Research on the Optimization Model of Railway Emergency Rescue Network Considering Space-Time Accessibility

Abstract

The capability of railway emergency rescue can be enhanced by maintaining the railway emergency rescue network and upgrading its technology. Nowadays, influenced by the factors, such as resource type, personnel distribution, line level, etc., space-time differences may be unavoidable. In the meantime, the general description method of the transportation network may lack the consideration of the rescue transportation particularity, so the strategies of resource allocation, maintenance, and upgrading could be illogical. Hence, in this paper, the gravity model is utilized to improve the classical travel time budget model and to construct the space-time accessibility model, firstly. Then, further exploring the space-time accessibility of nodes and edges of railway emergency rescue network and considering the randomness of travel time, a space-time accessibility measurement method for an emergency network is proposed. Moreover, a global optimization model with accessibility characteristics is then constructed for the maintenance allocation of the emergency rescue transportation network. The results show that the proposed method can solve the maintenance allocation problem of the large-scale rescue network effectively, reduce the risk of maintenance allocation strategy failure caused by unreasonable node index parameters, and provide an effective basis and theoretical support for the rational formulation of railway rescue transportation network maintenance allocation strategy.

1. Introduction

The maintenance and upgrade of the railway emergency rescue network can improve the railway emergency rescue capability significantly, which is of critical importance for guaranteeing the normal operation of the railway [1,2,3]. However, with the improvement of operation technology, the enrichment of operation modes, and the differences in emergency resource allocation, etc., the total maintenance cost cannot meet line maintenance and technical improvement of all rescue resource reserve nodes and lines at the same time, and, thus, the unreasonable maintenance cost allocation problems appear in which the maintenance scheme cannot fit the actual characteristics i.e., the space-time accessibility of the network nodes and edges mostly. Therefore, figuring out how to make full use of the node’s space-time accessibility to optimize the maintenance allocation strategy of the railway emergency rescue network is critical for assisting emergency rescue departments in rationally allocating maintenance costs and maximizing railway emergency rescue capabilities on a budget.

Currently, accessibility research includes two main components: (i) transport and (ii) facility. The transport component measures the accessibility of traffic travel modes and the accessibility differences between different travel modes, involving the convenience and fairness of traffic travel modes [4,5,6,7]. The facility component measures the accessibility from settlements to service facilities, involving the spatial distribution equality of public service facilities [8,9,10]. In particular, numerous studies are being conducted regarding accessibility to the emergency rescue facilities, such as fire stations [11,12], medical facilities [13,14], etc. Some scholars have analyzed the spatial accessibility of fire stations combined with the Geographic Information System (GIS) to determine the optimal locations of new fire stations [11]. Others have evaluated the spatial accessibility to healthcare and confirmed that the spatial allocation of resources was unbalanced [13]. Thus the application of accessibility has the potential for the improvement of maintenance strategies, which can help alleviate disparities in service provision across the emergency rescue network. However, as the emergency rescue of fire stations and medical facilities belongs to the social emergency type, it is mainly participated by the social emergency management department and completed by calling social emergency resources [14]. Although railway emergency rescue has its industry particularity and most of the emergency resources are deployed within the railway, thus studying the allocation of resources and the assessment of transport capacity can further enhance the railway emergency rescue capability. The specific management methods and processes of medical and fire stations can be used as a reference for the research methods in this paper. For the railway emergency rescue network, the corresponding description of the space-time accessibility is lacking, and only a few studies have attempted to focus on the space-time resources modeling method to investigate the train platform planning problem under railway emergencies [15]. In the previous research on the railway emergency rescue network, the focuses are mainly on the railway rescue location issues [16,17,18], railway emergency rescue decision-making [19,20], railway emergency rescue capacity assessment [21], and railway emergency rescue prediction [22]. The railway rescue network constructed based on space-time accessibility can better reflect the convenience of rescue sites, transport routes, etc., from the emergency rescue resource nodes in the region to the accident occurrence point, which can greatly close to the comprehensive performance of the railway emergency rescue network system.

