Fuzzy FMECA risk evaluation and its applications in Chinese train control systems based on cloud model

Abstract

In view of defects that traditional Fuzzy Analytical Hierarchy Process (FAHP) cannot accurately describe the ambiguity and randomness of the assessment, and as well as inconsistency existed in judgment process, in this paper a novel risk evaluation method is proposed using Fuzzy Failure Modes Effects and Criticality Analysis (FMECA) based on could model. The method firstly applies FMECA to identify the risk, and then uses FAHP to determine the subjection degree function with cloud model based. In the end, the group decision can be conducted with the synthetically aggregated cloud model, which can be directly observed through the distribution of the cloud pictures. Compared with traditional FAHP, the relevant practical examples in Chinese train control (CTC) systems show that the results of the two possess difference due to their original data coming from different 20-expert questionnaires, the reason is found that there exists inconsistence in 20-expert questionnaires on FAHP via t-examination method. Hence, though another 20-expert questionnaires and after inconsistence test, we obtain consistent result in both methods, but the Fuzzy-FMECA with cloud model based could implement the transformation between exact value and quantized one by incorporating the ambiguity and randomness, and provide more abundant information than subjection degree function of the conventional FAHP method, and possesses better consistency, and is a feasible and more effective decision method. In addition, the correlation coefficient method and center-of-gravity method are also applied to verify the correctness and effectiveness of the proposed method, and such that it can be widely applied to solve real-world practical issues.

Keywords

Risk fuzzy analytical hierarchy process (FAHP)cloud model Chinese train control(CTC) systems validation

1 Introduction

With high-speed railway distance extending and speed increasing in China, the requirements on railway signal security are becoming more and more significant. After “4.28” Jiaoji line and “7.23” Yongwen line particularly major traffic accidents occur in succession, Railway Ministry attaches importance to the security risk evaluation for high speed railway. For this reason, professional testing for high speed railway equipments and risk assessment are forcibly carried out. Considering that China does not have the comprehensive security certificate, it is urgent to study the risk assessment methods for signal equipments of high speed railway. The train control center is the core equipment of train operation control system, which plays a crucial role in protecting the safety operation of the train [1]. Implementing the risk assessment on train control center can help to improve the system security and safety certification for China railway signal equipments. However, it is hard to quantify the risk only depending on the collected data full of diverse sorts of fuzziness and randomness. Hence, all kinds of uncertainty methods are applied to assess railway signal system risk, e.g., Fault Tree Analysis (FTA) [2 –4], Event Tree Analysis(ETA) [5, 6], Failure Mode Effects and Criticality Analysis (FMECA) [7], Bayesian Network (BN) [8], Grey system theory [9], and Markov model [10], and also Fuzzy Analytical Hierarchy Process (FAHP) [11], and etc. However, the randomness is usually ignored during the assessment process in these methods, and meanwhile, an effective and suitable alteration model from quality to quantity does not be considered. Moreover, diverse danger elements reflect diverse states and significance, such that each cell possesses its own weight and risk degree in the index system [12, 13]. In terms of traditional analysis methods, Analytical Hierarchy Process (AHP) is a frequently-used mean to determinate the opposite weight of every element in the index frame, which makes use of pair-wise comparisons to obtain a scale of opposite significance for substitutes, and then establishes a pair-wise contrasting matrix applying it. But the ambiguities exist between the linguistic descriptive scene and the scale relationship, such that the qualitative language division is unscientific, which leads to the unconformity between AHP and swarm decision-making [14, 15].

Decision-making can be regarded as a process to implement the selecting or ranking alternatives from a universe based on the decision information under the acting conditions. Group decision-making (GDM) attempts to provide solutions to solve such complex real-world problems. However in real life, many decision-making problems in fields like industrial engineering, management science, or operational research, usually require multi-attribute decision-making (MADM), which can be considered as a process to select the optimal one from all objects [16]. Consequently, to tackle with the settings, multi-attribute group decision-making (MAGDM) pursues the selection or ranking for the feasible alternatives by a group, in the presence of conflicting and interactive criteria [17]. For MAGDM issues, rough set models with fuzzy theory based are usually suggested to applied [18].

As a main tool to tackle fuzzy issue, fuzzy numbers can unite the fuzziness of human determination, automatically. But once fuzzy sets are made clear by a subjection degree function, and the uncertainty of the fuzzy event can be then characterized via subjection degree. However, cloud model can perform the transition between accurate value and quality one to overcome the flaw above by combining the fuzziness and randomness in [19 –21]. Hence, it is more obvious to apply it to depict the fuzziness and randomness so as to lessen the decision-making subjectivity. In addition, in terms of swarm decision from AHP and fuzzy means, the risk values and the preference levels are represented and computed by the cloud shape. The cloud theory has been broadly applied in knowledge expression and data disposing [22], uncertainty inference [23], and validity evaluation of complex issues [24]. Combining the fuzziness and randomness, the subjection cloud performs the alteration between the accurate value and the quality one so that the cloud model possesses many excellent characteristics to depict reasoning and qualitative concept. Hence fuzzy comprehensive evaluation method with cloud model based can describe the fuzziness and randomness in pair-wise comparisons and assess the risk of the evaluated system.

