An intuitionistic fuzzy cloud model-based risk assessment method of failure modes considering hybrid weight information

Abstract

Failure mode and effects analysis (FMEA) is an effective tool utilized in various fields for discovering and eliminating potential failures in products and services, which is usually implemented based on experts’ linguistic assessments. However, incomprehensive weigh information of risk factors and experts, lacking the consideration of experts’ randomness and hesitation, and incomplete risk factor system is essential challenges for the traditional FMEA model. Therefore, to properly handle these challenges and further enhance the performance of the traditional FMEA, this study develops a new FMEA strategy for assessing and ranking failures’ risks. First, a novel concept of intuitionistic fuzzy clouds (IFCs) is developed by combining the merits of the intuitionistic fuzzy set theory and the cloud model theory in manipulating uncertain information. Some basic operations and the Minkowski-type distance measure of IFCs are also presented and discussed. Further, in the proposed FMEA model, two combination weighting methods are developed to determine the synthetic weights of experts and risk factors, respectively, which consider subjectivity and objectivity simultaneously. In addition, maintenance (M) is considered as a new risk factor to enrich the assessment factor system and facilitate a more reasonable risk assessment result. Finally, a case study is implemented along with comparisons to demonstrate the feasibility and superiority of the presented FMEA model.

Keywords

Failure mode and effects analysis (FMEA)intuitionistic fuzzy set theory cloud model risk analysis technique for order performance by similarity to ideal solution (TOPSIS)

Nomenclature

Acronyms and abbreviations

FMEA

Failure mode and effects analysis

IFS

Intuitionistic fuzzy set

IFN

Intuitionistic fuzzy number

TOPSIS

Technique for order performance by similarity to ideal solution

IFC

Intuitionistic fuzzy cloud

IFC-FMEA

Intuitionistic fuzzy cloud FMEA

Occurrence

Severity

Detection

Maintenance

MAGDM

Multi-attribute group decision making

RPN

Risk priority number

AHP

Analytic hierarchy process

IFHWED

Intuitionistic fuzzy hybrid weighted Euclidean distance

IFWA

Intuitionistic fuzzy weighted averaging

RDEA

Robust data envelopment analysis

PLTSs

linguistic term sets

VIFS

Variant of intuitionistic fuzzy set

VIFN

Variant of intuitionistic fuzzy number

IFC-PIS

IFC positive ideal solution

IFC-NIS

IFC negative ideal solution

Notations

Expectation

Entropy

Hyper entropy

Iex

IFC expectation

Ien

IFC entropy

Ihe

IFC hyper entropy

M_k

The kth expert

F_i

The ith failure mode

R_j

The jth risk factor

T_k

Linguistic assessment matrix given by kth expert

{\tilde{T}}_{k}

IFNs assessment matrix

IC_k

IFC assessment matrix given by kth expert

\bar{IC}

Synthetic IFC assessment matrix

\bar{\bar{IC}}

Synthetic weighted IFC assessment matrix

w_{k}^{s}

Subjective weight of expert kth expert

w_{k}^{o}

Objective weight of expert kth expert

w _k

Synthetic weight of expert kth expert

p_{j}^{s}

Subjective weight of jth risk factor

p_{j}^{o}

Objective weight of jth risk factor

p_{j}

Synthetic weight of jth risk factor

d_{i}^{+}

The distance of ith failure mode and cloud positive ideal solution

d_{i}^{-}

The distance of ith failure mode and cloud negative ideal solution

Effective interval

d (IC₁, IC₂)

The distance between IFCs IC₁ and IC₂

1 Introduction

As a systematic risk evaluation and reliability management instrument, failure mode and effect analysis (FMEA) has been commonly employed to identify and eliminate known or potential failures, errors, and defects damaging systems quality and performance [15, 20]. FMEA is usually performed by a group team composing of individual experts from different departments to assess and determine vital failure modes and thus assist managers in developing corresponding improvement measures for systems’ reliability [16]. Different from other quality management methods, its main concern is to extract and analyze possible failures and evaluate the effects of all failure modes, thus prevent the crucial failures through conducting pre-control measures rather than take the maintenances after the occurrence of failures. Due to its merits, FMEA has been widely utilized in numerous fields for systems’ quality, such as road transportation [8], steel industry [54], medical industry [60], and renewable energy system [2].

In the conventional FMEA process, the risk priority number (RPN) is applied to implement the risk assessment via calculating the product of the risk evaluation crisp value of three risk factors, i.e., occurrence (O), severity (S), and detection (D). A 10 point-scale system is used to assess the relative importance of each risk factor with respect to different failure modes in systems. The higher the product value of the three factors, the bigger the risk levels of the corresponding failure modes. Based on RPN calculation results, the risk orders of all failure modes are identified, and thus corrective actions are formulated to enhance the stability and quality of the product. However, it is generally imprecise and irrational for experts to assess the RPN value with crisp numbers in practical FMEA processes due to the fuzziness and vagueness of human knowledge. To dispose of the mentioned setbacks, a lot of improved researches based on uncertain theories, such as fuzzy set theory [62], grey theory [37], rough set theory [12], have been proposed by various researchers in recent years. In those improved FMEA methods, multi-attribute decision group making (MADGM) methods based on fuzzy set theory are the most widely applied. For instance, Yazdi et al. [62] extended classical FMEA to the fuzzy environment built a fuzzy developed FMEA model to manage the aircraft system’s accidents. Boral [3] combined the fuzzy analytic hierarchy process (AHP) with other fuzzy decision theories established some new integrated fuzzy-based MADGM models. Based on the limitations of the conventional RPN approach, Daneshvar et al. [9] proposed a fuzzy smart FMEA model by integrating fuzzy set theory, AHP method, and data envelopment analysis to address the vagueness in assessment procedure and enhance the reliability of evaluation information. Those fuzzy-based FMEA approaches allow experts to provide the evaluation information of risk factors with fuzzy linguistic variables. They enable them to quantify those linguistic variables by using the fuzzy set theory, which can effectively deal with the uncertainty and vague message in decision-making.

1.1 Motivation

Although numerous improved FMEA methods have been developed to enhance the performance of the traditional FMEA model, there still are some research gaps in the existing references, which are worth studying. The most important ones are shown as following contents [27 , 51]:

As two powerful tools in dealing with the intrapersonal perception uncertainty, the intuitionistic fuzzy sets (IFSs) theory and the cloud model have been applied to perform the improvement of FMEA, such as Mirghafoori et al. [40]; Wang et al. [56]. However, the previous IFS-based and clod model-based references have some limitations, e.g., the IFS-based methods lack the mechanism of dealing with the uncertainty of the interval, called randomness, and the cloud model-based methods may not precisely reflect the real preferences of decision-makers due to that adjacent clouds have numerous reduplicative scopes. These deficiencies may potentially affect the precision of the IFS-based and the cloud model-based methods.

Measuring the distance of two objects is a significant topic in the fuzzy decision-making fields. The most commonly applied distance measures include the Hamming, Euclidean, and Chebyshev distances. However, the previous FMEA methods usually only consider one of these measures, which may affect the risk rankings of failure modes because different measures have inconsistent measure mechanisms.

The combined weights of experts and risk factors, which consider the subjective and objective aspects, are important for risk assessment problems. However, the previous references usually consider one of the subjective weights or the objective weights of experts and risk factors or ignore the weights of experts or risk factors, which may lead to information loss.

Most of the existing improved FMEA methods often employ the three traditional risk aspects, i.e., O, S, and D. However, With the increase of the complexity of the decision circumstance and the scale of the factors system, this may lead to the ignorance of important information.

Based on the above discussions, there is a practical demand to develop an improved FMEA strategy to tackle these limitations, and thus to further facilitates the performance of FMEA strategy in facilitating reliability.

1.2 Novelties of the study

In this study, a new integrated IFC-FMEA risk evaluation model based on IFS theory, cloud model, and TOPSIS is proposed for improving the effectiveness and precision of the FMEA technique. The features of the study can be described in the following aspects:

A new concept of intuitionistic fuzzy clouds (IFCs) is developed by integrating the merits of the intuitionistic fuzzy sets in expressing the preferred and non-preferred degrees of decision-makers and the strengths of the cloud model theory in characterizing the randomness of experts’ judgments, and the basic operations of IFCs are also defined and discussed. Compared with the traditional intuitionistic fuzzy sets and the cloud model theories, the proposed IFCs can express decision-makers’ real preferences.

A novel Minkowski-type distance measure of IFCs, which is a parameter generalization of the Hamming, Euclidean, and Chebyshev, is developed, and some vital properties are also examined.

Two overall weighting methods are developed to compute the relative weights of FMEA experts and risk factors, respectively, which enhance the reliability and effectiveness of failure modes’ final ranking results.

In addition to the original risk factors O, S, and D, maintenance (M) is also considered as a novel risk factor to attain more reasonable risk assessments for failure modes. Additionally, the technique for order performance by similarity to ideal solution (TOPSIS) method is extended to the IFC environment to determine the risk orders of failure modes, which improves the performance of the traditional TOPSIS in capturing the uncertain preferences.

1.3 Structure of the study

The rest of this study is presented as follows: Section 2 presents a brief literature review for improved FMEA methods and introduces two essential uncertainty theories. In Section 3, the new IFC theory is developed by integrating the two uncertainty theories, and then its significant algorithms and a brief comparison between the IFC theory and the traditional normal cloud are presented. According to the developed IFC theory, a novel FMEA model, namely, the IFC-FMEA method, is proposed in Section 4. Section 5 provides an application case of the novel FMEA model. The result discussions are discussed in Section 6. Conclusions and future works are summarized in Section 7. The sequential steps of this study are shown in Fig. 1.

Fig. 1

The flowchart of the whole study.

2 Literature review

In this section, various improved FMEA strategies are mainly reviewed, and two essential uncertainty theories used in previous researches are briefly introduced and reviewed.

2.1 Improvements of FMEA

In recent years, many modified risk assessment methods have been developed to deal with the uncertain information in assessments provided by FMEA experts. For example, Liu et al. [28] described an FMEA approach under the intuitionistic fuzzy hybrid weighted Euclidean distance (IFHWED) operator and intuitionistic fuzzy weighted averaging (IFWA) operator. In this study, a linguistic term system and relative intuitionistic fuzzy numbers (IFNs) were built by FMEA team members to depict the risk levels of failure modes. Parameshwaran et al. [42] developed an integrated framework for new mechatronics product development and used an FMEA strategy based on fuzzy TOPSIS and fuzzy AHP to rank the potential failure modes that have been identified from the prototype model of the new developed mechatronic product. Except for the fuzzy TOPSIS, there are some other extended TOPSIS methods that have been performed to deal with the MADGM problems similar to FMEA [14 , 48]. Lo and Liou [37] evaluated the risk priority of failures on the basis of grey theory, in which the grey interval linguistic terms are used to handle the vagueness of assessment information, and the probability-based grey relational analysis method is applied to compute the risk priority number (RPN). Yousefi et al. [63] presented an integrated robust data envelopment analysis (RDEA) FMEA model to assess the safety and environment risks by considering both traditional input parameters (i.e., O, S, and D) and two extra output parameters (i.e., cost and duration of treatment). Wan et al. [53] proposed an FMEA model for coping with the fuzziness of FMEA members, in which the crisp assessment values of risk factors and failure modes are obtained based on experts’ experiences. Then, the crisp values are transformed into interval values by rough set theory. Finally, the extended rough TOPSIS approach is used to evaluate the closeness coefficient between the risk evaluation of each failure mode and the ideal solution. Li et al. [24] presented a comprehensive method by combining probabilistic linguistic term sets (PLTSs) and fuzzy Petri nets (FPNs) for the risk assessment and prioritization of failure modes. The PLTSs are employed to address the uncertainty of experts’ subjective evaluations, and the FPNs are structured to attain the risk rankings of the identified failure modes. Sang et al. [46] performed an improved fuzzy FMEA model using the genetic algorithm, fuzzy membership functions, and monotone fuzzy rules. For the FMEA model, the genetic algorithm is used to help the design of membership functions for RPN elements O, S, and D, and then combined through a set of monotone fuzzy rules to assess the risk rating of the discovered failure modes. Other studies were also conducted to cope with fuzzy and uncertain assessment by utilizing 2-tuple linguistic variables [31], fuzzy evidential reasoning rules [20, 27], grey relational analysis [52], and others.

Based on the reviews mentioned above, the existing approaches for handling risk evaluation uncertainties in the FMEA process can be mainly divided into three aspects, that is, membership degree-based approaches, ordinal scales-based linguistic calculation approaches, and linguistic two tuples-based approaches. However, the first approach can reflect vagueness but ignores randomness, and the last two approaches cannot yield an effective expression of either randomness or hesitation of linguistic assessment [55]. In contrast, the new extended IFC method developed in this paper can describe the vagueness, hesitation, and randomness of qualitative information, and thus can effectively and fully express the various uncertainties of experts’ judgments. Besides, lots of MADGM approaches, especially the TOPSIS approach, have been applied to improve the reliability of the traditional risk assessment method under uncertain environments. However, the single fuzzy set theory is s not powerful enough to treat various uncertainties in the application of the traditional TOPSIS. Hence, in this study, a novel risk assessment model based on the IFC method and TOPSIS method is developed to evaluate and rank all identified failure modes in FMEA. The novel FMEA model can not only effectively solve the vagueness, hesitation, and randomness of risk evaluations but also consider the comprehensive weights of experts and risk factors, and further extend the TOPSIS to IFC circumstance in dealing with complicated risk management issues.

2.2. IFS

The conception of IFS is first developed by Atanassov [1] as an expansion of the fuzzy sets theory, which is considered as a better method than fuzzy sets theory in dealing with vagueness information in practical decision-making problems by using membership function and non-membership function simultaneously rather than only utilizing membership function like fuzzy sets theory. In recent years, the IFS theory has become an important method in disposing of some decision-information systems because of its outstanding capability in handling imprecise and vagueness information under uncertain environment. The notions of IFS can be stated as follows.

Definition 1 [6, 32]: Suppose X is a fixed non-empty set, an IFS A in X can be expressed as follows: $A = {〈 x, μ_{A} (x), f_{A} (x) 〉 | x \in X}$ (1) where μ_A (x) and f_A (x) indicate the degrees of membership and non-membership of x to A, respectively; whilst μ_A (x) and f_A (x) are restrained by the conditions of μ_A (x) , f_A (x) ∈ [0, 1] and 0 ⩽ μ_A (x) + f_A (x) ⩽1, ∀ x ∈ X. Hence, the uncertainty (or called hesitancy degree) can be depicted by the function of ∂_A (x) =1 - μ_A (x) - f_A (x). Obviously, ∂_A (x) ∈ [0, 1] , ∀ x ∈ X. The relationships between μ_A (x), f_A (x), and ∂_A (x) are shown in Fig. 2.

Fig. 2

The relationship of μ_A (x), f_A (x) and ∂_A (x).

