ORIGINAL ARTICLE

A new FMEA method by integrating fuzzy belief structureand TOPSIS to improve risk evaluation process

Behnam Vahdani & M. Salimi & M. Charkhchian

Received: 26 April 2014 /Accepted: 3 October 2014 /Published online: 16 October 2014# Springer-Verlag London 2014

Abstract Failure mode and effect analysis (FMEA) model is atechnique used to evaluate the risk. This paper aimed to proposea new FMEA model combining technique for order of prefer-ence by similarity to ideal solution (TOPSIS) and belief structureto overcome the shortcomings of the traditional index of FMEA.In this paper, the fuzzy belief TOPSIS method is combined withFMEA to introduce a belief structure FMEA to describe theexpert knowledge by a number of linguists as a grammaticalphenomenon. Moreover, the weights of components in FMEAindex can be different from each other. Therefore, the flexibilityof assigning weight to each factor in this method is morecompatible to the real decision-making situation. In other word,TOPSIS method is applied to determine the preference of alter-natives versus risk criteria. Using linguistic terms in the fuzzybelief approach, the risk factors described a more meaningfulvalue and decision-makers’ judgment is assigned with beliefdegrees through evaluation of factors. Finally, a numerical casestudy about the preference of cause failures of steel productionprocess is provided to illustrate the process of proposed method,and then result and discussion is performed for each case.

Keywords Technique for order of preference by similarity toideal solution(TOPSIS) .Fuzzybeliefstructure (FBS) .Failuremode and effect analysis (FMEA) . Cause of failure (CF) .

Severity (S) . Occurrence (O) . Detection (D)

1 Introduction

Risk can be considered as a natural consequence productionactivity. There is no truth that all risks can be eliminated, so we

can reduce risks to an acceptable level. Risk assessment isapplied by suitable techniques to prevent the unexpectedfailure scenarios. In times of increasing global competition,the scientific methods have been applied frequently to identifyrisks and prevent failure of the project [1]. However, tech-niques for risk management have an important role to performbetter. One of the most famous mentioned methods to riskevaluation is the failure mode and effect analysis (FMEA)method [2]. FMEA is a technique used to evaluate the risk. Inthis method, risk assessment involves the failure occupancy(O), detectability (D), and severity (S) of the undesired failurescenario [3]. The first FMEAwas done as a systematic methodin the aerospace industry in the 1960s. Moreover, FMEA in1970s was used in nuclear establishments. Then, FMEAwasapplied in automotive industry for error and risk reduction.The FMEA was used in the health care industry in1990s formanufacture of drugs and in the prevention of medicationerrors in hospitals [4]. Because the application of FMEA inhealth care industry is very successful, it is named health carefailure mode and effect analysis (H-FMEA) [5]. A risk prioritynumber (RPN) is applied to assess the potential failure modesof a product or a process. The RPN is an aggregated indexwhich is obtained by finding the multiplication of three indi-cators of 1–10 (severity (S), occurrence (O), and detection(D)). S is applied as severity of the effect of risk; O is theprobability of the risk; and D is the ability of detect risk [6, 7,3]. The judgment about determination of risk criterion andfailure mode versus risk criteria are proposed by experts. Inorder to compare the factors of failure, an integrated risk indexis introduced based on the probability, detectability, and se-verity of the failure [8].

Recently, many studies have been published where FMEAis used together with fuzzy sets to overcome the weaknessesof traditional RPN ranking system [3]. A fuzzy logic-basedFMEA is implemented in diesel engine systems, chemical andvolume control system [9, 10]. The fuzzy logic system is

B. Vahdani (*) :M. Salimi :M. CharkhchianFaculty of Industrial and Mechanical Engineering, Qazvin Branch,Islamic Azad University, Qazvin, Irane-mail: b.vahdani@Gmail.com

Int J Adv Manuf Technol (2015) 77:357–368DOI 10.1007/s00170-014-6466-3

applied in the production of nuclear energy [11]. Tay and Linproposed a generic fuzzy logic-based FMEA method to sim-plify the traditional FMEA methodology by reducing thenumber of rules in the fuzzy RPN modeling process [12]. Inaddition, the traditional method of risk evaluation is simplebut it suffers from several weaknesses such as follows:

1. This is supposed to be a traditional method of risk evalua-tion, but it neglects the relative importance amongO, S, andD. The three factors are assumed to have the same impor-tance, whereas this assumption may not be the case whenthe FMEA is applied in a practical application [13, 14, 3].

2. The most critically debated disadvantage of the traditionalFMEA is that various values of O, S, and D may producean identical RPN value, whereas the risk implication maybe totally different. For case study, suppose two differentevents having values of 3, 5, 2 and 2, 3, 5 for O, S, and D,respectively. Both these events will have a same result(RPN1=3×5×2=30 and RPN2=2×3×5=30). However,the risk importance of these two events may not be nec-essarily the same [13, 14, 3].

3. The traditional FMEA neglect to human/expert knowl-edge. Therefore, the shortcoming of this approach is theneed of defining the risk index, which may be affected byexpert’s attitude toward risk [15]. Moreover, the groupdecision making is not considered in traditional FMEAmethod.

Recently, some concepts are combined with FMEA toimprove FMEA method such as fuzzy logic and multi-criteria decision-making (MCDM) methods. The behavior offuzzy inference techniques is expressed with a language that iseasily interpretable by humans [3]. The application of linguis-tic terms in the fuzzy approach improves the applicability ofthe FMEA, because the experts can assign a more meaningfulvalue for the factors considered [3]. Fuzzy logic allows impre-cise data usage, so it enables the treatment of many situationsin decision making. Furthermore, the studies about FMEAconsidering fuzzy approach use the experts who describe thequantitative data and qualitative information about risk factorsO, S, and D by using the fuzzy linguistic terms [16–18].

