Intelligent Decision Framework for Booster Fan Optimization in Underground Coal Mines: Hybrid Spherical Fuzzy-Cloud Model Approach Enhancing Ventilation Safety and Operational Efficiency

Abstract

Optimizing mine fan operations in underground coal mines is important for ensuring proper ventilation, enhancing safety, and improving operational efficiency. A single main ventilation fan is insufficient to meet the ventilation demands of the entire mine. Therefore, it is necessary to consider the addition of booster fans to ensure effective ventilation. However, the selection of booster fans involves multiple influencing factors, and the complex interrelationships among fans remain unclear, making solution selection and risk assessment more challenging. To address this issue, this study proposes an optimization and risk analysis method for booster fan selection based on an improved analytic hierarchy process. This method leverages spherical fuzzy sets to handle uncertainty in the ventilation parameters and cloud models to facilitate probabilistic decision making. Through this model, the important relationships of the influencing factors for fan selection can be systematically determined, allowing for a rational assessment of the performance scores of candidate solutions. It provides a ranking of the alternatives based on their superiority, along with the risk indicators and optimization potentials of the selected solution. Ultimately, the reliability of the chosen model was verified through comparison and validation. This method not only enhances the scientific and rational basis for booster fan selection, reducing the complexity of the selection process, but also provides theoretical support for the optimization of coal mine ventilation systems. This study demonstrates the model’s effectiveness at improving ventilation safety and cost efficiency, making it a valuable tool for modern underground mining operations.

1. Introduction

The method of coal mining is generally divided into open-pit mining and underground mining. Compared to underground mining, open-pit mining involves the direct exchange with surface air and lacks roof control [1,2], among other factors, leading to differences in the engineering complexity and associated risks [3,4]. For example, open-pit coal mining requires the consideration of the slope stability, ecological damage, and waste rock and water management [5,6,7,8]. In contrast, underground coal mining requires attention to the safety of the ventilation systems, gas management, mine pressure, and strata control, as well as water hazard prevention [9,10,11,12]. Among these, mine ventilation plays an indispensable role throughout the entire lifecycle of underground coal mine operations.

Mine ventilation provides essential functions, such as supplying fresh air to the underground, regulating temperature, diluting harmful gases, and removing hazardous dust. The surface airflow is introduced into the underground mine through the main ventilation fan. Booster fans are ventilation devices used in conjunction with the main surface fan to supplement the insufficient air pressure in certain areas of the mine [13]. In multi-layer coal seam designs, booster fans can effectively resolve ventilation challenges [14]. However, an unreasonable booster fan solution design may lead to risks, such as the spontaneous combustion of residual coal [15,16], the accumulation of pollutants [17], and gas explosions caused by local airflow turbulence. Therefore, the reliable optimization of alternative booster fan solutions and necessary risk analysis are key measures to ensure the safety of the mine ventilation and production efficiency [18].

The traditional booster fan solution designs face challenges due to varying tunnel layouts and production risks in different underground mines. When considering booster fan solutions for specific mines, it is necessary to rely on engineers’ experience, reference ventilation-related technical standards, and influencing factors to propose a solution, which should then be verified. This approach relies too heavily on subjective experience, which may lead to incomplete consideration and decision-making errors due to personal preferences. Additionally, the decision-making process often requires extensive trial and error, ultimately resulting in only feasible solutions without the ability to assess their superiority.

With the advancement of the research on fan-related behaviors and solution optimization theories, many potentially feasible approaches have been proposed. In fan behavior studies, the conclusions drawn from numerous research efforts have simplified the process of fan selection. Bayomi et al. suggested adding a straightener at the fan inlet to eliminate deformation, increase airflow, and reduce noise [19]. Bredel discussed the impact of the fan pressure and selection on the outcomes during the fan selection process [20]. De Villiers et al. evaluated the performances of underground fan systems and found that the total efficiency of the auxiliary fan system was 5%, concluding that a greater number of auxiliary fan combinations would lead to decreased energy efficiency, providing a valuable reference for fan selection [21]. Related studies provide conclusions that can serve as references during the selection of booster fans [22], helping to reduce the research and testing process and alleviate the challenges associated with the booster fan selection.

In related research on solution optimization, many theories have been applied in relevant engineering projects with certain effectiveness. Kursunoglu and Onder introduced the analytic hierarchy process (AHP) for fan selection, specifically for choosing a main fan for underground coal mines in Turkey [23]. Jiu et al. used a mixed-integer linear-programming (MILP) model to formulate the problem and employed a hybrid genetic algorithm (HGA) to calculate the solution, achieving nearly optimal results for scheduling equipment in coal production systems [24]. Li and Ding applied the fuzzy DEMATEL method to open-pit mining, discussing the relevant evaluation indicators and methods [25]. Mulumba et al. used an optimized PSO-BP neural network model to establish a safety risk assessment model for underground coal mines, evaluating and predicting the safety risk factors [26]. Can [27] applied failure mode effect analysis and Pareto analysis to assess risks in coal mine production and exploration measurements.

Although the research on fan-related behaviors and solution optimization theories [28,29,30,31] has advanced the progress of fan selection work, the studies on fan behavior tend to focus on the in-depth analysis of single indicators, resulting in scattered research outcomes. Related solution optimization theories, while achieving good results, are still limited by the insufficient expression of fuzzy information and the quality and scale of reference data (cases) for neural networks [32], leaving room for improvement in the efficiency and superiority of fan selection. Furthermore, the use of complex spherical fuzzy theory to rank multi-criteria, decision-making solutions based on the expression of two-dimensional fuzzy information has been widely discussed and is considered a reliable approach for multi-criteria decision making [33,34,35,36,37]. Moreover, the cloud model theory, with its rich information embedded in cloud droplets, has attracted significant attention in solution evaluation, risk analysis, and optimization potential research [38,39,40].

