Risk priorization for failure modes with extended MULTIMOORA method under interval type-2 fuzzy environment

Abstract

The Failure mode and effect analysis (FMEA) is an effective risk evaluation approach which has been widely used to assist in risk controlling in various workplaces. However, in practice, the conventional FMEA approach suffers from the drawbacks associated with the risk evaluation and priorization methods. In this paper, a novel risk priorization method for FMEA based on the extended MULTIMOORA (Multi-Objective Optimization by Ratio Analysis plus the Full Multiplicative Form) method is proposed. First, the interval type-2 fuzzy sets are applied to deal with the uncertainty of risk evaluation in FMEA. Second, the distance-based method is used to calculate the importance weight of each risk factor. Then, an extended MULTIMOORA method is presented to rank risk priority of each failure mode, in which the distance measure for interval type-2 fuzzy number is incorporated. Finally, a practical case in steel company is selected to illustrate the application and feasibility of the proposed approach. A comparative analysis is conducted to demonstrate the effectiveness of the developed risk priorization method.

Keywords

Failure mode and effect analysis risk prioritization interval type-2 fuzzy set MULTIMOORA method

1 Introduction

The Failure mode and effect analysis (FMEA) has been widely applied in various fields such as manufacturing [1], farming [2], chemical [3], marine [4], and railway [5]. The traditional FMEA approach evaluates risk of failure modes by using the Risk Priority Number (RPN) method. The conventional RPN method in FMEA has been proved to be a simple and effective way in practical application, however, it still possesses many drawbacks. The main ones can be described as follows [1 –3]: (i) the crisp number may be impossible to precisely determine the risk score of each risk factor within a real context; (ii) the different levels of importance for risk factors are not taken into account; (iii) the equal value of RPN can be derived by different combinations of the three risk factors, but the risk implications of these failure modes may be completely different.

To handle the drawback of crisp FMEA approach in risk evaluation process, the fuzzy theory has been widely applied to FMEA approach [6 –11]. For example, Zhao et al. [1] extended the FMEA approach with the interval-valued intuitionistic fuzzy sets. Liu et al. [12] developed a novel FMEA framework by using the intuitionistic fuzzy numbers. Mohsen and Fereshteh [13] combined the Z-number with FMEA approach. Moreover, more extended FMEA approaches under fuzzy environment can refer to the review implemented by Liu [14]. The type-1 fuzzy approaches based FMEA models have significantly improved the reliability of risk evaluation. However, the type-1 fuzzy set may be insufficient to model the intra-and inter-uncertainties because its membership function is crisp number [15, 16]. In such context, the type-2 fuzzy set can potentially be an effective approach to deal with the inter- and inter-uncertainty in risk evaluation problems [2]. The interval type-2 fuzzy set can be regarded as a special type of type-2 fuzzy sets, which has been widely used in various models [15 , 17–20]. The adoption of interval type-2 fuzzy set in FMEA is relatively less [2, 16]. Manuscripts must be written in English. Authors whose native language is not English are recommended to seek the advice of a native English speaker, if possible, before submitting their manuscripts. The pages in the manuscript should not be numbered and in the text no reference should be made to page numbers; if necessary, one may refer to sections. Try to avoid excessive use of italics and bold face.

In other way, the risk priorization for failure modes in FMEA should take multiple risk parameters into consideration. Therefore, the risk priorization in FMEA approach has been typically regarded as a group multi-criteria decision making (MCDM) problem. Accordingly, various MCDM approaches have been applied to FMEA. Chang and Cheng [21] reported a new risk assessment method in FMEA, the decision-making trial and evaluation laboratory (DEMATEL) method was introduced to analyze the interactions among failure modes. In [22], the FMEA approach was integrated with the Order Preference by Similarity to Ideal Solution (TOPSIS) MCDM method to determine the risk priority of each failure mode. In [23], an ELECTRE method was combined with the FMEA approach to determine the risk priorities of failure modes. Huang et al. [24] proposed an integrated FMEA framework in which the TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) was used to obtain the risk priority. Moreover, other MCDM techniques including analytic hierarchy process (AHP) [25], analytic network process (ANP) [26], VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [27], complex proportional assessment (COPRAS) [10] and the multi-attributive border approximation area comparison (MABAC) [28] were applied to prioritize the risk of failure modes. In contrast, the Multi-Objective Optimization by Ratio Analysis plus the Full Multiplicative Form (MULTIMOORA), developed by Brauers and Zavadskas [29], is a novel approach for resolving MCDM problems. The MULTIMOORA includes three methods, namely, the ratio system, the reference point method, and the full multiplicative form method, respectively. Consequently, incorporating the MULTIMOORA approach into risk assessment will help to obtain a more reasonable risk evaluation result. In the literature, there are some studies in which the MULTIMOORA method has been combined with FMEA model for risk evaluation and risk priorization. In [11], a trapezoidal fuzzy MULTI-MOORA was combined with FMEA model to evaluate the risk of preventing infant abduction. Zhao et al. [1] reported a hybrid FMEA framework by incorporating the MULTIMOORA with FMEA model. Fattahi and Khalilzadeh [8] introduced an integrated risk evaluation approach, in which the fuzzy AHP was combined with MULTIMOORA approach to identify the risk priority.

Although the MULTIMOORA approach has been adopted in the FMEA model, existing MULTIMOORA based FMEA approaches can-not deal with the intra-and inter-uncertainties in risk evaluation. Meanwhile, there are few studies in which the interval type-2 fuzzy set and the MULTIMOORA approach has been applied simultaneously to determine risk priority of failure modes. For these reasons, in this paper, we pro-pose an extended MULTIMOORA method for FMEA within an interval type-2 fuzzy context. Due to the uncertainty exists in practical application of FMEA, the modeling of uncertainty of words used by different FMEA team members has become an important problem in enhancing the reliability and validity of FMEA approach [2]. As the interval type-2 fuzzy set can better handle the expression of linguistic uncertainty [16], the linguistic terms applied for risk evaluation can be expressed by interval type-2 fuzzy set. This paper first develops a distance-based weighted method to objectively determine the importance weights of risk factors under interval type-2 fuzzy environment. Then, the weighted MULTIMOORA approach is proposed to obtain the risk priority of each failure mode within the interval type-2 fuzzy context, which will help to obtain a more reasonable risk priority ranking order. Finally, a case of sheet steel production procedure is used to illustrate the application and effectiveness of the proposed FMEA framework.

