}_{i, j}^{start} \cdot x_{i, j}^{wash} ⩾ T_{i}^{enter} + T^{turn} \\ \forall i = 1, 2, . . . n \end{matrix}$ (10) $\begin{matrix} T_{i}^{out} - T^{maint} - T^{turn} ⩾ \sum_{k = 1}^{m} {tm}_{i, k}^{start} \cdot x_{i, k}^{maint} ⩾ T_{i}^{enter} + T^{turn} \\ \forall i = 1, 2, . . . n \end{matrix}$ (11)

Equations (10) and (11) indicate that the start time of the operation should be later than each EMU enters the EMUD after completing the task of the day, and the end time should be earlier than each EMU is put into use the next day.

3.2.5 Constraints for track capacity

There is a limit to the number of EMU trains that can be operated simultaneously on each track, depending on the attributes of the tracks and the trains. $\begin{matrix} x_{i, j}^{wash} \cdot x_{i', j}^{wash} \cdot \\ [{tw}_{i, j}^{start} - ({tw}_{i', j}^{start} + T^{wash})] \cdot [({tw}_{i, j}^{start} + T^{wash}) - {tw}_{i', j}^{start}] ⩾ 0 \\ \forall i, i' = 1, 2, . . . n, i \neq i' \forall j = 1, 2, . . . w \end{matrix}$ (12) $\begin{matrix} (y_{i} + y_{i'}) x_{i, j}^{wash} \cdot x_{i', j}^{wash} \cdot \\ [{tw}_{i, j}^{start} - ({tw}_{i', j}^{start} + T^{wash})] \cdot [({tw}_{i, j}^{start} + T^{wash}) - {tw}_{i', j}^{start}] ⩾ 0 \\ \forall i, i' = 1, 2, . . . n, i \neq i' \forall k = 1, 2, . . . m \end{matrix}$ (13) $\begin{matrix} x_{i, k}^{maint} \cdot x_{i', k}^{maint} \cdot x_{i'', k}^{maint} \cdot \\ [max {{tm}_{i, k}^{start}, {tm}_{i', k}^{start}, {tm}_{i'', k}^{start}} - (min {{tm}_{i, k}^{start}, {tm}_{i', k}^{start}, {tm}_{i'', k}^{start}} + t_{M})] ⩾ 0 \\ \forall i, i', i'' = 1, 2, . . . n, i \neq i' \neq i'' \forall k = 1, 2, . . . m \end{matrix}$ (14)

Equation (12) indicates that one washing track can only be used for the operation of one train at the same time. Equation (13) indicates that one maintenance track cannot operate one long EMU train and any other EMU train at the same time. Equation (14) indicates that one maintenance track cannot carry out the operation of any three EMU trains at the same time. Consequently, Equation (13) and Equation (14) indicate that one maintenance track can be used for the operation of two short EMU trains or one long EMU train at the same time.

4 The modified genetic algorithm

The proposed FSTS model is a non-linear integer programming model, which is difficult to be solved by the linear programming solvers (such as ILOG CPLEX), as the objective function and some constraints (e.g. Equation (14)) are difficult to be described in the software. Therefore, a heuristic algorithm is needed to solve the FSTS model. The main decision variables are the order of the trains about washing and maintaining, which are two series of integers and the values cannot be repeated in each one. Based on the characteristics of the FSTS model, a modified genetic algorithm is constructed to solve this problem, which is effective and efficient. To follow the constraints, the immediate predecessor activity (IPA) principle is considered during the coding and decoding process and the correctness of order is considered during the selection, crossover and mutation process, which are the main modification in the algorithm. The related parameters in the modified genetic algorithm will be explained in Table 7 before introducing the algorithm.

