Effective Fuzzy Soft Expert Set Theory and Its Applications

Shawkat Alkhazaleh; Emadeddin Beshtawi

doi:10.5391/IJFIS.2023.23.2.192

Original Article

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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

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

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Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

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

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Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

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

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Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

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, I^U denote the set of all fuzzy subsets of U, and A ⊆ E. The pair (F, A) is said to be a fuzzy soft set over U, where F is a mapping given by F : A → I^U.

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 A ⊆ Z. The pair (F, A) is called a soft expert set over U, and the function F can be given by F : A → P(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}), I^U represent the power set of U, Z = E×X×O, and A ⊆ Z. The pair (F, A) is called a soft expert set over U, and the function F can be given by F : A → I^U.

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 I^U 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:E→IU

and defined as

F(ei)Λ={ujμU(uj)Λ:uj∈U ei∈E},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

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Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

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 Y ⊆ Z. A pair (F, Y )_Λ is said to be an EFSES over U provided that I^U 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:Y→IU

and defined as

F(Z)Λ={ujμU(uj)Λ:uj∈U},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):β∈E×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):β∈E×X×{0}}.

Example 3.1

Let U = {u₁, u₂, u₃} be a universal set. Let E = {e₁, e₂, e_e_₃} be a set of parameters, where e_i (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 (F_Λ, Y ) be an EFSES. Also, let Λ^c denote any fuzzy complement of Λ and F^c denote the fuzzy soft expert complement of F. For the pair (F_Λ, Y ), we have the following:

Total_complement is denoted by (FΛcc, Y ),
Soft – Expert_complement is denoted by (FΛc, Y ), and
Λ_complement is denoted by (F_Λ^c, 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 Λ^c:

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

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 Total_complement 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 – Expert_complement 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 Λ_complement 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 (F_Λ₁, Y₁) and (G_Λ₂, Y₂) be two EFSESs over the common universe U. The union of (F_Λ₁, Y₁) and (G_Λ₂, Y₂) is denoted by (FΛ1,Y1)∪Λ(GΛ2,Y2), which is the EFSES (H_{Λ_s}, Y ), where Y = Y₁ ∪ Y₂, Λ_s = s(Λ₁, Λ₂) and ∀ ɛ ∈ Y

HΛs(ɛ)={FΛs(ɛ)if ɛ∈Y1-Y2,GΛs(ɛ)if ɛ∈Y2-Y1,(F∪˜G)Λs(ɛ)if ɛ∈Y1∩Y2,

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 (F_Λ₁, Y₁), (G_Λ₂, Y₂), and (H_Λ₃, 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 ɛ∈Y2-Y3,HΛs′(ɛ)if ɛ∈Y3-Y2,(G∪˜H)Λs′(ɛ)if ɛ∈Y2∩Y3,

where Λs′=s(Λ2,Λ3).

Also, let

((FΛ1,Y1) ∪ΛKΛs′) (ɛ)={FΛs(ɛ)if ɛ∈Y1-(Y2∩Y3),KΛs(ɛ)if ɛ∈Y2∩Y3)-Y1,(F∪˜K)Λs(ɛ)if ɛ∈Y1∩(Y2∩Y3),

where Λ_s = s(Λ₁, Λ₂, Λ₃).

We consider the case in which ɛ ∈ Y₁ ∩ (Y₂ ∩ Y₃) because the other cases are trivial; thus,

((FΛ1,Y1) ∪ΛKΛs′) (ɛ)=((F∪˜K)(ɛ),Y1∪(Y2∪Y3), 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∪(Y2∪Y3),Λs=s(Λ1,Λ2,Λ3)).

Furthermore, for (FΛ1,Y1) (ɛ)∪ΛKΛs′(ɛ)

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

Definition 3.6

Let (F_Λ₁, Y₁) and (G_Λ₂, Y₂) be two EFSESs over a common universe U. The intersection of (F_Λ₁, Y₁) and (G_Λ₂, Y₂) is denoted by (FΛ1,Y1)∩Λ(GΛ2,Y2), which is the EFSES (K_{Λ_t}, Y ), where Y = Y₁ ∪ Y₂, Λ_t = t(Λ₁, Λ₂) and ∀ ɛ ∈ Y

KΛt(ɛ)={FΛt(ɛ)if ɛ∈Y1-Y2,GΛt(ɛ)if ɛ∈Y2-Y1,(F∩˜G)Λt(ɛ)if ɛ∈Y1∩Y2,

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 (F_Λ₁, Y₁), (G_Λ₂, Y₂), and (H_Λ₃, 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 ɛ∈Y2-Y3,HΛt′(ɛ)if ɛ∈Y3-Y2,(G∩˜H)Λt′(ɛ)if ɛ∈Y2∩Y3,

where Λt′=t(Λ2,Λ3). Also, let

((FΛ1,Y1) ∩ΛKΛt′) (ɛ)={FΛt(ɛ)if ɛ∈Y1-(Y2∩Y3),KΛt(ɛ)if ɛ∈(Y2∩Y3)-Y1,(F∩˜K)Λt(ɛ)if ɛ∈Y1∩(Y2∩Y3),

where Λ_t = t(Λ₁, Λ₂, Λ₃). We consider the case when ɛ ∈ Y₁ ∩ (Y₂ ∩ Y₃) because the other cases are trivial; thus,

((FΛ1,Y1) ∩ΛKΛt′) (ɛ)=((F∩˜K) (ɛ),Y1∪(Y2∪Y3),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∪(Y2∪Y3),Λt=t(Λ1,Λ2,Λ3)).

Furthermore, for (FΛ1,Y1) (ɛ) ∩ΛKΛt′(ɛ),

=(t(FΛ1(ɛ),t(G(ɛ),H(ɛ))),Y1∪(Y2∪Y3), Λt′=t(Λ2,Λ3))=(t(t(F(ɛ),G(ɛ)),HΛ3(ɛ)),(Y1∪Y2)∪Y3, Λt″=t(Λ1,Λ2))=((FΛ1,Y1) ∩Λ(GΛ2,Y2)) ∩Λ(HΛ3,Y3).

Definition 3.7

Let (F_Λ₁, Y₁) and (G_Λ₂, Y₂) be two EFSESs over a common universe U. We denote (F_Λ₁, Y₁) “AND” (G_Λ₂, Y₂) by (F_Λ₁, Y₁) ∧ (G_Λ₂, 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, Λ_t = t(Λ₁, 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 (F_Λ₁, Y₁), (G_Λ₂, Y₂), and (H_Λ₃, 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 (F_Λ₁, Y₁) and (G_Λ₂, Y₂) be two EFSESs over a common universe U. We denote (F_Λ₁, Y₁) “OR” (G_Λ₂, Y₂) by (F_Λ₁, Y₁) ∨ (G_Λ₂, 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, Λ_s = s(Λ₁, 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 (F_Λ₁, Y₁), (G_Λ₂, Y₂), and (H_Λ₃, 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

Go to

Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

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 u_i ∈ U 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 Λ_s or Λ_t from the effective sets Λ₁ and Λ₂.
Compute the corresponding resultant EFSES either H_{Λ_s} or K_{Λ_t}.
Find Cℓ=∑j=1nujℓ for the agree-EFSES and place it in a tabular form.
Compute Kℓ=∑j=1nujℓ for the disagree-EFSES and place it in a tabular form.
Compute R_ℓ = C_ℓ– K_ℓ.
Compute m, for which s_m = max_u_{_ℓ∈}_U {R}. Then, s_m 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 (H_{Λ_s}, Y )₁, and Table 2 shows the tabular representation of the disagree-EFSES (H_{Λ_s}, 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.

5. Conclusion

Go to

Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

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.

Acknowledgments

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Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

The authors would like to acknowledge the financial support received from Jadara University.

Conflict of Interest

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Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

No potential conflict of interest relevant to this article was reported.

Tables

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Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

Table. 1.

Table 1. Tabular representation of the agree-EFSES (H_{Λ_s}, Y )₁.

U	u₁	u₂	u₃
(e₁, p, 1)	0.82	0.88	0.96
(e₁, q, 1)	0.97	0.93	0.96
(e₁, r, 1)	0.85	0.88	0.92
(e₂, p, 1)	0.73	0.91	0.92
(e₂, q, 1)	0.91	0.93	0.64
(e₂, r, 1)	0.79	0.91	0.8
(e₃, p, 1)	0.97	0.95	0.88
(e₃, q, 1)	0.76	0.93	0.88
(e₃, r, 1)	0.85	0.84	0.64
Cℓ=∑j=1nujℓ	C₁ = 7.65	C₂ = 8.16	C₃ = 7.6

Table. 2.

Table 2. Tabular representation of the disagree-EFSES (H_{Λ_s}, Y)₀.

U	u₁	u₂	u₃
(e₁, p, 0)	0.97	0.91	0.84
(e₁, q, 0)	0.91	0.88	0.8
(e₁, r, 0)	0.94	0.95	0.84
(e₂, p, 0)	0.97	0.97	0.68
(e₂, q, 0)	0.73	0.86	0.96
(e₂, r, 0)	0.88	0.91	0.88
(e₃, p, 0)	0.94	0.88	0.92
(e₃, q, 0)	0.85	0.97	0.8
(e₃, r, 0)	0.73	0.82	0.96
Kℓ=∑j=1nujℓ	K₁ = 7.92	K₂ = 8.15	K₃ = 7.68

Table. 3.

Table 3. Score R_ℓ= C_ℓ– K_ℓ.