As illustrated in Fig. 2, When ∂_A (x) =0, ∀ x ∈ X, that is μ_A (x) =1 - f_A (x), the IFS degenerates to a classical fuzzy set. If either condition μ_A (x) =1 & f_A (x) =0 or condition μ_A (x) =0 & f_A (x) =1 is satisfied, the IFS A converts into a certain set. Therefore, the classical fuzzy set and certain set are the special forms of IFS. Usually, a set β (x) = (x, μ_β (x) , f_β (x)) is regarded as an intuitionistic fuzzy number (IFN) for an IFS, which can be simplified as β = (μ_β, f_β).

IFS theory has been wildly used to handle the vagueness of experts’ linguistic evaluations in numerous FMEA strategies. For example, Liu and Xiao [36] applied IFS to analyze the risks of failure modes. Efe et al. [11] integrated intuitionistic fuzzy sets with linear programming to overcome some limitations of traditional FMEA. Mirghafoori et al. [41] utilized intuitionistic fuzzy theory to predict and identify the failure modes in FMEA strategy.

2.3 Cloud model theory

The cloud model [22] is a novel artificial intelligence method for dealing with uncertainty based on probability theory and fuzzy sets theory. It can describe the vagueness and randomness of the concepts in human thought perfectly and make the transformation between qualitative concepts and quantitative values possible [50].

Definition 2 [22, 23]: Suppose T is a qualitative concept defined on a universe of discourse U, x (x∈U) is a random instantiation of the qualitative concept T, and G_T(x) indicates the membership degree of x belonging to T, which corresponds to a random number following a probability distribution rather than a specific crisp value. The distribution of x in the universe of discourse U is called a membership cloud, or simply, a cloud, and x is called a cloud drop.

Definition 3 [22, 50]: The characters of a cloud y are expressed by three quantitative values, i.e., expectation Ex, entropy En, and hyper entropy He. Ex is the center value of the qualitative concept domain, En reflects the vagueness and randomness of the qualitative concept, and He represents the uncertainty of the membership function and the dispersion degree of a cloud’s droplets. Usually, a cloud can be denoted as y=(Ex, En, He).

The normal cloud model, which is based on Gaussian membership function and normal distribution, is the most universal and applicable form and can reflect various uncertain and random phenomena in reality.

Definition 4 [43, 55]: For a qualitative concept T defined in a universe of discourse U, suppose a random instantiation x (x∈U) in T satisfies the following equations: $x \sim N (Ex, E n^{' 2})$ (2) ${En}^{'} \sim N (En, H e^{2})$ (3) That is, x follows the normal distribution with expectation Ex and variance En′; En′ follows the normal distribution with expectation En and variance He. In addition, if the certainty degree of x belonging to the concept T satisfies the following distribution: $y^{'} = e^{- \frac{{(x - Ex)}^{2}}{2 E {n^{'}}^{2}}}$ (4) where y′ is a probability distribution of x in U. En′ is an intermediate variable satisfying Equation (3), which is a bridge between y′ and En and He. Then, y′ is defined as a normal cloud and simply expressed as y′ = (Ex, En, He). Ex, En, and He are the characteristic values of the normal cloud y′, namely expectation, entropy, and hyper entropy, respectively.

Definition 5 [17, 59]: Let ${\tilde{y}}_{1}$ =(Ex₁, En₁, He₁) and ${\tilde{y}}_{2}$ =(Ex₂, En₂, He₂) be any two normal clouds, and λ > 0, then basic operational rules of clouds are presented as follows: ${\tilde{y}}_{1} + {\tilde{y}}_{2} = (E x_{1} + E x_{2}, \sqrt{{En}_{1}^{2} + {En}_{2}^{2}}, \sqrt{{He}_{1}^{2} + {He}_{2}^{2}})$ (5) $\begin{matrix} {\tilde{y}}_{1} \times {\tilde{y}}_{2} = \\ (E x_{1} E x_{2}, \sqrt{{(E n_{1} E x_{2})}^{2} + {(E n_{2} E x_{1})}^{2}}, \\ \sqrt{{(H e_{1} E x_{2})}^{2} + {(He 2 E x_{1})}^{2}}) \end{matrix}$ (6) $λ {\tilde{y}}_{1} = (λ E x_{1}, \sqrt{λ} E n_{1}, \sqrt{λ} H e_{1})$ (7)

Some researchers have adopted the cloud model to analyze the risks of failures in the FMEA model. For instance, For instance, in the aspect of risk priority judgment, Liu et al. [26] estimated the risk of failure modes by cloud model theory under the uncertain setting. Li et al. [23] applied cloud model theory to manipulate the randomness of experts’ risk assessments. Huang et al. [18] described a modified FMEA approach to assess the risk priority of failure modes. The cloud model is combined with rough number theory to handle the uncertainty of risk assessment information for failures.

3 The novel IFC theory

In this section, a novel IFC theory for supporting the proposed FMEA model is developed. First, the novel IFC theory and its relative definitions are introduced. Then various operational rules for the novel IFC theory are introduced. Finally, to illustrate the virtues of the IFC theory, a brief comparison between the IFC theory and the traditional cloud theory is discussed.

3.1 IFC theory

Although the intuitionistic fuzzy-based FMEA methods and the cloud model-based FMEA methods can effectively improve the performance of the conventional FMEA technique, they still exist some limitations. The intuitionistic fuzzy-based FMEA methods lack the mechanism of dealing with the uncertainty of the interval, called randomness. In the cloud model-based FMEA methods, any two adjacent clouds have numerous reduplicative scopes, which causes the difficulty of distinguishing adjacent clouds. The cloud droplets in the reduplicative scopes have a high certainty degree, which may weaken the precision of the final results. Besides, the cloud FMEA is not enough to deal with the hesitance of experts’ judgments. To cover these deficiencies, in this study, IFS is integrated into the cloud model to develop a novel linguistic calculation method, namely IFC theory, which can simultaneously reflect the fuzziness, hesitance, and randomness of assessment information. Thus, the novel integrated IFC model can enhance the performance of expressing experts’ judgments in the fuzzy decision-making environment and thus make the final risk rankings of failure modes more reliable and rational. The details of the integrated IFC model are stated as follows.

Definition 6. Assume T is a qualitative concept defined on a universe of discourse U, t (t∈U) be a random instantiation of the qualitative concept T. μ_X (x) ∈ [0, 1] indicates the membership degree of x belonging to T, and f_X (x) ∈ [0, 1] is the non-membership degree of t belonging to T. Then T can be translated as IFS $\tilde{T}$ : $\tilde{T} = {〈 t, μ_{T} (t), f_{T} (t) 〉 | t \in U}$ (8)

Let β (t) = (t, μ_β (t) , f_β (t)) be an IFN on IFS $\tilde{T}$ , which can be simplified as β = (μ_β, f_β). According to the relationship of μ_A (t), f_A (t) and ∂_A (t) noted in Definition 1, the IFS $\tilde{T} = {〈 t, μ_{T} (t), f_{T} (t) 〉 | t \in U}$ can be expressed as another form [34, 64], which is shown as follows: $T^{*} = {〈 t, \underline{μ_{T}^{*} (t)}, \bar{μ_{T}^{*} (t)} 〉 | t \in U}$ (9) where T^* is named as the variant of the intuitionistic fuzzy set (VIFS), and $\underline{μ_{T}^{*} (t)} = μ_{T} (t)$ , $\bar{μ_{T}^{*} (t)} = 1 - f_{T} (t)$ denote the lower limit and upper limit of membership degree of x to X, respectively. Similarly, the IFN β = (μ_β, f_β) also can be transformed to $β^{*} = (\underline{μ_{β}^{*}}, \bar{μ_{β}^{*}})$ , which is the variant of intuitionistic fuzzy number (VIFN). For the VIFN $β^{*} = (\underline{μ_{β}^{*}}, \bar{μ_{β}^{*}})$ , the non-membership degree and hesitancy degree are depicted as $f_{β}^{*} = 1 - \bar{μ_{β}^{*}}$ and $\partial_{β}^{*} = \bar{μ_{β}^{*}} - \underline{μ_{β}^{*}}$ , respectively. Obviously, VIFN β^* and IFN β are equivalent in terms of the amount of information they contained.

Definition 7. Let β = (μ_β, f_β) be an IFN, and its corresponding VIFN is $β^{*} = (\underline{μ_{β}^{*}}, \bar{μ_{β}^{*}})$ . Then, according to the studies developed by Yang et al. [61] and Li et al. [23], the IFC value IC_β = (IEx_β, IEn_β, IHe_β) can be obtained by the following formulas: $IE x_{β} = \frac{\underline{μ_{β}^{*}} + \bar{μ_{β}^{*}}}{2}$ (10) $I E n_{β} = \frac{\bar{μ_{β}^{*}} + \underline{μ_{β}^{*}}}{6}$ (11) $IH e_{β} = c_{β}$ (12) where 0 < c<1 is a predefined constant, which can be determined according to the degree of uncertainty. Then the IFC value IC_β = (IEx_β, IEn_β, IHe_β) can be constructed as: $\begin{matrix} I C_{β} = (IE x_{β}, IE n_{β}, IH e_{β}) \\ = (\frac{\underline{μ_{β}^{*}} + \bar{μ_{β}^{*}}}{2}, \frac{\bar{μ_{β}^{*}} - \underline{μ_{β}^{*}}}{6}, c_{β}) \end{matrix}$ (13)

According to the relationship between IFN β and VIFN β^*, the IFC valueIC_β = (IEx_β, IEn_β, IHe_β) also can be expressed as:

$\begin{matrix} I C_{β} = (IE x_{β}, IE n_{β}, IH e_{β}) \\ = (\frac{1 + μ_{β} - f_{β}}{2}, \frac{1 - μ_{β} - f_{β}}{6}, c_{β}) \end{matrix}$ (14) where IEx_β, IEn_β, IHe_β ∈ [0, 1].

Similarly, for β^c = (f_β, μ_β), the complement of β, its corresponding IFC value ${IC}_{β}^{c}$ can be represented by

$\begin{matrix} {IC}_{β}^{c} = (IE x_{β}, IE n_{β}, IH e_{β})^{c} \\ = (\frac{1 + f_{β} - μ_{β}}{2}, \frac{1 - f_{β} - μ_{β}}{6}, c_{β}) \end{matrix}$ (15) where ${IC}_{β}^{c}$ can be regarded as the complement of the IFC value IC_β. 3.2 Operation rules

For supporting the subsequent manipulation of multiple IFC information, some essential operation rules are needed to be established first. In this study, the essential operation rules of the IFC theory are derived from the operation rules of cloud model theory shown in Equations (5) —(7).

Definition 8: Let IC₁=(IEx₁, IEn₁, IHe₁) and IC₂=(IEx₂, IEn₂, IHe₂) be two arbitrary IFC values transformed from IFNs β₁ = (μ_β1, f_β1) and β₂ = (μ_β2, f_β2); and pre-giving constant c₁, c₂ and λ, let IHe₁ = c₁, IHe₁ = c₂, and c₁, c₂, λ > 0. Then basic operational laws of IFC are presented as follows: $\begin{matrix} I C_{1} \oplus I C_{2} = I C_{\oplus} = (IE x_{\oplus}, IE n_{\oplus}, I H e_{\oplus}) \\ = (IE x_{1} + I E x_{2}, \sqrt{{IEn}_{1}^{2} + {IEn}_{2}^{2}}, \sqrt{{IHe}_{1}^{2} + {IHe}_{2}^{2}}) \end{matrix}$ (16) in which $IE x_{\oplus} = 1 + \frac{μ_{β 1} + μ_{β 2} - f_{β 1} - f_{β 2}}{2}$ (17) $IE n_{\oplus} = \frac{1}{6} \sqrt{{(1 - μ_{β 1} - f_{β 1})}^{2} + {(1 - μ_{β 2} - f_{β 2})}^{2}}$ (18) $IE n_{\oplus} = \sqrt{{c_{1}}^{2} + {c_{2}}^{2}}$ (19) And $\begin{matrix} I C_{1} \otimes I C_{2} = I C_{\otimes} = (IE x_{\otimes}, IE n_{\otimes}, I H e_{\otimes}) \\ = (IE x_{1} \times I E x_{2}, | IE x_{1} IE x_{2} | \sqrt{{(\frac{IE n_{1}}{IE x_{1}})}^{2} + {(\frac{IE n_{2}}{IE x_{2}})}^{2}}, \\ | IE x_{1} IE x_{2} | \sqrt{{(\frac{IH e_{1}}{IE x_{1}})}^{2} + {(\frac{IH e_{2}}{IE x_{2}})}^{2}}) \end{matrix}$ (20) in which $IE x_{\otimes} = \frac{(1 + μ_{β 1} - f_{β 1}) \times (1 + μ_{β 2} - f_{β 2})}{4}$ (21) $\begin{matrix} IE n_{\otimes} = | \frac{(1 + μ_{β 1} - f_{β 1}) \times (1 + μ_{β 2} - f_{β 2})}{4} | \\ \times \sqrt{{(\frac{1 - μ_{β 1} - f_{β 1}}{1 + μ_{β 2} - f_{β 2}})}^{2} + {(\frac{1 - μ_{β 2} - f_{β 2}}{1 + μ_{β 2} - f_{β 2}})}^{2}} \end{matrix}$ (22) $\begin{matrix} IH e_{\otimes} = | (1 + μ_{β 1} - f_{β 1}) \times (1 + μ_{β 2} - f_{β 2}) | \\ \times \sqrt{{(\frac{c_{1}}{1 + μ_{β 1} - f_{β 1}})}^{2} + {(\frac{c_{2}}{1 + μ_{β 2} - f_{β 2}})}^{2}} \end{matrix}$ (23) And $\begin{matrix} λ I C_{1} = (λ IE x_{1}, \sqrt{λ} IE n_{1}, \sqrt{λ} IH e_{1}) \\ = [λ \frac{1 + μ_{β 1} - f_{β 1}}{2}, \frac{\sqrt{λ} (1 - μ_{β 1} - f_{β 1})}{6}, \sqrt{λ} c_{1}] \end{matrix}$ (24)

Theorem 1. Let IC₁ = (IEx₁, IEn₁, IHe₁) and IC₂ = (IEx₂, IEn₂, IHe₂) be two IFC values, and λ₁, λ₁ > 0 are two real numbers. Then,

IC₁ ⊕ IC₂ = IC₂ ⊕ IC₁;

IC₁ ⊗ IC₂ = IC₂ ⊗ IC₁;

λ₁IC₁ + λ₂IC₁ = (λ₁ + λ₂) IC₁;

Proof.

