Extended Fuzzy Sets and Their Applications

mathematics Article Extended Fuzzy Sets and Their Applications Bahram Farhadinia 1,* and Francisco Chiclana 2,3,*   Citation: Farhadinia, B.; Chiclana, F. Extended Fuzzy Sets and Their Applications. Mathematics 2021,9, 770. https://doi.org/10.3390/ math9070770 Received: 15 March 2021 Accepted: 30 March 2021 Published: 2 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Mathematics, Quchan University of Technology, Razavi Khorasan Province, Quchan 9477177870, Iran 2Institute of Artificial Intelligence, De Montfort University, Leicester LE1 9BH, UK 3Andalusian Research Institute on Data Science and Computational Intelligence (DaSCI), University of Granada, 18071 Granada, Spain *Correspondence: [email protected] (B.F.); [email protected] (F.C.) Abstract: This contribution deals with introducing the innovative concept of extended fuzzy set (E-FS), in which the S-norm function of membership and non-membership grades is less than or equal to one. The proposed concept not only encompasses the concept of the fuzzy set (FS), but it also includes the concepts of the intuitionistic fuzzy set (IFS), the Pythagorean fuzzy set (PFS) and the p-rung orthopair fuzzy set (p-ROFS). In order to explore the features of the E-FS concept, set and algebraic operations on E-FSs, average and geometric operations of E-FSs are studied and an E-FS score function is defined. The superiority of the E-FS concept is further confirmed with a score-based decision making technique in which the concepts of FS, IFS, PFS and p-ROFS do not make sense. Keywords: fuzzy set (FS); intuitionistic fuzzy set (IFS); pythagorean fuzzy set (PFS); p-rung orthopair fuzzy set (p-ROFS); extended fuzzy set (E-FS); decision making 1. Introduction Decision making is one of the most important and critical activities of the human being. However, human beings’ opinions or preferences are pervaded with vagueness and imprecision. Zadeh [ 1 ] proposed a new methodology to address vagueness and imprecision based on the concept of ‘fuzziness or gradual degree of membership’ to a set, which was term ’fuzzy set’ (FS). Since then, fuzzy sets have been extensively studied and extended with other types of fuzziness based sets: intuitionistic fuzzy set (IFS) [ 2 ], Pythagorean fuzzy set (PFS) [ 3 ] and p-rung orthopair fuzzy set (p-ROFS) [ 4 ] . The definitions of FS, IFS, PFS and p-ROFS rely on the concepts of membership and non-membership degrees of an element to a set. In the case of FSs, the sum of membership and non-membership degrees of an element to a set is constrained to be equal to one; for IFSs, this sum is constrained to be less than or equal to one; while constraints on the sum of powers of membership and non-membership degrees to be less than or equal to one are used in the case of PFS (powers of two) and p-ROFSs ( p -power). Recall that if p= 1 and p= 2, then a p-ROFS reduces to an IFS and a PFS, respectively, making IFS and PFS special cases of the p-ROFS. The concepts of FS, IFS, PFS and p-ROFS have been applied to a wide-range of decision making problems including aggregation-based studies, information measure-based research and ranking-based developments [ 5 – 8 ]. Among a large