Comprehensive evaluation model for coal mine safety based on uncertain random variables

Safety Science 68 (2014) 146–152

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Comprehensive evaluation model for coal mine safety based on uncertain random variables Jiqiang Chen a, Litao Ma a, Chao Wang b, Hong Zhang a, Minghu Ha a,⇑ a b

School of Science, Hebei University of Engineering, Handan 056038, PR China School of Economics and Management, Hebei University of Engineering, Handan 056038, PR China

a r t i c l e

i n f o

Article history: Received 17 July 2012 Received in revised form 18 December 2013 Accepted 27 March 2014

Keywords: Uncertain random variable Chance measure Coal mine safety Discriminatory weight

a b s t r a c t In the comprehensive evaluation of coal mine safety, the indices in the hierarchical structure are often expressed as uncertain random variables. Therefore, a new assessment approach based on uncertain random variables is proposed. Firstly, the values of the indices are described by uncertain random variables and the degree of an index belonging to a scale is described by the chance of an uncertain random event. Secondly, in order to show the different roles of different components of an index, the definition of discriminatory weight is introduced, and then an algorithm which is a generalization of the weighted average model is proposed based on discriminatory weight to realize the transformation from the scales of the underlying indices to the scales of the target. At last, a numerical evaluation example for coal mine safety is provided to illustrate the design methodology and give a better insight into the algorithmic details. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The apparent characteristic of coal mine production is underground operation affected by geological conditions, geological structures and coal seam construction. At the same time, the disasters such as coal dust, methane, water, and roof collapse are liable to occur under the mining process and seriously imperil the mining safety and lives of miners. Therefore, the production technology and management are required to be further improved so as to predict the accidents in advance and take appropriate measures timely. Among the methods to predict the accidents, comprehensive evaluation for the coal mine safety is an important method. At present, the main methods to evaluate the coal mine safety are expert evaluation method, data envelopment analysis, fuzzy comprehensive evaluation, grey relational analysis, analysis hierarchy process and so on. For example, a comprehensive assessment method for coal mine safety based on the analytic hierarchy process and grey analysis (Liu et al., 2007a,b; Fera and Macchiaroli, 2010) was studied; an evaluation model of coal mine safety based on rough set theory and AHP was investigated (Li and He, 2008; ⇑ Corresponding author. Tel.: +86 0310 2079773, +91 13323203786. E-mail addresses: [email protected] (J. Chen), [email protected] (L. Ma), [email protected] (C. Wang), [email protected] (H. Zhang), mhha@mail. hbu.edu.cn (M. Ha). http://dx.doi.org/10.1016/j.ssci.2014.03.013 0925-7535/Ó 2014 Elsevier Ltd. All rights reserved.

Cheng and Yang, 2012); a coal mine safety assessment model using fuzzy logic was proposed (Ataei et al., 2009). For the above methods, the assessment is based on the hierarchical structure of the comprehensive evaluation usually, and there are mainly three problems in the assessment: One is determining the hierarchical structure of the coal mine safety and the evaluation scales, another is determining the scales of the underlying indices according to the criteria, and the last one is realizing the transformation from the scales of underlying indices to the scales of the total target. For the first problem, the hierarchical structure of the coal mine safety and the evaluation scales can be determined by experts according to the criteria. For the second one, it should be noted that there is a kind of uncertainty in confirming the scales of the indices. That is, it is not the case that an index belongs to one scale absolutely and does not belong to another at all. Therefore, this kind of uncertainty is often treated as fuzziness (Chelgani et al., 2011; Zheng et al., 2012), grayness (Wang et al., 2011), roughness (Li and He, 2008; Cheng and Yang, 2012) and so on. However, in the application, a lot of survey shows that this kind of uncertainty is not fuzziness, grayness or roughness, but a mixture of randomness and uncertainty (Liu, 2013). For example, in order to obtain the thickness x of a coal seam, the experts will first select several places randomly (it is randomness), and then measure them respectively. At last, they may think that the coal seam is with the thickness of

