# TIFNCWBHG-MAGDM for System Evaluation Based on TIFNs for the Safety Input of Coal Enterprise

Authors: Chao Zhang; Qingjie Qi
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

Determining the safety input structure is essential to achieve efficient resource utilization and the safe production of coal enterprises. In this paper, the system evaluation method, TIFNCWBHG-MAGDM, is proposed to evaluate the safety input of coal enterprises. It is based on the integration of the intuitionistic triangular fuzzy numbers (TIFNs), TIFNs Compound Weight Bonferroni Hybrid Geometric (TIFNCWBHG) operator, and multiattribute group decision-making (MAGDM) theory. First, the judgment matrix of TIFNs is constructed from multiperspective: multitime points, multiattributes, and multiexperts. The TIFNCWBHG operator integrates the TIFNs score function, stability weight, and position weight. Then, the priorities for safety inputs are determined. The experimental results showed that safety inputs in industrial hygiene, propaganda, and education significantly impact the overall level of safety inputs. Also, it was proved that the efficiency of the safety input is important. The proposed TIFNCWBHG-MAGDM effectively coordinated the stability weight, position weight, and computation of TIFNs score function, taking the advantages of TIFNs. Accordingly, it was proved that it could optimize the safety input structure.


---

## Body

## 1. Introduction

The safety input structure aims to embody the safety production management level of coal enterprises, which is a crucial part of the safety production development strategy. In order to improve the safety input structure of coal enterprises, a timely scientific and reasonable evaluation is needed. However, the production system of coal enterprises is a dynamic and multivariate complex system constructed by stereoscopic multioverlapping factors in time and space. In addition, coal safety accidents occur as random, dynamic, and uncertain. The safety output is not necessarily correlated to the overall scale of the safety input of coal enterprises. Accordingly, the determination of the safety input structure should be thoroughly studied.Jiang et al. [1] established a new index system of safety investment based on the set theory. Also, the safety investment and accident control model was constructed using grey prediction theory. Xiao et al. [2] conducted a statistical analysis of accidents over the years to establish a safety investment optimization model, which was based on maximizing economic efficiency and in-depth analysis of safety investment. Dzonziundi [3] computed the total factor productivity (TFP) and technical efficiency (EF) of several enterprises and analyzed the impact of safety input and cleaner production input using the stochastic frontier analysis (SFA). Zhao et al. [4] employed grey theory, multicriteria group decision theory, Triangular Intuitionistic Fuzzy Numbers (TIFNs), and Analytic Hierarchy Process (AHP) to rank and evaluate the safety inputs of coal enterprises. Zhang et al. [5] conducted in-depth research on evaluating the safety resources structure of the coal production logistics system and coal mine safety status. Lu et al. [6] used an agent to study the safety investment of the construction industry under the influence of various factors. Noh and Chang [7] proposed an economic analysis method considering the cost of safety investment and applied it to the comparative evaluation of process plant design. Matthews et al. [8] designed software to support road safety practitioners in analyzing and making decisions. Roy and Gupta [9] proposed the framework of safety investment optimization (SIO) to decrease the accident risk that reduces the future costs under the given budget. Han et al. [10] used the safety investment theory model, questionnaire, and structural equation model (SEM) to study the relationship between safety investment, safety cognition, and construction personnel behavior. Hou and Zhoa [11] combined the Cobb–Douglas production function with the FTA probability model to establish a safety input structure risk-minimization model of petrochemical port enterprises, where the Gompertz curve model is used as a constraint. Lu et al. [12] studied the influence of different safety investments and parameters, such as human factors and environmental factors, on safety performance. Ma et al. [13] established an analysis model from the perspective of opportunity cost and analyzed the factors affecting the safety investment decision-making. Lopezalonso et al. [14] conducted a sample survey using a questionnaire and analyzed the health and safety investments of construction companies. Wu and Wemple [15] proposed a methodology for analyzing the costs and benefits of safety investment to optimize investments. Aven and Hiriart [16] studied the robust optimization of the basic safety input model. Unlike those methods, Zhang et al. [17] proposed a novel MABAC method for MAGDM under a linguistic environment. Wei et al. [18] proposed the MABAC based on the UPLTSs for green supplier selection. Many methods were synthesized in the system, such as the MAGDM, UPLTSs, and entropy method. The GRA method was proposed based on the PLTs for site selection of electric vehicle charging stations in [19] after several methods, including MAGDM, PLTs, GRA method, and CRITIC method, were compared. In [20], the VIKOR method was proposed based on the 2TLNNs and IV2TLNNs for green supplier selection.As described, the main research direction for enterprise safety input or safety investment has been focused on construction, transportation, and other industries [7–16]. Also, in many studies [1–4, 7–16], the quantitative information was too rough and inaccurate. The potential information and related weight information in the data could not be deeply mined, easily distorting important information. Further, the changes of state variables affecting the safety input are varied, and the influence factors are affected by each other. Thus, safety input needs to be systematically and comprehensively considered. However, few evaluation methods were studied to combine multiple-attribute group decision-making, fuzzy mathematics theory. In addition, the above methods had high computational complexity and were not conducive to the promotion and obtaining accurate evaluation results that adapt to various application environments.Uncertain multiattribute decision-making and linguistic decision-making are two significant branches of decision theory and technology. They were widely used since they were proposed, and recently latest variants were proposed, such as [17–21]. Motivated by the significant achievements in uncertain multiattribute decision-making, we propose integrating into the system the fuzzy mathematics theory, MAGDM, TIFNCWBHG operator, and other methods and theories. The proposed novel evaluation method is based on MAGDM theory and intuitionistic triangular fuzzy numbers [21]. TIFNs contain rich information, and thus it can better describe the uncertainty of the environment and the fuzziness of decision-makers. It is also flexible and practical so that it is easy to understand and use. It can supplement the lack of gravity center when the membership degree and non-membership degree of the interval-valued fuzzy-set are expressed by interval numbers and other similar situations. Each attribute value is first expressed by TIFNs, and then the TIFNs judgment matrix is built considering attributes, experts, time points, and other aspects. The TIFNCWBHG operator is given by fusing TIFNs score function [22], position weight [23], and stability weight [24]. The operator is used for integrated calculation, and then, the priority order of each safety input is determined. Finally, the TIFNCWBHG-MAGDM model evaluates the safety input categories of coal enterprises to provide the basis for safety input decision-making.