In summary, this study focused on the design of the booster fan solution for the Okaba coal mine. Based on the AHP, an evaluation system was established, and relevant experts were consulted to assess and create a judgment matrix to determine the weight of each indicator. Fuzzy theory was applied to conduct fuzzy evaluations of the indicators, and the qualitative fuzzy assessments were converted into quantitative superiority scores using the complex spherical fuzzy function. This allowed for the ranking of the alternative booster fan solutions based on their superiority. A reference cloud model for the solution was established using the cloud model theory, and the indicator cloud model for the solution was constructed based on the expert evaluation parameter data, revealing the risk associated with each indicator and completing the optimization of the solution. Finally, the validity of the proposed solution was confirmed by a comparison with other research theories. This study not only provides scientific decision-making support for the optimization and selection of the booster fan in the Okaba coal mine but also offers a transferable methodological framework for multi-criteria decision making in similar complex engineering systems.

2. Methodology Framework

This study aimed to improve and extend the AHP by integrating complex spherical fuzzy numbers and the cloud model theory to optimize the selection of booster fan alternatives for the Okaba coal mine and perform the corresponding risk analysis and optimization of the selected solutions. The AHP method served as the primary framework of this study, effectively decomposing multi-criteria decision-making problems into hierarchical levels and determining the weight of each factor through consistency judgments, thereby constructing a clear and structured decision system. The introduction of the hybrid spherical fuzzy membership function enhanced the capability of the fuzzy representation, improving the distinguishability and boundary clarity of the evaluation results. Furthermore, the integration of the cloud model theory not only bridged qualitative and quantitative analyses but also captured uncertainty through its cloud drop generator, offering strong visualization features and interpretability. The basic approach (Figure 1) was as follows: First, the evaluation indicators and their evaluation criteria were established through the AHP, and the weights of the indicators were obtained from the judgment matrix. Further, the fuzzy theory was applied to establish the evaluation criteria for the alternative solutions. Based on these criteria and the indicator weights, the superiority ranking of the alternative booster fan solutions was completed using the complex spherical fuzzy function, ultimately achieving the optimal selection of the booster fan solution. Finally, the cloud model theory was used to present the risks associated with each indicator of the selected solutions and identify any gaps, completing the analysis and optimization of the solutions’ risks.

2.1. Analytic Hierarchy Process

The AHP was proposed by Satty in 1971 as a quantitative analysis method to solve complex decision-making problems [41,42,43]. By breaking down complex issues into multiple levels, the AHP uses mathematical models to quantitatively evaluate and rank different decision alternatives, thereby assisting decisionmakers in making informed choices [44]. The specific process is outlined as follows:

(1)

Establishing the hierarchical structure

The first step in the analytic hierarchy process (AHP) is the establishment of a scientific and comprehensive evaluation system. The AHP is a method that integrates qualitative judgments with quantitative analysis to provide systematic and rational support for complex decision-making problems. The AHP evaluation system is generally divided into several levels, where an upper-level indicator typically encompasses several lower-level indicators;

(2)

Expert system evaluation

The expert evaluation system was designed to obtain the pairwise relationships of the relative importances between different indicators at the same level. The interaction matrix is a method used to determine the relative importances of evaluation criteria. At the same hierarchical level, the relative importances among different criteria are typically established through the interaction matrix, which is constructed based on pairwise comparisons made by experts. This results in a judgment matrix that reflects the expert knowledge and opinions.

The mutual importance relationship between two indicators is represented using the 1–9 scale proposed by Saaty [45]. The correspondence between the mutual importance relationship between the two indicators and the scale are shown in

Table 1. The 1–9 scale.

Scale	Relationship Between Two Factors
1	Equal importance
3	Slightly more important
5	Clearly more important
7	Extremely more important
9	Absolutely more important
2, 4, 6, 8	Intermediate values between the above levels

.

(3)

Constructing the judgment matrix

After establishing the evaluation system and evaluation criteria (Table 1), we invited researchers and engineering professionals closely related to the design field of the alternative solutions to participate in the evaluation. First, the importance of all the indicators in the evaluation system were compared pairwise to obtain qualitative evaluation information. Using Table 1, the qualitative language provided by the experts regarding the importance of each indicator was converted into quantitative scale data, and the judgment matrix was constructed as follows:

$$A={(a_{ij})}_{n\times n}=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \cdots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \cdots & a_{2n}\\ a_{31}& a_{32}& a_{33}& \cdots & a_{3n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& a_{n3}& \cdots & a_{nn}\end{array}\right]=\left[\begin{array}{ccccc}1& a_{12}& a_{13}& \cdots & a_{1n}\\ {\displaystyle \frac{1}{a_{12}}}& 1& a_{23}& \cdots & a_{2n}\\ {\displaystyle \frac{1}{a_{13}}}& {\displaystyle \frac{1}{a_{23}}}& 1& \cdots & a_{3n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{1}{a_{1n}}}& {\displaystyle \frac{1}{a_{2n}}}& {\displaystyle \frac{1}{a_{3n}}}& \cdots & 1\end{array}\right]$$

(1)

where A represents the expert judgment matrix; n denotes the total number of indicators in the established evaluation system, which also represents the order of the matrix; $a_{ij}$ represents the importance of the j-th indicator relative to the i-th indicator;

(4)

Calculation of the weight vector

First, the maximum eigenvalue of the judgment matrix is calculated:

$$AV={\lambda }_{\max }V$$

(2)

where $V$ represents the eigenvalue; ${\lambda }_{\max }$ max represents the maximum eigenvalue of the matrix.