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1000 m approximately (it is uncertainty). Thus, the thickness of the coal seam is a variable with both randomness and uncertainty. For this reason, the uncertain random variable is proposed (Liu, 2013) to describe this phenomena, and chance measure is introduced (Liu, 2013) to describe the chance of an uncertain random event occurring. For another thing, in order to confirm the degree of the total target belonging to each scale by the degree of the underlying indices belonging to each scale, we have to know how much effective information is offered by the underlying indices. For example, there are two students A and B. Let the characteristic vector (Sex, Age, Height, Weight) be (male, 20 years, 180 cm, 80 kg) and (male, 20 years, 175 cm, 70 kg) respectively. So we can differentiate them by ‘‘Weight’’ directly, instead of ‘‘Age’’ only. That’s to say, the indices ‘‘Weight’’ and ‘‘Age’’ play different roles in the differentiation. Therefore, we introduce the definition of discriminatory weight to show the different roles of different components. For the third problem, how do we realize the transformation from the scales of underlying indices to the scales of the total target? At present, the transformation are often realized by AHP method (Liu et al., 2007a,b; Woo and Lee, 2011; Zheng et al., 2012), the weighted-average method (Chen et al., 2006), fuzzy max–min method (Zhang et al., 2007), T  S fuzzy model (Kruszewski et al., 2008) and other improved methods (Chen et al., 2009; Chuang et al., 2009; Maiti et al., 2009). And they can tell us the correct results. But in application we are more concerned about the reasons and then we can improve the safety scale of the system. However, the above methods cannot show us the reasons directly. Therefore, these methods will encounter limitations in the practical application. If we can find a method to tell us the correct assessment results, and also the reason of the results at the same time, it will bring great convenience for us. Therefore, it is necessary for us to investigate new assessment models for the coal mine safety both in theory and application. As we all know, random and uncertainty are two kinds of basic uncertainties in real world, and probability theory (Kolmogorov, 1933) and uncertainty theory (Liu, 2007, 2011) are two branches of mathematics for dealing with random phenomena and uncertain phenomena respectively. As the improvement of understanding about uncertain phenomena, some researchers begin to study the complex system which includes both random and uncertainty. From 2011, this problem is paid much attention gradually and the uncertain random variable is introduced (Liu, 2013) to solve this problem. The uncertain random variable attempts to model the variable with both randomness and uncertainty. In nature, uncertain random variable is a random element taking values in the set of uncertain variables, and a generalization of uncertain variable and random variable. For dealing with the uncertain random variable, a chance measure on the basis of uncertain measure and probability measure is introduced (Liu, 2013), and it attempts to model the chance that an uncertain random event occurs. What is more, it is also a generalization of uncertain measure and probability measure. Thus, uncertain random theory (Liu, 2013) provides a rigorous mathematical foundation for the coal mine safety evaluation. In addition, there are many problems with both randomness and uncertainty in operational research, management science, reliability engineering, aerospace technology, risk analysis, weather forecasting, military behavior, economic behavior and many other areas. In this environment, the uncertainty random theory has a broad application prospects. Therefore, with the chance measure and uncertain random variable, this paper proposed a new assessment model for the coal mine safety evaluation, which generalizes the widely used weighted-average method. The structure of this article is as follows. In Section 2, we briefly review some relevant notations of uncertain random theory. In

Section 3, a simple evaluation example is given to illustrate the design methodology. In Section 4, we describe the values of indices by uncertain random variables and the degree of an index belonging to a scale by the chance of an uncertain random event occurring. In Section 5, in order to show the different roles of different components of an index, we introduce the definition of discriminatory weight. And then an algorithm based on discriminatory weight is proposed to realize the transformation from the scales of underlying indices to the scales of the total target. At last, a comprehensive evaluation model for coal mine safety is established in Sections 6 and 7 is the conclusions.

2. Preliminaries Let C be a nonempty set, L be a r-algebra over C, and M be a uncertain measure on L. Each element K in L is called an event, and MfKg indicates the possibility of event K occurring. Definition 1 (Liu, 2013). An uncertain random variable is a function n(x) from a probability space ðX; A; PrÞ to the set of uncertain variables such that MfnðxÞ 2 Bg is a measurable function of x for any Borel set B of R. For simplicity we often use n to describe an uncertain random variable. Roughly speaking, an uncertain random variable is a measurable function from a probability space to the set of uncertain variables. In other words, an uncertain random variable is a random element taking uncertain values. Then a random variable and an uncertain variable are both a degenerate uncertain random variable (see Liu, 2013). Theorem 1 (Liu, 2013). Assume that n is an uncertain random variable. Then for any Borel set B of R, the uncertain measure MfnðxÞ 2 Bg is a random variable.

