Mine safety assessment using gray relational analysis and bow tie model

Mine safety assessment is a precondition for ensuring orderly and safety in production. The main purpose of this study was to prevent mine accidents more effectively by proposing a composite risk analysis model. First, the weights of the assessment indicators were determined by the revised integrated weight method, in which the objective weights were determined by a variation coefficient method and the subjective weights determined by the Delphi method. A new formula was then adopted to calculate the integrated weights based on the subjective and objective weights. Second, after the assessment indicator weights were determined, gray relational analysis was used to evaluate the safety of mine enterprises. Mine enterprise safety was ranked according to the gray relational degree, and weak links of mine safety practices identified based on gray relational analysis. Third, to validate the revised integrated weight method adopted in the process of gray relational analysis, the fuzzy evaluation method was used to the safety assessment of mine enterprises. Fourth, for first time, bow tie model was adopted to identify the causes and consequences of weak links and allow corresponding safety measures to be taken to guarantee the mine’s safe production. A case study of mine safety assessment was presented to demonstrate the effectiveness and rationality of the proposed composite risk analysis model, which can be applied to other related industries for safety evaluation.


Introduction
Mine production provides the necessary material foundation for the economic and social development of China [1][2][3], but, at the same time, it also causes many accidental deaths [4][5][6]. Therefore, mine safety plays an important role in protecting and fostering the rapid development of the national economy [7][8][9]. To prevent mine accidents, risk assessments first need to be performed.
Some scholars focus on a variety of risk assessment methods, such as the fuzzy evaluation method [10][11][12], neural network [13,14], set pair analysis [15][16][17], cloud model [18][19][20] and gray system theory [21,22]. Niknejad [10] had set up a fuzzy arithmetic for strategic risk management in global production networks, and the risk impacts on the inoperability of alternative global production network configurations, considering different risk scenarios, had been analyzed. Zhang [14] had established a comprehensive assessment model based on a fuzzy neural network to evaluate the environmental impact of mining areas, and the results showed that a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 This study was organized as follows: The fundamental theories of the composite risk analysis model were summarized in Section 2. The applicability of this composite risk analysis model was illustrated, using a case study of mine safety practices, in Section 3. Discussion of the results was presented in Section 4 and conclusions presented in Section 5.

Methods
Theoretical knowledge of the basic model of the variation coefficient method and gray relational analysis and bow tie model included in the present composite risk analysis model are presented in this section.

Gray relational analysis
Gray relational analysis is an important part of gray system theory [32], which is used to measure the gray relational degree between different factors. The greater the gray relational degree between two factors is, the greater the correlation degree. The less the gray relational degree between two factors is, the less the correlation degree. Calculation of the gray relational degree is the core of gray relational analysis and the process as described below.
Let m be the number of programs evaluated, n be the number of assessment indicators of each program evaluated, and the original data matrix of the programs evaluated is where a ij is the original data of the jth assessment indicator of the ith program evaluated.
is the optimal index set, where d j is the optimal value of the jth assessment indicator of programs evaluated. For the larger, the better assessment indicators, d j is the maximum of the assessment indicators; for the smaller, the better assessment indicator, d j is the minimum of the assessment indicators.
If all the assessment indicators of a program being evaluated reached their optimal state, the program evaluated was the optimal index set and the best program. However, in reality, assessment indicators of each program evaluated did not reach optimal values at the same time and gray relational analysis was then used to evaluate the programs.
In the evaluation process, due to the different dimensions of assessment indicators, differences in numerical values in the original data might be large, such that gray relational analysis cannot be directly applied. Therefore, the assessment indicators needed to be rendered dimensionless. Here, a mean transformation was applied to nondimensionalize the assessment indicators, using formulas 1 and 2: After yielding assessment indicators dimensionless, the original data matrix was transferred 2   6  6  6  6  6  4   3   7  7  7  7  7  5 The optimal index set was transfered into . Then, with the optimal index set D Ã the reference sequence and B the sequence compared, the gray relational coefficient of the jth assessment indicator of the ith program evaluated was achieved according to gray relational analysis, and formula 3 was The significant differences between the gray relational coefficients was improved by weakening the influence on the data, due to the maximum absolute value being too large, through the addition of the resolution coefficient ρ in formula 3 and usually set to ρ = 0.5.
Let the weights of assessment indicators of the programs be evaluated by W = [w 1 ,w 2 ,Á Á Á,w n ]. Thus far, the gray relational degree of programs evaluated was achieved according to formula 4: The larger the gray relational degree r i , the closer the ith program evaluated was to the optimal index set D Ã . Accordingly, the order of the programs evaluated was determined.

