*Article* **Mining Method Optimization of Difficult-to-Mine Complicated Orebody Using Pythagorean Fuzzy Sets and TOPSIS Method**

**Shuai Li <sup>1</sup> , Qi Huang <sup>1</sup> , Boyi Hu 1,\*, Jilong Pan 1,\*, Junyu Chen <sup>2</sup> , Jianguo Yang <sup>2</sup> , Xinghui Zhou <sup>2</sup> , Xinmin Wang <sup>1</sup> and Haoxuan Yu <sup>3</sup>**


**Abstract:** In Suichang gold mine, the altered rock type gold deposits were cut by faults and joint fissures, leading to complex resource endowment characteristics, large changes in occurrence, a serious complex of ore vein branches and great difficulty in mining. In order to select a suitable mining method for such a difficult and complicated orebody, a multi-factor and multi-index comprehensive evaluation system involving benefits, costs, safety and other aspects was constructed by using the Pythagorean fuzzy sets and TOPSIS method. Taking Suichang gold mine as an example, the weighted aggregation evaluation matrix was constructed, the closeness index of the four mining schemes were 0.8436, 0.3370, 0.4296 and 0.4334, and the mechanized upward horizontal layering method was determined as the optimal scheme. This method overcame the fuzzy comparison of economic and technical indicators directly, but converted them into corresponding fuzzy numbers to obtain accurate closeness index for optimization. The application of this method not only ensured a safe, efficient and environment-friendly mining effect, but also provided a reference for the optimization of the mining scheme of the severely branched composite orebody.

**Keywords:** mining method optimization; difficult-to-mine complicated orebody; multiple attribute decision making; Pythagorean fuzzy sets and TOPSIS method; mechanized upward horizontal layering method

### **1. Introduction**

In the design process of new mines and reconstruction and upgrading of mines, the design of mining methods is an important part and is the core of the whole design work, which directly determines the subsequent technical personnel allocation, production organization management and industrial supporting facilities [1–3]. Choosing the most appropriate mining method to mine the deposit is very important to the safety, economy and environmental protection of mining operations and affects the benefits and long-term development of mining companies [4].

For a continuous orebody with regular shape, the most appropriate mining method can be selected by comparing the advantages and disadvantages and the economic and technical indicators. However, taking the difficult-to-mine complicated orebody (DCO) (see Figure 1) as example, it was greatly affected by faults and joint fissures, leading to obvious branching and compounding phenomenon. Since it is difficult to determine the optimal plan through traditional methods for DCO, more reliance is placed on the experience of design decision-makers and limited data, through the comprehensive comparison of various indicators. This is typical of Multiple Attribute Decision Making (MADM).

**Citation:** Li, S.; Huang, Q.; Hu, B.; Pan, J.; Chen, J.; Yang, J.; Zhou, X.; Wang, X.; Yu, H. Mining Method Optimization of Difficult-to-Mine Complicated Orebody Using Pythagorean Fuzzy Sets and TOPSIS Method. *Sustainability* **2023**, *15*, 3692. https://doi.org/10.3390/su15043692

Academic Editor: Jianjun Ma

Received: 4 January 2023 Revised: 8 February 2023 Accepted: 13 February 2023 Published: 16 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

(MADM).

experience of design decision-makers and limited data, through the comprehensive comparison of various indicators. This is typical of Multiple Attribute Decision Making

**Figure 1.** Plan of 140–260 m middle section of Suichang Gold Mine. **Figure 1.** Plan of 140–260 m middle section of Suichang Gold Mine.

Multiple Attribute Decision Making refers to the decision making problem of selecting the best alternative or ranking under a condition of considering multiple indicators [5]. In order to solve such problems, researchers have developed a variety of multi criteria decision-making methods; typical representative methods include the Analytic Hierarchy Process [6], Entropy method [7], CRITIC method [8], TOPSIS method [9], GST (Grey Target Decision) method [10], DEA method [11], VIKOR method [12], Fuzzy comprehensive Multiple Attribute Decision Making refers to the decision making problem of selecting the best alternative or ranking under a condition of considering multiple indicators [5]. In order to solve such problems, researchers have developed a variety of multi criteria decision-making methods; typical representative methods include the Analytic Hierarchy Process [6], Entropy method [7], CRITIC method [8], TOPSIS method [9], GST (Grey Target Decision) method [10], DEA method [11], VIKOR method [12], Fuzzy comprehensive evaluation method [13], etc.

evaluation method [13], etc. However, with increase in the complexity and scientific requirements of evaluation, a single decision-making method cannot guarantee optimal or accurate results [14,15]. Under this background, researchers have explored and developed mixed decision-making methods, for example, mixtures of Analytic Hierarchy Process and fuzzy, Entropy weight method and TOPSIS, Analytic Hierarchy Process, fuzzy set and VIKOR, fuzzy set However, with increase in the complexity and scientific requirements of evaluation, a single decision-making method cannot guarantee optimal or accurate results [14,15]. Under this background, researchers have explored and developed mixed decision-making methods, for example, mixtures of Analytic Hierarchy Process and fuzzy, Entropy weight method and TOPSIS, Analytic Hierarchy Process, fuzzy set and VIKOR, fuzzy set and TOPSIS, etc.

and TOPSIS, etc. In the field of mining engineering, many experts rely on these decision-making methods to optimize mining methods, for example, Karimnia [16] proposed the fuzzy analytical hierarchy process method, the most suitable method selected for Iran's Qapliq salt mine. Yavuz [17] used the AHP method and fuzzy multiple attribute decision-making method, respectively, in a lignite mine located in Istanbul, carried out sensitivity analysis of the two methods and concluded that the Room and pillar method with filling is the most appropriate method. Qinqiang Guo [18] used the mixed method of AHP to determine the index weight and TOPSIS to rank, selecting the most suitable mining method from the Soft Broken Complex Orebody, and achieved very good results, Iphar [19] is committed to developing a mobile application, integrating several decision-making meth-In the field of mining engineering, many experts rely on these decision-making methods to optimize mining methods, for example, Karimnia [16] proposed the fuzzy analytical hierarchy process method, the most suitable method selected for Iran's Qapliq salt mine. Yavuz [17] used the AHP method and fuzzy multiple attribute decision-making method, respectively, in a lignite mine located in Istanbul, carried out sensitivity analysis of the two methods and concluded that the Room and pillar method with filling is the most appropriate method. Qinqiang Guo [18] used the mixed method of AHP to determine the index weight and TOPSIS to rank, selecting the most suitable mining method from the Soft Broken Complex Orebody, and achieved very good results, Iphar [19] is committed to developing a mobile application, integrating several decision-making methods, and the optimal mining method can be obtained by inputting the original parameters for reference by engineering researchers.

ods, and the optimal mining method can be obtained by inputting the original parameters for reference by engineering researchers. However, in view of the complexity of mining method selection, simple expert scoring cannot fully reflect the fuzzy information, and the cognitive differences between different experts are easy to cause distortion of results. In terms of sorting, different ranging methods may have different results. In this case, the development of fuzzy set theory provides

a good idea for solving such problems [20]. Baji´c [21] transformed the indicators into triangular fuzzy numbers, constructed a fuzzy decision matrix and a fuzzy performance matrix, used to select the optimal alternative, and verified this through sensitivity analysis. Memori [22] uses the TOPSIS method based on intuitionistic fuzzy sets, providing an accurate sustainable ranking of suppliers and a relevant solution for sustainable sourcing decisions that is validated through a real-world case study. Narayanamoorthy [23] selected the best scheme for the selection of industrial robots by using a combination of Intervalvalued intuitionistic hesitant fuzzy entropy and VIKOR. vides a good idea for solving such problems [20]. Bajić [21] transformed the indicators into triangular fuzzy numbers, constructed a fuzzy decision matrix and a fuzzy performance matrix, used to select the optimal alternative, and verified this through sensitivity analysis. Memori [22] uses the TOPSIS method based on intuitionistic fuzzy sets, providing an accurate sustainable ranking of suppliers and a relevant solution for sustainable sourcing decisions that is validated through a real-world case study. Narayanamoorthy [23] selected the best scheme for the selection of industrial robots by using a combination of Interval-valued intuitionistic hesitant fuzzy entropy and VIKOR.

