An integrated FMEA approach using Best-Worst and MARCOS methods based on D numbers for prioritization of failures

Abstract

Failure modes and effects analysis (FMEA) is a useful reliability analysis technique to identify potential failure modes in a wide range of industries. However, the conventional FMEA method is deficient in dealing with the risk evaluation and prioritization method. To overcome the shortcomings, this paper presents a new risk priority model using Best-Worst Method based on D numbers (D-BWM) and the Measurement of Alternatives and Ranking according to COmpromise Solution based on D numbers (D-MARCOS). First, D numbers are used to deal with the uncertainty of FMEA team members’ subjective judgment. Second, the distance-based method is proposed to determine the objective weight of each team member. Then, the D-BWM was used to determine the weight of risk factors. The combination rule of D number theory combined the evaluation information of multiple members into group opinions. Finally, D-MARCOS method is proposed to obtain the risk priority of the failure modes. An example and the results of comparative analysis show the method is effective.

Keywords

Failure modes and effects analysis D numbers BWM MARCOS

1 Introduction

Failure mode and effect analysis (FMEA) [1], as a risk analysis tool, is used to identify potential failures to improve its security and reliability [2, 3]. FMEA is a preventative tool rather than finding a solution after a failure [4, 5]. Therefore, it can help risk managers take measures to reduce the possibility of failure and avoid dangerous accidents [6]. The aerospace industry first applied FMEA to assess safety and reliability. Today, the tool is also widely used in the electronics [7], car industry [8], manufacturing [9], and construction industry [10] to identify, prioritize, and mitigate potential failure patterns in systems.

Traditionally, the risk ranking of the failure modes is achieved using the risk priority number (RPN) [11], which is derived by multiplying the three risk factors: occurrence (O), severity (S), and detection (D). The traditional RPN method is proved to be a simple and effective method in the practical application of FMEA, but it still has many shortcomings [12, 13].

Considering the uncertainty and fuzziness of expert cognition, scholars extend the traditional FMEA to the uncertain environment [14, 15]. Bowles et al. [16] used fuzzy logic to describe the fuzziness in failure mode assessment information. Chang et al. [17] applied the fuzzy theory to the FMEA to express the team members’ verbal assessment of potential failure modes. Liu et al. [12] represented the uncertainty evaluation by the team members as a linguistic term by the intuitionistic fuzzy numbers (IFNs), and used the intuitionistic fuzzy hybrid weighted Euclidean distance (IFHWED) operator to sort the failure modes. Song et al. [18] used rough set theory to develop a risk assessment method to solve the ambiguity and uncertainty of the model. Deng et al. [19] converted the fuzzy evaluation of failure modes with different risk factors into D numbers and used the multi-sensor information fusion method to sort the failure modes. Guerrero et al. [20] researched that FMEA practice the quality of individual versus group evaluation in failure modes ordering. The results showed that groups surpass individuals and that combinative groups perform as well as group consensus. In order to overcome the shortcoming of the traditional FMEA method, which only allows individual experts to conduct evaluation, scholars had made many improvements. Faghih-Roohi et al. [21] proposed TOPSIS-based FMEA approach in the intuitionistic fuzzy environment. Chin et al. [22] presented an FMEA using the evidential reasoning (ER) approach. Yang et al. [23] applied evidence theory to obtain different evaluation information from multiple experts, which may be inconsistent, inaccurate and uncertain. But because of the complexity of Dempster-Shafer (D-S) evidence theory [24], it has been criticized in its application. D-S theory [25] only applies to the hypothesis set of exclusivity and exhaustiveness. Li et al. [26] used fuzzy belief structure to express experts’ opinions. Through grey relational projection (GRP) method, three different standards are combined with weights respectively to solve evaluation shortcomings. This method overcomes the limitation of traditional FMEA without considering risk factor weight.

Many academics have improved the FMEA method by improving decision making techniques. Liu et al. [27] used fuzzy analytic hierarchy process (FAHP) and entropy weight method to determine the weight of risk factors, and used the fuzzy VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method to rank the failure modes. Wang et al. [28] proposed a FMEA method using extended matter-element model and analytic hierarchy process (AHP). The scores of failure modes are obtained according to the conventional RPN method, and the weights of risk factors are derived from AHP. Ghoushchi et al. [29] combined the fuzzy BWM method and Multi-Objective Optimization by Ratio Analysis based on the Z-number theory (Z-MOORA) to obtain the priority of risk factors. Nie et al. [30] combined Bayesian fuzzy evaluation numbers with the extended gray relational analysis technique for order preference by similarity to ideal solution (GRA-TOPSIS) method to obtain the ranking failure modes. Bian et al. [31] proposed a FMEA risk assessment method based on D numbers and technique for the order of preference by similarity to ideal solution (D-TOPSIS). Liu et al. [32] combined D numbers and GRP to determine the risk priority order of identified failure modes.

BWM [33, 34] was proposed by Rezaei in 2015. The traditional BWM used crisp values to conduct the comparisons, human subjective judgment is usually characterized by uncertainty and ambiguity, it fails to determine weights under uncertain environment. Thus, fuzzy BWM [35, 36] had been developed. Guo et al. [37] proposed fuzzy BWM to solve the problem of lack of complete information in decision data and ambiguity caused by decision makers’ qualitative judgment. Tian et al. [38] used the fuzzy BWM method to solve the failure mode and effect analysis to obtain the weight of risk factors. Mousavi-Nasab et al. [39] extended the BWM method to D numbers to obtain more reliable weights and higher consistency.

Stević et al. [40] proposed MARCOS method for a sustainable supplier selection in the healthcare industry. Stankovic et al. [41] proposed fuzzy MARCOS method and verified the stability of the results obtained by the proposed method. Moreover, a new Measurement of Alternatives and Ranking according to COmpromise Solution extended to D numbers (D-MARCOS) was proposed in this paper to evaluate failure modes and obtain the ranking of failure modes. The D number is extended to the MARCOS method, which can express the uncertainty of expert evaluation and obtain more accurate ranking.