Definition 2 (Liu, 2013). (Arithmetic of the uncertain random variables) Let f:Rn ! R be a measurable function, and n1, n2, . . ., nn uncertain random variables on the probability space ðX; A; PrÞ. Then n = f(n1, n2, . . ., nn) is an uncertain random variable defined by

nðxÞ ¼ f ðn1 ðxÞ; n2 ðxÞ;    ; nn ðxÞÞ;

x2X

in the sense of operations of uncertain variables. The above definition providing the arithmetic of uncertain random variables, which tell us that the sum n = n1 + n2 and the product n = n1  n2 of two uncertain random variables n1 and n2 are also an uncertain random variable. Definition 3 (Liu, 2013). Let n be an uncertain random variable, and let B be a Borel set of R. Then the chance of uncertain random event n e B is defined by

Chfn 2 Bg ¼

Z

1

Prfx 2 XjMfn 2 Bg P r gdr;

0

and Ch{n e B} indicates the possibility that event n e B occurring. Note that Mfn 2 Bg is a measurable function of x and then it is a random variable. The chance measure is in fact an expected value of this random variable. And if an uncertain random variable n degenerates to a random variable g, then Ch{n e B} = Pr {g e B}; if an uncertain random variable n degenerates to an uncertain variable s, then Chfn 2 Bg ¼ Mfs 2 Bg. In the following, an example is provided to illustrate the calculation of Ch{n e B}.

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Example 1 (Liu, 2013). A linear uncertain variable Lða; bÞ is an uncertain variable whose uncertainty distribution is

8 if x 6 a > < 0; UðxÞ ¼ ðx  aÞ=ðb  aÞ; if a 6 x 6 b > : 1; if x P b: Consider an uncertain random variable taking values of linear uncertain variable,

 n¼

Lð0; 3Þ with probility 0:6 Lð2; 4Þ with probility 0:4:

At first, the uncertain measure that the uncertain variable n1  Lð0; 3Þ belongs to [2, 3] is Mfn1 2 ½2; 3g ¼ 13 : In addition, the uncertain measure that the uncertain variable n2  Lð0; 3Þ belongs to [2, 3] is Mfn2 2 ½2; 3g ¼ 12 : It follows from Definitions 2 and 3 that the chance measure is

Chfn 2 ½2; 3g ¼ Mfn1 2 ½2; 3g  0:6 þ Mfn2 2 ½2; 3g  0:4 ¼ 0:4: Next, in order to express the problem of comprehensive evaluation model for coal mine safety and the method clearly, we will give a simple example to illustrate the problem first, and then explain the idea to solve the problem. 3. A simple evaluation example to illustrate the design methodology Suppose that the system can be represented by a three layer hierarchical structure showing as Fig. 3.1, with the total target, the intermediate layer and the underlying layer. In order to obtain the scale of coal mine safety and find out the factors affecting coal mine safety, domain experts divide the degree of the security into three scales, namely ‘‘Light’’, ‘‘Medium’’ and ‘‘Heavy’’. As the system is very complex and experts cannot evaluate the level of environment pollution directly, they establish a hierarchical structure according to the actual situation of the system (see Fig. 3.1). Therefore, they can evaluate the scale of the underlying index which is easier and more convenient to evaluate than the total target. Then, obtain the safety degree of the intermediate indexes with a method, and then the safety degree of the total target. Therefore, in order to obtain the degree of the total target, there are four steps: (1) (2) (3) (4)

establish the hierarchical structure for the system; confirm the safety scales; determine the scales of the underlying indexes; construct a method to realize the transformation from the scales of the underlying indexes to the scales of the intermediate indexes, and then to the scales of the total target.

