A Lattice Boltzmann Method-like Algorithm for the Maximal Covering Location Problem on the Complex Network: Application to Location of Railway Emergency-Rescue Spot

Abstract

Inspired by the core idea of the lattice Boltzmann method (LBM), which is successfully used in complex and nonlinear processes, we developed a lattice Boltzmann method-like (LBM-like) algorithm to effectively solve the maximal covering location problem with continuous- and inhomogeneous-edge demand on the complex network. The LBM-like algorithm developed has three key components, including the basic map, transfer function and effect function. The basic map is responsible for reasonably mapping complex networks with multiple branches and circles. Transfer functions are used to describe the complex covering process of the facility on the network, by splitting the entire covering process into several single-step covering processes, while the effect function is responsible for recording and processing the coverage effect of each covering process, based upon the requirement of an objective function. This LBM-like algorithm has good applicability to a complex network, intuitiveness, relatively low computational complexity, and open developability. Furthermore, the idea of the greedy algorithm was coupled with the LBM-like algorithm, to form two types of hybrid algorithms for improving the computational efficiency for the location problem, with multiple facilities, on a large-scale network. Finally, we successfully applied the LBM-like algorithm to the location problem of an emergency rescue spot on a real railway network, to underline the practicality of the proposed algorithm.

1. Introduction

The maximal covering location problem (MCLP) was originally developed to maximize the coverage of demand within a limited-service level, such as a pre-specified set of facilities, service radius, or supplies available from stocks [1,2]. In practice, a series of variants of the MCLP were developed for diverse applications [3,4,5]. Among these variants, an important version is the MCLP on a network, where covering demand may be at some discrete points or distributed over the edge, and facility candidates can also be located at nodes, or anywhere [3,6]. In this study, we focus on the MCLP in which covering demand is continuously distributed on the edges and a facility candidate is located at the nodes. This type of the MCLP is called node-restricted MCLP with edge demand, and is usually present at real-world events, including locations of emergency rescue such as train derailing or collisions on railways, traffic accidents on highways, the incidence of crime on streets, and the contamination of rivers [7].

In terms of the node-restricted MCLP with edge demand, Church and Meadows [8] firstly defined the network intersect point set (NIPS) as the set of candidate points, and proposed the segment of equal coverage (SEC), whose endpoints can cover exactly the same demand point lying in the SEC. After that, Berman et al. [9] presented a nonlinear mixed-integer programming formulation (NMIPF) that has fewer variables and constraints, and designed a greedy heuristic solution for large-scale problems to reduce the computational time. On the basis of the node-restricted case, Berman et al. [10] identified a finite dominating set (FDS) for the unrestricted MCLP on the network, and developed an efficient-solution algorithm. It is only feasible for the context of constant demand density in tree networks, and it is a challenging problem to draw general theoretical conclusions for the multi-facility location problem in general networks. Fröhlich et al. [11] derived n FDS of exponential size for the absolute complementary edge-cover problem in general graphs, and developed a greedy algorithm based upon a polynomial bounded subset of the FDS for cactus graphs. Blanquero et al. [12] addressed mixed-integer nonlinear programming for the MCLP on networks with edge demand, in which the facilities can be a location anywhere on the edge, and used a developed branch-and-bound algorithm to determine the edge set of the network containing the facilities. Baldomero-Naranjo et al. [13] presented the feasibility of the designed polynomial algorithms for single-facility MCLP on networks in which demand is the interval (an unknown constant) or linear distribution. Mohri and Haghshenas [14] proposed a maximal-covering location problem with partial coverage of the facilities on the edges for location of the ambulance, in which the frequency and severity of crashes are considered to be random. Baldomero-Naranjo et al. [15] addressed a maximal-covering location problem with edge upgrading, in which the length of the edges can be reduced at a cost. This location problem can be potentially used in the location of a public service facility in the case of road upgrading. Recently, Blanco et al. [16] constructed a multi-type maximal-covering location problem, in which some types of facilities are to be located in finite sets or in continuous normed spaces.

To our knowledge, there are only two effective methods for the node-restricted MCLP with edge demand which have been reported, namely the SEC-based method and the NMIPF-based method. For the SEC-based method, a set of candidate points (NIPS) for an optimal solution need to be determined. And then, the maximal coverage demand and optimal facility location in the designed algorithms can be determined by using binary variables in the integer programming model. However, determination of NIPS for some specific networks with multiple node and complicated structures is relatively cumbersome. Compared with the SEC-based method, the NMIPF-based method has fewer variables and constraints, but must require the shortest distance of any two nodes. In fact, a facility on the node on a complex network usually goes through several nodes and edges to service (cover) a demand on the network, meaning that covering of a facility propagates through several edges in multiple steps (a step refers to the covering ability of a facility propagated through a node). If the edge demand is heterogeneous and the covering effect varies with the distance between demand point and facility location, identification of the covering effect for all the edge demand is important. From a perspective of the facility, the covering process of any facility needs to be identified. However, these existing SEC-based and NMIPF-based methods have no/weak ability for identifying the covering effect of any facility and/or the covered effect of any edge in each step.

Inspired by the lattice Boltzmann method, which is one of the extraordinarily successful fluid dynamics simulation approaches in physical fields, we developed an advanced lattice Boltzmann method-like (LBM-like) algorithm for the p-facility MCLP, with continuous and inhomogeneous demand on the complex network. Compared with the existing methods, the designed LBM-like method can not only decompose the complex covering process into several single-step covering processes, but also has a convenient adjustment for any specific research objective. In addition, it also has a good adaptability to any complex network in the intuitive way. And then, the feasibility and effectiveness of the LBM-like algorithm for different network sizes and inhomogeneous demand are also investigated systematically. The greedy algorithm was adopted to improve the computational efficiency of the LBM-like algorithm for a large-scale network with multiple facilities. Finally, we successfully applied the developed algorithm in the location of emergency rescue stations in a real railway network.

The rest of this paper is organized as follows. In Section 2, we define the problem in the case of continuous demand on the network, and give the notation throughout the paper. The problem is difficult to solve analytically, due to the complex topological structure of the network. Therefore, in Section 3, we propose the LBM-like algorithm, based on lattice Boltzmann theory. Several numerical experiments are illustrated with detailed analyses. Then, Section 4 presents a case study, which is an application of the introduced framework to the China Railway Nanchang Group, and the location of the railway emergency rescue bases is discussed. Finally, we provide our conclusions about the attributes and possible extensions of the LBM-like approach.

