An Investigation into the Rescue-Path Planning Algorithm for Multiple Mine Rescue Teams Based on FA-MDPSO and an Improved Force-Directed Layout

Abstract

It is noted that existing mine emergency-rescue algorithms have overlooked the requirement for multi-route sharing at critical nodes and have offered limited network visualisation. Consequently, a multi-team rescue-path-planning algorithm based on FA-MDPSO (Firefly Algorithm-Multiple Constraints Discrete Particle Swarm Optimisation) was proposed, and a graph-structure optimisation method combining a Force-Directed Layout with Breadth-First Search was introduced for node arrangement and visualisation. Methodologically, the superiority of the improved DPSO (Discrete Particle Swarm Optimisation) in route-planning precision was first validated on the DIMACS dataset. Subsequently, the hyperparameters of MDPSO (Multiple Constraints Discrete Particle Swarm Optimisation) were optimised by means of four intelligent algorithms—ACO (Ant Colony Optimization), FA (Firefly Algorithm), GWO (Grey Wolf Optimizer) and WOA (Whale Optimization Algorithm). Finally, simulations of one to three rescue-team deployments were conducted within a mine-fire scenario, and node-importance analysis was performed. Results indicated that FA-MDPSO achieved comprehensive superiority in route precision, search efficiency and convergence speed, with FA-based hyperparameter optimisation proving most effective in comparative experiments. The graph-structure optimisation was found to substantially reduce crossings and enhance hierarchical clarity. Moreover, the three-team deployment yielded the shortest equivalent path (56,357.02), and node-visitation frequency was observed to be highly concentrated on a small number of key nodes. This not only significantly improves the collaborative rescue efficiency but also provides intuitive and practical technical support for intelligent mine rescue operations. It lays an important foundation for optimising mine emergency rescue plans, ensuring the safety of underground personnel, and promoting the intelligent development of mines.

1. Introduction

When a mine fire occurs, fatalities and property losses are incurred, and the formulation of appropriate disaster-relief and disaster-avoidance strategies is of considerable practical significance for the protection of underground personnel and the coordination of rescue teams. With the advancement of mine automation and unmanned operations, numerous novel theories and technologies have been applied to mine disaster relief and personnel sheltering. From an algorithmic perspective, current mine-disaster mitigation efforts have been centred on graph-theoretic algorithms, intelligent optimisation algorithms and heuristic search algorithms, with corresponding avoidance strategies devised and escape routes planned for underground miners. However, in practice, during a fire outbreak, certain areas may experience insufficient oxygen supply and severe fire conditions, rendering standard sheltering procedures impracticable. Consequently, mine rescue teams are required to be deployed to assist in extricating trapped miners. In 2018, an underground conveyor-belt fire at the Palabora copper mine in South Africa resulted in five miners being killed and one reported missing. In 2021, a methane explosion at the Kuzbass coal mine in Russia triggered a fire that caused at least 52 fatalities and 57 injuries. In 2023, the spontaneous combustion of fragmented media at the Gengcun coal mine in China led to a fire causing five deaths. In 2024, a fire at a coal mine in Nagaland, north-eastern India, was responsible for at least six deaths and four injuries. Such mine-fire incidents have had an indelible impact on the development of the mining industry and have inflicted irreversible losses on the families of the victims. To date, mine rescue strategies have been primarily focused on the planning of personnel escape routes, with limited research addressing the coordinated deployment of multiple rescue teams for underground rescue operations. In situations where miner escape-route planning is rendered ineffective, disaster-relief algorithms requiring reduced response times and enhanced rescue efficiency must be developed, and rescue-team deployment strategies must be formulated to ensure safe evacuation and the protection of personnel.

Strategies for mine-personnel disaster avoidance and evacuation have been primarily concentrated on algorithmic research and the optimisation of emergency-rescue systems. Algorithmic studies have chiefly involved the refinement of graph-theoretic algorithms, intelligent-optimisation algorithms and heuristic-search algorithms, and the construction of methods for calculating tunnel equivalent lengths and for shortest-path searches tailored to mine topologies, ultimately enabling personnel disaster-avoidance and evacuation.

A dynamic-programming model for optimal mine emergency-rescue routing based on a Dijkstra-ACO hybrid algorithm was proposed by Lu Guoju and Shi Wenfang [1], in which the Dijkstra algorithm was utilised to identify an initial optimal path and subsequently optimised by the ACO algorithm to determine the final optimal route. A multiobjective real-time search method founded on dynamic programming was introduced by Lin Bi, Yulong Liu et al. [2], employing Pareto-optimal solution sets to rank identified evacuation paths and thus realise an evacuation model for complex mine networks that satisfies practical escape-route planning requirements. An interactive “human–environment–computer” system was constructed by Wei Li, Yongming Wang et al. [3] through the integration of the A*-3D Time algorithm with mixed-reality (MR) technology; mine spatial geometry and escape-time data were adopted, and Comsol was employed to simulate mine water-inrush scenarios, thereby implementing an assisted-escape system for miners. An evacuation-route planning method based on an improved A* algorithm was put forward by Yulun Zhu, Gui Zhang et al. [4], in which FAHP-CRITIC combined weighting analysis was applied to assess evacuation factors, five fire-ground zones were delineated, and dynamic fire-spread trends were incorporated to support personnel disaster-avoidance. A multiscenario, cross-scale smoke-control strategy for mine fires was advanced by Haiqing Hao, Shuguang Jiang et al. [5], on the basis of which an optimal-escape-route planning method was devised to provide guidance for mine-fire prevention, smoke extraction, and emergency rescue. A numerical-simulation-based evacuation-route planning method under a full-scale mine-fire scenario was designed by Menghui Xiao, Cuifeng Du et al. [6], predicated on smoke-diffusion characteristics and fire-induced pressure variations; it was demonstrated that CO concentration diminishes at tunnel corners and that both CO concentration and temperature decrease with depth, indicating that miners should evacuate along curved structures. A research framework combining mine-fire simulation with the Ford-Fulkerson algorithm was presented by Simo Lotero, Vasilis Androulakis et al. [7], in which fire-induced hazards were quantified according to MSHA safety standards to enable the planning and evaluation of safe evacuation routes. An underground escape-route planning system based on the MSPFA was developed by Zhenguo Yan, Zhixin Qin et al. [8], utilising DXF technology to visualise evacuation paths and to facilitate dynamic adjustment of routes and optimisation of emergency-avoidance strategies during a fire. An improved Dijkstra algorithm, combined with simulation-model techniques for the analysis of personnel disaster-avoidance strategies under mine-disaster conditions and for one-time determination of the optimal path, was proposed by Guoju Lu, Guofei Zhao et al. [9]. A TRTV obstacle-avoidance planning method was applied by Xin Zhang, Chunyu Yang et al. [10], and, through the design of the RGPSO algorithm, an improved obstacle-avoidance planning algorithm was advanced; four representative map benchmarks were employed to demonstrate TRTV’s capacity for narrow-passage navigation and dead-zone avoidance in complex environments. Finally, an underground-mine vehicle path-planning method based on an improved A* algorithm and a fuzzy-control dynamic window was proposed by Chuanwei Zhang, Xinyue Yang et al. [11], and its effectiveness and feasibility were experimentally validated in order to meet the requirements of underground-vehicle route planning. In the field of deep learning and multi-agent collaborative rescue strategies, an improved multi-agent reinforcement learning method was proposed by Qingqing Wang, Hong Liu and others [12]. By integrating it with an improved social force model, the problem of mutual influence among agents in crowd simulation was effectively addressed, and excellent results were obtained in crowd evacuation. A review of multi-robot systems for Search and Rescue (SAR) operations was carried out by Jorge Peña Queralta, Jussi Taipalmaa and others [13]. This review provided certain new technical ideas for robot collaborative operations under factors such as active perception, shared autonomy, and awareness of the victim’s situation. A multi-agent discrete search and rescue path-planning problem based on mixed-integer linear programming was put forward by Jean Berger and Nassirou Lo [14]. Network modelling and CPLEX technology were successfully utilised, and the calculation results verified its value and effectiveness.

