Stochastic Risk Assessment with a Lagrangian Solution for the Optimal Cost Allocation in High-Speed Rail Networks

In large-scale high-speed rail networks (HSRNs), the occurrence of occasional malfunctions or accidents is unavoidable. -e key issue considered in this study is the optimal allocation of the maintenance costs, based on the stochastic risk assessment for HSRNs. Inspired by the theoretical risk evaluation methods in the complex network, three major factors, including the local effects, global effects, and component self-effects are considered in the process of assessing the impact on the network components (nodes or lines). By introducing the component failure occurrence probability, which is considered to be an exponential function changing with the component maintenance costs, a feasible stochastic risk assessment model of the HSRNs together with the component impact assessment is proposed that can better unify the impact assessment of both the high-speed rail stations and railways. An optimal allocation algorithm based on a Lagrangian relaxation approach is designed. Correspondingly, the optimal cost allocation scheme can be determined using the algorithm to eliminate the various HSRN risks under the given costs. Furthermore, a real-world case study of the HSRNs in eastern China is illustrated. Compared with the genetic algorithm, the simulation shows that the approach can solve the optimal cost allocation problem tomore effectively reduce the risks of large-scale HSRNs in practice.


Introduction
Currently, high-speed rail (HSR) is considered as one of the prevailing intercity travel modes, providing brand-new and high-quality travel services for various passengers. Noteworthy progress has been made in the construction of HSR infrastructures worldwide [1,2]. Furthermore, an increasing number of HSR lines are located in the same region, leading to the implementation of large-scale high-speed rail networks (HSRNs), for instance, in Western Europe, Japan, and China [2,3].
Compared with common rail networks, HSRNs are vulnerable to most stochastic contingencies. In such a largescale HSRN, certain risks are unavoidable [4][5][6][7]. For example, the 7.23 China "Yongwen" railway accident, which occurred in 2011, was caused by an equipment failure, and it caused severe damage to the regional HSRN [8]. us, realizing the risk assessment to ensure the reliability and robustness in HSRNs is still a challenging issue [4,9,10], and it is essential to analyze both the research objects and methods.
HSRNs, which are special rail transportation networks with high-speed trains mainly for passenger transportation, always consist of numerous mechanical and electronic equipment. For HSRNs, Fiondella et al. [30] took the lead in adopting a game-theoretic risk assessment to study the impact of defending the network lines against their vulnerabilities, and they further explored the optimal assignment of the finite defensive resources to the railways. Subsequently, Zhang et al. [6] performed a detailed research on the structural vulnerability when a network was subjected to two different malicious attacks. Although the risk assessment of the HSRNs is considerably different from that of the other transportation networks, only a few studies have focused on this aspect.
In terms of the mainstream methods, many risk assessment methods for the transportation networks have been presented. In principle, these assessment methods mainly consist of four types, namely, complex network methods [6,17,22,23,31], linear or integer programming methods [9,20,26,32], sequence of critical nodes or lines [27,33,34], and data survey and analysis [35][36][37]. Furthermore, certain novel studies in this field have been reported. For instance, Cox et al. [38] presented the operational metrics to aid decision makers in rendering more informed judgments pertaining to resource allocation and the design of a portfolio of security and recovery strategies. Zhang et al. [39] proposed an analysis framework for the vulnerability of a network of networks subjected to failures. However, most of the aforementioned methods are partially inapplicable to the risk assessment of the HSRNs due to the specific characteristics of the HSRNs. e failure probability of each component in a network is significantly different. However, for the sake of feasibility, the prevalent risk assessment methods are inclined to directly remove the target nodes (or lines) from a transportation network instead of considering a stochastic failure probability of a component (node or line). Additionally, for the sake of convenience, these studies concentrated on either the node or line failures to conduct risk assessments based on the classical graph theory, and the nodes and lines in one network as a whole were not considered. In the HSRNs, the nodes and lines are interrelated and interact with each other. us, we attempt to simultaneously discuss the impact assessment of each component in the HSRNs and the risk assessment of the entire network. We effectively combine the risk assessment with the maintenance cost allocation as a united approach. In summary, we try to make the following three contributions to both the risk assessment and safeguard cost allocation scheme in the HSRNs: (1) For a new application field (HSRN), on the basis of the specific characteristics of HSR stations and railways, a united risk assessment model is formulated (see Section 3). Introducing the optimization theory and complex network methods, the assessment model better unifies the impact assessment of the nodes and lines.
(2) ree impact factors in the HSRNs, including the  local effects, global effects, and component self-effects, are considered simultaneously to formulate the  new impact assessment model, in order to solve the  issue more precisely (see Sections 3.1 and 3.2). (3) For this HSRN risk assessment model, a comprehensive HSRN maintenance cost allocation scheme can be developed by using a Lagrangian relaxation algorithm designed to solve the assessment model to minimize the network risk (see Section 4). e remainder of this paper is organized as follows. In Section 2, the major features of the HSRNs, pertaining to the risk assessment are analyzed. Section 3 formulates the risk assessment model for HSRNs, including the impact assessment of the nodes and lines and the risk assessment of the entire HSRNs. Section 4 describes the Lagrangian multiplier optimization algorithm for solving the proposed unified risk assessment model and explains the cost allocation approach. In Section 5, a real-world numerical case is provided to illustrate the proposed methodology. Finally, the conclusion of the research is summarized in Section 6.

