Research Article  Open Access
Jing Zuo, Jianwu Dang, Min Lyu, "Stochastic Risk Assessment with a Lagrangian Solution for the Optimal Cost Allocation in HighSpeed Rail Networks", Journal of Advanced Transportation, vol. 2020, Article ID 7160681, 12 pages, 2020. https://doi.org/10.1155/2020/7160681
Stochastic Risk Assessment with a Lagrangian Solution for the Optimal Cost Allocation in HighSpeed Rail Networks
Abstract
In largescale highspeed rail networks (HSRNs), the occurrence of occasional malfunctions or accidents is unavoidable. The 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 selfeffects 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 highspeed 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 realworld 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 to more effectively reduce the risks of largescale HSRNs in practice.
1. Introduction
Currently, highspeed rail (HSR) is considered as one of the prevailing intercity travel modes, providing brandnew and highquality 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 largescale highspeed 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 [4–7]. 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]. Thus, 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.
From the perspective of research objects, the existing research mainly focused on four aspects, namely, the road networks [11–14], rail networks [15–18], aviation networks [19, 20], and urban rail networks [21–23]. Additionally, several other interesting research objects have also been considered, such as the vulnerability of rural road networks [24, 25], resilience of freight transportation networks [26], public transport networks [10, 14, 27], and effectiveness of information network computing [28, 29].
HSRNs, which are special rail transportation networks with highspeed 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 gametheoretic 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 [35–37]. 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.
The 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.
Thus, 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)Three impact factors in the HSRNs, including the local effects, global effects, and component selfeffects, 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).
The 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 realworld numerical case is provided to illustrate the proposed methodology. Finally, the conclusion of the research is summarized in Section 6.
2. 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. The 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].
The two elements, namely, stations (nodes) and lines (links) interact with each other in the HSRNs. The 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, highspeed 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 smallworld network structure [42–45].
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. The 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.
The 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.
3. Risk Assessment Model for the HSRNs
The 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.
3.1. Impact Assessment of the HSRNs
The highspeed railway network consists of HSR stations and lines connecting stations. Therefore, it is necessary to establish the corresponding highspeed railway network model first and then make the corresponding impact assessment according to the overall characteristics of its stations and connecting stations.
represents an HSRN composed of a limited number of nodes, in which represents the node set and represents the line set; represents a node in the network; and represents a line in the network. represents an adjacent matrix to describe the relationships between the nodes in the network, and . If and are directly adjacent, ; otherwise, [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 selfeffects should be taken into account altogether, as shown in Figure 1. The 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 selfeffects lead to the failure of node 7.
represents the degree centrality of , which can be calculated using the following equation:
is a standardized value considering the network scale, with a greater indicating a higher level of degree centrality of . represents the betweenness centrality of , which can be calculated using the following equation:where represents the total of all the shortest paths from to and is the number of all these shortest paths (among shortest paths) through the given node . The 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 indicates a higher level of betweenness centrality of .
represents the selfeffects of , with a greater indicating a larger scale of . The value of can be achieved based on the station class of station , where , as shown in the following equation:where represents the maximum value of all the station classes; through normalization processing, .
Considering all the three factors mentioned above, we can define the impact of HSR station , as shown in the following equation:where , , and are the weight coefficients, reflecting the importance of each factor, and . Thus, .
3.1.2. Impact Assessment of the HSR Lines
The impact of the HSR line depends on the two impacts of the stations on both sides of the line. represents the impact of line , which can be calculated using the following equation:where represents the degree centrality of linked with line and represents the impact of linked with line . It can be seen that represents the degree of and .
3.2. Risk Assessment of the Complete HSRNs
If the failure occurrence probability of node and line is and , respectively, the corresponding impact of and is expected to be and , as shown in equations (6) and (7), respectively:where and 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]. Thus, the formula can be constructed as follows:where is the maintenance cost invested in station (node) and is the maintenance cost invested in line ; the coefficients and represent the change trend of the exponential function, as shown in Figure 2. According to equations (8) and (9), the relationship between and the two independent variables ( and ) is as shown in Figure 3(a), and the relationship between and the two independent variables ( and ) is as shown in Figure 3(b).
(a)
(b)
Next, the risk of the entire network 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:
4. 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 in the entire network is a realistic optimizing problem. If the total maintenance cost of the entire network is , the optimization model can be constructed as shown in the following equation:
The Lagrangian multipliers [50–52] can be adopted to solve the special nonlinear optimization problem, such as that defined in equation (11), to acquire the optimal solution. The Lagrangian optimization function can be constructed as shown in the following equation:
The expression is differentiated with respect to , , and , 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), can be substituted into equations (16) and (17) to calculate and , respectively. Therefore, we can obtain the optimal maintenance cost allocation of each station and line in the network, as shown in equations (20) and (21), respectively:
5. Examples and Results
We selected the HSRNs in MidEastern China, 2018, as the test object to establish the network model according to the appropriate abstraction on the actual HSRN structure. The aim of the simulation was to illustrate and validate the proposed optimization model of the risk assessment and maintenance cost allocation. The test network consisted of 25 HSR stations and 64 HSR lines, and the corresponding station names and line lengths are shown in Figure 4.
The parameters used in the simulation are listed in Table 1. These values are not actual values but serve to illustrate the abovementioned approach. The 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. The 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.
(a)
(b)
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. The 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 highspeed railway network maintenance is limited. The 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. Therefore, it can provide auxiliary support for the highspeed railway management department to make a reasonable routine maintenance plan.
Figure 6 illustrates the nonlinear numerical relationship between the entire network risk and total maintenance costs. To demonstrate the advantages of the proposed optimization algorithm, we compared this comprehensive optimization allocation approach with those of the genetic algorithm (GA) and average allocation method, in which the total maintenance costs are distributed evenly among all the stations and lines. As seen from the figure, irrespective of the 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.
The 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.

6. Conclusions
This paper proposes an effective approach to integrate the HSRN risk assessment and maintenance cost allocation for the actual situation of the HSRNs. The 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. The 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. The 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
The data used to support the findings of this study are owned by the Chinese Railway Company.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research work was partially supported by the National Social Science Foundation of China (Grant no. 14XGL011) and the National Natural Science Foundation of China (Grant no.61763025).
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Copyright © 2020 Jing Zuo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.