However, in view of the complexity of mining method selection, simple expert scoring cannot fully reflect the fuzzy information, and the cognitive differences between different experts are easy to cause distortion of results. In terms of sorting, different ranging methods may have different results. In this case, the development of fuzzy set theory pro-

Pythagorean fuzzy sets (PFS) generalized by Yager [24] is a new method to deal with fuzzy problems. Its main contribution is to go beyond the limit that the sum of membership and no membership of fuzzy sets is less than 1 (see Figure 2). Compared with other fuzzy sets such as intuitionistic fuzzy sets, it can more fully and accurately represent uncertain information [25]. In view of this, this paper introduces a TOPSIS method based on PFS [26]. The framework of this PFS–TOPSIS method is illustrated (see Figure 3), which is used for mining method decision-making for the Suichang Gold Mine in Zhejiang Province, China, and has achieved good results in actual production. Pythagorean fuzzy sets (PFS) generalized by Yager [24] is a new method to deal with fuzzy problems. Its main contribution is to go beyond the limit that the sum of membership and no membership of fuzzy sets is less than 1 (see Figure 2). Compared with other fuzzy sets such as intuitionistic fuzzy sets, it can more fully and accurately represent uncertain information [25]. In view of this, this paper introduces a TOPSIS method based on PFS [26]. The framework of this PFS–TOPSIS method is illustrated (see Figure 3), which is used for mining method decision-making for the Suichang Gold Mine in Zhejiang Province, China, and has achieved good results in actual production.

**Figure 2. Figure 2.** Spatial comparison between PFS and other fuzzy sets. Spatial comparison between PFS and other fuzzy sets.

*Sustainability* **2023**, *15*, x FOR PEER REVIEW 3 of 22

*where the functions μξ(x):X → [0, 1] and νξ(x): X → [0, 1] define the degree of membership and* 

**Definition 2.** For *the collection ξi = (μξ,νξ,) (i = 1,2,……n) of the of PFNs with the weight vector w = (w1 w2……wn) of ξi(i = 1,2,……n) such that ∑n i=1 wi = 1, the Pythagorean fuzzy weighted averaging (PFWA) operator and the Pythagorean fuzzy weighted geometric (PFWG) operator can* 

∈

= ൛ൣ, క(), క()หϵ൧ൟ (1)

*X.πξ(x) =* ඥ1 − (క())ଶ − (క())ଶ *is called the degree of indeter-*

*X. For convenience, they are called (μξ(x), νξ(x*)) *and a Pythagorean fuzzy* 

*X to the set ξ, respectively, with the condition that* 

(2)

(3)

**Figure 3.** Framework of PFS-TOPSIS method. **Figure 3.** Framework of PFS-TOPSIS method.

*the degree of non-membership of the element x*

∈

*number (PFN) denoted by ξ = (μξ,νξ,) [29].* 

∀*x*∈

*be defined as in Equations (2) and (3*), *respectively [30].* 

**2. Methods Introduction**  *2.1. Introduction of PFS Method* 

*0 ≤ (μξ(x))2 + (νξ(x))2 ≤ 1,*

*minacy of element x*

#### **2. Methods Introduction**

#### *2.1. Introduction of PFS Method*

**Definition 1.** *Let X be a universe of discourse [27,28]. The PFS ξ on X is given by Equation (1).*

$$\mathfrak{F} = \left\{ \left[ \mathbf{x}, \mu\_{\mathfrak{F}}(\mathbf{x}), \nu\_{\mathfrak{F}}(\mathbf{x}) \middle| \mathbf{x} \epsilon \mathbf{X} \right] \right\} \tag{1}$$

*where the functions µ<sup>ξ</sup> (x):X* → *[0, 1] and ν<sup>ξ</sup> (x): X* → *[0, 1] define the degree of membership and the degree of non-membership of the element x*∈*X to the set ξ, respectively, with the condition that 0* ≤ *(µ<sup>ξ</sup> (x))<sup>2</sup> + (ν<sup>ξ</sup> (x))<sup>2</sup>* <sup>≤</sup> *1,*∀*x*∈*X. <sup>π</sup><sup>ξ</sup> (x) =*<sup>q</sup> 1 − *µ<sup>ξ</sup>* (*x*) <sup>2</sup> <sup>−</sup> *ν<sup>ξ</sup>* (*x*) 2 *is called the degree of indeterminacy of element x*∈*X. For convenience, they are called (µ<sup>ξ</sup> (x), ν<sup>ξ</sup> (x*)) *and a Pythagorean fuzzy number (PFN) denoted by ξ = (µξ,νξ, )* [29].

**Definition 2.** *For the collection ξ<sup>i</sup> = (µξ,νξ, ) (i = 1,2,* . . . . . . *n) of the of PFNs with the weight vector w = (w<sup>1</sup> w<sup>2</sup>* . . . . . . *wn) of ξ<sup>i</sup> (i = 1,2,* . . . . . . *n) such that n* ∑ *i*=1 *w<sup>i</sup>* = 1*, the Pythagorean fuzzy weighted averaging (PFWA) operator and the Pythagorean fuzzy weighted geometric (PFWG) operator can be defined as in Equations (2) and (3), respectively [30].*

$$\begin{aligned} \text{PFWA}\_w(\mathfrak{f}\_1, \mathfrak{f}\_2, \dots, \mathfrak{f}\_n) &= w\_1 \mathfrak{f}\_1 \oplus w\_2 \mathfrak{f}\_2 \oplus \dots \oplus w\_n \mathfrak{f}\_n\\ &= \left( \sqrt{1 - \prod\_{i=1}^n \left( 1 - \left( \mu\_{\mathfrak{f}\_i} \right)^2 \right)}, \prod\_{i=1}^n \left( \upsilon\_{\mathfrak{f}\_i} \right)^{w\_i} \right), \end{aligned} \tag{2}$$

$$\begin{split} \text{PFWG}\_{w}(\mathfrak{f}\_{1}, \mathfrak{f}\_{2}, \dots, \mathfrak{f}\_{n}) &= w\_{1}\mathfrak{f}\_{1} \otimes w\_{2}\mathfrak{f}\_{2} \otimes \dots \otimes w\_{n}\mathfrak{f}\_{n} \\ &= \left(\prod\_{i=1}^{n} \left(\mu\_{\xi\_{i}}\right)^{w\_{i}} \sqrt{1 - \prod\_{i=1}^{n} \left(1 - \left(\upsilon\_{\xi\_{i}}\right)^{2}\right)^{w\_{i}}}\right). \end{split} \tag{3}$$

