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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 192-204

Published online June 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.2.192

© The Korean Institute of Intelligent Systems

Effective Fuzzy Soft Expert Set Theory and Its Applications

Shawkat Alkhazaleh and Emadeddin Beshtawi

Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid, Jordan

Correspondence to :
Shawkat Alkhazaleh (shmk79@gmail.com)

Revised: April 25, 2023; Accepted: May 2, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In 2022, Alkhazaleh introduced the concept of the effective fuzzy soft set (EFSS) as a new mathematical tool to address uncertain problems in decision-making and medical diagnosis. The virtue of this concept is its adaptability to deal with uncertain problems involving external effects. However, some uncertain decision-making problems, especially those with external effects, must be judged by several experts. To this end, this paper extends the concept of EFSS to the concept of an effective fuzzy soft expert set (EFSES). In particular, we provide a basic definition of EFSES and study its basic operations of complement, union, intersection, AND, and OR. We also present some of its properties. Finally, we present a new algorithm for solving decision-making problems.

Keywords: Fuzzy set, Soft set, Soft expert set, Fuzzy soft set, Fuzzy soft expert set, Effective fuzzy soft set

Many real-life problems involve uncertainty, which hampers classical methods to surmount Dicey problems of decision-making in economy, engineering, medicine, and other fields [1]. Zadeh [2] introduced the concept of fuzzy set theory as a mathematical tool for solving such problems. Furthermore, Zadeh [3] proposed the concept of an interval-valued fuzzy set, as the membership space consists of the set of all closed subintervals between 0 and 1. Atanassov [4] generalized the concept of fuzzy set to the concept of the intuitionistic fuzzy set, which shows both the degree of membership and degree of non-membership of each element in the universe using two functions. Furthermore, Molodtsov [5] presented the concept of soft set theory as a new approach to handling vague problems. In addition, Maji et al. [6] generalized the concept of soft set theory to a more general concept, the fuzzy soft set theory. Moreover, soft set theory has been applied to decision-making problems [7]. In addition, Maji et al. [8] studied soft set theory in detail and defined its basic operations of union, intersection, AND, and OR. Furthermore, Roy and Maji [9] applied the concept of fuzzy soft sets to decision-making problems. Additionally, Yang et al. [10] combined the concepts of an interval-valued fuzzy set and a soft set. Also, the concept of intuitionistic fuzzy sets has been extended to the concept of intuitionistic fuzzy soft sets [11]. In addition, Alkhazaleh and Salleh [12] introduced the concept of soft expert set theory, which enables the user to become familiar with the opinions of all experts in one model. Also, they presented some of its properties and defined its basic operations as union, intersection, AND, and OR.

Many difficult decision-making problems require multiple sets of universes and one set of parameters. Therefore, Alkhazaleh et al. [13] presented soft multisets theory. Furthermore, in [14], they introduced the concept of a possibility fuzzy soft set, which aims to define a fuzzy soft set by linking the possibility of each element in the universal set with the parameterization of the fuzzy sets. In addition, Alkhazaleh and Salleh [15] extended the concepts of a fuzzy set and soft multisets to the concept of fuzzy soft multisets. They introduced the concept of a generalized interval-valued fuzzy soft set, which shows both the membership degree and possibility degree of each element in the universe [16]. Bashir and Salleh [17] presented the concept of a fuzzy parameterized soft expert set, which essentially takes a subset of the set of fuzzy subsets of the set of parameters along with the set of experts and the set of opinions as a Cartesian product. Meanwhile, a combination of the fuzzy soft expert and possibility fuzzy soft set has been proposed [18]. The concept of a possibility fuzzy soft expert set involves defining a fuzzy soft expert set by binding the possibility of each element in a universal set with the parameterization of the fuzzy sets. Bashir et al. [19] extended the concept of the possibility fuzzy soft set to the possibility intuitionistic fuzzy soft set, which exposes the degree of membership and degree of possibility of each element in the universe. Additionally, the authors of [20] presented the concept of a fuzzy soft expert set as a combination of fuzzy and soft expert sets. Furthermore, Xu [21] presented the concept of a hesitant fuzzy set, which returns a subset of [0, 1] when applied to a fixed set X. Moreover, Broumi and Smarandache [22] extended the concept of possibility intuitionistic fuzzy soft sets to the concept of possibility intuitionistic fuzzy soft expert sets. In addition, [23] generalized the concept of a hesitant fuzzy set to the concept of a graded soft expert set and assumed a multivalued set of opinions. Furthermore, the concept of a neutrosophic set was proposed by [24], which consists of three functions and real standard or nonstandard subsets of the closed interval between 0 and 1. Subsequently, Maji [25] extended this concept to the concept of a neutrosophic soft set, Sahin et al. [26] introduced the concept of a neutrosophic soft expert set, Al-Quran and Hassan [27] introduced the concept of neutrosophic vague soft multisets, and Al-Sharqi et al. [28] presented the concept of an interval-valued complex neutrosophic soft set. Recently, Alkhazaleh [29] introduced the concept of effective fuzzy soft set (EFSS) theory and defined its basic operations as complement, union, and intersection. Moreover, he took the external effects into account in decision-making problems. In this study, we extended the concept of EFSS to effective fuzzy soft expert set (EFSES). In summary, the basic definition, operations, and properties of EFESS are presented. Finally, a new algorithm is established and applied to solve some decision-making problems. The merit of this extension in comparison with existing concepts, such as EFSS and fuzzy soft expert sets, is the combination of internal and external effects and the utilization of up to two opinions from each expert in solving decision-making problems.

Definition 2.1 ([5])

A pair (F, E) is called a soft set (over U) iff F is a mapping of E into the set of all subsets of U.

Definition 2.2 ([6])

Let U be a universal set, E be a set of parameters, IU denote the set of all fuzzy subsets of U, and AE. The pair (F, A) is said to be a fuzzy soft set over U, where F is a mapping given by F : AIU.

Definition 2.3 ([12])

Let U be a universal set, E a set of parameters, and X = {p, q, r} be a set of experts. Also, let O be a set of opinions (O = {0 = disagree, 1 = agree}), P(U) represent the power set of U, Z = E×X×O, and AZ. The pair (F, A) is called a soft expert set over U, and the function F can be given by F : AP(U).

Definition 2.4 ([20])

Let U be a universal set, E be a set of parameters, and X = {p, q, r} be a set of experts. Also, let O be a set of opinions (O = {0 = disagree, 1 = agree}), IU represent the power set of U, Z = E×X×O, and AZ. The pair (F, A) is called a soft expert set over U, and the function F can be given by F : AIU.

Definition 2.5 ([29])

Let U be a universal set and E be a set of parameters. A pair (F, E)Λ is said to be an EFSS over U provided that IU are all fuzzy subsets of U. Furthermore, is a set of effective parameters, and Λ is the effective set over , where F is a mapping given by

F:EIU

and defined as

F(ei)Λ={ujμU(uj)Λ:ujUeiE},μU(uj)Λ={μU(uj)+[(1-μU(uj))kδΛuj(ak)A],         if         μU(uj)(0,1),μU(uj),         O.W.,

, and | | is the cardinality of .

Definition 3.1

Let U be a universal set, E be a set of parameters, X be a set of experts (agents), and O = {1 = agree, 0 = disagree} be a set of opinions. Assume that Z = E × X × O and YZ. A pair (F, Y )Λ is said to be an EFSES over U provided that IU are all fuzzy subsets of U. Furthermore, is a set of effective parameters, and Λ is the effective set over , where F is a mapping given by

F:YIU

and defined as

F(Z)Λ={ujμU(uj)Λ:ujU},μU(uj)Λ={μU(uj)+[(1-μU(uj))kδΛuj(ak)A],         if         μU(uj)(0,1),μU(uj),         O.W.,

, and | | is the cardinality of .

Definition 3.2

A pair (F, Y )Λ1 is said to be an agree-effective fuzzy soft expert set over U, where (F, Y )Λ1 is an effective fuzzy soft expert subset of (F, Y )Λ and is defined as follows:

(F,Y)Λ1={F1(β):βE×X×{1}}.

Definition 3.3

A pair (F, Y )Λ0 is said to be a disagree-effective fuzzy soft expert set over U, where (F, Y )Λ0 is an effective fuzzy soft expert subset of (F, Y )Λ and is defined as follows:

(F,Y)Λ0={F0(β):βE×X×{0}}.

Example 3.1

Let U = {u1, u2, u3} be a universal set. Let E = {e1, e2, ee3} be a set of parameters, where ei (i = 1, 2, 3) denotes the parameters. Suppose that X = {p, q, r} is a set of experts. Assume that the effective sets over for all {u1, u2, u3} are as follows:

Λ(u1)={a10.7,a20,a30.3,a41},Λ(u2)={a10.6,a20.3,a30.8,a40},Λ(u3)={a10.4,a21,a30.9,a41}.

Let the fuzzy soft expert set F be defined as

F(e1,p,1)={u10.5u20.4u30.6},F(e1,q,1)={u10.1u20.6u30.3},F(e1,r,1)={u10.8u20.7u30.4},F(e2,p,1)={u10.2u20.5u30.3},F(e2,q,1)={u10.9u20.4u30.1},F(e2,r,1)={u10.2u20.3u30.7},F(e3,p,1)={u10.4u20.5u30.2},F(e3,q,1)={u10.7u20.1u30.4},F(e3,r,1)={u10.5u20.6u30.3},F(e1,p,0)={u10.4u20.4u30.6},F(e1,q,0)={u10.3u20.2u30.7},F(e1,r,0)={u10.6u20.7u30.1},F(e2,p,0)={u10.9u20.3u30.5},F(e2,q,0)={u10.1u20.5u30.6},F(e2,r,0)={u10.9u20.8u30.6},F(e3,p,0)={u10.5u20.5u30.4},F(e3,q,0)={u10.2u20.2u30.3},F(e3,r,0)={u10.7u20.3u30.2}.

By applying the definition of EFSES, we obtain

FΛ(e1,p,1)=[u10.5+0.5[0.7+0+0.3+14],u20.4+0.6[0.6+0.3+0.8+04],u30.6+0.4[0.4+1+0.9+14]].

Therefore, FΛ(e1,p,1)={u10.75u20.65u30.93}. Using a similar procedure for all given fuzzy soft expert sets, we obtain the following EFSES:

(F,Z)Λ={((e1,p,1),{u10.75u20.65u30.93}),((e1,q,1),{u10.55u20.77u30.87}),((e1,r,1),{u10.9u20.82u30.89}),((e2,p,1),{u10.6u20.71u30.87}),((e2,q,1),{u10.95u20.65u30.84}),((e2,r,1),{u10.6u20.59u30.94}),((e3,p,1),{u10.7u20.71u30.86}),((e3,q,1),{u10.85u20.48u30.89}),((e3,r,1),{u10.75u20.77u30.87}),((e1,p,0),{u10.7u20.65u30.93}),((e1,q,0),{u10.65u20.54u30.94}),((e1,r,0),{u10.8u20.82u30.84}),((e2,p,0),{u10.95u20.59u30.91}),((e2,q,0),{u10.55u20.71u30.93}),((e2,r,0),{u10.95u20.88u30.93}),((e3,p,0),{u10.75u20.71u30.89}),((e3,q,0),{u10.6u20.54u30.87}),((e3,r,0),{u10.85u20.59u30.86})}.

