ChapterPDF Available

Failure Mode and Effects Analysis Under Uncertainty: A Literature Review and Tutorial

Authors:

Abstract and Figures

The multidimensional nature of risks as well as substantial uncertainties and subjectivities inherent in the risk assessment process led a growing number of researchers to develop alternative approaches for failure mode and effects analysis . The purpose of this chapter is to provide a comprehensive review of the multi-criteria approaches proposed for failure mode and effects analysis under uncertainty and offer a brief tutorial for those who are interested in these approaches.
Content may be subject to copyright.
Chapter 10
Failure Mode and Effects Analysis Under
Uncertainty: A Literature Review
and Tutorial
Umut Asan and Ayberk Soyer
Abstract The multidimensional nature of risks as well as substanti al uncertainties
and subjectivities inherent in the risk assessment process led a growing number of
researchers to develop alternative approaches for failure mode and effects analysis.
The purpose of this chapter is to provide a comprehensive review of the
multi-criteria approaches proposed for failure mode and effects analysis under
uncertainty and offer a brief tutorial for those who are interested in these approaches.
Keywords Failure modes and effects analysis (FMEA)
Risk assessment
Multi-Criteria decision making (MCDM) approaches
Uncertainty
10.1 Introduction
Failure Modes and Effects Analysis (FMEA) is one of the rst structured, sys-
tematic and proactive techniques used for failure analysis. The purpose of FMEA is
to list out all possible failure modes (FMs) (i.e., the things that could go wrong in an
organization); evaluate the causes of each FM and their subsequent effects on the
performance of the system that is under consideration. By denition, FM refers to
the termination of the ability of a system to perform a required function or its
inability to perform within previously specied limits (ISO/IEC-15026-1 2013) and
includes both known and/or potential failures, problems, or errors that may affect
the customers an d thus endanger the reputation of the entire organization. Since
FMs are unavoidable for the majority of the systems, FMEA serves as an effective
tool to ensure that potential threats to the system have been considered and
addressed, and associated risks are minimized. The history of FMEA goes back to
U. Asan (&) A. Soyer
Industrial Engineering Department, Istanbul Technical University,
34357 Maçka/Istanbul, Turkey
e-mail: asanu@itu.edu.tr
© Springer International Publishing Switzerland 2016
C. Kahraman and S. Yanık (eds.), Intelligent Decision Making
in Quality Management, Intelligent Systems Reference Library 97,
DOI 10.1007/978-3-319-24499-0_10
265
the early 1950s and 1960s. In 1949, it was rst used in the United States military as
a reliability evaluation technique to determine the effect of system and equipment
failures. In 1963, National Aeronautics and Space Administration (NASA) used
FMEA during the Apollo missions to assure desired reliability of space systems
(Chang et al. 1999). Later in 1974, the US Navy developed MIL-STD-1629, which
discussed the proper use of the technique. In the late 1970s, Ford Motor Company
introduced FMEA to the automotive industry and then the automotive industry
collectively developed various standards in the 1990s. Over the years, FMEA
became a universally used technique in many different industries, such as, aero-
space, automotive, defense, medical, marine, nuclear power, semiconductor, etc.,
and it has been proven to be successful in any manufacturing or service industry
(Chang et al. 1999, 2001; Chen 2007; Welborn 2010; Arabian-Hoseynabadi et al.
2010).
Commonly, there are four types of FMEA: (i) System FMEA, (ii) Design
FMEA, (iii) Process FMEA, and (iv) Service FMEA. The general properties of
FMEA types are shortly summarized below (Stamatis 2003; Carlson 2012):
(i) System FMEA (sometimes referred as Concept FMEA) is used to analyze
systems and subsystems in the early concept and design stage. It focuses on
potential FMs between the functions of the system caused by system-related
deciencies (such as, system integration, interaction between systems and/or
subsystems, interaction with the external environment, etc. causing the system
not to work as intended).
(ii) Design FMEA (sometimes referred as Product FMEA) is used to analyze
products, early in the design phase to be able to identify potential design aws.
Therefore, it focuses on FMs caused by design-related deciencies to improve
the design and to ensure safety and reliability of the relevant product during its
lifetime.
(iii) Process FMEA is used to analyze processes required to produce a product or
service. It focuses on potential FMs caused by process-related or
assembly-related deciencies.
(iv) Service FMEA is used to analyze services before they reach to customer, and
focuses on potential FMs caused by system-related or process-related de-
ciencies to maximize customer satisfaction.
Today, FMEA is one of the most widely utilized and powerful techniques,
having several advantages for organizations that are trying to nd the ways of
improving quality and safety. Some of the major advantages of FMEA indicated in
the literature include:
Inclusion of people from different expertise areas in an organization, as each of
them views the system from various perspectives, responsibilities, and concerns.
By this means, it provides an opportunity to improve the communi cation and
cooperation between the different functions of an organization, and the rela-
tionships with external factors, such as suppliers and customers (Kostina et al.
2012).
266 U. Asan and A. Soyer
It provides a simple analysis procedure which is easy to learn and implement,
and makes even the evaluation of complex systems easy to do (Dhillon 2009;
Mozaffari et al. 2013).
It acts as a useful visibility tool for managers (Dhillon 2009; Braglia 2000) and
serves as an excellent instrument for learning.
It is a very structured, systematized and reliable method (Dhillon 2009;
Mozaffari et al. 2013; Kostina et al. 2012) that helps to identify the connections
between the FMs, the reasons of FMs, and the effects of FMs, as well.
It permits a realistic appreciation of the conformity of products and services with
the market and customer needs (Kerekes and Johanyák 1996); therefore,
increases the safety and reliability of products/services, reduces warranty and
service costs, shortens the development process, improves compliance with the
deadlines (Bujna and Prístavka 2013), and eventually improves the customer
satisfaction (Dhillon 2009).
As mentioned above, FMs (whether known or potential) are listed and prioritized
in FMEA to prepare for them in the best way possible and to prevent problems from
reaching the customer. To this end, FMEA uses Risk Priority Number
(RPN) methodology to analyze the risks associated with each identied FM. This
methodology consists of assessing the FMs with respect to their severity (S),
probability of occurrence (O), and like lihood of detection (D). For each FM, an
estimate is made of its S, O, and D on a numerical scale of 1 to 10, as described in
Tables 10.1, 10.2, and 10.3 (Chin et al. 2009a, b; Pillay and Wang 2003 ;
Seyed-Hosseini et al. 2006; Wang et al. 2009 ; Guimarães and Lapa 2007 ; Xu et al.
2002; Franceschini and Galetto 2001; Liu et al. 2013b; Chang and Cheng 2010).
The S, O, and D ratings are then multiplied together to get the RPN. In equation
form, RPN ¼ S O D. The FMs with higher RPNs are assumed to be more
important and should be given higher priorities (Wang et al. 2009; Liu et al.
2011).
Table 10.1 Severity scale for a FM
Rating Effect Severity of effect
10 Hazardous
without warning
Very high severity ranking when a potential FM effects safe
system operation without warning
9 Hazardous with
warning
Very high severity ranking when a potential FM effects safe
system operation with warning
8 Very high System inoperable with destructive failure without
compromising safety
7 High System inoperable with equipment damage
6 Moderate System inoperable with minor damage
5 Low System inoperable without damage
4 Very low System operable with signicant degradation of performance
3 Minor System operable with some degradation of performance
2 Very minor System operable with minimal interference
1 None No effect
10 Failure Mode and Effects Analysis Under Uncertainty 267
Whether applied to a system, process, product, or service, FMEA, basically,
consists of the implementation steps summarized in Fig. 10.1.
