Scheduling and Evaluation of a Power-Concentrated EMU on a Conventional Intercity Railway Based on the Minimum Connection Time

Abstract

Power-concentrated EMU trains have the advantages of being fast and comfortable, having a flexible formation and a short turn-back time, and so on. They can effectively release the transportation capacity of tense lines and hubs (the replacement of conventional trains with power-concentrated EMUs can reduce the time it takes to enter and exit locomotive yards by 40 min per train), optimize operating structures, improve the quality and efficiency of passenger products for conventional railways, and enhance the travel experience of passengers. Moreover, they have certain cost advantages and practical operational value for improving the market competitiveness of conventional railways. In this study, a two-stage, two-layer cycle method is adopted to solve the application plan of an EMU with the minimum total connection time. Through the decomposition of optimization objectives, the search space and the solution scale in each stage are reduced. In the first stage, the feasible number of routes and the number division plan of internal running lines are listed. In the second stage, an improved ant colony algorithm is designed to arrange and combine the internal running lines in each plan to improve the search quality and convergence speed, which changes the pheromone volatilization coefficient with iteration. The optimal number of routes, the number of internal routes, and the optimal sequence between routes are obtained. The study also puts forward a method of route division according to the passenger load factor, which can help railway bureaus adjust the capacity according to fluctuations in demand. A running diagram of six pairs of power-concentrated EMUs on an intercity railway is used as the background to solve the problem. The optimal connection plan with 14 groups of different route division plans was evaluated by using the entropy weight–TOPSIS method, and the optimal plan was obtained in the form of a route division method with two groups of routes with three pairs of trains in each group. Compared with the actual operation plan, the number of routes and the number of first-level repairs are reduced by 50%, respectively, which can effectively reduce the operation and maintenance costs of EMUs. Compared with the actual plan, the average operation mileage is increased by 100%, the average mileage loss is decreased by 54.6%, and the minimum distance traveled by EMUs is increased by 200%, which indicates that the mileage maintenance cycle of the actual operation plan is not fully used. The average number of tasks of EMUs is increased by 100%, indicating that the efficiency of EMUs in the actual operation plan needs to be improved. The traffic mileage balance is improved by 100%, indicating that the EMUs in different routes are more balanced.

1. Introduction

Electric multiple units (EMUs) are a type of multiple unit. A train is powered by electrical energy from an overhead contact line or a third rail to drive a traction motor to operate. According to the traction mode, EMUs are divided into the power-concentrated and power-distributed types. According to the electrical system, they can be further classified into DC and AC types. There is also a special type called a hybrid multiple unit that is designed by combining electric multiple units and diesel multiple units. In terms of EMU operation, Europe has both power-concentrated and power-distributed EMUs, and the main types of EMUs in France adopt the power-concentrated mode. In Europe, a power-concentrated EMU with a design speed above 250 km/h is typically represented by the French TGV series and the German ICE1/ICE2, and 200–230 km/h power-concentrated EMUs are typically represented by the Swedish X2000 (Bombardier, Dorval, Canada) and the Austrian/Czech Railjet (Siemens, Singapore). The Swedish X2000 is a high-speed train developed by Adtranz, a company active in the railway vehicle manufacturing industry during the 1990s. Adtranz was formed from the merger of the railway departments of Sweden’s ASEA and Switzerland’s Brown Boveri (BBC), subsequently becoming part of the ABB Group. Adtranz focused primarily on designing, manufacturing, and maintaining railway vehicles and systems, including metros, trams, commuter trains, high-speed trains, as well as related signaling and technical services, and was later acquired by Bombardier Transportation. The X2000 train was designed for Swedish State Railways (SJ) and first operated in the early 1990s. Its main feature is the ability to travel at speed up to 200 km/h with advanced tilting technology, allowing it to maintain higher speed on curves without causing discomfort to passengers. The X2000 mainly serves long-distance routes within Sweden, significantly enhancing the quality of Swedish railway passenger services by increasing speed and comfort. It has also been used on international routes, such as between Sweden and Denmark.Railjet is a high-speed train service operated by Austrian Federal Railways (ÖBB) that represents the high standards of modern European rail travel. Capable of reaching speed up to 230 km/h, Railjet trains serve as important means of transportation on domestic and international long-distance routes in Austria. While primarily serving major Austrian cities like Vienna, Salzburg, and Innsbruck, the Railjet network extends to neighboring countries including Germany, Switzerland, Hungary, and the Czech Republic. Railjet trains are manufactured by Siemens AG, a global technology company headquartered in Munich, Germany.

China began to explore the operation of 160 km/h power-concentrated EMUs in 2019, mainly for long-distance trains and short-distance intercity trains; 160 km/h power-concentrated EMU trains have the advantages of being fast and comfortable, having a flexible formation and short turn-back time, and so on. The integrated “locomotive + vehicle” formation mode can effectively release tight lines and hub transportation capacity, reduce the deduction coefficient, and improve the passing capacity when running on a conventional railway. Relevant research shows that the replacement of conventional trains with power-concentrated EMUs can reduce the time of exiting and entering locomotive yards by 40 min per train. The replacement of power-distributed EMUs on rapid intercity railways can improve the closeness of train speed classes and line speed classes, and it has certain cost advantages. Power-concentrated EMUs have practical operational value for improving the market competitiveness of conventional railways.

The preparation of an EMU routing plan occupies the core position in the field of EMU applications; it is the basis for the formulation of an EMU operation plan and maintenance plan, and it has a direct impact on the operational efficiency of railway transportation. An EMU routing plan can be defined as follows: based on the set train running diagram, the bottom succession logic between train tasks and the first-level maintenance location is clarified, and then the tasks in the train running diagram are integrated into a series of orderly operational routings to guide the actual operation of the production plan. At home and abroad, abundant research results have been obtained from the perspectives of EMU route planning, optimization models, and solution algorithms. Haahr et al. [1] compared the application of an optimization method based on mixed-integer linear programming in the operation planning and real-time scheduling stages, and they verified the effectiveness of their theoretical method in actual rail networks in Denmark and the Netherlands. In terms of solving algorithms, Giacco et al. [2] proposed a hybrid linear integer programming model for the network train turnover problem considering short-term maintenance plans, and they used a commercial solver to solve it so as to calculate an efficient solution in a short computing time. Erwin et al. [3] optimized the configuration of EMUs in a way that minimized capacity shortages during peak hours; they solved this with commercial software and verified it with Dutch railways. Arianna et al. [4] studied the problem of coupling or decoupling at stations according to changes in passenger demand during peak and off-peak hours; they solved this using a multi-commodity flow model based on integer programming and verified it with a Dutch timetable. Ralf et al. [5] proposed a generic hypergraph-based mixed-integer programming model for a rolling stock rotation problem and an integrated algorithm for its solution. In order to solve the turnover problem of EMUs, Valentina et al. [6] proposed a heuristic algorithm to solve the problem of the optimal allocation of EMU resources during peak hours. Sung et al. [7] proposed a two-stage algorithm to solve the minimum number of EMUs based on a weekly schedule. This was firstly to relax maintenance requirements, obtain the least costly EMU route, and then generate maintenance plans. Taking Korea’s high-speed railway as an example, a project is proposed to reduce the number of EMUs by 8.8% compared with the actual plan. Song et al. [8] proposed an autonomous route management system to enhance the efficiency of route management in stations. Rowan [9] presented a variable neighborhood search heuristic for the rolling stock rescheduling problem. Adolfo et al. [10] presented a model for optimizing fleet maintenance management with a particular application to train rolling stock fleets. Folco et al. [11] presented a novel flow formulation approach to schedule rolling stock maintenance activities by optimizing both maintenance costs and train availability. Prause et al. [12] studied the solution of the rolling stock rotation problem with predictive maintenance (RSRP-PdM) using an iterative refinement approach that was based on a state-expanded event graph. Pascariu et al. [13] addressed the optimized selection of alternative routes as a preliminary step for the real-time railway traffic management problem (rtRTMP). Satoshi et al. [14] proposed a two-phase rolling stock scheduling algorithm based on a mathematical programming method to cope with a temporary timetable. Pietro et al. [15] proposed an integer linear programming (ILP) model to solve a subproblem using a time-oriented model. Amorosi et al. [16] presented an integrated approach based on a mixed-integer mathematical formulation and computational results on a testbed of instances derived from a real-world case study. Pan et al. [17] formulated a novel stochastic integer programming model that was in an underlying space-time network to simultaneously optimize a train timetable and rolling stock circulation plan with a flexible train composition. Lin et al. [18] took the minimum number of train units required for a route as the main optimization objective and the minimum total time of the route as the secondary optimization objective, constructed an integer linear programming model, and took the Shanghai Bureau as a case study. Song et al. [19] uses model predictive control to dynamically search for routes and requests ground resources to complete route arrangements.

To sum up, the existing optimization studies of EMU routing plans mostly focus on distributed-power EMUs, and optimization studies on the application of planning algorithms for power-concentrated EMUs are lacking. At the same time, there is a lack of research on the integrated automatic scheduling and evaluation of EMU routes, and a system has not been formed for the evaluation index of EMU routes. In addition, existing studies rarely consider the impact of passenger flow demand on route scheduling. This study proposes a method of grouping routes according to the passenger load factor, which is conducive to the accurate adjustment of the daily capacity. Based on the characteristics of the maintenance system and the application of power-concentrated EMUs, this study puts forward a method for the automatic routing and optimization of power-concentrated EMUs, which can provide technical support for further expanding the operation and application of routing optimization plans.