TOPSIS is one of the MCDM methods that applied forproposing a new FMEA method. The TOPSIS and triangularfuzzy numbers are combined for evaluation of failure influ-ence. In this method, the indicators S, O, and D are assessedby fuzzy logic [17, 19]. The fuzzy version of TOPSIS ap-proach allows for the risk factors O, S, and D and their relativeimportance weights to be evaluated. The fuzzy TOPSIS areapplied to propose a new FMEA to overcome the shortcom-ings of traditional FMEA [20, 17]. In addition, the authorbelieves that the existing methods are not able to expresshuman beliefs excellently. Therefore, we utilize the beliefstructure to decide about risk properly.

This study aimed to propose a new FMEA combiningTOPSIS and belief structure and fuzzy logic to overcome theshortcomings of the traditional RPN. It is more proper thatdecision-makers’ judgment is assigned through evaluation offactors with belief degrees. TOPSIS is utilized with beliefstructure for decision making about alternatives versusmulti-criteria [21]. In this paper, the belief TOPSIS methodis combined with FMEA to introduce a belief structure FMEAto describe the expert knowledge by a number of linguists as agrammatical phenomenon. There are some advantages fornew proposed method such as follows:

1. The proposed group fuzzy belief structure (FBS) modelcontains both fuzzy evaluation grades and belief degrees;therefore, the application of linguistic terms in the fuzzybelief approach allows for the decision maker to assign amore meaningful value for the factors considered. There-fore, the expert knowledge and experience are combinedfor use in an FMEA study. Moreover, we can use theexperiences of several experts.

2. This approach assumes that the relative importanceamong probability, detectability, and severity of the un-desired failure scenario can be deferent from each other.Therefore, the flexibility of assigning weight to eachfactor in this method is more compatible to the realdecision-making situation.

3. The input factors (severity, occurrence, and detection) arecombined in this method according to their weightswhereby it is more applicable in industry. The steps ofthe proposed risk evaluation method are implemented in asteel production factory.

We apply TOPSIS to combine with belief structure becauseTOPSIS method has some advantages. This method isestablished upon the concept of positive ideal solution andnegative ideal solution concurrently. Moreover, we considerthis method to solve group decision-making problems withmulti-criteria and multi-judges with belief structure. A com-bined MCDM method for supplier selection is proposed byBhutia and Phipon [22]. They applied TOPSIS method forranking alternatives. They proposed most advantages ofTOPSIS as

1. It is simple to use.2. It takes into account all types of criteria (subjective and

objective).3. It is rational and understandable.4. The computation processes are straightforward.5. The concept permits the pursuit of best alternative criteri-

on depicted in a simple mathematical form.

Moreover, the other “advantage of TOPSIS is the ability toidentify the best alternative quickly” [23]. Other method such as

358 Int J Adv Manuf Technol (2015) 77:357–368

ELECTRE primarily focused on generating weights with onemethod including a different way to combine weights and dis-tance measures. Moreover, TOPSIS is better than analytic hier-archy process (AHP) in matching a base prediction model [24].

The remaining of this paper is organized as follows. Thetraditional FMEA method is presented in the next section. InSect. 3, the fundamental concepts about FBS model are intro-duced. In Sect. 4, the procedure of proposed new FMEAbased on group FBS model is described. In Sect. 5, a numer-ical case study about the preference of cause failures of steelproduction process is provided to illustrate the process of theproposed method; both complete and incomplete evaluationsare involved in the case study. The numerical case studydemonstrates the implementation process of the TOPSISFBS model step by step to find the high-risk alternatives.The last section is devoted to conclusion.

2 FMEA

FMEA is a step-by-step approach for failure modes and ef-fects analysis. This method is applied to identify the potentialfailure of a system and its effects in a design, a manufacturingor assembly process, or a product or service. In order to rankthe critical points, an aggregated index has been applied basedon three indicators of 1–10. They are related to their occur-rence probability, severity of the associated effects, and detec-tion to each failure mode [19, 3].

The steps of FMEA method are as follows [25]:

Step 1 Determine the function or process of service toevaluate.

Step 2 Recruit a multidisciplinary team. Be sure to considerall team members who are involved at any point inthe risk evaluation process.

Step 3 Have the teammeet to determine all the steps and tasksin the risk assessment process. It is important to knowhow the tasks and responsibilities of risk managementfit together and how each task relates to the others.

Step 4 Have the team list failure modes and causes. Listanything that could go wrong, including minor andrare problem. Then, identify all possible causes foreach failure mode listed.

Step 5 For each failure mode, the team has to determine theamount of RPN. The RPN is obtained by multiplyingthe three numerical value (severity, probability, anddetection) ratings:

RPN ¼ Severityð Þ � Probabilityð Þ � Detectionð Þ ð1Þ

where severity is importance of the effect on custom-er requirements, occurrence is the frequency with

which a given cause occurs and creates failure modes,and detection is the ability of the current controlscheme to detect or prevent a given cause. There is1 to 10 score for each of likelihood of occurrence,detection, and severity.