Despite its advantages mentioned above, there also exists several shortcomings
of the FMEA methodology indicated in the literature (Bowles and Peláez 1995;
Braglia 2000; Braglia et al. 2003, 2007; Chang 2009; Chang and Cheng 2010,
2011; Chang and Sun 2009; Chang and Wen 2010; Chang et al. 2001, 1999, 2010;
Chen 2007; Chen and Ko 2009; Chin et al. 2009a, b; Franceschini and Galetto
2001; Gargama and Chaturvedi 2011; Geum et al. 2011; Kutlu and Ekmekçioğlu
Table 10.2 Probability of occurrence scale for a FM
Rating Probability of occurrence Failure probability
10 Extremely high: failure is almost inevitable >1 in 2
9 Very high 1 in 3
8 Repeated failures 1 in 8
7 High 1 in 20
6 Moderately high 1 in 80
5 Moderate 1 in 400
4 Relatively low 1 in 2000
3 Low 1 in 15,000
2 Remote 1 in 150,000
1 Nearly impossible <1 in 1,500,000
Table 10.3 Likelihood of detection scale for a FM
Rating Detection Likelihood of detection
10 Absolute
uncertainty
Potential cause/mechanism and subsequent FM cannot be
detected
9 Very remote Very remote chance of detecting potential cause/mechanism and
subsequent FM
8 Remote Remote chance of detecting potential cause/mechanism and
subsequent FM
7 Very low Very low chance of detecting potential cause/mechanism and
subsequent FM
6 Low Low chance of detecting potential cause/mechanism and
subsequent FM
5 Moderate Moderate chance of detecting potential cause/mechanism and
subsequent FM
4 Moderately
high
Moderately high chance of detecting potential cause/mechanism
and subsequent FM
3 High High chance of detecting potential cause/mechanism and
subsequent FM
2 Very high Very high chance of detecting potential cause/mechanism and
subsequent FM
1 Almost certain Potential cause/mechanism and subsequent FM will be detected
268 U. Asan and A. Soyer
2012; Liu et al. 2011, 2012, 2013b, c; Pillay and Wang 2003, Seyed-Hosseini et al.
2006, Sharma et al. 2008a, b; Wang et al. 2009; Xiao et al. 2011; Xu et al. 2002;
Yang et al. 2008, 2011; Zammori and Gabbrielli 2012; Zhang and Chu 2011), some
of which can be summarized as follows:
Relative weights (importance) of the risk factors (i.e., S, O, and D) are not
considered, while different weights will result in different priorities.
Various sets of S, O, and D ratings may produce the same RPN value, although,
relevant risk implications may be different.
It is not possible to measure the amount of difference between the ranks in an
ordinal scale, as the intervals of an ordinal scale are determined subjectively,
and therefore, are not identical to each other. Because of this, mathematical
operations cannot be performed on ordinal scales. However, in FMEA, RPN
values are calculated by multiplying the numerical ratings of S, O, and D, which
are measured on ordinal scales.
The use of multiplication for RPN calculations, instead of other relationships, is
questionable, as multiplication is very sensitive to the variations in the ratings of
S, O, and D.
Review the process, design, product, or service
Identify potential FMs
Identify potential effect(s) of FMs
(Assign severity ratings for each effect)
Identify potential causes(s) of FMs
(Assign occurrence ratings for each FM)
Evaluate current controls
(Assign detectability ratings for each FM and/or effect)
Prioritize failure modes
(Calculate the RPNs for each FM)
Identify and implement actions leading to improvement
(Take action to eliminate or reduce the high-risk FMs)
Reassess risks with another FMEA cycle
Preparation
Identification
Prioritization
Risk Reduction
Reassessment
Fig. 10.1 Steps of FMEA process
10 Failure Mode and Effects Analysis Under Uncertainty 269
It considers only three factors (S, O, and D), ignoring other signicant factors
such as, costs, production quantities, quality, etc.
It considers only one FM at a time, interdependencies among various FMs and
their effects are not taken into account.
Since many of the points in the RPN scale, which ranges from 1 to 1000, cannot
be formed from the product of S, O, and D (only 120 of the 1000 numbers can
be generated), RPNs are not continuous. Furthermore, most of the unique points
in the scale can be formed in several different ways (e.g., 60 can be formed from
24 different combinations of S, O, and D).
The conversion of ratings for the three components of a FM is different. The
relation between O and Os probability scale is non-linear, while the relation
between D (S) and Ds(Ss) probability scale is linear.
It is difcult, or even impossible, to give a precise and direct numerical eval-
uation for intangible quantities, such as S, O, and D.
In their review of literature on FME A, Liu et al. (2013b) investigated the
shortcomings of the FMEA methodology (which some of them are mentioned
above) and discussed the approaches used in the FMEA literature. They proposed a
framework for classifying the reviewed articles according to the FM prioritization
method used, in which the relevant approaches were divided into ve main cate-
gories: (i) Multi-Criteria Decision Making (MCDM) Approaches, (ii) Mathematical
Programming (MP) Approaches, (iii) Articial Intelligence (AI) Approaches,
(iv) Integrated Approaches, and (v) Other Approaches.
Among others, some of the common approaches classied into these ve cate-
gories are:
i. Evidence Theory (Yang et al. 2011), Analytical Hierarchy Process (AHP) (Hu
et al. 2009), Analytical Network Process (ANP) (Zammori and Gabbrielli
2012), Grey Theory (Chang et al. 1999, 2001), and Intui tionistic Fuzzy Sets
(Chang and Cheng 2010; Chang et al. 2010)
ii. Linear Programming (Chen and Ko 2009), Data Envelopment Analysis
(DEA) (Chin et al. 2009a; Chang and Sun 2009), Fuzzy DEA (Garcia et al.
2005)
iii. Rule-base Systems and Fuzzy Rule-base Systems (Gargama and Chaturvedi
2011; Sharma et al. 2008a, b)
iv. Fuzzy Cognitive Maps (Pelaez and Bowles 1996), Fuzzy Evidential
Reasoning and Grey Theory (Liu et al. 2011), Fuzzy AHP and Fuzzy TOPSIS
(Kutlu and Ekmekçioğlu 2012), Intuitionistic Fuzzy Sets (IFS) and
DEMATEL (Chang and Cheng 2010)
v. Monte Carlo Simulation (Bevilacqua et al. 2000), Minimum Cut Sets Theory
(Xiao et al. 2011), Quality Function Deployment (QFD) (Braglia et al. 2007),
and Probability Theory (SantAnna 2012).
According to Liu et al. (2013b), the categories including the most frequently
used approaches for the prioritization of FMs, are AI and MCDM, respectively.
Particularly, fuzzy rule-base syst em in AI category is the most used approach,
270 U. Asan and A. Soyer
followed by grey theory and AHP/ANP in the MCDM category. Fuzzy rule-based
approaches, although applied extensively in the literature, have also been criticized,
since they have some drawbacks that will be discussed in detail, in Sect. 10.2.
This chapter will be focusing on studies addressing the issues related to mod-
elling, qualitative natur e of risk assessment, as well as subjectivities and substantial
uncertainties inherent in the assessment process. In other words, the approaches
dealing with both complexity and uncertainty of the risk assessmen t process will be
reviewed. The rest of this chapter is organized as follows. First, a comprehensive
review of the literat ure is provided which is followed by illustrative examples for
selected approaches. Then, the methodological differences of these alternative
approaches are examined. Finally, conclusions and further research opportunities
are presented.
10.2 Multi-criteria Risk Prioritization Under Uncertainty
Identifying and prioritizing potential failure modes and their effects generally
requires dealing with uncertain information (including incomplete, vague and/or
ambiguous information) as well as highly subjective judgments of experts. The
uncertainties and subjectivities that arise here may stem from different sources, such
as (1) lack of knowledge, limited attention and information processing capabilities
(Asan et al. 2013); (2) vague assessmen t and grading criteria whose meaning, value,
or boundaries vary considerably according to context or conditions; and (3) frag-
mented expert judgments. The last source, also known as inter-personal uncertainty
(see Wu and Mendel 2009), can even emerge in situations where sufcient
knowledge is available. This is related to the fact that FMEA is commonly per-
formed in a group decis ion environment where experts may provide different
judgments for the same risk factors because of their different expertise and back-
grounds (Chin et al. 2009b; Song et al. 2014).