Then, we can analysis the evaluation results and find out which factors cause the system in a low safety scale, so as to improve the safety of the coal mine. For the above four steps, the first one can be settled by domain experts according to the actual situation of the system. For the second one, assumed that the experts divide the evaluation scales into three scales (‘‘Light’’, ‘‘Medium’’ and ‘‘Heavy’’), which are represented by closed intervals S1 = [a0, a1], S1 = [a1, a2], S1 = [a2, a3] respectively. As the indexes are often exhibited as uncertain random variables n, we can describe the degree of the indices belonging to each scale by the chance of an uncertain event n e Si, i = 1, 2, 3. For the last one, which way can we use to realize the transformation from the scales of the underlying indexes to the scales of indexes immediately above, and then to the scales of the total target? In literature (Li and He, 2008; Liu et al., 2007a,b), the AHP method is used. However, we can only obtain the results and do not know the reason, so it is inconvenient for us to improve the safety of the coal mine in application. Therefore, in this paper we investigate a new method based on discriminatory weight to realize this transformation. And by this approach, the results and the reason can be obtained at the same time. In the following, suppose that the hierarchical structure has been established (shown as Fig. 3.2), and then we will confirm the evaluation scales and determine the scales of the underlying indexes. 4. Confirming the evaluation scales and determining the scales of the underlying indices 4.1. Confirming the evaluation scales of the system In order to give a comprehensive evaluation for coal mine safety, the evaluation scales should be determined firstly. For simplicity, denote M = {1, 2, . . ., m}, N = {1, 2, . . ., n}, P = {1, 2, . . ., 5}. Let the score of underlying indices Cj(j e M) be yj e [0, 5]. According to the evaluation criteria and the actual situation of the system, domain experts divide the scales into 5 scales, namely ‘‘Very heavy’’, ‘‘Heavy’’, ‘‘Medium’’, ‘‘Light’’, ‘‘Very light’’ represented by S1 = [0, 1], S2 = [1, 2], S3 = [2, 3], S4 = [3, 4], S5 = [4, 5] respectively. Then how do we describe the degree that underlying indices belonging to each scale is the next problem in the evaluation. In application, there are both randomness and uncertainty in describing the degree that underlying indices belonging to each scale, then according to the uncertain random theory, the value of an indexes can be viewed as uncertain random variables nj (see Definition 1), and the degree that underlying indices belonging to each scale can be described as an uncertain random event nj e Bk, j e M, k e P. In the following, we will explain it in detail. 4.2. Determining the scales of the underlying indexes As there are both uncertainty and randomness in the determining the values yj of index Cj, so we introduce the uncertain random variables nj to describe them. Let the uncertain random variable be

nj ¼

Fig. 3.1. A simple hierarchical structure.

8 sj1 > > > <

with probility pj1

sj2 with probility pj2

> > > :

...

; j 2 M:

sjn with probility pjn

Then the uncertain measure Mfsji 2 Sk g (k e P) can be calculated by uncertain distribution (the readers can consult Liu (2011) for more details), and similar to Example 1, the chance Ch{nj e Sk} can be calculated by Definition 3.

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J. Chen et al. / Safety Science 68 (2014) 146–152

Fig. 3.2. The hierarchical structure of the comprehensive evaluation of coal mine safety.

Therefore, the chances that index Cj belonging to each scale Sk(k e P) can be denoted by the chance vector

Chj ¼ ðChj1 ; Chj2 ;    ; Chj5 Þ

ð4:1Þ

where the element Chjk represents the chance of uncertain random event nj e Sk (k e P). We call the matrix

UðTÞ ¼ ðChjk Þm5 ;

j 2 M:

ð4:2Þ

the state-transition matrix of target T. In the following, we will construct the algorithm based on discriminatory weight to realize the transformation from the scales of underlying indices to the scales of the target.

0 6 xj ðTÞ 6 1 and

m X

xj ðTÞ ¼ 1

In order to confirm the scales of the total target by the scales of the underlying indices, we have to know how much effective information is offered by scales of the underlying indices. Therefore, to exhibit the different roles of different components of an index, we will give the definition of discriminatory weight first. 5.1. The definition of the discriminatory weight For simplicity, we denote Chk(T) the chance that uncertain random event T e Sk (k e P) and the vector Ch(T) = (Ch1(T), . . ., Ch5(T)) be the chance vector that T belongs to each scale. As the chance Chk(T) of target T is determined by the chance vectors of the underlying indexes, if the chance vectors of the underlying indexes are specified, then Chk(T) is certain and it just depends on the method we use. Therefore, we carry out the following calculations.

hold true. Note 1. The bigger xj(T) is, the bigger contribution Cj makes. When xj(T) = 1, Cj makes the biggest contribution regarding the T classification. When xj(T) = 0, the Cj does not have an effect to the classification of T.