This study aims to provide a new scoring method to compensate for the deficiency of the traditional FMEA method. This method is presented on the basis of the extended FMEA approach based on D-BWM and D-MARCOS. The main reasons for establishing the FMEA model are as follows. BWM can be viewed as an enhanced version of the traditional AHP method. Statistical results showed that BWM can require fewer comparison data and the results get more consistent. Moreover, D numbers theory, as an extension of D-S evidence theory, can better express various types of uncertainty, including complete information and incomplete information. Therefore, D-BWM, compared with the traditional BWM method, considers the uncertainty and fuzziness of decision making. D-MARCOS method, compared with the D-TOPSIS method, the ratio and the reference point are combined to obtain the comprehensive decision scheme, and the results obtained are more reliable.

The remainder of this paper is organized as follows. In Section 2, the basic concept of traditional FMEA and the D numbers are introduced. In Section 3, we introduce the main frame of proposed method and main steps to construct FMEA model based on D-BWM and D-MARCOS. In Section 4, an example is devoted to illustrate the proposed model. Furthermore, some comparisons and analysis with other methods are given to confirm the effectiveness of the proposed approach. Finally, some conclusions are provided in Section 5.

2 Preliminaries

2.1 Failure mode and effects analysis

FMEA is a crucial tool for realization of potential failure modes and their effects in order to increase the reliability and safety of a simple product or even complex system. In the traditional FMEA method [42], the prioritization of failure modes is determined by calculation the RPN, which is defined as follows: $RPN = S \times O \times D$ (1)

The first factor is the severity(S) of the failure representing the effect or hazard of failure occurrence. The second factor is occurrence (O) that is representation of how often the failure occurs. The third factor is the detection (D) which is represented by available methods and techniques that may detect the occurrence of a failure before the effect of this failure appears. A 1-10 qualitative scale is adopted for scoring the risk factors of each failure mode as shown in Table 1. The failure modes with larger RPN values would lead to greater risks to the system and thus should be given higher priorities for preventive measures.

Table 1

Suggested scale of failure mode

Rating	Probability of failure (O)	Effect (S)	Detection (D)
10	Extremely high: failure almost inevitable	Hazardous without warning	Absolute uncertainty
9	Very high	Hazardous with warning	Very remote
8	Repeated failures	Extreme	Remote
7	High	Major	Very low
6	Moderately high	Significant	Low
5	Moderate	Moderate	Moderate
4	Relatively low	Low	Moderate high
3	Low	Minor	High
2	Remote	Very minor	Very high
1	Nearly impossible	None	Almost certain

The traditional FMEA involves only one expert who does the following. First, give ratings to O, S, and D for the potential failure modes. Then, calculate RPN for each failure mode based on Equation (1). Finally, prioritize failure modes based on their RPNs.

Therefore, the main shortcomings of conventional FMEA are summarized as follows:

The traditional FMEA method only allows the expert to evaluate in integer from 1 to 10;

Weights of three risk factors occurrence, severity and detection are equal;

The traditional FMEA method can only be evaluated by a single expert or team, and does not provide a combination of multiple expert evaluations;

There is controversy over the mathematical formula for calculating RPN, which has no scientific basis.

2.2 D numbers

The D number theory [43 –46] is the promotion and improvement of the Dempster-Shafer (D-S) evidence theory [47 –49]. D-S evidence theory has limitations in its operation [50 –52]. First, any two elements in the D-S evidence theory identification framework must be completely independent, that is, mutually exclusive. In linguistic evaluation elements such as “very good", “good", “average", “bad” and “very bad", they are not completely independent and mutually exclusive [53], and there is inevitably overlap. The second is the sum of the basic probability assignment (BPA) must be 1, that is, the constraint of integrity must be satisfied. Thirdly, it is high that the time complexity of the combination rule calculation in evidence theory. When the evidence source increases linearly, the complexity of combinatorial rule algorithm increases exponentially. To overcome these deficiencies, Deng [46] proposed a new representation of uncertainty, called D numbers.

Definition 1 [46]. Suppose a finite nonempty set Ω, the D numbers is a mapping formulated by D: Ω → [0, 1], which is as follows: $\sum_{A \subseteq Ω} D (A) ⩽ 1, D (Φ) = 0$ (2) where A and Φ are an element of 2^Ω and an empty set, respectively. Clearly, the BPA can represent the “uncertainty” directly, the element in set Ω do not require mutually exclusive in D numbers and the completeness constraint is released as well. It is different from the concept of discernment frame in D-S evidence theory.

Definition 2 [46]. For a discrete set Ω = {b₁, b₂, . . . , b_i, . . . , b_n}, where b_i ∈ R. A special form of D numbers can be represented by: $\begin{matrix} D (b_{1}) = v_{1} \\ D (b_{2}) = v_{2} \\ \dots \dots \\ D (b_{i}) = v_{i} \\ \dots \dots \\ D (b_{n}) = v_{n} \end{matrix}$ (3) or simply writing as D = {(b₁, v₁) , (b₂, v₂) , . . . , (b_i, v_i) , . . . , (b_n, v_n)}, where v_i > 0 and $\sum_{i = 1}^{n} v_{i} ⩽ 1$ .