ℓ	U	C_ℓ	K_ℓ	R_ℓ
1	u₁	7.65	7.92	−0.27
2	u₂	8.16	8.15	0.01
3	u₃	7.6	7.68	−0.08

References

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Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

Singh, H, Gupta, MM, Meitzler, T, Hou, ZG, Garg, KK, Solo, AM, and Zadeh, LA (2013). Real-life applications of fuzzy logic. Advances in Fuzzy Systems. 2013. article no. 581879
Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Zadeh, LA (1975). The concept of a linguistic variable and its application to approximate reasoning—I. Information Sciences. 8, 199-249.
Atanassov, KT (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 20, 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3
Molodtsov, D (1999). Soft set theory—first results. Computers & Mathematics with Applications. 37, 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5
Maji, PK, Biswas, R, and Roy, AR (2001). Fuzzy soft sets. Journal of Fuzzy Mathematics. 9, 589-602.
Maji, PK, Roy, AR, and Biswas, R (2002). An application of soft sets in a decision making problem. Computers & Mathematics with Applications. 44, 1077-1083.
Maji, PK, Biswas, R, and Roy, AR (2003). Soft set theory. Computers & Mathematics with Applications. 45, 555-562. https://doi.org/10.1016/S0898-1221(03)00016-6
Roy, AR, and Maji, PK (2007). A fuzzy soft set theoretic approach to decision making problems. Journal of Computational and Applied Mathematics. 203, 412-418. https://doi.org/10.1016/j.cam.2006.04.008
Yang, X, Lin, TY, Yang, J, Li, Y, and Yu, D (2009). Combination of interval-valued fuzzy set and soft set. Computers & Mathematics with Applications. 58, 521-527. https://doi.org/10.1016/j.camwa.2009.04.019
Xu, YJ, Sun, YK, and Li, DF . Intuitionistic fuzzy soft set., Proceedings of 2010 2nd International Workshop on Intelligent Systems and Applications, 2010, Wuhan, China, Array, pp.1-4. https://doi.org/10.1109/IWISA.2010.5473444
Alkhazaleh, S, and Salleh, AR (2011). Soft expert sets. Advances in Decision Sciences. 2011. article no. 757868
Alkhazaleh, S, Salleh, AR, and Hassan, N (2011). Soft multi-sets theory. Applied Mathematical Sciences. 5, 3561-3573.
Alkhazaleh, S, Salleh, AR, and Hassan, N (2011). Possibility fuzzy soft set. Advances in Decision Sciences. 2011. article no. 479756
Alkhazaleh, S, and Salleh, AR (2012). Fuzzy soft multiset theory. Abstract and Applied Analysis. 2012. article no. 350603
Alkhazaleh, S, and Salleh, AR (2012). Generalised interval-valued fuzzy soft set. Journal of Applied Mathematics. 2012. article no. 870504
Bashir, M, and Salleh, AR (2012). Fuzzy parameterized soft expert set. Abstract and Applied Analysis. 2012. article no. 258361
Bashir, M, and Salleh, AR (2012). Possibility fuzzy soft expert set. Open Journal of Applied Sciences. 2, 208-211. http://dx.doi.org/10.4236/ojapps.2012.24B047
Bashir, M, Salleh, AR, and Alkhazaleh, S (2012). Possibility intuitionistic fuzzy soft set. Advances in Decision Sciences. 2012. article no. 404325
Alkhazaleh, S, and Salleh, AR (2014). Fuzzy soft expert set and its application. Applied Mathematics. 5, 1349. http://dx.doi.org/10.4236/am.2014.59127
Xu, Z (2014). Hesitant Fuzzy Sets Theory. Cham, Switzerland: Springer https://doi.org/10.1007/978-3-319-04711-9
Broumi, S, and Smarandache, F (2015). Intuitionistic fuzzy soft expert sets and its application in decision making. Journal of New Theory. 2015, 89-105.
Qayyum, A, and Shaheen, T (2020). Graded soft expert set as a generalization of hesitant fuzzy set. Journal of Intelligent Systems. 29, 223-236. https://doi.org/10.1515/jisys-2016-0312
Smarandache, F . Neutrosophic set-a generalization of the intuitionistic fuzzy set., Proceedings of 2006 IEEE International Conference on Granular Computing, 2005, Atlanta, GA, Array, pp.38-42. https://doi.org/10.1109/GRC.2006.1635754
Maji, PK (2013). Neutrosophic soft set. Annals of Fuzzy Mathematics and Informatics. 5, 157-168.
Sahin, M, Alkhazaleh, S, and Ulucay, V (2015). Neutrosophic soft expert sets. Applied Mathematics. 6, 116-127. http://dx.doi.org/10.4236/am.2015.61012
Al-Quran, A, and Hassan, N (2018). Neutrosophic vague soft multiset for decision under uncertainty. Songklanakarin Journal of Science and Technology. 40, 290-305. https://doi.org/10.14456/sjst-psu.2018.53
Al-Sharqi, F, Al-Quran, A, Ahmad, AG, and Broumi, S (2021). Interval-valued complex neutrosophic soft set and its applications in decision-making. Neutrosophic Sets and Systems. 40, 149-168.
Alkhazaleh, S (2022). Effective fuzzy soft set theory and its applications. Applied Computational Intelligence and Soft Computing. 2022. article no. 6469745

Biographies

Go to

Abstract
1. Introduction
2. Preliminaries
3. Effective Fuzzy Soft Expert Set
4. Applications
5. Conclusion
Acknowledgments
Conflict of Interest
Tables
References
Biographies

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

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

Abstract

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

1. Introduction

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])

Definition 2.3 ([12])

Definition 2.4 ([20])

Definition 2.5 ([29])

F : E \to I^{U}

and defined as

\begin{array}{l} F {(e_{i})}_{Λ} = {\frac{u_{j}}{μ_{U} {(u_{j})}_{Λ}} : u_{j} \in U e_{i} \in E}, \\ μ_{U} {(u_{j})}_{Λ} = {\begin{array}{l} μ_{U} (u_{j}) + & [\frac{(1 - μ_{U} (u_{j})) \sum_{k} δ_{Λ_{u_{j}}} (a_{k})}{∣ A ∣}], \\ if μ_{U} (u_{j}) \in (0, 1), \\ μ_{U} (u_{j}), & O.W., \end{array} \end{array}

, and | | is the cardinality of .

3. Effective Fuzzy Soft Expert Set

Definition 3.1

F : Y \to I^{U}

and defined as

\begin{array}{l} F {(Z)}_{Λ} = {\frac{u_{j}}{μ_{U} {(u_{j})}_{Λ}} : u_{j} \in U}, \\ μ_{U} {(u_{j})}_{Λ} = {\begin{array}{l} μ_{U} (u_{j}) + & [\frac{(1 - μ_{U} (u_{j})) \sum_{k} δ_{Λ_{u_{j}}} (a_{k})}{∣ A ∣}], \\ if μ_{U} (u_{j}) \in (0, 1), \\ μ_{U} (u_{j}), & O.W., \end{array} \end{array}

, 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)}_{Λ_{1}} = {F_{1} (β) : β \in E \times X \times {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)}_{Λ_{0}} = {F_{0} (β) : β \in E \times X \times {0}} .

Example 3.1

\begin{array}{l} Λ (u_{1}) = {\frac{a_{1}}{0.7}, \frac{a_{2}}{0}, \frac{a_{3}}{0.3}, \frac{a_{4}}{1}}, Λ (u_{2}) = {\frac{a_{1}}{0.6}, \frac{a_{2}}{0.3}, \frac{a_{3}}{0.8}, \frac{a_{4}}{0}}, \\ Λ (u_{3}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{1}, \frac{a_{3}}{0.9}, \frac{a_{4}}{1}} . \end{array}

Let the fuzzy soft expert set F be defined as

\begin{array}{l} F (e_{1}, p, 1) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.4} \frac{u_{3}}{0.6}}, F (e_{1}, q, 1) = {\frac{u_{1}}{0.1} \frac{u_{2}}{0.6} \frac{u_{3}}{0.3}}, \\ F (e_{1}, r, 1) = {\frac{u_{1}}{0.8} \frac{u_{2}}{0.7} \frac{u_{3}}{0.4}}, F (e_{2}, p, 1) = {\frac{u_{1}}{0.2} \frac{u_{2}}{0.5} \frac{u_{3}}{0.3}}, \\ F (e_{2}, q, 1) = {\frac{u_{1}}{0.9} \frac{u_{2}}{0.4} \frac{u_{3}}{0.1}}, F (e_{2}, r, 1) = {\frac{u_{1}}{0.2} \frac{u_{2}}{0.3} \frac{u_{3}}{0.7}}, \\ F (e_{3}, p, 1) = {\frac{u_{1}}{0.4} \frac{u_{2}}{0.5} \frac{u_{3}}{0.2}}, F (e_{3}, q, 1) = {\frac{u_{1}}{0.7} \frac{u_{2}}{0.1} \frac{u_{3}}{0.4}}, \\ F (e_{3}, r, 1) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.6} \frac{u_{3}}{0.3}}, F (e_{1}, p, 0) = {\frac{u_{1}}{0.4} \frac{u_{2}}{0.4} \frac{u_{3}}{0.6}}, \\ F (e_{1}, q, 0) = {\frac{u_{1}}{0.3} \frac{u_{2}}{0.2} \frac{u_{3}}{0.7}}, F (e_{1}, r, 0) = {\frac{u_{1}}{0.6} \frac{u_{2}}{0.7} \frac{u_{3}}{0.1}}, \\ F (e_{2}, p, 0) = {\frac{u_{1}}{0.9} \frac{u_{2}}{0.3} \frac{u_{3}}{0.5}}, F (e_{2}, q, 0) = {\frac{u_{1}}{0.1} \frac{u_{2}}{0.5} \frac{u_{3}}{0.6}}, \\ F (e_{2}, r, 0) = {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.6}}, F (e_{3}, p, 0) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.5} \frac{u_{3}}{0.4}}, \\ F (e_{3}, q, 0) = {\frac{u_{1}}{0.2} \frac{u_{2}}{0.2} \frac{u_{3}}{0.3}}, F (e_{3}, r, 0) = {\frac{u_{1}}{0.7} \frac{u_{2}}{0.3} \frac{u_{3}}{0.2}} . \end{array}

By applying the definition of EFSES, we obtain

F_{Λ} (e_{1}, p, 1) = [\frac{u_{1}}{0.5 + 0.5 [\frac{0.7 + 0 + 0.3 + 1}{4}]}, \frac{u_{2}}{0.4 + 0.6 [\frac{0.6 + 0.3 + 0.8 + 0}{4}]}, \frac{u_{3}}{0.6 + 0.4 [\frac{0.4 + 1 + 0.9 + 1}{4}]}] .