Optimisation of the emergency-rescue system has been primarily achieved through the application of evaluation methodologies and simulation-software analyses, and by the development of corresponding mathematical evaluation models to facilitate the allocation of mine emergency-rescue resources and the feasibility analysis of existing disaster-relief strategies, thereby refining current personnel-rescue procedures. A mine-fire emergency-rescue resource-allocation model was designed using Ventsim software by Nie Rongshan and Wang Zhen [15], in which factors influencing the allocation process were analysed, the relationship between resource-allocation volumes and underground resource availability was examined, and the linkage between mine incidents and command decisions was established. An emergency–rescue–capability evaluation model for mine water-inrush accidents, based on an improved combination-weighting method and an MPA-optimised BPNN algorithm, was proposed by Wei Wang, Xinchao Cui et al. [16] and applied in a Shanxi coal mine in China, thereby achieving accurate and effective evaluation of water-inrush rescue capabilities. Feedback on the usefulness of 21 proposed evacuation–intervention measures was collected by Eugene A. Gyawu, Danise A. Baker et al. [17], and the relationships between these measures and specific miner demographic parameters were assessed; the results indicated that SCSRs and RAs were the most beneficial interventions, and self-evacuation measures were improved via human–system integration. Multiple data-collection methods were employed by Moshood Onifade [18] to gather information on safety practices, mine rescue and safety management, and subsequent analyses demonstrated that self-rescue preparedness is among the most critical factors for an organised and timely response to mine emergencies. A digital-twin system for on-site mine-rescue environments was introduced by Hu Wen, Shengkai Liu et al. [19], in which the overall architecture, functional features and essential system–construction technologies were presented; the system was then applied to emergency rescue in a mine, resulting in enhanced rescue success rates. An assessment model based on grey system theory and a grey evaluation model was developed by Kejiang Lei, Dandan Qiu et al. [20] and deployed at the Lugou coal mine, culminating in the establishment of a mine-fire emergency-rescue indicator-system risk-assessment framework. Principal risk indicators—namely firefighting equipment, command decision-making, emergency education and training, and fire-alarm systems—were identified, and corresponding control measures were devised across four domains (rescue-organisation support, personnel support, material support and information support), thereby enhancing the fire-emergency rescue capability of the Lugou coal mine.

Currently, efforts in mine-personnel disaster-avoidance and relief are mainly concentrated on the planning of underground miners’ evacuation routes. However, research on the coordinated rescue of multiple teams at designated locations is relatively limited. Moreover, existing mine emergency-rescue algorithms have neglected the requirement for multi-route sharing at critical nodes, and the network visualisation they provide is limited. Consequently, a study has been undertaken to address the selection of rescue-team size and the efficient deployment of multiple teams to rescue numerous entrapped individuals underground, and an FA-MDPSO (Firefly Algorithm-Multiple Constraints Discrete Particle Swarm Optimisation) algorithm has been proposed for rapid multi-team rescue operations. In order to enhance the balance between fitness computation, node-search coverage, population diversity and weighted convergence time within the MDPSO (Multiple Constraints Discrete Particle Swarm Optimisation) framework, a fitness function that comprehensively incorporates these criteria has been designed and subsequently optimised via the FA (Firefly Algorithm) to improve MDPSO efficacy. To further refine rescue protocols, a node-importance analysis method has been introduced to facilitate the optimisation of mine emergency-rescue strategies. Moreover, to accommodate the constraints of mine-site environments, a graph-structure optimisation approach combining a force-directed layout with breadth-first search has been proposed to assist in on-site emergency-rescue command and control. Ultimately, the proposed methods are intended to provide technical support for the realisation of intelligent mine emergency-rescue operations.

2. Materials and Methods

Figure 1 is partitioned into two principal modules by means of dashed boxes, wherein the multi-team evacuation-path planning of MDPSO and the FA-based parameter-optimisation process are elucidated. In the upper module, the “Multiple Constraints DPSO for Rescue Path Planning” component presents a schematic of the particle-based optimal-path search, the velocity-update equations, the heuristic-scoring and phero-mone-update mechanisms, and the methods by which path cost and overall fitness are evaluated for single- and multi-team rescue deployments. To the right, the “Firefly Algo-rithm” component is employed to optimise parameters c_x, c_1_min, c_1_max, c_2, and c_3 by means of the optimal-fitness evaluation function, thereby enhancing the performance of the improved DPSO algorithm. In the lower module, the “Node Layout Optimisation Method” utilises a layout strategy combining a force-directed algorithm with breadth-first search: repulsive and attractive forces between nodes are computed, and, by way of velocity and position update equations alongside a system-energy evaluation, a visually layered mine-network graph is realised, as exemplified in the figure on the right. The complete flowchart thus systematically depicts the end-to-end technical framework from algorithmic optimisation through to result visualisation. The FA and the MDPSO are combined through five parameters, namely c_x, c_1_min, c_1_max, c_2, and c_3. The fitness function is used as the evaluation function to optimise these five parameters. The results of the optimisation are fed back to the MDPSO algorithm, and the five parameters within the algorithm are adjusted. Eventually, the optimisation of the parameters is achieved. Among them, the orange part represents the parameters to be optimised, the yellow part indicates the parameters that are currently being iteratively optimised, and the green part signifies that the parameter optimisation has been completed.

2.1. MDPSO Multi-Team Rescue Strategy Theory

Discrete Particle Swarm Optimisation (DPSO) is defined as a global-optimisation method directed at discrete spaces, in which collective behaviour is simulated to identify optimal solutions within a discrete domain [21]. On the basis of the DPSO algorithm, an MDPSO rescue algorithm suitable for multi-team rescue operations has been formulated; its fundamental principles are outlined below.

2.1.1. Definition of the Initial Graph-Structure Problem and Formulation of the Solution

The mine pathfinding process is commonly modelled as an undirected graph under the tunnel topology and is defined as $G=(V,E)$. The vertex set $V=\{1,2,\dots ,N\}$ is taken to represent the key nodes within the mine, while the edge set $E\subseteq V\times V$ denotes the directly accessible passages between nodes. The distance between nodes $i$ and $j$ is denoted by $d_{ij}$; should nodes $i$ and $j$ not be directly connected, $d_{ij}$ is assigned an infinite value.

In the multi-team pathfinding process, it is required that the set of requisite nodes be defined and that the overall planned route be configured as a closed loop: rescue teams are deployed from the shaft entrance (denoted $s$), personnel are extricated from within the mine, and the teams return to the shaft entrance, thereby finalising the rescue. The set of requisite nodes is denoted by $R\subset V$, comprising those nodes which must be traversed by at least one rescue team.

To enable coordinated scheduling of multiple rescue teams, the set of requisite nodes $R$ is partitioned in accordance with the number of available teams, $K$. By segmenting $R$, candidate requisite-node sequences $R^{\left(g\right)}(g=1,2,\dots ,K)$ are obtained; the route sequence for each rescue team—comprising the start node, the requisite nodes within the corresponding partition and the end node—is represented by Equation (1).

$$P^{\left(g\right)}=\{s,r_1^{\left(g\right)},r_2^{\left(g\right)},\dots ,r_{n_g}^{\left(g\right)},s\}$$

(1)

2.1.2. Principle of Single-Segment Path Planning

For each segment (whether between consecutive requisite nodes within a partition, from the start node to a requisite node, or from a requisite node to the end node), the MDPSO algorithm is employed to determine the optimal path.

(1)

Particle Representation and Initialisation

Each particle is used to represent a candidate path $\mathbf{x}$. The vector $\mathbf{x}$ is of length equal to the number of nodes; its non-zero entries sequentially record the nodes traversed, and any remaining positions are padded with zero. Initial paths are generated by a heuristic search function, namely a Random Depth-First Search (DFS). For a given path $\mathbf{x}=\left\{p_1,p_2,\dots ,p_n\right\}$, after the zero values have been removed, the fitness is defined as the total cost of the path, as given by Equation (2).

$$f(\mathbf{x})=\sum _{i=1}^{m-1} d_{p_i,p_{i+1}}$$

(2)

In Equation (2), $f\left(\mathbf{x}\right)$ denotes the fitness value, representing the total cost.