Definition 3 [46]. (the combination rule for two D numbers) If D₁ and D₂ are two D numbers, indicated by $D_{1} = {(b_{1}^{1}, v_{1}^{1}), (b_{2}^{1}, v_{2}^{1}), . . ., (b_{i}^{1}, v_{i}^{1}), . . ., (b_{n}^{1}, v_{n}^{1})}$ , $D_{2} = {(b_{1}^{2}, v_{1}^{2}), (b_{2}^{2}, v_{2}^{2}), . . ., (b_{i}^{2}, v_{i}^{2}), . . ., (b_{n}^{2}, v_{n}^{2})}$ , and the combination of D₁ and D₂ is expressed as D = D₁ ⊕ D₂, it may be determined by: $D (b) = v,$ (4)

with $b = \frac{b_{i}^{1} + b_{j}^{2}}{2},$ (5) $v = \frac{v_{i}^{1} + v_{j}^{2}}{2 C}$ (6) $C = {\begin{matrix} \sum_{j = 1}^{m} \sum_{i = 1}^{n} (\frac{v_{i}^{1} + v_{j}^{2}}{2}), \sum_{i = 1}^{n} v_{i}^{1} = 1 and \sum_{j = 1}^{m} v_{j}^{2} = 1; \\ \sum_{j = 1}^{m} \sum_{i = 1}^{n} (\frac{v_{i}^{1} + v_{j}^{2}}{2}) + \sum_{j = 1}^{m} (\frac{v_{c}^{1} + v_{j}^{2}}{2}), \sum_{i = 1}^{n} v_{i}^{1} 1 and \sum_{j = 1}^{m} v_{j}^{2} = 1; \\ \sum_{j = 1}^{m} \sum_{i = 1}^{n} (\frac{v_{i}^{1} + v_{j}^{2}}{2}) + \sum_{i = 1}^{n} (\frac{v_{i}^{1} + v_{c}^{2}}{2}), \sum_{i = 1}^{n} v_{i}^{1} = 1 and \sum_{j = 1}^{m} v_{j}^{2} < 1; \\ \sum_{j = 1}^{m} \sum_{i = 1}^{n} (\frac{v_{i}^{1} + v_{j}^{2}}{2}) + \sum_{j = 1}^{m} (\frac{v_{c}^{1} + v_{j}^{2}}{2}) \\ + \sum_{i = 1}^{n} (\frac{v_{i}^{1} + v_{c}^{2}}{2}) + \frac{v_{c}^{1} + v_{c}^{2}}{2}, \sum_{i = 1}^{n} v_{i}^{1} < 1 and \sum_{j = 1}^{m} v_{j}^{2} < 1; \end{matrix}$ (7) where $v_{c}^{1} = 1 - \sum_{i = 1}^{n} v_{i}^{1}$ and $v_{c}^{2} = 1 - \sum_{j = 1}^{m} v_{j}^{2}$ .

Definition 4 [31]. (Multiple D numbers’ rule of combination) Let D₁, D₂, . . . , D_n be n D numbers, where u_j is an order variable for each D_j, expressed as (u_j, D_{u
_j}). Then the combination operation of multiple D numbers is a mapping f_D, such that $f_{D} (D_{1}, D_{2}, . . ., D_{n}) = [. . . [D_{λ_{1}} \oplus D_{λ_{2}}] \oplus . . . D_{λ_{n}}]$ (8) where D_{λ
_j} is the D_{u
_j} of the tuple (u_j, D_{u
_j}) having the i^th lowest u_j.

Definition 5 [46]. (D numbers’ integration) Suppose that D = {(b₁, v₁) , (b₂, v₂) , . . . , (b_i, v_i) , . . . , (b_n, v_n)} is a D number, the integrating operation of D numbers is defined as $I (D) = \sum_{i = 1}^{n} b_{i} v_{i}$ (9) where $b_{i} \in R^{+}, v_{i} > 0, \sum_{i = 1}^{n} v_{i} ⩽ 1$ .

3 The proposed method for FMEA

This section presents a new risk priority model for FMEA, which is based on D-BWM and D-MARCOS methods. The flowchart in Fig. 1 shows the proposed FMEA model.

Fig. 1

Overall structure of proposed FMEA model.

Suppose that there are two parts evaluation teams, called team members and experts. There are K team members TM_k (k = 1, 2, . . . , K) are invited to judge the importance of risk factors and Q experts E_k (k = 1, 2, . . . , Q) are invited to evaluation the performance of failure modes. The FMEA team members E_k (k = 1, 2, . . . , Q) give assessments to m failure mode FM_i (i = 1, 2, . . . , m) with n risk factors RF_j (j = 1, 2, . . . , n). Considering that different team members and experts have different experiences and professional backgrounds, each team member and expert have a weight $λ_{k} (\sum_{k = 1}^{K} λ_{k} = 1)$ . In this paper, 10-point scale is used to describe the risk factors of each failure mode, as shown in Table 1. The proposed method is composed of the following parts.

3.1 Obtain the weights of team members and experts

Since it is the same to obtain the weights of team members and experts, take experts as an example to illustrate this method. The distance of D numbers is used to measure the degree of conflict evaluated by experts. The higher an expert’s evaluation results are supported by other experts, the less conflicts with other experts, the expert should be given more weight.

The degree of conflict between expert evaluation values can be represented by the distance between two D numbers. The distance of the D numbers is defined as follows: $d_{kl} = \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{n} | I (D)_{ij}^{k} - I (D)_{ij}^{l} |}{m \times n}$ (10) where k, l = 1, 2, . . . , Q and k ≠ l.

The distance between D numbers can be represented by the distance matrix DM: $DM = [\begin{matrix} 0 & d_{12} & \dots & d_{1 n} \\ d_{21} & 0 & \dots & d_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ d_{n 1} & d_{n 2} & L & 0 \end{matrix}]$

The greater the distance of two D numbers is, the greater the degree of conflict is, the less impact it has on the end.

