**Contents**


#### **Songtao Shao, Xiaohong Zhang**


## **About the Editors**

**Stefan Vladutescu** (Professor Dr.) is a Professor of Communication and Information at University of Craiova, Romania. He is a graduate of University of Craiova and University of Bucharest and obtained his doctorate from University of Bucharest. He is a member of International Association of Communication (ICA) (USA), a member of Neutrosophic Science International Association (Gallup, NM, USA), and serves on the board of the Web of Science journal *Neutrosophic Sets and Systems* (USA) and *Polish Journal of Management Studies* (Poland). He is also director of Social Sciences and Education Research Review (Romania) and a member of the editorial board of *European Scientific Journal* (Macedonia). He is author or co-author of 15 books and more than 100 scientific papers (including ISI/Web of Science articles) and proceedings of international seminars and conferences.

**Mihaela Colhon** (Assoc. Prof. Dr.) is Associate Professor at Department of Computer Science, University of Craiova, Romania. She received her Ph.D. in 2009 in Computer Science for her work at the Department of Computer Science, Faculty of Mathematics and Computer Science, University of Pites,ti, Romania. Her research field is artificial intelligence, with specialization in knowledge representation, natural language processing (NLP), and human language technologies (HLT) as well as computational statistics and data analysis with applications in NLP.

## **Preface to "New Challenges in Neutrosophic Theory and Applications"**

Neutrosophic theory has representatives on all continents and, therefore, it can be said to be a universal theory. On the other hand, according to the three volumes of "The Encyclopedia of Neutrosophic Researchers" (2016, 2018, 2019), plus numerous others not yet included in Encyclopedia book series, about 1200 researchers from 73 countries have applied both the neutrosophic theory and method.

Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics lies in the introduction of the degree of indeterminacy/neutrality (I) as an independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus, etc., and their applications in multiple fields have been extended and applied in various fields, such as communication, management, and information technology.

We believe that this book serves as useful guidance for learning about the current progress in neutrosophic theories. In total, 22 studies have been presented and reflect the call of the thematic vision. The contents of each study included in the volume are briefly described as follows.

The first contribution, authored by Wadei Al-Omeri and Saeid Jafari, addresses the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets in neutrosophic topological spaces. In the article "Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution", the authors Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, and Abdur Razzaque Mughal discuss the use of probability distribution function of Birnbaum–Saunders distribution as a proportion of defective items and the acceptance probability in a fuzzy environment.

Further, the authors Derya Bakbak, Vakkas Uluc¸ay, and Memet S¸ahin present the "Neutrosophic Soft Expert Multiset and Their Application to Multiple Criteria Decision Making" together with several operations defined for them and their important algebraic properties.

In "Neutrosophic Multigroups and Applications", Vakkas Uluc¸ay and Memet S¸ahin propose an algebraic structure on neutrosophic multisets called neutrosophic multigroups, deriving their basic properties and giving some applications to group theory.

Changxing Fan, Jun Ye, Sheng Feng, En Fan, and Keli Hu introduce the "Multi-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environment" and test the effectiveness of their new methods.

Another decision-making study upon an everyday life issue which empowered us to organize the key objective of the industry developing is given in "Neutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Method" written by Khaleed Alhazaymeh, Muhammad Gulistan, Majid Khan, and Seifedine Kadry.

In "Refined Neutrosophy and Lattices vs. Pair Structures and YinYang Bipolar Fuzzy Set", Florentin Smarandache presents the lattice structures of neutrosophic theories, classifies Zhang-Zhang's YinYang bipolar fuzzy sets, and shows that the number of types of neutralities (sub-indeterminacies) may be any finite or infinite number.

The linguistic neutrosophic environment is treated in the study of Changxing Fan, Sheng Feng, and Keli Hu entitled "Linguistic Neutrosophic Numbers Einstein Operator and Its Application in Decision Making".

Vasantha Kandasamy W.B., Ilanthenral Kandasamy, and Florentin Smarandache propose several properties of "Semi-Idempotents in Neutrosophic Rings" and also suggest some open problems.

This continuation of this study is presented in the next article entitled "Neutrosophic Triplets in Neutrosophic Rings" by the same authors.

An article about neutrosophic statistics applied in a variable sampling plan is proposed by Muhammad Aslam and Mohammed Albassam in "Inspection Plan Based on the Process Capability Index Using the Neutrosophic Statistical Method".

"Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes" are investigated by Songtao Shao and Xiaohong Zhang in their applicability as concerns investment problems.

In the article "Neutrosophic Quadruple Vector Spaces and Their Properties", Vasantha Kandasamy W.B., Ilanthenral Kandasamy, and Florentin Smarandache introduce, for the first time in the literature, the concept of neutrosophic quadruple (NQ) vector spaces and neutrosophic quadruple linear algebras.

In the next study, Muhammad Aslam and Osama Hasan Arif propose the use of "Classification of the State of Manufacturing Process under Indeterminacy" in an uncertainty environment in order to eliminate the non-conforming items and increase the profit of the company.

The neutrosophic statistics under the assumption that the product lifetime follows a Weibull distribution is studied by Muhammad Aslam, P. Jeyadurga, Saminathan Balamurali, and Ali Hussein AL-Marshadi in their article "Time-Truncated Group Plan under a Weibull Distribution based on Neutrosophic Statistics".

Muhammad Aslam, Ali Hussein AL-Marshadi, and Nasrullah Khan propose "A New X-Bar Control Chart for Using Neutrosophic Exponentially Weighted Moving Average" for monitoring data under an uncertainty environment. The modern portfolio theory is addressed by Marcel-Ioan Bolos, , Ioana-Alexandra Bradea, and Camelia Delcea in their paper "Neutrosophic Portfolios of Financial Assets. Minimizing the Risk of Neutrosophic Portfolios" using an innovative approach determined by the use of the neutrosophic triangular fuzzy numbers.

Next, Xiaogang An, Xiaohong Zhang, and Yingcang Ma propose the notion of "Generalized Abel-Grassmann's Neutrosophic Extended Triplet Loop" together with its properties.

Based on the theories of AG-groupoid, neutrosophic extended triplet and semigroup, Wangtao Yuan and Xiaohong Zhang present some important results in "Regular CA-Groupoids and Cyclic Associative Neutrosophic Extended Triplet Groupoids (CA-NET-Groupoids) with Green Relations".

In "Multi-Attribute Group Decision Making Based on Multigranulation Probabilistic Models with Interval-Valued Neutrosophic Information", the authors Chao Zhang, Deyu Li, Xiangping Kang, Yudong Liang, Said Broumi, and Arun Kumar Sangaiah present an approach intended to handle MAGDM issues with interval-valued neutrosophic information.

Nguyen Tho Thong, Luong Thi Hong Lan, Shuo-Yan Chou, Le Hoang Son, Do Duc Dong, and Tran Thi Ngan propose "An Extended TOPSIS Method with Unknown Weight Information in Dynamic Neutrosophic Environment" together with a practical example intended to illustrate the feasibility and effectiveness of the proposed method.

The last article included in this volume is dedicated to a popular fuzzy tool used to describe the deviation information in uncertain complex situations. The study "Single-Valued Neutrosophic Linguistic Logarithmic Weighted Distance Measures and Their Application to Supplier Selection of Fresh Aquatic Products", written by Jiefeng Wang, Shouzhen Zeng, and Chonghui Zhang, is based on SVNLS and also presents a case study for testing the performance of the proposed framework.

This book would not have been possible without the skills and efforts of many people: first, the advisory board who guided the editors through the editorial process; second, the contributors who have provided perspectives of their neutrosophic works; and third, the reviewers for their service in critically reviewing book chapters.

#### **Stefan Vladutescu, Mihaela Colhon**

*Editors*

## *Article* **On Generalized Closed Sets and Generalized Pre-Closed Sets in Neutrosophic Topological Spaces**

**Wadei Al-Omeri 1,\*,†,‡ and Saeid Jafari 2,‡**


Received: 17 November 2018; Accepted: 13 December 2018; Published: 20 December 2018

**Abstract:** In this paper, the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets are introduced. We also study relations and various properties between the other existing neutrosophic open and closed sets. In addition, we discuss some applications of generalized neutrosophic pre-closed sets, namely neutrosophic *pT*<sup>1</sup> 2 space and neutrosophic *gpT*<sup>1</sup> 2 space. The concepts of generalized neutrosophic connected spaces, generalized neutrosophic compact spaces and generalized neutrosophic extremally disconnected spaces are established. Some interesting properties are investigated in addition to giving some examples.

**Keywords:** neutrosophic topology; neutrosophic generalized topology; neutrosophic generalized pre-closed sets; neutrosophic generalized pre-open sets; neutrosophic *pT*<sup>1</sup> 2 space; neutrosophic *gpT*<sup>1</sup> 2 space; generalized neutrosophic compact and generalized neutrosophic compact

#### **1. Introduction**

Zadeh [1] introduced the notion of fuzzy sets. After that, there have been a number of generalizations of this fundamental concept. The study of fuzzy topological spaces was first initiated by Chang [2,3] in 1968. Atanassov [4] introduced the notion of intuitionistic fuzzy sets (IFs). This notion was extended to intuitionistic *L*-fuzzy setting by Atanassov and Stoeva [5], which currently has the name "intuitionistic *L*-topological spaces". Coker [6] introduced the notion of intuitionistic fuzzy topological space by using the notion of (IFs). The concept of generalized fuzzy closed set was introduced by Balasubramanian and Sundaram [7]. In various recent papers, Smarandache generalizes intuitionistic fuzzy sets and different types of sets to neutrosophic sets (*NSs*). On the non-standard interval, Smarandache, Peide and Lupianez defined the notion of neutrosophic topology [8–10]. In addition, Zhang et al. [11] introduced the notion of an interval neutrosophic set, which is a sample of a neutrosophic set and studied various properties.

Recently, Al-Omeri and Smarandache [12,13] introduced and studied a number of the definitions of neutrosophic closed sets, neutrosophic mapping, and obtained several preservation properties and some characterizations about neutrosophic of connectedness and neutrosophic connectedness continuity.

This paper is arranged as follows. In Section 2, we will recall some notions that will be used throughout this paper. In Section 3, we mention some notions in order to present neutrosophic generalized pre-closed sets and investigate its basic properties. In Sections 4 and 5, we study the neutrosophic generalized pre-open sets and present some of their properties. In addition, we provide an application of neutrosophic generalized pre-open sets. Finally, the concepts of generalized neutrosophic connected space, generalized neutrosophic compact space and generalized neutrosophic extremally disconnected spaces are introduced and established in Section 6 and some of their properties in neutrosophic topological spaces are studied.

This class of sets belongs to the important class of neutrosophic generalized open sets which is very useful not only in the deepening of our understanding of some special features of the already well-known notions of neutrosophic topology but also proves useful in neutrosophic multifunction theory in neutrosophic economy and also in neutrosophic control theory. The applications are vast and the researchers in the field are exploring these realms of research.

#### **2. Preliminaries**

**Definition 1.** *Let* Z *be a non-empty set. A neutrosophic set (NS for short) S*˜ *is an object having the form <sup>S</sup>*˜ <sup>=</sup> {*k*, *<sup>μ</sup>S*˜(*k*), *<sup>σ</sup>S*˜(*k*), *<sup>γ</sup>S*˜(*k*) : *<sup>k</sup>* <sup>∈</sup> <sup>Z</sup> }*, where <sup>γ</sup>S*˜(*k*)*, <sup>σ</sup>S*˜(*k*), *<sup>μ</sup>S*˜(*k*)*, and the degree of non-membership (namely γS*˜(*k*) *), the degree of indeterminacy (namely σS*˜(*k*)*), and the degree of membership function (namely <sup>μ</sup>S*˜(*k*)*), of each element k* <sup>∈</sup> <sup>Z</sup> *to the set S, see [* ˜ *14].*

A neutrosophic set *<sup>S</sup>*˜ <sup>=</sup> {*k*, *<sup>μ</sup>S*˜(*k*), *<sup>σ</sup>S*˜(*k*), *<sup>γ</sup>S*˜(*k*) : *<sup>k</sup>* <sup>∈</sup> <sup>Z</sup> } can be identified as *μS*˜(*k*), *<sup>σ</sup>S*˜(*k*), *<sup>γ</sup>S*˜(*k*) in 0−, 1+ on <sup>Z</sup> .

**Definition 2.** *Let <sup>S</sup>*˜ <sup>=</sup> *μS*˜(*k*), *<sup>σ</sup>S*˜(*k*), *<sup>γ</sup>S*˜(*k*) *be an NS on* <sup>Z</sup> *. [15] The complement of the set S*˜(*C*(*S*˜), *for short*) *may be defined as follows:*


Neutrosophic sets (*NSs*) 0*<sup>N</sup>* and 1*<sup>N</sup>* [14] in Z are introduced as follows:


2 − 1*<sup>N</sup>* can be defined as four types:


**Definition 3.** *Let k be a non-empty set, and generalized neutrosophic sets GNSs S*˜ *and R*˜ *be in the form <sup>S</sup>*˜ <sup>=</sup> {*k*, *<sup>μ</sup>S*˜(*k*), *<sup>σ</sup>S*˜(*k*), *<sup>γ</sup>S*˜(*k*)}*, <sup>B</sup>* <sup>=</sup> {*k*, *<sup>μ</sup>R*˜(*k*), *<sup>σ</sup>R*˜(*k*), *<sup>γ</sup>R*˜(*k*)}*. Then, we may consider two possible definitions for subsets* (*S*˜ <sup>⊆</sup> *<sup>R</sup>*˜) *[14]:*

*(i) <sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>* <sup>⇔</sup> *<sup>μ</sup>S*˜(*k*) <sup>≤</sup> *<sup>μ</sup>B*(*k*), *<sup>σ</sup>S*˜(*k*) <sup>≥</sup> *<sup>σ</sup>B*(*k*), *and <sup>γ</sup>S*˜(*k*) <sup>≤</sup> *<sup>γ</sup>B*(*k*)*, (ii) <sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>* <sup>⇔</sup> *<sup>μ</sup>S*˜(*k*) <sup>≤</sup> *<sup>μ</sup>B*(*k*), *<sup>σ</sup>S*˜(*k*) <sup>≥</sup> *<sup>σ</sup>B*(*k*), *and <sup>γ</sup>S*˜(*k*) <sup>≥</sup> *<sup>γ</sup>B*(*k*)*.*

**Definition 4.** *Let* {*S*˜ *<sup>j</sup>* : *j* ∈ *J*} *be an arbitrary family of NSs in* Z *. Then,*


**Definition 5.** *A neutrosophic topology (NT for short) [16] and a non empty set* Z *is a family* Γ *of neutrosophic subsets of* Z *satisfying the following axioms:*


*In this case, the pair* (Z , Γ) *is called a neutrosophic topological space (NTS for short) and any neutrosophic set in* Γ *is known as neutrosophic open set NOS* ∈ Z *. The elements of* Γ *are called neutrosophic open sets. A closed neutrosophic set R if and only if its C* ˜ (*R*˜) *is neutrosophic open.*

*Note that, for any NTS S in* ˜ (Z , Γ)*, we have NCl*(*S*˜*c*)=[*NInt*(*S*˜)]*<sup>c</sup> and N Int*(*S*˜*c*)=[*NCl*(*S*˜)]*c.*

**Definition 6.** *Let <sup>S</sup>*˜ <sup>=</sup> {*μS*˜(*k*), *<sup>σ</sup>S*˜(*k*), *<sup>γ</sup>S*˜(*k*)} *be a neutrosophic open set and <sup>B</sup>* <sup>=</sup> {*μB*(*k*), *<sup>σ</sup>B*(*k*), *<sup>γ</sup>B*(*k*)} *<sup>a</sup> neutrosophic set on a neutrosophic topological space* (Z , Γ)*. Then,*


**Definition 7.** *Let* (*k*, <sup>Γ</sup>) *be NT and <sup>S</sup>*˜ <sup>=</sup> {*k*, *<sup>μ</sup>S*˜(*k*), *<sup>σ</sup>S*˜(*k*), *<sup>γ</sup>S*˜(*k*)} *an NS in* <sup>Z</sup> *. Then,*


*It can be also shown that NCl*(*S*˜) *is an NCS and N Int*(*S*˜) *is an NOS in* Z *. We have*


**Definition 8.** *Let S be an NS and* ˜ (Z , Γ) *an NT. Then,*


The complement of *S*˜ is an NSOS, N*α*OS, NPOS, and NROS, which is called NSCS, N*α*CS, NPCS, and NRCS, resp.

**Definition 9.** *Let <sup>S</sup>*˜ <sup>=</sup> {*S*˜ 1, *S*˜ 2, *S*˜ <sup>3</sup>} *be an NS and* (<sup>Z</sup> , <sup>Γ</sup>) *an NT. Then, the* <sup>∗</sup>*-neutrosophic closure of <sup>S</sup>*˜ *(*∗ − *NCl*(*S*˜) *for short [12]) and* <sup>∗</sup>*-neutrosophic interior* (∗ − *NInt*(*S*˜) *for short [12]) of S are defined by* ˜


**Definition 10.** *An* (*NS*) *S*˜ *of an NT* (Z , Γ) *is called a generalized neutrosophic closed set [17] (GNC in short) if NCl*(*S*˜) <sup>⊆</sup> *B wherever* ˜ *<sup>S</sup>*˜ <sup>⊂</sup> *B and* ˜ *B is a neutrosophic closed set in* ˜ <sup>Z</sup> *.*

**Definition 11.** *An NS S*˜ *in an NT* Z *is said to be a neutrosophic α generalized closed set (NαgCS [18]) if <sup>N</sup>αNCl*(*S*˜) <sup>⊆</sup> *<sup>B</sup>*˜ *whensoever <sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜ *and <sup>B</sup>*˜ *is an NOS in* <sup>Z</sup> *. The complement <sup>C</sup>*(*S*˜) *of an <sup>N</sup>αgCS <sup>S</sup>*˜ *is an NαgOS in* Z *.*

#### **3. Neutrosophic Generalized Connected Spaces, Neutrosophic Generalized Compact Spaces and Generalized Neutrosophic Extremally Disconnected Spaces**

**Definition 12.** *Let* (Z , Γ) *and* (K , Γ1) *be any two neutrosophic topological spaces.*


*Equivalently g*−<sup>1</sup> *of every GN-open set in* (Z , Γ1) *is GN-open in* (Z , Γ)


**Definition 13.** *An NTS* (Z , Γ) *is said to be neutrosophic-T*<sup>1</sup> 2 *(NT*<sup>1</sup> 2 *in short) space if every GNC in* Z *is an NC in* Z *.*

**Definition 14.** *Let* (Z , Γ) *be any neutrosophic topological space.* (Z , Γ) *is said to be generalized neutrosophic disconnected (in shortly GN-disconnected) if there exists a generalized neutrosophic open and generalized neutrosophic closed set <sup>R</sup>*˜ *such that <sup>R</sup>*˜ <sup>=</sup> <sup>0</sup>*<sup>N</sup> and <sup>R</sup>*˜ <sup>=</sup> <sup>1</sup>*N.*(<sup>Z</sup> , <sup>Γ</sup>) *is said to be generalized neutrosophic connected if it is not generalized neutrosophic disconnected.*

**Proposition 1.** *Every GN-connected space is neutrosophic connected. However, the converse is not true.*

**Proof.** For a *GN*-connected (Z , Γ) space and let (Z , Γ) not be neutrosophic connected. Hence, there exists a proper neutrosophic set, *<sup>S</sup>*˜ <sup>=</sup> *μS*˜(*x*), *<sup>σ</sup>S*˜(*x*), *<sup>γ</sup>S*˜(*x*) *<sup>S</sup>*˜ <sup>=</sup> <sup>0</sup>*N*, *<sup>S</sup>*˜ <sup>=</sup> <sup>1</sup>*N*, such that *<sup>S</sup>*˜ is both neutrosophic open and neutrosophic closed in (Z , Γ). Since every neutrosophic open set is *GN*-open and neutrosophic closed set is *GN*-closed, Z is not *GN*-connected. Therefore, (Z , Γ) is neutrosophic connected.

**Example 1.** *Let* <sup>Z</sup> <sup>=</sup> {*u*, *<sup>v</sup>*, *<sup>w</sup>*}*. Define the neutrosophic sets <sup>S</sup>*˜, *<sup>R</sup>*˜ *and* <sup>Z</sup> *in* <sup>Z</sup> *as follows: <sup>S</sup>*˜ <sup>=</sup> *x*,( *<sup>a</sup>* 0.4 , *<sup>b</sup>* 0.5 , *<sup>c</sup>* 0.5 ),( *<sup>a</sup>* 0.4 , *<sup>b</sup>* 0.5 , *<sup>c</sup>* 0.5 ),( *<sup>a</sup>* 0.5 , *<sup>b</sup>* 0.5 , *<sup>c</sup>* 0.5 )*, <sup>R</sup>*˜ <sup>=</sup> *x*,( *<sup>a</sup>* 0.7 , *<sup>b</sup>* 0.6 , *<sup>c</sup>* 0.5 ),( *<sup>a</sup>* 0.7 , *<sup>b</sup>* 0.6 , *<sup>c</sup>* 0.5 ),( *<sup>a</sup>* 0.3 , *<sup>b</sup>* 0.4 , *<sup>c</sup>* 0.5 )*. Then, the family* <sup>Γ</sup> <sup>=</sup> {0*N*, 1*N*, *<sup>S</sup>*˜, *<sup>R</sup>*˜ } *is neutrosophic topology on* <sup>Z</sup> *. It is obvious that* (<sup>Z</sup> , <sup>Γ</sup>) *is NTS. Now,* (Z , Γ) *is neutrosophic connected. However, it is not a GN-connected for Z*˜ = *x*,( *<sup>a</sup>* 0.5 , *<sup>b</sup>* 0.6 , *<sup>c</sup>* 0.5 ),( *<sup>a</sup>* 0.5 , *<sup>b</sup>* 0.6 , *<sup>c</sup>* 0.5 ),( *<sup>a</sup>* 0.5 , *<sup>b</sup>* 0.6 , *<sup>c</sup>* 0.5 ) *is GN open and GN closed in* (Z , Γ)*.*

**Theorem 1.** *Let* (Z , Γ) *be a neutrosophic T*<sup>1</sup> 2 *space; then,* (Z , Γ) *is neutrosophic connected iff* (Z , Γ) *is GN-connected.*

**Proof.** Suppose that (Z , Γ) is not *GN*-connected, and there exists a neutrosophic set *S*˜ which is both *GN*-open and *GN*-closed. Since (Z , Γ) is neutrosophic *T*<sup>1</sup> 2 , *S*˜ is both neutrosophic open and neutrosophic closed. Hence, (Z , Γ) is *GN*-connected. Conversely, let (Z , Γ) is *GN*-connected. Suppose that (Z , Γ) is not neutrosophic connected, and there exists a neutrosophic set *<sup>S</sup>*˜ such that *<sup>S</sup>*˜ is both *NCs* and *NOs* <sup>∈</sup> (<sup>Z</sup> , <sup>Γ</sup>). Since the neutrosophic open set is *GN*-open and the neutrosophic closed set is *GN*-closed, (Z , Γ) is not *GN*-connected. Hence, (Z , Γ) is neutrosophic connected.

**Proposition 2.** *Suppose* (Z , Γ) *and* (K , Γ1) *are any two NTSs. If g* : (Z , Γ) −→ (K , Γ1) *is GN-continuous surjection and* (Z , Γ) *is GN-connected, then* (K , Γ1) *is neutrosophic connected.*

**Proof.** Suppose that (K , Γ1) is not neutrosophic connected, such that the neutrosophic set *S*˜ is both neutrosophic open and neutrosophic closed in (K , Γ1). Since *g* is *GN*-continuous, *g*−1(*S*˜) is *GN*-open and *GN*-closed in ((K , Γ). Thus, (K , Γ) is not *GN* connected. Hence, (K , Γ1) is neutrosophic connected.