Table 7
Explanation of parameters in the modified genetic algorithm

Notation Description

pop _max size of population

parentpop _set parent population set

newpop _set new population set

newpop _num number of chromosomes in new population set

pop1, pop2 pop1′, pop2′ chromosomes in population

pop1^′′, pop2^′′

Iter current iteration times

Iter _max maximum number of iteration times

Rbest _Iter optimal objective function value in current iteration

Sbest _Iter optimal solution in current iteration

Rbest _global global optimal objective function value

Sbest _global global optimal solution

Notation	Description
pop _max	size of population
parentpop _set	parent population set
newpop _set	new population set
newpop _num	number of chromosomes in new population set
pop1, pop2 pop1′, pop2′	chromosomes in population
pop1^′′, pop2^′′
Iter	current iteration times
Iter _max	maximum number of iteration times
Rbest _Iter	optimal objective function value in current iteration
Sbest _Iter	optimal solution in current iteration
Rbest _global	global optimal objective function value
Sbest _global	global optimal solution

4.1 The main processes of the modified genetic algorithm

4.1.1 Coding and decoding rule

Coding rule: The decision variables that include the trains’ washing order and the maintaining order are encoded as chromosomes to participate in the process of iterative solution. Specifically, as shown in Fig. 8, the length of a chromosome is 2n, the former n genes are maintenance gene segments and the latter n genes are washing gene segments. The coding rule is that the maintenance gene segment or the washing gene segment consists of randomly arranging from 1,2, ... ,n, which is the order of maintaining or washing the trains.

Fig. 8

The coding of a chromosome.

Decoding rule: Based on the process of coding the chromosome and the constraints of the FMOO model, the rule of decoding the chromosome is introduced. To improve the efficiency of the algorithm, the immediate predecessor activity (IPA) principle is used to arrange the trains to a specific maintaining track and schedule the maintaining time by the former trains’ occupied time on the track and the current trains’ arrival time. The IPA principle considers the former trains’ earliest finish time, which can confirm the earliest start time of the current train. In other words, this principle can avoid the trains’ unnecessary waiting time on the track, which may compress the finally finished time of all trains. The processes of decoding are shown as follows.

Step 1: Based on the order of maintenance gene segment, the trains are assigned to the maintenance tracks one by one, and the occupancy of tracks under different train lengths should be considered.

Step 2: Based on the train arrival time, the earliest start time on the distributed track and IPA principle, execute the maintenance operation and update the earliest start time on the selected track.

Step 3: Based on the order of washing gene segment, distribute the trains to the washing tracks one by one

Step 4: Considering the train arrival time and the earliest start time on the current washing track, check all trains on the washing tracks. If the train washing operation can be completed before it is transferred to the maintaining track for maintenance operation, the washing operation should be executed first (washing before maintaining); otherwise, the maintenance operation should be executed first (washing after maintaining). The specific washing operation time should be confirmed by order of washing and maintaining for the train and the order on the track by IPA principle.

4.1.2 Calculation of the objective function and fitness function

Based on the result of chromosome decoding, both the washing and maintenance operation process and the final finished time of all trains can be obtained, which comes from the objective function. What is more, a penalty of factors will be added in the objective function to meet the restrict of the train-set scheduling plan for every train (Equation (10)-(11)), if the Equation (10) or (11) is not satisfied in current solution. The fitness function is the reciprocal value of the objective function.

4.1.3 Selection, crossover and mutation process

Selection process: To avoid falling into the local optimal solution with a short iteration, the diversity of chromosomes in a population should be considered when selecting the chromosomes. Due to the limited number of chromosomes in the population, the selection method should balance the diversity and quantity of the chromosomes. The binary tournament selection (BTS) method can simultaneously consider the diversity and quantity of the chromosomes to complete the selection of population, which also easy to execute in parallel with low complexity. Specifically, in the BTS method, two chromosomes pop1, pop1′, pop2 and pop2′ from the parent population are randomly selected. Then, comparing the objective function value of pop1 and pop1′, and choose the better one. The better one from pop2 and pop2′ is selected in a similar way. pop1 and pop2 will execute the subsequent processes to generate a new population. The new population contains many better solutions, because the better one could be selected. Meanwhile, the worse two solutions are also selected, because every two chromosomes can be selected and compared, even they may both worser than others in the population. Therefore, the diversity of the population can be guaranteed, and avoids staying at the local optimal solution during the iteration of the algorithm.

Crossover process: To fit for the characteristic of the chromosomes, the same length segments are selected from the same site in two-parent generation chromosomes. Two chromosomes are randomly selected from the alternative pool. If the generated random number is smaller than the crossover rate, the two chromosomes will execute the crossover and join into the filial generation; otherwise, the two chromosomes join into the filial generation. Specifically, the random length of segments (within the feasible range) at a random site in two segments (maintaining and washing gene segments) of the parent generation chromosomes are selected and crossed over, respectively. Due to the repeated number that may exist in the chromosomes after crossing, the chromosome is an infeasible solution and the repair operation is necessary. To repair chromosomes, the uncrossed gene will be checked one by one, and the repeated gene with crossed part will be repaired by a random disappeared train number. The process of crossover and repair operation is shown in Fig. 9.