number of aggregationbased studies, we highlight those carried out by Xu and Yager [ 9 ], where geometric aggregation operators were introduced, and He et al. [ 10 ] where the geometric interaction average operators for IFSs was proposed. By using Einstein operations, Garg [ 11 ] presented the concept of information and geometric aggregations for PFSs; Liu and Wang [12] extended the weighted average and geometric operators to p-ROFSs, while Wei et al. [13] and Peng et al. [14] investigated the Heronian mean and an exponential operations for p-ROFSs, respectively. Abundant information measure-based research studies have been dedicated to the concepts of FS, IFS, PFS and p-ROFS. Wu et al. [ 15 ] exploited the isomorphism between IFSs and interval-valued FSs to formally develop an approach to consistency of IFSs. Mathematics 2021,9, 770. https://doi.org/10.3390/math9070770 https://www.mdpi.com/journal/mathematics Mathematics 2021,9, 770 2 of 19 Garg and Kumar [ 16 ] studied a class of similarity measures of IFSs, while Wu and Chiclana in [ 17 ] and Ureña et al. in [ 18 ] applied IFSs approaches to estimate missing preferences and consensus building. Zhang [ 19 ] worked on a multiple criteria group decision making technique of PFSs, and Guolin et al. [ 20 ] proposed a new decision-theoretic rough set model with p-ROFSs. An important topic worth attention is how to compare and rank FSs, IFSs, PFSs or p-ROFSs. Xu and Yager [ 9 ] presented a score and accuracy functions on IFSs. Peng [ 21 ] introduced a score function for PFSs and then investigated a number of its properties. Farhadinia and Liao [ 22 ] first reviewed the existing score functions for p-ROFSs, and then proposed a parametrised score function of p-ROFSs. It is worth noting that, as p value increases in p-ROFSs, the space of acceptable orthopairs increases and more orthopairs satisfy the boundary constraint. Therefore, we can express a wider range of fuzzy information by using p-ROFSs than using FSs, IFSs or PFSs. However, it is observed that their approaches are limited in nature. For example, if an expert provides the value 0.95 for membership degree and the value 0.9 for nonmembership degree, then 0.95 + 0.9 6= 1, 0.95 + 0.9 6≤ 1, ( 0.95 )2+ ( 0.9 )26≤ 1, while ( 0.95 )p+ ( 0.9 )p≤ 1 when p≥ 10, which restricts the option of selecting p in p-ROFS theory. Hence, FSs, IFSs, PFSs and p-ROFSs cannot describe this information properly and effectively. To manage such a situation, and releasing the restriction of p selection in p-ROFS theory, the concept of extended fuzzy set (E-FS) is introduced herein with a S-norm function of the membership and non-membership degrees being constrained to be less than or equal to 1. The proposed E-FS concept is therefore more effective and more general than the existing concepts for handling uncertain information in real-life decision processes. Indeed, our proposed E-FS concept expresses a wider range of fuzzy information than p-ROFS without needing to consider the extra parameter p , which is essential in defining a p-ROFS. In addition, the proposed E-FS diminishes the restriction that FS, IFS, PFS and p-ROFS impose on membership grades and it provides decision-makers with more