& Severity: 1=not severe, 10=very severe& Occurrence: 1=not likely, 10=very likely& Detection: 1=easy to detect, 10=not easy to detect

Step 6 Evaluate the results and use RPNs to plan improve-ment efforts (develop action plan). Then, determineappropriate activities to address potential failureswith high risk priority number. Identify the failuremodes and their causes with the top 10 highest RPNs.The minimum amount of score can be 1 and themaximum 1,000. Determination of high-risk failuremodes is the most important part of the risk reductionprocess. The low-risk failure modes do not affect theoverall process very much, and they should thereforebe at the bottom of the list of priorities.

3 The fundamental concepts about FBS model

Some fundamental concepts such as belief evaluation of al-ternatives, similarity between fuzzy evaluation grades, beliefdistance measure, and FBS TOPSIS method are described asfollows:

3.1 The belief evaluation of alternatives

The evaluation of alternative Ai with respect to criterion Cj

usually cannot be expressed by a single linguistic variable[21]. Therefore, the criteria on alternatives can be evaluated bya set of standards represented asH={H1, H2,…, Hn,…,HN},where Hn(n:1,2,…,N) is the nth evaluation grade. It is as-sumed thatHn is preferred to the following distribution show-ing the evaluation of criterion Ci mathematically:

S Cð Þ ¼ Hn;βnð Þ nj ¼ 1; 2;…;Nf g; ð2Þ

where βn≥0,∑n=1N βn≥ 1 and βn is a degree of belief. The

distribution S(C) means that the criterion C ∑n=1N βn=1 is

evaluated to the grade Hn with the belief degree βi. A distri-bution S(C) is completed if∑n=1

N βn=1, and it is in complementif ∑n=1

N βn < 1. In many real-world decision situations, theevaluation grade Hn may be presented by fuzzy concepts [21,26]. Therefore, we propose the belief structure model withfuzzy data to evaluate the alternatives with respect to criteria.The evaluation grades can be considered as each of type offuzzy sets like triangular or trapezoidal, etc. It is assumed that

Int J Adv Manuf Technol (2015) 77:357–368 359

the evaluation grades are represented with triangular fuzzynumbers as follows:

U Hnð Þ ¼ un1; un2; u

n3

� � ð3Þ

Therefore, in FBS, the alternatives are evaluated versuscriteria on the basis of N evaluation grades Hn with beliefdegree where evaluation grade can be either a fuzzy set.

3.2 Similarity between fuzzy evaluation grades

According to the definition in Eq. (2), S(C) demonstrate abelief structure model with N evaluation grades that each ofthem can be represented as triangular fuzzy numbers. Corre-sponding to each evaluation grade, the utilities to them can bealso represented by triangular fuzzy sets asU(Hn)=(u1

n,u2n,u3

n).That is called fuzzy grade utility where uk

n(k=1,2,3)∈[0,1].The similarity between Hi and Hj can be expressed by the

following equation.

bSi j Hi;H j

� � ¼ 1−

X 3

k¼1uik−u

jk

�� ��3

ð4Þ

bSi j∈ 0; 1½ � because ui belongs to [0, 1]. bSi j describes thedifference between two fuzzy evaluation grades (Hi, Hj).

The similarity matrix bS is constructed as bS ¼ bSi jh i

nxn,

where the bS is calculated in Eq. (4) [21].

3.3 Belief distance measure

Extending the TOPSIS method, the metric distance to deter-mine the preference of alternatives in FBS model is applied tocompare two numbers. B1 and B2 are supposed to be FBSnumbers. The distance between B1 and B2 is defined asfollows:

dBS B1;B2ð Þ ¼ 1

2B1−B2ð ÞbS B1−B2ð ÞT

� �12

ð5Þ

where bS is calculated with Eq. (4).

3.4 TOPSIS with FBS model

Suppose a FBS decision-making problem withM alternativesAi to the L criteriaCj; moreover, the judgments are representedby K decision maker. The TOPSIS method is applied to findthe best option. In other words, we want to determine thepreference of a group belief structure problem with TOPSISmethod in the following procedure [21]:

First, construct the fuzzy belief decision matrix based onthe expert opinion. Then, determine the weight of experts(WD) and criteria (WC), where

WD ¼ WDl ;…;WD

k ;…;WDk

� �: ð6Þ

Wc ¼ Wcl ;…;Wc

j;…;WcL

h i: ð7Þ

andMk is fuzzy belief decision matrix based on the judgmentof Kth decision maker. wj

c is the weight of criterion Cj. sijk=

{(Hn,βnk),n=1,…,N} is a fuzzy BS model. It means that thedecision maker Dk believes that Hn is the evaluation indicatorwith belief degree βnk.

In the second step, aggregate the assessment of decisionmakers in FB decision matrix M into fuzzy belief decisionmatrixMwith aggregated elements. Therefore, the element Sijis aggregated judgment on alternative Ai with respect to crite-rion Cj; in aggregated matrix, the aggregated element is ob-tained. According to the above concept sij

k(sijk={(Hn,βnk),n=1,

…,N} ) in FBS matrix with respect to Kth decision-maker’sassessment as

Si j ¼ Hn;βnð Þ; n ¼ 1;…;Nf g ð8Þ

There is a relation between βn and βnk, which is expressedas

βn ¼μ � πk

k¼1 wDK βn þ 1−wD

K

X N

j¼1β j ; n

� �−πkk¼1 1−wD

K

X N

j¼1β j ; n

� �

1−μ � ∏K

K¼1 1−WDK

� �

ð9Þ

μ ¼X N

n¼1πkk¼1 wD

K βn;K þ 1−wDK

X N

j¼1βj;K

� �− N−1ð Þπk

k¼1 1−wDK

X N

j¼1β j;K

� � −1ð10Þ

Then, normalize the incomplete FBS model. The FBSmodel is called incomplete if∑n=1

N βn<1. Therefore, the centerof gravity is introduced for FBS model to solve the problemswith ignorance. Moreover, βH is the degree of ignorance. The

center of gravity is defined as

SC Cð Þ ¼ Hn; βn þβH

N

� �; n ; 1; 2; …; Nð Þ

� �: ð11Þ

360 Int J Adv Manuf Technol (2015) 77:357–368

The center of gravity is applied for normalizing the prob-lems with incomplete FBS model.

