Modified Maintenance Network Model for Urban Rail Transit Systems Based on the Variable Coverage Radius: Evidence from Changchun City in China

Network-wide maintenance lacks strong theoretical support and practical cases. However, research on this topic has entered an extensive exploratory stage; for example, new network design methods are being sought, and successful practices from traditional maintenance by line and by profession are being incorporated. .is paper proposes a novel set-covering model with the variable coverage radius for the maintenance network of urban rail transit systems in the context of network-wide maintenance. .e concepts of network-wide maintenance follow principles that are similar to those of bio-geography-based optimization (BBO), i.e., patterns of migration, variation, and extinction of different populations in different habitats. .erefore, a BBO algorithm is implemented with combinatorial optimization programming. Experiments from Changchun city in China show that the proposed model and algorithm are effective in fulfilling network-wide requirements through a direct tradeoff between the coverage radius and maintenance response time. In addition, the maintenance capacity and variable coverage radius of each maintenance point influence both the maintenance timeliness and resource utilization of maintenance units.


Introduction
As of December 31, 2018, according to a statistical analysis of urban rail transit lines, 169 urban lines in 32 cities in mainland China were in operation, having a total length of 5494.9 kilometers. New lines with a length of 647.8 kilometers have created a record high. After the previous peak in construction in the past five years, the increasingly perfected rail transit network has entered a period of stable operation and maintenance. e first seminar on the topic of China's urban rail transit operation equipment maintenance innovation and practice was held in Shanghai in November 2015. e concept of "safe, reliable, and efficient operation of urban rail transit, with the maintenance guarantee as the top priority" was stated. In April 2016, the National Development and Reform Commission approved the founding of the first "National Engineering Laboratory for Urban Rail Transit System Safety and Maintenance" in China, stressing that the issue of urban rail transit system maintenance is just as important as planning and construction. In June 2019, the 3rd conference on vehicle operation and maintenance for Chinese urban rail transit systems was held by http://www. chinametro.net, focusing on security and intelligent and lean operation.
rough those series of events, network-wide maintenance has been a very hot topic of discussion. In particular, based on the concept of network-wide maintenance, in contrast to the modes of maintenance by line and by profession, the problem of how to construct a robust maintenance network for urban rail transit networks arises, which is an important part of the health management of such networks. e maintenance network is a complex network system consisting of maintenance demand points, maintenance points, resource supply points, and paths connecting these nodes, and it is an expansion network with service capabilities as the core. e urban rail transit maintenance network is a functionally complex network serving the urban rail transit physical network. As an implementation of network-wide maintenance, it consists of maintenance demand points, maintenance points, and maintenance paths, and it performs network operation maintenance and security protection.
In practice, the concept of network-wide maintenance comes from networked maintenance, which reflects the new demands of networked operation for the maintenance of urban rail transit systems: (1) e failure of a certain rail or running vehicle in urban rail transit will affect not only the traffic and passenger transport organization of the line but also those of relevant lines, even causing faults over a larger scope that will lead to new emergencies. Due to the urgency of functional enhancement, the concept of network-wide maintenance has shifted from the strategic level to the tactical level. (2) e efficiency of the traditional mode of maintenance by line needs to be improved. e modes of regional network maintenance and network-wide maintenance are more advantageous. (3) e sharing of maintenance resources between different maintenance points is a necessary condition to ensure that maintenance resources are immediately available, and professional maintenance personnel are also included as maintenance resources for sharing and scheduling. (4) e transfer stations are nodes that propagate faults with obviously high importance in network-wide maintenance, and more emphasis should be placed on them.
ese aspects are the focus of line management, and they need standardized and sufficient maintenance units to provide protection.
Based on the concept of network-wide maintenance, maintenance network design for urban rail transit systems aims to divide the whole maintenance network into several regional subnets; allocate and optimize various maintenance resources for regional subnets, lay out maintenance points, units, and warehouses; and, ultimately, form a safe, reliable, and timely response and a low-cost, low-consumption maintenance network to ensure the safe operation of urban rail transit systems.
is paper studies urban rail transit maintenance network design based on the concept of network-wide maintenance. We organize this paper as follows. Section 1 proposes problems and improvement directions based on a literature review. Both Sections 1 and 2 define the new problem of network-wide maintenance and quantify important parameters. Basic assumptions such as coverage and guarantee rules are constructed, and a maintenance network design model for urban rail transit systems based on the variable coverage radius is established and developed. Section 3 designs the bio-geography-based optimization (BBO) algorithm; Section 4 applies the model and analyzes the sensitivity of impact factors. Section 5 presents the conclusions and future research directions.

