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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 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.

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 ﬁrst 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 deﬁnition, FM refers to

the termination of the ability of a system to perform a required function or its

inability to perform within previously speciﬁed limits (ISO/IEC-15026-1 2013) and

includes both known and/or potential failures, problems, or errors that may affect

the customers and 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 ﬁrst 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

deﬁciencies (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 ﬂaws.

Therefore, it focuses on FMs caused by design-related deﬁciencies 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 deﬁciencies.

(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 deﬁ-

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 ﬁnd 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 communication 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 Johanyák1996); 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 identiﬁed FM. This

methodology consists of assessing the FMs with respect to their ‘severity (S)’,

‘probability of occurrence (O)’, and ‘likelihood 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 ¼SOD. 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 signiﬁcant 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 signiﬁcant 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 O’s probability scale is non-linear, while the relation

between D (S) and D’s(S’s) probability scale is linear.

•It is difﬁcult, 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 FMEA, 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 ﬁve main cate-

gories: (i) Multi-Criteria Decision Making (MCDM) Approaches, (ii) Mathematical

Programming (MP) Approaches, (iii) Artiﬁcial Intelligence (AI) Approaches,

(iv) Integrated Approaches, and (v) Other Approaches.

Among others, some of the common approaches classiﬁed into these ﬁve 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 Intuitionistic 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 (Sant’Anna 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 system 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 nature 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 assessment process will be

reviewed. The rest of this chapter is organized as follows. First, a comprehensive

review of the literature 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 assessment 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 sufﬁcient

knowledge is available. This is related to the fact that FMEA is commonly per-

formed in a group decision 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 literature 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 can’t handle imprecise data and subjective judgments of domain experts,

especially when the data set is small in size and its distribution is unknown.

•They can’t cope with incomplete assessments and total ignorance.

•They can’t deal with different types of assessment information simultaneously.

•They require prior information, such as, assumptions or pre-deﬁned functions to

deal with uncertainty.

10 Failure Mode and Effects Analysis Under Uncertainty …271

•They ignore the level of conﬁdence (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 accompanying 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 artiﬁcial 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 ﬂexible and realistic manner; and

ﬁnally, 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 signiﬁcant limitations. For

example, rule-based approaches require experts to design a sufﬁciently 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. Moreover, 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 deﬁne 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-

ﬁcation 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 (Deng 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 proposed approach in an

automobile PCB assembly case. Similarly, Chang et al. (1999), proposed a new

approach for prioritizing the risks related to FMs. They adopted the fuzzy linguistic

assessment to rate the risk factors, and applied grey theory to calculate the risk

priority numbers (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

ﬁshing vessel, they generated 35 fuzzy if-then rules in order to identify lacking

safety features, 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 ﬁrst quarter)

10 Failure Mode and Effects Analysis Under Uncertainty …273

Geum et al. (2011) developed a two-stage approach where a service-speciﬁc FMEA

was constructed in the ﬁrst 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 ﬁrstly

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 follows: (i) capability of dealing with incomplete information

(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 Classiﬁcation of multi-criteria methods developed for FMEA under uncertainty

Approach Literature Total

number*

Grey relational

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 Ekmekçioğlu (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 classiﬁed 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 ﬁnd 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 ranking 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 ﬂexible 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 conventional FMEA is TOPSIS

—a multi-criteria decision making approach used to rank alternatives on the basis

of the Euclidean 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 provide 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

conﬁrmed that the proposed approach gives a reasonable and robust ﬁnal 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 developed 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 coefﬁcient 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 developed in the 1990s to support the solution of multi-attribute

decision analysis problems with ignorance (see Yang and Singh (1994)). The recent

ER approaches can model both quantitative and qualitative attributes using a dis-

tributed modelling framework, in which each attribute is characterized 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 conﬁdent in

his assessments and may be willing to express beliefs to subsets of adjacent grades

(Liu et al. 2011). A belief structure, in the FMEA context, captures the performance

distribution of a subjective assessment of a FM. In one of the ﬁrst 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 ﬁnally ranked using the minimax regret approach. Inspired by the

work of Chin et al. (2009b); Yang et al. (2011) adopted the modiﬁed 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 difﬁcult 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 opinions, 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 (antecedents) 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 conﬁdence

276 U. Asan and A. Soyer

values can be well reﬂected 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 Ain a

universe U,isdeﬁned as (Atanassov 1986):

A¼u;lAðuÞ;mAðuÞðÞju2U

fg

A¼ðlA;mA

ðÞfor shortÞð10:1Þ

where the functions lA:U!0;1½and mA:U!0;1½deﬁne the grade of mem-

bership and the grade of non-membership of the each element of Uto A, respec-

tively. The functions lAuðÞand mAuðÞshould satisfy the condition:

0lAuðÞþmAuðÞ1ð8u2UÞð10:2Þ

and

pAuðÞ¼ð1mAuðÞlAuðÞÞ ð10:3Þ

where pAuðÞdenotes the uncertainty of u(also called as the hesitancy of u). Clearly,

in the case of ordinary fuzzy sets, pAuðÞ¼0 for 8u2U. 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

reﬂected in a more realistic and ﬂexible manner. In another study, Chang and

Cheng (2010), integrated the IFS and DEMATEL approach on risk assessment

providing a more ﬂexible 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 assessment 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 process (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

ﬁrst 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 identiﬁed 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 defuzziﬁed 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 (iii) S, O, and D can be com-

bined in a more ﬂexible 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 domain that, however, leads to loss of information and hence lack of

precision in the ﬁnal 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, prioritization 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 efﬁciency 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 ﬁnal 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, predeﬁned fuzzy membership functions or crisp rating scales in FMEA allow

only judgments in form of point or ﬁxed interval values and, hence, do not fully

reﬂect 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 ﬂexible 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 their study, Wang et al. (2009), ﬁrstly, evaluated the risk factors and their

importance weights in a linguistic manner; then computed FRPNs applying an

alpha cut based linear programming approach; and ﬁnally, defuzziﬁed FRPNs using

centroid defuzziﬁcation method for the ﬁnal 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 reﬂects 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 method of

partial ranking for evaluating and ranking the FMs. In this approach, ﬁrstly, 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 defuzziﬁed crisp risk

values are used to obtain ﬁnal 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 ﬁnal 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)

ﬁrstly, 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 conformance

guideline increases; on the other hand, as it gets closer to zero, then the possibility of

the relevant 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 identiﬁcation 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, adapted 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, identiﬁed 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 linguistic terms given in Table 10.5,

where each term corresponds to a triangular fuzzy number. If otherwise 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 following 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 uniﬁed framework. A further strength of the approach is its ability to reﬂect the

diversity in expert judgments. All these beneﬁts are achieved by incorporating

the experts’level of conﬁdence 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

ﬂexible 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 belief 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 using the linguistic terms provided in Table 10.5. For example,

the set of evaluation grades for Severity are represented as HFS ¼Very Low;

f

Low;Medium Low;Medium;Medium High;High;Very HighgFor 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 Hij ;bk

ijðFMn;RFlÞ

;i¼1;...;7;j¼1;...;7

no

be the belief

structure provided by expert E

k

on the assessment of failure mode FMnwith respect

to risk factor RFl, where Hii for i¼17 are the fuzzy assessment grades, Hij for

i¼16 and j¼iþ1 to 7 are the interval fuzzy assessment grades between Hii

and Hjj, and bk

ijðFMn;RFlÞare the belief degrees for the intervals Hij (Liu et al.

2011). The interval fuzzy assessment grades, Hij ,deﬁne trapezoidal fuzzy sets that

include the fuzzy assessment grades Hii ;Hðiþ1Þðiþ1Þ;...;Hjj as shown in Fig. 10.3.

For Severity, the grades Hii for i¼17 and the intervals Hij for i¼16 and

j¼iþ1 to7 all together can be expressed as ^

HFS ¼Hij;i¼1;...;7;j¼1; ::; 7

or equivalently as

284 U. Asan and A. Soyer

^

HFS ¼

H11 H12 H13 H14 H15 H16 H17

H22 H23 H24 H25 H26 H27

H33 H34 H35 H36 H37

H44 H45 H46 H47

H55 H56 H57

H66 H67

H77

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

H33;1:0ðÞ

fg

.

•Distribution. For example, a FM that is assessed as “High”with a conﬁdence

level of 0.3 and as “Very High”with a conﬁdence level of 0.7, can be expressed

as H66;0:3ðÞ;H77;0:7ðÞ

fg

. This is a complete distribution. When all conﬁdence

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 H46 ;1:0ðÞ

fg

.

•Total Ignorance. The FM can be assigned any grade between “Very Low”and

“Very High”and will be expressed as H17 ;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 conﬁdence levels

are more risky than those assessed to low values or intervals with high conﬁdence

levels (Chin et al. 2009b). Table 10.11 presents the assessment information (in form

of belief structures) on the eight FMs provided by three experts. The incomplete

assessments and ignorance information are shaded and highlighted, respectively.

For example, according to expert E

1

, the Severity of ‘using