## 2. TIFNCWBHG-MAGDM Method

### 2.1. TIFNs Judgment Matrix for MAGDM

In the various operations of TIFNs, we often need to use the score function, especially when using the operator, including TIFNs. The calculation formula of the score function of TIFNs [22] is defined as follows.Definition 1.
(see [22]). Let β˜=a,b,c,l,m,n be TIFNs. The score function of β˜ can be expressed as(1)S˜β˜=a+2b+c4+l+2m+n4.where S˜β˜∈−1,1. Equation (1) shows that there exists a corresponding relationship between S˜β˜ and β˜. The larger the value of S˜β˜, the larger the β˜. For example, when S˜β˜=1, β˜ is the maximum and β˜=1,1,1,0,0,0. When S˜β˜=−1, β˜ is the minimum and β˜=0,0,0,1,1,1.
Through (1), the sizes of two TIFNs are compared and sorted for computing TIFNs operators. However, sometimes, it is not easy to compare them with (1), because, in some special cases, two different TIFNs may get the same score. Thus, an accurate score function is required to calculate and compare the two TIFNs.
The score function is used to compare two TIFNs accurately, which is defined as follows.Definition 2.
(see [22]).(2)L˜β˜=a+2b+c42−a+2b+c4−l+2m+n4,where L˜β˜∈0,1 and the greater the value of L˜β˜, the greater the value of β˜. For example, when L˜β˜=1, then β˜ is the maximum and β˜=1,1,1,0,0,0. When L˜β˜=−1, then β˜ is the minimum and β˜=0,0,0,1,1,1.
When the score function is used to calculate the scores of two TIFNs in the actual evaluation process, the importance of membership degree and nonmembership degree of TIFNs may be different. In order to take into account the relative importance of membership and non-membership degrees, they are added to the score function of TIFNs. The reformulated score function is defined as follows.Definition 3.
(3)S˜β˜=4ξ1ζ1a+2ζ2b+ζ3c4+4ξ2ζ1′l+2ζ2′m+ζ3′n4=ξ1ζ1a+2ζ2b+ζ3c+ξ2ζ1′l+2ζ2′m+ζ3′n.
It is worth noting that, in (3), the coefficients of factors such as membership degree and nonmembership degree satisfy the following conditions: ξ1,ξ2≥0, ξ1,ξ2=1, ζ1,ζ2,ζ3,ζ1′,ζ2′,ζ3′≥0, and ζ1+ζ1′=ζ2+ζ2′=ζ3+ζ3′=1. Also, in order to maintain the consistency of calculations, the coefficient values of membership and nonmembership degrees remain unchanged once determined. Furthermore, it is clear that (3) still satisfies the following conditions: there is a corresponding relationship between S˜β˜ and β˜. The larger the value of S˜β˜, the larger the value of β˜. When S˜β˜=1, β˜ is the maximum value and β˜=1,1,1,0,0,0. When S˜β˜=−1, β˜ is the minimum value and β˜=0,0,0,1,1,1.
Similarly, the coefficients of membership and nonmembership degrees are added for the accurate score function, as follows.Definition 4.
(4)L˜β˜=4ξ1ζ1a+2ζ2b+ζ3c42−4ξ1ζ1a+2ζ2b+ζ3c4−4ξ2ζ1′l+2ζ2′m+ζ3′n4=ξ1ζ1a+2ζ2b+ζ3c2−ξ1ζ1a+2ζ2b+ζ3c−ξ2ζ1′l+2ζ2′m+ζ3′n.
Note that, in (4), the coefficients of membership and nonmembership degrees satisfy the following conditions: ξ1,ξ2≥0, ξ1+ξ2=1, ζ1,ζ2,ζ3,ζ1′,ζ2′,ζ3′≥0, and ζ1+ζ1′=ζ2+ζ2′=ζ3+ζ3′=1. Similarly, the coefficient values remain unchanged once determined. Also, (4) satisfies the following conditions: there exists a corresponding relationship between L˜β˜ and β˜. The larger the value of L˜β˜, the larger the value of β˜. When L˜β˜=1, β˜ is the maximum and β˜=1,1,1,0,0,0, while when L˜β˜=−1, β˜ is the minimum value and β˜=0,0,0,1,1,1.
Under an environment of uncertain multiattribute decision-making, experts' judgment often varies over different time points. Thus, TIFNs judgment matrixA for multiattribute group decision-making at time point t is constructed as follows, considering the influence of time:(5)B˜t=β˜ijtm∗n=β˜11tβ˜12t⋯β˜1ntβ˜21tβ˜21t⋯β˜2nt⋮⋮⋱⋮β˜m1tβ˜m2t⋯β˜mnt,where B˜t is the TIFNs judgment matrix, m is the number of experts, and n is the attributes at time point t. β˜ijt represents the TIFNs evaluation of the ith expert for the jth attribute at time point t. β˜ijt=μ˜ijt,γ˜ijt=aijt,bijt,cijt,oijt,pijt,qijt. TIFN for each time point and each factor is evaluated by experts.


