An Integrated Mathematical Attitude Utilizing Fully Fuzzy BWM and Fuzzy WASPAS for Risk Evaluation in a SOFC

: Nowadays, because of the energy crisis, combined heat and power systems have notable beneﬁts. One of the best devices is SOFC (Solid Oxide Fuel Cell) which joins heat and power frameworks. Some considerable failure modes arise that can affect these devices’ productivity. Generally, failure modes evaluations need an experts team to achieve uncertainties belongs to the risk assessment procedure. To improve the efﬁciency of the routine FMEA methodology and to represent a suitable hybrid fuzzy MCDM approach for FMEA, in this work, fully fuzzy best-worst method (FF-BWM) is employed to achieve the risk factors weights then fuzzy weighted aggregated sum product assessment (F-WASPAS) approach to detect the failure modes priorities is utilized. Ultimately, the sensitivity analyses demonstrate that the offered framework is veriﬁed and can make applicable data in risk management decision-making evaluation.


Introduction
The consistent improvement in industries causes a huge expansion in energy interest. In this way, expanding nature-accommodating devices with high proficiency is imperative to limit the adverse consequences of non-renewable energy sources [1]. The synthetic energy of vaporous or liquid reactants can be transformed into electricity employing sorts of devices called fuel cells [2]. Each fuel cell includes a special electrolyte layer segregating reactants from chemically reacting. This layer is regarding porous cathode and anode parts [3]. Fuel cells have various categories that the Solid Oxide Fuel Cells (SOFCs) have drawn into incredible consideration in recent years [4]. Guk et al. [5] expressed that for acquiring better knowledge into the SOFC efficiency, the most significant factor to consider is the electrode temperature dispensation. In this way, to gauge the cathode and anode temperature dispensation from a functioning SOFC, they introduced a new cell-incorporated multi-intersection thermocouple cluster. Gallo et al. [6] proposed a new strategy to prognosticate the SOFC staying valuable life. Kong et al. [7] investigated numerically a two dimensional configuration of a SOFC. They utilized a heat bar in their model and compared it to a model without that heat bar and analyzed the impact of heat bar on the cell proficiency. Xu et al. [8] simulated an SOFC with methanol as fuel. Their modeling was a 2D simulation and they represented the effect of some important factors Rezaei [15] introduced the BWM as a powerful MCDM method to obtain the criteria weight. The BWM is vector-based and used pairwise comparisons to evaluate the opinions of decision-makers [34]. The BWM compared to other similar methods requires fewer pairwise comparisons [35]. In recent years, the BWM has been developed by researchers the various fields, such as intuitionistic fuzzy multiplicative BWM [36], fuzzy BWM [37], interval-valued fuzzy-rough BWM [16], Z-BWM [38], piecewise linear fuzzy BWM [39], Interval-valued pythagorean hesitant fuzzy BWM [40], FMEA-BWM [39,41], Bayesian BWM [42], rough-fuzzy BWM [43], trapezoidal fuzzy BWM [44], BWM with D-number [45], modified fuzzy BWM [46], group BWM [47] and fully fuzzy BWM [48].
In the fully fuzzy methodology, all feasible pairwise comparisons were not required. The ff-BWM has provided high reliability to the results. This method is independent with its high capability to hybrid other MCDM methods. Therefore, the present work is an extended investigation of precise and hybrid fuzzy MCDM methods for risk evaluation by utilizing fully fuzzy BWM and fuzzy WASPAS to extract an appropriate risk ranking. The key contributions of this work are to deduce the risk factors' weight, fully fuzzy BWM as a linear mathematical model is used. The next novelty is that to achieve the risk scores of potential failure modes, fuzzy WASPAS technique is utilized to clarify the failure modes ranking. Finally, the suggested approach is employed to evaluate a ceramic anode solid oxide fuel cell and a sensitivity evaluation is derived.
Considering all mentioned points, this paper consists of several sections as follows: Section 2 is about methodology including FF-BWM and F-WASPAS. Sections 3 and 4 represent the suggested approach and introduced the main case study, respectively. Ultimately, Section 5 concluded the results.

