# A Potential Failure Mode and Effect Analysis Method of Electromagnet Based on Intuitionistic Fuzzy Number in Manufacturing Systems

Authors: Jihong Pang; Jinkun Dai; Faqun Qi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Failure mode and effect analysis (FMEA) is a systematic activity in the stage of product design and process design. However, the traditional FMEA has some shortcomings in practical application, such as too many evaluation languages, uncertain weights of influencing factors, and uncertain weights of evaluation members. This paper presents an FMEA evaluation method in manufacturing system based on similarity measure, nonlinear programming model, and intuitionistic fuzzy number (IFN). Firstly, the IFN is used to evaluate failure mode, which overcomes the defect of traditional FMEA evaluation value. Secondly, the weight of failure evaluation team members is solved according to the concept of similarity measure to make up for the blank of evaluation members’ weight aiming at the shortage of unknown weight. Then, the definition of consensus measure is introduced to make the evaluators reach a consensus, and the weights of influencing factors of failure modes (FMs) are calculated. Finally, the weights of evaluators and influencing factors are calculated by IFN algorithm and score function, and the score value of each FM is obtained to rank instead of risk priority number (RPN). The objectivity and practicability of the new method are verified by the example of failure mode for an attractive electromagnet manufacturing system.


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## Body

## 1. Introduction

FMEA is a systematic activity in product design stage and process design stage. The subsystems and parts of FMEA are analyzed one by one to identify all potential FMs and their possible consequences, so as to take necessary measures in advance to improve the quality and reliability of products. FMEA was originally carried out in the product design phase of the aerospace industry in the 1960s to help improve safety and reliability [1]. In addition, FMEA enables an organization to proactively prevent failure, not just respond to it [2]. FMEA is also a simple method to solve the complex failure mode evaluation process [3]. Because of its simple and easy operation, FMEA is widely used in transportation, energy, medical, and supply chain transportation [4, 5].On the other hand, the traditional FMEA process is used to evaluate the FMs of products or systems by team members. The severity, occurrence, and detection (S, O, D) of FMs according to their own professional knowledge are calculated by the scores of each failure mode [6]. Although FMEA is widely used, it has some inherent defects. To ensure the objectivity of FMEA, we must ensure that the evaluation language of failure mode should conform to the actual situation, so as to make a more comprehensive and specific description of failure mode. However, the traditional FMEA uses natural language to carry out evaluation, which does not conform to the psychological changes of the evaluation members in reality [7]. Secondly, the weight of factors is also the key to affect the final evaluation results [8]. For example, the weights of influencing factors are evenly distributed by traditional FMEA, which will lead to great difference between the calculation results and the actual situation [9]. In the actual production, the evaluation members in different working positions have different understanding of the definition and importance of failure; it is necessary to accurately calculate the weight of each evaluation member in the evaluation team to determine the failure mode sequencing [10, 11].In order to improve the shortcomings of the traditional FMEA, many scholars and researchers have carried out more rigorous research on this method and made it more objective and true to reflect the impact of failure. So, further research in this area would be needed. A comprehensive risk ranking method to improve the performance of FMEA was developed by using interval-valued intuitionistic fuzzy sets (IVIFSs) and multiattribute boundary approximation area comparison (MABAC). In addition, a linear programming model to obtain the optimal weight of risk factors was established under the condition of incomplete prior knowledge of weight information [12]. A new integrated fuzzy intelligent FMEA framework was proposed. The framework adopts the combination of fuzzy set theory, analytic hierarchy process (AHP), and data envelopment analysis (DEA) to deal with uncertainty and improve the reliability of risk assessment. All of these are achieved by employing a heterogeneous expert group. RPN, time, and cost are used as indicators to determine the efficiency of FMEA mode and determine the appropriate priority and corrective measures [13]. An interactive method to deal with the possible uncertainties in the evaluation process was proposed by using the fuzzy set theory. In order to deal with the weight of subjective and objective uncertainties, the AHP and entropy weight method were adopted [14]. A new method combining evidence theory, intuitive fuzzy set (IFS), and belief entropy has been proposed to determine the weight of team members to analyze FMs [15]. A hybrid FMEA framework has been developed which integrates the Portuguese abbreviation for interactive and multicriteria decision method with the Choquet integration method. Under this framework, the uncertainty in risk assessment is described by generalized trap type fuzzy numbers [16]. A consensus-based FMEA group decision-making framework was proposed. The purpose of this framework is to classify FMs into several ordinal risk classes. Assuming that FMEA participants provide their preferences in a linguistic manner using the possibility hesitant fuzzy linguistic information, a consensus driven method is proposed to generate the weight of risk factors in the FMEA framework. On this basis, the consensus rules based on optimization guided by the minimum adjustment distance strategy are calculated, and an interaction model for consensus is developed to generate consensus failure mode risk classes [17]. A triangular distribution-based basic probability assignment (TDBPA) method based on triangular distribution was presented within the framework of Dempster–Shafer evidence theory, the conflict risk levels assessed by different experts. The modified RPN model based on fusion assessment could calculate the ranking of FMs [18]. A new comprehensive multicriteria decision-making (MCDM) method and fuzzy multiattribute ideal real comparative analysis (FMAIRCA) were proposed. It combines the first method to calculate the fuzzy relative importance between risk factors by using quantitative method [19]. In order to improve the performance of classical FMEA, the weights of influencing factors are determined by using the fuzzy optimization method, and a fuzzy VIKOR approach was to evaluate the security policies and analyze the content of five press agencies in expert decision making [20].In this paper, the concept of similarity measurement is used to solve the weights of evaluation members and makes up for the blank of obtaining weights of evaluation members on the basis of existing research further to improve the FMEA. In the same time, the definition of consensus measurement is introduced to reach consensus on the evaluation of the evaluators, and the weights of influencing factors are solved by using the approved method. The remainder of this paper is arranged as follows: in Section2, a decision process of electromagnet FMEA is presented. Section 3 describes the IFN evaluation of failure mode with a form of influencing factors. In Section 4, the weight of evaluation members and influencing factors is determined by comprehensive calculation, and the IFN of FMs is obtained. The scores of each failure mode are obtained and sorted according to the score function. An illustrative example of FMEA of electromagnet is shown in Section 5. The last chapter summarizes the work done in this paper.


