Risk prioritization based on the combination of FMEA and dual hesitant fuzzy sets method

Paper aims: This paper proposes the combination of the quality tool FMEA (Failure Modes Effects and Analysis) with the DHFS (Dual Hesitant Fuzzy sets) technique to process judgements with hesitation and hence conduct the prioritization of failure modes considering a group decision making problem. Originality: There are no studies that combine the FMEA tool with the DHFS technique. Research method: Firstly, this paper presents a review of the current FMEA literature. Then, the group decision model is presented combining the FMEA and the DHFS. Finally, an illustrative example in the context of supplier failure modes is brought to guide future applications of the proposal. Main findings: The paper presents a model that combines the FMEA tool with the DHFS. It allows considering different risk factors weights in a group decision process with experts from several areas. The model is also able to deal with the different types of hesitations present in the judgements. Implications for theory and practice: The traditional FMEA does not deal with individual judgments of different decision makers. The new proposal can be easily applied in different contexts of potential failure modes analysis considering different types of hesitation in group decision making, such as medical and humanitarian.


FMEA
FMEA is an analytical method of risk assessment that seeks to identify, prioritize, and determine the causes and effects associated with failure modes (Fattahi & Khalilzadeh, 2018). Potential failure modes are defined as the way in which a component, a system, or a subsystem can potentially fail to achieve or deliver the functionality described for the item (Xu et al., 2002). FMEA is not only limited to failures, but includes errors in general, which are the inability to function in a certain way or to operate in an undesired way regardless of the cause (Jiang et al., 2017). In this way, the FMEA is presented as a powerful tool that has been applied in several industrial sectors (Liu et al., 2013), such as the aerospace sector (Yazdi et al., 2017;Chaudhuri et al., 2013), product development (Zhu et al., 2018;Chang, 2016), and several applications in healthcare Wang et al., 2016).
The most important indicator used by the traditional FMEA is the Risk Priority Number (RPN), calculated according to the following equation: RPN = O x S x D, where O, S and D respectively represent the risk factors of Occurrence, Severity, and Detection (Geramian et al., 2017). These factors are evaluated by experts that give scores from 1 to 10 for each factor related to each risk analyzed, according to the example presented in Table 1. The results obtained by multiplying the risk factors are used to prioritize the analyzed risks (Liu et al., 2012).
Some literature review studies were performed to analyze applications of the FMEA tool: Liu et al. (2013) presents the main limitations that are being addressed by the new proposals of combinations of techniques with the FMEA; Kabir & Papadopoulos (2018) present a review of the application of fuzzy representations with the FMEA tool for safety and reliability engineering; Ng et al. (2017) reviews the integration of FMEA with other quality tools for problem solving; Sutrisno et al. (2013) carried out a survey of Improvement strategy in FMEA; Chrysostom & Dwivedi (2013) raised the methodologies used in the FMEA and pointed the intuitionistic fuzzy as being the representation used to deal with the problem of group decision making. Despite the literature review work presented, no study has mapped the techniques used to deal with the problem of group decision making, which will be addressed in the next section.

Group decision making in the FMEA
As pointed out in the literature, the procedures for risk assessment and prioritization of failure modes by the FMEA tool can usually be considered as a multi criteria group decision making problem with the imprecision of information . In this way, the Fuzzy-Set-Based representations have been incorporated in several multi criteria decision-making techniques in order to overcome the limitations pointed out in the traditional FMEA method to deal with imprecise data (Hu et al., 2019;Huang et al., 2018;Liu et al. 2013). Table 2 presents the articles that address the group decision making problem in the FMEA tool. It shows the proposal of each article and the factors considered in the problem scope. Therefore, it is possible to identify the advantages of the proposed model when compared with the other studies. The proposed combination of FMEA with the DHFS representation can cope with all aspects analyzed in Table 2, such as: the definition of different weights for the risk factor; collection of individual judgments in order to consider a holistic problem view; fuzzy logic is used to deal with imprecisions; the decision makers use linguistic terms to represent their judgments; the hesitation in the definition of the linguistic terms parameterization and the hesitation in the judgments of the decision makers are addressed by the DHFS representation.
The studies in Table 2 present a variety of techniques, representations of information and application contexts. With the exception of TOPSIS, all techniques were applied only once. In addition, it is possible to perceive a predominance of techniques that aggregate different judgments from different decision makers and few consensus approaches that involve the minimization of divergence among decision makers. Concerning the aggregation operators, the most commonly applied are those based on weighted average, for instance, Huang et al. (2017) use the weighted averaging operator of linguistic distribution assessments; Chai et al. (2016) apply the linguistic weighted average operator; Wang et al. (2016) use the intuitionistic fuzzy ordered weighted average operator; Tooranloo (2016) employ the intuitionistic fuzzy weighted average operator; Chaudhuri et al. (2013) implement the ordered weighted averaging operator.
Regarding the information representations, a great predominance of the linguistic variables based on the fuzzy set theory was observed. However, it was not found studies that deals with Dual-Hesitant Fuzzy sets in failure modes analysis.

