Comprehensive decision-making with fuzzy combined weighting and its application on the order of gob management

Abstract

In order to make the comprehensive decision-making results more consistent with the actual situation, a fuzzy comprehensive decision-making model with fuzzy combined weighting is developed in this paper. The fuzzy combined weighting method (FCWM) is proposed in the model to determine the weights of decision factors (indexes). The subjective weight and objective weight are combined in the FCWM to reduce the decision error caused by either using subjective weight or objective weight alone. In addition, the fuzziness of combined weights is taken into account in FCWM to make the decision-making results more realistic. The fuzzy structured element is used to express fuzzy numbers in the model. The analysis algorithm of fuzzy comprehensive decision based on the expression of fuzzy structured element is given to solve the problem of complex operations among fuzzy numbers. The decision model was applied to the order of metal mine gob management. It provides guidance for gob management reasonably and orderly. The research results also provide a new idea for the determination of weights in other fuzzy analysis models.

Keywords

Fuzzy comprehensive decision-making fuzzy combined weighting fuzzy structured element theory order of gob management

1 Introduction

Fuzzy comprehensive decision-making (FCDM) is an effective decision-making method. Several FCDM models have been developed by many researchers. The study of those models mainly focused on developing a weight determination method on decision factors. Up till now, the methods of weight determination can be divided into two categories: subjective weighting method (SWM) and objective weighting method (OWM). In SWM, weights are obtained by experts’ subjective judgment based on their experiences. The SWM is a qualitative or semi-quantitative method, such as analytic hierarchy process (AHP) [3 , 25], Delphi method [4, 19], scoring models [9], power weighted average [21, 23], utility function method [2, 22], etc. In OWM, weights are determined based on relationships on decision objects about decision factors, such as entropy weight method [1, 8], analytic network process (ANP) [15, 16], TOPSIS model [11 , 20], grey relational analysis [6, 28], rough set model [26, 27], etc. The characteristics of these two weighting methods are different. Although the experts’ experiences play important role in SWM, the evaluation results of weight are inevitably disturbed by human factors. Therefore there will be a certain degree of subjective arbitrariness on the subjective weights. The objective weights can be obtained by objective statistical data, but the evaluation results of weight are inevitably affected by statistical error. In this regards, combined weighting method (CWM) that combined the SWM and OWM have been proposed by many researchers in recent years. The CWM combines the advantages of these two kinds of weighting methods and reduces the decision error caused by either using subjective weight or objective weight alone. Among them, the weighting method that combined FAHP and entropy is widely used. The AHP is used to determine the subjective weights, and the information entropy is used to determine the objective weight. Then the FCDM model was developed for hydropower project evaluation [10]. The entropy weight concept and FAHP are combined to build the multiple criteria decision-making model for the sequencing of semiconductor production plan [5]. FAHP with entropy weight is constructed to determine the combined weight for the risk analysis on constructors [24]. The FAHP-entropy weight decision method is used to optimize the energy-saving designs [17]. The combination method of FAHP and entropy is applied to the multi-attribute decision of contract quality sorting [12].

Although the fuzziness of subjective weight or objective weight was considered in the FAHP-entropy method, the fuzziness of combined weight was ignored. Zadeh’s extension principle was used to solve fuzzy operations in most of the FCDM models currently. However, there are three main problems during the application of extension principle: (1) the combination operation process of subjective weight and objective weight is very complicated. It is difficult to get fuzzy combination weights; (2) it is difficult to achieve the analytic expression of calculation results among fuzzy numbers due to the inherent ergodicity problem of extension principle; (3) Precise numbers rather than fuzzy numbers were obtained, which was not consistent with the actual situation.

Guo proposed the fuzzy structured element theory [13, 14]. The fuzzy structured element was used to express fuzzy numbers and operations among them, avoiding the ergodicity of the extension principle. Moreover, the fuzzy inheritance of calculation process and the analytic expression of calculation results can be realized. In this paper, the FCDM model with fuzzy combined weighting expressed by fuzzy structured element is proposed. The subjective weight and objective weight are combined in the developed model to comprehensively study the importance of each decision factor. The fuzziness of the combined weights is taken into account to make the decision results more realistic. In addition, the problem of complex fuzzy operations can be solved by using the fuzzy structured element to express fuzzy operations in the model.

The rest of the paper is organized as follows: in Section 2, the fuzzy structured element theory is briefly introduced. Section 3 describes the developing steps of the FCDM model with fuzzy combined weighting. The fuzzy numbers and operations among fuzzy numbers expressed by fuzzy structured element are in Section 4. Section 5 describes the application of the FCDM model to the decision-making of the order of gob management. Section 6 ends the paper with conclusion.

2 Fuzzy structured element theory [13, 14]

2.1 Fuzzy structured element

Let $\tilde{E}$ be the fuzzy set on the real set R, and $\tilde{E} (x)$ be the membership function of $\tilde{E}$ , x ∈ R. If the following properties are satisfied by $\tilde{E} (x)$ :

$\tilde{E} (0) = 1$ ;

$\tilde{E} (x)$ is a right continuous function that is monotonically increasing on [–1,0), and $\tilde{E} (x)$ is a left continuous function that is monotonically decreasing on (0,1];

When x ∈ (- ∞ , -1) or x ∈ (1, + ∞), then $\tilde{E} (x) = 0$ .