Problem Statement
Numerous assiduous works have addressed the network risk assessment in recent years. In general, unlike the network risk assessments of aviation, highway, and subway networks, the risk assessment of HSRNs is characterized as being increasingly complex. e network risk assessments of aviation and subway networks are mainly focused on the investigation of the node failures (as the number of line failures is less), while that of the road networks is based mainly on the analysis of the line failures (as the node effects are insignificant). Such a research approach is in full compliance with reality. Due to the features of high speed, high density, high technology, and large scale in the HSRNs, the risk assessment framework of the HSRNs should be different from that of the conventional rail networks [6,7,40,41]. e two elements, namely, stations (nodes) and lines (links) interact with each other in the HSRNs. e station failures affect the normal operation of the adjacent lines; similarly, the line faults, to a certain extent, restrict the normal functioning of the stations on both the ends. In addition, high-speed rail stations and lines are likely to be faulty and suffer from attacks. In other words, the HSRN risk assessment must consider both the stations and the lines. However, the traditional approaches that only consider the influence of either the station or the line fault do not reflect the actual situation. Furthermore, the HSRN risk assessment needs to consider the network topology (star, ring, mesh, etc.). Different network topology structures involve varying degrees of risk; for instance, the risk of the random topology structure is lower than that of the small-world network structure [42][43][44][45].
Considering the component (nodes or lines) failures, we should focus more on both the relative position of the component in the network and the importance of the component. For a specific node, the failure involves three types of influences: (1) local effects: the lines and nodes directly adjacent to the node are affected; (2) global effects: these effects influence the entire HSRNs; and (3) node selfeffects: the node itself cannot work properly. However, the overall impact of the line failure is similar to that of the node failure.
e former studies on the risk assessment of the traffic network, which focused on either local or global effects, were incomplete and did not conform to the actual situation. For example, if a failure occurs in the Sendai or Nagoya Station in central Japan, the global effects will be greater than the local effects. In contrast, if a failure occurs in the Albany or New Jersey Station in the eastern United States, the local effects will be significantly greater than the global effects.
e heterogeneity needs to be punctiliously considered due to the difference in the scales and levels between the various stations in the HSRNs when analyzing the effects of a node. For example, the Urumqi and Turpan stations located in the Xinjiang province of China are not far from each other; however, the impacts of the Urumqi Station are significantly greater than those of the Turpan Station. However, for simplifying the model, most of the previous investigations inappropriately regarded all the nodes as homogeneous. However, the analysis of the line effects must also take into account the length of a line and other factors. Moreover, the influence of a failure, regardless of the type of influence, in certain major hub stations in the HSRNs, such as in Germany's Frankfurt and Colin stations and France's Paris and Lille stations, can be extremely critical.
Consequently, it is still a realistic challenge to formulate an optimal maintenance cost allocation approach to ensure the safe operation in the HSRNs. A higher maintenance cost for a component (nodes or lines) in a network corresponds to stronger component robustness. Our study addresses how to accurately allocate the limited total maintenance costs to promote the safety and robustness of the HSRNs.

Risk Assessment Model for the HSRNs
e proposed HSRN risk assessment model involves two parts: (1) the impact assessment of the HSRNs, including the line between the HSR stations and the connecting stations, and (2) the risk assessment of the entire HSRN. For clarity, we define the influence caused by the failure of the HSR station (or line) as the impact and the failure occurrence probability of the entire HSRN as the network risk.