**Definition 3.** *Let A and B be PFSs of X = {x1, x2,* . . . *xn} [27]. Then, the sum of A and B is defined as Equation (4).*

$$A \oplus B = \begin{array}{c} \{ \langle \mathbf{x}, \sqrt{(\mu\_A(\mathbf{x}))^2 + (\mu\_B(\mathbf{x}))^2 - (\mu\_A(\mathbf{x}))^2(\mu\_B(\mathbf{x}))^2} \} \\ \nu\_A(\mathbf{x})\nu\_B(\mathbf{x}) \rangle \mid \mathbf{x} \in X \}, \end{array} \tag{4}$$

*The product of A and B is defined as Equation (5).*

$$A \otimes B = \begin{cases} \left< \mathbf{x}, \mu\_A(\mathbf{x}) \mu\_B(\mathbf{x}) \right> \\ \sqrt{(\nu\_A(\mathbf{x}))^2 + (\nu\_B(\mathbf{x}))^2 - (\nu\_A(\mathbf{x}))^2 (\nu\_B(\mathbf{x}))^2} \mid \mathbf{x} \in X \right>. \tag{5}$$

### *2.2. Introduction of TOPSIS Method*

The TOPSIS method was developed by Hwang [31], first put forward in 1981, and is a method of ranking according to the closeness of a limited number of evaluation objects to the ideal target. After years of development, dozens of derivative methods have been created by combining various mathematical theories, and have been widely used in various fields such as economic, management, engineering, medicine, etc. Its core ideas and steps are as follows [32]:

(1) Quantification of evaluation indicators, converting natural language into numbers, and ensuring a certain distinction between good and bad, *D* is the evaluation objective and *X* is the evaluation index. The characteristic matrix is defined as Equation (6).

$$D = \begin{bmatrix} \mathbf{x}\_{11} & \dots & \mathbf{x}\_{1j} & \dots & \mathbf{x}\_{1n} \\ \vdots & & \vdots & & \vdots \\ \mathbf{x}\_{i1} & \dots & \mathbf{x}\_{ij} & \dots & \mathbf{x}\_{in} \\ \vdots & & \vdots & & \vdots \\ \mathbf{x}\_{m1} & \dots & \mathbf{x}\_{mj} & \dots & \mathbf{x}\_{mn} \end{bmatrix} = \begin{bmatrix} D\_1(\mathbf{x}\_1) \\ \vdots \\ D\_i(\mathbf{x}\_j) \\ \vdots \\ D\_m(\mathbf{x}\_n) \end{bmatrix} = \begin{bmatrix} \mathbf{X}\_1(\mathbf{x}\_1), \dots, \mathbf{X}\_j(\mathbf{x}\_i), \dots, \mathbf{X}\_n(\mathbf{x}\_m) \end{bmatrix}. \tag{6}$$

(2) Normalize the characteristic matrix, obtain the normalized vector *rij* and establish the normalized matrix about the normalized vector. This is defined as Equation (7).

$$r\_{ij} = \frac{\mathfrak{x}\_{ij}}{\sqrt{\sum\_{i=1}^{m} \mathfrak{x}\_{ij}^{2}}}, i = 1, 2, \dots, m, j = 1, 2, \dots, n. \tag{7}$$

(3) Normalize the value *vij* by calculating the weight; weight normalization matrix is defined as Equation (8).

$$\mathbf{w}\_{\mathbf{i}\mathbf{j}} = w\_{\mathbf{j}} r\_{\mathbf{i}\mathbf{j}\prime} \mathbf{i} = 1, 2, \dots, m, \mathbf{j} = 1, 2, \dots, n. \tag{8}$$

(4) Determine ideal solution *A* <sup>+</sup> and anti-ideal solution *A* −; in the ideal solution and anti-ideal solution, *J*<sup>1</sup> is the optimal value of profitability index set expressed on the *i* index; *J*<sup>2</sup> is the worst value of the *i* index of the loss index set. *A* <sup>+</sup> and *A* − are defined as Equations (9) and (10).

$$A^{+} = \left(\max\_{i} \nu\_{i\bar{\imath}} \middle| j \in I\_1\right), \left(\min\_{i} \nu\_{i\bar{\jmath}} \middle| j \in I\_1\right), \left| i = 1, 2, \dots, m = \nu\_1^{+}, \nu\_2^{+}, \dots, \nu\_{\bar{\jmath}}^{+}, \dots, \nu\_n^{+} \right. \tag{9}$$

$$A^- = (\min\_i \nu\_{i\bar{j}} | j \in I\_1), (\max\_i \nu\_{i\bar{j}} | j \in J\_1), \left| i = 1, 2, \dots, m = \nu\_1^-, \nu\_2^-, \dots, \nu\_{\bar{j}}^-, \dots, \nu\_n^- \right. \tag{10}$$

(5) Calculate the distance *S* + from the target to the ideal solution *A* <sup>+</sup> and the distance *S* − from the target to the ideal solution *A* −. The distances are defined as Equation (11).

$$\mathcal{S}^{+} = \sqrt{\sum\_{j=1}^{n} \left(V\_{ij} - \boldsymbol{\nu}\_{j}^{+}\right)^{2}}, \mathcal{S}^{-} = \sqrt{\sum\_{j=1}^{n} \left(V\_{ij} - \boldsymbol{\nu}\_{j}^{-}\right)^{2}}, i = 1, 2, \dots, m. \tag{11}$$

(6) Calculate the closeness index of the ideal solution. It is defined as Equation (12).

$$\mathbf{C}\_{i}^{+} = \frac{\mathbf{S}\_{i}^{-}}{\left(\mathbf{S}\_{i}^{+} + \mathbf{S}\_{i}^{-}\right)}, i = 1, 2, \dots, m. \tag{12}$$

(7) Ranking according to the size of the ideal pasting progress.

#### *2.3. Distance Measures and Similarity Measures for PFS*

Distance measure for PFSs is a term that describes the difference between PFS. Let *A* and *B* be PFSs of *X* = {*x*1, *x*2, . . . *x*n} with three parameters *µ*(*x*), *ν*(*x*) and *π*(*x*). Here, some distance measures (DM) are presented for PFSs.

The normalized Hamming distance is defined as Equation (13).

$$d\_{\rm FFS}(A, \mathcal{B})\_{nH} = \frac{1}{2\pi} \sum\_{i=1}^{n} \left( |\left|\mu\_A^2(\mathbf{x}\_i) - \mu\_B^2(\mathbf{x}\_i)\right| + |\left|\nu\_A\right|^2(\mathbf{x}\_i) - \nu\_B^2(\mathbf{x}\_i)\left| + |\left|\pi\_A\right|^2(\mathbf{x}\_i) - \pi\_B^2(\mathbf{x}\_i)\right| \right) \tag{13}$$

The normalized Euclidean distance is defined as Equation (14).

$$d\_{\rm FFS}(A,B)\_{n\to} = \sqrt{\frac{1}{2n} \sum\_{i=1}^{n} \left( \left( \mu\_A 2 \left( \mathbf{x}\_i \right) - \mu\_B 2 \left( \mathbf{x}\_i \right) \right)^2 + \left( \nu\_A 2 \left( \mathbf{x}\_i \right) - \nu\_B 2 \left( \mathbf{x}\_i \right) \right)^2 + \left( \pi\_A 2 \left( \mathbf{x}\_i \right) - \pi\_B 2 \left( \mathbf{x}\_i \right) \right)^2 \right)} \tag{14}$$

The normalized Hausdorff distance is defined as Equation (15).