2. Problem Definition and Description

Let G = (V, E) be a network with node set V and edge set E, where $\left|V\right|=\overline{n}$, $\left|E\right|=\overline{m}$. Each edge $e_{ij}$ with endpoints $i,j\in V$ has a positive length $l_{ij}$ and is assumed to be rectifiable. The G is an undirected graph, and denotes the continuous set of all the points on the network. The $\overline{n}\times \overline{n}$ binary matrix $A(G)$ for the G is called the adjacency matrix, in which

$$a_{ij}=\left\{\begin{array}{l}1\hspace{1em}\mathrm{if} \mathrm{there} \mathrm{is} \mathrm{an} \mathrm{edge} \mathrm{from} \mathrm{node} j \mathrm{to} \mathrm{node} i\\ 0\hspace{1em}otherwise\end{array}\right.$$

(1)

We define $L(G)$ as the matrix of network edge length. The degree of a node is the number of its connected edges. In the undirected network, the degree for node $v_i$ is $\mathrm{deg}(v_i)={\displaystyle \sum _j{a}_{ij}}$, and its maximum degree refers to $\Delta (G)=\max \mathrm{deg}(v_i), v_i\in V(G)$ [17].

In this context, it is assumed that each edge e_ij has a non-negative continuous function ${f_e}_{ij}\in {ℝ}_0^{+}$ representing the demand distribution along the edge, and that the antiderivative ${F_e}_{ij}\in {ℝ}_0^{+}$ of $f_{e_{ij}}$ is known and continuous. And it is supposed that the ${f_e}_{ij}$ is bounded by setting some constants that may allow the objective function to have important structural properties, like Lipschitz continuity [18]. To be concise, we replace ${f_e}_{ij}$ and ${F_e}_{ij}$ with $f_{ij}$ and $F_{ij}$, respectively.

Let $X_p=\left\{x_1,\dots ,x_p\right\}\subset V$ be a set of nodes representing $P$ facilities with coverage radius $R>0$. $y_i(1\le i\le \overline{n})$ is a binary decision variable, which represents whether the facility is located at node $v_i$ or not.

We use $d(v_i,v_j)$ to represent the shortest network distance between two vertices $v_i$ and $v_j$. Let $s\in G$ be a point on the network and $s\in [v_i,v_j]$. Then, if the facility $x_k$ is located at a node, the distance between the demand point s and the facility $x_k$ can be defined as follows:

$$d(s,x_k)=\min \left\{d(s,v_i)+d(v_i,x_k),d(s,v_j)+d(v_j,x_k)\right\},\left(\forall i,j=1,\dots ,\overline{n}\right)$$

(2)

Generally, it is considered that $d(s,x_k$ and $d(x_k,s)$ are equal in the undirected network. The distance between a point $s$ on edge $e_{ij}$ and facility $x_k$ is illustrated in Figure 1. It can be said that a point $s\in G$ is covered by a facility $x_k\in X_p$ if $\underset{1\le k\le p}{\min }d(x_k,s)\le R$. Let $C_e(x_k)=\left\{s\in G\left|d(s,x_k)\le R\right.\right\}$ denote the set of all points that are covered by $x_k$ and call $C_e(x_k)$ as the coverage area of $x_k$. If a point s is covered by more than one facility, it is assumed that this point is only treated as a covered point.

Before the formulation, the common notations are summarized:

• i,j: i,j node index;
• k: k facility index;
• v ∈ V: Set of nodes on the network G;
• e ∈ E: Set of edges on the network G;
• l ∈ L: Set of edge lengths;
• $x_k\in X_p$: Set of p facilities;
• C_e: Set of all the points covered on the edge e;
• d: The shortest network distance between two vertices;
• R: Coverage radius of facility;
• s: Any point on the edge;
• P: Total facility number;
• f: Distribution function of demand;
• F: Anti-derivative of distribution function f;
• y_i: A binary variable representing whether or not a facility is located at point.

The total amount demand on an edge $e_{ij}$ covered by the facilities $X_p\subset V$ is given by

$$g_e(X_p)={\displaystyle \underset{C_e(X_p)}{\int }{f}_{ij}(s)}ds$$

(3)

The MCLP with edge demand engages p facilities to maximize the total covered demand along the networks, which can be formulated as follows:

$$\max {\displaystyle \sum _{e\in E}{g}_e(X_p)}={\displaystyle \sum _{e\in E}{\displaystyle \underset{C_e(X_p)}{\int }{f}_{ij}(s)}ds}$$

(4)

s.t.

$$\begin{gathered} {\displaystyle \sum _{i=1}^{\overline{n}}{y}_i}=p \\ {y}_i\in \{0,1\},\forall i\in V \end{gathered}$$

3. The Algorithm of MCLP with Edge Demand

3.1. Analysis of Location Problem with the Idea of LBM

The MCLP on the network is for maximizing the covering demand with p facilities. Actually, the covering process of multiple facilities spreads along the connected edges, and is analogical with some complex physical processes such as heat transfer and water wave propagation. The facility covering process and its two analogous processes are depicted in Figure 2.

Covering edge demand by the emergency rescue facility in Figure 2a can be treated as a transfer process, which is analogous with the heat-transfer process from high temperature to low temperature in Figure 2b and the water-wave-propagation process in Figure 2c. Obviously, many natural processes including heat transfer and water wave propagation are continuous in space and time.

These complex physical processes in computational physics are well modeled by the lattice Boltzmann method (LBM), which is based on the discretization of microscopic models and mesoscopic kinetic equations [19]. The LBM for the simulation of incompressible viscous flows was initially developed by Chen et al. [20] and Qian et al. [21]. Once a finite number of fictitious particles with given velocities (provided by the lattice velocity model) and density distribution functions (determined from a set of lattice Boltzmann equations) are known at a physical location, the macroscopic flow variables such as density and velocity can be computed from mass and momentum conservation [22]. Nonlinear convection and diffusion effects at the macroscopic level can be considered effectively as streaming and collision processes of fictitious particles. Specifically, the streaming step is to propagate values from the local site to adjacent lattice sites along all the lattice velocity directions. In the collision step, particles at each lattice site combine effects and redistribute across the lattice [23]. It is usually supposed that the mesh spacing for the uniform grid is the streaming distance per streaming step. The LBM with q velocities in d-dimensional space is usually denoted as a $D_d{Q}_q$ model [24]. The LBM has a good ability for modeling a complex iterative process in an efficient implementation. Inspired by this idea, we developed for the first time a lattice Boltzmann-like (LBM-like) method, to solve the MCLP, with continuous edge demand on the complex network.