The remainder of this paper is organized as follows. Preliminaries of the proposed risk priorization method are introduced in Section 2. The risk priorization approach based on MULTIMOORA method and interval type-2 fuzzy sets is described in Section 3. The FMEA approach with the extended MULTIMOORA approach under interval type-2 fuzzy environment is presented in Section 4. An illustrative example of steel production process is reported in Section 5. The conclusions and future research directions are provided in Section 6.

2. Preliminaries

In this section, we briefly introduce some basic concepts of interval type-2 fuzzy set and the MULTI-MOORA approach.

2.1 Interval type-2 fuzzy set

Definition 1. [18]: A type-2 fuzzy set $\tilde{\tilde{A}}$ in the universe of discourse X can be denoted by a type-2 membership function $μ_{\tilde{\tilde{A}}}$ , provided as follows: $\begin{matrix} \tilde{\tilde{A}} = {((x, u), μ_{\tilde{\tilde{A}}} (x, u)) | \forall x \in X, \forall u \in J_{x} \subseteq [0, 1]} \\ (1) \end{matrix}$ (1) In which J_x ⊆ [0, 1], and $0 \leq μ_{\tilde{\tilde{A}}} (x, u) \leq 1$ for all admissible x and u. Moreover, the type-2 fuzzy set $\tilde{\tilde{A}}$ can also be shown as the following form:

$\tilde{\tilde{A}} = \int_{x \in χ} \int_{u \in J_{x}} u_{\tilde{\tilde{A}}} (x, u) / (x, u)$ (2)

Where the symbol ∫ ∫ is union over all admissible x and u. The parameters x and u indicate the primary and secondary variables of the type-2 fuzzy set $\tilde{\tilde{A}}$ , respectively.

Definition 2. [18 , 31]: Let $\tilde{\tilde{A}}$ be a type-2 fuzzy set in the universe of discourse X represented by the type-2 membership function $μ_{\tilde{\tilde{A}}}$ . If all $u_{\tilde{\tilde{A}}} (x, u) = 1$ , then $\tilde{\tilde{A}}$ is defined as an interval type-2 fuzzy set (IT2FS). An IT2FS $\tilde{\tilde{A}}$ can be considered as a special case of the type-2 fuzzy set, and it can be expressed as follows:

$\tilde{\tilde{A}} = \int_{x \in X} \int_{u \in J_{x}} 1 / (x, u)$ (3)

Definition 3. [30, 31]: Let $\tilde{\tilde{A}}$ be an IT2FS in the universe of discourse X, $\tilde{\tilde{A}}$ fully depends on the lower membership function (LMF) and the upper membership function (UMF), then $\tilde{\tilde{A}}$ can be expressed by A^L and A^U , respectively.

$\tilde{\tilde{A}} = [A^{L}, A^{U}]$ (4)

Definition 4. [17, 18]: Generally, the interval type-2 trapezoidal fuzzy number (IT2TrFN) is selected to express the IT2FS. Let $\tilde{\tilde{A}}$ be an IT2TrFN, $\tilde{\tilde{A}} = [A^{L}, A^{U}] = [(a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}; h_{A}^{L}), (a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}; h_{A}^{U})]$ . Based on that, the visual representation of an IT2TrFN is provided in Fig. 1.

Fig.1

Visual representation of an IT2TrFN.

Definition 5. [32]: Let ${\tilde{\tilde{A}}}_{1}$ and ${\tilde{\tilde{A}}}_{2}$ be two IT2TrFNs as shown in follows: $\begin{matrix} {\tilde{\tilde{A}}}_{1} & = & [(a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L}; h_{1 A}^{L}), (a_{11}^{U}, a_{12}^{U}, a_{13}^{U}, a_{14}^{U}; h_{1 A}^{U})] \\ {\tilde{\tilde{A}}}_{2} & = & [(a_{21}^{L}, a_{22}^{L}, a_{23}^{L}, a_{24}^{L}; h_{2 A}^{L}), (a_{21}^{U}, a_{22}^{U}, a_{23}^{U}, a_{24}^{U}; h_{2 A}^{U})] \end{matrix}$ Then, the arithmetic operations between ${\tilde{\tilde{A}}}_{1}$ and ${\tilde{\tilde{A}}}_{2}$ can be expressed as follows:

(1) Addition operation

${\tilde{\tilde{A}}}_{1} \oplus {\tilde{\tilde{A}}}_{2} = [\begin{matrix} (a_{11}^{L} + a_{21}^{L}, a_{12}^{L} + a_{22}^{L}, a_{13}^{L} + a_{23}^{L}, a_{14}^{L} + a_{24}^{L}; \min {h_{1 A}^{L}, h_{2 A}^{L}}), \\ (a_{11}^{U} + a_{21}^{U}, a_{12}^{U} + a_{22}^{U}, a_{13}^{U} + a_{23}^{U}, a_{14}^{U} + a_{24}^{U}; \min {h_{1 A}^{U}, h_{2 A}^{U}}) \end{matrix}]$ (5)

(2) Subtraction operation

${\tilde{\tilde{A}}}_{1} ⊖ {\tilde{\tilde{A}}}_{2} = [\begin{matrix} (a_{11}^{L} - a_{24}^{L}, a_{12}^{L} - a_{23}^{L}, a_{13}^{L} - a_{22}^{L}, a_{14}^{L} - a_{21}^{L}; min {h_{1 A}^{L}, h_{2 A}^{L}}), \\ (a_{11}^{U} - a_{24}^{U}, a_{12}^{U} - a_{23}^{U}, a_{13}^{U} - a_{22}^{U}, a_{14}^{U} - a_{21}^{U}; \min {h_{1 A}^{U}, h_{2 A}^{U}}) \end{matrix}]$ (6)

(3) Multiplication operation

${\tilde{\tilde{A}}}_{1} \otimes {\tilde{\tilde{A}}}_{2} = [\begin{matrix} (a_{11}^{L} \cdot a_{21}^{L}, a_{12}^{L} \cdot a_{22}^{L}, a_{13}^{L} \cdot a_{23}^{L}, a_{14}^{L} \cdot a_{24}^{L}; \min {h_{1 A}^{L}, h_{2 A}^{L}}), \\ (a_{11}^{U} \cdot a_{21}^{U}, a_{12}^{U} \cdot a_{22}^{U}, a_{13}^{U} \cdot a_{23}^{U}, a_{14}^{U} \cdot a_{24}^{U}; \min {h_{1 A}^{U}, h_{2 A}^{U}}) \end{matrix}]$ (7)

