Solution of Multi-Crew Depots Railway Crew Scheduling Problems: The Chinese High-Speed Railway Case

Abstract

This paper presents a novel mathematical formulation in crew scheduling, considering real challenges most railway companies face such as roundtrip policy for crew members joining from different crew depots and stricter working time standards under a sustainable development strategy. In China, the crew scheduling is manually compiled by railway companies respectively, and the plan quality varies from person to person. An improved genetic algorithm is proposed to solve this large-scale combinatorial optimization problem. It repairs the infeasible gene fragments to optimize the search scope of the solution space and enhance the efficiency of GA. To investigate the algorithm’s efficiency, a real case study was employed. Results show that the proposed model and algorithm lead to considerable improvement compared to the original planning: (i) Compared with the classical metaheuristic algorithms (GA, PSO, TS), the improved genetic algorithm can reduce the objective value by 4.47%; and (ii) the optimized crew scheduling plan reduces three crew units and increases the average utilization of crew unit working time by 6.20% compared with the original plan.

1. Introduction

The sustainable development of transportation is a global focus. Railway transportation, especially high-speed railway, not only has fast travel speed and large capacity, but also has low energy consumption, so it has always been the focus of development in various countries such as the ICE in Germany, TGV in France, Shinkansen in Japan, and so on. Scientific studies show that railway transportation can reduce 20 million tons of carbon dioxide emissions under the same conditions as roads. For railway, achieving sustainable development not only requires updated transportation equipment to improve energy efficiency, but also needs to improve the quality of the transportation organization to ensure that transportation resources can be effectively used. Since infrastructure costs cannot be influenced substantially, personnel costs become a critical factor after rolling stock and energy. It is generally considered that the crew planning problem is an operational issue that is the task of building a work schedule for crew members to operate a planned timetable. Hence, the interest in using advanced optimization techniques for the complex crew scheduling task has arisen.

The remainder of this paper is organized as follows. A brief review of the relevant literature is presented in Section 2. Problem description and base mathematical model are given in Section 3. Section 4 proposes the solution algorithm for the crew scheduling problem. In Section 5, a real case study is presented, and the proposed solution algorithm is employed. Discussions are also given in the same section. Section 6 provides a conclusion and recommendations for future studies.

2. Literature Review

Crew planning problems in transportation originate in the airline and bus industries. Since the 1950s, foreign scholars, especially European scholars, began to study the Crew Planning Problem (CPP) [1] and gained greater momentum with the advances in computational power in 1980s [2,3]. Early studies were mainly concentrated on airlines [4,5], and followed by urban public transportation [6,7,8]. So far, many achievements have been made in the field of crew planning problems. Compared with other fields, the aviation field, which started earlier, has the most in-depth research and achieved fruitful results [9,10,11,12,13,14]. The research on the railway crew planning problem started relatively late. In the 1990s, the railway industry came to the fore in crew scheduling research activities. Several research groups from different countries contributed to the progress [7,15,16]. In previous studies, most scholars decomposed the crew planning problem into two steps: the crew scheduling problem (CSP) and the crew rostering problem (CRP). The first forms anonymous duties covering all the trips(tasks) for a defined period of time. Each duty represents a sequence of tasks and has to satisfy a large set of constraints. Next, the duties are combined to weekly, in some cases monthly, sequences that are subsequently assigned to individual crew members [17]. Feasible duty sequences typically follow rational/national/international legal and sustainable guidelines. Some studies have focused on the two sub-problems separately [18,19,20], and others have integrated the two steps into a unified optimization model for consideration [21,22,23]. Moreover, research on integrated planning came to the fore in the last year. A generic operational planning process [24] in railway transportation is illustrated in Figure 1. In most cases, they explore the integration of line planning and timetabling or timetabling and rolling stock scheduling [25,26]. The integration of crew scheduling with other planning tasks is mainly explored in freight transportation [27,28] or crew rostering. Furthermore, some of the work was done in the context of crew re-scheduling for disruption management [29,30,31].