(1) $I C_{1} \oplus I C_{2} = (IE x_{1} + I E x_{2}, \sqrt{{IEn}_{1}^{2} + {IEn}_{2}^{2}}, \sqrt{{IHe}_{1}^{2} + {IHe}_{2}^{2}})$

$= (IE x_{2} + I E x_{1}, \sqrt{{IEn}_{2}^{2} + {IEn}_{1}^{2}}, \sqrt{{IHe}_{2}^{2} + {IHe}_{1}^{2}})$

= IC₂ ⊕ IC₁;

(2) IC₁ ⊗ IC₂ $\begin{matrix} = (IE x_{1} \times I E x_{2}, | IE x_{1} IE x_{2} | \sqrt{{(\frac{IE n_{1}}{IE x_{1}})}^{2} + {(\frac{IE n_{2}}{IE x_{2}})}^{2}}, \\ | IE x_{1} IE x_{2} | \sqrt{{(\frac{IH e_{1}}{IE x_{1}})}^{2} + {(\frac{IH e_{2}}{IE x_{2}})}^{2}}) \end{matrix}$ $\begin{matrix} = (I E x_{2} \times IE x_{1}, | IE x_{2} IE x_{1} | \sqrt{{(\frac{IE n_{2}}{IE x_{2}})}^{2} + {(\frac{IE n_{1}}{IE x_{1}})}^{2}}, \\ | IE x_{2} IE x_{1} | \sqrt{{(\frac{IH e_{2}}{IE x_{2}})}^{2} + {(\frac{IH e_{1}}{IE x_{1}})}^{2}}) \end{matrix}$ = IC₂ ⊗ IC₁;

(3) λ₁IC₁ ⊕ λ₂IC₁

$= (λ_{1} IE x_{1}, \sqrt{λ_{1}} IE n_{1}, \sqrt{λ_{1}} IH e_{1}) \oplus (λ_{2} IE x_{1}, \sqrt{λ_{2}} IE n_{1}, \sqrt{λ_{2}} IH e_{1})$ $\begin{matrix} = ((λ_{1} + λ_{2}) IE x_{1}, \sqrt{{(\sqrt{λ_{1}} IE n_{1})}^{2} + {(\sqrt{λ_{2}} IE n_{1})}^{2}}, \\ \sqrt{{(\sqrt{λ_{1}} IH e_{1})}^{2} + {(\sqrt{λ_{2}} IH e_{1})}^{2}}) \end{matrix}$ $= ((λ_{1} + λ_{2}) IE x_{1}, \sqrt{λ_{1} + λ_{2}} IE n_{1}, \sqrt{λ_{1} + λ_{2}} IH e_{1})$ $= (λ_{1} + λ_{2}) I C_{1}$

Theorem 2. Let IC_i = (IEx_i, IEn_i, IHe_i) (i = 1, 2, 3, ... , z) be z IFCs, and then, $\begin{matrix} λ_{1} I C_{1} \oplus λ_{2} I C_{2} \oplus \dots \oplus λ_{z} I C_{z} \\ = (\sum_{i = 1}^{z} λ_{i} IE x_{i}, \sqrt{\sum_{i = 1}^{z} λ_{i} IE n_{i}}, \sqrt{\sum_{i = 1}^{z} λ_{i} IH e_{i}}) \end{matrix}$

Proof. According to Equations (16) and (24), the above equation is easy to derive.

3.3 Minkowski-type distance of IFC

In this section, some typical Minkowski-type distances of IFC are constructed and their properties are discussed.

Definition 9. Let IC₁ = (IEx₁, IEn₁, IHe₁) and IC₂ = (IEx₂, IEn₂, IHe₂) be two IFC values. The Minkowski distance between IC₁ and IC₂ are established as:

$\begin{matrix} d_{M} (I C_{1}, I C_{2}) \\ = {({| IE x_{1} - IE x_{2} |}^{p} + {| IE n_{1} - IE n_{2} |}^{p} + {| IH e_{1} - IH e_{2} |}^{p})}^{\frac{1}{p}} \end{matrix}$ (25) where p ⩾ 1. When p takes different values, the Minkowski distance d_M changes to different typical distances, the details are shown as follows.

When p = 1, the Minkowski distance d_M becomes the Hamming distance d_H between IC₁ and IC₂, i.e.,

$\begin{matrix} d_{H} (I C_{1}, I C_{2}) \\ = (| IE x_{1} - IE x_{2} | + | IE n_{1} - IE n_{2} | + | IH e_{1} - IH e_{2} |) \end{matrix}$ (26)

When p = 2, the Minkowski distance d_M is the Euclidean distance d_E between IC₁ and IC₂, that is,

$\begin{matrix} d_{E} (I C_{1}, I C_{2}) \\ = {({| IE x_{1} - IE x_{2} |}^{2} + {| IE n_{1} - IE n_{2} |}^{2} + {| IH e_{1} - IH e_{2} |}^{2})}^{\frac{1}{2}} \end{matrix}$ (27) When p =∞, the Minkowski distance d_M represents the Chebyshev distance d_C between IC₁ and IC₂,

$\begin{matrix} d_{C} (I C_{1}, I C_{2}) \\ = max {| IE x_{1} - IE x_{2} |, | IE n_{1} - IE n_{2} |, | IH e_{1} - IH e_{2} |} \end{matrix}$ (28)

Theorem 3. Assume IC₁ = (IEx₁, IEn₁, IHe₁) and IC₂ = (IEx₂, IEn₂, IHe₂) are two IFC values, and d_M (IC₁, IC₂) is the Minkowski distance between IC₁ and IC₂, then

0 ⩽ d_M (IC₁, IC₂) ⩽3;

d_M (IC₁, IC₂) =0, if and only IC₁ = IC₂;

d_M (IC₁, IC₂) = d_M (IC₂, IC₁)

Suppose IC₃ = (IEx₃, IEn₃, IHe₃), if IC₁ ⩽ IC₂ ⩽ IC₃, then d_M (IC₁, IC₃) ⩾ d_M (IC₁, IC₂) ∨ d_M (IC₂, IC₃);

d_M (IC₁, IC₂) = d_M ((IC₁) ^c, (IC₂) ^c);

d_M (IC₁, IC₃) ⩽ d_M (IC₁, IC₂) + d_M (IC₂, IC₃).

Proof.

(1) Due to IEx₁, IEn₁, IHe₁, IEx₂, IEn₂, IHe₂ ∈ [0, 1], we have $0 ⩽ | \tilde{E} x_{1} - \tilde{E} x_{2} | ⩽ 1, 0 ⩽ | \tilde{E} n_{1} - \tilde{E} n_{2} | ⩽ 1, 0 ⩽ | \tilde{H} e_{1} - \tilde{H} e_{2} | ⩽ 1$ . Hence, $\begin{matrix} 0 ⩽ d_{M} (I C_{1}, I C_{2}) \\ = {({| IE x_{1} - IE x_{2} |}^{p} + {| IE n_{1} - IE n_{2} |}^{p} + {| IH e_{1} - IH e_{2} |}^{p})}^{\frac{1}{p}} \\ ⩽ {(1)}^{\frac{1}{p}} \times 3^{\frac{1}{p}} ⩽ 3 \end{matrix}$

(2) According to definition 9, there is, $\begin{matrix} d_{M} (I C_{1}, I C_{2}) = 0 \\ \Leftrightarrow | IE x_{1} - IE x_{2} | = 0, | IE n_{1} - IE n_{2} | = 0, | IH e_{1} - IH e_{2} | = 0 \\ \Leftrightarrow I C_{1} = I C_{2} . \end{matrix}$

(3) d_M (IC₁, IC₂) $\begin{matrix} = {({| IE x_{1} - IE x_{2} |}^{p} + {| IE n_{1} - IE n_{2} |}^{p} + {| IH e_{1} - IH e_{2} |}^{p})}^{\frac{1}{p}} \\ = {({| IE x_{2} - IE x_{1} |}^{p} + {| IE n_{2} - IE n_{1} |}^{p} + {| IH e_{2} - IH e_{3} |}^{p})}^{\frac{1}{p}} \\ = d_{M} (I C_{2}, I C_{1}) \end{matrix}$

Similarly, d_M (IC₁, IC₃) ⩾ d_M (IC₂, IC₃) can be also proved.

(5) According to Equation (15) and Equation (25), $\begin{matrix} d_{M} (I C_{1}, I C_{2}) \\ = {({| IE x_{1} - IE x_{2} |}^{p} + {| IE n_{1} - IE n_{2} |}^{p} + {| IH e_{1} - IH e_{2} |}^{p})}^{\frac{1}{p}} \end{matrix}$ $\begin{matrix} = ({| \frac{1 + μ_{β 1} - f_{β 1}}{2} - \frac{1 + μ_{β 1} - f_{β 1}}{2} |}^{p} + \\ {{| \frac{1 - μ_{β 1} - f_{β 1}}{6} - \frac{1 - μ_{β 2} - f_{β 2}}{6} |}^{p} + {| c_{1} - c_{2} |}^{p})}^{\frac{1}{p}} \end{matrix}$ $\begin{matrix} = ({| \frac{1 + f_{β 1} - μ_{β 1}}{2} - \frac{1 + f_{β 2} - μ_{β 2}}{2} |}^{p} + \\ {{| \frac{1 - μ_{β 1} - f_{β 1}}{6} - \frac{1 - μ_{β 2} - f_{β 2}}{6} |}^{p} + {| c_{2} - c_{1} |}^{p})}^{\frac{1}{p}} \end{matrix}$ $= d_{M} ({(I C_{1})}^{c}, {(I C_{2})}^{c})$ (6) Denote IC₁ = (IEx₁, IEn₁, IHe₁) ^T, IC₂ = (IEx₂, IEn₂, IHe₂) ^T, and IC₃ = (IEx₃, IEn₃, IHe₃) ^T. Then the l_p norm of vector IC₁ - IC₃ is

IC₁ - IC₃

_P = (|IEx₁ - IEx₃|^p + |IEn₁ - IEn₃|^p + ${{| IH e_{1} - IH e_{3} |}^{p})}^{\frac{1}{p}}$ .}} Therefore, d_M (IC₁, IC₂) =

IC₁ - IC₂

_P. i.e., $\begin{matrix} d_{M} (I C_{1}, I C_{3}) = {∥ I C_{1} - I C_{3} ∥}_{P} \\ ⩽ {∥ I C_{1} - I C_{2} ∥}_{P} + {∥ I C_{2} - I C_{3} ∥}_{P} \\ = d_{M} (I C_{1}, I C_{2}) + d_{M} (I C_{2}, I C_{3}) \end{matrix}$

3.4 A brief comparison between IFC and normal cloud

Assume X = {x₁, x₂, . . . , x_2n+1} is a qualitative linguistic term set defined on the effective universe U = [u_min, u_max]. In importance evaluation items, 2n + 1 represents the number of importance levels described by linguistic terms. It is determined based on specific requirements and usually set as five, seven, and nine [19 , 57]. According to the study investigated by Fattahi and Khalilzadeh [13], nine-label linguistic terms can more delicately and smoothly depict the importance levels compared with five-label or seven-label linguistic variables. Therefore, let 2n+1 = 9, and then the linguistic term set X is established as X=(x₁ = extremely unimportant, x₂ = very unimportant, x₃ = unimportant, x₄ = slightly unimportant, x₅ = moderate important, x₆ = slightly important, x₇ = important, x₈ = very important, x₉ = extremely important). Suppose C₁ ∼ C₉ are the nine normal clouds corresponding to the nine linguistic term levels x₁ ∼ x₉, which represent the quantitative conversion values of the nine qualitative importance linguistic variables. The golden segmentation method [21, 30] can be used to calculate these nine normal clouds C₁ ∼ C₉, and its calculation principle is shown as follows: $C_{1} = (E x_{1}, E n_{1}, H e_{1}) = (u_{min} + 3 E n_{1}, E n_{2} / - 0.618, H e_{2} / - 0.618)$ (29) $\begin{matrix} C_{i} = (E x_{i}, E n_{i}, H e_{i}) = (E x_{i + 1} - 0.382 (E x_{i + 1} - E x_{1}), \\ E n_{i + 1} / - 0.618, H e_{i + 1} / - 0.618), where i \in [2, n - 1] \end{matrix}$ (30) $\begin{matrix} C_{n} = (E x_{n}, E n_{n}, H e_{n}) \\ = (\frac{u_{min} + u_{max}}{2}, 0.382 \frac{u_{min} - u_{max}}{3 (2 n + 2)}, H e_{n}) \end{matrix}$ (31) $\begin{matrix} C_{j} = (E x_{j}, E n_{j}, H e_{j}) = (E x_{j - 1} + 0.382 (E x_{2 n + 1} - E x_{j - 1}), \\ E n_{j - 1} / - 0.618, H e_{j - 1} / - 0.618), where j \in [n + 1, 2 n] \end{matrix}$ (32) $\begin{matrix} C_{2 n + 1} = (E x_{2 n + 1}, E n_{2 n + 1}, H e_{2 n + 1}) = (u_{max} - 3 E n_{2 n + 1}, \\ E n_{2 n} / - 0.618, H e_{2 n} / - 0.618) \end{matrix}$ (33)

Significantly, the effective universe U = [u_min, u_max] and the value of He_n are determined in prior. In this study, they are preset as U = [0, 1] and He_n = 0.002, respectively.

According to the computation principle elaborated above, the numerical values of the nine normal clouds are obtained as:

C₁ = (Ex₁, En₁, He₁) = (0.2616, 0.0872, 0.0136),

C₂ = (Ex₂, En₂, He₂) = (0.3179, 0.0539, 0.0084),

C₃ = (Ex₃, En₃, He₃) = (0.3527, 0.0333, 0.0052),

C₄ = (Ex₄, En₄, He₄) = (0.4098, 0.0206, 0.0032),

C₅ = (Ex₅, En₅, He₅) = (0.5, 0.0127, 0.002),

C₆ = (Ex₆, En₆, He₆) = (0.5911, 0.0206, 0.032),

C₇ = (Ex₇, En₇, He₇) = (0.6473, 0.0333, 0.0052),

C₈ = (Ex₈, En₈, He₈) = (0.6821, 0.0539, 0.84),

C₉ = (Ex₉, En₉, He₉) = (0.7384, 0.0872, 0.0136).

Figure 3 shows the distributions of the above nine normal clouds on an effective domain [0,1], which is produced by a normal cloud generator under the condition of cloud droplet number N = 2000. From Fig. 3, we can find that there is complex intersectional status among several clouds on both sides of cloud C₅, which makes it difficult to discriminate these clouds from each other clearly. Moreover, the valid domain [0.2616,0.7834] of Ex is more narrow than the global effective domain [0,1]. Moreover, En, the reflection of fuzziness and randomness, gradually increases in both sides of cloud C₅, which does not conform to the practical situations in decision-making issues.

Fig. 3

The conversional normal clouds.