In the next step, determine the positive ideal belief solution(Ai

+) and negative ideal belief solution (Ai−) for each criterion,

where fi+ is the maximum of FBS with respect to criteria and fi

−

is the minimum of FBS.

Aþ ¼ Aþ1 ;A

þ2 ;…;Aþ

L

�A− ¼ A−

1 ;A−2 ;…;A−

L

� ð12Þ

where Ai+ is the maximum of FBS with respect to criterion Cj

and Aj− is the minimum of FBS.

Then, calculate the values Di+ and Di

− as separationmeasure from positive ideal belief solution (PIBS) A+

and negative ideal belief solution (NIBS) A+ for eachalternative.

Dþi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX L

j¼1Wc

jdBS Si j; Sþj

� �2r

ð13Þ

D−i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX L

j¼1Wc

jdBS Si j; Sþj

� �2r

; i

¼ 1; 2; …; M ð14Þ

where the distance between S1,S2(dBS(S1,S2)) is defined as

dBS B1;B2ð Þ ¼ 1

2B1−B2ð ÞbS B1−B2ð ÞT

� �12

ð15Þ

and S~ is calculated as follows:

bSi j Hi;H j

� � ¼ 1−

X 3

k¼1uik−u

jk

�� ��3

ð16Þ

Now, calculate the relative closeness Ri for each alternativeAi as

Ri ¼ D−i

Dþi þ D−

i

ð17Þ

Propose the alternative Ai as a suitable option according tothe measure Ri. Note that Ri is a positive indicator; therefore,set the alternative Ai with largest Ai as the best candidatealternative.

4 New method

Suppose a FBS decision-making problem with M failuremodes (alternatives) Ai to the three risk evaluation criteria Cj

(S, O, D). Moreover, the judgments are represented by Kdecision maker. The TOPSIS method is applied to find theoption with high risk. In other words, we want to determinethe preference of a group belief structure problem withTOPSIS method in the following procedure. The proposedmethod is composed of the following eight steps:

Step 1 Using historical data, past experiences, and ex-pert opinion, list all failure modes (FMs) andcause of failure modes (CFs) throughout thesystem.

Step 2 Construct the fuzzy belief decision matrix to rank thecritical points as alternatives. The occurrence, proba-bility, and severity of the associated effects and de-tection to each failure mode are considered as criteriain the decision matrix. The judgment for each alter-native versus each criterion is proposed as fuzzybelief structure. Moreover, the judgments are repre-sented by K experts as

S O D

MK ¼

A1

⋮Ai

⋮AM

Sk11 Sk12 Sk13⋮ ⋮ ⋮Ski1 Ski2 Ski3⋮ ⋮ ⋮SkM1 SkM2 SkM3

266664

377775

ð18Þ

WD ¼ WD1 ;…;WD

k ;…;WDk

� �: ð19Þ

WC WC1 ;…;WC

j ;…;WCL

h i: ð20Þ

whereMk is the fuzzy belief decision matrix based onthe judgment of Kth decision maker. wj

c is the weightof criterion Cj. sj

c = {(Hn,βnk),n =1,…,N} is afuzzy BS model. It means that Kth decision maker(Dk) believes that there is βnk% insurance that MK isthe evaluation indicator.

Step 3 Aggregate the assessment of decision makers in FBSdecision matrix. A new FBS method is developed toaggregate the judgments of k experts using the infor-mation contained in the fuzzy belief matrix. There-fore, the element Sij in aggregated matrix is the ag-gregated judgment on alternative Ai with respect tocriterion Cj. According to the above concept, Sij

k is the

Int J Adv Manuf Technol (2015) 77:357–368 361

FBS model with respect to Kth decision-maker’sassessment as

Ski j ¼ Hn;βnkð Þ; n ¼ 1;…;Nf g

Furthermore, the aggregated element Sijk={(Hn,βn),

n=1,…,N}. There is a relation between βn and βnkthat is expressed as

βn ¼μ � πk

k¼1 wDK βn þ 1−wD

K

X N

j¼1β j ; n

� �−πk

k¼1 1−wDK

X N

j¼1β j ; n

� �

1−μ � ∏K

K¼1 1−WDK

� � ð21Þ

μ ¼X N

n¼1πkk¼1 wD

K βn;K þ 1−wDK

X N

j¼1βj;K

� �− N−1ð Þπk

k¼1 1−wDK

X N

j¼1β j;K

� � −1ð22Þ

There are D experts to express their views abouteach alternative versus each criterion. Using Eq. (21),the ideas are aggregate to an aggregated idea. Expertspropose their idea as a belief structure number. Eachbelief number has three levels which each level isproposed as a triangular fuzzy number. Decisionmak-er determines the fuzzy numbers of each level, andexperts determine the Mk as belief degree for fuzzybelief numbers.