Here, we focus on the first step of the crew planning problem on railways (crew scheduling problem). CSPs are typically formulated as mixed-integer problems. Most researchers have viewed it as a set covering problem or set partitioning problem [19,32,33], which are both path-flow formulations. Few researchers have applied the arc-flow formulation of the multi-commodity flow problem, here referred to as network flow problem [22,34]. In order to achieve an optimal crew schedule, an appropriate evaluation criterion should be specified. In most cases, researchers have evaluated the efficiency of the CSP by total cost, the total number of duties, total idle time, or a combination of these [35,36]. Few researchers have not only considered pure monetary foal criteria, but also the quality of the solution in terms of robustness [30,31,37] and employee satisfaction [38]. CSP is governed by a great variety of legal rules and regulations by the public transport authorities, so different countries or operators have their own rules and restrictions. In general, constraints apply to different levels of a schedule: to the duty itself, to a depot, to the total schedule, or to specific types of crew members. Since CSPs are NP-hard optimization problems [39], integer programming methods are only applied to small or sparse problem instances. The computational complexity of CSP has led to extensive research in heuristics [40,41], column generation techniques [35,42,43], or meta-heuristics [44,45].

Traditionally, the type of crew members varies depending on the type of railway transportation, and it is common practice to schedule each type of crew (driver, conductor, catering staff, security guard) separately because of variations in the rules and restrictions [46,47]. The driver operates the train and the conductor is responsible for operational and safety activities [48]. In general, subways require only one, or even no train driver, and regional trains are often managed by a driver and conductors, though almost all researchers have taken the driver as a subject and ignored others, where in some cases, a distinction makes sense. For example, in China, the work schedules between the driver and conductor are different for the following reasons. (1) Compared with drivers, conductors are less affected by the radiation range of the crew depots, because drivers can only undertake tasks on fixed sections while the conductors’ working routes are not fixed. In order to ensure driving safety, drivers can only work at a fixed section, while conductors can serve at any section according to the operation needs, which is more flexible. (2) Conductors can take on their duties for multiple trains, but for safety and other reasons, the drivers can only work on relatively fixed train types to guarantee that they are familiar with its performance. (3) The working time standard for train conductors is different from the drivers. Considering that fatigue driving affects safety, in general, the drivers must arrange a break for 4~5 h of continuous work while the train conductors usually continuously work for more than 10 h. (4) In comparison, conductors are greatly influenced by train types. For high-speed railway, one set of drivers can complete operations of different train types, while the number of conductors should be allocated according to the specific conditions. (5). In general, the schedule for drivers and conductors are made by different institutions. This paper focused on the conductors, with the crew appearing below all referring to conductors.

The existing literature has fully explored the model formulation and algorithm solution on the crew scheduling problem, however, there is only few research that assesses the effects of specific measures on the employees’ satisfaction and trade-offs between the conflicting objectives. The problem of how to attract talented professionals to rail organizations in an environment of increasing competition and choice is a real and ongoing challenge for the global rail industry. This topic has been on the workforce agenda for at least ten years and after a decade, is still widely debated. With plenty of discussion about attracting and retaining talent, skills shortages, mismatches of skills for new jobs, and jobs disappearing due to artificial intelligence and the increasing use of robots, few rail organizations acknowledge that the 21st century is a person-centered age. Many of the workforce issues that organizations cite about not finding the ‘right’ people or skills mix result from failure to communicate the right value proposition to the right people through the right channels [48,49]. Even without a strong work council, railway operators should aim for sustainable and fair crew schedules. Particularly in developed countries, labor markets become more competitive, so railway operators aim to make their jobs more appealing, which includes, besides payment and company culture, a satisfying and sustainable work schedule [17]. This paper plans to focus on the CSP against the background of the sustainable development of mankind. Although the main crew type discussed in this paper and the one investigated in the case studies is the conductor, the proposed framework is capable of dealing with the scheduling of the other types of crew members considering the cooperation of multi depots. The optimization of the crew scheduling problem from the perspective of sustainable development can increase the utilization of the crew members’ working time and reduce the cost investment in crew members.

3. Base Model Development

3.1. Problem Description and Assumption

In this section, we developed a mathematical model for crew scheduling considering the cooperation of multiple crew depots. The main assumptions used for this model deals with crew scheduling, focusing on the situation, conditions, and limitations of the companies dealing with passenger trains. The crew depots (also: crew bases) are used in the crew scheduling process to subdivide train journeys into smaller units, called trips or pieces, following the terminology in rolling stock scheduling. Trips are the smallest amount of work defined by a fixed starting and ending time at each location and can be assigned to a crew member. Each crew member is assigned to a crew depot, which is often located at terminals or stations with a high frequency of train services [17]. Typically, there are multiple depots in railway networks. A trip in a scheduled timetable generally contains these properties: train id, origin station, departure time, destination station, arrival time, stops, and required number of crew units [20]. Here, we regarded the group of a conductor and five stewards as a crew unit, and the group serves a train with 8-carriages. In such trips, the origin and destination can belong to different companies and the required number of crew units are different. Although some limitations may exist in assigning crew units to the trips (e.g., the crew units and trips must belong to the same company or the crew units should have licenses for the train type), we assumed that all crew units are eligible to perform any trips. Furthermore, from the perspective of the sustainable development of mankind, we limited dead-heading and night residents to reduce the fatigue of the crew and improve their work efficiency.