To overcome the defects of conventional FAHP method, in this paper, we propose a Fuzzy-FMECA method with cloud model based to evaluate risk of train control center, wherein FMECA is applied to generalize the potential failure modes of the systems, firstly, and then from the expert decision, fuzzy comprehensive evaluation method with the cloud model based is applied to map the investigation data to the corresponding fuzzy matrixes described by the cloud model of each failure mode, and also the weight distribution of every failure mode is obtained using AHP with cloud model based. Compared with the conventional FAHP, Fuzzy-FMECA method could demonstrate the data distribution characteristics more clearly and objectively based on cloud model, and can tackle with inconformity existed in FAHP experts’ comments, such that it is feasible and available in the risk estimation for the railway signal equipments.

2 Cloud theory

It is hard to distinguish ambiguity and vagueness between objective world and subjective thinking process. On the basis of vague and statistic theories, it is easy for cloud model to transit the uncertainty between qualitative concept and its numerical expression. In general, cloud model is an uncertainty transform tool between the qualitative and the quantitative, which fully incorporates ambiguity with randomness together, and such that a mapping is formed between both [25, 26].

2.1 Cloud concept

Definition 1. Cloud and Cloud drops [19]. Let U be a quantitative numerical domain, and C be a qualitative concept in U. If x ∈ U is a stochastic role in C, and thenμ (x) ∈ [0, 1], which stands for confidence level that x belongs to C, and is a stochastic variable of steady tendency. A subjection cloud denotes a mapping from U to [0, 1], that is, μ : U → [0, 1], ∀x ∈ U, x → μ (x) . The cloud is denoted as the distribution of x in U, and expressed by C (X), wherein x is called a cloud drop.

Single cloud drop might be insignificant, but a lot of cloud drops can generate cloud, the full shape of which depicts the significant features of the quantity concept [27]. The overall cloud can be depicted by several cloud features, which is respectively defined as Expected value Ex, Entropy En and Hyper Entropy He, and marked as C (Ex, En, He). This shows the quantitative features of qualitative concept [28, 29]. Figure 1 shows the normal cloud model.

Fig.1

Normal cloud distribution.

In Fig. 1, Ex expresses the concept’s centre in the domain, and also is its qualitative representation, and En is the surveying of qualitative notion vagueness, and depicts the numerical scope accepted by the notion in the domain, and shows the uncertainty margin of the qualitative notion, and He depicts the dispersion of the cloud drops.

Hence, with aid of three features number of the cloud, the ambiguity and randomness can be totally combined together, such that a mapping can be established between quality and quantity as the basis of knowledge representation.

2.2 Normal cloud concept

As a basic tool, the normal cloud be used to present the linguistic values, and exerted by the cloud (Ex, En, He).

(1) Generate $E_{ni}^{'} = G (En, He)$ , whereE_ni’ is a normal distributed stochastic number with its desired being En and standard deviation being He;

(2) Generate x_i = G (Ex, En′), where x_i is a normal distributed stochastic number with its desired being Ex and standard deviation being En;

(3) Compute

μ_{i} (x_{i}) = exp [- \frac{(x_{i} - Ex)^{2}}{2 (E_{ni}^{'})^{2}}]

(1)

and (x_i, μ_i (x_i)) is then defined as a cloud droplet.

(4) To repeat the procedure above, until the adequate cloud droplets emerge.

Cloud could be realized by Cloud Generator (CG) using programming or hardware solidified methods. From the numerical features of cloud, cloud drops could be achieved by forward cloud generator with normality as shown in Fig. 2, which means the transition realized from the qualitative to the quantitative, and corresponding algorithm is defined as cloud generator algorithm with forward direction.

Fig.2

Normal cloud generator with forward direction.

In another hand, if an aggregate of cloud droplets are given beforehand and accords with normal distributed, and then the three numerical features (Ex, En, He) of the cloud can be generated by cloud generator with backward direction of (CG^-1) as indicated in Fig. 3, whose transition is from quantitative to qualitative with converse property compared with CG. Combining the former CG and the later CG^-1 together, we may realize the transition between the two at any time.

Fig.3

Normal cloud generator with backward direction.

From statistic theory, CG^-1 mainly possesses two sorts of fundamental algorithms: the first one is based certain scale information, and the other is uncertain one. This paper takes the later [30].

Let us firstly define sample point x_i as input, wherein i = 1, 2, …, n, and the output be the (Ex, En, He), and then these arithmetic steps are shown below.

