# An Approach of Decision-Making under the Framework of Fermatean Fuzzy Sets

Authors: Muhammad Sarwar Sindhu; Imran Siddique; Muhammad Ahsan; Fahd Jarad; Taner Altunok
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Because of its influence on various elements of human life experiences and conditions, the building industry is a significant business. In the recent past, environmental considerations have been incorporated in the design and planning stages of building supply chains. The process of evaluating and selecting suppliers is one of the most important issues in supply chain management. A multicriteria decision-making (MCDM) problem can be utilized to handle such issues. The goal of this research is to present a new and efficient technique for selecting suppliers with ambiguous data. The suggested methodology’s structure is based on technology for order of preference by similarity to ideal solution (TOPSIS), with Fermatean fuzzy sets (Fr FSs) employed to cope with information uncertainty. In this article, authors modified the distance between Fr FSs to propose the similarity measure and implemented it to form the MCDM model to resolve the vague and uncertain data. Moreover, we used this similarity measure to choose the optimal alternative. A practical example for alternative selection is provided, along with a comparison of the acquired findings to existing approach. Finally, to strengthen the outcome obtained through the proposed model, sensitivity analysis and time complexity analysis are performed.


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## Body

## 1. Introduction

In real-world situations, we frequently encounter tasks and activities that necessitate the usage of decision-makingDMg procedures. DMg may be viewed as a problem-solving process that yields an ideal, or at the very least reasonable, solution. In general, DMg is a mental and reasoning process that leads to choose an ideal option from a collection of possible alternatives in a DMg circumstance. TOPSIS is a valuable method for MCDM issues in the real world. Hwang and Yoon [1] first proposed this strategy in 1981, with Yoon continuing the process in 1987. TOPSIS rates options and determines the best compromise between them and the ideal solution. TOPSIS is an effective approach for ranking and picking a number of generally recognized alternatives using distance metrics that is both practical and helpful. TOPSIS is the best compromise choice, having the lowest distance from the positive-ideal solution and the greatest distance from the negative-ideal solution [2–4]. So far, TOPSIS has been thoroughly investigated by explorers and experts, and it has been successfully applied to a wide range of DMg situations [5–8].TheDMg procedure demands the analysis of a small number of possibilities stated in terms of evaluative criteria for the most part. Instead, when analyzing all of the criteria at once, the issue may be to rank these possibilities in terms of how desirable they are to the decision-maker. If all parameters are assessed at the same time, another goal would be to discover the best choice or to estimate the relative over all preferences of each alternative. The basic goal of MCDM is to solve challenges like these: (1) PROMETHEE, (2) ELECTRE, (3) AHP, (4) VIKOR, (5) Fuzzy AHP, (6) TOPSIS, and (7) Fuzzy TOPSIS are the seven most significant MCDM approaches. Hundreds of experts have implemented TOPSIS in many domains, updated or modified the TOPSIS approach to meet unique issues.One of the inevitabilities of dealing withDMg challenges is the ambiguity of information. The opinions and expressions of decision-makers are frequently the source of this ambiguity. We may describe and capture information uncertainty in a variety of ways. Fuzzy sets (FSs) theory has been a popular method for dealing with uncertainty in DMg situations in recent years. Furthermore, the linear programming (LP) presented in [9] was used to calculate the weights of criteria [10–12] based on decision-makers’ evaluations. The Fr F-TOPSIS approach was created in a variety of fuzzy situations. The study’s key contribution is the use of Fr FSs to expand the Fr F-TOPSIS approach and apply the enlarged methodology to evaluating green building suppliers.