(4) Multiplication by crisp number operation

$\begin{matrix} k \cdot {\tilde{\tilde{A}}}_{1} = & [\begin{matrix} (k \cdot a_{11}^{L}, k \cdot a_{12}^{L}, k \cdot a_{13}^{L}, k \cdot a_{14}^{L}; h_{1 A}^{L}), \\ (k \cdot a_{11}^{U}, k \cdot a_{12}^{U}, k \cdot a_{13}^{U}, k \cdot a_{14}^{U}; h_{1 A}^{U}) \end{matrix}], \\ k > 0 \end{matrix}$ (8)

(5) Power operation

${\tilde{\tilde{A}}}_{1}^{α} = [\begin{matrix} ((a_{11}^{L})^{α}, (a_{12}^{L})^{α}, (a_{13}^{L})^{α}, (a_{14}^{L})^{α}; h_{1 A}^{L}), \\ ((a_{11}^{U})^{α}, (a_{12}^{U})^{α}, (a_{13}^{U})^{α}, (a_{14}^{U})^{α}; h_{1 A}^{U}) \end{matrix}]$ (9)

(6) Distance between ${\tilde{\tilde{A}}}_{1}$ and ${\tilde{\tilde{A}}}_{2}$ [15] $d ({\tilde{\tilde{A}}}_{1}, {\tilde{\tilde{A}}}_{2}) = | R_{d} ({\tilde{\tilde{A}}}_{1}, \tilde{\tilde{1}}) - R_{d} ({\tilde{\tilde{A}}}_{2}, \tilde{\tilde{1}}) |$ (10) Where $\tilde{\tilde{1}} = [(1, 1, 1, 1; 1), (1, 1, 1, 1; 1)]$ , and $\begin{matrix} R_{d} ({\tilde{\tilde{A}}}_{1}, \tilde{\tilde{1}}) \\ = & \frac{1}{2 h_{1 A}^{L} \cdot h_{1 A}^{U}} [h_{1 A}^{L} \cdot (a_{14}^{U} - a_{13}^{U} - a_{14}^{L} + a_{13}^{L})] \\ - \frac{1}{2 h_{1 A}^{L} \cdot h_{1 A}^{U}} [h_{1 A}^{U} \cdot (0.5 (a_{12}^{L} - a_{11}^{L} - a_{12}^{U} + a_{11}^{U}) \\ - (a_{14}^{L} - a_{13}^{L} - a_{12}^{L} + a_{11}^{L}))] \\ + 1 - a_{14}^{L} - 0.5 (a_{11}^{L} - a_{11}^{U} + a_{14}^{U} - a_{14}^{L}) \end{matrix}$ (11)

1.2 The MULTIMOORA approach

The MULTIMOORA approach, primely put forward by Brauers and Zavadskas [29], is a robust and flexible MCDM technique. This approach begins with the structuring of a decision matrix X. Let S ={ s₁, s₂, ⋯ , s_m } be an alternative set, and C ={ c₁, c₂, ⋯ , c_n } be a criterion set, then the decision matrix can be defined as X = (x_ij) _m×n. After the construction of decision matrix, there are three methods that should be constructed in the MULTIMOORA approach, namely, the ration system, the reference point method and the full multiplicative form method.

(i) The ration system

The ration system is the first method of MULTIMOORA approach. For accomplishing this method, the standardization of decision matrix X should be conducted as follows:

$x_{ij}^{*} = \frac{x_{ij}}{\sqrt{\sum_{i = 1}^{m} x_{ij}^{2}}}$ (12)

After standardization, the assessment value of each alternative under the ration system can be obtained by using Eq. (13). $y_{i}^{*} = \sum_{j = 1}^{g} x_{ij}^{*} - \sum_{j = g + 1}^{n} x_{ij}^{*}$ (13)

Where g indicates the numbers of benefit criteria, j = g + 1, g + 2, ⋯ , n are the criteria to be cost criteria; and the parameter $y_{i}^{*}$ is used to represent the assessment value of alternative s_i.

The optimal alternative $s_{rs}^{*}$ under ration system can be obtained as follows [33]: $s_{rs}^{*} = (s_{i} | max_{i} y_{i}^{*})$ (14)

(ii) The reference point method

The reference point method should be conducted on the base of the standardized decision matrix $X^{*} = {[x_{ij}^{*}]}_{m \times n}$ obtained by Eq. (12). At first, the reference points of all criteria are identified as follows:

$r_{j}^{*} = {\begin{matrix} max_{i} x_{ij}^{*}, if j \leq g \\ min_{i} x_{ij}^{*}, if j > g \end{matrix}$ (15)

After determination of reference points, the deviation between each assessment value and reference point can be denoted as $| x_{ij}^{*} - r_{j}^{*} |$ . Then, the assessment value of alternative s_i under reference point method can also be calculated as follows: $z_{i}^{*} = max_{j} | x_{ij}^{*} - r_{j}^{*} |$ (16)

Finally, the optimal alternative $s_{rp}^{*}$ under reference point method can be determined as follows: $s_{rp}^{*} = {s_{i} | min_{i} z_{i}^{*}}$ (17)

(iii) The full multiplicative form method

The full multiplicative form method was used by Brauers and Zavadskas [29] to develop the MOORA approach. It combines the minimization and maximization problems of purely multiplicative utility function. Based on that, the assessment value of alternatives_i under full multiplicative form method can be expressed as follows: $U_{i}^{*} = \frac{{\tilde{A}}_{i}}{{\tilde{B}}_{i}} = \frac{\prod_{j = 1}^{g} x_{ij}^{*}}{\prod_{j = g + 1}^{n} x_{ij}^{*}}$ (18)

In which, ${\tilde{A}}_{i}$ indicates the product of assessment values of all benefit criteria, and ${\tilde{B}}_{i}$ is the product of assessment values of all cost criteria.

The optimal alternative under the full multiplicative form method can be determined as the following form: $s_{fm}^{*} = {s_{i} | max_{i} U_{i}^{*}}$ (19) (iv) The final ranking of each alternative

The final optimal alternative can be determined by the dominance theory, initially developed by Brauers and Zavadskas [33], is an approach to embody the three ranking results obtained by the (i), (ii) and (iii). For detailed application of the dominance theory, readers can refer to the literature [33].