The aim of this mathematical model was to find a set of feasible duties that covers all the trips regarding the existing limitations and conditions. A duty is a sequence of scheduled trips that is to be assigned to a crew unit. A feasible set of duties needs to meet all contractual and legal requirements based on each country’s regulations. Most of the limitations observed in Chinese train companies form the constraints of the proposed model.

3.2. Parameters, Decision Variables, and Notations

The notations, input parameters, and variables for the optimization problems under consideration are listed as

Table 1. Parameters, decision variables, and notations.

Notations	Description
Sets
$D$	$\mathrm{Set}\mathrm{of}\mathrm{crew}\mathrm{depots},\mathrm{indexed}\mathrm{by}d\in D$
$Tr$	Set of complete trips
$K_d$	$\mathrm{Set}\mathrm{of}\mathrm{duties}\mathrm{in}\mathrm{crew}\mathrm{depot}d,\mathrm{indexed}\mathrm{by}k\in K_d$
$N$	$\mathrm{Set}\mathrm{of}\mathrm{crew}\mathrm{depots}\mathrm{and}\mathrm{complete}\mathrm{trips},\mathrm{indexed}\mathrm{by}i,j,\in N$
Parameters
$T{D}_i$	$\mathrm{The}$at the origin
$T{A}_i$	$\mathrm{The}$at the destination
$T{S}_i$	The duration of the trip i
$\underset{\_}{t}$	The minimum time needed for crew units to change the trip
$L$	The maximum allowed working time for a crew unit
$m_i$	The required number of crew units of the trip i
$s_{ij}$	The value is equal to 1 if the destination of the trip i is the same as the origin of trip j, 0 otherwise
$c_1$	The transit cost between trip i and trip j
$c_2$	The night resident cost
$c_3$	The duty fixed cost
M	An infinite number to ensure that the constraint is valid
Variables
$x_{ijk}^d$	The binary variable, which is equal to 1 if the duty k in crew depot d services trip j after trip i, 0 otherwise
$t_k^d$	Cumulative work time of duty k in crew depot d
$t_{ik}^d$	The starting time for duty k in crew depot d to service trip i
$y_{ik}^d$	The binary variable, which is equal to 1 if trip i is assigned to duty k in crew depot d

.

3.3. Mathematical Model

$$\begin{gathered} Min z=c_1\cdot {{\displaystyle \sum }}_{i\in Tr}{{\displaystyle \sum }}_{j\in Tr}{{\displaystyle \sum }}_{d\in D}{{\displaystyle \sum }}_{k\in K_d}{x}_{ijk}^d\left(T{D}_j-T{A}_i\right)+c_2\cdot \\ {{\displaystyle \sum }}_{d\in D}{{\displaystyle \sum }}_{k\in K_d}\left[{{\displaystyle \sum }}_{j\in T}{x}_{djk}^d\cdot {{\displaystyle \sum }}_{d^{\prime }\in D\backslash \left\{d\right\}}{{\displaystyle \sum }}_{j\in T}{x}_{j{d}^{\prime }k}^{d^{\prime }}\right]+c_3\cdot {{\displaystyle \sum }}_{i\in D}{{\displaystyle \sum }}_{k\in K_d}{{\displaystyle \sum }}_{j\in T}{x}_{ijk}^d \end{gathered}$$

(1)

$${{\displaystyle \sum }}_{k\in K_d}{{\displaystyle \sum }}_{j\in T}{x}_{ijk}^d\le K_d,\forall i\in D$$

(2)

$$t_k^d={{\displaystyle \sum }}_{i\in N}{{\displaystyle \sum }}_{j\in N\backslash \left\{i\right\}}{x}_{ijk}^d\left(T{D}_j-T{A}_i+T{S}_i\right),\forall d\in D,k\in K_d$$

(3)

$$t_k^d\le L,\forall d\in D,k\in K_d$$

(4)

$${{\displaystyle \sum }}_{d\in D}{{\displaystyle \sum }}_{k\in K_d}{y}_{ik}^d=m_i,\forall i\in Tr$$