Then the fuzzy set $\tilde{E}$ is the fuzzy structured element on R. Triangular fuzzy structured element is one of the commonly used fuzzy structured elements. The membership function of the triangular structured element is as follows: $\tilde{E} (x) = {\begin{matrix} 1 + x, & x \in [- 1, 0) \\ 1 - x, & x \in [0, 1] \\ 0, & otherwise \end{matrix}$ (1)

2.2 Fuzzy numbers expressed by fuzzy structured element

Theorem 1. Let $\tilde{E}$ be the fuzzy structured element on R. For any bounded closed fuzzy number $\tilde{A}$ , there is always a monotonic bounded function f (x) on [–1,1], which makes the relationship between $\tilde{A}$ and f (x) for $\tilde{A} = f (x)_{x = \tilde{E}} = f (\tilde{E})$ .

The simplest monotone function f is a linear function f (x) = a + bx. Let $\tilde{E}$ be the fuzzy structured element. If the fuzzy number $\tilde{A}$ can be expressed as $\tilde{A} = f (\tilde{E}) = a + b \tilde{E}$ (where a, b is a finite real number, b > 0), and then $\tilde{A}$ can be called that it is generated by the fuzzy structured element $\tilde{E}$ linearly. The membership function of fuzzy number $\tilde{A}$ is as follows: $μ_{\tilde{A}} (x) = \tilde{E} (\frac{x - a}{b})$

For the triangular fuzzy number $\tilde{A} = (a, b, c)$ , which can be generated by the piecewise monotone function f (x): $f (x) = {\begin{matrix} (b - a) x + b, x \in [- 1, 0) \\ (c - b) x + b, x \in [0, 1] \end{matrix}$ (2)

2.3 Comparison of fuzzy numbers

Let $\tilde{E}$ be the fuzzy structured element. f (x) and g (x) are the ordered monotone function on [–1, 1]. For any fuzzy numbers $\tilde{A}$ and $\tilde{B}$ which satisfies the relation, $\tilde{A} = f (\tilde{E})$ , $\tilde{B} = g (\tilde{E})$ . An operation is set as the formula (3): $d_{\tilde{E}} (\tilde{A}, \tilde{B}) = \int_{- 1}^{1} \tilde{E} (x) [f (x) - g (x)] dx$ (3)

$\tilde{E} (x)$ is the membership function of $\tilde{E}$ in the formula (3). If $d_{\tilde{E}} (\tilde{A}, \tilde{B}) > 0$ , then record as $\tilde{A} ≻_{\tilde{E}} \tilde{B}$ ; If $d_{\tilde{E}} (\tilde{A}, \tilde{B}) = 0$ , then record as $\tilde{A} \sim_{\tilde{E}} \tilde{B}$ . $≻_{\tilde{E}}$ is the structured element weighted order of fuzzy numbers.

3 FCDM with fuzzy combined weighting

3.1 Fuzzy index value matrix

Let U = (u₁, u₂, ⋯ , u_n) be the set of decision objects, and V = (v₁, v₂, ⋯ , v_m) be the set of decision indexes (factors). ${\tilde{x}}_{ij}$ was the fuzzy value of each object about each factor, which constitutes a fuzzy index value matrix ${\tilde{X}}_{m \times n}$ : ${\tilde{X}}_{m \times n} = [\begin{matrix} {\tilde{x}}_{11} & {\tilde{x}}_{12} & \dots & {\tilde{x}}_{1 n} \\ {\tilde{x}}_{21} & {\tilde{x}}_{22} & \dots & {\tilde{x}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{x}}_{m 1} & {\tilde{x}}_{m 2} & \dots & {\tilde{x}}_{mn} \end{matrix}]$

Usually, there are two types of the data ${\tilde{x}}_{ij}$ : ① ${\tilde{x}}_{ij}$ is positively correlated with the decision results. The greater the ${\tilde{x}}_{ij}$ , the higher the decision result; ② ${\tilde{x}}_{ij}$ is negatively correlated with the decision results. The greater the ${\tilde{x}}_{ij}$ , the lower the decision result.

Moreover, since ${\tilde{x}}_{ij}$ is the value of different objects about different factors, numerical size or dimension will be different on different values. Therefore, ${\tilde{x}}_{ij}$ should be standardized.

The triangular fuzzy numbers are used to the fuzzy value ${\tilde{x}}_{ij}$ in the matrix ${\tilde{X}}_{m \times n}$ . Assuming that the ${\tilde{x}}_{ij}$ is the triangular fuzzy number (a_ij, b_ij, c_ij), and let $\begin{matrix} a_{i}^{max} & = & max_{1 \leq j \leq n} (a_{ij}), a_{i}^{min} = min_{1 \leq j \leq n} (a_{ij}); \\ b_{i}^{max} & = & max_{1 \leq j \leq n} (b_{ij}), b_{i}^{min} = min_{1 \leq j \leq n} (b_{ij}); \\ c_{i}^{max} & = & max_{1 \leq j \leq n} (c_{ij}), c_{i}^{min} = min_{1 \leq j \leq n} (c_{ij}) . \end{matrix}$