The similarity between two D numbers is defined as follows: $Sim (d_{i}, d_{j}) = 1 - d_{D - number} (d_{i}, d_{j})$ (11)

Therefore, the similarity matrix SMM is: $SMM = [\begin{matrix} 1 & {Sim}_{12} & \dots & {Sim}_{1 n} \\ {Sim}_{21} & 1 & \dots & {Sim}_{2 n} \\ \dots & \dots & ⋱ & ⋮ \\ {Sim}_{n 1} & {Sim}_{n 2} & \dots & 1 \end{matrix}]$ (12)

The support degree of D numbers Sup (i) is defined as: $Sup (i) = \sum_{j = 1, j \neq i}^{n} Sim (d_{i}, d_{j})$ (13)

According to the support degree of the D number, the weight of the expert can be obtained. Then normalize the weights of experts. The normalization process is as follows: $W_{i} = \frac{Sup (i)}{\sum_{i = 1}^{Q} Sup (i)} (i = 1, 2, . . ., Q)$ (14)

The higher the weight of the expert is, the greater the influence of the corresponding D number on the final combination step.

3.2 D-BWM for weighting risk factors

BWM introduced by Rezaei [33] is a comparison-based multiple-criteria decision-making (MCDM) technique. Traditional BWM has aroused growing concerns on MCDM problem due to the flexibility and effectiveness. However, human subjective judgment is usually characterized by uncertainty and ambiguity. The criterion description in practical application has the disadvantages of incertitude and vagueness. To overcome the existing deficiencies in the traditional BWM, Mousavi-Nasab et al. [39] extended BWM based on D numbers to obtain more reliable weights and higher consistency.

With n risk factors, D-BWM performs a total of 2n-3 reference comparisons (n-2 Best-to-Others comparisons + n-2 Others-to-Worst comparisons + 1 Best-to-Worst comparison). In what follows, the detailed steps of D-BWM are described to determine the weights of risk factors. D-BWM is composed of the following steps:

Step 1. Determine the most important risk factor and least important risk factor

The K team members respectively evaluate the risk factors and list the most important risk factors RF_B and the least important risk factors RF_W.

Step 2. Specify the preference of the most important risk factor by D numbers

Using D numbers to execute reference information among risk factors, the best risk factor RF_B is determined in step 1. Then the Best-to-Others (BO) vector from team members TM_k is expressed as follows: $D_{B}^{k} = (d_{B 1}^{k}, d_{B 2}^{k}, . . ., d_{Bn}^{k})$ (15) where $d_{Bi}^{k}$ expresses the preference of the most important risk factor RF_B to others RF_j based on D numbers.

Then, according to Definition 3 and Definition 4, perform a combination operation on the preference evaluation of k team members to obtain the final $D_{B}$ vector: $d_{Bj} = [d_{Bj}]^{λ_{1}} \oplus [d_{Bj}]^{λ_{2}} \oplus . . . \oplus [d_{Bj}]^{λ_{k}}, j = 1, 2, . . ., n$ (16) where the order variables λ_s (s = 1, 2, . . . , k) are expressed by the weights of each team members. The overall evaluation of BO vector by all team members is as follows: $D_{B} = (d_{B 1}, d_{B 2}, . . ., d_{Bn})$ (17)

Step 3. Specify the preference of the least important risk factor by D numbers

Using D numbers to execute reference information among risk factors, the worst risk factor RF_W is determined in step 1. Then the Others-to-Worst (OW) vector from team member e_k are expressed as follows: $D_{W}^{k} = (d_{1 W}^{k}, d_{2 W}^{k}, . . ., d_{nW}^{k})$ (18) where $d_{jW}^{k}$ expresses the preference of the other risk factors RF_j to the least important risk factor RF_W based on D numbers.

Then, according to Definition 3 and Definition 4, perform a combination operation on the preference evaluation of k team members to obtain the final $D_{W}$ vector: $d_{jW} = [d_{jW}]^{λ_{1}} \oplus [d_{jW}]^{λ_{2}} \oplus . . . \oplus [d_{jW}]^{λ_{k}}, j = 1, 2, . . ., n$ (19) where the order variables λ_s (s = 1, 2, . . . , k) are expressed by the weights of each experts. The overall evaluation of OW vector by all team members is as follows: $D_{W} = (d_{1 W}, d_{2 W}, . . ., d_{nW})$ (20)

Step 4. Determine the optimal weights.

First, according to step2 and step3, we can obtain the BO vector and OW vector expressed by D numbers. D numbers are converted to crisp numbers using integration of D numbers by Equation (9). Then, we obtain the BO vector and OW vector as follows: $T_{B} = (t_{B 1}, t_{B 2}, . . ., t_{Bn})$ (21) $T_{W} = (t_{1 W}, t_{2 W}, . . ., t_{nW})$ (22)

Then, we use the nonlinearly constrained optimization model (23) to calculate criteria weights $(w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*})$ and ξ^*. $\begin{matrix} min max_{j} {| \frac{w_{B}}{w_{j}} - t_{Bj} |, | \frac{w_{j}}{w_{W}} - t_{jW} |} \\ s . t . {\begin{matrix} \sum_{j} w_{j} = 1 \\ w_{j} ⩾ 0 for all j \end{matrix} \end{matrix}$ (23)

For simplicity, Model (23) can be transferred to the Model (24): $\begin{matrix} min ξ \\ s . t . {\begin{matrix} | \frac{w_{B}}{w_{j}} - t_{Bj} | ⩽ ξ \\ | \frac{w_{j}}{w_{W}} - t_{jW} | ⩽ ξ \\ \sum_{j} w_{j} = 1 \\ w_{j} ⩾ 0 for all j \end{matrix} \end{matrix}$ (24)

Solving linear Model (24), the optimal weights $(w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*})$ and ξ are determined.

Step 5. Consistency check.

The value of the maximum value of ξ depends on the value of a_BW according to Table 2. Since d_BW is a continuous value, t_BW ∈ [1, 9]. a_BW is a natural number, therefore a_BW ≠ t_BW, in order to obtain its consistency index, we need to convert t_BW to a_BW. The conversion is as follows:

$a_{BW} = {\begin{matrix} [t_{BW}], t_{BW} - [t_{BW}] < 0.5 \\ [t_{BW}] + 1, t_{BW} - [t_{BW}] ⩾ 0.5 \end{matrix}$ (25) where [t_BW] = max {n ∈ Z|n ⩽ t_BW}.

