1. INTRODUCTION

In current society of production activities, risk is an unavoidable problem, nothing can completely eliminate the risk in the reality environment. Although it is inevitable, it can be achieved within the acceptable range by reducing the probability of its occurrence. The purpose of risk assessment is to prevent accidental failures through appropriate techniques, the most famous of which is the failure mode and effect analysis (FMEA) method [1]. FMEA was proposed by the NASA in the 1960s at the first time and applied in the aerospace industry to improve the reliability of military products [2]. Subsequently, FMEA has been widely applied in manufacture [3], agriculture [4], aerospace [5] and geothermal power plant [6]. The most important step of the conventional FMEA method is to rank the failure mode according to the value of the risk priority number (RPN) [7], where RPN is the product of the three risk factors: occurrence (O), severity (S) and detection (D). Although the conventional RPN method to evaluate and prioritize for failure modes is feasible, it still suffers from many drawbacks. Firstly, it is difficult to describe information with clear numbers in the actual environment. Secondly, the weight of each risk factor is considered equally important. Thirdly, the different combinations of O, S and D may produce exactly the same RPN value, but their hidden risk implications may be entirely different, which may lead to an incorrect ranking.

To overcome these limitations associated with the traditional RPN method, Zadeh (1965) [8] firstly proposed the theory of fuzzy sets, which was an important tool to narrate fuzzy information. Then, Atanassov (1986) [9] introduced the concept of intuitionistic fuzzy sets (IFSs) by increasing non-membership. Yager [10] proposed Pythagorean fuzzy sets(PFS) which it enlarged the range of the sum of membership degree and nonmembership degree. However, decision makers (DM) could be hesitant when describing the membership degree of an object. Torra [11] developed hesitant fuzzy sets (HFSs) to fully display DM’s hesitant. The aforementioned sets can only handle incomplete information but not the indeterminate information and inconsistent information which exist commonly in real situations. The emergence of the neutrosophic sets (NSs) just make up for the deficiency. Smarandache [12] proposed NSs, which are an extension of IFSs. Wang et al. [13] presented the notion of single-valued neutrosophic sets (SVNSs), which can use the truth-membership function, the indeterminacy-membership function and the falsity-membership function by three real numbers in the interval [0,1] to express the ambiguity of information, and these are independent of each other. Wang et al. [14] further extended SVNS to interval numbers, and proposed interval neutrosophic sets (INSs). Recently, interval-valued neutrosophic set (IVNS) has obtained extensive attention and was applied to the fields of multi-attribute decision-making (MADM). Broumi and Smarandache [15] investigated the correlation coefficient for INS. Bausys et al.[16] presented an VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) method in IVNS environment to address MADM problem. Garg et al.[17] used a nonlinear programming (NP) model based on the technique for order preference by similarity to ideal solution (TOPSIS) to solve decision-making problems in which information were given in the form of IVNNs. Ye et al. [18] was aim to study the distance and similarity of IVNS and proposed some entropy measures of IVNS based on the distances. Zhou et al. [19] investigated some Frank aggregation operators of IVNNs and applied to MADM. Moreover, some applications are focused on the aggregation operators [20–22] and information measures [23,24]. IVNS is relatively less compared with the IFSs ranges of application and research. Therefore, IVNSs are the more perfect and meaningful means to represent uncertainty information than FSs, IFSs and SVNSs.