The developed LBM-like method has three main components, including the basic map, transfer function, and effect function. The basic map is responsible for describing any complex network well, by using connection matrices and value matrices. The transfer function can decompose the complex and multi-step covering process (meaning that a facility may go through multiple nodes to cover a demand point) in the network into several single-step covering processes. The effect function is used to record the covering effect during each single-step covering process. Finally, covering effects of all the edges can be further processed to meet the requirement of the specific research objective. In this research, we use the $D_2{Q}_8$ square lattice on the basic map, responding to the maximum degree of the undirected network $\Delta (G)\le 8$. The basic map $B{M}_{z\times z}$ is actually a $z\times z$, $z\in {\mathbb{Z}}^{+}$ square grid, which can topologize any actual complex network by a flexible combination of nodes and edges.

Herein, we classified the edges in the basic map into four categories, including Horizontal edge $e^H$, Vertical edge $e^V$, Diagonal-Down edge $e^{DD}$, Diagonal-Up edge $e^{DU}$. We also define a cell in which there are one node and four edges, such as $cel{l}_{\theta }$ consisting of node $v_{\theta }$, and edges $e_{\theta }^H$, $e_{\theta }^V$, $e_{\theta }^{DD}$ and $e_{\theta }^{DU}$. It is worth noting that we add a dash region into the basic map in order to ensure all the elements in each cell have the same index as the order of their corresponding cell. There are $z\times z$ basic cells in the $B{M}_{z\times z}$.

For the basic map depicted in Figure 3a, all the nodes and four kinds of edges in the basic map are all numbered from 1 to 36. The designed complex network in Figure 3b contains some of the basic structures in the real network such as branch structure, closed cycle structure with odd nodes, and closed cycle structure with even nodes. Its topological structure is successfully depicted in Figure 3c, which indicates that the basic map has a good ability for drawing any generally complex network with closed cycles that have odd or even nodes. After the topological process of the actual network, the $G_{B{M}_{z\times z}}$, a form of G in the $B{M}_{z\times z}$ can be obtained. If any edge $e_{ou}$ on the $B{M}_{z\times z}$ is connected, the element $a_{ou}$ in its connection matrix $A\left(G_{B{M}_{z\times z}}\right)$ is equal to 1. Otherwise, it is 0. According to the edges $e_{ou}$ in four directions, we can obtain the four connection matrices $A^H$, $A^V$, $A^{DU}$ and $A^{DD}$ of the network $G_{B{M}_{z\times z}}$, and their corresponding value matrices $L^H$, $L^V$, $L^{DU}$ and $L^{DD}$. The length of all the edges can be obtained from the actual distances of the original network.

3.2. The LBM-like Algorithm

3.2.1. Basic LBM-like Algorithm

(a)

Building the basic map

Firstly, we need to analyze some characteristic parameters of the network. According to the maximum degree of the undirected network, the $D_2{Q}_8$ lattice model was designed. In order to topologize successfully all the elements in the network onto the basic map, we can find a path in the network which has the largest number of nodes $\eta (G)$. And then, the number $z$ of rows and columns of the basic map can be determined, to topologize the raw network. Generally, $z$ may be not less than $\eta (G)$. The edge number in the basic graph $2z(z+1)$ is greater than $\overline{m}$ in the undirected network. Moreover, node number in the basic map is greater than the number of nodes $\overline{n}$ in the network. After the topological process, all the nodes $v_i\in V$ in the network $G$ have a series of new subscripts of the nodes in the basic map. Further, we can obtain the connection matrices and value matrices.

(b)

Constructing the transfer function

The input of transfer function includes current active nodes and their influences (effective covering radius), while the outputs are next-generation active nodes and their current influences, as shown Figure 4. Obviously, the influences of new active nodes are less than those of the root nodes through the transfer process. If the new active nodes have transfer ability (the covering radius is not less than the length values of all the adjacent edges), these will act as input nodes in the next transfer process. The transfer function has a one-way property, so that if one active node transfers to adjacent nodes, all the new active nodes cannot transfer inversely to their root nodes.

There are eight lattice propagation directions for the $D_2{Q}_8$ lattice model $e_{\alpha }$, in which the direction index is $\alpha =1,2,\dots ,8$, displayed in Figure 5. The general transfer rule listed in

Table 1. General transfer rule in the eight directions.

Direction ${\mathit{e}}_{\mathit{\alpha }}({\mathit{v}}_{\mathit{n}}^{\mathit{t}})$	The Next Active Node ${\mathit{v}}_{\mathit{\theta }}^{\mathit{t}\mathbf{+}\mathbf{1}}$	Coverage Radius $\mathit{r}({\mathit{v}}_{\mathit{\theta }}^{\mathit{t}\mathbf{+}\mathbf{1}})$
$\alpha =1$	$\theta =n+z$	$\max \left\{R-l_{n,n+z},0\right\}$
$\alpha =2$	$\theta =n+z+1$	$\max \left\{R-l_{n,n+z+1},0\right\}$
$\alpha =3$	$\theta =n+1$	$\max \left\{R-l_{n,n+1},0\right\}$
$\alpha =4$	$\theta =n-z+1$	$\max \left\{R-l_{n,n-z+1},0\right\}$
$\alpha =5$	$\theta =n-z$	$\max \left\{R-l_{n,n-z},0\right\}$
$\alpha =6$	$\theta =n-z-1$	$\max \left\{R-l_{n,n-z-1},0\right\}$
$\alpha =7$	$\theta =n-1$	$\max \left\{R-l_{n,n-1},0\right\}$
$\alpha =8$	$\theta =n+z-1$	$\max \left\{R-l_{n,n+z-1},0\right\}$

is used to obtain the next active nodes and their coverage radius in the eight directions. We defined Corner node, $Co{r}_{v_{\theta }}$ with three transfer directions, Border node, $Bo{r}_{v_{\theta }}$ with five transfer directions and General node, $Ge{n}_{v_{\theta }}$ with eight transfer directions, based upon their node degrees on the basic map.

Correspondingly, different types of nodes have different transfer directions, as listed in

Table 2. Transfer rules of three types of nodes.

