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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods Introduction

#### 2.1. Introduction of PFS Method

**Definition**

**1.**

_{ξ}(x):X → [0, 1] and ν

_{ξ}(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

_{ξ}(x))

^{2}+ (ν

_{ξ}(x))

^{2}≤ 1,∀x∈X. π

_{ξ}(x) =$\sqrt{1-{\left({\mu}_{\xi}\left(x\right)\right)}^{2}-{\left({\nu}_{\xi}\left(x\right)\right)}^{2}}$is called the degree of indeterminacy of element x∈X. For convenience, they are called (μ

_{ξ}(x), ν

_{ξ}(x)) and a Pythagorean fuzzy number (PFN) denoted by ξ = (μ

_{ξ,}ν

_{ξ,}) [29].

**Definition**

**2.**

_{i}= (μ

_{ξ,}ν

_{ξ,}) (i = 1,2,……n) of the of PFNs with the weight vector w = (w

_{1}w

_{2}……w

_{n}) of ξ

_{i}(i = 1,2,……n) such that$\sum _{i=1}^{n}}{w}_{i}=1$, 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].

**Definition**

**3.**

_{1}, x

_{2}, … x

_{n}} [27]. Then, the sum of A and B is defined as Equation (4).

#### 2.2. Introduction of TOPSIS Method

- (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).

- (2)
- Normalize the characteristic matrix, obtain the normalized vector r
_{ij}and establish the normalized matrix about the normalized vector. This is defined as Equation (7).

- (3)
- Normalize the value v
_{ij}by calculating the weight; weight normalization matrix is defined as Equation (8).

- (4)
- Determine ideal solution A
^{+}and anti-ideal solution A^{−}; in the ideal solution and anti-ideal solution, J_{1}is the optimal value of profitability index set expressed on the i index; J_{2}is the worst value of the i index of the loss index set. A^{+}and A^{−}are defined as Equations (9) and (10).

- (5)
- Calculate the distance S
^{+}from the target to the ideal solution A^{+}and the distance S^{−}from the target to the ideal solution A^{−}. The distances are defined as Equation (11).

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

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

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

_{1}, x

_{2},…x

_{n}} with three parameters μ(x)

_{,}ν(x) and π(x). Here, some distance measures (DM) are presented for PFSs.

_{1}, d

_{2}and d

_{3}, and d

_{4}is defined as Equation (16).

#### 2.4. PFS–TOPSIS Method for MADM

_{Ai}(C

_{j})

^{k}, ν

_{Ai}(C

_{j})

^{k}, π

_{Ai}(C

_{j})

^{k}) is represented by (${\mu}_{ij}^{k}$, ${\nu}_{ij}^{k}$, ${\pi}_{ij}^{k}$). Therefore, the group decision matrix is obtained as Equation (21).

_{k}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 ${x}_{{\left(k\right)}_{ij}}$ into a Pythagorean fuzzy number x

_{ij}= (μ

_{Ai}(C

_{j}), ν

_{Ai}(C

_{j}), π

_{Ai}(C

_{j})). For convenience, this is expressed as x

_{ij}= (μ

_{ij}ν

_{ij}π

_{ij}); this transformation process is realized through the Python fuzzy aggregated averaging (PFWA) operator, which can be defined as Equation (22).

_{j}= (μ

_{j}ν

_{j}π

_{j}). This process is also implemented through the Python fuzzy aggregated averaging (PFWA) operator as in Equation (24).

_{1}w

_{2}…w

_{m}).

_{WA}= (x

_{Wij})

_{l×m}, where x

_{Wij}= (μ

_{Wij}, ν

_{Wij}, π

_{Wij}), the multiplication operator, is as in Equation (25).

_{1}and J

_{2}be the collection of benefit-type criteria and cost-type criteria. The Pythagorean fuzzy positive ideal solution (PFPIS) A

^{+}and the Pythagorean fuzzy negative ideal solution (PFNIS) A

^{−}are as in Equations (27)–(30).

_{i}, A

^{+}) and the worst solution D (A

_{i}, 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).

## 3. Results and Application

#### 3.1. Background of Suichang Gold Mine

^{2}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].

- (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 branch seriously.
- (2)
- The shrinkage method is not applicable to ore bodies with complex resource endowments such as large thickness changes and serious branching, the level of mechanized equipment is low, and the labor intensity of workers is high.
- (3)
- The technology of replacing ore pillar with concrete is complex, with low labor efficiency and high cost.