Inspired by Liu et al. [28] the IFS X^* corresponding to linguistic term set X can be denoted as: X^*=[<x₁, 0, 0.85>,<x₂, 0.15, 0.75>,<x₃, 0.25, 0.65>,<x₄, 0.35, 0.55>,<x₅, 0.45, 0.45>,<x₆, 0.55, 0.35>,<x₇, 0.65, 0.25>,<x₈, 0.75, 0.15>,<x₉, 0.85, 0 >], and let c₁ = c₂= ... =c₉ = 0.0075, then the IFC set transformed form IFS X^* can be obtained as IC=(IC₁, IC₂, ... , IC₉)=[(0.075, 0.025, 0.0075), (0.2, 0.017, 0.0075), (0.3, 0.017, 0.0075), (0.4, 0.017, 0.0075), (0.5, 0.017, 0.0075), (0.6, 0.017, 0.0075), (0.7, 0.017, 0.0075), (0.8, 0.017, 0.0075), (0.0925, 0.025, 0.0075)]. The corresponding of the nine IFC values in IFC set IC are graphically shown in Fig. 4.

As we can see from Fig. 4, the effective domain of Ex is expanded to [0.075, 0.925], and the intersections between one IFC to other IFCs are greatly reduced. As a result of the improvement of normal clouds to IFCs, the layout of nine IFCs in global effective domain [0,1] is more evenly, and every IFC can be distinguished clearly from each other. In addition, there is a minor discrepancy between the IEn values for the nine IFCs, which does more conform to reality.

Fig. 4

The conventional IFC.

4 The proposed FMEA strategy

In this section, a novel improved FMEA strategy supported by the developed IFC theory, namely, the IFC-FMEA model, is proposed to assess and determine the risk ranking of failure modes, in which the initial assessments of failure modes are described with linguistic terms by experts. Then, the IFC theory is used to convert these linguistic assessments into numerical values by considering various fuzzy properties of the assessments, such as fuzziness, uncertainty, and randomness.

The framework and procedure of the proposed IFC-FMEA method are expressed in Fig. 5, which is consisted of three phases. In the first phase, the linguistic risk evaluations of failure modes are provided by FMEA team members and then transformed to IFC values based on the developed IFC linguistic computation method. The second phase focuses on the analyses of the synthetic weights of FMEA members and risk factors, which consist of both subjective weight and objective weight. Finally, the traditional TOPSIS theory is extended on the basis of the IFC model to form a new IFC-based TOPSIS method. It is applied to estimate the risk priorities of failure modes. The detailed steps of this novel FMEA model are depicted as follows.

Fig. 5

The framework of the proposed IFC-FMEA approach.

4.1 Risk evaluation for failure modes using IFC-FMEA method

Step 1: Obtain the risk evaluations of failure modes with linguistic terms.

As previously mentioned, due to the fuzziness and uncertainty of human cognition, it is usually difficult for experts to express their judgment by crisp value. They prefer to describe their assessment for failure modes by linguistic variables. Thus, inspired by Liu et al. [28], in this study, experts are invited to conduct risk assessments of failure modes by applying linguistic terms set, which is shown in Table 1. In the developed risk evaluation method, the FMEA team composed of cross-position experts is built to estimate the risk ranking of failure modes. Assume that there are l members M_k (k = 1, 2, 3, ... , l) in the FMEA team, m failure modes F_i (i = 1, 2, 3, ... , m) and n risk factors R_j (j = 1, 2, 3, ... , n). The linguistic assessment of failure mode F_i under risk factor R_j provided by expert M_k can be marked as t_kij. Then the linguistic assessment matrix T_k = (t_kij) _m×n is defined as the risk rating set of all failure modes under different risk factors given by expert M_k.

Table 1
Linguistic terms and their IFNs for assessing the risk ratings of failure modes

Linguistic terms Symbols IFNs

Extremely low EL (0,0.85)

Very low VL (0.15,0.75)

Low L (0.25,0.65)

Slightly low SL (0.35,0.55)

Moderate M (0.45,0.45)

Slightly high SH (0.55,0.35)

High H (0.65.0.25)

Very high VH (0.75,0.15)

Extremely high EH (0.85,0)

Linguistic terms	Symbols	IFNs
Extremely low	EL	(0,0.85)
Very low	VL	(0.15,0.75)
Low	L	(0.25,0.65)
Slightly low	SL	(0.35,0.55)
Moderate	M	(0.45,0.45)
Slightly high	SH	(0.55,0.35)
High	H	(0.65.0.25)
Very high	VH	(0.75,0.15)
Extremely high	EH	(0.85,0)

Step 2: Transform linguistic assessments into IFNs

To quantitatively evaluate risk information of failure modes, the IFS theory is employed to convert the qualitative linguistic assessments into the corresponding quantitative IFNs. Based on Definition 1, the membership function μ_A (x) and the non-membership function f_A (x) satisfy 0 ⩽ μ_A (x) + f_A (x) ⩽1 for any elements. Besides, the fact that presetting the hesitancy degree ∂_A (x) =1 - μ_A (x) - f_A (x) as 0 ∼ 0.2 is appropriate for calculating the values of IFNs has been validated in industrial applications by studies [7 , 58]. Accordingly, this paper sets the corresponding transformational grades between linguistic variables and IFNs as shown in Table 1. According to the information in Table 1, the linguistic assessment matrix T_k = (t_kij) _m×n can be converted into the IFNs matrix, denoted as ${\tilde{T}}_{k} = {({\tilde{t}}_{kij})}_{m \times n} = {[(μ_{kij}, β_{kij})]}_{m \times n}$ , where ${\tilde{t}}_{kij} = (μ_{kij}, β_{kij})$ is the IFN corresponding to the linguistic assessment t_kij.

Step 3: Convert IFNs to IFC evaluations

Although IFS theory can effectively cope with the fuzziness and uncertainty of experts’ knowledge and evaluation environment, it is unable to manipulate the randomness of experts’ awareness. In this paper, IFNs are transformed into IFC evaluation values to establish IFC assessments for handling the vagueness and randomness of experts’ evaluations. The conversion result is constructed as follows: $I C_{k} = {(I C_{kij})}_{m \times n} = {[(IE x_{kij}, IE n_{kij}, IH e_{kij})]}_{m \times n}$ (34) where IC_k is the IFC assessment matrix of risk grade for all failure modes given by expert M_k (k = 1, 2, 3, ... , l). IC_kij = (IEx_kij, IEn_kij, IHe_kij) is the IFC assessment value of failure mode F_i (i = 1, 2, 3, ... , m) under risk factor R_j (j = 1, 2, 3, ... , n) provided by expert M_k (k = 1, 2, 3, ... , l), and its elements of IEx_kij, IEn_kij and IHe_kij are calculated as follows: $IE x_{kij} = \frac{μ_{kij} + (1 - f_{kij})}{2}$ (35) $IE n_{kij} = \frac{(1 - f_{kij}) - μ_{kij}}{6}$ (36) $IH e_{kij} = c$ (37) where c is a constant determined by the degree of uncertainty.

Step 4: Integrate the IFC assessments given by different FMEA team members

Considering the different importance of experts, the weighted averaging operator is used to aggregate all members’ IFC assessment matrices to obtain a synthetic IFC assessment matrix $\bar{IC}$ , which can be expressed by

$\begin{matrix} \bar{IC} = ({\bar{IC}}_{ij})_{m \times n} = (IE x_{ij}, IE n_{ij}, IH e_{ij})_{m \times n} = {(\sum_{k = 1}^{l} w_{k} I C_{kij})}_{m \times n} \\ = (\frac{1}{l} \sum_{k = 1}^{l} w_{k} IE x_{kij}, {\sqrt{\frac{1}{l} \sum_{k = 1}^{l} w_{k} {(IE n_{kij})}^{2}}, \sqrt{\frac{1}{l} \sum_{k = 1}^{l} w_{k} {(IH e_{kij})}^{2}})}_{m \times n} \end{matrix}$ (38) where w_k is the weight of kth expert. ${\bar{IC}}_{ij} = (IE x_{ij}, IE n_{ij}, IH e_{ij})$ is the synthetic IFC assessment of failure mode F_i under risk factor R _j.

4.2 Synthetic weights calculation of experts and risk factors

Step 1: Calculate the synthetic weights of FMEA members

The synthetic weight of each expert consists of subjective and objective weight. For gaining the subjective weight, a subject weight allocation table is established by considering the work experience, title, and professional relevance of different experts, which are presented in Table 2.

Table 2
The criterion system of experts’ weights

Aspects Classes Scores

Work experience Over 20 years 5

10–19 years 4

5–9 years 3

2–5 years 2

Under 2 yeas 1

Positional title Senior 5

Subsenior 4

Intermediate 3

Assistant 2

Junior 1

Professional relevance High 5

Medium High 4

Medium 3

Medium Low 2

Low 1

Aspects	Classes	Scores
Work experience	Over 20 years	5
	10–19 years	4
	5–9 years	3
	2–5 years	2
	Under 2 yeas	1
Positional title	Senior	5
	Subsenior	4
	Intermediate	3
	Assistant	2
	Junior	1
Professional relevance	High	5
	Medium High	4
	Medium	3
	Medium Low	2
	Low	1

By comparing the backgrounds of different experts, the subjective evaluations S_k (k = 1, 2, 3, ... , l) of experts can be determined based on Table 2. Then, the subjective weight $w_{k}^{s}$ of expert M_K (k = 1, 2, 3, ... , l) can be calculated by $w_{k}^{s} = \frac{S_{k}}{\sum_{k = 1}^{p} S_{k}}$ (39)

The objective weights of FMEA experts are obtained by comparing the discrepancy of the assessment information of failure modes between practice and experts’ evaluations. Suppose there are S failure modes collected from the historical fault database produced in the experimental and application stages of a mechanical product. g₁ ∼ g_s represent the risk degrees of these S failure modes, which can be further gathered into a risk degree set G = (g₁, g₂, . . . , g_S). The numerical values of g₁ ∼ g₂₀ can be determined based on the analysis of the failure characteristic information included in that historical fault data. Then, FMEA team members are asked to assess those failure modes and give evaluation values $G^{k} = (g_{1}^{k}, g_{2}^{k}, . . ., g_{S}^{k})$ . Thus, the discrepancy value D_k between G^k and G is calculated by $D_{k} = \sum_{s = 1}^{S} | g_{s} - g_{s}^{k} |$ (40) where s = (1, 2, 3, . . . , S),k = (1, 2, 3, . . . , l), $g_{s}^{k}$ is the evaluation value of risk level for sth collected failure mode gained from expert M_k.

Next, the objective weight $w_{k}^{s}$ for FMEA expert M_k is computed as $w_{k}^{o} = \frac{\sum_{k = 1}^{l} D_{k}}{D_{k}} / - \sum_{k = 1}^{l} (\frac{\sum_{k = 1}^{l} D_{k}}{D_{k}})$ (41)

Finally, based on the above calculations, based on the research of Li et al. [23], the synthetic weight w_k of expert M_k can be attained as $w_{k} = \frac{w_{k}^{o} \times w_{k}^{s}}{\sum_{k = 1}^{l} (w_{k}^{o} \times w_{k}^{s})}$ (42)

Step 2: Calculate the synthetic weights of risk factors

In this step, the FMEA team members conduct the importance evaluation of risk factors by using linguistic terms, as shown in Table 3. The initial importance matrix is marked as C = (c_kj) _l×n, where the element c_kj (k = 1, 2, 3, ... , l; j = 1, 2, 3, ... , n) is the linguistic value of the important evaluation for the risk factor R_j given by expert M_k. Based on Equations (35) –(37), initial importance matrix C can be transformed to IFC value matrix C′ as follows:

$C^{'} = {({c^{'}}_{kj})}_{l \times n} = {[(IE x_{kj}, IE n_{kj}, IH e_{kj})]}_{l \times n}$ (43) where $c_{kj}^{'}$ is the IFC value convoluted by linguistic value c_kj, the three elements of IEx_kj, IEn_kj and IHe_kj are eigenvalues of IFC value $c_{kj}^{'}$ .

Table 3

The linguistic terms system for initial importance evaluation of risk factors

Linguistic terms	Symbols	IFNs
Extremely unimportant	EU	(0,0.85)
Very unimportant	VU	(0.15,0.75)
unimportant	U	(0.25,0.65)
Slightly unimportant	SU	(0.35,0.55)
Moderate important	MI	(0.45,0.45)
Slightly important	SI	(0.55,0.35)
important	I	(0.65.0.25)
Very important	VI	(0.75,0.15)
Extremely important	EI	(0.85,0)

By using the weighted average method, the group importance evaluation for risk factor R_j is noted as $C_{j}^{'}$ (j = 1, 2, 3, ... , n) can be captured. The vector Z consisting of $C_{j}^{'}$ , named group importance IFC vector, is expressed, and the results are presented as follows: $\begin{matrix} {C^{'}}_{j} = (IE {x^{'}}_{j}, IE {n^{'}}_{j}, IH {e^{'}}_{j}) \\ = [\frac{1}{l} \sum_{k = 1}^{l} w_{k} IE x_{kj}, \sqrt{\frac{1}{l} \sum_{k = 1}^{l} w_{k} {(IE n_{kj})}^{2}}, \sqrt{\frac{1}{l} \sum_{k = 1}^{l} w_{k} {(IH e_{j})}^{2}}] \end{matrix}$ (44) $Z = (C_{1}^{'}, C_{2}^{'}, C_{3}^{'}, . . ., C_{n}^{'})^{T}$ (45)

The synthetic weight p_j for risk factor R_j can be constructed by synthesizing the objective weight and subjective weight. Similar to Equation (42), it can be denoted as $p_{j} = \frac{p_{j}^{s} \times p_{j}^{o}}{\sum_{j = 1}^{n} p_{j}^{s} \times p_{j}^{o}}$ (46) where $p_{j}^{s}$ and $p_{j}^{o}$ are the subjective and objective weight of R_j, respectively.

Based on Li et al. [23], the subjective weight $p_{j}^{s}$ can be calculated by $p_{j}^{s} = \frac{IE {x^{'}}_{j}}{\sum_{j = 1}^{n} IE {x^{'}}_{j}}$ (47) where $IE x_{j}^{'}$ refers to the eigenvalue of expectation in the group importance IFC value $C_{j}^{'}$ .

Inspired by the studies of Liu et al. [29] and Rao et al. [44], the objective weight $p_{j}^{o}$ can be calculated by using the concept of statistical variance that is a mathematical parameter to reflect the divergence information of a group data. Well-known that in the entropy weight method, the divergence information of a set of risk evaluations for failure modes under a risk factor can be used to analyze the objective weight of that factor since it can indirectly reflect the influence of that factor on the risk evaluations. Accordingly, the statistical variance used to compute the objective weights for risk factors is suitable, and it is more concise in computation than the entropy weight method [29].

Based on the above discussion, the objective weight of risk factor R _j can be calculated by $p_{j}^{o} = \frac{V_{j}}{\sum_{j = 1}^{n} V_{j}}$ (48) in which $V_{j} = \frac{1}{m} \sum_{i = 1}^{m} {[d ({\bar{IC}}_{ij}, mean ({\bar{IC}}_{ij}))]}^{2}$ (49) and $mean ({\bar{IC}}_{ij}) = \frac{1}{m} \sum_{i = 1}^{m} {\bar{IC}}_{ij}$ (50) where V_j is the statistical variance of the synthetic IFC assessment result of failure modes on the aspect of risk factor R _j, and mean $({\bar{IC}}_{ij})$ is the mean value of the synthetic IFC assessment results of all failure modes under risk factor R_j discussed in Step 4.