**Definition 15.** *Let* (K , Γ) *be an NT. If a family* {*k*, *μGi* (*k*), *σGi* (*k*), *γGi* (*k*) : *i* ∈ *J*} *of GN open sets in* (K , Γ) *satisfies the condition* -{*k*, *μGi* (*k*), *σGi* (*k*), *γGi* (*k*) : *i* ∈ *J*} = 1*N, then it is called a GN open cover of* (K , Γ)*. A finite subfamily of a GN open cover* {*k*, *μGi* (*k*), *σGi* (*k*), *γGi* (*k*) : *i* ∈ *J*} *of* (Z , Γ)*, which is also a GN open cover of* (K , Γ) *is called a finite subcover of*

$$\{ \langle k, \mu\_{G\_i}(k), \sigma\_{G\_i}(k), \gamma\_{G\_i}(k) : i \in J \rangle \}.$$

**Definition 16.** *An NT* (K , Γ) *is called GN compact iff every GN open cover of* (K , Γ) *has a finite subcover.*

**Theorem 2.** *Let* (K , Γ) *and* (K , Γ1) *be any two NTs, and g* : (Z , Γ) −→ (K , Γ1) *be GN continuous surjection. If* (K , Γ) *is GN-compact, hence so is* (K , Γ1).

**Proof.** Let *Gi* = {*y*, *μGi* (*x*), *σGi* (*x*), *γGi* (*x*) : *i* ∈ *J*} be a neutrosophic open cover in (K , Γ1) with

$$\widehat{\bigcup} \{ \langle y, \mu\_{G\_i}(\mathbf{x}), \sigma\_{G\_i}(\mathbf{x}), \gamma\_{G\_i}(\mathbf{x}) : i \in f \rangle \} = \widehat{\bigcup}\_{i \in f} G\_i = \mathbf{1}\_N.$$

Since *<sup>g</sup>* is *GN* continuous, *<sup>g</sup>*−1(*Gi*) = *Gi* <sup>=</sup> {*y*, *<sup>μ</sup>g*−1(*Gi*)(*x*), *<sup>σ</sup>g*−1(*Gi*)(*x*), *<sup>γ</sup>g*−1(*Gi*)(*x*) : *<sup>i</sup>* <sup>∈</sup> *<sup>J</sup>*} is *GN* open cover of (K , Γ). Now,

$$\overline{\bigcup\_{i \in J}} \mathcal{g}^{-1}(G\_i) = \mathcal{g}^{-1}(\overline{\bigcup\_{i \in J}} G\_i) = 1\_N.$$

Since (K , Γ) is *GN* compact, there exists a finite subcover *J*<sup>0</sup> ⊂ *J*, such that

$$\bigcup\_{i \in l\_0} \mathbb{g}^{-1}(\mathbf{G}\_i) = \mathbf{1}\_N.$$

Hence,

$$\mathfrak{g}\left(\bigcup\_{i\in J\_0} \mathfrak{g}^{-1}(G\_i) = 1\_N\right) \mathfrak{g}^{-1}\left(\bigcup\_{i\in J\_0} (G\_i) = 1\_N\right).$$

That is,

$$\bigcup\_{i \in I \cup} (\mathcal{G}\_i) = 1\_N.$$

Therefore, (K , Γ1) is neutrosophic compact.

**Definition 17.** *Let* (K , Γ) *be an NT and K be a neutrosophic set in* (Z , Γ)*. If a family* {*k*, *μGi* (*k*), *σGi* (*k*), *γGi* (*k*) : *i* ∈ *J*} *of GN open sets in* (K , Γ) *satisfies the condition K* ⊆ -{*k*, *μGi* (*k*), *σGi* (*k*), *γGi* (*k*) : *i* ∈ *J*} = 1*N, then it is called a GN open cover of K. A finite subfamily of a GN open cover* {*k*, *μGi* (*k*), *σGi* (*k*), *γGi* (*k*) : *i* ∈ *J*} *of K, which is also a GN open cover of K is called a finite subcover of* {*k*, *μGi* (*k*), *σGi* (*k*), *γGi* (*k*) : *i* ∈ *J*}*.*

**Definition 18.** *An NT* (K , Γ) *is called GN compact iff every GN open cover of K has a finite subcover.*

**Theorem 3.** *Let* (K , Γ) *and* (K , Γ1) *be any two NTs, and g* : (Z , Γ) −→ (K , Γ1) *be an GN continuous function. If K is GN-compact, then so is g*(*K*) *in* (K , Γ1)*.*

**Proof.** Let *Gi* = {*y*, *μGi* (*x*), *σGi* (*x*), *γGi* (*x*) : *i* ∈ *J*} be a neutrosophicopen cover of *g*(*K*) in (K , Γ1). That is,

$$\mathfrak{g}(K) \subseteq \overline{\bigcup\_{i \in I}} G\_i.$$

Since *<sup>g</sup>* is *GN* continuous, *<sup>g</sup>*−1(*Gi*) = {*x*, *<sup>μ</sup>g*−1(*Gi*)(*x*), *<sup>σ</sup>g*−1(*Gi*)(*x*), *<sup>γ</sup>g*−1(*Gi*)(*x*) : *<sup>i</sup>* <sup>∈</sup> *<sup>J</sup>*} is *GN* open cover of *K* in (Z , Γ). Now,

$$K \subseteq \mathcal{g}^{-1}(\bigcup\_{i \in J} G\_i) \subseteq \bigcup\_{i \in J} \mathcal{g}^{-1}(G\_i).$$

Since *K* is (Z , Γ) is *GN* compact, there exists a finite subcover *J*<sup>0</sup> ⊂ *J*, such that

$$\mathbb{K} \subseteq \overline{\bigcup\_{i \in \mathbb{J}}} \mathbb{g}^{-1}(\mathcal{G}\_i) = \mathbf{1}\_N.$$

Hence,

$$\mathfrak{g}(K) \subseteq \mathfrak{g}\left(\coprod\_{i \in J\_0} \mathfrak{g}^{-1}(G\_i)\right) \overline{\bigcup\_{i \in J\_0}}(G\_i).$$

Therefore, *g*(*K*) is neutrosophic compact.

**Proposition 3.** *Let* (Z , Γ) *be a neutrosophic compact space and suppose that K is a GN-closed set of* (Z , Γ)*. Then, K is a neutrosophic compact set.*

**Proof.** Let *Kj* == {*y*, *μGi* (*x*), *σGi* (*x*), *γGi* (*x*) : *i* ∈ *J*} be a family of neutrosophic open set in (Z , Γ) such that

$$\mathbb{K} \subseteq \overline{\bigcup\_{i \in \mathcal{J}} \mathbb{K}\_j}.$$

Since *K* is *GN*-closed, *NCl*(*K*) ⊆ - *<sup>i</sup>*∈*<sup>J</sup> Kj*. Since (<sup>Z</sup> , <sup>Γ</sup>) is a neutrosophic compact space, there exists a finite subcover *J*<sup>0</sup> ⊆ *J*. Now, *NCl*(*K*) ⊆ - *<sup>i</sup>*∈*J*<sup>0</sup> *Kj*. Hence, *<sup>K</sup>* ⊆ *NCl*(*K*) ⊆ - *<sup>i</sup>*∈*J*<sup>0</sup> *Kj*. Therefore, *K* is a neutrosophic compact set.

**Definition 19.** *Let* (Z , Γ) *be any neutrosophic topological space.* (Z , Γ) *is said to be GN extremally disconnected if NCl*(*K*) *neutrosophic open and K is GN open.*

**Proposition 4.** *For any neutrosophic topological space* (Z , Γ)*, the following are equivalent:*


#### **4. Generalized Neutrosophic Pre-Closed Set**

**Definition 20.** *An NS S*˜ *is said to be a neutrosophic generalized pre-closed set (GNPCS in short) in* (Z , Γ) *if pNCl*(*S*˜) <sup>⊆</sup> *<sup>B</sup>*˜ *whensoever <sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜ *and <sup>B</sup>*˜ *is an NO in* <sup>Z</sup> *. The family of all GNPCSs of an NT* (<sup>Z</sup> , <sup>Γ</sup>) *is defined by GNPC*(Z )*.*

**Example 2.** *Let* Z = {*a*, *b*} *and* Γ = {0*N*, 1*N*, *T*} *be a neutrosophic topology on* Z *, where T* = (0.2, 0.3, 0.5),(0.8, 0.7, 0.7)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> (0.2, 0.2, 0.2),(0.8, 0.7, 0.7) *is GNPCs* <sup>∈</sup> <sup>Z</sup> *.*

**Theorem 4.** *Every NC is a GNPC, but the converse is not true.*

**Proof.** Let *<sup>S</sup>*˜ be an *NC* in <sup>Z</sup> , *<sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜ and *<sup>B</sup>*˜ is *NOS* in (<sup>Z</sup> , <sup>Γ</sup>). Since *pNCl*(*S*˜) <sup>⊆</sup> *NCl*(*S*˜) and *<sup>S</sup>*˜ is *NCS* in <sup>Z</sup> , *pNCl*(*S*˜) <sup>⊆</sup> *NCl*(*S*˜) = *<sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜. Therefore, *<sup>S</sup>*˜ is *GNPCs* <sup>∈</sup> <sup>Z</sup> .

**Example 3.** *Let* Z = {*u*, *v*} *and* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where H* = (0.2, 0.3, 0.5),(0.8, 0.7, 0.7)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> (0.2, 0.2, 0.2),(0.8, 0.7, 0.7) *is a GNPC in* <sup>Z</sup> *but not an NCS* ∈ Z *.*

**Theorem 5.** *Every NαCS is GNPC, but the converse is not true.*

**Proof.** Let *<sup>S</sup>*˜ be an *<sup>N</sup>αCS* in <sup>Z</sup> and let *<sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜ and *<sup>B</sup>*˜ is an *NOS* in (<sup>Z</sup> , <sup>Γ</sup>). Now, *NCl*(*NInt*(*NCl*(*S*˜))) <sup>⊆</sup> *<sup>S</sup>*˜. Since *<sup>S</sup>*˜ <sup>⊆</sup> *NCl*(*S*˜), *NCl*(*NInt*(*S*˜)) <sup>⊆</sup> *NCl*(*NInt*(*NCl*(*S*˜))) <sup>⊆</sup> *<sup>S</sup>*˜. Hence, *pNCl*(*S*˜) <sup>⊆</sup> *<sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜. Therefore, *<sup>S</sup>*˜ is *GNPCs* <sup>∈</sup> <sup>Z</sup> .

**Example 4.** *Let* Z = {*u*, *v*} *and let* Γ = {0*N*, 1*N*, *H*} *is a neutrosophic topology on* Z *, where H* = (0.4, 0.2, 0.5),(0.6, 0.7, 0.6)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> (0.3, 0.1, 0.4),(0.7, 0.8, 0.7) *is a GNPC in* <sup>Z</sup> *but not <sup>N</sup>αCs in* <sup>Z</sup> *since NCl*(*NInt*(*NCl*(*S*˜))) = (0.5, 0.6, 0.5),(0.5, 0.3, 0.6) ⊂ *S.*˜

**Theorem 6.** *Every GNαC is a GNPC, but the converse is not true.*

**Proof.** Let *<sup>S</sup>*˜ be *GNαCs* <sup>∈</sup> <sup>Z</sup> , *<sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜, *<sup>B</sup>*˜ be an *NOs*in (<sup>Z</sup> , <sup>Γ</sup>). By Definition 6, *<sup>S</sup>*˜ <sup>∪</sup> *NCl*(*NInt*(*NCl*(*S*˜))) <sup>⊆</sup> *<sup>B</sup>*˜. This implies *NCl*(*NInt*(*NCl*(*S*˜))) <sup>⊆</sup> *<sup>B</sup>*˜ and *NCl*(*NInt*(*S*˜)) <sup>⊆</sup> *<sup>B</sup>*˜. Therefore, *pNCl*(*S*˜) = *<sup>S</sup>*˜ <sup>∪</sup> *NCl*(*NInt*(*S*˜)) <sup>⊆</sup> *<sup>B</sup>*˜. Hence, *<sup>S</sup>*˜ is *GNPCs* <sup>∈</sup> <sup>Z</sup> .

**Example 5.** *Let* Z = {*u*, *v*} *and* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where H* = (0.5, 0.6, 0.6),(0.5, 0.4, 0.4)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> (0.4, 0.5, 0.5),(0.6, 0.5, 0.5) *is GNPC in* <sup>Z</sup> *but not GNαC in* <sup>Z</sup> *since <sup>α</sup>NCl*(*S*˜) = <sup>1</sup>*<sup>N</sup>* ⊂ *H.*

**Definition 21.** *An NS S*˜ *is said to be a neutrosophic generalized pre-closed set (GNSCS ) in* (Z , Γ) *if SNCl*(*S*˜) <sup>⊆</sup> *<sup>B</sup>*˜ *whensoever <sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜ *and <sup>B</sup>*˜ *is an NO in* <sup>Z</sup> *. The family of all GNSCSs of an NT* (<sup>Z</sup> , <sup>Γ</sup>) *is defined by GNSC*(Z )*.*

**Proposition 5.** *Let S*˜, *B be a two GNPCs of an NT* (Z , Γ)*. NGSC and NGPC are independent.*

**Example 6.** *Let* Z = {*u*, *v*}*,* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where H* = (0.5, 0.4, 0.4),(0.5, 0.6, 0.5)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> *<sup>H</sup> is GNSC but not GNPC in* <sup>Z</sup> *since <sup>S</sup>*˜ <sup>⊆</sup> *<sup>H</sup> but pNCl*(*S*˜) = (0.5, 0.6, 0.4),(0.5, 0.4, 0.5) ⊂ *<sup>H</sup>*

**Example 7.** *Let* Z = {*u*, *v*}*,* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where H* = (0.7, 0.9, 0.7),(0.3, 0.1, 0.1)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> (0.6, 0.7, 0.6),(0.4, 0.3, 0.4) *is GNPC but not GNsC in* <sup>Z</sup> *since sNCl*(*S*˜) = <sup>1</sup>*<sup>N</sup>* <sup>⊆</sup> *H.*

**Proposition 6.** *NSC and GNPC are independent.*

**Example 8.** *Let* Z = {*a*, *b*}*,* Γ = {0*N*, 1*N*, *T*} *be a neutrosophic topology on* Z *, where T* = (0.5, 0.2, 0.3),(0.5, 0.6, 0.5)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> *<sup>T</sup> is an NSC but not GNPC in* <sup>Z</sup> *since <sup>S</sup>*˜ <sup>⊆</sup> *<sup>T</sup> but pNCl*(*S*˜) = <sup>1</sup>(0.5, 0.6, 0.5),(0.5, 0.2, 0.3) ⊂ *T.*

**Example 9.** *Let* Z = {*u*, *v*}*,* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where <sup>H</sup>* <sup>=</sup> (0.8, 0.8, 0.8),(0.2, 0.2, 0.2)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> (0.8, 0.8, 0.8),(0.2, 0.2, 0.2) *is GNPC but not an NSC in* <sup>Z</sup> *since N Int*(*NCl*(*S*˜)) ⊂ *S.*˜

*Mathematics* **2018**, *7*, 1

The following Figure 1 shows the implication relations between *GNPC* set and the other existed ones.

**Figure 1.** Relation between *GNPC* and others exists set.

**Remark 1.** *Let S*˜, *B be a two GNPCs of an NT* (Z , Γ)*. Then, the union of any two GNPCs is not a GNPC in general—see the following example.*

**Example 10.** *Let* (Z , Γ) *be a neutrosophic topology set on* Z *, where* Z = {*u*, *v*}*, T* = (0.6, 0.8, 0.6),(0.4, 0.2, 0.2)*. Then,* Γ = {0*N*, 1*N*, *T*} *is neutrosophic topology on* Z *and the NS <sup>S</sup>*˜ <sup>=</sup> (0.2, 0.9, 0.3),(0.8, 0.2, 0.6)*, <sup>B</sup>* <sup>=</sup> (0.6, 0.7, 0.6),(0.4, 0.3, 0.4) *are GNPCSs but <sup>S</sup>*˜ <sup>∪</sup> *<sup>B</sup> is not a GNPC in* Z *.*

#### **5. Generalized Neutrosophic Pre-Open Sets**

In this section, we present generalized neutrosophic pre-open sets and investigate some of their properties.

**Definition 22.** *An NS S*˜ *is said to be a generalized neutrosophic pre-open set (GNPOS ) in* (Z , Γ) *if the complement S*˜*<sup>c</sup> is a GNPCS in* Z *. The family of all GNPOSs of NTS* (Z , Γ) *is denoted by GNPO*(Z )*.*

**Example 11.** *Let* Z = {*u*, *v*} *and* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where <sup>H</sup>* <sup>=</sup> (0.8, 0.7, 0.8),(0.3, 0.4, 0.3)*. Then, the NS <sup>S</sup>*˜ <sup>=</sup> (0.9, 0.8, 0.8),(0.3, 0.3, 0.3) *is GNPO* <sup>∈</sup> <sup>Z</sup> *.*

**Theorem 7.** *Let* (<sup>Z</sup> , <sup>Γ</sup>) *be an NT. Then, for every <sup>S</sup>*˜ <sup>∈</sup> *GNPO*(<sup>Z</sup> ) *and for every <sup>R</sup>*˜ <sup>∈</sup> *NS*(<sup>Z</sup> )*, pNInt*(*S*˜) <sup>⊆</sup> *<sup>R</sup>*˜ <sup>⊆</sup> *S implies* ˜ *<sup>R</sup>*˜ <sup>∈</sup> *GNPO*(<sup>Z</sup> )*.*

**Proof.** By Theorem *<sup>S</sup>*˜*<sup>c</sup>* <sup>⊆</sup> *<sup>R</sup>*˜ *<sup>c</sup>* <sup>⊆</sup> (*pNInt*(*S*˜))*c*. Let *<sup>R</sup>*˜ *<sup>c</sup>* <sup>⊆</sup> *<sup>R</sup>*˜ and *<sup>R</sup>*˜ be *NOs*. Since *<sup>S</sup>*˜*<sup>c</sup>* <sup>⊆</sup> *<sup>B</sup>c*, *<sup>S</sup>*˜*<sup>c</sup>* <sup>⊆</sup> *<sup>R</sup>*˜. However, *<sup>S</sup>*˜*<sup>c</sup>* is a *GNPCs*, *pNCl*(*S*˜*c*) <sup>⊆</sup> *<sup>R</sup>*˜. In addition, *<sup>R</sup>*˜ *<sup>c</sup>* <sup>⊆</sup> (*pNInt*(*S*˜))*<sup>c</sup>* <sup>=</sup> *pNCl*(*S*˜*c*) (by theorem). Therefore, *pNCl*(*R*˜ *<sup>c</sup>*) <sup>⊆</sup> *pNCl*(*S*˜*c*) <sup>⊆</sup> *<sup>R</sup>*˜. Hence, *<sup>B</sup><sup>c</sup>* is *GNPC*. This implies that *<sup>R</sup>*˜ is a *GNPO* of <sup>Z</sup> .

**Remark 2.** *Let S*˜, *R*˜ *be two GNPOs of an NT* (Z , Γ)*. The intersection of any two GNPOSs is not a GNPO in general.*

**Example 12.** *Let* Z = {*u*, *v*} *and* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where <sup>H</sup>* <sup>=</sup> (0.6, 0.8, 0.6),(0.4, 0.2, 0.4)*. Then, the NSs, <sup>S</sup>*˜ <sup>=</sup> (0.9, 0.2, 0.1),(0.1, 0.8, 0.2) *and <sup>R</sup>*˜ <sup>=</sup> (0.4, 0.3, 0.4),(0.6, 0.7, 0.6) *is GNPO, but <sup>S</sup>*˜ <sup>∩</sup> *R is not GNPO* ˜ <sup>∈</sup> <sup>Z</sup> *.*

**Theorem 8.** *For any an NTS* (Z , Γ)*, the following hold:*

*(i) Every NO is GNPO, (ii) Every NSO is GNPO, (iii) Every NαO is GNPO, (iv) Every NPO is GNPO.*

**Proof.** The proof is clear, so it has been omitted.

The converses are not true in general.

**Example 13.** *Let* Z = {*u*, *v*} *and H* = (0.2, 0.3, 0.2),(0.8, 0.7, 0.7)*. Then,* Γ = {0*N*, 1*N*, *H*} *is a neutrosophic topology on* <sup>Z</sup> *, an NS <sup>S</sup>*˜ <sup>=</sup> (0.8, 0.7, 0.7),(0.2, 0.2, 0.2) *is an NSO in* (<sup>Z</sup> , <sup>Γ</sup>) *but not an NO* ∈ Z *.*

**Example 14.** *Let* Z = {*u*, *v*} *and* Γ = {0*N*, 1*N*, *H*} *be neutrosophic topology on* Z *, where <sup>H</sup>* <sup>=</sup> (0.6, 0.4, 0.7),(0.7, 0.4, 0.6)*. Then, an NS <sup>S</sup>*˜ <sup>=</sup> (0.2, 0.7, 0.7),(0.8, 0.3, 0.8) *is GNPO but not an NSO* ∈ Z *.*

**Example 15.** *Let* Z = {*u*, *v*} *and* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where <sup>H</sup>* <sup>=</sup> (0.4, 0.2, 0.4),(0.6, 0.7, 0.6)*. Then, an NS <sup>S</sup>*˜ <sup>=</sup> (0.8, 0.9, 0.8),(0.4, 0.2, 0.3) *is GNPO but not an NαO* ∈ Z *.*

**Example 16.** *Let* Z = {*u*, *v*} *and* Γ = {0*N*, 1*N*, *H*} *be a neutrosophic topology on* Z *, where <sup>H</sup>* <sup>=</sup> (0.6, 0.5, 0.6),(0.5, 0.6, 0.5)*. Then, an NS <sup>S</sup>*˜ <sup>=</sup> (0.8, 0.7, 0.8),(0.4, 0.5, 0.3) *is GNPO but not an NPO* ∈ Z *.*

**Theorem 9.** *Let* (<sup>Z</sup> , <sup>Γ</sup>) *be an NT. If <sup>S</sup>*˜ <sup>∈</sup> *GNPO*(<sup>Z</sup> )*, then <sup>R</sup>*˜ <sup>⊆</sup> *NInt*(*NCl*(*S*˜)) *whensoever <sup>R</sup>*˜ <sup>⊆</sup> *<sup>S</sup>*˜ *and <sup>R</sup>*˜ *is an NC in* Z *.*

**Proof.** Let *<sup>S</sup>*˜ <sup>∈</sup> *GNPO*(<sup>Z</sup> ). Then, *<sup>S</sup>*˜*<sup>c</sup>* is *GnPCS* in <sup>Z</sup> . Therefore, *pNCl*(*S*˜*c*) <sup>⊆</sup> *<sup>B</sup>*˜ whensoever *<sup>S</sup>*˜*<sup>c</sup>* <sup>⊆</sup> *<sup>B</sup>*˜ and *<sup>B</sup>*˜ is an *NO* in <sup>Z</sup> . That is, *NCl*(*NInt*(*S*˜*c*)) <sup>⊆</sup> *<sup>B</sup>*˜. This implies *<sup>B</sup>*˜ *<sup>c</sup>* <sup>⊆</sup> *NInt*(*NCl*(*S*˜)) whensoever *<sup>B</sup>*˜ *<sup>c</sup>* <sup>⊆</sup> *<sup>S</sup>*˜ and *<sup>B</sup>*˜ *<sup>c</sup>* is *NCs* in <sup>Z</sup> . Replacing *<sup>B</sup>*˜ *<sup>c</sup>*, by *<sup>R</sup>*˜, we get *<sup>R</sup>*˜ <sup>⊆</sup> *NInt*(*NCl*(*S*˜)) whensoever *<sup>R</sup>*˜ <sup>⊆</sup> *<sup>S</sup>*˜ and *<sup>R</sup>*˜ is an *NC* in Z .