Dual hesitant fuzzy
The fuzzy set theory initially proposed by Zadeh (1965) obtained wide acceptance in several fields of study, as well as in the risk management through the application of fuzzy FMEA. Many generalized forms of fuzzy sets have been proposed to deal with the imprecision of these problems such as: intuitionistic fuzzy (Tooranloo et al., 2018;Zhu et al., 2018) Zhu et al. (2012) proposed a new generalization of the fuzzy sets, the dual hesitant fuzzy sets (DHFS). This representation seeks to combine the intuitionist and hesitant concepts, integrating the advantage of each one of them. The Hesitant Fuzzy Sets were proposed by Torra (2010) and allow the membership degree of an element in a set to be associated with several possible values, enabling the decision maker's hesitation in the definition of the variables that represent his judgment (Zhang et al., 2017). As in intuitionistic fuzzy sets, DHFS also has degrees of membership and non-membership functions; however, these two functions are expressed by several determined numbers rather than a single number, modeling the real-world problems more accurately

Hesitation in the judgment
Tooranloo (2016) Proposed a model for failure mode and effects analysis based on intuitionistic fuzzy approach. (2016) Propose an integrated method, combining multiattribute failure mode analysis (MAFMA) and 2-tuple representation. (2016) Combines intuitionistic fuzzy sets (IFSs) with evidence theory to analyze the potential failure modes.
Applies linguistic distribution assessments and employs an improved TODIM in FMEA.
Proposes a consensus based group decision making framework for FMEA.
A risk evaluation and prioritization method for FMEA with prospect theory and Choquet integral.
Integrates the ordered weighted geometric (OWG) operator and hesitant fuzzy linguistic term sets. than other generalizations of fuzzy theory. Specifically, DHFS is very useful for group decision-making problems, when it is difficult to determine the membership and non-membership functions (Yu et al., 2016). Recently, this approach has been widely used for multicriteria decision-making problems, with the development of new models and theories (Zhang et al., 2017). Calache et al. (2021) present a literature review of dual hesitant fuzzy sets applications. Zhu et al. (2012) define the concept of dual hesitant fuzzy as an extension of hesitant fuzzy sets. Given a fixed set U, a Dual Hesitant Fuzzy set D in X is represented as: are two sets of some value in the range [0,1] denoting the membership and non-membership degrees of the element x U ∈ to the set D , respectively, with the conditions: The basic operations and properties of the DHFS sets were also presented by Zhu et al. (2012). Given three elements DHFS, , 1 d d and 2 d , and a non-negative integer n , then the basic operations can be presented as Yu et al. (2016): Union-Sum: Intersection-Multiplication: Multiplication by n: Potentiation by n: Based on the concepts proposed by Zhu et al. (2012), Wang et al. (2014) has developed some aggregation operators, including the Dual Hesitant Fuzzy Weighted Average (DHFWA), which is used to calculate weighted averages of judgments based on DHFS linguistic terms (Zeng et al., 2018;Li, 2014). , then the DHFWA aggregation operator can be represented as follows (Wang et al., 2014): be a set of DHFS elements, the score function ( ) j S d is defined as follows:  Characterization of failure modes: this step consists in defining the risks in focus. To contribute for a better evaluation of decision makers in relation to severity, occurrence and detection, failure modes are detailed, respectively, in relation to the effects of the failure, the potential causes of failure and the current control processes. Qualitative quality management tools, such as the relationship diagram and the Ishikawa diagram are typically applied for this step (Tummala & Schoenherr, 2011;International Electrotechnical Commission, 2009). This step also defines the decision makers who will be responsible for the analysis and judgments of failure modes; . The linguistic terms and respective intuitionist numbers used by decision makers to assess the importance of risk factors are presented in Table 3 as described in Liu et al. (2015a). The judgments collected following the DHFS approach can be presented according to the matrix given by Equation 7.

Proposed model
III) Aggregation of the judgments: the judgments collected in the previous phase will be aggregated through the application of the dual hesitant fuzzy aggregation operator (DHFWA), and the scores are calculated according to Equations 5 and 6, related to , , , k 1 2 n = … decision makers. As a result of this aggregation, a vector of weights is obtained, ( ) The results obtained by the aggregation are contained in the range between [-1, 1]. To enable RPN calculation, the aggregated results need to be converted to scales compatible with traditional FMEA.