(1) From x_i, to work out average of the sample set by $\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ ;

(2) To compute the central absolute moment of the samples with first-order by $\frac{1}{n} \sum_{i = 1}^{n} | x_{i} - \bar{X} |$ ;

(3) To compute the sample variance by $S^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - \bar{X})^{2}$ ;

(4) Let $Ex = \bar{X}$ , $En = \sqrt{\frac{π}{2}} \times \frac{1}{n} \sum_{i = 1}^{n} | x_{i} - Ex |$ , and then $He = \sqrt{S^{2} - {En}^{2}}$ .

The CG^-1 algorithm mentioned above is mainly based on statistic probability, such that its numerical feature is an estimate. As the more cloud drops are, and the smaller errors may become.

2.3 Operation rules of cloud

Let C₁ (Ex₁, En₁, He₁) and C₂ (Ex₂, En₂, He₂) be two clouds, respectively, and C be the operation results of the two with feature C (Ex, En, He), and then the operating laws are described in Table 1.

It is noted that C₁ and C₂ need to possess same domain, such that the cloud operations in Table 1 possesses significance.

Table 1
Cloud operation law

Operator Ex En He

+ Ex₁ + Ex₂ $\sqrt{{En}_{1}^{2} + {En}_{2}^{2}}$ $\sqrt{{He}_{1}^{2} + {He}_{2}^{2}}$

– Ex₁ - Ex₂ $\sqrt{{En}_{1}^{2} + {En}_{2}^{2}}$ $\sqrt{{He}_{1}^{2} + {He}_{2}^{2}}$

× Ex₁ × Ex₂ $| {Ex}_{1} {Ex}_{2} | \sqrt{{(\frac{{En}_{1}}{{Ex}_{1}})}^{2} + {(\frac{{En}_{2}}{{Ex}_{2}})}^{2}}$ $| {Ex}_{1} {Ex}_{2} | \sqrt{{(\frac{{He}_{1}}{{Ex}_{1}})}^{2} + {(\frac{{He}_{2}}{{Ex}_{2}})}^{2}}$

÷ Ex₁ ÷ Ex₂ $| \frac{{Ex}_{1}}{{Ex}_{2}} | \sqrt{{(\frac{{En}_{1}}{{Ex}_{1}})}^{2} + {(\frac{{En}_{2}}{{Ex}_{2}})}^{2}}$ $| \frac{{Ex}_{1}}{{Ex}_{2}} | \sqrt{{(\frac{{He}_{1}}{{Ex}_{1}})}^{2} + {(\frac{{He}_{2}}{{Ex}_{2}})}^{2}}$

Operator	Ex	En	He
+	Ex₁ + Ex₂	$\sqrt{{En}_{1}^{2} + {En}_{2}^{2}}$	$\sqrt{{He}_{1}^{2} + {He}_{2}^{2}}$
–	Ex₁ - Ex₂	$\sqrt{{En}_{1}^{2} + {En}_{2}^{2}}$	$\sqrt{{He}_{1}^{2} + {He}_{2}^{2}}$
×	Ex₁ × Ex₂	$\| {Ex}_{1} {Ex}_{2} \| \sqrt{{(\frac{{En}_{1}}{{Ex}_{1}})}^{2} + {(\frac{{En}_{2}}{{Ex}_{2}})}^{2}}$	$\| {Ex}_{1} {Ex}_{2} \| \sqrt{{(\frac{{He}_{1}}{{Ex}_{1}})}^{2} + {(\frac{{He}_{2}}{{Ex}_{2}})}^{2}}$
÷	Ex₁ ÷ Ex₂	$\| \frac{{Ex}_{1}}{{Ex}_{2}} \| \sqrt{{(\frac{{En}_{1}}{{Ex}_{1}})}^{2} + {(\frac{{En}_{2}}{{Ex}_{2}})}^{2}}$	$\| \frac{{Ex}_{1}}{{Ex}_{2}} \| \sqrt{{(\frac{{He}_{1}}{{Ex}_{1}})}^{2} + {(\frac{{He}_{2}}{{Ex}_{2}})}^{2}}$

3 Fuzzy FMECA method based on cloud model

A cloud model based Fuzzy-FMECA is proposed for risk assessment of Chinese train control (CTC) systems interface module for the train control center, wherein the cloud is applied to confirm the subjection degree function. Eventually, based on the presented algorithm and cloud mode, a group decision can be implemented which incorporates AHP and fuzzy together. Figure 4 shows the risk assessment flow chart by combining the Fuzzy-FMECA method with the cloud model.

Fig.4

Flow chart of Fuzzy-FMECA method with cloud model based.

The Fuzzy-FMECA method with cloud theorem based is firstly to perform the failure modes selection, and then the cloud model that combines the fuzziness and randomness together is applied to improve the evaluation and weight matrix of fuzzy comprehensive assessment means so as to be able to better evaluate failure modes influence. There are three elements that require to be considered in this means, that is, element aggregate (U), and weight aggregate (W), and evaluation aggregate (V).