Literature Review
In the academic field, the maintenance network design problem can be classified as a discrete multifacility location problem, and its theoretical model can be summarized as three classic models: the P-center model [1], the P-median model [2], and the set-covering model [3]. e set-covering model concerned in this paper is the most widely used in site selection or network location, and it is applicable to firefighting units, police stations, hospitals or emergency centers [4], telephone exchange centers, gas stations, and other research objects [5]. It covers a given level of demand with a minimum amount of facility or total construction cost, i.e., from a given set of alternatives, it chooses facility locations with the goal of minimizing costs based on certain coverage rules, minimizes the number of facilities, and maximizes profits or service. Farahani et al. offer a clear classification and review of the set-covering model for facility location and imply that the characteristics and functional location for maintenance points align with these goals just like normal facilities such as hospitals [6].
Berman et al. [7], Bai et al. [8], and Akella et al. [9] discussed the application of coverage models in site selection. A set-covering model using the variable coverage radius has obvious advantages in solving certain problems. erefore, this type of model has been explored and developed very quickly in real-world practice. For instance, Berman et al. [10] achieved a direct tradeoff between the coverage radius r and the facility variable cost φ (r). en, Berman and Krass emphasized three implicit assumptions for the set-covering model: full coverage or no coverage; individual coverage; and a fixed coverage radius [3]. ey summarized the extensive research on the maximum covering model, progressive covering model, cooperative covering model, and variable radius covering model and concluded that the set-covering model with the variable coverage radius is more in line with actual needs. Bashiri and Fotouhi used the maximum-minimum method to determine the coverage radius under the assumption of the existence and robustness of the optimal solution [11].
In contrast to the direct tradeoff between the coverage radius and facility cost in [10] and the robustness of the solution in [11], Davari et al. considered that the coverage radius is determined by the farthest distance between the specified node and facility [12]. If speed is not considered, the farthest distance is equivalent to the transit time between nodes, and the coverage radius is converted into a function related to the transit time between nodes and their facility. Since the transit time is uncertain, it can be considered a fuzzy variable. Our research considers many more characteristics of the rail transit maintenance network and covers the radius equivalent to the farthest distance between the maintenance center and the point under maintenance. is distance is determined by the speed and transit time of the maintenance unit. e variable coverage radius can thus be converted into a function related to the response time of the maintenance point.
To solve set-covering models, in addition to classic heuristic algorithms such as genetic algorithms, the BBO algorithm, which is a type of new artificial intelligence algorithm, has achieved much more far-reaching applications. Based on natural bio-geography and its mathematics, Simon first proposed the BBO algorithm and then discussed how it can be used to solve optimization problems [13]. To test the performance of the BBO algorithm, he used 14 benchmark functions to compare with seven population-based optimization algorithms, such as genetic algorithms and evolutionary algorithms. e BBO algorithm was shown to be the most effective. In its updating mechanism, the BBO algorithm is different from other group intelligent optimization algorithms such as the hybrid teaching-learning algorithm; it simulates the behaviors of species migration and mutation in bio-geography and designs the operators of migration, mutation, and clearance. e operators simulate the migration, variation, and extinction of species communities in biogeography, and thus, they are strong in problem solving; they solve complex combinatorial optimization problems through individual collaboration and competition among groups. In problem solving, globally optimal solutions can be found faster compared to the use of traditional optimization algorithms. erefore, we employ the BBO algorithm to solve the proposed model, paying more attention to make the heuristic method itself a suitable problem. Simon et al. followed this strategy to establish a multiobjective and heterogeneous indoor wireless network planning model and solved the model using the BBO algorithm [14].
We found three approaches for improving the BBO algorithm: (1) we can improve the original population emergence mechanism to enhance algorithm performance, e.g., Ergezer et al. proposed a new antithetic learning mechanism to increase the variety of populations [15]. (2) We can redesign the operators of migration, mutation, and clearance to expand the advantages of BBO. For instance, Ma and Simon explored the behavior of six different migration models in BBO and investigated their performance by integrating a hybrid crossover of genetic algorithms together with BBO to change the migration operator, which is blended bio-geography-based optimization (B-BBO) [16].
(3) We can improve the searching capability of the algorithm, i.e., Shukla and Singh enabled searching and discovered excellent individuals by importing gradient search and grid search methods into BBO [17]. In this paper, we prefer the second approach and redesign the operators to make them exactly fit the set-covering model established and the network-wide maintenance decision-making procedure we defined.