Also, the agree-EFSES (F, Y )Λ1 is

(F,Z)Λ1={((e1,p,1),{u10.75u20.65u30.93}),((e1,q,1),{u10.55u20.77u30.87}),((e1,r,1),{u10.9u20.82u30.89}),((e2,p,1),{u10.6u20.71u30.87}),((e2,q,1),{u10.95u20.65u30.84}),((e2,r,1),{u10.6u20.59u30.94}),((e3,p,1),{u10.7u20.71u30.86}),((e3,q,1),{u10.85u20.48u30.89}),((e3,r,1),{u10.75u20.77u30.87})},

and disagree-EFSES (F, Y )Λ0 is

(F,Z)Λ0={((e1,p,0),{u10.7u20.65u30.93}),((e1,q,0),{u10.65u20.54u30.94}),((e1,r,0),{u10.8u20.82u30.84}),((e2,p,0),{u10.95u20.59u30.91}),((e2,q,0),{u10.55u20.71u30.93}),((e2,r,0),{u10.95u20.88u30.93}),((e3,p,0),{u10.75u20.71u30.89}),((e3,q,0),{u10.6u20.54u30.87}),((e3,r,0),{u10.85u20.59u30.86})}.

Definition 3.4

Let the pair (FΛ, Y ) be an EFSES. Also, let Λc denote any fuzzy complement of Λ and Fc denote the fuzzy soft expert complement of F. For the pair (FΛ, Y ), we have the following:

  • Totalcomplement is denoted by (FΛcc, Y ),

  • Soft Expertcomplement is denoted by (FΛc, Y ), and

  • Λcomplement is denoted by (FΛc, Y ).

Example 3.2

Consider Example 3.1. Let

Λ(u1)={a10.7,a20,a30.3,a41},Λ(u2)={a10.6,a20.3,a30.8,a40},Λ(u3)={a10.4,a21,a30.9,a41}.

By using the basic fuzzy complement of the effective set Λ, we obtain the following effective set Λc:

Λc(u1)={a10.3,a21,a30.7,a40},Λc(u2)={a10.4,a20.7,a30.2,a41},Λc(u3)={a10.6,a20,a30.1,a40}.

Also, let

F(e1,p,1)={u10.5u20.4u30.6},F(e1,q,1)={u10.1u20.6u30.3},F(e1,r,1)={u10.8u20.7u30.4},F(e2,p,1)={u10.2u20.5u30.3},F(e2,q,1)={u10.9u20.4u30.1},F(e2,r,1)={u10.2u20.3u30.7},F(e3,p,1)={u10.4u20.5u30.2},F(e3,q,1)={u10.7u20.1u30.4},F(e3,r,1)={u10.5u20.6u30.3},F(e1,p,0)={u10.4u20.4u30.6},F(e1,q,0)={u10.3u20.2u30.7},F(e1,r,0)={u10.6u20.7u30.1},F(e2,p,0)={u10.9u20.3u30.5},F(e2,q,0)={u10.1u20.5u30.6},F(e2,r,0)={u10.9u20.8u30.6},F(e3,p,0)={u10.5u20.5u30.4},F(e3,q,0)={u10.2u20.2u30.3},F(e3,r,0)={u10.7u20.3u30.2}.

Using the fuzzy soft expert complement, we have

Fc(e1,p,1)={u10.5u20.6u30.4},Fc(e1,q,1)={u10.9u20.4u30.7},Fc(e1,r,1)={u10.2u20.3u30.6},Fc(e2,p,1)={u10.8u20.5u30.7},Fc(e2,q,1)={u10.1u20.6u30.9},Fc(e2,r,1)={u10.8u20.7u30.3},Fc(e3,p,1)={u10.6u20.5u30.8},Fc(e3,q,1)={u10.3u20.9u30.6},Fc(e3,r,1)={u10.5u20.4u30.7},Fc(e1,p,0)={u10.6u20.6u30.4},Fc(e1,q,0)={u10.7u20.8u30.3},Fc(e1,r,0)={u10.4u20.3u30.9},Fc(e2,p,0)={u10.1u20.7u30.5},Fc(e2,q,0)={u10.9u20.5u30.4},Fc(e2,r,0)={u10.1u20.2u30.4},Fc(e3,p,0)={u10.5u20.5u30.6},Fc(e3,q,0)={u10.8u20.8u30.7},F(e3,r,0)={u10.3u20.7u30.8}.

We have the Totalcomplement as follows:

(Fc,Z)Λc={((e1,p,1),{u10.75u20.83u30.50}),((e1,q,1),{u10.95u20.74u30.75}),((e1,r,1),{u10.6u20.70u30.67}),((e2,p,1),{u10.9u20.78u30.75}),((e2,q,1),{u10.55u20.83u30.91}),((e2,r,1),{u10.9u20.87u30.42}),((e3,p,1),{u10.8u20.78u30.83}),((e3,q,1),{u10.65u20.95u30.67}),((e3,r,1),{u10.75u20.74u30.75}),((e1,p,0),{u10.8u20.83u30.50}),((e1,q,0),{u10.85u20.91u30.42}),((e1,r,0),{u10.7u20.70u30.91}),((e2,p,0),{u10.55u20.87u30.58}),((e2,q,0),{u10.95u20.78u30.50}),((e2,r,0),{u10.55u20.66u30.50}),((e3,p,0),{u10.75u20.78u30.67}),((e3,q,0),{u10.9u20.91u30.75}),((e3,r,0),{u10.65u20.87u30.83})}.

Also, we have the Soft Expertcomplement as follows:

(Fc,Z)Λ={((e1,p,1),{u10.75u20.77u30.89}),((e1,q,1),{u10.95u20.65u30.94}),((e1,r,1),{u10.6u20.59u30.93}),((e2,p,1),{u10.9u20.71u30.94}),((e2,q,1),{u10.55u20.77u30.98}),((e2,r,1),{u10.9u20.82u30.87}),((e3,p,1),{u10.8u20.71u30.96}),((e3,q,1),{u10.65u20.94u30.93}),((e3,r,1),{u10.75u20.65u30.94}),((e1,p,0),{u10.8u20.77u30.89}),((e1,q,0),{u10.85u20.88u30.87}),((e1,r,0),{u10.7u20.59u30.98}),((e2,p,0),{u10.55u20.82u30.91}),((e2,q,0),{u10.95u20.71u30.89}),((e2,r,0),{u10.55u20.54u30.89}),((e3,p,0),{u10.75u20.71u30.93}),((e3,q,0),{u10.9u20.88u30.94}),((e3,r,0),{u10.65u20.82u30.96})}.

In addition, we have the Λcomplement as follows:

(F,Z)Λc={((e1,p,1),{u10.75u20.74u30.67}),((e1,q,1),{u10.55u20.83u30.42}),((e1,r,1),{u10.9u20.87u30.50}),((e2,p,1),{u10.6u20.78u30.42}),((e2,q,1),{u10.95u20.74u30.25}),((e2,r,1),{u10.6u20.70u30.75}),((e3,p,1),{u10.7u20.78u30.34}),((e3,q,1),{u10.85u20.61u30.50}),((e3,r,1),{u10.75u20.83u30.42}),((e1,p,0),{u10.7u20.74u30.67}),((e1,q,0),{u10.65u20.66u30.75}),((e1,r,0),{u10.8u20.87u30.25}),((e2,p,0),{u10.95u20.70u30.58}),((e2,q,0),{u10.55u20.78u30.67}),((e2,r,0),{u10.95u20.91u30.67}),((e3,p,0),{u10.75u20.78u30.50}),((e3,q,0),{u10.6u20.66u30.42}),((e3,r,0),{u10.85u20.70u30.34})}.

Definition 3.5

Let (FΛ1, Y1) and (GΛ2, Y2) be two EFSESs over the common universe U. The union of (FΛ1, Y1) and (GΛ2, Y2) is denoted by (FΛ1,Y1)Λ(GΛ2,Y2), which is the EFSES (HΛs, Y ), where Y = Y1Y2, Λs = s1, Λ2) and ∀ ɛY

HΛs(ɛ)={FΛs(ɛ)if ɛY1-Y2,GΛs(ɛ)if ɛY2-Y1,(F˜G)Λs(ɛ)if ɛY1Y2,

where s is any s-norm, and ∪̃ is the fuzzy soft expert union.

Example 3.3

Let Λ1(u1)={a10.7,a20,a30.3,a41},Λ1(u2)={a10.6,a20.3,a30.8,a40} and Λ3(u3)={a10.4,a21,a30.9,a41}, and let Λ2(u1)={a10.6,a20.9,a30.8,a40.5},Λ2(u2)={a10.4,a20.7,a30.6,a41} and Λ2(u3)={a10.5,a20.8,a30.7,a40.6}. Also, let

(F,Y1)={((e1,p,1),{u10.7u20.8u30.9}),((e1,q,1),{u10.4u20.6u30.5}),((e1,r,1),{u10.9u20.7u30.3}),((e3,p,1),{u10.2u20.9u30.1}),((e3,q,1),{u10.1u20.3u30.8}),((e3,r,1),{u10.6u20.1u30.4}),((e1,p,0),{u10.3u20.6u30.6}),((e1,q,0),{u10.7u20.2u30.8}),((e1,r,0),{u10.9u20.5u30.6}),((e3,p,0),{u10.2u20.9u30.2}),((e3,q,0),{u10.4u20.5u30.7}),((e3,r,0),{u10.1u20.1u30.9})},(G,Y2)={((e1,p,1),{u10.8u20.7u30.6}),((e1,q,1),{u10.9u20.5u30.7}),((e1,r,1),{u10.4u20.8u30.5}),((e2,p,1),{u10.7u20.9u30.8}),((e2,q,1),{u10.5u20.6u30.9}),((e2,r,1),{u10.6u20.5u30.7}),((e1,p,0),{u10.8u20.4u30.7}),((e1,q,0),{u10.6u20.9u30.6}),((e1,r,0),{u10.7u20.6u30.8}),((e2,p,0),{u10.8u20.8u30.9}),((e2,q,0),{u10.9u20.7u30.9}),((e2,r,0),{u10.6u20.9u30.7}),}.

Using the basic fuzzy union, we obtain the following effective sets: Λs(u1)={a10.7,a20.9,a30.8,a41},Λs(u2)={a10.6,a20.7,a30.8,a41}, and Λs(u3)={a10.5,a21,a30.9,a41}. Moreover, using the fuzzy soft expert union, we obtain

(H,Y)={((e1,p,1),{u10.8u20.8u30.9}),((e1,q,1),{u10.9u20.6u30.7}),((e1,r,1),{u10.9u20.8u30.5}),((e2,p,1),{u10.7u20.9u30.8}),((e2,q,1),{u10.5u20.6u30.9}),((e2,r,1),{u10.6u20.5u30.7}),((e3,p,1),{u10.2u20.9u30.1}),((e3,q,1),{u10.1u20.3u30.8}),((e3,r,1),{u10.6u20.1u30.4}),((e1,p,0),{u10.8u20.6u30.7}),((e1,q,0),{u10.7u20.9u30.8}),((e1,r,0),{u10.9u20.6u30.8}),((e2,p,0),{u10.8u20.8u30.9}),((e2,q,0),{u10.9u20.7u30.9}),((e2,r,0),{u10.6u20.9u30.7}),((e3,p,0),{u10.2u20.9u30.2}),((e3,q,0),{u10.4u20.5u30.7}),((e3,r,0),{u10.1u20.1u30.9})}.

