Comparative Analysis of AHP and TOPSIS Multi-Criteria Decision-Making Methods for Mining Method Selection

In this section, input data and a brief description of the method are introduced. The effective indexes in the selection of extraction methods are generally divided into two categories of internal indicators (related to the conditions of the depository, such as geometric and geomechanical indices) and external factors of the mine (executive and management indicators). These indicators are subdivided into criteria and sub-criteria, as presented in Table 1. These criteria are determined according to the views of well-qualified and experienced experts in the field of mining. Primary options will be determined according to the geometric and geomechanically parameters of the area and the use of Nicholas and UBC scoring methods.

Geometric indices are classified in terms of thickness, slope, shape and grade distribution, and geomechanical indices are classified in terms of RSS ratio (the ratio of uniaxial compressive strength to the pressure of cover rocks), distance of joints, shear strength of joints, RMR (rock mechanics classification) and primary mining options have been selected using Nicholas and UBC scoring methods.

2.1 Multi-criteria decision making method – AHP

The AHP hierarchy process was developed by Thomas Sathy in 1980. In this way, complex problems are decomposed and analyzed in a hierarchy. This method is based on the paired comparison. The AHP evaluation process consists of six steps: hierarchical structure, AHP decision matrix, paired comparisons, incompatibility rate calculation, non-scaling of the paired comparison matrix, and the calculation of special values [15-18].

2.1.1 Hierarchical construction

In the first step, the creation of a hierarchy of problem is where the purpose of the problem lies at the highest level and in the next step, the criteria and below are the criteria and finally, the decision options. There is no limit to the number of levels in a hierarchy. The following criteria for each criterion may be quantitative or qualitative [17-19].

2.1.2 Calculate the weight of the elements in AHP

Table 2. Classification for a paired comparison of criteria [16-18]

To calculate weight in a hierarchical analysis, if the importance of a criterion (i) is equal to the other criterion (j) equal to W_ij, In this case, the importance of criterion (j) relative to criterion (i) is $\frac{1}{W_{i j}}$. Also, the importance of each criterion in relation to the same criterion in the decision matrix is equal to one. To weigh the criteria, a n×n matrix is first formed, making up the decision criteria for the rows and columns of this matrix. The main diameter of this matrix is equal to one, and a pairwise comparison of the criteria is done according to Table 2. The pair comparison matrix is then used to determine the relative weight of the elements [17, 18].

The most important method for calculating relative weights based on the paired comparison matrix that is more accurate is the special vector method. In this method W_i is determined by relation 1 [17, 18]:

$A . W=\lambda . W$                     (1)

where, λ represents the special value and W denotes the special vector of the pair comparison matrix A. To calculate λ, the determinant value of the matrix A-λ.I is zero. Also, put the largest value λ in Eq. (2) to calculate W_i values [17, 18].

$A-\lambda_{\max }. I=0$                (2)

For anomalies No. 3 of Gol Gohar Iron Mine, according to the judgments of the experts and the elite of the matrix, the pair of matrices were made according to Table 2. In the next step, weighing of each criterion was done using MATLAB software and special vector method.

2.1.3 Calculating the final weight

The final weight of each option in the hierarchical process is equal to the sum of the product of the weight of each criterion in the option rating. Which is given in Appendix III. [17, 18]:

$A_{AHPscore}=\sum_{j=1}^n b_{i j}.W_{i j} \quad i=1,2, \ldots, m$                 (3)

$\sum_{i=1}^m b_{i j}=1 \quad j=1,2, \ldots, \mathrm{n}$                  (4)

$\sum_{i=1}^n W_j=1$                    (5)

where, b_ij represents the relative importance of the i-th option for the index a_j and W_j indicating the importance of the index a_j. Also, the values of the options and the weight of the indicators should be normalized [17, 18].

2.1.4 Calculating the inconsistency rate

In the AHP method, the calculation of the incompatibility rate is very important and one of the main advantages of this method. In a hierarchical analysis method, this rate should be less than 0.1 [17, 18]. With respect to the following relationships, the inconsistency index, the random inconsistency index and the hierarchical incompatibility rate are obtained.

$I.I.=\frac{\lambda_{\max }-n}{n-1}$                  (6)

$R . I . I=1.98 \frac{n-2}{n}$                  (7)

$I . R .=\frac{I . I .}{R . I . I}$                  (8)

where, I.I. refers to an inconsistency index, R.I.I represents an inconsistency index, I.R. indicates the inconsistency rate, λ_max denotes the largest special value of the matrix and n is the size of the matrix.