3. Risk priorization based on the extended MULTIMOORA approach under IT2TrFN environment

In practical FMEA application, due to the complexity of risk analysis system and lack of risk information, the risk factors in FMEA approach are often highly uncertain and difficult to evaluate by crisp numbers. It is reasonable that the risk assessment information of risk factors provided by FMEA team members will be expressed by IT2TrFN. On the other hand, risk factor weights determining and risk priority ranking are two key problems in the application of FMEA. So, in this section, we propose the following two algorithms (Algorithms I and II) to determine the weights of risk factors and the risk priority ranking order of each failure mode. The Algorithm I is applied to calculate the importance weights of risk factors. The Algorithm II is used to determine the risk priority ranking order of each failure mode by incorporating the weights in Algorithm I and the MULTIMOOR method. Thus, the final values ϖ_j (j = 1, 2, ⋯ , n) determined by Algorithm I are the initial values of ϖ_j in Algorithm II. The specific steps of Algorithms I and II are described as follows.

Algorithm I. The distance-based method for determining the risk factors’ weights

In the practical application of FMEA, due to the high uncertainty in risk assessment process, it is difficult to obtain the exact importance weight information of each risk factor. In such context, the objective weighting method is particularly appropriate for determining the relative importance of risk factors [13]. Compare with other objective weighted methods, the TOPSIS can consider the distance to both the negative and positive ideal solutions simultaneously. It determines relative importance of criteria by considering the two distances [34]. The calculation algorithm in TOPSIS is clear and without loss of evaluation information. So, Motivated by the calculating principle of TOPSIS method [34], a novel objective weight determining method based on the distance measures is developed within IT2TrFN context. The specific steps are described as follows:

Step 1: Construct the normalization risk evaluation matrix

The first step of the distance-based method for determining weight is to normalize the risk evaluation matrix $X = {({\tilde{\tilde{x}}}_{ij})}_{m \times n}$ , in which the element ${\tilde{\tilde{x}}}_{ij} = [(x_{ij 1}^{l}, x_{ij 2}^{l}, x_{ij 3}^{l}, x_{ij 4}^{l}; h_{ij}^{l}), (x_{ij 1}^{u}, x_{ij 2}^{u}, x_{ij 3}^{u}, x_{ij 4}^{u}; h_{ij}^{u})]$ is an IT2TrFN. The normalization matrix X can be conducted as the following form:

$\begin{matrix} {\tilde{\tilde{x}}}_{ij}^{'} = & [(\frac{x_{ij 1}^{l}}{{\tilde{d}}_{j}}, \frac{x_{ij 2}^{l}}{{\tilde{d}}_{j}}, \frac{x_{ij 3}^{l}}{{\tilde{d}}_{j}}, \frac{x_{ij 4}^{l}}{{\tilde{d}}_{j}}; h_{ij}^{l}), (\frac{x_{ij 1}^{u}}{{\tilde{d}}_{j}}, \frac{x_{ij 2}^{u}}{{\tilde{d}}_{j}}, \frac{x_{ij 3}^{u}}{{\tilde{d}}_{j}}, \frac{x_{ij 4}^{u}}{{\tilde{d}}_{j}}; h_{ij}^{u})] \\ = & [(x_{ij 1}^{' l}, x_{ij 2}^{' l}, x_{ij 3}^{' l}, x_{ij 4}^{' l}; h_{ij}^{l}), (x_{ij 1}^{' u}, x_{ij 2}^{' u}, x_{ij 3}^{' u}, x_{ij 4}^{' u}; h_{ij}^{u})] \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n \end{matrix}$ (20)

In which, the parameter ${\tilde{d}}_{j}$ can be defined as follows:

$\begin{matrix} {\tilde{d}}_{j} = & \sqrt{\sum_{i = 1}^{m} \sum_{p = 1}^{4} (x_{ijp}^{l})^{2} + \sum_{i = 1}^{m} \sum_{p = 1}^{4} (x_{ijp}^{u})^{2}} \\ p = 1, 2, 3, 4 \end{matrix}$ (21)

Step 2: Identify the optimistic and pessimistic risk scores of risk factors

For each risk factor, the optimistic and pessimistic risk scores are denoted as follows:

$Y = {\begin{matrix} Y^{+} = (y_{1}^{+}, y_{2}^{+}, \dots y_{n}^{+}) \\ Y^{-} = (y_{1}^{-}, y_{2}^{-}, \dots y_{n}^{-}) \end{matrix}$ (22)

Where the elements $y_{j}^{+}$ and $y_{j}^{-}$ can be defined as follows:

$y_{j}^{+} = {\begin{matrix} max_{i} {\tilde{\tilde{x}}}_{ij}^{'}, if j \leq g \\ min_{i} {\tilde{\tilde{x}}}_{ij}^{'}, if j > g \end{matrix}$ (23)

$y_{j}^{-} = {\begin{matrix} min_{i} {\tilde{\tilde{x}}}_{ij}^{'}, if j \leq g \\ max_{i} {\tilde{\tilde{x}}}_{ij}^{'}, if j > g \end{matrix}$ (24)

Step 3: Calculate the distance between each risk score and the optimistic and/or pessimistic risk scores

According to the distance definition of two IT2TrFNs in Eqs. (10) and (11), the distance between each risk score and the optimistic and/or pessimistic scores can be calculated as follows: $d_{j}^{+} = \sum_{i = 1}^{m} d | {\tilde{\tilde{x}}}_{ij}^{'}, y_{j}^{+} |$ (25) $d_{j}^{-} = \sum_{i = 1}^{m} d | {\tilde{\tilde{x}}}_{ij}^{'}, y_{j}^{-} |$ (26)

Step 4: Determine the dispersion values of risk factors Based on the result obtained by step 3, the dispersion value of each risk factor c_j (j = 1, 2, ⋯ , n) can be derived by the following form:

$δ_{j} = d_{j}^{+} / (d_{j}^{+} + d_{j}^{-})$ (27)

Remark: the greater the dispersion value of risk factor is, the more important of risk factor c_j will be. This is in agreement with the core rule of risk factors weight determination.