2. Problem Description

The repair class for power-concentrated EMUs ranges from D1 repair to D6 repair, of which D1 repair includes the maintenance of the motor car and the daily maintenance of the trailer according to the standard of every 4000 ± 400 km or every 72 h. Due to the limitations of service and maintenance mileage, the daily mileage of a train is usually not more than 2000 km. According to a survey, some railway bureaus often face challenges due to insufficient train units for the passenger flow during peak periods. In the face of this problem, these railway bureaus usually adopt a balanced use of the maintenance cycle, which is made as large as possible to use the maximum allowable mileage of D1 repair in order to reduce unnecessary mileage waste, improve the efficiency of train units, and effectively reduce the number of train units required.

However, in practice, railway bureaus mostly use manual route mapping to arrange train operation plans, which makes it difficult to ensure the optimal allocation of transport resources. Therefore, it is necessary to carry out research to explore how to maximize the operating efficiency of EMUs, minimize the use of train units on the basis of making full use of the maintenance mileage, and automatically compile an optimal EMU routing plan. Such research can not only help solve the problem of tight train units during peak periods but can also improve overall operational efficiency and provide a more scientific and reasonable solution for railway transportation.

3. Optimization Model for the Routing Plan of Power-Concentrated EMUs

The main problems to be solved in the optimization model for the power-concentrated EMU routing plan established in this study are as follows: based on the given running diagram and D1 repair mileage, several plans for dividing the number of routes and the number of running lines within a route are calculated, and a sub-plan of the running line connection sequence with the minimum total connection time is found in each plan to improve the operation efficiency of EMUs.

3.1. Symbol Specification

Table 1. Meanings of footmarks.

Symbol	Meaning
$p$	Route division plan number, $p\in P$, $P$ represents a set of routing plans
$s$	Sub-plan sequence number, $s\in S_p$
$k$	Routing number, $k\in K_{p,s}$, $K_{p,s}$ represents a routing set
$i,j$	Running line number, $i,j\in V_{p,s,k}$, $V_{p,s,k}$ represents a set of train lines, $i,j=1,2,\cdots ,numtrai{n}_{p,s,k}$
$e$	Edge set, where the available nodes are represented as $(i,j)$, $e\in E_{p,s,k}$, $E_{p,k}$ represents a set of node pairs $(i,j)$

shows meanings of footmarks.

Table 2. Meanings of input parameters.

Symbol	Meaning
$c_{p,s,k,i,j}$	Weight of the connection edge, that is, the connection time, min, $c_{p,s,k,i,j}\in C_{p,s,k}$, $C_{p,s,k}$ is a weight set
$numtrai{n}_{p,s,k}$	Number of running lines in a route
$numrout{e}_{p,s}$	Number of routes in a sub-plan $s$
$numplan$	Number of routing division plans
$\Delta l_{p,s,k,i}$	Mileage of running line $i$
$d_{p,s,k,i}$	Departure time of running line $i$
$a_{p,s,k,i}$	Arrival time of a running line $i$
$turn\_\min$	Minimum turn-back time including the necessary time for the train turnaround time and the operation preparation time, which is 15 min here
$M$	A sufficiently large number
$L\_cycle$	First-class repair (D1 repair) mileage cycle
$T\_cycle$	First-class repair (D1 repair) time cycle

shows meanings of input parameters.

Table 3. Meanings of decision variables.

Symbol	Meaning
$x_{p,s,k,i,j}$	If the running line $i$ is connected to the running line $j$, $x_{p,s,k,i,j}=1$, otherwise, $x_{p,s,k,i,j}=0$
$numuni{t}_{p,s}$	Total train units in sub-plan $s$
$T_{cycle,p,s,k}$	Total turnaround time for routing $k$
$\min \_ctim{e}_{p,s}$	The minimum total connection time of all routes in the sub-plan $s$
Auxiliary decision variable
$l_{p,s,k,i}$	The cumulative mileage of the route
$t_{p,s,k,i}$	The cumulative running time of the route

shows meanings of decision variables.

3.2. Model Assumptions

• Do not consider the limitations of the maintenance capacity of the depot.
• Do not consider EMU coupling and decoupling.
• Secondary repairs and above (D2-D6 repairs) are not considered.

3.3. Objective Function

Based on the above analysis, the objective of route optimization for power-concentrated EMUs is mainly concentrated in two aspects: the first is maximizing the utilization of the D1 repair maintenance cycle, reducing unnecessary mileage loss, and, thereby, improving the operational efficiency of EMUs. The second is alleviating the shortage of train units by reducing the number of train units required and, at the same time, reducing the procurement cost of EMUs. Specifically, the first goal can be translated into minimizing the frequency of D1 repair, which, in practice, means minimizing the number of routes. Considering the mathematical relationship between the number of train units, the turnaround time, and the total connection time (as shown in Formulas (1) and (2)), the second goal can be simplified to minimizing the total connection time. It is worth noting that the effective use of the D1 repair cycle to reduce mileage loss also helps to save the number of EMUs, so goal 1 can be achieved indirectly by achieving goal 2 to some extent. Since the train running time is a fixed value, the final objective function can be simplified to Formula (3), that is, the pursuit of the minimum total connection time.

$$numunit={\displaystyle \frac{{\displaystyle \sum Travel time of all trains in route+{\displaystyle \sum total connection time in route}}}{train diagram period}}$$

(1)

$$\begin{array}{l}numuni{t}_{p,s}={\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}\frac{⌈T_{cycle,p,s,k}⌉}{1440}}={\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}\frac{⌈{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}(a_{p,s,k,i}-d_{p,s,k,i})}+{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}{\displaystyle \sum _{j=1}^{numtrai{n}_{p,s,k}}{c}_{p,s,k,i,j}\times x_{p,s,k,i,j}}}⌉}{1440}},\\ \forall p\in P,s\in S_p\end{array}$$

(2)

$$\min \_ctim{e}_{p,s}=\min {\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}{\displaystyle \sum _{j=1}^{numtrai{n}_{p,s,k}}{c}_{p,s,k,i,j}\times x_{p,s,k,i,j}}}},\forall p\in P,s\in S_p,$$

(3)

3.4. Constraint

A connection network can be used to describe the connections of train routes. Connection networks can be divided into two categories according to the node type, namely, connection networks based on station nodes and connection networks based on running lines. Figure 1 shows a connection network based on station nodes. A connection network based on station nodes can be constructed by considering the starting and ending stations of train lines as nodes and the connection relationships between train lines as arcs. Figure 2 shows a connection network based on running lines. A connection network based on lines can be constructed by treating lines as nodes and the possible connection relationships as arcs. In this study, a connection network based on running lines is selected.

$${\displaystyle \sum _{j=1,i\ne j}^{numtrai{n}_{p,s,k}}{x}_{p,s,k,i,j}=1,}\forall p\in P,s\in S_p,k\in K_{p,s},i\in V_{p,s,k},(i,j)\in E_{p,s,k},$$

(4)

$${\displaystyle \sum _{i=1,i\ne j}^{numtrai{n}_{p,s,k}}{x}_{p,s,k,i,j}=1,}\forall p\in P,s\in S_p,k\in K_{p,s},j\in V_{p,s,k},(i,j)\in E_{p,s,k},$$

(5)

$$\begin{array}{c}{c}_{p,s,k,i,j}=\left\{\begin{array}{cc}\hfill d_{p,s,k,j}-a_{p,s,k,i},\hfill & \hfill e_{p,s,k,i}=s_{p,s,k,j}\wedge d_{p,s,k,j}\ge a_{p,s,k,i}\wedge d_{p,s,k,j}-a_{p,s,k,i}\ge turn\_\min \\ \hfill 1440+d_{p,s,k,j}-a_{p,s,k,i},\hfill & \hfill e_{p,s,k,i}=s_{p,s,k,j}\wedge d_{p,s,k,j}<a_{p,s,k,i}\wedge d_{p,s,k,j}-a_{p,s,k,i}\ge turn\_\min \\ \hfill M,\hfill & \hfill e_{p,s,k,i}\ne s_{p,s,k,j}\vee d_{p,s,k,j}-a_{p,s,k,i}\prec turn\_\min \hfill \end{array}\right.,\hfill \\ \forall p\in P,s\in S_p,k\in K_{p,s},i,j\in V_{p,s,k},(i,j)\in E_{p,s,k}\hfill \end{array}$$

(6)

$$l_{p,s,k,j}=\Delta l_{p,s,k,j}+{\displaystyle \sum _{i=1,i\ne j}^{numtrai{n}_{p,s,k}}{x}_{p,s,k,i,j}\times l_{p,s,k,i}},\forall p\in P,s\in S_p,k\in K_{p,s},j\in V_{p,s,k},(i,j)\in E_{p,s,k},$$

(7)

$$l_{p,s,k,j}\le L\_cycle,\forall p\in P,s\in S_p,k\in K_{p,s},j\in V_{p,s,k},(i,j)\in E_{p,s,k},$$

(8)

$$t_{p,s,k,j}=(a_{p,s,k,j}-d_{p,s,k,j})+{\displaystyle \sum _{i=1,i\ne j}^{numtrai{n}_{p,s,k}}{x}_{p,s,k,i,j}\times t_{p,s,k,i}},\forall p\in P,s\in S_p,k\in K_{p,s},j\in V_{p,s,k},(i,j)\in E_{p,s,k},$$

(9)

$$t_{p,s,k,j}\le T\_cycle,\forall p\in P,s\in S_p,k\in K_{p,s},j\in V_{p,s,k},(i,j)\in E_{p,s,k},$$

(10)

$$x_{p,s,k,i,j}\in \left\{0,1\right\},p\in P,s\in S_p,k\in K_{p,s},i,j\in V_{p,s,k},(i,j)\in E_{p,s,k},i\ne j,$$