In addition, a lot of multi-criteria decision-making (MCDM) methods have been applied in FMEA. In [25], ELECTRE method was employed to assign failure modes to predefined and ordered risk classes. Liu et al. [26] adopted the MULTIMOORA approach to analyze the rank of FMEA under fuzzy environment. Moreover, including analytic hierarchy process (AHP) [27], EDAS [28], and so on. Unfortunately, these are approaches based on completely rationalization. It is common phenomenon that a lot of experts have different attitudes toward uncertain and unknown information in reality environment. Therefore, considering the risk attitudes are significant for determining risk priority of FMEA. The TODIM method, proposed by Gomes and Lima [29], was an effective tool based on prospect theory for capturing psychological behavior of experts. The TODIM method has been widely used in aspects of MADM and risk assessment. Gomes et al. [30] presented an evaluation study for residential properties using TODIM method. Qin et al. [31] used triangular intuitionistic fuzzy numbers (TIFNs) and extended TODIM method to select renewable energy alternatives. A novel TODIM method, considering reference dependence and loss aversion at the same time, was proposed by Jiang et al. [32] to address interval MADM problems. Zhu et al. [33] established a comprehensive FMEA model based on the Bonferroni mean and TODIM method. Wang et al. [34] established a projection-based TODIM method with multi-valued neutrosophic sets (MVNSs) for personnel selection. Wang et al. [35] integrated Choquet integral and an extended generalized TODIM for risk evaluation and prioritization of failure modes. Yuan et al. [36] applied a combined ANP-Entropy method to determine weights of risk factors, and used the TODIM method to rank the overall risk level. Due to group emergency decision-making (GEDM) plays an important role in dealing with urgent situations, Wang et al. [37] used experts’ psychological behavior to GEDM process. Further, in the existing fuzzy TODIM method, converting fuzzy information to crisp value may lead to significant loss of information, Wang et al. [38] proposed a novel fuzzy TODIM method based on alpha level sets to solve it. Therefore, it is beneficial to utilize the superiority of TODIM method in representing the rationality of experts.

According to above literature review, we find that acquiring the weight of each risk factor is one of the key factors in FMEA, and few researches have been conducted to extend the TODIM method within the context of IVNSs for FMEA. Therefore, this paper proposes the following solutions: First, IVNNs are used to characterize the fuzzy and uncertainty evaluation information in FMEA. Secondly, a new similarity method is developed based on cross-entropy in IVNS environment. Then, we form a systematic thought to determine the final weights by combining similarity degree and entropy measures. Finally, the improved score function is applied in TODIM method, which reduce the fuzzy information in the process of decision, making the priority of failure modes are more reasonable and effective. Besides, a case study on steel company is used to demonstrate the application and effectiveness of the proposed method.

The rest of the paper is organized as follows: Section 2 reviews some basic concepts regarding IVNS and relevant decision methods, Section 3 proposes the improved FMEA model. Section 4 introduces the FMEA model into an illustrative example of steel company. Comparative analysis is presented in Section 5. Conclusion is finally drawn in Section 6.

2. PRELIMINARIES

3. THE PROPOSED FMEA MODEL

Assume that there are l cross-functional team member TMkk=1,2,⋯,l in a risk assessment group responsible for the risk evaluation of m failure modes FMii=1,2,⋯,m in terms of n risk factors RFjj=1,2,⋯,n. Because FMEA team members frequently come from different fields and professional experience, each team member should be given different weighting factor τk,k=1,2,⋯,lwhere∑k=1lτk=1 to reflect their relative importance. Figure 1 shows that the specific process of the proposed FMEA model.

Figure 1.

The flowchart of the risk priorization method based on extended TODIM.

• Step 1: Identify potential failure modes.

  The potential failure modes in a risk analysis system are determined according to different team members commented.

• Step 2: Evaluate failure modes using linguistic terms expressed in IVNNs.

  In the paper, we suppose that all the risk evaluation information about failure modes by using the linguistic terms shown in Table 1 (adopted from [28]), the risk scores can be converted into their corresponding IVNNs. the linguistic rating of the ith failure mode with respect to jth risk factor provided by the team member TMk can be denoted as rijk=TijkL,TijkU,IijkL,IijkU,FijkL,FijkU. Then, an interval-valued neutrosophic fuzzy assessment matrix is constructed as Rk=rijkm×nk=1,2,⋯,l for each team member of the FMEA.

• Step 3: Aggregate the FMEA team members’ assessment matrix R.