Methodology
In this part of the research, the principles of fuzzy logic, FF-BWM, and F-WASPAS are reviewed. Ref. [49], the first investigator, described the fuzzy set hypothesis. At that point, this concept was spread in various fields of study by numerous researchers. The membership function µ A : Y → [0, 1] is characterized for the TFNs A = (a 1 , a 2 , a 3 ) and is achieved as follows [50]: Main components of the TFNs A = (a 1 , a 2 , a 3 ) are the left bound (a 1 ), the center (a 2 ), and the right bound (a 3 ), individually. Some mathematical principles for two TFNs (a 1 1 , a 2 1 , a 3 1 ) and (a 1 2 , a 2 2 , a 3 2 ) are also introduced as follows [51]: (iv) Divition :(a 1 1 , a 2 1 , a 3 1 )%(a 1 2 , a 2 2 , a 3 2 ) = The mentioned operational principles are fuzzy number addition, subtraction, multiplication, division, and power, separately.

Fully Fuzzy Best Worst Method (FF-BWM)
During this procedure, the BWM is introduced in a fuzzy environment. This process includes four steps: step 1 consists of decision makers' team selection, step 2 is the best and worst criteria detection by decision makers' team, step 3 is pairwise comparisons covering the best criterion to other criteria and the other criteria to the worst criterion, and ultimately step 4 is fuzzy optimal weights calculation. It should be noted that after step 4, the mathematical model conversion to linear ranking function is performed. These steps are represented as follows: Step 1. Generating a system of decision objectives: An alternative according to some criteria must be chosen by the Decision Maker (DM) in an MCDM. Consider the set of n criteria is {c 1 , c 2 , . . . , c n }.
Step 2. The best and the worst criterion detection: During this stage, the best c B and the worst c W criteria should be chosen by DM.
Step 3. Fuzzy reference comparison: This step includes a comparison between the best criterion and other criteria and also, between the worst criterion and others. The results of this process are represented in Equations (7) and (8).
A W = ( a 1W , a 2W , · · · , a nW ) Step 4. Computation of the favorable fuzzy weights: The optimal quantity of w B % w j and w j % w W give the components of the comparison matrix w B % w j = a Bj , w j % w W = a jW . It is vital to minimize the maximum absolute value of | w B % w j ∃ a Bj | and | w j % w W ∃ a jW | for each j to achieve the optimal weights ( w * 1 , w * 2 , · · · , w * n ). This procedure is represented as follows: Min{Max{| w B % w j ∃ a Bj |, | w j % w W ∃ a jW |}} (9) subject to : Equation (9) can be linearized as follows [48] : where w j = (w 1 To take care of completely fuzzy linear programming issues, a novel technique with triangular fuzzy numbers was presented by [52]. To change the problem they utilized a linear ranking function. Along these lines, the accompanying condition was utilized to change the problem. Consequently, requirements (13)- (18) are changed into limitations (20)- (40).
s. to : After solving the model addressed in the imperatives (20)-(31), the ideal values of the triangular fuzzy loads are acquired.

Consistency Ratio
An ideal consistency of pairwise collation exists when a Bj ⊗ a jW = a BW . The values of a Bj and a jW describe the relative fluffy inclination of the better standard over basis and the relative fuzzy inclination of the criterion j over the worst criterion, separately. In addition, the relative fuzzy preference of the best criterion over the worst one is introduced as a BW . Since there is a feasibility for a criterion j not to be entirely reliable. In this way, it is essential to characterize a consistency proportion to assess the consistency level of pairwise correlation.
In a case that the worth of a Bj ⊗ a jW not equivalent to a BW , the level of consistency will be decreased. In addition, the greatest irregularity θ happens when both a Bj and a jW have acquired their most extreme conceivable worth that is equivalent a BW . In light of w B % w j ⊗ w j % w W = w B % w W and the previously mentioned issue, the worth of θ ought to be added to a BW and deducted from a Bj and a jW . This point is shown in Equation (32).
It is obvious that the high value of possible inconsistency happens when a Bj = a jW = a BW . Thus, Equation (32) can be rewritten as follows: The extended form of Equation (33) is described as Equation (34).
Equation (34) ought to be solved for various upsides of a ij ∈ {1, 2, · · · , 9} to track down the greatest conceivable worth of θ. Therefore, it is introduced as the consistency index (CI) and consistency ratio (CR) can be achieved by θ * , CI, and the Equation (35).
It is important to be noted that the fuzzy number θ * is changed into a number by taking the average of its components. Thereafter, the acquired number will be utilized in Equation (35).