$$d\_{\rm PFS}(A,B)\_{nHd} = \frac{1}{n} \sum\_{i=1}^{n} \max\left[ |\left.\mu\_A\right\|^2(\mathbf{x}\_i) - \left.\mu\_B\right\|^2(\mathbf{x}\_i)\left|\left.\nu\_A\right\|^2(\mathbf{x}\_i) - \left.\nu\_B\right\|^2(\mathbf{x}\_i)\right|\right] \tag{15}$$

For convenience, the above is called formula *d*1, *d*<sup>2</sup> and *d*3, and *d*<sup>4</sup> is defined as Equation (16).

$$d\_4(A, B) = \frac{1}{n} \sum\_{i=1}^n \frac{|\mu\_A^2(\mathbf{x}\_i) - \mu\_B^2(\mathbf{x}\_i)| + |\nu\_A^2(\mathbf{x}\_i) - \nu\_B^2(\mathbf{x}\_i)|}{\mu\_A^2(\mathbf{x}\_i) + \mu\_B^2(\mathbf{x}\_i) + \nu\_A^2(\mathbf{x}\_i) + \nu\_B^2(\mathbf{x}\_i)} \tag{16}$$

Like measure distance, similarity distance is also an important parameter between fuzzy sets. Let *f* be a monotone decreasing function. Then, the similarity measure between PFSs *A* and *B* can be defined as Equation (17).

$$s(A,B) = \frac{f(d(A,B)) - f(1)}{f(0) - f(1)}\tag{17}$$

By defining *f*, different similarity measures are obtained. Here, some simple methods are introduced. When *f*(*x*) = 1 − *x*, the similarity measure is defined as Equation (18).

$$s(A, B) = 1 - d(A, B) \tag{18}$$

When *f*(*x*) = 1/(1 + *x*), the similarity measure is defined as Equation (19).

$$s(A,B) = \frac{1 - d(A,B)}{1 + d(A,B)}\tag{19}$$

#### *2.4. PFS–TOPSIS Method for MADM*

The Pythagorean fuzzy set has a broader value space than the traditional fuzzy set and can represent uncertain information in more detail. In addition, with better adaptability, the combination with other MADM methods has achieved many successful cases. TOPSIS is a classic evaluation or ranking method. Based on this, this section introduced a TOPSIS method based on the PFS. The detailed procedure is presented in the following:

Step 1: First, determine alternatives, criteria and experts, and also determine the corresponding transformation relationship between natural language and fuzzy numbers.

Step 2: Establish a group decision matrix scored by experts R = (*x*(*k*) *ij*) *l*×*m* , which can be defined as Equation (20).

$$\mathbf{x}\_{ij}^{(k)} = \left\{ \left[ \boldsymbol{\mu}\_{A\_i} \left( \mathbf{C}\_j \right)^1 , \boldsymbol{\nu}\_{A\_i} \left( \mathbf{C}\_j \right)^1 , \boldsymbol{\pi}\_{A\_i} \left( \mathbf{C}\_j \right)^1 \right] \dots \left[ \boldsymbol{\mu}\_{A\_i} \left( \mathbf{C}\_j \right)^k , \boldsymbol{\nu}\_{A\_i} \left( \mathbf{C}\_j \right)^k , \boldsymbol{\pi}\_{A\_i} \left( \mathbf{C}\_j \right)^k \right] \dots \left[ \boldsymbol{\mu}\_{A\_i} \left( \mathbf{C}\_j \right)^u , \boldsymbol{\nu}\_{A\_i} \left( \mathbf{C}\_j \right)^u , \boldsymbol{\pi}\_{A\_i} \left( \mathbf{C}\_j \right)^u \right] \right\}. \tag{20}$$

This represents PFS formed by *n* experts' evaluation of a certain index of a certain scheme. For convenience, (*µA<sup>i</sup>* (*C<sup>j</sup>* ) *k* , *νA<sup>i</sup>* (*C<sup>j</sup>* ) *k* , *πA<sup>i</sup>* (*C<sup>j</sup>* ) *k* ) is represented by (*µ k ij*, *ν k ij*, *π k ij*). Therefore, the group decision matrix is obtained as Equation (21).

$$\begin{array}{c} \mathcal{C}\_{1} \\ \mathcal{C}\_{2} = \mathcal{A}\_{2} \\ \vdots \\ \mathcal{A}\_{l} \end{array} \left[ \begin{array}{ccccccccc} \mathcal{C}\_{2} & \cdots & \mathcal{C}\_{m} \\ \left( \left( \mu\_{11}^{1}, \nu\_{11}^{1}, \pi\_{11}^{1} \right) \cdots \left( \mu\_{1l}^{n}, \nu\_{1l}^{n}, \pi\_{1l}^{n} \right) \right) & \left( \left( \mu\_{12}^{1}, \nu\_{12}^{1}, \pi\_{22}^{1} \right) \right) & \cdots \left( \left( \mu\_{1n}^{1}, \nu\_{1n}^{n}, \pi\_{1n}^{n} \right) \right) \\ \left( \left( \mu\_{21}^{1}, \nu\_{21}^{1}, \pi\_{21}^{1} \right) \cdots \left( \mu\_{2l}^{n}, \nu\_{2l}^{n}, \pi\_{2l}^{n} \right) \right) & \left( \left( \mu\_{22}^{1}, \nu\_{22}^{1}, \pi\_{22}^{1} \right) \right) & \cdots \left( \left( \mu\_{2n}^{1}, \nu\_{2n}^{1}, \pi\_{2n}^{1} \right) \right) & \cdots \left( \left( \mu\_{2n}^{1}, \nu\_{2n}^{1}, \pi\_{2n}^{1} \right) \right) \\ \vdots & \vdots & \ddots & \vdots \\ \left( \left( \mu\_{l1}^{1}, \nu\_{l1}^{1}, \pi\_{l1}^{1} \right) \cdots \left( \mu\_{l1}^{n}, \nu\_{l1}^{n}, \pi\_{2l}^{n} \right) \right) & \left( \left( \mu\_{l2}^{1}, \nu\_{2$$

Step 3: Since the knowledge level, experience and focus of each expert are different, the importance of each expert is different, so the weight *σ<sup>k</sup>* of each expert should be determined according to certain standards. At the same time, for the evaluation of the same indicator of the same scheme, the individual opinions of all experts need to be aggregated into a general evaluation view, i.e., transforming a PFS *x*(*k*)*ij* into a Pythagorean fuzzy number *xij* = (*µA<sup>i</sup>* (*C<sup>j</sup>* ), *νA<sup>i</sup>* (*C<sup>j</sup>* ), *πA<sup>i</sup>* (*C<sup>j</sup>* )). For convenience, this is expressed as *xij* = (*µij νij πij*); this transformation process is realized through the Python fuzzy aggregated averaging (PFWA) operator, which can be defined as Equation (22).

$$\begin{array}{rcl} \mathbf{x}\_{ij} &=& \text{PFWA}\_{\mathcal{E}}\left(\mathbf{x}\_{ij}^{1}, \mathbf{x}\_{ij}^{2}, \dots, \mathbf{x}\_{ij}^{k}, \dots, \mathbf{x}\_{ij}^{n}\right) \\ &=& \sigma\_{1}\mathbf{x}\_{ij}^{1} \oplus \sigma\_{2}\mathbf{x}\_{ij}^{2} \oplus \dots \oplus \sigma\_{k}\mathbf{x}\_{ij}^{k} \oplus \dots \oplus \sigma\_{n}\mathbf{x}\_{ij}^{n} \\ &=& \left(\sqrt{1 - \prod\_{k=1}^{n} \left(1 - \left(\mu\_{ij}^{k}\right)^{2}\right)^{\sigma\_{k}}}, \prod\_{k=1}^{n} \left(\boldsymbol{\nu}\_{ij}^{k}\right)^{\sigma\_{k}}, \\ & \quad \sqrt{\prod\_{k=1}^{n} \left(1 - \left(\mu\_{ij}^{k}\right)^{2}\right)^{\sigma\_{k}} - \left(\prod\_{k=1}^{n} \left(\boldsymbol{\nu}\_{ij}^{k}\right)^{\sigma\_{k}}\right)^{2}}. \end{array} \tag{22}$$