elasticity to express their opinions according to the membership grades of an element than the concepts of FS, IFS, PFS and p-ROFS. Summarising, this study main research contributions are: 1. The introduction of the concept of E-FS, as an overarching concept of FS to include the concepts of IFS, PFS and p-ROFS; 2. The study of the fundamental principles of E-FSs in comparison to FSs, IFSs, PFSs and p-ROFSs; 3. The definition of some algebraic and set operations on E-FSs, including the average and geometric operations on E-FSs; 4. The presentation of a score function for E-FSs. The rest of this contribution is organised as follows—Section 2provides a brief review of some preliminaries needed before the concept of E-FS is introduced. Sections 3–5are devoted to the study of set and algebraic operations on E-FSs, average and geometric operations of E-FSs and a score function of E-FSs, respectively. Section 6presents the application of the proposed E-FS score function to solve a decision making problem. Conclusions are drawn in Section 7. 2. Fundamental Principles This section is devoted to reviewing some existing basic concepts and notions that are of importance in fuzzy set theory. Then, we deal with the main part of this contribution, which is the introduction of the concept of extended fuzzy set (E-FS). Formally, given a universal set X , a set A on X is characterised by two functions on X that measure the degree of membership ( µA(x) ) and the degree of non-membership ( νA(x) ) to Aof each element of the universal set x∈X. •Ais a classical set (CS) when µAand νArange is {0, 1}and verify the property µA(x) + νA(x) = 1, ∀x∈X. (1) Mathematics 2021,9, 770 3 of 19 In this case, the membership function is known as a characteristic function and usually denoted by δA; the non-membership function is uniquely defined by (1). •Ais a fuzzy set (FS) when µAand νArange is [0, 1]and verify the property µA(x) + νA(x) = 1, ∀x∈X. (2) Consequently, as with CSs, the non-membership function νA is uniquely defined from the membership function. •A is an intuitionistic fuzzy set (IFS) when µA and νA range is [ 0, 1 ] and verify the property µA(x) + νA(x)≤1, ∀x∈X. (3) In this case, given a membership function, multiple non-membership functions verifying (3) may exist. The concept of hesitancy is therefore present in an IFS, which is modelled in this framework via the hesitancy function πA= 1 −(µA(x) + νA(x)) . For IFSs, membership and non-membership functions are specifically denoted as µAIFS and νAIFS , respectively. •A is a Pythagorean fuzzy set (PFS) when µA and νA range is [ 0, 1 ] and verify the property µ2 A(x) + ν2 A(x)≤1, ∀x∈X. (4) As with IFSs, a membership function may have associated more than one nonmembership functions as per (4). For PFSs, membership and non-membership functions are specifically denoted as µAPFS and νAPFS , respectively. •A is a p-rung orthopair fuzzy set (p-ROFS) ( p≥ 1) when µA and νA range is [ 0, 1 ] and verify the property µp A(x) + νp A(x)≤1, ∀x∈X. (5) If p=2, then (5) becomes (4); while if p=1, then (5) becomes (3). It is obvious from examining the above set concept definitions that all of them follow the same pattern in that all impose a constraint to the membership and non-membership functions of the type φ:[0, 1]×[0, 1]−→ [0, 1] φ(µA(x),νA(x))∈[0, 1],∀x∈X.