The emergency rescue network is crucial in the transportation field, mainly involving road traffic and rail transit network, whose maintenance allocation strategies vary significantly depending on the characteristics of the transportation. Due to the standardization of road network construction and maintenance, the maintenance tasks can be entrusted to social organizations in addition to the road government departments [23]. It is well known that the maintenance of road traffic networks is mainly managed by each local government itself and there is a lack of network-wide coordination, so the existing research is mainly focused on improving road performance [24], emergency facility performance [25,26], and so on. Rail transit includes urban rail transit and mainline railways. Since the coverage area of urban rail transit is small, there is little difference in the space-time accessibility of resource demand points to obtain resources during the emergency rescue. Therefore, non-time-space factors (e.g., facility capacity and resource utilization [27,28,29]) are mainly considered in the existing maintenance allocation strategies for the urban rail transit emergency rescue networks, such as the optimal resource allocation based on the importance of each rescue resource reserve node and the number of demand points it covers [28]. With a large coverage area and strong rescue specialization of railway transportation, the emergency resource reserve nodes are required to transport resources within their jurisdiction during the emergency rescue process. The effectiveness of rescue operations will be greatly improved by investigating the space-time accessibility of rescue resource reserve nodes and optimizing the maintenance strategy for rescue resource reserve nodes. The current research on railway emergency rescue mainly focuses on predicting the probability of accidents occurring in the region under the jurisdiction and formulating the emergency rescue plan [30,31,32], with research on the optimal allocation of emergency rescue network based on the cost and comprehensiveness of rescue [31,32,33,34], and is deficient in studies that would maximize the effectiveness and utility of rescue resources in terms of time and space.

As a result, taking the timeliness of the railway emergency rescue network into account, this paper proposes a space-time accessibility model of node, in which the rescue characteristics of the railway emergency rescue network are fitted to the greatest extent. At the same time, based on the node time uncertainty parameters, the emergency rescue network maintenance allocation model that considers the space-time accessibility of railway emergency networks is constructed by improving the classical accessibility algorithm, which can provide theoretical support for maximizing the railway emergency rescue capability under resource availability conditions.

2. Analysis of Railway Rescue Transportation Network and Space-Time Accessibility

The railway emergency rescue transportation network is composed of nodes that have some unique characteristics compared to other points in the ordinary transportation network. The significant difference from the ordinary transportation network is that the railway emergency rescue transportation network’s physical nodes are rescue resource reserve nodes, and the logical rescue service range of the physical nodes define their service attributes, as shown in Figure 1.

While responding to an emergency, resources and equipment are called upon according to the location and level of response. Additionally, the sooner an emergency is effectively dealt with, the less likely it is to evolve, spread, and diffuse. Therefore, timeliness ought to be the first requirement while completing emergency transportation tasks. The utilization of space-time accessibility to depict the space-time aspects of emergency rescue transportation network nodes and edges can further improve the inaccurate description of the traditional transportation network, mainly the ease of arrival of emergency rescue transportation in time and space, and more closely match the characteristics of emergency transportation networks. As illustrated in Figure 2, space-time accessibility is calculated by the product mean value of an emergency resource reserve node’s capacity attribute size and the space-time impedance overcome value of all cover demand points from that node.

The establishment of a space-time accessibility identification model for nodes and edges in an emergency rescue transportation network can fully describe the time and space characteristics of the network, as well as objectively analyze the transportation efficiency under emergency conditions.

2.1. Construction of Railway Emergency Rescue Transportation Network

Railway emergency rescue transportation network is a special network in which the emergency resource reserve nodes are the network nodes, the transmission lines between nodes are the edges of the network, and the nodes’ demand for space-time accessibility is the attribute of the nodes. The railway emergency rescue transportation network can be defined as $G\left(K,I,L\right)$. $K$ represents the set of the logical subnet of rescue region divided in the network $G$, $K=\left\{K_u|u=1,2,3,\cdots , o\right\}$, $o$ is the total number of subnets, in which ${ K}_u \subseteq G$ and $K_1\cap K_2\cap \cdots \cap K_O=Ø$; $I$ represents the set of nodes composed of rescue resource reserve nodes,$I=\left\{i|1,2,\cdots ,v\right\}$, $v$ is the total number of nodes; $L$ denotes the set of directed connected edges connecting the rescue resource reserve nodes,$L=\left\{l|1,2,\cdots ,e\right\}$, $e$ is the total number of edges.

As shown in Figure 3, the example network is divided into 3 logical subnets of different rescue jurisdiction regions, i.e., $\left\{K_1{,K}_2{,K}_3\right\}$, where $I=\left\{i|1,2,3\right\}, i\in I$ are the nodes equipped with professional rescue teams, relief trains to be dispatched, and emergency resource reserves. Additionally,$L=\left\{l|1,2,3\right\}, l\in L$ are the edges composed of actual lines between adjacent nodes.