A velocity matrix is constructed as a binary matrix $V\in \{0,1{\}}^{N\times N}$, which characterises the “velocity” of a particle by indicating which node pairs are subject to adjustment. The initial velocity is determined by a random matrix that accounts for both symmetric and asymmetric constraints.

(2)

Velocity-Update Method

In the discrete domain, velocity updating is effected via the assembly of multiple components, each corresponding to particle inertia, convergence towards the personal best $P_{best}$, and convergence towards the global best $G_{best}$. The inertia weight is dynamically varied as a function of the iteration count, as expressed in Equation (3).

$$w=c_{1\_\mathrm{m}ax}-{\displaystyle \frac{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}{\mathrm{m}\mathrm{a}\mathrm{x}\_iter}}(c_{1\_\mathrm{m}ax}-c_{1\_\mathrm{m}in})$$

(3)

In Equation (3), $c_{1\_\mathrm{m}ax}$ and $c_{1\_\mathrm{m}in}$ are defined, respectively, as the upper-limit and lower-limit weights.

The velocity-update mechanism comprises three components: the inertia component, the personal-best convergence component and the global-best convergence component. The inertia component $V_1$ is defined by Equation (4).

$$V_1=PM2VM(V_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}},w)$$

(4)

In Equation (4), the function PM2VM is employed to generate a new matrix from the current velocity matrix and the weight $w$. The calculation method of the PM2VM function: Suppose the current path matrix is $V_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}$, the frequency is w, and the order of the matrix $V_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}$ is $n$. A random $n\times n$ matrix $R$ is generated and symmetrized to obtain $R={\displaystyle \frac{1}{2}}(R+R^T)$. Subsequently, two intermediate matrices $P_1=P\odot f$ and $P_2=P\odot R$ are calculated. Finally, $V_1$ is calculated. For each element $(i,j)$ in the matrix, $V_1\left(i,j\right)$ is determined as $V_1\left(i,j\right)=\left\{\begin{array}{c}1,P_1\left(i,j\right)>P_2\left(i,j\right)\\ 0,P_1\left(i,j\right)\le P_2\left(i,j\right)\end{array}\right.$.

The personal-best convergence component is given by Equation (5).

$$V_2=PM2VM\left(\mathrm{s}\mathrm{u}{\mathrm{b}}_{\mathrm{M}}\left(P_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}},V_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}\right),c_2\right)$$

(5)

In Equation (5), the function $\mathrm{s}\mathrm{u}{\mathrm{b}}_{\mathrm{M}}$ is used to compute the difference between the personal-best adjacency matrix $P_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ and the current velocity matrix, and this difference is then subjected to PM2VM and scaled by the coefficient $c_2$. The calculation method of the $\mathrm{s}\mathrm{u}{\mathrm{b}}_{\mathrm{M}}$ function: Suppose the matrices $P_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ and $V_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}$ are of the same order, and the order of the matrices is $n$. First, an $n\times n$ matrix filled with all 1s, denoted as $1_{n\times n}$, is created, so that $temp{M}_1=1_{n\times n}$. Then, the matrix difference $temp{M}_2=M_1-M_2$ is calculated. Finally, a logical AND operation is performed to obtain the result matrix $resul{t}_M$. For each element $(i,j)$ in the matrix, $resul{t}_{M\left(i,j\right)}$ is given by $temp{M}_1(i,j)\wedge temp{M}_2(i,j)$.

The global-best convergence component is represented by Equation (6).

$$V_3=PM2VM\left(\mathrm{s}\mathrm{u}{\mathrm{b}}_{\mathrm{M}}\left(G_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}},V_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}\right),c_3\right)$$

(6)

Finally, these components are combined via an intersection operation to produce the final velocity update $V$, as specified in Equation (7).

$$V=\mathrm{a}\mathrm{n}{\mathrm{d}}_{\mathrm{M}}(V_1,{\mathrm{a}\mathrm{n}\mathrm{d} }_{\mathrm{M}}(V_2,V_3\left)\right)$$

(7)

In this formulation, the function $\mathrm{a}\mathrm{n}{\mathrm{d}}_{\mathrm{M}}$ applies bitwise logic to determine which candidate edges are activated.

(3)

Path-Adjustment Method Based on Candidate Edges

From the updated velocity matrix $V$, a candidate-edge set $E$ is constructed, as specified by Equation (8).

$$\mathcal{E}=\left\{\right(i,j)\mid V(i,j)=1 \mathrm{and}{d}_{ij}<\mathrm{\infty }\}$$

(8)

$${\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}_{ij}={\left[{\tau }_{ij}\right]}^{\alpha }{\left({\displaystyle \frac{1}{d_{ij}}}\right)}^{\beta }\cdot C_{param}$$

(9)

For each candidate edge, the heuristic score is calculated in accordance with Equation (9), wherein ${\tau }_{ij}$ denotes the pheromone level on edge $(i,j)$ and $\alpha$ and $\beta$ govern the relative influence of pheromone intensity and distance heuristic, respectively. $C_{param}$ serves as an additional scaling factor. The selection probability for each candidate edge is then defined by Equation (10). An edge $e^{*}=(i^{*},j^{*})$ is randomly selected based on the cumulative probabilities; the chosen edge is incorporated into the current path, and the local node ordering is subsequently adjusted to yield the updated route.

$$p_{ij}={\displaystyle \frac{{\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}_{ij}}{\sum _{(k,l)\in \mathcal{E}} {\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}_{kl}}}$$

(10)

(4)

Pheromone-Update Mechanism

Upon the conclusion of each iteration, a global pheromone update is performed for all edges. Initially, pheromone evaporation is applied in accordance with Equation (11), where $\rho$ represents the evaporation coefficient. Thereafter, pheromone reinforcement is conducted along every edge of the global best path $P_{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}}=\{p_1,p_2,\dots ,p_m\}$, with a symmetric update of ${\tau }_{p_i,p_{i+1}}$ as defined by Equation (12).

$${\tau }_{ij}\leftarrow {\tau }_{ij}\cdot (1-\rho )$$

(11)

$${\tau }_{p_i,p_{i+1}}\leftarrow {\tau }_{p_i,p_{i+1}}+{\displaystyle \frac{1}{F_{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}}}}$$

(12)

2.1.3. Overall Fitness and Scheme Evaluation Method for Multi-Team Rescue

In order to facilitate effective multi-team rescue, each grouping scheme (i.e., the combination of routes assigned to each rescue team) is assessed by means of two components:

(1)

Single-Team Path Cost

$$f^g=\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}(P^{\left(g\right)})=\sum _{i=1}^{\left|P^{\left(g\right)}\right|-1} d_{p_i^{\left(g\right)},p_{i+1}^{\left(g\right)}}$$

(13)

The cost of each individual rescue-team path is denoted by $f^g=\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}\left(P^{\left(g\right)}\right)$, as given in Equation (13). The maximum of these costs across all $K$ teams is then used to define the bottleneck of the rescue strategy, namely $F_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}}=\underset{g=1,\dots ,K}{max} f^g$.

(2)

Requisite-Node Coverage

The union of nodes covered by all teams is denoted by $V_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}}$, as specified in Equation (14).

$$V_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}}=\bigcup _{g=1}^K \{p\in P^{\left(g\right)}\mid p>0\}$$

(14)

In Equation (14), should the set of requisite nodes $R$ not be entirely included within $V_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}}(i.e., \backslash R\notin V_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}})$, a penalty of ${10}^6$ is imposed. The overall fitness of the grouping scheme is then formulated in Equation (15).

$$F_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=F_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}}+\mathbb{I}\{R\not\subset V_{\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}}\}\times \mathrm{P}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{y}$$

(15)

In Equation (15), where $\mathbb{I}\{·\}$ denotes the indicator function.