### 2.2. Stability Weight of Judgement MatrixB˜t

In order to obtainBt, the scoring function of each element of the matrix B˜t is calculated:(6)Bt=S˜β˜11tS˜β˜12t⋯S˜β˜1ntS˜β˜21tS˜β˜22t⋯S˜β˜2nt⋮⋮⋱⋮S˜β˜m1tS˜β˜m2t⋯S˜β˜mnt.After normalizing the element of the matrixBt, a new matrix B¯t is obtained:(7)B¯t=S˜β˜11t∑i=1mS˜β˜i1tS˜β˜12t∑i=1mS˜β˜i2t⋯S˜β˜1nt∑i=1mS˜β˜intS˜β˜21t∑i=1mS˜β˜i1tS˜β˜22t∑i=1mS˜β˜i2t⋯S˜β˜2nt∑i=1mS˜β˜int⋮⋮⋱⋮S˜β˜m1t∑i=1mS˜β˜i1tS˜β˜m2t∑i=1mS˜β˜i2t⋯S˜β˜mnt∑i=1mS˜β˜int.Then, corresponding to time pointt and attribute j, the stability weights of expert evaluations are computed. From (6) and (7) [24], (8) and (9) are derived:(8)ψjt=1lnm∑i=1mβ˙ijtlnβ˙ijt,(9)φjt=ψjt∑j=1nψjt=1/lnm∑i=1mβ˙ijtlnβ˙ijt∑j=1n1/lnm∑i=1mβ˙ijtlnβ˙ijt=∑i=1mβ˙ijtlnβ˙ijt∑j=1n∑i=1mβ˙ijtlnβ˙ijt,where φjt represents the stability weight obtained by normalizing ψjt by column.The stability weight vector can be obtained by(10)φt=φ1t,φ2t,…,φnt.


### 2.3. TIFNCWBHG Operator

In order to use TIFNs, it is necessary to construct an operator corresponding to TIFNs. Motivated from [4], the following operators are proposed.Definition 5.
Let F be a mapping:Θn⟶Θ, if(11)Fw,φ,ωβ˜1,β˜2,…,β˜n=1p+q⊗k=1npβ˜k⊕q⊗t=1t≠kn−1β˜tn−1ξ1wt+ξ2φtωσt1/n,where Θ is the set of TIFNs. Let β˜1,β˜2,…,β˜n be a collection of TIFNs, and p,q>0. w=w1,w2,…,wnT is the weight vector, where wj∈0,1, ∑j=1nwj=1. φ=φ1,φ2,…,φnT is the stability weight vector, where φj∈0,1 and ∑j=1nφj=1. ξ1,ξ2≥0 are weight coefficients, where ξ1+ξ2=1. ω=ω1,ω2,…,ωnT represents an aggregation-associated vector such that ωj∈0,1 and ∑j=1nωj=1. β˜t is the σtth component in β˜1,β˜2,…,β˜n, and σ:1,2,…,n⟶1,2,…,n is a sort operator according to the size of the computed value by the score function in the array β˜1,β˜2,…,β˜n; then, ωσt is the position weight corresponding to β˜t. In β˜tn−1ξ1wt+ξ2φtωσt, n−1 is a balancing coefficient. Here, when vector w=w1,w2,…,wnT tends to 1/n−1,1/n−1,…,1/n−1T, vector φ=φ1,φ2,…,φnT tends to 1/n−1,1/n−1,…,1/n−1T, and ξ1=ξ2=0.5, and the vector β˜1n−1ξ1w1+ξ2φ1ωσ1,β˜2n−1ξ1w2+ξ2φ2ωσ2,…,β˜n−1n−1ξ1wn−1+ξ2φn−1ωσn−1T tends to β˜1ωσ1,β˜2ωσ2,…,β˜n−1ωσn−1T. ω can be determined by [23], and the calculation of score function of β˜t can be determined by (3).
When performing various operations on TIFNs in the operator, some algorithms [22] need to be used, defined as follows:(12)β˜1⊕β˜2=β˜2⊕β˜1,β˜1⊗β˜2=β˜2⊗β˜1,β˜1⊗β˜2λ=β˜2λ⊗β˜1λ,λ≥0,β˜1λ1⊗β˜1λ2=β˜1λ1+λ2;λ1,λ2≥0.