Variation coefficient method
The gray relational degree of the programs evaluated was achieved by first determining the assessment indicator weights. In determining these weights, the evaluation results deviated because of the influence of subjective factors in the subjective weight method. The variation coefficient method is an objective weight method [27], in which the weight is obtained based on original data, reflecting objective changes in the index information. A larger variation degree in the assessment indicators indicates that the programs evaluated have good uniqueness in this regard and, thus, a larger weight should be given to these assessment indicators; otherwise, a smaller weight should be given to the indicators. The process of calculating assessment indicator weights according to variation coefficient method is described below.
The expectation and standard deviation of assessment indicators were calculated based on formulas 5 and 6: The variation coefficient of assessment indicators was achieved based on formula 7: The weight of assessment indicators was obtained by normalization of the variation coefficient based on formula 8:

Bow tie model
Bow tie model is a safety assessment method that consists of a fault tree on the left side and an event tree on the right side and centered in it is a basic event (Fig 1). The causes of an event are indicated on the left of the bow tie and consequences on the right. The causes are the events that might lead to accidents and the consequences are the losses (including health and property) through accidents. To prevent basic accidents, safety barriers should be adopted. Preventive safety measures are set on the fault side and, thus, come before the basic event. At the same time, mitigative safety measures are set on the event tree side and, thus, come after the basic event.

Results
Assessment indicator weights are first calculated based on the variation coefficient method in this section and, then, gray relational analysis applied to evaluate mine safety practices.

Application
The primary critical factors in mine can be divided as follows according to the classification for casualty accidents of enterprise staff and workers: object strike, mechanical injury, electric shock, crashing from the high, roof fall and wall collapse, blasting, gas explosion, poisoning and asphyxia, vehicle injury, crane injury, fire hazard, collapse, permeable, gunpowder explosion and so on. The above accidents can cause serious injury and death, and the accidental injury and death rate can best reflect mine safety practices and is affected by various factors, mainly including staff training rate, post variation rate, safety management ability, and safety input from the safety management perspective. The staff training rate is the proportion of the number of staff who received job training to the total number of staff enrolled within the same year; post variation rate is the proportion of the number of staff who changed the post to the total number of staff enrolled within the same year; safety management ability refers to the knowledge and ability that safety administration should have, also including that whether the safety management organization and the rules and regulations are perfect, which can be determined according to the experts' scoring; safety input includes security costs, staff training costs and so on. Four mine enterprises [40] have been chosen for program evaluation and their basic information shown in Table 1.

Computational process and results
The staff training rate, safety management ability, and safety input were the larger and, clearly, the better assessment indicators ( Table 1). The assessment indicator of the post variation rate indicated both positive and negative impacts on mine enterprises. A reasonable and appropriate post variation rate can introduce new ideas and thoughts for an enterprise, thus preventing an inherent management model and rigidity of thinking, are conductive for maintaining enterprise vitality. Managers can reallocate staff work according to employee characteristics and outstanding staff can be absorbed to pursue creative work. From these points of view, the post variation rate should be the larger, the better assessment indicator. However, when the post variation rate is beyond normal range, the normal operation of enterprises is affected and management costs increased, resulting in negative effects. To be consistent with the original paper [40] and more easily compare the present evaluation results, here, the post variation rate was considered as the smaller but better assessment indicator. The original data matrix was shown as follows according to Table 1 A The expectation, standard deviation, variation coefficient, and assessment indicator weights were shown in Table 2 according to formulas 5-8. where the unit of safety input is ten thousand yuan.
A nondimensionalized original data matrix of assessment indicators follows according to formulas 1 and 2. The gray relational degrees of mine enterprises were obtained based on the gray relational coefficient and assessment indicator weights according to formula 4 According to the gray relational degree, mine safety from high to low was MC, MD, MB and MA, respectively.