Thus, we have the following EFSES:

(HΛs,Z)={((e1,p,1),{u10.97u20.95u30.98}),((e1,q,1),{u10.98u20.91u30.95}),((e1,r,1),{u10.98u20.95u30.92}),((e2,p,1),{u10.95u20.97u30.97}),((e2,q,1),{u10.92u20.91u30.98}),((e2,r,1),{u10.94u20.88u30.95}),((e3,p,1),{u10.88u20.97u30.86}),((e3,q,1),{u10.86u20.84u30.97}),((e3,r,1),{u10.94u20.79u30.91}),((e1,p,0),{u10.97u20.91u30.95}),((e1,q,0),{u10.95u20.97u30.97}),((e1,r,0),{u10.98u20.91u30.97}),((e2,p,0),{u10.97u20.95u30.98}),((e2,q,0),{u10.98u20.93u30.98}),((e2,r,0),{u10.94u20.97u30.95}),((e3,p,0),{u10.88u20.97u30.88}),((e3,q,0),{u10.91u20.88u30.95}),((e3,r,0),{u10.86u20.79u30.98})}.

Proposition 3.1

If (FΛ1, Y1), (GΛ2, Y2), and (HΛ3, Y3) are three EFSESs over U, then

(FΛ1,Y1)Λ((GΛ2,Y2)Λ(HΛ3,Y3))=((FΛ1,Y1)Λ(GΛ2,Y2))Λ(HΛ3,Y3).

Proof

Let

((GΛ2,Y2)Λ(HΛ3,Y3))   (ɛ)=KΛs(ɛ)={GΛs(ɛ)if ɛY2-Y3,HΛs(ɛ)if ɛY3-Y2,(G˜H)Λs(ɛ)if ɛY2Y3,

where Λs=s(Λ2,Λ3).

Also, let

((FΛ1,Y1)ΛKΛs)   (ɛ)={FΛs(ɛ)if ɛY1-(Y2Y3),KΛs(ɛ)if ɛY2Y3)-Y1,(F˜K)Λs(ɛ)if ɛY1(Y2Y3),

where Λs = s1, Λ2, Λ3).

We consider the case in which ɛY1 ∩ (Y2Y3) because the other cases are trivial; thus,

((FΛ1,Y1)ΛKΛs)   (ɛ)=((F˜K)(ɛ),Y1(Y2Y3),Λs=s(Λ1,Λ2,Λ3)).

As the fuzzy soft expert union is an s-norm, we can write the above relation as follows:

((FΛ1,Y1)ΛKΛs)   (ɛ)=((s(F(ɛ),K(ɛ))),Y1(Y2Y3),Λs=s(Λ1,Λ2,Λ3)).

Furthermore, for (FΛ1,Y1)(ɛ)ΛKΛs(ɛ)

,

=(s(FΛ1(ɛ),s(G(ɛ),H(ɛ))),Y1(Y2Y3),Λs=s(Λ2,Λ3))=(s(s(f(ɛ),G(ɛ)),HΛ3(ɛ)),(Y1Y2)Y3,Λs=s(Λ1,Λ2))=((FΛ1,Y1)Λ(GΛ2,Y2))Λ(HΛ3,Y3).

Definition 3.6

Let (FΛ1, Y1) and (GΛ2, Y2) be two EFSESs over a common universe U. The intersection of (FΛ1, Y1) and (GΛ2, Y2) is denoted by (FΛ1,Y1)Λ(GΛ2,Y2), which is the EFSES (KΛt, Y ), where Y = Y1Y2, Λt = t1, Λ2) and ∀ ɛY

KΛt(ɛ)={FΛt(ɛ)if ɛY1-Y2,GΛt(ɛ)if ɛY2-Y1,(F˜G)Λt(ɛ)if ɛY1Y2,

where t is any t-norm, and ∩̃ is the fuzzy soft expert intersection.

Example 3.4

Consider Example 3.3. Using the basic fuzzy intersection, we obtain the following effective sets:

Λt(u1)={a10.6,a20.3,a30,a40.5},Λt(u2)={a10.4,a20.3,a30.6,a40},Λt(u3)={a10.4,a20.8,a30.7,a40.6}.

Moreover, by using the fuzzy soft expert intersection, we obtain

(K,Y)={((e1,p,1),{u10.7u20.7u30.6}),((e1,q,1),{u10.4u20.5u30.5}),((e1,r,1),{u10.4u20.7u30.3}),((e2,p,1),{u10.7u20.9u30.8}),((e2,q,1),{u10.5u20.6u30.9}),((e2,r,1),{u10.6u20.5u30.7}),((e3,p,1),{u10.2u20.9u30.1}),((e3,q,1),{u10.1u20.3u30.8}),((e3,r,1),{u10.6u20.1u30.4}),((e1,p,0),{u10.3u20.4u30.6}),((e1,q,0),{u10.6u20.2u30.6}),((e1,r,0),{u10.7u20.5u30.6}),((e2,p,0),{u10.8u20.8u30.9}),((e2,q,0),{u10.9u20.7u30.9}),((e2,r,0),{u10.6u20.9u30.7}),((e3,p,0),{u10.2u20.9u30.2}),((e3,q,0),{u10.4u20.5u30.7}),((e3,r,0),{u10.1u20.1u30.9})}.

Thus, we have the following EFSES:

(KΛt,Z)={((e1,p,1),{u10.80u20.79u30.85}),((e1,q,1),{u10.61u20.66u30.81}),((e1,r,1),{u10.61u20.79u30.73}),((e2,p,1),{u10.80u20.93u30.86}),((e2,q,1),{u10.67u20.73u30.96}),((e2,r,1),{u10.74u20.66u30.88}),((e3,p,1),{u10.48u20.93u30.66}),((e3,q,1),{u10.41u20.52u30.92}),((e3,r,1),{u10.74u20.39u30.77}),((e1,p,0),{u10.54u20.59u30.85}),((e1,q,0),{u10.74u20.46u30.85}),((e1,r,0),{u10.80u20.66u30.85}),((e2,p,0),{u10.87u20.86u30.96}),((e2,q,0),{u10.93u20.79u30.96}),((e2,r,0),{u10.74u20.93u30.88}),((e3,p,0),{u10.48u20.93u30.7}),((e3,q,0),{u10.61u20.66u30.88}),((e3,r,0),{u10.41u20.39u30.96})}.

Proposition 3.2

If (FΛ1, Y1), (GΛ2, Y2), and (HΛ3, Y3) are three EFSESs over U, then

(FΛ1,Y1)Λ((GΛ2,Y2)Λ(HΛ3,Y3))=((FΛ1,Y1)Λ(GΛ2,Y2))Λ(HΛ3,Y3).

Proof

Let

((GΛ2,Y2)Λ(HΛ3,Y3))   (ɛ)=KΛt(ɛ)={GΛt(ɛ)if ɛY2-Y3,HΛt(ɛ)if ɛY3-Y2,(G˜H)Λt(ɛ)if ɛY2Y3,

where Λt=t(Λ2,Λ3). Also, let

((FΛ1,Y1)ΛKΛt)   (ɛ)={FΛt(ɛ)if ɛY1-(Y2Y3),KΛt(ɛ)if ɛ(Y2Y3)-Y1,(F˜K)Λt(ɛ)if ɛY1(Y2Y3),

where Λt = t1, Λ2, Λ3). We consider the case when ɛY1 ∩ (Y2Y3) because the other cases are trivial; thus,

((FΛ1,Y1)ΛKΛt)   (ɛ)=((F˜K)   (ɛ),Y1(Y2Y3),Λt=t(Λ1,Λ2,Λ3)).

Because the fuzzy soft expert intersection is a t-norm, we can write the above relation as

((FΛ1,Y1)ΛKΛt)   (ɛ)=((t(F(ɛ),K(ɛ))),Y1(Y2Y3),Λt=t(Λ1,Λ2,Λ3)).

Furthermore, for (FΛ1,Y1)(ɛ)ΛKΛt(ɛ),

=(t(FΛ1(ɛ),t(G(ɛ),H(ɛ))),Y1(Y2Y3),Λt=t(Λ2,Λ3))=(t(t(F(ɛ),G(ɛ)),HΛ3(ɛ)),(Y1Y2)Y3,Λt=t(Λ1,Λ2))=((FΛ1,Y1)Λ(GΛ2,Y2))Λ(HΛ3,Y3).

Definition 3.7

Let (FΛ1, Y1) and (GΛ2, Y2) be two EFSESs over a common universe U. We denote (FΛ1, Y1) “AND” (GΛ2, Y2) by (FΛ1, Y1) ∧ (GΛ2, Y2), and we define it by

(KΛt,Y1×Y2)=(FΛ1,Y1)(GΛ2,Y2),

such that

KΛt(α,β)=(F(α)˜G(β))Λt;(α,β)Y1×Y2,

where t is any t-norm, Λt = t1, Λ2), and ∩̃ is the fuzzy soft expert intersection.

Example 3.5

Let

Λ1(u1)={a10.1,a20.4,a31,a40.7},Λ1(u2)={a10.4,a21,a30,a40.8},Λ3(u3)={a10.9,a20.6,a30.7,a41},

and

Λ2(u1)={a10.5,a20.3,a30.8,a40.9},Λ2(u2)={a10.3,a20.7,a30.2,a41},Λ2(u3)={a11,a20.9,a30,a40.3}.

Also, let

Y1={(e1,p,1),(e2,p,0),(e1,r,0),(e2,r,1)},Y2={(e1,p,1),(e1,r,0),(e2,r,1)}.

In addition, let

(F,Y1)={((e1,p,1),{u10.8u20.5u30.3}),((e1,p,0),{u10.7u20.4u30.6}),((e1,r,0),{u10.9u20.8u30.5}),((e2,p,0),{u10.5u20.2u30.9})},

and

(G,Y2)={((e1,p,1),{u10.6u20.5u30.4}),((e1,r,0),{u10.8u20.9u30.5}),((e2,r,1),{u10.5u20.3u30.8})}.

Using the fuzzy soft expert intersection, we have

(K,Y1×Y2)={((e1,p,1),(e1,p,1),{u10.6u20.5u30.3}),((e1,p,1),(e1,r,0),{u10.8u20.5u30.3}),((e1,p,1),(e2,r,1),{u10.5u20.3u30.3}),((e2,p,0),(e1,p,1),{u10.6u20.4u30.4}),((e2,p,0),(e1,r,0),{u10.7u20.4u30.5}),((e2,p,0),(e2,r,1),{u10.5u20.3u30.6}),((e1,r,0),(e1,p,1),{u10.6u20.5u30.4}),((e1,r,0),(e1,r,0),{u10.8u20.8u30.5}),((e1,r,0),(e2,r,1),{u10.5u20.3u30.5}),((e2,r,1),(e1,p,1),{u10.5u20.2u30.4}),((e2,r,1),(e1,r,0),{u10.5u20.2u30.5}),((e2,r,1),(e2,r,1),{u10.5u20.2u30.8})}.

Furthermore, using the basic fuzzy intersection, we have the following effective sets:

Λt(u1)={a10.1,a20.3,a30.8,a40.7},Λt(u2)={a10.3,a20.2,a30,a40.8},Λt(u3)={a10.9,a20.6,a30,a40.3}.