Modeling Methodology
In this paper, we explore improving the classic set-covering model to construct a reliable and quick-response maintenance network for urban rail transit networks as follows: (1) e demand for each maintenance demand point of the maintenance network for urban rail transit systems is different. e service capability of each maintenance point can be either a standard service capability (same) or a hierarchical service capability (differentiated) to acclimate to changes in demand and supply. To that end, to distinguish the importance of maintenance demand points, employing the variable coverage radius to cover distinct maintenance points is more targeted and adaptable. (2) In selecting the location of maintenance points, a quick response after problems is the primary technical and safety requirement, compared to the tradeoff between the coverage radius and cost. For this reason, the coverage radius of each maintenance point depends on its effective response. In this paper, we directly weigh between the coverage radius and the maintenance response time, and the actual response time does not exceed the timely response time.

Problem Statements.
e maintenance demand point is the protection object of the maintenance network. In principle, all nodes and edges of the maintenance network are maintenance demand points, i.e., vehicles and rail tracks, thereby forming a set of maintenance demand points gf (station set V and track set E), or G � {V, E}. According to the type of station, set V is subdivided into three subsets: stop stations (V 1 ), same-way transfer stations (V 2 ), and multimodal transfer stations (V 3 ). e stations in V are denoted as v ni (n � 1, 2, 3), and the next station of station i is denoted by k; the track between these stations in set E is denoted as e ik . Vehicles must be moved to the next station for repair, while tracks require on-site repair. e set of stations as maintenance demand points V is also the set of alternative maintenance points. e set of maintenance points refers to the set of selected maintenance demand points, denoted as V ′ . e distance of the maintenance unit moving between a maintenance point v j ′ and the maintenance demand point f in the maintenance network is denoted as L fj , and their collection is defined as set Path � L fj (f, j) . Together, the maintenance points and paths constitute the maintenance network G ′ � V ′ , Path .
Once a fault occurs in an urban rail transit system, it is necessary for the maintenance unit and repair parts to arrive in a timely manner and for the fault to be handled. is process is completed by the maintenance point. e idea behind designing a maintenance network model for urban rail transit systems based on the set-covering model is as follows: first, we determine the repair Journal of Advanced Transportation demand stations set V; then, by changing the coverage radius R j , according to the coverage and rules of maintenance points gf established in this paper, we determine how the demand points can be covered. e set of maintenance points, together with the maintenance demand point f that can be covered by the maintenance point A(j), establishes the urban rail transit maintenance network 0-1 integer programming model B(f); thus, it meets dual goals of the minimum number of maintenance points and shortest total maintenance path.

Model Hypotheses
Hypothesis 1. An urban rail transit network is divided into several exclusive subareas called planned cells. e stations and tracks in each planned cell are assigned to a maintenance point.
e relevant planned cells are determined as the coverage of this point.

Hypothesis 2.
e requirements for all maintenance demand points are the same, the demand for each maintenance demand point is provided exclusively by one maintenance point, and there is no service overlap.