Delphi method.
Assessment indicator weights were determined by Delphi method in the original paper [40]. Delphi method is inviting experts in related fields to make comments on certain issues. Then the views of experts are scientifically integrated, collated and summarized, anonymously feedback the results to each expert to consult again. After many rounds of consult until the views of experts tend to be focused, getting a more consistent and reliable opinion. The assessment indicator weight determined by Delphi method was: [staff training rate, post variation rate, safety management ability, safety input] = [0.24, 0.19, 0.31, 0.26], and the gray relational degree of mine enterprises in the original paper [40]   Comparison of evaluation results from the two different methods were observed not to be consistent. MD has been evaluated to have the highest safety, followed by MC, based on the subjective weight method in the original paper [40]. Here, MC was evaluated to have the highest safety, followed by MD, based on the objective weight method. The main reason for this difference was different weight assignments of assessment indicators. For example, the weight of post variation rate was the largest based on the objective weight method, while it was the smallest based on the subjective weight method. In addition, the original paper did not compare evaluation results with the actual safety of mine enterprises. Therefore, to reduce the error caused by the subjective and objective weight methods in evaluation results, it was necessary to combine the two methods together to evaluate mine enterprise safety.
where w i is the subjective weight of the ith assessment indicator, θ i the objective weight of the ith assessment indicator, λ i the integrated weight of the ith assessment indicator, and n the number of assessment indicators.
There is a problem with the traditional integrated weight calculated by formula 9. If the subjective weight was not equal to the objective weight, the integrated weight should fall in between the subjective and objective weight. In particular, if the subjective and objective weights were equal, the integrated weight should be equal to these weights. However, the integrated weight calculated by formula 9 was outside the range of the subjective to objective weight or objective to subjective weight. For example, one program that was evaluated had three assessment indicators, with subjective weights at w 1 = 0.5, w 2 = 0.

Revised integrated weight method.
To solve this problem, the formula for calculating integrated weight was revised as follow.
where δ 2 [0,1] is the preference coefficient, with δ ! 0 indicating that the integrated weight was mainly determined by the objective weight, and δ ! 1 indicating that the integrated weight was mainly determined by the subjective weight. In practical application, if δ = 0.5, the integrated weight was in the range expected from the subjective and objective weights. Thus, the formula for calculating integrated weight was transformed into formula 11 [43]: The assessment indicator weights determined by subjective Delphi method and objective variation coefficient method were: The evaluation results based on revised integrated weights also indicated MC as having the highest safety, followed by MD, but the difference value of gray relational degree between MC and MD was smaller than with the objective variation coefficient method. Therefore, it was concluded that the evaluation result had a deviation in terms of the actual safety of mine enterprises in the original paper [40].

Compare with fuzzy evaluation method
To validate the revised integrated weight method adopted in the process of gray relational analysis, this section compare the results of safety assessment with fuzzy evaluation method [44]. The brief procedure of fuzzy evaluation method was as follows.
The nondimensionalize of the original data is calculated based on formulas 12 and 13.
where formula 12 is used for the smaller the better assessment indicators; formula 13 is used for the larger the better assessment indicators. After nondimensionalize, the membership of mine enterprises can be calculated based on the dimensionless data and revised integrated weight as follow. According to the fuzzy evaluation method, mine safety from high to low was MC, MD, MB and MA, respectively.
The safety assessment results of mine enterprises using fuzzy evaluation method were the same to the gray relational analysis, in which the assessment indicators weights were determined by revised integrated weight method, proved that the safety assessment method proposed in this paper was feasible.