### 2.4. Final Evaluation

Each column in the matrix is integrated using the TIFNCWBHG operator to compute the initial weight vector of evaluation factors:(13)η˜t=η˜1t,η˜2t,…,η˜nt,where η˜1t,η˜2t,…,η˜nt represent the integrated TIFNs evaluation value of each attribute by experts at time t. Then, the following matrix is constructed:(14)A˜=η˜11η˜21⋯η˜n1η˜12η˜22⋯η˜n2⋮⋮⋱⋮η˜1pη˜2p⋯η˜np.The score function of each element in the matrix is computed to form a new matrixA:(15)A=η11η21⋯ηn1η12η22⋯ηn2⋮⋮⋱⋮η1pη2p⋯ηnp.Let(16)λi=1pζi∑i=1pηit,i=1,2,…,n,where ζ1,ζ2,…,ζn are weight coefficients, ζ1,ζ2,…,ζn≥0, and ζ1+ζ2+⋯+ζn=1. Then, the final evaluation vector is obtained as follows:(17)h=λ1,λ2,…,λn.The flow chart of the proposed framework of TIFNCWBHG-MAGDM is shown in Figure1.Figure 1
The flow chart of the TIFNCWBHG-MAGDM method.


## 2.1. TIFNs Judgment Matrix for MAGDM

In the various operations of TIFNs, we often need to use the score function, especially when using the operator, including TIFNs. The calculation formula of the score function of TIFNs [22] is defined as follows.Definition 1.
(see [22]). Let β˜=a,b,c,l,m,n be TIFNs. The score function of β˜ can be expressed as(1)S˜β˜=a+2b+c4+l+2m+n4.where S˜β˜∈−1,1. Equation (1) shows that there exists a corresponding relationship between S˜β˜ and β˜. The larger the value of S˜β˜, the larger the β˜. For example, when S˜β˜=1, β˜ is the maximum and β˜=1,1,1,0,0,0. When S˜β˜=−1, β˜ is the minimum and β˜=0,0,0,1,1,1.
Through (1), the sizes of two TIFNs are compared and sorted for computing TIFNs operators. However, sometimes, it is not easy to compare them with (1), because, in some special cases, two different TIFNs may get the same score. Thus, an accurate score function is required to calculate and compare the two TIFNs.
The score function is used to compare two TIFNs accurately, which is defined as follows.Definition 2.
(see [22]).(2)L˜β˜=a+2b+c42−a+2b+c4−l+2m+n4,where L˜β˜∈0,1 and the greater the value of L˜β˜, the greater the value of β˜. For example, when L˜β˜=1, then β˜ is the maximum and β˜=1,1,1,0,0,0. When L˜β˜=−1, then β˜ is the minimum and β˜=0,0,0,1,1,1.
When the score function is used to calculate the scores of two TIFNs in the actual evaluation process, the importance of membership degree and nonmembership degree of TIFNs may be different. In order to take into account the relative importance of membership and non-membership degrees, they are added to the score function of TIFNs. The reformulated score function is defined as follows.Definition 3.
(3)S˜β˜=4ξ1ζ1a+2ζ2b+ζ3c4+4ξ2ζ1′l+2ζ2′m+ζ3′n4=ξ1ζ1a+2ζ2b+ζ3c+ξ2ζ1′l+2ζ2′m+ζ3′n.
It is worth noting that, in (3), the coefficients of factors such as membership degree and nonmembership degree satisfy the following conditions: ξ1,ξ2≥0, ξ1,ξ2=1, ζ1,ζ2,ζ3,ζ1′,ζ2′,ζ3′≥0, and ζ1+ζ1′=ζ2+ζ2′=ζ3+ζ3′=1. Also, in order to maintain the consistency of calculations, the coefficient values of membership and nonmembership degrees remain unchanged once determined. Furthermore, it is clear that (3) still satisfies the following conditions: there is a corresponding relationship between S˜β˜ and β˜. The larger the value of S˜β˜, the larger the value of β˜. When S˜β˜=1, β˜ is the maximum value and β˜=1,1,1,0,0,0. When S˜β˜=−1, β˜ is the minimum value and β˜=0,0,0,1,1,1.
Similarly, the coefficients of membership and nonmembership degrees are added for the accurate score function, as follows.Definition 4.
(4)L˜β˜=4ξ1ζ1a+2ζ2b+ζ3c42−4ξ1ζ1a+2ζ2b+ζ3c4−4ξ2ζ1′l+2ζ2′m+ζ3′n4=ξ1ζ1a+2ζ2b+ζ3c2−ξ1ζ1a+2ζ2b+ζ3c−ξ2ζ1′l+2ζ2′m+ζ3′n.
Note that, in (4), the coefficients of membership and nonmembership degrees satisfy the following conditions: ξ1,ξ2≥0, ξ1+ξ2=1, ζ1,ζ2,ζ3,ζ1′,ζ2′,ζ3′≥0, and ζ1+ζ1′=ζ2+ζ2′=ζ3+ζ3′=1. Similarly, the coefficient values remain unchanged once determined. Also, (4) satisfies the following conditions: there exists a corresponding relationship between L˜β˜ and β˜. The larger the value of L˜β˜, the larger the value of β˜. When L˜β˜=1, β˜ is the maximum and β˜=1,1,1,0,0,0, while when L˜β˜=−1, β˜ is the minimum value and β˜=0,0,0,1,1,1.
Under an environment of uncertain multiattribute decision-making, experts' judgment often varies over different time points. Thus, TIFNs judgment matrixA for multiattribute group decision-making at time point t is constructed as follows, considering the influence of time:(5)B˜t=β˜ijtm∗n=β˜11tβ˜12t⋯β˜1ntβ˜21tβ˜21t⋯β˜2nt⋮⋮⋱⋮β˜m1tβ˜m2t⋯β˜mnt,where B˜t is the TIFNs judgment matrix, m is the number of experts, and n is the attributes at time point t. β˜ijt represents the TIFNs evaluation of the ith expert for the jth attribute at time point t. β˜ijt=μ˜ijt,γ˜ijt=aijt,bijt,cijt,oijt,pijt,qijt. TIFN for each time point and each factor is evaluated by experts.