Thus, it becomes often unrealistic and impractical to acquire exact judgments in
risk assessment when distinct interpretations are present and/or available data is
incomplete or vague. Several authors have similarly reported that precision based
methods suggested in the literat ure have largely or totally failed to address these
certain sources of uncertainties. Below, the extensively criticized limitations of the
conventional FMEA methods in dealing with uncertainties associated with the
judgment process are summarized:
They cant handle imprecise data and subjective judgments of domain experts,
especially when the data set is small in size and its distribution is unknown.
They cant cope with incomplete assessments and total ignorance.
They cant deal with different types of assessment information simultaneously.
They require prior information, such as, assumptions or pre-de ned functions to
deal with uncertainty.
10 Failure Mode and Effects Analysis Under Uncertainty 271
They ignore the level of condence (belief degrees) experts are often willing to
express in their subjective assessments.
They ignore diversity in expert judgments.
They lack a framework to analyze complex structures.
They ignore other important factors (e.g., economical aspects).
Ideally, a complete theory and its accompanyin g tools used for identifying and
prioritizing potential FMs and their effects should therefore address not only
modeling issues, but also issues related to the qualitative nature of risk assessment
as well as the analysis of subjectivities and substantial uncertainties inherent in the
assessment process. To overcome some of the mentioned limitations of the con-
ventional FMEA methods, many alternative approaches have been suggested in the
literature. According to the literature review conducted by Liu et al. (2013b), the
most frequently studied class of approaches was found to be articial intelligence
in particular fuzzy rule-based approaches. There are several reasons why these
approaches have been more preferred. First of all, they can handle ambiguous,
qualitative as well as quantitative data in a consistent manner; second, they allow
combining risk factors (i.e. FMs) in a more exible and realistic manner; and
nally, the risk assessment function can be customized according to the particular
product, process or system under consideration (for more detail see Liu et al.
2013b). However, rule-based approaches have also signicant limitations. For
example, rule-based approaches require experts to design a sufciently rich set of
if-then rules and maintain them over time, which is often highly costly and
time-consuming. Otherwise, an incomplete rule base will produce biased or even
wrong inferences. Moreo ver, the rules with the same consequence but different
antecedents cannot be distinguished from one another, which makes a complete
prioritization or ranking of the failure modes impossible (Song et al. 2014). It is also
hard to dene proper membership functions for the risk factors and priority levels
(Liu et al. 2013b). Thus, rule based approaches, which tend to be highly subjective,
costly, and time consuming, should not necessarily be regarded as the best possible
method.
An alternative class developed for FMEA under uncertainty consists of multi-
criteria approaches. These approaches are able to handle both modelling issues
(e.g., scaling, structuring, aggregation, weighting, etc.) and issues related to the
analysis of subjectivities and substantial uncertainties inherent in the assessment
process. A review of the literature indicates a growing interest in these approaches,
especially in the past 5 years (see Fig. 10.2). Note that the source used for the
review was only academic journal articles published in the past 15 years. According
to the review, the most common theories and techniques employed in this class of
approaches are grey relational analysis, aggregation operators, fuzzy technique for
order preference by similarity to ideal solution (fuzzy TOPSIS), evidential rea-
soning (ER), intuitionistic fuzzy sets, type-1 fuzzy sets, 2-tuple fuzzy linguistic
representation, fuzzy analytic hierarchy process (fuzzy AHP), rough set theory,
fuzzy weighted geometric mean, fuzzy weighted least square, and possibility theory
272 U. Asan and A. Soyer
(see Table 10.4). Below, a summary of the reviewed papers based on this classi-
cation is provided.
Approaches Based on Grey Relational Analysis
Grey theory was initiated by Deng at the beginning of 1980s (Deng 1982, 1989).
Like fuzzy set theory, grey theory also deals with making decisions with poor,
incomplete, and multi-input information, and explores the behavior of a system
using relational analysis (De ng 1982, 1989; Sharma et al. 2008b; Kuo et al. 2008;
Geum et al. 2011; Chang et al. 2013). It provides a measure to analyze relationship
between discrete quantitative and qualitative series (Chang et al. 2001). As one of
the most applied techniques in FMEA, Grey Relational Analysis (GRA) is part of
grey system theory, which can easily handle complicated interactions between
multiple factors and variables. GRA provides a better distinction among decision
alternatives (Kuo et al. 2008), and it gives the opportunity to assign different
importance weights to S, O, and D. In one of the early studies, Chang et al. (2001)
used GRA where they assigned different weights to the risk factors, and eliminated
the need for a utility function and the conversion of ratings for the three components
of FMs. They demonstrated the applicability of the propos ed approach in an
automobile PCB assembly case. Similarly, Chang et al. (1999), proposed a new
approach for prioritizing the risks related to FMs. They adopte d the fuzzy linguistic
assessment to rate the risk factors, and applied grey theory to calculate the risk
priority numbe rs (RPNs) of potential causes of each FM. In another study, Pillay
and Wang (2003) presented an improved FMEA methodology utilizing fuzzy rule
base and grey relation theory. In their illustrative application to an ocean going
shing vessel, they generated 35 fuzzy if-then rules in order to identify lacking
safety featu res, and thus, to improve the operational safety of the vessel. Using this
fuzzy rule base, risk factor ratings for each FM were integrated to obtain linguistic
variables that were then used to rank FMs. As mentioned above, to address some of
the limitations of traditional FMEA (i.e., identical RPNs and equally weighted risk
factors) Sharma et al. (2008b), also proposed GRA to prioritize the causes of FMs.
Fig. 10.2 Distribution of the reviewed articles (*: only the rst quarter)
10 Failure Mode and Effects Analysis Under Uncertainty 273
Geum et al. ( 2011) developed a two-stage approach where a service-specic FMEA
was constructed in the rst stage; and GRA was applied to calculate the RPN of
each FM in the second stage. To represent the service characteristics, they deter-
mined 19 sub-dimensions for the three risk factors of FMEA. When performing
GRA in the second stage, risk scores for each sub-dimension were calculated rstly
to establish S, O, and D scores, respectively; and then, overall RPN of each FM
were obtained using these risk scores. Finally, in a recent study, Chang et al. (2013)
integrated the GRA and the DEMATEL method to rank FMs according to the risks
they represent for the organization, and presented an actual case of the TFT-LCD
cell process. They argued that their new approach provides a lower duplication rate,
generates more ideal rankings, and helps decision-makers to make more ideal
determinations. Consequently, the major advantages of applying GRA to FMEA
can be summarized as follo ws: (i) capability of dealing with incomplete infor mation
(ii) eliminating the need for a utility function, (iii) eliminating the need for the
conversion of ratings for the three components of FMs, (iv) capability of assigning
different importance weights to each risk factor, and (v) capability of providing a
better distinction among FMs.
Table 10.4 Classication of multi-criteria methods developed for FMEA under uncertainty
Approach Literature Total
number*
Grey relatio nal
analysis
Chang et al. (1999), (2001), (2013); Pillay and Wang
(2003); Geum et al. (2011); Liu et al. (2011)
6
Aggregation
operators
Chang (2009); Chang and Wen (2010); Chang and
Cheng (2011); Chang et al. (2012); Liu et al. (2014)
5
Fuzzy TOPSIS Braglia et al. (2003); Kutlu and Ekmekçioğlu (2012);
Song et al. (2013), (2014); Hadi-Vencheh and Aghajani
(2013)
5
Evidential reasoning Chin et al. (2009b); Liu et al. (2011); Yang et al.
(2011); Liu et al. (2013a)
4
Intuitionistic fuzzy
sets
Chang et al. (2010); Chang and Cheng (2010); Liu et al.
(2014)
3
Ordinary fuzzy sets Sharma et al. (2008b); Liu et al. (2012); Lin et al.
