Risk analysis using FMEA: Fuzzy similarity value and possibility theory based approach

Highlights

• The study enhances the capability of FMEA as a risk assessment tool.

• It uses fuzzy similarity value based measurement.

• It demonstrates the applicability of possibility theory in decision making.

• It considers two case studies.

• The results are compared with traditional methods.

Abstract

Fuzzy numerical technique for FMEA has been proposed to deal with the drawbacks of crisp FMEA and fuzzy rule based FMEA approaches. Fuzzy numerical approaches based on de-fuzzification also suffer from the drawback of providing arbitrary priority ranks of failure modes even when their membership functions overlap. To overcome this drawback we developed a new methodology integrating the concepts of similarity value measure of fuzzy numbers and possibility theory. Similarity value measure has been applied to group together failure modes having similar amount of risk value. The possibility theory has been used for checking for conformance guidelines. Two case studies have been shown to demonstrate the methodology thus developed. The proposed methodology is more robust in nature as it does not require arbitrary precise operations like de-fuzzification to prioritise the failure modes. Application of possibility theory is new to the domain of risk analysing using FMEA.

Introduction

The technique of Failure Mode and Effect Analysis (FMEA) was originally developed for systematic analysis of the failures modes and its subsequent effects for the defence related products particularly in the aviation sector (Bowles & Peláez, 1995). Initially FMEA techniques was mostly used for failure mode analysis of products only ranging from nuclear, automobile, chemical, mechanical, but with passage of time FMEA has been extensively used in the fault analysis of service industries like software industry (Shawulu, 2012, Stamatis, 1995; Guimaraes and Lapa, 2006; Kangari and Riggs, 1989). The major objective of application of FMEA is the identification of potential failure modes of the system components, evaluating their causes and their subsequent effects on the system behaviour, and as a result determination of the ways to eliminate or reduce either the chances of occurrence or severity or increase the detectibility of the particular failure mode. Traditionally, the risk computation of different failure modes using FMEA has been done by developing risk priority number (RPN). RPN is the value obtained by the product of three components, i.e. the occurrence probability of a failure mode (P), the severity of the failure mode (S) and the detectibility of the failure mode (D). Higher the value of the RPN higher is the risk associated with the corresponding failure mode. The purpose of RPN is to prioritize the failure modes of a product or system, so that the available resources can be effectively allocated. More risky failure modes will be tackled with more resources in terms of effort, time and cost. Mathematically the RPN can is represented as,

$$\text{RPN}=P\times S\times D$$

where, the risk parameters P, S and D are measured using 5, 7 or 10 point scale similar to Likert’s scale (Chen, Liu, & Liu, 2013). In Eq. (1) the probability, severity and detectability are crisp numbers, thus the RPN values is also crisp in nature. Several drawbacks of this crisp approach of calculating the RPN have been highlighted and it has been criticised by many authors. Yang et al., 2008, Gargama and Chaturvedi, 2011, Garcia et al., 2005 argued that it is very difficult for the experts to give precise numerical inputs for the three risk parameters as required in crisp model approach. Secondly, Liu et al., 2011, Zhang and Chu, 2011, Pillay and Wang, 2003 have criticised the fact that different combinations of the three risk parameters give rise to same RPN level which in reality may have very different risk implication altogether. Among the other drawbacks, another major drawback as pointed out by Liu et al., 2011, Gargama and Chaturvedi, 2011, Yang et al., 2008 pointed out that the relative importance among the risk parameters are not taken into account while calculating the RPN value. To remove the above mentioned drawbacks the application of fuzzy logic has been recommended.