#### 3.2. Primary Selection of Mining Method

#### 3.3. Mining Method Optimization

_{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.

_{1}= 0.3252, σ

_{2}= 0.3754 and σ

_{3}= 0.2994. The weighted aggregation of experts’ scores is conducted through PFWA and the aggregation evaluation matrix is obtained, as shown in Table 4.

^{+}and the Pythagorean fuzzy negative ideal solution (PFNIS) A

^{−}are given as in Equations (33) and (34).

_{1}is the best choice, that is, the MUH is the best scheme along with mechanized mining (see Figure 10).

## 4. Discussion

## 5. Conclusions

- (1)
- Through the PFS–TOPSIS method, based on the selection of technical and economic mining methods, a comprehensive evaluation system with multiple factors and indicators was constructed and an accurate closeness index was obtained to optimize mining methods. This overcomes the uncertainty and unpredictability of the traditional optimization system and provides a reference for the mining of the difficult-to-mine complicated orebody.
- (2)
- Taking Suichang Gold Mine as an example, according to the PFSTOPSIS method, a weighted aggregation evaluation matrix was constructed, and the closeness index of the four mining methods were calculated to be 0.8436, 0.3370, 0.4296 and 0.4334, respectively. The MUH has the highest closeness index, so this method was the best scheme.
- (3)
- There were many ranging methods and ranking methods for PFS and only one method could not ensure the accuracy and scientific nature of the results. This paper mainly used the first ranging method, which was ranked by the traditional closeness index. Finally, it discussed the three methods of traditional closeness index, revised closeness index and relative similarity values for comprehensive ranking under the four distance measures. When using the first distance measure, the revised closeness index of the four mining methods was 0, −3.8661, −3.1061 and −3.0775, and the relative similarity values were 0.5777, 0.46267, 0.48413 and 0.48499. It was concluded that MUH was the best scheme, which not only verified the accuracy of the results, but also showed that PFS was applicable to the selection of mining methods.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ma, D.; Duan, H.; Zhang, J.; Liu, X.; Li, Z. Numerical simulation of water-silt inrush hazard of fault rock: A three-phase flow model. Rock Mech. Rock Eng.
**2022**, 55, 5163–5182. [Google Scholar] [CrossRef] - Ma, D.; Duan, H.; Zhang, J. Solid grain migration on hydraulic properties of fault rocks in underground mining tunnel: Radial seepage experiments and verification of permeability prediction. Tunn. Undergr. Space Technol.
**2022**, 126, 104525. [Google Scholar] [CrossRef] - Ma, D.; Duan, H.; Zhang, J.; Bai, H. A state-of-the-art review on rock seepage mechanism of water inrush disaster in coal mines. Int. J. Coal Sci. Technol.
**2022**, 9, 50. [Google Scholar] [CrossRef] - Mijalkovski, S.; Peltechki, D.; Despodov, Z.; Mirakovski, D.; Adjiski, V.; Doneva, N. Methodology for underground mining method selection. Min. Sci.
**2021**, 28, 201–216. [Google Scholar] [CrossRef] - Ojha, M.; Singh, K.P.; Chakraborty, P.; Verma, S. A review of multi-objective optimisation and decision making using evolutionary algorithms. Int. J. Bio-Inspir. Comput.
**2019**, 14, 69–84. [Google Scholar] [CrossRef] - Saaty, T.L. What Is the Analytic Hierarchy Process? In Mathematical Models for Decision Support; Springer: Berlin/Heidelberg, Germany, 1988; pp. 109–121. [Google Scholar]
- Jiskani, I.M.; Han, S.; Rehman, A.U.; Shahani, N.M. An Integrated Entropy Weight and Grey Clustering Method-Based Evaluation to Improve Safety in Mines. Min. Metall. Explor.