Step 4 Normalize the incomplete FBS model in aggregatedmatrix. The FBS model is called incomplete if ∑n=

1Nβn< 1. Therefore, the center of gravity is intro-duced for FBS model to deal with this ignorance.

Moreover, βH is the degree of ignorance. Thecenter of gravity is defined as

SC Cð Þ ¼ Hn;βn þβH

N

� �; n ¼ 1; 2;…;Nð Þ

� �:

ð23Þ

The center of gravity is applied for normalizingthe problems with incomplete FBS model.

Step 5 Determine the positive ideal belief solution (Ai+) and

negative ideal belief solution (Ai−) for each criterion,

where Ai+ is the maximum of FBS with respect to

criteria and Ai− is the minimum of FBS.

Aþ ¼ Aþ1 ;A

þ2 ;…;Aþ

3

�A− ¼ A−

1 ;A−2 ;…;A−

3

� ð24Þ

where Cj is the maximum of FBS with respect tocriterion sij

k and sijk ={(Hn,βnk), n=1,…,N} is the

minimum of FBS.

Step 6 Calculate the values Sij={(Hn,βn),n=1,…,N} foreach alternative.

Dþi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX L

j¼1WC

j dBS Si j; Sþj

� �2r

ð25Þ

D−i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX L

j¼1WC

j dBS Si j; S−j

� �2r

; i

¼ 1; 2;…;M ð26Þ

where Di+ is distance from positive ideal belief

solution and Di− is the distance from negative ideal

belief solution.

Step 7 Calculate the value βn as

Ri ¼ D−i

Dþi þ D−

i

ð27Þ

Propose the alternative Ai as a suitable optionaccording to the measure Ri. Note that Ri is a negativeindicator; therefore, set the alternative Ai with largestRi as candidate alternative with most risk.

Step 8 Using the result of ranking, analyze the results andprovide suggestions to plan improvement efforts. Re-assess the severity, probability, and detection andreview the revised RPNs after provided action.

Finally, the flowchart of the proposed new FMEA methodbased on the FBS for group decision-making problems isdepicted in Fig. 1.

362 Int J Adv Manuf Technol (2015) 77:357–368

5 Numerical case study

The steps of above method are described in the following casestudy. In this case, ten options of sheet steel productionprocess in a steel factory (steel factory of guilan) are evaluatedby the proposed method with respect to the three criteria. Thealternatives of this case study are previously evaluated byDeshpande and Modak in a factory [27]. The judgment indecision matrix is taken by three experts. Three experts areassumed to have the same importance. The criteria are related

to their occurrence probability, severity of the associatedeffects, and detection to each failure mode as shown inFig. 2. The aim is to find high-risk options among the tenalternatives. The criteria are evaluated by a set of standardwith three fuzzy evaluation grades. The weights of the criteriapresent the amount of importance for one criterion than an-other, and their amount are as follows: 0.2, 0.3, and 0.1.Moreover, the weights of decision makers are equal to eachother. We utilize TOPSIS fuzzy belief presented by Jiangiet al. to rank our case study alternatives.

Fig. 1 The flowchart of proposedFBS FMEA method

Fig. 2 The preference of the CFs to find high-risk alternative

Int J Adv Manuf Technol (2015) 77:357–368 363

The proposed method is applied to evaluate ten options ofsteel production process as follows:

Step 1 List the CFs throughout the system versus threeindexes as shown in Table 1.

Step 2 Construct the group decision matrix based on theexpert opinion. The occurrence probability, severityof the associated effects, and detection to each failuremode are considered as criteria in the decision ma-trix. The judgments are represented by three expertsin this case study. Suppose there are ten alternativesCF1, CF2,⋯, CF10, three criteria (S, O, D), and threedecision makers D1, D2, D3. Each judgment isexpressed such as FBS with three evaluation grades{H1, H2, H3}={“good,” “average,” “poor”}. For thecase study, the first decision maker is 80 % sure thatthe assigned amount of alternative CF1 is good, 10%is average, and 10 % is poor with respect to the firstcriterion. The group FBS matrix is presented inTable 2.

Step 3 The assessment is aggregated by using the FBSaggregation method as given in Eqs. (20) and (21),as shown in Table 3.

Step 4 Normalize the incomplete FBS model. From Table 2,there are two incomplete FBS models S3,1

2 and S8,12 .

Therefore, S3,1 and S8,1 are incomplete in Table 3.We can calculate the center of gravity according toTable 3, in order to normalize the incomplete fuzzybelief model. The result of normalized FBS model isshown in Table 4.

Step 5 Determine the PIBS Ai+ and NIBS Ai

− for each crite-rion. Based on the concept of three fuzzy evaluationgrades, the PIBS and NIBS are

Aþ ¼ 1; 0; 0f gA− ¼ 0; 0; 1f g

Step 6 Calculate the values of separation measure from thePIBS Ai

+ and NIBS Ai− for each alternative as shown

in Table 5.Suppose the utilities of evaluation grades are

U H1ð Þ ¼ U ‘good’ð Þ ¼ 0:5; 0:7; 0:9ð ÞU H2ð Þ ¼ U ‘average’ð Þ ¼ 0:3; 0:5; 0:7ð ÞU H3ð Þ ¼ U ‘poor’ð Þ ¼ 0:1; 0:3; 0:5ð Þ

Moreover, the similarly matrix can be calculatedby Eq. (4):

bS ¼1 0:8 0:60:8 1 0:80:6 0:8 1

24

35

Table 1 The FMEA of the sheet steel production process in Guilan steelfactory

No. Failure mode (FM) Cause of failure (CF)