Thus, we have the following EFSES:

(KΛt,Y1×Y2)={((e1,p,1),(e1,p,1),{u10.79u20.66u30.61}),((e1,p,1),(e1,r,0),{u10.89u20.66u30.61}),((e1,p,1),(e2,r,1),{u10.73u20.52u30.61}),((e2,p,0),(e1,p,1),{u10.73u20.52u30.78}),((e2,p,0),(e1,r,0),{u10.79u20.66u30.67}),((e2,p,0),(e2,r,1),{u10.89u20.86u30.72}),((e1,r,0),(e1,p,1),{u10.73u20.52u30.72}),((e1,r,0),(e1,r,0),{u10.73u20.46u30.67}),((e1,r,0),(e2,r,1),{u10.73u20.52u30.72}),((e2,r,1),(e1,p,1),{u10.73u20.46u30.67}),((e2,r,0),(e2,r,0),{u10.73u20.46u30.72}),((e2,r,1),(e2,r,1),{u10.73u20.46u30.89})}.

Proposition 3.3

If (FΛ1, Y1), (GΛ2, Y2), and (HΛ3, Y3) are three EFSESs over U, then

(FΛ1,Y1)((GΛ2,Y2)(HΛ3,Y3))=((FΛ1,Y1)(GΛ2,Y2))(HΛ3,Y3).

Proof

Let

((GΛ2,Y2)(HΛ3,Y3))(α,β)=((G(α)~H(β))Λt)(α,β)Y2×Y3,Λt=t(Λ2,Λ3).

Because the fuzzy soft expert intersection is a t-norm, we can write the above relation as

((GΛ2,Y2)(HΛ3,Y3))(α,β)=(t(G(α),H(β)))(α,β)Y2×Y3,Λt=t(Λ2,Λ3).Thus,(FΛ1,Y1)(γ)((GΛ2,Y2)(HΛ3,Y3))(α,β)=(t(FΛ1(γ),t(G(α),H(β))),(γ,(α,β))Y1×(Y2×Y3),Λt=t(Λ2,Λ3))=(t(t(F(γ),G(α)),HΛ3(β)),((γ,α),β)(Y1×Y2)×Y3,Λt=t(Λ1,Λ2))=((FΛ1,Y1)(GΛ2,Y2))(HΛ3,Y3).

Definition 3.8

Let (FΛ1, Y1) and (GΛ2, Y2) be two EFSESs over a common universe U. We denote (FΛ1, Y1) “OR” (GΛ2, Y2) by (FΛ1, Y1) ∨ (GΛ2, Y2), and we define it by

(HΛs,Y1×Y2)=(FΛ1,Y1)(GΛ2,Y2),

such that

HΛs(α,β)=(F(α)˜G(β))Λs;(α,β)Y1×Y2,

where s is any s-norm, Λs = s1, Λ2), and ∪̃ is the fuzzy soft expert union.

Example 3.6

Consider Example 3.5. Using the fuzzy soft expert union, we obtain

(H,Y1×Y2)={((e1,p,1),(e1,p,1),{u10.8u20.5u30.4}),((e1,p,1),(e1,r,0),{u10.8u20.9u30.5}),((e1,p,1),(e2,r,1),{u10.8u20.5u30.8}),((e2,p,0),(e1,p,1),{u10.7u20.5u30.6}),((e2,p,0),(e1,r,0),{u10.8u20.9u30.6}),((e2,p,0),(e2,r,1),{u10.7u20.4u30.8}),((e1,r,0),(e1,p,1),{u10.9u20.8u30.5}),((e1,r,0),(e1,r,0),{u10.9u20.8u30.8}),((e1,r,0),(e2,r,1),{u10.9u20.8u30.8}),((e2,r,1),(e1,p,1),{u10.6u20.5u30.9}),((e2,r,1),(e1,r,0),{u10.8u20.9u30.9}),((e2,r,1),(e2,r,1),{u10.5u20.3u30.9})}.

Furthermore, using the basic fuzzy union, we have the following effective sets:

Λs(u1)={a10.5,a20.4,a31,a40.9},Λs(u2)={a10.4,a20.7,a31,a41},Λs(u3)={a11,a20.9,a30.7,a41}.

Thus, we have the following EFSES:

(HΛs,Y1×Y2)={((e1,p,1),(e1,p,1),{u10.94u20.88u30.94}),((e1,p,1),(e1,r,0),{u10.94u20.97u30.95}),((e1,p,1),(e2,r,1),{u10.94u20.88u30.98}),((e2,p,0),(e1,p,1),{u10.91u20.88u30.96}),((e2,p,0),(e1,r,0),{u10.94u20.97u30.96}),((e2,p,0),(e2,r,1),{u10.91u20.86u30.98}),((e1,r,0),(e1,p,1),{u10.97u20.95u30.95}),((e1,r,0),(e1,r,0),{u10.97u20.97u30.95}),((e1,r,0),(e2,r,1),{u10.97u20.95u30.98}),((e2,r,1),(e1,p,1),{u10.88u20.88u30.99}),((e2,r,0),(e2,r,0),{u10.94u20.97u30.99}),((e2,r,1),(e2,r,1),{u10.85u20.84u30.99})}.

Proposition 3.4

If (FΛ1, Y1), (GΛ2, Y2), and (HΛ3, Y3) are three EFSESs over U, then

(FΛ1,Y1)((GΛ2,Y2)(HΛ3,Y3))=((FΛ1,Y1)(GΛ2,Y2))(HΛ3,Y3).

Proof

Let

((GΛ2,Y2)(HΛ3,Y3))(α,β)=((G(α)˜H(β))Λs),(α,β)Y2×Y3,Λs=s(Λ2,Λ3).

Because the fuzzy soft expert union is an s-norm, we can write the above relation as

((GΛ2,Y2)(HΛ3,Y3))(α,β)=(s(G(α),H(β)),(α,β)Y2×Y3,Λs=s(Λ2,Λ3)).

Therefore,

(FΛ1,Y1)(γ)((GΛ2,Y2)(HΛ3,Y3))(α,β)=(s(FΛ1(γ),s(G(α),H(β))),(γ,(α,β))Y1×(Y2×Y3),Λs=s(Λ2,Λ3))=(s(s(F(γ),G(α)),HΛ3(β)),((γ,α),β)(Y1×Y2)×Y3,Λs=s(Λ1,Λ2))=((FΛ1,Y1)(GΛ2,Y2))(HΛ3,Y3).

In this section, we introduce an EFSES with a two-valued opinions theoretical approach to obtain a solution to some decision-making problems.

Example 4.1

Assume that a company wants to buy a car. Let U = {u1, u2, u3} be a set of model A5 Audi cars. There are four factories manufacturing this model in four countries and one of these factories is located in Germany, which is the main factory. Let E = {e1, e2, e3} be a the set of parameters, where e1 = Safety, e2 = Affordable, and e3 = Performance. Also, let X = {p, q, r} be the set of experts and be the set of effective parameters, where a1 = means all of its parts made in Germany, a2 = means it is reassembled in Germany, a3 = means salvaged, and a4 = means used for more than ten years. Let the effective sets over for all uiU be expressed as follows:

Λ1(u1)={a10.9,a20,a30.7,a40.5},Λ1(u2)={a10.8,a20.5,a31,a40.3},Λs(u3)={a11,a20.8,a30.4,a40},

and

Λ2(u1)={a10.7,a20.4,a31,a40.3},Λ2(u2)={a10.5,a20.2,a30.8,a40.8},Λ2(u3)={a10.9,a20.9,a30.1,a40.1}.

Furthermore, let

(F,Y1)={((e1,p,1),{u10.4u20.5u30.8}),((e1,q,1),{u10.6u20.7u30.9}),((e1,r,1),{u10.5u20.5u30.6}),((e3,p,1),{u10.9u20.8u30.7}),((e3,q,1),{u10.2u20.7u30.7}),((e3,r,1),{u10.5u20.3u30.1}),((e1,p,0),{u10.9u20.6u30.4}),((e1,q,0),{u10.7u20.1u30.5}),((e1,r,0),{u10.4u20.8u30.4}),((e3,p,0),{u10.8u20.5u30.8}),((e3,q,0),{u10.5u20.9u30.5}),((e3,r,0),{u10.1u20.2u30.9})},

and

(G,Y2)={((e1,p,1),{u10.3u20.2u30.9}),((e1,q,1),{u10.9u20.5u30.7}),((e1,r,1),{u10.3u20.5u30.8}),((e2,p,1),{u10.1u20.6u30.8}),((e2,q,1),{u10.7u20.7u30.1}),((e2,r,1),{u10.3u20.6u30.5}),((e1,p,0),{u10.6u20.2u30.6}),((e1,q,0),{u10.5u20.5u30.3}),((e1,r,0),{u10.8u20.5u30.6}),((e2,p,0),{u10.9u20.9u30.2}),((e2,q,0),{u10.1u20.4u30.9}),((e2,r,0),{u10.6u20.6u30.7})}.

We give a new algorithm that may be followed to buy the car:

  • Select the fuzzy soft expert sets (F, Y1) and (G, Y2).

  • Select the effective sets of parameters A.

  • Select the effective sets Λ1 and Λ1 over A for the fuzzy soft expert sets (F, Y1) and (G, Y2), respectively.

  • Compute the corresponding resultant fuzzy soft expert set either (H, Y ) or (K, Y ) from the fuzzy soft expert sets (F, Y1) and (G, Y2).

  • Compute the corresponding resultant effective set either Λs or Λt from the effective sets Λ1 and Λ2.

  • Compute the corresponding resultant EFSES either HΛs or KΛt.

  • Find C=j=1nuj for the agree-EFSES and place it in a tabular form.

  • Compute K=j=1nuj for the disagree-EFSES and place it in a tabular form.

  • Compute R = CK.

  • Compute m, for which sm = maxuU {R}. Then, sm is the optimal choice object. If m has more than one value, then any one of them could be chosen.

In addition, using the basic fuzzy union, we obtain the following effective sets:

Λs(u1)={a10.9,a20.4,a30.1,a40.5},Λs(u2)={a10.8,a20.5,a31,a40.8},Λs(u3)={a11,a20.9,a30.4,a40.1}.

Moreover, by using the fuzzy soft expert union, we obtain

(H,Y)={((e1,p,1),{u10.4u20.5u30.9}),((e1,q,1),{u10.9u20.7u30.9}),((e1,r,1),{u10.5u20.5u30.8}),((e2,p,1),{u10.1u20.6u30.8}),((e2,q,1),{u10.7u20.7u30.1}),((e2,r,1),{u10.3u20.6u30.5}),((e3,p,1),{u10.9u20.8u30.7}),((e3,q,1),{u10.2u20.7u30.7}),((e3,r,1),{u10.5u20.3u30.1}),((e1,p,0),{u10.9u20.6u30.6}),((e1,q,0),{u10.7u20.5u30.5}),((e1,r,0),{u10.8u20.8u30.6}),((e2,p,0),{u10.9u20.9u30.2}),((e2,q,0),{u10.1u20.4u30.9}),((e2,r,0),{u10.6u20.6u30.7}),((e3,p,0),{u10.8u20.5u30.8}),((e3,q,0),{u10.5u20.9u30.5}),((e3,r,0),{u10.1u20.2u30.9})},

Thus, we have the following EFSES:

(HΛs,Y)={((e1,p,1),{u10.82u20.88u30.96}),((e1,q,1),{u10.97u20.93u30.96}),((e1,r,1),{u10.85u20.88u30.92}),((e2,p,1),{u10.73u20.91u30.92}),((e2,q,1),{u10.91u20.93u30.64}),((e2,r,1),{u10.79u20.91u30.0.8}),((e3,p,1),{u10.97u20.95u30.88}),((e3,q,1),{u10.76u20.93u30.88}),((e3,r,1),{u10.85u20.84u30.64}),((e1,p,0),{u10.97u20.91u30.84}),((e1,q,0),{u10.91u20.88u30.8}),((e1,r,0),{u10.94u20.95u30.84}),((e2,p,0),{u10.97u20.97u30.68}),((e2,q,0),{u10.73u20.86u30.96}),((e2,r,0),{u10.88u20.91u30.88}),((e3,p,0),{u10.94u20.88u30.92}),((e3,q,0),{u10.85u20.97u30.8}),((e3,r,0),{u10.73u20.82u30.96})}.