## 2. Flow Chart of FMEA of Electromagnet

FMEA methodically breaks down the analysis of complex manufacturing processes of electromagnet into manageable steps. FMEA can be used to identify potential failure modes for electromagnet and determine their impacts during production and fabrication. The flow chart of FMEA of electromagnet is shown in Figure1.Figure 1
Flow chart of FMEA of electromagnet.To achieve the desired effect of this paper, the following key steps are taken. Step 1: determine the target and risk level of the electromagnet product risk analysis, collect the potential failure mode and failure impact, and the evaluation team evaluates the failure mode by using the IFN. Step 2: according to the incomplete weight information in FMEA, the similarity measure is used to determine the weight of evaluation team members. Step 3: for the influencing factors of failure, the evaluation members use IFNs for comparative evaluation and form a comparison matrix. In order to find the optimal solution of the influencing factors, the exact weight of the influencing factors is derived from the preference relation of the comparison matrix. Step 4: after the weight information in FMEA is all determined, simple intuitionistic fuzzy weighted geometry (SIFWG) is used to aggregate the evaluation information and weight information, and the score of each failure mode is calculated according to the definition of intuitionistic fuzzy score function, and the order is made.


## 3. Evaluation of IFNs on FMs and Influencing Factors

In 1965, Zadeh defined the mathematical meaning of IFSs for the first time [21] and opened a new chapter of fuzzy mathematics. In 1986, Atanassov popularized fuzzy sets, and it was also the first time to express the relationship between an element and a specific set with the three indexes of membership degree, nonmembership degree, and hesitation, and he put forward the concept of IFSs [22]. Let the set X be a universe of discourse, the fuzzy set F on X is represented by membership degree μF,μF:X⟶0,1, and then the membership degree of x in the set F is denoted by μFx. IFS is an object that has the following form:(1)A=x,μAx,νAx|x∈X,whereμAx, νAx means the membership degree and nonmembership degree of the element x∈X to A, respectively, with the condition that for all x∈X, then(2)0≤μAx≤1,0≤νAx≤1,0≤νAx+μAx≤1.


### 3.1. Basic Operation Law of IFN

Leta=μa,va be IFN, then the basic operation law of IFN is displayed as follows [23]:(3)λa=1−1−μaλ,νaλ,(4)aλ=μaλ,1−1−νaλ,(5)⊕j=1naj=1−Πj=1n1−μa,Πj=1nνaj,(6)⊕j=1naj=Πj=1nμaj,1−Πj=1n1−νa.


### 3.2. Ranking Rules of IFNs

Leta=μa,va be IFN; its score functionSa and accuracy function Ha are expressed, respectively, by the following formulas:(7)Sa=μa−va,(8)Ha=μa+va.Letai=μai,νai and aj=μaj,νaj be two IFNs, then the procedure of comparing the magnitude of two IFNs is introduced as follows:(9)ifSai<Saj,thenai<aj.IfSai=Saj, then we can get the following conclusions:(10)IfHai<Haj,thenai<aj.(11)IfHai=Haj,thenai=aj.Because the similarity function is improved on the basis of score function, all similarity functionsLa conform to the rules of score function.


## 3.1. Basic Operation Law of IFN

Leta=μa,va be IFN, then the basic operation law of IFN is displayed as follows [23]:(3)λa=1−1−μaλ,νaλ,(4)aλ=μaλ,1−1−νaλ,(5)⊕j=1naj=1−Πj=1n1−μa,Πj=1nνaj,(6)⊕j=1naj=Πj=1nμaj,1−Πj=1n1−νa.


## 3.2. Ranking Rules of IFNs

Leta=μa,va be IFN; its score functionSa and accuracy function Ha are expressed, respectively, by the following formulas:(7)Sa=μa−va,(8)Ha=μa+va.Letai=μai,νai and aj=μaj,νaj be two IFNs, then the procedure of comparing the magnitude of two IFNs is introduced as follows:(9)ifSai<Saj,thenai<aj.IfSai=Saj, then we can get the following conclusions:(10)IfHai<Haj,thenai<aj.(11)IfHai=Haj,thenai=aj.Because the similarity function is improved on the basis of score function, all similarity functionsLa conform to the rules of score function.


## 4. Failure Mode and Effect Analysis

FMEA is a comprehensive calculation of decision information of failure mode by comprehensive decision personnel. The total proportion of influencing factors by each evaluation member can directly affect the ranking of final failure mode. Therefore, the weights of objective evaluation members and influencing factors are determined before the final comprehensive calculation.


### 4.1. Determination of the Weight of Evaluation Members

Considering all possible options, we select the risk assessment matricesXk=xijkm×n=μijk,νijkm×n,wherek=1,…,l, and adopt many ways to estimate the evaluation model. Next, the ideal assessment of the failure mode can be the average of a single risk assessment matrix, and the expression of ideal matrix is shown as follows [24]:(12)X∗=xij∗m×n=μij∗,νij∗m×n,wherex∗=1/lxijk,k=1,…,l, and μij∗=1−∏k=1l1−μijk1/l,νij∗=∏k=1lνijk1/l,i=1,…,m,j=1,…,n.Leta1=μa1,νa1 and a2=μa2,νa2 be two IFNs, then the interval-valued intuitionistic fuzzy Euclidean distance is calculated as(13)da1,a2=12μ1−μ22+ν1−ν22.LetX1=xij1m×n=μij1,νij1,X2=xij2m×n=μij2,νij2 be two interval-valued intuitionistic fuzzy matrixes, then the similarity measure of the two matrixes is defined as(14)simX1,X2=∑i=1m∑j=1ndxij1,xij2c∑i=1m∑j=1ndxij1,xij2c+dxij1,xij2,where d is the Euclidean distance, xij2c=νij2,μij2, 0≤simX1,X2≤1,whenX1=X2,simX1,X2=1.Next, the similarity between the evaluation results of each evaluator and the ideal evaluation results is obtained(15)simXk,X∗=∑i=1m∑j=1ndxijk,xij∗c∑i=1m∑j=1ndxijk,xij∗c+dxijk,xij∗,where X∗c=xij∗cm×n=νij∗,μij∗m×n.At last, the weights of the FMEA team members are determined as follows:(16)λk=simXk,X∗∑k=1lsimXk,X∗.