4.3 Identification of the risk ranking by IFC-based TOPSIS method

Step 1: Obtain the synthetic weighted IFC assessment matrix.

The purpose of this step is to obtain the synthetic weighted IFC assessment matrix by executing the data processing on the synthetic IFC assessment matrix $\bar{IC}$ .

Due to the different importance of risk factors in the risk estimation, the synthetic IFC assessment matrix $\bar{IC}$ should add the consideration of the weights of risk factors. Thus, the synthetic weighted IFC assessment matrix $\bar{\bar{IC}}$ can be calculated as follows: $\bar{\bar{IC}} = ({\bar{\bar{IC}}}_{ij})_{m \times n} = {[(\bar{\bar{IE x_{ij}}}, \bar{\bar{IE n_{ij}}}, \bar{\bar{IH e_{ij}}})]}_{m \times n} = (p_{j} {\bar{IC}}_{ij})_{m \times n}$ (51) where p_j is the weight of jth risk factor, ${\bar{IC}}_{ij}$ is the synthetic weighted IFC assessment of failure mode F_i under risk factor R_j.

Step 2: Determine the positive and negative ideal IFC solutions.

A new conception of IFC-TOPSIS, extended by traditional TOPSIS theory under the IFC environment, is proposed to rank failure modes. Similar to the traditional TOPSIS method, the IFC-TOPSIS approach needs to gain the IFC positive ideal solution (IFC-PIS) and IFC negative ideal solution (IFC-NIS) of the synthetic weighted IFC assessment matrix, respectively. Inspired by the calculations of the positive ideal solution and the negative ideal solution in the traditional TOPSIS method [5], the expressions of C-PIS and C-NIS are defined as $C S_{P} = (t_{j}^{+})_{1 \times n} = ({max}_{i = 1}^{m} {\bar{\bar{IC}}}_{ij})_{1 \times n}$ (52) $C S_{N} = (t_{j}^{-})_{1 \times n} = ({min}_{i = 1}^{m} {\bar{\bar{IC}}}_{ij})_{1 \times n}$ (53) where $t_{j}^{+} = {max}_{i = 1}^{m} {\bar{\bar{IC}}}_{ij} = ({max}_{i = 1}^{m} \bar{\bar{IE x_{ij}}}, {min}_{i = 1}^{m} \bar{\bar{IE n_{ij}}}, {min}_{i = 1}^{m} \bar{\bar{IH e_{ij}}})$ (54) $t_{j}^{-} = {min}_{i = 1}^{m} {\bar{\bar{IC}}}_{ij} = ({min}_{i = 1}^{m} \bar{\bar{IE x_{ij}}}, {max}_{i = 1}^{m} \bar{\bar{IE n_{ij}}}, {max}_{i = 1}^{m} \bar{\bar{IH e_{ij}}})$ (55) and CS_P and CS_N represent the C-PIS vector and C-NIS vector, respectively.

Step 3: Calculate the closeness coefficient of each failure mode.

Based on Equation (25), the distances between ith failure mode and the two types of ideal solution are computed as $\begin{matrix} d_{i}^{+} = \sum_{j = 1}^{n} d_{M} ({\bar{\bar{IC}}}_{ij}, t_{j}^{+}) \\ = \sum_{j = 1}^{n} ({| \bar{\bar{IE x_{ij}}} - {max}_{i = 1}^{m} \bar{\bar{IE x_{ij}}} |}^{p} + {| \bar{\bar{IE n_{ij}}} - {min}_{i = 1}^{m} \bar{\bar{IE n_{ij}}} |}^{p} \\ {{| \bar{\bar{IH e_{ij}}} - {min}_{i = 1}^{m} \bar{\bar{IH e_{ij}}} |}^{p})}^{\frac{1}{p}} \end{matrix}$ (56) $\begin{matrix} d_{i}^{-} = \sum_{j = 1}^{n} d_{M} ({\bar{\bar{IC}}}_{ij}, t_{j}^{-}) \\ = \sum_{j = 1}^{n} ({| \bar{\bar{IE x_{ij}}} - {min}_{i = 1}^{m} \bar{\bar{IE x_{ij}}} |}^{p} + {| \bar{\bar{IE n_{ij}}} - {max}_{i = 1}^{m} \bar{\bar{IE n_{ij}}} |}^{p} \\ {{| \bar{\bar{IH e_{ij}}} - {max}_{i = 1}^{m} \bar{\bar{IH e_{ij}}} |}^{p})}^{\frac{1}{p}} \end{matrix}$ (57) where $d_{i}^{+}$ is the distance of ith failure mode and C-PIS, $d_{i}^{-}$ is the distance of ith failure mode and C-NIS.

Then, according to Silva et al. [49], the closeness coefficient of ith failure mode, noted as Ct_i which represents the relative distance between the ith failure mode and the C-PIS and the C-NIS, can be calculated by $C t_{i} = \frac{d_{i}^{-}}{d_{i}^{-} + d_{i}^{+}}$ (58)

Step 4: Obtain the risk ranking of failure modes by IFC-based TOPSIS method

The closeness coefficient vector Ct is presented to express the closeness coefficients of all failure modes. The risk ranking of each failure mode is obtained by comparing the Ct_i (i = 1, 2, 3, . . . , m) values in Ct vector. The bigger the value of the Ct_i (i = 1, 2, 3, . . . , m), the higher the risk rating of ith failure mode. $Ct = (C t_{1}, C t_{2}, . . ., C t_{m})$ (59)

5 Case study

In this part, the risk evaluation of failure modes for a winding engine is implemented as a case study to demonstrate the effectiveness of the proposed IFC-FMEA approach. The winding engine, as the essential member of engineering machinery, is a common complex system that consists of connecting and supporting parts, gripper lifting mechanism, hand drive device, grasping device, and electric control system.

A potential failure mode set of the winding engine is established by the FMEA team based on the function and structure analysis of the winding engine, which is shown in Table 4.

Table 4
The potential failure modes of the winding engine

Serial number The potential failure of the winding engine

F₁ Location control error

F₂ Winding drum and Wirerope abrasion

F₃ Fastener and connector loosening

F₄ Clutch failure

F₅ Lubricant leakage

F₆ Gripper inoperation or deformation or fracture

F₇ Worm gear reducer failure to transmit power

F₈ Hand drive device failed

F₉ Supporting components fracture or deformation

F₁₀ Circuit anomaly

F₁₁ Wirerope Guide apparatus failure

F₁₂ Position sensor of lift and transposition failed

Serial number	The potential failure of the winding engine
F₁	Location control error
F₂	Winding drum and Wirerope abrasion
F₃	Fastener and connector loosening
F₄	Clutch failure
F₅	Lubricant leakage
F₆	Gripper inoperation or deformation or fracture
F₇	Worm gear reducer failure to transmit power
F₈	Hand drive device failed
F₉	Supporting components fracture or deformation
F₁₀	Circuit anomaly
F₁₁	Wirerope Guide apparatus failure
F₁₂	Position sensor of lift and transposition failed

To obtain reasonable and fair risk assessment information for potential failure modes, five cross-department experts are invited to form the FMEA team and required to provide their risk evaluation for each failure mode under every risk factor. These experts are engineers who are closely related to the analyzed product (i.e., the winding machine). They come from different departments related to the whole life cycle of the product, such as the design department, manufacturing department, quality department, and after-sales department. Therefore, they know the potential failures in different stages clearly and can conduct the risk assessments of these failures from different professional perspectives, i.e., design, manufacturing, assembly, testing, and product services, etc. Based on the above discussions, it can be considered that the obtained risk assessments of failure modes are considered comprehensive and highly credible. Because of the vagueness and uncertainness in the context of evaluation, it is difficult for experts to give a crisp value for each failure mode. Thus the original assessments are expressed with linguistic terms, and the assessment results are shown in Table 5.

Table 5

The risk evaluation for failure modes with linguistic terms

Failure modes	Risk factors
	O	S	D	M
F₁	SH,H,SH,H,SH	VH,VH,VH,H,VH	SH,VH,SH,M,H	L,L,SL,SL,L
F₂	H,H,H,VH,H	H,H,H,SH,H	SL,SL,L,VL,L	L,SL,SL,SL,L
F₃	EL,EL,VL,EL,VL	VL,VL,EL,L,VL	L,SL,L,SL,SL	L,L,L,EL,EL
F₄	SH,SH,M,SH,SH	M,SH,M,SH,M	H,M,SH,SH,SH	SL,SL,M,SL,M
F₅	L,VL,L,VL,L	L,L,L,L,VL	EL,EL,EL,VL,VL	SL,L,L,M,L
F₆	H,SH,H,H,SH	VH,H,EH,VH,EH	EH,EH,VH,EH,VH	H,VH,H,VH,H
F₇	M,SL,SH,M,M	SL,SL,SH,SH,SH	H,SH,H,H,H	EL,L,VL,VL,VL
F₈	VH,H,H,H,H	VH,VH,VH,VH,VH	M,H,SH,H,SH	SH,SH,VH,SH,H
F₉	L,L,L,L,VL	VL,L,L,VL,L	SH,SH,H,M,SH	M,SL,M,SH,SH
F₁₀	M,SL,M,M,M	SL,SL,M,M,SH	SH,SH,SH,H,H	VL,L,VL,L,VL
F₁₁	M,M,M,M,SL	SL,L,SL,SL,L	H,SH,SH,SH,SH	SL,M,M,M,SL
F₁₂	SH,SH,H,SH,SH	H,H,H,VH,H	H,H,H,SH,SH	L,VL,L,L,L

According to the developed linguistic computation method, the original assessments can be transformed into IFC values by Equations (35)–(37). Inspired by Li et al. [23], the hyper entropy He can be set as 0.0075. Limit to space, and Table 6 only gives the IFC assessment matrix transformed from linguistic evaluation given by expert M₁. FMEA members with different backgrounds, e.g., work experience, positional title, and professional relevance, are considered to possess different weights due to their diverse knowledge and experiences.

Table 6

The IFC value matrix of the risk evaluation for failure modes given by expert M₁

Failure modes	Risk factors
	O	S	D	M
F₁	(0.6,0.0167,0.0075)	(0.8,0.0167,0.0075)	(0.6,0.0167,0.0075)	(0.3,0.0167,0.0075)
F₂	(0.7,0.0167,0.0075)	(0.7,0.0167,0.0075)	(0.4,0.0167,0.0075)	(0.3,0.0167,0.0075)
F₃	(0.075,0.025,0.0075)	(0.2,0.0167,0.0075)	(0.3,0.0167,0.0075)	(0.3,0.0167,0.0075)
F₄	(0.6,0.0167,0.0075)	(0.5,0.0167,0.0075)	(0.7,0.0167,0.0075)	(0.4,0.0167,0.0075)
F₅	(0.3,0.0167,0.0075)	(0.3,0.0167,0.0075)	(0.075,0.025,0.0075)	(0.4,0.0167,0.0075)
F₆	(0.7,0.0167,0.0075)	(0.8,0.0167,0.0075)	(0.925,0.025,0.0075)	(0.7,0.0167,0.0075)
F₇	(0.5,0.0167,0.0075)	(0.4,0.0167,0.0075)	(0.7,0.0167,0.0075)	(0.075,0.025,0.0075)
F₈	(0.8,0.0167,0.0075)	(0.8,0.0167,0.0075)	(0.5,0.0167,0.0075)	(0.6,0.0167,0.0075)
F₉	(0.3,0.0167,0.0075)	(0.2,0.0167,0.0075)	(0.6,0.0167,0.0075)	(0.5,0.0167,0.0075)
F₁₀	(0.5,0.0167,0.0075)	(0.4,0.0167,0.0075)	(0.6,0.0167,0.0075)	(0.2,0.0167,0.0075)
F₁₁	(0.5,0.0167,0.0075)	(0.4,0.0167,0.0075)	(0.7,0.0167,0.0075)	(0.4,0.0167,0.0075)
F₁₂	(0.6,0.0167,0.0075)	(0.7,0.0167,0.0075)	(0.7,0.0167,0.0075)	(0.3,0.0167,0.0075)

The subjective weight scores S_k (k = 1, 2, 3, 4, 5) of FMEA experts are determined with the criteria of work experience, positional title, and professional relevance, and the results are shown in Table 7. Then, based on the importance assessment for the failure modes, the objective weight of each expert is computed by using Equations (40) –(41). Based on the above steps, the composite calculation based on Equation (42) is performed to gain the synthetic weights of FMEA members, and the final results are provided as a vector shown as: $\begin{matrix} W = (w_{1}, w_{2}, w_{3}, w_{4}, w_{5}) \\ = (0.160, 0.237, 0.280, 0.128, 0.195) . \end{matrix}$

Table 7

The weights information of FMEA team members

Experts	Weights information
	S_k	$W_{k}^{s}$	D_k	$W_{k}^{0}$
M₁	9	0.167	28	0.195
M ₂	12	0.222	25	0.218
M ₃	13	0.241	23	0.237
M ₄	9	0.167	35	0.156
M ₅	11	0.204	28	0.195

On the basis of the individual IFC value matrix supplied by each expert and the experts’ weight vector, the synthetic IFC assessment matrix $\bar{IC}$ is attained by utilizing Equation (38), and the result is revealed in Table 8. For calculating the risk factors’ synthetic weights, the individual initial importance matrix for each risk factor is provided by each expert according to the linguistic terms shown in Table 2. Then, the initial importance matrix is transformed to IFC value matrix C′

Table 8

The synthetic IFC assessment matrix of the risk evaluation for failure modes

Failure modes	Risk factors
	O	S	D	M
F₁	(0.6365,0.0075,0.0034)	(0.7872,0.0075,0.0034)	(0.6541,0.0075,0.0034)	(0.3408,0.0075,0.0034)
F₂	(0.7128,0.0075,0.0034)	(0.6872,0.0075,0.0034)	(0.3269,0.0075,0.0034)	(0.3645,0.0075,0.0034)
F₃	(0.1344,0.0096,0.0034)	(0.1778,0.0087,0.0034)	(0.356,0.0075,0.0034)	(0.2273,0.0088,0.0034)
F₄	(0.572,0.0075,0.0034)	(0.5365,0.0075,0.0034)	(0.5923,0.0075,0.0034)	(0.4475,0.0075,0.0034)
F₅	(0.2635,0.0075,0.0034)	(0.2805,0.0075,0.0034)	(0.1154,0.0101,0.0034)	(0.3416,0.0075,0.0034)
F₆	(0.6568,0.0075,0.0034)	(0.8357,0.0094,0.0034)	(0.8656,0.0096,0.0034)	(0.7365,0.0075,0.0034)
F₇	(0.5043,0.0075,0.0034)	(0.5206,0.0075,0.0034)	(0.6763,0.0075,0.0034)	(0.2037,0.0082,0.0034)
F₈	(0.716,0.0075,0.0034)	(0.8,0.0075,0.0034)	(0.6205,0.0075,0.0034)	(0.6755,0.0075,0.0034)
F₉	(0.2805,0.0075,0.0034)	(0.2712,0.0075,0.0034)	(0.6152,0.0075,0.0034)	(0.5086,0.0075,0.0034)
F₁₀	(0.4763,0.0075,0.0034)	(0.4798,0.0075,0.0034)	(0.6323,0.0075,0.0034)	(0.2365,0.0075,0.0034)
F₁₁	(0.4805,0.0075,0.0034)	(0.3568,0.0075,0.0034)	(0.616,0.0075,0.0034)	(0.4645,0.0075,0.0034)
F₁₂	(0.628,0.0075,0.0034)	(0.7128,0.0075,0.0034)	(0.6677,0.0075,0.0034)	(0.2763,0.0075,0.0034)

$Z = (C_{1}^{'}, C_{2}^{'}, C_{3}^{'}, . . ., C_{12}^{'})^{T} = [\begin{matrix} (0.1536, 0.0051, 0.0004) \\ (0.155, 0.0037, 0.0004) \\ (0.0902, 0.0057, 0.0004) \\ (0.1067, 0.0052, 0.0004) \end{matrix}] .$

After the obtainment of the group importance IFC vector Z, by employing Equations (46)–(50), the weight information of risk factors O, S, D, M can be obtained and shown in Table 9.