Therefore, $F_{Λ} (e_{1}, p, 1) = {\frac{u_{1}}{0.75} \frac{u_{2}}{0.65} \frac{u_{3}}{0.93}}$ . Using a similar procedure for all given fuzzy soft expert sets, we obtain the following EFSES:

\begin{array}{l} {(F, Z)}_{Λ} = & {((e_{1}, p, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.65} \frac{u_{3}}{0.93}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.77} \frac{u_{3}}{0.87}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.82} \frac{u_{3}}{0.89}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.71} \frac{u_{3}}{0.87}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.65} \frac{u_{3}}{0.84}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.59} \frac{u_{3}}{0.94}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.71} \frac{u_{3}}{0.86}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.48} \frac{u_{3}}{0.89}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.77} \frac{u_{3}}{0.87}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.65} \frac{u_{3}}{0.93}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.65} \frac{u_{2}}{0.54} \frac{u_{3}}{0.94}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.82} \frac{u_{3}}{0.84}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.59} \frac{u_{3}}{0.91}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.71} \frac{u_{3}}{0.93}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.88} \frac{u_{3}}{0.93}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.71} \frac{u_{3}}{0.89}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.54} \frac{u_{3}}{0.87}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.59} \frac{u_{3}}{0.86}})} . \end{array}

Also, the agree-EFSES (F, Y )_Λ₁ is

\begin{array}{l} {(F, Z)}_{Λ_{1}} = & {((e_{1}, p, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.65} \frac{u_{3}}{0.93}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.77} \frac{u_{3}}{0.87}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.82} \frac{u_{3}}{0.89}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.71} \frac{u_{3}}{0.87}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.65} \frac{u_{3}}{0.84}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.59} \frac{u_{3}}{0.94}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.71} \frac{u_{3}}{0.86}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.48} \frac{u_{3}}{0.89}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.77} \frac{u_{3}}{0.87}})}, \end{array}

and disagree-EFSES (F, Y )_Λ₀ is

\begin{array}{l} {(F, Z)}_{Λ_{0}} = & {((e_{1}, p, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.65} \frac{u_{3}}{0.93}}), ((e_{1}, q, 0), {\frac{u_{1}}{0.65} \frac{u_{2}}{0.54} \frac{u_{3}}{0.94}}), \\ ((e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.82} \frac{u_{3}}{0.84}}), ((e_{2}, p, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.59} \frac{u_{3}}{0.91}}), \\ ((e_{2}, q, 0), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.71} \frac{u_{3}}{0.93}}), ((e_{2}, r, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.88} \frac{u_{3}}{0.93}}), \\ ((e_{3}, p, 0), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.71} \frac{u_{3}}{0.89}}), ((e_{3}, q, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.54} \frac{u_{3}}{0.87}}), \\ ((e_{3}, r, 0), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.59} \frac{u_{3}}{0.86}})} . \end{array}

Definition 3.4

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

Total_complement is denoted by ( $F_{Λ^{c}}^{c}$ , Y ),
Soft – Expert_complement is denoted by ( $F_{Λ}^{c}$ , Y ), and
Λ_complement is denoted by (F_Λ^c, Y ).

Example 3.2

Consider Example 3.1. Let

\begin{array}{l} Λ (u_{1}) = {\frac{a_{1}}{0.7}, \frac{a_{2}}{0}, \frac{a_{3}}{0.3}, \frac{a_{4}}{1}}, Λ (u_{2}) = {\frac{a_{1}}{0.6}, \frac{a_{2}}{0.3}, \frac{a_{3}}{0.8}, \frac{a_{4}}{0}}, \\ Λ (u_{3}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{1}, \frac{a_{3}}{0.9}, \frac{a_{4}}{1}} . \end{array}

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

\begin{array}{l} Λ^{c} (u_{1}) = {\frac{a_{1}}{0.3}, \frac{a_{2}}{1}, \frac{a_{3}}{0.7}, \frac{a_{4}}{0}}, Λ^{c} (u_{2}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{0.7}, \frac{a_{3}}{0.2}, \frac{a_{4}}{1}}, \\ Λ^{c} (u_{3}) = {\frac{a_{1}}{0.6}, \frac{a_{2}}{0}, \frac{a_{3}}{0.1}, \frac{a_{4}}{0}} . \end{array}

Also, let

\begin{array}{l} F (e_{1}, p, 1) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.4} \frac{u_{3}}{0.6}}, F (e_{1}, q, 1) = {\frac{u_{1}}{0.1} \frac{u_{2}}{0.6} \frac{u_{3}}{0.3}}, \\ F (e_{1}, r, 1) = {\frac{u_{1}}{0.8} \frac{u_{2}}{0.7} \frac{u_{3}}{0.4}}, F (e_{2}, p, 1) = {\frac{u_{1}}{0.2} \frac{u_{2}}{0.5} \frac{u_{3}}{0.3}}, \\ F (e_{2}, q, 1) = {\frac{u_{1}}{0.9} \frac{u_{2}}{0.4} \frac{u_{3}}{0.1}}, F (e_{2}, r, 1) = {\frac{u_{1}}{0.2} \frac{u_{2}}{0.3} \frac{u_{3}}{0.7}}, \\ F (e_{3}, p, 1) = {\frac{u_{1}}{0.4} \frac{u_{2}}{0.5} \frac{u_{3}}{0.2}}, F (e_{3}, q, 1) = {\frac{u_{1}}{0.7} \frac{u_{2}}{0.1} \frac{u_{3}}{0.4}}, \\ F (e_{3}, r, 1) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.6} \frac{u_{3}}{0.3}}, F (e_{1}, p, 0) = {\frac{u_{1}}{0.4} \frac{u_{2}}{0.4} \frac{u_{3}}{0.6}}, \\ F (e_{1}, q, 0) = {\frac{u_{1}}{0.3} \frac{u_{2}}{0.2} \frac{u_{3}}{0.7}}, F (e_{1}, r, 0) = {\frac{u_{1}}{0.6} \frac{u_{2}}{0.7} \frac{u_{3}}{0.1}}, \\ F (e_{2}, p, 0) = {\frac{u_{1}}{0.9} \frac{u_{2}}{0.3} \frac{u_{3}}{0.5}}, F (e_{2}, q, 0) = {\frac{u_{1}}{0.1} \frac{u_{2}}{0.5} \frac{u_{3}}{0.6}}, \\ F (e_{2}, r, 0) = {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.6}}, F (e_{3}, p, 0) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.5} \frac{u_{3}}{0.4}}, \\ F (e_{3}, q, 0) = {\frac{u_{1}}{0.2} \frac{u_{2}}{0.2} \frac{u_{3}}{0.3}}, F (e_{3}, r, 0) = {\frac{u_{1}}{0.7} \frac{u_{2}}{0.3} \frac{u_{3}}{0.2}} . \end{array}

Using the fuzzy soft expert complement, we have

\begin{array}{l} F^{c} (e_{1}, p, 1) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.6} \frac{u_{3}}{0.4}}, F^{c} (e_{1}, q, 1) = {\frac{u_{1}}{0.9} \frac{u_{2}}{0.4} \frac{u_{3}}{0.7}}, \\ F^{c} (e_{1}, r, 1) = {\frac{u_{1}}{0.2} \frac{u_{2}}{0.3} \frac{u_{3}}{0.6}}, F^{c} (e_{2}, p, 1) = {\frac{u_{1}}{0.8} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}, \\ F^{c} (e_{2}, q, 1) = {\frac{u_{1}}{0.1} \frac{u_{2}}{0.6} \frac{u_{3}}{0.9}}, F^{c} (e_{2}, r, 1) = {\frac{u_{1}}{0.8} \frac{u_{2}}{0.7} \frac{u_{3}}{0.3}}, \\ F^{c} (e_{3}, p, 1) = {\frac{u_{1}}{0.6} \frac{u_{2}}{0.5} \frac{u_{3}}{0.8}}, F^{c} (e_{3}, q, 1) = {\frac{u_{1}}{0.3} \frac{u_{2}}{0.9} \frac{u_{3}}{0.6}}, \\ F^{c} (e_{3}, r, 1) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.4} \frac{u_{3}}{0.7}}, F^{c} (e_{1}, p, 0) = {\frac{u_{1}}{0.6} \frac{u_{2}}{0.6} \frac{u_{3}}{0.4}}, \\ F^{c} (e_{1}, q, 0) = {\frac{u_{1}}{0.7} \frac{u_{2}}{0.8} \frac{u_{3}}{0.3}}, F^{c} (e_{1}, r, 0) = {\frac{u_{1}}{0.4} \frac{u_{2}}{0.3} \frac{u_{3}}{0.9}}, \\ F^{c} (e_{2}, p, 0) = {\frac{u_{1}}{0.1} \frac{u_{2}}{0.7} \frac{u_{3}}{0.5}}, F^{c} (e_{2}, q, 0) = {\frac{u_{1}}{0.9} \frac{u_{2}}{0.5} \frac{u_{3}}{0.4}}, \\ F^{c} (e_{2}, r, 0) = {\frac{u_{1}}{0.1} \frac{u_{2}}{0.2} \frac{u_{3}}{0.4}}, F^{c} (e_{3}, p, 0) = {\frac{u_{1}}{0.5} \frac{u_{2}}{0.5} \frac{u_{3}}{0.6}}, \\ F^{c} (e_{3}, q, 0) = {\frac{u_{1}}{0.8} \frac{u_{2}}{0.8} \frac{u_{3}}{0.7}}, F (e_{3}, r, 0) = {\frac{u_{1}}{0.3} \frac{u_{2}}{0.7} \frac{u_{3}}{0.8}} . \end{array}