2.1.4. Construction of the Overall Rescue Scheme and Node-Importance Statistics

(1)

Construction of the Overall Scheme

Upon solution and evaluation of each grouping scheme, the scheme exhibiting the minimum overall fitness is selected. The individual rescue-team routes are denoted $\{P^{\left(1\right)},P^{\left(2\right)},\dots ,P^{\left(K\right)}\}$ and these routes are concatenated to form the final closed-loop path, as specified in Equation (16).

$$P_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}}=P^{\left(1\right)}\oplus P^{\left(2\right)}\oplus \cdots \oplus P^{\left(K\right)}$$

(16)

(2)

Node-Visit Counting

Throughout the search process, the frequency with which each node $v$ is covered by all rescue teams is recorded for subsequent analysis of node importance, as defined in Equation (17).

$$\mathrm{VisitCount}(v)=\sum _{g=1}^K \mathbb{I}\{v\in P^{\left(g\right)}\}$$

(17)

2.2. Firefly Algorithm Heuristic Optimisation Theory

The Firefly Algorithm (FA) is a heuristic method inspired by the flashing behaviour of fireflies, in which their bioluminescent signals are treated as an attraction mechanism [22]. Since MDPSO is capable of identifying a final solution, a composite fitness function that integrates node-search coverage, diversity and convergence-time weighting has been devised to enhance the optimisation performance of MDPSO in rescue-path planning.

2.2.1. Fitness-Function Computation Method for Mine-Tunnel Topologies

(1)

Search Coverage Computation Method

In order to effectively improve search coverage, the coverage metric has been defined as Equation (18).

$$Coverage={\displaystyle \frac{\left|V\right|}{N_{node}}}$$

(18)

In Equation (18), where $N_{node}$ denotes the total number of nodes in the mine tunnels, $\left|V\right|$ represents the count of unique nodes visited during the rescue-path search, and Coverage is the search-coverage ratio.

(2)

Path Diversity Computation Method

Path diversity quantifies the degree of dissimilarity among previously explored routes. Within the algorithm, the Jaccard similarity coefficient is employed to measure the similarity between any two paths $P_1$ and $P_2$:

$$J\left(P_1,P_2\right)={\displaystyle \frac{\left|P_1\cap P_2\right|}{|P_1\cup P_2|}}$$

(19)

In Equation (19), where $\left|P_1\cap P_2\right|$ is the number of nodes common to both paths, and $|P_1\cup P_2|$ is the cardinality of their union. Given a historical set of paths $\left\{P_1,P_2,\dots ,P_n\right\}$, the cumulative similarity $S$ between the latest path $P_n$ and all preceding paths is defined as Equation (20).

$$S=\sum _{i=1}^{n-1} J(P_i,P_n)$$

(20)

To normalise for the total number of possible pairwise comparisons, ${\displaystyle \frac{n(n-1)}{2}}$, a diversity metric is constructed as Equation (21), indicating that diversity decreases as path similarity increases.

$$Diversity=1-{\displaystyle \frac{S}{\frac{n(n-1)}{2}}}$$

(21)

(3)

Convergence–Time Computation Method

The convergence–time evaluation comprises three elements: the recording of the best fitness values, the definition of the relative change rate, and the specification of convergence–judgement criteria, as expressed in Equations (22)–(24). Let $\Delta f^{\left(t\right)}$ denote the best-fitness change at iteration $t$, and $\epsilon$ the convergence threshold (set to 0.001). If $\Delta f^{\left(t\right)}<\epsilon$, $\Delta f^{(t+1)}<\epsilon$, $\Delta f^{(t+k-1)}<\epsilon$ holds for $k$ successive iterations, convergence is deemed to have occurred, and the convergence time $T_c$ is assigned the value of the initial iteration $t$.

$$\Delta f^{\left(t\right)}={\displaystyle \frac{|f^{(t+1)}-f^{\left(t\right)}|}{\left|f^{\left(t\right)}\right|}}$$

(22)

$$\Delta f^{\left(t\right)}<\epsilon$$

(23)

$$\Delta f^{\left(t\right)}<\epsilon ,\Delta f^{(t+1)}<\epsilon ,\dots ,\Delta f^{(t+k-1)}<\epsilon$$

(24)

$$T_c=min\left\{t\in \{1,2,\dots ,ma{x}_{i}ter-k+1\}\mid \Delta f^{(t+i)}<\epsilon ,\forall i=0,1,\dots ,k-1\right\}$$

(25)

In Equation (25), if convergence is detected, otherwise, $T_c=\mathrm{m}\mathrm{a}\mathrm{x}\_iter$.

By combining the three criteria above, the final fitness-value computation is expressed as Equation (26).

$$fitness=w_{cov}\left(1-Coverage\right)+w_{div}\left(1-Diversity\right)+w_{conv}\left(T_c\right)$$

(26)

In Equation (26), where the weighting coefficients are $w_{cov}=0.4$, $w_{div}=0.3$, $w_{conv}=0.3$.

2.2.2. Firefly Algorithm Optimisation Principles

(1)

Fitness-Related Calculation Method

Let the solution at firefly $i$ be denoted by ${\mathbf{x}}_i$. To avoid division by zero, a small positive constant $\delta$ was introduced $(\delta ={10}^{-6})$. The brightness $I_i$ is then defined in Equation (27).

$$I_i={\displaystyle \frac{1}{f\left({\mathbf{x}}_i\right)+\delta }}$$

(27)

In Equation (27), $I_i$ denotes the brightness of firefly $i$, so that, in a minimisation context, fireflies with lower objective-function values exhibit higher brightness. The term $f\left({\mathbf{x}}_i\right)$ represents the fitness of the solution at firefly $i$.

(2)

Attractiveness Function Calculation Method

The attractiveness between fireflies is computed as per Equation (28).

$${\beta }_{ij}={\beta }_0\cdot {\displaystyle \frac{I_j}{I_i}}\mathrm{e}\mathrm{x}\mathrm{p}(-\gamma r_{ij}^2)={\beta }_0\cdot {\displaystyle \frac{f\left({\mathbf{x}}_i\right)+\delta }{f\left({\mathbf{x}}_j\right)+\delta }}\mathrm{e}\mathrm{x}\mathrm{p}(-\gamma r_{ij}^2)$$

(28)

In Equation (28), ${\beta }_0$ is the base attractivity constant, representing the attractive force when two fireflies are co-located and their brightness ratio equals one. The symbols $I_i$ and $I_j$ denote the brightnesses of fireflies $i$ and $j$, respectively, while $f\left({\mathbf{x}}_i\right)$ and $f\left({\mathbf{x}}_j\right)$ are their fitness values. The light-absorption coefficient $\gamma$ (together with ${\beta }_0$) determines the rate at which attractiveness decays with distance. The Euclidean distance $r_{ij}=\parallel x_i-x_j\parallel =\sqrt{{\sum }_{k=1}^d (x_{i,k}-x_{j,k}{)}^2}$, where $d$ is the dimensionality of the search space, influences the attractiveness such that greater distances yield weaker attraction. The exponential term $\mathrm{e}\mathrm{x}\mathrm{p}(-\gamma r_{ij}^2)$ serves as the distance-attenuation factor.

(3)

Position Update Equation

According to Equation (29), whenever the fitness of firefly $j$ exceeds that of firefly $i$, firefly iii is moved towards firefly $j$. A random-perturbation factor ${\alpha }_0$ and a random vector ${\mathit{\epsilon }}_i$ uniformly distributed in $[0,1]$ are included to maintain stochastic exploration.

$${\mathbf{x}}_i^{t+1}={\mathbf{x}}_i^t+\beta \left(r_{ij}\right)\left(\begin{array}{c}{\mathbf{x}}_j^t-{\mathbf{x}}_i^t\end{array}\right)+{\alpha }_0\left(\begin{array}{c}{\mathit{\epsilon }}_i-0.5\end{array}\right)$$

(29)

2.3. Force-Directed + Breadth-First Search Graph-Structure Optimisation Theory

In order to more effectively present rescue-team routes in alignment with conventional mine ventilation-network diagrams, a Force-Directed + BFS(Breadth-First Search) graph-structure optimisation method was developed to enable efficient command of rescue teams [23,24].