The Fuzzy-FMECA with cloud theorem based is firstly to implement the failure modes selection, and the cloud model combining the fuzziness and randomness is then applied to improve the fuzzy evaluation matrix and weight matrix of fuzzy comprehensive assessment means to evaluate failure modes influence. Three elements should be considered in this means, that is, element aggregate (U), and weight aggregate (W), and evaluation aggregate (V).

3.1 To determine the factor set

Let U ={ u₁, u₂, ⋯ , u_m } be evaluation factor set, where u_i (i = 1, 2, ⋯ , m) shows the ith factor influential to assessment object. To construct index set should meet the comparable and feasible principles. After Delphi questionnaires repeatedly inquiries, index set may be classified into many levels from their attributes. In most cases, they can be divided as three levels below.

The top level: In this layer there is one factor alone, usually being the intended target or the expected results as analyzing issues, such that it is also denoted as the aim level.

The middle level: In this layer there are some intermediate links between the top and the bottom level, which is composed of many levels, and so also denoted as criteria level.

The bottom level: In this layer there are diverse sorts of measures and decision-making strategies to arrive at the target, and such that it is also denoted as scheme level.

3.2 To determine the weight set

The weight set is based on the expert investigation, all weight subsets are presented by qualitative language, which are transformed into normal cloud. The degree of the importance could be presented using diverse normal cloud digital characters. The weight set could be described by W ={ w₁, w₂, ⋯ , w_m }. Usually, the grade of the weight element subsets is in range of three to nine.

3.3 Establishment of evaluation criterion based on cloud model

We here let the subject clouds take instead of the membership functions of the standard risk states in the conventional fuzzy means. The qualitative scores on elements are limited in range of C_min to C_max, such that the cloud processing serve Ex as the median of restraints to approximate the score, and we could then compute the eigenvalues of the cloud [31]. ${\begin{matrix} Ex = (C_{min} + C_{max}) / 2 \\ En = (C_{max} - C_{min}) / 6 \\ He = i \end{matrix}$ (2)

The i used in (2) can be modified from blur threshold of the variable.

The evaluation aggregate of risk is presented by V ={ V₁, V₂, ⋯ , V_m }. Suppose that k experts are employed to evaluate the significance of every element in the AHP index system, and then we will get k scores for each element. Thus $V_{i} = {C_{i}^{1}, C_{i}^{2}, \dots, C_{i}^{k}}$ can be used to describe the judgment set of factor i, among which $C_{i}^{j}$ is the cloud model of the ith factor with the score of the expert j (i = 1, 2, 3, ⋯ , m ; j = 1, 2, 3 ⋯ , k).

For the comments with single-side restrain C_min or C_max, we will firstly access the expectation missing, and then work out cloud parameters from (2), and describe it by semi-up and down.

3.4 Cloud model based comprehensive evaluation

Suppose that the evaluation factor set C = (c₁, c₂, ⋯ , c_q) be a set of m factors for evaluation unit, and the weight set be A = [a₁, a₂, ⋯ , a_q], and comprehensive assessment matrix of every evaluation unit be R = [r₁, r₂, ⋯ , r_q] ^T. Cloud model is adopted to work out the weights and evaluation matrix instead of constructing the subjection degree function. As for weight calculation, this can be calculated in a statistical form, based on the collation of the questionnaires, experts are found to score the weight of the evaluation factors. Scoring results can be a fraction. Then the parameters (Ex, En, He) can be obtained using backward cloud generator to calculate the statistical samples [32].

The weight set described by the cloud model of evaluation factors is $A = [a_{1}, a_{2}, \dots, a_{q}] = {[\begin{matrix} {Ex}_{a_{1}} & {En}_{a_{1}} & {He}_{a_{1}} \\ {Ex}_{a_{2}} & {En}_{a_{2}} & {He}_{a_{2}} \\ ⋮ & ⋮ & ⋮ \\ {Ex}_{a_{q}} & {En}_{a_{q}} & {He}_{a_{q}} \end{matrix}]}^{T}$ (3)

The fuzzy comprehensive evaluation matrix is then described by $R = [\begin{matrix} r_{1} \\ r_{2} \\ ⋮ \\ r_{q} \end{matrix}] = [\begin{matrix} {Ex}_{1} & {En}_{1} & {He}_{1} \\ {Ex}_{2} & {En}_{2} & {He}_{2} \\ ⋮ & ⋮ & ⋮ \\ {Ex}_{q} & {En}_{q} & {He}_{q} \end{matrix}]$ (4)

For the factor set is C = (c₁, c₂, ⋯ , c_q), the weight coefficient a_i (Ex_{a
_i}, En_{a
_i}, He_{a
_i}) can be understood as each weight coefficient having certain fuzziness and randomness. The evaluation factor can be calculated as Ex_{a
_i}, the evaluation results of different people are defined in the scope of [Ex_{a
_i} - 3En_{a
_i}, Ex_{a
_i} + 3En_{a
_i}] [33]. He_{a
_i} further reflects the randomness of subjective evaluation. Comprehensive assessment matrixe from the cloud model has the same mathematical meaning.