Then, the weight vector is obtained by normalizing the eigenvector:

$$w_i={\displaystyle \frac{v_i}{{\displaystyle {\sum }_{i=1}^n{v}_i}}}$$

(3)

$$W=(w_1,w_2,\cdots ,w_n)$$

(4)

where $w_i$ represents the weight of the i-th indicator; $v_i$ represents the value of the i-th eigenvector; $W$ represents the weight vector of the evaluation matrix;

(5)

Consistency test

A consistency test is performed on the obtained judgment matrix to ensure its rationality. First, the consistency index ($CI$) is calculated:

$$CI={\displaystyle \frac{{\lambda }_{\max }-n}{n-1}}$$

(5)

where ${\lambda }_{\max }$ represents the largest eigenvalue of the matrix, and n represents the order of the matrix.

Then, the consistency ratio ($CR$) is calculated:

$$CR={\displaystyle \frac{CI}{RI}}$$

(6)

where $RI$ represents the random consistency index (selected from

Table 2. Values of the random consistency index.

n	1	2	3	4	5	6	7	8	9	10
$RI$	0	0	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49

); $CI$ represents the consistency index; $CR$ represents the consistency ratio.

(6)

Obtainment of the overall weights of the indicators

First, assume there are m experts. After performing the consistency check, the expert judgment matrices are merged to form a comprehensive judgment matrix:

$$W_f={(w_{ij})}_{m\times n}=\left[\begin{array}{cccc}{w}_{11}& w_{12}& \cdots & w_{1n}\\ w_{21}& w_{22}& w_{23}& w_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ w_{m1}& w_{m2}& w_{m3}& w_{mn}\end{array}\right]$$

(7)

where m represents the number of expert matrices that passed the consistency check; n represents the number of indicators involved in the evaluation.

Next, based on the professional background and work experience of each expert, a reliability coefficient matrix for each expert is provided:

$$R=\left[\begin{array}{cccc}{r}_1& r_2& \cdots & r_m\end{array}\right]$$

(8)

where R is the reliability coefficient matrix; $r_m$ represents the reliability coefficient of the m-th expert, i.e., the weight of the judgment matrix provided by the i-th expert.

Finally, the comprehensive indicator weight is obtained by calculating the expert weight through a comprehensive evaluation:

$$B=R·W_f=\left[\begin{array}{cccc}{b}_1& b_2& \cdots & b_n\end{array}\right]$$

(9)

where B represents the comprehensive indicator weight vector; $b_n$ represents the weight of the n-th indicator.

2.2. Complex Spherical Fuzzy Number

The complex spherical fuzzy number is a further extension of the spherical fuzzy number, designed to handle more complex fuzzy information, particularly suitable for multi-criteria decision-making problems with multiple fuzzy dimensions and intricate uncertainties. By introducing higher-dimensional fuzzy information, the complex spherical fuzzy number allows for a more precise representation of the relationships between different fuzzy factors [46]:

(1)

Basic concepts

In the domain Z, the complex spherical fuzzy set (C) can be represented as follows:

$$C=\left\{(z,{\mu }_c(z)e^{i{\alpha }_c(z)},{\varphi }_c(z)e^{i{\beta }_c(z)},v_c(z)e^{i{\gamma }_c(z)},),z\in Z\right\}$$

(10)

where $i=\sqrt{-1}$ for $\forall z\in Z$, and there exists ${\mu }_c(z),{\varphi }_c(z),v_c(z)\in \left[0,1\right]$ and $({\alpha }_c(z),{\beta }_c(z),{\gamma }_c(z)\in \left[0,2\pi \right]$. The membership degree ($f_c(z)$), neutrality degree ($g_c(z)$), and non-membership function ($h_c(z)$) are all confined within a unit circle. If $\forall z\in Z$ exists, and there are corresponding ${\mu }_c^2(z)+{\varphi }_c^2(z)+v_c^2(z)\le 1$ and ${({\displaystyle \frac{{\alpha }_c(z)}{2\pi }})}^2+{({\displaystyle \frac{{\beta }_c(z)}{2\pi }})}^2+{({\displaystyle \frac{{\gamma }_c(z)}{2\pi }})}^2\le 1$, then $({\mu }_c(z)e^{i{\alpha }_c(z)},{\varphi }_c(z)e^{i{\beta }_c(z)},v_c(z)e^{i{\gamma }_c(z)})$ is referred to as a complex spherical fuzzy number;

(2)

Scoring function

Although the complex spherical fuzzy number contains higher-level mathematical information, which can significantly enhance the extraction of fuzzy information features, its complexity makes the representation results difficult to interpret. In this context, referring to relevant research [47], the scoring function is defined as follows:

$$S(A)={\displaystyle \frac{1}{7}}[{(2{\mu }_c-{\displaystyle \frac{{\varphi }_c}{2}})}^2-{(v_c-{\displaystyle \frac{{\varphi }_c}{2}})}^2+{\displaystyle \frac{1}{4{\pi }^2}}({\alpha }_c^2-{\beta }_c^2-{\gamma }_c^2)+2]$$

(11)

2.3. Cloud Model

The cloud model is a mathematical model used to handle uncertainty and fuzziness [48,49]. It is intuitive, easy to understand, and exhibits strong fault tolerance and robustness, making it effective at simplifying the fuzzy reasoning process. It has been widely applied in fields such as fuzzy systems, pattern recognition, and decision support. The core idea is to model the fuzziness of the objects being processed using the concept of a cloud, which is represented by numerous cloud droplets.

For the forward cloud model, the generation of cloud droplets is primarily controlled by three parameters (see Figure 2): Expectation (Ex), Entropy (En), and Hyperentropy (He). The Ex controls the symmetry axis of the cloud model and is often considered the evaluator’s assessment opinion. The En controls the width (degree of dispersion) of the cloud model, while the He controls the thickness (uncertainty) of the cloud layer. These two parameters are typically regarded as reflecting the concentration and clarity of the decisionmakers’ assessment opinions.