We have the Total_complement as follows:

\begin{array}{l} {(F^{c}, Z)}_{Λ^{c}} = & {((e_{1}, p, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.83} \frac{u_{3}}{0.50}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.74} \frac{u_{3}}{0.75}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.70} \frac{u_{3}}{0.67}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.78} \frac{u_{3}}{0.75}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.83} \frac{u_{3}}{0.91}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.87} \frac{u_{3}}{0.42}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.78} \frac{u_{3}}{0.83}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.65} \frac{u_{2}}{0.95} \frac{u_{3}}{0.67}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.74} \frac{u_{3}}{0.75}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.83} \frac{u_{3}}{0.50}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.91} \frac{u_{3}}{0.42}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.70} \frac{u_{3}}{0.91}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.87} \frac{u_{3}}{0.58}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.78} \frac{u_{3}}{0.50}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.66} \frac{u_{3}}{0.50}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.78} \frac{u_{3}}{0.67}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.91} \frac{u_{3}}{0.75}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.65} \frac{u_{2}}{0.87} \frac{u_{3}}{0.83}})} . \end{array}

Also, we have the Soft – Expert_complement as follows:

\begin{array}{l} {(F^{c}, Z)}_{Λ} = & {((e_{1}, p, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.77} \frac{u_{3}}{0.89}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.65} \frac{u_{3}}{0.94}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.59} \frac{u_{3}}{0.93}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.71} \frac{u_{3}}{0.94}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.77} \frac{u_{3}}{0.98}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.82} \frac{u_{3}}{0.87}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.71} \frac{u_{3}}{0.96}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.65} \frac{u_{2}}{0.94} \frac{u_{3}}{0.93}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.65} \frac{u_{3}}{0.94}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.77} \frac{u_{3}}{0.89}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.88} \frac{u_{3}}{0.87}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.59} \frac{u_{3}}{0.98}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.82} \frac{u_{3}}{0.91}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.71} \frac{u_{3}}{0.89}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.54} \frac{u_{3}}{0.89}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.71} \frac{u_{3}}{0.93}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.88} \frac{u_{3}}{0.94}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.65} \frac{u_{2}}{0.82} \frac{u_{3}}{0.96}})} . \end{array}

In addition, we have the Λ_complement as follows:

\begin{array}{l} {(F, Z)}_{Λ^{c}} = & {((e_{1}, p, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.74} \frac{u_{3}}{0.67}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.83} \frac{u_{3}}{0.42}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.87} \frac{u_{3}}{0.50}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.78} \frac{u_{3}}{0.42}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.74} \frac{u_{3}}{0.25}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.70} \frac{u_{3}}{0.75}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.78} \frac{u_{3}}{0.34}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.61} \frac{u_{3}}{0.50}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.83} \frac{u_{3}}{0.42}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.74} \frac{u_{3}}{0.67}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.65} \frac{u_{2}}{0.66} \frac{u_{3}}{0.75}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.87} \frac{u_{3}}{0.25}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.70} \frac{u_{3}}{0.58}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.55} \frac{u_{2}}{0.78} \frac{u_{3}}{0.67}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.91} \frac{u_{3}}{0.67}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.75} \frac{u_{2}}{0.78} \frac{u_{3}}{0.50}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.66} \frac{u_{3}}{0.42}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.70} \frac{u_{3}}{0.34}})} . \end{array}

Definition 3.5

Let (F_Λ₁, Y₁) and (G_Λ₂, Y₂) be two EFSESs over the common universe U. The union of (F_Λ₁, Y₁) and (G_Λ₂, Y₂) is denoted by $(F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cup} (G_{Λ_{2}}, Y_{2})$ , which is the EFSES (H_{Λ_s}, Y ), where Y = Y₁ ∪ Y₂, Λ_s = s(Λ₁, Λ₂) and ∀ ɛ ∈ Y

H_{Λ_{s}} (ɛ) = {\begin{array}{l} F_{Λ_{s}} (ɛ) & if ɛ \in Y_{1} - Y_{2}, \\ G_{Λ_{s}} (ɛ) & if ɛ \in Y_{2} - Y_{1}, \\ {(F \tilde{\cup} G)}_{Λ_{s}} (ɛ) & if ɛ \in Y_{1} \cap Y_{2}, \end{array}

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

Example 3.3

Let $Λ_{1} (u_{1}) = {\frac{a_{1}}{0.7}, \frac{a_{2}}{0}, \frac{a_{3}}{0.3}, \frac{a_{4}}{1}}, Λ_{1} (u_{2}) = {\frac{a_{1}}{0.6}, \frac{a_{2}}{0.3}, \frac{a_{3}}{0.8}, \frac{a_{4}}{0}}$ and $Λ_{3} (u_{3}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{1}, \frac{a_{3}}{0.9}, \frac{a_{4}}{1}}$ , and let $Λ_{2} (u_{1}) = {\frac{a_{1}}{0.6}, \frac{a_{2}}{0.9}, \frac{a_{3}}{0.8}, \frac{a_{4}}{0.5}}, Λ_{2} (u_{2}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{0.7}, \frac{a_{3}}{0.6}, \frac{a_{4}}{1}}$ and $Λ_{2} (u_{3}) = {\frac{a_{1}}{0.5}, \frac{a_{2}}{0.8}, \frac{a_{3}}{0.7}, \frac{a_{4}}{0.6}}$ . Also, let

\begin{array}{l} \begin{array}{l} (F, Y_{1}) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.8} \frac{u_{3}}{0.9}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.6} \frac{u_{3}}{0.5}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.7} \frac{u_{3}}{0.3}}), ((e_{3}, p, 1), {\frac{u_{1}}{0.2} \frac{u_{2}}{0.9} \frac{u_{3}}{0.1}}), \\ ((e_{3}, q, 1), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.3} \frac{u_{3}}{0.8}}), ((e_{3}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.1} \frac{u_{3}}{0.4}}), \\ ((e_{1}, p, 0), {\frac{u_{1}}{0.3} \frac{u_{2}}{0.6} \frac{u_{3}}{0.6}}), ((e_{1}, q, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.2} \frac{u_{3}}{0.8}}), \\ ((e_{1}, r, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.5} \frac{u_{3}}{0.6}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.2} \frac{u_{2}}{0.9} \frac{u_{3}}{0.2}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.1} \frac{u_{3}}{0.9}})}, \end{array} \\ \begin{array}{l} (G, Y_{2}) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.7} \frac{u_{3}}{0.6}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.8} \frac{u_{3}}{0.5}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.9} \frac{u_{3}}{0.8}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.6} \frac{u_{3}}{0.9}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}), \\ ((e_{1}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.4} \frac{u_{3}}{0.7}}), ((e_{1}, q, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.9} \frac{u_{3}}{0.6}}), \\ ((e_{1}, r, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.6} \frac{u_{3}}{0.8}}), ((e_{2}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.8} \frac{u_{3}}{0.9}}), \\ ((e_{2}, q, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.7} \frac{u_{3}}{0.9}}), ((e_{2}, r, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.9} \frac{u_{3}}{0.7}}),} . \end{array} \end{array}

Using the basic fuzzy union, we obtain the following effective sets: $Λ_{s} (u_{1}) = {\frac{a_{1}}{0.7}, \frac{a_{2}}{0.9}, \frac{a_{3}}{0.8}, \frac{a_{4}}{1}}, Λ_{s} (u_{2}) = {\frac{a_{1}}{0.6}, \frac{a_{2}}{0.7}, \frac{a_{3}}{0.8}, \frac{a_{4}}{1}}$ , and $Λ_{s} (u_{3}) = {\frac{a_{1}}{0.5}, \frac{a_{2}}{1}, \frac{a_{3}}{0.9}, \frac{a_{4}}{1}}$ . Moreover, using the fuzzy soft expert union, we obtain

\begin{array}{l} (H, Y) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.8} \frac{u_{3}}{0.9}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.6} \frac{u_{3}}{0.7}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.5}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.9} \frac{u_{3}}{0.8}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.6} \frac{u_{3}}{0.9}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.2} \frac{u_{2}}{0.9} \frac{u_{3}}{0.1}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.3} \frac{u_{3}}{0.8}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.1} \frac{u_{3}}{0.4}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.6} \frac{u_{3}}{0.7}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.9} \frac{u_{3}}{0.8}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.6} \frac{u_{3}}{0.8}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.8} \frac{u_{3}}{0.9}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.7} \frac{u_{3}}{0.9}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.9} \frac{u_{3}}{0.7}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.2} \frac{u_{2}}{0.9} \frac{u_{3}}{0.2}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.1} \frac{u_{3}}{0.9}})} . \end{array}