Category	Number of Nodes and Degree	Position	Index $\mathit{\theta }$	Transfer Directions ${\mathit{e}}_{\mathit{\alpha }}({\mathit{v}}_{\mathit{\theta }})$
$Co{r}_{v_{\theta }}$	4 $\mathrm{deg}(v_{\theta })=3$	Top-left	$\theta =1$	$\alpha =1\sim 3$
		Bottom-left	$\theta =z$	$\alpha =1,7,8$
		Top-right	$\theta =z^2-z+1$	$\alpha =3\sim 5$
		Bottom-right	$\theta =z^2$	$\alpha =5\sim 7$
$Bo{r}_{v_{\theta }}$	$4\ast (z-2)$ $\mathrm{deg}(v_{\theta })=5$	Upper boundary	$\begin{array}{l}\theta =1,1+z,1+2z,\\ \dots ,1+(z-1)z\end{array}$	$\alpha =1\sim 5$
		Bottom boundary	$\theta =z,2z,\dots ,z\cdot z$	$\alpha =1,5\sim 8$
		Left boundary	$\theta =1\sim z$	$\alpha =1\sim 3,7,8$
		Right boundary	$\theta =z^2-z+1 \sim z$	$\alpha =3\sim 7$
$Ge{n}_{v_{\theta }}$	${(z-2)}^2$ $\mathrm{deg}(v_{\theta })=8$	Interior	$\theta =others$	$\alpha =1\sim 8$

. The transfer rule for each type of node can be determined if the node category is known. If one of the initial active nodes $v_n^t$ is a general node $Ge{n}_{v_{\theta }}$ with coverage radius $r(v_n^t)=R$, we may obtain the next active node and its coverage radius.

The operational rule of the transfer function is as follows:

Stage 0:

With the initial stage $t=0$, there is no propagation for facility nodes. We find the current active node set $V^0$, $\left|V^0\right|=p$. Every active node $v_o^0\in V^0$ has the coverage radius $r(v_o^0)=R$.

Stage 1:

Identify the category of the transfer node $v_o^t$. The coverage ability could transfer along all the transfer directions $e_{\alpha }(v_o^t)$, leading to the adjacent nodes of $v_o^t$, which may become the next-generation active node.

Stage 2:

At the transfer process t, the active node $v_o^t$ has transfer ability with coverage radius $r(v_o^t)$. All the nodes $v_u$ adjacent to node $v_o^t$ in the basic map $B{M}_{z\times z}$ can obtain transfer ability if $r(v_o^t)-l_{ou}>0$. We define $V_u^{t+1}(v_o^t)$ as the active-node set transferred by node $v_o^t$ in the next stage. The coverage radius of the active node at each sub-process in successive transfer processes is determined by calculating the remaining transfer capacity. If we number states from $t=0$ (initial stage) to $t=q_{v_o}$, $q_{v_o}\in {\mathbb{Z}}^{+}$ (the last transfer stage for initial active node $v_o^t\in V^t,t=0$), then transfer ability $f(v_u^{t+1})=\max \left\{tf(v_o^t)-l_{ou},0\right\}$. Herein, $f(v_o^t)=r(v_o^t)$, which is $r(v_u^{t+1})=\max \left\{r(v_o^t)-l_{ou},0\right\}$.

By obeying this transfer rule, $v_o^0\in V_p^0$ terminates its transfer process after $q_{v_o^t}$ stage, meaning that the coverage radius of the nodes $v_u^t$ is equal to 0.

(c)

Establishing the effect function

Another important procedure is to develop a reasonable and feasible effect function. In terms of the objective function of the MCLP, the effect function aims to determine the covering distance of the active nodes on their connected edges. For the research objective, total covering distance is the sum of the effective covering distances of all the edges. And then, the set of facility candidates corresponding to the maximal-covering distance could be found.

In an undirected network, the demand on an edge can receive coverage from multiple facilities. Diverse effect functions based on the requirements of the objective function can be designed. Based on the above objective function in Equation (4), we design a specific effect function, as follows.

If one edge or any of its parts is covered by more than one active node, the effective covering distance is the set of all the points covered one time or more. Each initial node $v_o^t$ can propagate at most $q_{v_o}\in {\mathbb{Z}}^{+}$ times until $r(v_u^t)-l_{ou}\le 0$. For an edge $e_{ou}$, it can receive coverage of facilities from two directions through the end points $v_o$ and $v_u$. We denote that the coverage effect on the edge $e_{ou}$ from $v_o$ is positive, and write $e{f}_{ouk}\ge 0$ as $e{f}_{ouk}^{+}$, and the largest positive effect is $\max e{f}_{ouk}^{+}$. In the same way, the coverage effect $e{f}_{uok}^{-}$ on the edge $e_{ou}$ from $v_u$ is negative. The smallest negative effect is $\min e{f}_{uok}^{-}$. As depicted in Figure 6, the coverage effects with eight directions can be regulated as: ${\overrightarrow{e}}_{\alpha }(v_o^t)·r(v_o^t)$, where ${\overrightarrow{e}}_{\alpha }(v_o^t)=\left\{\begin{array}{l}1,\alpha =1\sim 4\\ -1,\alpha =5\sim 8\end{array}\right.$ in Figure 5, left. When the demand on each edge of the network is uniform, the length of the covered edge is calculated by the following: ${\psi }_{e_{ou}}=\min (\max e{f}_{ouk}^{+}-\min e{f}_{uok}^{-},l_{ou})$, as shown in Figure 6.

If the demand is non-uniformly distributed on the edge, the coverage-effect function on the edge $e_{ou}$ can be expressed as

$${\psi }_{e_{ou}}=\left\{\begin{array}{l}{\displaystyle \underset{0}{\overset{\max e{f}_{ouk}^{+}}{\int }}f(s)ds+{\displaystyle \underset{-\min e{f}_{uok}^{-}}{\overset{l_{ou}}{\int }}f(s)ds\hspace{1em}if \max e{f}_{ouk}^{+}-\min e{f}_{uok}^{-}<l_{ou}}}\\ {\displaystyle \underset{0}{\overset{l_{ou}}{\int }}f(s)ds\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}if \max e{f}_{ouk}^{+}-\min e{f}_{uok}^{-}>l_{ou}}\end{array}\right.$$

(5)

The total coverage effect of network $G=(V,E)$ is

$$\Psi (X_p)={\displaystyle \sum _{e_{ou}\in E}{\psi }_{e_{ou}}}(\forall o,u\in V)$$

(6)

To find an optimal solution (a specific facility set) for MCLP on a complex network $G$, we find the coverage effects of all combined sets $X_p$, and achieve the optimal objective $g(X_p)=\underset{X_p}{\max }\Psi (X_p)$.