**2021**, 38, 1773–1787. [Google Scholar] [CrossRef] - Krishnan, A.R.; Kasim, M.M.; Hamid, R.; Ghazali, M.F. A Modified CRITIC Method to Estimate the Objective Weights of Decision Criteria. Symmetry
**2021**, 13, 973. [Google Scholar] [CrossRef] - Yang, W.G. Ingenious Solution for the Rank Reversal Problem of TOPSIS Method. Math. Probl. Eng.
**2020**, 2020, 9676518. [Google Scholar] [CrossRef] [Green Version] - Luo, D.; Wang, X. The multi-attribute grey target decision method for attribute value within three-parameter interval grey number. Appl. Math. Model.
**2012**, 36, 1957–1963. [Google Scholar] [CrossRef] - Omrani, H.; Alizadeh, A.; Naghizadeh, F. Incorporating decision makers’ preferences into DEA and common weight DEA models based on the best-worst method (BWM). Soft Comput.
**2020**, 24, 3989–4002. [Google Scholar] [CrossRef] - Yang, W.G.; Wu, Y.J. A New Improvement Method to Avoid Rank Reversal in VIKOR. IEEE Access
**2020**, 8, 21261–21271. [Google Scholar] [CrossRef] - Zhang, J.; Li, H.; Liu, Y.; Feng, X. The improvement and application of fuzzy comprehensive evaluation method under the hybrid information. In Proceedings of the 25th Chinese Control and Decision Conference (CCDC), Guiyang, China, 25–27 May 2013; p. 20131612-1615. [Google Scholar]
- Samanta, S.; Jana, D.K. A multi-item transportation problem with mode of transportation preference by MCDM method in interval type-2 fuzzy environment. Neural Comput. Appl.
**2019**, 31, 605–617. [Google Scholar] [CrossRef] - Bera, A.K.; Jana, D.K.; Banerjee, D.; Nandy, T. A two-phase multi-criteria fuzzy group decision making approach for supplierevaluation and order allocation considering multi-objective, multi-product and multi-period. Ann. Data Sci.
**2021**, 8, 577–601. [Google Scholar] [CrossRef] - Karimnia, H.; Bagloo, H. Optimum mining method selection using fuzzy analytical hierarchy process–Qapiliq salt mine, Iran. Int. J. Min. Sci. Technol.
**2015**, 25, 225–230. [Google Scholar] [CrossRef] - Yavuz, M. The application of the analytic hierarchy process (AHP) and Yager’s method in underground mining method selection problem. Int. J. Min. Reclam. Environ.
**2015**, 29, 453–475. [Google Scholar] [CrossRef] - Guo, Q.Q.; Yu, H.X.; Dan, Z.Y.; Li, S. Mining Method Optimization of Gently Inclined and Soft Broken Complex Orebody Based on AHP and TOPSIS: Taking Miao-Ling Gold Mine of China as an Example. Sustainability
**2021**, 13, 12503. [Google Scholar] [CrossRef] - Iphar, M.; Alpay, S. A mobile application based on multi-criteria decision-making methods for underground mining method selection. Int. J. Min. Reclam. Environ.
**2019**, 33, 480–504. [Google Scholar] [CrossRef] - Atanassov, K.; Vassilev, P. Intuitionistic fuzzy sets and other fuzzy sets extensions representable by them. J. Intel. Fuzzy Syst.
**2020**, 38, 525–530. [Google Scholar] [CrossRef] - Bajic, S.; Bajic, D.; Gluscevic, B.; Vakanjac, V.R. Application of Fuzzy Analytic Hierarchy Process to Underground Mining Method Selection. Symmetry
**2020**, 12, 192. [Google Scholar] [CrossRef] [Green Version] - Memari, A.; Dargi, A.; Jokar, M.; Ahmad, R.; Abdul Rahim, A.R. Sustainable supplier selection: A multi-criteria intuitionistic fuzzy TOPSIS method. J. Manuf. Syst.