Table 2

Consistency index for BWM

a _BW	1	2	3	4	5	6	7	8	9
Consistency index(maxξ)	0.00	0.44	1.00	1.63	2.30	3.00	3.73	4.47	5.23

The consistency ratio can be obtained, based on ξ^* and consistency index given in Table 2 as follows: $η_{c} = \frac{ξ^{*}}{max {ξ}}$ (26)

If η_c is closer to 0, the consistency is higher. If η_c is closer to 1, the consistency is lower.

3.3 Obtain the ranking of failure modes.

Stevi’c et al. [40] first proposed the MARCOS method in 2019. This method is based on a compromise solution to measure and sort alternatives, which is simple, effective and easy to sort. However, in real life, experts have various ambiguities and inaccuracies in the evaluation process. The traditional MARCOS method cannot solve the problems in this situation. D number theory is an effective tool for solving uncertainty. Therefore, this paper extends the MARCOS method to D numbers, which effectively solves the problem of uncertainty in real life and obtains more reliable conclusions. The specific steps of this method are as follows:

Step 1. Construct a group decision matrix.

Suppose that Q experts E_k (k = 1, 2, . . . , Q) evaluate the O, S, and D of m failure modes FM_i (i = 1, 2, . . . , m) using D numbers, and then integrate the evaluation results of the Q experts to form an initial group decision matrix.

Step 2. Combine the evaluation results of q experts

In this step, considering there are q experts, the evaluation results of q experts were combination by using Equation (8) and then obtain the overall evaluation results of O, S and D of the expert group.

Step 3. Covert the extended D numbers initial extended matrix to a crisp matrix

In this step, the extended initial matrix using D numbers can be converted to a crisp matrix by using the integration representation of D numbers by Equation (9) and construction initial matrix. Then, an extended crisp matrix X = (x_ij) is derived as follows:

$\begin{matrix} X = I (D) = [\begin{matrix} X_{D - NI} \\ X_{1} \\ ⋮ \\ X_{m} \\ X_{D - AI} \end{matrix}] \\ = [\begin{matrix} I (D_{ni 1}) & I (D_{ni 2}) & \dots & I (D_{nin}) \\ I (D_{11}) & I (D_{12}) & \dots & I (D_{1 n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \begin{matrix} I (D_{m 1}) \\ I (D_{ai 1}) \end{matrix} & \begin{matrix} I (D_{m 2}) \\ I (D_{ai 2}) \end{matrix} & \begin{matrix} \dots \\ \dots \end{matrix} & \begin{matrix} I (D_{mn}) \\ I (D_{ain}) \end{matrix} \end{matrix}] \end{matrix}$ (27)

The negative ideal solution X_D-NI is the worst alternative in the criteria j, and the ideal solution X_D-AI is the best alternative in the criteria j. Depending on type of the criteria, X_D-NI and X_D-AI are defined by applying Equations (28) and (29). $X_{D - NI} = {\begin{matrix} min_{i} D_{ij}, if j \in benefit criteria \\ max_{i} D_{ij}, if j \in cos t criteria \end{matrix}$ (28) $X_{D - AI} = {\begin{matrix} max_{i} D_{ij}, if j \in benefit criteria \\ min_{i} D_{ij}, if j \in cos t criteria \end{matrix}$ (29)

Step 4. Normalize the crisp matrix(X).

Normalize the decision by Equations (30) and (31). $n_{ij} = \frac{I (D_{ai})}{I (D_{ij})}, if j \in cos t criteria$ (30) $n_{ij} = \frac{I (D_{ij})}{I (D_{ai})}, if j \in benefit criteria$ (31) where elements I (D_ai) and I (D_ij) represent the elements of the matrix X. Then the normalized decision matrix N = [n_ij] _m×n are obtained.

Step 5. Compute the weighted decision matrix.

The weighted decision matrix V = [v_ij] _m×n is obtained by multiplying the standardized matrix N with the weight coefficients of the criterion w_j, which is constructed based on Equation (32) as follows. $v_{ij} = n_{ij} \times w_{j}$ (32)

Step 6. Calculate the utility degree of failure modes

Calculate the utility of the failure modes FM_i by applying Equations (33) and (34). $k_{i}^{-} = \frac{\sum_{j = 1}^{n} v_{ij}}{\sum_{j = 1}^{n} v_{nij}}$ (33) $k_{i}^{+} = \frac{\sum_{j = 1}^{n} v_{ij}}{\sum_{j = 1}^{n} v_{aij}}$ (34)

Among them, $k_{i}^{+}$ represents the positive utility of the failure modes FM_i, $k_{i}^{-}$ represents the negative utility of the failure modes FM_i.

Step 7. Determine of the utility function of the failure mode f (k_i)

Calculate the utility function relative to the negative ideal solution by applying Equation (35) and calculate the utility function relative to the ideal solution by applying Equation (36). $f (k_{i}^{-}) = \frac{k_{i}^{+}}{k_{i}^{+} + k_{i}^{-}}$ (35) $f (k_{i}^{+}) = \frac{k_{i}^{-}}{k_{i}^{+} + k_{i}^{-}}$ (36)

Then, the utility function of failure modes is defined by Equation (37). $f (k_{i}) = \frac{k_{i}^{+} + k_{i}^{-}}{1 + \frac{1 - f (k_{i}^{+})}{f (k_{i}^{+})} + \frac{1 - f (k_{i}^{-})}{f (k_{i}^{-})}}$ (37)

Obviously, the utility function is the compromise of the observed failure modes in relation to the ideal and negative ideal solution.

Step 8. Rank the failure mode

The ranking can be obtained based on the values of utility function by descending these crisp values.