(5)

$${{\displaystyle \sum }}_{i\in N\backslash \left\{j\right\}}{{\displaystyle \sum }}_{d\in D}{{\displaystyle \sum }}_{k\in K_d}{x}_{ijk}^d=y_{jk}^d,\forall j\in Tr$$

(6)

$${{\displaystyle \sum }}_{j\in N\backslash \left\{i\right\}}{{\displaystyle \sum }}_{d\in D}{{\displaystyle \sum }}_{k\in K_d}{x}_{ijk}^d=y_{ik}^d,\forall i\in Tr$$

(7)

$${{\displaystyle \sum }}_{j\in Tr}{x}_{ijk}^d\le 1,\forall i\in D,k\in K_d$$

(8)

$${{\displaystyle \sum }}_{i\in D}{{\displaystyle \sum }}_{j\in Tr}{x}_{jik}^d\le 1,\forall k\in K$$

(9)

$$t_{jk}^d\ge t_{ik}^d+T{S}_i+\underset{\_}{t}-M\left(1-x_{ijk}^d\right),\forall i,j\in Tr,i\ne j,d\in D,k\in K_d$$

(10)

$$t_{jk}^d\le T{D}_i,\forall d\in D,k\in K_d,i\in Tr$$

(11)

$$x_{ijk}^d\le y_{ik}^d\cdot s_{ij},\forall i,j\in Tr,i\ne j,d\in D,k\in K_d$$

(12)

$$x_{ijk}^d\le y_{jk}^d\cdot s_{ij},\forall i,j\in Tr,i\ne j,d\in D,k\in K_d$$

(13)

$${\mathrm{x}}_{i,j,k}^d\ge y_{ik}^d\cdot s_{ij}+y_{jk}^d\cdot s_{ij}-1,\forall i,j\in Tr,i\ne j,d\in D,k\in K_d$$

(14)

The objective function (1) minimizes the total system cost including the transit cost between trips, night resident cost, and duty cost. Constraint (2) ensures that the number of duties starting from each crew depot will not deviate from the number of crew units available in the crew depot. Constraints (3) and (4) ensure that each duty’s working time is less than the maximum allowed working time. Constraint (5) ensures that each trip must be assigned to the required number of duties. Constraints (6) and (7) ensure that each duty forms a closed tour. Each duty’s origin and destination is guaranteed by Constraints (8) and (9). Constraints (10) and (11) ensure that the crew unit must make preparations before the specified departure time of the trip. The prevention of crew unit traveling as passengers between the trips is guaranteed by Constraints (12), (13), and (14).

4. Solution Algorithm

As described above, the proposed model contains a nonlinear item in the objective function. Although some optimization solvers can handle quadratic programming if the objective and constraints are positive semi-definite, the resultant model becomes significantly large, which makes it difficult to obtain the optimal solution in moderate and large scales of a problem using commercial solvers. Thus, we improved the genetic algorithm (GA) [50] to obtain a better solution. GA has a good global search ability and has been applied to solve various combinatorial optimization problems. However, crossover and mutation operations usually cause large infeasible solutions and reduce the viability of the population, leading to the advanced convergence of the algorithm. Inspired by the simulated evolutionary algorithm [51] and gene recombination [52], in this paper, the improved GA uses the elimination and regeneration method to repair the infeasible gene fragments to improve the quality of the population. The improved GA is shown in Figure 2.

4.1. Representation and Fitness Function

As shown in Figure 3, TSP-like permutation chromosomes with trip delimiters were used. The segment between two adjacent crew points is the work content of a crew unit, for example, the crew unit of duty 1 departures from crew depot D1 and serves Tr1, Tr2, Tr3, and Tr4, and finally returns back to crew depot D1. A crew unit will not go back to their own crew depot after accomplishing the duty if the two adjacent crew points are not the same, for example, the crew unit of duty 2 has to reside in crew depot D2, which is not the same as the departure crew depot. We can easily convert the chromosome to duties as shown in Figure 4.

Moreover, we can see that some trips appeared twice (i.e., Tr6 and Tr7 in Figure 4) if the train with 16-carriages needs two crew units. Note that Figure 4 is only a schematic diagram. The origin and destination of some duties may not be the same.

The fitness of a chromosome is directly related to the target of problem optimization. In this paper, fitness is defined as the reciprocal of the objective function (Equation (15)).

$$f_i\left(n\right)=Z_{max}\left(n\right)-Z_i\left(n\right)$$

(15)

$\mathrm{where} f_i\left(n\right)$ is the fitness of chromosome i at iteration n; $Z_i\left(n\right)$ is the objective value chromosome i at iteration n; and $Z_{max}\left(n\right)$ is the max objective value of the population at iteration n.

