Application of the FUZZY TOPSIS method for selecting an underground mining method

Selecting the underground mining method is one of the most difficult decisions any mining engineer makes when designing a new mine or opening new parts in an existing mine. The underground mining method selection depends on many mining-geological, technical and economic factors. The process of selecting a mine excavation method is a multi-criteria decision-making process since many factors are considered when selecting a mining method. The selection of the most suitable mining method is of great importance for each mine, and this is especially evident in the phase of preparation and excavation of the ore deposit and optimization of the total cost of excavation. This paper uses the Fuzzy TOPSIS method to select the underground mining method for metallic mineral resources as one of the most important multi-criteria decision-making methods. The methodology presented will enable the application of other fuzzy methods to solve problems related to the mining method selection.


Introduction
When opening a new underground mine or opening new sections in an existing mine, it is very important to select the most appropriate mining method.The mining method has a major impact on the mine capital and operating costs, the safety and health conditions for employees, achieving the necessary rate of production, etc.
The mining method selection depends on many factors, which can be quantitative (can be measured or calculated) or qualitative (cannot be measured and defined by descriptive values; they need to be transformed into numerical values so that they can be used for calculation).The factors influencing the mining method selection can be divided into three groups (Hartman, 1992): − The mining-geological factors, such as the geometry of the deposit (general shape, ore thickness, dip, plunge, depth), rock quality (ore zone, hangingwall and footwall, i.e., rock strength, fracture spacing, fracture shear strength, rock quality designation, presence and strength of faults or other structures, in-situ stress), ore variability (ore boundaries, ore uniformity, continuity, grade distribution), quality of the resource, etc. − The mining-technical factors, such as the annual productivity, equipment used, health and safety of the workforce, environmental impact, ore dilution, mine recovery, the flexibility of the mining methods, the mining rate, and − The economic factors, such as the capital cost, operating cost, mineable ore tons, orebody grades and ore value.
There are cases where mining and geological factors allow the application of a particular mining method, but its application is not economically justified.There are also cases where a particular mining method allows the application of certain mechanization, but this is not allowed by mining and technical factors (Hartman, 1992).
All the factors influencing the mining method selection are not the same weightsome are permanent, and some may change.The main factors for the mining method selection are the natural conditions of the deposit and its surroundings, on the basis of which the mining method is selected, the stope is constructed, the mining technology is adopted, the work is organized according to the planned production capacity and analyzes mining cost.The mining method selection is made between several possible variants, sometimes diametrically opposed to each other.The final decision on the mining method selection is made on the basis of economic analysis; that is, the mining method that provides the fastest return on investment and the highest profit is chosen.
The methodologies for mining method selection can be divided into three groups: qualitative methods, numerical methods, and decision-making methods (Nourali et al., 2012).This paper discusses the application of the fuzzy TOPSIS method for mining method selection, which is one of the most important methods in the group of multi-criteria decision-making methods.
Until today, many scientists dealt with the question of underground mining method selection by using different multi-criteria decision-making methods (MCDM), such as the Analytic Hierarchy Process (AHP), the Elimination Et Choix Traduisant la Realite (ELECTRE), the Preference Ranking Organization Method for Enrichment of Evaluations (PROMETHEE), the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Visekriterijumsko Kompromisno Rangiranje (VIKOR), fuzzy logic and others, separately and together.The combination of MCDM methods and the fuzzy extensions of methods are often applied for the same purpose.Bitarafan et al. (2004) used multiple criteria decision-making tools for mining method selection.Shahriar et al. (2007) used a new numerical Shahriar and Bakhtavar (Sh&B) approach and the AHP.The method is a combined and modified system of Nicholas, Modified Nicholas and UBC for mining method selection.Alpay et al. (2007) used a decision support system and AHP for the selection of an underground mining method at the Eskisehir-Karaburun chromite mine.Namin et al. (2008) used fuzzy TOPSIS for mining method selection and examined the model for GEG anomaly No. 3 at the Chahar Gonbad mine.Ataei et al. (2008b) used the TOPSIS method with 13 criteria to develop a suitable mining method for Golbini No. 8 of the Jajarm bauxite mine in Iran.Also, Ataei et al. (2008a) used the AHP method to select a mining method for the same mine.Karadogan et al. (2008) used fuzzy set theory to select the underground mining method.Namin et al. (2009) used AHP, TOPSIS and PROMETHEE to solve the mining method selection problem.Jamshidi et al. (2009) used the AHP to select the optimal underground mining method