(6) A CS requires that µA(x) , νA(x)∈ { 0, 1 } , and can be seen as a subclass of FS, IFS, PFS or p-ROFS. Although, the concept of p-ROFS has been widely studied by researchers, very little is known about the pre-determination of the parameter p . This motivates us to propose the concept of extended fuzzy set (E-FS), which is not dependent on any value of p. Definition 1. Consider the referential set X . An extended fuzzy set (E-FS) AE−FS on X is characterised by two functions, µAE−FS :X→[ 0, 1 ] and νAE−FS :X→[ 0, 1 ] , called the membership and non-membership functions of AE−FS, respectively, that verify the property 0≤µAE−FS (x)νAE−FS (x)≤1, ∀x∈X, (7) where  is a S-norm or union function. Using set theoretic notation, an extended fuzzy set (E-FS) on X will be denoted as follows AE−FS =hx,µAE−FS (x),νAE−FS (x)i:x∈X. Recall that an S-norm is a binary function :[ 0, 1 ]×[ 0, 1 ]→[ 0, 1 ] that satisfies the following properties (see e.g., [23]) 1. (x, 0) = x(boundary condition); 2. ∀x,y,z∈[0, 1], if y≤z, then (x,y)≤ (x,z)(monotonicity); Mathematics 2021,9, 770 4 of 19 3. ∀x,y∈[0, 1],(x,y) = (y,x)(commutativity); 4. ∀x,y,z∈[0, 1],(x,(y,z)) = ((x,y),z)(associativity). The following S-norms are respectively known as Algebraic, Einstein, Hamacher and Frank norms: 1(x,y) = x+y−xy; 2(x,y) = x+y 1+xy; e 3(x,y) = x+y−xy −(1−e)xy 1−(1−e)xy ,e>0; e 4(x,y) = 1−loge(1+(e1−x−1)(e1−y−1) e−1),e>1. Proposition 1. Any FS, IFS, PFS and p-ROFS on X is an E-FS on X. Proof of Proposition 1is in Appendix A.1. Remark 1. The converse is not true. Indeed, an E-FS may not necessarily be a p-ROFS for all values of p∈[ 1, ∞) . For instance, the E-FS µAE−FS (x),νAE−FS (x)={( 0.95, 0.9 )} is not a p-ROFS for any p ∈[1, 10). Remark 2. In order to simplify the following discussions, µAE−FS (x),νAE−FS (x) is called an extended fuzzy number (E-FN). This is nothing else than a special case of E-FS. Furthermore, since the same treatment strategy will be applied for all types of S-norm  , we only consider :=1 in what follows. 3. Set and Algebraic Operations on E-FNs The usual operations of addition and multiplication between E-FSs are denoted by ⊕ and ⊗ while the operation relating the membership and non-membership degrees of an E-FS is denoted by 1 . We now propose a number of set and algebraic operations on E-FNs. Definition 2. For any E-FNs AE−FN = (µAE−FN , νAE−FN ) and BE−FN = (µBE−FN , νBE−FN ) , the following operations are defined: Ac E−FN = (µAc E−FN ,νAc E−FN ) = (νAE−FN ,µAE−FN ); (8) AE−FN ∩BE−FN = (µAE−FN∩BE−FN ,νAE−FN ∪BE−FN ) =min{µAE−FN ,µBE−FN }, max{νAE−FN ,νBE−FN };(9) AE−FN ∪BE−FN = (µAE−FN∪BE−FN ,νAE−FN ∩BE−FN ) =max{µAE−FN ,µBE−FN }, min{νAE−FN ,νBE−FN };(10) AE−FN⊕BE−FN = (µAE−FN ⊕BE−FN ,νAE−FN⊕BE−FN ) =1−(1−µAE−FN )(1−µBE−FN ),νAE−FN νBE−FN ;(11) AE−FN⊗BE−FN = (µAE−FN ⊗BE−FN ,νAE−FN⊗BE−FN ) =µAE−FN µBE−FN , 1 −(1−νAE−FN )(1−νBE−FN );(12) λAE−FN =µλAE−FN ,νλAE−FN =1−(1−µAE−FN )λ,(νAE−FN )λ; (13) Aλ E−FN =µAλ E−FN ,νAλ E−FN =(µAE−FN )λ, 1 −(1−νAE−FN )λ,λ>0. (14) Proof of Definition 2operations are well defined is in Appendix A.2. Mathematics 2021,9, 770 5 of 19 Definition 3. For any E-FNs AE−FN = (µAE−FN ,νAE−FN )and BE−FN = (µBE−FN ,νBE−FN ), AE−FN ⊆BE−FN ⇐⇒ µAE−FN ≤µBE−FN ∧νAE−FN ≥νBE−FN . (15) An early source of representation of fuzzy subsets can be found in [24,25]. 