In Table 1, we show the tabular representation of the agree-EFSES (HΛs, Y )1, and Table 2 shows the tabular representation of the disagree-EFSES (HΛs, Y )0. Finally, from Table 3 indicates that the maximum choice is 0.01. Therefore, the decision is to buy Car 2. Obviously, we can see that the convergence between some values in the effective sets affected the determination of the maximum choice in Table 3. Thus, more reasonable values for the effective sets may be conducive to obtaining a more precise choice.

We introduced the concept of EFSES theory as a new mathematical tool to deal with uncertainty. Furthermore, we presented some of its properties and defined its basic operations as complement, union, intersection, AND, and OR. In addition, we established a new algorithm to solve some decision-making problems. As a future direction, researchers can develop this concept into an effective neutrosophic vague soft expert set.

Table. 1.

Table 1. Tabular representation of the agree-EFSES (HΛs, Y )1.

Uu1u2u3
(e1, p, 1)0.820.880.96
(e1, q, 1)0.970.930.96
(e1, r, 1)0.850.880.92
(e2, p, 1)0.730.910.92
(e2, q, 1)0.910.930.64
(e2, r, 1)0.790.910.8
(e3, p, 1)0.970.950.88
(e3, q, 1)0.760.930.88
(e3, r, 1)0.850.840.64
C=j=1nujC1 = 7.65C2 = 8.16C3 = 7.6

Table. 2.

Table 2. Tabular representation of the disagree-EFSES (HΛs, Y)0.

Uu1u2u3
(e1, p, 0)0.970.910.84
(e1, q, 0)0.910.880.8
(e1, r, 0)0.940.950.84
(e2, p, 0)0.970.970.68
(e2, q, 0)0.730.860.96
(e2, r, 0)0.880.910.88
(e3, p, 0)0.940.880.92
(e3, q, 0)0.850.970.8
(e3, r, 0)0.730.820.96
K=j=1nujK1 = 7.92K2 = 8.15K3 = 7.68

Table. 3.

Table 3. Score R= CK.

UCKR
1u17.657.92−0.27
2u28.168.150.01
3u37.67.68−0.08

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Shawkat Alkhazaleh is a professor of Mathematics at Jadara University in Jordan. He received his M.A. and Ph.D. degrees from the National University of Malaysia (UKM). He specializes in fuzzy sets, soft fuzzy sets, and topics related to uncertainty and has conducted extensive research in this field. He is currently working as a dean for the Deanship of Scientific Research at Jadara University, in addition to his work as a faculty member at the College of Science and Information Technology. E-mail: s.alkhazaleh@jadara.edu.jo

Emadeddin Beshtawi obtained his Bachelor’s and Master’s degrees in Mathematics from Jadara University in 2020 and 2023, respectively. In the final phase of his studies for the Master’s degree, he completed his thesis in the field of fuzzy sets under the supervision of Prof. Dr. Shawkat Alkhazaleh. E-mail: bemadeddin@yahoo.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 192-204

Published online June 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.2.192

Copyright © The Korean Institute of Intelligent Systems.

Effective Fuzzy Soft Expert Set Theory and Its Applications

Shawkat Alkhazaleh and Emadeddin Beshtawi

Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid, Jordan

Correspondence to:Shawkat Alkhazaleh (shmk79@gmail.com)

Revised: April 25, 2023; Accepted: May 2, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In 2022, Alkhazaleh introduced the concept of the effective fuzzy soft set (EFSS) as a new mathematical tool to address uncertain problems in decision-making and medical diagnosis. The virtue of this concept is its adaptability to deal with uncertain problems involving external effects. However, some uncertain decision-making problems, especially those with external effects, must be judged by several experts. To this end, this paper extends the concept of EFSS to the concept of an effective fuzzy soft expert set (EFSES). In particular, we provide a basic definition of EFSES and study its basic operations of complement, union, intersection, AND, and OR. We also present some of its properties. Finally, we present a new algorithm for solving decision-making problems.

Keywords: Fuzzy set, Soft set, Soft expert set, Fuzzy soft set, Fuzzy soft expert set, Effective fuzzy soft set

1. Introduction

Many real-life problems involve uncertainty, which hampers classical methods to surmount Dicey problems of decision-making in economy, engineering, medicine, and other fields [1]. Zadeh [2] introduced the concept of fuzzy set theory as a mathematical tool for solving such problems. Furthermore, Zadeh [3] proposed the concept of an interval-valued fuzzy set, as the membership space consists of the set of all closed subintervals between 0 and 1. Atanassov [4] generalized the concept of fuzzy set to the concept of the intuitionistic fuzzy set, which shows both the degree of membership and degree of non-membership of each element in the universe using two functions. Furthermore, Molodtsov [5] presented the concept of soft set theory as a new approach to handling vague problems. In addition, Maji et al. [6] generalized the concept of soft set theory to a more general concept, the fuzzy soft set theory. Moreover, soft set theory has been applied to decision-making problems [7]. In addition, Maji et al. [8] studied soft set theory in detail and defined its basic operations of union, intersection, AND, and OR. Furthermore, Roy and Maji [9] applied the concept of fuzzy soft sets to decision-making problems. Additionally, Yang et al. [10] combined the concepts of an interval-valued fuzzy set and a soft set. Also, the concept of intuitionistic fuzzy sets has been extended to the concept of intuitionistic fuzzy soft sets [11]. In addition, Alkhazaleh and Salleh [12] introduced the concept of soft expert set theory, which enables the user to become familiar with the opinions of all experts in one model. Also, they presented some of its properties and defined its basic operations as union, intersection, AND, and OR.

Many difficult decision-making problems require multiple sets of universes and one set of parameters. Therefore, Alkhazaleh et al. [13] presented soft multisets theory. Furthermore, in [14], they introduced the concept of a possibility fuzzy soft set, which aims to define a fuzzy soft set by linking the possibility of each element in the universal set with the parameterization of the fuzzy sets. In addition, Alkhazaleh and Salleh [15] extended the concepts of a fuzzy set and soft multisets to the concept of fuzzy soft multisets. They introduced the concept of a generalized interval-valued fuzzy soft set, which shows both the membership degree and possibility degree of each element in the universe [16]. Bashir and Salleh [17] presented the concept of a fuzzy parameterized soft expert set, which essentially takes a subset of the set of fuzzy subsets of the set of parameters along with the set of experts and the set of opinions as a Cartesian product. Meanwhile, a combination of the fuzzy soft expert and possibility fuzzy soft set has been proposed [18]. The concept of a possibility fuzzy soft expert set involves defining a fuzzy soft expert set by binding the possibility of each element in a universal set with the parameterization of the fuzzy sets. Bashir et al. [19] extended the concept of the possibility fuzzy soft set to the possibility intuitionistic fuzzy soft set, which exposes the degree of membership and degree of possibility of each element in the universe. Additionally, the authors of [20] presented the concept of a fuzzy soft expert set as a combination of fuzzy and soft expert sets. Furthermore, Xu [21] presented the concept of a hesitant fuzzy set, which returns a subset of [0, 1] when applied to a fixed set X. Moreover, Broumi and Smarandache [22] extended the concept of possibility intuitionistic fuzzy soft sets to the concept of possibility intuitionistic fuzzy soft expert sets. In addition, [23] generalized the concept of a hesitant fuzzy set to the concept of a graded soft expert set and assumed a multivalued set of opinions. Furthermore, the concept of a neutrosophic set was proposed by [24], which consists of three functions and real standard or nonstandard subsets of the closed interval between 0 and 1. Subsequently, Maji [25] extended this concept to the concept of a neutrosophic soft set, Sahin et al. [26] introduced the concept of a neutrosophic soft expert set, Al-Quran and Hassan [27] introduced the concept of neutrosophic vague soft multisets, and Al-Sharqi et al. [28] presented the concept of an interval-valued complex neutrosophic soft set. Recently, Alkhazaleh [29] introduced the concept of effective fuzzy soft set (EFSS) theory and defined its basic operations as complement, union, and intersection. Moreover, he took the external effects into account in decision-making problems. In this study, we extended the concept of EFSS to effective fuzzy soft expert set (EFSES). In summary, the basic definition, operations, and properties of EFESS are presented. Finally, a new algorithm is established and applied to solve some decision-making problems. The merit of this extension in comparison with existing concepts, such as EFSS and fuzzy soft expert sets, is the combination of internal and external effects and the utilization of up to two opinions from each expert in solving decision-making problems.

2. Preliminaries

Definition 2.1 ([5])

A pair (F, E) is called a soft set (over U) iff F is a mapping of E into the set of all subsets of U.

Definition 2.2 ([6])

Let U be a universal set, E be a set of parameters, IU denote the set of all fuzzy subsets of U, and AE. The pair (F, A) is said to be a fuzzy soft set over U, where F is a mapping given by F : AIU.

Definition 2.3 ([12])

Let U be a universal set, E a set of parameters, and X = {p, q, r} be a set of experts. Also, let O be a set of opinions (O = {0 = disagree, 1 = agree}), P(U) represent the power set of U, Z = E×X×O, and AZ. The pair (F, A) is called a soft expert set over U, and the function F can be given by F : AP(U).

Definition 2.4 ([20])

Let U be a universal set, E be a set of parameters, and X = {p, q, r} be a set of experts. Also, let O be a set of opinions (O = {0 = disagree, 1 = agree}), IU represent the power set of U, Z = E×X×O, and AZ. The pair (F, A) is called a soft expert set over U, and the function F can be given by F : AIU.

Definition 2.5 ([29])

Let U be a universal set and E be a set of parameters. A pair (F, E)Λ is said to be an EFSS over U provided that IU are all fuzzy subsets of U. Furthermore, is a set of effective parameters, and Λ is the effective set over , where F is a mapping given by

F:EIU

and defined as

F(ei)Λ={ujμU(uj)Λ:ujUeiE},μU(uj)Λ={μU(uj)+[(1-μU(uj))kδΛuj(ak)A],         if         μU(uj)(0,1),μU(uj),         O.W.,

, and | | is the cardinality of .

3. Effective Fuzzy Soft Expert Set

Definition 3.1

Let U be a universal set, E be a set of parameters, X be a set of experts (agents), and O = {1 = agree, 0 = disagree} be a set of opinions. Assume that Z = E × X × O and YZ. A pair (F, Y )Λ is said to be an EFSES over U provided that IU are all fuzzy subsets of U. Furthermore, is a set of effective parameters, and Λ is the effective set over , where F is a mapping given by

F:YIU

and defined as

F(Z)Λ={ujμU(uj)Λ:ujU},μU(uj)Λ={μU(uj)+[(1-μU(uj))kδΛuj(ak)A],         if         μU(uj)(0,1),μU(uj),         O.W.,

, and | | is the cardinality of .

Definition 3.2

A pair (F, Y )Λ1 is said to be an agree-effective fuzzy soft expert set over U, where (F, Y )Λ1 is an effective fuzzy soft expert subset of (F, Y )Λ and is defined as follows:

(F,Y)Λ1={F1(β):βE×X×{1}}.