If ${\tilde{x}}_{ij}$ is positively correlated with the decision result, then ${\tilde{x}}_{ij}$ can be standardized as the formula (4): ${\tilde{r}}_{ij} = (\frac{a_{ij}}{c_{i}^{max}}, \frac{b_{ij}}{b_{i}^{max}}, \frac{c_{ij}}{a_{i}^{max}} \land 1)$ (4)

If ${\tilde{x}}_{ij}$ is negatively correlated with the decision result, then ${\tilde{x}}_{ij}$ can be standardized as the formula (5): ${\tilde{r}}_{ij} = (\frac{a_{i}^{min}}{c_{ij}}, \frac{b_{i}^{min}}{b_{ij}}, \frac{c_{i}^{min}}{a_{ij}} \land 1)$ (5)

To ensure the values of decision results falling within [0,1], the ${\tilde{r}}_{ij}$ can be normalized as the formula (6): ${\tilde{t}}_{ij} = \frac{{\tilde{r}}_{ij}}{\sum_{j = 1}^{n} {\tilde{r}}_{ij}}$ (6)

In summary, the fuzzy index value matrix is as follows: ${\tilde{T}}_{m \times n} = [\begin{matrix} {\tilde{t}}_{11} & {\tilde{t}}_{12} & \dots & {\tilde{t}}_{1 n} \\ {\tilde{t}}_{21} & {\tilde{t}}_{22} & \dots & {\tilde{t}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{t}}_{m 1} & {\tilde{t}}_{m 2} & \dots & {\tilde{t}}_{mn} \end{matrix}]$

3.2 Fuzzy combined weighting

Fuzzy combined weighting is a method that combines the subjective weight and objective weight. Fuzziness of combined weight is also taken into account. In this paper, the fuzzy analytical hierarchy process (FAHP) [7, 25] is chosen as the subjective weighting method, and the entropy [1] weight method is chosen as the objective weighting method.

3.2.1 Subjective weighting

The steps to determine fuzzy subjective weights of the decision factors based on the FAHP are as follows:

Giving the fuzzy evaluation matrix of the relative importance of the decision factors is determined by experts. Let $\tilde{Y} = {({\tilde{y}}_{ij}^{k})}_{m \times m}$ be the fuzzy evaluation matrix of the decision factors, v₁, v₂, ⋯ , v_m. Supposing that the number of the experts was q, and the ${\tilde{y}}_{ij}^{k}$ represented the fuzzy evaluation value of the factor v_i relative to the factor v_j that is given by the kth expert, i, j = 1, 2, ⋯ , m; k = 1, 2, ⋯ , q. ${\tilde{y}}_{ij}^{k}$ can be expressed by triangular fuzzy numbers, namely ${\tilde{y}}_{ij}^{k} = (a_{ij}^{k}, b_{ij}^{k}, c_{ij}^{k})$ . Data of matrix $\tilde{Y}$ are shown in Table 1.

Integrating and unifying data of the experts’ opinions. According to the formula (7), collating the data in the matrix $\tilde{Y}$ , and then the fuzzy matrix ${\tilde{Y}}^{'} = {({\tilde{y}}_{ij})}_{m \times m}$ is obtained.

$\begin{matrix} {\tilde{y}}_{ij} & = & (\frac{1}{q} \sum_{k = 1}^{q} a_{ij}^{k}, \frac{1}{q} \sum_{k = 1}^{q} b_{ij}^{k}, \frac{1}{q} \sum_{k = 1}^{q} c_{ij}^{k}) \\ = & (a_{ij}, b_{ij}, c_{ij}) \end{matrix}$ (7)

Calculating fuzzy weight.

Table 1

Data of matrix $\tilde{Y}$

$\tilde{Y}$	v ₁	v ₂	…	v _m
v ₁	(1,1,1)	$\begin{matrix} {\tilde{y}}_{12}^{1} \\ ⋮ \\ {\tilde{y}}_{12}^{q} \end{matrix}$		$\begin{matrix} {\tilde{y}}_{1 m}^{1} \\ ⋮ \\ {\tilde{y}}_{1 m}^{q} \end{matrix}$
v ₂	$\begin{matrix} {\tilde{y}}_{21}^{1} \\ ⋮ \\ {\tilde{y}}_{21}^{q} \end{matrix}$	(1,1,1)	᠁	$\begin{matrix} {\tilde{y}}_{12}^{1} \\ ⋮ \\ {\tilde{y}}_{2 m}^{q} \end{matrix}$
᠁	᠁	᠁	᠁	᠁
v _m	$\begin{matrix} {\tilde{y}}_{m 1}^{1} \\ ⋮ \\ {\tilde{y}}_{m 1}^{q} \end{matrix}$	$\begin{matrix} {\tilde{y}}_{m 2}^{1} \\ ⋮ \\ {\tilde{y}}_{m 2}^{q} \end{matrix}$	᠁	(1,1,1)

Let ${\tilde{w}}_{i}^{(1)}$ be the subjective weight of the factor v_i, i = 1, 2, ⋯ , m. And then ${\tilde{w}}_{i}^{(1)}$ is obtained by the formulas (8): ${\tilde{w}}_{i}^{(1)} = \sum_{j = 1}^{m} {\tilde{y}}_{ij} / \sum_{i = 1}^{m} \sum_{j = 1}^{m} {\tilde{y}}_{ij}$ (8)