Step 5: Obtain the importance weight of each risk factor

According to the result of step 4, we can obtain the final importance weight of each risk factor c_j as follows:

$ϖ_{j} = δ_{j} / \sum_{j = 1}^{n} δ_{j}$ (28)

Algorithm II. Determine risk priority by the extended MULTIMOORA approach

The FMEA based on conventional MULTIMOORA approach cannot calculate the relative importance of risk factors in the risk prioritization procedure, which may lead to a biased result for decision makers [8]. In order to obtain a reasonable risk priority ranking order, the extended MULTIMOORA based risk priorization method should take into account the relative importance of risk factors. Thus, motivated by the weighted MULTIMOORA approach [1], we propose an extended MULTIMOORA approach to determine the risk priority ranking order of each failure mode under interval type-2 fuzzy environment. The risk priorization method based on extended MULTIMOORA approach can be constructed as follows:

Step 1: Construct the extended ratio system

According to the normalization risk evaluation matrix and importance weight of each risk factor, the weighted-based ration system can be defined as follows:

${\tilde{y}}_{i}^{*} = \sum_{j = 1}^{g} ϖ_{j} {\tilde{\tilde{x}}}_{ij}^{'} - \sum_{j = g + 1}^{n} ϖ_{j} {\tilde{\tilde{x}}}_{ij}^{'}$ (29)

Where $y_{i}^{*}$ represents the risk evaluation value of failure mode FM_i (i = 1, 2, ⋯ , m). The parameter $y_{i}^{*}$ is an IT2TrFN, and it is denoted as:

$\begin{matrix} {\tilde{y}}_{i}^{*} = [({\tilde{y}}_{i 1}^{l}, {\tilde{y}}_{i 2}^{l}, {\tilde{y}}_{i 3}^{l}, {\tilde{y}}_{i 4}^{l}; h_{iy}^{l}), ({\tilde{y}}_{i 1}^{u}, {\tilde{y}}_{i 2}^{u}, {\tilde{y}}_{i 3}^{u}, {\tilde{y}}_{i 4}^{u}; h_{iy}^{u})] . \end{matrix}$

The parameter g indicates the number of risk factors to be maximized and n - g is the number of risk factors to be minimized.

Then, according to Eq. (14), we can identify the critical failure mode in the extended ratio system under IT2TrFN environment as follows:

$s_{It 2 s}^{*} = {{FM}_{i} | max_{i} {\tilde{y}}_{i}^{*}}$ (30)

Step 2: Construct the extended reference point approach

According to conventional MULTIMOORA approach, the first step of reference point approach is to identify reference points of overall risk factors. It can be accomplished by using Eqs. (11) and (31).

$r_{j}^{*} = {\begin{matrix} r_{j}^{+} = max_{i} {\tilde{\tilde{x}}}_{ij}^{'}, if j \leq g \\ r_{j}^{-} = min_{i} {\tilde{\tilde{x}}}_{ij}^{'}, if j > g \end{matrix}$ (31)

Then, the deviation between ${\tilde{\tilde{x}}}_{ij}^{'}$ and $r_{j}^{*}$ can be calculated as follows:

${\tilde{d}}_{ij}^{*} = d (ϖ_{j} r_{j}^{*}, ϖ_{j} {\tilde{\tilde{x}}}_{ij}^{'})$ (32)

After that, the risk evaluation value of each failure mode can be determined by the weight-based reference point approach under IT2TrFN environment as follows: $z_{i}^{*} = max_{j} {\tilde{d}}_{ij}^{*}$ (33)

The critical failure mode under the extended reference point approach can be identified by the following form: $s_{IT 2 p}^{*} = {{FM}_{i} | min_{i} z_{i}^{*}}$ (34)

Step 3: Construct the extended full multiplicative form approach

According to the conventional MULTIMOORA approach, the weight-based full multiplicative form approach under IT2TrFN environment can be defined as follows: $U_{i}^{*} = \frac{\prod_{j = 1}^{g} {({\tilde{\tilde{x}}}_{ij}^{'})}^{ϖ_{j}}}{\prod_{j = g + 1}^{n} ({\tilde{\tilde{x}}}_{ij}^{'})^{ϖ_{j}}}$ (35)

In which, the parameter $U_{i}^{*}$ is the risk evaluation value of failure mode FM_i (i = 1, 2, ⋯ , m). It should note that $U_{i}^{*}$ is also an IT2TrFN, which can be expressed as:

$\begin{matrix} U_{i}^{*} = [({\tilde{u}}_{i 1}^{l}, {\tilde{u}}_{i 2}^{l}, {\tilde{u}}_{i 3}^{l}, {\tilde{u}}_{i 4}^{l}; h_{iU}^{l}), ({\tilde{u}}_{i 1}^{u}, {\tilde{u}}_{i 2}^{u}, {\tilde{u}}_{i 3}^{u}, {\tilde{u}}_{i 4}^{u}; h_{iU}^{u})] \end{matrix}$

Then, the critical failure mode under the extended full multiplicative form approach can be found out by using Eqs. (11) and (36): $s_{IT 2 m}^{*} = {{FM}_{i} | max_{i} U_{i}^{*}}$ (36)

Step 4: Determine the final risk priority

Based on the results obtained by the aforementioned extended ratio system, the extended reference point approach, and the extended full multiplicative form approach, the final risk priority ranking order of each failure mode under the extended MULTIMOORA approach can be determined by the principle described in Section 2.2.

4. The risk priorization for FMEA based on extended MULTIMOORA under interval type-2 environment

In this section, we develop a risk priorization method for FMEA based on the extended MULTIMOORA within interval type-2 fuzzy context. First, we present a description of the FMEA based risk evaluation problem under interval type-2 fuzzy environment. Then, we present the risk priorization for FMEA based on the extended MULTIMOORA method within interval type-2 fuzzy context.

In this paper, we suppose that all the FMEA team members provide their risk evaluation information about failure modes by using the linguistic terms (see Table 1). Table 1 (adopted from [19]) provides the linguistic terms for determining the risk scores of risk factors and their corresponding IT2TrFNs.

Table 1
Linguistic terms and corresponding IT2TrFNs for risk scores [19]

Linguistic terms Interval type-2 trapezoid fuzzy number

Very high (VH) [(0.9,1,1,1;1);(0.95,1,1,1;0.9)]

High (H) [(0.7,0.9,0.9,1;1);(0.8,0.9,0.9,0.95;0.9)]

Slightly high (SH) [(0.5,0.7,0.7,0.9;1),(0.6,0.7,0.7,0.8;0.9)]

Medium (M) [(0.3,0.5,0.5,0.7;1),(0.4,0.5,0.5,0.6;0.9)]

Slight low (SL) [(0.1,0.3,0.3,0.5;1),(0.2,0.3,0.3,0.4;0.9)]

Low (L) [(0,0.1,0.1,0.3;1),(0.05,0.1,0.1,0.2;0.9)]

Very low (VL) [(0,0,0,0.1;1),(0,0,0,0.05;0.9)]

Linguistic terms	Interval type-2 trapezoid fuzzy number
Very high (VH)	[(0.9,1,1,1;1);(0.95,1,1,1;0.9)]
High (H)	[(0.7,0.9,0.9,1;1);(0.8,0.9,0.9,0.95;0.9)]
Slightly high (SH)	[(0.5,0.7,0.7,0.9;1),(0.6,0.7,0.7,0.8;0.9)]
Medium (M)	[(0.3,0.5,0.5,0.7;1),(0.4,0.5,0.5,0.6;0.9)]
Slight low (SL)	[(0.1,0.3,0.3,0.5;1),(0.2,0.3,0.3,0.4;0.9)]
Low (L)	[(0,0.1,0.1,0.3;1),(0.05,0.1,0.1,0.2;0.9)]
Very low (VL)	[(0,0,0,0.1;1),(0,0,0,0.05;0.9)]