(11)

According to Figure 2, it can be seen that the connection problem of running lines is a network flow problem that follows the principle of flow balance, and there is only one outflow arc and inflow arc for each node. Formulas (4) and (5) indicate that for any train line, there is one and only one subsequent train line. For any train line, there is one and only one preceding train line. Formula (6) indicates that only when the terminal station of the train line is equal to the starting station of the train line, and the minimum return time between the two is satisfied, the two can be connected. The minimum connection time is based on experience. Extending the minimum train connection time at a station will reduce the number of possible train connection solutions, that is, reduce the solution space, so the quality of the solution may decline. Shortening the time of D1 repair work can help reduce the connection time. Formula (7) indicates that the accumulated mileage of a running line is equal to the sum of the accumulated mileage of the running line and the accumulated mileage of the previous running line. Formula (8) indicates that the accumulated mileage of the operating line must meet the limitation of the mileage cycle of the corresponding first-level repair. The maintenance mileage cycle is obtained in accordance with the “Notice of the National Railway Group on the issuance of the Maintenance Rules for the use of Power Concentrated EMUs”. Shortening the maintenance mileage cycle may make the original routing plan unfeasible, thus narrowing the solution space and reducing the quality of the routing plan. Formula (9) indicates that the accumulated running time of a running line is equal to the sum of the running time of the running line and the accumulated running time of the previous running line. Formula (10) indicates that the accumulated running time of the running line must meet the limitation of the corresponding time cycle of the first-level repair. Shortening the maintenance time cycle may make the original routing plan unfeasible, thereby reducing the solution space and reducing the quality of the routing plan. Extending the maintenance time cycle can help reduce the number of train units or shorten the total turnaround time for a route. Formula (11) is the constraint of a 0–1 variable.

4. Route Optimization for Power-Concentrated EMUs Based on the Improved Ant Colony Algorithm

4.1. Algorithm Description

The model established in Section 3 is a nonlinear mixed integer programming model, which is difficult to solve accurately with mathematical programming solvers. According to the characteristics of the problem, a hierarchical ant colony optimization method is designed. The core idea of the ant colony algorithm is that the pheromone concentration on the short path gradually increases, and the path with the highest pheromone concentration is most likely to be selected. The evaporation mechanism of pheromones ensures that the algorithm will not fall into a local optimum. As it is a parallel algorithm, each ant searches independently of the others, communicating only through pheromones. Therefore, the ant colony algorithm can be regarded as a distributed multi-agent system, and it starts to conduct an independent solution search at multiple points in the problem space at the same time, which not only increases its reliability but also endows it with a strong global search ability. Furthermore, an improved ant colony algorithm, which changes pheromone volatilization coefficient with iteration, has been adopted to improve the search quality and convergence speed.

Since intercity trains usually run in pairs, and each line has the same mileage, the maximum number of lines within each route can be determined based on the first-level repair mileage cycle. The process of solving this problem can be divided into two stages: an outer-layer problem and an inner-layer problem. Through the decomposition of optimization objectives, the search space and the solution scale in each stage are reduced. The outer layer determines all feasible plans for the number of routes and the number of operating lines within each route, while the inner layer involves finding the best connection plan for the internal lines based on the given outer-layer plan. By using a two-stage, two-layer cycle method, the number of routes and connection plans can be calculated simultaneously, and all route numbers and connection plans can be exhausted. In this way, the routing arrangement of EMUs can be optimized effectively using an iterative solution of the two-layer loop.

Stage 1: Route—train line division.

List all possible plans of routing–operating line divisions, with $numplan$ kinds in total. The basic requirement of route division is that the number of operating lines within a route is even, and the total mileage of operating lines within a route does not exceed the first-level repair mileage cycle.

Stage 2: The solution is based on the improved ant colony algorithm.

This problem is divided into two parts; the first part determines the running line selection plan for each route, and the second part determines the plan with the minimum total connection time of all routes.

Step 1: Determination of the selection plan of each route (external sub-plan cycle).

For the $p$th plan in the first stage, it is necessary to determine which running lines are selected in each route. The method is as follows: we divide $numtrain$ daily running lines into $numrout{e}_{p,s}$ routes, in which the number of running lines of the $k$th route is $numtrai{n}_{p,s,k}$, thus forming $numsubpla{n}_p$ sub-plans. The counting variable of the sub-plans is initialized as 0.

Step 2: Improvement of the ant colony algorithm to find the shortest path (internal iteration loop).

For the $s$th sub-plan, the problem of finding the shortest total connection time is analogous to the ant travel problem, which is equivalent to several ants visiting each route; ants are analogous to EMUs, access nodes are analogous to train lines, and the ant path is the connection between train lines. One EMU corresponds to one route, and each EMU needs to traverse all of the operating lines within each route.

Step 2-1: Initialization of the parameters. The distance matrix between nodes ($C_{p,s,k}$), pheromone importance factor ($\alpha$), heuristic information importance factor ($\beta$), pheromone volatilization coefficient ($\rho$), number of nodes at each route ($numtrai{n}_{p,s,k}$), number of ants at each route ($numan{t}_{p,s,k}$) (the number of ants is 1.5 times the number of nodes), total concentration of ant pheromones ($Q$), and cycle maximum ($iter\max$) pheromone matrix of each route ($pheromon{e}_{p,s,k}$) are initialized. The heuristic function matrix ($heuris$) (reciprocal of the distance matrix) is computed. The shortest-path-length matrix encountered by each generation and ever ($lengthbes{t}_{p,s,k}$), the total shortest-path-length matrix of each route in each generation ($lengthbest\_tota{l}_{p,s}$), the total shortest-path-length matrix of each sub-plan ($lengthbest\_s\_tota{l}_p$), the path table matrix of each route ($pathtabl{e}_{p,s,k}$), the shortest-path matrix of each route encountered by each generation and ever ($pathbes{t}_{p,s,k}$), the total shortest-path matrix of each route in each generation ($pathbest\_tota{l}_{p,s}$), and the total shortest-path matrix of each sub-plan ($pathbest\_s\_tota{l}_p$) are initialized.

Step 2-2: The number of initialization cycles is set to 0. The list of nodes not accessed by ants is initialized as $allowe{d}_{p,s,k,a,i}=[1,2,\cdots ,numtrai{n}_{p,s,k}]$, and the matrix of the ant path length table ($lengt{h}_{p,s,k}$) is initialized. The array randomly generates ants in the initial city, as recorded in the first column of the path table. The initial node is removed from the list of unvisited nodes, and the selection probability of ants from the initial node to all unvisited nodes is calculated. The calculation formula is as follows:

$$\begin{array}{l}{p}_{p,s,k,i,j}^a(iter)={\displaystyle \frac{{\left[{\tau }_{p,s,k,i,j}(iter)\right]}^{\alpha }\times {\left[\frac{1}{c_{p,s,k,i,j}}\right]}^{\beta }}{{\displaystyle \sum _{j\in allowe{d}_{p,s,k,a,i}}{\left[{\tau }_{p,s,k,i,j}(iter)\right]}^{\alpha }\times {\left[\frac{1}{c_{p,s,k,i,j}}\right]}^{\beta }}}},\\ \forall a\in A_{p,s,k},i\in V_{p,s,k},j\in allowe{d}_{p,s,k,a,i},iter,p\in P,s\in S_p,k\in K_{p,s}\end{array}$$

(12)

Step 2-3: The roulette method is used to generate random numbers $rand\in [0,1)$, and it is compared with the cumulative probability of the unvisited nodes. The first node whose cumulative probability exceeds the random number is selected as the next access node, and this node is deleted from unvisited list. This step reflects the improvement of the ant colony algorithm; the traditional ant colony algorithm generally takes the node with the highest probability as the next access node.

Step 2-4: Steps 2-2 and 2-3 are repeated to traverse the unvisited list of each ant until the unvisited list is empty.

Step 2-5: The pheromones on each side are updated by using the ant cycle model, and the calculation formulas for pheromone concentration and the increase in pheromone concentration are shown in Equations (13)–(15). Formula (13) indicates that after each iteration, the pheromone concentration on each path needs to be updated to the sum of the remaining value of the original pheromone concentration after volatilization over time and the new pheromone concentration on each path after an iteration. Formula (14) indicates that when the number of iterations is less than the threshold, the volatilization coefficient is constant, and when the number is greater than the threshold, the volatilization coefficient increases. Formula (15) indicates that the pheromone concentration increment of the path with ant selection is inversely proportional to the total path length, and the pheromone concentration left by the same ant on each path is equal, while the pheromone increment of the path without ant selection is 0. Finally, the path length matrix and total path length matrix of each route are updated. This step is the embodiment of the improved ant colony algorithm. Firstly, a low volatilization coefficient is set to expand the search range, and the volatilization coefficient is gradually increased with the increase in generation to accelerate the convergence speed. The search quality and convergence speed of the algorithm are both enhanced through the dynamic volatilization mechanism.

$$\begin{array}{l}{\tau }_{p,s,k,i,j}(iter+1)=(1-\rho ){\tau }_{p,s,k,i,j}(iter)+{\displaystyle \sum _{a=1}^{numan{t}_{p,s,k}}\Delta {\tau }_{p,s,k,i,j}^a}(iter),\\ \forall a\in A_{p,s,k},i\in V_{p,s,k},j\in allowe{d}_{p,s,k,a,i},iter,p\in P,s\in S_p,k\in K_{p,s}\end{array},$$

(13)

$$\rho (\nu )=\left\{\begin{array}{cc}{\rho }_0& \nu \le {\nu }_1\\ 1-{\displaystyle \frac{\nu \max -\nu }{\nu \max }}\times (1-{\rho }_0)& \nu >{\nu }_1\end{array}\right.,$$