in the Jajarm bauxite mine.Alpay et al. (2009) have suggested a combination of AHP and fuzzy logic methods for underground mining method selection.Naghadehi et al. (2009) used fuzzy AHP for mining method selection at the Jajarm Bauxite mine.Mikaeil et al. (2009) developed a decision support system using Fuzzy AHP, and TOPSIS approaches to select the optimum underground mining method.Azadeh et al. (2010) used fuzzy AHP for mining method selection by modifying the Nicholas technique for the Choghart iron mine.Liu et al. (2010) used an optimization model of unascertained measurement for underground mining method selection and its application.Gupta et al. (2012) developed an AHP model for underground mining method selection.Bogdanovic et al. (2012) used the PROMETHEE and AHP methods to choose an appropriate mining method in the Coka Marin mine in Serbia.Chamzini et al. (2012) used the integrated MCDM model for mining method selection in the presence of uncertainty (fuzzy AHP and fuzzy TOPSIS).Namin et al. (2012) used the hybrid fuzzy-based decision support system for MMS in order to estimate interrelationships between criteria.Mijalkovski et al. (2013) used AHP, PROMETHEE and AHP-PROMETHEE integrated methods for mining method selection for the Sasa mine in Macedonia.Shariati et al. (2013) used fuzzy AHP and TOPSIS for mining method selection for the Angouran mine in Iran.Ataei et al. (2013) proposed a Monte Carlo-based AHP (MAHP) technique for the mining method selection of a bauxite ore deposit in Iran.Gelvez et al. (2014) applied the AHP and VIKOR methods to select the optimum mining method in a coal mine in Colombia.Yavuz (2015) used AHP and Yager's method for the selection of an underground mining method for the Ciftalan lignite mine in Istanbul.Karimnia et al. (2015) used AHP to choose the better mining method at a salt mine in Iran.Chen et al. (2015) applied AHP and PROMETHEE methods to select the most suitable technique for mechanized mining in a thin coal mine in China.Mahase et al. (2016) gave a good overview of the MCDM methods applied in mine planning and similar cases.Javanshirgiv et al. (2017b) used fuzzy TOPSIS for mining method selection at the Kamar Mahdi fluorine mine in Iran.Dehghani et al. (2017) used a new model for mining method selection based on grey and TODIM methods.Balusa et al. (2018a) used fuzzy AHP for mining method selection at the Tummalapalle and Turamdih uranium mines in India.Chander et al. (2018) used AHP and VIKOR to select the optimal underground bauxite mining method.Balusa et al. (2018b) used AHP, WPM and PROMETHEE to determine the effective mining method for a bauxite mine.Liang et al. (2018) used the selection of optimal mining method with extended multi-objective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA) approach.Ooriad et al. (2018) used a novel model for mining method selection in a fuzzy environment at the Tazareh coal mine, Semnan province, Iran.Balusa et al. (2019a) used AHP, TOPSIS, VIKOR, ELECTRE, PROMETHEE, and WPM for mining method selection at the Tummalapalle uranium mine, India.Balusa et al. (2019b) analyze the sensitivity in decisionmaking, which results in selecting an appropriate underground metal mining method using the fuzzy-AHP (FAHP) model.Wang et al. (2019) used Monte Carlo AHP for the selection of the longwall mining method in thin coal seams.Popovic et al. (2019) used the underground mining method selection methodology based on the Extended Pivot Pairwise Relative Criteria Importance Assessment (PIPRECIA-E) and group decision-making.Bajic et al. (2020) used fuzzy AHP for mining method selection at the Borska Reka copper mine in Serbia.Mijalkovski et al. (2021) used the ELECTRE, PROMETHEE, AHP and AHP-PROMETHEE methods for mining method selection.
As discussed earlier, this paper will apply fuzzy multi-criteria decision-making (the fuzzy TOPSIS method) to the mining method selection.The mining method selection will be made between 4 mining methods or alternatives, and the comparison will be made according to 22 parameters or criteria.The main advantage of this paper will be the application of fuzzy multi-criteria decision-making methods in mining to solve complex problems, such as the mining method selection while taking into account a large number of influential parameters.

Fuzzy multi-criteria decision-making
Over the years, numerous fuzzy multi-criteria decision-making (FMCDM) methods have been proposed in the literature (Liao and Xu, 2017), which differ in subject areas, such as the type of questions asked the theoretical background, and the type of obtained results.A number of methods have been designed for a particular problem.Therefore, they are not applicable to other problems.Recently, a number of FMCDM methods have been introduced to select the best compromise options (Liao and Xu, 2017;Kore et al., 2017).The FMCDM approaches have been developed not only by the motivation received from various real-life problems that require the consideration of multiple criteria but also by the desire of practitioners to enhance decisionmaking techniques through recent developments occurred in computer technology, scientific computing, and mathematical optimization (Mardani et al., 2015).All methods aim to make the decision-making process better informed and more formalized.The multi-criteria decision-making (MCDM) approach falls into two categories: classical MCDM and FMCDM (Bashiri et al., 2011).In the FMCDM approach, alternatives are ranked and selected from among a set of feasible alternatives.FMCDM can be categorized as fuzzy multi-objective decision-making (FMODM) and fuzzy multi-attribute decision-making (FMADM) approach (Liou et al., 2012).

