Extended fuzzy sets and their applications

: This contribution deals with introducing the innovative concept of extended fuzzy set 1 (E-FS) in which the S-norm function of membership and non-membership grades is less than or 2 equal to one. The proposed concept not only encompasses the concept of fuzzy set (FS), but it 3 also includes the concepts of intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PFS) and p-rung 4 orthopair fuzzy set (p-ROFS). In order to explore the features of the E-FS concept, set and algebraic 5 operations on E-FSs, average and geometric operations of E-FSs are studied, and an E-FS score 6 function is deﬁned. The superiority of the E-FS concept is further conﬁrmed with a score-based 7 decision making technique in which the concepts of FS, IFS, PFS and p-ROFS do not make


Introduction 11
Decision making is one of the most important and critical activities of the human 12 being.However, human beings' opinions or preferences are pervaded with vagueness 13 and imprecision.Zadeh [32] proposed a new methodology to address vagueness and 14 imprecision based on the concept of 'fuzziness or gradual degree of membership' to a In this case, the membership function is known as characteristic function and usually denoted by δ A ; the non-membership function is uniquely defined by (1).

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A is a fuzzy set (FS) when µ A and ν A range is [0, 1] and verify the property Consequently, as with CSs, the non-membership function ν A is uniquely defined from the membership function.

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A is an intuitionistic fuzzy set (IFS) when µ A and ν A range is [0, 1] and verify the property 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 π A = 1 − (µ A (x) + ν A (x)).
For IFSs, membership and non-membership functions are specifically denoted as µ A IFS and ν A IFS , respectively.
• A is a Pythagorean fuzzy set (PFS) when µ A and ν A range is [0, 1] and verify the property 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 µ A PFS and ν A PFS , respectively.

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• A is a p-rung orthopair fuzzy set (p-ROFS) (p ≥ 1) when µ A and ν A range is [0, 1] and verify the property If p = 2, then (5) becomes (4); while if p = 1, then (5) becomes (3). 101 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 A CS requires that µ A (x), ν A (x) ∈ {0, 1}, and can be seen as a subclass of FS, IFS, PFS 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 Recall that a S-norm is a binary function : ∀x, y, z ∈ [0, 1], (x, (y, z)) = ( (x, y), z) (associativity).

113
The following S-norms are respectively known as Algebraic, Einstein, Hamacher, and Frank norms: Proposition 1.Any FS, IFS, PFS and p-ROFS on X is an E-FS on X.
114 115 Remark 1.The converse is not true.Indeed, an E-FS may not necessarily be a p-ROFS for all the following operations are defined: An early source of representation of fuzzy subsets can be found in [1,3].

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 : given by: where ω i ≥ 0 for any 1 ≤ i ≤ m, and ∑ m i=1 ω i = 1.
Theorem 1.The output of the weighted average operator E-IFWA is an E-FN.
The above-proposed aggregation operator satisfy the impotency, boundary and monotonicity properties, which are stated below.
for a set of equal E-FNs A E−FN := If we set then we have the following result.
Definition 5.The weighted geometric operator for a set of E-FNs is a mapping E − IFWA : given by: where ω i ≥ 0 for any 1 ≤ i ≤ m, and ∑ m i=1 ω i = 1. 140 The weighted geometric operator E − IFWG also satisfies idempotency, monotonic-141 ity and boundedness properties.
147 the score function Sc 1 decreases from the value 1 − ν A E−FN (for λ = 0) to the value 149 One of the superiorities of the proposed E-FS score function with respect to the 150 existing p-ROFS score functions, given next in Eq. ( 25)-( 31), is that it is defined based on 157 Thus, the maximum score is obtained for full membership, while complete non-membership Theorem 6.  from Average Solution (EDAS) method is a worthy topic for this evaluation study.

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Peng and Dai's [22] score function: The above score functions may not provide good performance, even when con-  1-3).This is not the case 182 with the proposed new score function (denoted Sc FC in Table 4).Furthermore, it is not 183 possible to apply the existing score functions of p-ROFSs to E-FSs that are not p-ROFSs.