2.2. Accessibility Identification of Rescue Resource Reserve Nodes in Emergency Rescue Network

The space-time accessibility $A_i$ of rescue resource reserve node $i$ is defined as the degree of convenience in time and space for emergency resources to reach the rescue demand points $j$ in the logical subnet k under its jurisdiction when carrying out a rescue, taking into account the fact that the reserve node $i$ in the logical subnet k of the rescue region covers all rescue demand points j, where $j=\left\{1,2,\cdots ,n\right\}$,$n$ is the number of demand points $j$ in the logical subnet $k$. The greater the space-time accessibility of the nodes, the better the convenience of emergency resources and the more efficient the transport reaching the rescue demand points in the region under their jurisdiction. The traditional travel time budget model only considers a single demand node as the research object. To measure the space-time accessibility of a rescue resource reserve node $i$, it is necessary to analyze it in conjunction with the full range of demand points in the rescue jurisdiction region, so this paper improves the travel time budget model by using the gravity model to construct an accessibility identification algorithm that covers multiple demand points.

Equation (1) is obtained by introducing the travel time budget model into the gravity model, where $A_i$ is the space-time accessibility of the rescue resource reserve nodes in the regional logical subnet $k$, then the equation can be given as follows:

$$A_i=\frac{1}{n}{{\displaystyle \sum }}_{j=1}^n{S}_i^{\left(k\right)}·b_{i,j}^{-\beta }$$

(1)

where $n$ is the number of rescue demand points in the logical subnet $k$; $S_i^{\left(k\right)}$ is the rescue resource allocation capacity of the node $i$ in the logical subnet $k$; $\beta$ is the traffic friction coefficient, the railway emergency rescue process is managed by using line blockade for relief trains passing only, so selected $\beta$ to 1; $b_{i,j}$ is the travel time budget value from node $i$ to rescue demand point $j$, which is negatively correlated with the space-time accessibility of the node. Due to the process being affected by factors such as the speed limit of relief trains and regional lines, preparation of rescue resources, etc., the value of $b_{i,j}$ will appear arbitrary changes, which can be described as follows:

$$b_{i,j}=E\left(T_{i,j}\right)+\lambda ·{\theta }_{T_{i,j}}$$

(2)

where $E\left(T_{i,j}\right)$ denotes the expected travel time from node $i$ to the rescue demand point $j$, influenced by factors such as line speed limit, relief train speed limit, and distance; ${\theta }_{T_{i,j}}$ is the standard time deviation, determined by the standard deviation of $E\left(T_{i,j}\right)$ and the actual travel time;$\lambda$ is the parameter that arrives at the demand point $j$ on time from the node $i$ within the time range of $b_{i,j}$.

Set $b_{i,j}$ to obey the normal distribution, by the principle of the cumulative distribution function of normal distribution then we have:

$$\mathrm{P}\left\{T_{i,j} \le { b}_{i,j}\right\}=\mathsf{\Phi }\left(\frac{b_{i,j}-\mu }{\sigma }\right)=\mathsf{\Phi }\left(\frac{b_{i,j } - E\left(T_{i,j}\right)}{{\theta }_{T_{i,j}}}\right)$$

(3)

where $\mathsf{\Phi }(·)$ denotes the cumulative distribution function of the normal distribution. Let the probability of arriving at the rescue demand point on time be $\alpha$, in combination with $P\left\{T_{i,j} \le { b}_{i,j}\right\}=\alpha$, then:

$$\mathsf{\Phi }\left(\frac{E\left(T_{i,j}\right)+\lambda ·{\theta }_{T_{i,j} }- E\left(T_{i,j}\right)}{{\theta }_{T_{i,j}}}\right)=\alpha$$

(4)

$${\mathsf{\Phi }}^{-1}\left(\alpha \right)=\mathsf{\lambda }$$

(5)

Hence,$b_{i,j}$ can be rewritten as Equation (6):

$$b_{i,j}=E\left(T_{i,j}\right){+\mathsf{\Phi }}^{-1}\left(\alpha \right)·{\theta }_{T_{i,j}}$$

(6)

Combining Equations (2) and (6) gives Equation (7) as follows:

$$A_i=\frac{1}{n}{{\displaystyle \sum }}_{j=1}^n{S}_i^{\left(k\right)}·{\left(E\left(T_{i,j}\right){+\mathsf{\Phi }}^{-1}\left(\alpha \right)·{\theta }_{T_{i,j}}\right)}^{-\beta }$$

(7)

2.3. Space-Time Accessibility Identification of Emergency Rescue Network Edges

The space-time accessibility $A_l$ of the edge $l$ in an emergency rescue reserve network is defined as the correlation degree of the interaction between two directly adjacent rescue resource reserve nodes $i$ connecting this edge. The higher the correlation degree, the more convenient the transportation of emergency resources between reserve nodes, as well as the smoother the regional allocation of resources. Combined with the actual distribution of the lines, the relevant expressions for the edges can be obtained as follows:

$$A_l=\frac{1}{2\left(m - 1\right)}·\left(A_i{+A}_a\right)$$

(8)

where the rescue reserve node $i$ is directly adjacent to the reserve node $a$, $a \ne i\in l$; $m$ is the total number of the logical subnet associated with the edge $l$.