A1 Non-acceptable formation Non-conductive scrap

A2 Nipple thread pitted Proper coverage not obtained

A3 Arc formation loss Leakage of water, proper grippingloss

A4 Burn-out electrode Cooler not working properly

A5 Breaking of house of pipe Wearing of pipe due to use

A6 Problem in movement of arm Sever leakage

A7 Refractory damage Due to slag

A8 Formation of steam Roof leak

A9 Refractory line damage By hot gas

A10 Movement of roof stop Jam of plunger in un loader valve

Table 2 Group belief structure judgment of the sheet steel productionprocess

Alternatives Experts Severity Occurrence Detectability

CF1 D1 (0.8, 0.1, 0.1) (0.1, 0.2, 0.7) (0.2, 0.5, 0.3)

D2 (0.7, 0.0, 0.3) (0.0, 0.4, 0.6) (0.3, 0.4, 0.3)

D3 (0.8, 0.7, 0.0) (0.1, 0.4, 0.5) (0.2, 0.5, 0.3)

CF2 D1 (0.7, 0.1, 0.2) (0.1, 0.2, 0.7) (0.8, 0.1, 0.1)

D2 (0.7, 0.0, 0.3) (0.0, 0.4, 0.6) (0.7, 0.0, 0.3)

D3 (0.6, 0.4, 0.0) (0.1, 0.4, 0.5) (0.8, 0.2, 0.0)

CF3 D1 (0.8, 0.1, 0.1) (0.0, 0.1, 0.9) (0.2, 0.5, 0.3)

D2 (0.9, 0.0, 0.0) (0.0, 0.2, 0.8) (0.3, 0.4, 0.3)

D3 (0.7, 0.3, 0.0) (0.1, 0.0, 0.9) (0.2, 0.5, 0.3)

CF4 D1 (0.4, 0.4, 0.2) (0.0, 0.1, 0.9) (0.1, 0.2, 0.7)

D2 (0.3, 0.5, 0.2) (0.0, 0.2, 0.8) (0.0, 0.4, 0.6)

D3 (0.4, 0.4, 0.2) (0.1, 0.0, 0.9) (0.1, 0.4, 0.5)

CF5 D1 (0.4, 0.4, 0.2) (0.2, 0.4, 0.4) (0.7, 0.0, 0.3)

D2 (0.3, 0.5, 0.2) (0.2, 0.4, 0.4) (0.8, 0.2, 0.0)

D3 (0.4, 0.4, 0.2) (0.1, 0.5, 0.4) (0.6, 0.3, 0.1)

CF6 D1 (0.4, 0.4, 0.2) (0.2, 0.4, 0.4) (0.7, 0.0, 0.3)

D2 (0.3, 0.5, 0.2) (0.2, 0.4, 0.4) (0.8, 0.2, 0.0)

D3 (0.4, 0.4, 0.2) (0.1, 0.5, 0.4) (0.6, 0.3, 0.1)

CF7 D1 (0.4, 0.4, 0.2) (0.2, 0.4, 0.4) (0.1, 0.2, 0.7)

D2 (0.5, 0.5, 0.0) (0.2, 0.4, 0.4) (0.0, 0.4, 0.6)

D3 (0.6, 0.4, 0.0) (0.1, 0.5, 0.4) (0.1, 0.4, 0.5)

CF8 D1 (0.8, 0.1, 0.1) (0.0, 0.1, 0.9) (0.2, 0.5, 0.3)

D2 (0.9, 0.0, 0.0) (0.0, 0.2, 0.8) (0.3, 0.4, 0.3)

D3 (0.7, 0.3, 0.0) (0.1, 0.0, 0.9) (0.2, 0.5, 0.3)

CF9 D1 (0.4, 0.4, 0.2) (0.2, 0.4, 0.4) (0.7, 0.0, 0.3)

D2 (0.3, 0.5, 0.2) (0.2, 0.4, 0.4) (0.8, 0.2, 0.0)

D3 (0.4, 0.4, 0.2) (0.1, 0.5, 0.4) (0.6, 0.3, 0.1)

CF10 D1 (0.7, 0.0, 0.3) (0.2, 0.4, 0.4) (0.7, 0.5, 0.3)

D2 (0.8, 0.2, 0.0) (0.4, 0.0, 0.6) (0.3, 0.4, 0.3)

D3 (0.6, 0.3, 0.1) (0.4, 0.0, 0.6) (0.2, 0.5, 0.3)

364 Int J Adv Manuf Technol (2015) 77:357–368

The distances between decision judgment and ide-al solution are measured by Eq. (5). The result isshown in Table 5. Moreover, calculateDi

+ as distance

from positive ideal belief solution and Ri ¼ D−i

Dþi þD−

ias

distance from negative ideal belief solution byEqs. (25) and (26).

Step 7 Calculate the valueAi by Eq. (27) as shown in Table 5.Because risk is a negative concept, the alternative A4

is ranked first; alternatives A3 and A8 are rankedsecond; A7 is ranked third; A1 is ranked fourth; A10

is ranked fifth; A5, A6, and A9 are ranked sixth; and A2

is ranked seventh.Step 8 Using the results fromTable 5, analyze the results and

provide suggestions to plan improvement efforts. Re-assess the severity, probability, and detection andreview the revised RPNs after provided action.