## 2.2. Stability Weight of Judgement MatrixB˜t

In order to obtainBt, the scoring function of each element of the matrix B˜t is calculated:(6)Bt=S˜β˜11tS˜β˜12t⋯S˜β˜1ntS˜β˜21tS˜β˜22t⋯S˜β˜2nt⋮⋮⋱⋮S˜β˜m1tS˜β˜m2t⋯S˜β˜mnt.After normalizing the element of the matrixBt, a new matrix B¯t is obtained:(7)B¯t=S˜β˜11t∑i=1mS˜β˜i1tS˜β˜12t∑i=1mS˜β˜i2t⋯S˜β˜1nt∑i=1mS˜β˜intS˜β˜21t∑i=1mS˜β˜i1tS˜β˜22t∑i=1mS˜β˜i2t⋯S˜β˜2nt∑i=1mS˜β˜int⋮⋮⋱⋮S˜β˜m1t∑i=1mS˜β˜i1tS˜β˜m2t∑i=1mS˜β˜i2t⋯S˜β˜mnt∑i=1mS˜β˜int.Then, corresponding to time pointt and attribute j, the stability weights of expert evaluations are computed. From (6) and (7) [24], (8) and (9) are derived:(8)ψjt=1lnm∑i=1mβ˙ijtlnβ˙ijt,(9)φjt=ψjt∑j=1nψjt=1/lnm∑i=1mβ˙ijtlnβ˙ijt∑j=1n1/lnm∑i=1mβ˙ijtlnβ˙ijt=∑i=1mβ˙ijtlnβ˙ijt∑j=1n∑i=1mβ˙ijtlnβ˙ijt,where φjt represents the stability weight obtained by normalizing ψjt by column.The stability weight vector can be obtained by(10)φt=φ1t,φ2t,…,φnt.