**Theorem 10.** *For NS S,*˜ *S is an NO and GNPC in* ˜ Z *if and only if S is an NRO in* ˜ Z *.*

**Proof.** <sup>=</sup><sup>⇒</sup> Let *<sup>S</sup>*˜ be an *NO* and a *GNPCS* in <sup>Z</sup> . Then, *pNCl*(*S*˜) <sup>⊆</sup> *<sup>S</sup>*˜. This implies *NCl*(*NInt*(*S*˜)) <sup>⊆</sup> *<sup>S</sup>*˜. Since *<sup>S</sup>*˜ is an *NO*, it is an *NPO*. Hence, *<sup>S</sup>*˜ <sup>⊆</sup> *NInt*(*NCl*(*S*˜)). Therefore, *<sup>S</sup>*˜ <sup>=</sup> *NInt*(*NCl*(*S*˜)). Hence, *<sup>S</sup>*˜ is an *NRO* in Z .

⇐<sup>=</sup> Let *<sup>S</sup>*˜ be an *NRO* in <sup>Z</sup> . Therefore, *<sup>S</sup>*˜ <sup>=</sup> *NInt*(*NCl*(*S*˜)). Let *<sup>S</sup>*˜ <sup>⊆</sup> *<sup>B</sup>*˜ and *<sup>B</sup>*˜ be an *NO* in <sup>Z</sup> . This implies *pNCl*(*S*˜) <sup>⊆</sup> *<sup>S</sup>*˜. Hence, *<sup>S</sup>*˜ is *GNPC* in <sup>Z</sup> .

**Theorem 11.** *An NS <sup>S</sup>*˜ *of an NT* (<sup>Z</sup> , <sup>Γ</sup>) *is a GNPO iff <sup>H</sup>* <sup>⊆</sup> *pNInt*(*S*˜)*, whensoever <sup>H</sup> is an NC and <sup>H</sup>* <sup>⊆</sup> *<sup>S</sup>*˜*.*

**Proof.** <sup>=</sup><sup>⇒</sup> Let *<sup>S</sup>*˜ be *GNPO* in <sup>Z</sup> . Let *<sup>H</sup>* be an *NCs* and *<sup>H</sup>* <sup>⊆</sup> *<sup>S</sup>*˜. Then, *<sup>H</sup><sup>c</sup>* is an *NOS* in <sup>Z</sup> such that *<sup>S</sup>*˜*<sup>c</sup>* <sup>⊆</sup> *<sup>H</sup>c*. Since *<sup>S</sup>*˜*<sup>c</sup>* is *GNPC*, we have *pNCl*(*S*˜*c*) <sup>⊆</sup> *<sup>H</sup>c*. Hence, (*pNInt*(*S*˜))*<sup>c</sup>* <sup>⊆</sup> *<sup>H</sup>c*. Therefore, *<sup>H</sup>* <sup>⊆</sup> *pNInt*(*S*˜).

⇐<sup>=</sup> Suppose *<sup>S</sup>*˜ is an *NS* of <sup>Z</sup> and let *<sup>H</sup>* <sup>⊆</sup> *pNInt*(*S*˜) whensoever *<sup>H</sup>* is an *NC* and *<sup>H</sup>* <sup>⊆</sup> *<sup>S</sup>*˜. Then, *<sup>S</sup>*˜*<sup>c</sup>* <sup>⊆</sup> *<sup>H</sup><sup>c</sup>* and *<sup>H</sup><sup>c</sup>* is an *NO*. By assumption, (*pNInt*(*S*˜))*<sup>c</sup>* <sup>⊆</sup> *<sup>H</sup>c*, which implies *pNCl*(*S*˜*c*) <sup>⊆</sup> *<sup>H</sup>c*. Therefore, *S*˜*<sup>c</sup>* is *GNPCs* of Z . Hence, *S*˜ is a *GNPOS* of Z .

**Corollary 1.** *An NS <sup>S</sup>*˜ *of an NTS* (<sup>Z</sup> , <sup>Γ</sup>) *is GNPO iff <sup>H</sup>* <sup>⊆</sup> *NInt*(*NCl*(*S*˜))*, whensoever <sup>H</sup> is an NC and <sup>H</sup>* <sup>⊆</sup> *S.*˜

**Proof.** <sup>=</sup><sup>⇒</sup> Let *<sup>S</sup>*˜ is a *GNPOS* in <sup>Z</sup> . Let *<sup>H</sup>* be an *NCS* and *<sup>H</sup>* <sup>⊆</sup> *<sup>S</sup>*˜. Then, *<sup>H</sup><sup>c</sup>* is an *NOS* in <sup>Z</sup> such that *<sup>S</sup>*˜*<sup>c</sup>* <sup>⊆</sup> *<sup>H</sup>c*. Since *<sup>S</sup>*˜*<sup>c</sup>* is *GNPC*, we have *pNCl*(*S*˜*c*) <sup>⊆</sup> *<sup>H</sup>c*. Therefore, *NCl*(*NInt*(*S*˜*c*)) <sup>⊆</sup> *<sup>H</sup>c*. Hence, (*NInt*(*NCl*(*S*˜)))*<sup>c</sup>* <sup>⊆</sup> *<sup>H</sup>c*. This implies *<sup>H</sup>* <sup>⊆</sup> *NInt*(*NCl*(*S*˜)).

⇐<sup>=</sup> Suppose *<sup>S</sup>*˜ be an *NS* of <sup>Z</sup> and *<sup>H</sup>* <sup>⊆</sup> *NInt*(*NCl*(*S*˜)), whensoever *<sup>H</sup>* is an *NC* and *<sup>H</sup>* <sup>⊆</sup> *<sup>S</sup>*˜. Then, *<sup>S</sup>*˜*<sup>c</sup>* <sup>⊆</sup> *<sup>H</sup><sup>c</sup>* and *<sup>H</sup><sup>c</sup>* is an *NO*. By assumption, (*NInt*(*NCl*(*S*˜)))*<sup>c</sup>* <sup>⊆</sup> *<sup>H</sup><sup>c</sup>* . Hence, *NCl*(*NInt*(*S*˜ *<sup>c</sup>*)) <sup>⊆</sup> *<sup>H</sup><sup>c</sup>* . This implies *pNCl*(*S*˜ *<sup>c</sup>*) <sup>⊆</sup> *<sup>H</sup><sup>c</sup>* . Hence, *S*˜ is a *GNPOS* of Z .

#### **6. Applications of Generalized Neutrosophic Pre-Closed Sets**

**Definition 23.** *An NTS* (Z , Γ) *is said to be neutrosophic-pT*<sup>1</sup> 2 *(NpT*<sup>1</sup> 2 *in short) space if every GNPC in* Z *is an NCs* ∈ Z *.*

**Definition 24.** *An NTS* (Z , Γ) *is said to be neutrosophic-gpT*<sup>1</sup> 2 *(NgpT*<sup>1</sup> 2 *in short) space if every GNPC in* Z *is an NPCs* ∈ Z *.*

**Theorem 12.** *Every N pT*<sup>1</sup> 2 *space is an NgpT*<sup>1</sup> 2 *space.*

**Proof.** Let Z be an *NpT*<sup>1</sup> 2 space and *<sup>S</sup>*˜ be *GNPC* <sup>∈</sup> <sup>Z</sup> . By assumption, *<sup>S</sup>*˜ is *NCs* in <sup>Z</sup> . Since every *NC* is an *NPC*, *S*˜ is an *NPC* in Z . Hence, Z is an *NgpT*<sup>1</sup> space.

2

The converse is not true.

**Example 17.** *Let* Z = {*u*, *v*}*, H* = (0.9, 0.9, 0.9),(0.1, 0.1, 0.1) *and* Γ = {0*N*, 1*N*, *H*}*. Then,* (Z , Γ) *is an NgpT*<sup>1</sup> 2 *space, but it is not NpT*<sup>1</sup> 2 *since an NS H* = (0.2, 0.3, 0.3),(0.8, 0.7, 0.7) *is GNPC but not an NCS* ∈ Z *.*

**Theorem 13.** *Let* (Z , Γ) *be an NT and* Z *is an N pT*<sup>1</sup> 2 *space; then,*


**Proof.** (i) Let {*S*˜ *<sup>i</sup>*}*i*∈*<sup>J</sup>* be a collection of *GNPCs* in an *NpT*<sup>1</sup> 2 space (Z , Γ). Thus, every *GNPCs* is an *NCS*. However, the union of an *NC* is an *NCS*. Therefore, the Union of *GNPCs* is *GNPCs* in Z . (ii) Proved by taking complement in (i).

**Theorem 14.** *An NT* Z *is an NgpT*<sup>1</sup> 2 *space iff GNPO*(Z ) = *NPO*(Z )*.* **Proof.** <sup>=</sup><sup>⇒</sup> Let *<sup>S</sup>*˜ be a *GNPOs* in <sup>Z</sup> ; then, *<sup>S</sup>*˜*<sup>c</sup>* is *GNPCs* in <sup>Z</sup> . By assumption, *<sup>S</sup>*˜*<sup>c</sup>* is an *NPCs* in <sup>Z</sup> . Thus, *S*˜ is *NPOs* in Z . Hence, *GNPO*(Z ) = *NPO*(Z ).

⇐<sup>=</sup> Let *<sup>S</sup>*˜ be *GNPC* <sup>∈</sup> <sup>Z</sup> . Then, *<sup>S</sup>*˜*<sup>c</sup>* is *GNPO* in <sup>Z</sup> . By assumption, *<sup>S</sup>*˜*<sup>c</sup>* is an *NPO* in <sup>Z</sup> . Thus, *<sup>S</sup>*˜ is an *NPC* ∈ Z . Therefore, Z is an *NgpT*<sup>1</sup> 2 space.

**Theorem 15.** *For an NTS* (Z , Γ), *the following are equivalent:*


**Proof.** (*i*) =⇒ (*ii*). Suppose that (Z , Γ) is a neutrosophic pre-*T*<sup>1</sup> 2 space. Suppose that {*x*} is not an *NPCS* for some *x* ∈ Z . Then, Z − {*x*} is not an *NPOS* and hence Z is the only an *NPOS* containing Z − {*x*}. Hence, Z − {*x*} is an *NPGCS* in (Z , Γ). Since (Z , Γ) is a neutrosophic pre-*T*<sup>1</sup> 2 space, then Z − {*x*} is an *NPCS* or equivalently {*x*} is an *NPOS*. (*ii*) =⇒ (*i*). Let every singleton set of <sup>Z</sup> be either *NPCS* or *NPOS*. Let *<sup>S</sup>*˜ be an *NPGCS* of (<sup>Z</sup> , <sup>Γ</sup>). Let *<sup>x</sup>* <sup>∈</sup> <sup>Z</sup> . We show that *<sup>x</sup>* <sup>∈</sup> <sup>Z</sup> in two cases.

Case (i): Suppose that {*x*} is *NPCS*. If *<sup>x</sup>* <sup>∈</sup>/ *<sup>S</sup>*˜, then *<sup>x</sup>* <sup>∈</sup> *pNCl*(*S*˜) <sup>−</sup> *<sup>S</sup>*˜. Now, *pNCl*(*S*˜) <sup>−</sup> *<sup>S</sup>*˜ contains a non—empty *NPCS*. Since *<sup>S</sup>*˜ is *NPGCS*, by Theorem 7, we arrived to a contradiction. Hence, *<sup>x</sup>* <sup>∈</sup> <sup>Z</sup> .

Case (ii): Let {*x*} be *NPOS*. Since *<sup>x</sup>* <sup>∈</sup> *pNCl*(*S*˜), then {*x*} ∩ *<sup>S</sup>*˜ <sup>=</sup> *<sup>φ</sup>*. Thus, *<sup>x</sup>* <sup>∈</sup> <sup>Z</sup> . Thus, in any case *<sup>x</sup>* <sup>∈</sup> <sup>Z</sup> . Thus, *PNCl*(*S*˜) <sup>⊆</sup> *<sup>S</sup>*˜. Hence, *<sup>S</sup>*˜ <sup>=</sup> *pNCl*(*S*˜) or equivalently *<sup>S</sup>*˜ is an *NPCS*. Thus, every *NPGCS* is an *NCS*. Therefore, (Z , Γ) is neutrosophic pre-*T*<sup>1</sup> space.

2

#### **7. Conclusions**

We have introduced generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets over neutrosophic topology space. Many results have been established to show how far topological structures are preserved by these neutrosophic pre-closed. We also have provided examples where such properties fail to be preserved. In this paper, we have studied a few ideas only; it will be necessary to carry out more theoretical research to establish a general framework for decision-making and to define patterns for complex network conceiving and practical application.

**Author Contributions:** All authors have contributed equally to this paper. The individual responsibilities and contribution of all authors can be described as follows: the idea of this paper was put forward by W.A.-O. W.A.-O. completed the preparatory work of the paper. S.J. analyzed the existing work. The revision and submission of this paper was completed by W.A.-O.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


c 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution**

**Muhammad Zahir Khan 1,\*, Muhammad Farid Khan 1, Muhammad Aslam <sup>2</sup> and Abdur Razzaque Mughal <sup>1</sup>**


Received: 18 November 2018; Accepted: 14 December 2018; Published: 21 December 2018

**Abstract:** Acceptance sampling is one of the essential areas of quality control. In a conventional environment, probability theory is used to study acceptance sampling plans. In some situations, it is not possible to apply conventional techniques due to vagueness in the values emerging from the complexities of processor measurement methods. There are two types of acceptance sampling plans: attribute and variable. One of the important elements in attribute acceptance sampling is the proportion of defective items. In some situations, this proportion is not a precise value, but vague. In this case, it is suitable to apply flexible techniques to study the fuzzy proportion. Fuzzy set theory is used to investigate such concepts. It is observed there is no research available to apply Birnbaum-Saunders distribution in fuzzy acceptance sampling. In this article, it is assumed that the proportion of defective items is fuzzy and follows the Birnbaum-Saunders distribution. A single acceptance sampling plan, based on binomial distribution, is used to design the fuzzy operating characteristic (FOC) curve. Results are illustrated with examples. One real-life example is also presented in the article. The results show the behavior of curves with different combinations of parameters of Birnbaum-Saunders distribution. The novelty of this study is to use the probability distribution function of Birnbaum-Saunders distribution as a proportion of defective items and find the acceptance probability in a fuzzy environment. This is an application of Birnbaum-Saunders distribution in fuzzy acceptance sampling.

**Keywords:** fuzzy operating characteristic curve; fuzzy OC band; Birnbaum-Sunders distribution; single acceptance sampling plan

#### **1. Introduction**

An acceptance sampling plan is used to determine how many units can be selected from a lot, or consignment, and how many defective units are allowed in that sample. If the number of defective units is above the preset number of defective items, the lot is excluded. According to the rule of acceptance sampling, quality can be monitored by checking a few units from the whole lot. The plan that mentions guidelines for sampling and the associated criteria for accepting or rejecting a lot is called the acceptance sampling plan. This acceptance sampling plan can be implemented to check raw material, the material in a process or finished goods. An acceptance sampling plan can be classified as an attribute acceptance sampling plan and a variable acceptance sampling plan. An acceptance sampling plan can be classified with further attributes as a single sampling plan, double sampling plan, multiple sampling plans, and sequential sampling plan. An elementary acceptance sampling plan is a single sampling plan. In a single sampling plan, we select (n) units from the entire lot. This consists of (N) units. After selection of n units they are examined; if the number of damaged units

(d) is more than the specified number of defective items (c), the lot will be disallowed. Otherwise, it will be passed. The performance of any acceptance sampling plan can be judged by its operating characteristic (OC) curve. It determines how well an acceptance sampling plan distinguishes between good and bad lots. This OC curve has two parameters (n, c), where n is sample size and c is acceptance number. In an acceptance sampling plan, two groups are involved: the supplier and buyer. The supplier desires to avoid rejection of a good lot (producer's risk) and the buyer tries to avert acceptance of a bad lot (consumer's risk). In case a bad lot is accepted, it is the responsibility of the consumer. The producer's risk is denoted by α. This is the probability of rejection of the lot having an average quality level (AQL). Similarly, the consumer's risk is denoted with β. This shows the probability of acceptance of the lot, having low quality (LQL) [1]. The proportion of defective items is denoted by p and treated as a precise number. However, in some situations, it is not possible to get the precise numerical value of p. Mostly this value is determined by the expert, based on his judgment. It is used to calculate fuzzy acceptance probability. Further, this fuzzy p value and fuzzy acceptance probability are used to design a fuzzy OC curve [1]. In the study presented in Reference [2], the authors suggested a double acceptance sampling plan, based on assumption that lifetime of the product follows a generalized logistic distribution with known shape parameters, and analyzed the operating characteristic curve to several ratios of the true median life to the specified life. In the study presented in Reference [3], the authors proposed the double sampling plan and specified the design parameters fulfilling both the producer's and consumer's risks at the same time for a stated reliability, in the form of the mean ratio to the specific life. Moreover, double sampling and group sampling plans are constructed using the two-point technique, with the assumption that the lifetime of the product follows the Birnbaum-Saunders distribution. In the study presented in Reference [4], the pioneer of fuzzy set theory gave scientific structure to study imprecise and ambiguous concepts that are based on human judgment; comprising verbal expressions, contentment degree and significance degree, that are often fuzzy. A linguistic variable consists of expressions in a natural language, but not the number. In Reference [5] the authors applied fuzzy set theory to help explain complex and not easy to express linguistic terms, in traditional measurable terms. In Reference [6] the authors proposed a single acceptance sampling plan with a fuzzy parameter and explained the single acceptance sampling plan with fuzzy probability theory. In Reference [7] the authors used the expression for the OC curve and various values to help accept or reject a lot for a particular number of defective items. Proficiency of different acceptance sampling plans can be assessed by using the OC curve. These OC curves are used to determine the producer's risk, as well as the consumer's risk [8]. In Reference [9] the authors suggested using acceptance sampling in the fuzzy environment using Poisson distribution. In Reference [10] the authors explored if N is large, then the defective items will have a fuzzy binomial distribution. In Reference [11] the authors applied parameters of the acceptance sampling plan, sample size n, and acceptance number c, in a fuzzy environment. Acceptance probabilities of two major discrete distributions were also derived. The multiple deferred sampling plans and characteristic curves were proposed—where (p) proportion of defective items was treated as a fuzzy number—and also proposed fuzzy OC curves with different combinations of parameters [12]. Multiple deferred acceptance sampling plans with inspection errors were proposed by the authors of Reference [13]. In the study presented in Reference [14], the authors investigated the inspection errors and their impact on a single acceptance sampling plan, when the proportion of defective items was not known exactly. In Reference [15] the authors proposed an acceptance sampling plan for geospatial data with uncertainty in the proportion of defective items. In Reference [16] the authors investigated a double acceptance sampling plan with the fuzzy parameter. Average outgoing quality (AOQ) and average total inspection (ATI) in a double acceptance sampling plan with the imprecise proportion of defective items were presented [17]. In Reference [18] the authors suggested the fuzzy parameter for quality interval acceptance sampling plan, applying Poisson distribution. The fuzzy double acceptance sampling plan for Poisson distribution was proposed by the authors in Reference [19]. In Reference [20] the authors proposed an application of Weibull distribution in an acceptance sampling plan in the fuzzy

environment and calculated fuzzy acceptance probabilities for different sample sizes using real-life data. In Reference [21] the authors proposed truncated life time, based on the Birnbaum-Saunders (BS) distribution. This distribution is used to define the number of stress cycles until failure of the material. In Reference [22] the authors applied the concept of the failure process of materials due to weariness, to design the BS distribution. Estimation of parameters based on crack length data was proposed in Reference [23]. In Reference [24] the authors presented a literature review of the BS distribution and discussed in detail the importance of this distribution and its application in different fields. In this study [25], they developed an acceptance sampling plan using the BS distribution to get the minimum sample size, n.

The aim of this article was to apply a single acceptance sampling plan when data were fuzzy and the proportion of defective items followed the BS distribution. According to the best of our knowledge, there is no work on the fuzzy plan using the BS distribution in the literature. In this paper, we will develop the fuzzy sampling plan using this distribution. The application of the proposed sampling will be given with the aid of a real example.

#### **2. Materials and Methods**

#### *Design of Proposed Plan*

Probability distribution function (Pdf) of BS distribution

$$F\_T(t, a, \lambda) = \Phi\left(\frac{1}{a}\xi\left(\frac{t}{\lambda}\right)\right), \ 0 < \ t < \infty, \lambda > 0 \tag{1}$$

where *α* is the shape parameter and *λ* is the scale parameter, Φ(.) is the standard normal cumulative function and *ξ*(*t*/*λ*) = *<sup>t</sup> λ* − *<sup>λ</sup> <sup>t</sup>* . It can be shown that the median of the BS distribution is equal to the scale parameter and the mean of the BS distribution is

$$
\mu = \lambda \left( 1 + \alpha^2 / 2 \right) \tag{2}
$$

Here we write the assumptions for the BS distribution.

Let *t*<sup>0</sup> = *aμ*0; *a* be called the termination ratio. The cumulative distribution function (Cdf) given in Equation (5) can be rewritten as

$$F\_T(t\_0, a, \lambda) = \Phi\left(\frac{1}{a}\zeta\_5^x \left(\frac{a\left(1 + a^2/2\right)}{\mu/\mu\_0}\right)\right),\tag{3}$$

#### **The acceptance probability**

According to [26], the acceptance probability of sampling plans can be obtained by using the binomial distribution. The lot acceptance probability of a lot in a single acceptance sampling plan (SASP) case is given as

$$L(p) = \left[ \sum\_{i=0}^{n} \binom{n}{i} p^i (1-p)^{n-i} \right] \tag{4}$$

#### **The proportion of defective items in the fuzzy form.**

According to the equation proposed by the authors of Reference [27].

$$
\hat{p}K = (K, b\_2 + K, b\_3 + K, b\_4 + K).
\\
p\_K \in \hat{p}\mathbb{K}[a], q\_K \in \hat{q}\mathbb{K}[a], p\_K + q\_K = 1\tag{5}
$$

*bi* = *ai* − *a*2, i = 2, 3, 4 and *K* = [0, 1 − *b*4].

<sup>α</sup>**-cut of** *pK*

$$
\hat{p}K(\mathfrak{a}) = (K + (b\_2 + K - K)\mathfrak{a}, b\_3 + K + (b\_3 - b\_4)\mathfrak{a}) \tag{6}
$$

*Mathematics* **2019**, *7*, 9

<sup>α</sup>**-cut of** *pK* **at** <sup>α</sup> **= 0**

$$
\widetilde{p}\mathcal{K}(0) = (\mathcal{K}\_\prime b\_4 + \mathcal{K})\_\prime
$$

where *p* is the *FT*(*t*0, *α*, *λ*) in Equation (4).