## 2. Literature Review

In the middle of the 1960s, Zadeh [13] proposed the concept of FSs, which ushered in a new era for scholars. In real-world situations, FSs typically reflect uncertainty and ambiguity. The majority of the experts have concentrated on FS expansions and applications. In 1986, Atanassov [14] proposed the notion of intuitionistic fuzzy sets (IFSs), which is one of the most important extensions of FSs and have two number of degrees named, membership degree (MD), and nonmembership degree (NMD) such that 0≤MD+NMD≤1.Recently, Pythagorean fuzzy sets (Pg FS) [15] have gotten more concentration from the experts and implemented in different fields of DMg procedures. When comparing two items based on their unequal content, distance measures are quite useful. Zeng et al. [16] demonstrated the use of various Pg F distance and similarity measurements in MCDM. Hussain and Yang [17] provided various Hausdorff metric-based Pg F distance and similarity measures with Pg F-TOPSIS applicability. Li and Lu [18] presented some generalized distance measurements and their continuous versions for Pg FSs. Ejegwa [19] provided several distance and similarity measurements for Pg FSs based on membership grades. Wei and Wei [20] proposed some cosine function-based Pg F similarity measurements. Peng et al. [21] presented 12 Pg F distance and similarity measurements, along with their applicability (2017). Although Pg FSs have a wide spectrum of uses, they are unable to handle circumstances, where MD2+NMD2>1, for instance, if MD = 0.8 and NMD = 0.7, then 0.82+0.72=0.64+0.49=1.13>1. To overcome such situations, Senapati and Yager [22] introduced as a new sort of FSs recently, named Fr FSs. Fr FSs make up of both MD and NMD which satisfies the condition MD3+NMD3<1, so it handles the abovementioned circumstances accurately. Fr FSs are derived from the ideas of IFS and Pg FS. Fr FSs, on the other hand, use novel concepts to manage uncertain data that make them more flexible and efficient than IFSs and Pg FS [23, 24]. Because they are all confined within the space of Fr FSs, Fr FSs are more powerful than FSs, IFSs, and Pg FSs. Senapati and Yager [24] presented certain Fr FS aggregation operators and their application in decision-making. Mishra and Rani [25] proposed the weighted aggregated sum product assessment (WASPAS) method in the Fermatean fuzzy (Fr F) environment. Garg et al. [26] demonstrated the use of FF aggregating functions in the COVID-19 testing facility. The continuities and derivatives of FF functions were investigated by Yang et al. [27]. Sergi and Sari [28] proposed some FF capital budgeting approaches. Sahoo [29] suggested some FFS scoring functions and their application to transportation issues and decision-making.The major reason we usedFr FSs in designing the current study’s strategy is because of its flexibility in dealing with unclear information. The goal of this research is to develop a new and efficient system for evaluating and selecting green suppliers in a building supply chain where there is uncertainty. In the evaluation process, the technique described in this study takes into account the ambiguity of information given by decision-makers. To deal with information uncertainty, we employed Fr FSs. The suggested technique is based on the extended TOPSIS (E-TOPSIS) and LP methods, which is both efficient and helpful.Failure mode effect analysis (FMEA) is a common and effective technique that may be used to assess risk and improve the safety of a repairable engineering system, according to Kushwaha et al. [30]. Yorulmaz et al. [31] proposed TOPSIS based on modified Mahalanobis distance measure to rank the 81 Turkish provinces by considering distinct levels of development. One of the most important activities in the purchasing department is supplier selection. By assisting in the selection of the most suitable supplier, choosing the correct supplier makes a strategic difference in an organization’s capacity to decrease costs and improve product quality. Cakar and Cavus [32] implemented fuzzy TOPSIS to select the best supplier. The criteria for choosing an air traffic control (ATC) radar station that effectively fulfills the job of radar in air traffic management are developed and assessed in [33]. Picture fuzzy set and rough setbased approaches are proposed in this study to consider the unclear concerns linked with students’ job decision since they are shown to be appropriate due to their inherent qualities to cope with incomplete and imprecise information [34]. To select the construction machinery, Bozanic et al. [35] offered the Neuro-Fuzzy System as a decision-making aid.There has been no previous study employing theFr F-TOPSIS approach with Fr FSs to deal with MCDM, to the best of the authors’ knowledge. The primary contributions of this study can be summarized as follows:(1)
To tackle MCDM situations with ambiguous knowledge that may be stated by a number of decision-makers, a novel DMg technique based on Fr F-TOPSIS and Fr FSs is proposed.(2)
An example demonstrates the effectiveness of the proposed technique for evaluating green building providers.The remainder of the paper is arranged as follows: Section2 contains some fundamental and relevant knowledge. In Section 3, the features of novel Fr FSs are thoroughly examined. To address the ambiguous information, an MCDM model based on Fr F-TOPSIS is created. An MCDM issues relevant to select the supplier is provided in Section 5. The validity of the suggested model is explored in Section 6. Subsection 7.1 examines a complete comparison based on TC. Figure 1 represents the research process of this article.Figure 1
Research process.