**2019**, 50, 9–24. [Google Scholar] [CrossRef] - Narayanamoorthy, S.; Geetha, S.; Rakkiyappan, R.; Joo, Y.H. Interval-valued intuitionistic hesitant fuzzy entropy based VIKOR method for industrial robots selection. Expert Syst. Appl.
**2019**, 121, 28–37. [Google Scholar] [CrossRef] - Yager, R.R. Pythagorean Membership Grades in Multicriteria Decision Making. IEEE Trans. Fuzzy Syst.
**2014**, 22, 958–965. [Google Scholar] [CrossRef] - Peng, X.D.; Selvachandran, G. Pythagorean fuzzy set: State of the art and future directions. Artif. Intel. Rev.
**2019**, 52, 1873–1927. [Google Scholar] [CrossRef] - Akram, M.; Dudek, W.A.; Ilyas, F. Group decision-making based on pythagorean fuzzy TOPSIS method. Int. J. Intel. Syst.
**2019**, 34, 1455–1475. [Google Scholar] [CrossRef] - Yager, R.R. Pythagorean Fuzzy Subsets. In Proceedings of the Joint World Congress of the International-Fuzzy-Systems-Association (IFSA)/Annual Meeting of the North-American-Fuzzy-Information-Processing-Society (NAFIPS), Edmonton, AB, Canada, 24–28 June 2013; p. 201357-61. [Google Scholar]
- Yager, R.R.; Abbasov, A.M. Pythagorean Membership Grades, Complex Numbers, and Decision Making. Int. J. Intel. Syst.
**2013**, 28, 436–452. [Google Scholar] [CrossRef] - Zhang, X.L.; Xu, Z.S. Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets. Int. J. Intel. Syst.
**2014**, 29, 1061–1078. [Google Scholar] [CrossRef] - Zhang, X.L. A Novel Approach Based on Similarity Measure for Pythagorean Fuzzy Multiple Criteria Group Decision Making. Int. J. Intel. Syst.
**2016**, 31, 593–611. [Google Scholar] [CrossRef] - Hwang, C.L.; Yoon, K. Methods for Multiple Attribute Decision Making. In Multiple Attribute Decision Making; Springer: Berlin/Heidelberg, Germany, 1981; pp. 58–191. [Google Scholar]
- Wu, Z.; Li, Q.; Kong, D.; Chen, G.; Luo, D. The ANP-Fuzzy-TOPSIS model for the optimization of the scheme of large-section blasting. Arab. J. Geosci.
**2020**, 13, 1–9. [Google Scholar] [CrossRef] - Feasibility Study Report on Safe, Efficient and Low Lean Filling Mining of High Grade Gold and Silver Resources in Soft Rock Strata; Central South University: Changsha, China, 2021.
- Yu, H.X.; Li, S.; Yu, L.F.; Wang, X. The Recent Progress China Has Made in Green Mine Construction, Part II: Typical Examples of Green Mines. Int. J. Environ. Res. Public Health
**2022**, 19, 8166. [Google Scholar] [CrossRef] - Ejegwa, P.A.; Awolola, J.A. Novel distance measures for Pythagorean fuzzy sets with applications to pattern recognition problems. Granular Comput.
**2021**, 6, 181–189. [Google Scholar] [CrossRef] - Mahanta, J.; Panda, S. Distance measure for Pythagorean fuzzy sets with varied applications. Neural Comput. Appl.
**2021**, 33, 17161–17171. [Google Scholar] [CrossRef] - Ejegwa, P.A. Distance and similarity measures for Pythagorean fuzzy sets. Granular Comput.
**2020**, 5, 225–238. [Google Scholar] [CrossRef] - Hussian, Z.; Yang, M.S. Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS. Int. J. Intel. Syst.
**2019**, 34, 2633–2654. [Google Scholar] [CrossRef] - Hadi-Vencheh, A.; Mirjaberi, M. Fuzzy inferior ratio method for multiple attribute decision making problems. Inform. Sci.
**2014**, 277, 263–272. [Google Scholar] [CrossRef]