6 Result and discussion

In our proposed method, the FBS TOPSIS method isemployed to determine the preference of alternatives in risk

assessment problems which can be easily solved by thismethod step by step. This research has conducted a perfor-mance analysis on a case study using FBS MCDM approach.In this case study, there are three criteria for ranking thealternatives. The judgments are represented by three expertsD1,D2, andD3. Each judgment is expressed such as FBS withthree evaluation grades {H1, H2, H3}={“good,” “average,”“poor”}. For the case study, the first decision maker is 80 %sure that the assigned amount of alternative Ai is good and10 % is average with respect to criterion CF1. The experts areassumed to have equal importance in this case. Moreover, theweights of criteria are as follows: 0.2, 0.3, and 0.1. The CFthat has a higher value in decision-making process is assumedto be more important and is given a higher priority. FromTable 5, the value ofRi for the first alternative (non-conductivescrap) is 0.5674 and the value of Ri for the second alternative(proper coverage not obtained) is 0.6645. This result demon-strates that according to the FBS TOPSIS method, CF2 has ahigher priority than CF1. From Table 1, the first expert is 80%sure that the assigned amount of first cause failure mode (non-conductive scrap) is good, 10 % is average, and 10 % is poorwith respect to the first criterion (severity of the associatedeffects). Moreover, the first expert is 80 % sure that theoccurrence probability of the associated effects for the first

Table 3 Aggregated fuzzy beliefdecision matrix Alternatives Severity Occurrence Detectability

A1 (0.8193, 0.0771, 0.1033) (0.0545, 0.3105, 0.6346) (0.2191, 0.4894, 0.2914)

A2 (0.7224, 0.1373, 0.1399) (0.0545, 0.3104, 0.6346) (0.8250, 0.0776, 0.1040)

A3 (0.8708, 0.1038, 0.0252) (0.0233, 0.0722, 0.2042) (0.2191, 0.4892, 0.2914)

A4 (0.3669, 0.4475, 0.1854) (0.0233, 0.0722, 0.9043) (0.0545, 0.3104, 0.6346)

A5 (0.3669, 0.4475, 0.1854) (0.1504, 0.4446, 0.4045) (0.7546, 0.1373, 0.1075)

A6 (0.3669, 0.4475, 0.1854) (0.1504, 0.4446, 0.4045) (0.7546, 0.1373, 0.1075)

A7 (0.7227, 0.1374, 0.1400) (0.1504, 0.4446, 0.4045) (0.0545, 0.3104, 0.6346)

A8 (0.8708, 0.1038, 0.0252) (0.0233, 0.0722, 0.9042) (0.2191, 0.4894, 0.2914)

A9 (0.3669, 0.4475, 0.1854) (0.1504, 0.4446, 0.4045) (0.7546, 0.1373, 0.1075)

A10 (0.7966, 0.1070, 0.1135) (0.3092, 0.1103, 0.5671) (0.3365, 0.4158, 0.2475)

Table 4 Normalized fuzzy beliefdecision matrix Alternatives Severity Occurrence Detectability

A1 (0.8193, 0.0771, 0.1033) (0.0545, 0.3105, 0.6346) (0.2191, 0.4894, 0.2914)

A2 (0.7224, 0.1373, 0.1399) (0.0545, 0.3104, 0.6346) (0.8250, 0.0776, 0.1040)

A3 (0.8709, 0.1039, 0.0253) (0.0233, 0.0722, 0.2042) (0.2191, 0.4892, 0.2914)

A4 (0.3669, 0.4475, 0.1854) (0.0233, 0.0722, 0.9043) (0.0545, 0.3104, 0.6346)

A5 (0.3669, 0.4475, 0.1854) (0.1504, 0.4446, 0.4045) (0.7546, 0.1373, 0.1075)

A6 (0.3669, 0.4475, 0.1854) (0.1504, 0.4446, 0.4045) (0.7546, 0.1373, 0.1075)

A7 (0.7227, 0.1374, 0.1400) (0.1504, 0.4446, 0.4045) (0.0545, 0.3104, 0.6346)

A8 (0.8709, 0.1039, 0.0253) (0.0233, 0.0722, 0.9042) (0.2191, 0.4894, 0.2914)

A9 (0.3669, 0.4475, 0.1854) (0.1504, 0.4446, 0.4045) (0.7546, 0.1373, 0.1075)

A10 (0.7966, 0.1070, 0.1135) (0.3092, 0.1103, 0.5671) (0.3365, 0.4158, 0.2475)

Int J Adv Manuf Technol (2015) 77:357–368 365

cause failure mode is good, 10% is average, and 10% is poor;also, the first expert believes that 10 % of the rate of detectionto the first cause failure mode is good, 10 % is average, and10 % is poor. From Table 3, the aggregated values of Ri foralternatives A1 (non-conductive scrap), A2 (proper coveragenot obtained) versus criterion C1 are proposed as follows:(0.8193, 0.0771, 0.1033), (0.7224, 0.1373, 0.1399), respec-tively. In fact, there is 0.8193 insurance that the value of thefirst cause failure mode (non-conductive scrap) is good,whereas expert is 0.7224 sure that value of the second causefailure mode (proper coverage not obtained) is good. There-fore, CF1 has a higher priority than CF2 about goodness. Itmeans that the risk priority of CF2 is higher than CF1 becauserisk is a negative concept. Moreover, there is 0.0771 insurancethat the severity of the associated effects to the first causefailure mode is average, whereas expert is 0.1373 sure thatvalue of the second cause failure mode is average. Also, thesecond alternative is poorer than the first alternative. We candescribe this belief priority for other alternatives with respect

to each criterion. The preferences of CFs with respect toseverity are depicted in Fig. 3.