Fig. 9

The process of crossover and repair operation.

Mutation process: To avoid the mutated chromosome becoming an infeasible solution, a site exchange mutation rule is proposed. Select chromosome one by one from the filial generation, and if the generated random number is smaller than the mutation rate, mutation will be executed and filial generation replaced. Specifically, two random sites are selected from the same segment (maintaining or washing gene segment) and the values are exchanged. The process of mutation operation is shown in Fig. 10.

Fig. 10

The Process of Mutation Operation.

4.2 The framework of the modified genetic algorithm

Based on the main processes of the modified genetic algorithm, the framework is proposed, and the pseudocode is shown as follows.

Input:
the number of emus n;
the number of washing tracks w;
the number of maintenance tracks m;
other basic operation parameter T^wash, T^maint, T^turn, $T_{i}^{enter}$ , $T_{i}^{out}$ , y_i;
size of population pop_max;
maximum iteration times Iter_max;
crossover rate P_c and mutation rate P_m;
Output:
the optimal objective function value Rbest_global;
the optimal solution Sbest_global;
the optimal operation plan $x_{i, j}^{wash}$ , $x_{i, k}^{maint}$ .
1: initialize: initial iteration Iter ← 1, initial optimal objective function value Rbest_global ← 0, initial optimal solution Sbest_global ← 0;
2: randomly generate initial population newpop_set by Coding rule and calculate the objective function value of every chromosome in the population;
3: while (Iter ⩽ Iter_max) do
4: parentpop_set ← newpop_set, newpop_set← ∅, newpop_num ← 0;
5: while (newpop_num ⩽ pop_max) do
6: execute the selection process and obtain pop1and pop2 from the parent population parentpop_set by BTS method;
7: if (Rand (1) ⩽ P_c) then
8: pop1 and pop2execute the Crossover process and obtain pop1 and pop2;
9: end if
10: if (Rand (1) ⩽ P_m) then
11: pop1 and pop2 execute the Mutation process and obtain newpop1and newpop2;
12: end if
13: put newpop1and newpop2into the newpop_set;
14: update the size of new population newpop_num ← newpop_num + 2;
15: end while
16: calculate the objective function value of chromosomes in the newpop_set;
17: find out the optimal value Rbest_Iter and the optimal solution Sbest_Iter in current iteration;
18: if Rbest_global > Rbest_Iter then
19: Rbest_global ← Rbest_Iter, Sbest_global ← Sbest_Iter;
20: end if
21: update iteration: Iter ← Iter + 1;
22: end while
23: generate optimal plan $x_{i, j}^{wash}$ , $x_{i, k}^{maint}$ from Sbest_global by Decoding rule;
24: returnRbest_global, Sbest_global, $x_{i, j}^{wash}$ , $x_{i, k}^{maint}$ .

5 Case study

In this section, we further conduct a case study with real data from an EMUD of China Railway Corporation to verify the proposed FSTS model. In addition, we also verified the feasibility and effectiveness of the Flexible-JSS mode by analyzing and comparing with the traditional Fixed-JSS modes.

5.1 Introduction of the case

We take China’s TaiYuan EMUD as an example. There are two washing tracks and six maintenance tracks in the depot. The storage yard is located between the washing yard and the maintenance yard. Both the storage track and the maintenance track are arranged in two sections, as shown in Fig. 11. A total of 19 EMU trains need to complete the first-level maintenance task on a certain day, and their basic situation and attributes are shown in Table 8. The washing operation takes 30 minutes, the maintenance operation takes 180 minutes, and the switching yard takes 30 minutes. The parameter setting values are shown in Table 9.

Fig. 11

Yard distribution in TaiYuan EMUD.