(2014)
3
2-Tuple fuzzy
linguistic
representation
Chang and Wen (2010); Chang et al. (2012)2
Fuzzy AHP Hu et al. (2009); Kutlu and Ekmekçioğ
lu (2012)2
Rough set theory Song et al. (2014)1
Fuzzy weighted
geometric mean
Wang et al. (2009)1
Fuzzy weighted least
square
Zhang and Chu (2011)1
Possibility theory Mandal and Maiti (2014)1
*Studies that involve more than one method are classied in more than one category in the table
274 U. Asan and A. Soyer
Approaches Based on Aggregation Operators
Aggregation operators weight values according to their ordering. In other words,
these techniques are used to nd optimal weights of the risk factors based on the
ranks of the weighting vectors after an aggregation process (Chang et al. 2012). At
the end, more accurate and reasonable ranki ng of the risk of failures may be
obtained. In this category, Chang (2009) and Chang and Cheng (2011) proposed
methodologies, which combine aggregation operators, such as, the ordered
weighted geometric averaging and fuzzy ordered weighted averaging
(OWA) operator, respectively, with the DEMATEL approach to evaluate the
orderings of FMs. Findings suggest that it is more suitable to consider preferences
in form of linguistic variables rather than numerical ones (Chang et al. 2012).
Chang and Wen (2010) proposed a technique, where the OWA operator and 2-tuple
fuzzy linguistic modelling is integrated, to prioritize failures in product design.
They showed that the proposed approach, in comparison to the conventional RPN
method, provides a more exible structure for combining S, O, and D factors.
Finally, Liu et al. (2014) proposed a new operator (intuitionistic fuzzy hybrid
weighted Euclidean distance) that takes into account both subjective and objective
weights of risk factors during the assessment process. The fragmented and uncertain
assessments provided by a group of experts are treated as linguistic terms expressed
in intuitionistic fuzzy numbers. The proposed operator allows reducing the impact
of disproportionately large (or small) deviations on the results by assigning them
low (or high) weights.
Approaches Based on Fuzzy TOPSIS
Another powerful method suggested to improve the convent ional FMEA is TOPSIS
a multi-criteria decision making approach used to rank alternatives on the basis
of the Euc lidean distance of an alternative from both the positive and negative ideal
solutions. Here, FMs are considered as the alternatives to be ranked with respect to
the risk factors S, O, and D, which correspond to the criteria. Braglia et al. (2003)
developed a fuzzy version of TOPSIS to provi de a framework that allows dealing
with imprecise quantities, such as those deriving from linguistic evaluations or
subjective and qualitative assessments. By performing a sensitivity analysis of the
fuzzy judgment weights and comparing results with the conventional method, they
conrmed that the proposed approach gives a reasonable and robust nal ranking of
FMs. In another study, Kutlu and Ekmekçioğlu (2012) integrated fuzzy AHP with
fuzzy TOPSIS in order to determine more realistic weights for the risk factors.
Fuzzy AHP allows experts weighting the risk factors in linguistic variables. Song
et al. (2013) also suggested a fuzzy weighted TOPSIS for FMEA under uncertainty.
However, they develo ped a novel weighting approach where subjective weights
derived from experts and objective weights obtained from an entropy-based method
are integrated to avoid any underestimation or overestimation of the FMs. In
another study of Song et al. (2014) a rough group TOPSIS method was proposed.
The method integrates the strength of rough set theory in handling vagueness and
the advantages of TOPSIS in modeling multi-criteria problems. Finally,
Hadi-Vencheh and Aghajani (2013) proposed a fuzzy TOPSIS method based on
10 Failure Mode and Effects Analysis Under Uncertainty 275
α-level sets and the fuzzy extension principle. They formulated a new relative
closeness coefcient in form of nonlinear programming (NLP) models and solved
them in a series of linear programming models. Consequently, all these studies have
shown that fuzzy TOPSIS is capable of: (1) assigning relative importance to risk
factors, (2) introducing a potentially larger number of risk factors, and (3) using
imprecise data in the form of fuzzy numbers.
Approaches Based on Evidential Reasoning
Evidential reasoning, as another popular approach in FMEA under uncertainty has
been originally d eveloped in the 1990s to support the solution of multi-attribute
decision analysis problems with ignorance (see Yang and Singh (1994)). The recent
ER approac hes can model both quanti tative and q ualitative attributes using a dis-
tributed modelling framework, in which each attribute is characteri zed by a set of
collectively exhaustive assessment grades (including incomplete information,
complete ignorance and/or fuzzy uncertainty) with different degrees of belief (Wang
et al. 2006). Experiences show that an expert may not always be fully condent in
his assessments and may be willing to express beliefs to subsets of adjacent grades
(Liu et al. 2011). A belief structure, in the FME A context, captures the performance
distribution of a subjective assessment of a FM. In one of the rst studies in this
group, Chin et al. (2009b) proposed a group-based ER approach, which can capture
diversity in FMEA team members opinions and prioritize FMs under different
types of uncertainties, such as, incomplete assessment, ignorance and intervals.
They calculate the overall belief structures and convert them into expected risk
scores, which are nally ranked using the minimax regret approach. Inspired by the
work of Chin et al. (2009b); Yang et al. (2011) adopte d the modied Dempster
Shafer evidence theory to aggregate the different opinions about FMs, which may
be inconsistent and uncertain. However, in the proposed model, the three risk
factors are regarded as discrete random variables and all assessment grades are
assumed to be crisp and independent of each other . The ER approach is further
developed by Liu et al. (2011) to deal with risk evaluation problems which involve
both probabilistic and fuzzy uncertainties. These are problems, where some of the
assessment grades are difcult to be expressed as clearly distinctive crisp sets, but
easier as overlapping fuzzy sets (Yang et al. 2006). The most recent study in this
group, by Liu et al. (2013a), combined the fuzzy evidential reasoning
(FER) approach with belief rule-based (BRB) methodology. The FER approach is
used to capture and aggregate expert opini ons, while the BRB methodology is used
to model the uncertain causal relationships between risk factors and the risk level.
A belief rule-base is a collection of expert knowledge that represents functional
mappings between risk factors (ante cedents) and risk levels (conclusions), possibly
with uncertainty. According to Yang et al. (2008), BRB provides a more infor-
mative and realistic scheme than a simple if-then rule base on uncertain knowledge
representation. To sum up, in comparison with the traditional FMEA and its
variants, an ER approach to FMEA yields the following advantages (see also Chin
et al. (2009b)): (1) the relative importance of risk factors are considered, (2) the
diversity and uncertainty of experts assessment information and related condence
276 U. Asan and A. Soyer
values can be well reected and modelled using belief structures, (3) FMs can be
fully ranked and well distinguished from each other, (4) the expected risk score is a
continuous number, and (5) risk factors are aggregated in a highly nonlinear
manner.
Approaches Based on Intuitionistic Fuzzy Sets
Intuitionistic Fuzzy Set (IFS), which is an extension of fuzzy set, was introduced by
Atanassov in 1983 (Atanassov 1986). In fuzzy set theory, the degree of
non-membership is calculated by subtracting the degree of membership from one.
However, this is not the case for IFSs. IFS adds an extra degree of uncertainty to
classic fuzzy sets for modelling the hesitation and uncertainty about the degree of
membership (Da Costa et al. 2010). Therefore, IFS can represent the imprecision of
data in a more comprehensive manner than fuzzy sets (Xu 2011). An IFS A in a
universe U,isdened as (Atanassov 1986):
A ¼ u; l
A
ðuÞ; m
A
ðuÞðÞju 2 U
fg
A ¼ðl
A
; m
A
ðÞfor shortÞð10:1Þ
where the functions l
A
: U ! 0; 1½and m
A
: U ! 0; 1½dene the grade of mem-
bership and the grade of non-membership of the each element of U to A, respec-
tively. The functions l
A
uðÞand m
A
uðÞshould satisfy the condition:
0 l
A
uðÞþm
A
uðÞ1 ð8u 2 UÞð10:2Þ
and
p
A
uðÞ¼ð1 m
A
uðÞl
A
uðÞÞ ð10:3Þ
where p
A
uðÞdenotes the uncertainty of u (also called as the hesitancy of u). Clearly,
in the case of ordinary fuzzy sets, p
A
uðÞ¼0 for 8u 2 U. For further detail, the
reader should refer to Atanassov (1986).