Hypothesis 3.
e stations are location alternatives for setting up maintenance points.

Hypothesis 4.
e number of maintenance points and the allocation plan for each specific maintenance point are uncertain; we constrain the service capacity of each maintenance point, and calculation is performed under these assumptions.

Hypothesis 5.
e service capacity of a maintenance point is bound by its own mobile maintenance units, and the number of initial maintenance units for each maintenance point is the same. Hypothesis 6. Since stations are closely related to tracks, if the maintenance demand station v ni (n � 1, 2, 3) is assigned to a maintenance point v j ′ for maintenance, track L fj on the path is also assigned to the same point v j ′ .

Hypothesis 7.
e repair parts inventory of each maintenance point is highly shared. If the repair parts at a maintenance point are insufficient, the nearest maintenance point should supply them in time, and the spare parts inventory should be replenished in a timely manner.

Hypothesis 8.
A faulty vehicle can be driven to the next station on its route at any time or can be transported to a maintenance point in a timely manner.

Notations.
e notations used in this section are listed as follows: provides the maintenance actions for maintenance demand system gf, Y fj � 1; otherwise, 0 Y kj If v j ′ provides the maintenance actions for the vehicles of rail station k, Y kj � 1; otherwise, 0 Y ikj If v j ′ provides the maintenance actions for rail e ik , Y ikj � 1; otherwise, 0.

Descriptions of Important Parameters
(1) We differentiate the importance of maintenance demand points. e coverage of maintenance points is differentiated by a variable coverage radius. For V 1 , V 2 , and V 3 , there is the following: Here, R is the average coverage radius of maintenance points within the maintenance network. e elastic coefficients η and φ of R j are rooted on two main influence factors in theory, i.e., the urban rail transit network size and the average spacing between stations. In our research, we take the latter as reference because, in practice, the average spacing is one of the key design parameters for urban rail transit network.
(3) e rules of resource allocation for maintenance demand stations v ni (n � 1, 2, 3) are as follows: For the three types of stations, V1, V2, and V3, the value of the maintenance resource property factor α i is different in consideration of station function type and design capacity.
When a new line passes v ni (n � 1, 2, 3), i.e., considering the complex effect of line increase, when the number of lines β i increases by one, the number of its preassigned maintenance units, q imax , increases by 0.5 standard units, q max : (4) Coverage radius R j for maintenance point j e actual response time t includes maneuver time t 1 , maintenance time t 2 , and maintenance vehicle time on trip t 3 ; thus, t � t 1 + t 2 + t 3 . To meet the timely response requirement, the actual response time must be less than or equal to the expected response time, i.e., t ≤ t ′ . e maximum excepted response time t ′ is the minimum requirement for the maintenance time.
If the traveling speed of maintenance units is v 0 , the relationship among the distance between maintenance demand point gf and maintenance point v j ′ (denoted as L fj ), t ′ , and v 0 is as follows: According to the set-covering model, the relationship between the maintenance point v j ′ 's coverage radius R j and L fj is as follows: When v 0 is constant, t ′ determines the value of R j . To ensure a timely response capacity, t ′ and R j can be interconverted according to the needs of the study. To simplify the calculation, the shortest fault maintenance time is defined as a unit maintenance time t 0 ; thus, the maintenance time of other faults can be defined as a multiple of the unit maintenance time.

Objective Functions and Constraints.
e objective functions and constraints of the maintenance network design model for urban rail transit systems (referred to as the design model) proposed in this paper are as follows: Subject to the constraints Journal of Advanced Transportation 5 Equation (8) is the objective function of the model; it aims to ensure the minimum number of maintenance support points, which means the minimum construction cost. As another objective function of the model, equation (9) ensures the shortest total maintenance support distance that can achieve a timely response of the maintenance support network. Equation (10) is the constraint of the selection of maintenance support routes; based on this constraint, every single maintenance support route selected should be within the coverage of maintenance support point v j ′ , i.e., the distance L fj between gf and v j ′ should be no more than the coverage radius R j of v j ′ . Equation (11) is the service capacity constraint of the maintenance support point. e total maintenance demands assigned to the maintenance support point v j ′ cannot exceed its service capacity Q jmax . Equations (12) and (13) indicate that every single rail e ik or rail station v ni (n � 1, 2, 3) accepts service from only one maintenance support point v j ′ . Equations (14) and (15) indicate that only when alternative location j is selected as a maintenance support point can it provide maintenance actions for rail station k; additionally, only when alternative location v j ′ is selected as a maintenance support point can it provide support for the rails on route L fj . Equation (16) presents the constraint on the number of maintenance support points, which means that the number of maintenance support points cannot exceed the number of rail stations because the maintenance support points are located at the rail stations. Equations (17)(19) are all 0-1 decision variables.