According to the state-transition matrix U(T) of target T and the weights kj ðTÞðj 2 MÞ, the procedure to calculate target credibility Chk(T) is as follows. Step 1. In order to know how much effective information is offered by Chj, we have to calculate the discriminatory weight xj(T) of indexes Cj (j e M) by virtue of formulas (5.1)–(5.3). Step 2. As xj(T) show us how much effective information offered by Chj, then calculate

xj ðTÞ  Chjk ; ðj 2 M; k 2 PÞ;

ð5:1Þ

ð5:5Þ

Step 3. Multiply xj(T)  Chjk by kj ðTÞ to make sure that the value kj ðTÞ  xj ðTÞ  Chjk is comparable and added directly, then we have

kj ðTÞ  xj ðTÞ  Chjk ; ðj 2 MÞ;

ð5:4Þ

j¼1

5.2. The transformation algorithm of the scales based on discriminatory weight

5. The algorithm based on discriminatory weight

5 X Hj ðTÞ ¼  Chjk  lg Chjk ;

We call xj(T) the discriminatory weight of index Cj with respect to T. And by (5.3) we have

ðj 2 M; k 2 PÞ;

ð5:6Þ

Step 4. Calculate

k¼1

V j ðTÞ ¼ 1 

1 Hj ðTÞ; lg 5

M k ðTÞ ¼ ðj 2 MÞ;

, m X xj ðTÞ ¼ V j ðTÞ V t ðTÞ; t¼1

ðj 2 MÞ:

ð5:2Þ

ð5:3Þ

m X kj ðTÞ  xj ðTÞ  Chjk ;

ðk 2 PÞ:

ð5:7Þ

j¼1

Mk(T) represents the possibility that target T belonging to Sk. And the bigger Mk(T) is, the more possible T belongs to the state of Sk (k e P).

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Step 5. Normalize Mk(T), we have

, Chk ðTÞ ¼ Mk ðTÞ

5 X

M t ðTÞ;

ðk 2 PÞ

ð5:8Þ

t¼1

Therefore, Chk(T) satisfies

0 6 Chk ðTÞ 6 1;

5 X Chk ðTÞ ¼ 1:

ð5:9Þ

k¼1

We call Chk(T) the chance that target T belongs to scale Sk, and the bigger Chk(T) is, the more possibility T belonging to Sk. The transformation method is denoted by M(1, 2, 3) method. Note 2. For every row vector (Chj1, Chj2, . . ., Chj5) in the statetransition matrix U(T), if there is one component Chjk = 1, k e P and the rest components are 0, then Hj(T) = 0, Vj(T) = 1, xj ðTÞ ¼ 1n : Therefore, by (5.8) and (5.9) we have n X Chk ðTÞ ¼ kj  Chjk ;

ðk 2 PÞ:

ð5:10Þ

Then the model M(1, 2, 3) degenerates to the weighted-average model. In other words, the weighted-average method is a special case of the model M(1, 2, 3).

In order to determining the scales of the total target, an example is given first. Let the situation of air pollution be divided into three scales, namely ‘‘Light, Medium and Heavy’’. Let the chance vector (ChL(x), ChM(x), ChH(x)) of the sample x belonging to each scale be (0.4, 0.3, 0.3). According to the maximum principle (namely the scales of x is the scale that x with the biggest chance), x is considered to be the scale ‘‘Light’’ with the chance 0.4. But the sum of the chance of x belonging to ‘‘Medium’’ and ‘‘Heavy’’ is 0.6. Thus, defining x to be the scale of ‘‘Light’’ is not reasonable. Therefore, the confidence degree principle is proposed in literature (Cheng, 1997). Confidence degree principle (Cheng, 1997). Let C1, C2, . . ., CK be an order division, k ð0:5 < k < 1Þ be the confidence level and ‘‘  ’’be a partial relation. If C1  C2  . . .  CK,

)

k X

Take the calculation of the chance vector of B1 (Geological conditions) as an example. r Presides B1 over the four underlying indexes C11, . . ., C14, the state transition matrix is

0 B 0 B UðB1 Þ ¼ B @ 0

0:2 0:8

0

0:7 0:3 C C C: 0:3 0:7 0 A

0

0

1

0

0

0:1 0:8 0:1

0

0

s The weight vector of C11, . . ., C14 is

5.3. Identification principle

k0 ¼ min k :

6.2. Calculation of the chance vectors of the intermediate indices

0

j¼1

(

scales(‘‘Very low’’, ‘‘Low’’, ‘‘Medium’’, ‘‘High’’ and ‘‘Very high’’) denoted by S1 = [0, 1], S2 = [1, 2], S3 = [2, 3], S4 = [3, 4], S5 = [4, 5] separately and the chance vectors of underlying indices (see the last column in Table 6.1) are obtained according to the score and the uncertain random variables by Definition 3. After obtaining chance vectors of underlying indexes (see the last column in Table 6.1), we can use M(1, 2, 3) model to calculate chance vectors of the intermediate indexes immediately. Then calculate chance vectors of the uncertain random event A e Sk in the top layer. Hence, realize the transformation of the chance vectors from the underlying indexes to the top goal finally.