## 2.3. TIFNCWBHG Operator

In order to use TIFNs, it is necessary to construct an operator corresponding to TIFNs. Motivated from [4], the following operators are proposed.Definition 5.
Let F be a mapping:Θn⟶Θ, if(11)Fw,φ,ωβ˜1,β˜2,…,β˜n=1p+q⊗k=1npβ˜k⊕q⊗t=1t≠kn−1β˜tn−1ξ1wt+ξ2φtωσt1/n,where Θ is the set of TIFNs. Let β˜1,β˜2,…,β˜n be a collection of TIFNs, and p,q>0. w=w1,w2,…,wnT is the weight vector, where wj∈0,1, ∑j=1nwj=1. φ=φ1,φ2,…,φnT is the stability weight vector, where φj∈0,1 and ∑j=1nφj=1. ξ1,ξ2≥0 are weight coefficients, where ξ1+ξ2=1. ω=ω1,ω2,…,ωnT represents an aggregation-associated vector such that ωj∈0,1 and ∑j=1nωj=1. β˜t is the σtth component in β˜1,β˜2,…,β˜n, and σ:1,2,…,n⟶1,2,…,n is a sort operator according to the size of the computed value by the score function in the array β˜1,β˜2,…,β˜n; then, ωσt is the position weight corresponding to β˜t. In β˜tn−1ξ1wt+ξ2φtωσt, n−1 is a balancing coefficient. Here, when vector w=w1,w2,…,wnT tends to 1/n−1,1/n−1,…,1/n−1T, vector φ=φ1,φ2,…,φnT tends to 1/n−1,1/n−1,…,1/n−1T, and ξ1=ξ2=0.5, and the vector β˜1n−1ξ1w1+ξ2φ1ωσ1,β˜2n−1ξ1w2+ξ2φ2ωσ2,…,β˜n−1n−1ξ1wn−1+ξ2φn−1ωσn−1T tends to β˜1ωσ1,β˜2ωσ2,…,β˜n−1ωσn−1T. ω can be determined by [23], and the calculation of score function of β˜t can be determined by (3).
When performing various operations on TIFNs in the operator, some algorithms [22] need to be used, defined as follows:(12)β˜1⊕β˜2=β˜2⊕β˜1,β˜1⊗β˜2=β˜2⊗β˜1,β˜1⊗β˜2λ=β˜2λ⊗β˜1λ,λ≥0,β˜1λ1⊗β˜1λ2=β˜1λ1+λ2;λ1,λ2≥0.


## 2.4. Final Evaluation

Each column in the matrix is integrated using the TIFNCWBHG operator to compute the initial weight vector of evaluation factors:(13)η˜t=η˜1t,η˜2t,…,η˜nt,where η˜1t,η˜2t,…,η˜nt represent the integrated TIFNs evaluation value of each attribute by experts at time t. Then, the following matrix is constructed:(14)A˜=η˜11η˜21⋯η˜n1η˜12η˜22⋯η˜n2⋮⋮⋱⋮η˜1pη˜2p⋯η˜np.The score function of each element in the matrix is computed to form a new matrixA:(15)A=η11η21⋯ηn1η12η22⋯ηn2⋮⋮⋱⋮η1pη2p⋯ηnp.Let(16)λi=1pζi∑i=1pηit,i=1,2,…,n,where ζ1,ζ2,…,ζn are weight coefficients, ζ1,ζ2,…,ζn≥0, and ζ1+ζ2+⋯+ζn=1. Then, the final evaluation vector is obtained as follows:(17)h=λ1,λ2,…,λn.The flow chart of the proposed framework of TIFNCWBHG-MAGDM is shown in Figure1.Figure 1
The flow chart of the TIFNCWBHG-MAGDM method.