3. Maintenance Allocation Strategy Model of Railway Emergency Rescue Network

Due to capital cost constraints, capital investment in batches and stages is frequently used in daily maintenance and technology upgrades of resource nodes and lines. Therefore, with limited maintenance costs, the existing emergency rescue network should be optimized to the greatest extent possible, thus maximizing the utility of maintenance and upgrading at this stage, which will not only improve the level and capability of railway emergency management but also reduce the rescue risk caused by insufficient resource allocation at this node in emergency rescue.

3.1. Risk Model Considering Space-Time Accessibility

The space-time accessibility $A_l$ of the edge $l$ in an emergency rescue network is defined as the correlation degree of the interaction between two directly adjacent rescue resource reserve nodes.

In this paper, the nonlinear characteristics of the exponential distribution are used to describe the emergency rescue capability, maintenance cost, and risk relationship of emergency rescue network nodes and edges in conjunction with the input and maintenance characteristics, and the following assumptions are made:

Assumption: let the exponential distribution be how maintenance costs and the risk of maintenance failure probability are distributed. Therefore, the risk can be described as:

$$p_i{=e}^{-\phi ·\frac{g_i^z}{C_i^{\left(k\right)}}}$$

(9)

$$p_l{=e}^{-\varphi ·\frac{g_l^z}{B_l}}$$

(10)

where $g_i^z$ and $g_l^z$ are the maintenance input cost for nodes $i$ and edges $l$; $C_i^{\left(k\right)}$ represents the configuration status level of nodes; $B_l$ represents the line status level of edges;${ C}_i^{\left(k\right)}$ and $B_l$ are divided into I, II, III, and IV levels, respectively, in which I is the highest level; $\phi$ and $\varphi$ are the trend parameters for nodes and edges to be obtained in combination with daily maintenance data.

Let $p_i$ and $p_l$ be the risk probabilities of node and edge input maintenance failure, which represents the probability of failure after maintenance is carried out, respectively; $E_i$ and $E_l$ are the maintenance failure expectations for emergency rescue network nodes and edges, respectively. According to the concept of expectation, the following non-linear relationship can be obtained.

$$E_i{=p}_i·A_i+\left({1 - p}_i\right)·0$$

(11)

$$E_l{=p}_l·A_l+\left({1 - p}_l\right)·0$$

(12)

where $A_i$ and $A_l$ are the space-time accessibility of node $i$ and edge $l$ in the railway emergency rescue network.

Combined with the description of node and edge maintenance risks, the risk model of maintenance failure of emergency rescue regional network nodes and edges under the certain maintenance input is constructed.

$$R={{\displaystyle \sum }}_{i=1}^v{E}_i+{{\displaystyle \sum }}_{l=1}^e{E}_l={{\displaystyle \sum }}_{i=1}^v\left(p_i·A_i\right)+{{\displaystyle \sum }}_{l=1}^e\left(p_l·A_l\right)$$

(13)

3.2. Objective Function of Maintenance Allocation Strategy Model

The railway emergency rescue network is distinguished by specific routes, numerous rescue nodes, and an uneven distribution of rescue resources, necessitating a significant investment in maintenance and upgrade. Under the constraint of a limited total maintenance cost, this paper developed a network maintenance allocation optimization model that took into account the space-time accessibility of emergency rescue network nodes and edges, with the research goal of minimizing the failure risk of emergency rescue network maintenance allocation inputs. This model provides a theoretical research foundation for rationally determining the emergency rescue network’s maintenance objectives and maintenance intensity.