Table 9

The weight information of risk factors

Risk factor	Weights information
	$P_{j}^{s}$	V_j	$P_{j}^{0}$	P _j
O	0.319	0.0234	0.212	0.262
S	0.333	0.0380	0.344	0.445
D	0.153	0.0290	0.262	0.155
M	0.195	0.0202	0.182	0.138

Next, based on the synthetic weight p_j (j = 1,2,3,4) for risk factors and synthetic IFC assessment matrix, the synthetic weighted IFC assessment matrix $\bar{\bar{IC}}$ (cf. Table 10) is described by using Equation (51). Finally, based on the matrix $\bar{\bar{IC}}$ , the C-PIS CS_p and the C-NIS CS_N can be defined by Equations (52) –(55), and the results of CS_p and CS_N are identified in Table 10.

Table 10

The synthetic weighted IFC assessment matrix and the vector of CS_p and CS_N

Failure modes	Risk factors
	O	S	D	M
F₁	(0.1668,0.0038,0.0017)	(0.3503,0.005,0.0022)	(0.1014,0.0029,0.0013)	(0.047,0.0028,0.0012)
F₂	(0.1868,0.0038,0.0017)	(0.3058,0.005,0.0022)	(0.0507,0.0029,0.0013)	(0.0503,0.0028,0.0012)
F₃	(0.0352,0.0049,0.0017)	(0.0791,0.0058,0.0022)	(0.0552,0.0029,0.0013)	(0.0314,0.0033,0.0012)
F₄	(0.1499,0.0038,0.0017)	(0.2387,0.005,0.0022)	(0.0918,0.0029,0.0013)	(0.0618,0.0028,0.0012)
F₅	(0.069,0.0038,0.0017)	(0.1248,0.005,0.0022)	(0.0179,0.004,0.0013)	(0.0471,0.0028,0.0012)
F₆	(0.1721,0.0038,0.0017)	(0.3719,0.0063,0.0022)	(0.1342,0.0038,0.0013)	(0.1016,0.0028,0.0012)
F₇	(0.1321,0.0038,0.0017)	(0.2317,0.005,0.0022)	(0.1048,0.0029,0.0013)	(0.0281,0.003,0.0012)
F₈	(0.1876,0.0038,0.0017)	(0.356,0.005,0.0022)	(0.0962,0.0029,0.0013)	(0.0932,0.0028,0.0012)
F₉	(0.0735,0.0038,0.0017)	(0.1207,0.005,0.0022)	(0.0954,0.0029,0.0013)	(0.0702,0.0028,0.0012)
F₁₀	(0.1248,0.0038,0.0017)	(0.2135,0.005,0.0022)	(0.098,0.0029,0.0013)	(0.0326,0.0028,0.0012)
F₁₁	(0.1259,0.0038,0.0017)	(0.1588,0.005,0.0022)	(0.0955,0.0029,0.0013)	(0.0641,0.0028,0.0012)
F₁₂	(0.1645,0.0038,0.0017)	(0.3172,0.005,0.0022)	(0.1035,0.0029,0.0013)	(0.0381,0.0028,0.0012)
CS_P	(0.1876,0.0049,0.0017)	(0.3719,0.0063,0.0022)	(0.1342,0.004,0.0013)	(0.1016,0.0033,0.0012)
CS_N	(0.0352,0.0038,0.0017)	(0.0791,0.005,0.0022)	(0.0179,0.0029,0.0013)	(0.0281,0.0028,0.0012)

Then, the distances $d_{i}^{+}$ and $d_{i}^{-}$ between failure modes and the C-PIS and C-NIS are obtained with Equations (56)–(57), respectively, and in this case the p is set to 2. Through Equation (58), the closeness coefficient Ct_i (i = 1,2,3 ... ,12) of each failure mode is obtained. After that, the risk ranking of each failure mode can be determined by sorting Ct_i values. The distance value, closeness coefficient, and risk ranking for each failure mode are shown in Table 11.

Table 11

The results of distances values, closeness coefficient, and risk ranking for each failure mode

Failure modes	$d_{i}^{+}$	$d_{i}^{-}$	Ct_i	Ranking
F₁	0.1299	0.5052	0.7955	3
F₂	0.2023	0.4332	0.6817	5
F₃	0.5944	0.0425	0.0667	12
F₄	0.2531	0.3818	0.6013	6
F₅	0.5364	0.0996	0.1566	11
F₆	0.0163	0.6195	0.9744	1
F₇	0.2986	0.3367	0.5300	7
F₈	0.0635	0.5727	0.9002	2
F₉	0.4356	0.1994	0.3140	10
F₁₀	0.3264	0.3086	0.4860	8
F₁₁	0.3511	0.2839	0.4471	9
F₁₂	0.1720	0.4630	0.7292	4

6 Results discussion

6.1 Sensitivity analysis

In this portion, the sensitivity analysis of the risk assessment result on the parameter p and risk factor weights are discussed to validate the robustness of the proposed IFC-FMEA strategy.

A. The sensitivity analysis of risk assessments to the parameter p

The parameter p in Equations (56) –(57) is an essential variable, which reflects the differences of the decision-making spaces. Different p values may conduct different final risk assessments of failure modes. Thus, it is meaningful to perform the sensitivity analysis of the parameter p to investigate the effectiveness and robustness of the risk assessment result of failure modes. In this part, ten trials are implemented to discuss the changes of final risk assessments with the change of p values, where, the ps are set as p = 1, 2, 3, ... , 10, respectively. The closeness coefficients Cts for the 12 failure modes under different p values are illustrated in Fig. 6. As shown in Fig. 6, the Cts change little with the increase of p, which demonstrates that the risk assessments of failure modes are stable in different decision-making circumstances. In addition, the non-intersecting state of the lines in Fig. 6 indicates that the risk ranking of the failures has not changed. Therefore, it can be claimed that the proposed IFC-FMEA strategy is robust in dealing with the failures’ risk assessment of the case mentioned in Section 5. On the other hand, apparently, the above results only can be applicable to the initial decision-making matrices in Section 5. For other decision-matrices, the changes in the parameter p may affect the final risk rankings of failure modes.

B. The sensitivity analysis of risk assessments to the risk factor weight

The weight information of risk factors represents the relative importance of different factors. It is practical to consider it in the FMEA process and can improve the risk assessment accuracy of failure modes. To investigate the play of factor weight in the FMEA model, a sensitivity analysis of the failures’ risk ranking to the factor weight is implemented in this portion, where four risk assessment trials (noted as T1-T4) for the case mentioned in Section 5 are performed under different factor weight conditions. The T1 corresponds to the weight condition in Section 5. In T2, all the four risk factors are empowered equal weights, i.e., p_j = 0.25 (j = 1, 2, 3, 4), where p_j is the weight of the risk factor R₁. In T3, the weights of the two most important risk factors are interchanged, that is, interchanging the weights of risk factors R₁ and R₂. In T4, the weights of the most important factor and the least factor are exchanged, i.e., exchanging the weights of risk factors R₂ and R₄. The results of the four trials are illustrated in Table 12. The Cts and risk ranking shown in Table 12 can be further graphically expressed by Figs. 7 and 8, respectively. As we can see in Fig. 7, the Cts of most failure modes are different in different trials. It indicates that the risk assessment of failures is greatly affected by the weight assignment of risk factors. In Fig. 8, it can be found that the risk ranking in T2 and T4 are greatly different from that in T1. This is because, with the adjustment of the factors’ weights, some unimportant factors obtain higher weights than before while the weights of some important factors are reduced. Accordingly, the rankings of the failure modes which have high-risk estimates under unimportant factors are improved, and the rankings of the high-risk failure modes under important factors are dropped. This is inconsistent with the practice. Base on the above discussion, it can be found that the risk ranking of failure modes is greatly sensitive to the risk factors, and thus a reasonable weight assignment strategy in the FMEA model will improve the accuracy of the risk assessments and risk ranking for failure modes.

Table 12
The risk assessment results of the four trials

Failure modes T1 T2 T3 T4

Cts Risk ranking Cts Risk ranking Cts Risk ranking Cts Risk ranking

F₁ 0.7955 3 0.7085 3 0.7817 3 0.6037 3

F₂ 0.6817 5 0.5783 6 0.7172 5 0.5463 5

F₃ 0.0667 12 0.1072 12 0.0684 12 0.0826 12

F₄ 0.6013 6 0.6014 5 0.6381 6 0.5809 4

F₅ 0.1566 11 0.1484 11 0.1679 11 0.1848 11

F₆ 0.9744 1 0.9750 1 0.9564 1 0.9721 1

F₇ 0.5300 7 0.5052 8 0.5499 7 0.3880 9

F₈ 0.9002 2 0.8629 2 0.9080 2 0.8808 2

F₉ 0.3140 10 0.4139 10 0.3366 10 0.4431 8

F₁₀ 0.4860 8 0.4732 9 0.5087 8 0.3788 10

F₁₁ 0.4471 9 0.5100 7 0.5064 9 0.5180 7

F₁₂ 0.7292 4 0.6554 4 0.7334 4 0.5382 6

Failure modes	T1	T2	T3	T4
F₁	0.7955	3	0.7085	3	0.7817	3	0.6037	3
F₂	0.6817	5	0.5783	6	0.7172	5	0.5463	5
F₃	0.0667	12	0.1072	12	0.0684	12	0.0826	12
F₄	0.6013	6	0.6014	5	0.6381	6	0.5809	4
F₅	0.1566	11	0.1484	11	0.1679	11	0.1848	11
F₆	0.9744	1	0.9750	1	0.9564	1	0.9721	1
F₇	0.5300	7	0.5052	8	0.5499	7	0.3880	9
F₈	0.9002	2	0.8629	2	0.9080	2	0.8808	2
F₉	0.3140	10	0.4139	10	0.3366	10	0.4431	8
F₁₀	0.4860	8	0.4732	9	0.5087	8	0.3788	10
F₁₁	0.4471	9	0.5100	7	0.5064	9	0.5180	7
F₁₂	0.7292	4	0.6554	4	0.7334	4	0.5382	6

Fig. 6

The Cts of failure modes under different p values.

Fig. 7

The Cts of 12 failure modes under different trials.

Fig. 8

The risk ranking of failure modes under different trials.

On the other hand, Fig. 8 shows that the risk ranking of failure modes in T3 is the same as that in T1, which demonstrates that the interchange of the weights of both very important factors does not change the risk ranking of failure modes. In other words, although the proposed method is sensitive to the risk factors, it still has a certain tolerance for inaccurate weight allocation.

6.2 Comparative discussion

In this part, four representative FMEA models, including conventional FMEA [4], fuzzy FMEA [65], rough FMEA [38], and intuitionistic fuzzy FMEA [10], are implemented to the above exemplification as comparisons to prove the effectiveness of the propose IFC-FMEA model. The detailed execution procedures of those methods are revealed in the relevant literature.

Table 13 provides the execution results and the priority ratings of the twelve failure modes by utilizing the proposed FMEA model and the four contrastive methods. The ranking diversities of failure modes between the IFC-FMEA model and the other four comparative FMEA models are displayed in Fig. 9. It can be seen from Fig. 9, a large proportion of ranking orders of failure modes in the proposed method coincide with those in the comparative FMEA model, which illustrates that the FMEA model presented in this paper is effective. The differences in the ranking results obtained by different methods are detailed compared as follows.

Fig. 9

Ranking comparison between IFC-FMEA model with representative FMEA model.

Table 13

The risk rankings under different FMEA strategies

Failure modes	Conventional					Fuzzy		Rough		Intuitionistic		Proposed
	FMEA					FMEA		FMEA		fuzzy FMEA		method
	O	S	D	RPN	Ranking	Ct_i	Ranking	Ct_i	Ranking	Ct_i	Ranking	Ct_i	ranking
F₁	6	8	8	384	4	0.308	4	0.700	3	0.539	4	0.8165	3
F₂	8	7	4	224	5	0.375	3	0.661	5	0.561	3	0.6997	5
F₃	5	3	4	60	12	0.133	12	0.367	12	0.278	12	0.0976	12
F₄	6	6	5	180	7	0.273	6	0.586	6	0.474	6	0.5862	6
F₅	6	4	3	72	11	0.137	11	0.449	10	0.291	11	0.1654	11
F₆	8	9	7	504	1	0.593	1	0.837	1	0.681	1	0.9576	1
F₇	6	5	5	150	8	0.148	10	0.395	11	0.449	7	0.5408	7
F₈	6	9	9	486	2	0.417	2	0.742	2	0.657	2	0.8875	2
F₉	2	9	4	72	10	0.173	9	0.536	8	0.324	10	0.2719	10
F₁₀	6	4	7	158	6	0.201	8	0.487	9	0.407	8	0.4905	8
F₁₁	5	4	6	120	9	0.242	7	0.574	7	0.375	9	0.3932	9
F₁₂	6	8	9	432	3	0.288	5	0.676	4	0.496	5	0.7551	4

The differences comparison between the developed FMEA model and the conventional FMEA method is executed first. Based on Fig. 9, the risk priority orders of all failure modes in the proposed approach and the classical FMEA method can be identified as F₆ _> F₈ _> F₁ _> F₁₂ > F₂ _> F₄ _> F₇ _> F₁₀ _> F₁₁ _> F₉ _> F₅ _> F₃ and F₆ _> F₈ _> F₁₂ _> F₁ _> F₂ _> F₁₀ _> F₄ _> F₇ _> F₁₁ _> F₉ _>F₅ _> F₃, respectively. From the orders, it can be found that the risk priorities of failure modes F₁, F₄, F₇, F₁₀, and F₁₂ are different in the two ranking sequences. These disparities can be explicated by the inherent weaknesses of the traditional FMEA model. For instance, the priorities of the failure mode F₁ and F₁₂ are located at fourth and third positions in the conventional FMEA method, respectively; however, it is exactly the opposite in the proposed method. This variation is mainly due to that the maintenance difficulty of F₁ is greater than that of F₁₂. Thus, the risk priority of F₁ is higher than that of F₁₂ in the proposed FMEA model by adding maintenance into the risk evaluation system. Besides, the RPN values of F₇, and F₁₀ are assigned 150 and 158 in the traditional FMEA model, respectively. Hence, F₁₀ obtains a higher risk ranking than F₇ in the conventional RPN method. However, in the proposed FMEA model, F₁₀ is ranked behind failure mode F₇. The cause for the change is that the traditional FMEA method is not enough to consider the relative importance of risk factors and the difficulty of maintenance, which may produce unrealistic risk orders of failure modes. Furthermore, the neglect of the weights of FMEA members and the incorrect crisp assessment values of RPN elements also may be the reasons for the above differences in risk ranking between the traditional FMEA approach and the proposed IFC-FMEA model.