(d)

Pseudocode of the basic LBM-like algorithm

Pseudocode of basic LBM-like algorithm for MCLP is presented in Algorithm 1.

Algorithm 1: Basic LBM-like Algorithm for MCLP

Input: Undirected network $G=(V,E)$, $\left|V\right|=\overline{n}$, $\left|E\right|=\overline{m}$; each edge $e_{ij}$ with a positive length $l(e_{ij})$; a predefined coverage radius $R$ of all the facilities. Output: Find an optimal set $X_p$, corresponding to maximum total covering distance.  Step1. Build the Basic Map  Determine the size $z$ and dimension of the Basic Map. If $\Delta (G)\le 8$, then use the $D_2{Q}_8$ lattice model. Find a positive integer $z$ so that it satisfies the condition $z>\eta (G)$, $2z(z+1)>\overline{m}$ and $z^2>\overline{n}$  Step2. Topologize the network $G$ into the Basic Map  Obtain the Basic Map codes of network $G$, and the connection matrices $A^H$, $A^V$, $A^{DD}$, $A^{DU}$ and value matrices $L^H$, $L^V$, $L^{DD}$, $L^{DU}$ of the four directions. Step3. Main procedure  Calculate the total covering distance on the network $G$   For each mother node set $X_p$ do $V^0=X_p\left(i\right)$    if $V^t\ne \left\{ \right\}$ do     For each $v_o^t\in V^t$ do      a. Choose the transfer function category according to the $e_{\alpha }(v_o^t)$ b. Obtain the next generation active nodes $V_u^{t+1}$       and their transfer capacity $r(v_u^{t+1})$ c. Record the coverage effect ${\overrightarrow{e}}_{\alpha }(v_o^t)\cdot r(v_o^t)$ in the transfer process end $V^t\leftarrow V_u^{t+1}$ else break end  end  Obtain ${\psi }_{e_{ou}}$  $\Psi (X_p)={\displaystyle \sum _{e_{ou}\in E}{\psi }_{e_{ou}}}(\forall o,u\in V)$ end Return the optimal facility set $X_p^{\ast }$, $X_p^{\ast }=\arg \underset{X_p}{\max }\Psi (X_p)$ and the maximum covering area $g(X_p^{\ast })=\underset{X_p}{\max }\Psi (X_p)$

3.2.2. Hybrid Algorithm of LBM-like and Greedy Idea

Usually, the computational time is quite long for a relatively larger-sized network when the facility number is more than a specific one. This problem is caused by the increasing number of combinations of the potential facilities. Some mathematical methods can be used to overcome this problem completely or partially. Berman et al. [9] used a heuristic procedure to improve the efficiency of solving the large-scale location problem. In this study, we also applied the idea of the greedy algorithm to the reduction of computational time in the large-size network with a greater facility number. The general procedure of this greedy algorithm is to determine individually the optimal location of each facility on the basis of the optimal location of the facility that has been determined. For example, on the basis of the incumbent location of the first facility, the location of the second facility was determined. The rest of the facility locations also need to obey this rule. It is worth noting that the computational time of locating p facilities is approximately equal to p-fold the time of a single facility location. However, the application of this greedy idea cannot ensure the absolutely optimal location all the time. Therefore, we designed a series of numerical experiments to investigate the effectiveness and errors of this hybrid algorithm. Considering that the greedy idea may reduce the computational time at the expense of calculation accuracy, we defined two types of hybrid algorithms, including LBM-like+Greedy1 and LBM-like+Greedy2, to investigate the interaction between time reduction and calculation accuracy.

LBM-like+Greedy1 algorithm: Firstly, the basic LBM-like algorithm is used to determine the optimal locations of the first two facilities at the same time. And then, the greedy idea is applied, to sequentially determine the optimal location of the following facilities.

LBM-like+Greedy2 algorithm: Firstly, the basic LBM-like algorithm is used to determine the optimal locations of the first four facilities at the same time. And then, the greedy idea is activated, to sequentially determine the optimal location of the following facilities.

3.3. Numerical Experiments

In this section, a series of numerical experiments were designed to test the effectiveness of the LBM-like algorithm for the MCLP. All the computations are carried out using the MATLAB R2016b parallel compute pool on a mobile work station in Window 10 ThinkPad made in China (6 dual-threaded Intel Core i7-10750H CPU with 2.6 GHz, and 64G RAM Memory). Four sizes of networks with a given number of nodes and edges were generated randomly on a two-dimensional basic map, in which the length of the edges is between 150 and 300. In addition, the effects of coverage radius and demand density on computational cost were also investigated systematically. For the large-scale location problem, the greedy algorithm was coupled into the LBM-like algorithm, to determine the solution at a sub-optimal condition.

The number of nodes and edges for four networks (N1, N2, N3, and N4) are 22 (16) nodes and 41 edges, 40 (25) nodes and 72 edges, 39 (32) nodes and 89 edges, and 50 (41) nodes and 103 edges, where the number in the bracket refers to the effective node, excluding the nodes on the branch endpoint. The coverage radius of each network is set as 1.5 times the average edge length. The computational results are depicted in

Table 3. Computational results of four sizes of network.

Network: Size (Effective Number) * Edge	Facility Number	Coverage Radius	TCD 1	CR (%) 2	CT (min) 3
N1: 22(16) * 41	2	346	8220	87.03	0.04
	3	346	8776	92.92	0.07
	4	346	9044	95.75	0.26
	5	346	9367	99.17	0.75
	6	-	-	-	-
N2: 40(25) * 72	2	340	11,518	70.65	0.07
	3	340	13,137	80.58	0.27
	4	340	14,473	88.78	1.71
	5	340	15,190	93.17	9.04
	6	340	15,616	95.79	36.09
N3: 39(32) * 89	2	336	12,802	64.28	0.10
	3	336	15,795	79.30	0.71
	4	336	17,628	88.51	6.06
	5	336	18,415	92.46	41.88
	6	336	18,954	95.16	224.12
N4: 50(41) * 103	2	333	11,287	49.33	0.10
	3	333	15,372	67.18	1.06
	4	333	17,784	77.72	12.68
	5	333	19,550	85.44	119.28
	6	333	20,920	91.43	859.80

¹ TCD is total coverage distance without dimension. ² CR is ratio of coverage distance and total distance. ³ CT is computational time.

.