  After obtaining the risk assessment information and relative weights of the FMEA team members, to aggregate all individual risk assessment matrices Rkk=1,2,⋯,l into the collective risk assessment matrix R=rijm×n by the INWA operator as follows:

  Rij=INWArij1,rij2,⋯,rijn=∑k=1lτkrijk13
  where rij=TijL,TijU,IijL,IijU,FijL,FijUi=1,2,⋯,m;j=1,2,⋯,n and TijL=1−∏k=1l1−TijLτk, TijU=1−∏k=1l1−TijUτk, IijL=∏k=1lIijLτk,IijU=∏k=1lIijUτk, FijL=∏k=1lFijLτk, FijU=∏k=1lFijUτki=1,2,⋯,m;j=1,2,⋯,n

• Step 4: Normalized risk assessment decision matrix.

  Normalize decision matrix Rk=rijkm×n into Rk=r~ijkm×n=T~ijkL,T~ijkU,I~ijkL,I~ijkU,F~ijkL,F~ijkUm×n by Eq. (14):

  r~ijk=TijkL,TijkU,IijkL,IijkU,FijkL,FijkUCjisbenefittypeFijkL,FijkU,1−IijkU,1−IijkL,TijkL,TijkUCjiscosttype(14)

• Step 5: Determine the weights of risk factors in similarity degree based on cross-entropy.

  1. Determine absolute positive ideal solution.

     Let R˜+=χ1,χ2,⋯,χni=1,2,⋯,n is called an interval-valued neutrosophic absolute positive ideal solution (INAPIS), where χi=1,1,0,0,0,0 is the nth largest IVNN.

  2. Compute cross-entropy of r˜ij from INSr˜ij,χi by Eq. (8).

  3. Compute the similarity degree of each factor.

     Usually, Sr˜ij,χi is used to represent the similarity between r˜ij and χi. Therefore, the similarity degree Sr˜ij,χi is calculated as

     Sr˜ij,χi=1−INSr˜ij,χi∑j=1nINSr˜ij,χi15

     The similarity between R+=χ1,χ2,⋯,χn and R=r~1j,r~2j,⋯,r~mj is calculated as follows:

     Sj=∑i=1mSr˜ij,χi16

  4. Determine weights of risk factors.

     It is obvious that there is a greater degree of similarity with the interval absolute positive ideal solution, the attribute should have a bigger weight. Hence, the weight of the jthj=1,2,⋯,n risk factor can be obtained as follows:

     ωj=Sj∑j=1nSj17

• Step 6: Determine the entropy value of risk factors.

  Shannon entropy [43] is a measure of information uncertainty. It is well suitable for measuring the relative contrast intensities of criteria to represent the importance.

  1. Compute the improved score function S'r˜ij of r˜ij by Eq. (6).

  2. Normalize the score function S'r˜ij by Pij=S'r˜ij/∑i=1mS'r˜ij

  3. Calculate the entropy measure of risk factors

     ej=−1lnm∑i=1mPijlnPij(18)
     ϖj=1−ejn−∑j=1nej(19)

• Step 7: Calculate the comprehensive weight of each risk factor.

  wj=λωj+(1−λ)ϖjj=1,2,⋯,n(20)
  where λ is a parameter, 0 < λ < 1 (in this paper, we set λ = 0.5).

• Step 8: Calculate the relative weight of each risk factor.

  wjt=wjwt(21)
  where wt=maxwj | j=1,2,⋯,n.

• Step 9: Calculate the dominance of alternative Ri to Rt.

  ϑRi,Rt=∑j=1nϕjRi,Rt,∀i,t(22)

  Among them,

  ϕjRi,Rt=wjtdwhwjr~ij,wjr~tj/∑j=1nwjt0-1θ∑j=1nwjtdwhwjr~ij,wjr~tj/wjtifS'r~ij−S'r~tj>0ifS'r~ij−S'r~tj=0ifS'r~ij−S'r~tj<0(23)
  where dwhwjr˜ij,wjr˜tj is the weighted distance of the IVNNs r˜ij and r˜tj, S'r˜ij and S'r˜tj are improved scoring functions of r˜ij and r˜tj.