Thus, we have the following EFSES:

\begin{array}{l} (H_{Λ_{s}}, Z) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.95} \frac{u_{3}}{0.98}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.98} \frac{u_{2}}{0.91} \frac{u_{3}}{0.95}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.98} \frac{u_{2}}{0.95} \frac{u_{3}}{0.92}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.97} \frac{u_{3}}{0.97}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.92} \frac{u_{2}}{0.91} \frac{u_{3}}{0.98}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.88} \frac{u_{3}}{0.95}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.88} \frac{u_{2}}{0.97} \frac{u_{3}}{0.86}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.86} \frac{u_{2}}{0.84} \frac{u_{3}}{0.97}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.79} \frac{u_{3}}{0.91}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.91} \frac{u_{3}}{0.95}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.95} \frac{u_{2}}{0.97} \frac{u_{3}}{0.97}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.98} \frac{u_{2}}{0.91} \frac{u_{3}}{0.97}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.95} \frac{u_{3}}{0.98}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.98} \frac{u_{2}}{0.93} \frac{u_{3}}{0.98}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.97} \frac{u_{3}}{0.95}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.88} \frac{u_{2}}{0.97} \frac{u_{3}}{0.88}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.91} \frac{u_{2}}{0.88} \frac{u_{3}}{0.95}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.86} \frac{u_{2}}{0.79} \frac{u_{3}}{0.98}})} . \end{array}

Proposition 3.1

If (F_Λ₁, Y₁), (G_Λ₂, Y₂), and (H_Λ₃, Y₃) are three EFSESs over U, then

(F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cup} ((G_{Λ_{2}}, Y_{2}) \overset{Λ}{\cup} (H_{Λ_{3}}, Y_{3})) = ((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cup} (G_{Λ_{2}}, Y_{2})) \overset{Λ}{\cup} (H_{Λ_{3}}, Y_{3}) .

Proof

Let

\begin{array}{l} ((G_{Λ_{2}}, Y_{2}) \overset{Λ}{\cup} (H_{Λ_{3}}, Y_{3})) (ɛ) = K_{Λ_{s}^{'}} (ɛ) \\ = {\begin{array}{l} G_{Λ_{s}^{'}} (ɛ) & if ɛ \in Y_{2} - Y_{3}, \\ H_{Λ_{s}^{'}} (ɛ) & if ɛ \in Y_{3} - Y_{2}, \\ {(G \tilde{\cup} H)}_{Λ_{s}^{'}} (ɛ) & if ɛ \in Y_{2} \cap Y_{3}, \end{array} \end{array}

where $Λ_{s}^{'} = s (Λ_{2}, Λ_{3})$ .

Also, let

((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cup} K_{Λ_{s}^{'}}) (ɛ) = {\begin{array}{l} F_{Λ_{s}} (ɛ) & if ɛ \in Y_{1} - (Y_{2} \cap Y_{3}), \\ K_{Λ_{s}} (ɛ) & if ɛ \in Y_{2} \cap Y_{3}) - Y_{1}, \\ {(F \tilde{\cup} K)}_{Λ_{s}} (ɛ) & if ɛ \in Y_{1} \cap (Y_{2} \cap Y_{3}), \end{array}

where Λ_s = s(Λ₁, Λ₂, Λ₃).

We consider the case in which ɛ ∈ Y₁ ∩ (Y₂ ∩ Y₃) because the other cases are trivial; thus,

((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cup} K_{Λ_{s}^{'}}) (ɛ) = ((F \tilde{\cup} K) (ɛ), Y_{1} \cup (Y_{2} \cup Y_{3}), Λ_{s} = s (Λ_{1}, Λ_{2}, Λ_{3})) .

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

((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cup} K_{Λ_{s}^{'}}) (ɛ) = ((s (F (ɛ), K (ɛ))), Y_{1} \cup (Y_{2} \cup Y_{3}), Λ_{s} = s (Λ_{1}, Λ_{2}, Λ_{3})) .

Furthermore, for $(F_{Λ_{1}}, Y_{1}) (ɛ) \overset{Λ}{\cup} K_{Λ_{s}^{'}} (ɛ)$

\begin{array}{l} = (s (F_{Λ_{1}} (ɛ), s (G (ɛ), H (ɛ))), Y_{1} \cup (Y_{2} \cup Y_{3}), Λ_{s}^{'} = s (Λ_{2}, Λ_{3})) \\ = (s (s (f (ɛ), G (ɛ)), H_{Λ_{3}} (ɛ)), (Y_{1} \cup Y_{2}) \cup Y_{3}, Λ_{s}^{″} = s (Λ_{1}, Λ_{2})) \\ = ((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cup} (G_{Λ_{2}}, Y_{2})) \overset{Λ}{\cup} (H_{Λ_{3}}, Y_{3}) . \end{array}

Definition 3.6

Let (F_Λ₁, Y₁) and (G_Λ₂, Y₂) be two EFSESs over a common universe U. The intersection of (F_Λ₁, Y₁) and (G_Λ₂, Y₂) is denoted by $(F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cap} (G_{Λ_{2}}, Y_{2})$ , which is the EFSES (K_{Λ_t}, Y ), where Y = Y₁ ∪ Y₂, Λ_t = t(Λ₁, Λ₂) and ∀ ɛ ∈ Y

K_{Λ_{t}} (ɛ) = {\begin{array}{l} F_{Λ_{t}} (ɛ) & if ɛ \in Y_{1} - Y_{2}, \\ G_{Λ_{t}} (ɛ) & if ɛ \in Y_{2} - Y_{1}, \\ {(F \tilde{\cap} G)}_{Λ_{t}} (ɛ) & if ɛ \in Y_{1} \cap Y_{2}, \end{array}

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:

Λ_{t} (u_{1}) = {\frac{a_{1}}{0.6}, \frac{a_{2}}{0.3}, \frac{a_{3}}{0}, \frac{a_{4}}{0.5}}, Λ_{t} (u_{2}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{0.3}, \frac{a_{3}}{0.6}, \frac{a_{4}}{0}}, Λ_{t} (u_{3}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{0.8}, \frac{a_{3}}{0.7}, \frac{a_{4}}{0.6}} .

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

\begin{array}{l} (K, Y) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.7} \frac{u_{3}}{0.6}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.5} \frac{u_{3}}{0.5}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.7} \frac{u_{3}}{0.3}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.9} \frac{u_{3}}{0.8}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.6} \frac{u_{3}}{0.9}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.2} \frac{u_{2}}{0.9} \frac{u_{3}}{0.1}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.3} \frac{u_{3}}{0.8}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.1} \frac{u_{3}}{0.4}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.3} \frac{u_{2}}{0.4} \frac{u_{3}}{0.6}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.2} \frac{u_{3}}{0.6}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.5} \frac{u_{3}}{0.6}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.8} \frac{u_{3}}{0.9}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.7} \frac{u_{3}}{0.9}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.9} \frac{u_{3}}{0.7}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.2} \frac{u_{2}}{0.9} \frac{u_{3}}{0.2}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.1} \frac{u_{3}}{0.9}})} . \end{array}

Thus, we have the following EFSES:

\begin{array}{l} (K_{Λ_{t}}, Z) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.80} \frac{u_{2}}{0.79} \frac{u_{3}}{0.85}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.61} \frac{u_{2}}{0.66} \frac{u_{3}}{0.81}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.61} \frac{u_{2}}{0.79} \frac{u_{3}}{0.73}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.80} \frac{u_{2}}{0.93} \frac{u_{3}}{0.86}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.67} \frac{u_{2}}{0.73} \frac{u_{3}}{0.96}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.74} \frac{u_{2}}{0.66} \frac{u_{3}}{0.88}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.48} \frac{u_{2}}{0.93} \frac{u_{3}}{0.66}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.41} \frac{u_{2}}{0.52} \frac{u_{3}}{0.92}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.74} \frac{u_{2}}{0.39} \frac{u_{3}}{0.77}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.54} \frac{u_{2}}{0.59} \frac{u_{3}}{0.85}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.74} \frac{u_{2}}{0.46} \frac{u_{3}}{0.85}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.80} \frac{u_{2}}{0.66} \frac{u_{3}}{0.85}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.87} \frac{u_{2}}{0.86} \frac{u_{3}}{0.96}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.93} \frac{u_{2}}{0.79} \frac{u_{3}}{0.96}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.74} \frac{u_{2}}{0.93} \frac{u_{3}}{0.88}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.48} \frac{u_{2}}{0.93} \frac{u_{3}}{0.7}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.61} \frac{u_{2}}{0.66} \frac{u_{3}}{0.88}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.41} \frac{u_{2}}{0.39} \frac{u_{3}}{0.96}})} . \end{array}

Proposition 3.2

If (F_Λ₁, Y₁), (G_Λ₂, Y₂), and (H_Λ₃, Y₃) are three EFSESs over U, then

(F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cap} ((G_{Λ_{2}}, Y_{2}) \overset{Λ}{\cap} (H_{Λ_{3}}, Y_{3})) = ((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cap} (G_{Λ_{2}}, Y_{2})) \overset{Λ}{\cap} (H_{Λ_{3}}, Y_{3}) .