4 Application

In this section, the proposed FMEA model will be applied to a case of rotor blades for aircraft turbines [23]. Rotor blades are the key rotating parts of aircraft turbines [53]. The rotor blades rotate at high speed during the working process, and the working environment is complicated. At the same time, the aviation industry has developed rapidly this year, the thrust-to-weight ratio (TWR) of aircraft turbines has also become increasingly higher, and the stress level of rotor blades has also been greatly improved. The rotor blade is the most vulnerable component in the aircraft turbine, so improving the reliability of the rotor blade has a great impact on the safety of the aircraft turbine. In order to improve its safety and reliability, use FMEA to evaluate before its design.

4.1 Illustration of the proposed FMEA

In this case, supposing that there are three FMEA team members, TM₁, TM₂, TM₃, and three experts, E₁, E₂, E₃, to evaluate risk factors of eight failure modes (FM_m, m = 1, 2, . . . , 8) of rotor blades using the values defined in Table 1. The three team members believe that S is the most important risk factor, and D is the least important risk factor, and give the BO vector and OW vector using D numbers respectively, as shown in Tables 3 and 4. The evaluation results of eight failure modes by three experts using D numbers are shown in Table 5.

Table 3
The pairwise vector for the best risk factor using D numbers

Best risk factor S O D

E ₁ S {(1,1)} {(2,0.3), (3,0.6)} {(3,0.8), (4,0.2)}

E ₂ S {(1,1)} {(3,1)} {(3,0.4), (4,0.6)}

E ₃ S {(1,1)} {(3,1)} {(3,0.9), (4,0.1)}

	Best risk factor	S	O	D
E ₁	S	{(1,1)}	{(2,0.3), (3,0.6)}	{(3,0.8), (4,0.2)}
E ₂	S	{(1,1)}	{(3,1)}	{(3,0.4), (4,0.6)}
E ₃	S	{(1,1)}	{(3,1)}	{(3,0.9), (4,0.1)}

Table 4

The pairwise vector for the worst risk factor using D numbers

	E ₁	E ₂	E ₃
Worst risk factor	D	D	D
S	{(3,0.8), (4,0.2)}	{(3,0.4), (4,0.6)}	{(3,0.9), (4,0.1)}
O	{(3,0.9)}	{(3,1)}	{(3,1)}
D	{(1,1)}	{(1,1)}	{(1,1)}

Table 5

Evaluation information on eight failure modes by three experts

	E ₁			E ₂			E ₃
	S	O	D	S	O	D	S	O	D
FM ₁	(7,0.8)	(2,0.9), (3,0.1)	(4,0.8)	(7,1)	(3,0.8), (4,0.2)	(3,0,7), (4,0.3)	(7,1)	(4,0.6), (5,0.4)	(2,0.9), (3,0.1)
FM ₂	(8,1)	(3,0.8), (4,0.2)	(3,0.9), (4,0.1)	(8,0.7), (9,0.3)	(4,0.8)	(4,1)	(8,1)	(3,0.2), (4,0.8)	(4,1)
FM ₃	(1,1)	(1,0.9)	(10,1)	(1,1)	(1,1)	(1,0.9)	(1,1)	(1,1)	(2,1)
FM ₄	(4,1)	(1,1)	(4,1)	(5,1)	(1,1)	(2,0.3), (3,0.7)	(4,0.8), (5,0.2)	(1,1)	(3,1)
FM ₅	(3,1)	(4,1)	(3,1)	(3,1)	(4,0.6)	(1,0.7), (2,0.3)	(2,0.2), (3,0.8)	(3,0.6)	(1,1)
FM ₆	(6,1)	(2,1)	(6,1)	(6,1)	(2,1)	(5,1)	(6,1)	(2,1)	(5,1)
FM ₇	(7,1)	(1,0.7)	(7,1)	(7,0.8), (8,0.1)	(1,1)	(3,1)	(7,1)	(1,1)	(3,1)
FM ₈	(5,1)	(3,1)	(5,1)	(4,1)	(3,1)	(1,1)	(4,0.8), (5,0.2)	(3,0.9), (4,0.1)	(1,1)

Step 1. Obtain the weights of team members and experts

We compute the distance among D numbers based on Equation (10). According to Equations (11)–(13), the support degrees for different team members are [1.689, 1.700, 1.767]. Then the weights of team members are [0.328, 0.330, 0.342] according to Equation (14). In the same way as above, the weight of experts can be obtained as [0.2820, 0.3595, 0.3585].

Step 2. Derive the weights of risk factors

Then we combined the individual assessments of team members into a group assessment by using Equation (16) and Equation (19) according to the expert weight ranking obtained in the first step, we can show the BO vector and OW vector are as follows: $A_{B}^{D} = (\begin{matrix} {(1, 1)}, {(2.75, 0.33), (3, 0.35)}, \\ {(3, 0.24), (3.25, 0.28), \\ (3.5, 0.3), (3.75, 0.12), (4, 0.06)} \end{matrix})$ $A_{W}^{D} = (\begin{matrix} {\begin{matrix} (3, 0.24), (3.25, 0.28), (3.5, 0.3), \\ (3.75, 0.12), (4, 0.06) \end{matrix}}, \\ {(3, 0.544)}, {(1, 1)} \end{matrix})$

The crisp matrices are aggregated using Equation (9), we can show as follows. $A_{B}^{I} = (1, 1.9575, 3.37)$ $A_{W}^{I} = (3.37, 1.632, 1)$

Establish a nonlinear constraint model, as shown in Model (38). $\begin{matrix} min ξ \\ s . t . \\ | \frac{w_{S}}{w_{O}} - 1.9575 | ⩽ ξ \\ | \frac{w_{S}}{w_{D}} - 3.37 | ⩽ ξ \\ | \frac{w_{O}}{w_{D}} - 1.632 | ⩽ ξ \\ w_{S} + w_{O} + w_{D} = 1 \\ w_{S} ⩾ 0, w_{O} ⩾ 0, w_{D} ⩾ 0 \end{matrix}$ (38)

The optimal weights of the three risk factors can be determined by solving the model (38) as $w_{S}^{*} = 0.55, w_{O}^{*} = 0.28, w_{D}^{*} = 0.17, ξ^{*} = 0.0074$ . Therefore, t_BW = t_SD = 3.37, a_BW can be calculated by Equation (25) and calculated as a_BW = 3. Further, the consistency index can be obtained as 1 in the Table 2. The consistency ratio can be calculated as 0.0074/1 = 0.0074, which is close to zero means consistency well.