• Step 10: Compute the overall dominance value of ϑRi as follows:

  ϑ(Ri)=∑t=1mϑ(Ri,Rt)−mini{∑t=1mϑ(Ri,Rt)}maxi{∑t=1mϑ(Ri,Rt)}−mini{∑t=1mϑ(Ri,Rt)}(24)

• Step 11: Determine the risk priority order of failure modes.

  For FMEA, the greater the value of ϑRi, the higher the risk priority of the failure modes will be. As the result, all the failure modes that have been listed in FMEA can be ranked according to the ascending order of the overall dominance value.

4. AN ILLUSTRATIVE EXAMPLE

In this section, we introduced a previous case to demonstrate the feasibility of the proposed approach for risk evaluation in FMEA. A case about steel production process risk management has ten options of sheet steel production process(adopted from [44]). The failure modes of this case were evaluated by Deshpande and Modak in a factory in the past which had three team members TMk(k=1,2,3) have given the risk evaluation. The criteria are related to their occurrence probability, severity of the associated effects and detection to each failure mode as shown in Figure 2.

Figure 2.

Hierarchical structure of the risk analysis problem.

• Step 1: Identify all potential failure modes.

  There are ten failure modes have been identified according to Figure 2. As shown in Table 2.

• Step 2: Evaluate failure modes using linguistic terms expressed in IVNNs.

  The evaluate information for each failure mode in terms of the risk factors provided by FMEA team members are shown in Table 3 [45]. Therefore, it is simple to transform linguistic information into IVNNs according to Table 1 which are shown in Appendix A Table 13–15.

• Step 3: Aggregate each team member’s evaluation information into a group evaluation matrix.

  Due to these team members have different background knowledge and experience, the relative importance weights of the three FMEA team members are known as τk=0.35,0.4,0.25 according to literature [45]. Therefore, the aggregated the risk assessment information for each team member can be obtained according to Eq. (13) and the result is shown in Table 4.

• Step 4: Three risk factors are benefit criterion and do not need to be normalized. Thus, the normalized operation should not be conducted.

• Step 5: Determine the similarity degree based on the cross-entropy of risk factors.

  1. Because INAPIS is FMi+=χ1,χ2,⋯,χni=1,2,⋯,n, where χi=1,1,0,0,0,0.

  2. Based on Eq. (8), we can calculate the cross-entropy. The results are shown in Table 5.

  3. Combining the data in Table 5 with Eqs. (15) and (16), we can calculate the similarity of failure modes under each risk factor, and the result is denoted as Sj=(5.6795, 7.357, 6.9662).

  4. Finally, according to Eq. (17), the weight of each risk factor is ω1=0.28,ω2=0.37,ω3=0.35.

• Step 6: Determine the entropy value of risk factors.

  Based on Eqs. (4), (18) and (19), the weight of each risk factor can be derived as ϖ1=0.48,ϖ2=0.09,ϖ3=0.43.

• Step 7: Calculate the final weight of each risk factor.

  Combined with above weights, we can get the final weight of each risk factor according to Eq. (20) (suppose λ = 0.5), The results are w1=0.403,w2=0.21,w3=0.387.

• Step 8: Calculate the relative weight of each risk factor.

  Since wmax=maxwj | j=1,2,3=w1=0.403, the reference weight is wt=0.403. So, we can calculate the relative weight wjt of each risk factor wj to reference weight wt based on Eq. (21), The result is as follows: w1t=1,w2t=0.521,w3t=0.96.

• Step 9: Calculate the dominance of each failure mode FMii=1,2,⋯,10 over each failure mode FMtt=1,2,⋯,10 with respect to RFjj=1,2,3 via Eqs. (3), (4) and (23) by taking θ=2.5. The results are listed in Tables 6–8.

• Step 10: Based on Eq. (22), the overall dominance degree matrix is shown in Table 9.

• Step 11: Derive the risk priorization.

  According to Eq. (24), we can compute the overall dominance value of each failure modeϑFMii=1,2,…,10, the final risk priority ranking of overall failure modes based on numerical result is shown in Table 10.

5. COMPARATIVE ANALYSIS AND DISCUSSION