The continual confirmation of risk evaluation with rules and with their development in decision making cause the expansion of theoretical base, methods, and tools that are scientific and practical (Haimes et al., 2015; Jafarzadeh Ghoushchi et al., 2019). Among the methods, in the literature FMEA is used as a general method, in which the application of this method is a tool to define, identify, eliminate probable defect and create a competitive advantage; the widespread use of this method is due to its ease of use. First, the FMEA technique was reported in 1920, but since the early 1960s the use of it has widely increased in different industries including food, energy, automotive, chemical, medical, and mechanical (Bao et al., 2017; Liu et al., 2019; Wang et al., 2019). Despite the high utilization of the FMEA method, prioritization of failures in this method relies on the common Risk Priority Number (RPN) (Multiplication of the Severity (S), Occurrence (O) and Detection (D) criteria) deficiencies that have been studied in previous studies (Rezaee et al., 2017; Spreafico et al., 2017). In this regard, the researchers have tried to cover some of the disadvantages of the common RPN index using alternative approaches, including Multidisciplinary decision making (MCDM) (Liu et al., 2015). MCDM approaches reflect natural behaviour and human thinking and are used when decision making is considered with a few options and quantitative and qualitative criteria (Liu et al., 2015). One of the applications of interchangeable approaches mixed with the FMEA method is the AHP hierarchy analysis that is used by Bevilacqua and Braglia (2000) for prioritizing the reasons of failure in a company in Italy which produce refrigerators. An analysis method of the ANP network process was also used in Zammori and Gabbrielli (2012) research, to check the relationship among risk criteria in the process of decision making. Prioritization method based on similarity to the TOPSIS ideal solution was used by Rau et al. (2007) for prioritizing risks at the packaging stage of a computer company. Garcia and Schirru (2005) used the Data Envelopment Analysis (DEA) method to determine ranking indices in failure modes. Gray Relationship Analysis method (GRA) was suggested in Sharma and Sharma (2012) study for prioritizing and evaluating issues which are expensive in the production process, and Seyed-Hosseini et al. (2006) use the DEMATEL Decision Measurement and Testing Laboratory method for re-evaluating the failure modes in the FMEA. Other methods combined with the FMEA method are the VIKOR (Liu et al., 2012), PROMETHEE (Maheswaran and Loganathan, 2013), COPRAS (Adhikary et al., 2014) methods. It needs to be explained that, despite the improvement of FMEA reliance on decision-making methods, the opinions of the members of the FMEA team can be uncertain regarding the type of experience and base; therefore, the use of existing approaches to solve uncertainty problems, containing fuzzy theories, is presented to cover some of the traditional RPN deficiencies (Wang et al., 2016). Also, in terms of developing the risk assessment methods by the matrix approach, the most pragmatic matrix (qualitative) risk assessment methods, such as a 3×3 matrix (OHSAS), a 4×4 matrix (AS/NZS 4360), and a 5×5 matrix (MIL-STD-882B) was presented (Kovačević et al., 2019). Moreover, a study for emphasizing the importance of involving Multiple-Criteria Decision-Making (MCDM) methods was conducted which aims to select the optimal type of hotel for investment. The usage of Single-Valued Intuitionistic Fuzzy Numbers (SVIFN) is proposed in which three decision-makers estimated five alternative types of hotels corresponding to the five evaluation criteria. The results are valid and confirm that the introduction of suitable multiple-criteria models minimizes the possibility of making the wrong decisions (Karabasevic et al., 2019).