. 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), i.e. it is useful to deal with conflicting attributes [13].
Suppose that {R 1 , R 2 , ..., R m } is the set of alternatives, {C 1 , C 2 , ..., C n } a set of criteria, and {E 1 , E 2 , ..., E l } a set of experts who evaluate alternatives against criteria using E-FNs 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 1, 2, ..., l.
Step 2. Apply a weighted aggregation operator, for instance operator E − IFWA (16), to compute the E-FN group matrix A E−FN = A ij E−FN m×n .
Step 3. Compute the following average value for each alternative Step 4. Apply Sc 1 (23) to derive the positive distance (PD) and negative distance (ND) from A E−FN : where Sc and Sc stand for the score function of E-FNs and their average E-FN, respectively.

An EDAS-based case study
A refrigerator is an essential appliance that contributes most in the living standards of a household, although it also consumes a considerable amount of energy due to its continue 24-hour running.Therefore, the adequate selection of refrigerator can benefit positively to the individual household by reducing its energy-consumption cost, and to the general environment sustainability on the other side.Below, we consider a decision-making process related to the purchasing of a refrigerator within the context of information dealt with in this paper.
In a recent contribution, Li et al. [15] proposed a MCGDM technique based on the EDAS model with p-ROFS data, which they applied to the application problem of buying an adequate refrigerator based on the following 6 criteria: C 1 ≡ safety, C 2 ≡ the performance of refrigeration, C 3 ≡ designation, C 4 ≡ the reliability scale, C 5 ≡ the economical benefit, C 6 ≡ aesthetics for deciding which of 5 available refrigerators R j (j = 1, 2, 3, 4, 5) should be bought.All the criteria are of benefit-type, and their corresponding weights are considered as ω C = (ω C 1 , ω C 2 , ..., ω C 6 ) = (0.20, 0.15, 0.25, 0.17, 0.13, 0.10), and obtained the below p-ROFS collective decision matrix D by applying the q-ROFHA operator, with entry d ij the decision value of the pair (C i , R j ): With the above p-ROFS collective decision matrix at hand, the average p-ROFS of each of its rows with p = 3 in (32) results in the following alternatives average p-ROFSs Table 6 presents the score values of the average p-ROFSs based on the score functions ( 25)- (31), and the proposed one (23).
If we denote the ij-th entry of  The positive and negative weighted distances, with (ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , ω 6 ) = (0.2, 0.15, 0.25, 0.17, 0.13, 0.1) adopted from [15], are computed and the corresponding integrative appraisal scores are obtained and reported in Table 7, from which we observe slightly different arrangements of possible refrigerators, based on the existing p-ROFS score functions and the E-FS one, although the best refrigerator is same.Needless to say that if the decision matrix D elements are E-FSs, then existing p-ROFS-based methods would be inapplicable.Case 2 ( := 2 ): are: The partial derivatives of Sc 1 (A E−FN ) with respect to λ is: It is obvious that ∂Sc 1 (A E−FN ) ∂λ ≤ 0 for any λ ∈ [0, 1].

102of p. 106 Definition 1 .
or p-ROFS.Although, the concept of p-ROFS has been widely studied by researchers, 103 very little is known about the pre-determination of the parameter p.This motivates us to 104 propose the concept of extended fuzzy set (E-FS), which is not dependent on any value 105 Consider the referential set X.An extended fuzzy set (E-FS) A E−FS on X is characterised by two functions, µ A E−FS : X → [0, 1] and ν A E−FS : X → [0, 1], called the membership and non-membership functions of A E−FS , respectively, that verify the property 0

142 5 .Definition 6 .
Score function of E-FNs143In what follows, we propose a score function for E-FNs based on the S-norm 1 , 144 and prove its properties.145Given an E-FN A E−FN , we define its score function Sc while 152 existing p-ROFS scores are mainly based on the difference µ A E−FN − ν A E−FN , which is 153 meaningless in the case µ A E−FN = ν A E−FN .In addition, for all λ ∈ [0, 1], it can be easily 154 observed from Definition 6 and the S-norm 1 that 155

158Theorem 5 .
has associated a score value of 0. We provide below other interesting properties for the 159 proposed score function.160For any two E-FNs A E−FN and B