Impact Assessment of the HSRNs.
e high-speed railway network consists of HSR stations and lines connecting stations.
erefore, it is necessary to establish the corresponding high-speed railway network model first and then make the corresponding impact assessment according to the overall characteristics of its stations and connecting stations.
G(V, E, A) represents an HSRN composed of a limited number of nodes, in which V � v i | i � 1, 2, . . . , n represents the node set and E � e k | k � 1, 2, . . . , m represents the line set; v i represents a node in the network; and e k represents a line in the network. A represents an adjacent matrix to describe the relationships between the nodes in the network, and A � a i,j | a i,j � 0 , 1 . If v i and v j are directly adjacent, a i,j � 1; otherwise, a i,j � 0 [46,47].
3.1.1. Impact Assessment of the HSR Stations. To assess the impact of an HSR station comprehensively and accurately, all the three factors including the local effects, global effects, and component self-effects should be taken into account altogether, as shown in Figure 1. e degree centrality can be adopted to measure the local effects of the stations [48], betweenness centrality is used to measure the global effects [49], and the station grade is used to measure the node selfeffects. As indicated in Figure 1, if node 7 fails, the local effects directly influence the nodes 4, 5, 6, 8, and 9, as well as all the HSR lines linked with node 7; the global effects correspond to an indirect influence on all the nodes in the entire network, and the node self-effects lead to the failure of node 7.
D v i represents the degree centrality of v i , which can be calculated using the following equation: (1) is a standardized value considering the network scale, with a greater D i indicating a higher level of degree centrality of v i . B v i represents the betweenness centrality of v i , which can be calculated using the following equation: where δ s,t represents the total of all the shortest paths d s,t from v s to v t and δ i s,t is the number of all these shortest paths (among δ s,t shortest paths) through the given node v i . e betweenness centrality is the ratio of the shortest paths in all the node pairs through the given node in the network, reflecting the transfer and cohesive function of the given node in the entire network. A greater B v i indicates a higher level of betweenness centrality of v i .
where c v max represents the maximum value of all the station classes; through normalization processing, C v i ∈ (0, 1]. Considering all the three factors mentioned above, we can define the impact H v i of HSR station v i , as shown in the following equation: where β 1 , β 2 , and β 3 are the weight coefficients, reflecting the importance of each factor, and

Impact Assessment of the HSR Lines.
e impact of the HSR line depends on the two impacts of the stations on both sides of the line. H l k represents the impact of line l k , which can be calculated using the following equation: where D v,k i1 represents the degree centrality of v i1 linked with line l k and H v,k i1 represents the impact of v i1 linked with line l k . It can be seen that (n − 1)D v,k i1 represents the degree of v i1 and H l k ∈ [0, 1].

Risk Assessment of the Complete HSRNs.
If the failure occurrence probability of node v i and line l k is p v i and p l k , respectively, the corresponding impact of v i and l k is expected to be E v i and E l k , as shown in equations (6) and (7), respectively: where p v i and p l k denote the nonlinear function of the maintenance costs for the corresponding node and unit length line, respectively. For a station, a larger amount of maintenance costs invested in a station corresponds to a lower occurrence probability of the failure station. For different stations, to reduce the same risk probability, a larger station requires the investment of higher maintenance costs. Similarly, for a line, a larger amount of maintenance costs invested in the unit length of the certain line corresponds to a lower failure probability of the section. Without loss of generality, we assume that the occurrence failure probability of the station and the maintenance costs accord with the exponential function and that of the line and the maintenance costs in the unit length is the same as that of the station [32].
us, the formula can be constructed as follows: where g v i is the maintenance cost invested in station (node) i and g l k is the maintenance cost invested in line k; the coefficients θ and φ represent the change trend of the exponential function, as shown in Figure 2. According to equations (8) and (9), the relationship between p v i and the two independent variables (g v i and C v i ) is as shown in Figure 3(a), and the relationship between p l k and the two independent variables (g l k and u k ) is as shown in Figure 3(b).
Next, the risk R of the entire network G(V, L, A) can be defined as the sum of the expected values of the impacts of the network elements (nodes and lines), representing the relative probability of the failure occurring in the entire network, as shown in the following equation:

Optimization Algorithm for the Risk Minimization Costs
Under the constraint of the limited total maintenance costs, the allocation of the maintenance costs to minimize the risk R in the entire network is a realistic optimizing problem. If the total maintenance cost of the entire network is g m , the optimization model can be constructed as shown in the following equation: e Lagrangian multipliers [50][51][52] λ can be adopted to solve the special nonlinear optimization problem, such as that defined in equation (11), to acquire the optimal solution. e Lagrangian optimization function Φ can be constructed as shown in the following equation: e expression is differentiated with respect to g v i , g l k , and λ, as shown in equations (13), (14), and (15), respectively: To solve equations (13) and (14), the results of equations (13) and (14) can be expressed as shown in equations (16) and (17), respectively: Next, equations (16) and (17) can be substituted into equation (15), from which the following equation is derived: Furthermore, the following equation can be derived from equation (18): According to equation (19), ln(−λ) can be substituted into equations (16) and (17) to calculate g v i and g l k , respectively. erefore, we can obtain the optimal maintenance cost allocation of each station and line in the network, as shown in equations (20) and (21), respectively:

Examples and Results
We selected the HSRNs in Mid-Eastern China, 2018, as the test object to establish the network model according to the appropriate abstraction on the actual HSRN structure. e aim of the simulation was to illustrate and validate the proposed optimization model of the risk assessment and maintenance cost allocation. e test network consisted of 25 HSR stations and 64 HSR lines, and the corresponding station names and line lengths are shown in Figure 4. e parameters used in the simulation are listed in Table 1. ese values are not actual values but serve to illustrate the abovementioned approach. e shortest path can be solved using the Floyd algorithm.
On the basis of the proposed risk assessment model and cost allocation optimization algorithm, the 25 test stations and 64 lines in the network were evaluated. e approach could obtain the basic attribute information and impact values of these stations and lines, as given in Tables 2 and 3.
From Table 2, it can be seen that the attribute values vary considerably for different nodes. Among these nodes, the degree centrality of nodes 8,9,13,19, and 20 is higher, while the betweenness centrality of nodes 6,8,9,13, and 20 is higher. Overall, nodes 6,8,9,13, and 20 are more important in the network because their impact values are higher.
From Table 3, it can be seen that the impact values of the lines are also considerably different. Among these lines, the impact values of lines 1, 2, 27, 38, 39, 51, etc., are larger, while those of lines 45, 46, 47, 49, 60, etc., are smaller.
Taking the total maintenance costs (20000, 35000, and 50000, respectively) as an example, the allocation scheme of the optimal maintenance costs among the various stations and lines is described in detail in Figure 5. Figure 5(a) shows the maintenance cost allocation scheme among different stations, and Figure 5(b) shows the corresponding scheme among different lines.
Based on the impact values of the stations and the lines and the optimization algorithm for the risk minimization costs, the corresponding input allocation is completed. e model can achieve the specific allocation of maintenance input for each station and line under the goal of minimum maintenance failure risk when the input of high-speed railway network maintenance is limited. e calculated maintenance strategy is the optimal allocation scheme in network maintenance with a global optimal solution provided by the model. When the maintenance cost is fixed, the maximum effect of the investment can be achieved. erefore, it can provide auxiliary support for the high-speed railway management department to make a reasonable routine maintenance plan.      type of allocation scheme, with an increase in the total maintenance costs, the risk of the entire network is significantly reduced. A comparison of the three allocation schemes indicates that the proposed optimization allocation scheme exhibits notable advantages, and it is better than the other two allocation schemes in various situations. When the total maintenance costs are higher than 100000, the network risk corresponding to the optimal allocation scheme is close to zero. e calculation results of this example indicate that the risk assessment model and the maintenance cost allocation algorithm proposed in this paper can help realize the risk assessment and cost allocation in various conditions to determine the optimal maintenance cost allocation scheme. In addition, the total cost of 25000 was considered as an example to illustrate the concrete computation process of the three optimization allocation schemes, as presented in Table 4.

Conclusions
is paper proposes an effective approach to integrate the HSRN risk assessment and maintenance cost allocation for the actual situation of the HSRNs. e actual test results show that the comprehensive optimization approach can be suitably applied for the HSRN risk assessment. Under the conditions of the given limited maintenance costs, the approach can solve the optimization allocation of the maintenance costs to minimize the entire HSRN risk in practice. In addition, the approach can also be used to measure the total amount of maintenance costs in a certain range of network risk control.
In the specific usage of the approach to allocate maintenance costs, the model parameters must be calibrated according to the actual situation in each region to ensure that the approach has a high fitting degree. e example provided in this paper is aimed at illustrating the feasibility and effectiveness of the approach. In reality, certain other risk factors related to the HSRNs exist. e next step of our research will be to introduce several important risk factors into the proposed assessment model to make the model and algorithm more precise and practical.

Data Availability
e data used to support the findings of this study are owned by the Chinese Railway Company.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.