# Advanced FMECA Method Based on Intuitionistic 2-Tuple Linguistic Variables and the Triangular Fuzzy Analytic Hierarchy Process

Authors: Guangze Pan; Dan Li; Qian Li; Yaqiu Li; Yuanhang Wang
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Failure mode effects and criticality analysis (FMECA) is a commonly adopted approach to defining, assessing, and reducing possible failures in designs, systems, processes, products, and services. Traditional FMECA ranks the failure modes of products based on a risk priority number (RPN), which is obtained by multiplying the risk elements. Conventional FMECA has the shortcomings of badly handling unknown information and unreasonably assessing RPNs. To deal with these issues, an advanced FMECA method based on intuitionistic 2-tuple linguistic variables (I2LVs) and the triangular fuzzy analytic hierarchy process (TFAHP) is proposed. In this method, the fuzzy evaluation of risk elements given by different FMECA members is represented by I2LVs, which can efficiently handle unknown information. The TFAHP method is adopted to assess the weights of risky elements and rank the risk priorities of different failure modes. Finally, an application case of an insulated-gate bipolar transistor is used to verify the effectiveness and robustness of the proposed method.


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## Body

## 1. Introduction

Failure mode effects and criticality analysis (FMECA) is typically adopted to identify, evaluate, and reduce existing or underlying errors and failures in system designs or processes [1]. Due to its simplicity and high efficiency, FMECA has been widely applied in some industries, such as the nuclear, aerospace, transportation, and manufacturing industries [2–9]. In traditional FMECA [10–13], every failure mode is assessed using three risk elements: detection (D), severity (S), and occurrence (O). The risk factors for the failure modes are integers between 1 and 10. By multiplying the D, S, and O values, a risk priority number (RPN) can be acquired. Although traditional FMECA has been proven to be a useful way to assess possible product failures in various areas, some shortcomings and limitations remain. For example, the hazard analysis is highly subjective and ignores the uncertainties of the actual intermediate state and the fuzzy nature of the language information. Moreover, risk factors are not weighed, meaning that they are taken to be equally valuable. These shortcomings lead to errors between traditional FMECA evaluations and actual results, which significantly limits their effectiveness.To overcome these shortcomings and limitations, a reasonable evaluation can be made by using fuzzy numbers and linguistic variables instead of exact values. George et al. [14] and Mangeli et al. [15] converted language descriptions into triangular and trapezoidal fuzzy numbers or linguistic variables in FMECA. Xiao [16] and Liu et al. [17] used D numbers to more flexibly and intuitively represent the attribute information in an FMECA multiple criteria decision. Wang et al. [18] identified an exceptional fuzzy number and proposed a corresponding fuzzy RPN to identify the risk priority of failures. However, these evaluations also have shortcomings; e.g., the evaluation implies that the linguistic description of the subject is affiliated with the set and cannot use linguistic descriptions that are not affiliated with the set, such as the hesitation that a decision-maker cannot judge. Therefore, intuitionistic 2-tuple linguistic variables (I2LVs) [19, 20] consisting of language terms and explicit numbers can be used to describe the risk factors. I2LVs use qualitative linguistic terms to express criterion membership and nonmembership. This can summarize the ambiguity of linguistic information better than intuitionistic fuzzy numbers and linguistic variables can, thereby reflecting the actual situation more reliably and truthfully.In addition, given the limitations of traditional FMECA (which does not weigh the risk elements), many researchers have tried to use the multiple-criteria decision-making (MCDM) methods instead of typical RPN methods to prioritize failure modes. Liu et al. [21] combined vague numbers and the VIKOR method to propose a new FMEA method that is particularly applicable to MCDM problems with confusing and incommensurable (comprising various units) standards. Alencar et al. [22] proposed an MCDM model to rank the possible causes of failure by considering more attributes and using the same methodological support as in MCDM. Ju et al. [23] and Geum et al. [24] used grey relation projection (GRP) and grey relation analysis (GRA) methods to decide the weight and risk priority of failure modes. The triangular fuzzy analytic hierarchy process (TFAHP) [25] is a widely used comprehensive evaluation method. Compared with traditional evaluation methods, TFAHP introduces triangular fuzzy numbers in an expert scoring process, making the evaluation results more reasonable, accurate, and operable.To clarify the fuzzy information of FMECA and improve the accuracy of the analysis, a new FMECA method based on I2LVs and TFAHP is proposed. The I2LVs are adopted to describe the vague evaluation of the risk elements by FMECA members, while TFAHP is used to assess the weights of the risk elements and comprehensively rank the risk priorities of the failures. This method is suitable for the FMECA of various products that have multiple failure modes.