Then the fuzzy synthesis operator M (• , ⊕) is adopted to calculate comprehensive evaluation result which is described by (5).

B = A ∘ R = (Ex, En, He) is the characteristic value of comprehensive evaluation, which is an evaluation of cloud model. Ex is compared with reviews various expectations of cloud model, the closest evaluation is the evaluation result. Evaluation criterion based on cloud model and evaluation model are simulated using MATLAB, respectively, and the nearest evaluation criterion based on cloud model has the greatest influence on evaluation of cloud model, which is the final risk rating. The risk assessment results can be evaluated and shown with the MATLAB tools.

$\begin{matrix} B = A \circ R = {[\begin{matrix} {Ex}_{a_{1}} & {En}_{a_{1}} & {He}_{a_{1}} \\ {Ex}_{a_{2}} & {En}_{a_{2}} & {He}_{a_{2}} \\ ⋮ & ⋮ & ⋮ \\ {Ex}_{a_{q}} & {En}_{a_{q}} & {He}_{a_{q}} \end{matrix}]}^{T} \circ [\begin{matrix} {Ex}_{1} & {En}_{1} & {He}_{1} \\ {Ex}_{2} & {En}_{2} & {He}_{2} \\ ⋮ & ⋮ & ⋮ \\ {Ex}_{q} & {En}_{q} & {He}_{q} \end{matrix}] = \\ = [\begin{matrix} {Ex}_{a_{1}} \times {Ex}_{1} + {Ex}_{a_{2}} \times {Ex}_{2} + \dots + {Ex}_{a_{q}} \times {Ex}_{q} \\ \sqrt{{(| {Ex}_{a_{1}} {Ex}_{1} \sqrt{{(\frac{{En}_{a_{1}}}{{Ex}_{a_{1}}})}^{2} + {(\frac{{En}_{1}}{{Ex}_{1}})}^{2}} |)}^{2} + {(| {Ex}_{a_{2}} {Ex}_{2} \sqrt{{(\frac{{En}_{a_{2}}}{{Ex}_{a_{2}}})}^{2} + {(\frac{{En}_{2}}{{Ex}_{2}})}^{2}} |)}^{2} + \dots + {(| {Ex}_{a_{q}} {Ex}_{q} \sqrt{{(\frac{{En}_{a_{q}}}{{Ex}_{a_{q}}})}^{2} + {(\frac{{En}_{q}}{{Ex}_{q}})}^{2}} |)}^{2}} \\ \sqrt{{(| {Ex}_{a_{1}} {Ex}_{1} \sqrt{{(\frac{{He}_{a_{1}}}{{Ex}_{a_{1}}})}^{2} + {(\frac{{He}_{1}}{{Ex}_{1}})}^{2}} |)}^{2} + {(| {Ex}_{a_{2}} {Ex}_{2} \sqrt{{(\frac{{He}_{a_{2}}}{{Ex}_{a_{2}}})}^{2} + {(\frac{{He}_{2}}{{Ex}_{2}})}^{2}} |)}^{2} + \dots + {(| {Ex}_{a_{q}} {Ex}_{q} \sqrt{{(\frac{{He}_{a_{q}}}{{Ex}_{a_{q}}})}^{2} + {(\frac{{He}_{q}}{{Ex}_{q}})}^{2}} |)}^{2}} \end{matrix}] \\ = (Ex, En, He)^{'} \end{matrix}$ (5)

The computing rules between two clouds can be seen in Table 1 [34].

4 Example analysis

4.1 Cloud model application

FMECA method is used to identify the risk of CTC interface module, based on the characteristics of signal transmission between the functions and the knowledge and expertise of the expert, the failure modes of CTC interface module are summarized as five types, the first one is that the system can’t receive information from the CTC interface module, the second one is that the system receives incorrect information from the CTC interface module, the third one is that the system can’t send information to the CTC interface module, the fourth one is that the system sends incorrect information to the CTC interface module, the fifth one is that the information communicated with CTC interface module is always the same.

The basic idea of security risk classification is based on the risk theory and mathematical relationship related to it, that is, risk equals risk probability is multiplied by risk severity degree. The risk rating can be obtained according to the degree of risk levels. But in the actual process of risk management, it is hard to calculate the risk accurately and quantitatively, so qualitative or half quantitative methods commonly are used in the risk level classification. In the process of the risk assessment of failure modes for CTC interface module, expert assignment or index can be used to express the likelihood and the severity of the accident. Risk matrix rating table is presented in Table 2.