From Table 3, it can be seen that computational cost increases with the increasing facility number and network size. With the increase in facility number from 2 to 6, the computational times for N2, N3 and N4 increase from 0.07 min to 36.09 min, from 0.10 min to 224.12 min, and from 0.10 min to 859.80 min, respectively. This indicates that a larger network size increases more obviously the computational time. This trend is related to the size of the combination of p (a given facility number) from the potential locations. In addition, total coverage distance for a larger-size network is more than that for a smaller one at a same facility number, which is due to there being a smaller back-up distance for a larger-size network. These results indicate that it is possible that we may cut one big- size network into several small-size networks, controlling a reasonable facility number, to obtain an accepted computational time.

The effect of coverage radius on the computational time and total coverage distance was also evaluated by using a group of numerical experiments (N2 with four facilities), in which the coverage radius is from 0.5 to 3.5 times the average edge length, as shown in Figure 7.

From Figure 7, with the increasing coverage radius, computational time and total coverage distance both increase. For the MCLP, total coverage distance can be treated as output, while the computational time is actually cost. Based on the input–output analysis, the whole process can be further divided into two regions: economic region and consumed region. The economic region means that the unit cost (computational time) in this area can bring a huge output (total coverage distance). In the consumed region, a large increase in input just leads to a tiny change in output. Based on the above study, it can be known that the reasonable selection of coverage radius is important for the high-efficiency operation of the LBM-like algorithm. In the current study, the critical point between two regions for N2 is 2.5 times the average edge length, which is attributed to the fact that the total covering ratio is approaching 1 when the coverage radius becomes larger. In this case, larger coverage radius may lead to more transfer and covering processes, and, consequently, a large percentage of duplicate coverage. Actually, large coverage radius and a greater facility number both lead to high coverage rate, but both require more computational time. In some contexts, at least one of them needs to be selected. As said above, a large coverage radius requires more transfer and covering processes, while the larger facility number leads to more combinations of p. Therefore, an enlightenment from the related numerical experiment is that achieving an optimal balance between facility number and coverage radius is of great importance.

In some cases, the demand is non-uniformly distributed along the edge, which can be described by a certain kind of demand density. We assumed that the demand density is linear along an edge. Of course, other types of demand functions, including exponential function, step-wise function, multiple-peak function, and so on, can be also evaluated by using the same research method. To demonstrate the effectiveness of the proposed LBM-like algorithm, a series of linear demand-density functions were given randomly to all the edges. The values of slope and intercept range, for linear functions, from 0.8 to 1.2, and from 0.5 to 1.5. The result is depicted in Figure 8.

From Figure 8, we can see that the addition of the linear demand-density function can make the computational times at several times the average edge length increase, especially for more than four facilities. For six facilities, the computational times at 1.5 times and 1 time the average edge length increase by 25.54% and 102.91%, respectively. This is because the integral of the demand-density function on the covering distance requires extra computational cost. At this point, we have applied the numerical integration method to conduct the integral of the demand function to reduce the computational time.

Obviously, the computational time is quite long for a relatively larger-size network when the facility number is more than four. The sharp increase in computational time is caused by the increasing number of combinations of the potential facilities. To address this problem, two types of hybrid algorithms, including LBM-like+Greedy1 and LBM-like+Greedy2 were used to investigate the interaction between time reduction and calculation accuracy. The related results are listed in

Table 4. Performance of the Basic LBM-like and two hybrid algorithms.

Network Size + Facility	Basic LBM-like			LBM-like+Greedy1			LBM-like+Greedy2
	TCD	CR (%)	CT (min)	TCD	CR (%)	CT (min)	TCD	CR (%)	CT (min)
N3 + 5	18,415	92.46	41.88	17,965	90.20	0.23	18,415	92.46	6.10
N3 + 6	18,954	95.16	224.12	18,479	92.78	0.28	18,954	95.16	6.15
N4 + 5	19,550	85.44	119.28	18,947	82.80	0.23	19,172	83.79	12.73
N4 + 6	20,920	91.43	859.80	20,012	87.46	0.27	20,254	88.51	12.76

.

From Table 4, it can be seen that the two kinds of hybrid algorithms can both dramatically reduce the computational time, especially for the LBM-like+Greedy1. Compared to the LBM-like+Greedy1, the computational time for the LBM-like+Greedy2 is relatively long, while the total coverage distance becomes larger. It is necessary to note that this trend for all cases is not absolute, but is general. This is consistent with the fact that, although an exact algorithm can obtain more optimal results, it also requires more computational cost. In view of effectiveness of hybrid algorithms, we can see that the total coverage distance on the N3 for the LBM-like+Greedy1 reduces by 2.50%, while that for the LBM-like+Greedy2 has no change. For the N4, the total coverage distances of LBM-like+Greedy1 and LBM-like+Greedy2 reduce by 3.71% and 5.56%, respectively. These errors, caused by introducing the greedy algorithm, are relatively small and acceptable, which indicates that the developed hybrid algorithm with a high-quality near-optimal solution is effective for reducing computational time in some cases. The related research results may enable the achievement of an appropriate balance of accuracy and computational cost, limited to the limited resources.

3.4. Advantage and Discussion

At present, some algorithms, including heuristics and metaheuristics, have been explored and developed for the MCLP. In the work of Berman et al. [9], an alternative formulation for the maximal arc-covering location model based on the formulation of Church and Meadows was developed, to avoid double counting for covering, which further optimized the structure of an emergency-response network. It is necessary to note that the existing developed formulations are hardly used or are ineffective for complex networks; this is because seeking NIPS and SEC in the complex network with multiple circles is a difficult task, especially for the non-uniform and continuous edge demand on a complex network.

Inspired by the successful application of the lattice Boltzmann method in the heat- and mass-transfer complex process that is analogous to the covering process of the facility, we creatively proposed the LBM-like method that belongs to an exact algorithm, to solve this general location problem with complex network structures. In the LBM-like method, any complex network (linear, branch, circle, and dual-circle structures, as well as any combination of all the structures) can be easily topologized to a predefined basic map, where the maximum degree of node is equal to 8. Firstly, this topological process can not only intuitively depict the complex network structure, but also easily obtain the connection and matrix value of a network, which is used for further computation. Secondly, it can actually simplify the connection relationship among all the nodes. For example, the connection relationship and shortest distance between any two nodes in the common methods for the MCLP need to be obtained during the computational process, while only the relationship between one node and its adjacent nodes in the proposed method is required. This advantage, to some extent, may reduce the computational complexity. In addition, the topology process is open, meaning that the user can topologize any network into a basic map in your own style, which means this method has good potential to develop easy-to-use interaction software. More importantly, the LBM-like method can easily capture information about all the covering process of a facility, which is useful for the development of a specific objection function. For example, we can identify the covering relationship of any point on the edge from all the potential facilities, which is a key element for building complex objective functions, considering the covering relationship.