lx ðC i Þ P k; 1 6 k 6 K ;

ðk11 ; k12 ; k13 ; k14 Þ ¼ ð0:0667; 0:1333; 0:2667; 0:5333Þ: t Calculating the chance vector Ch(B1) of B1, we have

ChðB1 Þ ¼ ð0:0522; 0:4178; 0:1478; 0:3419; 0:0403Þ: Similarly, we can obtain the chance vectors

ChðB2 Þ ¼ ð0; 0:0196; 0:2299; 0:4503; 0:3002Þ; ChðB3 Þ ¼ ð0:0098; 0:1497; 0:4799; 0:3370; 0:0236Þ; ChðB4 Þ ¼ ð0; 0; 0:5989; 0:2623; 0:1388Þ; ChðB5 Þ ¼ ð0; 0:0532; 0:3862; 0:2985; 0:2621Þ; ChðB6 Þ ¼ ð0; 0:0108; 0:2068; 0:4051; 0:3773Þ:

i¼1

6.3. Calculation of the chance vector of the total target

if C1  C2  . . .  CK,

( k0 ¼ max k :

k X

)

lx ðC i Þ P k; 1 6 k 6 K ;

i¼1

then x belongs to C k0 with the confidence level k at least. 6. An application example In this part, we will provide a numerical evaluation example to illustrate the design methodology. 6.1. The system of comprehensive evaluation model for coal mine safety and some relevant data Using the above model, we can carry on the appraisal to comprehensive evaluation for coal mine safety. In the following, take the above coal mine safety evaluation as an example to explain the application of the model. The index system of the coal mine safety assessment (see Table 6.1) and the weights in the brackets are from literature (Liu et al., 2007a,b). Domain experts divide the safety scale into five

A By the chance vectors Ch(B1), . . ., Ch(B6) of indices B1, . . ., B6, we can obtain the state transition matrix of target A as

0

0:0522 0:4178 0:1478 0:3419

0:0403

1

B 0 0:0196 0:2299 0:4503 0:3002 C C B C B B 0:0098 0:1497 0:4799 0:3370 0:0236 C C: UðAÞ ¼ B B 0 0 0:5989 0:2623 0:1388 C C B C B @ 0 0:0532 0:3862 0:2985 0:2621 A 0

0:0108

0:2068

ð5:11Þ

0:4051 0:3773

Calculating as the same procedure in Section 6.2, by virtue of matrix U(A) we can obtain the chance vector of A as

ChðAÞ ¼ ð0:0153; 0:1445; 0:2824; 0:3658; 0:1920Þ: 6.4. Identification In this example, the confidence level of A belonging to S3 (Medium) is not higher than 0.44 (0.0153 + 0.1415 + 0.2824 6 0.44), and the confidence level of A belonging to S4 (High) is no less than 0.8

151

J. Chen et al. / Safety Science 68 (2014) 146–152 Table 6.1 The evaluation system of coal mine safety. The total target

The intermediate indexes

The underlying indexes

Chance vectors {S1, S2, S3, S4, S5}

The evaluation system of coal mine security A

Geological conditions B1 (0.3626)

Coal seam C11 (0.06667) Hydrogeological C12 (0.1333) Roof and floor structure C13 (0.2667) Gas conditions C14 (0.5333) Mechanization level C21 (0.4551)

{0, 0, 0.20, 0.80, 0} {0, 0, 0, 0.70, 0.30} {0, 0, 0.30, 0.70, 0} {0.10, 0.80, 0.10, 0, 0} {0, 0, 0.20, 0.80, 0}

Technology and equipment B2 (0.2309)

Equipment configuration status C22 (0.1411) Equipment maintenance C23 (0.2627) Research and innovation C24 (0.1411)

{0, 0, 0, 0.30, 0.70}

The quality of personnel B3 (0.0899)

Personnel structure C31 (0.1111) Technical level C32 (0.2222) Cultural level C33 (0.2222) Safety awareness C34 (0.4445)

{0.12, 0.68, 0.20, 0, 0} {0, 0, 0.11, 0.88, 0.11} {0, 0, 0.30, 0.70, 0} {0, 0.20, 0.80, 0, 0}