As mentioned above, due to its capability to deal with uncertainty, IFS has
recently been used in the FMEA literature. Chang et al. (2010) proposed a new
approach utilizing the IFS ranking technique for reprioritization of FMs and pre-
sented an illustrative example of a silane supply system in a TFT-LCD process.
According to Chang et al. (2010), their new approach reduces the occurrence of
duplicate RPNs and provides more accurate information, and real situations are
reected in a more realistic and exible manner. In another study, Chang and
Cheng (2010), integrated the IFS and DEMATEL approach on risk assessment
providing a more exible structure for combining risk factors. They claim that, the
proposed approach provides a more reasonable ranking where FMs are better
distinguished. Finally, Liu et al. (2013c) developed a methodology using
Intuitionistic Fuzzy Hybrid Weighted Euclidean Distance (IFHWED) operator. In
this methodology, linguistic terms were used for the asses sment of risk factors. In
order to aggregate multiple experts assessments into a group assessment, fuzzy
weighted averaging operator was used, and then, IFHWED operator was applied to
rank FMs, considering the weights of risk factors.
10 Failure Mode and Effects Analysis Under Uncertainty 277
Approaches Based on Ordinary Fuzzy Sets
One of the prominent area of application of fuzzy set theory is in modeling where
typically the available information contains various kinds of uncertainty due to
internal and external disturbances and limitation of human knowledge and under-
standing (Liu and Lin 2010). As experts from different expertise areas and skill
levels are included in FMEA proces s (Chin et al. 2009a, b), there usually exists an
imprecise information to be treated as an input of this process. Additionally,
complexity of the systems/products under investigation also increases the impre-
cision and uncertainty in FMEA. Therefore, as an effective tool providing a means
for representing the uncertainty, fuzzy set theory has been extensively employed in
FMEA literature. Bowles and Peláez (1995)s study, in which the risk factors used
in FMEA (i.e., S, O, and D) were represented as members of a fuzzy set, was the
rst study using the fuzzy sets theory for criticality analysis. In this study, linguistic
variables were used to assess the S, O, and D of FMs. Following the determination
of the degree of membership of each FM assessment to the corresponding fuzzy
sets, which were identied as a guide for ranking S, O, and D; these fuzzy inputs
were then evaluated using a linguistic rule base and fuzzy logic operations. Finally,
the results were defuzzied and all FMs were ranked according to their criticality
levels. Sharma et al. (2008b) established a framework based on fuzzy methodology
and grey relation analysis to evaluate and assess system failure behavior, and
presented a case from a process industry to demonstrate the applicability of the
proposed framework. They concluded that their framework provides an effective
way to combine expert knowledge and experience as well as to deal with uncer-
tainty and imprecision in a more realistic manner. In an another study, Liu et al.
(2012) used linguistic variables to assess the ratings and weights of the risk factors
S, O, and D, and proposed a new risk priority model that extends VIKOR method to
determine the risk priorities of FMs. They applied this model to the assessment of
risks in general anesthesia process and claimed that they address some of the
shortcomings of the traditional FMEA. Finally, in a very recent study, Lin et al.
(2014) proposed an assessment model for human reliability in the risk assessment
of medical devices, which applies the fuzzy linguistic theory to deal with the
subjective assessments of experts. They noted that their proposed model, differing
from the qualitative and quantitative methods used in human reliability analysis,
considers some critical aspects, such as, context related factors, organizational
factors, and errors in FMEA team members assessments. Consequently, fuzzy set
theory yields the following advantages over traditional FMEA: (i) qualitative, as
well as quantitative, data can be used in the assessment, (ii) risk factors of FMs can
be directly assessed using the linguistic terms, and (ii i) S, O, and D can be com-
bined in a more exible manner.
Approaches Based on 2-Tuple Fuzzy Linguistic Representation
As indicated above, several authors have applied the fuzzy linguistic approach to
FMEA problems with uncertain data. In these studies, the FMs are evaluated with
respect to S, O, and D using a linguistic domain treated as discrete. However,
operations (most notably multiplication) on fuzzy numbers produce results that
278 U. Asan and A. Soyer
usually do not exactly match any of the initial linguistic terms. To resolve this issue,
an approximation process is used to express the results in the discrete initial
expression domai n that, however, leads to loss of information and hence lack of
precision in the nal results (Herrera and Martínez 2000). To overcome this critical
shortcoming, Chang and Wen (2010) have suggested an FMEA model based on the
2-tuple fuzzy linguistic representation developed by Herrera and Martínez (2000).
This model represents the crisp or linguistic information with a pair of values, called
2-tuple, which is composed by a linguistic term and a numeric value assessed in
[0.5, 0.5). In this way, any information obtained in an aggregation process can be
represented on its domain. In their case study, Chang and Wen (2010) showed that
their fuzzy linguistic representation model combined with the OWA operator
effectively solves the problem of measurement scales (i.e., information loss in
aggregation). In another study, Chang et al. (2012) have integrated 2-tuple fuzzy
linguistic representation and the Linguistic Ordered Weighted Geometric Averaging
(LOWGA) operator in process FMEA. This approach, as in Chang and Wen (2010 ),
provides reasonable rankings for cases including FMs having the same RPN.
Approaches Based on Fuzzy AHP
Some recent studies have suggested using fuzzy AHP to explicitly accommodate
the inherent uncertainty and complexity associated with risk assessment.
Fuzzy AHP involves several concepts and techniques, such as, hierarchical struc-
turing, pairwise comparison, prior itization principles for deriving weights , consis-
tency considerations, and priority synthesis (see Saaty 1988). The hierarchical
models developed in these studies typically consist of a goal (risk assessment),
criteria (risk factors) and alternatives (FMs). Hu et al. (2009), for example, sug-
gested a hierarchical risk assessment model to evaluate the risk of green compo-
nents. They used triangular fuzzy numbers to express the comparative judgments of
decision-makers. The resulting global priority values, i.e., the green component
RPNs, are used to identify high-risk components and provide insight to the
incoming quality control staff for improving the efciency of inspection and miti-
gating risk. Kutlu and Ekmekçioğlu (2012) have also suggested applying fuzzy
AHP to determine the weight vector of the three risk factors (S, O, and D).
However, differently from the former study, they preferred Chen (2000)s fuzzy
TOPSIS to prioritize the nal risk scores of the FMs.
Approaches Based on Rough Set Theory
Rough set theory, proposed by Pawlak in the early 1980s, is a formal approximation
of the classical set theory that can handle imprecise and subjective judgments
without any assumption and additional information (e.g., membership functions). In
fact, predened fuzzy membership functions or crisp rating scales in FMEA allow
only judgments in form of point or xed interval values and, hence, do not fully
reect the subjectivity and preference differences of experts. However, the rough set
approach to FMEA, proposed by Song et al. (2014), provides a more rational risk
evaluation framework where exible intervals (i.e., rough intervals) are used to
represent the inter-personal uncertainty. Here, a larger rough interval indicates a
higher inconsistency among the experts. In this respect, the proposed rough FMEA
10 Failure Mode and Effects Analysis Under Uncertainty 279
not only provides an improved representation of the subjectivity and uncertainty in
the evaluations, but also maintains the objectivity of original information (Zhai
et al. 2007).
Approaches Based on Fuzzy Weighted Geometric Mean
As discussed in previous sections, traditional FMEA has limitations in terms of
acquiring precise assessment information on the three components of FMs. In
response to this limitation, it has been suggested in the literature to evaluate these
risk factors by using linguistic scales and to use fuzzy FMEA, which utilizes a
fuzzy rule-based reasoning approach to obtain RPN ratings. However, building up a
complete and accurate rule base is a tedious and time-consuming task, particularly
for the complex systems/products. Additionally, relative importance weights of the
risk factors are not taken into consideration in traditional FMEA. Therefore, to
overcome these limitations, Wang et al. (2009) suggested using Fuzzy Weighted
Geometric Mean (FWGM) method for the calculation of FRPNs to prioritize FMs.