## 3. Example Analysis

TIFNCWBHG-MAGDM is applied to the evaluation of various safety inputs in a coal mine. There were a few accidents in the mine where all equipment was brand-new and in good condition. Many new miners were recently hired due to the expansion of the production. However, the dust concentration in the workplace was high. Also, they were not well prepared to prevent harmful gases and dust from the lack of attention, and the personal safety protection of miners was poor. In this use-case,t = (1, 2, 3), m = 3, n = 4. Attributes are auxiliary equipment input, safety technology input, propaganda, education input, and industrial hygiene input. Via experts consulting, the multipoint and multiattribute group decision-making TIFNs judgment matrices B˜1,B˜2,B˜3 were established as follows:(18)0.5,0.5,0.6,0.1,0.2,0.20.6,0.7,0.7,0.1,0.2,0.20.6,0.7,0.7,0.1,0.1,0.20.7,0.8,0.8,0.2,0.2,0.20.5,0.6,0.6,0.2,0.2,0.20.6,0.7,0.7,0.1,0.2,0.20.6,0.7,0.7,0.1,0.2,0.20.7,0.8,0.9,0.1,0.1,0.10.5,0.6,0.7,0.1,0.2,0.20.6,0.6,0.7,0.1,0.2,0.20.6,0.6,0.7,0.1,0.1,0.10.7,0.8,0.8,0.1,0.1,0.2,0.5,0.6,0.6,0.1,0.2,0.20.6,0.7,0.7,0.1,0.2,0.20.6,0.6,0.7,0.1,0.2,0.20.7,0.8,0.8,0.2,0.2,0.20.5,0.5,0.6,0.1,0.1,0.20.6,0.6,0.7,0.2,0.2,0.20.6,0.6,0.7,0.1,0.1,0.10.8,0.8,0.8,0.1,0.2,0.20.5,0.5,0.5,0.2,0.2,0.20.5,0.7,0.7,0.1,0.1,0.10.6,0.7,0.7,0.1,0.1,0.10.7,0.8,0.8,0.1,0.2,0.2,0.5,0.5,0.6,0.2,0.2,0.20.6,0.6,0.7,0.2,0.2,0.20.6,0.7,0.8,0.1,0.1,0.10.7,0.8,0.8,0.1,0.2,0.20.5,0.5,0.5,0.1,0.1,0.20.6,0.7,0.8,0.1,0.2,0.20.6,0.7,0.7,0.1,0.2,0.20.7,0.7,0.9,0.1,0.1,0.10.5,0.5,0.6,0.1,0.1,0.10.6,0.6,0.8,0.1,0.1,0.10.6,0.6,0.7,0.2,0.2,0.20.7,0.8,0.8,0.1,0.2,0.2.Then, matrixBtB1 for expert 1 is obtained as(19)B1=0.350.50.550.5750.3750.50.50.70.4250.450.5250.65.The score function value of TIFNs evaluation data for safety input items given by expert 1 (dates obtained from (19)) was analyzed using the columnar diagram, broken line diagram, and radar diagram, which are depicted in Figures 2–4, respectively.Figure 2
Columnar diagram of score function value of TIFNs evaluation data for safety input items (expert 1).Figure 3
Broken line diagram of score function value of TIFNs evaluation data for safety input items (expert 1).Figure 4
Radar diagram of score function value of TIFNs evaluation data for safety input items (expert 1).The matrixBtB2 for expert 2 is obtained as(20)B2=0.40.50.450.5750.40.4250.5250.6250.30.550.5750.6.The score function value of TIFNs evaluation data for safety input items given by expert 2 (dates obtained from (20)) was also analyzed using the columnar diagram, broken line diagram, and radar diagram, which are depicted in Figures 5–7, respectively.Figure 5
Columnar diagram of score function value of TIFNs evaluation data for safety input items (expert 2).Figure 6
Broken line diagram of score function value of TIFNs evaluation data for safety input items (expert 2).Figure 7
Radar diagram of score function value of TIFNs evaluation data for safety input items (expert 2).The computed matrixBtB3 for expert 3 is as follows:(21)B3=0.3250.4250.60.60.3750.5250.50.650.4250.550.4250.6.The score function value of TIFNs evaluation data for safety input items given by expert 3 (dates obtained from (21)) was also analyzed using the columnar diagram, broken line diagram, and radar diagram, which are depicted in Figures 8–10, respectively.Figure 8
Columnar diagram of score function value of TIFNs evaluation data for safety input items (expert 3).Figure 9
Broken line diagram of score function value of TIFNs evaluation data for safety input items (expert 3).Figure 10
Radar diagram of score function value of TIFNs evaluation data for safety input items (expert 3).It can be seen from the data diagrams, including columnar diagrams, broken line diagrams, and radar diagrams, that the score function values of the evaluation of safety input items given by each expert are not consistent. Also, each expert evaluates differently over different time points. It shows the need to consider the impact of timing factors and the influence of different experts on the final evaluation result in the evaluation process. Thus, they are integrated into the final evaluation results in the following.The matricesB¯t are recomputed as follows:(22)B¯1=0.30430.3448034920.29870.32610.34480.31750.36360.36960.31030.33330.3377,B¯2=0.36360.3390.29030.31940.36360.28810.33870.34720.27270.37290.3710.3333,B¯3=0.28890.28330.39340.32430.33330.350.32790.35140.37780.36670.27870.3243.Then, the stability weight vectorφt is calculated using (8), (9), and (10), respectively, as follows: φ1=(0.2497, 0.2502, 0.2503, 0.2497), φ2 = (0.2492, 0.2499, 0.25, 0.251), and φ3 = (0.2499, 0.2499, 0.249, 0.2511). The TIFNCWBHG operator is used to calculate the initial weight vector of evaluation factors:η˜1 = (([0.4678, 0.5103, 0.5963], [0.1532, 0.1895, 0.1995]), ([0.4875, 0.6123, 0.6235], [0.6921, 0.7585, 0.7826]), ([0.1824, 0.1907, 0.2008], [0.6023, 0.7115, 0.7288]), ([0.7316, 0.7601, 0.7723], [0.2006, 0.2019, 0.2218])).η˜2 = (([0.4678, 0.5103, 0.5963], [0.1532, 0.1895, 0.1995]), ([0.4875, 0.6123, 0.6235], [0.6921, 0.7585, 0.7826]), ([0.1824, 0.1907, 0.2008], [0.6023, 0.7115, 0.7288]), ([0.7316, 0.7601, 0.7723], [0.2006, 0.2019, 0.2218])).η˜3 = (([0.5237, 0.6038, 0.7521], [0.1198, 0.1982, 0.2012]), ([0.6607, 0.6621, 0.7127], [0.1733, 0.1987, 0.2057]), ([0.606, 0.6679, 0.712],[0.1006, 0.1078, 0.1257]), ([0.7023, 0.7652, 0.7895], [0.1572, 0.1589, 0.2102])).η˜1,η˜2,η˜3 are combined to obtain the matrix A˜, and the matrix A is obtained by calculating the score function value of each element in matrix A˜:(23)A=0.33830.55680.49880.54950.38170.49960.49930.77580.44150.48030.5530.5843.The score function value of TIFNs evaluation data for safety input items from (23) was analyzed using the columnar diagram, broken line diagram, and radar diagram, which are depicted in Figures 11–13, respectively.Figure 11
Columnar diagram of score function value of TIFNs evaluation data for safety input items, obtained from (23).Figure 12
Broken line diagram of score function value of TIFNs evaluation data for safety input items, obtained from (23).Figure 13
Radar diagram of score function value of TIFNs evaluation data for safety input items, obtained from (23).Finally,h is computed as h = (0.1886, 0.2495, 0.2519, 0.3101).Each component in the vectorh corresponds to the evaluation items (auxiliary equipment input, safety technology input, propaganda and education input, and industrial hygiene input). Therefore, it can be seen from the results that industrial hygiene, propaganda, and education have a great impact on safety input. The auxiliary equipment has the least influence. In the practical application, it was known that the dust concentration in the workplace was very high, and personal protection needed to be improved, and the attention and input in industrial hygiene should be increased. Miners generally lacked the protection awareness of the hazards of dust and harmful gases. The dust and harmful gases should be prevented through careful attention, and it is necessary to carry out extensive safety education and publicity for miners. Since the equipment was relatively new, it is not required to be repaired or changed. The comparison analysis shows that the evaluation results are consistent with the practical situation. Also, the final ranking obtained by using TIFNCWBHG-MAGDM is consistent with that obtained by the method in [4], further verifying the proposed method.The proposed TIFNCWBHG-MAGDM integrates multiexperts, multitime points, multifactors, stability weights, intuitionistic triangular fuzzy score function, and other information for integrated calculation, reflecting more information. In order to validate the performance of the proposed TIFNCWBHG-MAGDM, it is compared to G-INFNs-MCGDM. Table1 summarizes the comparison results, showing that the two methods provide consistent ranking results. The columnar diagram, broken line diagram, and radar diagram are depicted in Figures 14–16, respectively. Note that the G-INFNs-MCGDM requires the accuracy-weight of expert evaluation information and the overall evaluation weight at all levels. Therefore, compared with G-INFNs-MCGDM, TIFNCWBHG-MAGDM is more concise and easy to calculate. On the other hand, the proposed TIFNCWBHG-MAGDM is more feasible and straightforward, which is conducive to the evaluators to carry out various analyses and comprehensive evaluation for the correct judgment. The comparison of the calculation results of TIFNCWBHG-MAGDM and G-INFNs-MCGDM is shown in Table 1.Table 1
Comparison of TIFNCWBHG-MAGDM and G-INFNs-MCGDM.
Evaluation methodAuxiliary equipment inputSafety technology inputPropaganda and education inputIndustrial hygiene inputG-INFNs-MCGDM0.18510.24010.24650.3283TIFNCWBHG-MAGDM0.18860.24950.25190.3101Figure 14
Columnar diagram obtained from Table1.Figure 15
Broken line diagram obtained from Table1.Figure 16
Radar diagram obtained from Table1.