(14)

$$\begin{array}{c}\Delta {\tau }_{p,s,k,i,j}^a(iter)=\left\{\begin{array}{cc}{\displaystyle \frac{Q}{L_{p,s,k,a}(iter)}},& the ath \mathrm{ant} \mathrm{passes} \mathrm{the} \mathrm{edge}(i,j) \mathrm{in} \mathrm{this} \mathrm{iteration} \mathrm{round}\\ 0,& else\end{array}\right.\hfill \\ \forall a\in A_{p,s,k},i\in V_{p,s,k},j\in allowe{d}_{p,s,k,a,i},iter,p\in P,s\in S_p,k\in K_{p,s}\hfill \end{array},$$

(15)

Step 2-6: The iteration is stopped when the maximum number of iterations is reached. After each iteration round, the shortest-path-length matrix of each generation and the previous generation of each route ($lengthbes{t}_{p,s,k}$), the total shortest-path-length matrix of each route ($lengthbest\_tota{l}_{p,s}$), the total shortest-path-length matrix of each generation and the previous generation of each route ($pathbes{t}_{p,s,k}$), and the shortest-path table of each route of each generation ($pathbest\_tota{l}_{p,s}$) are updated. In this method, if the total shortest path length of the current round is greater than or equal to that of the previous round, the total shortest path length of the current round is equal to the total shortest path length of the previous round. Otherwise, the current total shortest path length equals the total shortest path length of the current round.

Step 2-7: All sub-plans are traversed, steps 2-2–2-6 are repeated for each sub-plan, and the total shortest-path-length matrix of each plan ($lengthbest\_s\_tota{l}_p$) and total shortest-path matrix of each plan ($pathbest\_s\_tota{l}_p$) are updated. The update method is the same as that in step 2-6, and the shortest path iteration diagram, shortest path length, and total shortest path of each plan are provided as output.

The Algorithm 1 pseudocode is as follows.

Algorithm 1: Improved Ant Colony Algorithm for route optimization

Input: citydis, alpha, belta, ro, numcity, Q, itermax, iter1, heuris, listcity Output: lengthbest_s_total, pathbest_s_total Procedure for running line division select_city for s in select_city:   for t in s:      remove t from listcity   obtain numcity Procedure for running line division   Input numant     iter=0     while iter<itermax:       for i in range(numant):          visiting=pathtable[i,0]          remove visiting from unvisited          for j in range(numcity):           for k in range(listunvisited):              choose city according to roulette rule              update lengthbest_s_total and pathbest_s_total            update pheromone according to Formula (13)       iter+1    s_num+1

A flowchart of the improved ant colony algorithm is shown in Figure 3.

4.2. Case Study

4.2.1. Basic Experimental Data

In order to evaluate the performance of the proposed algorithm, stations A and B on an intercity railway line in western China are selected as experimental objects. The time span for this study is set to 1440 min, during which a total of 12 trains are running. It is assumed that the time required for a power-concentrated EMU to turn back immediately after arriving at the terminal is 15 min, the mileage cycle of the first-level repair of a power-concentrated EMU is 4000 km, the repair time cycle is 72 h, and the distance between station A and station B is 471 km.

Table 4. Schedule of power-concentrated EMU trains from station A to B.

Train Number	Arrive/Depart	Station		Train Number	Arrive/Depart	Station
		Station A	Station B			Station A	Station B
1	Arrive	—	12:55	2	Arrive	—	12:38
	Depart	9:00	—		Depart	8:45	—
3	Arrive	—	16:08	4	Arrive	—	14:10
	Depart	12:08	—		Depart	10:10	—
5	Arrive	—	16:48	6	Arrive	—	16:50
	Depart	13:20	—		Depart	13:24	—
7	Arrive	—	18:28	8	Arrive	—	20:48
	Depart	14:29	—		Depart	16:50	—
9	Arrive	—	21:22	10	Arrive	—	21:50
	Depart	17:20	—		Depart	17:50	—
11	Arrive	—	22:29	12	Arrive	—	23:03
	Depart	18:18	—		Depart	19:00	—

shows schedule of power-concentrated EMU trains from station A to B. Figure 4 shows the actual routes of power-concentrated EMUs between stations A and B. Figure 5 is a Gantt chart of an EMU plan for an actual two-day route plan (The ordinate represents the route number, the number in the square is the train number, the first train in each route is shown in red, the second train is shown in pink, and so on).

4.2.2. Experimental Results

Python 3.12 was used on a Dell computer equipped with a 13th Gen Intel(R) Core(TM) i7-13620H 2.40GHz processor (manufactured by Intel, which is located in Santa Clara, America), 16 GB RAM, and the Windows 11 Home operating system (Chinese version) to solve the above algorithm to test the operation of power-concentrated EMUs in 2880 min and automatically compile a route plan for EMUs. After several rounds of experiments, the optimal parameter values were finally selected, as shown in

Table 5. Optimal parameter values.

Initial Pheromone Concentrations for All Paths	Pheromone Importance Factor	Heuristic Function Importance Factor	Volatilization Coefficient	Total Pheromone	Cycle Maximum
1	1	5	0.1	1	200

. According to the number of routes and the number of operating lines within the route division, a total of seven different route division plans were tested, and the experimental results are summarized in

Table 6. Minimum total connection times and best routes.

Plan Number	Route Number	Line Division	Minimum Total Connection Time/min	Optimum Routing
1	2	(8, 4)	2372	3−8−1−6−9−2−5−10, 11−4−7−12
2	3	(8, 2, 2)	1698	7−12−1−6−11−2−5−10, 3−8, 4−9
3	2	(6, 6)	2327	4−7−12−1−6−9, 11−2−5−10−3−8
4	3	(6, 4, 2)	1469	4−7−12−1−6−9, 11−2−5−10, 3−8
5	4	(6, 2, 2, 2)	1101	1−6−11−2−5−10, 3−8, 4−9, 7−12
6	3	(4, 4, 4)	2342	3−8−1−6, 9−2−5−10, 11−4−7−12
7	4	(4, 4, 2, 2)	1437	12−1−6−9, 11−2−5−10, 3−8, 4−7
Actual plan	4	(6, 2, 2, 2)	1679	1−6−11−2−5−10, 3−8, 7−12, 9−4

. As can be seen from the data in Table 6, plan 6 achieved the smallest total connection time of only 1101 min, and its corresponding routing arrangements were 1−6−11−2−5−10, 3−8, 4−9, and 7−12. In addition, the evolution process of the optimal path length for the sub-plan of plan 5 is shown in Figure 6. In Figure 6, the horizontal coordinate is the iteration of all internal running line connection plans for all routes, and the vertical coordinate is the optimal value of each plan after iteration with the ant colony algorithm. A “plan” here refers to a plan for the number of routes and the selection of running lines within a route. For example, 12 running lines are divided into two routes, each of which has eight running lines and four running lines. The first route has running lines 1, 2, 3, 4, 5, 6, 7, and 8, and the second route has running lines 9, 10, 11, and 12.

In order to study the sensitivity of parameters in the ant colony algorithm, plan 1 is taken as the research object to explore the effects of different ant numbers, pheromone importance factors, heuristic function importance factors, and iteration numbers on the minimum connection time and calculation time.

Table 7. Algorithm parameter sensitivity analysis for plan 1.

Parameter Analysis Plan	Parameter				Calculation Index
	Ant Number	Pheromone Importance Factor	Heuristic Function Importance Factor	Iteration Number	Minimum Connection time/min	Calculation Time/s
P1	18	1	5	200	2372	242.6850514412
P2	6	1	5	200	2372	51.7229347229
P3	12	1	5	200	2372	389.9291911125
P4	24	1	5	200	2372	634.6604983807
P5	18	0.5	5	200	2372	654.1037375927
P6	18	2	5	200	2372	914.2404844761
P7	18	3	5	200	2372	432.2556602955
P8	18	4	5	200	No solution	--
P9	18	5	5	200	No solution	--
P10	18	1	2	200	2372	885.7629337311
P11	18	1	3	200	2372	491.3823523521
P12	18	1	4	200	2372	601.4855833054
P13	18	1	6	200	2372	247.1675834656
P14	18	1	5	10	2372	8.1696150303
P15	18	1	5	50	2372	36.9381115437
P16	18	1	5	100	2372	73.2007288933
P17	18	1	5	150	2372	109.4171283245

shows algorithm parameter sensitivity analysis for plan 1. P1-P4 analyze the ant number while fixing the pheromone importance factor, heuristic function importance factor, and iteration number. Figure 7 shows influences of different ant colony algorithm parameters on the calculation time. The ant number is increased from 6 to 24, and calculation time with different ants is shown in Figure 7a. P1 and P5-P9 analyze the pheromone importance factor while fixing the ant number, heuristic function importance factor, and iteration number. The pheromone importance factor is increased from 0.5 to 3. Calculation time with different pheromone importance factors is shown in Figure 7b. P1 and P10-P13 analyze the heuristic function importance factor while fixing the ant number, pheromone importance factor, and iteration number. The heuristic function importance factor is increased from 2 to 6. Calculation time with different heuristic function importance factors is shown in Figure 7c. P1 and P14-P17 analyze the iteration number while fixing the ant number, pheromone importance factor, and heuristic function importance factor. The iteration number is increased from 10 to 200. Calculation time with different iteration numbers is shown in Figure 7d.