As for the computational process, the two key functions, namely transfer function and effect function, were firstly defined, and are used to discretize the covering process of any facility node with a fixed coverage radius on a complex network, and to record all the covering nodes (transfer nodes) and the covering effect of each transfer process. The transfer function has a one-way propagation characteristic, to avoid the invalid time-consuming transfer process. In addition, the output of the effect function can be selectively defined as distance or service time, and so on. Finally, the covering effects of all the edges can be further handled, according to the objective function. Compared to the existing exact algorithms, the computational process in this novel method has a clear and simple step, avoiding the complex mathematical derivation, and has excellent ability with respect to multiple transfer processes and the crossed covering process in the complex network.

Of course, the current basic map in the LBM-like algorithm is available only for the maximum node degree of 8 in a 2-D plane, which is similar to the D₂Q₉ lattice scheme in the common LBM. This limitation may be not suitable for the complex network with a maximum node degree of more than 8. In further work, we will develop the basic map with a greater node degree in the 3-D space, to deal with more complex networks.

4. Application to Location of Railway Emergency-Service Facilities

At present, the MCLP has been successfully applied to the location problem of single or multiple emergency-service facilities (natural disaster, emergency medical services, medical evacuation, railway emergency rescue, and so on) on a network or plane within a limited capacity, resources, or budget [25,26]. Locating humanitarian relief stations after a large-scale natural disaster is an important application scenario for the MCLP. Some descriptions about location problems of emergency cases can be found by referring to the reviews by Simpson and Hancock [27], Feng and Cui [28], Caunhye et al. [29], and Dönmez et al. [30]. For the ambulance-location problem, the MCLP is also widely applied, to locate a set of fixed sites for health-care systems [31]. Brotcorne et al. [32] and Bélanger et al. [33] provided a comprehensive review on location, relocation, and dispatching models, which took into account the uncertainty and dynamism inherent in the emergency medical services. Eaton et al. [34] used the MCLP to design the emergency medical service in Austin, Texas, which successfully decreased the average response time. Yin et al. [35] introduced a modular MCLP to optimize the location of emergency vehicles. Moreover, researchers also applied the MCLP to reduce the hazardous-material transportation risk related to the accidental release of dangerous goods during shipment [36,37].

In recent years, railway transportation has been dramatically developed in China, because of its safety, sustainability and economy. Although most railway technology and equipment has been improved, railway accidents, including breakdown and drop, collision, and derailment, inevitably occur sometimes. In cases of high speed and large carrying weight, railway accidents usually lead to disastrous consequences. Pre-event planning is important for timely rescue in all railway accidents, and is determined by the distance between the rescue spot and accident site. The location layout of the rescue spot is crucial for effectively and timely alleviating the rescue resources. The railway, especially the high-speed railway, is characterized by extremely fast running speed, heavy load, and complex systems, which makes railway emergency-rescue activities more timely and professional [38]. The optimal distribution of railway relief equipment can provide an available, quick and fair rescue. Wu and Wang [39] noted that first-aid times for a traffic accident within 90, 60, and 30 min have an injury survival rate of 10%, 40%, and 80%, respectively. Most of the relief trains in the Chinese railway rescue centers have an average radius of 180 km to 250 km, meaning that the time for some rescue activities is hardly able to meet the requirements of the high-efficiency rescue. Cheng and Liang [40] figured out a fuzzy multi-objective model to optimize railway emergency systems, combined with urban ambulances. Bababeik et al. [41] studied the optimal location and allocation of relief trains to enhance the resilience level of the rail network, and proposed a cooperative pattern of relief equipment, where the coverage of critical links was achieved by multiple rescue stations. Tang and Sun [42] established a single objective model that aims at minimizing the time of emergency resource dispatch, and formulated a multi-objective model to minimize the emergency-resource dispatch time and the number of emergency rescue bases. Wang et al. [43] defined a distributed conditional vertex p-center problem, and developed a robust chance-constrained model for optimizing the location of the high-speed railway emergency rescue stations. Wang and Zhou [38] examined the location problem of railway rescue centers in relation to uncertain demand, and described a hierarchical rescue network including fast rescue centers and regional rescue centers, based on the diverse rescue demands in various emergency events.

We assumed each railway segment has same potential accident probability. In addition, for a quick response and cost-effective procedure, the existing stations can be promptly converted to the emergency rescue stations, rather than establishing new emergency centers [20]. In order to maximize the rescue ability of the regional railway, we can apply the MCLP model, with a finite number of potential facility locations, to the design of a railway emergency-rescue-base network. In this case, we selected the railway network of the China Railway Nanchang Group, with diverse basic structures (crossed branch, single cycle, and coupled cycles), to validate the effectiveness of the proposed LBM-like algorithm. The network is composed of railway stations and railway lines between stations. The railway network structure, as depicted in Figure 9a, consists of 44 nodes and 47 edges, with a total of 3926 km. We numbered all the nodes, and represented the lengths of all the edges; this is presented in

Table A1. Distance between two connected nodes.

Node Pair	Distance (km)	Node Pair	Distance (km)	Node Pair	Distance (km)	Node Pair	Distance (km)
(1,2)	55	(7,12)	109	(14,41)	24	(25,44)	369
(2,36)	106	(8,9)	30	(15,16)	134	(27,32)	92
(2,3)	12	(9,10)	48	(15,18)	153	(28,30)	179
(3,4)	20	(9,29)	47	(16,17)	119	(28,29)	228
(3,37)	65	(10,11)	47	(19,20)	7	(31,37)	155
(4,5)	104	(10,29)	23	(19,44)	8	(32,33)	50
(4,34)	68	(11,32)	97	(20,21)	150	(34,35)	58
(5,6)	120	(12,13)	16	(21,23)	29	(37,38)	24
(5,43)	31	(12,20)	7	(21,24)	20	(39,44)	107
(6,7)	289	(13,14)	54	(21,22)	40	(40,41)	24
(6,28)	29	(13,19)	5	(25,26)	148	(41,42)	45
(7,11)	21	(14,15)	97	(25,34)	297

.