### 4.2. Determination of the Weight of Influencing Factors

#### 4.2.1. Construction of Perfect Product Consistency Matrix

In the traditional AHP, the consistency ratio (CR) is used to measure the consistency of the product relation. In this paper, the intuitionistic fuzzy matrix ofR=rikm×m=μik,νik is constructed into a perfect product consistent intuitionistic fuzzy matrix Rp=rikpm×m by using the following algorithm:(1)
Ifk>i+1,rikp=μikp,νikp, where(17)μikp=Πt=i+1k−1μitμtkk−i−1Πt=i+1k−1μitμtkk−i−1+Πt=i+1k−11−μit1−μtkk−i−1,(18)νikp=Πt=i+1k−1νitνtkk−i−1Πt=i+1k−1νitνtkk−i−1+Πt=i+1k−11−νit1−νtkk−i−1.(2)
If(19)k=i+1,rikp=rik.(3)
If(20)k<i+1,rikp=νikp,μikp.


#### 4.2.2. Determination and Iterative Modification of Consistency Measure

Generally, the level of evaluation members is basically the same, and the evaluation results obtained should coincide. However, the inconsistency of evaluation results cannot be avoided in real life. The process of group consensus is to find the final solution accepted by the evaluated personnel. The following gives an algorithm to reach consensus.Step 1: calculating the consensus degree of evaluation members,(21)Cl=1−∑1≤i≤j≤mmμijp−μij∗+νijp−νij∗+πijp−πij∗m−1m−2.If the consensus degree reaches the critical valueγ of group consensus, the result of SIFWG will be output; otherwise, turn to step 2.Step 2: revising according to the formula,(22)μijlp′=μijlpξ×μij∗1−ξ,νijlp′=νijlpξ×νij∗1−ξ,whereRlp=riklpm×n=μijlp,νijlpm×n is the perfect product consistent intuitionistic fuzzy matrix of evaluator l.Step 3: calculating the consensus degree.In this paper, we use the new modified perfect product consistent intuitionistic fuzzy matrix to calculate the consensus degree. If it meets the critical value, the new modified perfect product consistent intuitionistic fuzzy matrix is aggregated with SIFWG operator to form a new aggregation matrix. If it does not meet the critical value, turn to Step 2 until it meets the critical value.


#### 4.2.3. Aggregation of Evaluation Information and Derivation of Influencing Factors’ Weight

According to abovementioned algorithm, the perfect product consistency intuitionistic fuzzy matrixRp=rikpm×n is obtained. For information aggregation, there are many operators that can be used. This paper uses SIFWG operator, because the matrix before aggregation has consistency, and because the after aggregation through SIFWG operator has the same consistency. Then, the matrix R∗=rij∗m×n=μij∗,νij∗ of SIFWG aggregation is obtained, which is shown in the following:(23)μij∗=∏k=1lμijpλk,νij∗=∏k=1lνijpλk,i=1,…,m,j=1,…,n.According to the accurate priority derivation method of IFNs, the nonlinear programming model is solved by LINGO software, and the weight of influencing factors is obtained directly [25]. Based on the standard operation of LINGO software, we take the matrix as an example to show the nonlinear programming model:(24)minf=∑i=1m−1∑k=i+1mρik+δik,s.tωiωi+ωk+ρik−μik≥0,ωiωi+ωk−δik+νik≥0,∑i=1mωi=1,ωi∈0,1,ρik≥0,δik≥0,ρik∗δik=0.Exportω=ωs,ωo,ωd, where i=1,…,m−1;k=i+1,…,m, ρik, δik is the relaxation variable,μik−ρik≤ωi/ωi+ωk≤1−νik+δik.


### 4.3. Potential Failure Mode Score and Ranking

In this paper, we present the integrate evaluator and influence factor weights with evaluation information, which is shown as(25)FMi=∑k=1lλkμijk,μijkωS+∑k=1lλkμijk,μijkωO+∑k=1lλkμijk,μijkωd,where the evaluation information of k assessor is μijk,μijki=1,…,m,j=1,…,n.Finally, the IFNμFM,νFM of failure mode is obtained. According to the score function formula of IFN, the score of each failure mode is sorted.


## 4.1. Determination of the Weight of Evaluation Members

Considering all possible options, we select the risk assessment matricesXk=xijkm×n=μijk,νijkm×n,wherek=1,…,l, and adopt many ways to estimate the evaluation model. Next, the ideal assessment of the failure mode can be the average of a single risk assessment matrix, and the expression of ideal matrix is shown as follows [24]:(12)X∗=xij∗m×n=μij∗,νij∗m×n,wherex∗=1/lxijk,k=1,…,l, and μij∗=1−∏k=1l1−μijk1/l,νij∗=∏k=1lνijk1/l,i=1,…,m,j=1,…,n.Leta1=μa1,νa1 and a2=μa2,νa2 be two IFNs, then the interval-valued intuitionistic fuzzy Euclidean distance is calculated as(13)da1,a2=12μ1−μ22+ν1−ν22.LetX1=xij1m×n=μij1,νij1,X2=xij2m×n=μij2,νij2 be two interval-valued intuitionistic fuzzy matrixes, then the similarity measure of the two matrixes is defined as(14)simX1,X2=∑i=1m∑j=1ndxij1,xij2c∑i=1m∑j=1ndxij1,xij2c+dxij1,xij2,where d is the Euclidean distance, xij2c=νij2,μij2, 0≤simX1,X2≤1,whenX1=X2,simX1,X2=1.Next, the similarity between the evaluation results of each evaluator and the ideal evaluation results is obtained(15)simXk,X∗=∑i=1m∑j=1ndxijk,xij∗c∑i=1m∑j=1ndxijk,xij∗c+dxijk,xij∗,where X∗c=xij∗cm×n=νij∗,μij∗m×n.At last, the weights of the FMEA team members are determined as follows:(16)λk=simXk,X∗∑k=1lsimXk,X∗.


## 4.2. Determination of the Weight of Influencing Factors

### 4.2.1. Construction of Perfect Product Consistency Matrix

In the traditional AHP, the consistency ratio (CR) is used to measure the consistency of the product relation. In this paper, the intuitionistic fuzzy matrix ofR=rikm×m=μik,νik is constructed into a perfect product consistent intuitionistic fuzzy matrix Rp=rikpm×m by using the following algorithm:(1)
Ifk>i+1,rikp=μikp,νikp, where(17)μikp=Πt=i+1k−1μitμtkk−i−1Πt=i+1k−1μitμtkk−i−1+Πt=i+1k−11−μit1−μtkk−i−1,(18)νikp=Πt=i+1k−1νitνtkk−i−1Πt=i+1k−1νitνtkk−i−1+Πt=i+1k−11−νit1−νtkk−i−1.(2)
If(19)k=i+1,rikp=rik.(3)
If(20)k<i+1,rikp=νikp,μikp.