As shown in the Fig. 4, the result demonstrates that accord-ing to the FBS TOPSIS method, the occurrence probabilities(second criterion) of CF1 and CF2 have same goodness. Con-sidering assigned goodness, this means that CF1 and CF2 havesame risk with respect to the second criterion; moreover, thereis 0.3105 insurance that the occurrence probability to both ofCF1 and CF2 is average; also, the occurrence probability ofCF1 and CF2 are same poor. Similarly, the severity of otheralternatives can be compared. Using this structure, the expertcan make decisions based on a full understanding of the risks.Therefore, decision maker can assign a more meaningfulvalue for the factors considered whereas in traditional FMEA,three index of risk (S, O, and D) are multiplied with eachother. Moreover, the weights of indexes are considered differ-ent from each other (0.4, 0.2, and 0.4). This advantage of newmethod does not exist in the traditional way.

The rate of detection (third criterion) for CF2 has a highergoodness than CF1. Moreover, the average rate of directivity

Table 5 Belief distances measureand preference order ranking byFBS TOPSIS

Alternatives PIS/NIS Severity Occurrence Detectability Di+/Di

− Ri Rank

A1 A+ 0.0928 0.5092 0.3404 0.3188 0.5674 4A− 0.5432 0.2523 0.3317 0.4181

A2 A+ 0.1390 0.5092 0.0917 0.2509 0.6645 7A− 0.5375 0.2523 0.5448 0.4970

A3 A+ 0.0589 0.5952 0.3404 0.3444 0.5527 2A− 0.5846 0.0440 0.3317 0.4256

A4 A+ 0.2950 0.5952 0.3191 0.3826 0.4174 1A− 0.3995 0.0440 0.1652 0.2741

A5 A+ 0.2950 0.4207 0.1198 0.2756 0.6110 6A− 0.3995 0.2729 0.5212 0.4329

A6 A+ 0.2950 0.4207 0.1198 0.2756 0.6110 6A− 0.3995 0.2729 0.5212 0.4329

A7 A+ 0.1390 0.4207 0.3191 0.2896 0.5649 3A− 0.5375 0.2729 0.1652 0.3760

A8 A+ 0.0589 0.5952 0.3404 0.3444 0.5527 2A− 0.5846 0.0440 0.3317 0.4256

A9 A+ 0.2950 0.4207 0.1198 0.2756 0.6110 6A− 0.3995 0.2729 0.5212 0.4329

A10 A+ 0.1036 0.3992 0.3167 0.2760 0.6063 5A− 0.5334 0.2508 0.3686 0.4251

Fig. 3 The preference of the alternatives versus severity Fig. 4 The preference of the alternatives versus occurrence

366 Int J Adv Manuf Technol (2015) 77:357–368

for CF1 is more than CF2, and the rate of directivity for CF2 ispoorer than CF1 as shown in Fig. 5. Note that detection is apositive criterion in our method. Therefore, the rate of detec-tion is inversely related to risk value. It means that for the rateof detection for CF1 (non-conductive scrap), 0.2191 is good,whereas for the rate of detection for CF2 (proper coverage notobtained), 0.8250 is good. Considering goodness, the firstalternative has a higher risk versus third criteria. This beliefpriority of other alternatives with respect to each criterion canbe described for every evaluation grade (good, average, andpoor). Because the risk indexes of the various projects canhave different degrees of importance, our new risk evaluationmethod that has criteria with different weights is moreappropriate.

To reduce the risk of CFs, all criteria have consideredsimultaneously. Therefore, alternative A4 is ranked first; alter-natives A3 and A8 are ranked second; A7 is ranked third; A1 isranked fourth; A10 is ranked fifth; A5, A6, and A9 are rankedsixth; and A2 is ranked seventh. In comparing the results of thenewmethod and the traditional FMEAmethod, there are somedifferences between our method and the other techniques.Using FBS MCDM method, the experts can assign a moremeaningful value for the factors considered. Moreover, theproposed approach can reduce the occurrence of duplicate riskrankings, whereas the traditional risk evaluation has highduplication rate. The proposed approach follows the orderedweighted criteria of the S, O, and D indicators whereas theimportance of S, O, and D is assumed to be equal in traditionalFMEA method. We propose this result to experts of thisindustry, and experts decide about methods of preventing oreliminating the risk by considering the priority of action.

7 Conclusion

We have presented a new method for ranking of failure modesand cause of failures modes step by step.

Because the judgment on alternatives with respect tocriteria is not usually crisp to deal with the real-world situationproblems, this method is useful to provide the stability ofproduct and process assurance. In this paper, the evaluationof alternatives where introduced as group FBS model with

precise data, ignorance, and fuzziness environment. This pro-posed method is applied to eliminate the conversion debate byevaluating the linguistic assessment of factors to obtain RPNby assigning relative weighting coefficient. The multiple in-dividual FBS where aggregated. The TOPSIS method wasapplied to ranking the group belief structure model. Theincomplete evaluation grades where normalized with centerof gravity of FBS. In addition, the result demonstrated that A4

has maximum risk value and A2 has minimum risk value.Therefore, A4 (cooler not working properly) was introducedas the first alternative in order to reduce the amount of risk.The relative importance among probability, detectability, andseverity of the undesired failures can be different from eachother. Therefore, the flexibility of assigning weight to eachfactor in this method is more compatible to the real decision-making situation. In future research, we expect to furtherdevelop this approach by using other risk evaluation methodssuch as HAZOP to propose a new risk evaluation method.Moreover, the other MCDM methods with robust data can besuggested for the future research.

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