Table 2
Risk matrix rating table

Risk severity degree \ Risk probability Risk rating

V (negligible) IV (mild) III (moderate) II (serious) I (disaster)

frequent 6 6 7 8 8

sometimes 4 5 6 7 8

occasional 2 4 5 7 7

rare 1 2 4 6 7

remote 1 1 3 5 6

Risk severity degree \ Risk probability	Risk rating
frequent	6	6	7	8	8
sometimes	4	5	6	7	8
occasional	2	4	5	7	7
rare	1	2	4	6	7
remote	1	1	3	5	6

The evaluation criteria can be defined as four levels in this paper, acceptable, conditional acceptable, undesired, unacceptable [35], the value range of these four levels are defined as, [0, 3), (3, 5), (5, 7) and (7, 8], use these evaluation criteria to describe the evaluation results. According to (2), these evaluation criteria can be transformed into normal cloud. Evaluation criteria based on cloud model presented in Table 3.

Table 3

Evaluation criterion based on cloud model

Risk level	Acceptable	Conditional acceptable	Undesirable	Unacceptable
Evaluation indicators	(1.5,0.5,0.1)	(4,0.33,0.1)	(6,0.33,0.1)	(7.5,0.167,0.1)

20 experts are selected to assess the risk of failure modes of CTC interface module, according to (4), the expectations, entropy and hyper entropy of each failure mode can be calculated using backward normal cloud generator, the fuzzy comprehensive evaluation matrix based on cloud model is then obtained by $R = [\begin{matrix} (2.000, 0.8773, 0.1812) \\ (6.9500, 0.8335, 0.1143) \\ (7.000, 0.7520, 0.2571) \\ (4.6500, 1.2345, 0.4816) \\ (4.8000, 1.0779, 0.2144) \end{matrix}]$ (6)

Also twenty experts are selected to assess the relative importance ratings of these five failure modes for CTC interface module. From the corresponding comparison matrix, the corresponding weight coefficient matrix is obtained based on cloud model by $A = [\begin{matrix} (0.0383, 0.0093, 0.0045) \\ (0.4136, 0.0150, 0.0053) \\ (0.4136, 0.0150, 0.0053) \\ (0.0728, 0.0252, 0.0128) \\ (0.0616, 0.0198, 0.0075) \end{matrix}]$

Then according to (5), the evaluation result can be obtained by B = A ∘ R = (6.4369, 0.5191, 0.1403).

Finally, we obtain the assessment result cloud for the CTC interface module as displayed in Fig. 5.

Fig.5

Risk assessment result of CTC interface module.

From Fig. 5, it can be seen that the assessment results cloud (the second in the right-side) is more close to the third of all the four ranks of the evaluation criterion based on cloud model. In fact, it roundly locates at the middle of the two evaluation criterion clouds, that is, the undesired and the unacceptable, and more close to the third. Hence we can easily and visually determine that the potential failure risk of the CTC interface module tends to be undesired to be accepted.

4.2 Conventional FAHP method

To verify the correctness of the above model, based on the recycling of 20 experts’ questionnaire, by counting the number of occurrences for each comment, fuzzy comprehensive evaluation matrix can be get by

$R = [\begin{matrix} 0.65 & 0 & 0 & 0.35 \\ 0 & 0.65 & 0.35 & 0 \\ 0 & 0.7 & 0.3 & 0 \\ 0 & 0 & 0.55 & 0.45 \\ 0 & 0 & 0.65 & 0, 35 \end{matrix}]$ (7)

At the same time the relative weight of the five failure modes for CTC interface module can be calculated by $W = (0.0383, 0.4139, 0.4139, 0.0728, 0.0616)$

Fuzzy comprehensive evaluation matrix can be described by $B = W \circ R$ (8) wherein “∘” is synthetically operator making use of weighted average model, for instance, M (• , ⊕). For fuzzy mathematics, there are four types of the operators to be often used, such as M (∧ , ∨), M (• , ∨), M (∧ , ⊕) and M (• , ⊕), and the practical applications show M (• , ⊕) has broadly recommended [36].

The evaluation result can be then gotten by $B = W \circ R = (0.0250, 0.5587, 0.2689, 0.1474)$

According to the results of the fuzzy comprehensive evaluation, we can get the conclusions. The comprehensive assessment results of the CTC interface module is taken into account, based on the calculation, the risk ratio are calculated as follows: 2.5%, 55.87%, 26.89%, and 14.74%. According to the maximum membership principle, the risk rating for the failure modes of the CTC interface module is conditional acceptable.