4.1 The description of the FMEA based risk evaluation under interval type-2 fuzzy environment

Let us consider FMEA based risk evaluation problem as a decision making problem within interval type-2 fuzzy context. Suppose that there are l team members TM_k (k = 1, 2, ⋯ , l) in a risk assessment group responsible for the risk evaluation of m failure modes FM_i (i = 1, 2, ⋯ , m) in terms of n risk factors c₃ (j = 1, 2, ⋯ , n). Assume that the vector τ_k > 0 (k = 1, 2, ⋯ , l) satisfying $\sum_{k = 1}^{l} τ_{k} = 1$ is the importance weight assigned to each FMEA team member. Let ${\tilde{\tilde{χ}}}^{k} = {({\tilde{\tilde{x}}}_{ij}^{k})}_{m \times n}$ be an IT2TrFN risk evaluation matrix provided by member TM_k, where ${\tilde{\tilde{x}}}_{ij}^{k} = [(x_{ij 1}^{kl}, x_{ij 2}^{kl}, x_{ij 3}^{kl}, x_{ij 4}^{kl}; h_{ij}^{kl}), (x_{ij 1}^{ku}, x_{ij 2}^{ku}, x_{ij 3}^{ku}, x_{ij 4}^{ku}; h_{ij}^{ku})]$ is an IT2TrFN for the failure mode FM_i in terms of risk factor c_j.

4.2. The risk priorization for FMEA based on the extended MULTIMOORA method

The proposed risk priorization with extended MULTIMOORA method under interval type-2 fuzzy environment is illustrated in Fig. 2, which includes the following specific steps:

Fig.2

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

Step 1: Identify the potential failure modes

In this step, the potential failure modes of a risk analysis system are determined according to FMEA team members’ analysis result. The FMEA team, encompassing who possess related background knowledge and professional experience, is set up for identifying potential failure modes.

Step 2: Determine risk score of each risk factor

The scope of this step is to obtain the risk scores of risk factors. Once the failure modes are identified, the risk score of each risk factor in FMEA model can be determined. In the light of identified failure modes, the FMEA team members are invited to provide the risk rating for three risk factors, O, S and D by adopting the fuzzy linguistic variables shown in Table 1.

Step 3: Aggregate the risk evaluation information

Based on the risk evaluation information provided by FMEA team members, the aggregated risk evaluation matrix X can be obtained as follows:

$X = \sum_{k = 1}^{l} τ_{k} {\tilde{\tilde{χ}}}^{k} = {[{\tilde{\tilde{x}}}_{ij}]}_{m \times n}$ (37)

Step 4: Determine the weights of risk factors

According to the aggregated risk evaluation matrix X obtained by Step 3, the importance weight vector ϖ_j (j = 1, 2, ⋯ , n) of each risk factor c_j can be calculated by using Algorithm I.

Step 5: Obtain the risk priorities of failure modes.

Based on the result of Step 3, the final risk priority ranking order of each failure mode FM_i can be determined by employing Algorithm II.

5. An illustrative example

In this section, an illustrative risk evaluation case study in a steel company is presented to demonstrate the application and feasibility of the proposed FMEA approach. Furthermore, a comparison study is also conducted to validate the effectiveness of the proposed risk priorization method in FMEA approach.

5.1 Implementation

Step 1: Identify the potential failure modes

To illustrate the proposed hybrid approach for risk priorization in FMEA model, a real case of the risk evaluation in a steel company [1] is adopted in this subsection. The specific failure modes of the steel production process are provided in Table 2.

Table 2
Failure modes of the steel production process [1]

The nodes of failure modes The description of failure modes

FM1 Non acceptable formation

FM2 Nipple thread pitted

FM3 Arc formation loss

FM4 Burn-out electrode

FM5 Breaking of house of pipe

FM6 Problem in movement of arm

FM7 Refractory damage

FM8 Formation of steam

FM9 Refractory line damage

FM10 Movement of roof stop

The nodes of failure modes	The description of failure modes
FM1	Non acceptable formation
FM2	Nipple thread pitted
FM3	Arc formation loss
FM4	Burn-out electrode
FM5	Breaking of house of pipe
FM6	Problem in movement of arm
FM7	Refractory damage
FM8	Formation of steam
FM9	Refractory line damage
FM10	Movement of roof stop

Step 2: Determine risk score of each risk factor

In order to obtain the risk evaluation information of three risk factors c_j (j = 1, 2, 3) under each failure mode, a group of FMEA team including three members TM_k (k = 1, 2, 3) is set up to perform the risk evaluation. The FMEA team members are invited to determine the risk scores of risk factors under each failure mode by using the linguistic terms shown in Table 1.

Step 3: Aggregate the risk evaluation information

According to the risk evaluation information aggregation method, the importance weight vector for the three FMEA team members is set as τ_k = (0.35, 0.4, 0.25) by considering the relative importance of each FMEA team member. Then, the aggregated risk evaluation information can be obtained according to Eq. (37) and the result is shown in Table 3.