### 4.2.2. Determination and Iterative Modification of Consistency Measure

Generally, the level of evaluation members is basically the same, and the evaluation results obtained should coincide. However, the inconsistency of evaluation results cannot be avoided in real life. The process of group consensus is to find the final solution accepted by the evaluated personnel. The following gives an algorithm to reach consensus.Step 1: calculating the consensus degree of evaluation members,(21)Cl=1−∑1≤i≤j≤mmμijp−μij∗+νijp−νij∗+πijp−πij∗m−1m−2.If the consensus degree reaches the critical valueγ of group consensus, the result of SIFWG will be output; otherwise, turn to step 2.Step 2: revising according to the formula,(22)μijlp′=μijlpξ×μij∗1−ξ,νijlp′=νijlpξ×νij∗1−ξ,whereRlp=riklpm×n=μijlp,νijlpm×n is the perfect product consistent intuitionistic fuzzy matrix of evaluator l.Step 3: calculating the consensus degree.In this paper, we use the new modified perfect product consistent intuitionistic fuzzy matrix to calculate the consensus degree. If it meets the critical value, the new modified perfect product consistent intuitionistic fuzzy matrix is aggregated with SIFWG operator to form a new aggregation matrix. If it does not meet the critical value, turn to Step 2 until it meets the critical value.


### 4.2.3. Aggregation of Evaluation Information and Derivation of Influencing Factors’ Weight

According to abovementioned algorithm, the perfect product consistency intuitionistic fuzzy matrixRp=rikpm×n is obtained. For information aggregation, there are many operators that can be used. This paper uses SIFWG operator, because the matrix before aggregation has consistency, and because the after aggregation through SIFWG operator has the same consistency. Then, the matrix R∗=rij∗m×n=μij∗,νij∗ of SIFWG aggregation is obtained, which is shown in the following:(23)μij∗=∏k=1lμijpλk,νij∗=∏k=1lνijpλk,i=1,…,m,j=1,…,n.According to the accurate priority derivation method of IFNs, the nonlinear programming model is solved by LINGO software, and the weight of influencing factors is obtained directly [25]. Based on the standard operation of LINGO software, we take the matrix as an example to show the nonlinear programming model:(24)minf=∑i=1m−1∑k=i+1mρik+δik,s.tωiωi+ωk+ρik−μik≥0,ωiωi+ωk−δik+νik≥0,∑i=1mωi=1,ωi∈0,1,ρik≥0,δik≥0,ρik∗δik=0.Exportω=ωs,ωo,ωd, where i=1,…,m−1;k=i+1,…,m, ρik, δik is the relaxation variable,μik−ρik≤ωi/ωi+ωk≤1−νik+δik.


## 4.2.1. Construction of Perfect Product Consistency Matrix

In the traditional AHP, the consistency ratio (CR) is used to measure the consistency of the product relation. In this paper, the intuitionistic fuzzy matrix ofR=rikm×m=μik,νik is constructed into a perfect product consistent intuitionistic fuzzy matrix Rp=rikpm×m by using the following algorithm:(1)
Ifk>i+1,rikp=μikp,νikp, where(17)μikp=Πt=i+1k−1μitμtkk−i−1Πt=i+1k−1μitμtkk−i−1+Πt=i+1k−11−μit1−μtkk−i−1,(18)νikp=Πt=i+1k−1νitνtkk−i−1Πt=i+1k−1νitνtkk−i−1+Πt=i+1k−11−νit1−νtkk−i−1.(2)
If(19)k=i+1,rikp=rik.(3)
If(20)k<i+1,rikp=νikp,μikp.


## 4.2.2. Determination and Iterative Modification of Consistency Measure

Generally, the level of evaluation members is basically the same, and the evaluation results obtained should coincide. However, the inconsistency of evaluation results cannot be avoided in real life. The process of group consensus is to find the final solution accepted by the evaluated personnel. The following gives an algorithm to reach consensus.Step 1: calculating the consensus degree of evaluation members,(21)Cl=1−∑1≤i≤j≤mmμijp−μij∗+νijp−νij∗+πijp−πij∗m−1m−2.If the consensus degree reaches the critical valueγ of group consensus, the result of SIFWG will be output; otherwise, turn to step 2.Step 2: revising according to the formula,(22)μijlp′=μijlpξ×μij∗1−ξ,νijlp′=νijlpξ×νij∗1−ξ,whereRlp=riklpm×n=μijlp,νijlpm×n is the perfect product consistent intuitionistic fuzzy matrix of evaluator l.Step 3: calculating the consensus degree.In this paper, we use the new modified perfect product consistent intuitionistic fuzzy matrix to calculate the consensus degree. If it meets the critical value, the new modified perfect product consistent intuitionistic fuzzy matrix is aggregated with SIFWG operator to form a new aggregation matrix. If it does not meet the critical value, turn to Step 2 until it meets the critical value.


## 4.2.3. Aggregation of Evaluation Information and Derivation of Influencing Factors’ Weight

According to abovementioned algorithm, the perfect product consistency intuitionistic fuzzy matrixRp=rikpm×n is obtained. For information aggregation, there are many operators that can be used. This paper uses SIFWG operator, because the matrix before aggregation has consistency, and because the after aggregation through SIFWG operator has the same consistency. Then, the matrix R∗=rij∗m×n=μij∗,νij∗ of SIFWG aggregation is obtained, which is shown in the following:(23)μij∗=∏k=1lμijpλk,νij∗=∏k=1lνijpλk,i=1,…,m,j=1,…,n.According to the accurate priority derivation method of IFNs, the nonlinear programming model is solved by LINGO software, and the weight of influencing factors is obtained directly [25]. Based on the standard operation of LINGO software, we take the matrix as an example to show the nonlinear programming model:(24)minf=∑i=1m−1∑k=i+1mρik+δik,s.tωiωi+ωk+ρik−μik≥0,ωiωi+ωk−δik+νik≥0,∑i=1mωi=1,ωi∈0,1,ρik≥0,δik≥0,ρik∗δik=0.Exportω=ωs,ωo,ωd, where i=1,…,m−1;k=i+1,…,m, ρik, δik is the relaxation variable,μik−ρik≤ωi/ωi+ωk≤1−νik+δik.