2.2 Multi-criteria decision making method– TOPSIS

In multi-criteria decision-making issues, we need to analyze some options. Each issue also has several indicators that are analyzed and analyzed in relation to each of the options, and should carefully identify these indicators in the issues [6, 19]. Among the multi-criteria decision-making methods, the TOPSIS method, which was presented in 1981, was pointed out. This model is one of the best multi-criteria decision making models [9, 20]. In this method, the decision matrix m×n, which has m option and n is the criterion and the measurement, is evaluated. In this model, for mathematical calculations, all quantities assigned to the criteria must be quantitative and, if qualitatively attributed to the criteria, they should be converted into small quantities. In the TOPSIS method, you must have the option of choosing the ideal positive solution with the least distance and with the ideal negative solution is the greatest distance [15, 21, 22]. The TOPSIS technique has six steps to solve the decision problem. Which is described below [5, 23, 24]:

1. Without scaling the decision matrix according to the 9th relationship

$n_{i j}=\frac{r_{i j}}{\sqrt{\sum_{i=1}^m r_{i j}^2}}$                    (9)

2. Creating a weightless matrix as the input of the algorithm, assuming the vector W derived from expert opinions

Weightless matrix $=N_D \cdot W_{n * n}$                  (10)

So that N_D represents a matrix whose scores of indices are scalar and comparable, and W_n*n denotes a diagonal matrix whose nonzero elements are only non-zero.

3. Determine the positive ideal solution and the negative ideal solution

A⁺ denotes the positive ideal option and A^- denotes the negative ideal option.

$v^{+}=\left\{\left(\max _{i j} \mid j \in J^{\prime}\right),\left(minv_{i j} \mid j \in J^{\prime}\right) \mid i=1,2, \ldots, n\right\}=\left\{v_1^{+}, v_2^{+}, \ldots, v_j^{+}, \ldots, v_n^{+}\right\}$                  (11)

$v^{-}=\left\{\left(minv_{i j} \mid j \in J^{\prime}\right),\left(\max _{i j} \mid j \in J^{\prime}\right) \mid i=1,2, \ldots, m\right\}=\left\{v_1^{-}, v_2^{-}, \ldots, v_j^{-}, \ldots, v_n^{-}\right\}$              (12)

4. Determine the separation distance or size

$d_{i+}=\left\{\sum_{j=1}^n\left(v_{i j}-v_j^{+}\right)^2\right\}^{0.5} ; i=1,2, \ldots, m$                     (13)

$d_{i-}=\left\{\sum_{j=1}^n\left(v_{i j}-v_j^{-}\right)^2\right\}^{0.5} ; i=1,2, \ldots, m$                  (14)

5. Determine the relative v_iproximity to the ideal solution

$C l_{i+}=\frac{d_{i-}}{\left(d_{i+}+d_{i-}\right)} ; 0 \leq C l_{i+} \leq 1 ; i=1,2, \ldots \ldots, m$                    (15)

6. Ratings options (Descending order Cl_i+)

In the TOPSIS method, it is possible to make a decision if there are positive and negative criteria (even together in the same issue) and a significant number of criteria can be examined to determine the best option. This method is simple and fast, and it responds well to a large number of criteria. In the TOPSIS method, it is easy to quantify qualitative criteria, and decision-making is possible despite qualitative and quantitative criteria. In this method, the output of the system is quantitative and in addition to determining the best option, the ranking of other options is expressed numerically. The TOPSIS method has appropriate mathematical foundations and deals with distances. Another advantage of this method is the involvement of the weight of all options and criteria in decision-making, and no weight is ignored in this method.

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Figure 2. Rated first stage of selective mining methods

Table 7. Comparison of paired a₂

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Figure 3. The rate of production of each method and

Table 8. Comparison of paired a₃

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Figure 4. Flexibility in changing the extraction method

Table 9. Comparison of paired a₄

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Figure 5. Life of mine

Table 10. Comparison of paired a₅

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Figure 6. Minimum time access to mineral

Table 11. Comparison of paired a₆

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Figure 7. Capital cost

Table 12. Comparison of paired a₇

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Figure 8. Operational cost

Table 13. Comparison of paired a₈

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Figure 9. Revenue and net present method

Table 14. Comparison of paired a₉

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Figure 10. Minimum return on investment and internal rate of return method

Table 15. Comparison of paired a₁₀

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Figure 11. The impact of each method on the environment

Table 16. Comparison of paired a₁₁

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Figure 12. Potential capital lost in each method

Table 17. Comparison of paired a₁₂

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Figure 13. Loss in case of failure of the project and the strategy set

Table 18. Comparison of paired a₁₃

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Figure 14. Efficient use of financial human resources facilities

Table 19. Comparison of paired a₁₄

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Figure 15. Willingness to invest in each method

Table 20. Comparison of paired a₁₅

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Figure 16. Absorption or non-absorption of labor and skilled labor supply conditions

Table 21. Comparison of paired a₁₆

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Figure 17. Having a machine of any of the company

Table 22. Comparison of paired a₁₇

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Figure 18. Empirical technology and the ability to participate in each method

Table 23. Comparison of paired a₁₈

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Figure 19. Having the human resources of each of the company

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Figure 20. Appropriate extraction method with AHP method

Table 24. Calculate the final weight of the options

Table 25. Values λ_max, I.I., R.I.I. and I.R. for different matrices

Table 26. Normalized decision matrix used in the TOPSIS method

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Figure 21. Proper extraction method with TOPSIS method