Step 3. Determine the ranking of failure modes

Same as step 1 and step 2, transform the individual expert evaluation results into group evaluation results, then covert to a crisp number matrix.

Take the expert’s comprehensive evaluation result of the failure mode as the initial matrix, and obtain the extended initial decision matrix D through Equations (28) and (29) to form matrix D. $D = \begin{matrix} \begin{matrix} {FM}_{D - NI} \\ {FM}_{1} \\ {FM}_{2} \\ {FM}_{3} \\ {FM}_{4} \\ {FM}_{5} \\ {FM}_{6} \end{matrix} \\ {FM}_{7} \\ {FM}_{8} \\ {FM}_{D - AI} \end{matrix} [\begin{matrix} 1.00 & 0.52 & 1.72 \\ 3.73 & 3.39 & 2.18 \\ 8.22 & 2.55 & 3.86 \\ 1.00 & 0.54 & 2.22 \\ 4.62 & 1.00 & 3.03 \\ 2.88 & 1.06 & 1.72 \\ 6.00 & 2.00 & 5.25 \\ 6.95 & 0.52 & 4.00 \\ 4.37 & 3.11 & 2.00 \\ 8.22 & 3.39 & 5.25 \end{matrix}]$

The risk factors O, S, and D are all benefit criteria, so the matrix D is standardized using Equation (31) to form a standardized matrix I. The next step is to use Equation (32) to multiply all the values in the standardized matrix with the weight of each risk factor to obtain a weighted standardized matrix V. By applying Equations (33) and (34) to calculate the utility degree of the negative ideal solution and the positive ideal solution, and then calculating the utility function of the negative ideal solution and the positive ideal solution by Equations (35) and (36), use Equation (37) to calculate the utility function of each failure mode, and finally rank the failure modes in descending order. The calculation results are shown in Table 6.

Table 6

Results of D-MARCOS method

	$k_{i}^{-}$	$k_{i}^{+}$	$f (k_{i}^{-})$	$f (k_{i}^{+})$	f (k_i)	Rank
FM ₁	3.623	0.600	0.142	0.858	0.586	5
FM ₂	5.347	0.886	0.142	0.858	0.866	1
FM ₃	1.108	0.184	0.142	0.858	0.179	8
FM ₄	2.957	0.490	0.142	0.858	0.479	6
FM ₅	2.030	0.336	0.142	0.858	0.329	7
FM ₆	4.448	0.737	0.142	0.858	0.720	2
FM ₇	3.851	0.638	0.142	0.858	0.623	3
FM ₈	3.709	0.615	0.142	0.858	0.601	4

4.2 Comparison with other approaches

In order to further illustrate the effectiveness of the proposed approach, a comparative analysis is conducted between the proposed method and some similar risk prioritization methods existing on the FMEA, including D-TOPSIS method [31], D-VIKOR method [38], D-EDAS method [54] and conventional FMEA method [55].

After calculation, these results were compared are shown in Fig. 2. The model in this article is consistent with the ranking order obtained by D-TOPSIS method, D-VIKOR method D-EDAS method, and the ranking order is FM₂ ≻ FM₆ ≻ FM₇ ≻ FM₈ ≻ FM₁ ≻ FM₄ ≻ FM₅ ≻ FM₃. However, the ranking order is different from that using traditional method. The traditional FMEA ranking result is FM₂ ≻ FM₆ ≻ FM₁ ≻ FM₈ ≻ FM₇ ≻ FM₄ ≻ FM₅ ≻ FM₃. The same thing is, the failure mode with the highest risk level is FM₂, the lowest risk level is FM₃.

Fig. 2

Failure modes ranking in different failure mode and effects analysis models.

Compared with the FMEA based on D numbers and TOPSIS, the ranking of failure modes obtained by two methods are the same. These two methods are both based on positive ideal solution and negative ideal solution, which are simple, effective and easy to sort. MARCOS method combines ratio method and reference point generation to obtain the comprehensive decision scheme, and the conclusion obtained is more stable and reliable than that obtained by TOPSIS method. In this paper, the D number is extended to the MARCOS method, which is helpful to solve the problem of uncertain real situation and uncertain expert evaluation.

Compared with the traditional FMEA based on RPN method, the failure mode ranking is different in FM₁ and FM₇, where the proposed method ranks FM₇ higher than FM₁, while the conventional FMEA gives a higher priority to FM₁. The main reason of such difference is that the traditional FMEA is the failure mode level obtained by multiplying three risk factors O, S and D, therefore the method fails to carefully consider the relative importance of risk factors for failure modes. While FM₇ has a higher value in terms of the severity, and its weight in this paper is higher. The D-BWM is adopted in this paper to obtain the weights of risk factors which overcome the defects of traditional FMEA and yield reasonable results.

4.3 The influence of reverse rank matrices

Because the method proposed in this paper is the same as the ranking results obtained by the D-TOPSIS method, D-VIKOR method, and D-EDAS method, the influence of the dynamic matrix on the rank inversion is studied. When the elements of the decision matrix change, we will test the validity of the decision result. When the decision matrix changes dynamically, the final failure mode rankings may show logical contradictions. These contradictions are expressed in the form of inconsistent failure mode rankings, which will raise concerns about the reliability of the method.