**Figure 4.**Plan of the Suichang gold mine (accessed on 10 July 2022. https://www.fengyunditu.com/?ver=bd-wx-1604).

**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).

**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).

Linguistic Variables | PFNs |
---|---|

Very important (VI) | (0.90, 0.20, 0.39) |

Important (I) | (0.75, 0.30, 0.59) |

Medium (M) | (0.60, 0.50, 0.62) |

Unimportant (U) | (0.45, 0.70, 0.55) |

Very unimportant (VU) | (0.20, 0.90, 0.39) |

Linguistic Variables | PFNs |
---|---|

Perfect (VI) | (1.00, 0.00, 0.00) |

Very very good (VVG) | (0.90, 0.20, 0.39) |

Very good (VG) | (0.80, 0.30, 0.52) |

Good (G) | (0.70, 0.35, 0.62) |

Medium (M) | (0.60, 0.50, 0.62) |

Medium bad (MB) | (0.50, 0.60, 0.62) |

Bad (B) | (0.40, 0.70, 0.59) |

Very bad (VB) | (0.25, 0.80, 0.55) |

Very very bad (VVB) | (0.10, 0.90, 0.42) |

Criteria | Alternatives | Expert | ||
---|---|---|---|---|

E_{1} | E_{2} | E_{3} | ||

C_{1} | A_{1} | VG (0.80, 0.30, 0.52) | VG (0.80, 0.30, 0.52) | VVG (0.90, 0.20, 0.39) |

A_{2} | G (0.70, 0.35, 0.62) | M (0.60, 0.50, 0.62) | G (0.70, 0.35, 0.62) | |

A_{3} | VVG (0.90, 0.20, 0.39) | VG (0.80, 0.30, 0.52) | VG (0.80, 0.30, 0.52) | |

A_{4} | M (0.60, 0.50, 0.62) | G (0.70, 0.35, 0.62) | M (0.60, 0.50, 0.62) | |

C_{2} | A_{1} | VVG (0.90, 0.20, 0.39) | VG (0.80, 0.30, 0.52) | VG (0.80, 0.30, 0.52) |

A_{2} | M (0.60, 0.50, 0.62) | G (0.70, 0.35, 0.62) | M (0.60, 0.50, 0.62) | |

A_{3} | G (0.70, 0.35, 0.62) | M (0.60, 0.50, 0.62) | M (0.60, 0.50, 0.62) | |

A_{4} | VG (0.80, 0.30, 0.52) | G (0.70, 0.35, 0.62) | VG (0.80, 0.30, 0.52) | |

C_{3} | A_{1} | G (0.70, 0.35, 0.62) | G (0.70, 0.35, 0.62) | VG (0.80, 0.30, 0.52) |

A_{2} | G (0.70, 0.35, 0.62) | M (0.60, 0.50, 0.62) | G (0.70, 0.35, 0.62) | |

A_{3} | VG (0.80, 0.30, 0.52) | VG (0.80, 0.30, 0.52) | G (0.70, 0.35, 0.62) | |

A_{4} | M (0.60, 0.50, 0.62) | M (0.60, 0.50, 0.62) | M (0.60, 0.50, 0.62) | |

C_{4} | A_{1} | VG (0.80, 0.30, 0.52) | G (0.70, 0.35, 0.62) | VG (0.80, 0.30, 0.52) |

A_{2} | VG (0.80, 0.30, 0.52) | G (0.70, 0.35, 0.62) | G (0.70, 0.35, 0.62) | |

A_{3} | G (0.70, 0.35, 0.62) | M (0.60, 0.50, 0.62) | M (0.60, 0.50, 0.62) | |

A_{4} | G (0.70, 0.35, 0.62) | M (0.60, 0.50, 0.62) | M (0.60, 0.50, 0.62) | |

C_{5} | A_{1} | VB (0.25, 0.80, 0.55) | VB (0.25, 0.80, 0.55) | VB (0.25, 0.80, 0.55) |

A_{2} | M (0.60, 0.50, 0.62) | MB (0.50, 0.60, 0.62) | M (0.60, 0.50, 0.62) | |

A_{3} | VB (0.25, 0.80, 0.55) | B (0.40, 0.70, 0.59) | VB (0.25, 0.80, 0.55) | |

A_{4} | B (0.40, 0.70, 0.59) | VB (0.25, 0.80, 0.55) | B (0.40, 0.70, 0.59) | |

C_{6} | A_{1} | B (0.40, 0.70, 0.59) | MB (0.50, 0.60, 0.62) | MB (0.50, 0.60, 0.62) |

A_{2} | B (0.40, 0.70, 0.59) | B (0.40, 0.70, 0.59) | B (0.40, 0.70, 0.59) | |