Multi-Objective Optimization based on Ratio Analysis is a multi-criteria optimization method that can be used to solve various types of complex decision problems (Shihab et al., 2018; Abdulhasan et al., 2019; Kamaruzzaman et al., 2018; Stanujkic et al., 2019). The MOORA was first used in Brauers and Zavadskas (2006) research. This technique provides the possibility of subjective assessments due to the unwillingness to use the weighting method. This method was used to overcome weighing problems such as ELECTRE, AHP, TOPSIS, and PROMETHEE in previous optimization models (Baležentis et al., 2012). This method relies on the relative analysis theory and the method of reference point, and in the forthcoming development, a complete multiplicative formation has been added to this method, which has led to the formation of the MULTI MOORA method for multi-objective optimization as a strong method (Brauers and Zavadskas, 2012). Simplicity and ease of implementation of this method for multi-objective optimization, low mathematical computing, good stability, and very low solving time are among the most important features of the MOORA method, which has made it preferable to other methods in this paper. Yet, the MULTIMOORA and MOORA method has been used in many studies for solving wrapped problems of decision making, including the Karande and Chakraborty (2012) research on the selection of materials to reduce defects in products, in order to select personnel at the factory to prevent using customary methods in fuzzy environments by Baležentis et al. (2012). In the research of Attri and Grover (2014), the MOORA method was used to decide on product design, facility location, suppliers, type of materials and technology in a production system. Mavi (2015) used a third-party logistics provider (3PRLP) to use the MOORA method in fuzzy environments. Chand et al. (2018) have identified and analysed the issues selected in green supply chain management such as waste reduction, conservation of natural resources, and reducing the consumption of hazardous materials in industries. MOORA’s approach has also been considered in the risk assessment (Arabsheybani et al., 2018; Liu et al., 2014; Jafarzadeh Ghoushchi, 2018). Also, to precedence the failures, the FMEA approach relies on the fuzzy BWM and new approach, Z-MOORA, improved by Ghoushchi et al. (2019). The proposed approach has been implemented in an active automotive parts company, and results demonstrate a complete prioritization of the failures in comparison with other customary methods such as fuzzy MOORA and FMEA.

Among the various MADM methods, the SWARA method is a method which implies the implicit ideas and experiences of experts and is not as time consuming and complex (Zolfani and Saparauskas, 2013), as proposed by Keršuliene et al. (2010). Several factors, such as inaccessible information and incomplete information, make decision making inaccurate; therefore, fuzzy multiple attribute decision making methods were created due to the lack of ambiguity in their characteristics, because conventional MADM methods can’t solve the problem of incomplete information effectively (Mavi, 2015). Hence, the SWARA method was extended to fuzzy SWARA by Mavi et al. (2017). Fuzzy stepwise weight assessment ratio analysis (Fuzzy SWARA) was used for weighing the evaluation criteria in the research in case of combining the sustainability and risk factors for evaluation of the third-party reverse logistic provider (3PRLP), also, for ranking the sustainable third-party reverse logistic providers, the fuzzy multi-objective optimization based on ratio analysis (Fuzzy MOORA) was applied in the plastic industry (Mavi et al., 2017). In this research, in order to increase reliability of the SWARA method, this method has been converted to the Z-SWARA method. Furthermore, a new hybrid model includes a combination of the Delphi, SWARA (Step-Wise Weight Assessment Ratio Analysis), and MABAC (Multi-Attributive Border Approximation Area Comparison) methods considering railway models on the basin of Bosnia and Herzegovina. This study contains three phases: determining the criteria ranking using the Delphi Method, determining the mutual criteria impact and the relative weight values of the criteria, and evaluating the most appropriate variant using the MABAC Method (Vesković et al., 2018). Also, in Mukhametzyanov and Pamucar (2018) research, a model for result consistency evaluation of multi-criteria decision-making (MDM) methods and selection of the optimal one was presented. The results of the sensitivity of decision-making with various methods like SAW, MOORA, VIKOR, COPRAS, CODAS, TOPSIS, D’IDEAL, MABAC, PROMETHEE-I, II, ORESTE-II with changing in elements of matrix is proposed.

The Z-Number concept is used to calculate the numbers that are not completely reliable. Z-Number was first proposed by Zadeh (2011), as a general characteristic of the theory of uncertainty. The Z-Number in combination with AHP was used to identify reliable assessment criteria from the best universities in adverse environmental conditions (Sahrom and Dom, 2015), using the AHP-Fuzzy DEA and Z-Number method and integrating the concepts of reliability and fuzzy numbers, and worked on the priority assessment of 20 bridge structures. Also, Azadeh and Kokabi (2016) used a Z-DEA model to choose the IT project to deal with uncertainty, interaction between projects and trust. Yaakob and Gegov (2016) proposed a method for solving MCDM problems based on the concept of Z-Number, Z-TOPSIS for solving stock choice issues. Shen and Wang (2018) developed the VIKOR method relying on the theory of the Z-Number and used the regional circular economy development plan.

Zadeh (1965) introduced the first notion of fuzzy sets. A fuzzy set is generally determined as a membership function that shows the degree of membership of the elements in a certain interval, usually in the [0,1] interval. The basic descriptions of the fuzzy set of numbers, which have been used in this study, are as follows. Definition 1.

A defined fuzzy set A˜ in reference to X is an equation (1): (1) A˜={(x,μA˜(x))|x∈X}.

Fig. 1

Triangular fuzzy number.