The method proposed in this paper and D-TOPSIS, D-VIKOR and D-EDAS methods are respectively used to change the number of failure modes for each case, eliminating the worst alternative from the decision matrix to avoid further consideration of the worst failure mode. At the same time, a new decision matrix is formed, and the remaining schemes are sorted by four methods respectively. The purpose of this analysis is to analyze the performance of the four methods under dynamic initial decision conditions. We then perform this step further to get a ranking of the remaining scenarios. The results obtained by the four methods are shown in Fig. 3 –6.

Fig. 3

Effects of dynamic decision matrices in proposed method

Fig. 4

Effects of dynamic decision matrices in D-TOPSIS method

Fig. 5

Effects of dynamic decision matrices in D-VIKOR method

Fig. 6

Effects of dynamic decision matrices in D-EDAS method

As can be seen from the Fig. 3 –6, in the method proposed in this paper, when the initial matrix of MARCOS model is changed, there is no rank inversion in the whole scenario (Fig. 3). When the D-TOPSIS method is used, the positions of the third and fourth level failure modes change for scenarios S5-S6, that is, rank inversion occurs (Fig. 4). When D-VIKOR is used, for scenario S2–S4, the position of the fifth and sixth failure modes changes, that is, rank inversion occurs; for scenarios S6-S7, the position of the second level alternative and the third level failure mode changes, that is, rank inversion occurs (Fig. 5). When the D-EDAS method is used, the positions of the first-level to the sixth-level fault modes change, and after scenario S2, FM2 is no longer the first-level failure mode, with a strong rank inversion (Fig. 6).

According to the results obtained, the robustness and stability of the FMEA model established in this paper are stronger than the other three methods in the dynamic environment. For the other three models, removing the worst scheme from the decision matrix would lead to the ordering inversion problem, but this situation does not occur in the proposed model. Therefore, the results of this model in the dynamic environment have significant reliability and stability.

5 Conclusion

This paper presents a new FMEA risk priority model. The model integrates D-BWM, distance-based method and D-MARCOS method. The method based on D numbers is used to express various types of uncertainty existing in the risk assessment, such as incomplete assessment opinion, uncertain assessment opinion and complete assessment opinion. This model presents a combination method, which combines the opinions of multiple team members into group opinions by using D numbers combination operation. D-BWM is used to obtain the weight of risk factors, which requires less pairwise comparison than D-AHP method, and achieves high consistency and more reliable results. Although the process of solving mathematical programming is complicated, LINGO can be used to accomplish this task efficiently. The distance-based method of the D number is used to obtain the weight of FMEA team members. The D-MARCOS method is used to obtain the risk priority of failure mode, which further improves the accuracy and reliability of decision results, and extends the problem to the uncertain field, providing a logical choice for multi-criteria decision problems. Finally, through the rotor blades of an aircraft turbine as an example, introduces the application of this method and a comparison is made to prove the effectiveness of the method. In addition, considering the knowledge and experience of the team members, the proposed model can fully reflect the relative importance of each FMEA team member. To sum up, this model is suitable for the FMEA problem in which multiple experts with different knowledge accumulation and experience evaluate the failure mode under the dynamic environment, and each risk factor has different weight in the engineering problem.

The advantages of the FMEA model are summarized as follows:

the proposed method adopts the combination operation of D numbers to fuse the evaluations of experts into group assessment.

According to the expert evaluation results, D number distance–based method is adopted to determine the weight of each expert.

The risk priority of failure modes is obtained by applying D-BWM, which can not only enable experts to express uncertain evaluation, but also make the evaluation results more consistent.

MARCOS method is extended to D numbers for the first time, which can express various types of uncertainties in expert evaluation process. Moreover, the sorting method is simple and effective, which shows higher robustness and effectiveness compared with TOPSIS method.

At the same time, the model still has some problems that have not been solved, such as fail to consider the risk preference of decision-makers in the environment of risk and uncertainty. The model is not smart enough to fill in the missing data. At the same time, when different experts have different opinions, how to reach a consensus is also a problem that needs further discussion.

Future research mainly focuses on the following aspects. First, we can use different decision-making methods and compare the method proposed in this paper with other MCDM methods to find a more accurate solution. Second, we can establish FMEA model based on distributed linguistic representations [56]. Compared with D numbers, distributed linguistic representations can better express the uncertainty and complexity of preference information in linguistic decision making. Third, we can further study the psychological and behavioral characteristics of decision makers under risk conditions, and the problem of team members’ uncertainty about their own preferences [57, 58]. In addition, we can also combine D numbers with linguistic terms to meet the evaluation team’s needs for linguistic evaluation [59].

Footnotes

Acknowledgments

This work was supported in part by the Fund for Shanxi “1331 Project” Key Innovative Research Team (TD201710), and in part by “The Discipline Group Construction Plan for Serving Industries Innovation”, Shanxi, China: The Discipline Group Program of Intelligent Logistics Management for Serving Industries Innovation (XKQ201801).

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An integrated FMEA approach using Best-Worst and MARCOS methods based on D numbers for prioritization of failures

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Failure mode and effects analysis

4.1 Illustration of the proposed FMEA

Table 3 The pairwise vector for the best risk factor using D numbers Best risk factor S O D E 1 S {(1,1)} {(2,0.3), (3,0.6)} {(3,0.8), (4,0.2)} E 2 S {(1,1)} {(3,1)} {(3,0.4), (4,0.6)} E 3 S {(1,1)} {(3,1)} {(3,0.9), (4,0.1)}

Footnotes

Acknowledgments

References

Table 3
The pairwise vector for the best risk factor using D numbers

Best risk factor S O D

E ₁ S {(1,1)} {(2,0.3), (3,0.6)} {(3,0.8), (4,0.2)}

E ₂ S {(1,1)} {(3,1)} {(3,0.4), (4,0.6)}

E ₃ S {(1,1)} {(3,1)} {(3,0.9), (4,0.1)}