**Fuzzy acceptance probability**

According to Reference [11], the fuzzy acceptance probability can be calculated as

$$\widetilde{p}\_k \neq \left\{ \binom{n}{k} p^k \neq^{n-k} \middle| p \in p \ a, q \in q \ a \right\}, \ 0 \le a \le 1 \tag{7}$$

$$\tilde{p}\_k \, a = [p\_{kl}, p\_{kr}]$$

$$p\_{kl} \, a = \min \left\{ \binom{n}{k} p^k \, q^{n-k} \, \middle| \, p \in p \, a, q \in q \, a \right\} \tag{8}$$

$$p\_{kr} \ a = \max \left\{ \binom{n}{k} p^k \ q^{n-k} \middle| p \in p \ a, q \in q \ a \right\} \tag{9}$$

**The fuzzy acceptance probability when the number of defective items,** *c* **= 0, and** *α* **= 0**

$$
\left[\widetilde{p}\_{K}(0)\right][0] = \left(1 - \widetilde{p}\_{K}^{\ \!\!\/ L}[a]\right)^{n}, \left(1 - \widetilde{p}\_{K}^{\ \!\!\/ L}[a]\right)^{n} \tag{10}
$$

$$
\left.\right.
\left.\right.
\left.\right.
\left.\right.
\left.\right.}
$$

$$
\widetilde{p}\_{\mathbb{K}}{}^L = \mathbb{K} \\
\widetilde{p}\_{\mathbb{K}}{}^{\mathbb{U}} = b\_4 + \mathbb{K}
$$

**The fuzzy acceptance probability based when the number of defective items,** *c* **= 1 and** *α* **= 0**

$$\tilde{p}\_{\mathbf{K}}(1)[0] = \left(1 - \tilde{p}\_{\mathbf{K}}^{\ \ \ \ \mathbf{I}}[a]\right)^{n} + n\left(1 - \tilde{p}\_{\mathbf{K}}^{\ \ \ \ \mathbf{I}}[a]\right)^{n-1}, \left(1 - \tilde{p}\_{\mathbf{K}}^{\ \ \ \ \mathbf{I}}[a]\right)^{n} + n\left(1 - \tilde{p}\_{\mathbf{K}}^{\ \ \ \ \ \mathbf{I}}[a]\right)^{n-1} \tag{11}$$

Here *<sup>p</sup>KL*[*α*], *<sup>p</sup>KU*[*α*] are calculated using CDF of the BS distribution

#### **The design for a single acceptance sampling plan (ASP) to generate a fuzzy operating characteristic curve (FOC)**


The advantage of the fuzzy OC curve is that it is flexible and can be applied when the proportion of defective items is fuzzy. Secondly, the width of the fuzzy OC curve indicates the quality. Where the wider the width, the lesser the quality, and vice versa. In this study, the width of the fuzzy OC curve is influenced by the mean ratio. When the mean ratio is higher, the width of the band decreases. When the mean ratio is lower, the width increases. The advantage of this approach is that it can be applied to study any fuzzy data, which follows the BS distribution. This approach is more flexible than conventional p because it considers intermediate values of the fuzzy curve.

The fuzzy proportion of defective item k at <sup>α</sup> = 0 is denoted by *<sup>p</sup>K*[0] and the fuzzy acceptance probability as *PK*(0)[0]. Values for sample size n = 5 and acceptance number c = 0, will therefore be (0.00, 0.001), and (0.95, 0.96), respectively at k = 0.01. Similarly, values of proportion and acceptance probability for sample size n = 5 and acceptance number c = 0, will be (0.052, 0.054) and (0.77, 0.78), respectively at k = 0.01. Furthermore, the fuzzy proportion of defective item k at α = 0 is denoted by *<sup>p</sup>K*[0] and the fuzzy acceptance probability as *<sup>P</sup>K*(1)[0]. These values for sample size n = 5 and acceptance number c = 1, will be the proportion of defective items (0.01, 0.029), and the acceptance probability (0.99, 0.995) at K = 0.01.

#### **3. Real Life Example**

In this section, we will discuss the application of the proposed sampling plan using real data selected from [28] and [29]. As mentioned above, the BS distribution is also known as the fatigue life distribution. It is used extensively in reliability applications to model failure times. The BS distribution is used in circumstances where occurring of events is independent of each other, from one cycle to another cycle, with same random distribution [29]. In this study, the failure life data given in Reference [28] is used. The authors of Reference [28] found that the data follow the BS distribution. The fatigue life data of aluminum coupons having n = 101 observations are shown in Table 1.


**Table 1.** Fatigue data of aluminum in hours.

We assume that data follows the BS distribution, the proportion of defective items p is fuzzy and the shape parameter is taken as the trapezoidal fuzzy number, *α*ˆ = (0.15, 0.16, 0.17, 0.18). When the actual mean is *μ*<sup>0</sup> = 134, the termination ratio a = 0.5 is then truncated, time will be t = 67 for c = 0, and at *<sup>μ</sup>*<sup>0</sup> *<sup>μ</sup><sup>T</sup>* <sup>=</sup> 1, the proportion of defective items is *<sup>p</sup>* <sup>=</sup> 0.0211673. The acceptance probabilities are calculated by using Equations (9) and (10), for different sample sizes n = (5,25,75,100) and K = (0.0, 0.01,0.02, 0.03, 0.04, 0.05). The fuzzy OC curve is designed using fuzzy p values and fuzzy acceptance probabilities for c = 0. Similarly, when *μ*<sup>0</sup> = 134, termination ratio a = 0.67 is then truncated, time will be t = 89.7 for c = 1. In this case, acceptance number c = 1, and *<sup>μ</sup>*<sup>0</sup> *<sup>μ</sup><sup>T</sup>* = 1, *<sup>p</sup>* = 0.01923. The acceptance probabilities are calculated using Equation (11) for different sample sizes n = (5,25,75,100) and K = (0.0, 0.01, 0.02, 0.03, 0.04, 0.05). The fuzzy OC curve is developed using p values and acceptance probabilities for c = 1. Fuzzy acceptance probability for Birnbaum-Saunders distribution is presented in Tables 2 and 3 using real life data and their respective fuzzy OC curves are shown in Figures 1–3. Entire calculations and graphs were completed using R software and codes were given in Appendix A. Acceptance probability is influenced by mean ratio, when mean ratio increases it reduces Uuncertainty and bandwidth of fuzzy OC curve become narrow while decreasing mean ration increases the width of fuzzy OC curve. The fuzzy OC curves show more convexity when sample size n increases. Fuzzy OC curve withc=0 shows less uncertainty than c = 1. We presented acceptance probabilities and fuzzy OC curves with c = 0, it is almost equal to conventional OC curve. The fuzzy OC curve forc=0 and c = 1 is more convex at large sample size as compared to small sample size.




**Table 3.** The fuzzy acceptance probability of Birnbaum-Saunders distribution for c = 0.

**Figure 1.** The fuzzy operating characteristic (OC) Curve of the Birnbaum-Saunders distribution atc=0 (**a**) n = 20, (**b**) n = 40, (**c**) n = 70, and (**d**) n = 100.

**Figure 2.** The fuzzy operating characteristic (OC) curve of Birnbaum-Saunders distribution at c = 1, mean ratio = 0.5, (**a**) n = 20, (**b**) n = 40, (**c**) n = 70, and (**d**) n = 100.

**Figure 3.** The fuzzy operating characteristic (OC) curve of Birnbaum-Saunders distribution at c = 1, mean ratio = 0.5, (**a**) n = 5, (**b**) n = 15, (**c**) n = 30, and (**d**) n = 50.

#### **4. Conclusions**

Acceptance sampling is one of the important aspects of statistical quality control. When the data follow the Birnbaum-Saunders distribution and the proportion of defective items is fuzzy, acceptance probability and the OC curve can be presented in a fuzzy form. In this article, the fuzzy OC curve of the Birnbaum-Saunders distribution is presented in a single acceptance sampling plan, using the binomial distribution. The fuzzy OC curve has a band with two bounds, lower and upper. The width of the band depends upon the uncertainty in the proportion of defective items in the fuzzy environment. Less uncertainty will give a narrow width. The fuzzy OC curves also show more convexity at a large sample size. The mean ratio in the Birnbaum-Saunders distribution is another important factor in quality. Here, a lower value of the mean ratio causes the width of the band of the fuzzy OC curve to increase. This indicates more uncertainty. The advantage of this approach is that it can be used to calculate the proportion of defective items when fuzzy data follows a Birnbaum-Saunders distribution, because mostly we assume the value of the proportion of defective items without using any distribution. Secondly, the fuzzy acceptance probability based on the Birnbaum-Saunders distribution is calculated. The fuzzy OC curve of the Birnbaum-Saunders distribution is constructed based on fuzzy p and the acceptance probability. The OC curve is more convex at large sample sizes, as compared to small sample sizes. It was concluded that when data followed the Birnbaum-Saunders distribution, this proposed approach was suitable to calculate the proportion (p), the acceptance probability, and the OC curve in both conventional and fuzzy form. In the future, we will apply the same concept to group acceptance sampling and chain acceptance sampling in a fuzzy environment.

**Author Contributions:** Conceptualization, M.F.K. and M.Z.K.; methodology, M.Z.K.; software, M.A.; validation, A.R.M.

**Funding:** This research received no funding.

**Acknowledgments:** The authors are deeply thankful to the reviewers and editor for their valuable suggestions to improve the quality of the paper.

**Conflicts of Interest:** The authors declare that there is no conflict of interest regarding the publication of this paper.

#### **Abbreviations**


#### **Appendix A**

#### **R codes**

#When n = 5, 20, 30, 30,c=0 rm(list=ls ()) windows () par(mfrow=c(2,2)) a = 0.5 #For c = 0 at t = 67 alpha = c(0.15, 0.16, 0.17, 0.18) K = seq(0, 0.05, 0.01) x = a\*((1 + alphaˆ2)/2) #Here b is teated as Alpha y = 0.5# 1, 2, 3, 4, 5, 6 values of ratio of X = c((1/alpha)\*(sqrt(x/y)−sqrt(y/x))) FX = pnorm(X, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE) FX p = FX p = 2.470246e-16 K = seq(0,0.05,0.01) W=p+K p = 0.226627400 W=p+K #p = 0.226627400 p = 0.019 W=p+K K = seq(0,0.05,0.01) c = o, a = 0.5) and (c = 1 when a = 0.67) for a = 0.5, p = 2.470246e-16, for a = 0.67 p = 0.01923 B = dbinom(0,10,K) # B = (1−K)ˆ5 When n = 5, c = 0 (1) A = dbinom(0,10, W)# A = (1−(K+p))ˆ5#When n = 5, c = 0 (2) data.frame(A,B) data.frame(K,W,A,B) #B = (1-K)ˆ5 # THIS WILL GIVE US UPPER BAND HIGHER PROBABILITY #A = (1-(K+p))ˆ5#THIS WILL GIVE US LOWER BAND LOWER PROBABILITY plot(K,A,type = "l", col = "red", xlab = "K", ylab = "Pa ", main = "fuzzy OC curve") par(new = TRUE) plot(W,B,type = "l", col = "blue", xlab = "k", ylab = " ", main = "") legend("topright",c(expression(paste(alpha==0.15,",",a==0.67)),expression(paste(alpha==0.16,",",a== 0.67)),expression(paste(alpha==0.17,",",a==0.1)),expression(paste(alpha==0.18,",",a==0.67)),expression (paste(n==5,",",c==0))))

#### **References**


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Neutrosophic Soft Expert Multiset and Their Application to Multiple Criteria Decision Making**

#### **Derya Bakbak 1, Vakkas Uluçay 2,\* and Memet ¸Sahin <sup>2</sup>**


Received: 11 December 2018; Accepted: 2 January 2019; Published: 6 January 2019

**Abstract:** In this paper, we have investigated neutrosophic soft expert multisets (NSEMs) in detail. The concept of NSEMs is introduced. Several operations have been defined for them and their important algebraic properties are studied. Finally, we define a NSEMs aggregation operator to construct an algorithm for a NSEM decision-making method that allows for a more efficient decision-making process.

**Keywords:** aggregation operator; decision making; neutrosophic soft expert sets; neutrosophic soft expert multiset

#### **1. Introduction**

Multiple criteria decision making (MCDM) is an important part of modern decision science and relates to many complex factors, such as economics, psychological behavior, ideology, military and so on. For a proper description of objects in an uncertain and ambiguous environment, indeterminate and incomplete information has to be properly handled. Intuitionistic fuzzy sets were introduced by Atanassov [1], followed by Molodtsov [2] on soft set and neutrosophy logic [3] and neutrosophic sets [4] by Smarandache. The term neutrosophy means knowledge of neutral thought and this neutral represents the main distinction between fuzzy and intuitionistic fuzzy logic and set. Presently, work on soft set theory is progressing rapidly. Various operations and applications of soft sets were developed rapidly, including multi-adjoint t-concept lattices [5], signatures, definitions, operators and applications to fuzzy modelling [6], fuzzy inference system optimized by genetic algorithm for robust face and pose detection [7], fuzzy multi-objective modeling of effectiveness and user experience in online advertising [8], possibility fuzzy soft set [9], soft multiset theory [10], multiparameterized soft set [11], soft intuitionistic fuzzy sets [12], Q-fuzzy soft sets [13–15], and multi Q-fuzzy sets [16–18], thereby opening avenues to many applications [19,20]. Later, Maji [21] introduced a more generalized concept, which is a combination of neutrosophic sets and soft sets and studied its properties. Alkhazaleh and Salleh [22] defined the concept of fuzzy soft expert sets, which were later extended to vague soft expert set theory [23], generalized vague soft expert set [24], and multi Q-fuzzy soft expert set [25]. Sahin et al. [ ¸ 26] introduced neutrosophic soft expert sets, while Hassan et al. [27] extended it further to Q-neutrosophic soft expert sets. Broumi et al. [28] defined neutrosophic parametrized soft set theory and its decision making. Deli [29] introduced refined neutrosophic sets and refined neutrosophic soft sets.

Since membership values are inadequate for providing complete information in some real problems which has different membership values for each element, different generalizations of fuzzy sets, intuitionistic fuzzy sets and neutrosophic sets have been introduced called the multi fuzzy set [30], intuitionistic fuzzy multiset [31] and neutrosophic multiset [32,33], respectively. In the multisets, an element of a universe can be constructed more than once with possibly the same or different membership values. Some work on the multi fuzzy set [34,35], on the intuitionistic fuzzy multiset [36–39] and on the neutrosophic multiset [40–43] have been studied. The above set theories have been applied to many different areas including real decision-making problems [44–47]. The aim of this paper is allow the neutrosophic set to handle problems involving incomplete, indeterminacy and awareness of inconsistency knowledge, and this is further developed to neutrosohic soft expert sets.

The initial contributions of this paper involve the introduction of various new set-theoretic operators on neutrosophic soft expert multisets (NSEMs) and their properties. Later, we intend to extend the discussion further by proposing the concept of NSEMs and its basic operations, namely complement, union, intersection AND and OR, along with a definition of a NSEMs-aggregation operator to construct an algorithm of a NSEMs decision method. Finally we provide an application of the constructed algorithm to solve a decision-making problem.

#### **2. Preliminaries**

In this section we review the basic definitions of a neutrosophic set, neutrosophic soft set, soft expert sets, neutrosophic soft expert sets, and NP-aggregation operator required as preliminaries.

**Definition 1** ([4])**.** *A neutrosophic set A on the universe of discourse* U *is defined as A* = {*u*, (*μA*(*u*), *vA*(*u*), *wA*(*u*)) : *u* ∈ *U*, *μA*(*u*), *vA*(*u*), *wA*(*u*) ∈ [0, 1]}. *There is no restriction on the sum of <sup>μ</sup>A(u); vA(u) and wA(u), so* <sup>0</sup><sup>−</sup> <sup>≤</sup> *<sup>μ</sup>*A(*u*) <sup>+</sup> *<sup>v</sup>*A(*u*) <sup>+</sup> *<sup>w</sup>*A(*u*) <sup>≤</sup> <sup>3</sup>+*.*

**Definition 2** ([21])**.** *Let* U *be an initial universe set and E be a set of parameters. Consider A* ⊆ *E. Let NS*(U) *denotes the set of all neutrosophic sets of* U*. The collection* (*F*, *A*) *is termed to be the neutrosophic soft set over* U*, where F is a mapping given by F* : *A* → *NS*(U)*.*

**Definition 3** ([22])**.** U *is an initial universe, E is a set of parameters X is a set of experts (agents), and O* = {*agree* = 1, *disagree* = 0} *a set of opinions. Let Z* = *E* × *X* × *O and A* ⊆ *Z. A pair* (*F*, *A*) *is called a soft expert set over* U*, where F is mapping given by F* : *A* → *P*(U) *where P*(U) *denote the power set of* U*.*

**Definition 4** ([26])**.** *A pair* (*F*, *A*) *is called a neutrosophic soft expert set over* U*, where F is mapping given by*

$$F: A \to P(\mathcal{U}) \tag{1}$$

*where P*(U) *denotes the power neutrosophic set of U.*

**Definition 5** ([26])**.** *The complement of a neutrosophic soft expert set* (*F*, *A*) *denoted by* (*F*, *A*) *<sup>c</sup> and is defined as* (*F*, *A*) *<sup>c</sup> <sup>=</sup>* (*Fc*, *<sup>A</sup>*) *where <sup>F</sup><sup>c</sup>* <sup>=</sup> <sup>¬</sup>*<sup>A</sup>* <sup>→</sup> *<sup>P</sup>*(U) *is mapping given by <sup>F</sup>c*(*x*) *= neutrosophic soft expert complement with μF<sup>c</sup>* (*x*) = *wF*(*x*), *vFc* (*x*) = *vF*(*x*), *wFc* (*x*) = *μF*(*x*)*.*

**Definition 6** ([26])**.** *The agree-neutrosophic soft expert set* (*F*, *A*)<sup>1</sup> *over* U *is a neutrosophic soft expert subset of* (*F*, *A*) *is defined as*

$$(F,A)\_1 = \{F\_1(m) : m \in E \times X \times \{1\}\}.\tag{2}$$

**Definition 7** ([26])**.** *The disagree-neutrosophic soft expert set* (*F*, *A*)<sup>0</sup> *over* U *is a neutrosophic soft expert subset of* (*F*, *A*) *is defined as*

$$\{(F, A)\_0 = \{F\_0(m) : m \in E \times X \times \{0\}\}.\tag{3}$$

**Definition 8** ([26])**.** *Let* (*H*, *A*) *and* (*G*, *B*) *be two NSESs over the common universe U. Then the union of* (*H*, *A*) *and* (*G*, *B*) *is denoted by "*(*H*, *A*) ∼ ∪ (*G*, *B*)*" and is defined by* (*H*, *A*) ∼ ∪ (*G*, *B*) = (*K*, *C*)*, where* *C* = *A* ∪ *B and the truth-membership, indeterminacy-membership and falsity-membership of* (*K*, *C*) *are as follows:*

$$\mu\_{K(\varepsilon)}(m) = \begin{cases} \mu\_{H(\varepsilon)}(m), & \text{if } \varepsilon \in A - B, \\ \mu\_{G(\varepsilon)}(m), & \text{if } \varepsilon \in B - A, \\ \max\left(\mu\_{H(\varepsilon)}(m), \mu\_{G(\varepsilon)}(m)\right), & \text{if } \varepsilon \in AB. \end{cases}$$

$$w\_{K(\varepsilon)}(m) = \begin{cases} \begin{aligned} v\_{H(\varepsilon)}(m), & \text{if } \varepsilon \in A - B, \\ v\_{G(\varepsilon)}(m), & \text{if } \varepsilon \in B - A, \\ \frac{v\_{H(\varepsilon)}(m) + v\_{G(\varepsilon)}(m)}{2}, & \text{if } \varepsilon \in AB. \end{cases} \\ w\_{K(\varepsilon)}(m) = \begin{cases} \begin{array}{c} w\_{H(\varepsilon)}(m), \\ w\_{G(\varepsilon)}(m), \end{cases} & \text{if } \varepsilon \in A - B, \\ \min\left(w\_{H(\varepsilon)}(m), w\_{G(\varepsilon)}(m)\right), & \text{if } \varepsilon \in B - A. \end{cases} \end{cases} \tag{4}$$

**Definition 9** ([26])**.** *Let* (*H*, *A*) *and* (*G*, *B*) *be two NSESs over the common universe U. Then the intersection of* (*H*, *A*) *and* (*G*, *B*) *is denoted by "*(*H*, *A*) ∼ ∩ (*G*, *B*)*" and is defined by* (*H*, *A*) ∼ ∩ (*G*, *B*) = (*K*, *C*)*, where C* = *A* ∩ *B and the truth-membership, indeterminacy-membership and falsity-membership of* (*K*, *C*) *are as follows:*

$$\begin{aligned} \varepsilon \mu\_{K(\varepsilon)}(m) &= \min \left( \mu\_{H(\varepsilon)}(m), \mu\_{G(\varepsilon)}(m) \right), \\ \upsilon\_{K(\varepsilon)}(m) &= \frac{\upsilon\_{H(\varepsilon)}(m) + \upsilon\_{G(\varepsilon)}(m)}{2}, \\ \upsilon\_{K(\varepsilon)}(m) &= \max \left( w\_{H(\varepsilon)}(m), w\_{G(\varepsilon)}(m) \right), \end{aligned} \tag{5}$$

*if e* ∈ *AB*.

**Definition 10** ([29])**.** *Let* U *be a universe. A neutrosophic multiset set (Nms) A on* U *can be defined as follows:*

$$\mathcal{A} = \left\{ \prec u, \left( \mu\_A^1(u), \mu\_A^2(u), \dots, \mu\_A^p(u) \right), \left( \upsilon\_A^1(u), \upsilon\_A^2(u), \dots, \upsilon\_A^p(u) \right), \left( \upsilon\_A^1(u), \upsilon\_A^2(u), \dots, \upsilon\_A^p(u) \right) \succcolon u \in \mathcal{U} \right\}$$

*where,*

$$\begin{aligned} &c\,\mu\_A^1(\mathfrak{u}), \mu\_A^2(\mathfrak{u}), \dots, \mu\_A^p(\mathfrak{u}): \mathcal{U} \to [0,1], \\ &\upsilon\_A^1(\mathfrak{u}), \upsilon\_A^2(\mathfrak{u}), \dots, \upsilon\_A^p(\mathfrak{u}): \mathcal{U} \to [0,1], \end{aligned}$$

*and*

$$w\_A^1(\mu), w\_A^2(\mu), \dots, w\_A^p(\mu) : \mathcal{U} \to [0, 1],$$

*such that*

$$0 \le \sup \mu\_A^i(\mu) + \sup \upsilon\_A^i(\mu) + \sup \upsilon\_A^i(\mu) \le 3$$

(*i* = 1, 2, . . . , *P*) *and*

$$\left(\mu\_A^1(u), \mu\_A^2(u), \dots, \mu\_A^p(u)\right), \left(v\_A^1(u), v\_A^2(u), \dots, v\_A^p(u)\right) \text{ and } \left(w\_A^1(u), w\_A^2(u), \dots, w\_A^p(u)\right)$$

This is the truth-membership sequence, indeterminacy-membership sequence and falsity-membership sequence of the element *u*, respectively. Also, P is called the dimension (cardinality) of Nms *A*, denoted *d*(*A*). We arrange the truth-membership sequence in decreasing order but the corresponding indeterminacy-membership and falsity-membership sequence may not be in decreasing or increasing order.

The set of all neutrosophic multisets on U is denoted by NMS(U).