## 4.3. Potential Failure Mode Score and Ranking

In this paper, we present the integrate evaluator and influence factor weights with evaluation information, which is shown as(25)FMi=∑k=1lλkμijk,μijkωS+∑k=1lλkμijk,μijkωO+∑k=1lλkμijk,μijkωd,where the evaluation information of k assessor is μijk,μijki=1,…,m,j=1,…,n.Finally, the IFNμFM,νFM of failure mode is obtained. According to the score function formula of IFN, the score of each failure mode is sorted.


## 5. An Example

This chapter is divided into two sections.Section 5.1 shows the calculation process and precautions of each part of the new method through an evaluation example of the attractive electromagnet. Section 5.2 illustrates the objectivity and practicability of this method by comparing with the other three methods.


### 5.1. An Example in Manufacturing Systems

The attractive electromagnet itself has a plunger. When the coil is energized, it will attract the plunger to a certain distance and transmit the movement to other loads requiring mechanical action. The structure is shown in Figure2. They are widely used in electromagnetic relays, solenoid valves, and other products. When the suction electromagnet works, it not only needs to complete the specified movement but also needs to be in the position. The iron reaches the load area, so the consequence of its failure mode is very serious. This paper takes the FMEA of suction electromagnet in the early stage of production as an example to verify the practicability and objectivity of this paper. Since the evaluation information of evaluation members in this paper is in natural language, it is necessary to convert natural language into IFN for calculation. The corresponding table of natural language and IFN is shown as Table 1, and the main FMs are shown in Table 2.Figure 2
Schematic diagram of suction electromagnet structure.Table 1
Table of natural language and IFNs for failure mode assessment.
Natural languageIFNExtremely low (EL)(0.10, 0.90)Very low (VL)(0.10, 0.75)Low (L)(0.25, 0.60)Medium low (ML)(0.40, 0.50)Medium (M)(0.50, 0.50)Medium high (MH)(0.60, 0.30)High (H)(0.70, 0.20)Very high (VH)(0.80, 0.10)Extremely high (EH)(0.90, 0.10)Definitely high (DH)(1.00, 0.00)Table 2
Failure mode analysis table.
No.Failure modeFailure effectFailure reasonF1Poor insulationProduct leakageInsufficient insulation level of materialF2The strength of the spool is not enough, and the product splits after windingMagnetic flux leakage products(i)Low strength of plastic materials(ii)Improper design of spool(iii)Excessive tension designF3Short circuit/open circuit of product after power on and heatingCannot drive the action of customer organization, customer cannot use itInsufficient temperature resistance of materialsF4Insufficient strength of end footAffect customer installationImproper soldering and assembly designF5Poor surface roughnessAffect product lifeImproper design of dimensional toleranceF6Poor concentricityCustomer unable to install or not suitable for installationImproper design of dimensional toleranceF7The magnetic force of the product is insufficient after being electrifiedUnable to drive customer organization action(i)The magnetic permeability of the material is insufficient, and the attraction is insufficient after being electrified(ii)Improper designF8Residual magnetic field after power onUnable to drive customer organization action(i)The magnetic permeability of the material is insufficient, and the attraction is insufficient after being electrified(ii)The thickness of the lower wall is thinF9Appearance corrosion and rustPoor appearance, customer dissatisfiedThe antirust layer is too thin and the process is improperThe company organized three evaluators to evaluate theS, O, and D of the fault in natural language mode according to the evaluation criteria of influencing factors in Tables 3–5.Table 3
S assessment criteria.
InfluenceAssessment criteria: severity of consequences (customer impact)GradeFailure of laws and regulations affecting safety or governmentWork safety is affected or government regulations are not met without any warningDHIn case of warning, it will affect work safety or not comply with government regulationsEHLoss or reduction of expected functionLoss of basic function (electromagnet does not work and does not affect safety)VHExpected function reduction (electromagnet can work, performance level is reduced)HLoss or reduction of secondary functionLoss of secondary function (failure of comfort and convenience function)MHLower secondary function (lower comfort and convenience function level)MCustomer feedbackElectromagnets work, and most customers (>75%) perceive noise and appearanceMLElectromagnets work, and some customers (50%) perceive noise and appearanceLThe electromagnet works, and a small number of customers (<25%) are aware of noise and appearanceVLNo impactNo detectable impactELTable 4
O assessment criteria.
Possibility of failureEvaluation criteria: the possibility of causesGradeVery highNew technology and new design without corresponding historyDHHighInevitable failure due to new design, new application, or change of operating conditionsEHNew design, new application, or change of operating conditions may lead to failureVHUncertain failures due to new design, new application, or change of operating conditionsHMediumFailure of similar design and design testMHOccasional failures in design and testingMIsolated failures in similar designs and design testsMLLowAlmost identical designs and isolated failures in design testingLAlmost the same design and undetected failure in design testingVLVery lowBy preventing controllable failuresELTable 5
D assessment criteria.
Opportunity discoveryEvaluation criteria: possibility of discovery through design controlGradeVery very lowThere is no current design control that cannot be found or analyzedELVery lowDesign analysis has weak discovery abilityVLLowVerification of a product that has passed testing (subsystem or system testing, e.g., noise and shipping) prior to releaseLMedium lowVerification of products using fault testing before product release (testing of subsystem or system failure)MLMediumVerification of products using failure testing before product release (durability testing subsystem or system, such as function check)MMedium highProduct validation by testing before design freezingMHHighProduct confirmation by fault test before design freezingHVery highUse old test to confirm products before design freezingVHVery very highDesign data analysis has strong ability of discoveryEHExtremely highFMs cannot occur because of a fully preventive designDHAlso, the results of the computations are shown in Table6.Table 6
Evaluation table of evaluators in natural language.
Failure modeEvaluatorsE1E2E3SODSODSODF1VHELVLVHLVLVHVLELF2VHLELHLVLHELVLF3MHVLMLVHELVLVHVLVLF4MHLELVHVLMVHLELF5MVLMLMHLELMHLLF6EHELVLEHLMEHELELF7VHLVLMHMLMHLLF8MHMLVHELELHVLVLF9MVLVLMVLVLMVVLVLThen, according to the corresponding table of natural language and IFN in Table1, the evaluation table in natural language is transformed into the form of IFN; calculated results are given in Table 7.Table 7
Evaluation table under IFN.
Failure modeEvaluatorsE1E2E3SODSODSODF10.80.10.10.750.10.750.80.10.250.60.10.750.80.10.10.750.10.9F20.80.10.250.60.10.90.70.20.250.60.10.750.70.20.10.750.10.75F30.60.30.10.750.50.50.80.10.10.90.10.750.80.10.10.750.10.75F40.60.30.250.60.10.90.80.10.10.750.50.50.80.10.250.60.10.9F50.50.50.10.750.40.50.60.30.250.60.10.90.60.30.250.60.250.6F60.90.10.10.90.10.750.90.10.50.50.50.50.90.10.10.90.10.9F70.80.10.250.60.10.750.60.30.250.60.250.60.60.30.250.60.250.6F80.60.30.50.50.10.90.80.10.10.90.10.90.70.20.10.750.10.75F90.50.50.10.750.10.750.50.50.10.750.10.750.50.50.10.90.10.75According to formulas (14)–(20), the weight of evaluation members is determined, λ1=0.316, λ2=0.34, and λ3=0.344.The comparison matrix is obtained by comparing the influence factors of the evaluation members, as shown in Table8.Table 8
Comparison of influencing factors.
Evaluator E1Evaluator E2Evaluator E3OSODOSODOSODSMHMHSMVHMSMVHHOH′MMLOVH′MMOVH′MHDMH′ML′MDM′M′MDH′H′MHere, H′ is the complement of H natural language. For example, the IFN corresponding to H natural language is (0.70, 0.20), and then the IFN corresponding to H´ natural language is (0.20, 0.70).The natural language in Table8 is transformed into IFN according to the corresponding table of natural language and IFN, and then according to algorithm 1, the perfect product consistent intuitionistic fuzzy matrix is constructed, as shown in Table 9.Table 9
Perfect product consistency intuitionistic fuzzy matrix for every evaluator.
Evaluator E1SODS0.50.50.70.20.6670.25O0.20.70.50.50.40.5D0.250.6670.50.40.50.5Evaluator E2SODS0.50.50.80.10.6670.25O0.10.80.50.50.50.5D0.250.6670.50.50.50.5Evaluator E3SODS0.50.50.80.10.6670.25O0.10.80.50.50.70.2D0.250.6670.20.70.50.5According to the SIFWG aggregation operator of formula (23), the weight of all evaluators, λ, has been obtained, λ1=0.316, λ2=0.34, and λ3=0.344, for example, the aggregation result of membership degree of influencing factor S compared with that of influencing factor O=0.70.316∗0.80.34∗0.80.344=0.7669. The aggregation result of nonmembership degree =0.20.316∗0.10.34∗0.10.344=0.1245. Similarly, other polymerization results are shown in Table 10.Table 10
Evaluation information of influencing factors after aggregation.
SODS0.50000.50000.76690.12450.66700.2500O0.12450.76690.50000.50000.52310.3648D0.25000.66700.36480.52310.50000.5000According to formula (21), the consensus degree of each evaluator is C1=0.5722,C2=0.7422,C3=0.8537. If the critical value of consensus degree C is set to 0.5, then all three evaluators meet the critical value, and it is generally considered that consensus can be reached only when the critical value C=0.85. Then, it is modified according to formula (22), where the iterative operation is taken as ξ=0.5. After three iterations, the consensus reached is C1′=0.8682,C2′=0.8528,C3′=0.9544. When consensus is reached, the matrix aggregated by SIFWG operators is shown in Table 11.Table 11
Aggregation matrix after iterative operation.
SODS0.5000.5000.7900.1070.6670.250O0.1070.7900.5000.5000.5750.323D0.2500.6670.3230.5750.5000.500According to algorithm 3, the nonlinear programming model is constructed as follows:(26)minf=ρ12+δ12+ρ13+δ13+ρ23+δ23,(27)s.tω1ω1+ω2+ρ12−0.79≥0,ω1ω1+ω3+ρ13−0.667≥0,ω2ω2+ω3+ρ23−0.575≥0,ω1ω1+ω2−δ12+0.107≥0,ω1ω1+ω3−δ13+0.25≥0,ω2ω2+ω3−δ23+0.323≥0,ω1+ω2+ω3=1,0≤ω1≤1,0≤ω2≤1,0≤ω3≤1,ρ12≥0,ρ13≥0,ρ23≥0,δ12≥0,δ13≥0,δ23≥0,ρ12∗δ12=0,ρ13∗δ13=0,ρ23∗δ23=0.In this paper, we use LINGO software to solve formulas (26) and (27) and then derive the weight of each influencing factor in Figure 3.Figure 3
Influencing factor weight derivation chart.From Figure3, we can see ω1=0.684, ω2=0.182, and ω3=0.134; in turn, the corresponding is ωS=0.684, ωO=0.182, and ωd=0.134.According to formula (25) and the operation rules of IFNs, such as the final score calculation in FM1, the score after the weight of comprehensive influencing factors is calculated first, and the calculation formula is as follows:(28)0.684∗0.9,0.1+0.684∗0.8,0.1+0.684∗0.8,0.1+0.182∗0.1,0.8+0.182∗0.1,0.9+0.182∗0.1,0.8+0.134∗0.1,0.9+0.134∗0.1,0.8+0.134∗0.1,0.9=6.658.Similarly, the final scores for other FMs are calculated and ranked according to the final scores, as shown in Table12.Table 12
Summary of final results.
Failure modeScoreSortF16.6589F26.8128F37.0186F47.2995F57.6642F66.9437F77.4824F87.4973F98.0111