Bow tie analysis of weak links
It can be seen from the above analysis that evaluation results based on the objective and integrated weight methods were not consistent with the subjective weight method, mainly because the weight of the post variation rate was not assigned in the same manner. Post variation is an important factor affecting mine safety, having great influence on evaluation results. In previous studies, only a simple analysis of the identified weak links have been carried out after the safety evaluation of mine enterprises, which failed to identify the causes and consequences of the weak links and could not effectively improve the safety situation of these enterprises [40][41][42]. Therefore, post variation was taken as the basic event for bow tie analysis (Fig 2), identifying the causes and consequences led by post variation and taking corresponding safety measures to prevent the basic event.
The left of a bow tie was a fault tree analysis that included five causes that can lead to post variation and the right of the bow tie was an event tree analysis that included four consequences of post variation (Fig 2). Twelve preventive safety measures were set on the left to prevent the occurrence of post variation and eight mitigative safety measures set on the right to mitigate the consequences of post variation. The risk of post variation was thus further reduced by this bow tie analysis.

Brief summary of discussion
The present results confirmed that the composite risk analysis model proposed here was successfully applied to the assessment of mine safety practices, in which a new formula for integrated weight was adopted and identified weak links analyzed by bow tie analysis for the first time. The advantages of the composite risk analysis model proposed in this paper were as follows. First, the safety of mines was evaluated so as to compare the safety of different mines. Second, the objective variation coefficient method was first applied to calculate the weight of assessment indicators, avoiding the interference of human factors and also saving a lot of manpower. Third, the bow tie analysis can identify the causes and consequences of the weak links, Safety assessment using gray relational analysis and bow tie model which is easy to understand the critical event in-depth and provide reasonable suggestions for the safe production of mine. Also, to validate the revised integrated weight method adopted in the process of gray relational analysis, the fuzzy evaluation method was used to the safety assessment of mine enterprises.
Motivated by the previous studies on gray relational analysis [32][33][34][35] and bow tie model [36][37][38][39], these two methods were used for the safety assessment of mine for the first time. In order to eliminate the influence of human factors, such as subjective weight method, the objective variation coefficient method [27] was adopted in the process of gray relational analysis, which also saves a lot of manpower. In previous studies [28][29][30][31], formula 9 for calculating integrated weight had a problem in which the integrated weight was outside the ranges of paired subjective and objective weights. To solve this problem, an expectation formula that ameliorated this problem was adopted. In addition, only a simple analysis of the identified weak links have been carried out after the safety evaluation of mine enterprises [40][41][42], which failed to identify the causes and consequences of the weak links and could not effectively improve the safety situation of these enterprises. Therefore, post variation was taken as the basic event for bow tie analysis for the first time, identifying the causes and consequences led by post variation and taking corresponding safety measures to prevent the basic event. The approach proposed in this paper can be applied to other related industries for safety evaluations.
To simplify the discussion, the perference coefficient δ in formula 10 was chosen as only 0.5. Future research should focus on the influence of this preference coefficient in safety evaluation results.

Conclusion
A composite risk analysis model of mine safety practices based on gray relational analysis and bow tie model was proposed. The main conclusions were as follows.
First, assessment indicator weights were determined by the objective variation coefficient method, and the safety evaluation results for mine enterprises were obtained based on gray relational analysis, mine safety from high to low was MC, MD, MB and MA, respectively.
Second, a new formula was adopted to calculate the integrated weights, and mine safety from high to low was also MC, MD, MB and MA. The evaluation results based on integrated weights also indicated MC as having the highest safety, followed by MD, but the difference value of gray relational degree between MC and MD was smaller than with the objective weight method.
Third, to validate the revised integrated weight method adopted in the process of gray relational analysis, the fuzzy evaluation method was used to the safety assessment of mine enterprises. Both methods had the same safety assessment results, showed that the safety assessment method proposed in this paper was feasible.
Fourth, the identified weak link post variation was analyzed using bow tie model for the first time, in which twelve preventive safety measures were set on the left to prevent the occurrence of post variation and eight mitigative safety measures set on the right to mitigate the consequences of post variation. The risk of post variation was thus further reduced by this bow tie analysis.