In thei r study, Wang et al. (2009), rstly, evaluated the risk factors and their
importance weights in a linguistic manner; then computed FRPNs applying an
alpha cut based linear programming approach; and nally, defuzzied FRPNs using
centroid defuzzication method for the nal ranking of FMs. According to the
authors, besides the above-mentioned advantages, the proposed methodology has
the potential to fully prioritize FMs and hence to distinguish each FM from one
another, and is not limited to the risk factors, S, O, and D.
Approaches Based on Fuzzy Weighted Least Square
As mentioned before, generally, people from different expertise areas are included
in FMEA process. Therefore, members of these cross-functional FMEA teams,
usually view the system/product under investigation from various perspectives,
responsibilities and concerns, as they have different levels of knowledge, skills,
experiences and personalities (Liu et al. 2013c). For that reason, FMEA team
members may use different linguistic term sets when evaluating and weighting the
relevant risk factors (Herrera et al. 2000); in other words, they may give their
judgments in different forms. In order to ensure that the aggregated assessment of
the FMEA team reects all members viewpoints and priorities, Zhang and Chu
(2011) claimed that, Fuzzy Weighted Least Squares Model (FWLSM) can be used.
By using FWLSM, the total deviation degree between each individual assessment
information and the aggregated assessment information can be easily determined.
As being the only study in this category, Zhang and Chu ( 2011) proposed a new
approach, integrating FWLSM, the method of imprecision (MOI) and the met hod of
partial ranking for evaluating and ranking the FMs. In this approach, rstly, a
FMEA team evaluates each FM by using linguistic term sets with different cardi-
nalities (i.e., multi-granularity linguistic term sets). Then the individual assessments
are aggregated by means of FWLSM. Following the aggregation step, nonlinear
programming method incorporated with the MOI is used for calculating the fuzzy
RPNs in order to address the compensation levels among risk factors. Finally, by
using the Hamming distance between each two fuzzy RPNs, the partial ranking
method that is based on fuzzy preference relations is applied to rank FMs. For
280 U. Asan and A. Soyer
illustrative purposes, Zhang and Chu (2011) have applied their approach to the case
of a new product development application, and have concluded that their approach
provides more precisely expressed individual assessments, more accurate fuzzy
RPNs, and thus, more robust results.
Approaches Based on Possibility Theory
Fuzzy numerical technique for FMEA, as well as traditional FMEA technique and
fuzzy rule-based technique, have some limitations. When defuzzied crisp risk
values are used to obtain nal ranking of FMs, as in fuzzy numerical technique, the
entropy present in fuzzy sets is ignored (Mandal and Maiti 2014). Therefore, like
other techniques, fuzzy numerical technique also suffers from the limitation of
providing arbitrary nal ranking of FMs. In response to this limitation, Mandal and
Maiti (2014) developed a robust methodology that integrates the similarity value
measure of fuzzy numbers and possibility and necessity measures of possibility
theory. Similar to the fuzzy set theory, possibility theory is also an uncertainty theory
devoted to the handling of incomplete, imprecise, and uncertain information. As it
uses the possibility and necessity measures, it has the capability to capture partial
ignorance (Dubois and Prade 2011). In their recent study, Mandal and Maiti (2014)
rstly, used similarity measure approach to obtain FRPNs, and subsequently, rele-
vant priority values are clustered by means of comparison with a standard linguistic
scale. After partially ordering FRPNs, they used possibility theory for making
comparison with conformance guidelines. To this end, after calculating the possi-
bility and necessity measures, they combined these two dual measures to obtain
credibility measure, and consequently, used this measure to compare FRPNs with
compliance guidelines. Here, as the credibility measure gets closer to one, the
possibility of the relevant risk being lower than or equal to the confor mance
guideline increases; on the other hand, a s it gets closer to zero, then the possibil ity of
the relev ant risk being lower than or equal to the conformance guideline decreases.
From the review above, it can be concluded that although all approaches deal
with uncertainty and subjectivity associated with risk assessment, each one
addresses only a particular set of shortcomings of the conventional FMEA. In the
following section, the most frequently studied and promising approaches are
illustrated by a simple example.
10.3 Illustrative Example for Selected Approaches
In the previous section, many of the new approaches mentioned in the literature are
reviewed, but here only six well-known ones (based on ordinary fuzzy sets, grey
relational analysis, evidential reasoning, intuitionistic fuzzy sets, 2-tuple fuzzy
linguistic representation, and rough set theory) will be discussed in detail. The main
concern will not be the identi cation of risk factors, but their assessment and
aggregation. First, a summary of their theoretical underpinnings will be presented.
Then, to illustrate their basic steps and ability to deal with uncertainty, a simple
10 Failure Mode and Effects Analysis Under Uncertainty 281
example will be worked out for all six approaches. The example, adapte d from
Kutlu and Ekmekçioğlu (2012), involves the prioritization of risks in an assembly
process at a manufacturing facility operating in the automotive industry. The
potential failure modes (FMs) in the assembly process, identied by a group of
experts, are: non-conforming material (FM
1
), wrong die (FM
2
), wrong program
(FM
3
), excessive cycle time (FM
4
), wrong process (FM
5
), damaged goods (FM
6
),
wrong part (FM
7
), and incorrect forms (FM
8
). For the rating of these FMs, with
respect to three risk factors, experts use the linguist ic terms given in Table 10.5,
where each term corresponds to a triangular fuzzy number. If other wise not stated,
the relative importance of the risk factors S, O, and D are assessed by pairwise
comparisons using the linguistic scale provided in Table 10.6. Note that, for the
approaches using crisp ratings in the assessment of FMs or risk factors, only the
midvalues of the triangular fuzzy numbers will be considered in the analyses.
The assessments of the FMs and risk factors using linguistic terms and numerical
values were obtained from three experts as presented in Tables 10.7, 10.8, 10.9 and
10.10, respectively. For example, as shown in Table 10.7, the assessments of the
three experts of FM
1
with respect to severity are Medium, Medium, and
Medium Low. These linguistic terms can be converted into the follow ing crisp
values 5, 5, and 3 as shown in Table 10.9. In another example, it can be seen from
Table 10.8 that the comparison of the risk factors severity and occurrence is in
favour of the former, as Strongly Important, Strongly Important, and Very
Important (3/2, 3/2, and 2 in crisp values as given in Table 10.10).
Table 10.5 Linguistic scales used for rating FMs
Linguistic scale Fuzzy scale
Severity Occurrence Detection
Very low (VL) Very low (VL) Very high (VH) (0, 0, 1)
Low (L) Low (L) High (H) (0, 1, 3)
Medium low (ML) Medium low (ML) Medium high (MH) (1, 3, 5)
Medium (M) Medium (M) Medium (M) (3, 5, 7)
Medium high (MH) Medium high (MH) Medium low (ML) (5, 7, 9)
High (H) High (H) Low (L) (7, 9, 10)
Very high (VH) Very high (VH) Very low (VL) (9, 10, 10)
Table 10.6 Linguistic scale used for pairwise comparisons
Linguistic scale Fuzzy scale Fuzzy reciprocal scale
Equally important (EI) (1, 1, 1) (1, 1, 1)
Weakly important (WI) (1, 1, 3/2) (2/3, 1, 1)
Strongly important (SI) (1, 3/2, 2) (1/2, 2/3, 1)
Very important (VI) (3/2, 2, 5/2) (2/5, 1/2, 2/3)
Absolutely important (AI) (2, 5/2, 3) (1/3, 2/5, 1/2)
282 U. Asan and A. Soyer
Table 10.7 Linguistic scores of FMs with respect to each risk factor
Failure mode S O D
E
1
E
2
E
3
E
1
E
2
E
3
E
1
E
2
E
3
FM
1
M M ML M MH MH L ML L
FM
2
L ML ML VH H VH MH MH H
FM
3
ML L ML VH H H VH MH H
FM
4
ML M ML M MH MH L ML L
FM
5
M M ML MH MH H L VL L
FM
6
MH MH M MH H MH MH MH M
FM
7
L MLVLVHVHVHVHMHH
FM
8
VL VL L VL VL VL VH VH VH
Table 10.8 Pairwise comparisons of risk factors using the linguistic scale (R: Reciprocal)
SOD
E
1
E
2
E
3
E
1
E
2
E
3
E
1
E
2
E
3
Severity EI EI EI SI SI VI WI WI WI
Occurrence RRREIEIEIWIR EI
Detection RRRRSIREIEIEI
Table 10.9 Crisp scores of
FMs with respect to each risk
factor
Failure mode S O D
E
1
E
2
E
3
E
1
E
2
E
3
E
1
E
2
E
3
FM
1
553577979
FM
2
13310910331
FM
3
3131099031
FM
4
353577979
FM
5
5537799109
FM
6
775797335
FM
7
1 3 0 10 10 10 0 3 1
FM
8
001000000
Table 10.10 Pairwise
comparisons of risk factors
using crisp values
SOD
E
1
E
2
E
3
E
1
E
2
E
3
E
1
E
2
E
3
Severity 1 1 1 3/2 3/2 2 1 1 1
Occurrence 2/3 2/3 1/2 1 1 1 1 2/3 1
Detection 11113/21111
10 Failure Mode and Effects Analysis Under Uncertainty 283
10.3.1 FMEA Using Fuzzy Evidential Reasoning
The illustrated fuzzy evidential reasoning based approach, suggested by Liu et al.