Fuzzy WASPAS
Since by utilizing a fuzzy methodology, it is possible to allocate the relative significance of qualities using fuzzy numbers instead of exact numbers, this part stretches out WASPAS to the fuzzy environment. MADM technique, specifically WASPAS, was presented in 2012 [22]. Afterward, alteration of the strategy WASPAS-IFIV was presented [31]. The WASPAS method includes two major steps: 1.
The Weighted Production Model (WPM).
The WSM addresses the general score of an option as a weighted amount of the property estimations. The WPM is created to stay away from the alternatives with poor property estimations. It decides the score of every option as a product of the scale rating of each attribute to power equivalent to the significant weight of the characteristic [53]. Given the momentarily summed up fuzzy hypothesis above, WASPAS-F steps can be illustrated as follows: Step 1. Fuzzy Decision-Making Matrix (FDMM) generation: A DMM consists of the performance values m ij and the attribute loads w j . An expert as DM decides the attributes and initial weights of them. The discrete enhancement issue is addressed by the inclinations for m sensible alternatives (rows) appraised on n ascribes (columns): A tilde ( ) is set over a parameter if that parameter addresses a fuzzy set. So, m ij is a fuzzy value that describes the proficiency value of the i alternative in terms of the j criteria. Afterward, the assurance of the ranks of choices is completed in a few stages.
Step 2. Normalization of attributes By introducing values m ij of normalized decision-making matrix M = [ m ij ] (mn) , the initial values of all the attributes x ij are reported as normalized values.
Step 3. Weighted normalization The fuzzy decision matrix in a weighted and normalized form and values of the optimality function calculation is as follows: (a) According to WSM, Equation (38) can be written for each alternative: (b) According to WPM, Equation (39) can be written for each alternative: Step 4. Defuzzification Defuzzification can be performed utilizing an average of components as follows: Step 5. Utility function Fuzzy WASPAS methodology utility function for an alternative can be described as follows: α is Utility Function Coefficient (UFC) and determined as Equation (43): Step 6. Rank An alternative with a maximal value of F i must be chosen as the first rank. Meaning that the more F i , the better rank.

Offered Approach
This section consists of all steps that must be obeyed to solve the proposed problem. As mentioned before, this approach is based on FF-BWM and F-WASPAS. The flowchart of the offered FMEA model is represented in Figure 1. This figure shows the proposed approach in three main layers. The first layer consists of triple factors detection, pair-wise comparison for SOD, and failure modes detection by decision-maker. The second layer includes weighting by FF-BWM and ranking by F-WASPAS and the final layer is the final rank of failure modes and sensitivity analysis. Since the use of linguistic terms is essential in fuzzy logic, therefore Table 1 includes these terms for FF-BWM. The expert has to utilize these terms to perform a pair-wise comparison. In this table, CIs show consistency indexes. Table 2 also presents failure modes linguistic variables. Furthermore, evaluation scales to determine the SOD values are collected in Table 3.   (5,7,9) 13.77 Absolutely significant (AS) (7,9,9) 13.77