Table 8

Basic situation and attributes of EMU trains

Trip	y _i	$T_{i}^{enter}$	$T_{i}^{out}$
G1953	1	18 : 00	11 : 45
G697	0	18 : 15	5 : 20
D5342	0	18 : 20	8 : 05
D1959	0	19 : 20	10 : 00
D2539	0	19 : 35	4 : 40
G2605	0	19 : 45	7 : 10
D5355	0	20 : 00	5 : 55
D2561	0	20 : 35	6 : 55
D5327	0	20 : 40	5 : 40
D5331	0	20 : 50	9 : 20
D2527	0	21 : 20	8 : 15
G621	1	21 : 30	7 : 25
G623	1	22 : 20	4 : 35
D5335	0	22 : 30	6 : 50
D2533	0	23 : 00	6 : 35
G625	1	23 : 05	7 : 55
D5337	1	23 : 10	5 : 00
D1633	0	23 : 20	8 : 20
D5333	0	23 : 40	8 : 35

Table 9

List of variables

Parameter	Value
n	19
w	2
m	6
T ^wash	30 minutes
T ^maint	180 minutes
T ^turn	30 minutes

5.2 Results

The example calculation results calculated by the model and algorithm proposed above are as follows. Solve the model on the Matlab R2012b platform according to the proposed genetic algorithm, and run it on a laptop with 8 GB memory and i7-6500 CPU. Set the number of population to 50, the maximum number of iterations to 500, the crossover probability P_c = 0.8, the mutation probability P_m = 0.01, and the computation time is 4.23 s. The iteration process is shown in Fig. 12. The blue line indicates the iteration of the optimal solution for each generation, and the orange line indicates the iteration of global optimal solution. The calculated first-level maintenance arrangement of EMU trains is shown in Table 10. The objective function value is 640 (18 : 00 is the origin time 0, so 640 is 4 : 10), which means all operations are completed at 4 : 10. The corresponding EMU-to-track arrangement is shown in Fig. 13. The different colors in the figure represent different trains.

Fig. 12

Diagram of the Optimal Iteration Trend.

Table 10

The arrangement of EMU trains with flexible-JSS mode

Trip	Start time of washing	End time of washing	Start time of maintenance	End time of maintenance
G1953	22 : 00	22 : 30	18 : 30	21 : 30
G697	22 : 15	22 : 45	18 : 45	21 : 45
D5342	22 : 30	23 : 00	18 : 50	21 : 50
D1959	23 : 20	23 : 50	19 : 50	22 : 50
D2539	23 : 35	0 : 05	20 : 05	23 : 05
G2605	0 : 20	0 : 50	20 : 15	23 : 15
D5355	0 : 05	0 : 35	20 : 30	23 : 30
D2561	0 : 35	1 : 05	21 : 05	0 : 05
D5327	0 : 50	1 : 20	21 : 10	0 : 10
D5331	1 : 05	1 : 35	21 : 20	0 : 20
D2527	1 : 20	1 : 50	21 : 50	0 : 50
G621	1 : 35	2 : 05	22 : 00	1 : 00
G623	22 : 50	23 : 20	0 : 20	3 : 20
D5335	2 : 30	3 : 00	23 : 00	2 : 00
D2533	3 : 00	3 : 30	23 : 30	2 : 30
G625	3 : 05	3 : 35	23 : 35	2 : 35
D5337	23 : 50	0 : 20	1 : 00	4 : 00
D1633	3 : 30	4 : 00	23 : 50	2 : 50
D5333	3 : 40	4 : 10	0 : 10	3 : 10

Fig. 13

Gantt chart of EMU-to-track with flexible-JSS mode.

5.3 Comparison and analysis

Due to the washing operation takes a short time, in practice, the EMUD generally adopts the fixed “washing-maintenance” job-shop sequence with first-level maintenance. Aiming to increase the operation efficiency, a manual washing method is added on the basis of the fixed job-shop sequence at some EMUDs, which is defined as fixed job-shop sequence combined with manual washing (Fixed-JSS&MW) mode. That is, some EMU trains skip the washing operation and enter the maintenance yard directly when they need a waiting time before they can be washed. During the maintenance process, these EMU trains are manually washed, as shown in Fig. 14.

Fig. 14

The flow chart of the fixed-JSS&MW mode.

The calculation results, which are calculated according to the two fixed job-shop sequence modes, are 680 and 690 (all first-level maintenance operations are completed at 4 : 50 and 5 : 00), respectively, and the Gantt charts for the corresponding EMU-to-track arrangement are shown in Figs. 15 and 16.