4. Average and Geometric Operators of E-FNs The weighted average operator and the weighted geometric operator of a set of E-FNs are defined. Some of their properties are also outlined. Definition 4. The weighted average operator for a set of E-FNs is a mapping E−IFWA : E−FN × · · · × E−FN →[0, 1]given by: E−IFWA(A1E−FN, . . . , AmE−FN) = m M i=1 ωiAiE−FN = 1− m ∏ i=11−µAiE−FN ωi, m ∏ i=1 νωi AiE−FN !(16) where ωi≥0for any 1≤i≤m, and ∑m i=1ωi=1. Theorem 1. The output of the weighted average operator E-IFWA is an E-FN. Proof of Theorem 1is in Appendix A.3. The above-proposed aggregation operator satisfies the impotency, boundary and monotonicity properties, which are stated below. Theorem 2. (Idempotency property) The weighted average operator E −IFWA satisfies E−IFWA(A1E−FN, . . . , AmE−FN) = AE−FN (17) for a set of equal E-FNs AE−FN :=A1E−FN =· · · =AmE−FN. Proof of Theorem 2is in Appendix A.4. If we set AlE−FN =µAlE−FN ,νAlE−FN =min 1≤i≤m{µAiE−FN }, max 1≤i≤m{νAiE−FN }(18) AuE−FN =µAuE−FN ,νAuE−FN =max 1≤i≤m{µAiE−FN }, min 1≤i≤m{νAiE−FN }, (19) then we have the following result. Theorem 3. (Boundedness property) The weighted average operator E −IFWA satisfies AlE−FN ⊆E−IFWA(A1E−FN, . . . , AmE−FN)⊆AuE−FN (20) for a set of E-FNs A1E−FN, . .., AmE−FN. Proof of Theorem 3is in Appendix A.5. Theorem 4. (Monotonicity property) Suppose that for any two classes of E-FNs A1E−FN , ..., AmE−FN and B1E−FN ,..., BmE−FN , it holds that AiE−FN ⊆BiE−FN for all 1 ≤i≤m . Then, the weighted average operator E −IFWA satisfies E−IFWA(A1E−FN, . . . , AmE−FN)⊆E−IFWA(B1E−FN, . . . , BmE−FN). (21) Proof of Theorem 4is in Appendix A.6. Mathematics 2021,9, 770 6 of 19 Definition 5. The weighted geometric operator for a set of E-FNs is a mapping E−IFWA : E−FN(X)× · · · × E−FN(X)→[0, 1]given by: E−IFWG(A1E−FN, . . . , AmE−FN) = m O i=1 Aωi iE−FN = m ∏ i=1 µωi AiE−FN , 1 − m ∏ i=11−νAiE−FN ωi!(22) where ωi≥0for any 1≤i≤m, and ∑m i=1ωi=1. The weighted geometric operator E−IFWG also satisfies idempotency, monotonicity and boundedness properties. 5. Score Function of E-FNs In what follows, we propose a score function for E-FNs based on the S-norm 1 , and prove its properties. Definition 6. Given an E-FN AE−FN, we define its score function Sc1(AE−FN) = 1−νAE−FN −λ1−µAE−FN 1νAE−FN  =1−νAE−FN −λ1−1−1−µAE−FN 1−νAE−FN ,(23) where λ∈[0, 1]. It is Sc1∈[ 0, 1 ] for any µAE−FN , νAE−FN , λ∈[ 0, 1 ] . When λ increases in [ 0, 1 ] , the score function Sc1 decreases from the value 1 −νAE−FN (for λ= 0) to the value µAE−FN (1−νAE−FN )(for λ=1). One of the superiorities of the proposed E-FS score function with respect to the existing p-ROFS score functions, given next in Equations (25)–(31), is that it is defined based on the multiplication of µAE−FN and ( 1 −νAE−FN ) , ( 1 −νAE−FN )(1−λ) + λµAE−FN  , while existing p-ROFS scores are mainly based on the difference µAE−FN −νAE−FN , which is meaningless in the case µAE−FN =νAE−FN . In addition, for all λ∈[ 0, 1 ] , it can be easily observed from Definition 6and the S-norm 1that •Sc1(AE−FN) = 