Let $g^z$ be the total maintenance cost of the emergency rescue network; $g_i^z$ and $g_l^z$ be the input cost of the emergency rescue network node $i$ and the edge $l$, respectively; hence the objective function of the optimal model for network maintenance allocation is as follows

$$\begin{array}{c} \mathit{minR}\left(g^z\right)={\displaystyle \sum }_{i=1}^v\left(e^{-\phi ·\frac{g_i^z}{C_i^{\left(k\right)}}}·A_i\right)+{\displaystyle \sum }_{l=1}^e\left(e^{-\varphi ·\frac{g_l^z}{B_l}}·A_l\right)\\ s.t.\{\begin{array}{c}{\displaystyle \sum }_{i=1}^v{g}_i^z+{\displaystyle \sum }_{l=1}^e{g}_l^z{=g}^z\\ 0 \le { g}_i^{z }\le { g}^z\\ 0 \le { g}_l^z \le { g}^z\end{array}\end{array}$$

(14)

3.3. Methodology for Solving the Maintenance Allocation Policy Model

Optimization models usually contain multiple non-linear complex problems, and it is difficult to determine the analytical solution of the model directly, and local Pareto solutions can only be obtained through simulation and simplification. In this paper, the Lagrangian multiplier is introduced to complete the analysis of this model to obtain the global optimal solution. The above equation is varied by Lagrangian relaxation to obtain the corresponding optimization function as follows.

$$\mathsf{\Phi }\left(g_i^z{,g}_l^z,\lambda \right)={{\displaystyle \sum }}_{i=1}^v\left(e^{-\phi ·\frac{g_i^z}{C_i^{\left(k\right)}}}·A_i\right)+{{\displaystyle \sum }}_{l=1}^e\left(e^{-\varphi ·\frac{g_l^z}{B_l}}·A_l\right) - \lambda ·\left({{\displaystyle \sum }}_{i=1}^v{g}_i^z+{{\displaystyle \sum }}_{l=1}^e{g}_l^z{ - g}^z\right)$$

(15)

Based on the Lagrangian algorithm, $g_i^z$ and $g_l^z$ can be calculated, respectively, as shown in the following equations:

$$\frac{\partial \mathsf{\Phi }\left(g_i^z{,g}_l^z,\lambda \right)}{\partial g_i^z}=-\frac{\phi }{c_i^{\left(k\right)}}{A}_i·e^{-·\frac{\phi }{C_i^{\left(k\right)}}·g_i^z}-\lambda =0$$

(16)

$$\frac{\partial \mathsf{\Phi }\left(g_i^z{,g}_l^z,\lambda \right)}{\partial g_l^z}=-\frac{\varphi }{B_l}{A}_l·e^{-·\frac{\varphi }{B_l}·g_l^z}-\lambda =0$$

(17)

$$\frac{\partial \mathsf{\Phi }\left(g_i^z{,g}_l^z,\lambda \right)}{\partial \lambda }={{\displaystyle \sum }}_{i=1}^v{g}_i^z+{{\displaystyle \sum }}_{l=1}^e{g}_l^z {- g}^z=0$$

(18)

From the above we can get:

$$g_i^{z }=-\frac{c_i^{\left(k\right)}}{\phi }\left(\left(\mathit{ln}\frac{c_i^{\left(k\right)}}{{\phi A}_i}\right)+\mathit{ln}\left(-\lambda \right)\right)$$

(19)

$$g_l^z=- \frac{B_l}{\varphi }\left(\left(\mathit{ln}\frac{B_l}{\varphi A_l}\right)+\mathit{ln}\left(-\lambda \right)\right)$$

(20)

Bringing Equations (19) and (20) into the constraints gives:

$${{\displaystyle \sum }}_{i=1}^v\left(\frac{c_i^{\left(k\right)}}{\phi }\left(\left(\mathit{ln}\frac{{\phi A}_i}{c_i^{\left(k\right)}}\right)-\mathit{ln}\left(-\lambda \right)\right)\right)+{{\displaystyle \sum }}_{l=1}^e\left(\frac{B_l}{\varphi }\left(\left(\mathit{ln}\frac{\varphi A_l}{B_l}\right)-\mathit{ln}\left(-\lambda \right)\right)\right){=g}^z$$

(21)

Collation gives:

$$\mathit{ln}\left(-\lambda \right)=\frac{{{\displaystyle \sum }}_{i=1}^v\left(\frac{c_i^{\left(k\right)}}{\phi }\left(\mathit{ln}\frac{{\phi A}_i}{c_i^{\left(k\right)}}\right)\right)+{{\displaystyle \sum }}_{l=1}^e\left(\frac{B_l}{\varphi }\left(\mathit{ln}\frac{\varphi A_l}{B_l}\right)\right){-g}^z}{{{\displaystyle \sum }}_{i=1}^v\left(\frac{c_i^{\left(k\right)}}{\phi }\right)+{{\displaystyle \sum }}_{l=1}^e\left(\frac{B_l}{\varphi }\right)}$$

(22)