With the increase in the ant number, the calculation time generally increases, but it does not affect the calculation results. The pheromone importance factor represents the influence of the pheromone concentration on ant path selection. The probability of path selection is completely determined by the path length when the pheromone importance factor is too small, and the randomness is weakened when the pheromone importance factor is too large. The pheromone importance factor has no direct relationship with the calculation time, and the calculation time is the shortest when the pheromone importance factor is 1. The heuristic function importance factor represents the influence of the path distance on ant path selection, and convergence may be too slow when the heuristic function importance factor is too small. As shown in Figure 7c, the calculation time decreases with the increase in the heuristic function importance factor. The calculation time is shortest when the heuristic function importance factor is 5. As can be seen in Figure 7d, the calculation time increases with the increase in the iteration number.

4.2.3. Extended Experiment—Route Optimization for EMUs Considering Passenger Flow Demand

In view of the significant time-segment imbalance in intercity railway passenger flow demand, some intercity railways have been exploring the implementation of “daily charts”; “daily charts” are divided into peak charts, weekend charts, and daily charts. A peak chart is a full chart, a weekend chart is implemented on the basis of the peak chart, and a daily chart is implemented on the basis of the weekend chart. When operation lines are removed, this is often carried out with the unit of a route; otherwise, other running lines in the route after the line cannot be connected. Even if the running lines are removed according to the route, a train with a high load factor may be removed from the route, which will affect the income of the railway department. This problem can be avoided if running lines with a consistent load factor level are divided into one route. This section assumes that the train diagram mentioned in Section 4.2.1 is a full chart, and it explores how to combine route optimization with the passenger load factor. The section intends to study the grouping of routes according to different passenger load factor range thresholds of running lines within a route, determine the number of routes and the number of routes within a route, and then solve the optimal connection plan of running lines within the route. The step of grouping according to the passenger load factor is equivalent to eliminating an unreasonable plan with too large of a range of passenger load factors in the outer loop of the algorithm described in Section 4.1, reducing the number of plans in the outer loop of the algorithm, and greatly saving calculation time. A flowchart of the routing optimization algorithm considering passenger flow demand is shown in Figure 8. Imported parameters are shown in

Table 8. Imported parameters.

Symbol	Meaning
$r_i$	Load rate of $i$

. The load factor is shown in

Table 9. Table of load factors.

Train Number	Load Factor/%	Train Number	Load Factor/%	Train Number	Load Factor/%	Train Number	Load Factor/%	Train Number	Load Factor/%	Train Number	Load Factor/%
1	50	2	60	3	90	4	80	5	50	6	50
7	80	8	80	9	50	10	60	11	80	12	80

. The route division plan and the corresponding optimal routes with different ranges are shown in

Table 10. The route division plan and the corresponding optimal route with different ranges.

Plan Number	Route Number	Line Division	Route Load Factor Range Threshold	Number of Plans Meeting the Load Factor Connection Condition	Minimum Total Connection Time/min	Optimal Routing	Route Average Range of Passenger Load Factor
1−1	2	(8, 4)	10%	0	—	—	—
1−2			20%	0	—	—	—
1−3			30%	36	2556	4−11−2−5−10−1−6−9, 7−12−3−8	20%
1−4			40%	495	2372	3−8−1−6−9−2−5−10, 11−4−7−12	20%
1−5			100%	495	2372	3−8−1−6−9−2−5−10, 11−4−7−12	20%
2−1	3	(8, 2, 2)	10%	—	—	—	—
2−2			20%	—	—	—	—
2−3			30%	636	1698	7−12−1−6−11−2−5−10, 3−8, 4−9	23%
2−4			40%	2970	1698	7−12−1−6−11−2−5−10, 3−8, 4−9	23%
2−5			100%	2970	1698	7−12−1−6−11−2−5−10, 3−8, 4−9	23%
3−1	2	(6, 6)	10%	2	2412	2−5−10−1−6−9, 11−4−7−12−3−8	10%
3−2			20%	2	2412	2−5−10−1−6−9, 11−4−7−12−3−8	10%
3−3			30%	42	2412	2−5−10−1−6−9, 11−4−7−12−3−8	10%
3−4			40%	924	2327	4−7−12−1−6−9, 11−2−5−10−3−8	35%
3−5			100%	924	2327	4−7−12−1−6−9, 11−2−5−10−3−8	35%
4−1	3	(6, 4, 2)	10%	30	1627	2−5−10−1−6−9, 11−4−7−12, 3−8	7%
4−2			20%	35	1627	2−5−10−1−6−9, 11−4−7−12, 3−8	7%
4−3			30%	2765	1469	4−7−12−1−6−9, 11−2−5−10, 3−8	23%
4−4			40%	13,860	1469	4−7−12−1−6−9, 11−2−5−10, 3−8	23%
4−5			100%	13,860	1469	4−7−12−1−6−9, 11−2−5−10, 3−8	23%
5−1	4	(6, 2, 2, 2)	10%	180	1155	2−5−10−1−6−9, 3−8, 4−11, 7−12	5%
5−2			20%	270	1155	2−5−10−1−6−9, 3−8, 4−11, 7−12	5%
5−3			30%	28,350	1101	1−6−11−2−5−10, 3−8, 4−9, 7−12	18%
5−4			40%	83,160	1101	1−6−11−2−5−10, 3−8, 4−9, 7−12	18%
5−5			100%	83,160	1101	1−6−11−2−5−10, 3−8, 4−9, 7−12	18%
6−1	3	(4, 4, 4)	10%	—	—	—	—
6−2			20%	60	—	—	—
6−3			30%	7350	2347	8−1−6−9, 11−2−5−10, 4−7−12−3	23%
6−4			40%	34,650	2342	3−8−1−6, 9−2−5−10, 11−4−7−12	17%
6−5			100%	34,650	2342	3−8−1−6, 9−2−5−10, 11−4−7−12	17%
7−1	4	(4, 4, 2, 2)	10%	900	1565	10−1−6−9, 11−4−7−12, 2−5, 3−8	8%
7−2			20%	1380	1565	10−1−6−9, 11−4−7−12, 2−5, 3−8	8%
7−3			30%	73,500	1437	12−1−6−9, 11−2−5−10, 3−8, 4−7	18%
7−4			40%	207,900	1437	12−1−6−9, 11−2−5−10, 3−8, 4−7	18%
7−5			100%	207,900	1437	12−1−6−9, 11−2−5−10, 3−8, 4−7	18%
Actual plan	4	(6, 2, 2, 2)	—	—	1679	1−6−11−2−5−10, 3−8, 7−12, 9−4	18%

(when the range is 100%, the experimental results are the same without considering the passenger load factor).

The Algorithm 2 pseudocode is as follows.

Algorithm 2: Improved Ant Colony Algorithm for route optimization considering the load factor rate

Input: citydis, alpha, belta, ro, numcity, Q, itermax, iter1, heuris, listcity, kezuolv Output: lengthbest_s_total, pathbest_s_total Procedure for running line division select_city for s in select_city:   maxkezuolv=max(kezuolv1)   minkezuolv=min(kezuolv1)   jicha=maxkezuolv-minkezuolv   if(jicha>threshold):     continue   for t in s:     remove t from listcity   obtain numcity Procedure for running line division   Input numant     iter=0     while iter<itermax:       for i in range(numant):         visiting=pathtable[i,0]         remove visiting from unvisited         for j in range(numcity):             for k in range(listunvisited):                choose city according to roulette rule                update lengthbest_s_total and pathbest_s_total              update pheromone according to Formula (13)       iter+1       s_num+1

The following conclusions can be drawn from Table 10. (1) When the passenger load factor range threshold is too low, there may be no routing plan that meets the passenger load factor continuity conditions. (2) With the increase in the passenger load factor range threshold, the number of routing plans that meet the passenger load factor connection conditions gradually increases, and the minimum connection time gradually decreases. That is, after the passenger load factor connection conditions are added, the optimal connection time will be longer than the minimum connection time. (3) Since the maximum range of the passenger load factor of the running lines is 40%, when the passenger load factor threshold is 40% or higher, the experimental results are the same as those without considering the passenger load factor (the passenger load factor threshold is 100%), so experiments using values between 40% and 100% are not carried out. (4) The minimum load factor threshold of different feasible line-grouping plans is different. (In plan 6−2, there are 60 plans that meet the conditions of passenger load factor connection, but they do not meet the condition of the minimum connection time, making them unreasonable plans, so there is no plan that meets the requirements.) According to the best connection time and best route, 14 plans are listed in Table 10; these are 1−3, 1−4, 2−3, 3−1, 3−4, 4−1, 4−3, 5−1, 5−3, 6−3, 6−4, 7−1, 7−3, and the actual plan. These 14 plans are evaluated below.

5. Route Evaluation for Power-Concentrated EMUs

The entropy–TOPSIS method integrates the objectivity, comprehensiveness, and good explanatory ability of the entropy weight method, and it is very suitable for the evaluation and selection of multiple plans. In this section, we will apply the entropy weight–TOPSIS method to evaluate routing plans for power-concentrated EMUs. Specifically, the entropy weight method is used to construct a weighted standardized matrix, and then the TOPSIS method is used to determine the priority ranking of each plan. This method can not only objectively reflect the advantages and disadvantages of each plan but can also provide a clear decision basis to help select the best routing plan.