1 if AE−FN = (1, 0); •Sc1(AE−FN) = 0 if AE−FN = (0, 1). Thus, the maximum score is obtained for full membership, while complete nonmembership has associated a score value of 0. We provide below other interesting properties for the proposed score function. Theorem 5. For any two E-FNs AE−FN and BE−FN, if AE−FN ⊆BE−FN, then Sc1(AE−FN)≤Sc1(BE−FN). (24) Proof of Theorem 5is in Appendix A.7. Theorem 6. For any E-FN AE−FN = (µAE−FN , νAE−FN ) , the score function Sc1(AE−FN) is monotonically increasing with respect to µAE−FN and monotonically decreasing with respect to νAE−FN . Proof of Theorem 6is in Appendix A.8. Lemma 1. For any E-FN AE−FN , the score function Sc1(AE−FN) is a decreasing function of λ . Proof of Lemma 1is in Appendix A.9. Mathematics 2021,9, 770 7 of 19 6. Decision Making with E-FSs Through this section, we first compare the performance of the proposed E-FS score function and the existing p-ROFS scores in the same setting on the pairs of p-ROFS and E-FS datasets. A classical multiple attribute group decision making (MCGDM) problem in which the ranking order of alternatives based on the Evaluation based on Distance from Average Solution (EDAS) method is a worthy topic for this evaluation study. 6.1. A Critical Analysis of all Existing p-ROFS Score Functions In what follows, we first present a set of existing score functions of p-ROFSs, and then compare their outcomes with the proposed E-FS score function from a point of view of finding weaknesses in them. Notice that in all cases discussed the higher the value of a score the more preferable the p-ROFS is. Let Ap−ROFS ={hx , µAp−ROFS (x) , νAp−ROFS (x)i:x∈X} , the following existing p-ROFS score functions in the literature are: • Yager’s [4] score function: ScY(Ap−ROFS) = µp Ap−ROFS (x)−νp Ap−ROFS (x); (25) • Wei et al.’s [13] score function: ScW(Ap−ROFS) = 1 21+µp Ap−ROFS (x)−νp Ap−ROFS (x); (26) • Peng et al.’s [14] score function: ScPDG(Ap−ROFS) = µp Ap−ROFS (x)−νp Ap−ROFS (x) +   eµp Ap−ROFS (x)−νp Ap−ROFS (x) eµp Ap−ROFS (x)−νp Ap−ROFS (x)+1 −1 2  ×(1−µp Ap−ROFS (x)−νp Ap−ROFS (x)); (27) • Mi et al.’s [26] score function: ScMLL(Ap−ROFS) = 2+µp Ap−ROFS (x)−νp Ap−ROFS (x) 2−µp Ap−ROFS (x) + νp Ap−ROFS (x)×2−µp Ap−ROFS (x)−νp Ap−ROFS (x); (28) • Farhadinia and Liao’s [22] score function: ScFL(Ap−ROFS) = µp Ap−ROFS (x) + λ1−µp Ap−ROFS (x)−νp Ap−ROFS (x), 0 ≤λ≤1. (29) • Peng and Huang’s [27] score function: ScPH(Ap−ROFS) = µp Ap−ROFS (x)−νp Ap−ROFS (x) + ln2−µp Ap−ROFS (x)−νp Ap−ROFS (x); (30) • Peng and Dai’s [14] score function: ScPD(Ap−ROFS) = µp Ap−ROFS (x)−2νp Ap−ROFS (x)−1 3+λ 3(µp Ap−ROFS (x) + νp Ap−ROFS (x) + 2); (31) The above score functions may not provide good performance, even when considering different p-ROFSs. For instance, if we consider p-ROFSs A= ( 0.3, 0.3 ) and B= ( 0.2, 0.2 ) , then the first, second, third and seventh score functions are not able to distinguish between them when p= 1, 2, 3 (refer to Tables 1–3). This is not the case with the proposed new score function (denoted ScFC in Table 4). Furthermore, it is not possible to apply the existing score functions of p-ROFSs to E-FSs that are not p-ROFSs. For instance, if we have A= ( 0.9, 0.8 ) and B= ( 0.2, 0.2 ) , then the existing