2.3.1. Inter-Node Force Computation Method

For the original graph, the initial positions of the nnn nodes, ${\mathbf{p}}_i^{\left(0\right)}=(x_i^{\left(0\right)},y_i^{\left(0\right)})$, were generated at random within an ellipse of semi-major axis $a$ and semi-minor axis $b$. Polar coordinates were sampled via $r_i=\sqrt{\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(0,1)}$, ${\theta }_i=2\pi \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(0,1)$, and converted to Cartesian coordinates by $x_i^{\left(0\right)}=a{r}_i\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_i,y_i^{\left(0\right)}=b{r}_i\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_i$. All nodes were initialised with zero velocity, $\mathrm{i}.\mathrm{e}., \backslash {\mathbf{v}}_i^{\left(0\right)}=(0,0),i=1,2,\dots ,n$.

(1)

Repulsive-Force Computation

During each iteration, nodes were subjected to both repulsive and attractive forces. For any node pair $(i,j)$, the displacement vector was defined as $\Delta {\mathbf{p}}_{ij}=(x_j-x_i,y_j-y_i)$, and the squared distance was computed as $r_{ij}^2=(x_j-x_i{)}^2+(y_j-y_i{)}^2+ϵ$, where $ϵ$ is a small constant introduced to prevent division by zero. Consequently, the distance was given by $r_{ij}=\sqrt{r_{ij}^2}$. Adopting a Coulomb-law formulation, the scalar repulsive force was defined by Equation (30).

$$F_{\mathrm{r}\mathrm{e}\mathrm{p}}\left(r_{ij}\right)={\displaystyle \frac{k\_rep}{r_{ij}^2}}$$

(30)

In Equation (30), where $\mathrm{k}\_rep$ is the repulsion constant (set to 3000).

The repulsive force acting on node iii was then obtained as per Equation (31), and node $j$ was subjected to an equal and opposite force (Equation (32)). By superposing the repulsive forces over all node pairs, the total repulsive force on each node was acquired, as shown in Equation (33).

$${\mathbf{F}}_{\mathrm{r}\mathrm{e}\mathrm{p},i}^{\left(ij\right)}=-F_{\mathrm{r}\mathrm{e}\mathrm{p}}\left(r_{ij}\right)\cdot {\displaystyle \frac{\Delta {\mathbf{p}}_{ij}}{r_{ij}}}$$

(31)

$${\mathbf{F}}_{\mathrm{r}\mathrm{e}\mathrm{p},j}^{\left(ij\right)}=+F_{\mathrm{r}\mathrm{e}\mathrm{p}}\left(r_{ij}\right)\cdot {\displaystyle \frac{\Delta {\mathbf{p}}_{ij}}{r_{ij}}}$$

(32)

$${\mathbf{F}}_{\mathrm{r}\mathrm{e}\mathrm{p},i}=\sum _{j\ne i} {\mathbf{F}}_{\mathrm{r}\mathrm{e}\mathrm{p},i}^{\left(ij\right)}$$

(33)

(2)

Attractive-Force Computation Method

For any node pair $(i,j)$ connected by an edge in the graph, the attractive force is computed using a linear-spring model. The scalar form of the attraction is defined by Equation (34):

$$F_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r}}\left(r_{ij}\right)=k\_attr\times r_{ij}$$

(34)

In Equation (34), where $\mathrm{k}\_attr$ is the attraction constant, set to 0.05.

For the edge $(i,j)$, the force acting on node $i$ is given by

$${\mathbf{F}}_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r},i}^{\left(ij\right)}=+F_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r}}\left(r_{ij}\right)\cdot {\displaystyle \frac{\Delta {\mathbf{p}}_{ij}}{r_{ij}}}$$

(35)

And the force acting on node $j$ is

$${\mathbf{F}}_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r},j}^{\left(ij\right)}=-F_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r}}\left(r_{ij}\right)\cdot {\displaystyle \frac{\Delta {\mathbf{p}}_{ij}}{r_{ij}}}$$

(36)

The total attractive force on node $i$ is obtained by summing over all incident edges:

$${\mathbf{F}}_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r},i}=\sum _{j:A(i,j)=1} {\mathbf{F}}_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r},i}^{\left(ij\right)}$$

(37)

Finally, the net force on node $i$ is expressed as the sum of its repulsive and attractive components:

$${\mathbf{F}}_i={\mathbf{F}}_{\mathrm{r}\mathrm{e}\mathrm{p},i}+{\mathbf{F}}_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r},i}$$

(38)

2.3.2. Velocity and Position Update Method

On this basis, a discrete time-step $dt$ and a damping coefficient $damp$ were used to update each node’s velocity and position. The velocity update was given by Equation (39):

$${\mathbf{v}}_i^{(t+1)}=\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}\cdot {\mathbf{v}}_i^{\left(t\right)}+dt\cdot {\mathbf{F}}_i^{\left(t\right)}$$

(39)

In this expression, $damp$ denotes the velocity–decay coefficient (set to 0.85) and $dt$ is the time step (taken to be 1.0). The new position was then obtained from the updated velocity, as shown in Equation (40):

$${\mathbf{p}}_i^{(t+1)}={\mathbf{p}}_i^{\left(t\right)}+dt\cdot {\mathbf{v}}_i^{(t+1)}$$

(40)

2.3.3. Node Constraints and Adjustment Method

To ensure that all nodes remained within the ellipse, the constraint ${\displaystyle \frac{x_i^2}{a^2}}+{\displaystyle \frac{y_i^2}{b^2}}\le 1$ was imposed. If the updated node position ${\mathbf{p}}_i=(x_i,y_i)$ violated this condition (i.e., \${\displaystyle \frac{x_i^2}{a^2}}+{\displaystyle \frac{y_i^2}{b^2}}>1$), it was projected back onto the ellipse boundary by scaling with $s_i={\displaystyle \frac{1}{\sqrt{\frac{x_i^2}{a^2}+\frac{y_i^2}{b^2}}}}$ and the node coordinates were updated as $(x_i,y_i)\leftarrow s_i\left(x_i,y_i\right)$.

A BFS algorithm was then employed to compute the hierarchical level $L_i$ of each node relative to a reference node, within the interval $[L_{min},L_{max}]$. An ideal vertical coordinate was defined by $y_i^{\mathrm{d}\mathrm{e}\mathrm{s}}=-b+2b\cdot {\displaystyle \frac{L_i-L_{\mathrm{m}\mathrm{i}\mathrm{n}}}{L_{\mathrm{m}\mathrm{a}\mathrm{x}}-L_{\mathrm{m}\mathrm{i}\mathrm{n}}}}$. Each node’s $y$-coordinate was interpolated towards this target via $y_i\leftarrow \left(1-{\alpha }_y\right)y_i+{\alpha }_y{y}_i^{\mathrm{d}\mathrm{e}\mathrm{s}}$, where ${\alpha }_y$ (set to 0.2) is an adjustment factor.

After each iteration, the system’s kinetic energy was calculated as per Equation (41):

$$E_{\mathrm{k}\mathrm{i}\mathrm{n}}={\displaystyle \frac{1}{2}}\sum _{i=1}^n \left(v_{x,i}^2+v_{y,i}^2\right)$$

(41)

In Equation (41), convergence was deemed to have been achieved, and iterations were terminated once $E_{\mathrm{k}\mathrm{i}\mathrm{n}}<{10}^{-6}$.

To ensure that certain nodes remained within the graphical bounds, the values $y_{\mathrm{t}\mathrm{o}\mathrm{p}}=\underset{i}{max} y_i+10$, $y_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{m}}=\underset{i}{min} y_i-10$ were defined. Nodes exceeding $y_{\mathrm{t}\mathrm{o}\mathrm{p}}=\underset{i}{max} y_i+10$, and those below $y_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{m}}=\underset{i}{min} y_i-10$ were assigned $y_i=y_{top}$, $y_i=y_{bottom}$, thus finalising the node-layout optimisation.