At the same time, the aggregation evaluation matrix can be obtained as Equation (23).

$$\begin{array}{c} \mathcal{C}\_{1} \qquad \mathcal{C}\_{2} \qquad \cdots \qquad \mathcal{C}\_{m} \\ A\_{1} = A\_{2} \\ \vdots \\ A\_{l} \end{array} \quad \begin{bmatrix} (\mu\_{11}, \nu\_{11}, \pi\_{11}) & (\mu\_{12} \cdot \nu\_{12}, \pi\_{12}) & \cdots & (\mu\_{1m}, \nu\_{1m}, \pi\_{1m}) \\ (\mu\_{21}, \nu\_{21}, \pi\_{21}) & (\mu\_{22}, \nu\_{22}, \pi\_{22}) & \cdots & (\mu\_{2m}, \nu\_{2m}, \pi\_{2m}) \\ \vdots & \vdots & \ddots & \vdots \\ (\mu\_{l1}, \nu\_{l1}, \pi\_{l1}) & (\mu\_{l2}, \nu\_{l2}, \pi\_{l2}) & \cdots & (\mu\_{lm}, \nu\_{lm}, \pi\_{lm}) \end{array} \tag{23}$$

Step 4: All criteria may not have equal importance, so it is necessary to assign weight to indicators. This step is also determined by experts. Let *w k <sup>j</sup>* = [*µ k j ν k j π k j* ] be a PFN, which is used to indicate the evaluation and scoring of the *j* indicator by the *k* expert. Different experts' evaluation and scoring of an indicator also need to be aggregated into a Pythagorean fuzzy number *w<sup>j</sup>* = (*µ<sup>j</sup> ν<sup>j</sup> π<sup>j</sup>* ). This process is also implemented through the Python fuzzy aggregated averaging (PFWA) operator as in Equation (24).

$$\begin{array}{rcl} w\_{j} &=& \operatorname{PFWA}\_{\sigma} \Big( w\_{j}^{1}, w\_{j}^{2}, \dots, w\_{j}^{k}, \dots, w\_{j}^{n} \Big) \\ &=& \sigma\_{1} w\_{j}^{1} \oplus \sigma\_{2} w\_{j}^{2} \oplus \dots \oplus \sigma\_{k} w\_{j}^{k} \oplus \dots \oplus \sigma\_{n} w\_{j}^{n} \\ &=& \left( \sqrt{1 - \prod\_{k=1}^{n} \left( 1 - \left( \mu\_{j}^{k} \right)^{2} \right)}, \prod\_{k=1}^{n} \left( \nu\_{j}^{k} \right)^{\sigma\_{k}} \right) \\ &=& \sqrt{\prod\_{k=1}^{n} \left( 1 - \left( \mu\_{j}^{k} \right)^{2} \right)^{\sigma\_{k}} - \left( \prod\_{k=1}^{n} \left( \nu\_{j}^{k} \right)^{\sigma\_{k}} \right)^{2}} \right) \end{array} \tag{24}$$

At the same time, construct a weight matrix for all indicators *W*, *W*= (*w<sup>1</sup> w<sup>2</sup>* . . . *wm*). Step 5: After the aggregation evaluation matrix and index weight matrix are obtained, the weighted aggregation Python fuzzy decision matrix (PFDM) can be obtained through the multiple operator *RWA* = (*xWij*)*l*×*m*, where *xWij* = (*µWij*, *νWij*, *πWij*), the multiplication operator, is as in Equation (25).

$$\begin{array}{rcl} \mathbf{x}\_{\text{Wij}} &=& \mathbf{x}\_{ij} \otimes \mathbf{w}\_{j} \\ &=& \left(\mu\_{ij} \cdot \mu\_{j\prime} \sqrt{\left(\upsilon\_{ij}\right)^{2} + \left(\upsilon\_{j}\right)^{2} - \left(\upsilon\_{ij}\right)^{2} \cdot \left(\upsilon\_{j}\right)^{2}}\right) \\ & \sqrt{1 - \left(\mu\_{ij} \cdot \mu\_{j}\right)^{2} - \left(\upsilon\_{ij}\right)^{2} - \left(\upsilon\_{j}\right)^{2} + \left(\upsilon\_{ij}\right)^{2} \cdot \left(\upsilon\_{j}\right)^{2}}} \\ \end{array} \tag{25}$$

The weighted aggregated PFDM can be constructed as in Equation (26).

$$\begin{array}{c} \mathsf{C}\_{1} \\ \mathsf{R}\_{wA} = A\_{2} \\ \vdots \\ \mathsf{A}\_{I} \end{array} \begin{bmatrix} (\mu\_{\mathrm{W11}}, \nu\_{\mathrm{W11}}, \pi\_{\mathrm{W11}}) & (\mu\_{\mathrm{W12}}, \nu\_{\mathrm{W12}}, \pi\_{\mathrm{W12}}) & \cdots & (\mu\_{\mathrm{W1m}}, \nu\_{\mathrm{W1m}}, \pi\_{\mathrm{W1m}}) \\ (\mu\_{\mathrm{W21}}, \nu\_{\mathrm{W21}}, \pi\_{\mathrm{W21}}) & (\mu\_{\mathrm{W22}}, \nu\_{\mathrm{W22}}, \pi\_{\mathrm{W22}}) & \cdots & (\mu\_{\mathrm{W2m}}, \nu\_{\mathrm{W2m}}, \pi\_{\mathrm{W2m}}) \\ \vdots & \vdots & \ddots & \vdots \\ (\mu\_{\mathrm{W11}}, \nu\_{\mathrm{W11}}, \pi\_{\mathrm{W11}}) & (\mu\_{\mathrm{W12}}, \nu\_{\mathrm{W22}}, \pi\_{\mathrm{W12}}) & \cdots & (\mu\_{\mathrm{W1m}}, \nu\_{\mathrm{W1m}}, \pi\_{\mathrm{W1m}}) \end{bmatrix} \tag{26}$$

Step 6: Let *J*<sup>1</sup> and *J*<sup>2</sup> be the collection of benefit-type criteria and cost-type criteria. The Pythagorean fuzzy positive ideal solution (PFPIS) *A* <sup>+</sup> and the Pythagorean fuzzy negative ideal solution (PFNIS) *A* − are as in Equations (27)–(30).