The Z-Numbers concept was first proposed by Zadeh (2011), which was proposed as a generic version of the theory of uncertainty that is used to compute non-reliable numbers. Z-Numbers is a pair of fuzzy numbers Z=(A,B) , the first component, A, is a fuzzy subset of the domain X, and the second component, B, is a fuzzy subset of the unit interval and shows the reliability of component A. For example, if a wrong detection is assumed to be a Z-number, the first part can be of the “bottom” type and the second part is “not sure”. The triple (X,A,B) is known as Z-VALUATION, which is equivalent to an assignment description and determined as a general limitation on X as equation (8): (8) Prob(XisA)isB. This constraint is known as a probability restriction, which shows a probability distribution function. In specific, it can be described in equation (9): (9) R(x):Xis→poss(X=u)=μA˜(u). In the above equation, μA˜ is a membership function of A, and u is a generic value of X. μA˜ can be considered as a constraint in relation with R(x) . This means that μA˜(u) covers to what degree u is satisfied. Therefore, X is a casual variable with possibility distribution that plays a possible constraint role on X. Probable constraints and the probability density function are defined in (10) and (11): (10) R(x):Xisp, (11) R(x):Xisp→Prob(u⩽X⩽u+du)=p(u)du.

In equation (11), du shows the component of “u” derivative.

The step-wise weight assessment ratio analysis (SWARA) method was proposed by Keršuliene et al. (2010). Several factors such as unquantifiable information, incomplete information, unobtainable information, and partial ignorance cause the imprecision in decision making. Since conventional MADM methods cannot effectively handle problems with such imprecise information, therefore, fuzzy multiple attribute decision-making methods have been developed owing to the imprecision in assessing the relative importance of attributes and the performance ratings of alternatives concerning attributes (Mavi, 2015). Hence, this paper aims to extend SWARA to Z-SWARA. This paper assumes that all criteria are independent. The process of determining the relative weights of criteria using the Z-SWARA method is the same as using the SWARA method, such as the following steps:

Step 1. Sort the evaluation factors in descending order of expected significance.

Step 2. Conversion rules for z-numbers linguistic variables.

In this step (Step 2), the verbal variables for factors that are in the form of Z-Numbers are converted to triangular fuzzy verbal variables. The process of this conversion is as follows:

Assume that Z=(A,B) , is a z number (A is the verbal variable presented in Table 1 and B is the verbal variable presented in Table 2) and assume that, A˜={(x,μA˜(x))|x∈[0,1]} and B˜={(x,μB˜(x))|x∈[0,1]} are triangular membership functions. In this case, the second part of Z-Number (Reliability) is converted into a crisp number using equations (12) and (13): (12) α=∫xμB˜(x)dx∫μB˜(x)dx, (13) Z˜α={(X,μA˜α)|μA˜α(x)=αμA˜,X∈[0,1]}.

In the above equations, α represents the weight of reliability, μB˜(x) indicates the degree of dependence x∈X in B and μA˜α(x) indicates the degree of dependence x∈X in  Aα .

Then, by combining the Linguistics variable for evaluating the factors (Table 1) and the conversion rules of linguistics variables of reliability (Table 2), the roles of transforming verbal variables of decision-makers are obtained for the Z-SWARA method.

For instance, assume that Z=(A,B) is a Z-Number, whose first component is A˜=(MOL) and the second component is R˜=(H) , so Z-Number is described as Z=[(2/3,1,3/2),(0.5,0.7,0.9)] . Initially, the second component of Z-Number converts to a definite crisp due to equation (12) and (13). According to equation (12), the value of α is 0.7, then, this value is used in equation (13), Z˜α=(2/3,1,3/2;0.7) . Now, the Z-number weight is converted to the triangular fuzzy number using equation (13), Z˜′=(23×0.7,1×0.7,32×0.7)=(0.561,0.837,1.255) , other conversions are presented in Table 3 according to Tables 1 and 2.

Step 3. According to Table 3, state the relative importance of the factor j about the previous (j−1) factor according to Z-Number, which has higher importance, and follow to the last factor. After determining all relative importance scores by all experts, in order to aggregate their judgments, the geometric mean of corresponding scores was obtained.

Step 4. Obtain the coefficient k˜j as Eq. (14): (14) k˜j=1˜j=1,s˜j+1˜j>1. 1.\end{array}\right.\]]]>

Step 5. Calculate the fuzzy weight q˜j as Eq. (15): (15) q˜j=1˜j=1,x˜j−1k˜jj>1. 1.\end{array}\right.\]]]>

Step 6. Calculate the relative weights of the evaluation criteria as (16): (16) w˜j=q˜j∑k=1nq˜k. where w˜j=(wjl,wjm,wju) is the relative fuzzy weight of the jth criterion and n shows the number of evaluation criteria.