## 3. Basic Concepts

Some basic ideas connected to the present work such as FSs, IFSs,Fr FSs, and LP are briefly penned in this section.Definition 1.
[13] A FS ℱ over Y=y1,y2,…,yn can be illustrated as follows:(1)ℱ=y,μℱy|y∈Y.where μℱy:X⟶0,1 is a MD so that y∈Y to ℱ.Definition 2.
[14] Let Y be a fixed set, an IFS ℐ on Y is characterized as follows:(2)ℐ=y,αℐy,βℐy|y∈Y,where αℐy, βℐy∈0,1 are called the MD and NMD of y∈Y to set ℐ with the following condition: 0≤αℐy+βℐy≤1, for all y∈Y.
For ally∈Y, ωℐy is known as hesitancy degree of y∈ℐ, where ωℐy=1−αℐy−βℐy.Definition 3.
[36] A Pg FS P over Y is given by(3)P=y,<αPy,βPy>|y∈Y,where αPy,βPy∈0.1 are the MD and NMD of y to P such that 0≤αP2y+βP2y≤1. The degree of hesitancy or indeterminacy represented by ηPy is written as ηPy=1−αP2y−βP2y.Definition 4.
[22] A Fermatean fuzzy set over the set Y=y1,y2,…,yn is defined as follows:(4)F=y,αFy,ηFy|y∈Y,where αFy,ηFy∈0,1 and are called the MD, NMD of y∈Y to the set F, respectively and αFy,ηFy fulfil the condition: 0≤αF3y+ηF3y≤1, for all y∈Y. Also ζFy=1−αF3y−ηF3y3, then ζFy is supposed to be an indeterminacy membership degree (IMD) of y∈Y in F. For simplicity, Fr FSs over Y is read as Fr FSs(Y).Definition 5.
Reference [9]. The following is the formula of an LP model:(5)Maximize:S=c1t1+c2t2+c3t3+⋯+cntn,Subject to:a11t1+a12t2+a13t3+⋯+a1ntn≤b1,a21t1+a22t2+a23t3+⋯+a2ntn≤b2,⋮am1t1+am2t2+am3t3+⋯+amntn≤bm,t1,t2,…,tn≥0.
In LP model,m indicates the cardinality of constraints and n shows the number of decision variables.