score functions when p= 1, 2, 3 cannot be applied because, for such p -values, A= ( 0.9, 0.8 ) is not a p-ROFS. Therefore, existing score functions of p-ROFSs are not useful in ranking A and B , but we can use the proposed Mathematics 2021,9, 770 8 of 19 new score function (see Table 5). All of these findings indicate that the proposed E-FS score function Sc1 allows the decision-maker to effectively discount the influence of other score-based decisions. Table 1. The ranking of p−ROFS A = ( 0.3, 0.3 ) and B= ( 0.2, 0.2 ) with existing score functions with p=1. Score Function A–Score B–Score Ranking ScY0 0 A=B ScW0.5 0.5 A=B ScPDG 0 0 A=B ScMLL 0.7143 0.625 A>B ScFL (λ=1)0.7 0.8 A<B ScPH −0.3365 −0.47 A>B ScPD 0 0 A=B Table 2. The ranking of p−ROFS A = ( 0.3, 0.3 ) and B= ( 0.2, 0.2 ) with existing score functions with p=2. Score Function A–Score B–Score Ranking ScYA=B ScW0.5 0.5 A=B ScPDG 0 0 A=B ScMLL 0.5495 0.5208 A>B ScFL (λ=1)0.91 0.96 A<B ScPH −0.5988 −0.6523 A>B ScPD 0 0 A=B Table 3. The ranking of p−ROFS A = ( 0.3, 0.3 ) and B= ( 0.2, 0.2 ) with existing score functions with p=3. Score Function A–Score B–Score Ranking ScY0 0 A=B ScW0.5 0.5 A=B ScPDG 0 0 A=B ScMLL 0.5139 0.504 A>B ScFL (λ=1)0.973 0.992 A<B ScPH −0.6658 −0.6851 A>B ScPD 0 0 A=B Table 4. The ranking of p−ROFS A = ( 0.3, 0.3 ) and B= ( 0.2, 0.2 ) as extended fuzzy sets (E-FSs) with new score function. ScFC (λ)A–Score B–Score Ranking ScFC (λ=1)0.21 0.16 A>B ScFC (λ=1/2)0.455 0.48 A<B ScFC (λ=0)0.7 0.8 A<B Mathematics 2021,9, 770 9 of 19 Table 5. The ranking of A= (0.9, 0.8)and B= (0.2, 0.2)as E-FSs with new score function. ScFC (λ)A–Score B–Score Ranking ScFC (λ=1)0.18 0.16 A>B ScFC (λ=1/2)0.19 0.48 A<B ScFC (λ=0)0.2 0.8 A<B 6.2. E-FS-Based EDAS Technique for MCGDM The EDAS model is useful to determine the best alternative(s) corresponding to the biggest value of positive distance from average solution (PDAS) and the smallest value of negative distance from average solution (NDAS), that is, it is useful to deal with conflicting attributes [28]. Suppose that {R1 , R2 , ..., Rm} is the set of alternatives, {C1 , C2 , ..., Cn} a set of criteria, and {E1 , E2 , ..., El} a set of experts who evaluate alternatives against criteria using E-FNs AijtE−FN =µAijtE−FN ,νAijtE−FN  for i= 1, 2, ..., m , j= 1, 2, ..., n , t= 1, 2, ..., l . Assume that {ωE1 , ωE2 , ..., ωEl} and {ωC1 , ωC2 , ..., ωCn} are these of weights of experts and criteria, respectively, subject to the constraints: 0 ≤ωEt , ωCj≤ 1, ∑l t=1ωEt=∑n j=1ωCj= 1. The E-FS-based EDAS technique for MCGDM problem may be carried out by the following steps: Step 1. Construct the individual experts’ evaluation matrices At=hAijtE−FN im×nt= 1, 2, ..., l. Step 2. Apply a weighted aggregation operator, for instance operator E−IFWA (16), to compute the E-FN group matrix AE−FN =hAijE−FN im×n. Step 3. Compute the following average value for each alternative µAE−FN ,νAE−FN i= p v u u t1− n ∏ j=11−µp AE−FN 1 n, n ∏ j=1νAE−FN 1 n ,i=1, 2, ..., m.