Table 3

The interval type-2 trapezoid fuzzy risk evaluation matrix

Failure modes	O	S	D
FM1	[(0,0.06,0.06,0.22;1), (0.03,0.06,0.06,0.14;0.9)]	[(0.605,0.74,0.74,0.825;1), (0.673,0.74,0.74,0.783;0.9)]	[(0.04,0.18,0.18,0.38;1), (0.11,0.18,0.18,0.28;0.9)]
FM2	[(0,0.06,0.06,0.22;1), (0.03,0.06,0.06,0.14;0.9)]	[(0.65,0.85,0.85,0.975;1), (0.75,0.85,0.85,0.913;0.9)]	[(0.82,0.96,0.96,1;1), (0.89,0.96,0.96,0.98;0.9)]
FM3	[(0,0,0,0.1;1), (0,0,0,0.05;0.9)]	[(0.85,0.975,0.975,1;1), (0.913,0.975,0.975,0.988;0.9)]	[(0.065,0.23,0.23,0.43;1), (0.148,0.23,0.23,0.33;0.9)]
FM4	[(0,0,0,0.1;1), (0,0,0,0.05;0.9)]	[(0.1,0.3,0.3,0.5;1), (0.2,0.3,0.3,0.4;0.9)]	[(0,0.035,0.035,0.17;1), (0.018,0.035,0.035,0.103;0.9)]
FM5	[(0,0,0,0.1;1), (0,0,0,0.05;0.9)]	[(0.85,0.975,0.975,1;1), (0.913,0.975,0.975,0.988;0.9)]	[(0.1,0.3,0.3,0.5;1), (0.2,0.3,0.3,0.4;0.9)]
FM6	[(0,0,0,0.1;1), (0,0,0,0.05;0.9)]	[(0.85,0.975,0.975,1;1), (0.913,0.975,0.975,0.988;0.9)]	[(0.1,0.3,0.3,0.5;1), (0.2,0.3,0.3,0.4;0.9)]
FM7	[(0,0.1,0.1,0.3;1), (0.05,0.1,0.1,0.2;0.9)]	[(0.28,0.48,0.48,0.68;1), (0.38,0.48,0.48,0.58;0.9)]	[(0,0.06,0.06,0.22;1), (0.03,0.06,0.06,0.14;0.9)]
FM8	[(0,0.1,0.1,0.3;1), (0.05,0.1,0.1,0.2;0.9)]	[(0.1,0.3,0.3,0.5;1), (0.2,0.3,0.3,0.4;0.9)]	[(0.73,0.89,0.89,0.975;1), (0.81,0.89,0.89,0.933;0.9)]
FM9	[(0,0,0,0.1;1), (0,0,0,0.05;0.9)]	[(0.85,0.975,0.975,1;1), (0.913,0.975,0.975,0.988;0.9)]	[(0.1,0.3,0.3,0.5;1), (0.2,0.3,0.3,0.4;0.9)]
FM10	[(0.065,0.23,0.23,0.43;1), (0.148,0.23,0.23,0.33;0.9)]	[(0.73,0.89,0.89,0.975;1), (0.81,0.89,0.89,0.933;0.9)]	[(0.31,0.51,0.51,0.675;1), (0.41,0.51,0.51,0.593;0.9)]

Step 4: Determine the weights of risk factors

In order to obtain the weights of risk factors, we should standardize the risk evaluation matrix. However, all of the three risk factors are to be maximized and are presented by the generalized IT2TrFN, so, the standardized operation should not be conducted. Then, according to the distance-based weighted method and the Eq. (27), we can calculate the dispersion value of each risk factor, and the result is denoted as δ_j = (0.399, 0.479, 0.530). Finally, according to Eq. (28), we can calculate the importance weight of each risk factor as ϖ_j = (0.283, 0.341, 0.376).

Step 5: Obtain the risk priorities of failure modes

Based on the weights of risk factors obtained by Step 4, then, we can use the risk priorization method based on extended MULTIMOORA to obtain the risk priority ranking order of each failure mode. First, the risk evaluation values and risk priority ranking order of each failure mode under the extended ratio system can be obtained by using Eqs. (29) and (30). The result is provided in Table 4. Second, the distance of each risk factor can be obtained by using Eqs. (31) and (32). And, the risk evaluation value and priority of each failure mode can be determined by Eqs. (33) and (34) and the result is shown in Table 5. Third, the risk evaluation values and risk priority ranking orders of overall failure modes under the extended full multiplicative form approach can be derived by applying Eqs. (35) and (36). The result is shown in Table 6.

Table 4

The risk evaluation and priorization of failure modes under the extended ratio system

Failure modes	Risk evaluation values	Ranking
FM1	[(0.221,0.337,0.337,0.486;1),(0.279,0.337,0.337,0.412;0.9)]	8
FM2	[(0.530,0.668,0.668,0.771;1),(0.599,0.668,0.668,0.719;0.9)]	1
FM3	[(0.314,0.419,0.419,0.531;1),(0.367,0.419,0.419,0.475;0.9)]	4
FM4	[(0.034,0.115,0.115,0.263;1),(0.075,0.115,0.115,0.189;0.9)]	10
FM5	[(0.327,0.445,0.445,0.557;1),(0.386,0.445,0.445,0.501;0.9)]	5
FM6	[(0.327,0.445,0.445,0.557;1),(0.386,0.445,0.445,0.501;0.9)]	5
FM7	[(0.095,0.215,0.215,0.400;1),(0.155,0.215,0.215,0.307;0.9)]	9
FM8	[(0.309,0.465,0.465,0.622;1),(0.387,0.465,0.465,0.544;0.9)]	3
FM9	[(0.327,0.445,0.445,0.557;1),(0.386,0.445,0.445,0.501;0.9)]	5
FM10	[(0.384,0.560,0.560,0.708;1),(0.472,0.560,0.560,0.634;0.9)]	2

Table 5

The risk evaluation and priorization of failure modes under the extended reference point approach

Failure modes	Distance of each risk factor			Risk evaluation value	Ranking
	d_i1	d_i2	d_i3
FM1	0.027	0.123	0.067	0.123	2
FM2	0.027	0.168	0.313	0.313	10
FM3	0.000	0.188	0.084	0.188	5
FM4	0.000	0.000	0.000	0.000	1
FM5	0.000	0.188	0.108	0.188	5
FM6	0.000	0.188	0.108	0.188	5
FM7	0.046	0.180	0.015	0.180	4
FM8	0.046	0.000	0.299	0.299	9
FM9	0.000	0.188	0.108	0.188	5
FM10	0.079	0.173	0.177	0.177	3

Table 6

The risk evaluation and priorization of failure modes under the extended full multiplicative form approach

Failure modes	Risk evaluation values	Ranking
FM1	[(0.000,0.214,0.214,0.424;1),(0.141,0.214,0.214,0.327;0.9)]	6
FM2	[(0.000,0.420,0.420,0.646;1),(0.322,0.420,0.420,0.551;0.9)]	2
FM3	[(0.000,0.000,0.000,0.379;1),(0.000,0.000,0.000,0.281;0.9)]	4
FM4	[(0.000,0.000,0.000,0.211;1),(0.000,0.000,0.000,0.133;0.9)]	10
FM5	[(0.000,0.000,0.000,0.402;1),(0.000,0.000,0.000,0.302;0.9)]	8
FM6	[(0.000,0.000,0.000,0.402;1),(0.000,0.000,0.000,0.302;0.9)]	8
FM7	[(0.000,0.141,0.141,0.353;1),(0.082,0.141,0.141,0.251;0.9)]	7
FM8	[(0.000,0.331,0.331,0.556;1),(0.229,0.331,0.331,0.452;0.9)]	3
FM9	[(0.000,0.000,0.000,0.402;1),(0.000,0.000,0.000,0.302;0.9)]	8
FM10	[(0.267,0.492,0.492,0.674;1),(0.387,0.492,0.492,0.586;0.9)]	1

Finally, according to the risk priority ranking orders of overall failure modes obtained by the three methods in extended MULTIMOORA approach and the dominance theory, the final risk priority ranking orders of overall failure modes under the extended MULTIMOORA approach can be summarized in Table 7.