### 5.2. Discussions

This approved method is compared with the other three methods in terms of final scores. The three methods for comparison are traditional FMEA, only calculating the weight of influencing factors, and only calculating the weight of evaluating personnel. Except for the weight, the other steps are the same as the new method. The score table is shown in Figure4.Figure 4
Comparison score table.It is obvious from the two tables that there is a big difference between the traditional FMEA ranking and the score of this new method. This is because the traditional FMEA does not specify the weight of influencing factors and the weight of evaluation members; so, when the score function is used to calculate the score, the value of evaluation content is all involved in the calculation. However, from the above calculation, the result isωS=0.684, ωO=0.182, ωd=0.134,ωE1=0.316,ωE2=0.34,ωE3=0.344. According to the operation rules of IFNs, the rule calculation is carried out twice, and the proportion of each calculation is the weight. On the other hand, the calculated results are quite different from the default weight, especially the weight of influencing factors, so there will be a lot of differences in scores. If we evaluate according to the 10-point system, the final score difference may be more crucial.


## 5.1. An Example in Manufacturing Systems

The attractive electromagnet itself has a plunger. When the coil is energized, it will attract the plunger to a certain distance and transmit the movement to other loads requiring mechanical action. The structure is shown in Figure2. They are widely used in electromagnetic relays, solenoid valves, and other products. When the suction electromagnet works, it not only needs to complete the specified movement but also needs to be in the position. The iron reaches the load area, so the consequence of its failure mode is very serious. This paper takes the FMEA of suction electromagnet in the early stage of production as an example to verify the practicability and objectivity of this paper. Since the evaluation information of evaluation members in this paper is in natural language, it is necessary to convert natural language into IFN for calculation. The corresponding table of natural language and IFN is shown as Table 1, and the main FMs are shown in Table 2.Figure 2
Schematic diagram of suction electromagnet structure.Table 1
Table of natural language and IFNs for failure mode assessment.
Natural languageIFNExtremely low (EL)(0.10, 0.90)Very low (VL)(0.10, 0.75)Low (L)(0.25, 0.60)Medium low (ML)(0.40, 0.50)Medium (M)(0.50, 0.50)Medium high (MH)(0.60, 0.30)High (H)(0.70, 0.20)Very high (VH)(0.80, 0.10)Extremely high (EH)(0.90, 0.10)Definitely high (DH)(1.00, 0.00)Table 2
Failure mode analysis table.
No.Failure modeFailure effectFailure reasonF1Poor insulationProduct leakageInsufficient insulation level of materialF2The strength of the spool is not enough, and the product splits after windingMagnetic flux leakage products(i)Low strength of plastic materials(ii)Improper design of spool(iii)Excessive tension designF3Short circuit/open circuit of product after power on and heatingCannot drive the action of customer organization, customer cannot use itInsufficient temperature resistance of materialsF4Insufficient strength of end footAffect customer installationImproper soldering and assembly designF5Poor surface roughnessAffect product lifeImproper design of dimensional toleranceF6Poor concentricityCustomer unable to install or not suitable for installationImproper design of dimensional toleranceF7The magnetic force of the product is insufficient after being electrifiedUnable to drive customer organization action(i)The magnetic permeability of the material is insufficient, and the attraction is insufficient after being electrified(ii)Improper designF8Residual magnetic field after power onUnable to drive customer organization action(i)The magnetic permeability of the material is insufficient, and the attraction is insufficient after being electrified(ii)The thickness of the lower wall is thinF9Appearance corrosion and rustPoor appearance, customer dissatisfiedThe antirust layer is too thin and the process is improperThe company organized three evaluators to evaluate theS, O, and D of the fault in natural language mode according to the evaluation criteria of influencing factors in Tables 3–5.Table 3
S assessment criteria.
InfluenceAssessment criteria: severity of consequences (customer impact)GradeFailure of laws and regulations affecting safety or governmentWork safety is affected or government regulations are not met without any warningDHIn case of warning, it will affect work safety or not comply with government regulationsEHLoss or reduction of expected functionLoss of basic function (electromagnet does not work and does not affect safety)VHExpected function reduction (electromagnet can work, performance level is reduced)HLoss or reduction of secondary functionLoss of secondary function (failure of comfort and convenience function)MHLower secondary function (lower comfort and convenience function level)MCustomer feedbackElectromagnets work, and most customers (>75%) perceive noise and appearanceMLElectromagnets work, and some customers (50%) perceive noise and appearanceLThe electromagnet works, and a small number of customers (<25%) are aware of noise and appearanceVLNo impactNo detectable impactELTable 4
O assessment criteria.
Possibility of failureEvaluation criteria: the possibility of causesGradeVery highNew technology and new design without corresponding historyDHHighInevitable failure due to new design, new application, or change of operating conditionsEHNew design, new application, or change of operating conditions may lead to failureVHUncertain failures due to new design, new application, or change of operating conditionsHMediumFailure of similar design and design testMHOccasional failures in design and testingMIsolated failures in similar designs and design testsMLLowAlmost identical designs and isolated failures in design testingLAlmost the same design and undetected failure in design testingVLVery lowBy preventing controllable failuresELTable 5