$$A^{+} = \{ \langle \mathbb{C}\_{j'} \mu^{+}\_{\text{Wi}j'} \nu^{+}\_{\text{Wi}j} \rangle | \mathbb{C}\_{j} \in \mathbb{C}, j = 1, 2, \dots, m \}, \tag{27}$$

$$A^{-} = \{ \langle \mathcal{C}\_{j\prime} \mu\_{\text{Wij}}^{-} \nu\_{\text{Wij}}^{-} \rangle | \mathcal{C}\_{j} \in \mathbb{C}, j = 1, 2, \cdots, m \ \}, \tag{28}$$

$$\mu\_{\mathit{Wij}}^{+} = \left\{ \begin{array}{l} \max \mu\_{\mathit{Wij}} \text{ if } \mathsf{C}\_{j} \in I\_{1} \\ \min\_{1 \le i \le l} \mu\_{\mathit{Wij}} \text{ if } \mathsf{C}\_{j} \in I\_{2} \end{array} \right. \\ \left. \begin{array}{l} \min \nu\_{\mathit{Wij}} \text{ if } \mathsf{C}\_{j} \in I\_{1} \\ \max \nu\_{\mathit{Wij}} \text{ if } \mathsf{C}\_{j} \in I\_{2} \end{array} \right. \\ \left. \begin{array}{l} \min \nu\_{\mathit{Wij}} \text{ if } \mathsf{C}\_{j} \in I\_{1} \\ \max \nu\_{\mathit{Wij}} \text{ if } \mathsf{C}\_{j} \in I\_{2} \end{array} \right. \tag{29}$$

$$\mu\_{\mathit{Wij}}^{-} = \begin{cases} \min\_{1 \le i \le l} \mu\_{\mathit{Wij}} \text{ if } \mathsf{C}\_{j} \in \mathsf{J}\_{1} \\ \max\_{1 \le i \le l} \mu\_{\mathit{Wij}} \text{ if } \mathsf{C}\_{j} \in \mathsf{J}\_{2} \end{cases} \\ \text{\textit{\textit{\textit{\tiny}}}} \\ \text{\textit{\textit{\tiny}}}^{-} \text{\textit{\textit{\tiny}}} \text{ if } \mathsf{C}\_{j} \in \mathsf{J}\_{2} \text{ \textit{\tiny}}^{-} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\}}^{\text{\tiny}} \text{\textit{\tiny}}^{\text{\tiny}} \text{\textit{\}}^{\text{\tiny}} \text{\textit{\}}^{\text{\tiny}} \text{\textit{\}}^{\text{\tiny}$$

Step 7: After obtaining PFPIS and PFNIS, the next step is to calculate the distance between each scheme and the optimal solution *D* (*A<sup>i</sup>* , *A* + ) and the worst solution *D* (*A<sup>i</sup>* , *A* −). The normalized hamming distance formula is used. Then, the proximity between the alternatives and PFPIS is obtained and the calculation formula is as in Equation (31).

$$\mathcal{C}(A\_i) = \frac{D(A\_{i\prime}A^-)}{D(A\_{i\prime}A^+) + D(A\_{i\prime}A^+)}\tag{31}$$

Step 8: According to the calculated closeness, rank each alternative from high to low and select the best one.

#### **3. Results and Application**

#### *3.1. Background of Suichang Gold Mine*

Suichang Gold Mine, the largest state-owned gold mining enterprise in Zhejiang Province (see Figure 4), the backbone of national gold system production and the first member of the Shanghai Gold Exchange, won the honorary titles of "National Green Mine", "National Excellent Mining Enterprise for Saving and Comprehensive Utilization of Mine Resources" and "National 4A Tourist Attraction" (see Figure 5). The mining rights area of Suichang Gold Mine is 2.3729 km<sup>2</sup> and the design production scale is 91,800 t/a. There are two gold and silver ore bodies in the main mining area, which are distributed in layers and veins, with an obvious branching compound phenomenon. The ore veins are 27~190 m long, with occurrence elevation of 125~317 m, dip angle of 35~85◦ , average thickness of 1~4 m and average grade of Au 15 g/t and Ag 400 g/t. The surrounding rock of the roof of the orebody in the middle section is relatively stable, while the roof of the orebody in the west section is controlled by the compressive torsional fracture, the surrounding rock is relatively broken and the joints are developed, often resulting in the collapse of the surrounding rock of the roof in the goaf [33].

At present, the mine faces the following three main technical problems.


lapse of the surrounding rock of the roof in the goaf [33].

and select the best one.

**3. Results and Application** 

*3.1. Background of Suichang Gold Mine* 

**Figure 4.** Plan of the Suichang gold mine (accessed on 10 July 2022. https://www.fengyunditu.com/?ver=bd-wx-1604). **Figure 4.** Plan of the Suichang gold mine (accessed on 10 July 2022. https://www.fengyunditu.com/ ?ver=bd-wx-1604). *Sustainability* **2023**, *15*, x FOR PEER REVIEW 10 of 22

Step7: After obtaining PFPIS and PFNIS, the next step is to calculate the distance between each scheme and the optimal solution *D* (*Ai*, *A*+) and the worst solution *D* (*Ai*, *A*−). The normalized hamming distance formula is used. Then, the proximity between the alternatives and PFPIS is obtained and the calculation formula is as in Equation (31).

Step8: According to the calculated closeness, rank each alternative from high to low

Suichang Gold Mine, the largest state-owned gold mining enterprise in Zhejiang Province (see Figure 4), the backbone of national gold system production and the first member of the Shanghai Gold Exchange, won the honorary titles of "National Green Mine", "National Excellent Mining Enterprise for Saving and Comprehensive Utilization of Mine Resources" and "National 4A Tourist Attraction" (see Figure 5). The mining rights area of Suichang Gold Mine is 2.3729 km2 and the design production scale is 91,800 t/a. There are two gold and silver ore bodies in the main mining area, which are distributed in layers and veins, with an obvious branching compound phenomenon. The ore veins are 27~190 m long, with occurrence elevation of 125~317 m, dip angle of 35°~85°, average thickness of 1~4 m and average grade of Au 15 g/t and Ag 400 g/t. The surrounding rock of the roof of the orebody in the middle section is relatively stable, while the roof of the orebody in the west section is controlled by the compressive torsional fracture, the surrounding rock is relatively broken and the joints are developed, often resulting in the col-

(31)

**Figure 5.** Construction of green mine at SuiChang gold mine: (**a**) exterior view of SuiChang National Mine Park; (**b**) the gold grottoes of the Dang Dynasty; (**c**) the gold grottoes of the Song Dynasty; (**d**) mining disaster size from the Ming Dynasty [29]. (cc by-sa 4.0). large production capacity (see Figure 6). **Figure 5.** Construction of green mine at SuiChang gold mine: (**a**) exterior view of SuiChang National Mine Park; (**b**) the gold grottoes of the Dang Dynasty; (**c**) the gold grottoes of the Song Dynasty; (**d**) mining disaster size from the Ming Dynasty [29]. (cc by-sa 4.0).

#### At present, the mine faces the following three main technical problems. *3.2. Primary Selection of Mining Method*

branch seriously.

ciency and high cost.

*3.2. Primary Selection of Mining Method* 

and shrinkage filling method (SFM).

(1) The stability of ore and rock in the altered zone is poor and mining technology is difficult. The endowment characteristics of altered rock type gold deposits are complex, the occurrence, grade and dip angle vary greatly and the ore veins intersect and For the further development of the enterprise, Suichang Gold Mine decided to upgrade and reconstruct the mine, optimize mining methods, improve supporting facilities and

(2) The shrinkage method is not applicable to ore bodies with complex resource endow-

(3) The technology of replacing ore pillar with concrete is complex, with low labor effi-

For the further development of the enterprise, Suichang Gold Mine decided to upgrade and reconstruct the mine, optimize mining methods, improve supporting facilities and working environment, and increase production capacity. After the preliminary analysis of technical conditions, investigation of engineering rock mechanics and investigation of mineability, four mining methods were preliminarily selected four mining methods. These are the mechanized upward horizontal layering method (MUH), general upward horizontal layering method (GUH), upward horizontal approach filling method (UHA)

MUH adopts trackless mechanized equipment such as drilling jumbos and scrapers for production, which can realize strong mining, strong extraction and strong filling, with

equipment is low, and the labor intensity of workers is high.

working environment, and increase production capacity. After the preliminary analysis of technical conditions, investigation of engineering rock mechanics and investigation of mineability, four mining methods were preliminarily selected four mining methods. These are the mechanized upward horizontal layering method (MUH), general upward horizontal layering method (GUH), upward horizontal approach filling method (UHA) and shrinkage filling method (SFM).