5.1. Selection of Route Evaluation Indexes for Power-Concentrated EMUs

5.1.1. Operational Efficiency Index

• Number of routes ($U_1$):

  $$U_1=numrout{e}_{p,s},\forall p\in P,s\in S_p,$$

  (16)
• Number of D1 repairs ($U_2$):

  $$U_2=numrout{e}_{p,s},\forall p\in P,s\in S_p,$$

  (17)
• Minimum total connection time ($U_3$) (excluding the connection time when returning to the starting point, i.e., the D1 repair time):

  $$U_3=lengthbest\_s\_tota{l}_p,\forall p\in P,$$

  (18)
• Total turnaround time for routing ($U_4$):

  $$U_4={\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}{T}_{cycle,p,s,k}},\forall p\in P,s\in S_P,$$

  (19)
• Average turnaround time for routing ($U_5$):

  $$U_5={\displaystyle \frac{{\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}{T}_{cycle,p,s,k}}}{U_1}},\forall p\in P,s\in S_p,$$

  (20)
• Number of days ($U_6$):

  $$U_6=numuni{t}_{p,s},\forall p\in P,s\in S_p,$$

  (21)
• Number of train units ($U_7$):

  $$U_7=numuni{t}_{p,s},\forall p\in P,s\in S_p,$$

  (22)
• Average route mileage ($U_8$):

  $$U_8={\displaystyle \frac{{\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}\Delta l_{p,s,k,i}}}}{U_1}},\forall p\in P,s\in S_p,$$

  (23)
• Average route distance loss ($U_9$):

  $$U_9=L\_cycle-U_8,$$

  (24)
• Vehicle kilometers per day ($U_{10}$):

  $$U_{10}={\displaystyle \frac{{\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}\Delta l_{p,s,k,i}}}}{U_6}},\forall p\in P,s\in S_p,$$

  (25)
• Average daily occupancy time ($U_{11}$):

  $$U_{11}={\displaystyle \frac{{\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}(a_{p,s,k,i}-d_{p,s,k,i})}}}{U_6}},\forall p\in P,s\in S_p,$$

  (26)
• Utilization rate of EMUs ($U_{12}$):

The utilization rate of EMUs refers to the ratio of the occupied time of EMUs to the total time of the day.

$$U_{12}={\displaystyle \frac{U_{11}}{1440}},$$

(27)

13.

Average connection time between trains ($U_{13}$):

$$U_{13}={\displaystyle \frac{U_3}{{\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}numtrai{n}_{p,s,k}-U_1}}},\forall p\in P,s\in S_p,$$

(28)

14.

Average number of tasks performed by EMUs ($U_{14}$):

$$U_{14}={\displaystyle \frac{{\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}numtrain}}{U_1}},\forall p\in P,s\in S_p,$$

(29)

15.

Minimum mileage of EMUs ($U_{15}$):

$$U_{15}={\min }_{k\in K_{p,s}}{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}\Delta l_{p,s,k,i}},\forall p\in P,s\in S_p,$$

(30)

16.

Maximum mileage of EMUs ($U_{16}$):

$$U_{16}={\max }_{k\in K_{p,s}}{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}\Delta l_{p,s,k,i}},\forall p\in P,s\in S_p,$$

(31)

5.1.2. Balance Indicators

1.

Route mileage balance ($U_{17}$):

Route mileage balance refers to the difference in the maximum loss mileage and the minimum loss mileage.

$$U_{17}=U_{16}-U_{15},$$

(32)

2.

Variance of route utilization ($U_{18}$):

The variance of route utilization refers to the ratio of the occupied running time of the route to the total turnover time of the route.

$$U_{18}={\displaystyle \frac{{\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}(\frac{{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}(a_{p,s,k,i}-d_{p,s,k,i})}}{T_{cycle,p,s,k}}-\frac{{\displaystyle \sum _{k=1}^{numrout{e}_{p,s}}\frac{{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}(a_{p,s,k,i}-d_{p,s,k,i})}}{T_{cycle,p,s,k}}}}{k}}{)}^2}{k}},\forall p\in P,s\in S_p,$$

(33)

5.1.3. Load Factor Consistency

1.

Route average load factor range ($U_{19}$):

The average range of the route load factor reflects the average fluctuation degree of the passenger load factor of each route. The smaller the average range is, the higher the horizontal consistency of the running lines within each route is, and the more consistent it is with the expectation of grouping the routes according to the level of the passenger load factor.

$$o={\displaystyle \frac{{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}(\max r_i-\min r_i)}}{numrout{e}_{p,s}}},\forall p\in P,s\in S_p,$$

(34)

2.

Route mean standard deviation of the load factor ($U_{20}$):

The route mean standard deviation of the load factor reflects the average dispersion degree of the passenger load factor of each route, and it also reflects the degree of coincidence when grouping the running lines according to the passenger load factor.

$$\varpi ={\displaystyle \frac{\sqrt{\frac{{\displaystyle \sum _{i=1}^{numtrai{n}_{p,s,k}}{(r_i-\overline{r})}^2}}{numtrai{n}_{p,s,k}}}}{numrout{e}_{p,s}}},\forall p\in P,s\in S_p,$$

(35)

5.2. Plan Evaluation Based on the Entropy Weight–TOPSIS Method

Taking six pairs (12 train lines) from station A to station B of a western railway bureau in the second quarter of 2024 as an example, fourteen different route grouping plans are designed, as shown in Table 10. These are denoted as plans 1−3, 1−4, 2−3, 3−1, 3−4, 4−1, 4−3, 5−1, 5−3, 6−3, 6−4, 7−1, 7−3, and 8. Based on this, an evaluation model for power-concentrated EMUs is constructed. Route evaluation index system for power-concentrated EMUs is shown in

Table 11. Route evaluation index system for power-concentrated EMUs.

Primary Index	Secondary Index	Index Type
Operation Efficiency	$\mathrm{Number} \mathrm{of} \mathrm{routes} U_1$	Negative
	Number of D1 repairs $U_2$	Negative
	Minimum total connection time $U_3$	Negative
	Total turnaround time for routing $U_4$	Negative
	Average turnaround time for routing $U_5$	Negative
	Number of days $U_6$	Negative
	Number of train units $U_7$	Negative
	Average route mileage $U_8$	Positive
	Average route distance loss $U_9$	Negative
	Vehicle kilometers per day $U_{10}$	Positive
	Average daily occupancy time $U_{11}$	Positive
	Utilization rate of EMUs $U_{12}$	Positive
	Average connection time between trains $U_{13}$	Negative
	Average number of tasks performed by EMUs $U_{14}$	Positive
	Minimum mileage of EMUs $U_{15}$	Positive
	Maximum mileage of EMUs $U_{16}$	Positive
Operating Balance	Route mileage balance $U_{17}$	Negative
	Variance of route utilization $U_{18}$	Negative
Load Factor Consistency	Route average load factor range $U_{19}$	Negative
	Route mean standard deviation of the load factor $U_{20}$	Negative

. The specific evaluation index values are shown in

Table 12. Evaluation index values.

Plan Num	1−3	1−4	2−3	3−1	3−4	4−1	4−3
$U_1$	2	2	3	2	2	3	3
$U_2$	2	2	3	2	2	3	3
$U_3$ /min	2556	2372	1698	2412	2327	1627	1469
$U_4$ /min	5371	5187	4513	5227	5142	4442	4284
$U_5$ /min	2685.5	2593.5	1504.333	2613.5	2571	1480.667	1428
$U_6$	5	5	5	5	5	5	5
$U_7$	5	5	5	5	5	5	5
$U_8$ /km	2826	2826	1884	2826	2826	1884	1884
$U_9$ /km	1174	1174	2116	1174	1174	2116	2116
$U_{10}$ /km	1130.4	1130.4	1130.4	1130.4	1130.4	1130.4	1130.4
$U_{11}$ /min	563	563	563	563	563	563	563
$U_{12}$	0.390972222	0.390972	0.390972	0.390972	0.390972	0.390972	0.390972
$U_{13}$ /min	255.6	237.2	188.6667	241.2	232.7	180.7778	163.2222
$U_{14}$	6	6	4	6	6	4	4
$U_{15}$ /km	1884	1884	942	2826	2826	942	942
$U_{16}$ /km	3768	3768	3768	2826	2826	2826	2826
$U_{17}$	1884	1884	2826	0	0	1884	1884
$U_{18}$	0.002307112	0.00196	0.002052	0.004742	0.006517	0.004282	0.005705
$U_{19}$	0.2	0.2	0.23	0.1	0.35	0.07	0.23
$U_{20}$	0.03	0.03	0.06	0.02	0.06	0.02	0.05
Plan Num	5−1	5−3	6−3	6−4	7−1	7−3	8
$U_1$	4	4	3	3	4	4	4
$U_2$	4	4	3	3	4	4	4
$U_3$ /min	1155	1101	2347	2342	1437	1437	1679
$U_4$ /min	3970	3916	5162	5157	4252	4252	4494
$U_5$ /min	992.5	979	1720.667	1719	1063	1063	1123.5
$U_6$	5	5	6	6	6	6	5
$U_7$	5	5	6	6	6	6	5
$U_8$ /km	1413	1413	1884	1884	1413	1413	1413
$U_9$ /km	2587	2587	2116	2116	2587	2587	2587
$U_{10}$ /km	1130.4	1130.4	942	942	942	942	1130.4
$U_{11}$ /min	563	563	469.1667	469.1667	469.1667	469.1667	563
$U_{12}$	0.390972	0.390972	0.32581	0.32581	0.32581	0.32581	0.390972222
$U_{13}$ /min	144.375	137.625	260.7778	260.2222	179.625	179.625	209.875
$U_{14}$	3	3	4	4	3	3	3
$U_{15}$ /km	942	942	1884	1884	942	942	942
$U_{16}$ /km	2826	2826	1884	1884	1884	1884	2826
$U_{17}$	1884	1884	0	0	942	942	1884
$U_{18}$	0.003568	0.003832	3.62 × 105	7.28 × 105	2.42 × 105	2.42 × 105	0.003832247
$U_{19}$	0.05	0.18	0.23	0.17	0.08	0.18	0.18
$U_{20}$	0.02	0.05	0.05	0.04	0.02	0.04	0.05

.