## 4. Conclusion

In order to reduce the difference of the traditional single-value evaluation, this paper proposes the TIFNCWBHG-MAGDM based on the intuitionistic triangular fuzzy number theory and MAGDM theory. First, taking into account the factors such as multitime points, multiattributes, and multiexperts, the judgment matrix of TIFNs is constructed from the comprehensive consideration of multiperspectives. The TIFNCWBHG operator integrating the information of TIFNs score function, stability weight, and position weight is proposed for integrated calculation. Finally, the priorities for safety inputs are determined. The proposed method was evaluated on the practical example of coal enterprise safety inputs, showing that the result is consistent with the actual situation. The main advantages, characteristics, and summary of the proposed method were found through the comparison and in-depth analysis, as follows:(a)
The proposed method improves the accuracy, simplicity, and rationality of the safety input evaluation of coal enterprises. It systematically uses the evaluation matrix, operator, score function formula, and accurate score function formula, which provides a new idea and method for comprehensive, systematic, and in-depth evaluation of coal enterprise safety inputs.(b)
In the TIFNCWBHG operator, a variety of weights are integrated into the system to more accurately describe the complex internal relationship between various factors and attributes to improve the accuracy of the final evaluation further.(c)
In specific decision-making, the stability of expert evaluation information will affect the ranking and selection of alternatives. Therefore, considering the stability of expert evaluation information, the evaluation can be more accurate.(d)
The proposed TIFNCWBHG-MAGDM has the advantages of lower computational complexity and broad applicability. Therefore, it can be used to compare, analyze, and sort the factors in other safety input structure models.Future works will include the application of the proposed method to linguistic decision-making, which is another one of the most active research topics. In addition, we will further optimize the whole evaluation method to make it more accurate and convenient.



---

*Source: 6644806-2021-03-22.xml*