Definition 3.3

A pair (F, Y )Λ0 is said to be a disagree-effective fuzzy soft expert set over U, where (F, Y )Λ0 is an effective fuzzy soft expert subset of (F, Y )Λ and is defined as follows:

(F,Y)Λ0={F0(β):βE×X×{0}}.

Example 3.1

Let U = {u1, u2, u3} be a universal set. Let E = {e1, e2, ee3} be a set of parameters, where ei (i = 1, 2, 3) denotes the parameters. Suppose that X = {p, q, r} is a set of experts. Assume that the effective sets over for all {u1, u2, u3} are as follows:

Λ(u1)={a10.7,a20,a30.3,a41},Λ(u2)={a10.6,a20.3,a30.8,a40},Λ(u3)={a10.4,a21,a30.9,a41}.

Let the fuzzy soft expert set F be defined as

F(e1,p,1)={u10.5u20.4u30.6},F(e1,q,1)={u10.1u20.6u30.3},F(e1,r,1)={u10.8u20.7u30.4},F(e2,p,1)={u10.2u20.5u30.3},F(e2,q,1)={u10.9u20.4u30.1},F(e2,r,1)={u10.2u20.3u30.7},F(e3,p,1)={u10.4u20.5u30.2},F(e3,q,1)={u10.7u20.1u30.4},F(e3,r,1)={u10.5u20.6u30.3},F(e1,p,0)={u10.4u20.4u30.6},F(e1,q,0)={u10.3u20.2u30.7},F(e1,r,0)={u10.6u20.7u30.1},F(e2,p,0)={u10.9u20.3u30.5},F(e2,q,0)={u10.1u20.5u30.6},F(e2,r,0)={u10.9u20.8u30.6},F(e3,p,0)={u10.5u20.5u30.4},F(e3,q,0)={u10.2u20.2u30.3},F(e3,r,0)={u10.7u20.3u30.2}.

By applying the definition of EFSES, we obtain

FΛ(e1,p,1)=[u10.5+0.5[0.7+0+0.3+14],u20.4+0.6[0.6+0.3+0.8+04],u30.6+0.4[0.4+1+0.9+14]].

Therefore, FΛ(e1,p,1)={u10.75u20.65u30.93}. Using a similar procedure for all given fuzzy soft expert sets, we obtain the following EFSES:

(F,Z)Λ={((e1,p,1),{u10.75u20.65u30.93}),((e1,q,1),{u10.55u20.77u30.87}),((e1,r,1),{u10.9u20.82u30.89}),((e2,p,1),{u10.6u20.71u30.87}),((e2,q,1),{u10.95u20.65u30.84}),((e2,r,1),{u10.6u20.59u30.94}),((e3,p,1),{u10.7u20.71u30.86}),((e3,q,1),{u10.85u20.48u30.89}),((e3,r,1),{u10.75u20.77u30.87}),((e1,p,0),{u10.7u20.65u30.93}),((e1,q,0),{u10.65u20.54u30.94}),((e1,r,0),{u10.8u20.82u30.84}),((e2,p,0),{u10.95u20.59u30.91}),((e2,q,0),{u10.55u20.71u30.93}),((e2,r,0),{u10.95u20.88u30.93}),((e3,p,0),{u10.75u20.71u30.89}),((e3,q,0),{u10.6u20.54u30.87}),((e3,r,0),{u10.85u20.59u30.86})}.

Also, the agree-EFSES (F, Y )Λ1 is

(F,Z)Λ1={((e1,p,1),{u10.75u20.65u30.93}),((e1,q,1),{u10.55u20.77u30.87}),((e1,r,1),{u10.9u20.82u30.89}),((e2,p,1),{u10.6u20.71u30.87}),((e2,q,1),{u10.95u20.65u30.84}),((e2,r,1),{u10.6u20.59u30.94}),((e3,p,1),{u10.7u20.71u30.86}),((e3,q,1),{u10.85u20.48u30.89}),((e3,r,1),{u10.75u20.77u30.87})},

and disagree-EFSES (F, Y )Λ0 is

(F,Z)Λ0={((e1,p,0),{u10.7u20.65u30.93}),((e1,q,0),{u10.65u20.54u30.94}),((e1,r,0),{u10.8u20.82u30.84}),((e2,p,0),{u10.95u20.59u30.91}),((e2,q,0),{u10.55u20.71u30.93}),((e2,r,0),{u10.95u20.88u30.93}),((e3,p,0),{u10.75u20.71u30.89}),((e3,q,0),{u10.6u20.54u30.87}),((e3,r,0),{u10.85u20.59u30.86})}.

Definition 3.4

Let the pair (FΛ, Y ) be an EFSES. Also, let Λc denote any fuzzy complement of Λ and Fc denote the fuzzy soft expert complement of F. For the pair (FΛ, Y ), we have the following:

  • Totalcomplement is denoted by (FΛcc, Y ),

  • Soft Expertcomplement is denoted by (FΛc, Y ), and

  • Λcomplement is denoted by (FΛc, Y ).

Example 3.2

Consider Example 3.1. Let

Λ(u1)={a10.7,a20,a30.3,a41},Λ(u2)={a10.6,a20.3,a30.8,a40},Λ(u3)={a10.4,a21,a30.9,a41}.

By using the basic fuzzy complement of the effective set Λ, we obtain the following effective set Λc:

Λc(u1)={a10.3,a21,a30.7,a40},Λc(u2)={a10.4,a20.7,a30.2,a41},Λc(u3)={a10.6,a20,a30.1,a40}.

Also, let

F(e1,p,1)={u10.5u20.4u30.6},F(e1,q,1)={u10.1u20.6u30.3},F(e1,r,1)={u10.8u20.7u30.4},F(e2,p,1)={u10.2u20.5u30.3},F(e2,q,1)={u10.9u20.4u30.1},F(e2,r,1)={u10.2u20.3u30.7},F(e3,p,1)={u10.4u20.5u30.2},F(e3,q,1)={u10.7u20.1u30.4},F(e3,r,1)={u10.5u20.6u30.3},F(e1,p,0)={u10.4u20.4u30.6},F(e1,q,0)={u10.3u20.2u30.7},F(e1,r,0)={u10.6u20.7u30.1},F(e2,p,0)={u10.9u20.3u30.5},F(e2,q,0)={u10.1u20.5u30.6},F(e2,r,0)={u10.9u20.8u30.6},F(e3,p,0)={u10.5u20.5u30.4},F(e3,q,0)={u10.2u20.2u30.3},F(e3,r,0)={u10.7u20.3u30.2}.

Using the fuzzy soft expert complement, we have

Fc(e1,p,1)={u10.5u20.6u30.4},Fc(e1,q,1)={u10.9u20.4u30.7},Fc(e1,r,1)={u10.2u20.3u30.6},Fc(e2,p,1)={u10.8u20.5u30.7},Fc(e2,q,1)={u10.1u20.6u30.9},Fc(e2,r,1)={u10.8u20.7u30.3},Fc(e3,p,1)={u10.6u20.5u30.8},Fc(e3,q,1)={u10.3u20.9u30.6},Fc(e3,r,1)={u10.5u20.4u30.7},Fc(e1,p,0)={u10.6u20.6u30.4},Fc(e1,q,0)={u10.7u20.8u30.3},Fc(e1,r,0)={u10.4u20.3u30.9},Fc(e2,p,0)={u10.1u20.7u30.5},Fc(e2,q,0)={u10.9u20.5u30.4},Fc(e2,r,0)={u10.1u20.2u30.4},Fc(e3,p,0)={u10.5u20.5u30.6},Fc(e3,q,0)={u10.8u20.8u30.7},F(e3,r,0)={u10.3u20.7u30.8}.

We have the Totalcomplement as follows:

(Fc,Z)Λc={((e1,p,1),{u10.75u20.83u30.50}),((e1,q,1),{u10.95u20.74u30.75}),((e1,r,1),{u10.6u20.70u30.67}),((e2,p,1),{u10.9u20.78u30.75}),((e2,q,1),{u10.55u20.83u30.91}),((e2,r,1),{u10.9u20.87u30.42}),((e3,p,1),{u10.8u20.78u30.83}),((e3,q,1),{u10.65u20.95u30.67}),((e3,r,1),{u10.75u20.74u30.75}),((e1,p,0),{u10.8u20.83u30.50}),((e1,q,0),{u10.85u20.91u30.42}),((e1,r,0),{u10.7u20.70u30.91}),((e2,p,0),{u10.55u20.87u30.58}),((e2,q,0),{u10.95u20.78u30.50}),((e2,r,0),{u10.55u20.66u30.50}),((e3,p,0),{u10.75u20.78u30.67}),((e3,q,0),{u10.9u20.91u30.75}),((e3,r,0),{u10.65u20.87u30.83})}.

Also, we have the Soft Expertcomplement as follows:

(Fc,Z)Λ={((e1,p,1),{u10.75u20.77u30.89}),((e1,q,1),{u10.95u20.65u30.94}),((e1,r,1),{u10.6u20.59u30.93}),((e2,p,1),{u10.9u20.71u30.94}),((e2,q,1),{u10.55u20.77u30.98}),((e2,r,1),{u10.9u20.82u30.87}),((e3,p,1),{u10.8u20.71u30.96}),((e3,q,1),{u10.65u20.94u30.93}),((e3,r,1),{u10.75u20.65u30.94}),((e1,p,0),{u10.8u20.77u30.89}),((e1,q,0),{u10.85u20.88u30.87}),((e1,r,0),{u10.7u20.59u30.98}),((e2,p,0),{u10.55u20.82u30.91}),((e2,q,0),{u10.95u20.71u30.89}),((e2,r,0),{u10.55u20.54u30.89}),((e3,p,0),{u10.75u20.71u30.93}),((e3,q,0),{u10.9u20.88u30.94}),((e3,r,0),{u10.65u20.82u30.96})}.

In addition, we have the Λcomplement as follows:

(F,Z)Λc={((e1,p,1),{u10.75u20.74u30.67}),((e1,q,1),{u10.55u20.83u30.42}),((e1,r,1),{u10.9u20.87u30.50}),((e2,p,1),{u10.6u20.78u30.42}),((e2,q,1),{u10.95u20.74u30.25}),((e2,r,1),{u10.6u20.70u30.75}),((e3,p,1),{u10.7u20.78u30.34}),((e3,q,1),{u10.85u20.61u30.50}),((e3,r,1),{u10.75u20.83u30.42}),((e1,p,0),{u10.7u20.74u30.67}),((e1,q,0),{u10.65u20.66u30.75}),((e1,r,0),{u10.8u20.87u30.25}),((e2,p,0),{u10.95u20.70u30.58}),((e2,q,0),{u10.55u20.78u30.67}),((e2,r,0),{u10.95u20.91u30.67}),((e3,p,0),{u10.75u20.78u30.50}),((e3,q,0),{u10.6u20.66u30.42}),((e3,r,0),{u10.85u20.70u30.34})}.

Definition 3.5

Let (FΛ1, Y1) and (GΛ2, Y2) be two EFSESs over the common universe U. The union of (FΛ1, Y1) and (GΛ2, Y2) is denoted by (FΛ1,Y1)Λ(GΛ2,Y2), which is the EFSES (HΛs, Y ), where Y = Y1Y2, Λs = s1, Λ2) and ∀ ɛY

HΛs(ɛ)={FΛs(ɛ)if ɛY1-Y2,GΛs(ɛ)if ɛY2-Y1,(F˜G)Λs(ɛ)if ɛY1Y2,

where s is any s-norm, and ∪̃ is the fuzzy soft expert union.