**Definition 11** ([28])**.** *Let* <sup>Ψ</sup>*<sup>K</sup>* <sup>∈</sup> *NP-soft set. Then an NP-aggregation operator of* <sup>Ψ</sup>*K*, *denoted by* <sup>Ψ</sup>*agg <sup>K</sup> is defined by*

$$\Psi\_K^{\mathfrak{g}\underline{\otimes} \mathcal{S}} = \left\{ \left( \langle \mu\_\prime \operatorname{\mathbf{T}}\_K^{\mathfrak{g}\underline{\otimes} \mathcal{S}}, \operatorname{\mathbf{I}}\_K^{\mathfrak{g}\underline{\otimes} \mathcal{S}}, \operatorname{\mathbf{F}}\_K^{\mathfrak{g}\underline{\otimes} \mathcal{S}} \rangle \right) : u \in \mathcal{U} \right\},$$

*which is a neutrosophic set over U,*

$$\begin{aligned} \mathbf{T}\_{K}^{\rm g\mathcal{K}\mathcal{S}} : \mathcal{U} \to [0, 1] & \quad \mathbf{T}\_{K}^{\rm g\mathcal{K}\mathcal{S}}(u) = \underset{\mathbf{u} \in E}{\operatorname{\mathcal{U}}} & \quad \sum\_{e \in E} \quad \mathbf{T}\_{K}(u). \lambda f\_{K(x)}(u), \\ & u \in E \\ \mathbf{I}\_{K}^{\rm g\mathcal{K}\mathcal{S}} : \mathcal{U} \to [0, 1] & \quad \mathbf{I}\_{K}^{\rm g\mathcal{K}\mathcal{S}}(u) = \underset{\mathbf{u} \in E}{\operatorname{\mathcal{U}}} & \quad \mathbf{I}\_{K}(u). \lambda f\_{K(x)}(u), \\ & e \in E \\ \mathbf{F}\_{K}^{\rm g\mathcal{K}\mathcal{S}} : \mathcal{U} \to [0, 1] & \quad \mathbf{F}\_{K}^{\rm g\mathcal{K}\mathcal{S}} = \underset{\mathbf{I}[\mathcal{U}]}{\operatorname{\mathcal{U}}} & \quad \sum\_{e \in E} \quad \mathbf{F}\_{K}(u). \lambda f\_{K(x)}(u) \\ & e \in E \\ & u \in \mathcal{U} \end{aligned} \tag{6}$$

*and where,*

$$
\lambda f\_{\mathbf{K}(\mathbf{x})}(\boldsymbol{u}) = \begin{cases} \ 1, & \mathbf{x} \in f\_{\mathbf{K}(\mathbf{x})}(\boldsymbol{u}), \\ \ 0, & \text{otherwise.} \end{cases} \tag{7}
$$


#### **3. Neutrosophic Soft Expert Multiset (NSEM) Sets**

This section introduces neutrosophic soft expert multiset as a generalization of neutrosophic soft expert set. Throughout this paper, *V* is an initial universe, *E* is a set of parameters *X* is a set of experts (agents), and *O* = {agree = 1, disagree = 0} a set of opinions. Let *Z* = *E* × *X* × *O* and *G* ⊆ *Z* and *u* is a membership function of *G*; that is, Ω : *G* →= [0, 1].

**Definition 12.** *A pair F*Ω, *G is called a neutrosophic soft expert multiset over V, where F*<sup>Ω</sup> *is mapping given by*

$$F^{\Omega}: G \to \mathcal{N}(V) \times\_{\prime} \tag{8}$$

*where* N (*V*) *be the set of all neutrosophic soft expert subsets of U. For any parameter e* ∈ *G*, *F*(*e*) *is referred as the neutrosophic value set of parameter e, i.e.,*

$$F(\varepsilon) = \left\{ (\frac{\upsilon}{\left(D\_{F(\varepsilon)}^1(\upsilon), \dots, D\_{F(\varepsilon)}^n\right) / \left(I\_{F(\varepsilon)}^1(\upsilon), \dots, I\_{F(\varepsilon)}^n\right) / \left(Y\_{F(\varepsilon)}^1(\upsilon), \dots, Y\_{F(\varepsilon)}^n\right)})\right\},\tag{9}$$

*where D<sup>i</sup>* , *<sup>i</sup>* ,*Y<sup>i</sup>* : *<sup>U</sup>* <sup>→</sup> [0, 1] *are the membership sequence of truth, indeterminacy and falsity respectively of the element v* ∈ *V. For any v* ∈ *V, e* ∈ *G and i* = 1, 2, . . . , *n*.

$$0 \le D^i\_{\,^F(\mathfrak{e})}(\mathfrak{v}) + ^i\_{\,^F(\mathfrak{e})}(\mathfrak{v}) + \mathcal{Y}^i\_{\,^F(\mathfrak{e})}(\mathfrak{v}) \le 3$$

*In fact F*<sup>Ω</sup> *is a parameterized family of neutrosophic soft expert multisets on V, which has the degree of possibility of the approximate value set which is prepresented by* Ω(*e*) *for each parameter e. So we can write it as follows:*

$$F^{\Omega}(\varepsilon) = \left\{ \left( \frac{\upsilon\_1}{F(\varepsilon)(\upsilon\_1)}, \frac{\upsilon\_2}{F(\varepsilon)(\upsilon\_2)}, \frac{\upsilon\_3}{F(\varepsilon)(\upsilon\_3)}, \dots, \frac{\upsilon\_n}{F(\varepsilon)(\upsilon\_n)} \right), \Omega(\varepsilon) \right\}.\tag{10}$$

**Example 1.** *Suppose that V* = {*v*1} *is a set of computers and E* = {*e*1,*e*2} *is a set of decision parameters. Let X* = {*p*,*r*} *be set of experts. Suppose that*

$$\begin{aligned} \ \_CF^{\Omega}(\mathbf{e}\_1, p, 1) &= \left\{ \left( \frac{v\_1}{(0.4, 0.3, \ldots, 0.2), (0.5, 0.7, \ldots, 0.2), (0.6, 0.1, \ldots, 0.3)}{(0.3, 0.2, \ldots, 0.5), (0.8, 0.1, \ldots, 0.2)} \right) , 0.8 \right\} \\ \ \_FF^{\Omega}(\mathbf{e}\_1, p, 1) &= \left\{ \left( \frac{v\_1}{(0.7, 0.3, \ldots, 0.6), (0.3, 0.2, \ldots, 0.6), (0.8, 0.2, \ldots, 0.1)} \right) , 0.5 \right\} \\ \ \_FF^{\Omega}(\mathbf{e}\_2, p, 1) &= \left\{ \left( \frac{v\_1}{(0.8, 0.3, \ldots, 0.4), (0.3, 0.1, \ldots, 0.5), (0.2, 0.3, \ldots, 0.4)} \right) , 0.4 \right\} \\ \ \_FF^{\Omega}(\mathbf{e}\_1, p, 0) &= \left\{ \left( \frac{v\_1}{(0.5, 0.1, \ldots, 0.2), (0.6, 0.3, \ldots, 0.4), (0.7, 0.2, \ldots, 0.4)} \right) , 0.1 \right\} \\ \ \_FF^{\Omega}(\mathbf{e}\_1, r, 0) &= \left\{ \left( \frac{v\_1}{(0.4, 0.2, \ldots, 0.1), (0.6, 0.1, \ldots, 0.3), (0.7, 0.2, \ldots, 0.4)} \right) , 0.4 \right\} \\ \ \_FF^{\Omega}(\mathbf{e}\_2, p, 0) &= \left\{ \left( \frac{v\_1}{(0.8, 0.1, \ldots, 0.5), (0.2, 0.1, \ldots, 0.4), (0.6,$$

*The neutrosophic soft expert multiset* (*F*, *Z*) *is a parameterized family* {*F*(*ei*), *i* = 1, 2, . . .} *of all neutrosophic multisets of V and describes a collection of approximation of an object.*

**Definition 13.** *For two neutrosophic soft expert multisets (NSEMs) F*Ω, *G and* (*Hη*, *R*) *over U, F*Ω, *G is called a neutrosophic soft expert subset of* (*Hη*, *R*) *if*

*i. R* ⊆ *G, ii. for all <sup>ε</sup>* <sup>∈</sup> *<sup>H</sup>*, *<sup>H</sup>η*(*ε*) *is neutrosophic soft expert subset F*Ω(*ε*)*.*

**Example 2.** *Consider Example 1. Suppose that G and R are as follows.*

$$\begin{aligned} cG &= \{ (e\_1, p, 1), (e\_2, p, 1), (e\_2, p, 0), (e\_2, r, 1) \}, \\ R &= \{ (e\_1, p, 1), (e\_2, r, 1) \} \end{aligned}$$

*Since <sup>R</sup> is a neutrosophic soft expert subset of G, clearly <sup>R</sup>* <sup>⊂</sup> *G. Let* (*Hη*, *<sup>R</sup>*) *and F*Ω, *G be defined as follows:*

$$\begin{split} c\left(F^{\Omega},G\right) &= \left\{ \left[ (\varepsilon\_1,p,1), \left( \frac{v\_1}{(0.4,0.3,\ldots,0.2), (0.5,0.7,\ldots,0.2), (0.6,0.1,\ldots,0.3)} \right), 0.4 \right], \\ & \left[ (\varepsilon\_2,p,1), \left( \frac{v\_1}{(0.7,0.3,\ldots,0.6), (0.3,0.2,\ldots,0.6), (0.8,0.2,\ldots,0.1)} \right), 0.5 \right], \\ & \left[ (\varepsilon\_2,p,0), \left( \frac{v\_1}{(0.8,0.1,\ldots,0.5), (0.2,0.1,\ldots,0.4), (0.6,0.3,\ldots,0.1)} \right), 0.6 \right], \\ & \left[ (\varepsilon\_2,r,1), \left( \frac{v\_1}{(0.8,0.3,\ldots,0.4), (0.3,0.1,\ldots,0.5), (0.2,0.3,\ldots,0.4)} \right), 0.4 \right] \right\}. \\ & \left(H^{\eta},R\right) = \left\{ \left[ (\varepsilon\_1,p,1), \left( \frac{v\_1}{(0.4,0.3,\ldots,0.2), (0.5,0.7,\ldots,0.2), (0.6,0.1,\ldots,0.3)} \right), 0.4 \right], \\ & \left[ (\varepsilon\_2,r,1), \left( \frac{v\_1}{(0.8,0.3,\ldots,0.4), (0.3,0.1,\ldots,0.5), (0.2,0.3,\ldots,0.4)} \right), 0.4 \right] \right\}. \end{split}$$

*Therefore* (*Hη*, *<sup>R</sup>*) <sup>⊆</sup> *F*Ω, *G .* **Definition 14.** *Two NSEMs F*Ω, *G and* (*Gη*, *B*) *over V are said to be equal if F*Ω, *G is a NSEM subset of* (*Hη*, *R*) *and* (*Hη*, *R*) *is a NSEM subset of F*Ω, *G .*

**Definition 15.** *Agree-NSEMs F*Ω, *G* <sup>1</sup> *over V is a NSEM subset of F*Ω, *G defined as follows.*

$$\left(\left(F^{\Omega},G\right)\_1\right)\_1 = \{F\_1(\Delta) : \Lambda \in E \times X \times \{1\}\}.\tag{11}$$

**Example 3.** *Consider Example 1. The agree- neutrosophic soft expert multisets* (*F*Ω, *<sup>Z</sup>*)<sup>1</sup> *over V is*

$$\begin{split} \operatorname{c}(F^{\Omega},Z)\_{1} &= \left\{ \left[ (\varepsilon\_{1},p,1), \left( \frac{v\_{1}}{(0.4,0.3,\ldots,0.2),(0.5,0.7,\ldots,0.2),(0.6,0.1,\ldots,0.3)} \right),0.4 \right], \\ & \left[ (\varepsilon\_{1},r,1), \left( \frac{v\_{1}}{(0.3,0.2,\ldots,0.5),(0.8,0.1,\ldots,0.4),(0.5,0.6,\ldots,0.2)} \right),0.8 \right], \\ & \left[ (\varepsilon\_{2},p,1), \left( \frac{v\_{1}}{(0.7,0.3,\ldots,0.6),(0.3,0.2,\ldots,0.6),(0.8,0.2,\ldots,0.1)} \right),0.5 \right], \\ & \left[ (\varepsilon\_{2},r,1), \left( \frac{v\_{1}}{(0.8,0.3,\ldots,0.4),(0.3,0.1,\ldots,0.5),(0.2,0.3,\ldots,0.4)} \right),0.4 \right] \end{split}$$

**Definition 16.** *A disagree-NSEMs F*Ω, *G* <sup>0</sup> *over V is a NSES subset of F*Ω, *G is defined as follows:*

$$(F^{\Omega}, A)\_0 = \{ F\_0(\Delta) : \Delta \in E \times X \times \{0\} \}. \tag{12}$$

**Example 4.** *Consider Example 1. The disagree- neutrosophic soft expert multisets* (*F*Ω, *<sup>Z</sup>*)<sup>0</sup> *over V are*

$$\begin{split} (F^{\Omega}, Z)\_{0} &= \left\{ \left[ (\varepsilon\_{1}, p, 0), \left( \frac{v\_{1}}{(0.5, 0.1, \ldots, 0.2), (0.6, 0.3, \ldots, 0.4), (0.7, 0.2, \ldots, 0.6)} \right), 0.1 \right] \right\}, \\ &\left[ (\varepsilon\_{1}, r, 0), \left( \frac{v\_{1}}{(0.4, 0.2, \ldots, 0.1), (0.6, 0.1, \ldots, 0.3), (0.7, 0.2, \ldots, 0.4)} \right), 0.4 \right], \\ &\left[ (\varepsilon\_{2}, p, 0), \left( \frac{v\_{1}}{(0.8, 0.1, \ldots, 0.5), (0.2, 0.1, \ldots, 0.4), (0.6, 0.3, \ldots, 0.1)} \right), 0.6 \right], \\ &\left[ (\varepsilon\_{2}, r, 0), \left( \frac{v\_{1}}{(0.7, 0.2, \ldots, 0.3), (0.4, 0.1, \ldots, 0.6), (0.3, 0.2, \ldots, 0.1)} \right), 0.2 \right] \end{split}$$

#### **4. Basic Operations on NSEMs**

**Definition 17.** *The complement of a neutrosophic soft expert multiset* (*F*Ω, *G*) *is denoted by* (*F*Ω, *G*) *<sup>c</sup> and is defined by* (*F*Ω, *G*) *<sup>c</sup>* = *<sup>F</sup>*Ω(*c*), <sup>¬</sup>*<sup>G</sup> where Fu*(*c*) : <sup>¬</sup>*<sup>G</sup>* → N (*V*)<sup>×</sup> *is mapping given by*

$$\mathcal{F}^{\Omega(\varepsilon)}(\Delta) = \left\{ D^{i}\_{\,\,F(\Delta)^{(\varepsilon)}} = \mathcal{Y}^{i}\_{\,\,F(\Delta)^{\*}} \, \, I^{i}\_{\,\,F\_{\{\Delta\}}(\varepsilon)} = \overline{1} - I^{i}\_{\,\,F(\Delta)^{\*}} \, \, \mathcal{Y}^{i}\_{\,\,F(\Delta)} = \mathcal{D}^{i}\_{\,\,F(\Delta)} \, \text{and } \, \Omega^{\varepsilon}(\Delta) = \overline{1} - \Omega(\Delta) \right\} \tag{13}$$

*for each* Δ ∈ *E*.

**Example 5.** *Consider Example 1. The complement of the neutrosophic soft expert multiset F*<sup>Ω</sup> *denoted by F*Ω(*c*) *is given by as follows:*

*c*(*F*Ω(*c*) , *<sup>Z</sup>*) = (¬*e*1, *<sup>p</sup>*, 1), *v*<sup>1</sup> (0.2, 0.7, . . . , 0.4),(0.2, 0.3, . . . , 0.5),(0.3, 0.9, . . . , 0.6) , 0.6 , (¬*e*1,*r*, 1), *v*<sup>1</sup> (0.5, 0.8, . . . , 0.3),(0.4, 0.9, . . . , 0.8),(0.2, 0.4, . . . , 0.5) , 0.2 , (¬*e*2, *p*, 1), *v*<sup>1</sup> (0.6, 0.7, . . . , 0.7),(0.6, 0.8, . . . , 0.3),(0.1, 0.8, . . . , 0.8) , 0.5 , (¬*e*2,*r*, 1), *v*<sup>1</sup> (0.4, 0.7, . . . , 0.8),(0.5, 0.9, . . . , 0.3),(0.4, 0.7, . . . , 0.2) , 0.6 , (¬*e*1, *p*, 0), *v*<sup>1</sup> (0.2, 0.9, . . . , 0.5),(0.4, 0.7, . . . , 0.6),(0.6, 0.8, . . . , 0.7) , 0.9 , (¬*e*1,*r*, 0), *v*<sup>1</sup> (0.1, 0.8, . . . , 0.4),(0.3, 0.9, . . . , 0.6),(0.4, 0.8, . . . , 0.7) , 0.6 , (¬*e*2, *p*, 0), *v*<sup>1</sup> (0.5, 0.9, . . . , 0.8),(0.4, 0.9, . . . , 0.2),(0.1, 0.7, . . . , 0.6) , 0.4 , (¬*e*2,*r*, 0), *v*<sup>1</sup> (0.3, 0.8, . . . , 0.7),(0.6, 0.9, . . . , 0.4),(0.1, 0.8, . . . , 0.3) , 0.8 .

**Proposition 1.** *If* (*F*Ω, *G*) *is a neutrosophic soft expert multiset over V, then*

*1.* ((*F*Ω, *G*) *c* ) *c* = (*F*Ω, *G*) *2.* ((*F*Ω, *<sup>G</sup>*)1) *<sup>c</sup>* = (*F*Ω, *<sup>G</sup>*)<sup>0</sup> *3.* ((*F*Ω, *<sup>G</sup>*)0) *<sup>c</sup>* = (*F*Ω, *<sup>G</sup>*)<sup>1</sup>

**Proof.** (1) From Definition 17, we have (*F*Ω, *G*) *<sup>c</sup>* = *<sup>F</sup>*Ω(*c*), <sup>¬</sup>*<sup>G</sup>* where *F*Ω(*c*)(Δ) = *D<sup>i</sup> F*(Δ) (*c*) = *Y<sup>i</sup> F*(Δ) , *Ii F*(Δ) (*c*) <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>I</sup><sup>i</sup> F*(Δ) , *Y<sup>i</sup> F*(Δ) (*c*) = *D<sup>i</sup> <sup>F</sup>*(Δ) and <sup>Ω</sup>*c*(Δ) <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>Ω</sup>(Δ) for each <sup>Δ</sup> <sup>∈</sup> *<sup>E</sup>*. Now ((*F*Ω, *<sup>G</sup>*) *c* ) *<sup>c</sup>* = *<sup>F</sup>*Ω(*c*) *c* , *G* where

$$\begin{array}{rclcrcl} \left(\mathsf{F}^{\Omega(\mathsf{c})}\right)^{\mathsf{c}}(\Delta) &=& [\mathsf{D}^{i}\_{\mathsf{F}(\mathsf{A})^{(c)}} = \mathsf{Y}\_{\mathsf{F}(\mathsf{A})^{r}}^{i} & \mathsf{I}^{i}\_{\mathsf{F}(\mathsf{A})^{(c)}} & = \mathsf{T} - \mathsf{I}^{i}\_{\mathsf{F}(\mathsf{A})^{r}} & \mathsf{Y}^{i}\_{\mathsf{F}(\mathsf{A})^{(c)}} & = \mathsf{D}^{i}\_{\mathsf{F}(\mathsf{A})^{r}} & (\mathsf{D}^{i})^{c}(\Delta) & = \mathsf{T} - \Omega^{i}(\Delta)^{c}(\Delta) \\ &=& \mathsf{D}^{i}\_{\mathsf{F}(\Delta)} = \mathsf{Y}\_{\mathsf{F}(\Delta)}^{i} & \mathsf{I}^{i}\_{\mathsf{F}(\Delta)} & = \mathsf{T} - \mathsf{I}^{i}\_{\mathsf{F}(\Delta)^{(c)}} & \mathsf{Y}\_{\mathsf{F}(\Delta)}^{i} & = \mathsf{D}^{i}\_{\mathsf{F}(\Delta)^{(c)}} & \Omega^{i}(\Delta) & = \mathsf{T} - \left(\Omega^{i}\right)(\Delta) \\ & & & & & & & & & \left(\mathsf{T} - \left(\mathsf{T} - \Omega^{i}\right)\right) \\ & & & & & & & & & & & \left(\mathsf{T} - \left(\mathsf{T} - \Omega^{i}\right)\right) \\ & & & & & & & & & & & & \left(\mathsf{T} - \left(\mathsf{T} - \Omega^{i}\right)\right) \end{array}$$

Thus (*F*Ω, *G*) *c c* = *<sup>F</sup>*Ω(*c*) *c* , *G* = (*F*Ω, *<sup>G</sup>*), for all <sup>Δ</sup> <sup>∈</sup> *<sup>E</sup>*. The Proofs (2) and (3) can proved similarly. -

**Definition 18.** *The union of two NSEMs* (*F*Ω, *G*) *and* (*Kρ*, *L*) *over V, denoted by* (*F*Ω, *G*) ∼ <sup>∪</sup> (*Kρ*, *<sup>L</sup>*) *is a NSEMs* (*Hσ*, *<sup>C</sup>*) *where C* <sup>=</sup> *<sup>G</sup>* <sup>∪</sup> *L and* <sup>∀</sup> *<sup>e</sup>* <sup>∈</sup> *<sup>C</sup>*,

$$(H^{\sigma}, \mathbb{C}) = \begin{cases} \max\left(D^{i}\_{\left(F^{\Omega}(\varepsilon)\right)}(m), D^{i}\_{\left(K^{\rho}(\varepsilon)\right)}(m)\right) & \text{if } \Delta \in G \cap L\\ \min\left(I^{i}\_{\left(F^{\Omega}(\varepsilon)\right)}(m), I^{i}\_{\left(K^{\rho}(\varepsilon)\right)}(m)\right) & \text{if } \Delta \in G \cap L\\ \min\left(Y^{i}\_{\left(F^{\Omega}(\varepsilon)\right)}(m), Y^{i}\_{\left(K^{\rho}(\varepsilon)\right)}(m)\right) & \text{if } \Delta \in G \cap L \end{cases} \tag{14}$$

*where <sup>σ</sup>*(*m*) <sup>=</sup> *max* Ω(*e*)(*m*), *ρ*(*e*)(*m*) *.* **Example 6.** *Suppose that* (*F*Ω, *G*) *and* (*Kρ*, *L*) *are two NSEMs over V, such that*

$$\begin{split} c(F^{\Omega},G) &= \left\{ \left[ (e\_1, p, 1), \left( \frac{v\_1}{(0.7, 0.3, \ldots, 0.6), (0.5, 0.2, \ldots, 0.4), (0.7, 0.6, \ldots, 0.3)} \right), 0.3 \right], \\ &\left[ (e\_2, q, 1), \left( \frac{v\_1}{(0.4, 0.3, \ldots, 0.6), (0.8, 0.2, \ldots, 0.4), (0.5, 0.1, \ldots, 0.7)} \right), 0.6 \right], \\ &\left[ (e\_3, r, 1), \left( \frac{v\_1}{(0.8, 0.2, \ldots, 0.3), (0.6, 0.3, \ldots, 0.7), (0.4, 0.2, \ldots, 0.8)} \right), 0.5 \right] \right\}, \\ (K^{\rho}, L) &= \left\{ \left[ (e\_1, p, 1), \left( \frac{v\_1}{(0.4, 0.3, \ldots, 0.1), (0.7, 0.2, \ldots, 0.3), (0.5, 0.4, \ldots, 0.7)} \right), 0.6 \right], \\ &\left[ (e\_3, r, 1), \left( \frac{v\_1}{(0.8, 0.3, \ldots, 0.2), (0.6, 0.1, \ldots, 0.2), (0.3, 0.5, \ldots, 0.3)} \right), 0.7 \right] \right\} \end{split}$$