3.2.2 Objective weighting

Assuming that the fuzzy index value matrix is ${\tilde{T}}_{m \times n} = {({\tilde{t}}_{ij})}_{m \times n}$ after normalizing. Let fuzzy number ${\tilde{e}}_{i}$ be the fuzzy entropy of decision factor v_i, i = 1, 2, ⋯ , m, and then the ${\tilde{e}}_{i}$ can be obtained by the formula (9): ${\tilde{e}}_{i} = - \frac{1}{ln n} \sum_{j = 1}^{n} {\tilde{t}}_{ij} ln {\tilde{t}}_{ij}$ (9)

Stipulate: if ${\tilde{t}}_{ij} = 0$ , then ${\tilde{t}}_{ij} ln {\tilde{t}}_{ij} = 0$ .

According to the nature of the entropy, the smaller the entropy of the decision factors, the greater the weight should be. Let fuzzy number ${\tilde{w}}_{i}^{(2)}$ be the fuzzy entropy weight of decision factor v_i: ${\tilde{w}}_{i}^{(2)} = \frac{1}{{\tilde{e}}_{i}}$ (10)

Stipulate: if ${\tilde{e}}_{i} = 0$ , then ${\tilde{w}}_{i}^{(2)} = 1$ .

3.2.3 Combined weighting

Let fuzzy number ${\tilde{w}}_{i}^{(1, 2)}$ be the fuzzy combined weight of the decision factor v_i, which can be obtained by combining weights ${\tilde{w}}_{i}^{(1)}$ with ${\tilde{w}}_{i}^{(2)}$ : ${\tilde{w}}_{i}^{(1, 2)} = \frac{{\tilde{w}}_{i}^{(1)} \cdot {\tilde{w}}_{i}^{(2)}}{\sum_{i = 1}^{m} {\tilde{w}}_{i}^{(1)} \cdot {\tilde{w}}_{i}^{(2)}}$ (11)

The weight ${\tilde{w}}_{i}^{(1, 2)}$ is normalized to be ${\tilde{w}}_{i}$ : ${\tilde{w}}_{i} = \frac{{\tilde{w}}_{i}^{(1, 2)}}{\sum_{i = 1}^{m} {\tilde{w}}_{i}^{(1, 2)}}$ (12)

3.3 FCDM model with fuzzy combined weighting

The FCDM with fuzzy combined weighting is as follows: $\begin{matrix} \tilde{P} & = & \tilde{W} \cdot {\tilde{T}}_{m \times n} \\ = & ({\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{m}) \cdot [\begin{matrix} {\tilde{t}}_{11} & {\tilde{t}}_{12} & \dots & {\tilde{t}}_{1 n} \\ {\tilde{t}}_{21} & {\tilde{t}}_{22} & \dots & {\tilde{t}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{t}}_{m 1} & {\tilde{t}}_{m 2} & \dots & {\tilde{t}}_{mn} \end{matrix}] \\ = & ({\tilde{p}}_{1}, {\tilde{p}}_{2}, \dots, {\tilde{p}}_{n}) \end{matrix}$

The FCDM can be realized by comparing the size of the fuzzy numbers ${\tilde{p}}_{1}, {\tilde{p}}_{2}, \dots, {\tilde{p}}_{n}$ .

4 FCDM with combined weighting expressed by fuzzy structured element

The fuzzy structured element is used to express the fuzzy numbers and the fuzzy calculation among fuzzy numbers in the FCDM model. Fuzzy inheritance of fuzzy computation can be successfully realized, and the analytic solution of fuzzy operation can be obtained.

Given the fuzzy structured element $\tilde{E}$ , and let the functions f_ij (x), g_ij (x), y_ij (x), $h_{i}^{(1)} (x)$ , $h_{i}^{(2)} (x)$ , $h_{i}^{(1, 2)} (x)$ and H_i (x) be the same order monotone function in [–1,1] [14]. Then according to theorem 1, let the functions f_ij (x), g_ij (x), y_ij (x), $h_{i}^{(1)} (x)$ , $h_{i}^{(2)} (x)$ , $h_{i}^{(1, 2)} (x)$ and H_i (x) be the generated functions of the fuzzy numbers ${\tilde{r}}_{ij}$ , ${\tilde{t}}_{ij}$ , ${\tilde{y}}_{ij}$ , ${\tilde{w}}_{i}^{(1)}$ , ${\tilde{e}}_{i}$ , ${\tilde{w}}_{i}^{(1, 2)}$ and ${\tilde{w}}_{i}$ in the formulas (5–9), (11) and (12) respectively.

On the basis of the literature [14], supposing that there is the same order monotonic transformation τ₂ of f (x) in [–1,1], then the relationship of the formula (13) can be satisfied by f (x) and τ₂: $f^{τ_{2}} (x) = \frac{1}{f (- x)}$ (13)

The fuzzy numbers ${\tilde{w}}_{i}^{(1)}$ can be expressed by using the fuzzy structure element $\tilde{E}$ and the function y_ij (x), which is based on the Theorem 1 and the formulas (8) and (13).