Next, the priority order produced by the IFC-FMEA model is relatively dissimilar from those obtained by the fuzzy FMEA and the rough FMEA. For the comparison between fuzzy FMEA and IFC-FMEA, as exhibited in Fig. 9, the ranks of failure modes F₁, F₂, F₇, F₉, F₁₁, F₁₂ obtained by fuzzy FMEA are different from those gained by the proposed IFC-FMEA method. The main cause of those disparities is that the fuzzy FMEA model ignores the randomness of experts’ evaluations.

In addition, the fuzzy FMEA model also ignores both subjective and objective weights for FMEA members and risk factors. According to the comparison of rough FMEA and IFC-FMEA, the differences can be determined as that the ranks of failure modes of F₅, F₇, F₉, F₁₀, F₁₁ are changed in the rough FMEA model compared with the proposed method. The reason which causes these distinctions can be represented from two aspects. The one is that the subjective and objective weights and the randomness are also not considered. The other one can be explained as that the rough FMEA does not take the randomness of each experts’ evaluations into account. Nevertheless, those kinds of uncertainties (such as subjectivity, objectivity, randomness, uncertainty, and fuzziness) are handled in the presented IFC-FMEA model. From this aspect, the methodology proposed in this study may produce a more accurate risk ranking of failure modes than that fuzzy FMEA model and rough FMEA model.

Finally, the comparison of disparity is enacted between the intuitionistic fuzzy FMEA model and the developed method. As shown in Table 13, the priority orders of failure modes calculated by the two FMEA models are coincident except for those of failure modes F₁, F₂, and F₁₂. For instance, the failure mode F₁, F₂, and F₁₂ are assigned the ranks of 3, 5, and 4 by intuitionistic fuzzy FMEA model, respectively; however, they have the serial numbers of 4, 3, and 5 in the developed method. One of the main causes of those disparities can be explained as that the developed IFC-FMEA model integrates cloud model theory and intuitionistic fuzzy set to address the randomness and uncertainness of FMEA members’ judgments simultaneously. Via the novel developed IFC-FMEA method, the IFN can be transformed into IFC value containing three characteristic values IEx, IEn, and IHe, where IEn and IHe can reflect the randomness of experts’ evaluations. The lower the values of IEn and IHe, the lower the randomness of experts’ evaluations, and thus the higher the credibility of the evaluation results. Take the comparison between failure mode F₂ and F₁₂ as an example, and it can be observed that the closeness coefficient Ct of failure mode F₂ is higher than that of F₁₂ in the intuitionistic fuzzy FMEA approach, which makes the risk ranking of F₂ is higher than that of F₁₂.

However, due to the influence of IEn and IHe, the Ct value of F₂ is exceeded by that of F₁₂ in the proposed method, which changes the relative priorities of F₂ and F₁₂. From the above comparative analysis, it can be convinced that the randomness of experts’ assessments is an important item that cannot be ignored in the risk ranking of failure modes. In addition to the major reason mentioned above, both subjective and objective weights of risk factors and FMEA members are also considered in the IFC-FMEA model. Hence, the proposed FMEA model may produce a more practical risk ranking order for failure modes than the intuitionistic fuzzy FMEA approach.

Base on the above statements, the capabilities of different methods in dealing with risk evaluations given by FMEA team are compared in Table 14.

Table 14

The capability comparisons of different FMEA models

Methods	Dealing with fuzziness	Implementing linguistic values	Dealing with synthetic weights for risk factors	Dealing with synthetic weights for experts	Considering randomness	Considering maintenance as a risk factor	The discernibility among failure modes
Conventional FMEA	×	×	×	×	×	×	Slightly High
Fuzzy FMEA	√	√	×	×	×	×	Medium
Rough FMEA	√	√	×	×	×	×	Low
Intuitionistic fuzzy FMEA	√	√	√	×	×	×	Low
Proposed method	√	√	√	√	√	√	High

6.3 Simulation discussion

In this section, to reveal the superiority and generalizability of the presented novel IFC-FMEA framework, a simulation analysis is executed in four risk priority cases.

The simulation data for case 1 are obtained from Table 13, and that of the other three cases are calculated from a set of qualitative assessments randomly produced by a computer. In this study, these qualitative assessments are produced in MATLAB R2017b, and the corresponding pseudocode is shown in Fig. 10. Then, the assessment information processing and graphics rendering are completed in Excel 2016 under Windows 10 system.

Fig. 10

The pseudocode for the generation of random assessments.

Suppose R_d is a criterion, which is established to reflect the relative distance of risk degree between each failure mode and reference failure mode. The computation of Rd can be defined as $R_{di} = | C t_{i} - {Ct}_{i}^{-} | / - {Ct}_{i}^{-}$ , where Ct_i is the closeness coefficient of ith failure mode and ${Ct}_{i}^{-}$ is the closeness coefficient of the reference failure mode which has the lowest risk grade among all failure modes. Figure 11 illustrates the simulation results generated by the IFC-FMEA model and other comparative FMEA methods. As can be seen from Fig. 11, undulates among twelve failure modes produced by the IFC-FMEA approach are significantly stronger than those generated by other comparative FMEA methods. Besides, most of the R_d values obtained by the proposed IFC-FMEA model are bigger than those by the four comparative FMEA models. Moreover, according to Fig. 11, we can find that the fluctuation of the simulation line of the IFC-FMEA model is stronger than that of other FMEA models. That is to say, the presented FMEA strategy possesses higher discrimination than other representative FMEA models and thus enhances the risk identification in FMEA technical. The simulation results for the case implemented in this research and the other three random cases further illustrate the effectiveness and superiorities of the proposed IFC-FMEA model.

Fig. 11

The relative distances between the reference failure mode and the other failure modes.

6.4 Validity discussion

The risk evaluation result obtained by the proposed FMEA model indicates that the failure modes F₆ (gripper inoperation or deformation or fracture), F₈ (hand drive device failed), F₁ (Location control error), F₁₂ (position sensor of lift and transposition failed), and F₂ (winding drum and wirerope abrasion) are the top five items in the risk ranking. In the case study, the winding engine is used to elevate and transfer important assemblies, and its life requirement is 40 years. The gripper is to grip and clamp the assemblies at a specific position where the shape is certain. Thus, the gripper is a quite important component for the winding engine, and it has a specific design shape to fit the shape of assembles. When the gripper fails, it will seriously threaten the workers’ safety and needs a lot of time and cost to produce replacement parts. In addition, continuous grasping and relaxation make the failure frequency of the gripper is comparatively high. Therefore, the failure mode F₆ has the highest risk level, which is in good agreement with the real situation. The hand drive device is an emergency standby system used to control the lifting and falling of the assemblies in case of electric drive failure states. Although its use is infrequency, its failure will lead to severe consequences. In addition, the low utilization makes its failures difficult to be found in time, i.e., the detectability of the failures is low. Except that, the maintenance of the hand drive device is relatively difficult since its complex mechanical structure. Therefore, the failure mode F₈ has a high-risk level. Similarly, failure modes F₁ and F₁₂ may cause the transferred assemblies to collide in error locations and thus be damaged. Therefore, the failure modes F₁ and F₁₂ are ranked in front of the most failures. On the other hand, the location control system contains a complete control circuit and multiple control elements making its failures more difficult to maintain than position senor. Thus, the risk of F₁ is higher than that of F₁₂. The frequency of failure mode F₂ is higher than that of F₁ and F₁₂, but its severity is lower than F₁ and F₁₂. Besides, it is easier to be detection and maintenance than F₁ and F₁₂. Hence, F₂ is ranked behind F₁ and F₁₂. The above discussions reveal the risk ranking obtained by the proposed FMEA strategy is in line with reality.

Table 13 shows that the risk ranks of failure modes F₇, F₉, F₁₀, and F₁₁ are quite different in different FMEA methods. Most methods think the risk of F₇ is lower than F₁₀. However, although F₁₀ is more difficult to detect than F₇, its severity is lower than F₇, and the weight of severity is less than that of detection. Furthermore, since the complexity of assembly structure and the costs of time and economics in manufacturing replace parts, the maintenance of F₇ is more complicated than that of F₁₀. Therefore, the risk of F₇ is higher than that of F₁₀ is conforms to the realistic. Besides, some methods believe the failure mode F₉ has a higher risk level than F₇, which is regarded as an error decision by experts since there is a pretty low failure probability and detection difficulty of F₉. According to the above statement, it can be found that the risk ranking produced by the proposed FMEA framework is more reasonable than that gained by other methods mentioned in Subsection 6.1. To further verify this conclusion, two experts are invited to judge the validity of the risk ranking of failure modes generated by the proposed FMEA strategy. They are unanimous that the risk ranking obtained by the proposed method more conforms to the actual circs. Based on the discussions mentioned above, it can be concluded that the evaluation result of risk priorities for failure modes provided by the proposed FMEA model is more reasonable and practical than those produce by other comparison methods. It exhibits the effectiveness and validity of the proposed FMEA framework.

7 Managerial insights

The study can help the engineers and manufacturers in the following aspects:

The study can help engineers obtain high accuracy risk rankings of failure modes, and then facilitates manufacturers select the essential failure modes to priority control. In this way, the product reliability can be effectively improved with low time cost and economic cost.

This study provides a novel uncertainty processing tool, i.e., IFC theory, which can be utilized in FMEA to address various uncertainties in risk assessment of failure modes.

The sensitivity analysis can help engineers understand the impact of the parameter p and the risk factors’ weights on risk assessment and risk ranking of failure modes.

The proposed evaluation framework has strong flexibility, which can be extended to other MADGM environments for improving the decision accuracy.

8 Conclusions and future work

FMEA is a crucial implement in risk evaluation and reliability management. However, the conventional FMEA model and existing developed FMEA strategies have paid little attention to considering the fuzziness and randomness simultaneously. In this study, a novel FMEA model, called IFC-FMEA, is presented based on IFC theory and TOPSIS method to help engineers estimate and determine the risk priorities of failure modes. It can cope with the fuzziness and randomness in the conversion procedure between qualitative linguistic assessments to quantitative numerical information. Furthermore, our developed model considers a more completed risk factor system and deals with more comprehensive weight decisions of risk factors and FMEA members. The validity and effectiveness of the IFC-FMEA model are verified with a realistic case. Then, the risk evaluation result is compared with that obtained by several previous FMEA models. Via the comparative analysis and discussion implemented in Subsections 6.1–6.3, the main contributions of the IFC-FMEA model can be presented as follows:

The proposed method combines the IFS theory cloud model to produce a novel IFC methodology. It is able to overcome the shortcomings of conventional normal cloud, i.e., some characteristics (cf. Subsection 3.3) lead it is not greatly corresponding to the realistic in the risk decision-making process. Furthermore, the relative algorithms of the new IFC theory have been discussed in this study.

The new IFC-FMEA model developed based on the developed IFC theory integrates the virtues of the IFS theory and cloud mode in dealing with the decision-making problem, which makes it can help managers manipulate several uncertainties in risk decision of failure modes, such as fuzziness, hesitation, and randomness.

The IFC-FMEA framework takes the integrated weights consisted of both subjective and objective weights into account by using an integrated algorithm. These weights are able to illustrate the relative importance of risk factors and experts comprehensively. Moreover, the maintenance is considered as a supplement risk factor in the IFC-FMEA model to enhance the comprehensiveness of the factor system of risk evaluation. The application of the proposed IFC-FMEA framework and its result discussions illustrate that the proposed FMEA model can make the risk ranking decision of failure modes more reasonable, and it has higher discrimination than other compared FMEA models in risk identification of failure modes.

Although some advantages can be achieved by applying IFC-FMEA in risk assessment, there are still some limitations that should be taken into account in future research. Liu [25] summarized numerous significant shortcomings for FMEA methods of literature, most of them are considered in our study, but several items are still ignored. Except for the shortcomings presented by Liu [25], some other issues of the proposed IFC theory should be identified and manipulated in further study. The details of those drawbacks and future works are expressed as follows:

First, the relationships between failure modes and causes are neglected; hence the corrective actions and their effectiveness cannot be further proposed and evaluated in FMEA strategies. In the next researches, direct and indirect relations should be taken into consideration. A set of corrective actions for failure modes and an effective evaluation system for these corrective actions should be established to improve the practicability of the proposed FMEA model.

Second, most of the FMEA models only consider O, S, and D as risk factors in the failure modes evaluation process. Although a complete risk assessment system is built by adding M (maintenance) into the risk factor set in the proposed IFC-FMEA strategy, the factors O, S, D, and M are the main aspects of safety. Other essential criteria, like economic and impact, may be required for different practical problems. Thus, there is a need to establish a targeted and comprehensive risk assessment criteria system for different problems to improve the practicability and effectiveness of the proposed FMEA framework.

Third, the availability and effectiveness of the newly developed IFC theory in dealing with the risk decision of failure modes has been proved by a case study. However, the universality, an important metric for a new methodology, cannot be fully illustrated in this study. Hence, the IFC theory should be employed in broader realms in the future to validate its universality in decision-making and other spheres.

Footnotes

Acknowledgment

This work is supported in part by the National Natural Science Foundation of China under Grant 5183000158, and in part by the National Major Scientific and Technological Special Project of China under Grant 2018ZX04032-001.

References

Atanassov

K.T.

, Intuistionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Bhardwaj

, Teixeira

A.P.

and Soares

C.G.

, Reliability prediction of an offshore wind turbine gearbox, Renewable Energy 141 (2019), 693–706.

Boral

, Howard

, Chaturvedi

S.K.

, McKee

and Naikan

V.N.A.

, An integrated approach for fuzzy failure modes and effects analysis using fuzzy AHP and fuzzy MAIRCA, Engineering Failure Analysis 108 (2020).

Chang

K.H.