Illustrative application
The analysis of potential failure modes of a supplier is an important activity of supply chain management, due to the impact that potential failure modes have on the performance of a supply chain (He & Yang, 2018;Wu et al., 2006). For the evaluation of supplier potential failure modes, the FMEA tool is widely applied, for example, Valinejad & Rahmani (2018) evaluate the failure modes in the internet service providers; Foroozesh et al. (2018) assess sustainable-suppliers for manufacturing services; Ghadge et al. (2017) present an application in a printed circuit board supply chain; among others. In this way, an illustrative application related to supplier potential failure modes was developed following the steps proposed by the model described in Section 5.
Characterization of failure modes: assume a group of decision makers, composed of a production manager (DM1), a purchasing manager (DM2) and a quality manager (DM3) with respective weights [0.4; 0.3; 0.3], which evaluated 6 potential failure modes, presented in Table 5, in relation to the traditional FMEA risk factors (Severity, Occurrence and Detection). DMs used quality tools such as relationship diagram and Ishikawa diagram to analyze failure modes. These tools complement the FMEA in managing information and improving the failure modes understanding; II) Decision-makers' judgment: Table 6 presents the judgments of each decision maker on the risk factors importance using the linguistic terms in Table 3.
Each of the decision makers made judgments on the levels of failure modes according to risk factors using the linguistic terms in Table 4. The results of these assessments are presented in Tables 7 to 9 with the abbreviations of the linguistic terms provided by the decision makers.
In this case, each of the decision makers has the autonomy to judge the failure modes according to their respective points of view. In this way, differences between judgments can happen. The procedure of judgment aggregation, presented in the next step, seeks to consider these different points of view from different areas of the organization to obtain a more systemic view of the impacts of failure modes. In addition, the process of collecting judgments and presenting information in tables can contribute to the presentation of these different points of view that can be used in a consensus search for the development of improvement action plans. . The same procedure is carried out for the calculation of the other risk factors weights, and the results are exhibit in Table 10.          and the respective priority order of mitigation is given in Table 12 below. For this illustrative application failure mode of Lack of integration between customer and supplier ( 5 FM ) is presented with higher mitigation priority.
It should be noted that the proposed model has a compensatory nature in which it considers the aggregated judgments of the decision makers and the risk factors weights. Thus, all the risk factors are used for the priority definition. For example, 1 FM has a higher score in the severity and detection factors than 4 FM , but the score of occurrence in 4 FM is much bigger than in 1 FM . Although the value of the RPN of the two failure modes are very close, the 4 FM RPN is higher due to the compensation between the evaluated criteria. Explanation of the problem using FMEA enables team discussions in the search for consensus for continuous improvement actions. For example, the most critical failure mode was 5 FM with the highest RPN, presented in Table 12. The results in Table 11 show that the most critical risk factors are related to potential cause and control. Thus, based on the information in Table 5, DMs should discuss these results and action plans aimed at 1) improving communication with suppliers to address the potential cause of the failure and 2) creating a control process for this failure mode.

Conclusion
Current quality management is highly concerned with managing the potential failures of a process, system, or product. This concern was reaffirmed with the update of ISO 9001: 2015. To deal with the evaluation and prioritization of potential failures, the FMEA tool is widely used in the literature. The techniques based on hesitant fuzzy and intuitionistic fuzzy representations have been applied to deal with the group decision making problem, however, each one deals with a type of hesitation. Therefore, the Dual Hesitant Fuzzy technique was proposed to combine the advantages of Hesitant Fuzzy and Intuitionistic Fuzzy, providing a greater ability to deal with hesitations.
Thus, this article presented a new proposal that integrates the FMEA tool with the Dual Hesitant Fuzzy technique for group decision making to deal with the hesitation in the evaluation and prioritization of failure modes. It provides a more appropriate treatment of expert hesitation than other FMEA approaches found in the literature, based on Hesitant Fuzzy or Intuitionistic Fuzzy. Models based on Hesitant Fuzzy deal only with hesitation related to the activation of one or more linguistic terms for a variable, while those based on Intuitionistic Fuzzy deal only with the hesitation regarding the definition of the fuzzy number representing the linguistic term. This proposal based on the Dual Hesitant Fuzzy addresses the hesitation by means of these two combined ways. In addition, an application model was proposed following four steps: characterization of failure modes, judgments of decision makers, aggregation of judgments and ranking of risks based on the RPN weighted calculation. Finally, an illustrative application for failure modes in the context of supply risk management was developed to elucidate the steps of the proposed model.
Through the illustrative application, it is possible to verify that the proposed model combining FMEA and DHFS can easily be replicated in several real problems. The main practical implications of this combination are: • ability to deal with group decision making considering individual judgments; • the treatment of the subjectivity generated by hesitation in the evaluation of risk factors; • the use of linguistic terms to represent the judgments of decision makers; • it deals with decision makers with different importance and; • it considers risk factors with varying weights according to the addressed context.
As a limitation of the study, the proposed model does not deal with the interrelation between failure modes and does not verify the interrelationship between the risk factors. In this way, the proposed model can be improved in future studies to overcome these limitations. Finally, additional studies can be conducted in different application contexts in order to further explore the applicability of the proposed model.