$\begin{matrix} {\tilde{w}}_{i}^{(1)} & = & {[(\sum_{j = 1}^{m} y_{ij} (x)) \cdot {(\sum_{i = 1}^{m} \sum_{j = 1}^{m} y_{ij} (x))}^{τ_{2}}]}_{x = \tilde{E}} \\ = & {[h_{i}^{(1)} (x)]}_{x = \tilde{E}} \end{matrix}$ (14)

The fuzzy numbers ${\tilde{t}}_{ij}$ can be expressed by using the $\tilde{E}$ and the function f_ij (x), which is based on the Theorem 1 and the formulas (6) and (13).

$\begin{matrix} {\tilde{t}}_{ij} & = & {(f_{ij} (x) \cdot {(\sum_{j = 1}^{n} f_{ij} (x))}^{τ_{2}})}_{x = \tilde{E}} \\ = & {[g_{ij} (x)]}_{x = \tilde{E}} \end{matrix}$ (15)

The fuzzy numbers ${\tilde{e}}_{i}$ can be expressed by using the $\tilde{E}$ and the function g_ij (x) in the formula (15), which is based on the Theorem 1 and the formula (9).

$\begin{matrix} {\tilde{e}}_{i} & = & {(- \frac{1}{ln n} \sum_{j = 1}^{n} g_{ij} (x) ln g_{ij} (x))}_{x = \tilde{E}} \\ = & {[h_{i}^{(2)} (x)]}_{x = \tilde{E}} \end{matrix}$ (16)

The fuzzy numbers ${\tilde{w}}_{i}^{(1, 2)}$ can be expressed by using the fuzzy structure element $\tilde{E}$ and the function $h_{i}^{(1)} (x)$ , $h_{i}^{(2)} (x)$ , which is based on the Theorem 1 and the formulas (11) and (13).

$\begin{matrix} {\tilde{w}}_{i}^{(1, 2)} & = & {(Φ_{i} (x) \cdot {(\sum_{i = 1}^{m} Φ_{i} (x))}^{τ_{2}})}_{x = \tilde{E}} \\ = & {[h_{i}^{(1, 2)} (x)]}_{x = \tilde{E}} \end{matrix}$ (17) where $Φ_{i} (x) = h_{i}^{(1)} (x) \cdot {[h_{i}^{(2)} (x)]}^{τ_{2}}$ , and $h_{i}^{(1)} (x)$ is in the formula (14), $h_{i}^{(2)} (x)$ is in the formula (16).

The fuzzy numbers ${\tilde{w}}_{i}$ can be expressed by using the fuzzy structured element $\tilde{E}$ and the function $h_{i}^{(1, 2)} (x)$ in the formula (17), which is based on the Theorem 1 and the formulas (12) and (13).

$\begin{matrix} {\tilde{w}}_{i} & = & {(h_{i}^{(1, 2)} (x) \cdot {(\sum_{i = 1}^{m} h_{i}^{(1, 2)} (x))}^{τ_{2}})}_{x = \tilde{E}} \\ = & {[H_{i} (x)]}_{x = \tilde{E}} \end{matrix}$ (18)

In summary, the FCDM with fuzzy combined weighting expressed by fuzzy structured element is as follows:

$\begin{matrix} \tilde{P} & = & {[(H_{1} (x), H_{2} (x), \dots, H_{m} (x)) \cdot G (x)]}_{x = \tilde{E}} \\ = & {[p_{1} (x), p_{2} (x), \dots, p_{n} (x)]}_{x = \tilde{E}} \end{matrix}$ (19) where the G (x) in formula (19) is $G (x) = [\begin{matrix} g_{11} (x) & g_{12} (x) & \dots & g_{1 n} (x) \\ g_{21} (x) & g_{22} (x) & \dots & g_{2 n} (x) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ g_{m 1} (x) & g_{m 2} (x) & \dots & g_{mn} (x) \end{matrix}]$

The size relationship among the fuzzy numbers $p_{1} (\tilde{E}), p_{2} (\tilde{E}), \dots, p_{n} (\tilde{E})$ can be obtained by using the comparison method of formula (3), and then FCDM can be realized.

5 Engineering application

Taking gob of underground metal mine as a case study, the FCDM method with fuzzy combined weighting was used to evaluate the stability order of gob.

Based on professional knowledge and data analysis, the decision factors that affected the stability of gobs were given. The qualitative factors include rock mass structure (v₁), hydrological factors (v₂), mining disturbance (v₃) and impact of adjacent gob (v₄). The quantitative factors include burial depth (v₅), empty volume (v₆), exposure time (v₇), uniaxial compressive strength R_c (v₈) and tensile strength R_t (v₉).