2.4. Tunnel Equivalent-Length Computation Under Fire Conditions

The computation of tunnel equivalent length during fire was recognised as a critical component for multi-team rescue-path planning. Based upon the methodology of Zhenguo Yan, Zhixin Qin et al. [8], the fire-period tunnel equivalent length was determined in accordance with Equations (42)–(45).

$$\tau \left(U_{ij}\right)={\displaystyle \frac{{\alpha }_4}{{\alpha }_1{\alpha }_2{\alpha }_{3}v\left(U_{ij}\right)}}+\sum _{\epsilon =1}^n{\beta }_{\epsilon }\left[t\left(U_{ij}\right)\right]{\mu }_i{\mu }_j+t$$

(42)

$$t\left(U_{ij}\right)={\displaystyle \frac{{\alpha }_4}{{\alpha }_1{\alpha }_2{\alpha }_{3}u\left(U_{ij}\right)}}$$

(43)

$$ϵ=601.85{\alpha }_2{e}^{-0.08\theta }$$

(44)

$${ϵ}_{max}=1812e^{-0.046\theta }$$

(45)

In these four Equations, ${\alpha }_1$ is the influence coefficient of the roadway height. ${\alpha }_2$ is the influence coefficient of the roadway slope. ${\alpha }_3$ is the influence coefficient of the obstacles in the roadway. ${\alpha }_4$ is the roadway length, m. $v\left(U_{ij}\right)$ is the predicted escape speed of personnel without smoke spreading in the roadway, m/s. $u\left(U_{ij}\right)$ is the predicted escape speed of personnel when smoke spreads along roadways, m/s. $t\left(U_{ij}\right)$ is the predicted time needed for personnel pass through the roadway in the case of smoke spreading, s. $t$ is the temperature threshold. ${\beta }_{\epsilon }$ is the influence coefficient of different toxic and harmful gas concentrations in the roadway. ${\mu }_i,{\mu }_j$ is the coefficient of the end points of the roadway, $i$ and $j,$ when there are toxic and harmful gases in the connecting roadway; the coefficient is a multiple of 1.5 when the same toxic and harmful gases exist in the connecting roadway. $ϵ$ is the time when the roadway is allowed to reach high temperatures, min. $\theta$ is the temperature, °C. ${ϵ}_{max}$ is the maximum endurance time of the human body in a high-temperature environment.

3. Results

In order to more effectively verify the effectiveness of the model, performance tests were conducted on large-scale networks as a supplement to verify the scalability of the algorithm in complex scenarios. In this section, multiple datasets from the DIMACS graph-theoretic benchmark suite, namely Dsjc250.5, Dsjc500.1, flat300_28_0, r250.5, and r1000.5 were selected for model performance evaluation. Among them, Dsjc250.5, Dsjc500.1, flat300_28_0, r250.5, and r1000.5 have 250, 500, 300, 250, and 1000 nodes, respectively. Finally, a simplified mine model derived from an actual Shaanxi mine was employed as test data to validate the optimality of both the node-layout optimisation and the multi-team rescue-route planning results.

3.1. Analysis of the Improved DPSO Algorithm’s Path-Planning Capability

To abstract the mine galleries, the prevailing approach of converting mine-excavation engineering plans into ventilation-system diagrams was adopted, thereby yielding a mine-ventilation network graph that maps the physical tunnels onto a network topology. Given that the number of internal tunnel nodes does not exceed 1000, the DIMACS instances Dsjc250.5, Dsjc500.1, flat300_28_0, r250.5, and le450_25c were assembled to formulate shortest-path problems spanning from the smallest to the largest numbered nodes. Path planning was then performed under both persistent-pheromone and reset-pheromone conditions, with analyses conducted to assess the impact of varying edge-weight perturbation and particle counts on planning performance. Optimal model parameters were subsequently chosen to enhance the accuracy of multi-team rescue-route planning within the mine.

Four algorithms—the improved DPSO, GA (Genetic Algorithm), TS (Tabu Search) and SA (Simulated Annealing)—were compared across the five case studies (Dsjc250.5, Dsjc500.1, flat300_28_0, r250.5 and le450_25c). Each case was subjected to ten perturbation events, with edge-weight disturbances set at 10%, 30%, 50%, and 70%. Performance under persistent- and reset-pheromone scenarios was contrasted by means of absolute error, and the resulting error distributions were visualised via boxplots and heatmaps [25,26,27,28].

In Figure 2, SA results are depicted in purple, TS in blue, GA in orange and DPSO in green. Across all disturbance levels (0.1, 0.3, 0.5, and 0.7), the median marker for DPSO (solid green) remains consistently lower than those of SA, TS, and GA, indicating that DPSO typically yields solutions with smaller deviations. The DPSO box (green outline) exhibits the smallest IQR (interquartile range) and shortest whiskers, demonstrating minimal fitness variability and strong robustness. Furthermore, DPSO produces the fewest outliers (circles beyond the whiskers) at each disturbance level, signifying a reduced sensitivity to edge-weight fluctuations and resilience against single, large-scale perturbations.

In the heatmap presented in Figure 3, colour intensity (dark blue → yellow) corresponds to median magnitude. The DPSO row is predominantly dark blue (lowest range), whereas GA, SA, and TS display lighter blues and occasional yellow cells (higher ranges). Across all four disturbance-level columns, DPSO’s cells remain at similarly low colour grades, evidencing that its performance degrades only marginally with increasing perturbation strength.

It is reasoned that, in dynamic scenarios, DPSO maintains and continually reinforces pheromone information along the global best path, enabling rapid recovery to near-optimal regions following each disturbance. By combining the global search capability of particle swarms with local refinement mechanisms, DPSO circumvents entrapment in local optima while preserving convergence speed. In contrast to GA’s recombination/mutation, TS’s tabu-neighbourhood strategy and SA’s stochastic jumps, DPSO’s velocity-and-position update scheme is shown to sustain superior stability and faster adaptive response under dynamic conditions.

3.2. Node-Layout Optimisation

In order to perform effective node-layout optimisation, a Force-Directed + BFS graph-structure optimisation theory was applied. A method for transforming the original mine-network diagram into an elliptical-layout network was developed, thereby rendering the mine diagram more visually appealing and conducive to command-and-control decisions during rescue operations. Furthermore, this transformation methodology may provide algorithmic support for the layout optimisation of large-scale network-diagram datasets.

To conduct a more effective analysis, a simplified model of a real mine system was selected, based on the fire escape routes in mine planning, so that the path-finding process inside the mine roadway can be presented more clearly and meticulously. For the verification of large-scale roadways, the roadway topological structure of DIMACS was analysed.

Figure 4 shows the composite topology of airflow and rescue routes within the coal mine tunnel network, which is presented under the state of fire–smoke propagation. All nodes are depicted as grey circles; the rescue-start node is highlighted by a green circle (node 1), and the locations of trapped personnel are indicated by red circles (nodes 15, 19 and 21). Arrow directions denote airflow: green arrows signify fresh air being pumped in from the entrance towards each node, whereas red arrows indicate the backflow of contaminated air along the routes. The numerical labels on each edge correspond to the equivalent length of that tunnel segment. The network is laid out from top to bottom and left to right in accordance with the hierarchical order of the rescue path, making the fresh-air channels from the start node to each trapped node immediately apparent. Along the main trunk path (1→3→5→6→13→14→17→18→21), contiguous green arrows ensure adequate fresh-air supply; on the branches (e.g., 6→19 and 13→12→11→10→8→4→2→…), red arrows reveal the routes of contaminated-air recirculation [29]. The locations of the pollutant location are situated in the middle of node 2, node 12 and node 13, as well as at the location of node 15. Therefore, the airflow in this area is considered to be stale air.

In the node-layout optimisation process, the efficacy of the force-directed layout was demonstrated by comparison with a conventional top-down layered layout. The parameters used for the force-directed optimisation are listed in the accompanying table. In the left panel of Figure 5, the top-down layered layout exhibits numerous node crossings, which are generally avoided to improve graphical clarity. In contrast, the right panel shows the force-directed, ellipse-style arrangement: nodes 20 and 21 are positioned accurately at the top of the diagram, node 1 at the bottom, and all nodes follow an elliptical distribution. The rescue route is drawn in red in the left panel and in blue in the right panel. Overall, the force-directed layout yields a visually appealing and unambiguous node arrangement, thereby supporting effective on-site rescue command.

In

Table 1. Parameter settings for the force-directed layout.