Substituting Equation (22) into Equations (19) and (20) to calculate $g_i^z$ and $g_l^z$ respectively, as shown in Equations (23) and (24) below:

$$g_i^z=-\frac{c_i^{\left(k\right)}}{\phi }\left(\left(\mathit{ln}\frac{c_i^{\left(k\right)}}{{\phi A}_i}\right)+\frac{{{\displaystyle \sum }}_{i=1}^v\left(\frac{c_i^{\left(k\right)}}{\phi }\left(\mathit{ln}\frac{{\phi A}_i}{c_i^{\left(k\right)}}\right)\right)+{{\displaystyle \sum }}_{l=1}^e\left(\frac{B_l}{\varphi }\left(\mathit{ln}\frac{\varphi A_l}{B_l}\right)\right){-g}^z}{{{\displaystyle \sum }}_{i=1}^v\left(\frac{c_i^{\left(k\right)}}{\phi }\right)+{{\displaystyle \sum }}_{l=1}^e\left(\frac{B_l}{\varphi }\right)}\right)$$

(23)

$$g_l^z=-\frac{B_l}{\varphi }\left(\left(\mathit{ln}\frac{B_l}{\varphi A_l}\right)+\frac{{{\displaystyle \sum }}_{i=1}^v\left(\frac{c_i^{\left(k\right)}}{\phi }\left(\mathit{ln}\frac{{\phi A}_i}{c_i^{\left(k\right)}}\right)\right)+{{\displaystyle \sum }}_{l=1}^e\left(\frac{B_l}{\varphi }\left(\mathit{ln}\frac{\varphi A_l}{B_l}\right)\right)-g^z}{{{\displaystyle \sum }}_{i=1}^v\left(\frac{c_i^{\left(k\right)}}{\phi }\right)+{{\displaystyle \sum }}_{l=1}^e\left(\frac{B_l}{\varphi }\right)}\right)$$

(24)

From the above, we can obtain the optimal solution for the global comprehensive maintenance cost allocation of each node and edge, with a certain maintenance cost for the emergency rescue network.

4. Examples and Results

4.1. Case Overview

This section uses data for emergency resource reserve nodes from China Railway Xi’an Bureau Group Co., Ltd. as the experimental basis. This group currently consists of a total of 9 rescue resource reserve nodes. The different reserve nodes are divided into different rescue regions in the daily management. Figure 4 below shows the emergency rescue network of this railway group company, different colors represent different regions.

4.2. Maintenance Allocation Model Construction

4.2.1. Space-Time Accessibility of Emergency Rescue Network

The railway emergency rescue network is constructed using the ArcGIS platform. Additionally, the resource reserve nodes and edges data of China Railway Xi’an Bureau Group Co., Ltd. were sorted out and the results are shown in Figure 5a,b. After completing the construction of the railway emergency rescue network with the actual data, Equations (2) to (6) are used to calculate the transport travel time budget value $b_{i,j}$ of each resource reserve node $i$ in the logical subnet $k$ of the rescue region for this group company. Additionally, the value of $S_i^{\left(k\right)}$ can be obtained by combining the evaluation of factors, such as personnel management, resource allocation, and emergency technology of the resource reserve nodes, where $b_{i,j}$, $S_i^{\left(k\right)}$ need to be normalized.

Thus, the space-time accessibility $A_i$ of the node $i$ is obtained according to Equation (1) and the results are presented in

Table 1. Related variables and accessibility measurements for the railway rescue network.

Node $i$	Lintong	Yan’an	Lueyang	Yanliang	Jingbian	Baoji	Hanzhong	Wanyuan	Ankang
$A_i$	9.6700	3.9302	10.1202	9.3524	7.4984	6.1247	5.8570	7.3292	8.4631
Edge $l$	Lintong-Ankang	Lintong-Yanliang	Lintong-Baoji	Lintong- Hanzhong	Ankang- Hanzhong	Ankang-Wanyuan	Yan’an- Jingbian	Yan’an-Yanliang	Baoji- Lueyang	Baoji- Yanliang	Lueyang-Hanzhong
$m$	2	2	2	2	2	2	2	3	2	3	2
$A_l$	9.0643	9.5112	7.8974	7.7635	7.1578	7.8776	5.7143	3.3207	8.1225	3.8693	7.9886

. Then the space-time accessibility values of $A_l$ for the edge $l$ are measured through Equation (8) and all these results are presented in Table 1.