Obviously, this is inconsistent with the former result based cloud model. By observing the evaluation matrix R under the two cases given by the experts, we can find the two are inconsistent, practically. Taking the first line of the two matrixes as instance, in evaluation based on cloud model, the scores given by all 20 experts basically situates in range of [0, 3), namely, acceptable, but in conventional FAHP model, only there are 65% of all 20 experts consistent with the former, and roundly 35% experts tend to unacceptable, that is, the scores given by them should be in range of (7, 8]. In other words, the scores given by the 20 experts in FAHP are inconsistent. To show this case, we apply stochastic simulation method to generate the 20 random data, wherein 13 data (65% experts) distribute in range of [0, 3), uniformly, and 7 data in range of (7, 8], namely, 1.97, 0.11, 2.55, 2.80, 2.04, 2.27, 2.23, 1.18, 1.97, 0.51, 2.12, 0.10, 0.83, 7.05, 7.82, 7.69, 7.32, 7.10, 7.95, 7.03. Clearly, it is easy to obtain their average value E(x)=3.6313, and variance S² = 8.7440, and standard deviation σ=2.9579. 3σ law is applied to judge whether the 20 data contain crassitude errors or not, namely, if one data exceeds the scope of |x - E (x) |>3σ, we have 99.73% reason to consider it stochastic error [37]. Obviously, all the 20 data do not contain such error. Below we adopt student test (t-distribution) to examine the consistence of two groups of data given by experts separately applied in two evaluation models [38]. Let us take the 1st lines as an example in (6) and (7) to show the issue. From (6), we can obtain the sample average value μ=2, and variance unknown. Let us select significant level a = 0.05, so that t_a (n - 1) =1.7921. Then we have $\begin{matrix} t & = & \frac{\bar{X} - μ}{S / \sqrt{n}} = \frac{3.6313 - 2}{\sqrt{8.7440} / \sqrt{20}} \\ = & 2.4671 > 1.7921 = t_{a} (n - 1) \end{matrix}$

Clearly, t falls in the rejected domain, such that the two groups of data are inconsistent given by experts, which directly leads to a mismatch result. In the same way, we also show the rest four groups data in (6) and (7) are inconsistent. See from it, to implement the comparisons on the two models of FMECA and traditional FAHP, we firstly should perform the consistence examination on their evaluation matrixes, and otherwise we have no way to implement the comparisons on them. Due to this reason, another 20 experts are issued questionnaires to give out data below. $R = [\begin{matrix} 0.85 & 0.15 & 0 & 0 \\ 0 & 0 & 0.7 & 0.3 \\ 0 & 0 & 0.65 & 0.35 \\ 0 & 0.5 & 0.45 & 0.05 \\ 0 & 0.5 & 0.5 & 0 \end{matrix}]$ (9)

Similarly, we randomly generate 20 data to express the concrete scores given by experts according to the first line in (9) by x={1.69, 2.19, 1.88, 2.43, 3.09, 1.96, 1.17, 3.24, 2.17, 2.83, 1.92, 2.98, 2.01, 1.45, 3.51, 2.47, 0.86, 2.21, 1.74, 2.05}. Applying same method with the former, we can obtain the expected value E(x)=2.2221, and variance s² = 0.4786, and standard deviation σ=0.6918. It is easy to show that there are no crassitude errors contained in (9). In addition, applying t-test method, we have $\begin{matrix} t & = & \frac{\bar{X} - μ}{S / \sqrt{n}} = \frac{2.2221 - 2}{\sqrt{0.4786} / \sqrt{20}} \\ = & 1.4357 < 1.7921 = t_{a} (n - 1) \end{matrix}$

Clearly, it does not fall in the rejected domain, so that we conclude it is consistent with the 1st line in (6). In the same way, we can conclude that all other four groups of data in (9), t all does not fall into the rejection domain.

According to (8), the fuzzy comprehensive evaluation can be obtained by $B = W \circ R = (0.0325, 0.0729, 0.6223, 0.2727)$

According to fuzzy comprehensive evaluation, we can get the following conclusions. The comprehensive evaluation results of the CTC interface module are shown by 3.25%, 7.29%, 62.23%, and 27.27%. According to the maximum membership principle, the risk rating for the failure modes of the CTC interface module is undesirable to acceptable. This is consistent with judge result based cloud model. Finally, through on-site inspection, the risk rating for the failure modes of the CTC interface module tends to be undesirable to acceptable.

Compared with evaluation results between two methods, the results of the two are consistent, but the precondition is that the original data must be consistent given by relative experts, and otherwise there will be mismatch result. In addition, the FMECA provide more abundant information by the distribution of clouds, which gives us an intuitive feeling. By the comparisons between the two models, we also obtain inner connections between the two, this help us to understand the issue more perfect and deep.

4.3 Statistics methods

To further verify the correctness and effectiveness of the proposed method in this paper, the correlation coefficient method and the center-of-gravity method are applied to solve the issue again [36].

The correlation coefficient method obtains the final result from the correlation levels between the cloud to expect to be evaluated and the other four evaluation standard clouds. If the larger the correlation coefficient between the compared two objects is, the closer the relationship between them is. Hence, it should belong to the class of the largest correlation coefficient with it. The correlation coefficient can be expressed by $ρ_{XY} = \frac{E (XY) - E (X) E (Y)}{\sqrt{D (X)} \sqrt{D (Y)}}$ (10) where ρ _XY is the correlation coefficient, and X is the cloud to be evaluated, and Y is the evaluation standard cloud, and E (•) expresses the sample average, and D (•) expresses the sample variance.

To calculate ρ XY used in (10), based on the eigenvalues of the each cloud, we can generate enough cloud drops though stochastic simulation under Matlab environment. And then we can obtain the sample average and variance required in (10).