1.

Construction of a standard matrix.

The original evaluation matrix $U$ was extracted from Table 12, and the indicators were divided into positive indicators and negative indicators. The positive indicators included the average route mileage ($U_8$), vehicle kilometers per day ($U_{10}$), average daily occupancy time ($U_{11}$), utilization rate of EMUs ($U_{12}$), average number of tasks performed by EMUs ($U_{14}$), minimum mileage of EMUs ($U_{15}$), and maximum mileage of EMUs ($U_{16}$). The negative indicators included the number of routes ($U_1$), the number of D1 repairs ($U_2$), minimum total connection time ($U_3$), total turnaround time for routing ($U_4$), average turnaround time for routing ($U_5$), number of days ($U_6$), number of train units ($U_7$), average route distance loss ($U_9$), average connection time between trains ($U_{13}$), route mileage balance ($U_{17}$), variance of route utilization ($U_{18}$), route average load factor range ($U_{19}$), and route mean standard deviation of the load factor ($U_{20}$).

$$x_{ij}=\left\{\begin{array}{l}{\displaystyle \frac{u_{ij}-\min u_j}{\max u_j-\min u_j}},u_j is positive index\\ {\displaystyle \frac{\max u_j-u_{ij}}{\max u_j-\min u_j}},u_j is negative index\end{array}\right.,$$

(36)

The optimal value of each indicator is chosen to form a standard matrix:

$$X=\left[\begin{array}{cccc}1.0000& 1.0000& \cdots & 0.7500\\ 1.0000& 1.0000& \cdots & 0.7500\\ 0.5000& 0.5000& \cdots & 0.0000\\ 1.0000& 1.0000& \cdots & 1.0000\\ 1.0000& 1.0000& \cdots & 0.0000\\ 0.5000& 0.5000& \cdots & 1.0000\\ 0.5000& 0.5000& \cdots & 0.2500\\ 0.0000& 0.0000& \cdots & 1.0000\\ 0.0000& 0.0000& \cdots & 0.2500\\ 0.5000& 0.5000& \cdots & 0.2500\\ 0.5000& 0.5000& \cdots & 0.5000\\ 0.0000& 0.0000& \cdots & 1.0000\\ 0.0000& 0.0000& \cdots & 0.5000\\ 0.0000& 0.0000& \cdots & 0.2500\end{array}\right],$$

(37)

2.

The ratio of each index is found for each plan.

$$p_{ij}={\displaystyle \frac{x_{ij}}{{\displaystyle \sum _{i=1}^{plannumber}{x}_{ij}}}},$$

(38)

The ratio of each index in each plan ($P$) is calculated using Equation (39):

$$P=\left[\begin{array}{cccc}0.1538& 0.1538& \cdots & 0.1000\\ 0.1538& 0.1538& \cdots & 0.1000\\ 0.0769& 0.0769& \cdots & 0.0000\\ 0.1538& 0.1538& \cdots & 0.1333\\ 0.1538& 0.1538& \cdots & 0.0000\\ 0.0769& 0.0769& \cdots & 0.1333\\ 0.0769& 0.0769& \cdots & 0.0333\\ 0.0000& 0.0000& \cdots & 0.1333\\ 0.0000& 0.0000& \cdots & 0.0333\\ 0.0769& 0.0769& \cdots & 0.0333\\ 0.0769& 0.0769& \cdots & 0.0667\\ 0.0000& 0.0000& \cdots & 0.1333\\ 0.0000& 0.0000& \cdots & 0.0667\\ 0.0000& 0.0000& \cdots & 0.0333\end{array}\right],$$

(39)

3.

The information entropy of each index is found.

The smaller the information entropy, the more discrete the index, and the greater the influence (weight) of the index on the comprehensive evaluation.

$$e_j=-{\displaystyle \frac{1}{\ln plannum ber}}{\displaystyle \sum _{i=1}^{plannumber}(p_{ij}\times \ln p_{ij})},$$

(40)

The information entropy of each index is calculated using Equation (41):

$$E=[0,8106,0.8106,\cdots ,0.8905],$$

(41)

4.

The weight of each indicator is determined.

The redundancy of indicator information is

$$g_j=1-e_j,$$

(42)

The weight of each indicator is

$$w_j={\displaystyle \frac{g_j}{{\displaystyle \sum _{i=1}^{plannumber}{g}_j}}},$$

(43)

The weight of each index can be calculated using Equation (43):

$$W=\left[0.0645,0.0645,\cdots ,0.0373\right],$$

(44)

5.

The weighted standardized matrix is constructed.

The weighted standardized matrix $Z={(z_{ij})}_{plannumber\times n}$ is calculated, where

$$z_{ij}=w_j\times x_{ij},$$

(45)

$$Z=\left[\begin{array}{cccc}0.0645& 0.0645& \cdots & 0.0280\\ 0.0645& 0.0645& \cdots & 0.0280\\ 0.0322& 0.0322& \cdots & 0.0000\\ 0.0645& 0.0645& \cdots & 0.0373\\ 0.0645& 0.0645& \cdots & 0.0000\\ 0.0322& 0.0322& \cdots & 0.0373\\ 0.0322& 0.0322& \cdots & 0.0093\\ 0.0000& 0.0000& \cdots & 0.0373\\ 0.0000& 0.0000& \cdots & 0.0093\\ 0.0322& 0.0322& \cdots & 0.0093\\ 0.0322& 0.0322& \cdots & 0.0186\\ 0.0000& 0.0000& \cdots & 0.0373\\ 0.0000& 0.0000& \cdots & 0.0186\\ 0.0000& 0.0000& \cdots & 0.0093\end{array}\right],$$

(46)

6.

The positive and negative ideal solutions are found.

$$Z^{+}=[\max z_1,\max z_2,\cdots ,\max z_n],$$

(47)

Through Equation (47), the positive ideal solution of each index ($Z^{+}$) can be calculated. The positive ideal solution is the optimal solution of each plan for each index.

$$Z^{+}=\left[0.0645,0.0645,\cdots ,0.0373\right],$$

(48)

$$Z^{-}=[\min z_1,\min z_2,\cdots ,\min z_n],$$

(49)

The negative ideal solution of each index ($Z^{-}$) is calculated using Equation (49). The negative ideal solution is the worst solution of each plan for each index.

$$Z^{-}=\left[0,0,\cdots ,0\right],$$

(50)

7.

The positive and negative ideal solution distances of objects are calculated.

$$D_i^{+}=\sqrt{{\displaystyle \sum _{j=1}^m{(Z_j^{+}-z_{ij})}^2}},$$

(51)

$$D_i^{-}=\sqrt{{\displaystyle \sum _{j=1}^m{(Z_j^{-}-z_{ij})}^2}},$$

(52)

The positive ideal solution distance of each plan ($D_i^{+}$) is calculated using Equation (51):

$$D_i^{+}={\left[0.1002,0.0930,0.1622,0.0776,0.0860,0.1585,0.1607,0.1997,0.2018,0.1738,0.1723,0.2262,0.2271,0.2046\right]}^T$$

(53)

The negative ideal solution distance of each plan ($D_i^{-}$) is calculated using Equation (52):

$$D_i^{-}={\left[0.2042,0.2046,0.1364,0.2261,0.2227,0.1359,0.1335,0.1331,0.1291,0.0962,0.0978,0.0797,0.0720,0.1137\right]}^T$$

(54)

8.

The relative proximity is calculated and sorted.

$$C_i={\displaystyle \frac{D_i^{-}}{D_i^{+}+D_i^{-}}},$$

(55)

The relative proximity degree of each plan is calculated using Equation (55):

$$C_i={\left[0.6709,0.6875,0.4568,0.7445,0.7215,0.4616,0.4537,0.4000,0.3902,0.3564,0.3622,0.2606,0.2408,0.3572\right]}^T$$

(56)

The larger $C_i$ is, the larger the distance between the plan and the negative ideal solution is, and the better the plan is. In the same way, the first-level index of “EMU operation efficiency” and the indicators of the first-level index of “EMU operation balance” are evaluated. $C_{i\_efficiency}$, $C_{i\_balance}$, and $C_{i\_loadfactor}$ are obtained.

Table 13. Maximum and minimum values of relative proximity.

Index Range	Total	Operation Efficiency of EMU	EMU Operation Balance	Load Factor Consistency
Maximum Value	0.7445	0.8216	0.5505	0.5589
Plan Number	3−1	1−3	6−3 and 6−4	3−1
The Route Corresponding to the Plan	1−4−9−0−5−8, 10−3−6−11−2−7	4−11−2−5−10−1−6−9, 7−12−3−8	8−1−6−9, 11−2−5−10, 4−7−12−3 and 3−8−1−6, 9−2−5−10, 11−4−7−12	2−5−10−1−6−9, 11−4−7−12−3−8
Minimum Value	0.2408	0.1183	0.3276	0.0000
Plan Number	7−3	7−1 and 7−3	4−3	3−4
The Route Corresponding to the Plan	11−0−5−8, 10−1−4−9, 2−7, 3−6	10−1−6−9, 11−4−7−12, 2−5, 3−8 and 12−1−6−9, 11−2−5−10, 3−8, 4−7	4−7−12−1−6−9, 11−2−5−10, 3−8	4−7−12−1−6−9, 11−2−5−10−3−8

shows maximum and minimum values of relative proximity.

$$C_{i\_efficiency}={\left[0.8216,0.4048,0.2667,0.4274,0.4284,0.2584,0.2634,0.2195,0.2219,0.1975,0.1975,0.1183,0.1183,0.1998\right]}^T$$

(57)

$$C_{i\_balance}={\left[0.4969,0.5053,0.4368,0.5284,0.5001,0.4182,0.3276,0.4534,0.4413,0.5505,0.5505,0.5473,0.5473,0.4413\right]}^T$$

(58)

$$C_{i\_loadfactor}={\left[0.5487,0.5487,0.1899,0.5589,0.0000,0.5576,0.3670,0.5568,0.3981,0.3670,0.5054,0.5581,0.5037,0.3981\right]}^T$$

(59)

Table 14. Comparison of indicators of each plan and the actual operation plan.