## 2. Preliminaries

### 2.1. Intuitionistic 2-Tuple Linguistic Variables

The 2-tuple linguistic variables (2LVs) refer to 2-tuple group evaluation datali,εi, where li is the language term in the language term set L=l0,l1,⋯,l2τ, and εi∈−0.5,0.5 is the symbol transfer value, which indicates the error between the integrated language term and the closest original one.Definition 1.
LetL=l0,l1,⋯,l2r be the language term set and L^ be the extended language term set of L; then, the I2LVs are as follows:
(1)H=lαx,lβxx∈X,lαx,lβx∈L^,where the language terms lαx and lβx indicate the membership and nonmembership degrees of element x, respectively, and satisfy 0≤αx+βx≤2τ.Definition 2.
LetU and V be continuous and strictly monotonic utility functions of L^. If, for any I2LV, U and V satisfy the following functions:
(2)Ulαx=Vneglαx,Vlβx=Uneglβx,Ul0=Vl2τ=0,Ul2t=Vl0=1,
thenU and V are called the utility functions of membership degree lαx and nonmembership degree lβx, respectively.Definition 3.
Lethi=lαhi,lβhii=1,2,⋯,m be an I2LV and w=w1,w2,⋯,wnT be its weight vector, satisfying ∑i=1nwi=1 and wi≥0. Then, the intuitionistic 2-tuple weighted average variables (I2WAVs) are
(3)h=I2WAVh1,h2,⋯,hm=U−1∑i=1mwiUlαh,V−1∑i=1mwiVlβk.Definition 4.
Leth=lαh,lβh be an I2LV; then, the expected utility function of h is
(4)Eh=Ulαh+Vlβh2.Definition 5.
Leth=lαh,lβh be an I2LV; then, the hesitation utility function of h is
(5)Eh=Ulαh+Vlβh2.Definition 6.
Leth1 and h2 be two I2LVs; then
(i)
ifEh1>Eh2, then h1>h2(ii)
ifEh1=Eh2 and Qh1<Qh2, then h1>h2(iii)
ifEh1=Eh2 and Qh1=Qh2, then h1=h2


### 2.2. TFAHP Method

The core of the analytic hierarchy process (AHP) is to construct a judgment matrix by using an integer between 1 and 9, with its inverse number used as a scale. This evaluation often does not consider the fuzzy nature of the subjective judgment. If the ratio of the weights of the two factors is not easy to determine, it is only known that the range of change isl−u, and the maximum possible value is m; then, the triangular fuzzy number l,m,u can be used in the evaluation.In comparison with the traditional AHP method, TFAHP uses triangular fuzzy numbers in the expert scoring process, which makes the scoring relatively reasonable and accurate [26]. Moreover, during the weight calculation, the problem of judging and adjusting the consistency of the matrix in the traditional AHP is skillfully solved by adopting a possibility matrix [27].First of all, a set of triangular fuzzy judgment matrices is established as follows:(6)Ak=aijkn×naijk=lijk,mijk,uijks.t.,0⩽lijk⩽mijk⩽uijk⩽1where n is the number of factors, k is the expert serial number, K is the total number of experts (where k=1,2,⋯,K), aijk is the ratio of the importance of factor i to factorj, lijk and uijk are the upper and lower bounds of the triangular fuzzy number, respectively (where lijk+ujik=1, uijk+ljik=1, liik=0.5, uiik=0.5), and mij is the median of the triangular fuzzy number mijk+mjik=1.The set of triangular fuzzy judgment matrices is merged according to the following function:(7)aij=aij1+aij2+⋯+aijKK.Secondly, the single-level triangular fuzzy weights of the judgment matrix are calculated.(8)ci=∑j=1naij∑i=1n∑j=1naij=∑j=1nlij∑i=1n∑j=1nμij,∑j=1nmij∑i=1n∑j=1nmij,∑j=1nμij∑i=1n∑j=1nlij.The single-level triangular fuzzy weightsci are compared with each other. Let ci1=cl1,cm1,cμ1 and ci2=cl2,cm2,cμ2; then, the likelihood matrix P=pijn×n is solved as follows:
(9)pci1≥ci2=0.5max1−maxcm2−cl1cm1−cl1+cm2−cl2,0,0+0.5max1−maxcμ2−cm1cμ1−cm1+cμ2−cm2,0,0.Then, the likelihood matrix is transformed into a fuzzy matrix with consistent features.(10)rij=ri−rj2n−1+0.5,(11)ri=∑k=1nPik.Finally, the final weights are calculated using the following formula:(12)wi=∑j=1nrij+n/2−1nn−1.