, Generalized multi-attribute failure mode analysis, Neurocomputing 175 (2016), 90–100.

Chen

T.L.

, Hsu

H.M.

, Pan

S.Y.

and Chiang

P.C.

, Advances and challenges of implementing carbon offset mechanism for a low carbon economy: The Taiwanese experience, Journal of Cleaner Production 239 (2019), 15.

Cheng

, Xiao

and Cao

, A New Distance for Intuitionistic Fuzzy Sets Based on Similarity Matrix, Ieee Access 7 (2019), 70436–70446.

Chu

Y.T.

, Sun

L.Y.

and Li

L.J.

, Lightweight scheme selection for automotive safety structures using a quantifiable multi-objective approach, Journal of Cleaner Production 241 (2019), 15.

Dadsena

K.K.

, Sarmah

S.P.

and Naikan

V.N.A.

, Risk evaluation and mitigation of sustainable road freight transport operation: a case of trucking industry, International Journal of Production Research 57 (2019), 6223–6245.

Daneshvar

, Yazdi

and Adesina

K.A.

, Fuzzy smart failure modes and effects analysis to improve safety performance of system: Case study of an aircraft landing system, Quality and Reliability Engineering International 36 (2020), 890–909.

10.

Efe

, Analysis of operational safety risks in shipbuilding using failure mode and effect analysis approach, Ocean Engineering 187 (2019), 9.

11.

Efe

, Kurt

and Efe

O.F.

, An integrated intuitionistic fuzzy set and mathematical programming approach for an occupational health and safety policy, Gazi University Journal of Science 30 (2017), 73–95.

12.

Fang

, Li

and Song

W.Y.

, Failure mode and effects analysis: an integrated approach based on rough set theory and prospect theory, Soft Computing 24 (2020), 6673–6685.

13.

Fattahi

and Khalilzadeh

, Risk evaluation using a novel hybrid method based on FMEA, extended MULTIMOORA, and AHP methods under fuzzy environment, Safety Science 102 (2018), 290–300.

14.

Huang

G.Q.

, Xiao

L.M.

and Zhang

G.B.

, Improved failure mode and effect analysis with interval-valued intuitionistic fuzzy rough number theory, Engineering Applications of Artificial Intelligence 95 (2020), 103856.

15.

Huang

G.Q.

, Xiao

L.M.

and Zhang

G.B.

, Decision-making model of machine tool remanufacturing alternatives based on dual interval rough number clouds, Engineering Applications of Artificial Intelligence 104 (2021), 104392.

16.

Huang

G.Q.

, Xiao

L.M.

and Zhang

G.B.

, Risk evaluation model for failure mode and effect analysis using intuitionistic fuzzy rough number approach, Soft Computing 25 (2021), 4875–4897.

17.

Huang

, Xiao

and Zhang

, Assessment and prioritization method of key engineering characteristics for complex products based on cloud rough numbers, Advanced Engineering Informatics 49 (2021), 101309.

18.

Huang

G.Q.

, Xiao

L.M.

, Zhang

, Li

, Zhang

G.B.

and Ran

, An improving approach for failure mode and effect analysis under uncertainty environment: A case study of critical function component, Quality and Reliability Engineering International 36 (2020), 2119–2145.

19.

Huang

G.Q.

and Xiao

L.M.

, Failure mode and effect analysis: An interval-valued intuitionistic fuzzy cloud theory-based method, Applied Soft Computing 98 (2021), 106834.

20.

Huang

G.Q.

, Chen

B.K.

, Xiao

L.M.

, Ran

and Zhang

G.B.

, Cascading fault analysis and control strategy for computer numerical control machine tools based on meta action, IEEE Access 7 (2019), 91202–91215.

21.

C.B.

, Qi

Z.Q.

and Feng

, A multi-risks group evaluation method for the informatization project under linguistic environment, Journal of Intelligent & Fuzzy Systems 26 (2014), 1581–1592.

22.

, Liu

and Gan

, A New Cognitive Model: Cloud Model, International Journal of Intelligent Systems 24 (2009), 357–375.

23.

, Fang

and Song

, Modified failure mode and effects analysis under uncertainty: A rough cloud theory-based approach, Applied Soft Computing 78 (2019), 195–208.

24.

X.-Y.

, Wang

Z.-L.

, Xiong

and Liu

H.-C.

, A Novel Failure Mode and Effect Analysis Approach Integrating Probabilistic Linguistic Term Sets and Fuzzy Petri Nets, Ieee Access 7 (2019), 54918–54928.

25.

Liu

H.-C.

, FMEA using uncertainty theories and MCDM methods, (2016).

26.

Liu

H.-C.

, Li

, Song

and Su

, Failure Mode and Effect Analysis Using Cloud Model Theory and PROMETHEE Method, Ieee Transactions on Reliability 66 (2017), 1058–1072.

27.

Liu

H.-C.

, Liu

, Bian

Q.-H.

, Lin

Q.-L.

, Dong

and Xu

P.-C.

, Failure mode and effects analysis using fuzzy evidential reasoning approach and grey theory, Expert Systems with Applications 38 (2011), 4403–4415.

28.

Liu

H.-C.

, Liu

and Li

, Failure mode and effects analysis using intuitionistic fuzzy hybrid weighted Euclidean distance operator, International Journal of Systems Science 45 (2014), 2012–2030.

29.

Liu

H.-C.

, Liu

and Wu

, Material selection using an interval 2-tuple linguistic VIKOR method considering subjective and objective weights, Materials & Design 52 (2013), 158–167.

30.

Liu

H.-C.

, Wang

L.-E.

, Li

and Hu

Y.-P.

, Improving Risk Evaluation in FMEA With Cloud Model and Hierarchical TOPSIS Method, Ieee Transactions on Fuzzy Systems 27 (2019), 84–95.

31.

Liu

H.-C.

, You

J.-X.

, Chen

and Chen

Y.-Z.

, An integrated failure mode and effect analysis approach for accurate risk assessment under uncertainty, Iie Transactions 48 (2016), 1027–1042.

32.

Liu

H.-C.

, You

J.-X.

and Duan

C.-Y.

, An integrated approach for failure mode and effect analysis under interval-valued intuitionistic fuzzy environment, International Journal of Production Economics 207 (2019), 163–172.

33.

Liu

H.-C.

, You

J.-X.

, You

X.-Y.

and Shan

M.-M.

, A novel approach for failure mode and effects analysis using combination weighting and fuzzy VIKOR method, Applied Soft Computing 28 (2015), 579–588.

34.

Liu

H.W.

, New similarity measures between intuitionistic fuzzy sets and between elements, Mathematical and Computer Modelling 42 (2005), 61–70.

35.

Liu

, Bi

J.W.

and Fan

Z.P.

, Ranking products through online reviews: a method based on sentiment analysis technique and intuitionistic fuzzy set theory, Information Fusion 36 (2017), 149–161.

36.

Liu

Z.Y.

and Xiao

F.Y.

, An Intuitionistic Evidential Method for Weight Determination in FMEA Based on Belief Entropy, Entropy 21 (2019), 16.

37.

H.-W.

and Liou

J.J.H.

, A novel multiple-criteria decision-making-based FMEA model for risk assessment, Applied Soft Computing 73 (2018), 684–696.

38.

H.W.

, Liou

J.J.H.

, Huang

C.N.

and Chuang

Y.C.

, A novel failure mode and effect analysis model for machine tool risk analysis, Reliability Engineering & System Safety 183 (2019), 173–183.

39.

Maghsoodi

A.I.

, Rasoulipanah

, Martinez Lopez

, Liao

and Zavadskas

E.K.

, Integrating interval-valued multi-granular 2-tuple linguistic BWM-CODAS approach with target-based attributes: Site selection for a construction project, Computers & Industrial Engineering 139 (2020).

40.

Mirghafoori

S.H.

, Izadi

M.R.

and Daei

, Analysis of the barriers affecting the quality of electronic services of libraries by VIKOR, FMEA and entropy combined approach in an intuitionistic-fuzzy environment, Journal of Intelligent & Fuzzy Systems 34 (2018), 2441–2451.

41.

Mirghafoori

S.H.

, Tooranloo

H.S.

and Saghafi

, Diagnosing and routing electronic service quality improvement of academic libraries with the FMEA approach in an intuitionistic fuzzy environment, Electronic Library 38 (2020), 597–631.

42.

Parameshwaran

, Baskar

and Karthik

, An integrated framework for mechatronics based product development in a fuzzy environment, Applied Soft Computing 27 (2015), 376–390.

43.

Peng

H.-G.

, Zhang

H.-Y.

, Wang

J.-Q.

and Li

, An uncertain Z-number multicriteria group decision-making method with cloud models, Information Sciences 501 (2019), 136–154.

44.

Rao

R.V.

, Patel

B.K.

and Parnichkun

, Industrial robot selection using a novel decision making method considering objective and subjective preferences, Robotics and Autonomous Systems 59 (2011), 367–375.

45.

Ren

J.Z.

, Sustainability prioritization of energy storage technologies for promoting the development of renewable energy: A novel intuitionistic fuzzy combinative distance-based assessment approach, Renewable Energy 121 (2018), 666–676.

46.

Sang

A.J.

, Tay

K.M.

, Lim

C.P.

and Nahavandi

, Application of a Genetic-Fuzzy FMEA to Rainfed Lowland Rice Production in Sarawak: Environmental, Health and Safety Perspectives, Ieee Access 6 (2018), 74628–74647.

47.

Xiao

L.M.

, Huang

G.Q.

and Zhang

G.B.

, An integrated risk assessment method using Z-fuzzy clouds and generalized TODIM, Quality and Reliability Engineering International (2022), 1–35. https://doi.org/10.1002/qre.3062

48.

Xiao

L.M.

, Huang

G.Q.

and Zhang

G.B.

, Toward an action-granularity-oriented modularization strategy for complex mechanical products using a hybrid GGACGA method, Neural Computing & Applications (2022). https://doi.org/10.1007/s00521-021-06796-9

49.

Silva

D.F.D.

, Ferreira

and De Almeida-Filho

A.T.

, A new preference disaggregation TOPSIS approach applied to sort corporate bonds based on financial statements and expert’s assessment, Expert Systems with Applications 152 (2020), 13.

50.

Song

and Zhu

, Three-reference-point decision-making method with incomplete weight information considering independent and interactive characteristics, Information Sciences 503 (2019), 148–168.

51.

Tang

, Zhou

and Chan

F.T.S.

, AMWRPN: Ambiguity Measure Weighted Risk Priority Number Model for Failure Mode and Effects Analysis, Ieee Access 6 (2018), 27103–27110.

52.

Tsai

T.-N.

and Yeh

J.-H.

, Identification and risk assessment of soldering failure sources using a hybrid failure mode and effect analysis model and a fuzzy inference system, Journal of Intelligent & Fuzzy Systems 28 (2015), 2771–2784.

53.

Wan

, Li

, Ye

and Wang

, Risk Assessment in Intelligent Manufacturing Process: A Case Study of an Optical Cable Automatic Arranging Robot, Ieee Access 7 (2019), 105892–105901.

54.

Wang

, Deng

, Zhang

and Jiang

, A New Failure Mode and Effects Analysis Method Based on Dempster-Shafer Theory by Integrating Evidential Network, Ieee Access 7 (2019), 79579–79591.

55.

Wang

J.-Q.

, Peng

J.-J.

, Zhang

H.-Y.

, Liu

and Chen

X.-H.

, An Uncertain Linguistic Multi-criteria Group Decision-Making Method Based on a Cloud Model, Group Decision and Negotiation 24 (2015), 171–192.

56.

Wang

, Yan

, Wang

and Li

, FMEA-CM based quantitative risk assessment for process industries-A case study of coal-to-methanol plant in China, Process Safety and Environmental Protection 149 (2020), 299–311.

57.

Wang

W.Z.

, Liu

X.W.

and Liu

S.L.

, Failure Mode and Effect Analysis for Machine Tool Risk Analysis Using Extended Gained and Lost Dominance Score Method, Ieee Transactions on Reliability 69 (2020), 954–967.

58.

Xiao

, Huang

, Zhang

, Zhu

and Jin

, A failure tracing method with hierarchical digraph and meta action for complex electromechanical systems, Quality and Reliability Engineering International (2022), 1–27. https://doi.org/10.1002/qre.3095

59.

Xiao

, Huang

and Zhang

, Improved assessment model for candidate design schemes with an interval rough integrated cloud model under uncertain group environment, Engineering Applications of Artificial Intelligence 104 (2021), 104352.

60.

Z.Z.

, Lee

, Albani

, Dobbins

, Ellis

R.J.

, Biswas

, Machtay

and Podder

T.K.

, Evaluating radiotherapy treatment delay using Failure Mode and Effects Analysis (FMEA), Radiotherapy and Oncology 137 (2019), 102–109.

61.

Yang

, Yan

and Zeng

, How to handle uncertainties in AHP: The Cloud Delphi hierarchical analysis, Information Sciences 222 (2013), 384–404.

62.

Yazdi

, Daneshvar

and Setareh

, An extension to Fuzzy Developed Failure Mode and Effects Analysis (FDFMEA) application for aircraft landing system, Safety Science 98 (2017), 113–123.

63.

Yousefi

, Alizadeh

, Hayati

and Baghery

, HSE risk prioritization using robust DEA-FMEA approach with undesirable outputs: A study of automotive parts industry in Iran, Safety Science 102 (2018), 144–158.

64.

Zhang

Q.H.

, Zhao

, Yang

and Wang

G.Y.

, Three-way decisions of rough vague sets from the perspective of fuzziness, Information Sciences 523 (2020), 111–132.

65.

Zhao

X.S.

, Chen

T.H.

, Zhang

and Wang

M.J.J.

, Applying an improved failure mode effect analysis method to evaluate the safety of a three-in-one machine tool, Human Factors and Ergonomics in Manufacturing & Service Industries (2020), 12.

Failure modes	T1		T2		T3		T4
	Cts	Risk ranking	Cts	Risk ranking	Cts	Risk ranking	Cts	Risk ranking
F₁	0.7955	3	0.7085	3	0.7817	3	0.6037	3
F₂	0.6817	5	0.5783	6	0.7172	5	0.5463	5
F₃	0.0667	12	0.1072	12	0.0684	12	0.0826	12
F₄	0.6013	6	0.6014	5	0.6381	6	0.5809	4
F₅	0.1566	11	0.1484	11	0.1679	11	0.1848	11
F₆	0.9744	1	0.9750	1	0.9564	1	0.9721	1
F₇	0.5300	7	0.5052	8	0.5499	7	0.3880	9
F₈	0.9002	2	0.8629	2	0.9080	2	0.8808	2
F₉	0.3140	10	0.4139	10	0.3366	10	0.4431	8
F₁₀	0.4860	8	0.4732	9	0.5087	8	0.3788	10
F₁₁	0.4471	9	0.5100	7	0.5064	9	0.5180	7
F₁₂	0.7292	4	0.6554	4	0.7334	4	0.5382	6