Table 7

The final ranking order of failure modes under the extended MULTIMOORA approach

Failure modes	The ratio system	The reference point approach	The full multiplicative approach	Final ranking
FM1	8	2	6	8
FM2	1	10	2	2
FM3	4	5	4	4
FM4	10	1	10	10
FM5	5	5	8	5
FM6	5	5	8	5
FM7	9	4	7	9
FM8	3	9	3	3
FM9	5	5	8	5
FM10	2	3	1	1

5.2 Comparisons and discussions

In order to further illustrate the effectiveness of the proposed risk priorization method in FMEA model, the case mentioned above is adopted to analyze some similar risk priorization methods existing on the FMEA model, including the traditional RPN method (method 1), the crisp MULTIMOORA method (method 2) and the MULTIMOORA method based on interval-valued intuitionistic fuzzy numbers (IVIF-MULTIMOORA) [1](method 3). The final risk priority ranking order of each failure mode determined by these risk priorization methods is provided in Table 8.

Table 8
Risk priorities of failure modes by different approaches

Failure mode method 1 method 2 method 3 The proposed method

FM1 8 8 8 8

FM2 5 2 2 2

FM3 6 4 6 4

FM4 10 10 10 10

FM5 2 5 2 5

FM6 2 5 2 5

FM7 9 9 9 9

FM8 6 3 6 3

FM9 2 5 2 5

FM10 1 1 1 1

Failure mode	method 1	method 2	method 3	The proposed method
FM1	8	8	8	8
FM2	5	2	2	2
FM3	6	4	6	4
FM4	10	10	10	10
FM5	2	5	2	5
FM6	2	5	2	5
FM7	9	9	9	9
FM8	6	3	6	3
FM9	2	5	2	5
FM10	1	1	1	1

As it can be seen from Table 8, the failure mode FM10 has the highest risk priority, the failure mode FM4 has the lowest risk priority and the failure mode FM7 ranks ninth in the four methods. Moreover, the ranking order of each failure mode is completely consistent with the result obtained by IVIF-MULTIMOORA approach. These results indicate that there is a consensus among the four risk priorization methods. This implies that the proposed FMEA approach is effective to determine risk priority.

From Table 8, there are also some differences between the proposed method and other two methods (except the IVIF-MULTIMOORA). These differences can be explained as follows: (i) The traditional RPN approach and the Crisp MULTIMOORA do not take into account the uncertainty in the risk evaluation. (ii) The different important weights of risk factors are not considered in the traditional RPN approach and the Crisp MULTIMOORA approach. In addition, Table 8 exhibits that the failure modes FM3 and FM8 have the same risk priority ranking orders in the traditional RPN approach and the Crisp MULTIMOORA approach. However, it can be seen from Table 3, the risk score of each risk factor is evidently different under failure modes FM3 and FM8. On the other hand, the proposed risk prioritization method based with the extended MULTIMOORA and distance-based method can distinguish the two failure modes from each other, which can provide more reasonable and reliable risk evaluation information for decision makers than the traditional FMEA method.

The comparison analysis aforementioned indicates that the risk priority ranking order derived by the proposed FMEA framework is more reasonable and robust. The application of IT2FSs to reflect higher uncertainty on the risk evaluation, which demonstrates the need of considering the uncertainty and fuzziness in risk evaluation for FMEA approach. More importantly, the risk priority ranking order determined by the extended MULTIMOORA approach is more perceptible in case of risk assessment.

6 Conclusion

In order to overcome the weaknesses of the traditional FMEA approach, a hybrid FMEA based on the extended MULTIMOORA method is proposed. In the proposed framework, the interval type-2 fuzzy set is applied to model the higher uncertainty in risk evaluation. Next, the objective importance weight of each risk factor is determined by using the distance-based method, which can help to derive a more reliable weight vector of risk factors. Then, the risk priority of each risk factor is determined by the extended MULTIMOORA approach. Finally, the practical case study and comparison analysis are conducted to illustrate the effectiveness of the proposed framework. The analysis results demonstrate the effectiveness and reliability of the extended MULTIMOORA over other risk priorization approaches. The proposed approach can provide more reasonable risk priority ranking order, which can assist the steel company in identifying the most serious risk failure modes during the procedures of sheet steel production. Moreover, the proposed risk prioritization method can also be adopted to analyze risk of failure modes in other fields, especially the complex systems.

In this paper, the interaction relationships among risk factors do not consider in the risk prioritization procedure. Thus, in the future research, it is recommended to model the interaction relationships among risk factors in the risk evaluation procedure. Besides, there are only three risk factors in this paper, therefore, more risk factors should be considered in the further research to develop a more reliable FMEA.

Footnotes

Acknowledgement

The work is supported by the National Science Foundation of China (NSFC) (71771051, 71371049 and 71701158) and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (17YJC630114) and the Scientific Research Foundation of Graduate School of Southeast University (YBPY1876) and Fundamental Research Funds for the Central Universities under the Projects 2018IVB036 and 2019VI030.

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Risk priorization for failure modes with extended MULTIMOORA method under interval type-2 fuzzy environment

Abstract

Keywords

1 Introduction

2. Preliminaries

2.1 Interval type-2 fuzzy set

4.2. The risk priorization for FMEA based on the extended MULTIMOORA method

5.1 Implementation

Table 8 Risk priorities of failure modes by different approaches Failure mode method 1 method 2 method 3 The proposed method FM1 8 8 8 8 FM2 5 2 2 2 FM3 6 4 6 4 FM4 10 10 10 10 FM5 2 5 2 5 FM6 2 5 2 5 FM7 9 9 9 9 FM8 6 3 6 3 FM9 2 5 2 5 FM10 1 1 1 1

Footnotes

Acknowledgement

References

Table 8
Risk priorities of failure modes by different approaches

Failure mode method 1 method 2 method 3 The proposed method

FM1 8 8 8 8

FM2 5 2 2 2

FM3 6 4 6 4

FM4 10 10 10 10

FM5 2 5 2 5

FM6 2 5 2 5

FM7 9 9 9 9

FM8 6 3 6 3

FM9 2 5 2 5

FM10 1 1 1 1