## 2.1. Intuitionistic 2-Tuple Linguistic Variables

The 2-tuple linguistic variables (2LVs) refer to 2-tuple group evaluation datali,εi, where li is the language term in the language term set L=l0,l1,⋯,l2τ, and εi∈−0.5,0.5 is the symbol transfer value, which indicates the error between the integrated language term and the closest original one.Definition 1.
LetL=l0,l1,⋯,l2r be the language term set and L^ be the extended language term set of L; then, the I2LVs are as follows:
(1)H=lαx,lβxx∈X,lαx,lβx∈L^,where the language terms lαx and lβx indicate the membership and nonmembership degrees of element x, respectively, and satisfy 0≤αx+βx≤2τ.Definition 2.
LetU and V be continuous and strictly monotonic utility functions of L^. If, for any I2LV, U and V satisfy the following functions:
(2)Ulαx=Vneglαx,Vlβx=Uneglβx,Ul0=Vl2τ=0,Ul2t=Vl0=1,
thenU and V are called the utility functions of membership degree lαx and nonmembership degree lβx, respectively.Definition 3.
Lethi=lαhi,lβhii=1,2,⋯,m be an I2LV and w=w1,w2,⋯,wnT be its weight vector, satisfying ∑i=1nwi=1 and wi≥0. Then, the intuitionistic 2-tuple weighted average variables (I2WAVs) are
(3)h=I2WAVh1,h2,⋯,hm=U−1∑i=1mwiUlαh,V−1∑i=1mwiVlβk.Definition 4.
Leth=lαh,lβh be an I2LV; then, the expected utility function of h is
(4)Eh=Ulαh+Vlβh2.Definition 5.
Leth=lαh,lβh be an I2LV; then, the hesitation utility function of h is
(5)Eh=Ulαh+Vlβh2.Definition 6.
Leth1 and h2 be two I2LVs; then
(i)
ifEh1>Eh2, then h1>h2(ii)
ifEh1=Eh2 and Qh1<Qh2, then h1>h2(iii)
ifEh1=Eh2 and Qh1=Qh2, then h1=h2


## 2.2. TFAHP Method

The core of the analytic hierarchy process (AHP) is to construct a judgment matrix by using an integer between 1 and 9, with its inverse number used as a scale. This evaluation often does not consider the fuzzy nature of the subjective judgment. If the ratio of the weights of the two factors is not easy to determine, it is only known that the range of change isl−u, and the maximum possible value is m; then, the triangular fuzzy number l,m,u can be used in the evaluation.In comparison with the traditional AHP method, TFAHP uses triangular fuzzy numbers in the expert scoring process, which makes the scoring relatively reasonable and accurate [26]. Moreover, during the weight calculation, the problem of judging and adjusting the consistency of the matrix in the traditional AHP is skillfully solved by adopting a possibility matrix [27].First of all, a set of triangular fuzzy judgment matrices is established as follows:(6)Ak=aijkn×naijk=lijk,mijk,uijks.t.,0⩽lijk⩽mijk⩽uijk⩽1where n is the number of factors, k is the expert serial number, K is the total number of experts (where k=1,2,⋯,K), aijk is the ratio of the importance of factor i to factorj, lijk and uijk are the upper and lower bounds of the triangular fuzzy number, respectively (where lijk+ujik=1, uijk+ljik=1, liik=0.5, uiik=0.5), and mij is the median of the triangular fuzzy number mijk+mjik=1.The set of triangular fuzzy judgment matrices is merged according to the following function:(7)aij=aij1+aij2+⋯+aijKK.Secondly, the single-level triangular fuzzy weights of the judgment matrix are calculated.(8)ci=∑j=1naij∑i=1n∑j=1naij=∑j=1nlij∑i=1n∑j=1nμij,∑j=1nmij∑i=1n∑j=1nmij,∑j=1nμij∑i=1n∑j=1nlij.The single-level triangular fuzzy weightsci are compared with each other. Let ci1=cl1,cm1,cμ1 and ci2=cl2,cm2,cμ2; then, the likelihood matrix P=pijn×n is solved as follows:
(9)pci1≥ci2=0.5max1−maxcm2−cl1cm1−cl1+cm2−cl2,0,0+0.5max1−maxcμ2−cm1cμ1−cm1+cμ2−cm2,0,0.Then, the likelihood matrix is transformed into a fuzzy matrix with consistent features.(10)rij=ri−rj2n−1+0.5,(11)ri=∑k=1nPik.Finally, the final weights are calculated using the following formula:(12)wi=∑j=1nrij+n/2−1nn−1.