Based on the procedure, the railway network is firstly topological on a basic map with 9 × 9 nodes, presented in Figure 9b. It is obvious that this basic map has excellent ability for describing this complex structure. All the station nodes have their basic map codes, which are also summarized in

Table A2. Nodes of all the stations on the basic map.

Station	Node	Basic Map Code	Station	Number	Basic Map Code
Xiamen	1	54	Ruichang	23	14
Zhangzhoudong	2	53	Jiujiangbei	24	15
Meishuikeng	3	52	Ganzhou	25	33
Zhangping	4	43	Dingnan	26	25
Yongan	5	42	Yingli	27	47
Waiyang	6	41	Nanpingnan	28	50
Yingtan	7	40	Yanshanxi	29	59
Yushan	8	76	Mawei	30	60
Shangrao	9	67	Shicuo	31	62
Hengfeng	10	58	Lepingshi	32	48
Guixi	11	49	Xiangtun	33	57
Liangjiadong	12	39	Longyan	34	34
Tangang	13	30	Yongding	35	35
Zhangjiashan	14	21	Zhangzhou	36	44
Fenyi	15	20	Changji	37	61
Liling	16	29	Xiayang	38	70
Chaling	17	28	Jiangbiancun	39	24
Wenzhu	18	19	Shangtang	40	11
Jiangjia	19	22	Dongjia	41	12
Xiangtang	20	31	Jianshan	42	3
Jiujiangxi	21	23	.iafu	43	51
Konglong	22	13	Sanjiangzhen	44	32

. Additionally, all the edge lengths exhibited in Table A1 are represented by using four kinds of edges in $B{M}_{9\times 9}$. The value matrices of four directions for the Nanchang railway network are as follows:

Moreover, the candidates of emergency rescue spots were selected from the existing railway station or hubs, to reduce construction cost. Not all the station nodes will be candidates for the emergency rescue spot, which is because the optimal location of an emergency rescue spot is almost always not located at the branch end of the railway network, under the general MCLP. Therefore, the candidate locations of emergency rescue spots can be obtained.

$$L^H=\left[\begin{array}{ccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 134& 0& 0& 0& 0& 0& 0\\ 45& 24& 54& 16& 0& 50& 0& 0& 0\\ 0& 0& 7& 0& 21& 47& 10& 30& 0\\ 0& 29& 0& 0& 29& 228& 0& 0& 0\\ 0& 0& 0& 0& 31& 0& 0& 0& 0\\ 0& 0& 0& 68& 20& 65& 24& 0& 0\\ 0& 0& 0& 0& 12& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]L^V=\left[\begin{array}{ccccccccc}0& 0& 153& 119& 0& 0& 0& 0& 0\\ 0& 24& 97& 0& 0& 92& 0& 0& 0\\ 0& 0& 0& 0& 109& 97& 0& 0& 0\\ 0& 0& 0& 0& 289& 0& 23& 0& 0\\ 0& 0& 0& 369& 120& 0& 0& 0& 0\\ 0& 0& 0& 297& 104& 0& 0& 0& 0\\ 0& 0& 0& 58& 0& 106& 155& 0& 0\\ 0& 0& 0& 0& 0& 55& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

$$L^{DD}=\left[\begin{array}{ccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 40& 8& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 179& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]L^{DV}=\left[\begin{array}{ccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 5& 7& 0& 0& 0& 0& 0\\ 0& 0& 150& 0& 0& 0& 47& 0& 0\\ 0& 24& 107& 0& 0& 0& 0& 0& 0\\ 0& 0& 148& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

After introducing of railway network and potential locations, we need to further determine the optimal location of emergency rescue spots, corresponding to the maximum covering distance. The LBM-like method for obtaining the effective covering distance was used. The mother nodes with an initial covering radius (it is set as 240 km) are input nodes. The transfer node refers to the nodes with less than the initial covering radius, which originated from the transfer process of the mother nodes. The covering effects of all the nodes, including mother nodes and transfer nodes, are recorded in covering-effect output matrices. The effect of facility number (from 2 to 6) on covering effect was also evaluated, and is listed in

Table 5. Total coverage distance and rate at different facility numbers.

Facility Number	Basic Map Code	TCD * (km)	Coverage Ratio
2	<39,42>	2648	67.4%
3	<33,39,42>	3176	80.9%
4	<30,33,41,43>	3553	90.5%
5	<20,33,39,41,43>	3797	96.7%
6	<20,22,33,40,41,43>	3913	99.7%

* TCD is total coverage distance.

.

From Table 5, we can see that only two facilities can cover the 67.4% total railway distance. As the facility number reaches four, the covering ratio can be up to 90.5%. All the railway network will be covered completely by using six facilities. The optimal locations of emergency rescue depots are Zhangping, Waiyang, Yingtai, Fenyi, Jiangjia, and Ganzhou. These results can provide the useful and basic support for the decision-makers. This method can be applied selectively to the other similar location problem of the railway system, with total and partial covering at a specific economic cost. The computational results confirmed that the proposed model showed appropriate performance for disaster management.

5. Conclusions

In this study, we developed a new algorithm called the LBM-like algorithm, to solve effectively the MCLP with continuous edge demand on a complex network. Compared with other algorithms, the LBM-like algorithm not only has better applicability on complex networks and relatively low computational complexity (ignoring all the shortest distances between any two nodes), but also has open developability, due to its easily editable effect function for matching diverse objective functions. The results of numerical experiments show that increasing the coverage radius in a specific range may contribute to an obvious enhancement in output (total coverage distance) in an economic way, while it will become uneconomic when the coverage radius is more than the critical value. Considering that non-uniform distribution of demand, we also assigned a linear demand-density function for each edge. The LBM-like algorithm can deal effectively with this location problem. To further improve the efficiency of the LBM-like algorithm on large-size networks, we proposed two hybrid algorithms by coupling the greedy idea with the LBM-like algorithm; this exhibits a good effect on the reduction in computational time for large-scale networks. Furthermore, we successfully applied the developed LBM-like algorithm in the location of emergency rescue spots on a real railway network in China.

The main contribution of this study is to develop a new LBM-like algorithm, based on the core idea of the lattice Boltzmann method, which is based on the similarity of a covering process in a facility location and transfer processes in physical fields. In the future, the LBM-like algorithm can be further developed for other location problems, such as the decay covering-location problem, back-up covering-location problem, the location problem without a facility candidate, and so on. It is believed that there are many expansions of the LBM-like algorithm, due to its open editable characteristic.