161Lemma 1 .165 6 .
monotonically increasing with respect to µ A E−FN and monotonically decreasing with respect to 162 ν A E−FN .163 For any E-FN A E−FN , the score function Sc 1 (A E−FN ) is a decreasing function of 164 λ.Decision making with E-FSs 166 Through this section, we first compare the performance of the proposed E-FS score 167 function and the existing p-ROFS scores in the same setting on the pairs of p-ROFS and 168 E-FS datasets.A classical multiple attribute group decision making (MCGDM) problem 169 in which the ranking order of alternatives based on the Evaluation based on Distance 170

Set and algebraic operations on E-FNs
[1,10)9)} is not 117 a p-ROFS for any p ∈[1,10).118 119 Remark 2. In order to simplify the following discussions, µ A E−FS (x), ν A E−FS (x) is called an 120 extended fuzzy number (E-FN).This is nothing else than a special case of E-FS.Furthermore, 121 since the same treatment strategy will be applied for all types of S-norm , we only consider 122 := 1 in what follows.123 3. 124 The usual operations of addition and multiplication between E-FSs are denoted by 125 ⊕ and ⊗ while the operation relating the membership and non-membership degrees of 126 an E-FS is denoted by 1 .We now propose a number of set and algebraic operations on 127 E-FNs.128 Definition 2. For any E-FNs A 6.1.A critical analysis of all existing p-ROFS score functions 172In what follows, we first present a set of existing score functions of p-ROFSs, and 173 then compare their outcomes with the proposed E-FS score function from a point of view 174 of finding weaknesses in them.Notice that in all cases discussed the higher the value of 175a score the more preferable the p-ROFS is.176LetA p−ROFS = { x, µ A p−ROFS (x), ν A p−ROFS (x) : x ∈ X},the following existing p-177 ROFS score functions in the literature are: 178 • Yager's [31] score function:

Table 1 :
2, 0.2), then the existing score func-The ranking of p − ROFS A = (0.3, 0.3) and B = (0.2, 0.2) with existing score functions with p = 1 187A and B, but we can use the proposed new score function (see Table5).All of these 188 findings indicate that the proposed E-FS score function Sc 1 allows the decision-maker 189 to effectively discount the influence of other score-based decisions.190

Table 6 :
Table 6 by µ A p−ROFS , ν A p−ROFS ij , then the score Score values of alternatives average p-ROFSs Score function (µ A In order to save space, we only provide the PD and ND matrices for Sc Y and the proposed Sc FC (λ = 1).

Table 7 :
Ranking of alternatives with considered score functions> R 1 > R 3 > R 4 > R 5 > R 1 > R 3 > R 4 > R 5 > R 1 > R 4 > R 3 > R 5 > R 1 > R 4 > R 5 > R 2Sc PH (Eq.(30)) -6.6229 23.0165 -9.6394 -6.9174 -20.1849R 2 > R 1 > R 4 > R 3 > R 5 > R 3 > R 1 > R 4 > R 5To deal with the complexity of decision making techniques in practice, this contribution introduced the concept of E-FS, which extends the concept of FS, and the concepts of IFS, PFS together with p-ROFS.The prominent role of E-FS concept is apparent in that the concepts of FS, IFS, PFS and p-ROFS do not make sense in all the situations with uncertainty.Other main contributions of the work are summarised in the following: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.inwhichthemembership degree µ A p−ROFS ∈ [0, 1] and the non-membership degreeν A p−ROFS ∈ [0,1] satisfy 0 ≤ µ p A p−ROFS (x) + ν p A p−ROFS (x) ≤ 1 for any x ∈ X and p ∈ [1, ∞).Now, if we set µ A E−FS (x), ν A E−FS (x) = µ [16]direction of future work of this research may be be focused on different forms of E-FS information aggregation operators, and E-FS information measures, decision makers' preference information[4]of E-FSs, individual consensus and group consensus measures[16]of E-FSs.Author Contributions: Conceptualization, B.F.; methodology, B.F. and F.C.; formal analysis, B.F.; validation, B.F. and F.C.; visualization, B.F and F.C..; writing-original draft preparation, B.F.; writing-review and editing, B.F. and F.C.; All authors have read and agreed to the published version of the manuscript.Conflicts of Interest: ) .It is obvious that ∂Sc 1 (A E−FN ) ∂µ A E−FN ≥ 0 while ∂Sc 1 (A E−FN ) ∂ν A E−FN ≤ 0 for any λ ∈ [0, 1].