## 3. Methods

Supposing that there arep FMECA team experts TE=TE1,TE2,⋯,TEp evaluating m failure modes FM=FM1,FM2,⋯,FMm with respect to n risk factors RF=RF1,RF2,⋯,RFn, the expert weight vector is w=w1,w2,⋯,wpT. The expert TEk gives an evaluation of risk factor RFj of failure mode FMi as a I2LV bijk=lαijk,lβijk, with the I2LV matrix being Bk=bijkm×nk=1,2,⋯,p. The set of triangular fuzzy judgment matrices of risk factors is Ak=aijkn×n.The proposed FMECA method includes the steps shown in Figure1.Step 1.
According to I2WAV and the expert weight vectorw, the I2LV matrix Bk is integrated into the group I2LV matrix B=bijm×n=lαij,lβijm×n.
(13)bij=I2WAVbij1,bij2,⋯,bijp=U−1∑k=1pwkUlαik,V−1∑k=1pwkVlβijk.Step 2.
The TFAHP method is adopted to assess the weight vector of risk elementsϖ=ϖ1,ϖ2,⋯,ϖnT.Step 3.
According to the group I2WAV and the weight vector of risk elements, the group I2WAV for every risk element of failure modesAi is integrated to obtain the comprehensive I2WAV bi=la,lb.
(14)bi=I2WAVbi1,bi2,⋯,bin=U−1∑j=1nϖjUlαij,V−1∑j=1nϖjVlβij.Step 4.
According to Definitions4 and 5, the expected utility value Ebi and the hesitation utility valueQbiof the comprehensive I2WAV bi are calculated.Step 5.
According to Definition6, the expected utility value and the hesitation utility value are sorted to acquire the failure mode’s risk priorities.Figure 1
Implementation process of the advanced FMECA method.