Parameters	a	b	p	k_rep	k_attr	damp	dt	maxIter	minEnergy
Value	150	150	0.5	3000	0.05	0.85	1.0	2000	1 × 10−6

, the parameters $a$, $b,$ and $p$ are defined as follows: $a$ is taken as the semi-major axis length of the ellipse used for layout; $b$ is taken as the semi-minor axis length; and $p$ denotes the scaling exponent applied during the secondary positional adjustment within the ellipse. The force-directed algorithm parameters comprise k_rep, k_attr, damp, dt, and k_rep. The constant k_rep governs the magnitude of the repulsive force between any two nodes, while k_attr governs the magnitude of the attractive force for each actual edge in the graph. The damping coefficient damp is applied to node-velocity updates to simulate frictional or resistive forces and to inhibit excessive oscillations. The time step dt is employed for the integration of velocity and position updates. The maximum iteration count, maxIter, specifies the maximum number of adjustment cycles permitted, and the minimum kinetic-energy threshold, minEnergy, provides an early-termination criterion for the layout process: if $\sum {\displaystyle \frac{1}{2}}\left(v_x^2+v_y^2\right)<\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{E}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}$ convergence is assumed and iterations cease. Moreover, node 1 was assigned to the lowest BFS layer and nodes 20 and 21 to the highest BFS layer, thereby ensuring that nodes are arranged in accordance with their connectivity and designated importance.

To optimise the node layout position more effectively, the grid search algorithm is employed. Three key parameters, namely k_rep, k_attr, and damp, are utilised to optimise the node layout position. Three parameters, namely the optimal distribution distance, the node deviation value, and the average node deviation value, are adopted as the parameters for node layout optimisation. The optimal distribution distance $distance=\sqrt{{\displaystyle \frac{Acr}{N_{node}}}}$, where $Acr$ represents the area of the ellipse and $N_{node}$ is the number of nodes. The node deviation value $deviation={\displaystyle \frac{\sqrt{{\left|D_{x,i}-D_{x,i+1}\right|}^2+{\left|D_{y,i}-D_{y,i+1}\right|}^2}}{distance}}$, in which $D_{x,i}$ and $D_{y,i}$ are the coordinates of the $i$th node in the $x$ and $y$ directions, respectively, and $distance$ is the optimal distribution distance calculated above [30]. The average node deviation value $avg\_deviation={\displaystyle \frac{{\sum }_{i=1}^{n-1}{deviation}_i}{n-1}}$, where ${deviation}_i$ is the deviation value between all adjacent nodes, and then it is divided by the number of nodes minus 1. During the grid search process, after calculating the average node deviation value $avg\_deviation$ for each set of parameters (k_rep, k_attr, damp), it is compared with the currently recorded minimum node deviation value $best\_avg\_deviation$. If the current $avg\_deviation$ is less than $best\_avg\_deviation$, $best\_avg\_deviation$ is updated, and the corresponding parameters k_rep, k_attr and damp are recorded. The selection range of the parameter k_rep is [1000, 5000], the selection range of the parameter k_attr is [0.01, 0.1], and the selection range of the parameter damp is [0.8, 0.9]. The final parameter selection results are $k\_rep=3000$, $k\_attr=0.05$, $damp=0.85$, and the average node deviation value $deviation=0.74$.

3.3. Comparative Analysis of Optimisation Algorithms

To facilitate a more effective optimisation of algorithmic performance, the fitness function $fitness=w_{cov}\left(1-Coverage\right)+w_{div}\left(1-Diversity\right)+w_{conv}\left(T_c\right)$ was employed, and four widely used heuristic optimisers—ACO (Ant Colony Optimization), FA (Firefly Algorithm), GWO (Grey Wolf Optimizer) and WOA (Whale Optimization Algorithm)—were contrasted. The respective parameter configurations are detailed in

Table 2. Optimisation-algorithm parameter settings.

Algorithm	Parameters	Default Value	Description
ACO	numAnts	20	Number of ants
	q0	0.9	Greediness threshold
FA	numFireflies	30	Firefly population size
	beta0	1	Initial attractiveness constant
	gamma	1	Light absorption coefficient
	alpha0	0.2	Initial randomization factor
GWO	numWolves	30	Grey wolf population size
	a_max	2	Initial value of coefficient a
WOA	numWhales	30	Whale population size
	a_max	2	Initial value of parameter a in encircling prey
	b	1	Spiral update constant

.

In Table 2, the number of iterations (numIterations) was unified at 100 for all algorithms. For ACO, the number of ants (numAnts) determines the count of solutions constructed in parallel during each iteration, and the greediness threshold q0 controls whether, when a random number is less than or equal to q0, the ant selects the edge with maximum pheromone or otherwise resorts to roulette-wheel selection. In FA, the quantity of fireflies (numFireflies) governs the number of solutions updated; the base attractiveness constant beta0 dictates the strength of inter-firefly attraction; the light-absorption coefficient gamma modulates the decay of attractiveness with distance; and the random-perturbation factor alpha0 preserves population diversity. In GWO, the size of the wolf pack (numWolves) sets the number of solutions updated, and the parameter a_max determines the balance between exploration and exploitation, analogous to a probabilistic selection mechanism. In WOA, the number of whales (numWhales) specifies the solution-set size; a_max controls the step length of the search; and the parameter b defines the spiral-shaped movement characteristic exhibited by whales when encircling prey. In the FA, beta0 is set to 1. An excessively large value of beta0 can lead to the algorithm becoming trapped in a local optimal solution, while an excessively small value of beta0 can cause the convergence to be too slow. Gamma is set to 1; a too large value of gamma will result in a smaller interaction range of fireflies, which is more conducive to local search, whereas a too small value of gamma will lead to an overly large interaction range of fireflies, making it more suitable for global search. Alpha is set to 0.2; an excessively small value of alpha can cause the algorithm to be trapped in a local optimum, and an excessively large value of alpha will make it difficult for the algorithm to converge. Through multiple comparisons, it has been found that such parameter settings can better balance the local optimum and the global optimum [31].

For effective optimisation, the MDPSO hyperparameters c_x, c_1_min, c_1_max, c_2, c_3 were subjected to tuning using the constructed fitness function. The optimisation outcomes are illustrated in Figure 6: the ACO algorithm is represented by the blue curve; the GWO algorithm by the orange curve; the WOA algorithm by the yellow curve; and the FA by the purple curve. Among these, the ACO algorithm did not fully converge, whereas GWO and WOA achieved slightly higher fitness values yet still outperformed ACO. The FA was found to be optimal, exhibiting a comparatively low fitness level from the initial iteration.

Figure 7 presents a line-heatmap of the FA-driven parameter-optimisation process for MDPSO. The colormap on the right indicates the mapping between fitness values and colours, with red denoting larger fitness values and blue denoting smaller ones. During the FA optimisation, the optimal parameter trajectory is depicted as a deep-blue line. The first axis corresponds to fitness values; the second to c_x; the third to c_1_min; the fourth to c_2; and the fifth to c_3. The optimal parameter set was determined to be: $c\_x=0.737872$, $c\_1\_min=0.135363$, $c\_1\_max=0.521081$, $c\_2=0.6337$, $c\_3=0.692428$.

In Figure 8, The FA is characterised by the clearest parameter-fitness relationships: the Pearson correlation coefficient between c_1_min and fitness reaches +0.85, whereas those between c_2 and fitness and between c_3 and fitness are approximately −0.71 and −0.81, respectively. It may thus be inferred that an increase in c_1_min together with reductions in c_2 and c_3 results in a marked improvement in final fitness—a unidirectional, strongly linear relationship that greatly eases parameter tuning. By contrast, within the WOA algorithm, near-unity collinearity among parameters renders their individual effects indistinguishable, thereby inhibiting targeted adjustment; under ACO, parameter-fitness correlations are negligible (coefficients near zero), indicating that modifications to any single parameter produce only marginal performance changes; and although GWO exhibits moderate correlations (circa ± 0.3–0.7), it lacks the simultaneous presence of strong positive and strong negative sensitivities that would enable a focused optimisation strategy. In FA, however, the distribution of parameter sensitivities is balanced—some parameters exert strong positive influence, others strong negative, and some remain neutral—thereby establishing a “push-pull” dynamic in which raising certain parameters “pulls” performance upwards while suppressing others “pushes” inferior solutions away, leading to an efficient and stable route to performance enhancement.