Safety education B4 (0.0486)

Education plans C41 (0.1111) Pre-job safety training C42 (0.2222) Daily safety education C43 (0.2222) Special types of training C44 (0.4445)

{0, 0, 0.20, 0.80, 0} {0, 0, 0, 0.30, 0.70} {0, 0, 0.80, 0.20, 0} {0, 0, 0.85, 0.15, 0}

Environmental Safety B5 (0.1601)

Noise control C51 (0.0660) Lighting C52 (0.1322) Dust control C53 (0.4200) Temperature and humidity C54 (0.1322) Air Quality C55 (0.2496)

{0, 0, 0.20, 0.80, 0} {0, 0, 0, 0.30, 0.70} {0, 0.15, 0.70, 0.15, 0} {0, 0, 0, 0.10, 0.90} {0, 0, 0.50, 0.50, 0}

Management level B6 (0.1079)

Leader’ safety awareness C61 (0.3754) Safety input C62 (0.1389) Safety culture C63 (0.1281) Safety warning C64 (0.2105) Accident processing C65 (0.1471)

{0, 0, 0.20, 0.80, 0} {0, 0, 0, 0.30, 0.70} {0, 0.10, 0.80, 0.10, 0} {0, 0, 0, 0, 1.00} {0, 0, 0.41, 0.59, 0}

6.6. Results comparison

Table 6.2 The results comparison with other two methods. Methods

The M(1,2,3) model AHP method Fuzzy max–min method

Pollution scales Very low S1

Low S2

Medium S3

High S4

Very high S5

0.0153 0.0306 0.0586

0.1445 0.1568 0.1807

0.2824 0.2952 0.2804

0.3658 0.3065 0.2901

0.1920 0.2109 0.1902

Table 6.3 The differences between two adjacent scales in the above three methods. Methods

The M(1,2,3) model AHP method Fuzzy max–min method

{0, 0.10, 0.70, 0.20, 0} {0, 0, 0, 0, 1.00}

Differences S2 and S1

S3 and S2

S4 and S3

S5 and S4

0.1292 0.1262 0.1211

0.1379 0.1384 0.0997

0.0834 0.0113 0.0097

0.1738 0.0956 0.0999

With the same index system and the same data, we calculate the chance vectors with the AHP method and the fuzzy weighted-average model respectively. The results are shown as Table 6.2. Table 6.2 illustrates that the above three methods can all get the same classification results. But there is some difference in the confidence level of each scale. And also, it is well known that the larger the difference between two adjacent scales is, the stronger classification ability of the method is. Therefore, we analysis the differences between two adjacent scales in Table 6.3. From Table 6.3 we can see that the differences between S5 and S4 with M(1, 2, 3) method is much bigger than the other two methods; the remaining three differences are almost the same. It shows that the M(1, 2, 3) method has a stronger classification ability. What is more, the M(1, 2, 3) method points out the weak factor needs to be further improved without any other calculations at the same time. 7. Conclusions

(0.0153 + 0.1415 + 0.2824 + 0.3658 P 0.8). Comparatively, it is more reasonable to define A in grade S4. Therefore, the safety scale of this coal mine is ‘‘High’’ with the confidence level 0.8 at least. 6.5. Results analysis The confidence level of A belonging to S4 (High) is no less than 0.8, which is a relatively good result. It indicates that the coal mine system is in a relatively safe state. From matrix U(A) we have: B1, . . ., B6 belong to S4 with the confidence level about 0.95, 0.7, 0.97, 0.86, 0.73, 0.63 respectively. This illustrates that the indexes B2, B5 and B6 are need to be further improved. A lot of surveys (Liu et al., 2007a,b) show that this result is consistent with the unit’s actual situation.

Based on uncertain random theory, a new assessment method for coal mine safety based on chance measure and uncertain random variable is proposed in this paper. And this new method generalizes the weighted average method. The numerical evaluation example illustrates that we can obtain the accurate results with this new method, and the indices need to be further improved at the same time. The comparison with other two methods shows that the new method has a stronger classification ability. Therefore, this new method brings great convenience for the improvement of the coal mine safety. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 60773062 and 61073121), the Natural Science

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Foundation of Hebei Province of China (Nos. F2012402037 and A2012201033), the Natural Science Foundation of Hebei Education Department (No. Q2012046). The authors also thank anonymous reviewers for their constructive comments and suggestions.

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