The fuzzy subjective weights of decision factors were determined by applying FAHP. The fuzzy evaluation values of decision factors were given by five professional experts, which constituted the fuzzy evaluation matrix $\tilde{Y}$ . The simplified fuzzy evaluation matrix ${\tilde{Y}}^{'}$ was obtained after the processing of the data in the matrix $\tilde{Y}$ based on the formula (7). The simplified fuzzy evaluation matrix ${\tilde{Y}}^{'}$ was as follws:

${\tilde{Y}}^{'} = [\begin{matrix} (1.0, 1.0, 1.0) (2.3, 3.5, 4.3) (0.3, 0.4, 0.7) (0.4, 0.7, 1.0) (1.3, 2.4, 3.3) (0.3, 0.6, 0.8) (0.4, 1.0, 1.3) (0.3, 0.5, 0.9) (0.4, 0.7, 1.0) \\ (0, 2.0.5, 0.7) (1.0, 1.0, 1.0) (1.3, 2.2, 3.1) (0.4, 0.7, 0.9) (0.2, 0.5, 0.7) (0.2, 0.4, 0.7) (0.4, 0.7, 0.9) (0.3, 0.4, 0.7) (0, 2.0.5, 0.7) \\ (1.6, 2.8, 3.7) (0.3, 0.6, 0.8) (1.0, 1.0, 1.0) (0.6, 1.2, 1.7) (1.3, 2.4, 3.3) (0.3, 0.6, 0.9) (1.3, 1.7, 2.3) (1.2, 1.8, 2.1) (0.4, 0.7, 1.0) \\ (1.0, 1.3, 2.3) (0.6, 1.2, 2.0) (0.4, 0.7, 1.0) (1.0, 1.0, 1.0) (1.3, 2.2, 3.1) (0.4, 0.7, 0.9) (1.0, 2.2, 3.1) (0.4, 0.7, 0.9) (0.3, 0.4, 0.8) \\ (0.3, 0.4, 0.8) (1.6, 2.5, 3.7) (0.3, 0.4, 0.8) (0.3, 0.6, 0.8) (1.0, 1.0, 1.0) (0.2, 0.4, 0.7) (0.5, 0.7, 0.9) (0.5, 0.7, 0.8) (0.4, 0.7, 0.9) \\ (2.6, 3.4, 4.6) (2.3, 3.5, 4.3) (1.6, 2.5, 3.4) (1.6, 2.5, 3.7) (2.3, 3.5, 4.3) (1.0, 1.0, 1.0) (1.6, 2.8, 4.1) (1.5, 2.7, 3.8) (1.7, 2.6, 3.7) \\ (1.0, 2.2, 3.1) (1.5, 2.2, 3.0) (0.6, 0.9, 1.0) (0.4, 0.7, 0.9) (0.7, 1.5, 2.2) (0.4, 0.6, 0.8) (1.0, 1.0, 1.0) (1.0, 1.8, 2.5) (0.4, 0.6, 0.9) \\ (0.7, 1.5, 2.2) (1.6, 2.8, 3.7) (0.4, 0.7, 1.0) (1.0, 2.2, 3.1) (1.0, 1.5, 2.1) (0.3, 0.5, 0.8) (0.4, 0.6, 0.7) (1.0, 1.0, 1.0) (0.5, 0.7, 1.0) \\ (1.1, 1.6, 2.8) (2.3, 3.5, 4.3) (1.0, 2.2, 3.1) (1.3, 2.4, 3.3) (1.6, 2.5, 3.4) (0.5, 0.7, 1.0) (1.5, 2.2, 2.9) (1.3, 2.0, 2.5) (1.0, 1.0, 1.0) \end{matrix}]$

According to the matrix ${\tilde{Y}}^{'}$ and the formula (8), the fuzzy subjective weights of decision factors v₁, v₂, ⋯ , v₉ were determined: $\begin{matrix} {\tilde{w}}_{1}^{(1)} & = & (0 . 13, 0 . 19, 0 . 27), {\tilde{w}}_{2}^{(1)} = (0 . 07, 0 . 11, 0 . 20), \\ {\tilde{w}}_{3}^{(1)} & = & (0 . 44, 0 . 58, 0 . 74), {\tilde{w}}_{4}^{(1)} = (0 . 22, 0 . 40, 0 . 51), \\ {\tilde{w}}_{5}^{(1)} & = & (0 . 14, 0 . 33, 0 . 48), {\tilde{w}}_{6}^{(1)} = (0 . 67, 0 . 81, 0 . 88), \\ {\tilde{w}}_{7}^{(1)} & = & (0 . 51, 0 . 64, 0 . 71), {\tilde{w}}_{8}^{(1)} = (0 . 39, 0 . 51, 0 . 64), \\ {\tilde{w}}_{9}^{(1)} & = & (0 . 59, 0 . 71, 0 . 83) . \end{matrix}$

It was known that the value of every decision object about each factor was fuzzy, such as rock structure and other qualitative factors, which could only be determined by the fuzzy value through experts analysis or other qualitative description methods. Even for the quantitative factors, the exact values could not be given. For example, the burial depth, based on the geological drilling data, it could only be a fuzzy value due to the fluctuating surface and the variability of the ore body.

According to the statistical data, the fuzzy values of four gobs on each decision factor were shown in Table 2, and the fuzzy values were expressed by triangular fuzzy numbers. In the table, the values of gobs on qualitative factors were the fuzzy comprehensive scores that were given by five experts according to the geological data and the mining technology condition of gobs. The scores range was [0, 10]. The data of burial depth of gobs were obtained according to the geological drilling data. Data of the gobs volume and exposure time were obtained according to the measurement statistics. Values of R_c and R_t were obtained by tension and compression experiment on rock samples from the surrounding rock of the gobs.