Fuzzy TOPSIS method
The TOPSIS method was first proposed by Hwang et al. (1981).The basic concept of this method is that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS).PIS minimizes the cost criteria and maximizes the benefit criteria, whereas the NIS minimizes the benefit criteria and maximizes the cost criteria.TOPSIS is an easy-toapply method with uncomplicated computations (Lashgari et al., 2011).In the classical TOPSIS method, the criteria weights and the ratings of alternatives are known precisely, and crisp values are used in the evaluation process.In the TOPSIS method, decision-makers' judgments are represented with crisp values.According to the problems associated with determining the precise preference rating for an alternative for the criteria under consideration, decision-makers are keen on using fuzzy numbers instead of precise numbers.For this reason, the fuzzy TOPSIS method is appropriate for solving real-world problems under a fuzzy environment (Alavai et al., 2011;Chamzini et al., 2012;Ebrahimabadi et al., 2013;Kacprzak, 2018;Kore et al., 2017;Nadaban et al., 2016).
The TOPSIS and fuzzy TOPSIS methods have been extensively applied to engineering and management fields over the last two decades, and there have been plenty of studies related to the TOPSIS method in the literature (Huang et al., 2020;Javanshirgiv et al., 2017a;Kumar et al. 2013;Lashgari et al., 2012;Sitorus et al., 2019).The fuzzy TOPSIS method is based on the fuzzy theory used for decisions hindered by uncertainty.The fuzzy theory is one of the modern techniques which can deal with the impreciseness of input data and domain knowledge by giving quick, simple and often sufficiently good approximations of the desired solutions.This theory is able to convert most incorrect and enigmatic concepts, variables and systems into a mathematical form and set the context for reasoning, deduction and decision-making in uncertain conditions.
In this paper, the fuzzy TOPSIS method is considered, which involves the following steps (Alavi et al., 2011;Javanshirgiv et al., 2017b;Kacprzak, 2018;Kore et al., 2017;Kusumawardani et al., 2015;Nadaban et al., 2016;Sun, 2010): Step 1. Identify the evaluation criteria and alternatives.We assume that we have a decision group with K members.The fuzzy rating of the k th decision maker about alternative Ai and criterion Cj is denoted: ~   = (   ;    ;    ) and the weight of criterion Cj is denoted ~   = ( 1  ;  2  ;  3  ).
Step 2. Choose the appropriate linguistic variable.
Triangular fuzzy numbers can be used to represent linguistic variables, which can be used for the importance weight of the criteria (Tab. 1) and the evaluation of alternatives with respect to each criterion (Tab.2).
A fuzzy multi-criteria group decision-making problem can be concisely expressed in a matrix format as follows: Where, x are linguistic variables that can be shown by triangular fuzzy numbers: x = (  ;   ;   ) The aggregated fuzzy rating x = (  ;   ;   ) of i th alternative and j th criterion is obtained as follows: Step 4. Establish a criteria-weighted matrix.
It cannot be assumed that each evaluation criterion is equally important because the criteria have various weights. w where w are linguistic variables that can be shown by fuzzy triangular numbers: w = ( 1 ;  2 ;  3 ).
The aggregated fuzzy weights w = ( 1 ;  2 ;  3 ) for the criterion Cj are calculated by these formulas: Step 5. Normalize the fuzzy decision matrix.
The normalized fuzzy-decision matrix denoted by R is shown as the following formula: Then, the normalization process can be performed by the following formula: ) and Step 6. Compute the weighted normalized fuzzy decision matrix.
The weighted normalized fuzzy decision matrix is shown as the following matrix Ṽ: where: ṽ = r • w Step 7. Compute the Fuzzy Positive Ideal Solution (FPIS) and Fuzzy Negative Ideal Solution (FNIS).
The FPIS indicates the preferable alternative, and the negative ideal solution indicates the least preferable alternative.The FPIS and FNIS are calculated as follows: Step 8. Compute the distance from each alternative to the FPIS and to the FNIS.The distance of each alternative from the FPIS and FNIS is calculated using the following equations: where d (.,.) is the distance measured between two fuzzy numbers.
Step 9. Compute the closeness coefficient of each alternative.
For each alternative Ai we calculate the closeness coefficient CCi as follows: Step 10.Rank the alternatives.
The alternative with the highest closeness coefficient represents the best alternative.