A_{3} | MB (0.50, 0.60, 0.62) | M (0.60, 0.50, 0.62) | MB (0.50, 0.60, 0.62) | |

A_{4} | B (0.40, 0.70, 0.59) | VB (0.25, 0.80, 0.55) | B (0.40, 0.70, 0.59) | |

C_{7} | A_{1} | B (0.40, 0.70, 0.59) | B (0.40, 0.70, 0.59) | VB (0.25, 0.80, 0.55) |

A_{2} | M (0.60, 0.50, 0.62) | G (0.70, 0.35, 0.62) | G (0.70, 0.35, 0.62) | |

A_{3} | MB (0.50, 0.60, 0.62) | MB (0.50, 0.60, 0.62) | M (0.60, 0.50, 0.62) | |

A_{4} | M (0.60, 0.50, 0.62) | M (0.60, 0.50, 0.62) | G (0.70, 0.35, 0.62) | |

C_{8} | A_{1} | MB (0.50, 0.60, 0.62) | M (0.60, 0.50, 0.62) | MB (0.50, 0.60, 0.62) |

A_{2} | B (0.40, 0.70, 0.59) | B (0.40, 0.70, 0.59) | MB (0.50, 0.60, 0.62) | |

A_{3} | M (0.60, 0.50, 0.62) | G (0.70, 0.35, 0.62) | M (0.60, 0.50, 0.62) | |

A_{4} | B (0.40, 0.70, 0.59) | MB (0.50, 0.60, 0.62) | MB (0.50, 0.60, 0.62) |

A_{1} | A_{2} | A_{3} | A_{4} | |
---|---|---|---|---|

C_{1} | (0.838, 0.266, 0.476) | (0.667, 0.400, 0.629) | (0.841, 0.263, 0.473) | (0.642, 0.437, 0.630) |

C_{2} | (0.841, 0.263, 0.473) | (0.642, 0.437, 0.630) | (0.637, 0.445, 0.629) | (0.768, 0.318, 0.556) |

C_{3} | (0.735, 0.334, 0.590) | (0.667, 0.400, 0.629) | (0.775, 0.314, 0.548) | (0.600, 0.500, 0.624) |

C_{4} | (0.768, 0.318, 0.556) | (0.738, 0.333, 0.587) | (0.637, 0.445, 0.629) | (0.637, 0.445, 0.629) |

C_{5} | (0.250, 0.800, 0.545) | (0.566, 0.535, 0.627) | (0.317, 0.761, 0.566) | (0.353, 0.736, 0.578) |

C_{6} | (0.471, 0.631, 0.616) | (0.400, 0.700, 0.591) | (0.542, 0.560, 0.627) | (0.353, 0.736, 0.578) |

C_{7} | (0.363, 0.729, 0.580) | (0.672, 0.393, 0.628) | (0.534, 0.568, 0.626) | (0.634, 0.449, 0.630) |

C_{8} | (0.542, 0.560, 0.627) | (0.434, 0.668, 0.604) | (0.642, 0.437, 0.630) | (0.471, 0.631, 0.616) |

Criteria | Experts | ||
---|---|---|---|

E_{1} | E_{2} | E_{3} | |

C_{1} | (0.90, 0.20, 0.39) | (0.90, 0.20, 0.39) | (0.75, 0.30, 0.59) |

C_{2} | (0.75, 0.30, 0.59) | (0.90, 0.20, 0.39) | (0.60, 0.50, 0.62) |

C_{3} | (0.75, 0.30, 0.59) | (0.75, 0.30, 0.59) | (0.60, 0.50, 0.62) |

C_{4} | (0.90, 0.20, 0.39) | (0.90, 0.20, 0.39) | (0.90, 0.20, 0.39) |

C_{5} | (0.60, 0.50, 0.62) | (0.90, 0.20, 0.39) | (0.60, 0.50, 0.62) |

C_{6} | (0.75, 0.30, 0.59) | (0.90, 0.20, 0.39) | (0.60, 0.50, 0.62) |

C_{7} | (0.60, 0.50, 0.62) | (0.75, 0.30, 0.59) | (0.90, 0.20, 0.39) |

C_{8} | (0.60, 0.50, 0.62) | (0.90, 0.20, 0.39) | (0.60, 0.50, 0.62) |

A_{1} | A_{2} | A_{3} | A_{4} | |
---|---|---|---|---|

C_{1} | (0.729, 0.344, 0.592) | (0.580, 0.450, 0.679) | (0.731, 0.342, 0.590) | (0.558, 0.482, 0.676) |

C_{2} | (0.674, 0.391, 0.627) | (0.514, 0.514, 0.687) | (0.510, 0.520, 0.685) | (0.615, 0.427, 0.663) |

C_{3} | (0.525, 0.469, 0.710) | (0.476, 0.513, 0.714) | (0.553, 0.457, 0.697) | (0.428, 0.585, 0.689) |