Proof

Let

\begin{array}{l} ((G_{Λ_{2}}, Y_{2}) \overset{Λ}{\cap} (H_{Λ_{3}}, Y_{3})) (ɛ) = K_{Λ_{t}^{'}} (ɛ) \\ = {\begin{array}{l} G_{Λ_{t}^{'}} (ɛ) & if ɛ \in Y_{2} - Y_{3}, \\ H_{Λ_{t}^{'}} (ɛ) & if ɛ \in Y_{3} - Y_{2}, \\ {(G \tilde{\cap} H)}_{Λ_{t}^{'}} (ɛ) & if ɛ \in Y_{2} \cap Y_{3}, \end{array} \end{array}

where $Λ_{t}^{'} = t (Λ_{2}, Λ_{3})$ . Also, let

((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cap} K_{Λ_{t}^{'}}) (ɛ) = {\begin{array}{l} F_{Λ_{t}} (ɛ) & if ɛ \in Y_{1} - (Y_{2} \cap Y_{3}), \\ K_{Λ_{t}} (ɛ) & if ɛ \in (Y_{2} \cap Y_{3}) - Y_{1}, \\ {(F \tilde{\cap} K)}_{Λ_{t}} (ɛ) & if ɛ \in Y_{1} \cap (Y_{2} \cap Y_{3}), \end{array}

where Λ_t = t(Λ₁, Λ₂, Λ₃). We consider the case when ɛ ∈ Y₁ ∩ (Y₂ ∩ Y₃) because the other cases are trivial; thus,

((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cap} K_{Λ_{t}^{'}}) (ɛ) = ((F \tilde{\cap} K) (ɛ), Y_{1} \cup (Y_{2} \cup Y_{3}), Λ_{t} = t (Λ_{1}, Λ_{2}, Λ_{3})) .

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

((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cap} K_{Λ_{t}^{'}}) (ɛ) = ((t (F (ɛ), K (ɛ))), Y_{1} \cup (Y_{2} \cup Y_{3}), Λ_{t} = t (Λ_{1}, Λ_{2}, Λ_{3})) .

Furthermore, for $(F_{Λ_{1}}, Y_{1}) (ɛ) \overset{Λ}{\cap} K_{Λ_{t}^{'}} (ɛ)$ ,

\begin{array}{l} = (t (F_{Λ_{1}} (ɛ), t (G (ɛ), H (ɛ))), Y_{1} \cup (Y_{2} \cup Y_{3}), Λ_{t}^{'} = t (Λ_{2}, Λ_{3})) \\ = (t (t (F (ɛ), G (ɛ)), H_{Λ_{3}} (ɛ)), (Y_{1} \cup Y_{2}) \cup Y_{3}, Λ_{t}^{″} = t (Λ_{1}, Λ_{2})) \\ = ((F_{Λ_{1}}, Y_{1}) \overset{Λ}{\cap} (G_{Λ_{2}}, Y_{2})) \overset{Λ}{\cap} (H_{Λ_{3}}, Y_{3}) . \end{array}

Definition 3.7

Let (F_Λ₁, Y₁) and (G_Λ₂, Y₂) be two EFSESs over a common universe U. We denote (F_Λ₁, Y₁) “AND” (G_Λ₂, Y₂) by (F_Λ₁, Y₁) ∧ (G_Λ₂, Y₂), and we define it by

(K_{Λ_{t}}, Y_{1} \times Y_{2}) = (F_{Λ_{1}}, Y_{1}) \land (G_{Λ_{2}}, Y_{2}),

such that

K_{Λ_{t}} (α, β) = {(F (α) \tilde{\cap} G (β))}_{Λ_{t}}; \forall (α, β) \in Y_{1} \times Y_{2},

where t is any t-norm, Λ_t = t(Λ₁, Λ₂), and ∩̃ is the fuzzy soft expert intersection.

Example 3.5

Let

Λ_{1} (u_{1}) = {\frac{a_{1}}{0.1}, \frac{a_{2}}{0.4}, \frac{a_{3}}{1}, \frac{a_{4}}{0.7}}, Λ_{1} (u_{2}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{1}, \frac{a_{3}}{0}, \frac{a_{4}}{0.8}}, Λ_{3} (u_{3}) = {\frac{a_{1}}{0.9}, \frac{a_{2}}{0.6}, \frac{a_{3}}{0.7}, \frac{a_{4}}{1}},

and

Λ_{2} (u_{1}) = {\frac{a_{1}}{0.5}, \frac{a_{2}}{0.3}, \frac{a_{3}}{0.8}, \frac{a_{4}}{0.9}}, Λ_{2} (u_{2}) = {\frac{a_{1}}{0.3}, \frac{a_{2}}{0.7}, \frac{a_{3}}{0.2}, \frac{a_{4}}{1}}, Λ_{2} (u_{3}) = {\frac{a_{1}}{1}, \frac{a_{2}}{0.9}, \frac{a_{3}}{0}, \frac{a_{4}}{0.3}} .

Also, let

\begin{array}{l} Y_{1} = {(e_{1}, p, 1), (e_{2}, p, 0), (e_{1}, r, 0), (e_{2}, r, 1)}, \\ Y_{2} = {(e_{1}, p, 1), (e_{1}, r, 0), (e_{2}, r, 1)} . \end{array}

In addition, let

\begin{array}{l} (F, Y_{1}) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.5} \frac{u_{3}}{0.3}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.4} \frac{u_{3}}{0.6}}), \\ ((e_{1}, r, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.5}}), ((e_{2}, p, 0), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.2} \frac{u_{3}}{0.9}})}, \end{array}

and

\begin{array}{l} (G, Y_{2}) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.5} \frac{u_{3}}{0.4}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.9} \frac{u_{3}}{0.5}}), \\ ((e_{2}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.3} \frac{u_{3}}{0.8}})} . \end{array}

Using the fuzzy soft expert intersection, we have

\begin{array}{l} (K, Y_{1} \times Y_{2}) = & {((e_{1}, p, 1), (e_{1}, p, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.5} \frac{u_{3}}{0.3}}), \\ ((e_{1}, p, 1), (e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.5} \frac{u_{3}}{0.3}}), \\ ((e_{1}, p, 1), (e_{2}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.3} \frac{u_{3}}{0.3}}), \\ ((e_{2}, p, 0), (e_{1}, p, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.4} \frac{u_{3}}{0.4}}), \\ ((e_{2}, p, 0), (e_{1}, r, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.4} \frac{u_{3}}{0.5}}), \\ ((e_{2}, p, 0), (e_{2}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.3} \frac{u_{3}}{0.6}}), \\ ((e_{1}, r, 0), (e_{1}, p, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.5} \frac{u_{3}}{0.4}}), \\ ((e_{1}, r, 0), (e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.8} \frac{u_{3}}{0.5}}), \\ ((e_{1}, r, 0), (e_{2}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.3} \frac{u_{3}}{0.5}}), \\ ((e_{2}, r, 1), (e_{1}, p, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.2} \frac{u_{3}}{0.4}}), \\ ((e_{2}, r, 1), (e_{1}, r, 0), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.2} \frac{u_{3}}{0.5}}), \\ ((e_{2}, r, 1), (e_{2}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.2} \frac{u_{3}}{0.8}})} . \end{array}

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

Λ_{t} (u_{1}) = {\frac{a_{1}}{0.1}, \frac{a_{2}}{0.3}, \frac{a_{3}}{0.8}, \frac{a_{4}}{0.7}}, Λ_{t} (u_{2}) = {\frac{a_{1}}{0.3}, \frac{a_{2}}{0.2}, \frac{a_{3}}{0}, \frac{a_{4}}{0.8}}, Λ_{t} (u_{3}) = {\frac{a_{1}}{0.9}, \frac{a_{2}}{0.6}, \frac{a_{3}}{0}, \frac{a_{4}}{0.3}} .

Thus, we have the following EFSES:

\begin{array}{l} (K_{Λ_{t}}, Y_{1} \times Y_{2}) = & {((e_{1}, p, 1), (e_{1}, p, 1), {\frac{u_{1}}{0.79} \frac{u_{2}}{0.66} \frac{u_{3}}{0.61}}), \\ ((e_{1}, p, 1), (e_{1}, r, 0), {\frac{u_{1}}{0.89} \frac{u_{2}}{0.66} \frac{u_{3}}{0.61}}), \\ ((e_{1}, p, 1), (e_{2}, r, 1), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.52} \frac{u_{3}}{0.61}}), \\ ((e_{2}, p, 0), (e_{1}, p, 1), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.52} \frac{u_{3}}{0.78}}), \\ ((e_{2}, p, 0), (e_{1}, r, 0), {\frac{u_{1}}{0.79} \frac{u_{2}}{0.66} \frac{u_{3}}{0.67}}), \\ ((e_{2}, p, 0), (e_{2}, r, 1), {\frac{u_{1}}{0.89} \frac{u_{2}}{0.86} \frac{u_{3}}{0.72}}), \\ ((e_{1}, r, 0), (e_{1}, p, 1), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.52} \frac{u_{3}}{0.72}}), \\ ((e_{1}, r, 0), (e_{1}, r, 0), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.46} \frac{u_{3}}{0.67}}), \\ ((e_{1}, r, 0), (e_{2}, r, 1), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.52} \frac{u_{3}}{0.72}}), \\ ((e_{2}, r, 1), (e_{1}, p, 1), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.46} \frac{u_{3}}{0.67}}), \\ ((e_{2}, r, 0), (e_{2}, r, 0), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.46} \frac{u_{3}}{0.72}}), \\ ((e_{2}, r, 1), (e_{2}, r, 1), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.46} \frac{u_{3}}{0.89}})} . \end{array}

Proposition 3.3

If (F_Λ₁, Y₁), (G_Λ₂, Y₂), and (H_Λ₃, Y₃) are three EFSESs over U, then

(F_{Λ_{1}}, Y_{1}) \land ((G_{Λ_{2}}, Y_{2}) \land (H_{Λ_{3}}, Y_{3})) = ((F_{Λ_{1}}, Y_{1}) \land (G_{Λ_{2}}, Y_{2})) \land (H_{Λ_{3}}, Y_{3}) .

Proof

Let

((G_{Λ_{2}}, Y_{2}) \land (H_{Λ_{3}}, Y_{3})) (α, β) = ({(G (α) \tilde{\cap} H (β))}_{Λ_{t}^{'}}) \forall (α, β) \in Y_{2} \times Y_{3}, Λ_{t}^{'} = t (Λ_{2}, Λ_{3}) .