(2011), offers a unique way for aggregating expert judgments and prioritizing FMs
in FMEA. One of the main advantages of this approach is its ability to coherently
model both accurate data and subjective judgments with various types of uncer-
tainties (such as, incomplete and fuzzy information as well as complete ignorance)
in a uni ed fram ework. A furt her strength of the approach is its ability to reect the
diversity in expert judgments. All these benets are achieved by incorporating
the experts level of condence in their assessments (i.e., belief degrees) into the
analysis. Here, a subjective assessment is characterized by a belief structure that
describes the intensity of the belief for each possible assessment value. The
aggregation of such structures allows one to merge multiple sources of evidence
(numerical or linguistic) for the same risk factor or FM (Yang et al. 2006).
Unquestionably, the belief structures provide experts with an easy-to-use and
exible way to express their opinions and can better quantify risk factors than the
traditional FMEA methods (Liu et al. 2011).
Below, it will be illustrated how the beli ef structures of each FM provided by
each expert can be aggregated into a group belief structure and how the group belief
structures of each FM with respect to the three risk factors can be synthesized into
an overall belief structure.
Stage 1: Assessment of FMs Using Belief Structures
FMs are assessed u sing the linguistic terms provided in Table 10.5. For example,
the set of evaluation grades for Severity are represented as H
FS
¼ Very Low;
f
Low; Medium Low; Medium; Medium High; High; Very Highg For the sake of
simplicity without losing generality, all seven individual assessment grades are
approximated by triangular fuzzy numbers of which only two adjacent ones
intersect. Then, let H
ij
; b
k
ij
ðFM
n
; RF
l
Þ

; i ¼ 1; ...; 7; j ¼ 1; ...; 7
no
be the belief
structure provided by expert E
k
on the assessment of failure mode FM
n
with respect
to risk factor RF
l
, where H
ii
for i ¼ 1 7 are the fuzzy assessment grades, H
ij
for
i ¼ 1 6 and j ¼ i þ 1 to 7 are the interval fuzzy assessment grades between H
ii
and H
jj
, and b
k
ij
ðFM
n
; RF
l
Þ are the belief degrees for the intervals H
ij
(Liu et al.
2011). The interval fuzzy assessment grades, H
ij
,dene trapezoidal fuzzy sets that
include the fuzzy assessment grades H
ii
; H
ði þ 1Þði þ 1Þ
; ...; H
jj
as shown in Fig. 10.3.
For Severity, the grades H
ii
for i ¼ 1 7 and the intervals H
ij
for i ¼ 1 6 and
j ¼ i þ 1 to7 all together can be expressed as
^
H
FS
¼ H
ij
; i ¼ 1; ...; 7; j ¼ 1; ::; 7

or equivalently as
284 U. Asan and A. Soyer
^
H
FS
¼
H
11
H
12
H
13
H
14
H
15
H
16
H
17
H
22
H
23
H
24
H
25
H
26
H
27
H
33
H
34
H
35
H
36
H
37
H
44
H
45
H
46
H
47
H
55
H
56
H
57
H
66
H
67
H
77
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
This type of formulation allows experts to provide their subjective judgments in
four possible ways (Liu et al. 2011):
Certain. For example, Medium as a certain grade can be written as
H
33
; 1:0ðÞ
fg
.
Distribution. For example, a FM that is assessed as High with a condence
level of 0.3 and as Very High with a condence level of 0.7, can be expressed
as H
66
; 0:3ðÞ; H
77
; 0:7ðÞ
fg
. This is a complete distribution. When all condence
levels do not sum to one, the distribution is known to be incomplete. The
missing information in such cases is called local ignorance and it can be
assigned any grade between Very Low and Very High.
Interval. For example, a FM that is assessed between Medium and High,
can be expressed as H
46
; 1:0ðÞ
fg
.
Total Ignorance. The FM can be assigned any grade between Very Low and
Very High and will be expressed as H
17
; 1:0ðÞfg. In such cases the expert is
whether unable or unwilling to provide an assessment.
Notice that FMs assessed to high values or intervals with high condence levels
are more risky than those assessed to low values or intervals with high condence
levels (Chin et al. 2009b). Table 10.11 presents the assessment information (in form
of belief structures) on the eight FMs provided by three expert s. The incomplete
assessments and ignorance information are shaded and highlighted, respectively.
For example, according to expert E
1
, the Severity of using non-conforming
Fig. 10.3 Interval fuzzy grade H
ij
(shown in dashed line)
10 Failure Mode and Effects Analysis Under Uncertainty 285
material (FM
1
) is Medium with high condence level (90 %). This assessment is
incomplete with 10 % missing information.