*Then* (*F*Ω, *G*) ∼ <sup>∪</sup> (*Kρ*, *<sup>L</sup>*) *<sup>=</sup>* (*Hσ*, *<sup>C</sup>*) *where*

$$\begin{split} \mathcal{L}(H^{\sigma},\mathbb{C}) &= \left\{ \left[ (c\_{1}, p, 1), \left( \frac{v\_{1}}{(0.7, 0.3, \ldots, 0.1), (0.7, 0.2, \ldots, 0.3), (0.7, 0.4, \ldots, 0.3)} \right), 0.6 \right] \right\}, \\ &\left[ (c\_{2}, q\_{1}), \left( \frac{v\_{1}}{(0.4, 0.3, \ldots, 0.6), (0.8, 0.2, \ldots, 0.4), (0.5, 0.1, \ldots, 0.7)} \right), 0.6 \right], \\ &\left[ (c\_{3}, r, 1), \left( \frac{v\_{1}}{(0.8, 0.2, \ldots, 0.2), (0.6, 0.1, \ldots, 0.2), (0.4, 0.2, \ldots, 0.3)} \right), 0.7 \right] \right\}. \end{split}$$

**Proposition 2.** *If* (*F*Ω, *G*)*,* (*Kρ*, *L*) *and H*Ω, *C are three NSEMs over V, then*

$$\begin{array}{ll} 1. & \left( (F^{\Omega}, G) \stackrel{\sim}{\cup} (K^{\rho}, L) \right) \stackrel{\sim}{\cup} (H^{\sigma}, \mathbb{C}) = (F^{\Omega}, G) \stackrel{\sim}{\cup} \left( (K^{\rho}, L) \stackrel{\sim}{\cup} (H^{\sigma}, \mathbb{C}) \right) \\ 2. & (F^{\Omega}, G) (F^{\Omega}, G) \subseteq (F^{\Omega}, G). \end{array}$$

**Proof.** (1) We want to prove that

$$\left( (F^{\Omega}, \mathcal{G}) \stackrel{\sim}{\cup} (K^{\rho}, L) \right) \stackrel{\sim}{\cup} (H^{\sigma}, \mathcal{C}) = (F^{\Omega}, \mathcal{G}) \stackrel{\sim}{\cup} \left( (K^{\rho}, L) \stackrel{\sim}{\cup} (H^{\sigma}, \mathcal{C}) \right).$$

by using Definition 18, we consider the case when if *e* ∈ *G* ∩ *L* as other cases are trivial. We will have

$$\begin{aligned} & \left( \mathcal{F}^{\Omega}, \mathcal{G} \right) \stackrel{\sim}{\cup} \left( \mathcal{K}^{\rho}, \mathcal{L} \right) \\ &= \left\{ \left( v/\max \left( \mathcal{D}^{i}\_{F^{\Omega}(\varepsilon)}(m), D^{i}\_{G^{\rho}(\varepsilon)}(m) \right), \min \left( \mathcal{I}^{i}\_{F^{\Omega}(\varepsilon)}(m), I^{i}\_{G^{\rho}(\varepsilon)}(m) \right), \min \left( \mathcal{Y}^{i}\_{F^{\Omega}(\varepsilon)}(m), Y^{i}\_{G^{\rho}(\varepsilon)}(m) \right) \right), \ & \begin{aligned} & \left( \mathcal{I}^{i}\_{F^{\Omega}(\varepsilon)}(m), D^{i}\_{G^{\rho}(\varepsilon)}(m) \right) \right\}, \ & \left( \mathcal{I}^{i}\_{G^{\rho}(\varepsilon)}(m), D^{i}\_{G^{\rho}(\varepsilon)}(m) \right) \end{aligned} \right. \\ & \left. \max \left( \mathcal{Q}\_{\{e\}}(m), \rho\_{\{e\}}(m) \right), \nu \in V \right\} \end{aligned}$$

Also consider the case when *e* ∈ *H* as the other cases are trivial. We will have

 (*Fu*, *A*) ∼ <sup>∪</sup> (*Gη*, *<sup>B</sup>*) ∼ ∪  *H*Ω, *C* = *v*/*max Di <sup>F</sup>*Ω(*e*)(*m*), *D<sup>i</sup> <sup>G</sup>ρ*(*e*)(*m*) , *min Ii <sup>F</sup>*Ω(*e*)(*m*), *I<sup>i</sup> <sup>G</sup>ρ*(*e*)(*m*) , *min Yi F*Ω(*e*)(*m*),*Y<sup>i</sup> <sup>G</sup>ρ*(*e*)(*m*) , *v*/*D<sup>i</sup> <sup>H</sup>*Ω(*e*)(*m*), *I<sup>i</sup> H*Ω(*e*)(*m*),*Y<sup>i</sup> <sup>H</sup>*Ω(*e*)(*m*) , *max u*(*e*)(*m*), *η*(*e*)(*m*), Ω(*m*) , *v* ∈ *V* = ⎧ ⎨ ⎩ *v*/*D<sup>i</sup> <sup>F</sup>*Ω(*e*)(*m*), *I<sup>i</sup> F*Ω(*e*)(*m*),*Y<sup>i</sup> <sup>F</sup>*Ω(*e*)(*m*) , *v*/*max Di <sup>G</sup>u*(*e*)(*m*), *D<sup>i</sup> <sup>H</sup><sup>η</sup>* (*e*)(*m*) , *min Ii <sup>G</sup>u*(*e*)(*m*), *I<sup>i</sup>* (*m*) , *min Yi Gu*(*e*)(*m*),*Y<sup>i</sup>* (*m*) *max* Ω(*e*)(*m*), *ρ*(*e*)(*m*), σ(*m*) , *v* ∈ *V* = (*F*Ω, *G*) ∼ ∪ (*Kρ*, *L*) ∼ <sup>∪</sup> (*H*σ, *<sup>C</sup>*) .

(2) The proof is straightforward. -

**Definition 19.** *The intersection of two NSEMs* (*F*Ω, *G*) *and* (*Kρ*, *L*) *over V, denoted by* (*F*Ω, *G*) ∼ <sup>∩</sup> (*Kρ*, *<sup>L</sup>*) = *Pδ*, *C where C* = *G* ∩ *L and* ∀ *e* ∈ *C*,

$$\mathbf{P}\left(P^{\mathcal{S}},\mathbb{C}\right) = \begin{cases} \min\left(\mathbf{D}^{i}\_{\left(F^{\Omega}\left(\varepsilon\right)\right)}\left(m\right), \mathbf{D}^{i}\_{\left(K^{\mathcal{S}}\left(\varepsilon\right)\right)}\left(m\right)\right) & \text{if } \varepsilon \in \mathcal{G} \cap L\\\max\left(\mathbf{I}^{i}\_{\left(F^{\Omega}\left(\varepsilon\right)\right)}\left(m\right), \mathbf{I}^{i}\_{\left(K^{\mathcal{S}}\left(\varepsilon\right)\right)}\left(m\right)\right) & \text{if } \varepsilon \in \mathcal{G} \cap L\\\max\left(\mathbf{Y}^{i}\_{\left(F^{\Omega}\left(\varepsilon\right)\right)}\left(m\right), \mathbf{Y}^{i}\_{\left(K^{\mathcal{S}}\left(\varepsilon\right)\right)}\left(m\right)\right) & \text{if } \varepsilon \in \mathcal{G} \cap L \end{cases} \tag{15}$$

*where <sup>δ</sup>*(*m*) <sup>=</sup> *min* Ω(*e*)(*m*), *ρ*(*e*)(*m*) *.*

**Example 7.** *Suppose that* (*F*Ω, *G*) *and* (*Kρ*, *L*) *are two NSEMs over V, such that*

$$\begin{split} c(F^{\Omega},G) &= \left\{ \left[ (e\_3,r,1), \left( \frac{v\_1}{(0.8,0.3,\ldots,0.2),(0.6,0.1,\ldots,0.2),(0.3,0.5,\ldots,0.3)} \right), 0.4 \right] \right\}, \\ & \left[ (e\_1,q,1), \left( \frac{v\_1}{(0.8,0.2,\ldots,0.2),(0.7,0.3,\ldots,0.2),(0.4,0.2,\ldots,0.3)} \right), 0.7 \right], \\ & \left[ (e\_3,q,0), \left( \frac{v\_1}{(0.4,0.3,\ldots,0.6),(0.8,0.2,\ldots,0.4),(0.5,0.1,\ldots,0.7)} \right), 0.6 \right] \right\}. \\ (K^{\rho},L) &= \left\{ \left[ (e\_1,p,1), \left( \frac{v\_1}{(0.7,0.3,\ldots,0.1),(0.7,0.2,\ldots,0.3),(0.7,0.4,\ldots,0.3)} \right), 0.3 \right], \\ & \left[ (e\_3,r,1), \left( \frac{v\_1}{(0.4,0.7,\ldots,0.8),(0.5,0.9,\ldots,0.3),(0.4,0.7,\ldots,0.2)} \right), 0.8 \right] \right\} \end{split}$$

*Then* (*F*Ω, *G*) ∼ <sup>∩</sup> (*Kρ*, *<sup>L</sup>*) =  *Pδ*, *C where*

$$\mathbb{P}\left(P^{\mathcal{S}}, \mathbb{C}\right) = \left\{ \left[ (\varepsilon\_{3}, r, 1), \left( \frac{\upsilon\_{1}}{(0.4, 0.3, \dots, 0.2), (0.6, 0.9, \dots, 0.3), (0.4, 0.7, \dots, 0.3)} \right), 0.4 \right] \right\}.$$

**Proposition 3.** *If* (*F*Ω, *G*)*,* (*Kρ*, *L*) *and H*Ω, *C are three NSEMs over V, then*

$$\begin{aligned} 1. \quad & \left( (F^{\Omega}, G) \stackrel{\sim}{\cap} (K^{\rho}, L) \right) \stackrel{\sim}{\cap} (H^{\sigma}, \mathbb{C}) = (F^{\Omega}, G) \stackrel{\sim}{\cap} \left( (K^{\rho}, L) \stackrel{\sim}{\cap} (H^{\sigma}, \mathbb{C}) \right) \\ 2. \quad & (F^{\Omega}, G) \stackrel{\sim}{\cap} (F^{\Omega}, G) \subseteq (F^{\Omega}, G). \end{aligned}$$

**Proof.** (1) We want to prove that

$$\left( \left( F^{\Omega}, \mathcal{G} \right) \widetilde{\cap} \left( K^{\rho}, L \right) \right) \widetilde{\cap} \left( H^{\sigma}, \mathcal{C} \right) = \left( F^{\Omega}, \mathcal{G} \right) \widetilde{\cap} \left( \left( K^{\rho}, L \right) \widetilde{\cap} \left( H^{\sigma}, \mathcal{C} \right) \right)$$

by using Definition 19, we consider the case when if *e* ∈ *G* ∩ *L* as other cases are trivial. We will have

$$\begin{aligned} & \left( \left( \boldsymbol{I}^{\Omega}, \boldsymbol{G} \right) \widetilde{\cap} \left( \boldsymbol{K}^{p}, \boldsymbol{L} \right) \\ &= \left\{ \left( \boldsymbol{v} / \min \left( \boldsymbol{D}^{i}\_{\mathrm{F}^{\Omega}(\boldsymbol{\varepsilon})} (\boldsymbol{m}), \boldsymbol{D}^{i}\_{\mathrm{G}^{p}(\boldsymbol{\varepsilon})} (\boldsymbol{m}) \right), \max \left( \boldsymbol{I}^{i}\_{\mathrm{F}^{\Omega}(\boldsymbol{\varepsilon})} (\boldsymbol{m}), \boldsymbol{I}^{i}\_{\mathrm{G}^{p}(\boldsymbol{\varepsilon})} (\boldsymbol{m}) \right), \max \left( \boldsymbol{Y}^{i}\_{\mathrm{F}^{\Omega}(\boldsymbol{\varepsilon})} (\boldsymbol{m}), \boldsymbol{Y}^{i}\_{\mathrm{G}^{p}(\boldsymbol{\varepsilon})} (\boldsymbol{m}) \right) \right\}, \\ & \quad \min \left( \boldsymbol{\Omega}\_{\{\boldsymbol{\varepsilon}\}} (\boldsymbol{m}), \boldsymbol{\rho}\_{\{\boldsymbol{\varepsilon}\}} (\boldsymbol{m}) \right), \boldsymbol{\nu} \in V \right\} \end{aligned}$$

Also consider the case when Δ ∈ *H* as the other cases are trivial. We will have

 (*Fu*, *A*) ∼ <sup>∩</sup> (*Gη*, *<sup>B</sup>*) ∼ ∩  *H*Ω, *C* = *v*/*max Di <sup>F</sup>*Ω(*e*)(*m*), *D<sup>i</sup> <sup>G</sup>ρ*(*e*)(*m*) , *min Ii <sup>F</sup>*Ω(*e*)(*m*), *I<sup>i</sup> <sup>G</sup>ρ*(*e*)(*m*) , *min Yi F*Ω(*e*)(*m*),*Y<sup>i</sup> <sup>G</sup>ρ*(*e*)(*m*) , *v*/*D<sup>i</sup> <sup>H</sup>*Ω(*e*)(*m*), *I<sup>i</sup> H*Ω(*e*)(*m*),*Y<sup>i</sup> <sup>H</sup>*Ω(*e*)(*m*) , *min u*(*e*)(*m*), *η*(*e*)(*m*), Ω(*m*) , *v* ∈ *V* = ⎧ ⎨ ⎩ *v*/*D<sup>i</sup> <sup>F</sup>*Ω(*e*)(*m*), *I<sup>i</sup> F*Ω(*e*)(*m*),*Y<sup>i</sup> <sup>F</sup>*Ω(*e*)(*m*) , *v*/*min Di <sup>G</sup>u*(*e*)(*m*), *D<sup>i</sup> <sup>H</sup><sup>η</sup>* (*e*)(*m*) , *max Ii <sup>G</sup>u*(*e*)(*m*), *I<sup>i</sup>* (*m*) , *max Yi Gu*(*e*)(*m*),*Y<sup>i</sup>* (*m*) *min* Ω(*e*)(*m*), *ρ*(*e*)(*m*), σ(*m*) , *v* ∈ *V* = (*F*Ω, *G*) ∼ ∩ (*Kρ*, *L*) ∼ <sup>∩</sup> (*H*σ, *<sup>C</sup>*) .

(2) The proof is straightforward. -

**Proposition 4.** *If* (*F*Ω, *G*)*,* (*Kρ*, *L*) *and H*Ω, *C are three NSEMs over V. Then*

$$\begin{array}{ll} 1. & \left( (F^{\Omega}, G) \stackrel{\sim}{\cup} (\mathbb{K}^{\rho}, L) \right) \stackrel{\sim}{\cap} (H^{\sigma}, \mathbb{C}) = \left( (F^{\Omega}, G) \stackrel{\sim}{\cap} (H^{\sigma}, \mathbb{C}) \right) \stackrel{\sim}{\cup} \left( (\mathbb{K}^{\rho}, L) \stackrel{\sim}{\cap} (H^{\sigma}, \mathbb{C}) \right). \\ 2. & \left( (F^{\Omega}, G) \stackrel{\sim}{\cap} (\mathbb{K}^{\rho}, L) \right) \stackrel{\sim}{\cup} (H^{\sigma}, \mathbb{C}) = \left( (F^{\Omega}, G) \stackrel{\sim}{\cup} (H^{\sigma}, \mathbb{C}) \right) \stackrel{\sim}{\cap} \left( (\mathbb{K}^{\rho}, L) \stackrel{\sim}{\cup} (H^{\sigma}, \mathbb{C}) \right). \end{array}$$

**Proof.** The proofs can be easily obtained from Definitions 18 and 19. -

#### **5. AND and OR Operations**

**Definition 20.** *Let* (*F*Ω, *G*) *and* (*Kρ*, *L*) *be any two NSEMs over V, then* (*F*Ω, *G*)*AND*(*Kρ*, *L*)" *denoted* (*F*Ω, *<sup>G</sup>*) <sup>∧</sup> (*Kρ*, *<sup>L</sup>*) *is defined by*

$$(F^{\Omega}, G) \wedge (K^{\rho}, L) = (H^{\sigma}, G \times L) \tag{16}$$

*where* (*Hσ*, *<sup>G</sup>* <sup>×</sup> *<sup>L</sup>*) <sup>=</sup> *<sup>H</sup>σ*(*α*, *<sup>β</sup>*) *such that <sup>H</sup>σ*(*α*, *<sup>β</sup>*) <sup>=</sup> *<sup>F</sup>*Ω(*α*) <sup>∩</sup> *<sup>K</sup>ρ*(*β*) *for all* (*α*, *<sup>β</sup>*) <sup>∈</sup> *<sup>G</sup>* <sup>×</sup> *<sup>L</sup> where* <sup>∩</sup> *represent the basic intersection.*

**Example 8.** *Suppose that* (*F*Ω, *G*) *and* (*Kρ*, *L*) *are two NSEMs over V, such that*

$$\begin{split} \mathbf{c}(F^{\Omega},\mathbf{G}) &= \left\{ \left[ (\mathbf{c}\_{1},p,1), \left( \frac{\upsilon\_{1}}{(0.2,0.3,\ldots,0.6),(0.2,0.1,\ldots,0.8),(0.3,0.2,\ldots,0.6)} \right), 0.1 \right] \right\}, \\ & \left[ (\mathbf{c}\_{2},r,0), \left( \frac{\upsilon\_{1}}{(0.5,0.3,\ldots,0.4),(0.6,0.5,\ldots,0.4),(0.2,0.4,\ldots,0.3)} \right), 0.5 \right] \right\} \\ (K^{p},L) &= \left\{ \left[ (\mathbf{c}\_{1},p,1), \left( \frac{\upsilon\_{1}}{(0.3,0.2,\ldots,0.1),(0.5,0.2,\ldots,0.3),(0.8,0.3,\ldots,0.4)} \right), 0.2 \right] \right\} \\ & \left[ (\mathbf{c}\_{2},q,0), \left( \frac{\upsilon\_{1}}{(0.6,0.4,\ldots,0.7),(0.3,0.4,\ldots,0.2),(0.6,0.1,\ldots,0.5)} \right), 0.6 \right] \right\}. \end{split}$$

*Then* (*F*Ω, *<sup>G</sup>*) <sup>∧</sup> (*Kρ*, *<sup>L</sup>*) = (*Hσ*, *<sup>G</sup>* <sup>×</sup> *<sup>L</sup>*) *where*

$$\begin{split} \mathcal{L}(H^{\sigma},\mathbb{G}\times\mathbb{L}) &= \left\{ \left[ (\epsilon\_{1},p,1), (\epsilon\_{1},p,1) \left( \frac{v\_{1}}{(0.2,0.2,\ldots,0.1),(0.5,0.2,\ldots,0.8),(0.8,0.3,\ldots,0.6)} \right),0.1 \right] \right\}, \\ &\left[ (\epsilon\_{1},p,1), (\epsilon\_{2},q,0), \left( \frac{v\_{1}}{(0.2,0.3,\ldots,0.6),(0.3,0.4,\ldots,0.8),(0.6,0.2,\ldots,0.6)} \right),0.1 \right] \end{split}$$

$$\begin{split} \left[ (\epsilon\_{2},r,0), (\epsilon\_{1},p,1), \left( \frac{v\_{1}}{(0.3,0.2,\ldots,0.1),(0.6,0.5,\ldots,0.4),(0.8,0.4,\ldots,0.4)} \right),0.2 \right] \end{split}$$

$$\left[ \left( \epsilon\_{2},r,0), (\epsilon\_{2},q,0), \left( \frac{v\_{1}}{(0.5,0.3,\ldots,0.4),(0.6,0.5,\ldots,0.4),(0.6,0.4,\ldots,0.5)} \right),0.5 \right] \right] \end{split}$$

**Definition 21.** *Let* (*F*Ω, *G*) *and* (*Kρ*, *L*) *be any two NSEMs over V, then* (*F*Ω, *G*)*OR*(*Kρ*, *L*)" *denoted* (*F*Ω, *<sup>G</sup>*) <sup>∨</sup> (*Kρ*, *<sup>L</sup>*) *is defined by*

$$(F^{\Omega}, G) \vee (K^{\rho}, L) = (H^{\sigma}, G \times L) \tag{17}$$

*where* (*Hσ*, *<sup>G</sup>* <sup>×</sup> *<sup>L</sup>*) <sup>=</sup> *<sup>H</sup>σ*(*α*, *<sup>β</sup>*) *such that <sup>H</sup>σ*(*α*, *<sup>β</sup>*) <sup>=</sup> *<sup>F</sup>*Ω(*α*) <sup>∪</sup> *<sup>K</sup>ρ*(*β*) *for all* (*α*, *<sup>β</sup>*) <sup>∈</sup> *<sup>G</sup>* <sup>×</sup> *<sup>L</sup> where* <sup>∪</sup> *represent the basic union.*

**Example 9.** *Suppose that* (*F*Ω, *G*) *and* (*Kρ*, *L*) *are two NSEMs over V, such that*

$$\begin{split} \operatorname{c}(\boldsymbol{F}^{\Omega},\mathcal{G}) &= \left\{ \left[ (\boldsymbol{e}\_{1},\boldsymbol{p},\boldsymbol{1}), \left( \frac{\boldsymbol{v}\_{1}}{(0.2,0.3,\ldots,0.6),(0.2,0.1,\ldots,0.8),(0.3,0.2,\ldots,0.6)} \right), 0.1 \right], \\ & \left[ (\boldsymbol{e}\_{2},\boldsymbol{r},\boldsymbol{0}), \left( \frac{\boldsymbol{v}\_{1}}{(0.5,0.3,\ldots,0.4),(0.6,0.5,\ldots,0.4),(0.2,0.4,\ldots,0.3)} \right), 0.5 \right] \right\}, \\ (\boldsymbol{k}^{\rho},\boldsymbol{L}) &= \left\{ \left[ (\boldsymbol{e}\_{1},\boldsymbol{p},\boldsymbol{1}), \left( \frac{\boldsymbol{v}\_{1}}{(0.3,0.2,\ldots,0.1),(0.5,0.2,\ldots,0.3),(0.8,0.3,\ldots,0.4)} \right), 0.2 \right], \\ & \left[ (\boldsymbol{e}\_{2},\boldsymbol{q},\boldsymbol{0}), \left( \frac{\boldsymbol{v}\_{1}}{(0.6,0.4,\ldots,0.7),(0.3,0.4,\ldots,0.2),(0.6,0.1,\ldots,0.5)} \right), 0.6 \right] \right\}. \end{split}$$

*Then* (*F*Ω, *<sup>G</sup>*) <sup>∨</sup> (*Kρ*, *<sup>L</sup>*) = (*Hσ*, *<sup>G</sup>* <sup>×</sup> *<sup>L</sup>*) *where*

$$\begin{split} \mathcal{L}(H^{\sigma},\mathbb{G}\times\mathbb{L}) &= \left\{ \left[ (\varepsilon\_{1},p,1), (\varepsilon\_{1},p,1) \left( \frac{v\_{1}}{(0.3,0.3,\ldots,0.6), (0.2,0.1,\ldots,0.3), (0.3,0.2,\ldots,0.4)} \right) , 0.2 \right] \right\}, \\ &\left[ (\varepsilon\_{1},p,1), (\varepsilon\_{2},q,0), \left( \frac{v\_{1}}{(0.6,0.4,\ldots,0.7), (0.2,0.1,\ldots,0.2), (0.3,0.1,\ldots,0.5)} \right) , 0.6 \right] \\ &\left[ (\varepsilon\_{2},r,0), (\varepsilon\_{1},p,1), \left( \frac{v\_{1}}{(0.5,0.3,\ldots,0.4), (0.5,0.2,\ldots,0.3), (0.2,0.3,\ldots,0.3)} \right) , 0.2 \right] \end{split}$$

$$\left[ (\varepsilon\_{2},r,0), (\varepsilon\_{2},q,0), \left( \frac{v\_{1}}{(0.6,0.4,\ldots,0.7), (0.3,0.4,\ldots,0.2), (0.2,0.1,\ldots,0.3)} \right) , 0.6 \right] \right\}.$$

**Proposition 5.** *Let* (*F*Ω, *G*) *and* (*Kρ*, *L*) *be NSEMs over V. Then*

*1.* ((*F*Ω, *<sup>G</sup>*) <sup>∧</sup> (*Kρ*, *<sup>L</sup>*))*<sup>c</sup>* = (*Fu*, *<sup>A</sup>*) *<sup>c</sup>* <sup>∨</sup> (*Gη*, *<sup>B</sup>*) *c 2.* ((*F*Ω, *<sup>G</sup>*) <sup>∨</sup> (*Kρ*, *<sup>L</sup>*))*<sup>c</sup>* = (*Fu*, *<sup>A</sup>*) *<sup>c</sup>* <sup>∧</sup> (*Gη*, *<sup>B</sup>*) *c*

**Proof.** (1) Suppose that (*F*Ω, *G*) and (*Kρ*, *L*) be NSEMs over *V* defined as:

$$\begin{array}{rcl} \left(F^{\Omega},G\right)\wedge\left(K^{\rho},L\right) &=& \left(F^{\Omega}(\mathfrak{a})\wedge K^{\rho}(\mathfrak{f})\right)^{c} \\ &=& \left(F^{\Omega}(\mathfrak{a})\cap K^{\rho}(\mathfrak{f})\right)^{c} \\ &=& \left(F^{\Omega}(\mathfrak{a})\cap K^{\rho}(\mathfrak{f})\right)^{c} \\ &=& \left(F^{\Omega\left(c\right)}(\mathfrak{a})\cup K^{\rho\left(c\right)}(\mathfrak{f})\right) \\ &=& \left(F^{\Omega\left(c\right)}(\mathfrak{a})\vee K^{\rho\left(c\right)}(\mathfrak{f})\right) \\ &=& \left(F^{\Omega\left(c\right)}\vee \left(G^{\eta},B\right)^{c}\right) \end{array}$$

(2) The proofs can be easily obtained from Definitions 20 and 21. -

#### **6. NSEMs-Aggregation Operator**

In this section, we define a NSEMs-aggregation operator of NSEMs to construct a decision method by which approximate functions of a soft expert set are combined to produce a neutrosophic set that can be used to evaluate each alternative.