Example 3.3

Let Λ1(u1)={a10.7,a20,a30.3,a41},Λ1(u2)={a10.6,a20.3,a30.8,a40} and Λ3(u3)={a10.4,a21,a30.9,a41}, and let Λ2(u1)={a10.6,a20.9,a30.8,a40.5},Λ2(u2)={a10.4,a20.7,a30.6,a41} and Λ2(u3)={a10.5,a20.8,a30.7,a40.6}. Also, let

(F,Y1)={((e1,p,1),{u10.7u20.8u30.9}),((e1,q,1),{u10.4u20.6u30.5}),((e1,r,1),{u10.9u20.7u30.3}),((e3,p,1),{u10.2u20.9u30.1}),((e3,q,1),{u10.1u20.3u30.8}),((e3,r,1),{u10.6u20.1u30.4}),((e1,p,0),{u10.3u20.6u30.6}),((e1,q,0),{u10.7u20.2u30.8}),((e1,r,0),{u10.9u20.5u30.6}),((e3,p,0),{u10.2u20.9u30.2}),((e3,q,0),{u10.4u20.5u30.7}),((e3,r,0),{u10.1u20.1u30.9})},(G,Y2)={((e1,p,1),{u10.8u20.7u30.6}),((e1,q,1),{u10.9u20.5u30.7}),((e1,r,1),{u10.4u20.8u30.5}),((e2,p,1),{u10.7u20.9u30.8}),((e2,q,1),{u10.5u20.6u30.9}),((e2,r,1),{u10.6u20.5u30.7}),((e1,p,0),{u10.8u20.4u30.7}),((e1,q,0),{u10.6u20.9u30.6}),((e1,r,0),{u10.7u20.6u30.8}),((e2,p,0),{u10.8u20.8u30.9}),((e2,q,0),{u10.9u20.7u30.9}),((e2,r,0),{u10.6u20.9u30.7}),}.

Using the basic fuzzy union, we obtain the following effective sets: Λs(u1)={a10.7,a20.9,a30.8,a41},Λs(u2)={a10.6,a20.7,a30.8,a41}, and Λs(u3)={a10.5,a21,a30.9,a41}. Moreover, using the fuzzy soft expert union, we obtain

(H,Y)={((e1,p,1),{u10.8u20.8u30.9}),((e1,q,1),{u10.9u20.6u30.7}),((e1,r,1),{u10.9u20.8u30.5}),((e2,p,1),{u10.7u20.9u30.8}),((e2,q,1),{u10.5u20.6u30.9}),((e2,r,1),{u10.6u20.5u30.7}),((e3,p,1),{u10.2u20.9u30.1}),((e3,q,1),{u10.1u20.3u30.8}),((e3,r,1),{u10.6u20.1u30.4}),((e1,p,0),{u10.8u20.6u30.7}),((e1,q,0),{u10.7u20.9u30.8}),((e1,r,0),{u10.9u20.6u30.8}),((e2,p,0),{u10.8u20.8u30.9}),((e2,q,0),{u10.9u20.7u30.9}),((e2,r,0),{u10.6u20.9u30.7}),((e3,p,0),{u10.2u20.9u30.2}),((e3,q,0),{u10.4u20.5u30.7}),((e3,r,0),{u10.1u20.1u30.9})}.

Thus, we have the following EFSES:

(HΛs,Z)={((e1,p,1),{u10.97u20.95u30.98}),((e1,q,1),{u10.98u20.91u30.95}),((e1,r,1),{u10.98u20.95u30.92}),((e2,p,1),{u10.95u20.97u30.97}),((e2,q,1),{u10.92u20.91u30.98}),((e2,r,1),{u10.94u20.88u30.95}),((e3,p,1),{u10.88u20.97u30.86}),((e3,q,1),{u10.86u20.84u30.97}),((e3,r,1),{u10.94u20.79u30.91}),((e1,p,0),{u10.97u20.91u30.95}),((e1,q,0),{u10.95u20.97u30.97}),((e1,r,0),{u10.98u20.91u30.97}),((e2,p,0),{u10.97u20.95u30.98}),((e2,q,0),{u10.98u20.93u30.98}),((e2,r,0),{u10.94u20.97u30.95}),((e3,p,0),{u10.88u20.97u30.88}),((e3,q,0),{u10.91u20.88u30.95}),((e3,r,0),{u10.86u20.79u30.98})}.

Proposition 3.1

If (FΛ1, Y1), (GΛ2, Y2), and (HΛ3, Y3) are three EFSESs over U, then

(FΛ1,Y1)Λ((GΛ2,Y2)Λ(HΛ3,Y3))=((FΛ1,Y1)Λ(GΛ2,Y2))Λ(HΛ3,Y3).

Proof

Let

((GΛ2,Y2)Λ(HΛ3,Y3))   (ɛ)=KΛs(ɛ)={GΛs(ɛ)if ɛY2-Y3,HΛs(ɛ)if ɛY3-Y2,(G˜H)Λs(ɛ)if ɛY2Y3,

where Λs=s(Λ2,Λ3).

Also, let

((FΛ1,Y1)ΛKΛs)   (ɛ)={FΛs(ɛ)if ɛY1-(Y2Y3),KΛs(ɛ)if ɛY2Y3)-Y1,(F˜K)Λs(ɛ)if ɛY1(Y2Y3),

where Λs = s1, Λ2, Λ3).

We consider the case in which ɛY1 ∩ (Y2Y3) because the other cases are trivial; thus,

((FΛ1,Y1)ΛKΛs)   (ɛ)=((F˜K)(ɛ),Y1(Y2Y3),Λs=s(Λ1,Λ2,Λ3)).

As the fuzzy soft expert union is an s-norm, we can write the above relation as follows:

((FΛ1,Y1)ΛKΛs)   (ɛ)=((s(F(ɛ),K(ɛ))),Y1(Y2Y3),Λs=s(Λ1,Λ2,Λ3)).

Furthermore, for (FΛ1,Y1)(ɛ)ΛKΛs(ɛ)

,

=(s(FΛ1(ɛ),s(G(ɛ),H(ɛ))),Y1(Y2Y3),Λs=s(Λ2,Λ3))=(s(s(f(ɛ),G(ɛ)),HΛ3(ɛ)),(Y1Y2)Y3,Λs=s(Λ1,Λ2))=((FΛ1,Y1)Λ(GΛ2,Y2))Λ(HΛ3,Y3).

Definition 3.6

Let (FΛ1, Y1) and (GΛ2, Y2) be two EFSESs over a common universe U. The intersection of (FΛ1, Y1) and (GΛ2, Y2) is denoted by (FΛ1,Y1)Λ(GΛ2,Y2), which is the EFSES (KΛt, Y ), where Y = Y1Y2, Λt = t1, Λ2) and ∀ ɛY

KΛt(ɛ)={FΛt(ɛ)if ɛY1-Y2,GΛt(ɛ)if ɛY2-Y1,(F˜G)Λt(ɛ)if ɛY1Y2,

where t is any t-norm, and ∩̃ is the fuzzy soft expert intersection.

Example 3.4

Consider Example 3.3. Using the basic fuzzy intersection, we obtain the following effective sets:

Λt(u1)={a10.6,a20.3,a30,a40.5},Λt(u2)={a10.4,a20.3,a30.6,a40},Λt(u3)={a10.4,a20.8,a30.7,a40.6}.

Moreover, by using the fuzzy soft expert intersection, we obtain

(K,Y)={((e1,p,1),{u10.7u20.7u30.6}),((e1,q,1),{u10.4u20.5u30.5}),((e1,r,1),{u10.4u20.7u30.3}),((e2,p,1),{u10.7u20.9u30.8}),((e2,q,1),{u10.5u20.6u30.9}),((e2,r,1),{u10.6u20.5u30.7}),((e3,p,1),{u10.2u20.9u30.1}),((e3,q,1),{u10.1u20.3u30.8}),((e3,r,1),{u10.6u20.1u30.4}),((e1,p,0),{u10.3u20.4u30.6}),((e1,q,0),{u10.6u20.2u30.6}),((e1,r,0),{u10.7u20.5u30.6}),((e2,p,0),{u10.8u20.8u30.9}),((e2,q,0),{u10.9u20.7u30.9}),((e2,r,0),{u10.6u20.9u30.7}),((e3,p,0),{u10.2u20.9u30.2}),((e3,q,0),{u10.4u20.5u30.7}),((e3,r,0),{u10.1u20.1u30.9})}.

Thus, we have the following EFSES:

(KΛt,Z)={((e1,p,1),{u10.80u20.79u30.85}),((e1,q,1),{u10.61u20.66u30.81}),((e1,r,1),{u10.61u20.79u30.73}),((e2,p,1),{u10.80u20.93u30.86}),((e2,q,1),{u10.67u20.73u30.96}),((e2,r,1),{u10.74u20.66u30.88}),((e3,p,1),{u10.48u20.93u30.66}),((e3,q,1),{u10.41u20.52u30.92}),((e3,r,1),{u10.74u20.39u30.77}),((e1,p,0),{u10.54u20.59u30.85}),((e1,q,0),{u10.74u20.46u30.85}),((e1,r,0),{u10.80u20.66u30.85}),((e2,p,0),{u10.87u20.86u30.96}),((e2,q,0),{u10.93u20.79u30.96}),((e2,r,0),{u10.74u20.93u30.88}),((e3,p,0),{u10.48u20.93u30.7}),((e3,q,0),{u10.61u20.66u30.88}),((e3,r,0),{u10.41u20.39u30.96})}.

Proposition 3.2

If (FΛ1, Y1), (GΛ2, Y2), and (HΛ3, Y3) are three EFSESs over U, then

(FΛ1,Y1)Λ((GΛ2,Y2)Λ(HΛ3,Y3))=((FΛ1,Y1)Λ(GΛ2,Y2))Λ(HΛ3,Y3).

Proof

Let

((GΛ2,Y2)Λ(HΛ3,Y3))   (ɛ)=KΛt(ɛ)={GΛt(ɛ)if ɛY2-Y3,HΛt(ɛ)if ɛY3-Y2,(G˜H)Λt(ɛ)if ɛY2Y3,

where Λt=t(Λ2,Λ3). Also, let

((FΛ1,Y1)ΛKΛt)   (ɛ)={FΛt(ɛ)if ɛY1-(Y2Y3),KΛt(ɛ)if ɛ(Y2Y3)-Y1,(F˜K)Λt(ɛ)if ɛY1(Y2Y3),

where Λt = t1, Λ2, Λ3). We consider the case when ɛY1 ∩ (Y2Y3) because the other cases are trivial; thus,

((FΛ1,Y1)ΛKΛt)   (ɛ)=((F˜K)   (ɛ),Y1(Y2Y3),Λt=t(Λ1,Λ2,Λ3)).

Because the fuzzy soft expert intersection is a t-norm, we can write the above relation as

((FΛ1,Y1)ΛKΛt)   (ɛ)=((t(F(ɛ),K(ɛ))),Y1(Y2Y3),Λt=t(Λ1,Λ2,Λ3)).

Furthermore, for (FΛ1,Y1)(ɛ)ΛKΛt(ɛ),

=(t(FΛ1(ɛ),t(G(ɛ),H(ɛ))),Y1(Y2Y3),Λt=t(Λ2,Λ3))=(t(t(F(ɛ),G(ɛ)),HΛ3(ɛ)),(Y1Y2)Y3,Λt=t(Λ1,Λ2))=((FΛ1,Y1)Λ(GΛ2,Y2))Λ(HΛ3,Y3).