Case study
In this paper, the Fuzzy TOPSIS method will be used for mining method selection for an underground leadzinc ore mine.The total proven reserves of lead-zinc ore are approximately 945,000 tonnes, with an average grade of 5 per cent of lead and 4 per cent of zinc, an average ore thickness of 10 to 15 meters, an average ore length of 800 to 900 meters and an average ore plunge of 40 degrees.The planned production capacity is 500,000 tonnes per year, the number of working days in a year is 350, and the number of shifts per day is 3.An average lead price of 1,650 US$ per tonne and an average zinc price of 1850 US$ per tonne is used.
The mining method selection will be based on several mining-geological, mining-technical and economic factors.The most important mining-geological characteristics (Darling, 2011;Nicholas, 1992)

Alternatives
According to the mining-geological characteristics of the deposit, four mining methods are being considered for the selection of the mining method for underground mining of lead-zinc ore, which is considered as an alternative (Tab.4).These mining methods have in the past been used in the same mine, and some of them are still used today to mine some parts of the mine.Each mining method has advantages and disadvantages in terms of mining-technical and economic factors, as well as recommendations for application in certain mining-geological conditions.In Tab. 5, qualitative information on the conditions applicable to each mining method is given (Hartman, 1992;Nicholas, 1992;Darling, 2011;Tatiya, 2013).

Criteria
As already mentioned, the mining method selection depends on many parameters or criteria (Bogdanovic et al., 2012).This paper presents a total of 22 mining-geological, mining-technical and economic characteristics, which are the criteria against which the alternatives will be compared (Tab.7).Each criterion has a different weight, that is, influence on alternative solutions.In this paper, criterion weight was adopted by vote (Nourali et al., 2012), i.e., in consultation with a group of 15 underground mining experts, to minimize the subjectivity of optimization.Those values were then converted into equivalent fuzzy values.Also, Tab. 7 gives the criteria goal (max or min) and classification category (quantitative or qualitative).

Numerical example
For the determination of the best mining method out of the four proposed alternatives, the fuzzy TOPSIS method involves the following steps: In the first step of the fuzzy TOPSIS analysis, three decision makers use the linguistic variables (Tab. 1 and Tab. 2) to evaluate the relative importance or weights of criteria and the ratings of alternatives for various attributes.There may be more decision-makers.The final results of the outcome of decision makers' views are presented in the fuzzy decision matrix (Tab.8) and normalized fuzzy decision matrix (Tab.9).By using Eq. 2, the combined normalized fuzzy decision matrix is obtained (Tab.10).By using Eqs. 5 to 7, the combined normalized fuzzy decision matrix is normalized.The corresponding matrix is presented in Tab.11.Next, by using Eq. 8, a weighted normalized fuzzy decision matrix is calculated, and the result is given in Tab.12.

Conclusion
One of the biggest problems a mining engineer faces when designing a new mine or developing new parts in an existing mine is the selection of the mining method.It is very important to select the appropriate mining method selection as the mining method has a direct impact on the costs incurred during the deposit mining phase and thus directly impacts the financial operations of the mine itself.
When choosing the method of mining excavation, it is necessary to consider as many parameters as possible that influence the mining method selection because, in that case, the most suitable method of mining excavation in a specific case will be chosen.That was the main goal of the research in this paper as well as the contribution of the research.
From the previous research on solving the problem of mining method selection, we notice that several authors have applied different methods from the group of multi-criteria decision-making methods.A more advanced step in the research of decision-making methods is the application of fuzzy multi-criteria decisionmaking methods.The application of fuzzy multi-criteria decision-making methods enables the selection of the most appropriate way of mining excavation, considering a large number of influential parameters.For this purpose, several fuzzy multi-criteria decision-making methods can be applied, such as fuzzy AHP, fuzzy ELECTRE, fuzzy TOPSIS, fuzzy VIKOR and others.
In this paper, the underground mining method selection was carried out using the fuzzy TOPSIS method.The selection was made out of four mining methods, which were compared according to 22 criteria.According to the mining-geological characteristics of the ore deposit under consideration and the planned capacity, it was concluded that the most appropriate mining method is Cut and Fill Stoping.
The next step in researching this problem is the application of three or more fuzzy multi-criteria decisionmaking methods.After obtaining the ranking of the mining methods according to each fuzzy method, it is necessary to compare the results.In this way, the most suitable method will be mining method selection, which is important for solving this very complex issue.

Fig. 1 .
Fig. 1.The closeness coefficient (Rank) of alternatives for this deposit are listed in Tab. 3.