4.2. Initial Population

The convergence of GA and the final solution quality are influenced by its initial population [53]. Therefore, it is essential to provide a good initial solution. The method of generating the initial solution is as follows:

Step 1. Initialization: k = 0, D = ∅;

Step 2. k = k + 1, randomly select a trip from Tr as the first trip of duty k, then mark it;

Step 3. For each trip in Tr, if the trip is not marked but meets the Constraints (3)–(14), put it at the end of duty k, then mark it;

Step 4. If the departure crew depot and arrival crew depot of duty k are the same, put duty k in D, delete the marked trips from Tr. If the departure crew depot and arrival crew depot of duty k are not the same, and can back to the trip which can form a new closed duty k’, put duty k’ in D, then delete marked trips from Tr. Otherwise, put duty k in D, delete the marked trips from Tr, and the duty k will be punished.

Step 5. If Tr = ∅, go to Step 6, otherwise, go to Step 2.

Step 6. Repeat Steps (2)–(5) until obtaining the population of the given size.

4.3. Selection

Selection aims at selecting dominant descendants from a population. Various selection methods have been proposed such as wheel selection, stochastic tournament selection, stochastic universal selection, etc. In this paper, the roulette wheel selection was adopted.

4.4. Crossover

Crossover is one of the core parts of GA, aiming at obtaining better individuals from the previous generation population. Otman [54] studied the effects of different crossover operations on the genetic search process in TSP. The results show that the quality of the solution can be the best using the ordered crossover (OX) crossover operation. In this paper, the OX operation was adopted. The steps are as follows.

Step 1. Randomly select two parents (i.e.,$P_1=\left\{1,2,3,4,5,6,7,8\right\},P_2=\left\{8,7,6,1,2,3,4,5\right\})$.

Step 2. Randomly determine two crossover points, ${\mathrm{Index}}_1=3,{\mathrm{Index}}_2=5$.

Step 3. Descendants inherit fathers’ genes, $C_1=\left\{\ast ,\ast ,3,4,5,\ast ,\ast ,\ast \right\},C_2=\left\{\ast ,\ast ,6,1,2,\ast ,\ast ,\ast \right\}$.

Step 4. Sequentially insert the remaining genes of $P_2$ into $C_1$, sequentially insert the remaining genes of $P_1$ into $C_2$, $C_1=\left\{8,7,3,4,5,6,1,2\right\}$,$C_2=\left\{3,4,6,1,2,5,7,8\right\}$.

4.5. Mutation

After the crossover, the descendants are randomly adjusted by mutation. The mutation enlarges the diversity of solutions and searches space by small adjustments, which is helpful to find better solutions. This paper adopted the reverse sequence mutation (RSM) method. In RSM, two mutation points are randomly generated, then the genes are exchanged at two points.

4.6. Elimination and Regeneration

The crossover and mutation do not consider the connection time between two trips so that the descendants may violate the duty constraints. In this paper, the method of elimination and regeneration was used to solve this problem. There are two main stages (Figure 5): elimination and regeneration.

(i) Elimination: First, judge individuals in the population, if one individual contains an infeasible chromosome fragment, delete this chromosome fragment and release the occupied resources. As shown in Figure 4, trip $T{r}_6$ and trip $T{r}_7$ are not satisfied with the constraints in the duty $k_1$ of crew depot $B_1$; trip $T{r}_9$ and trip $T{r}_{10}$ are not satisfied with the constraints in the duty $k_2$ of crew depot $B_2$. Therefore, duty $k_1$ and duty $k_2$ are eliminated, and their occupied trips are released.

(ii) Regeneration: New chromosome fragments are obtained by using the method above, and new individuals are formed by replacing the original unfeasible chromosome fragments with new chromosome fragments. As shown in Figure 4, new duty $k_1^{\prime }=\left\{B_1,T{r}_5,T{r}_6,T{r}_9,T{r}_7,B_1\right\}$ and $k_2^{\prime }=\left\{B_2,T{r}_8,T{r}_{10},T{r}_7,B_2\right\}$ are obtained, and new crew scheduling is formed by replacing the duty $k_1$ and duty $k_2$ with the duty $k_1^{\prime }$ and duty $k_2^{\prime }$.

5. Case Study

To validate the proposed model as well as analyze the efficiency of the proposed solution algorithm, the Wuhan–Guangzhou high-speed railway as a whole case study is provided in this section.