Taking the CTC interface module assessment as an example, according to (10) and the former descriptions, we can obtain the evaluation result as (0.0054, 0.0363, 0.6805, 0.2778). Obviously, the correlation coefficient between the cloud to be evaluated and the 3rd criterion cloud is the largest. From it, it can be concluded that the risk rating for the failure mode of the CTC interface module is undesirable to acceptable.

The center-of-gravity method performs the judgment by comparing the distance between the two centers of gravity [39]. The distance between the two classes can be defined by $D_{pq}^{2} = d_{{\bar{X}}_{p} {\bar{X}}_{q}}^{2} = {({\bar{X}}_{p} - {\bar{X}}_{q})}^{'} ({\bar{X}}_{p} - {\bar{X}}_{q})$ (11) where D_pq expresses the distance of the two centers of gravity, and $\bar{X} p$ , $\bar{X} q$ are respectively gravity centers of the two classes.

According to (11), the distance between the cloud to be evaluated and each evaluation criterion cloud is acquired by (4.938, 2.444 · 0.442, 1.057). Clearly, the distance between the cloud to be evaluated and the third criterion cloud is the smallest. Therefore, it can be said that the risk rating of the failure mode of the CTC interface module is undesirable to acceptable.

In the end, we obtain the consistent results with the former, but the former method based on cloud model is more visual and possesses stronger information disposing ability. This shows that the proposed method in this paper is correct and credible.

4.4 Results analysis

From the risk results of Fuzzy-FMECA based on cloud model with the risk results of conventional FAHP method, we can see risk assessment results of the two methods are consistent in consistence of original data: the risk rating for the failure modes of the CTC interface module tends to undesirable to accept. But the conventional FAHP method has the following defects:

(1) As shown from the comprehensive assessment results, the results of conventional FAHP method largely depend on the expert’s subjective judgment, is not objective. Subjection degree is given on the basis of experience, and according to maximum membership principle, the results of conventional FAHP method lack volatility and randomness.

(2) For simplicity to get the assessment result, the final evaluation result of conventional FAHP is transformed into a fractional simply, so conventional FAHP method is difficult to embody the essence of fuzzy. And cloud model cannot only rate the evaluation results, but also the internal fuzzy characteristics of evaluation objects can be analyzed, such that it can provide more abundant information than subjection degree function of the conventional FAHP.

(3) The results will lose precision as the fuzzy value is transformed into the accurate value, so the conventional FAHP is not suitable for the raw data which is accurate. And the Fuzzy-FMECA method from cloud model is suitable for the unitary data (accurate or fuzzy value) or the binary data. So the Fuzzy-FMECA method with cloud model based is more widely applicable.

(4) The results with cloud model based are more visual and direct-viewing, and possesses better consistence for experts’ knowledge, whereas fuzzy evaluation matrix given out by experts requires to perform consistence examination, and mistaken results possibly occurs otherwise. Moreover, the method with could model based can be used to validity the consistence of the data given out by the experts.

5 Conclusion

This paper represents a risk evaluation approach from Fuzzy-FMECA with cloud model, which can effectively make comprehensive decision on group evaluation of the risk, and reduce the influence of expert factors on the evaluation results. In this means, the cloud model with normal distributed is applied to construct the subjection degree function, such that the risk degree of fuzziness and randomness of the risk factor can be fully characterized, and the transformation between qualitative and quantitative evaluation is realized, easily. At the same time, the correlation coefficient and the center-of-gravity methods are also applied to verify the correctness and effectiveness of the proposed method. Hence the Fuzzy-FMECA method with cloud model is more objective and accurate. Clearly, the applications of the cloud model should be able to develop and improve the risk assessment method.

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Risk severity degree \ Risk probability	Risk rating
	V (negligible)	IV (mild)	III (moderate)	II (serious)	I (disaster)
frequent	6	6	7	8	8
sometimes	4	5	6	7	8
occasional	2	4	5	7	7
rare	1	2	4	6	7
remote	1	1	3	5	6

Fuzzy FMECA risk evaluation and its applications in Chinese train control systems based on cloud model

Abstract

Keywords

1 Introduction

2 Cloud theory

2.1 Cloud concept

3.2 To determine the weight set

3.3 Establishment of evaluation criterion based on cloud model

4.1 Cloud model application

Table 2 Risk matrix rating table Risk severity degree \ Risk probability Risk rating V (negligible) IV (mild) III (moderate) II (serious) I (disaster) frequent 6 6 7 8 8 sometimes 4 5 6 7 8 occasional 2 4 5 7 7 rare 1 2 4 6 7 remote 1 1 3 5 6

5 Conclusion

References

Table 2
Risk matrix rating table

Risk severity degree \ Risk probability Risk rating

V (negligible) IV (mild) III (moderate) II (serious) I (disaster)

frequent 6 6 7 8 8

sometimes 4 5 6 7 8

occasional 2 4 5 7 7

rare 1 2 4 6 7

remote 1 1 3 5 6