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

\begin{array}{l} ((G_{Λ_{2}}, Y_{2}) \land (H_{Λ_{3}}, Y_{3})) (α, β) = (t (G (α), H (β))) \\ \forall (α, β) \in Y_{2} \times Y_{3}, Λ_{t}^{'} = t (Λ_{2}, Λ_{3}) . Thus, \\ (F_{Λ_{1}}, Y_{1}) (γ) \land ((G_{Λ_{2}}, Y_{2}) \land (H_{Λ_{3}}, Y_{3})) (α, β) \\ = (t (F_{Λ_{1}} (γ), t (G (α), H (β))), \forall (γ, (α, β)) \in Y_{1} \times (Y_{2} \times Y_{3}), Λ_{t}^{'} = t (Λ_{2}, Λ_{3})) \\ = (t (t (F (γ), G (α)), H_{Λ_{3}} (β)), \forall ((γ, α), β) \in (Y_{1} \times Y_{2}) \times Y_{3}, Λ_{t}^{″} = t (Λ_{1}, Λ_{2})) \\ = ((F_{Λ_{1}}, Y_{1}) \land (G_{Λ_{2}}, Y_{2})) \land (H_{Λ_{3}}, Y_{3}) . \end{array}

Definition 3.8

Let (F_Λ₁, Y₁) and (G_Λ₂, Y₂) be two EFSESs over a common universe U. We denote (F_Λ₁, Y₁) “OR” (G_Λ₂, Y₂) by (F_Λ₁, Y₁) ∨ (G_Λ₂, Y₂), and we define it by

(H_{Λ_{s}}, Y_{1} \times Y_{2}) = (F_{Λ_{1}}, Y_{1}) \lor (G_{Λ_{2}}, Y_{2}),

such that

H_{Λ_{s}} (α, β) = {(F (α) \tilde{\cup} G (β))}_{Λ_{s}}; \forall (α, β) \in Y_{1} \times Y_{2},

where s is any s-norm, Λ_s = s(Λ₁, Λ₂), and ∪̃ is the fuzzy soft expert union.

Example 3.6

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

\begin{array}{l} (H, Y_{1} \times Y_{2}) = & {((e_{1}, p, 1), (e_{1}, p, 1), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.5} \frac{u_{3}}{0.4}}), \\ ((e_{1}, p, 1), (e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.9} \frac{u_{3}}{0.5}}), \\ ((e_{1}, p, 1), (e_{2}, r, 1), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.5} \frac{u_{3}}{0.8}}), \\ ((e_{2}, p, 0), (e_{1}, p, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.5} \frac{u_{3}}{0.6}}), \\ ((e_{2}, p, 0), (e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.9} \frac{u_{3}}{0.6}}), \\ ((e_{2}, p, 0), (e_{2}, r, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.4} \frac{u_{3}}{0.8}}), \\ ((e_{1}, r, 0), (e_{1}, p, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.5}}), \\ ((e_{1}, r, 0), (e_{1}, r, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.8}}), \\ ((e_{1}, r, 0), (e_{2}, r, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.8}}), \\ ((e_{2}, r, 1), (e_{1}, p, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.5} \frac{u_{3}}{0.9}}), \\ ((e_{2}, r, 1), (e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.9} \frac{u_{3}}{0.9}}), \\ ((e_{2}, r, 1), (e_{2}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.3} \frac{u_{3}}{0.9}})} . \end{array}

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

Λ_{s} (u_{1}) = {\frac{a_{1}}{0.5}, \frac{a_{2}}{0.4}, \frac{a_{3}}{1}, \frac{a_{4}}{0.9}}, Λ_{s} (u_{2}) = {\frac{a_{1}}{0.4}, \frac{a_{2}}{0.7}, \frac{a_{3}}{1}, \frac{a_{4}}{1}}, Λ_{s} (u_{3}) = {\frac{a_{1}}{1}, \frac{a_{2}}{0.9}, \frac{a_{3}}{0.7}, \frac{a_{4}}{1}} .

Thus, we have the following EFSES:

\begin{array}{l} (H_{Λ_{s}}, Y_{1} \times Y_{2}) = & {((e_{1}, p, 1), (e_{1}, p, 1), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.88} \frac{u_{3}}{0.94}}), \\ ((e_{1}, p, 1), (e_{1}, r, 0), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.97} \frac{u_{3}}{0.95}}), \\ ((e_{1}, p, 1), (e_{2}, r, 1), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.88} \frac{u_{3}}{0.98}}), \\ ((e_{2}, p, 0), (e_{1}, p, 1), {\frac{u_{1}}{0.91} \frac{u_{2}}{0.88} \frac{u_{3}}{0.96}}), \\ ((e_{2}, p, 0), (e_{1}, r, 0), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.97} \frac{u_{3}}{0.96}}), \\ ((e_{2}, p, 0), (e_{2}, r, 1), {\frac{u_{1}}{0.91} \frac{u_{2}}{0.86} \frac{u_{3}}{0.98}}), \\ ((e_{1}, r, 0), (e_{1}, p, 1), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.95} \frac{u_{3}}{0.95}}), \\ ((e_{1}, r, 0), (e_{1}, r, 0), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.97} \frac{u_{3}}{0.95}}), \\ ((e_{1}, r, 0), (e_{2}, r, 1), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.95} \frac{u_{3}}{0.98}}), \\ ((e_{2}, r, 1), (e_{1}, p, 1), {\frac{u_{1}}{0.88} \frac{u_{2}}{0.88} \frac{u_{3}}{0.99}}), \\ ((e_{2}, r, 0), (e_{2}, r, 0), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.97} \frac{u_{3}}{0.99}}), \\ ((e_{2}, r, 1), (e_{2}, r, 1), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.84} \frac{u_{3}}{0.99}})} . \end{array}

Proposition 3.4

If (F_Λ₁, Y₁), (G_Λ₂, Y₂), and (H_Λ₃, Y₃) are three EFSESs over U, then

(F_{Λ_{1}}, Y_{1}) \lor ((G_{Λ_{2}}, Y_{2}) \lor (H_{Λ_{3}}, Y_{3})) = ((F_{Λ_{1}}, Y_{1}) \lor (G_{Λ_{2}}, Y_{2})) \lor (H_{Λ_{3}}, Y_{3}) .

Proof

Let

((G_{Λ_{2}}, Y_{2}) \lor (H_{Λ_{3}}, Y_{3})) (α, β) = ({(G (α) \tilde{\cup} H (β))}_{Λ_{s}^{'}}), \forall (α, β) \in Y_{2} \times Y_{3}, Λ_{s}^{'} = s (Λ_{2}, Λ_{3}) .

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

((G_{Λ_{2}}, Y_{2}) \lor (H_{Λ_{3}}, Y_{3})) (α, β) = (s (G (α), H (β)), \forall (α, β) \in Y_{2} \times Y_{3}, Λ_{s}^{'} = s (Λ_{2}, Λ_{3})) .

Therefore,

\begin{array}{l} (F_{Λ_{1}}, Y_{1}) (γ) \lor ((G_{Λ_{2}}, Y_{2}) \lor (H_{Λ_{3}}, Y_{3})) (α, β) = (s (F_{Λ_{1}} (γ), s (G (α), H (β))), \forall (γ, (α, β)) \in Y_{1} \times (Y_{2} \times Y_{3}), Λ_{s}^{'} = s (Λ_{2}, Λ_{3})) \\ = (s (s (F (γ), G (α)), H_{Λ_{3}} (β)), \forall ((γ, α), β) \in (Y_{1} \times Y_{2}) \times Y_{3}, Λ_{s}^{″} = s (Λ_{1}, Λ_{2})) \\ = ((F_{Λ_{1}}, Y_{1}) \lor (G_{Λ_{2}}, Y_{2})) \lor (H_{Λ_{3}}, Y_{3}) . \end{array}

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

Λ_{1} (u_{1}) = {\frac{a_{1}}{0.9}, \frac{a_{2}}{0}, \frac{a_{3}}{0.7}, \frac{a_{4}}{0.5}}, Λ_{1} (u_{2}) = {\frac{a_{1}}{0.8}, \frac{a_{2}}{0.5}, \frac{a_{3}}{1}, \frac{a_{4}}{0.3}}, Λ_{s} (u_{3}) = {\frac{a_{1}}{1}, \frac{a_{2}}{0.8}, \frac{a_{3}}{0.4}, \frac{a_{4}}{0}},

and

Λ_{2} (u_{1}) = {\frac{a_{1}}{0.7}, \frac{a_{2}}{0.4}, \frac{a_{3}}{1}, \frac{a_{4}}{0.3}}, Λ_{2} (u_{2}) = {\frac{a_{1}}{0.5}, \frac{a_{2}}{0.2}, \frac{a_{3}}{0.8}, \frac{a_{4}}{0.8}}, Λ_{2} (u_{3}) = {\frac{a_{1}}{0.9}, \frac{a_{2}}{0.9}, \frac{a_{3}}{0.1}, \frac{a_{4}}{0.1}} .