The generator function for the cloud drops based on eigenvalues is defined as follows:

$$y=e^{(-{\displaystyle \frac{{(x-Ex)}^2}{2·E{n}^2}})}$$

(12)

where x represents the value of a cloud drop generated from a normal distribution; y represents the membership degree of x belonging to the concept center.

2.4. Expert Decision Making Based on Fuzzy Theory

By combining the characteristics of both the complex spherical fuzzy theory and the cloud model theory, qualitative languages divided into five levels were organized, and quantitative information was then mapped to these levels through the fuzzy theory, converting the expert’s fuzzy qualitative assessments into complex spherical fuzzy parameters and cloud model parameters, respectively (

Table 3. Expert opinion correspondence.

Superiority Language	Superiority Levels	Complex Spherical Fuzzy Parameters	Cloud Model Parameters
Excellent	I	$(0.9,0.05,0.05,0,{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{2}})$	$(1,0.1031,0.013)$
Good	II	$(0.8,0.1,0.1,{\displaystyle \frac{\pi }{6}},{\displaystyle \frac{\pi }{3}},{\displaystyle \frac{\pi }{2}})$	$(0.691,0.0637,0.008)$
Average	III	$(0.7,0.2,0.2,{\displaystyle \frac{\pi }{3}},{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{5\pi }{6}})$	$(0.5,0.0394,0.005)$
Below average	IV	$(0.6,0.3,0.3,{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{2\pi }{3}},{\displaystyle \frac{4\pi }{3}})$	$(0.309,0.0637,0.008)$
Poor	V	$(0.5,0.4,0.4,\pi ,{\displaystyle \frac{5\pi }{6}},{\displaystyle \frac{7\pi }{6}})$	$(0,0.1031,0.013)$

).

3. Introduction to the Study Area

The validation project selected for the application of the improved hierarchical analysis method optimized by the complex spherical fuzzy and cloud models in the selection and risk analysis of booster fan solutions was the booster fan optimization and risk analysis at the Okaba coal mine. The Okaba coal mine is located in one of Nigeria’s key coal production regions, with production commencing as early as the 1950s. The mine primarily extracts bituminous coal and high-rank coal. Situated at a latitude of 7°20′ N and a longitude of 7°25′ E (see Figure 3), the region’s underlying strata are predominantly composed of sedimentary rocks, with some folds and faults present. The groundwater geological conditions are relatively complex [50,51].

The main shaft diameter of the Okaba underground coal mine (see Figure 4) is 3 m and is responsible for the transportation of personnel and related equipment, with fresh air flowing into the mine through this shaft. The shaft also functions to transport fresh air from the surface to the underground, where four booster fans (denoted as BF1, BF2, BF3, and BF4) are installed. These booster fans further deliver air from the lower part of the ventilation shaft to various working faces and other air-using locations. The circulated exhaust air is discharged through the ventilation shaft, with the airflow powered by the fans [52]. Due to the different positions of the four booster fans (BF1, BF2, BF3, and BF4), the airflow experiences varying degrees of loss before reaching each fan. Therefore, under different influencing conditions, the relative importance of each fan varies accordingly.

4. Case Application of the Method

4.1. Determining Indicator Weights (AHP)

(1)

Establishing the hierarchical structure

Based on the basic information about the Okaba coal mine described in Section 3, based on the relevant literature [52,53] and expert opinions, an evaluation indicator system for the selection of fan solutions was established. The system consists of three levels (see Figure 5). The first level is the solution layer, which includes information on the four fans: BF1, BF2, BF3, and BF4. The second level comprises the main criteria, which include technical (M1), operational (M2), environmental (M3), and economical (M4) indicators. These four criteria cover all aspects that need to be considered in the decision-making process and provide a comprehensive reflection of the actual situation of the alternative solutions. The third level is the sub-criteria, which include the air quality (S1), pressure (S2), air power (S3), and efficiency (S4) under the technical (M1) criterion; production (S5), safety (S6), and flexibility (S7) under the operational (M2) criterion; noise (S8) and vibration (S9) under the environmental (M3) criterion; and operation cost (S10) under the economical (M4) criterion. The experts’ evaluations of the importances of these indicators are conducted at the third (sub-criteria) level, which contains a total of 10 indicators. The solution selection is based on the first-level solutions.

This study conducted a comprehensive evaluation of the optimal layout scheme for mine booster fans through the establishment of four main criteria (M1–M4), which were graded across multiple dimensions [54,55]. M1 was used to assess the core performance of the fan, serving as the fundamental and crucial criterion. S1 reflects the fan’s impact on the air quality improvement, ensuring compliance with the air standards in the mine; S2 determines the fan’s ability to overcome tunnel resistance, ensuring effective airflow transmission; S3 represents the power consumption of the fan, reflecting the power requirements; and S4 influences the fan’s energy efficiency and economic viability. M2 focuses on the stability and adaptability of the fan during actual operation. S5 reflects whether the fan can stably meet the production needs of the mine, S6 assesses the fan’s reliability under extreme operating conditions, and S7 measures the fan’s ability to adapt to different working conditions. M3 addresses the impact of the fan on the working environment. S8 and S9 focus on the noise and vibration generated by the fan during operation, which affect the health of the miners and the working environment and require careful attention. M4 focuses on S10, which evaluates the economic performance of the equipment throughout its lifecycle, ensuring that the selected fan offers not only a good performance and favorable working environment but also sustainable economic advantages. Through the comprehensive consideration of these indicators, the optimal fan layout scheme can be scientifically assessed to meet the overall requirements of the mine ventilation system [56,57].