(32) Step 4. Apply Sc1 (23) to derive the positive distance (PD) and negative distance (ND) from AE−FN: PDµAE−FN ,νAE−FN ij =maxn0, ScµAE−FN ,νAE−FN ij−ScµAE−FN ,νAE−FN io ScµAE−FN ,νAE−FN i, (33) NDµAE−FN ,νAE−FN ij =maxn0, ScµAE−FN ,νAE−FN i−ScµAE−FN ,νAE−FN ijo ScµAE−FN ,νAE−FN i, (34) where Sc and Sc stand for the score function of E-FNs and their average E-FN, respectively. Step 5. Compute the positive weighted distance Pi(i= 1, 2, ..., m) and the negative weighted distance Ni(i=1, 2, ..., m): Pi= n ∑ j=1 ωCjPDµAE−FN ,νAE−FN ij, (35) Ni= n ∑ j=1 ωCjNDµAE−FN ,νAE−FN ij, (36) Step 6. Normalise Pi(i=1, 2, ..., m)and Ni(i=1, 2, ..., m) Pi=Pi max{P1,P2, ..., Pm}; (37) Mathematics 2021,9, 770 16 of 19 in which 0≤µLm i=1ωiAiE−FN⊕ωm+1Am+1E−FN 1νLm i=1ωiAiE−FN⊕ωm+1Am+1E−FN = 1− m ∏ i=11−µAiE−FN ωi1−µAm+1E−FN ωm+1!1 m ∏ i=1 νωi AiE−FN ×νωm+1 Am+1E−FN ! =1− 1− 1− 1− m ∏ i=11−µAiE−FN ωi1−µAm+1E−FN ωm+1!!! × 1− m ∏ i=1 νωi AiE−FN ×νωm+1 Am+1E−FN !! =1− 1− m+1 ∏ i=11−µAiE−FN ωi!× 1− m+1 ∏ i=1 νωi AiE−FN !≤1. This completes the proof. Appendix A.4. Proof of Theorem 2 Proof. Taking the equality relationship between all the unified E-FNs, that is, AE−FN := A1E−FN =· · · =AmE−FN together with the condition ∑m i=1ωi= 1 in which ωi≥ 0 for any 1≤i≤m, we conclude that E−IFWA(A1E−FN, . . . , AmE−FN) = E−IFWA(AE−FN, . . . , AE−FN) = m M i=1 Aωi E−FN = 1− m ∏ i=1 (1−µAE−FN )ωi, m ∏ i=1 νωi AE−FN ! =1−1−µAE−FN ∑m i=1ωi,ν∑m i=1ωi AE−FN  =1−1−µAE−FN ,νAE−FN  =AE−FN. Appendix A.5. Proof of Theorem 3 Proof. Let ∑m i=1ωi=1 such that ωi≥0∀i. Denoting Minµ=min1≤i≤m{µAiE−FN };Maxµ=max1≤i≤m{µAiE−FN }; Minν=min1≤i≤m{νAiE−FN };Maxν=max1≤i≤m{νAiE−FN }. Since ωi≥0 , it is: (1−Minµ)ωi≥(1−µAiE−FN )ωi≥(1−Maxµ)ωi; (Minν)ωi≤(νAiE−FN )ωi≤(Maxν)ωi, which implies (1−Minµ)∑m i=1ωi≥∏m i=1(1−µAiE−FN )ωi≥(1−Maxµ)∑m i=1ωi; (Minν)∑m i=1ωi≤∏m i=1(νAiE−FN )ωi≤(Maxν)∑m i=1ωi. Since ∑m i=1ωi=1, it is Minµ≤1−∏m i=1(1−µAiE−FN )ωi≤Maxµ; Minν≤∏m i=1(νAiE−FN )ωi≤Maxν. Mathematics 2021,9, 770 17 of 19 From Minµ≤1− m ∏ i=1 (1−µAiE−FN )ωiand m ∏ i=1 (νAiE−FN )ωi≤Maxν, we deduce that AlE−FN ⊆E−IFWA(A1E−FN, . . . , AmE−FN), while from Minµ≤1− m ∏ i=1 (1−µAiE−FN )ωiand m ∏ i=1 (νAiE−FN )ωi≤Maxν, we deduce that E−IFWA(A1E−FN, . . . , AmE−FN)⊆AuE−FN, which completes the proof. Appendix A.6. Proof of Theorem 4 Proof. Let ∑m i=1ωi=1 such that ωi≥0∀i. From Definition 3 AE−FN ⊆BE−FN ⇐⇒ µAE−FN ≤µBE−FN ∧νAE−FN ≥νBE−FN . Since ωi≥0 , it is: 1− m ∏ i=11−µAiE−FN ωi≤1− m ∏ i=11−µBiE−FN ωi∧ m ∏ i=1νAiE−FN ωi≥ m ∏ i=1νBiE−FN ωi. Therefore, we conclude that E−IFWA(A1E−FN, . . . , AmE−FN)⊆E−IFWA(B1E−FN, . . . , BmE−FN). Appendix A.7. Proof of Theorem 5 Proof. Since AE−FN ⊆BE−FN, it is µAE−FN ≤µBE−FN ∧νAE−FN ≥νBE−FN , which implies −(1−µAE−FN )≤ −(1−µBE−FN )∧1−νAE−FN ≤1−νBE−FN . Algebraic manipulation lead to the following: 1−(1−µAE−FN )(1−νAE−FN )≤1−(1−µBE−FN )(1−νBE−FN ), 1−µAE−FN 1νAE−FN ≥1−µBE−FN 1νBE−FN , (1−νAE−FN )−λ(1−µAE−FN 1νAE−FN )≤(1−νBE−FN )−λ(1−µBE−FN 1νBE−FN ). The latter inequality implies that Sc1(AE−FN)≤Sc1(BE−FN). Appendix A.8. Proof of Theorem 6 Proof. The first partial derivatives of Sc1(AE−FN) = 1−νAE−FN −λ1−1−1−µAE−FN 1−νAE−FN  Mathematics 2021,9, 770 18 of 19 are: ∂Sc1(AE−FN) ∂µAE−FN =λ(1−νAE−FN ); ∂Sc1(AE−FN) ∂νAE−FN =−νAE−FN −λ(−νAE−FN )(1−µAE−FN ) =−νAE−FN 1−λ(1−µAE−FN ). It is obvious that ∂Sc1(AE−FN) ∂µAE−FN ≥0 while ∂Sc1(AE−FN) ∂νAE−FN ≤0 for any λ∈[0, 1]. Appendix A.9. Proof of Lemma 1 Proof. 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