5.1. Case Description

The Wuhan–Guangzhou high-speed railway, which opened in 2009, is one of the busiest high-speed railway trunk lines in China. It has a total length of 1069 km and 16 stations (Figure 6). It operates more than 100 pairs of high-speed trains every day. This paper considered 75 high-speed trains on the Wuhan–Guangzhou high-speed railway and some cross-line trains (Wuhan–Shenzhen North) as input.

Table 2. Timetable.

Train ID	Departure Station	Arrival Station	Departure Time	Arrival Time	Train Types	Duration
G1003	Wuhan	Guangzhou South	07:55	12:01	11	04:06
G1005	Wuhan	Guangzhou South	08:12	12:24	22	04:12
G1007	Wuhan	Guangzhou South	09:30	13:37	2	04:07
G1013	Wuhan	Shenzhen North	12:07	16:26	1	04:19
G1015	Wuhan	Guangzhou South	13:45	18:16	2	04:31
G1017	Wuhan	Guangzhou South	14:39	18:51	2	04:12
G1019	Wuhan	Guangzhou South	15:45	20:00	2	04:15
G1021	Wuhan	Guangzhou South	16:58	21:22	1	04:24
G1101	Wuhan	Guangzhou South	06:45	11:11	1	04:26
G1103	Wuhan	Guangzhou South	06:53	11:21	1	04:28
G1105	Wuhan	Guangzhou South	07:37	11:31	1	03:54
G1107	Wuhan	Guangzhou South	08:26	13:09	2	04:43
G1117	Wuhan	Guangzhou South	11:45	16:06	1	04:21
G1123	Wuhan	Guangzhou South	13:58	18:23	1	04:25
G1125	Wuhan	Guangzhou South	15:07	19:03	1	03:56
G1127	Wuhan	Guangzhou South	15:23	19:34	1	04:11
⁝	⁝	⁝	⁝	⁝	⁝	⁝
G1129	Wuhan	Guangzhou South	15:50	19:47	1	03:57

¹ Train with 8-carriages that needs one crew unit; ² Train with 16-carriages that needs two crew units.

lists some train schedule information. Fourteen high-speed trains are operated by the China Railway Wuhan Bureau Group Co. Ltd. and 61 high-speed trains are operated by China Railway Guangzhou Bureau Group Co. Ltd.; both companies are the crew depots. Each railway company independently formulates crew scheduling based on their respective train conditions.

Table 3. Original train conductors crew scheduling.

Operators	ID	Duty	Form	Total Number of Crew Units
Wuhan Bureau Group Co. Ltd.	1 2 3 4 5 6 7 8	G1003-G10121 G1005 G1008-G1019 G1101-G1116 G1103-G1140 G1105-G1120-G1133 G1107-G1124 G1007-G1018	A3 A-A4 A-A A A A A-A A-A	12
Guangzhou Bureau Group Co. Ltd.	9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28	G1002-G1013-G60242 G1004-G1015 G1006-G1017 G6013-G1010-G1021 G1102-G1117-G1134 G1108-G1123 G6101-G1110-G1125-6116 G1112-G1127-G6118 G6105-G1114-G1129-G6120 G1122-G1135 G1014 G6011-G6016-G6001G6026 G6132-G6131-G6110-G6119 G6015-G6018-G6031 G6029-G6034-G6033 G6017-G6020-G6117-G6122 G6021-G6022-G6027 G6012-G6019-G6002-G6025 G6111-G6112 G6103-G6030	A A-A A-A A A A A A A A A-A A A A-A A A A-A A-A A A	26

¹ The train operated by Wuhan Bureau Group Co. Ltd.; ² The train operated by Guangzhou Bureau Group Co. Ltd.; ³ Each train in the duty needs one crew unit; ⁴ Each train in the duty needs two crew units.

is the original crew scheduling.

The time needed for the crew unit to perform preparatory activities before each duty and shift handover after each duty are both two hours. The standard transit time between trips is 20 min. The maximum allowed work time for a crew unit is 12 h.

The case is solved through the improved GA using Python 3.6 on a PC with a Intel (R) Core (TM) i7-6700 CPU 3.40 and 8.0 GB of RAM.

5.2. Parameter Tune and Performance Analysis

First, the experiment is repeated 50 times with different crossover rates (pc) and mutation rates (pm), setting the population size (pop size) of each generation to 100 [55]. Figure 7 shows the distributions of the objective value with different combinations of pc and pm, and the optimal objective values are listed in

Table 4. The optimal objective value of experiments with different combinations of pc and pm.