BBO Algorithm Design
is research uses the BBO algorithm to solve the model proposed above. e BBO algorithm designs the migration operator, mutation operator, and clearance operator to simulate the migration of species communities between habitats, species variation, and species extinction in biogeography, and it solves complex combinatorial optimization problems through individual collaboration and competition among groups. Compared to other algorithms, the BBO algorithm has an advantage in its updating mechanism and global searching ability. Regarding the model in this research, the BBO algorithm is defined as follows.
In an urban rail transit maintenance support network, we define the maintenance demand of vehicles and rails as a collection of species and denote the maintenance support points as islands that are colonies of species. According to the principle of the algorithm, maintenance support points with more maintenance resources are more attractive to vehicles and rails with maintenance demands, i.e., these maintenance support points have a higher adaptability index. By adjusting the immigration rate λ j , emigration rate μ j , and adaptation index HSI j , we identify the relationship between the maintenance support point and vehicles and rails. e mutation operator is introduced to make the maintenance network more stable and robust, and finally, the optimal solution is obtained. We define the adaptation index HSI j as the ratio of the number of rail stations and rails Q jr supported by v j ′ up to their max number Q jmax , which is formulated as follows: (1) When no maintenance support points have maintenance tasks, i.e., the number of rail stations s and the number of railroad tracks n that need maintenance are both 0, the emigration rate is 0, and the immigration rate λ j is the highest, which is denoted as I. We can formulate λ j as follows: (2) As the maintenance demands increase, the maintenance of more vehicles and rails will be assigned to other maintenance support points, leading to an increase in the emigration rate μ j . Meanwhile, the available maintenance support resources will decrease, which will reduce the immigration rate λ j . (3) When the maintenance resources of a maintenance support point are fully utilized, i.e., when the number of stations and rails served by a maintenance support point is its maximum service capacity Q jmax , the emigration rate is the highest, which is denoted as E, and the immigration rate μ j drops to 0. μ j can be formulated as follows: (4) When the service capacity of v j ′ and the number of stations and rails supported reach a balance, their relationship tends to be stable. To avoid local stability rather than global stability, the mutation operator α is used to obtain the global optimal solution. We set α � 0.0005. e process of the algorithm is shown in Figure 1.

Urban Rail Transit Network and Basic Data.
A case study using the urban rail transit operation network in Changchun, the capital city of Jilin Province in northeastern China, is employed to test the model. Compared with other same medium-sized cities in China, Changchun urban rail transit network steps in fast-speed developing stage and focuses on improving the accessibility between districts to solve the problem of difficult travel for the masses in the surrounding areas of the city. Its maintenance experience at present is not rich and lacks targeted solution for such a kind of network. Its maintenance network design thereby comes to one of the essential decisions of operation safety management. Figure 2 shows the layout of the network, which consists of 4 lines, i.e., line 1, line 2, line 3, and line 4, and 77 rail stations, including 8 interchange stations. e 77 rail stations are the alternatives of the maintenance support points. We number the 77 rail stations as shown in Figure 2. We define v 2 � 4, 5, 8, 12, 28, 34 { } and v 3 � 23, 53 { }; additionally, the remaining rail stations belong to v 1 . Accordingly, we code the rails and collect the length data of each rail segment