(2)

Obtaining the expert judgment matrix

Based on the standards in Table 1, two research-oriented experts (E1 and E2) and one technical expert (E3) were invited to evaluate the 10 sub-criteria indicators (S1–S10) in the sub-criterion layer of the hierarchical structure shown in Figure 5.

To fully account for the evaluation opinions and areas of expertise of the different experts, and to enhance the scientific validity and rationality of the assessment results, this study considered that the research-oriented experts possessed stronger theoretical judgment capabilities in comprehensive analyses compared to front-line technical personnel. Accordingly, the reliability coefficients were set at 0.35 for the two research experts and at 0.30 for the technical personnel. Based on these values, the expert reliability matrix was constructed as follows:

$$R=\left[0.35\hspace{1em}0.35\hspace{1em}0.3\right]$$

(13)

Based on the Expert System Evaluation Form (

Table A6. Expert system evaluation form.

No.	Comparison	Which Is More Important?	Importance Level (1–9)
1	Operating Cost vs. Air Quality
2	Operating Cost vs. Pressure
3	Operating Cost vs. Air Power
4	Operating Cost vs. Efficiency
5	Operating Cost vs. Productivity
6	Operating Cost vs. Safety
7	Operating Cost vs. Flexibility
8	Operating Cost vs. Noise
9	Operating Cost vs. Vibration
10	Air Quality vs. Pressure
11	Air Quality vs. Air Power
12	Air Quality vs. Efficiency
13	Air Quality vs. Productivity
14	Air Quality vs. Safety
15	Air Quality vs. Flexibility
16	Air Quality vs. Noise
17	Air Quality vs. Vibration
18	Pressure vs. Air Power
19	Pressure vs. Efficiency
20	Pressure vs. Productivity
21	Pressure vs. Safety
22	Pressure vs. Flexibility
23	Pressure vs. Noise
24	Pressure vs. Vibration
25	Air Power vs. Efficiency
26	Air Power vs. Productivity
27	Air Power vs. Safety
28	Air Power vs. Flexibility
29	Air Power vs. Noise
30	Air Power vs. Vibration
31	Efficiency vs. Productivity
32	Efficiency vs. Safety
33	Efficiency vs. Flexibility
34	Efficiency vs. Noise
35	Efficiency vs. Vibration
36	Productivity vs. Safety
37	Productivity vs. Flexibility
38	Productivity vs. Noise
39	Productivity vs. Vibration
40	Safety vs. Flexibility
41	Safety vs. Noise
42	Safety vs. Vibration
43	Flexibility vs. Noise
44	Flexibility vs. Vibration
45	Noise vs. Vibration

) developed, three experts were invited to fill it out. According to the feedback from Expert E1, the expert judgment matrix 1 is organized as follows:

$$\left[\begin{array}{llllllllll}1\hfill & 3\hfill & 4\hfill & 3\hfill & 2\hfill & 2\hfill & 1\hfill & 5\hfill & 6\hfill & 2\hfill \\ 1/3\hfill & 1\hfill & 2\hfill & 1\hfill & 1/2\hfill & 1/2\hfill & 1/3\hfill & 3\hfill & 4\hfill & 1/2\hfill \\ 1/4\hfill & 1/2\hfill & 1\hfill & 1/2\hfill & 1/3\hfill & 1/3\hfill & 1/4\hfill & 2\hfill & 4\hfill & 1/2\hfill \\ 1/3\hfill & 1\hfill & 2\hfill & 1\hfill & 1/2\hfill & 1/2\hfill & 1/3\hfill & 3\hfill & 4\hfill & 1/2\hfill \\ 1/2\hfill & 2\hfill & 3\hfill & 2\hfill & 1\hfill & 1\hfill & 1/2\hfill & 4\hfill & 5\hfill & 1\hfill \\ 1/2\hfill & 2\hfill & 3\hfill & 2\hfill & 1\hfill & 1\hfill & 1/2\hfill & 4\hfill & 5\hfill & 1\hfill \\ 1\hfill & 3\hfill & 4\hfill & 3\hfill & 2\hfill & 2\hfill & 1\hfill & 5\hfill & 6\hfill & 2\hfill \\ 1/5\hfill & 1/3\hfill & 1/2\hfill & 1/3\hfill & 1/4\hfill & 1/4\hfill & 1/5\hfill & 1\hfill & 2\hfill & 1/4\hfill \\ 1/6\hfill & 1/4\hfill & 1/4\hfill & 1/4\hfill & 1/5\hfill & 1/5\hfill & 1/6\hfill & 1/2\hfill & 1\hfill & 1/5\hfill \\ 1/2\hfill & 2\hfill & 2\hfill & 2\hfill & 1\hfill & 1\hfill & 0.5\hfill & 4\hfill & 5\hfill & 1\hfill \end{array}\right]$$

(14)

Based on the opinions of E2, expert judgment matrix 2 is organized as follows:

$$\left[\begin{array}{llllllllll}1\hfill & 3\hfill & 4\hfill & 3\hfill & 2\hfill & 2\hfill & 1\hfill & 5\hfill & 6\hfill & 2\hfill \\ 1/3\hfill & 1\hfill & 2\hfill & 1\hfill & 1/2\hfill & 1/2\hfill & 1/3\hfill & 3\hfill & 4\hfill & 1/2\hfill \\ 1/4\hfill & 1/2\hfill & 1\hfill & 1/2\hfill & 1/3\hfill & 1/3\hfill & 1/4\hfill & 2\hfill & 4\hfill & 1/2\hfill \\ 1/3\hfill & 1\hfill & 2\hfill & 1\hfill & 1/2\hfill & 1/2\hfill & 1/3\hfill & 3\hfill & 4\hfill & 1/2\hfill \\ 1/2\hfill & 2\hfill & 3\hfill & 2\hfill & 1\hfill & 1\hfill & 1/2\hfill & 4\hfill & 5\hfill & 1\hfill \\ 1/2\hfill & 2\hfill & 3\hfill & 2\hfill & 1\hfill & 1\hfill & 1/2\hfill & 4\hfill & 5\hfill & 1\hfill \\ 1\hfill & 3\hfill & 4\hfill & 3\hfill & 2\hfill & 2\hfill & 1\hfill & 5\hfill & 6\hfill & 2\hfill \\ 1/5\hfill & 1/3\hfill & 1/2\hfill & 1/3\hfill & 1/4\hfill & 1/4\hfill & 1/5\hfill & 1\hfill & 2\hfill & 1/4\hfill \\ 1/6\hfill & 1/4\hfill & 1/4\hfill & 1/4\hfill & 1/5\hfill & 1/5\hfill & 1/6\hfill & 1/2\hfill & 1\hfill & 1/5\hfill \\ 1/2\hfill & 2\hfill & 2\hfill & 2\hfill & 1\hfill & 1\hfill & 0.5\hfill & 4\hfill & 5\hfill & 1\hfill \end{array}\right]$$

(15)

Based on the opinions of E3, expert judgment matrix 3 is organized as follows:

$$\left[\begin{array}{llllllllll}1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 4\hfill & 1\hfill & 1\hfill & 4\hfill & 1\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 4\hfill & 1\hfill & 1\hfill & 4\hfill & 1\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 4\hfill & 1\hfill & 1\hfill & 4\hfill & 1\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 4\hfill & 1\hfill & 1\hfill & 4\hfill & 1\hfill \\ 1/2\hfill & 1/2\hfill & 1/2\hfill & 1/2\hfill & 1\hfill & 3\hfill & 1/2\hfill & 1/2\hfill & 3\hfill & 1/2\hfill \\ 1/4\hfill & 1/4\hfill & 1/4\hfill & 1/4\hfill & 1/3\hfill & 1\hfill & 1/4\hfill & 1/4\hfill & 1\hfill & 1/4\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 4\hfill & 1\hfill & 1\hfill & 4\hfill & 1\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 4\hfill & 1\hfill & 1\hfill & 4\hfill & 1\hfill \\ 1/4\hfill & 1/4\hfill & 1/4\hfill & 1/4\hfill & 1/3\hfill & 1\hfill & 1/4\hfill & 1/4\hfill & 1\hfill & 1/4\hfill \\ 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 4\hfill & 1\hfill & 1\hfill & 4\hfill & 1\hfill \end{array}\right]$$

(16)

(3)

Calculation of indicator weights

Referring to Equations (2)–(4) and using the judgment matrices provided by the three experts in Matrices (14–16), the expert weights (EWs) were calculated and are presented in

Table 4. The indicator weights.

Indicator	EW1	EW2	EW3	CIW
S1	0.199	0.097	0.125	0.141
S2	0.072	0.170	0.125	0.120
S3	0.050	0.097	0.125	0.090
S4	0.072	0.170	0.125	0.120
S5	0.120	0.097	0.068	0.095
S6	0.120	0.026	0.030	0.060
S7	0.199	0.097	0.125	0.143
S8	0.031	0.038	0.125	0.066
S9	0.022	0.038	0.030	0.030
S10	0.115	0.170	0.125	0.135

Note: EW_i represents the evaluation weight of the indicator assigned by the i-th expert; CIW represents the comprehensive indicator weight calculated from all the experts’ evaluations.

. Based on the obtained indicator weights from the three experts and the expert reliability coefficients (Matrix (13)), the comprehensive indicator weights (CIWs) were then calculated and are presented in Table 4.

(4)

Consistency test

Based on Equations (5) and (6) and Table 3, the consistency parameters (CI, RI, ${\lambda }_{\max }$) corresponding to the three matrices provided by the experts were calculated. The final consistency ratio (CR) was then determined. As shown in

Table 5. The consistency parameters.

Parameter	EW1	EW2	EW3
CI	0.022	0.010	0.003
RI	1.49	1.49	1.49
${\lambda }_{max}$	10.198	10.093	10.023
CR	0.015	0.007	0.002

, the CR for E1 is 0.015, that for E2 is 0.007, and that for E3 is 0.002. Since the consistency ratios for all three experts are less than 0.1, the results pass the consistency test. Therefore, the indicator weights obtained from the three experts (Table 5) are considered reliable.

(5)

Analysis of comprehensive indicator weights

Based on the indicator weights obtained in Table 5, the distribution of the comprehensive indicator weights for the sub-criterion layer was analyzed, and the weights for the main criterion layer were further calculated (Figure 6). For the sub-criterion layer, the weight of S7 (flexibility) reached 0.143, and the weight of S1 (quality) was 0.141. This shows that, according to the invited experts, the quality of the booster fan and its impact on the complexity of the ventilation system are highly valued. This is because the ventilation system must remain stable over the long term, and frequent damage to the booster fan or an overly complex ventilation system could lead to airflow short-circuiting, which could damage equipment and personnel, affecting the safe production of the mine. Additionally, the weights for S2 (pressure), S4 (efficiency), and S10 (operation costs) are all above 0.1, indicating their relative importances, as they influence mine production from the technical, production efficiency, and economic perspectives. In the main criterion layer, M1 (technical) is the most important indicator. This is because the technical parameters represent the essential requirements that the fan must meet to satisfy production needs, and they serve as the decisive indicators to ensure the feasibility of the plan.

4.2. Fan Solution Selection Based on Complex Spherical Fuzzy Theory

(1)

Solution evaluation based on expert system

Similarly, these three experts provided evaluations of the effectiveness of the alternative fan configurations for the four booster fan locations (see Figure 4) based on the expert opinion reference table in Table 3. The evaluation results are shown in

Table A1. Evaluation of alternative solutions based on expert system.