## 4. A Modified Distance Measure betweenFr FSs

A modified Hamming distance measure between twoFr FSs is presented to tackle the vague data in this section.Definition 6.
Suppose thatF1 and F2 be two Fr FSs defined on a fixed set Y=y1,y2,y3, then the distance DFF1,F2 is defined as follows:(6)DFF1,F2=13n∑i=1nαF13yi−αF23yi+ηF13yi−ηF23yi+maxαF13yi−αF23yi,ηF13yi−ηF23yi.Example 1.
LetF1 and F2 be two Fr FSs over Y=y1,y2,y3 given by F1=y1,0.8,0.7, y2,0.9,0.8,y3,0.5,0.9 and F2=y1,0.8,0.62,y2,0.7,0.6,y3,0.9,0.6, based on Definition 6, we get, DFF1,F2=0.2974.Theorem 1.
LetD be a mapping such that D:FrFSsX×FrFSsX⟶0,1. If the requirements below are achieved, then DFF1,F2 is a distance measure.(1)
0≤DFF1,F2≤1;(2)
DFF1,F2=0b2−4ac iff F1=F2;(3)
DFF1,F2=DFF2,F1;(4)
DFF1,F3≥DFF1,F2 and DFF1,F3≥DFF2,F3, for any F1,F2,F3∈Fr FSs(X).Proof.
As, (6) is easy to prove, however, the last condition (4) is proved as follows: For any F1,F2,F3∈Fr FSs(X), and F1⊆F2⊆F3, then on the basis of Definition 5, we get(7)αF13xi−αF33xi≥αF13xi−αF23xi,αF13xi−ηF33xi≥αF1xi−ηF23xi.
By adding equation (7), we get(8)αF13xi−αF33xi+αF13xi−ηF33xi≥αF13xi−αF23xi+αF13xi−ηF23xi,⇒αF13xi−αF33xi+αF13xi−ηF33xi+maxαF13xi−αF33xi,αF13xi−ηF33xi≥αF13xi−αF23xi+αF13xi−ηF23xi+maxαF13xi−αF23xi,αF13xi−ηF23xi,
⇒DFF1,F3≥DFF1,F3, similarly, we can show, DFF1,F3≥DFF2,F3.Since, criteria’s weights have great impact inDMg, we transform the Definition 2.6 into a weighted distance measure (WDM) between two Fr FSs as follows:where wj1≤j≤m denotes the m criteria weights such that ∑j=1mwj=1.Definition 7.
Suppose thatF1 and F2 are two Fr FSs over Y=y1,y2,…,yn and wj are the m criteria’s weights satisfying the condition ∑j=1mwj=1. Then the WDM DFwF1,F2 is penned as below:(9)DFwF1,ℬ=∑i=1nwjαF13yi−αF23yi+ηF13yi−ηF23yi+maxαF13yi−αF23yi,ηF13yi−ηF23yi.Example 2.
Let.F1. and F1 be two Fr FSs on a set Y=y1,y2,y3. Example 1 takes the result by using the weights of y1,y2 and y3 as w1=0.25,w2=0.35 and w3=0.4, respectively. based on Definition 2.7, DFwF1,F2=0.7539.Theorem 2.
The WDMDFwF1,F2 between two Fr FSs F1 and F2 satisfy the following four conditions:(1)
0≤DFwF1,F2≤1;(2)
DFwF1,F2=0 iff F1=F2;(3)
DFwF1,F2=DFF2,F1;(4)
DFwF1,F3≥DFwF1,F2 and DFwF1,F3≥DFwF2,F3, for any F1,F2,F3∈Fr FSs(X).Proof.
In order to prove Theorem2, follow the same strategy as Theorem 1.Definition 8.
Suppose thatF1 and F2 are two Fr FSs over Y=y1,y2,…,yn. Then measure of similarity SpF1,F2 on the basis of Definition 7 is penned as follows:(10)SFF1,F2=1−∑i=1nwjαF13yi−αF23yi+ηF13yi−ηF23yi+maxαF13yi−αF23yi,ηF13yi−ηF23yi.Definition 9.
A mappingS:FrFSsX×FrFSsX⟶0,1. SFF1,F2 is supposed to be a measure of similarity if SFF1,F2 fulfills the following four axioms:(1)
0≤SFF1,F2≤1;(2)
SFF1,F2=1 iff F1=F2;(3)
SFF1,F2=SFF2,F1;(4)
SFF1,F3≤SFF1,F2 and SFF1,F3≤SFF2,F3, for any F1,F2,F3∈Fr FSs(X) and F1⊆F2⊆F3.


## 5. MCDM Model Based on Fermatean Fuzzy TOPSIS (Fr F-Topsis)

We suggested an MCDM usingFr F information based on TOPSIS employing LP methodology in this part. The LP model is used to assess the weights of criteria under various restrictions. Suppose that H=H1,H2,…,Hn be a collection of alternatives, and G=G1,G2,…,Gm be the collection of criteria with μ=μ1,μ2,…,μm, where ∑j=1mμj=1 as the weight vector of the criteria Gj, where j=1,2,3,…,m. A Fr F decision matrix denoted by ℱ=Ωijn×m=αij,ηijn×m with αij as MD and ηij NMD that the alternatives Aii=1,2,…,n fulfills, respectively. To reach the optimal solution, follow the steps of proposed MCDM model.Step 1.
Developed aFr F decision matrix denoted by ℱ=Ωijn×m according to the given information presented by the DM.Step 2.
Figure out theFr F positive-ideal solution (Fr FPIS), Ωp+ and Fr F negative-ideal solution (Fr FNIS), Ωp− as follows: (11)Ωp+=αij+,ηij+=maxjαij,maxjηij:Uj∈ξ1minjαij,minjηij:Uj∈ξ2.(12)Ωp−=αij−,ηij−=minjαij,minjηij:Uj∈ξ1minjαij,minjηij:Uj∈ξ2,where ξ1 and ξ2 are subcollections of beneficial and cost criteria, respectively, so that ξ1∩ξ2=∅.Step 3.
Compute the weighted similarity degree (WSD)SFi+w between Fr FPIS ΩF+ and each alternative likewise the WSD SFi−w between Fr FNIS ΩF− by using equation (12), respectively:(13)SFi+wHi,ΩF+=1−∑j=1mwjαF1xi−αij++ηF1xi−ηij++maxαF1xi−αij+,ηF1xi−ηij+.(14)SFi−wHi,ΩF−=1−∑j=1mwjαF1xi−αij−+ηF1xi−ηij−+maxαF1xi−αij−,ηF1xi−ηij−,where, 1≤i≤n.Step 4.
Based on equations (15) and (16), construct the model to find the objective function Z for the weights of criteria as follows:(15)Z=SFi+wHi,ΩF+−SFi−wHi,ΩF−.Step 5.
We derive the weightsμj of the criterion Gjj=1,2,3,…,m by solving the LP model described in [30], so that the objective function Z produced in Step 4 is maximized.Step 6.
Based on equations (15) and (16), calculate the degree of similarity and evaluate SFi+w and SFi−w on the basis of equations (9) and (10) between each option and the components achieved in Fr FPIS Ωp+ and Fr FNIS Ωp−, respectively.Step 7.
Determine the coefficient of relative closenessℛCi of each alternative Hi with respect to the Fr FPIS ΩF+ as follows:(16)ℛiC=SFi+wSFi+w+SFi−w.
The greater the valueℛCi of the alternatives to Fr FPIS Ωp+w, the more likely we are to find the greatest choice from a set of alternatives Hi, where 1≤i≤n.