**Definition 22.** *Let* <sup>Γ</sup>*<sup>G</sup>* <sup>∈</sup> *NSEMs. Then NSEMs-aggregation operator of* <sup>Γ</sup>*G*, *denoted by* <sup>Γ</sup>*agg <sup>G</sup>* , *is defined by*

$$\Gamma\_G^{\mathfrak{g}\mathfrak{g}} = \left\{ \left( \langle v, \left( \mathbf{D}^i \right)\_G^{\mathfrak{g}\mathfrak{g}}(v), \left( \mathbf{I}^i \right)\_G^{\mathfrak{g}\mathfrak{g}}(v), \left( \mathbf{Y}^i \right)\_G^{\mathfrak{g}\mathfrak{g}}(v) \rangle \right) : v \in V \right\},$$

*which are NSEMs over V*,

$$\begin{pmatrix} \left(\mathbf{D}^{\mathsf{i}}\right)\_{G}^{\operatorname{agg}} : V \to [0, 1] & \left(\mathbf{D}^{\mathsf{i}}\right)\_{G}^{\operatorname{agg}}(\upsilon) = \begin{pmatrix} 1 & \Sigma & \mathbf{D}^{\mathsf{i}}\_{G}(\upsilon) \\ \frac{1}{|V|} & \varepsilon \in E \\ & \upsilon \in V \end{pmatrix} \Omega, \\\\ \left(\mathbf{Y}^{\operatorname{i}}\right)\_{G}^{\operatorname{agg}} : V \to [0, 1] & \left(\mathbf{Y}^{\operatorname{i}}\right)\_{G}^{\operatorname{agg}}(\upsilon) = \begin{pmatrix} 1 & \Sigma & \mathbf{Y}^{\operatorname{i}}\_{G}(\upsilon) \\ \frac{1}{|V|} & \sum\_{\varepsilon \in E} & \mathbf{Y}^{\operatorname{i}}\_{G}(\upsilon) \\ \upsilon \in V \end{pmatrix} \Omega, \\\\ \left(\mathbf{\dot{I}}^{\operatorname{i}}\right)\_{G}^{\operatorname{agg}} : V \to [0, 1] & \left(\mathbf{I}^{\operatorname{i}}\right)\_{G}^{\operatorname{agg}}(\upsilon) = \begin{pmatrix} 1 & \Sigma & \mathbf{Y}^{\operatorname{i}}\_{G}(\upsilon) \\ \frac{1}{|V|} & \sum\_{\varepsilon \in E} & \mathbf{Y}^{\operatorname{i}}\_{G}(\upsilon) \\ \upsilon \in V \end{pmatrix} . \end{pmatrix}. \tag{18}$$

*where* <sup>|</sup>*V*<sup>|</sup> *is the cardinality of V and* <sup>Ω</sup>*<sup>i</sup> is defined below*

$$
\Omega = \frac{1}{n} \cdot \sum\_{i=1}^{n} \Omega(e\_i). \quad \text{ } (e\_i, i = 1, 2, 3, \dots, n) \tag{19}
$$

**Definition 23.** *Let* <sup>Γ</sup>*<sup>G</sup>* <sup>∈</sup> *NSEMs,* <sup>Γ</sup>*agg <sup>G</sup> be NSEMs. Then a reduced fuzzy set of* <sup>Γ</sup>*agg <sup>G</sup> is a fuzzy set over is denoted by*

$$\Gamma\_G^{\mathfrak{g}\mathfrak{g}} = \left\{ \frac{\lambda \Gamma\_G^{\mathfrak{g}\mathfrak{g}}(\upsilon)}{\upsilon} : \upsilon \in V \right\}\_{\prime} \tag{20}$$

*where λ*Γ*agg <sup>G</sup>* (*v*) : *V* → [0, 1] *and vi* = *Di agg Gi* <sup>−</sup> *Yi agg Gi* <sup>−</sup> *Ii agg Gi* .

#### **7. An Application of NSEMs**

In this section, we present an application of NSEMs theory in a decision-making problem. Based on Definitions 22 and 23, we construct an algorithm for the NSEMs decision-making method as follows:

**Step 1**-Choose a feasible subset of the set of parameters.

**Step 2**-Construct the NSEMs for each opinion (agree, disagree) of expert.

**Step 3**-Compute the aggregation NSEMS Γ*agg <sup>G</sup>* of Γ*<sup>G</sup>* and the reduced fuzzy set D*i agg* , Y*i agg* , I *i agg* of Γ*agg G*

*Gi Gi Gi* **Step 4**-Score (*v*˙ <sup>I</sup>) = (max − *agree*(*vi*)) − (min − *disagree*(*vi*))

**Step 5**-Choose the element of *vi* that has maximum membership. This will be the optimal solution.

**Example 10.** *In the architectural design process, let us assume that the design outputs used in the design of moving structures are taken by a few experts at certain time intervals. So, let us take the samples at three different timings in a day (in 08:30, 14:30 and 20:30) The design of moving structures consists of the architectural design, the design of the mechanism and the design of the surface covering membrane. Architectural design will be evaluated from these designs., V* = {*v*1, *v*2, *v*3}. *Suppose there are three parameters E* = {*e*1,*e*2,*e*3} *where the parameters ei* (*i* = 1, 2, 3) *stand for "time", "temperature" and "spatial needs" respectively. Let X* = {*p*, *q*} *be a set of experts. After a serious discussion, the experts construct the following NSEMs.*

**Step 1**-Choose a feasible subset of the set of parameters:

**Step 2**-Construct the neutrosophic soft expert tables for each opinion (agree, disagree) of expert.

**Step 3**-Now calculate the score of agree (*vi*) by using the data in Table 1 to obtain values in Table 2.

 D1 *agg G*1 = *D*<sup>1</sup> *G*1 +*D*<sup>1</sup> *G*2 +*D*<sup>1</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 <sup>=</sup>  0.3+0.6+0.8 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.34 D2 *agg G*1 = *D*<sup>2</sup> *G*1 +*D*<sup>2</sup> *G*2 +*D*<sup>2</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 = 0.1+0.4+0.1 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.12 D3 *agg G*1 = *D*<sup>3</sup> *G*1 +*D*<sup>3</sup> *G*2 +*D*<sup>3</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 = 0.4+0.2+0.5 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.22 (D) *agg <sup>G</sup>*<sup>1</sup> (*p*, *<sup>v</sup>*1) <sup>=</sup> 0.34+0.12+0.22 <sup>3</sup> <sup>=</sup> 0.2267 I 1 *agg G*1 = *I*<sup>1</sup> *G*1 +*I*<sup>1</sup> *G*2 +*I*<sup>1</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 <sup>=</sup>  0.2+0.3+0.2 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.1404 I 2 *agg G*1 = *I*<sup>2</sup> *G*1 +*I*<sup>2</sup> *G*2 +*I*<sup>2</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 = 0.1+0.1+0.3 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.1002 I 3 *agg G*1 = *I*<sup>3</sup> *G*1 +*I*<sup>3</sup> *G*2 +*I*<sup>3</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 = 0.5+0.4+0.4 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.2604 (I) *agg <sup>G</sup>*<sup>1</sup> (*p*, *<sup>v</sup>*1) <sup>=</sup> 0.1404+0.1002+0.2604 <sup>3</sup> <sup>=</sup> 0.167 Y1 *agg G*1 = *Y*<sup>1</sup> *G*1 +*Y*<sup>1</sup> *G*2 +*Y*<sup>1</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 <sup>=</sup>  0.5+0.8+0.5 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.36 Y2 *agg G*1 = *Y*<sup>2</sup> *G*1 +*Y*<sup>2</sup> *G*2 +*Y*<sup>2</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 <sup>=</sup>  0.2+0.2+0.2 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.12 Y3 *agg G*1 = *Y*<sup>3</sup> *G*1 +*Y*<sup>3</sup> *G*2 +*Y*<sup>3</sup> *G*3 3 . <sup>Ω</sup>1+Ω2+Ω<sup>3</sup> 3 <sup>=</sup>  0.6+0.5+0.3 <sup>3</sup> .  0.7+0.8+0.3 <sup>3</sup> = 0.2802 (Y) *agg <sup>G</sup>*<sup>1</sup> (*p*, *<sup>v</sup>*1) <sup>=</sup> 0.36+0.12+0.2802 <sup>3</sup> <sup>=</sup> 0.2534 *v*<sup>1</sup> = (D) *agg <sup>G</sup>*<sup>1</sup> − (I) *agg <sup>G</sup>*<sup>1</sup> − (Y) *agg G*1 <sup>=</sup> <sup>|</sup>0.2267 <sup>−</sup> 0.167 <sup>−</sup> 0.2534<sup>|</sup> <sup>=</sup> 0.1937

**Table 1.** Agree-neutrosophic soft expert multiset.



**Table 2.** Degree table of agree- neutrosophic soft expert multiset.

Now calculate the score of disagree (*vi*) by using the data in Table 3 to obtain values in Table 4.

**Table 3.** Disagree-neutrosophic soft expert multiset.


**Table 4.** Degree table of disagree-neutrosophic soft expert multiset.


**Step 4**-The final score of *vi* is computed as follows:

Score(*v*1) = 0.1142 − 0.1155 = −0.0013, Score(*v*2) = 0.1267 − 0.0933 = 0.0334, Score(*v*3) = 0.093 − 0.04 = 0.053.

**Step 5**-Clearly, the maximum score is the score 0.053, shown in the above for the *v*3. Hence the best decision for the experts is to select worker *v*<sup>2</sup> as the company's employee.

#### **8. Comparison Analysis**

The NSEMs model give more precision, flexibility and compatibility compared to the classical, fuzzy and/or neutrosophic models.

In order to verify the feasibility and effectiveness of the proposed decision-making approach, a comparison analysis using neutrosophic soft expert decision method, with those methods used by Alkhazaleh and Salleh [18], Maji [17], Sahin et al. [22], Hassan et al. [23] and Ulucay et al. [40] are given in Table 5, based on the same illustrative example as in An Application of NSEMs. Clearly, the ranking order results are consistent with those in [17,18,22,23,40].

**Table 5.** Comparison of fuzzy soft set and its extensive set theory.


#### **9. Conclusions**

In this paper, we reviewed the basic concepts of neutrosophic set, neutrosophic soft set, soft expert sets, neutrosophic soft expert sets and NP-aggregation operator before establishing the concept of neutrosophic soft expert multiset (NSEM). The basic operations of NSEMs, namely complement, union, intersection AND and OR were defined. Subsequently a definition of NSEM-aggregation operator is proposed to construct an algorithm of a NSEM decision method. Finally an application of the constructed algorithm to solve a decision-making problem is provided. This new extension will provide a significant addition to existing theories for handling indeterminacy, and spurs more developments of further research and pertinent applications.

**Author Contributions:** All authors contributed equally.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Neutrosophic Multigroups and Applications**

#### **Vakkas Uluçay \* and Memet ¸Sahin**

Department of Mathematics, Gaziantep University, 27310 Gaziantep, Turkey; mesahin@gantep.edu.tr **\*** Correspondence: vulucay27@gmail.com; Tel.: +90-537-643-5034

Received: 12 December 2018; Accepted: 14 January 2019; Published: 17 January 2019

**Abstract:** In recent years, fuzzy multisets and neutrosophic sets have become a subject of great interest for researchers and have been widely applied to algebraic structures include groups, rings, fields and lattices. Neutrosophic multiset is a generalization of multisets and neutrosophic sets. In this paper, we proposed a algebraic structure on neutrosophic multisets is called neutrosophic multigroups which allow the truth-membership, indeterminacy-membership and falsity-membership sequence have a set of real values between zero and one. This new notation of group as a bridge among neutrosophic multiset theory, set theory and group theory and also shows the effect of neutrosophic multisets on a group structure. We finally derive the basic properties of neutrosophic multigroups and give its applications to group theory.

**Keywords:** neutrosophic sets; neutrosophic multisets; neutrosophic multigroups; neutrosophic multisubgroups

#### **1. Introduction**

In the real world, there are much uncertainty information which cannot be handled by crisp values. The fuzzy set theory [1] has been an age old and effective tool to tackle uncertainty information by introduced Zadeh but it can be applied only on random process. Therefore, on the basis of fuzzy set theory, Sebastian and Ramakrishnan [2] introduced Multi-Fuzzy Sets, Atanassov [3] proposed intuitionistic fuzzy set theory, Shinoj and John [4] initiated intuitionistic fuzzy multisets. Recently, the above theories have developed in many directions and found its applications in a wide variety of fields including algebraic structures. For example, on fuzzy sets [5–7], on fuzzy multi sets [8–10], on intuitionistic fuzzy sets [11–19], on intuitionistic fuzzy multi sets [20] are some of the selected works.

But these theories cannot manage the all types of uncertainties, such as indeterminate and inconsistent information some decision-making problems. For instance, "when we ask the opinion of an expert about certain statement, he or she may that the possibility that the statement is true is 0.5 and the statement is false is 0.6 and the degree that he or she is not sure is 0.2" [21]. In order to overcome this shortage, Smarandache [22] introduced neutrosophic set theory to makes the theory of Atanassov [3] very convenient and easily applicable in practice. Then, Wang et al. [21] gave the some operations and results of single valued neutrosophic set theory. In order to establish the algebraic structures of neutrosophic sets, some authors gave definition of neutrosophic groups [23–26] that is actually a example of a group. To develop the neutrosophic set theory, the concept of neutrosophic multi sets was initiated by Deli et al. [27] and Ye [28,29] for modeling vagueness and uncertainty. Using their definitions, in this paper, we define a new type of neutrosophic group on a neutrosophic multi set, which we call neutrosophic multi set group. Since this new concept a brings the neutrosophic multi set theory, set theory and the group theory together, it is very functional in the sense of improving the neutrosophic multi set theory with respect to group structure. Rosenfeld [30] extended the classical group theory to fuzzy set. By using the definitions and results on fuzzy sets in [6,30] and on intuitionistic fuzzy multiset in [20], we applied the definitions and results to

neutrosophic multi set theory.The above set theories have been applied to many different areas including neutrosophic environments have been studied by many researchers in [31–39]. In this paper the notion of neutrosophic multigroup along with some related properties have been introduced by follow the results of intuitionistic fuzzy group theory. This concept will bring a new opportunity in research and development of neutrosophic sets theory.

The paper is organized as follows. In Section 2, we briefly review some preliminary concepts that will be used in the paper. In Section 3, we introduce the concept of neutrosophic multi group and give several basic properties and operations. In Section 4, we give some applications to the group theory with respect to neutrosophic multi groups. In Section 5, we make some concluding remarks and suggest.

#### **2. Preliminary**

In this section, we present basic definitions of fuzzy set theory, multi fuzzy set theory, intuitionistic fuzzy set theory, intuitionistic fuzzy multi set theory, neutrosophic set theory and neutrosophic multi set theory. For more detailed explanations related to this section, we refer to the earlier studies [1,2,4,6,20,22,27,30].

**Definition 1** ([1])**.** *Let E be a universe.*

*Then, a fuzzy set X over E is defined by*

$$X = \{ (\mu\_X(\mathbf{x})/\mathbf{x}) : \mathbf{x} \in E \},\tag{1}$$

*where μ<sup>X</sup> is called membership function of X and defined by μ<sup>X</sup>* : *E* → [0, 1]*. For each x* ∈ *E, the value μX*(*x*) *represents the degree of x belonging to the fuzzy set X.*

**Definition 2** ([2])**.** *Let X be a non-empty set. A multi-fuzzy set A on X is defined as:*

$$A = \{ <\text{x}, \mu\_1(\mathbf{x}), \mu\_2(\mathbf{x}), \mu\_3(\mathbf{x}), \dots, \mu\_i \dots : \mathbf{x} \in E \mid \},\tag{2}$$

*where μ<sup>i</sup>* : *X* → [0, 1] *for all i* ∈ {1, 2, ..., *p*} *and x* ∈ *E.*

**Definition 3** ([4])**.** *Let X be a nonempty set. An Intuitionistic Fuzzy Multi-set A denoted by IFMS drawn from X is characterized by two functions: 'count membership' of A*(*CMA*) *and 'count non membership' of A*(*CNA*) *given respectively by A*(*CMA*) : *X* → *Q and A*(*CNA*) : *X* → *Q where Q is the set of all crisp multi-sets drawn from the unit interval* [0, 1] *such that, for each x* ∈ *X, the membership sequence is defined as a decreasingly ordered sequence of elements in CMA*(*x*), *which is denoted by* (*μ*<sup>1</sup> *<sup>A</sup>*(*x*), *<sup>μ</sup>*<sup>2</sup> *<sup>A</sup>*(*x*), ..., *<sup>μ</sup><sup>P</sup> <sup>A</sup>*(*x*)) *where μ*<sup>1</sup> *<sup>A</sup>*(*x*) <sup>≥</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>A</sup>*(*x*) <sup>≥</sup> ... <sup>≥</sup> *<sup>μ</sup><sup>P</sup> <sup>A</sup>*(*x*) *and the corresponding non membership sequence will be denoted by* (*ν*<sup>1</sup> *<sup>A</sup>*(*x*), *<sup>ν</sup>*<sup>2</sup> *<sup>A</sup>*(*x*), ..., *<sup>ν</sup><sup>P</sup> <sup>A</sup>*(*x*)) *such that* <sup>0</sup> <sup>≤</sup> *<sup>μ</sup><sup>i</sup> <sup>A</sup>*(*x*) + *<sup>ν</sup><sup>i</sup> <sup>A</sup>*(*x*) ≤ 1 *for every x* ∈ *X and i* = (1, 2, 3, ..., *p*)*. An IFMS A is denoted by*

$$A = \{ \langle \mathbf{x} : (\mu\_A^1(\mathbf{x}), \mu\_A^2(\mathbf{x}), \dots, \mu\_A^P(\mathbf{x})), (\nu\_A^1(\mathbf{x}), \nu\_A^2(\mathbf{x}), \dots, \nu\_A^P(\mathbf{x})) \rangle : \mathbf{x} \in X \}. \tag{3}$$

**Definition 4** ([4])**.** *Length of an element x in an IFMS*. *A defined as the Cardinality of CMA*(*x*) *or CNA*(*x*) *for which* <sup>0</sup> <sup>≤</sup> *<sup>μ</sup><sup>j</sup> <sup>A</sup>*(*x*) + *ν j <sup>A</sup>*(*x*) ≤ 1 *and it is denoted by L*(*x* : *A*). *That is,*

$$L(\mathbf{x} : \mathcal{A}) = |\mathbb{C}M\_A(\mathbf{x})| = |\mathbb{C}N\_A(\mathbf{x})|.\tag{4}$$

**Proposition 1** ([20])**.** *Let A*, *B*, *Ai* ∈ *IFMS*(*X*)*; then, the following results hold:*


*Mathematics* **2019**, *7*, 95

*4.* [ )*n <sup>i</sup>*=<sup>1</sup> *Ai*] <sup>−</sup><sup>1</sup> = )*<sup>n</sup> i*=1[*A*−<sup>1</sup> *<sup>i</sup>* ]. *5.* (*<sup>A</sup>* ◦ *<sup>B</sup>*)−<sup>1</sup> <sup>=</sup> *<sup>B</sup>*−<sup>1</sup> ◦ *<sup>A</sup>*−1. *6. CMA*◦*B*(*x*) = <sup>∨</sup>*y*∈*X*{*CMA*(*y*) <sup>∧</sup> *CMB*(*y*−1*x*)} ∀ *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>* <sup>=</sup> <sup>∨</sup>*y*∈*X*{*CMA*(*xy*−1) <sup>∧</sup> *CMB*(*y*)} ∀ *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*. *CNA*◦*B*(*x*) = <sup>∧</sup>*y*∈*X*{*CNA*(*y*) <sup>∨</sup> *CNB*(*y*−1*x*)} ∀ *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>* <sup>=</sup> <sup>∧</sup>*y*∈*X*{*CNA*(*xy*−1) <sup>∨</sup> *CNB*(*y*)} ∀ *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*.