Stage 2: Synthesis of Individual Belief Structures into Group Belief Structures
In this stage, the belief structures provided by the experts for each FM are syn-
thesized into a group belief structure. Suppose that K experts, each given a weight
k
k
[ 0(k =1,, K) satisfying the condition
P
K
k¼1
k
k
¼ 1toreect his/her relative
importance, assess N failure modes with respect to L risk factors. Then, the
aggregate assessment value, i.e., the fuzzy group belief structure, for each FM with
respect to each risk factor is derived as follows (Liu et al. 2011, Chin et al. 2009b)
~
X
n
lðÞ¼ H
ij
; b
ij
ðFM
n
; RF
l
Þ

; i ¼ 1; ...; 7; j ¼ 1; ...; 7

; n ¼ 1; ...; N; l ¼ 1; ...; L
ð10:4Þ
where the group belief degree, b
ij
ðFM
n
; RF
l
Þ, is calculated as
b
ij
FM
n
; RF
l
ðÞ¼
X
k
k¼1
k
k
b
k
ij
FM
n
; RF
l
ðÞ; i ¼ 1; ...; 7; j ¼ 1; ...; 7;
n ¼ 1; ::; N; l ¼ 1; ...; L
ð10:5Þ
The relative importance of each expert shoul d reect the experts experience and
domain knowledge (Chin et al. 2009b). In the current example, the weights are
supposed to be k
1
¼ 0:2, k
2
¼ 0:5, k
3
¼ 0:3 and Table 10.12 presents the resulting
group assessment values calculated using these weights. For example, given the
individual belief structures ðH
44
; 0:90Þfg, ðH
44
; 1:00Þfg, ðH
33
; 1:00Þfgof three
experts for FM
1
with respect to the risk factor Severity (see Table 10.11), the
corresponding group belief structure is obtained as H
33
; 0:30ðÞ; H
44
; 0:68ðÞ;
f
H
17
; 0:02ðÞgwhere
Table 10.11 Assessment information on eight FMs by three experts
Risk factors Experts Failure modes
FM
1
FM
2
FM
3
FM
4
FM
5
FM
6
FM
7
FM
8
Severity E
1
(H
44
, 0.9) H
22
H
33
H
33
H
44
H
55
H
22
H
11
E
2
H
44
H
33
H
22
H
44
H
44
H
55
H
33
H
11
E
3
H
33
H
33
H
33
H
33
H
33
H
44
H
11
H
22
Occurrence E
1
H
44
H
77
H
77
H
44
H
55
H
55
H
77
H
11
E
2
H
55
H
66
H
66
H
55
H
55
H
66
H
77
H
11
E
3
H
55
H
77
H
66
H
55
H
66
H
55
H
77
H
11
Detection E
1
H
66
H
33
H
11
H
66
H
66
H
33
H
11
?*
E
2
H
55
H
33
H
33
H
55
H
77
H
33
H
33
H
11
E
3
H
66
(H
22
, 0.8) H
22
H
66
H
66
H
44
H
22
H
11
*:? Refers to total ignorance
286 U. Asan and A. Soyer
Table 10.12 Fuzzy group belief structures
Failure mode Severity Occurrence Detection
FM
1
H
33
; 0 :30ðÞ; H
44
; 0 :68ðÞ; H
17
; 0:02ðÞfgH
44
; 0 :20ðÞ; H
55
; 0 :80ðÞfgH
55
; 0 :50ðÞ; H
66
; 0 :50ðÞfg
FM
2
H
22
; 0 :20ðÞ; H
33
; 0 :80ðÞfg H
66
; 0 :50ðÞ; H
77
; 0 :50ðÞfgH
22
; 0 :24ðÞ; H
33
; 0 :70ðÞ; H
17
; 0:06ðÞfg
FM
3
H
22
; 0 :50ðÞ; H
33
; 0 :50ðÞfg H
66
; 0 :80ðÞ; H
77
; 0 :20ðÞfgH
11
; 0 :20ðÞ; H
22
; 0 :30ðÞ; H
33
; 0:50ðÞfg
FM
4
H
33
; 0 :50ðÞ; H
44
; 0 :50ðÞfg H
44
; 0 :20ðÞ; H
55
; 0 :80ðÞfgH
55
; 0 :50ðÞ; H
66
; 0 :50ðÞfg
FM
5
H
33
; 0 :30ðÞ; H
44
; 0 :70ðÞfg H
55
; 0 :70ðÞ; H
66
; 0 :30ðÞfgH
66
; 0 :50ðÞ; H
77
; 0 :50ðÞfg
FM
6
H
44
; 0 :30ðÞ; H
55
; 0 :70ðÞfg H
55
; 0 :50ðÞ; H
66
; 0 :50ðÞfgH
33
; 0 :70ðÞ; H
44
; 0 :30ðÞfg
FM
7
H
11
; 0 :30ðÞ; H
22
; 0 :20ðÞ; H
33
; 0:50ðÞfgH
77
; 1 :00ðÞfg H
11
; 0 :20ðÞ; H
22
; 0 :30ðÞ; H
33
; 0:50ðÞfg
FM
8
H
11
; 0 :70ðÞ; H
22
; 0 :30ðÞfg H
11
; 1 :00ðÞfg H
11
; 0 :80ðÞ; H
17
; 0 :20ðÞfg
10 Failure Mode and Effects Analysis Under Uncertainty 287
b
33
FM
1
; RF
1
ðÞ¼
X
3
k¼ 1
k
k
b
k
33
FM
1
; RF
1
ðÞ¼0:20 0 þ 0:50 0 þ 0:30 1:00
¼ 0:30
b
44
FM
1
; RF
1
ðÞ¼
X
3
k¼ 1
k
k
b
k
44
FM
1
; RF
1
ðÞ¼0:20 0:90 þ 0:50 1 :00 þ 0:30 0
¼ 0:68
For the missing information in b
44
FM
1
; RF
1
ðÞany grade between Very Low
and Very High can be assigned, thus
b
17
FM
1
; RF
1
ðÞ¼
X
3
k¼ 1
k
k
b
k
17
FM
1
; RF
1
ðÞ¼0:20 0:10 þ 0:50 0 þ 0:30 0
¼ 0:02
Stage 3: Defuzzication
Before a group belief structure is aggregated into an overall belief structure, a
defuzzication process is applied to convert fuzzy numbers into appropriate crisp
values. Various defuzzication methods have been developed in the literature,
including various forms of the centroid method, rst (or last) of maxima, mean max
membership, total integral value met hod, among others (see Ross 2009; Ramli and
Mohamad 2009). Liu et al. (2011 ) suggests using the defuzzi cation method
developed by Chen and Klein (1997) which is quite simple to carry out. The
formula is as follows
h
ij
¼
P
n
i¼0
ðb
i
cÞ
P
n
i¼0
b
i
cðÞ
P
n
i¼0
ða
i
dÞ
; i ¼ 1; ...; 7; j ¼ 1; ...; 7 ð10:6Þ
where the values c and d denote the lower and upper limits of the linguistic scale,
the values a
0
and b
0
(for a triangular membership function) represent the extreme
limits of each linguistic term where the membership function is 0, and a
1
and b
1
are
the values where the membership function is 1 (Pillay and Wang 2003). Here, h
ij
is
the defuzzied crisp value of H
ij
. Accordingly, the crisp group belief structure can
be represented as follows:
X
n
lðÞ¼ h
ij
; b
ij
ðFM
n
; RF
l
Þ

; i ¼ 1; ...; 7; j ¼ 1; ...; 7

; n ¼ 1; ...; N; l ¼ 1; ...; L
ð10:7Þ
For example, the linguistic term Very LowVery High, i.e., H
17
, can be
defuzzied as shown below (see Fig. 10.4):
h
17
¼
b
0
c½þb
1
c½
b
0
c½þb
1
c½fga
0
d½þa
1
d½fg
ð10:8Þ
288 U. Asan and A. Soyer
h
17
¼
10 0½þ10 0½
10 0½þ10 0½
fg
0 10½þ0 10½
fg
¼ 0:5
After calculating h
33
and h
44
in a similar way, the crisp group belief structure of
FM
1
with respect to Severity can be stated as follows
X
1
1ðÞ¼ 0:33; 0:30ðÞ; 0:5; 0:68ðÞ; ð0:5; 0:02Þ
fg
Table 10.13 presents all FMs crisp group belief structures.
Stage 4: Aggregation of Defuzzied Group Belief Structures into Overall Belief
Structure
Once the group belief structures of the FMs are defuzzied, they are aggregated into
overall belief structures by using the following equation:
X
n
lðÞ¼
X
7
i¼1
X
7
j¼1
h
ij
b
ij
FM
n
; RF
l
ðÞ; n ¼ 1; ...; N; l ¼ 1; ...; L ð10:9Þ
As an example, the overall belief structure of FM
1
with respect to the risk factor
Severity is obtained as given below:
X
1
1ðÞ¼
X
7
i¼1
X
7
j¼1
h
ij
b
ij
FM
1
; RF
1
ðÞ¼0:333 0:30 þ 0:5 0:68 þ 0:5 0:02
¼ 0:450
Note that, in the examp le above, values with zero belief degrees are omitted. The
results for all FMs are presented in Table 10.14.
Stage 5: Introduction of Group Weights
In this stage, the relative importance of the risk factors, which will later be used to
calculate the nal priorities of the FMs, are determined. Instead of presenting the
direct estimation method suggested by Liu et al. (2011), an alternative approach is
introduced here. According to this method, the weight information can be elicited
Fig. 10.4 Defuzzication of the linguistic term Very LowVery High
10 Failure Mode and Effects Analysis Under Uncertainty 289
Table 10.13 Crisp group belief structures
Failure mode Severity Occurrence Detection
FM
1
0:333; 0:30ðÞ; 0:5; 0:68ðÞ;
0:5; 0:02ðÞ

0:5; 0:20ðÞ; 0:667; 0:80ðÞ
fg
0:667; 0:50ðÞ; 0:826; 0:50ðÞ
fg
FM
2
0:174; 0:20ðÞ; 0:333; 0:80ðÞ
fg
0:826; 0:50ðÞ; 0:952; 0:50ðÞ