## 6. Solution of Problems Based onFr F-Topsis

The authors used the proposed MCDM model to recognize the pattern and breakout of dengue disease in this section.Step 1.Fr F decision matrix Pc=Ωij4×5 denoted in Table 1.Step 2. The ideal solutionΩF+  = {(y1,0.9000, 0.8000), (y2,0.8000,0.6000), (y3,0.7000, 0.5000), (y4,0.8000, 0.5000), (y5,0.9000, 0.5000)} ΩF−  = {(y1,0.5000, 0.3000), (y2,0.4000, 0.1000), (y3,0.4000, 0.2000), (y4, 0.5000, 0.2000), (y5,0.6000, 0.3000)}Step 3. The WSDSFi+w between Fr FPIS ΩF+ and each alternative as well as the WSD SFi−w between Fr FNIS ΩF− by using equations (13) and (14), respectively, in terms of weights.Step 4. Based on equations (13) and (14), evaluate Z=−0.0250w1−0.3650w2−0.1450w3−0.0550w4 which is written in equation (17).Step 5. Based on LP model penned in [9], the weights wj of the criteria Pj, where j=1,2,3,4,5 are obtained as follows:(17)w1=0.2,w2=0.3,w3=0.25,w4=0.1andw5=0.15.Step 6. Degree of positive and negative weighted similaritiesS+fri and S−fri are obtained by using equations (7) and (8) as follows:(18)S+fr1Q1,ΩF+=0.5100,S+fr2Q2,ΩF+=0.7000,S+fr3Q3,ΩF+=0.5700,S+fr4Q4,ΩF+=0.5550,andS−fr1Q1,ΩF−=0.4850,S−fr2Q2,ΩF−=0.3350,S−fr3Q3,ΩF−=0.4250,S−fr4Q4,ΩF−=0.5000.Step 7. Values ofℛCi of each alternative is the following:(19)ℛ1C=0.5126,ℛ2C=0.6763,ℛ3C=0.5729,ℛ4C=0.5261.Step 8. Arrange the alternatives according to the values ofRiC as obtained in Step 4. We get, Q2≺Q3≺Q4≺Q1. Hence, the optimal alternative attained is Q2 which is illustrated in Figure 2.Example 3.
A construction company wanted to select four suppliers,Q1, Q2, Q3, and Q5 according to certain criteria. Suppliers are evaluated against five parameters, P1, P2, P3, and p5. Weights of criteria have great impact in decisions, authors have used LP model to compute the weights. Assume that the evaluation values of the alternatives in relation to each criterion provided by the committee are represented by Fr FNs, as shown in the Fr F decision matrix given in Table 1.Table 1
Fr F decision matrix.
AlternativesQ1{(y1, 0.7, 0.3), (y2, 0.4, 0.6), (y3, 0.5, 0.5), (y4, 0.8, 0.2), (y5, 0.8, 0.4)}Q2{(y1, 0.5, 0.8), (y2, 0.8, 0.6), (y3, 0.4, 0.5), (y4, 0.7, 0.4), (y5, 0.6, 0.5)}Q3{(y1, 0.9, 0.6), (y2, 0.8, 0.1), (y3, 0.6, 0.4), (y4, 0.7, 0.5), (y5, 0.9, 0.3)}Q4{(y1, 0.6, 0.7), (y2, 0.8, 0.3), (y3, 0.7, 0.2), (y4, 0.5, 0.3), (y5, 0.7, 0.3)}Figure 2
Ranking of alternatives.