Model Optimal Solution.
In the optimal solutions, all demand nodes are fulfilled, as shown in Tables 1 and 2, which is the basis of scenario analysis. Take maintenance support point 29 as an example. e service coverage of a single point is shown in Figure 3.
e analysis and revision of the optimal solution are as follows: (1) e numbers of stations and rails served by each maintenance point are 11 and 11.5, respectively; the actual numbers of stations and rails served by each point are 11-12 and 10-14, respectively. Maintenance demand is averagely distributed, and any maintenance point is bound not to be too idle or congested. Performance experiment 2: in scenario 2, Q max � 15 and the coverage range is 8/5/2 km, 7.5/5/2.5 km, 7/5/3 km, 6/5/ 4 km, and 5/5/5 km. e objective of the experiment is to examine the sensitivity of changes in R j to the optimal solution to the design model, given a fixed Q jmax . is examination also reveals the impact of the maintenance timeliness constraint on the maintenance network program.
Under various Q jmax and R j settings, the optimal solutions to the design model can be summarized in Tables 3 and  4.
For performance experiment 1, the sensitivity analysis of Q jmax is as follows:  Table 4, when Q jmax 's service capacity increases from 12 units to 18, the number of maintenance points decreases from 11 to 7 and remains unchanged thereafter. e continuous increase in the number of maintenance points does not necessarily cause a decrease in the number of maintenance points or an increase in the total length of support routes. Additionally, the standard deviations of the support routes' average length L and average arrival time t are 0.072 and 0.055, respectively. e change is relatively small, and the timeliness of maintenance can be fully achieved.
For performance experiment 2, the sensitivity analysis of R j is shown in Table 5   by a decrease in the utilization rate from 90.48% to 58.79%. e maintenance capacity and construction cost of the maintenance points are sensitive to the range change of the radius R 1 /R 2 /R 3 .

Conclusions
e maintenance network of urban rail transit systems serves the physical network of urban rail transit. rough the scientific deployment of maintenance points, maintenance units, and maintenance part warehouses and the optimization of the configuration of various maintenance resources, we can obtain a safe, reliable, and timely response and a low-cost system, guaranteeing the safe operation of the urban rail transit system. In this paper, we study the design of a maintenance network for urban rail transit systems to provide methodological support for the construction and improvement of these systems. is paper obtains the following results: (1) e number of maintenance units for maintenance points (the service capacity for maintenance points) should be considered. Additionally, the upper limit of the service capacity for maintenance points should be set. e layout plan for the maintenance network should also be obtained. Furthermore, the demand points and support points should be evenly distributed. Maintenance requirements and guarantees should be fulfilled as well. Finally, the ability of individual maintenance points to match each other will not cause congestion or idleness.
(2) e network-wide maintenance solution obtained by the variable coverage radius can not only meet the maintenance requirements but also set a larger coverage radius for areas where maintenance needs are dispersed, thereby reducing the number of maintenance points.
(3) Although in the fixed coverage radius solution, the maintenance barrier network layout setting can also meet the maintenance requirements, more maintenance points are set because the demand distribution in the network is relatively scattered, which increases the construction cost. e above two points show that the set-coverage model with the variable coverage radius is advantageous in solving the problem, and the obtained network arrangement is more reasonable and feasible.
(4) When constructing the maintenance network, we need to balance between the farthest service distance of maintenance points and the number of maintenance resource nodes. As the farthest service   distance of a maintenance point increases, the number of maintenance points required for construction gradually decreases. When the farthest protection distance of a maintenance point decreases, the number of maintenance points that must be built increases faster. Conversely, the greater the farthest protection distance of a maintenance point, the more slowly the number of required maintenance points is reduced.
It is worth pointing out the following: Both performance experiments suffer from the problem of unevenly distributed maintenance points; future research should emphasize the causes of this phenomenon and whether it is the result of a change in the radius of coverage or of the principles or performance of the algorithm.
In actual scenarios in real-world practice, maintenance networks for urban rail transit systems also have to consider other technical and professional requirements. Future research should consider this design problem by taking professional maintenance into account to reach a more practical solution.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.