## 4. Case Study

The section uses the new FMEA method according to I2LV and TFAHP (I2LV-TFAHP method) to assess the failure modes of a crimp-type insulated-gate bipolar transistor (IGBT).Step 1.
The FMECA team consists of four experts,TE1,TE2,TE3,TE4. The risk elements are detection (D), severity (S), and occurrence (O). The weight of the experts in each risk factor is w=0.2,0.35,0.15,0.3T, and the linguistic term set is L=l0,l1,l2,l3,l4,l5,l6=extremelylow,verylow,low,moderate,high,veryhigh,extremelyhigh. Through detailed analysis of the IGBT, the FMECA team identifies seven potential failure modes: fretting wear (FM1), short circuit (FM2), open circuit (FM3), microcorrosion (FM4), boundary warpage (FM5), gate oxide-layer destruction (FM6), and spring failure (FM7).Step 2.
The experts use I2LV to evaluate the failure modes. The I2LV matricesBk=bijkm×nk=1,2,3,4;i=1,2,3;j=1,2,3,4,5,6,7 are provided.Step 3.
According to I2WAV and the experts’ weight vectorw, the group I2LV matrix is obtained.Step 4.
Experts use triangular fuzzy numbers to score and average the risk factors to obtain a judgment matrix (Table1). The risk element weights calculated by the TFAHP method are ϖ=0.319,0.485,0.196T.Step 5.
According to Equation (14), the comprehensive I2WAV is calculated.
(15)b1=l0.68,l0.48,b2=l2.59,l2.89,b3=l2.43,l2.96,b4=l3.18,l2.04,b5=l1.85,l3.49,b6=l3.36,l1.70,b7=l2.95,l1.94.Step 6.
The expected utility value and hesitation utility value of the comprehensive I2WAVbii=1,2,3,4,5,6,7 are calculated according to Equations (4) and (5).Step 7.
The failure mode risk priority ranks are determined based onEbi and Qbi, as shown in the last column of Table 2.Table 1
Judgment matrix.
Risk factorOSDO0.5,0.5,0.50.35,0.43,0.510.54,0.63,0.72S0.49,0.57,0.650.5,0.5,0.50.63,0.72,0.82D0.28,0.37,0.450.18,0.28,0.370.5,0.5,0.5Table 2
Results of the I2LV-TFAHP method.
Failure modeEQRankFM10.6810.1261FM20.4760.0875FM30.4560.1016FM40.5950.1303FM50.3630.1097FM60.6380.1572FM70.5840.1854Table3 shows that FM1 has the highest expected utility degree in fretting wear failure and, therefore, should be offered a top risk priority. The order of the risk priorities of the seven failure modes is FM1>FM6>FM4>FM7>FM2>FM3>FM5.Table 3
Priority ranks evaluated by the five FMECA methods.
Failure modeI2LV-TFAHPTFAHPI2LVI2LV-TOPSISTraditional methodFM111112FM256764FM367554FM434325FM575677FM623431FM742243By comparing TFAHP, I2LV, I2LV-TOPSIS, and the traditional method, the rationality of the I2LV-TFAHP method is verified. The evaluation outcomes are displayed in Table3 and Figure 2.Figure 2
Comparison of the results of the five FMEA methods.Table4 and Figure 2 demonstrate the following facts.Table 4
Expert weights in different cases.
CasesTE1TE2TE3TE4Case 00.20.350.150.3Case 10.10.40.30.2Case 20.250.250.250.25Case 30.30.20.350.15Case 40.40.10.40.1TFAHP, I2LV, and I2LV-TOPSIS all indicate that the failure mode with the highest risk priority isFM1 (fretting wear), which is consistent with the conclusion of I2LV-TFAHP.I2LV-TFAHP ranksFM6 (gate oxide-layer destruction) in second place, while TFAHP and I2LV rank FM7 (spring failure) in second place and the I2LV-TOPSIS method ranks FM4 (microcorrosion) in second place. The TFAHP, I2LV, and I2LV-TOPSIS methods rank FM6 in fifth, fourth, and third places, respectively. Using engineering knowledge during the operation of IGBT, the failure frequency of the gate oxide-layer destruction is higher than that of the spring failure as well as microcorrosion, as shown in the risk element O. Thus, in the I2LV-TFAHP method, it is reasonable to rank the FM6 risk priority in second place.The I2LV-TFAHP ranksFM5 (boundary warpage) in seventh place, while the TFAHP and I2LV methods rank FM3 (open circuit) and FM2 (short circuit) in seventh place. The open circuit and short circuit of the IGBT make it unable to work, and the effect of boundary warpage on the IGBT function is small, as reflected in the risk factors. Therefore, it is also reasonable for the I2LV-TFAHP method to rank FM5 in seventh place.The first four failure modes with the highest risk, as evaluated by typical methods, are not the same as those of the four other methods, particularly when all of the new methods selectFM1 as the highest-risk failure mode. Moreover, the traditional method ranks FM2 and FM3 in equal fourth place, meaning that it cannot distinguish between their risks.The above information indicates that the proposed I2LV-TFAHP method is more reasonable and accurate than other methods.Sensitivity analysis is performed by changing the expert weights, as shown in Table4, where case 0 demonstrates the previously mentioned application, and cases 1 to 4 indicate cases with different weights.The ranking outcomes of the failure mode risk priorities in various cases are displayed in Figure3 and Table 5. They show that a change in expert weight has a very slight influence on the risk priority rank, which means that the suggested method is sufficiently robust in ranking the risk priorities of the failure modes identified in the FMECA.Figure 3
Sensitivity analysis for the I2LV-TFAHP method.Table 5
Failure mode priority ranks under various cases.
Failure modeCase 0Case 1Case 2Case 3Case 4FM111112FM256764FM367554FM434325FM575677FM623431FM742243


## 5. Conclusions

In this study, an advanced FMECA method based on I2LV and TFAHP is proposed to deal with the shortcomings and limitations of the conventional FMECA method. It also suggests a different way to prioritize the risks of different failure modes. I2LV can effectively deal with the fuzzy nature of linguistic variables, while TFAHP is adopted to evaluate the risk element weights and the comprehensive ranking of the failure mode risk priorities. An application example is provided to demonstrate the failure mode risk priorities in the FMECA provided by the IGBT. Compared with TFAHP, I2LV, I2LV-TOPSIS, and the traditional method, the proposed I2LV-TFAHP method is more accurate and reasonable and has greater robustness when analysing risk priority ranks.



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