MUH adopts trackless mechanized equipment such as drilling jumbos and scrapers for production, which can realize strong mining, strong extraction and strong filling, with large production capacity (see Figure 6). *Sustainability* **2023**, *15*, x FOR PEER REVIEW 11 of 22

**Figure 6.** Mining method diagram of MUH. **Figure 6.** Mining method diagram of MUH.

GUH faces problems of small production capacity, low mechanization, high labor intensity, large amount of preparation work, large amount of reserved space pillar and bottom pillar and low ore drawing efficiency of its two electric harrows (see Figure 7). GUH faces problems of small production capacity, low mechanization, high labor intensity, large amount of preparation work, large amount of reserved space pillar and bottom pillar and low ore drawing efficiency of its two electric harrows (see Figure 7).

UHA adopts trackless mechanized equipment such as the drilling jumbo and scraper, with large production capacity (see Figure 8).

SFM has low drilling efficiency, large amount of reserved space pillar and bottom pillar, difficulty of recovery, low recovery rate and large resource loss (see Figure 9).

UHA adopts trackless mechanized equipment such as the drilling jumbo and scraper,

**Figure 7.** Mining method diagram of GUH.

with large production capacity (see Figure 8).

GUH faces problems of small production capacity, low mechanization, high labor intensity, large amount of preparation work, large amount of reserved space pillar and bottom pillar and low ore drawing efficiency of its two electric harrows (see Figure 7).

**Figure 7.** Mining method diagram of GUH. **Figure 7.** Mining method diagram of GUH.

**Figure 6.** Mining method diagram of MUH.

**Figure 8.** Mining method diagram of UHA. **Figure 8.** Mining method diagram of UHA.

SFM has low drilling efficiency, large amount of reserved space pillar and bottom

pillar, difficulty of recovery, low recovery rate and large resource loss (see Figure 9).

**Figure 9.** Mining method diagram of SFM. **Figure 9.** Mining method diagram of SFM.

#### *3.3. Mining Method Optimization 3.3. Mining Method Optimization*

The specific steps of the PFS–TOPSIS model for mining method optimization are as follows: The specific steps of the PFS–TOPSIS model for mining method optimization are as follows:

Step 1: In the optimization of mining methods, we selected three parties as scoring experts, namely the designer (*E*1), the mining enterprise (*E*2) and the operator (*E*3). The four alternatives are MUHSM (*A*1), GUHLM (*A*2), UHAFM (*A*3) and SFM (*A*4). The indexes considered are ore recovery rate (*C*1), stope production capacity (*C*2), flexibility and adaptability (*C*3), stope safety conditions (*C*4), ore dilution rate (*C*5), mining and cutting quantities (*C*6), construction organization and labor intensity (*C*7) and comprehensive total cost (*C*8). Obviously, *C*1–*C*4 belongs to benefit index (*J*1), and *C*5–*C*8 belongs to cost index (*J*2). Next, the corresponding relationship between natural evaluation language and fuzzy number is defined. Table 1 defines the conversion criteria between the relative importance of indicators and PFN and Table 2 defines the conversion criteria between the relative superiority of the scheme and PFN. Step 1: In the optimization of mining methods, we selected three parties as scoring experts, namely the designer (*E*1), the mining enterprise (*E*2) and the operator (*E*3). The four alternatives are MUHSM (*A*1), GUHLM (*A*2), UHAFM (*A*3) and SFM (*A*4). The indexes considered are ore recovery rate (*C*1), stope production capacity (*C*2), flexibility and adaptability (*C*3), stope safety conditions (*C*4), ore dilution rate (*C*5), mining and cutting quantities (*C*6), construction organization and labor intensity (*C*7) and comprehensive total cost (*C*8). Obviously, *C*1–*C*<sup>4</sup> belongs to benefit index (*J*1), and *C*5–*C*<sup>8</sup> belongs to cost index (*J*2). Next, the corresponding relationship between natural evaluation language and fuzzy number is defined. Table 1 defines the conversion criteria between the relative importance of indicators and PFN and Table 2 defines the conversion criteria between the relative superiority of the scheme and PFN.

**Table 1.** The conversion criteria for the relative importance of indicators and PFN. Step 2: Three party experts will evaluate and score the superiority of the four schemes under each indicator, as shown in Table 3.

**Linguistic Variables PFNs Table 1.** The conversion criteria for the relative importance of indicators and PFN.




**Table 3.** Superiority evaluation result and fuzzy number.


Step 3: Determine the importance of all experts, that is, assign a certain weight. Muhammad Akram proposed a method to determine the weight according to the three elements of PFN and established a corresponding relationship between natural language variables and weight. For convenience, this section directly refers to *σ*<sup>1</sup> = 0.3252, *σ*<sup>2</sup> = 0.3754 and *σ*<sup>3</sup> = 0.2994. The weighted aggregation of experts' scores is conducted through PFWA and the aggregation evaluation matrix is obtained, as shown in Table 4.


**Table 4.** Aggregate evaluation matrix.

Step 4: Each expert will evaluate the superiority of various indicators, as shown in Table 5 for the indicator evaluation. The results will be weighted and aggregated into the indicator weight matrix as Equation (32).

$$W = \begin{bmatrix} (0.8695, 0.2258, 0.4393) \\ (0.8010, 0.3002, 0.5180) \\ (0.7140, 0.3496, 0.6066) \\ (0.9000, 0.2000, 0.3873) \\ (0.7709, 0.3545, 0.5292) \\ (0.8010, 0.3002, 0.5180) \\ (0.7838, 0.3137, 0.5360) \\ (0.7709, 0.3545, 0.5292) \end{bmatrix}^T \tag{32}$$


Step 5: After the aggregation evaluation matrix and index weight matrix are obtained, the weighted aggregation evaluation matrix can be obtained through the multiplication operator, as shown in Table 6.

**Table 6.** Weighted aggregate evaluation matrix.


Step 6: The Pythagorean fuzzy positive ideal solution (PFPIS) *A* <sup>+</sup> and the Pythagorean fuzzy negative ideal solution (PFNIS) *A* − are given as in Equations (33) and (34).


Step 7: Calculate the distance between each scheme and the best solution and the worst solution and rank the schemes by calculating the closeness index to obtain the best scheme, as shown in Table 7.


It can be seen from the above table that *A*<sup>1</sup> is the best choice, that is, the MUH is the best scheme along with mechanized mining (see Figure 10). *Sustainability* **2023**, *15*, x FOR PEER REVIEW 17 of 22

**Figure 10.** Mechanized mining in the Suichang gold mine: (**a**) filling material; (**b**) filling station; (**c**) resource survey; (**d**) mechanized mining [34]. (cc by-sa 4.0). **Figure 10.** Mechanized mining in the Suichang gold mine: (**a**) filling material; (**b**) filling station; (**c**) resource survey; (**d**) mechanized mining [34]. (cc by-sa 4.0).