Brauers and Zavadskas (2006) presented the MOORA method. This method is a multi-objective optimization method that can be used to perform a variety of complex decision-making problems in any environment. In the following, the MOORA fuzzy method was presented by Akkaya et al. (2015) to consider the uncertainty in the decision matrix. But this method, despite the improvement of the traditional MOORA method, makes it impossible to consider the reliability of the decision. To this end, in this research, the decision making based on Z-Numbers is expressed in terms of increasing reliability in decision-making by experts. The final ranking options can be affected by considering the reliability part of the decision-making matrix. There are three different approaches for solving problems with the Z MOORA: Z ratio method, Z reference point approach, and full multiplicative form. In this paper, we use the Z ratio method.

In order to check the suggested approach’s capacity in the current research, it tries to prioritize defeats as one of the basic production progression elements in an automotive supplier firm using this approach. The firm started work in the north of Iran in 1994, intending to design and produce industrial molds and work in the plastic sector. After a short time, it became successful in the automobile industry of Iran, due to the capabilities and modern technical knowledge of machinery it has been able to rank among the manufacturers of plastic parts of cars, including Peugeot 206, Peugeot 207, Peugeot 405, Samand and so on. The overall production procedure of the company consists of 6 stages: Mold Making, Paint Operation Assembly Line, Ultrasonic Injection Production Line, and Warehouse. In this research, intending to evaluate the abilities of the proposed approach, we examine the failures of one important step, Warehouse. This research seeks to identify and prioritize the failure of the warehouse process in producing the Axle Crossmember arm 405 pieces using the suggested approach. In this regard, with the collaboration of the expert team, the available failures, identified by the FMEA method, are identified. The list of 8 identified failures due to the FMEA method, along with the five criteria of severity, occurrence, diagnosis, cost, and time of each failure, with the help of a team of experts, is presented in Table 8.

In this section, the results of implementing the proposed research approach to assess the failure of the Warehouse process in the company are studied and presented. In the first stage, the fault scenarios are first recognized by the team of the FMEA and the five-factor amounts for each failure are specified by the team (Table 9). Then, given the uncertainty of these factors, Z-Number theory is used. This theory, in addition to considering the uncertainty of the factors as a fuzzy number, considers their reliability as well. The values of the Z-Number triple factors for failure scenarios are shown in Table 9, due to opinions of the FMEA team.

In the following, based on the second stage of the suggested approach, the weight of the five factors is specified by the Z-SWARA method. For this, the team of FMEA has identified the relative importance of each factor as compared to the previous factor in verbal variables, as shown in Table 10.

Then, in the second phase of the research method, and also according to the SWARA method expressed, the values of coefficient k and the weight of q and w are calculated on the basis of equations (14) to (16) for each decision-maker in examining the failures in Table 12. In this step, the linguistic variables are converted into triangular fuzzy numbers, based on the equations shown in Tables 1 and 2. For example, the fuzzy numbers corresponding to the linguistic variable LI-M are (0.283,0.354,0.474) , respectively. After the conversion of linguistic variables into fuzzy numbers, the coefficient kj from the equation (14), the fuzzy weight qj from equation (15), and the final weight of the factors in the form of fuzzy numbers wj from equation (16) are obtained. Final fuzzy weight of main criteria by each decision maker is shown in Table 11.

According to Table 12, weights of factors in the form of triangular fuzzy numbers are as follows: w˜S=(0.322,0.377,0.445),w˜O=(0.077,0.123,0.183),w˜D=(0.109,0.166,0.238),w˜C=(0.231,0.291,0.369),w˜T=(0.073,0.116,0.173).

In the third stage of the proposed approach, based on the results of the first and second stages, priority is given to the failure scenarios using the developed Z-MOORA. Initially, the decision-making matrix of the Z-MOORA is organized in the form of Z-number strings. The rows show the evaluated options, or the failure scenarios, and the columns show the evaluation criteria or the five factors. Subsequently, the decision-making matrix is transformed into a decision matrix in the form of triangular fuzzy numbers using the transformations presented in Table 6, which is presented in Table 13.

Now, after normalizing the matrix shown in Table 13, the weighted normalized matrix is obtained by considering the weights of the risk factors (Table 14).

In the proposed Z-MOORA approach, according to Table 14, the results are shown with uncertainty in risk factors and reliability in failures. The results are presented in Table 15.