Fig. 15

Gantt chart of EMU-to-track with fixed-JSS mode.

Fig. 16

Gantt chart of EMU-to-track with fixed-JSS&MW mode.

According to the data of Figs. 13, 15, 16 and Table 10, the waiting time of all EMU trains, utilization efficiency of tracks and latest completion time under three different operation modes can be calculated, as shown in Table 11.

Table 11

Calculation results under different modes

Mode	Total waiting time	NONW	UWT	UMT	Latest completion time
Fixed-JSS	155min	11	43.55%	60.48%	4 : 50
Fixed-JSS&MW	115min	2	30.95%	64.02%	5 : 00
Flexible-JSS	140min	8	46.55%	68.82%	4 : 10

Note: NONW-Number of operations need to wait; UWT-Utilization of washing tracks; UMT-Utilization of maintenance tracks.

5.3.1 Waiting time for all EMU trains

According to the calculation results of the above figures and table, in the Fixed-JSS mode, the waiting time of all operations is 155 minutes, and 11 operations out of 38 that need to wait. Due to the fixed and single operation sequence, when the density of arriving trains increases, there may be a large number of trains waiting in line.

In the Fixed-JSS&MW mode, some trains skip the washing operation link, which reduces the waiting link of the train. In this case, only the G623 and G625 need to wait before the maintenance operation, and the total waiting time is 115 minutes. In this mode, although the number of trains that need to wait is reduced, the waiting operation is concentrated on individual trains, and the waiting time for individual trains will be very long. As the EMU train-set scheduling plan has time requirements for the next day, some trains may not meet the time requirements for completion time.

In the Flexible-JSS mode, the waiting time of all operations is 140 minutes, and 8 operations need to wait. In this mode, the operation distribution can be more balanced and the EMU-to-track arrangement more coordinated, which reduces the time-consuming of the overall first-level maintenance, and the advantage will be more obvious when the trains arrive intensively.

5.3.2 Track utilization efficiency

We use Q_W and Q_M to represent the utilization rate of the washing tracks and the maintenance tracks, and the calculation formulas are shown in Equation 15 and Equation 16. $Q_{W} = \frac{T_{W}^{occupy}}{\sum T_{W}}$ (15) $Q_{M} = \frac{T_{M}^{occupy}}{\sum T_{M}}$ (16)

$T_{W}^{occupy}$ represents the entire time that the washing tracks are occupied, $T_{M}^{occupy}$ represents the entire time that the maintenance tracks are occupied. ∑T_W represents the entire time that washing tracks can be used, ∑T_M represents the entire time that maintenance tracks can be used.

As shown in Table 10, the utilization rate of the maintenance tracks is high while that of washing tacks is low. This is mainly due to the long maintenance operation time, which affects the completion efficiency of the entire first-level maintenance operation. When the number of trains increases in the future, it is necessary to continuously improve the maintenance technical means, so as to shorten the maintenance operation time and improve the operation efficiency.

The utilization rate of the tracks in the Flexible-JSS mode is improved compared to the other two modes. In the Fixed-JSS mode, there are only two washing tracks for the train to choose after entering the EMUD. In the Flexible-JSS mode, the train has eight options after entering the EMUD, including two washing tracks and six maintenance tracks. The increase in the choice of track selection leads to a more reasonable EMU-to-track arrangement, and the track utilization rate is also improved.

In the Fixed-JSS&MW, some EMU trains skip the washing operation and directly enter the maintenance yard, resulting in a waste of washing track resources and low utilization rate. In addition, manual washing during the maintenance operation will increase the cost, as some links in the maintenance process need to be tested by power-on, and there are certain safety risks during the manual washing process.

In the case of this article, there are few cases where EMU trains arrive densely within half an hour, and the effect of improving the track utilization rate is not particularly obvious. In the future, when the number of trains increases and the trains arrive at EMUD intensively, the Flexible-JSS mode may better improve the track utilization rate.

5.3.3 Completion time of all EMU trains

In this case, all EMU trains can complete the first-level maintenance operations before being put into use the next day, but the latest completion time is different in different modes.