4.2.2. Reserve Node and Edge Maintenance Strategy Allocation Results

To test the maintenance allocation strategy model developed in this paper, the total maintenance cost (250,000, 1,000,000, and 1,750,000, respectively) was employed as an example. The trend parameters $\mathsf{\phi }$ and $\mathsf{\varphi }$ of nodes and edges used in the maintenance model are obtained as 0.0001 and 0.0002, respectively, with daily maintenance data first. Then, according to the space-time accessibility $A_i$ and $A_l$ of nodes $i$ and edges $l$ as in Table 1, the optimal allocation scheme of node and edge maintenance strategy aiming at the minimum failure risk is obtained by applying Equations (23) to (24), and the results are shown in

Table 2. Maintenance cost allocation strategy with different inputs.

$\mathbf{Node} \mathit{i}$	$\mathbf{Allocation} \mathbf{Results} {\mathit{g}}_{\mathit{i}}^{\mathit{z}}$ [CNY]			$\mathbf{Edge} \mathit{l}$	$\mathbf{Allocation} \mathbf{Results} {\mathit{g}}_{\mathit{l}}^{\mathit{z}}$ [CNY]
	250,000(gz)	1,000,000(gz)	17,500,000(gz)		250,000(gz)	1,000,000(gz)	17,500,000(gz)
Lintong	16,585	40,779	64,972	Lintong-Ankang	15,938	40,132	64,325
Yan’an	1300	49,688	98,075	Lintong-Yanliang	18,547	54,838	91,128
Lueyang	18,162	90,743	163,323	Lintong-Baoji	10,746	22,843	34,939
Yanliang	15,795	88,376	160,956	Lintong-Hanzhong	10,660	22,757	34,854
Jingbian	14,220	62,608	110,995	Ankang-Hanzhong	13,577	37,770	61,964
Baoji	10,173	58,560	106,947	Ankang-Wanyuan	14,535	38,729	62,922
Hanzhong	9279	57,667	106,054	Yan’an-Jingbian	11,325	35,518	59,712
Wanyuan	8349	80,929	153,510	Yan’an-Yanliang	5897	30,090	54,284
Ankang	16,630	65,017	113,405	Baoji-Lueyang	16,180	52,470	88,760
				Baoji-Yanliang	7426	31,619	55,813
				Lueyan-Hanzhong	14,675	38,869	63,062

.

From the above calculation results, we can see that the amount invested in each emergency reserve node and edge varies depending on the capital investment. Taking the Yan’an emergency reserve node as an example, its allocation strategy is 1300, 49,688, and 98,075 in that sequence when the total maintenance cost is 250,000, 1,000,000, and 1,750,000, respectively.

4.3. Analysis of Results

4.3.1. Reserve Node and Edge Maintenance Allocation Results

The comparison of allocation algorithms for each node and edge with various financial inputs allows for a more detailed analysis of the amount allocated in the node under different inputs, which can be used as a decision aid for selecting the optimal allocation of nodes and edges. The algorithms have been depicted in Figure 6. As can be seen, when the amount of inputs increases, the allocation volume of each emergency reserve node and edge increases continuously while the value of the allocation ratio has changed continuously.

To further analyze the changing trend of the allocation strategy under different inputs, the allocation results obtained by each node and edge under different inputs were counted, and the results are shown in Figure 7 below. It is also observed that the allocation ratio of each emergency reserve node and edge tends to vary in different trends as input cost increases.

The change rate of the allocation ratio acquired at distinct nodes and edges progressively converges to zero as the input cost increases, and the allocation ratio might be indefinitely near a specific definite value. This is depicted in Figure 7. It can also be found that the change rate of the input allocation ratio is positively correlated with the change rate of the minimum failure risk. The reason behind this conclusion is that when the input ratio at this node is 0.064, it means that the risk of transporting emergency resources due to insufficient allocation at this node is infinitely close to the minimum.

4.3.2. Comparative Analysis

To verify the applicability and accuracy of the method proposed in this paper, a comparative analysis of the method proposed in this paper with available analytical solutions, including the genetic algorithm (GA) and the average allocation algorithm was carried out, under an equal increasing series of inputs with a tolerance of 100,000, as shown in Figure 8 below. All three algorithms were simulated and verified in the MATLAB R2020a software environment. The hardware environment chosen was Windows 10 and executed on a core i7 CPU with 8 GB of RAM. Additionally, the CPU time is recorded by the command “tic-toc”.

In the execution of the GA allocation algorithm, the objective function of Equation (14) is transformed into the fitness function. Additionally, the parameters are set as follows: crossover probability = 0.6, mutation probability = 0.2, population size = 75 and maximum algebra = 300. For the average allocation algorithm, the total maintenance costs are distributed evenly among all the nodes and edges. The value of $g_i^z{=g}_l^z{=g}^z/(v+e)$ is used as the maintenance cost per node and edge, and the remaining parameters are all set as in the optimization allocation algorithm of this paper.