ID	pc	pm	Objective Value	ID	pc	pm	Objective Value
1	0.3	0.01	13,800	14	0.5	0.04	13,571
2	0.3	0.02	13,764	15	0.5	0.05	13,560
3	0.3	0.03	13,700	16	0.6	0.01	13,640
4	0.3	0.04	13,680	17	0.6	0.02	13,600
5	0.3	0.05	13,674	18	0.6	0.03	13,574
6	0.4	0.01	13,750	19	0.6	0.04	13,472
7	0.4	0.02	13,700	20	0.6	0.05	13,472
8	0.4	0.03	13,680	21	0.7	0.01	13,600
9	0.4	0.04	13,605	22	0.7	0.02	13,580
10	0.4	0.05	13,600	23	0.7	0.03	13,564
11	0.5	0.01	13,680	24	0.7	0.04	13,472
12	0.5	0.02	13,650	25	0.7	0.05	13,472
13	0.5	0.03	13,635	-	-	-	-

. From Table 4, we can see that the optimal objective value was obtained with pc = 0.7 and pm = 0.04. The objective value no longer decreased with the increase in pc and pm, indicating that the algorithm has reached convergence. In addition, the optimal value was found within fifty iterations.

Based on the above results, we used variable mutation rates to improve the solution quality. The specific parameter setting of the improved GA are shown in

Table 5. Mutation rate experimental parameters.

Experiment ID	Pop size	N	pc	pm
				$\mathit{N}\le 30$	$\mathit{N}\ge 30$
1	100	60	0.6	0.04	0.04
2	100	60	0.6	0.04	0.05
3	100	60	0.6	0.05	0.05
4	100	60	0.6	0.05	0.06
5	100	60	0.6	0.06	0.06
6	100	60	0.6	0.06	0.07

. The results are shown in Figure 8. We can conclude that: (1) compared with the fixed pm, the variable pm can avoid premature convergence of GA; and (2) the objective value is optimized by 1.65% with pm = 0.05(N $\le$ 30) and pm = 0.06(N $>$30).

Next, we compared the improved GA with the classical GA, PSO, and TS under the same maximum iteration. Fifty experiments were conducted. As shown in

Table 6. Comparison results of classical GA, classical PSO, classical TS, and improved GA.

Algorithm	Best Objective Value	Average Feasible Solution Rate	Average Running Time
Classical GA	13,871	83%	41 s
Classical PSO	14,515	74%	24 s
Classical TS	14,017	81%	31 s
Improved GA	13,250	100%	58 s

, we can conclude the following. (1) The classical PSO costs the least time, but finds the worst solution. The objective values obtained by classical GA is better than classical PSO and TS, but needs more time. Overall, the performance of the classical TS is between the classical GA and classical PSO. This may be because PSO is better at dealing with continuous optimization problems, and TS depends on the neighborhood search rules. As for GA, it needs more time for global search. (2) The improved GA took 58 s on average to find the optimal objective value, and the objective value was reduced by 4.47% compared with the classical GA. Compared with the classical GA, classical PSO, and classical TS, although the improved GA needed more time to find the optimal solution, it is acceptable, and the objective value was optimized by more than 4.47%.

The results show that the improved measures proposed in this paper can effectively remedy the shortcomings of the classical GA and improve spatial searchability. A better solution was obtained with finite computing time.

5.3. Results, Discussion, and Analysis

The optimal crew scheduling plan is shown in

Table 7. The optimized crew scheduling based on multi-crew depots.

Operators	ID	Duty	Form	Total Number of Crew Units
Wuhan Bureau Group Co. Ltd.	1 2 3 4 5 6 7	G1101-G1116-G1021 G1103-G1010-G1133 G1105-G1120-G1135 G1003-G1122 G1005-G1124 G1107-G1018 G1007-G1014	A A A A A-A A-A A-A	10
Guangzhou Bureau Group Co. Ltd.	8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29	G1102 -G1117-G1134 G6132-G6131-G6110-G6119 G6011-G6016-G6001-G6030 G6101-G1110-G1125 G6013-G1112-G1127 G6103-G1012 G1002-G1013-G6024 G6105-G1008-G1019 G6012-G6019-G6022-G6027 G6015-G6034-G6033 G6015-G6018-G6031-G6026 G6029-G6018-G6031 G1004-G1015-G1140-G6116 G1108-G1123-G6120 G6017-G6020-G6117-G6122 G1006-1017-G6118 G1006-1017 G6024-G6110 G6021-6002-6025 G1008-G1019 G1114-G1129 G6111-G6112	A A A A A A A A A-A A A A A-A A A A A A A-A A A A	25