Case Study
One of the electrochemical transformation gadgets that generate power straightforwardly from oxidizing a fuel is a Solid Oxide Fuel Cell (SOFC). The SOFC includes a solid oxide or ceramic electrolyte. Benefits of this type of fuel cells incorporate high joined warmth and power productivity, fuel adaptability, low emanations, and somewhat minimal expense. The biggest hindrance is the high working temperature which brings about longer beginning up occasions and mechanical and chemical adaptability similarity issues [54]. Figure 2 shows a schematic of SOFC and its reactions. SOFCs comprise numerous parts that each have their failure modes. The main part of this fuel cell is the anode. This section fostered the FMEA strategy for ceramic anodes. Figure 3 represented the most notable failure modes based on the Ishikawa fishbone diagram [55]. As it is obvious in literature, the main failure modes are considered: Interfacial delamination, Coke deposition, Sulfur adsorption onto the metal catalyst, Reduction in catalyst porosity, Corrosion of anode, and Crack. The offered process in this study is simultaneously the investigation of SOD factors and weight of criteria in a fully fuzzy environment and ultimately obtaining the rank of proposed failure modes.  Based on the previous mention, this work utilizes the FF-BWM method to weigh the factors. In this study, multiple DM methodology is used namely, three decision-makers give their ideas. Table 4 includes these opinions.
The SODs weights based on TFNs, using Equations (20) to (31), will be as follows: s.to: Similarly, the triple factors' ultimate optimal weight for all experts is presented in Table 5.   Table 5 represents the SOD factors optimal weights considering the optimal value of the objective function and CR factor that it is found that the results are acceptable. Herein, to show the reliability of the proposed approach in comparison to other weighting methods, the achieved CR factors are compared with those of F-BWM and F-AHP. Based on the mentioned issues, six main failure modes are considered in this study. Table 6 shows all these failure modes and their circumstances and end results. In the next step, failure modes scores are required. These scores are gathered in Table 7. Then, Table 8 consists of assessments of failure modes concerning risk factors evaluated by the DM team. These data are essential for failure modes ranking in fuzzy WASPAS. Tables 9-11 are the converted group decision matrix to TFNs, WSM, and WPM, respectively.  Ultimately, the final rank of failure modes can be achieved. Table 12 shows the final failures modes prioritization using of Risk Priority Number (RPN). As it is clear, traditional RPN can not give a complete rank but fuzzy WASPAS is able to extract acceptable complete rank for considered failure modes.  Table 12 represents, as indicated by the Traditional RPN, failure mode FM2 and FM 3 with RPN = 125 has been situated in the fourth rank. Investigation of failure ranking dependent on the conventional RPN represented that during the time spent failure ranking, failures are assembled into five classifications. This shows that the ranking because of this conventional RPN doesn't completely prioritize the failure modes and confuses the DM in risk management. As indicated by the examinations made in Table 12, the non-complete ranking of the failure modes may be because of avoiding the weight of SOD factors.
Contrasting the aftereffects of the fuzzy WASPAS rank and the traditional one represented that the full class rating of failure modes has been performed and failures that have been situated in a similar rank dependent on the traditional RPN are isolated into six classes dependent on the suggested approach. A considerable point is that FM1 and FM4 with the second and third ranks, respectively, in traditional RPN ranking, changes its rank to the third and second. Also, it can be found that FM6 namely Crack is the most important failure mode, and FM5, namely Corrosion of anode, is the least important one.
As Figure 3 mentions that the proposed approach is more reliable and accurate, herein, the effect of approach selection in final ranking for failure modes has been gathered in Table 13. This Table shows that failure modes rank by proposed methodology is different from two other methods and of course is closer to reality and includes decision makers' preferences. Triple factor sensitivity is analyzed by varying the weight of them based on the data given in Table 14. In this table, eight cases are studied. Case 0 shows the extracted weight values of the triple factors in the FF-BWM weighting process while the other cases represent distinctive weights for conceivable situations. The outcomes for positioning the failure modes for various cases are addressed in Table 15 and Figure 4.  FM1  3  4  2  4  5  5  1  1  FM2  5  5  5  5  4  4  5  4  FM3  4  3  4  3  3  3  3  2  FM4  2  2  3  2  2  2  4  3  FM5  6  6  6  6  6  6  6  5  FM6  1  1  1  1  1  1  2  1 Based on Table 15, FM6 always comes in the first rank, but when the D factor has been increased by 4.38 times in Case 6 , the rank varies to second. Also, Case 7 demonstrates that when the weight of D is 0.608 and the weights of S and O are 0.196, the approach will not represent complete rank, and FM1 and FM6 will both have the same rank.

Conclusions
In this investigation, the main failure modes of a solid oxide fuel cell have been evaluated based on FMEA. The methodology consists of fully fuzzy BWM and fuzzy WASPAS. The uncertain risk assessment information is expressed by the fuzzy set-based method. To achieve the risk factors weights, the fully fuzzy BWM is employed. Then, the fuzzy WASPAS method is utilized to rank the failure modes. The results of the proposed approach based on the CR factor show that this method is more reliable and accurate than other methods like F-BWM and F-AHP. The final rank for considered failure modes shows that the achieved results in newly developed methodology is different from the other methods and based on having higher consistency is closer to reality. According to the obtained rank, crack (FM6) and reduction in catalyst porosity (FM4) are came in first and second rank, respectively, which means these failures should be considered by solid oxide fuel cell designers. Ultimately, sensitivity analyses are further extracted firstly by triple factors' weight changing in seven cases and secondly by alpha changing from 0.0 to 1.0 with unit step. The results of this section show that the final rank is more dependent on alpha rather than triple factors' weight. So, DM can use these results to choose suitable strategies among total existence strategies.
Some recommendations can be represented as follows. A hybrid system including spherical fuzzy sets, linguistics Z-number in the MCDM area, and interval type 2 fuzzy sets can be employed in various sciences such as economics, social issues, agriculture, energy systems, and so forth. Additionally, instead of FWASPAS, other ranking techniques like MOORA and VIKOR and also other MCDM methods such as SWARA can be utilized.