According to Table 15 and the proposed approach, F2, F8, and F4 failures are ranked from first to third with values 2.675, 2.335, and 2.257, respectively. These defeats are considered as serious failures and require planning to implement reformative/suppressive measures. In other words, these defeats are considered as vital defeats and require planning to implement reformative/suppressive measures. With this approach, it is observed that the failure of F1 is the last priority, and, given the limited resources of the organization, no corrective action is currently required. In general, by prioritizing the observed, the output of the proposed approach has made a complete difference between defeat scenarios. So, decision-makers can concentrate on the company’s corrective/preventive measures on failures with complete prioritization of downtime and resource constraints, which will result in system improvements if their negative effects are eliminated.

In the following, the final ranking of failures relying on the Z-MOORA approach is compared with other conventional methods consisting of fuzzy MOORA and the usual FMEA method in Table 16.

According to Table 16, due to traditional RPN, failures of F2, F8 with RPN = 3375 are considered jointly in the prior precedence. Also, defeats of F2 and F5 with RPN = 1440 are shared in the last precedence. With a general review of the prioritization of failures based on traditional RPNs, it can be seen that the prioritization of failures has been done in such a way that defeats are grouped into six classes. This suggests that prioritization relying on this customary index is not quite realized and confounds the decision-maker in risk management and planning of corrective/preventive measures. Regarding the comparisons in Table 16, according to expert’s opinions and organizational conditions, prioritizing inadequate failures can be attributed to five factors as a result of not allocating different weights, as well as the lack of attention to the uncertainty of the value of these five agents.

In the following, a more detailed analysis of Table 16 shows that when the fuzzy MOORA method is used, the problem of defective prioritization of failures has been improved. Given the prioritization of the fuzzy MOORA rating, F2 and F8 failures are in the first and second precedencies for analysis. Also, F1 defeats are in the last precedence. In this method, the uncertainty of the five factors and the weight of these factors are considered, using the fuzzy theory and the FBWM method, respectively, to cover some of the disadvantages of the usual RPN index. But in this method, contrary to the increased differentiation in prioritizing failures in conventional FMEA, there is a problem with decision-making reliability. For this reason, this research has developed an improved approach to dominate this deficiency, which is based on the prioritization performed based on the developed FMEA-based Z-MOORA method, which detects all recognized defeats are in distinctive priorities. In other words, the proposed method of this study by merging the meanings of uncertainty and the reliability of the defeat points by using the theory of Z-Number has tried to decrease some of the principal weaknesses of the traditional RPN, which is reliability of the decision making by the method. The customary MOORA was introduced, which makes the decision according to the actual output. By examining the results of this method, compared to the other two approaches, it can be said that the prioritization of defeats is complete and that defeats relying on the traditional RPN in the same priorities is divided into eight classes relying on the suggested approach of the research (Fig. 3). By comparing the results of the two Z-MOORA and usual FMEA approaches, F2 and F8 failures, which were ranked jointly in the priority due to the usual RPN index, were ranked first and second according to the traditional RPN index. This suggests that the Z-MOORA approach, in addition to making a distinction in priority, prioritizes failures which have more values in the most important determinants of RPN (in this study, S, C, O, D and T are the most important factors respectively), thus, these are located in higher priorities.

Sensitivity analysis by changing the weight of risk factors is calculated according to the information given in Table 17. For example, Case 0 shows the original weight values of the risk factors while the other cases show different weight values for possible situations. In fact, in different cases, one of the five risk factors weight increases, and in proportion to this amount, we reduce the weight of other factors so that the weight of that factor is the highest. The effects of weight changes on the five factors in the ranking of fashion fillers are presented in Table 18 and Fig. 4. In Case 1, 0.1 is added to the weight of C, and 0.025 is deducted from the weight of S, O, D, and T. Also 0.2 is added to the weight of D in Case 2, 0.24 is added to the weight of O in Case 3 and 0.28 to the weight of T in Case 4, and the same proportion is deducted from the initial weight of the others in each case. The results for ranking the Warehouse process for different cases are represented in Fig. 4 and Table 18 indicates that in three of five cases the most important failure mode is Pallet falling when loading transportation. In Cases 2 and 4, as the weights of cost and time are the highest, “Pallet falling when load transportation” failure mode is the second most important failure mode. Meanwhile, “Dealing with people and equipment when driving lift trucks” is the first most important failure mode. “Falling parts when lifting them” failure mode is also ranked the last in Cases 0, 1, and 2.