Table 2
Fuzzy index values of four gobs on decision factors

Gobs gob 1 gob 2 gob 3 gob 4

Factors

Rock structure (2,3,4) (4,5,6) (4,5,6) (3,4,5)

Hydrological factors (2,3,4) (7,8,9) (4,5,6) (2,3,4)

Mining disturbance (2,3,4) (2,3,4) (4,5,6) (4,5,6)

Adjacent gobs (6,7,8) (2,3,4) (4,5,6) (2,3,4)

Burial depth/m (140,170,190) (250,270,320) (300,360,390) (400,440,460)

Empty volume/m³ (1800,2000,2300) (5000,5300,5600) (9500,10000,11000) (8300,8500,9000)

Exposure time/d (25,28,30) (35,40,42) (33,36,38) (47,50,55)

R_c/MPa (28.6,35.3,38.7) (35.7,42.2,47.4) (52.8,60.6,64.3) (75.4,83.3,90.8)

R_t/MPa (2.2,3.4,4.8) (5.4,7.9,11.3) (6.0,9.3,12.6) (7.2,11.5,15.3)

Gobs	gob 1	gob 2	gob 3	gob 4
Factors
Rock structure	(2,3,4)	(4,5,6)	(4,5,6)	(3,4,5)
Hydrological factors	(2,3,4)	(7,8,9)	(4,5,6)	(2,3,4)
Mining disturbance	(2,3,4)	(2,3,4)	(4,5,6)	(4,5,6)
Adjacent gobs	(6,7,8)	(2,3,4)	(4,5,6)	(2,3,4)
Burial depth/m	(140,170,190)	(250,270,320)	(300,360,390)	(400,440,460)
Empty volume/m³	(1800,2000,2300)	(5000,5300,5600)	(9500,10000,11000)	(8300,8500,9000)
Exposure time/d	(25,28,30)	(35,40,42)	(33,36,38)	(47,50,55)
R_c/MPa	(28.6,35.3,38.7)	(35.7,42.2,47.4)	(52.8,60.6,64.3)	(75.4,83.3,90.8)
R_t/MPa	(2.2,3.4,4.8)	(5.4,7.9,11.3)	(6.0,9.3,12.6)	(7.2,11.5,15.3)

The data of Table 2 could be normalized according to the formulas (4) and (5), and then the matrix $\tilde{R} = {({\tilde{r}}_{ij})}_{9 \times 4}$ was obtained: $\tilde{R} = [\begin{matrix} (0.33, 0.60, 1.00) & (0.67, 1.00, 1.00) & (0.67, 1.00, 1.00) & (0.50, 0.80, 1.00) \\ (0.22, 0.38, 0.57) & (0.78, 1.00, 1.00) & (0.44, 0.63, 0.86) & (0.22, 0.38, 0.57) \\ (0.33, 0.60, 1.00) & (0.33, 0.60, 1.00) & (0.67, 1.00, 1.00) & (0.67, 1.00, 1.00) \\ (0.75, 1.00, 1.00) & (0.25, 0.43, 0.67) & (0.50, 0.71, 1.00) & (0.25, 0.43, 0.67) \\ (0.74, 1.00, 1.00) & (0.44, 0.63, 0.76) & (0.36, 0.47, 0.63) & (0.30, 0.39, 0.48) \\ (0.78, 1.00, 1.00) & (0.32, 0.38, 0.46) & (0.16, 0.20, 0.24) & (0.20, 0.24, 0.28) \\ (0.83, 1.00, 1.00) & (0.60, 0.70, 0.86) & (0.66, 0.78, 0.91) & (0.45, 0.56, 0.64) \\ (0.31, 0.42, 0.51) & (0.39, 0.51, 0.63) & (0.58, 0.72, 0.85) & (0.83, 1.00, 1.00) \\ (0.13, 0.27, 0.57) & (0.33, 0.64, 1.00) & (0.40, 0.82, 1.00) & (0.47, 1.00, 1.00) \end{matrix}]$

The fuzzy number ${\tilde{r}}_{ij}$ in the matrix $\tilde{R}$ could be expressed by the fuzzy structured element $\tilde{E}$ and function f_ij (x), which was based on the Theorem 1, i = 1, 2, ⋯ , 9; j = 1, 2, 3, 4. Calculation process of f₁₁ (x) and ${\tilde{t}}_{11}$ was taken as an example to illustrate the calculation process of fuzzy numbers calculation that expressed by the fuzzy structured element.

In the matrix $\tilde{R}$ , ${\tilde{r}}_{11} = (0.33, 0.60, 1.00)$ , f₁₁ (x) could be obtained using the formula (2): $\begin{matrix} f_{11} (x) & = & {\begin{matrix} (0.6 - 0.33) x + 0.6 \\ (1 - 0.6) x + 0.6 \end{matrix} \\ = & {\begin{matrix} 0.27 x + 0.6, x \in [- 1, 0) \\ 0.4 x + 0.6, x \in [0, 1] \end{matrix} \end{matrix}$

Then, ${\tilde{r}}_{11} = {[f_{11} (x)]}_{x = \tilde{E}}$ .