As shown in Figure 8, the risk of input failure for the entire emergency rescue network is significantly reduced as the total maintenance cost increases. By analyzing the three allocation schemes, the optimization allocation scheme proposed in this paper can reduce the input failure risk to a greater extent than the other two methods. Additionally, when the total maintenance cost is higher than 2 million, the network risk corresponding to the optimization allocation scheme is close to zero.

Additionally, the optimization allocation algorithm suggested in this work was contrasted and analyzed via GA allocation and average allocation algorithms to verify its computational complexity. The allocation model was evaluated and estimated with a total cost of 1.25 million, as shown in

Table 3. Comparison of allocation algorithms at a fixed total cost (1.25 million).

Algorithm	${{\displaystyle \sum }}_{\mathit{i}=1}^{\mathit{v}}{\mathit{g}}_{\mathit{i}}^{\mathit{z}}$	${{\displaystyle \sum }}_{\mathit{l}=1}^{\mathit{e}}{\mathit{g}}_{\mathit{l}}^{\mathit{z}}$	Risk R	CPU Time
Optimization allocation algorithm	75.5656	49.4344	2.2675	1 s
Average allocation algorithm	56.25	68.75	5.1124	1 s
GA allocation algorithm	63.3879	61.6121	4.0681	23 s

.

The results show that the optimization allocation algorithm in this paper yields a lower risk of failure value than the other two algorithms and a shorter CPU run time, i.e., a more desirable cost allocation result in a shorter time. Additionally, the detailed allocation schemes of these three algorithms are shown in

Table 4. Maintenance cost allocation results with different algorithms.

$\mathbf{Node} \mathit{i}$	$\mathbf{Allocation} \mathbf{Results} {\mathit{g}}_{\mathit{i}}^{\mathit{z}}$ [CNY]			$\mathbf{Edge} \mathit{l}$	$\mathbf{Allocation} \mathbf{Results} {\mathit{g}}_{\mathit{l}}^{\mathit{z}}$ [CNY]
	Optimization	Average	GA		Optimization	Average	GA
Lintong	48,843	62,500	71,239	Lintong-Ankang	48,196	62,500	56,323
Yan’an	65,817	62,500	69,971	Lintong-Yanliang	66,934	62,500	52,377
Lueyang	114,936	62,500	94,448	Lintong-Baoji	26,875	62,500	60,421
Yanliang	112,569	62,500	66,254	Lintong-Hanzhong	26,789	62,500	62,454
Jingbian	78,737	62,500	65,872	Ankang-Hanzhong	45,835	62,500	50,482
Baoji	74,689	62,500	61,112	Ankang-Wanyuan	46,793	62,500	51,207
Hanzhong	73,796	62,500	49,856	Yan’an-Jingbian	43,583	62,500	70,992
Wanyuan	105,123	62,500	93,767	Yan’an-Yanliang	38,155	62,500	50,273
Ankang	81,146	62,500	61,361	Baoji-Lueyang	64,567	62,500	60,319
				Baoji-Yanliang	39,684	62,500	59,023
				Lueyang-Hanzhong	46,933	62,500	42,251

.

5. Conclusions

This paper investigates the maintenance investment allocation strategy for the nodes and edges of the railway emergency rescue network. The space-time accessibility is employed in the service performance description of emergency rescue network nodes and edges, and the rescue network space-time accessibility identification model including temporal uncertainty is created by combining with the uncertainty of arrival time. At the same time, Lagrangian multipliers are introduced to construct a maintenance cost allocation strategy for the railway emergency rescue network with space-time accessibility. Unlike most of the previous optimization methods that can only obtain the locally optimal solution, this method can obtain the global optimal solution with minimal failure risk of global input costs under the condition of limited maintenance costs. Moreover, the model is solved with actual railway rescue network data, and the effectiveness and accuracy of the proposed method are analyzed and discussed through experimental comparisons in this paper. The experimental results show that the proposed method can obtain a more accurate global optimal solution in the allocation process. Additionally, the maintenance investment allocation strategy for the nodes and edges of the railway emergency rescue network can provide support for the management to develop a reasonable maintenance plan.

This model also has limitations, which opens up a direction for future work. For the limitations, it is necessary to redefine the accessibility model when analyzing the accessibility of non-regional emergency rescue networks. It should be noted that in the specific application of this method, the model parameters must be calibrated according to the actual situation of each region to ensure that the method has a high degree of accuracy. Moreover, future work can expand the application objects, including the implementation of medical, fire, and other emergency rescue networks.