D assessment criteria.
Opportunity discoveryEvaluation criteria: possibility of discovery through design controlGradeVery very lowThere is no current design control that cannot be found or analyzedELVery lowDesign analysis has weak discovery abilityVLLowVerification of a product that has passed testing (subsystem or system testing, e.g., noise and shipping) prior to releaseLMedium lowVerification of products using fault testing before product release (testing of subsystem or system failure)MLMediumVerification of products using failure testing before product release (durability testing subsystem or system, such as function check)MMedium highProduct validation by testing before design freezingMHHighProduct confirmation by fault test before design freezingHVery highUse old test to confirm products before design freezingVHVery very highDesign data analysis has strong ability of discoveryEHExtremely highFMs cannot occur because of a fully preventive designDHAlso, the results of the computations are shown in Table6.Table 6
Evaluation table of evaluators in natural language.
Failure modeEvaluatorsE1E2E3SODSODSODF1VHELVLVHLVLVHVLELF2VHLELHLVLHELVLF3MHVLMLVHELVLVHVLVLF4MHLELVHVLMVHLELF5MVLMLMHLELMHLLF6EHELVLEHLMEHELELF7VHLVLMHMLMHLLF8MHMLVHELELHVLVLF9MVLVLMVLVLMVVLVLThen, according to the corresponding table of natural language and IFN in Table1, the evaluation table in natural language is transformed into the form of IFN; calculated results are given in Table 7.Table 7
Evaluation table under IFN.
Failure modeEvaluatorsE1E2E3SODSODSODF10.80.10.10.750.10.750.80.10.250.60.10.750.80.10.10.750.10.9F20.80.10.250.60.10.90.70.20.250.60.10.750.70.20.10.750.10.75F30.60.30.10.750.50.50.80.10.10.90.10.750.80.10.10.750.10.75F40.60.30.250.60.10.90.80.10.10.750.50.50.80.10.250.60.10.9F50.50.50.10.750.40.50.60.30.250.60.10.90.60.30.250.60.250.6F60.90.10.10.90.10.750.90.10.50.50.50.50.90.10.10.90.10.9F70.80.10.250.60.10.750.60.30.250.60.250.60.60.30.250.60.250.6F80.60.30.50.50.10.90.80.10.10.90.10.90.70.20.10.750.10.75F90.50.50.10.750.10.750.50.50.10.750.10.750.50.50.10.90.10.75According to formulas (14)–(20), the weight of evaluation members is determined, λ1=0.316, λ2=0.34, and λ3=0.344.The comparison matrix is obtained by comparing the influence factors of the evaluation members, as shown in Table8.Table 8
Comparison of influencing factors.
Evaluator E1Evaluator E2Evaluator E3OSODOSODOSODSMHMHSMVHMSMVHHOH′MMLOVH′MMOVH′MHDMH′ML′MDM′M′MDH′H′MHere, H′ is the complement of H natural language. For example, the IFN corresponding to H natural language is (0.70, 0.20), and then the IFN corresponding to H´ natural language is (0.20, 0.70).The natural language in Table8 is transformed into IFN according to the corresponding table of natural language and IFN, and then according to algorithm 1, the perfect product consistent intuitionistic fuzzy matrix is constructed, as shown in Table 9.Table 9
Perfect product consistency intuitionistic fuzzy matrix for every evaluator.
Evaluator E1SODS0.50.50.70.20.6670.25O0.20.70.50.50.40.5D0.250.6670.50.40.50.5Evaluator E2SODS0.50.50.80.10.6670.25O0.10.80.50.50.50.5D0.250.6670.50.50.50.5Evaluator E3SODS0.50.50.80.10.6670.25O0.10.80.50.50.70.2D0.250.6670.20.70.50.5According to the SIFWG aggregation operator of formula (23), the weight of all evaluators, λ, has been obtained, λ1=0.316, λ2=0.34, and λ3=0.344, for example, the aggregation result of membership degree of influencing factor S compared with that of influencing factor O=0.70.316∗0.80.34∗0.80.344=0.7669. The aggregation result of nonmembership degree =0.20.316∗0.10.34∗0.10.344=0.1245. Similarly, other polymerization results are shown in Table 10.Table 10
Evaluation information of influencing factors after aggregation.
SODS0.50000.50000.76690.12450.66700.2500O0.12450.76690.50000.50000.52310.3648D0.25000.66700.36480.52310.50000.5000According to formula (21), the consensus degree of each evaluator is C1=0.5722,C2=0.7422,C3=0.8537. If the critical value of consensus degree C is set to 0.5, then all three evaluators meet the critical value, and it is generally considered that consensus can be reached only when the critical value C=0.85. Then, it is modified according to formula (22), where the iterative operation is taken as ξ=0.5. After three iterations, the consensus reached is C1′=0.8682,C2′=0.8528,C3′=0.9544. When consensus is reached, the matrix aggregated by SIFWG operators is shown in Table 11.Table 11
Aggregation matrix after iterative operation.
SODS0.5000.5000.7900.1070.6670.250O0.1070.7900.5000.5000.5750.323D0.2500.6670.3230.5750.5000.500According to algorithm 3, the nonlinear programming model is constructed as follows:(26)minf=ρ12+δ12+ρ13+δ13+ρ23+δ23,(27)s.tω1ω1+ω2+ρ12−0.79≥0,ω1ω1+ω3+ρ13−0.667≥0,ω2ω2+ω3+ρ23−0.575≥0,ω1ω1+ω2−δ12+0.107≥0,ω1ω1+ω3−δ13+0.25≥0,ω2ω2+ω3−δ23+0.323≥0,ω1+ω2+ω3=1,0≤ω1≤1,0≤ω2≤1,0≤ω3≤1,ρ12≥0,ρ13≥0,ρ23≥0,δ12≥0,δ13≥0,δ23≥0,ρ12∗δ12=0,ρ13∗δ13=0,ρ23∗δ23=0.In this paper, we use LINGO software to solve formulas (26) and (27) and then derive the weight of each influencing factor in Figure 3.Figure 3
Influencing factor weight derivation chart.From Figure3, we can see ω1=0.684, ω2=0.182, and ω3=0.134; in turn, the corresponding is ωS=0.684, ωO=0.182, and ωd=0.134.According to formula (25) and the operation rules of IFNs, such as the final score calculation in FM1, the score after the weight of comprehensive influencing factors is calculated first, and the calculation formula is as follows:(28)0.684∗0.9,0.1+0.684∗0.8,0.1+0.684∗0.8,0.1+0.182∗0.1,0.8+0.182∗0.1,0.9+0.182∗0.1,0.8+0.134∗0.1,0.9+0.134∗0.1,0.8+0.134∗0.1,0.9=6.658.Similarly, the final scores for other FMs are calculated and ranked according to the final scores, as shown in Table12.Table 12
Summary of final results.
Failure modeScoreSortF16.6589F26.8128F37.0186F47.2995F57.6642F66.9437F77.4824F87.4973F98.0111