. Without the restriction of the crew depot, the trips of the Wuhan Bureau Group Co. Ltd. can be connected with the trips of the Guangzhou Bureau Group Co. Ltd. and form a duty together. However, this is not allowed in original crew scheduling. The G1004 of Guangzhou Bureau Group Co. Ltd. can only connect with the G1015 in the original crew scheduling (Table 3), and the duration of the duty is 520 min, which is 200 min less than the required work time. In the improved crew scheduling plan, the G1004, G1015, and G6116 of the Guangzhou Bureau Group Co. can be connected with the G1140 of Wuhan Bureau Group Co. Ltd., and the duration of the duty was up to 645 min, increasing the duration of the duty by 25.96% compared with the original crew scheduling plan within the maximum allowed work time. The utilization of the crew units working time was obviously improved.

Furthermore, this paper solves the crew scheduling problem with different train types by disassembling the train with 16-carriages into the standard train (a trip need a crew unit) with 8-carriages, as shown in Figure 9. Thus, the crew scheduling can be more flexible, and more train types can also be handled in this way.

The left of Figure 9 is a part of the original crew scheduling plan, where G1006 and G1007 are both trains with 16-carriages and form duty 11 together, which need two crew units. The right of Figure 9 is a part of the modified crew scheduling plan, where G1006 and G1017 are considered as two trains with 8-carriages, respectively. Each trip can connect with any other trip, satisfying the constraints to form a duty. Therefore, duty 11 is split into duty 23 and duty 24. Each duty is the content of one crew unit. It is worth noting that a new trip of G6118 was added on duty 23. Because the proposed method can solve the connection of different train types, there was an increase in the crew unit working time not exceeding the maximum allowed work time.

The comparison results between the optimized crew scheduling plan and the original one are shown in

Table 8. Results between original and improved train conductors crew scheduling.

Crew Scheduling	Number of crew units	Average Transit Time (min)	Average Duty Time (min)	Average Duty Efficiency (%)
original	38	70.41	694.50	80.14%
improved	35	54.38	640.67	86.34%

. From Table 8, we can draw the following conclusions: (1) 35 crew units were required to carry out the operation of 75 trips, which reduced three crew units compared with the original crew scheduling plan; (2) The average transit time between trips in duty was 54.38 min in this paper, which was 22.76% lower than the original crew scheduling plan; (3) The average duty time was 640.67 min, which was 7.75% shorter than the original crew scheduling plan; (4) The utilization of crew unit work was 87.34% in the optimized crew scheduling plan, which was 6.20% higher than the original scheme. In summary, the method proposed in this paper completed the same task with fewer crew units, and the crew’s working time was effectively utilized, which is conducive to people’s sustainable development.

6. Conclusions

After the analysis of the differences in the operational policies between the drivers and conductors, a model on the railway crew scheduling problem was proposed. The proposed method breaks the restriction of the crew depot from the perspective of the sustainable development of mankind, and allows the crew unit to be on duty for all trips on the premise of meeting the work requirements. Therefore, the working time of the crew unit can be effectively utilized and sustainable. Furthermore, the proposed method can be used to solve the crew scheduling problem with more complex train types.

An improved GA was proposed and used to solve the problem. In the improved GA, the repair mechanism of infeasible genes transferred the infeasible solution to a feasible solution, improving the search space of solutions and the quality of the solutions. By using the improved GA, the optimal solution can be obtained within an acceptable time cost.

The proposed algorithm was applied to the Wuhan–Guangzhou high-speed railway, and the results showed that the modified crew scheduling plan reduced three crew units and increased the average utilization of crew unit working time by 6.20%. Besides, compared with the classical GA, the classical PSO and the classical TS, the improved GA effectively alleviated premature convergence and reduced the objective value of the solution by about more than 4.47%. All of the results verified the applicability and validity of the model and the algorithm proposed, which will significantly improve the service of the high-speed railway.

In future work, the cooperation mode between multiple crew depots will be studied in detail, for example, the cooperation mode on crew bases of the same or the different railway companies, the distribution problem of trips of the different crew bases, and the cost allocation problem. Besides, more powerful models and effective solution algorithms will be developed. Furthermore, research on real-time scheduling could lead to higher customer satisfaction in a fast-moving world with information available instantly, so the robust crew schedules could also be our focus for future studies.