Indicator	Solutions	E1	E2	E2
S1	BF1	V	IV	IV
	BF2	IV	IV	IV
	BF3	III	III	III
	BF4	III	III	III
S2	BF1	III	V	V
	BF2	III	IV	III
	BF3	III	III	III
	BF4	II	III	III
S3	BF1	II	III	IV
	BF2	III	IV	III
	BF3	I	III	III
	BF4	II	IV	III
S4	BF1	III	IV	V
	BF2	III	IV	IV
	BF3	III	III	II
	BF4	III	III	III
S5	BF1	III	III	III
	BF2	III	III	IV
	BF3	III	III	II
	BF4	II	II	II
S6	BF1	III	II	II
	BF2	II	II	II
	BF3	III	III	IV
	BF4	III	II	III
S7	BF1	V	III	III
	BF2	III	III	IV
	BF3	V	IV	IV
	BF4	III	IV	III
S8	BF1	II	II	V
	BF2	I	II	III
	BF3	II	II	II
	BF4	II	II	II
S9	BF1	I	II	II
	BF2	I	II	II
	BF3	I	II	II
	BF4	I	II	II
S10	BF1	III	V	IV
	BF2	III	III	III
	BF3	III	IV	III
	BF4	III	IV	III

;

(2)

Fuzzy information transformation

The complex spherical fuzzy parameters from Table 3 were computed using Equations (10) and (11) to obtain the fuzzy scores corresponding to the evaluation levels (

Table 6. Correspondence between evaluation levels and fuzzy scores.

Level	I	II	III	IV	V
Score	0.691	0.578	0.395	0.131	0.113

). It should be noted that the levels from I to V represent a classification of the indicator importances, ranging from the most important to the least important, with the scores corresponding accordingly from high to low.

Based on the correspondence between the levels and scores in Table 6, the qualitative evaluations provided by the experts in Table A1 were converted into quantitative evaluation results (

Table A2. The complex spherical fuzzy scores for each indicator.

Indicator	Solutions	E1	E2	E2
S1	BF1	0.113	0.131	0.131
	BF2	0.131	0.131	0.131
	BF3	0.395	0.395	0.395
	BF4	0.395	0.395	0.395
S2	BF1	0.395	0.113	0.113
	BF2	0.395	0.131	0.395
	BF3	0.395	0.395	0.395
	BF4	0.578	0.395	0.395
S3	BF1	0.578	0.395	0.131
	BF2	0.395	0.131	0.395
	BF3	0.691	0.395	0.395
	BF4	0.578	0.131	0.395
S4	BF1	0.395	0.131	0.113
	BF2	0.395	0.131	0.131
	BF3	0.395	0.395	0.578
	BF4	0.395	0.395	0.395
S5	BF1	0.395	0.395	0.395
	BF2	0.395	0.395	0.131
	BF3	0.395	0.395	0.578
	BF4	0.578	0.578	0.578
S6	BF1	0.395	0.578	0.578
	BF2	0.578	0.578	0.578
	BF3	0.395	0.395	0.131
	BF4	0.395	0.578	0.395
S7	BF1	0.113	0.395	0.395
	BF2	0.395	0.395	0.131
	BF3	0.113	0.131	0.131
	BF4	0.395	0.131	0.395
S8	BF1	0.578	0.578	0.113
	BF2	0.691	0.578	0.395
	BF3	0.578	0.578	0.578
	BF4	0.578	0.578	0.578
S9	BF1	0.691	0.578	0.578
	BF2	0.691	0.578	0.578
	BF3	0.691	0.578	0.578
	BF4	0.691	0.578	0.578
S10	BF1	0.395	0.113	0.131
	BF2	0.395	0.395	0.395
	BF3	0.395	0.131	0.395
	BF4	0.395	0.131	0.395

);

(3)

Solution selection

The complex spherical fuzzy scores for each indicator corresponding to each solution (Table A2) were multiplied by the comprehensive indicator weights (Table 5) to obtain the final scores for each solution. Further, the average of the three experts’ opinions was taken as the final complex spherical fuzzy score for each solution (

Table 7. The complex spherical fuzzy scores for different solutions.

	ES1	ES2	ES3	ES
BF1	0.35234	0.282734	0.228554	0.287876
BF2	0.397172	0.299204	0.279734	0.325370
BF3	0.402272	0.339176	0.398321	0.379923
BF4	0.471773	0.343781	0.429953	0.415169

Note: ES_i represents the evaluation scores of the indicator assigned by the i-th expert; ES represents the comprehensive scores calculated from all the experts’ evaluations.

).

To ensure the validity of the evaluation verification method and assess the stability of the model, a comparative study was conducted with the fuzzy TOPSIS (FT) method, which was also applied in the selection of booster fans for the Okaba mine [58]. It was observed that the hybrid spherical fuzzy scores exhibited a similar trend to that of the FT-based scheme (see Figure 7a). Whether considering the individual experts’ complex spherical score rankings, the comprehensive score ranking, or the rankings from other related studies [58], the order remained consistent: BF4 > BF3 > BF2 > BF1. This high consistency validates the effectiveness of the method. It indicates that when considering different booster fan solutions, BF4 should be prioritized, followed by BF3, BF2, and, finally, BF1 (see Figure 7b).

4.3. Analysis of Alternative Indicators Based on Cloud Model Theory

(1)

Evaluation reference cloud

Based on the expert evaluation opinions and the corresponding relationship of the cloud model parameters (Table 3), the cloud droplet generation for the different levels was calculated using Equation (12) (see Figure 8). This served as the reference cloud for the indicator analysis.