In the Fixed-JSS&MW mode, some trains skip the washing process, but the final completion time take 10 minutes longer than the Fixed-JSS mode, which can be attributed to the JSS&MW mode making G625 cannot be able to complete the first-level maintenance operation before put into use the next day, and the sequence adjustment of G625 and G623 operations will lead to an increase in overall operation time.

The latest completion time of the Flexible-JSS mode is 40 minutes and 50 minutes ahead of the other two modes, respectively. This phenomenon can be explained as the overall arrangement of the EMU-to-track rationally utilizes the track resources, reduces unnecessary waiting time, improves the utilization efficiency of the tracks, and then completes the first-level maintenance of all the EMU trains in a shorter time smoothly.

5.3.4 Discussion

From the results and analysis in the previous section, it can be observed that using the modified genetic algorithm to solve the FSTS model can quickly obtain the EMU-to-track arrangement results of the first-level maintenance with flexible-JSS mode. The FSTS model comprehensively considers various factors that affect the first-level maintenance operation, including the length of EMU trains, the operation sequence, the track section, and the EMU train-set scheduling plan.

Compared with the other two Fixed-JSS modes, the Flexible-JSS mode can effectively improve the efficiency of the first-level maintenance operation. In the example of this article, the completion time of the Flexible-JSS mode is 40 minutes and 50 minutes earlier than the other two modes, respectively. The utilization rate of tracks has also been improved, and the arrangement of the EMU-to-track is more reasonable. The advantages will be more prominent as the maintenance workload increases, especially many EMU trains arrive intensively at the EMUD.

In this paper, only through type yard of EMUD is selected for research, and stub-end type yard is not selected, which is inferior to the literature [12]. Because this paper considers the constraint of the EMU train-set scheduling plan, when the depot type is stub-end, the sequence of EMU trains entering and leaving the yard is more complicated, and the movement of trains will conflict, which can easily lead to the train not meeting the train-set scheduling plan. A possible future research direction is to consider the train scheduling of different types of EMUDs when the trains move in conflict. In addition, it is worth to take the capacity of the storage yard into account, when the number of EMU trains is increased.

6 Conclusions

Aiming at the first-level maintenance with flexible-JSS mode for EMU trains, we proposed a FSTS model, which is solved by the modified genetic algorithm based on the binary tournament method. By comparing the proposed Flexible-JSS mode with the other two Fixed-JSS modes, the conclusions are as follows.

The proposed FSTS model is effective for the Flexible-JSS mode of EMU trains, which integrates trains with different length, different train sequence, single train operation process, and train-set scheduling plan requirements of EMU trains.

The modified genetic algorithm takes the length of the EMU train into account. When decoding the chromosomes into a first-level maintenance plan, the long EMU train is regarded as two short EMU trains working simultaneously. On this basis, the matching relationship between different formations and track capacity is solved.

The applicability of the proposed methodology is proved by the actual instance of CHSR, and all results have been verified by dispatchers. Besides, the validity of the proposed method is proved by a comparative analysis. Computational results indicate our method outperforms the other modes on both aspects of time saving and track utilization.

Footnotes

Acknowledgments

This work was partially supported by the National key research and development plan of China under grant fund (2016YFE0201700), and Beijing Natural Science Foundation (L201013) and partially supported by The project of reform research and practice of higher education teaching in Hebei Province (2019GJJG640).

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Model and algorithm for the first-level maintenance operation optimization of EMU trains

Abstract

Keywords

1 Introduction

2.1 Problem description

Table 4 Information and attributes of the trains N y i T i enter N y i T i enter 1 1 7 : 30 5 1 9 : 00 2 1 8 : 00 6 1 9 : 30 3 1 8 : 10 7 1 9 : 40 4 2 8 : 30

3.1 Objective function

3.2.1 Constraints for the completion time

4.1.1 Coding and decoding rule

4.1.3 Selection, crossover and mutation process

5 Case study

5.1 Introduction of the case

5.3.2 Track utilization efficiency

5.3.4 Discussion

6 Conclusions

Footnotes

Acknowledgments

References

Table 4
Information and attributes of the trains

N y _i $T_{i}^{enter}$ N y _i $T_{i}^{enter}$

1 1 7 : 30 5 1 9 : 00

2 1 8 : 00 6 1 9 : 30

3 1 8 : 10 7 1 9 : 40

4 2 8 : 30