Definition 3.7

Let (FΛ1, Y1) and (GΛ2, Y2) be two EFSESs over a common universe U. We denote (FΛ1, Y1) “AND” (GΛ2, Y2) by (FΛ1, Y1) ∧ (GΛ2, Y2), and we define it by

(KΛt,Y1×Y2)=(FΛ1,Y1)(GΛ2,Y2),

such that

KΛt(α,β)=(F(α)˜G(β))Λt;(α,β)Y1×Y2,

where t is any t-norm, Λt = t1, Λ2), and ∩̃ is the fuzzy soft expert intersection.

Example 3.5

Let

Λ1(u1)={a10.1,a20.4,a31,a40.7},Λ1(u2)={a10.4,a21,a30,a40.8},Λ3(u3)={a10.9,a20.6,a30.7,a41},

and

Λ2(u1)={a10.5,a20.3,a30.8,a40.9},Λ2(u2)={a10.3,a20.7,a30.2,a41},Λ2(u3)={a11,a20.9,a30,a40.3}.

Also, let

Y1={(e1,p,1),(e2,p,0),(e1,r,0),(e2,r,1)},Y2={(e1,p,1),(e1,r,0),(e2,r,1)}.

In addition, let

(F,Y1)={((e1,p,1),{u10.8u20.5u30.3}),((e1,p,0),{u10.7u20.4u30.6}),((e1,r,0),{u10.9u20.8u30.5}),((e2,p,0),{u10.5u20.2u30.9})},

and

(G,Y2)={((e1,p,1),{u10.6u20.5u30.4}),((e1,r,0),{u10.8u20.9u30.5}),((e2,r,1),{u10.5u20.3u30.8})}.

Using the fuzzy soft expert intersection, we have

(K,Y1×Y2)={((e1,p,1),(e1,p,1),{u10.6u20.5u30.3}),((e1,p,1),(e1,r,0),{u10.8u20.5u30.3}),((e1,p,1),(e2,r,1),{u10.5u20.3u30.3}),((e2,p,0),(e1,p,1),{u10.6u20.4u30.4}),((e2,p,0),(e1,r,0),{u10.7u20.4u30.5}),((e2,p,0),(e2,r,1),{u10.5u20.3u30.6}),((e1,r,0),(e1,p,1),{u10.6u20.5u30.4}),((e1,r,0),(e1,r,0),{u10.8u20.8u30.5}),((e1,r,0),(e2,r,1),{u10.5u20.3u30.5}),((e2,r,1),(e1,p,1),{u10.5u20.2u30.4}),((e2,r,1),(e1,r,0),{u10.5u20.2u30.5}),((e2,r,1),(e2,r,1),{u10.5u20.2u30.8})}.

Furthermore, using the basic fuzzy intersection, we have the following effective sets:

Λt(u1)={a10.1,a20.3,a30.8,a40.7},Λt(u2)={a10.3,a20.2,a30,a40.8},Λt(u3)={a10.9,a20.6,a30,a40.3}.

Thus, we have the following EFSES:

(KΛt,Y1×Y2)={((e1,p,1),(e1,p,1),{u10.79u20.66u30.61}),((e1,p,1),(e1,r,0),{u10.89u20.66u30.61}),((e1,p,1),(e2,r,1),{u10.73u20.52u30.61}),((e2,p,0),(e1,p,1),{u10.73u20.52u30.78}),((e2,p,0),(e1,r,0),{u10.79u20.66u30.67}),((e2,p,0),(e2,r,1),{u10.89u20.86u30.72}),((e1,r,0),(e1,p,1),{u10.73u20.52u30.72}),((e1,r,0),(e1,r,0),{u10.73u20.46u30.67}),((e1,r,0),(e2,r,1),{u10.73u20.52u30.72}),((e2,r,1),(e1,p,1),{u10.73u20.46u30.67}),((e2,r,0),(e2,r,0),{u10.73u20.46u30.72}),((e2,r,1),(e2,r,1),{u10.73u20.46u30.89})}.

Proposition 3.3

If (FΛ1, Y1), (GΛ2, Y2), and (HΛ3, Y3) are three EFSESs over U, then

(FΛ1,Y1)((GΛ2,Y2)(HΛ3,Y3))=((FΛ1,Y1)(GΛ2,Y2))(HΛ3,Y3).

Proof

Let

((GΛ2,Y2)(HΛ3,Y3))(α,β)=((G(α)~H(β))Λt)(α,β)Y2×Y3,Λt=t(Λ2,Λ3).

Because the fuzzy soft expert intersection is a t-norm, we can write the above relation as

((GΛ2,Y2)(HΛ3,Y3))(α,β)=(t(G(α),H(β)))(α,β)Y2×Y3,Λt=t(Λ2,Λ3).Thus,(FΛ1,Y1)(γ)((GΛ2,Y2)(HΛ3,Y3))(α,β)=(t(FΛ1(γ),t(G(α),H(β))),(γ,(α,β))Y1×(Y2×Y3),Λt=t(Λ2,Λ3))=(t(t(F(γ),G(α)),HΛ3(β)),((γ,α),β)(Y1×Y2)×Y3,Λt=t(Λ1,Λ2))=((FΛ1,Y1)(GΛ2,Y2))(HΛ3,Y3).

Definition 3.8

Let (FΛ1, Y1) and (GΛ2, Y2) be two EFSESs over a common universe U. We denote (FΛ1, Y1) “OR” (GΛ2, Y2) by (FΛ1, Y1) ∨ (GΛ2, Y2), and we define it by

(HΛs,Y1×Y2)=(FΛ1,Y1)(GΛ2,Y2),

such that

HΛs(α,β)=(F(α)˜G(β))Λs;(α,β)Y1×Y2,

where s is any s-norm, Λs = s1, Λ2), and ∪̃ is the fuzzy soft expert union.

Example 3.6

Consider Example 3.5. Using the fuzzy soft expert union, we obtain

(H,Y1×Y2)={((e1,p,1),(e1,p,1),{u10.8u20.5u30.4}),((e1,p,1),(e1,r,0),{u10.8u20.9u30.5}),((e1,p,1),(e2,r,1),{u10.8u20.5u30.8}),((e2,p,0),(e1,p,1),{u10.7u20.5u30.6}),((e2,p,0),(e1,r,0),{u10.8u20.9u30.6}),((e2,p,0),(e2,r,1),{u10.7u20.4u30.8}),((e1,r,0),(e1,p,1),{u10.9u20.8u30.5}),((e1,r,0),(e1,r,0),{u10.9u20.8u30.8}),((e1,r,0),(e2,r,1),{u10.9u20.8u30.8}),((e2,r,1),(e1,p,1),{u10.6u20.5u30.9}),((e2,r,1),(e1,r,0),{u10.8u20.9u30.9}),((e2,r,1),(e2,r,1),{u10.5u20.3u30.9})}.

Furthermore, using the basic fuzzy union, we have the following effective sets:

Λs(u1)={a10.5,a20.4,a31,a40.9},Λs(u2)={a10.4,a20.7,a31,a41},Λs(u3)={a11,a20.9,a30.7,a41}.

Thus, we have the following EFSES:

(HΛs,Y1×Y2)={((e1,p,1),(e1,p,1),{u10.94u20.88u30.94}),((e1,p,1),(e1,r,0),{u10.94u20.97u30.95}),((e1,p,1),(e2,r,1),{u10.94u20.88u30.98}),((e2,p,0),(e1,p,1),{u10.91u20.88u30.96}),((e2,p,0),(e1,r,0),{u10.94u20.97u30.96}),((e2,p,0),(e2,r,1),{u10.91u20.86u30.98}),((e1,r,0),(e1,p,1),{u10.97u20.95u30.95}),((e1,r,0),(e1,r,0),{u10.97u20.97u30.95}),((e1,r,0),(e2,r,1),{u10.97u20.95u30.98}),((e2,r,1),(e1,p,1),{u10.88u20.88u30.99}),((e2,r,0),(e2,r,0),{u10.94u20.97u30.99}),((e2,r,1),(e2,r,1),{u10.85u20.84u30.99})}.

Proposition 3.4

If (FΛ1, Y1), (GΛ2, Y2), and (HΛ3, Y3) are three EFSESs over U, then

(FΛ1,Y1)((GΛ2,Y2)(HΛ3,Y3))=((FΛ1,Y1)(GΛ2,Y2))(HΛ3,Y3).

Proof

Let

((GΛ2,Y2)(HΛ3,Y3))(α,β)=((G(α)˜H(β))Λs),(α,β)Y2×Y3,Λs=s(Λ2,Λ3).

Because the fuzzy soft expert union is an s-norm, we can write the above relation as

((GΛ2,Y2)(HΛ3,Y3))(α,β)=(s(G(α),H(β)),(α,β)Y2×Y3,Λs=s(Λ2,Λ3)).

Therefore,

(FΛ1,Y1)(γ)((GΛ2,Y2)(HΛ3,Y3))(α,β)=(s(FΛ1(γ),s(G(α),H(β))),(γ,(α,β))Y1×(Y2×Y3),Λs=s(Λ2,Λ3))=(s(s(F(γ),G(α)),HΛ3(β)),((γ,α),β)(Y1×Y2)×Y3,Λs=s(Λ1,Λ2))=((FΛ1,Y1)(GΛ2,Y2))(HΛ3,Y3).

4. Applications

In this section, we introduce an EFSES with a two-valued opinions theoretical approach to obtain a solution to some decision-making problems.

Example 4.1

Assume that a company wants to buy a car. Let U = {u1, u2, u3} be a set of model A5 Audi cars. There are four factories manufacturing this model in four countries and one of these factories is located in Germany, which is the main factory. Let E = {e1, e2, e3} be a the set of parameters, where e1 = Safety, e2 = Affordable, and e3 = Performance. Also, let X = {p, q, r} be the set of experts and be the set of effective parameters, where a1 = means all of its parts made in Germany, a2 = means it is reassembled in Germany, a3 = means salvaged, and a4 = means used for more than ten years. Let the effective sets over for all uiU be expressed as follows:

Λ1(u1)={a10.9,a20,a30.7,a40.5},Λ1(u2)={a10.8,a20.5,a31,a40.3},Λs(u3)={a11,a20.8,a30.4,a40},

and

Λ2(u1)={a10.7,a20.4,a31,a40.3},Λ2(u2)={a10.5,a20.2,a30.8,a40.8},Λ2(u3)={a10.9,a20.9,a30.1,a40.1}.

Furthermore, let

(F,Y1)={((e1,p,1),{u10.4u20.5u30.8}),((e1,q,1),{u10.6u20.7u30.9}),((e1,r,1),{u10.5u20.5u30.6}),((e3,p,1),{u10.9u20.8u30.7}),((e3,q,1),{u10.2u20.7u30.7}),((e3,r,1),{u10.5u20.3u30.1}),((e1,p,0),{u10.9u20.6u30.4}),((e1,q,0),{u10.7u20.1u30.5}),((e1,r,0),{u10.4u20.8u30.4}),((e3,p,0),{u10.8u20.5u30.8}),((e3,q,0),{u10.5u20.9u30.5}),((e3,r,0),{u10.1u20.2u30.9})},

and

(G,Y2)={((e1,p,1),{u10.3u20.2u30.9}),((e1,q,1),{u10.9u20.5u30.7}),((e1,r,1),{u10.3u20.5u30.8}),((e2,p,1),{u