Index	U1	U2	U3	U4	U5	U6	U7	U8	U9	U10
1-3	50.0%	50.0%	−52.2%	−19.5%	−139.0%	0.0%	0.0%	100.0%	54.6%	0.0%
1-4	50.0%	50.0%	−41.3%	−15.4%	−130.8%	0.0%	0.0%	100.0%	54.6%	0.0%
2-3	25.0%	25.0%	−1.1%	−0.4%	−33.9%	0.0%	0.0%	33.3%	18.2%	0.0%
3-1	50.0%	50.0%	−43.7%	−16.3%	−132.6%	0.0%	0.0%	100.0%	54.6%	0.0%
3-4	50.0%	50.0%	−38.6%	−14.4%	−128.8%	0.0%	0.0%	100.0%	54.6%	0.0%
4-1	25.0%	25.0%	3.1%	1.2%	−31.8%	0.0%	0.0%	33.3%	18.2%	0.0%
4-3	25.0%	25.0%	12.5%	4.7%	−27.1%	0.0%	0.0%	33.3%	18.2%	0.0%
5-1	0.0%	0.0%	31.2%	11.7%	11.7%	0.0%	0.0%	0.0%	0.0%	0.0%
5-3	0.0%	0.0%	34.4%	12.9%	12.9%	0.0%	0.0%	0.0%	0.0%	0.0%
6-3	25.0%	25.0%	−39.8%	−14.9%	−53.2%	−20.0%	−20.0%	33.3%	18.2%	−16.7%
6-4	25.0%	25.0%	−39.5%	−14.8%	−53.0%	−20.0%	−20.0%	33.3%	18.2%	−16.7%
7-1	0.0%	0.0%	14.4%	5.4%	5.4%	−20.0%	−20.0%	0.0%	0.0%	−16.7%
7-3	0.0%	0.0%	14.4%	5.4%	5.4%	−20.0%	−20.0%	0.0%	0.0%	−16.7%
Actual Plan	1	1	1	1	1	1	1	1	1	1
Index	U11	U12	U13	U14	U15	U16	U17	U18	U19	U20
1-3	0.0%	0.0%	−21.8%	100.0%	100.0%	33.3%	0.0%	39.8%	−11.1%	40.0%
1-4	0.0%	0.0%	−13.0%	100.0%	100.0%	33.3%	0.0%	48.9%	−11.1%	40.0%
2-3	0.0%	0.0%	10.1%	33.3%	0.0%	33.3%	−50.0%	46.5%	−27.8%	−20.0%
3-1	0.0%	0.0%	−14.9%	100.0%	200.0%	0.0%	100.0%	−23.8%	44.4%	60.0%
3-4	0.0%	0.0%	−10.9%	100.0%	200.0%	0.0%	100.0%	−70.1%	−94.4%	−20.0%
4-1	0.0%	0.0%	13.9%	33.3%	0.0%	0.0%	0.0%	−11.7%	61.1%	60.0%
4-3	0.0%	0.0%	22.2%	33.3%	0.0%	0.0%	0.0%	−48.9%	−27.8%	0.0%
5-1	0.0%	0.0%	31.2%	0.0%	0.0%	0.0%	0.0%	6.9%	72.2%	60.0%
5-3	0.0%	0.0%	34.4%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%
6-3	−16.7%	−16.7%	−24.3%	33.3%	100.0%	−33.3%	100.0%	99.1%	−27.8%	0.0%
6-4	−16.7%	−16.7%	−24.0%	33.3%	100.0%	−33.3%	100.0%	98.1%	5.6%	20.0%
7-1	−16.7%	−16.7%	14.4%	0.0%	0.0%	−33.3%	50.0%	99.4%	55.6%	60.0%
7-3	−16.7%	−16.7%	14.4%	0.0%	0.0%	−33.3%	50.0%	99.4%	0.0%	20.0%
Actual Plan	1	1	1	1	1	1	1	1	1	1

Note: Positive numbers in the table indicate that the plan is better than the actual operation plan in terms of this index; the percentage represents the degree of goodness, and vice versa for negative numbers.

shows comparison of indicators of each plan and the actual operation plan. From the perspective of the relative proximity of the overall index, plan 3−1 has the best performance, mainly because it has the lowest number of routes, the highest number of daily vehicle kilometers, and the most evenly distributed number of operating lines within a route, which indicates that the operation efficiency and balance of EMUs are relatively high, although the connection time is not the shortest. When considering the relative closeness of the EMU operation balance, plan 6−3 and 6−4 stand out because plan 6−3 and 6−4 ensure the same number of operating lines within each route, making each route more balanced in terms of mileage and time compared with the actual operation plan. In contrast, the number of running lines within each route in the actual operation plan is different, resulting in a relatively low ranking in terms of balance.

From the comparison between each plan and the actual operation plan, we can draw the following conclusions.

The number of routes and number of D1 repairs in plan 1−3 are reduced by 50%. Reducing the number of routes can effectively reduce the operating cost and maintenance cost of EMUs. From the point of view of time, the minimum total connection time, total turnaround time for routing, and average connection time between trains in plan 4−1, 4−3, 5−1, and 5−3 are lower than in those in actual operation (plan 8). In particular, plan 5−3 reduced the minimum total connection time by 34.4%, the total turnaround time by 12.9%, and the average connection time between trains by 12.9%. Shortening the connection time can effectively reduce the idle time of EMUs and improve their operation efficiency.

From the point of view of mileage, the average route mileage and average route distance loss in plan 1−3 are better than those in actual operation (plan 8). The average route mileage in plan 1−3 increases by 100%, and the average route distance loss decreases by 54.6%. In addition, the minimum mileage of EMUs in plan 1−3 is greater than that in actual operation (plan 8), and the minimum mileage of EMUs in plan 3−1 is increased by 200%. The maximum mileage of EMUs in plan 1−3 also exceeds that in actual operation (plan 8), increasing by 33.3%. These data show that the actual operation plan for the maintenance mileage cycle is not sufficient.

From the perspective of the task commitment of EMUs, the average number of tasks of EMUs in plan 1−3 is more than that in actual operation (plan 8). The average number of tasks of EMUs in plan 1−3 is increased by 100%, which shows that the actual operation plan needs improvement in terms of the efficiency of EMUs.

From the perspective of balance, plan 3−1 are superior to the actual operation plan (plan 8) in terms of route mileage balance, which is improved by 100% for plan 3−1. Plan 1−3 is also superior to the actual operation plan (plan 8) in terms of the variance of route utilization, especially plan 7−1 and 7−3, for which it is reduced by 99.4%.

From the perspective of load rate consistency, plan 3−1 is superior to the actual operation (plan 8) in terms of route mileage balance, and the route mileage balance of plan 3−1 is improved by 100%. Plan 1−3 is also superior to the actual operation (plan 8) in terms of the variance of route utilization, especially plan 7−1 and 7−3, whose variance of route utilization is reduced by 99.4%.

In summary, by optimizing the routing plan, not only can the operation efficiency of EMUs be improved, but so can the cost control, time management, mileage utilization, and balance of task allocation.

6. Conclusions

Power-concentrated EMUs have the advantages of speed, comfort, flexible formations, and a short turn-back time. They can effectively release the transportation pressure of tight lines and hubs, optimize train operation structures, improve the travel experience of passengers, and have certain cost advantages. In order to solve the problem of insufficient train units in the current application of power-concentrated EMUs, an improved ant colony algorithm based on an internal and external double-layer circulation mechanism is proposed in this study, with the aim of constructing an optimization model for the operation of power-concentrated EMUs and the goal of minimizing the total connection time. The model is solved by an improved ant colony algorithm to improve the search quality and convergence speed, which changes pheromone volatilization coefficient with iteration. The automatic routing problem is divided into two parts: an outer loop and an inner loop. The outer loop is responsible for exploring the number of different routes and the number of internal lines, while the inner loop focuses on solving the minimum connection time and optimal route layout with the given outer plan. Through the decomposition of optimization objectives, the search space and the solution scale in each stage are reduced.

Based on an actual power-concentrated EMU operating diagram from a railway bureau, in this study, a comprehensive evaluation of 20 index dimensions, covering efficiency and equalization, was performed based on 13 different plans for the number of routing/running lines. The evaluation results show that the best overall plan involves dividing the operating lines into two groups, with each group containing six operating lines. The plan with the best operational efficiency involves dividing the lines into two groups that contain eight and four lines, respectively. The plan with the best use of balance involves dividing the lines into three groups that each contain four lines. The plan with the best load rate consistency involves dividing the lines into two groups with six lines in each group, which is also the plan with the best overall performance. Compared with the existing operational plans, these optimized plans evidently still have room for improvement with regard to the number of routes and the connection time of routes.

The algorithm proposed in this study has important application and research value for improving the practical operation efficiency of power-concentrated EMUs, rationally utilizing the maintenance mileage cycle, and promoting the balanced allocation of resources. Moreover, considering the daily changes in train operation plans, this study puts forward a method of route division according to the passenger load factor, which can help railway bureaus adjust their capacity according to fluctuations in demand.