(2)

Analysis of solution indicators

The corresponding information on the most superior booster fan solution (BF4) for solution indicator analysis was selected. Based on the expert evaluation opinions and the correspondence of the cloud model parameters (Table 3), the cloud parameters for the evaluation of different indicators by three experts (E1, E2, E3) were organized (

Table A3. Cloud model parameters for E1’s evaluation of the BF4 solution indicators.

Indicator	Ex1	En1	He1
S1	0.5	0.0394	0.005
S2	0.691	0.0637	0.008
S3	0.691	0.0637	0.008
S4	0.5	0.0394	0.005
S5	0.691	0.0637	0.008
S6	0.5	0.0394	0.005
S7	0.5	0.0394	0.005
S8	0.691	0.0637	0.008
S9	1	0.1031	0.013
S10	0.5	0.0394	0.005

,

Table A4. Cloud model parameters for E2’s evaluation of the BF4 solution indicators.

Indicator	Ex2	En2	He2
S1	0.5	0.0394	0.005
S2	0.5	0.0394	0.005
S3	0.309	0.0637	0.008
S4	0.5	0.0394	0.005
S5	0.691	0.0637	0.008
S6	0.691	0.0637	0.008
S7	0.309	0.0637	0.008
S8	0.691	0.0637	0.008
S9	0.691	0.0637	0.008
S10	0.309	0.0637	0.008

and

Table A5. Cloud model parameters for E3’s evaluation of the BF4 solution indicators.

Indicator	Ex3	En3	He3
S1	0.5	0.0394	0.005
S2	0.5	0.0394	0.005
S3	0.5	0.0394	0.005
S4	0.5	0.0394	0.005
S5	0.691	0.0637	0.008
S6	0.5	0.0394	0.005
S7	0.5	0.0394	0.005
S8	0.691	0.0637	0.008
S9	0.691	0.0637	0.008
S10	0.5	0.0394	0.005

);

Combining Expert Opinions:

Based on Equation (12), Matrix (13), and

Table 8. Final cloud model parameters for the BF4 solution indicator evaluation.

Indicator	Ex	En	He
S1	0.5	0.0394	0.005
S2	0.56685	0.04791	0.00605
S3	0.5	0.05641	0.0071
S4	0.5	0.0394	0.005
S5	0.691	0.0637	0.008
S6	0.56685	0.04791	0.00605
S7	0.43315	0.04791	0.00605
S8	0.691	0.0637	0.008
S9	0.79915	0.07749	0.00975
S10	0.43315	0.04791	0.00605

, a large number of cloud droplets were generated (Figure 9), and the superiority of each indicator for the BF4 solution was then analyzed.

The results shown in Figure 9 reveal significant differences in the performance of the BF4 booster fan solution across the various indicators. Firstly, the cloud plots of the eight indicators (S1–S6, S8, S9) have symmetry axes greater than or equal to 0.5, indicating a high success rate for these indicators. This means that the BF4 booster fan solution performs prominently in these areas, demonstrating strong competitiveness. However, the symmetry axes of indicators S7 and S10 are less than 0.5, suggesting that the superiority of these two indicators is lower, with significant differences in their expert evaluations. There is considerable potential for optimization in these areas, and further exploration of their improvement space may be required.

Another noteworthy observation is the larger thicknesses and widths of the cloud plots for S5 and S10, indicating substantial disagreement and inconsistency among the experts when evaluating these indicators. The thickness and width of the cloud plots typically reflect the degree of dispersion in the expert evaluations. The greater the thickness and width, the larger the differences in the expert opinions, suggesting significant variation in the evaluation criteria and emphasis. To address these inconsistencies, further expert discussions or data support may be necessary to clarify and unify the evaluations of S5 and S10. The complexity of S5 and S10 may arise from multiple factors. Therefore, resolving these issues requires a multi-faceted approach, addressing data, technology, or process aspects to further refine the evaluation criteria for these indicators and reduce the evaluation discrepancies among the experts, ultimately improving the overall consistency and accuracy of the assessment.

In summary, further optimization should focus on the potential improvements of S7 and S10, particularly by refining the evaluation standards and methods for S5 and S10 to reduce discrepancies among the experts and enhance the overall performance of the solution.

5. Conclusions

This paper establishes a scientific fuzzy qualitative evaluation language system for indicators and solutions, along with the corresponding quantitative parameters, based on an improved AHP and the complex spheroidal fuzzy and cloud model theories. A new method for selecting the optimal underground booster fan solution and conducting risk assessment and optimization in coal mines is proposed. By referring to the Okaba booster fan arrangement solution, a fuzzy language system for evaluation was established using the AHP, and through fuzzy functions, it was transformed into quantitative parameters, further determining the weights of the indicators involved in the evaluation. The complex spheroidal fuzzy theory was used to complete the superiority ranking of the four alternative solutions, and a comparison with relevant studies demonstrated the validity of the superiority. The cloud model theory is applied to present the risk and superiority of the solution’s indicators through cloud droplets, thereby analyzing and optimizing the layout of underground booster fans in coal mines. The main conclusions are as follows.

(1)

A booster fan solution evaluation system based on the improved AHP was established, further determining the weights of the indicators involved in the evaluation and extracting the decision preferences of the experts;

(2)

A booster fan solution optimization model based on the complex spherical fuzzy theory was established, and the effectiveness of the model was validated through a comparison with related studies;

(3)

A risk analysis and optimization model for booster fan solution indicators based on the cloud model theory was established. The model visually presented the rich risk information and optimization potential of the solution indicators through cloud droplets, providing valuable insights for solution optimization.

The scalability of the designed method remains one of the main challenges when optimizing and risk managing the underground booster fans in coal mines. Although this paper aims to extract the implicit information of the solutions primarily using fuzzy logic and compares the conclusions with other studies under the same conditions to verify the effectiveness of the proposed model, the generalization capability of the model still requires further exploration under more complex and variable engineering conditions.