Furthermore, let

\begin{array}{l} (F, Y_{1}) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.5} \frac{u_{3}}{0.8}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.7} \frac{u_{3}}{0.9}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.5} \frac{u_{3}}{0.6}}), ((e_{3}, p, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.7}}), \\ ((e_{3}, q, 1), {\frac{u_{1}}{0.2} \frac{u_{2}}{0.7} \frac{u_{3}}{0.7}}), ((e_{3}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.3} \frac{u_{3}}{0.1}}), \\ ((e_{1}, p, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.6} \frac{u_{3}}{0.4}}), ((e_{1}, q, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.1} \frac{u_{3}}{0.5}}), \\ ((e_{1}, r, 0), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.8} \frac{u_{3}}{0.4}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.5} \frac{u_{3}}{0.8}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.9} \frac{u_{3}}{0.5}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.2} \frac{u_{3}}{0.9}})}, \end{array}

and

\begin{array}{l} (G, Y_{2}) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.3} \frac{u_{2}}{0.2} \frac{u_{3}}{0.9}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.5} \frac{u_{3}}{0.7}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.3} \frac{u_{2}}{0.5} \frac{u_{3}}{0.8}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.6} \frac{u_{3}}{0.8}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.7} \frac{u_{3}}{0.1}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.3} \frac{u_{2}}{0.6} \frac{u_{3}}{0.5}}), \\ ((e_{1}, p, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.2} \frac{u_{3}}{0.6}}), ((e_{1}, q, 0), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.5} \frac{u_{3}}{0.3}}), \\ ((e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.5} \frac{u_{3}}{0.6}}), ((e_{2}, p, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.9} \frac{u_{3}}{0.2}}), \\ ((e_{2}, q, 0), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.4} \frac{u_{3}}{0.9}}), ((e_{2}, r, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.6} \frac{u_{3}}{0.7}})} . \end{array}

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 Λ_s or Λ_t from the effective sets Λ₁ and Λ₂.
Compute the corresponding resultant EFSES either H_{Λ_s} or K_{Λ_t}.
Find $C_{ℓ} = \sum_{j = 1}^{n} u_{j ℓ}$ for the agree-EFSES and place it in a tabular form.
Compute $K_{ℓ} = \sum_{j = 1}^{n} u_{j ℓ}$ for the disagree-EFSES and place it in a tabular form.
Compute R_ℓ = C_ℓ– K_ℓ.
Compute m, for which s_m = max_u_{_ℓ∈}_U {R}. Then, s_m 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:

Λ_{s} (u_{1}) = {\frac{a_{1}}{0.9}, \frac{a_{2}}{0.4}, \frac{a_{3}}{0.1}, \frac{a_{4}}{0.5}}, Λ_{s} (u_{2}) = {\frac{a_{1}}{0.8}, \frac{a_{2}}{0.5}, \frac{a_{3}}{1}, \frac{a_{4}}{0.8}}, Λ_{s} (u_{3}) = {\frac{a_{1}}{1}, \frac{a_{2}}{0.9}, \frac{a_{3}}{0.4}, \frac{a_{4}}{0.1}} .

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

\begin{array}{l} (H, Y) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.4} \frac{u_{2}}{0.5} \frac{u_{3}}{0.9}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.7} \frac{u_{3}}{0.9}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.5} \frac{u_{3}}{0.8}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.6} \frac{u_{3}}{0.8}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.7} \frac{u_{3}}{0.1}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.3} \frac{u_{2}}{0.6} \frac{u_{3}}{0.5}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.8} \frac{u_{3}}{0.7}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.2} \frac{u_{2}}{0.7} \frac{u_{3}}{0.7}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.3} \frac{u_{3}}{0.1}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.6} \frac{u_{3}}{0.6}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.7} \frac{u_{2}}{0.5} \frac{u_{3}}{0.5}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.8} \frac{u_{3}}{0.6}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.9} \frac{u_{2}}{0.9} \frac{u_{3}}{0.2}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.4} \frac{u_{3}}{0.9}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.6} \frac{u_{2}}{0.6} \frac{u_{3}}{0.7}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.8} \frac{u_{2}}{0.5} \frac{u_{3}}{0.8}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.5} \frac{u_{2}}{0.9} \frac{u_{3}}{0.5}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.1} \frac{u_{2}}{0.2} \frac{u_{3}}{0.9}})}, \end{array}

Thus, we have the following EFSES:

\begin{array}{l} (H_{Λ_{s}}, Y) = & {((e_{1}, p, 1), {\frac{u_{1}}{0.82} \frac{u_{2}}{0.88} \frac{u_{3}}{0.96}}), ((e_{1}, q, 1), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.93} \frac{u_{3}}{0.96}}), \\ ((e_{1}, r, 1), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.88} \frac{u_{3}}{0.92}}), ((e_{2}, p, 1), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.91} \frac{u_{3}}{0.92}}), \\ ((e_{2}, q, 1), {\frac{u_{1}}{0.91} \frac{u_{2}}{0.93} \frac{u_{3}}{0.64}}), ((e_{2}, r, 1), {\frac{u_{1}}{0.79} \frac{u_{2}}{0.91} \frac{u_{3}}{0.0.8}}), \\ ((e_{3}, p, 1), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.95} \frac{u_{3}}{0.88}}), ((e_{3}, q, 1), {\frac{u_{1}}{0.76} \frac{u_{2}}{0.93} \frac{u_{3}}{0.88}}), \\ ((e_{3}, r, 1), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.84} \frac{u_{3}}{0.64}}), ((e_{1}, p, 0), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.91} \frac{u_{3}}{0.84}}), \\ ((e_{1}, q, 0), {\frac{u_{1}}{0.91} \frac{u_{2}}{0.88} \frac{u_{3}}{0.8}}), ((e_{1}, r, 0), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.95} \frac{u_{3}}{0.84}}), \\ ((e_{2}, p, 0), {\frac{u_{1}}{0.97} \frac{u_{2}}{0.97} \frac{u_{3}}{0.68}}), ((e_{2}, q, 0), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.86} \frac{u_{3}}{0.96}}), \\ ((e_{2}, r, 0), {\frac{u_{1}}{0.88} \frac{u_{2}}{0.91} \frac{u_{3}}{0.88}}), ((e_{3}, p, 0), {\frac{u_{1}}{0.94} \frac{u_{2}}{0.88} \frac{u_{3}}{0.92}}), \\ ((e_{3}, q, 0), {\frac{u_{1}}{0.85} \frac{u_{2}}{0.97} \frac{u_{3}}{0.8}}), ((e_{3}, r, 0), {\frac{u_{1}}{0.73} \frac{u_{2}}{0.82} \frac{u_{3}}{0.96}})} . \end{array}

Table 1 . Tabular representation of the agree-EFSES (H_{Λ_s}, Y )₁.

U	u₁	u₂	u₃
(e₁, p, 1)	0.82	0.88	0.96
(e₁, q, 1)	0.97	0.93	0.96
(e₁, r, 1)	0.85	0.88	0.92
(e₂, p, 1)	0.73	0.91	0.92
(e₂, q, 1)	0.91	0.93	0.64
(e₂, r, 1)	0.79	0.91	0.8
(e₃, p, 1)	0.97	0.95	0.88
(e₃, q, 1)	0.76	0.93	0.88
(e₃, r, 1)	0.85	0.84	0.64
Cℓ=∑j=1nujℓ	C₁ = 7.65	C₂ = 8.16	C₃ = 7.6

Table 2 . Tabular representation of the disagree-EFSES (H_{Λ_s}, Y)₀.

U	u₁	u₂	u₃
(e₁, p, 0)	0.97	0.91	0.84
(e₁, q, 0)	0.91	0.88	0.8
(e₁, r, 0)	0.94	0.95	0.84
(e₂, p, 0)	0.97	0.97	0.68
(e₂, q, 0)	0.73	0.86	0.96
(e₂, r, 0)	0.88	0.91	0.88
(e₃, p, 0)	0.94	0.88	0.92
(e₃, q, 0)	0.85	0.97	0.8
(e₃, r, 0)	0.73	0.82	0.96
Kℓ=∑j=1nujℓ	K₁ = 7.92	K₂ = 8.15	K₃ = 7.68

Table 3 . Score R_ℓ= C_ℓ– K_ℓ.

ℓ	U	C_ℓ	K_ℓ	R_ℓ
1	u₁	7.65	7.92	−0.27
2	u₂	8.16	8.15	0.01
3	u₃	7.6	7.68	−0.08

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SCOPUS 4

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Effective Fuzzy Soft Expert Set Theory and Its Applications

Abstract

1. Introduction

2. Preliminaries

Definition 2.1 ([5])

Definition 2.2 ([6])

Definition 2.3 ([12])

Definition 2.4 ([20])

Definition 2.5 ([29])

3. Effective Fuzzy Soft Expert Set

Definition 3.1

Definition 3.2

Definition 3.3

Example 3.1

Definition 3.4

Example 3.2

Definition 3.5

Example 3.3

Proposition 3.1

Proof

Definition 3.6

Example 3.4

Proposition 3.2

Proof

Definition 3.7

Example 3.5

Proposition 3.3

Proof

Definition 3.8

Example 3.6

Proposition 3.4

Proof

4. Applications

Example 4.1

5. Conclusion

Acknowledgments

Conflict of Interest

Tables

References

Biographies

Article

Original Article

Effective Fuzzy Soft Expert Set Theory and Its Applications

Abstract

1. Introduction

2. Preliminaries

Definition 2.1 ([5])

Definition 2.2 ([6])

Definition 2.3 ([12])

Definition 2.4 ([20])

Definition 2.5 ([29])

3. Effective Fuzzy Soft Expert Set

Definition 3.1

Definition 3.2

Definition 3.3

Example 3.1

Definition 3.4

Example 3.2

Definition 3.5

Example 3.3

Proposition 3.1

Proof

Definition 3.6

Example 3.4

Proposition 3.2

Proof

Definition 3.7

Example 3.5

Proposition 3.3

Proof

Definition 3.8

Example 3.6

Proposition 3.4

Proof

4. Applications

Example 4.1

5. Conclusion

References

International Journal
of Fuzzy Logic and
Intelligent Systems