## 7. Sensitivity Analysis

Because the information for MCDM problems is frequently uncertain and ambiguous, there is a need for a tool that can assist us make more correct decisions. Sensitivity analysis (SA) can help in this regard. In this part, weighted SA is used to evaluate the impact of changing the weights of criteria on the results provided by the proposed model. A formula described in [37] is used to generate a new vector for criterion weights, and the behavior of the findings obtained by the suggested model is then examined. We changed the weights of individual criteria by raising different ratios and looked at the effect on the final findings. Table 2 shows the outcomes achieved by varying the weights of criterion. Figures 3 to 8 show that raising 0.05 to 0.1 in each weight results in a little change in the numeric values, but the ranking orders remain same, demonstrating the usefulness and strength of our suggested model.Table 2
Results obtained for altering the weights of criteria.
AlternativesOriginalIncrement inw1Increment inw2Increment inw3Increment inw4Increment inw5Q10.51260.50130.52140.51800.51270.5210Q20.67630.66000.65080.66910.67070.6700Q30.57290.56300.56210.55920.57300.5745Q40.52610.51870.52010.51500.52710.5268Figure 3
Original result.Figure 4
0.05 to 0.1 raise inw1.Figure 5
0.05 to 0.1 raise inw2.Figure 6
0.05 to 0.1 raise inw3.Figure 7
0.05 to 0.1 raise inw4.Figure 8
0.05 to 0.1 raise inw5.


### 7.1. Comparison Based on Time Complexity (TC)

In order to strengthen the results obtained from the proposed MCDM model, TC analysis is performed in the present subsection. TC is the time required to execute an algorithm to reach the final result. TC is measured among the proposed and the existing techniques presented by Senapati and Yager [23]. The executing time of each technique is evaluated with the help of MATLAB which is presented in Table 3 and its graphical view is illustrated in Figure 9. From Table 3, it can be seen that our approach takes less time as compared to others; hence, the proposed MCDM model is more effective and resolves the issues rapidly.Table 3
TC among the proposed and existing technique.
TechniquesExecuting timeProposed model0.4513 secondsSenapati and Yager [23]0.510 secondsFigure 9
Graphical view of TC analysis.


## 7.1. Comparison Based on Time Complexity (TC)

In order to strengthen the results obtained from the proposed MCDM model, TC analysis is performed in the present subsection. TC is the time required to execute an algorithm to reach the final result. TC is measured among the proposed and the existing techniques presented by Senapati and Yager [23]. The executing time of each technique is evaluated with the help of MATLAB which is presented in Table 3 and its graphical view is illustrated in Figure 9. From Table 3, it can be seen that our approach takes less time as compared to others; hence, the proposed MCDM model is more effective and resolves the issues rapidly.Table 3
TC among the proposed and existing technique.
TechniquesExecuting timeProposed model0.4513 secondsSenapati and Yager [23]0.510 secondsFigure 9
Graphical view of TC analysis.


## 8. Conclusions

TOPSIS is one of the most well-known MCDM approaches. The focus of this research was on TOPSIS extensions namedFr. F-TOPSIS is used in complicated decision scenarios with uncertainty. The total of squares of MD and NMD to which an item meeting a criteria supplied by expert is subjected in some real-world situations may be greater than one, but their cube sum may be less than or equal to one. As a result, Pg FS is unable to handle such a situation. From this perspective, the Fr FS might be used to mimic some D Mg scenarios that Pg FS cannot handle. In this study, we offer an MCDM technique based on TOPSIS in a Fr FS environment. Finally, we provided an example to demonstrate how this method might be utilized efficiently.In light of the foregoing, future research could concentrate on:(1)
Using other traditional objective and subjective multicriteria decision-making methods in conjunction withFr FS to determine and evaluate criteria for the selection of the alternative.(2)
Aside from that, the benefits of the current strategy can be enhanced by considering the objective weight of risk factors, which are not taken into account in this study.



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