## 5.2. Discussions

This approved method is compared with the other three methods in terms of final scores. The three methods for comparison are traditional FMEA, only calculating the weight of influencing factors, and only calculating the weight of evaluating personnel. Except for the weight, the other steps are the same as the new method. The score table is shown in Figure4.Figure 4
Comparison score table.It is obvious from the two tables that there is a big difference between the traditional FMEA ranking and the score of this new method. This is because the traditional FMEA does not specify the weight of influencing factors and the weight of evaluation members; so, when the score function is used to calculate the score, the value of evaluation content is all involved in the calculation. However, from the above calculation, the result isωS=0.684, ωO=0.182, ωd=0.134,ωE1=0.316,ωE2=0.34,ωE3=0.344. According to the operation rules of IFNs, the rule calculation is carried out twice, and the proportion of each calculation is the weight. On the other hand, the calculated results are quite different from the default weight, especially the weight of influencing factors, so there will be a lot of differences in scores. If we evaluate according to the 10-point system, the final score difference may be more crucial.


## 6. Conclusions

To avoid the disadvantages of traditional FMEA, this paper presents IFN to reflect the final ranking of evaluation team for electromagnet in manufacturing systems. The failure evaluation mode is a general and comprehensive decision of the consequences and effects of failure. Based on the distinguishing features of subordination degree and nonsubordination degree of IFN, the error of final calculation failure mode score can be reduced to a more manageable level. In addition, this paper supplements the incomplete determination of weight information with traditional FMEA and calculates the best evaluation information by using the concept of similarity measure.On the other hand, in view of the unknown weight of influencing factors, this paper compares and evaluates influencing factors by many evaluators forming a contrast matrix. A nonlinear programming model based on the contrast matrix of influencing factors was built to simulate the evaluation results of the weight of influencing factors. Then, the absolute distance between each evaluator’s decision information and the best decision information are calculated in two different ways. The contributions of this paper lie in that we put forward a potential FMEA method of electromagnet based on IFN in empirical study. This paper also discusses the further improving measures of IFN and presents future research directions for other mechanical and electrical products.



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*Source: 9998526-2021-08-23.xml*