According to the formulas (6), (13) and (15), ${\tilde{t}}_{11}$ could be obtained: $\begin{matrix} \sum_{j = 1}^{4} f_{1 j} (x) = {\begin{matrix} 1.23 x + 3.4, x \in [- 1, 0) \\ 0.6 x + 3.4, x \in [0, 1] \end{matrix} \\ {(\sum_{j = 1}^{4} f_{1 j} (x))}^{τ_{2}} \\ = {\begin{matrix} 1 / (3.4 - 0.6 x), x \in [- 1, 0) \\ 1 / (3.4 - 1.23 x), x \in [0, 1] \end{matrix} \\ {\tilde{t}}_{11} = g_{11} (\tilde{E}) \\ = {\begin{matrix} (0.27 \tilde{E} + 0.6) / (3.4 - 0.6 \tilde{E}), x \in [- 1, 0) \\ (0.4 \tilde{E} + 0.6) / (3.4 - 1.23 \tilde{E}), x \in [0, 1] \end{matrix} \end{matrix}$

Similarly, the operations of fuzzy numbers ${\tilde{t}}_{ij}$ , ${\tilde{w}}_{i}^{(1)}$ , ${\tilde{e}}_{i}$ , ${\tilde{w}}_{i}^{(1, 2)}$ , ${\tilde{w}}_{i}$ and ${\tilde{p}}_{j}$ could be obtained through formulas (14∼19) with the application of the symbol calculation in the Matlab.

The size relationship of ${\tilde{p}}_{j}$ could be compared by formula (3), j = 1, 2, 3, 4, and the integral values were obtained through the integral function of the symbol calculation in the Matlab: $\begin{matrix} d_{E} ({\tilde{p}}_{1}, {\tilde{p}}_{2}) & = & \int_{- 1}^{1} E (x) [p_{1} (x) - p_{2} (x)] dx \\ = & 0.158 > 0, \\ d_{E} ({\tilde{p}}_{1}, {\tilde{p}}_{3}) & = & \int_{- 1}^{1} E (x) [p_{1} (x) - p_{3} (x)] dx \\ = & 0.14 > 0, \\ d_{E} ({\tilde{p}}_{1}, {\tilde{p}}_{4}) & = & \int_{- 1}^{1} E (x) [p_{1} (x) - p_{4} (x)] dx \\ = & 0.156 > 0, \\ d_{E} ({\tilde{p}}_{2}, {\tilde{p}}_{3}) & = & \int_{- 1}^{1} E (x) [p_{2} (x) - p_{3} (x)] dx \\ = & - 0.018 < 0, \\ d_{E} ({\tilde{p}}_{2}, {\tilde{p}}_{4}) & = & \int_{- 1}^{1} E (x) [p_{2} (x) - p_{4} (x)] dx \\ = & - 0.002 < 0, \\ d_{E} ({\tilde{p}}_{3}, {\tilde{p}}_{4}) & = & \int_{- 1}^{1} E (x) [p_{3} (x) - p_{4} (x)] dx \\ = & 0.016 > 0 . \end{matrix}$

Therefore, the order of stability of gobs was u₁ ≻ u₃ ≻ u₄ ≻ u₂. It was also the order of gob management.

In order to verify the correctness and practicality of the decision results, the roof subsidence, subsidence velocity and roof caving degree of four gobs were monitored. The monitoring results showed that the damage degree of the gob u₁ was more severe than that of the other three gobs, further proving the validity of this method for metal mine gob management.

Considering the surface subsidence, environmental protection and other factors, the filling method was commonly used to deal with gobs currently in metal mines. However, a lot of manpower, material and time was required by filling systems and filling operations. Therefore, this method, based on FCDM with fuzzy combined weighting, can be used for managing gobs reasonably and orderly. It is also important for mines’ safety control and environmental protection.

6 Conclusions

In this paper, the fuzzy comprehensive decision-making model, based on fuzzy structured element, with fuzzy combined weighting was developed.

The method of fuzzy combined weighting was proposed to determine the weights of decision factors by the combination of the subjective weight and objective weight. It could not only reduce the deviation of subjective weights caused by personal subjectivity, but also the deviation of objective weights caused by statistical error of data. Moreover, the fuzziness of combined weights was taken into account in the method that made the decision results more realistic. A new idea was provided by the fuzzy combined weighting method for the determination of weights of factors in fuzzy comprehensive evaluation, fuzzy grade judgment and other fuzzy analysis models.

In the fuzzy combined weighting comprehensive decision model, fuzzy numbers were expressed by fuzzy structured element. The operations among fuzzy numbers were developed based on the fuzzy structured element, which solved the problem of the complex operations of fuzzy numbers in the model. In addition, the fuzzy inheritance of fuzzy computation was successfully realized, and the analytic solution of fuzzy operation was obtained.

The fuzzy comprehensive decision model with fuzzy combined weighting was used to determine the order of metal mine gobs management, providing guidance for controling gobs reasonably and orderly.

Footnotes

Acknowledgments

The authors express their thankfulness to the editors and the anonymous reviewers for their constructive comments and suggestions. The research described in this paper was financially supported by the Open Projects of Research Center of Coal Resources Safe Mining and Clean Utilization, Liaoning (LNTU15KF14), National Natural Science Foundation of Surface project, China (51774172). The authors would like to thank for all the supports to publish this paper.

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