C_{4} | (0.691, 0.370, 0.621) | (0.664, 0.383, 0.642) | (0.573, 0.480, 0.664) | (0.573, 0.480, 0.664) |

C_{5} | (0.193, 0.828, 0.526) | (0.436, 0.613, 0.659) | (0.244, 0.795, 0.555) | (0.272, 0.774, 0.572) |

C_{6} | (0.377, 0.672, 0.637) | (0.320, 0.732, 0.601) | (0.434, 0.613, 0.660) | (0.283, 0.764, 0.580) |

C_{7} | (0.285, 0.760, 0.584) | (0.527, 0.488, 0.696) | (0.419, 0.624, 0.660) | (0.497, 0.529, 0.688) |

C_{8} | (0.418, 0.632, 0.653) | (0.335, 0.718, 0.610) | (0.495, 0.541, 0.680) | (0.363, 0.688, 0.628) |

Alternatives | D(A_{i}, A^{+}) | D(A_{i}, A^{−}) | C(A_{i}) | Ranks |
---|---|---|---|---|

A_{1} | 0.0352 | 0.1897 | 0.8436 | 1 |

A_{2} | 0.1501 | 0.0763 | 0.3370 | 4 |

A_{3} | 0.1270 | 0.0956 | 0.4296 | 3 |

A_{4} | 0.1261 | 0.0964 | 0.4334 | 2 |

A_{1} | A_{2} | A_{3} | A_{4} | ||
---|---|---|---|---|---|

d_{1} | D(A,A^{+}) | 0.03517 | 0.15011 | 0.12696 | 0.12611 |

D(A,A^{−}) | 0.18968 | 0.07630 | 0.09560 | 0.09645 | |

d_{2} | D(A,A^{+}) | 0.05468 | 0.16732 | 0.13444 | 0.13656 |

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 |

A_{1} | A_{2} | A_{3} | A_{4} | Rank | ||
---|---|---|---|---|---|---|

C(A_{i}) | d_{1} | 0.84359 | 0.33702 | 0.42955 | 0.43336 | A_{1} > A_{4} > A_{3} > A_{2} |

d_{2} | 0.76956 | 0.36300 | 0.47783 | 0.45919 | A_{1} > A_{3} > A_{4} > A_{2} | |

d_{3} | 0.84361 | 0.33703 | 0.42920 | 0.43361 | A_{1} > A_{4} > A_{3} > A_{2} | |

d_{4} | 0.85523 | 0.35822 | 0.44652 | 0.43577 | A_{1} > A_{3} > A_{4} > A_{2} | |

RC(A_{i}) | d_{1} | 0 | −3.8661 | −3.1061 | −3.0775 | A_{1} > A_{4} > A_{3} > A_{2} |

d_{2} | 0 | −2.5379 | −1.7851 | −1.8625 | A_{1} > A_{3} > A_{4} > A_{2} | |

d_{3} | 0 | −3.8669 | −3.1078 | −3.0747 | A_{1} > A_{4} > A_{3} > A_{2} | |

d_{4} | 0 | −4.0280 | −3.3029 | −3.3947 | A_{1} > A_{3} > A_{4} > A_{2} | |

S_{r}(A_{i}) | d_{1} | 0.57777 | 0.46267 | 0.48413 | 0.48499 | A_{1} > A_{4} > A_{3} > A_{2} |

d_{2} | 0.56460 | 0.46343 | 0.49420 | 0.48953 | A_{1} > A_{3} > A_{4} > A_{2} | |

d_{3} | 0.57778 | 0.46267 | 0.48406 | 0.48505 | A_{1} > A_{4} > A_{3} > A_{2} | |

d_{4} | 0.60768 | 0.45645 | 0.48360 | 0.48027 | A_{1} > A_{3} > A_{4} > A_{2} |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 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/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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(4):3692.
https://doi.org/10.3390/su15043692

**Chicago/Turabian Style**

Li, Shuai, Qi Huang, Boyi Hu, Jilong Pan, Junyu Chen, Jianguo Yang, Xinghui Zhou, Xinmin Wang, and Haoxuan Yu.
2023. "Mining Method Optimization of Difficult-to-Mine Complicated Orebody Using Pythagorean Fuzzy Sets and TOPSIS Method" *Sustainability* 15, no. 4: 3692.
https://doi.org/10.3390/su15043692