In Figure 9, from the FA scatter matrix, it can be observed that, in the five-dimensional space defined by c_1_min, c_x, c_1_max, c_2, and c_3, the samples maintain a relatively uniform coverage while exhibiting weak to moderate pairwise linear correlations. The histograms on the diagonal indicate that c_1_min and c_2 tend to cluster in the mid-to-high value ranges, whereas the distribution of c_3 is slightly skewed towards mid-to-low values; such a distribution is conducive to preserving sufficient exploration during the global search phase. The off-diagonal scatter plots reveal, for instance, a mild negative correlation between c_1_min and c_3, and a weak positive correlation between c_x and c_2. These relationships suggest that the cognitive-learning and social-learning components of the FA algorithm can be tuned relatively independently, thereby mitigating the risk of premature convergence arising from strong inter-parameter coupling.

Based on this, the t-test method and p-value test were adopted to analyse the comparison results of the four algorithms from a statistical perspective. The predicted fitness values of different algorithms were used as the inspection criteria, and relevant analyses were carried out to verify the correlations among the four algorithms. The results show that when comparing WOA with ACO, the t-statistic is −25.33 and the p-value is 0.00050. When comparing WOA with FA, the t-statistic is 22.32 and the p-value is 0.00059. When comparing WOA with GWO, the t-statistic is −19.86 and the p-value is 0.00052. When comparing ACO with FA, the t-statistic is 28.07 and the p-value is 0.00061. When comparing ACO with GWO, the t-statistic is 18.41 and the p-value is 0.00090. When comparing FA with GWO, the t-statistic is −29.48 and the p-value is 0.00024. Through comprehensive comparison, it is found that the t-value of FA is the smallest, and the p-values of the four algorithms are all less than 0.05, which meets the requirements of the hypothesis test and verifies the high efficiency of the FA.

3.4. Rescue-Route Planning Results

An experiment was conducted to plan the underground rescue routes for multiple teams, with nodes 19, 15, and 21 selected as the trapped-person locations. The number of rescue teams was varied from one to three, and the FA-optimised parameters $c\_x=0.737872$, $c\_1\_min=0.135363$, $c\_1\_max=0.521081$, $c\_2=0.6337$, $c\_3=0.692428$ were employed.

Using the MDPSO algorithm, the rescue routes for each grouping were computed, and the results are summarised in

Table 3. Results of rescue routes for multiple rescue-team configurations.

Number of Teams	Group Routes	Equivalent Path Length
Group 1	1→3→5→6→13→14→15→14→13→6→19→6→13→14→17→18→ 21→18→17→14→13→6→5→3→1	72,242.02
Group 2	1→3→5→6→13→14→15→14→13→6→19→6→5→3→1	60,639.78
	1→3→5→6→13→14→17→18→21→18→17→14→13→6→5→3→1	66,294.86
Group 3	1→3→5→6→13→14→15→14→13→6→5→3→1	56,357.02
	1→3→5→6→19→6→5→3→1	58,975.38
	1→3→5→6→13→14→17→18→21→18→17→14→13→6→5→3→1	66,294.86

. In Table 3, the longest path length for each configuration is reported. For the single-team scenario, the route length of 72,242.02 corresponds to the only team visiting nodes 15, 19 and 21 in sequence. In the two-team configuration, the maximum route length was 66,294.86: the first team traversed nodes 15 and 19, while the second team covered node 21. In the three-team case, the longest route remained 66,294.86, with each team assigned to a single node (team 1 to node 15, team 2 to node 19 and team 3 to node 21). Although two teams or three teams both yielded feasible routes, the three-team deployment achieved the shortest equivalent path of 56,357.02 and was therefore selected as the optimal configuration.

Figure 10 is presented as the outcome of the multi-team rescue-route planning. The network is arranged using a hybrid Force-Directed + BFS layout: all 21 nodes are shown as white circles with black borders, and the underlying topology is depicted with thin black arcs. Four high-contrast colours are employed to distinguish each rescue team’s route allocation: The deep-red trunk path (nodes 1→3→5→6→13) is rendered as the shared backbone route for all teams. The blue essential segment (nodes 14→15) is highlighted as the mandatory route for the first team. The orange branch arc (nodes 6→19) marks the route assigned to the second team. The yellow extension segment (nodes 14→17→18→21) indicates the third team’s path along the northern branch.

3.5. Node Importance Analysis

In Figure 11, the “Node vs. Team-Count Heatmap” (top-left), a colour gradient from deep indigo (lowest visit frequency) through pale green (moderate) to bright yellow (highest) is employed. When only one rescue team is deployed (first column), the vast majority of nodes appear in deep indigo or indigo, indicating very low visit counts. With two teams (second column), certain hub nodes transition to green hues, reflecting a marked increase in visit frequency. Under three-team deployment (third column), a number of critical nodes (e.g., \ nodes 1, 3, 6, and 12) are rendered bright yellow, signifying highly concentrated visitation. This visualisation intuitively demonstrates that, as team count increases, both the overall visit frequency and its concentration on a small set of key nodes rise.

In the “Grouped Bar Chart” (top-right), three shades of blue distinguish the deployment scenarios: dark blue for “1 Team”, teal blue for “2 Teams”, and steel blue for “3 Teams”. For each node, the three adjacent bars form a stepped increase from dark blue through steel blue: peripheral nodes exhibit near-zero dark-blue bars, slight teal-blue increments, and prominent steel-blue bars at principal hubs (nodes 1, 3, 6 and 12). This chart quantitatively confirms that increased team numbers shift visitation towards core nodes, thereby intensifying visit concentration.

The “Rank-Frequency Curve” (bottom-left) depicts descending order visit counts under the three scenarios, again using dark blue, teal blue, and steel blue. All three curves display long-tail distributions; however, the steel-blue curve lies highest overall, followed by the teal-blue curve, and finally the dark-blue curve, indicating that three teams achieve the broadest coverage, two teams the next, and one team the least. Beyond validating the rise in total visits with team number, this plot also reveals the “long-tail” nature of node importance, wherein a few nodes account for the majority of visits.

The “Radial Plot of Visit Counts” (bottom-right) arranges nodes uniformly around a polar coordinate system, with dark blue, teal blue, and steel blue radial lines representing the visit counts for one, two, and three teams, respectively. Pronounced peaks appear on the steel-blue curve at nodes 6, 12, and 3, with diminishing peak magnitudes on the teal-blue and dark-blue curves. This polar visualisation enables immediate comparison of node importance and distribution patterns across deployment scenarios, further illustrating the trend of visit focus consolidating on the same core nodes as team number increases.

4. Discussion

An MDPSO-based multi-team rescue-path planning algorithm, optimised via the Firefly Algorithm (FA), has been developed. Specifically

(1)

An improved DPSO algorithm was constructed and evaluated on five DIMACS instances (Dsjc250.5, Dsjc500.1, flat300_28_0, r250.5 and le450_25c). By comparing persistent and reset pheromone strategies across DPSO, GA, SA, and TS, the superior accuracy of the improved DPSO was confirmed.

(2)

A composite fitness function tailored to the MDPSO multi-team rescue scenario was formulated, and four heuristic optimisers (ACO, FA, GWO, and WOA) were employed to tune the MDPSO hyperparameters. The FA-driven optimisation and the fitness-function design were demonstrated to be effective in refining MDPSO performance.

(3)

In a mine-fire smoke scenario, the MDPSO-based multi-team rescue routes were computed, and a force-directed + BFS network-layout method was applied to optimise node positioning. Three trap nodes (15, 19 and 21) were selected, and the results indicated that deploying three rescue teams yielded the fastest underground evacuation.

The work presented herein provides an algorithmic foundation for the coordinated rescue of multiple teams in underground mining emergencies and contributes technical support for intelligent mine-rescue systems. Moreover, the network-layout optimisation approach may serve as a novel method for the visualisation of large-scale graph datasets. In the future, based on the existing work, methods for updating time steps and the equivalent length of roadways in discrete roadway situations will be added. These methods will be combined with the underground fire situation and sensor data for a more in-depth analysis of dynamic rescue plans, providing certain support for the intelligent construction of mines.