After the formal introduction of fuzzy set theory by Zadeh (1965), fuzzy logic has found application is various domains. one of the major application of fuzzy set theory has been in the area of modelling where epistemic uncertainty comes into play. Fuzzy if–then rule has been applied by many authors for FMEA purpose (Yang et al., 2008). Although fuzzy if–then rule is capable enough to deal with the problems encountered in the crisp RPN but they also suffered some drawbacks of their own. The major drawback of the fuzzy if–then rule is the cost and time involved in building the complete rule base before any inference process can be started. The affair is costly as it requires large amount of decision making by the experts in developing the complete if–then rule base (Wang, Chin, Poon Gary Ka, & Yang, 2009). To deal with this problem many researchers proposed the use of numerical approach using fuzzy numbers to calculate fuzzy RPN, also called FRPN, values of the failure modes. Chang and Sun, 2009, Garcia et al., 2005, Lertworasirikul, Fang, Joines, & Nuttle (2003) showed the application of data envelopment analysis (DEA) technique to improve the assessment capability of the FMEA. However, Chang and Sun (2009) had used crisp numbers to model the risk parameters, thus the model was not capable to deal with the inherent vagueness of the input values. Garcia’s model although captures well the ambiguity associated with the values of the risk parameters but Chen and Ko (2009) criticised the approach for being very complicated in terms of computation and argued that it not possible to provide a full ranking of the RPN values for prioritisation purpose. Apart from DEA the mathematical programming approach for FMEA was suggested by some researchers. Wang et al. (2009) demonstrated the use of linear programming approach using alpha level sets to calculate the RPN values. Zhang and Chu (2011) used the concept of fuzzy preference relations using hamming distance concept to partially order the FRPN values. This concept required pairwise comaparison of each of the FRPN values as a result it becomes computationally inefficient with increase in number of failure modes. Liu, Liu, Mao, and Liu (2012) demonstrated risk evaluation in the used of FMEA using extended VIKOR method. Wang et al. (2009) captured the expert opinions on the risk parameters by using trapezoidal membership function for occurrence probability and triangular membership function for other two risk parameters including their relative weights. The risk parameters were modelled within the range of 1–10 whereas the relative weights were modelled in the range of 0–1. The FRPN value was calculated as the fuzzy weighted geometric mean of the fuzzy rating of the risk parameters. The computation was done using alpha level sets and linear programming approach and then, the fuzzy RPN values were de-fuzzified using a de-fuzzification method developed by them using the concepts of piecewise linearity of the fuzzy sets. Finally, the de-fuzzified crisp values were ranked and the failure modes are prioritised accordingly. Although, traditionally the de-fuzzification method has been used for decision making (Yager and Filev, 1993), but it should be kept in mind that the decisions based on the crisp analogous of a fuzzy sets ignores the entropy present in the fuzzy sets under consideration.

For example, let us consider a system where we provide X as input and get Y as output after some transformation on the input variable represented by F. Therefore, we can say Y = F(X). Thus, we require precise knowledge of two things firstly about X and then knowledge about the transformation F to get to a precise representation for the Y variable. Now suppose due to limitation of our knowledge about X we can only capture about 60% of X, and for simplicity sake assume that we have complete knowledge about F. Thus, given the present situation howsoever robust system we build we have an upper limit on the knowledge we can acquire about Y, which is less than 60%. But this logic does not seem to be followed by the de-fuzzification process being employed by Wang et al. (2009) method. On a similar note the VIKOR method (Liu et al., 2012) can be criticized for being conceptually more demanding for use in real life situations as the amount of indepth knowledge about the parameters required for computation in the VIKOR method is not always possible for real life cases. Apart from this Liu et al. (2012) advocates the use of defuzzification method to reach to the final ordered values of the FRPN scores, thus their method also suffers from the same tendency to provide arbitrary precise values for the FRPN scores and subsequent ordering.The de-fuzzification process represents the fuzzy set with a single crisp number, and subsequent decision with respect to the ordering scheme is made on the basis of the values of the crisp numbers only. It is analogous to making decision for a probability distribution with using only its mean value. Thus, the de-fuzzification process brings in arbitrary precision level which does not represents the real life situation prevalent; as a result the similarity measure of fuzzy numbers should be preferred for partial ordering as this method does not resort to finding the arbitrary precise value for the fuzzy sets. To deal with these drawbacks we propose the use of similarity value of fuzzy numbers developed by Chen and Chen (2003) and subsequent application of possibility theory approach by Kentel and Aral (2007) for risk analysis using FMEA.

The rest of the paper is organised as follows. In Section 2 we have described the proposed methodology. Section 3 deals with the application of the proposed methodology for two cases studies. In Section 4 the comparison between the results obtained from the proposed methodology and from the traditional approach is made. Section 5 contains the conclusion.