#### **4. Discussion 4. Discussion**

The selection of mining method is very important and complex. In this paper, through a TOPSIS method based on PFS, the MUH is selected as the final scheme among the four mining methods suitable for Suichang Gold Mine. By combining the advantages The selection of mining method is very important and complex. In this paper, through a TOPSIS method based on PFS, the MUH is selected as the final scheme among the four mining methods suitable for Suichang Gold Mine. By combining the advantages of PFS

of PFS which can fully represent fuzzy information with the advantages of TOPSIS rank-

the results. In fact, as TOPSIS methods that rank according to the proximity of good and bad solutions, the core influencing factors are the distance between the scheme and the positive and negative ideal solutions, as well as the ranking method. As mentioned above, there are many methods to measure the distance between PFS and each method has its own advantages and disadvantages and the most suitable application [35–38]. In terms of sorting, Hadi-Vencheh [39] believes that the traditional closeness index rank may not be able to produce an optimal alternative, being close to the PIS and far from the NIS. Con-

However, due to the importance and particularity of mining method decision mak-

Mahanta [31] adopted a method of ranking by similarity in their article. It is defined

(35)

(36)

, (37)

sequently, they introduced the revised closeness index as Equation (35).

ing science, an ideal result is achieved.

in Equations (29) and (30).

C(Ai)

RC(Ai)

Sr(Ai)

which can fully represent fuzzy information with the advantages of TOPSIS ranking science, an ideal result is achieved. *D(A,A−)* 0.18968 0.07630 0.09560 0.09645 *d*2 *D(A,A+)* 0.05468 0.16732 0.13444 0.13656

*A***<sup>1</sup>** *A***<sup>2</sup>** *A***<sup>3</sup>** *A***<sup>4</sup>**

*D(A,A+)* 0.03517 0.15011 0.12696 0.12611

In view of this, the stability of the above results is analyzed through the four distance measures and three ranking methods mentioned above (see Figures 11–13), as shown in

*Sustainability* **2023**, *15*, x FOR PEER REVIEW 18 of 22

Tables 8 and 9, and the data in Figure 12 have been normalized.

**Table 8.** Results of distance measures.

*d*1

However, due to the importance and particularity of mining method decision making, the above models and calculations cannot fully guarantee the scientific accuracy of the results. In fact, as TOPSIS methods that rank according to the proximity of good and bad solutions, the core influencing factors are the distance between the scheme and the positive and negative ideal solutions, as well as the ranking method. As mentioned above, there are many methods to measure the distance between PFS and each method has its own advantages and disadvantages and the most suitable application [35–38]. In terms of sorting, Hadi-Vencheh [39] believes that the traditional closeness index rank may not be able to produce an optimal alternative, being close to the PIS and far from the NIS. Consequently, they introduced the revised closeness index as Equation (35). *D(A,A−)* 0.18260 0.09535 0.12303 0.11595 *d*3 *D(A,A+)* 0.03516 0.15013 0.12699 0.12600 *D(A,A−)* 0.18969 0.07632 0.09548 0.09646 *d*4 *D(A,A+)* 0.04340 0.19304 0.16600 0.16947 *D(A,A−)* 0.25637 0.10775 0.13392 0.13089 **Table 9.** Results of ranking methods. *A***<sup>1</sup>** *A***<sup>2</sup>** *A***<sup>3</sup>** *A***4 Rank**  *d*1 0.84359 0.33702 0.42955 0.43336 *A*1 > *A*4 > *A*3 > *A*<sup>2</sup>

$$\text{RC}(A\_i) = \frac{D(A\_{i\prime}A^-)}{D\_{\text{max}}(A\_{i\prime}A^-)} - \frac{D(A\_{i\prime}A^+)}{D\_{\text{min}}(A\_{i\prime}A^+)} \tag{35}$$

Mahanta [31] adopted a method of ranking by similarity in their article. It is defined in Equations (29) and (30). *d*1 0 −3.8661 −3.1061 −3.0775 *A*1 > *A*4 > *A*3 > *A*<sup>2</sup> *d*2 0 −2.5379 −1.7851 −1.8625 *A*1 > *A*3 > *A*4 > *A*<sup>2</sup>

$$S\_r = \frac{\mathcal{S}(D\_+^i)}{\mathcal{S}(D\_+^i) + \mathcal{S}(D\_-^i)}\tag{36}$$

$$D\_{+}^{j} = D(A\_{i\cdot}A^{+})\_{\prime}D\_{-}^{j} = D(A\_{i\cdot}A^{-})\_{\prime}S(D) = (1 - D)/(1 + D) \tag{37}$$

In view of this, the stability of the above results is analyzed through the four distance measures and three ranking methods mentioned above (see Figures 11–13), as shown in Tables 8 and 9, and the data in Figure 12 have been normalized. *d*2 0.56460 0.46343 0.49420 0.48953 *A*1 > *A*3 > *A*4 > *A*<sup>2</sup> *d*3 0.57778 0.46267 0.48406 0.48505 *A*1 > *A*4 > *A*3 > *A*<sup>2</sup> *d*4 0.60768 0.45645 0.48360 0.48027 *A*1 > *A*3 > *A*4 > *A*<sup>2</sup>

**Figure 11. Figure 11.** Closeness index rank based on different distance measures. Closeness index rank based on different distance measures.

**Figure 12.** Revised closeness index rank based on different distance measures. **Figure 12.** Revised closeness index rank based on different distance measures. **Figure 12.** Revised closeness index rank based on different distance measures.

**Figure 13.** Relative similarity values rank based on different distance measures. **Figure 13.** Relative similarity values rank based on different distance measures.

**Figure 13.** Relative similarity values rank based on different distance measures.



fuzzy information, e.g., the recovery rate, cut ratio and other factors have certain empirical values. Therefore, the perfect solution is to build a corresponding transformation relationship between the exact value and the PFS and to build a mining method optimization

ship between the exact value and the PFS and to build a mining method optimization

model that combines fuzzy information with accurate data.

model that combines fuzzy information with accurate data.


**Table 9.** Results of ranking methods.

It can be seen from the analysis results that the best scheme obtained by changing the distance measures and ranking methods is still the MUH, so it can be considered that the results are accurate. In fact, in the actual production of Suichang Gold Mine, the application of this method has also achieved the ideal results of safety, efficiency and environmental protection.

Whether it can be considered that the model is universal and can be generalized in mining method optimization, the answer is obviously unknown. As is known, the factors that need to be considered in the optimization of mining methods are not completely fuzzy information, e.g., the recovery rate, cut ratio and other factors have certain empirical values. Therefore, the perfect solution is to build a corresponding transformation relationship between the exact value and the PFS and to build a mining method optimization model that combines fuzzy information with accurate data.

#### **5. Conclusions**


Of course, there are still limitations to the mathematical approach used in this study, i.e., indicators with definite values need to be evaluated first in natural language and then converted to fuzzy sets. Even so, the method still performs well in problems with a large amount of fuzzy information, such as the selection of mining methods. Therefore, we

advocate the application of such methods to more mines and encourage more and more researchers to test and optimize them in practice.

**Author Contributions:** Conceptualization, S.L.; resources, S.L.; writing—original draft preparation, Q.H.; writing—review and editing, Q.H., B.H. and J.P.; visualization, Q.H., B.H. and J.P.; supervision, J.C., J.Y. and. X.Z.; project administration, X.W. and H.Y.; funding acquisition, S.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** Authors thank the financial support from the Natural Science Foundation of Hunan Province (Grant No. 2021JJ40745) and the National Natural Science Foundation of China (Grant No. 51804337).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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