**Definition 5** ([20])**.** *Let X be a group. An intuitionistic fuzzy multiset G over X is an intuitionistic fuzzy multi group* (*IFMG*) *over X if the counts(count membership and non membership) of G satisfies the following four conditions:*


**Definition 6** ([22])**.** *Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set(N-set) A in X is characterized by a truth-membership function TA, a indeterminacy-membership function IA and a falsity-membership function FA. TA*(*x*)*, IA*(*x*) *and FA*(*x*) *are real standard or nonstandard subsets of* [ <sup>−</sup>0, 1+]*.*

*It can be written as*

$$A = \{<\mathbf{x}, (T\_A(\mathbf{x}), I\_A(\mathbf{x}), F\_A(\mathbf{x}))> \colon \mathbf{x} \in X, \ T\_A(\mathbf{x}), I\_A(\mathbf{x}), F\_A(\mathbf{x}) \in [0, 1] \}. \tag{5}$$

*There is no restriction on the sum of TA*(*x*)*; IA*(*x*) *and FA*(*x*)*, so* <sup>−</sup>0 ≤ *supTA*(*x*) + *supIA*(*x*) + *supFA*(*x*) <sup>≤</sup> <sup>3</sup>+*.*

*Here, 1*<sup>+</sup> *= 1+ε, where 1 is its standard part and ε its non-standard part. Similarly,* <sup>−</sup>0 *= 1+ε, where 0 is its standard part and ε its non-standard part.*

**Definition 7** ([27])**.** *Let E be a universe. A neutrosophic multiset set(Nms) A on E can be defined as follows:*

$$A\_{\quad} = \{<\mathbf{x}, (T\_A^1(\mathbf{x}), T\_A^2(\mathbf{x}), \dots, T\_A^P(\mathbf{x})), (I\_A^1(\mathbf{x}), I\_A^2(\mathbf{x}), \dots, I\_A^P(\mathbf{x})),\\(F\_A^1(\mathbf{x}), F\_A^2(\mathbf{x}), \dots, F\_A^P(\mathbf{x})) > \colon \mathbf{x} \in E\},\tag{6}$$

*where*

$$T\_A^1(\mathbf{x}), T\_A^2(\mathbf{x}), \dots, T\_A^P(\mathbf{x}): E \to [0, 1],$$

$$I\_A^1(\mathbf{x}), I\_A^2(\mathbf{x}), \dots, I\_A^P(\mathbf{x}): E \to [0, 1],$$

*and*

$$F\_A^1(\mathfrak{x}), F\_A^2(\mathfrak{x}), \dots, F\_A^P(\mathfrak{x}): E \to [0, 1]$$

*such that*

$$0 \le \sup T\_A^i(\mathbf{x}) + \sup I\_A^i(\mathbf{x}) + \sup F\_A^i(\mathbf{x}) \le 3$$

*(i* = 1, 2, ..., *P) and*

$$T\_A^1(\mathfrak{x}) \le T\_A^2(\mathfrak{x}) \le \dots \le T\_A^P(\mathfrak{x})$$

*for any x* ∈ *E.*

(*T*<sup>1</sup> *<sup>A</sup>*(*x*), *<sup>T</sup>*<sup>2</sup> *<sup>A</sup>*(*x*), ..., *<sup>T</sup><sup>P</sup> <sup>A</sup>*(*x*))*,* (*I*<sup>1</sup> *<sup>A</sup>*(*x*), *<sup>I</sup>*<sup>2</sup> *<sup>A</sup>*(*x*), ..., *<sup>I</sup><sup>P</sup> <sup>A</sup>*(*x*)) *and* (*F*<sup>1</sup> *<sup>A</sup>*(*x*), *<sup>F</sup>*<sup>2</sup> *<sup>A</sup>*(*x*), ..., *<sup>F</sup><sup>P</sup> <sup>A</sup>*(*x*)) *is the truth-membership sequence, indeterminacy-membership sequence and falsity-membership sequence of the element x, respectively. In addition, P is called the dimension(cardinality) of Nms A, denoted d(A).* *We arrange the truth-membership sequence in decreasing order, but the corresponding indeterminacy-membership and falsity-membership sequence may not be in decreasing or increasing order.*

**Definition 8** ([27,28])**.** *Let A, B be two Nms. Then,*


$$\mathbb{C}\_{\mathbf{x}} = \{ < \mathbf{x}, (T\_{\mathbf{C}}^1(\mathbf{x}), T\_{\mathbf{C}}^2(\mathbf{x}), \dots, T\_{\mathbf{C}}^p(\mathbf{x})), (I\_{\mathbf{C}}^1(\mathbf{x}), I\_{\mathbf{C}}^2(\mathbf{x}), \dots, I\_{\mathbf{C}}^p(\mathbf{x})), (F\_{\mathbf{C}}^1(\mathbf{x}), F\_{\mathbf{C}}^2(\mathbf{x}), \dots, F\_{\mathbf{C}}^p(\mathbf{x})) > \text{\textquotedblleft } \ge \varepsilon \right\},$$

*where T<sup>i</sup> <sup>C</sup>* = *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>T</sup><sup>i</sup> B*(*x*)*, I<sup>i</sup> <sup>C</sup>* = *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>I</sup><sup>i</sup> B*(*x*)*, F<sup>i</sup> <sup>C</sup>* = *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>F</sup><sup>i</sup> <sup>B</sup>*(*x*)*,* ∀*x* ∈ *E and i* = 1, 2, ..., *P. 4. The intersection of A and B is denoted by A*∩*<sup>B</sup>* = *D and is defined by*

$$D\_{-}=\{<\mathbf{x}, (T\_{\mathbf{D}}^{1}(\mathbf{x}), T\_{\mathbf{D}}^{2}(\mathbf{x}), \dots, T\_{\mathbf{D}}^{p}(\mathbf{x})), (I\_{\mathbf{D}}^{1}(\mathbf{x}), I\_{\mathbf{D}}^{2}(\mathbf{x}), \dots, I\_{\mathbf{D}}^{p}(\mathbf{x})), (F\_{\mathbf{D}}^{1}(\mathbf{x}), F\_{\mathbf{D}}^{2}(\mathbf{x}), \dots, F\_{\mathbf{D}}^{p}(\mathbf{x}))>: \mathbf{x} \in E\}, \quad \mathbf{x} \in E, \quad \mathbf{x} \in E$$

$$\text{where } T\_D^i = T\_A^i(\mathbf{x}) \land T\_B^i(\mathbf{x}), I\_D^i = I\_A^i(\mathbf{x}) \lor I\_B^i(\mathbf{x}), F\_D^i = F\_A^i(\mathbf{x}) \lor F\_B^i(\mathbf{x}), \forall \mathbf{x} \in \mathcal{E} \text{ and } \mathbf{i} = 1, 2, \dots, \text{P.}$$

#### **3. Neutrosophic Multigroups**

In this section, we introduce neutrosophic multigroups and investigate their basic properties. Throughout this section,


**Definition 9.** *Let X be a group A* <sup>∈</sup> *NMS*(*X*). *Then, A*−<sup>1</sup> *is defined as*

$$A^{-1} \ = \{ < \infty, \left( T\_A^{1-1}(\mathbf{x}), T\_A^{2-1}(\mathbf{x}), \dots, T\_A^p(\mathbf{x}) \right)^{-1}, \left( I\_A^1(\mathbf{x})^{-1}, I\_A^{2-1}(\mathbf{x}), \dots, I\_A^{p-1}(\mathbf{x}) \right), \\ \ (F\_A^{1-1}(\mathbf{x}), F\_A^{2-1}(\mathbf{x}), \dots, F\_A^{p-1}(\mathbf{x})) > \text{\textquotedblleft } \ge \varepsilon \right\}, \tag{7}$$

*where Ti*−<sup>1</sup> *<sup>A</sup>* (*x*) = *T<sup>i</sup> A*(*x*−1)*, Ii*−<sup>1</sup> *<sup>A</sup>* (*x*) = *I<sup>i</sup> <sup>A</sup>*(*x*−1) *and Fi*−<sup>1</sup> *<sup>A</sup>* (*x*) = *F<sup>i</sup> <sup>A</sup>*(*x*−1) *for all i* = 1, 2, ..., *P.*

**Definition 10.** *Let X be a classical group A* ∈ *NMS*(*X*). *Then, A is called a neutrosophic multi groupoid over X if*


*for all x*, *y* ∈ *X and i* = 1, 2, ..., *P.*

*A is called a neutrosophic multi group(NM-group) over X if the neutrosophic multi groupoid satisfies*


*for all x* ∈ *X and i* = 1, 2, ..., *P.*

**Example 1.** *Assume that* (*Z*3, +) *is a classical group. Then,*

*A* = {0;(0.8, 0.7, 0.6, 0.4),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.5),1;(0.7, 0.6, 0.4, 0.3), (0.2, 0.3, 0.2, 0.3),(0.3, 0.4, 0.5, 0.6),2;(0.8, 0.6, 0.6, 0.4),(0.1, 0.2, 0.2, 0.3),(0.2, 0.4, 0.4, 0.5)} *is a NM-group. However,*

*B* = {0;(0.8, 0.7, 0.6, 0.4),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.5),1;(0.9, 0.5, 0.4, 0.3),(0.2, 0.1, 0.2, 0.3), (0.3, 0.3, 0.5, 0.4),2;(0.8, 0.7, 0.6, 0.4),(0.1, 0.3, 0.2, 0.3),(0.2, 0.4, 0.4, 0.6)}

*is not a NM-group because T<sup>i</sup> <sup>B</sup>*(1−1) *is not greater than or equal to T<sup>i</sup> <sup>B</sup>*(1)*.*

From the Definition 10 and Example 1, it is clear that a *NM*-group is a generalized case of fuzzy group and intuitionistic fuzzy multi group.

**Proposition 2.** *Let X be a classical group and A* ∈ *NMS*(*X*)*. If A* ∈ *NMG*(*X*)*; then,*


*for all x* ∈ *X and i* = 1, 2, ..., *P.*

**Proof.** Since A an *NM* − *group* over X, then

1.

$$\begin{array}{ll} T^{i}\_{A}(\mathfrak{e}) &= T^{i}\_{A}(\mathfrak{x}.\mathfrak{x}^{-1}) \\ &\geq T^{i}\_{A}(\mathfrak{x}) \wedge T^{i}\_{A}(\mathfrak{x}^{-1}) \\ &\geq T^{i}\_{A}(\mathfrak{x}) \wedge T^{i}\_{A}(\mathfrak{x}) \\ &= T^{i}\_{A}(\mathfrak{x}) \end{array}$$

for all *x* ∈ *X* and *i* = 1, 2, ..., *P*.

2.

3.

$$\begin{array}{ll} I^{\bar{i}}\_{A}(\mathfrak{e}) &= I^{\bar{i}}\_{A}(\mathfrak{x}.\mathfrak{x}^{-1}) \\ &\leq I^{\bar{i}}\_{A}(\mathfrak{x}) \vee I^{\bar{i}}\_{A}(\mathfrak{x}^{-1}) \\ &\leq I^{\bar{i}}\_{A}(\mathfrak{x}) \vee I^{\bar{i}}\_{A}(\mathfrak{x}) \\ &= I^{\bar{i}}\_{A}(\mathfrak{x}) \end{array}$$

for all *x* ∈ *X* and *i* = 1, 2, ..., *P*.

$$\begin{array}{lcl}F^{i}\_{A}(\boldsymbol{e})&=F^{i}\_{A}(\boldsymbol{\mathfrak{x}}.\boldsymbol{x}^{-1})\\ &\leq F^{i}\_{A}(\boldsymbol{\mathfrak{x}})\vee F^{i}\_{A}(\boldsymbol{\mathfrak{x}}^{-1})\\ &\leq F^{i}\_{A}(\boldsymbol{\mathfrak{x}})\vee F^{i}\_{A}(\boldsymbol{\mathfrak{x}})\\ &= F^{i}\_{A}(\boldsymbol{\mathfrak{x}})\end{array}$$

for all *x* ∈ *X* and *i* = 1, 2, ..., *P*.

**Proposition 3.** *Let X be a classical group and A* ∈ *NMS*(*X*)*. If A* ∈ *NMG*(*X*)*, then*

*1. T<sup>i</sup> <sup>A</sup>*(*xn*) <sup>≥</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) ∀ *x* ∈ *X*, *2. I<sup>i</sup> <sup>A</sup>*(*xn*) <sup>≤</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*) ∀ *x* ∈ *X*, *3. F<sup>i</sup> <sup>A</sup>*(*xn*) <sup>≤</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*x*) ∀ *x* ∈ *X*,

*for all x* ∈ *X and i* = 1, 2, ..., *P.*

**Proof.** Since A an *NM* − *group* over X, then

1.

$$\begin{array}{ll} T^{\dot{\imath}}\_{A}(\mathfrak{x}^{n}) & \geq T^{\dot{\imath}}\_{A}(\mathfrak{x}) \wedge T^{\dot{\imath}}\_{A}(\mathfrak{x}^{n-1}) \\ & \geq T^{\dot{\imath}}\_{A}(\mathfrak{x}) \wedge T^{\dot{\imath}}\_{A}(\mathfrak{x}) \wedge \dots \wedge T^{\dot{\imath}}\_{A}(\mathfrak{x}) \\ & = T^{\dot{\imath}}\_{A}(\mathfrak{x}) \end{array}$$

for all *x* ∈ *X* and *i* = 1, 2, ..., *P*.

2.

$$\begin{array}{rcll}I^{\dot{i}}\_{A}(\mathfrak{x}^{n}) & \leq I^{\dot{i}}\_{A}(\mathfrak{x}) \vee I^{\dot{i}}\_{A}(\mathfrak{x}^{n-1})\\ & \leq I^{\dot{i}}\_{A}(\mathfrak{x}) \vee I^{\dot{i}}\_{A}(\mathfrak{x}) \vee \dots \vee I^{\dot{i}}\_{A}(\mathfrak{x})\\ & = I^{\dot{i}}\_{A}(\mathfrak{x})\end{array}$$

for all *x* ∈ *X* and *i* = 1, 2, ..., *P*.

3.

$$\begin{array}{ll} F^{i}\_{A}(\mathfrak{x}^{n}) & \leq F^{i}\_{A}(\mathfrak{x}) \vee F^{i}\_{A}(\mathfrak{x}^{n-1}) \\ & \leq F^{i}\_{A}(\mathfrak{x}) \vee F^{i}\_{A}(\mathfrak{x}) \vee \dots \vee F^{i}\_{A}(\mathfrak{x}) \\ & = F^{i}\_{A}(\mathfrak{x}) \end{array}$$

for all *x* ∈ *X* and *i* = 1, 2, ..., *P*.

**Definition 11.** *Let <sup>Y</sup> be a subgroup of <sup>X</sup>*, *<sup>B</sup>* <sup>∈</sup> *NMG*(*Y*), *<sup>B</sup>*⊆˜ *<sup>A</sup> and <sup>A</sup>* <sup>∈</sup> *NMG*(*X*)*. If <sup>B</sup>* <sup>∈</sup> *NMG*(*Y*)*, then B is called a neutrosophic multi subgroup of A over X and denoted by B*≤˜ *<sup>A</sup>*.

**Example 2.** *Assume that* (*Z*3, +) *is a classical group. We define A and B neutrosophic multi group over* (*Z*3, +) *by*

> *A* = {0;(0.4, 0.3, 0.3, 0.2),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.6), 1;(0.6, 0.5, 0.3, 0.2),(0.2, 0.4, 0.2, 0.3),(0.3, 0.2, 0.5, 0.6), 2;(0.8, 0.7, 0.5, 0.4),(0.1, 0.3, 0.2, 0.3),(0.2, 0.1, 0.4, 0.5)}.

> *B* = {0;(0.4, 0.3, 0.3, 0.2),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.6), 1;(0.6, 0.5, 0.3, 0.2),(0.2, 0.4, 0.2, 0.3),(0.3, 0.2, 0.5, 0.6)}.

*Then, B is a neutrosophic multi subgroup of A over* (*Z*3, +) *and denoted by B*≤˜ *<sup>A</sup>*.

**Theorem 1.** *Let <sup>X</sup> be a group <sup>A</sup>* <sup>∈</sup> *NMS*(*X*)*. Then, <sup>A</sup> is an NM* <sup>−</sup> *group if and only if <sup>T</sup><sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≥</sup> *Ti <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*y*), *I<sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≤</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*y*) *and F<sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≤</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*y*) *for all x*, *y* ∈ *X.*

**Proof.** Assume that A is an *NM* − *group* over X. Then,

$$\begin{array}{ll} T^i{}\_A(xy^{-1}) & \geq T^i{}\_A(x) \land T^i{}\_A(y^{-1})\\ & \geq T^i{}\_A(x) \land T^i{}\_A(y) \end{array}$$

for all *x*, *y* ∈ *X* and *i* = 1, 2, ..., *P*.

$$\begin{array}{rcl} I^{i}\_{A}(\mathfrak{x}y^{-1}) & \leq I^{i}\_{A}(\mathfrak{x}) \vee I^{i}\_{A}(\mathfrak{y}^{-1})\\ & \leq I^{i}\_{A}(\mathfrak{x}) \vee I^{i}\_{A}(\mathfrak{y}) \end{array}$$

for all *x*, *y* ∈ *X* and *i* = 1, 2, ..., *P*.

$$\begin{array}{rcl} F^{i}\_{\mathcal{A}}(\mathfrak{x}\mathfrak{y}^{-1}) & \leq F^{i}\_{\mathcal{A}}(\mathfrak{x}) \vee F^{i}\_{\mathcal{A}}(\mathfrak{y}^{-1}) \\ & \leq F^{i}\_{\mathcal{A}}(\mathfrak{x}) \vee F^{i}\_{\mathcal{A}}(\mathfrak{y}) \end{array}$$

for all *x*, *y* ∈ *X* and *i* = 1, 2, ..., *P*.

Conversely, the given condition be satisfied. Firstly,

$$\begin{array}{ll} T^{\dot{\mathbf{i}}}A(\mathbf{x}^{-1}) &= T^{\dot{\mathbf{i}}}A(\mathbf{c}\mathbf{x}^{-1}) \\ &\geq T^{\dot{\mathbf{i}}}A(\mathbf{c}) \wedge T^{\dot{\mathbf{i}}}A(\mathbf{x}) \\ &= T^{\dot{\mathbf{i}}}A(\mathbf{x}) \end{array}$$

$$\begin{array}{ll} T^{\dot{\mathbf{i}}}A(\mathbf{x}y) &\geq T^{\dot{\mathbf{i}}}A(\mathbf{x}) \wedge T^{\dot{\mathbf{i}}}A(y^{-1}) \\ &\geq T^{\dot{\mathbf{i}}}A(\mathbf{x}) \wedge T^{\dot{\mathbf{i}}}A(y). \end{array}$$

$$\begin{array}{ll} I^{\dot{\mathbf{i}}}\_{A}(\mathbf{x}^{-1}) &= I^{\dot{\mathbf{i}}}\_{A}(\mathbf{c}\mathbf{x}^{-1}) \\ &\leq I^{\dot{\mathbf{i}}}\_{A}(\mathbf{c}) \vee I^{\dot{\mathbf{i}}}A(\mathbf{x}) \\ &= I^{\dot{\mathbf{i}}}\_{A}(\mathbf{x}) \\ &\leq I^{\dot{\mathbf{i}}}\_{A}(\mathbf{x}) \vee I^{\dot{\mathbf{i}}}\_{A}(\mathbf{y}). \end{array}$$

Thirdly,

Secondly,

$$\begin{array}{rcl} F^{i}\_{A}(\mathbf{x}^{-1}) &= F^{i}\_{A}(\boldsymbol{\varepsilon}\mathbf{x}^{-1}) \\ &\leq F^{i}\_{A}(\boldsymbol{\varepsilon}) \vee F^{i}\_{A}(\mathbf{x}) \\ &= F^{i}\_{A}(\mathbf{x}) \end{array}$$

$$\begin{array}{rcl} F^{i}\_{A}(\mathbf{x}y) &\leq F^{i}\_{A}(\mathbf{x}) \vee F^{i}\_{A}(y^{-1}) \\ &\leq F^{i}\_{A}(\mathbf{x}) \vee F^{i}\_{A}(y) \end{array}$$

so the proof is complete.

**Definition 12.** *Let A*, *B* ∈ *NMS*(*X*). *Then, their "AND" operation is denoted by A*∧˜ *B and is defined by*

$$A \vec{\wedge} B = \{ (\mathbf{x}, \mathbf{y}), \, T^i{}\_{A \vec{\wedge} B} (\mathbf{x}, \mathbf{y}), \, I^i{}\_{A \vec{\wedge} B} (\mathbf{x}, \mathbf{y}), \, F^i{}\_{A \vec{\wedge} B} (\mathbf{x}, \mathbf{y}) : (\mathbf{x}, \mathbf{y}) \in \mathbf{X} \times \mathbf{X} \}, \tag{8}$$

*where T<sup>i</sup> <sup>A</sup>*∧˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>T</sup><sup>i</sup> B*(*y*)*, I<sup>i</sup> <sup>A</sup>*∧˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>I</sup><sup>i</sup> B*(*y*)*, F<sup>i</sup> <sup>A</sup>*∧˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>F</sup><sup>i</sup> <sup>B</sup>*(*y*)*.*

**Theorem 2.** *Let A*, *B* ∈ *NMG*(*X*). *Then, A*∧˜ *B is a neutrosophic multi group over X*.

**Proof.** Let (*x*1, *y*1),(*x*2, *y*2) ∈ *X* × *X*. Then,

$$\begin{array}{ll} T^{i}{}\_{A \wedge B}((\mathbf{x}\_{1}, y\_{1}), (\mathbf{x}\_{2}, y\_{2})^{-1}) &= T^{i}{}\_{A \wedge B}(\mathbf{x}\_{1} \mathbf{x}\_{2}^{-1}, y\_{1} y\_{2}^{-1}) \\ &= T^{i}{}\_{A}(\mathbf{x}\_{1} \mathbf{x}\_{2}^{-1}) \wedge T^{i}{}\_{B}(y\_{1} y\_{2}^{-1}) \\ &\geq (T^{i}{}\_{A}(\mathbf{x}\_{1}) \wedge T^{i}{}\_{A}(\mathbf{x}\_{2})) \wedge (T^{i}{}\_{B}(y\_{1}) \wedge T^{i}{}\_{B}(y\_{2})) \\ &= (T^{i}{}\_{A}(\mathbf{x}\_{1}) \wedge T^{i}{}\_{B}(y\_{1})) \wedge (T^{i}{}\_{A}(\mathbf{x}\_{2}) \wedge T^{i}{}\_{B}(y\_{2})) \\ &= T^{i}{}\_{A \wedge B}(\mathbf{x}\_{1}, y\_{1}) \wedge T^{i}{}\_{A \wedge B}(\mathbf{x}\_{2}, y\_{2}) \end{array}$$

$$\begin{array}{ll} \left( \begin{array}{l} \left( \left( \mathbf{x}\_{1}, y\_{1} \right), \left( \mathbf{x}\_{2}, y\_{2} \right) \right)^{-1} \right) & = \, \right. \\ & \left( \left( \mathbf{x}\_{1} \mathbf{x}\_{2}^{-1} \right) \lor I\_{B}^{i} \left( y\_{1} y\_{2}^{-1} \right) \\ & = \, \right. \\ & \left( \left( \mathbf{I}\_{A} \left( \mathbf{x}\_{1} \right) \lor I\_{A}^{i} \left( \mathbf{x}\_{2} \right) \right) \lor \left( I\_{B}^{i} \left( y\_{1} \right) \lor I\_{B}^{i} \left( y\_{2} \right) \right) \\ & = \left( \left( \mathbf{I}\_{A} \left( \mathbf{x}\_{1} \right) \lor I\_{B}^{i} \left( y\_{1} \right) \right) \lor \left( I\_{A}^{i} \left( \mathbf{x}\_{2} \right) \lor I\_{B}^{i} \left( y\_{2} \right) \right) \right) \\ & = \, \right. \\ & \left( \mathbf{I}\_{A \land B}^{i} \left( \mathbf{x}\_{1}, y\_{1} \right) \lor I\_{A \land B}^{i} \left( \mathbf{x}\_{2}, y\_{2} \right) \right) \end{array}$$

and

$$\begin{array}{ll} F^i\_{A \wedge B}((\mathbf{x}\_1, y\_1), (\mathbf{x}\_2, y\_2)^{-1}) &= F^i\_{A \wedge B}(\mathbf{x}\_1 \mathbf{x}\_2^{-1}, y\_1 y\_2^{-1}) \\ &= F^i\_{A}(\mathbf{x}\_1 \mathbf{x}\_2^{-1}) \vee F^i\_B(y\_1 y\_2^{-1}) \\ &\leq (F^i\_{A}(\mathbf{x}\_1) \vee F^i\_{A}(\mathbf{x}\_2)) \vee (F^i\_B(y\_1) \vee F^i\_B(y\_2)) \\ &= (F^i\_{A}(\mathbf{x}\_1) \vee F^i\_B(y\_1)) \vee (F^i\_{A}(\mathbf{x}\_2) \vee F^i\_B(y\_2)) \\ &= F^i\_{A \wedge B}(\mathbf{x}\_1, y\_1) \vee F^i\_{A \wedge B}(\mathbf{x}\_2, y\_2) \end{array}$$

for all (*x*1, *y*1),(*x*2, *y*2) ∈ *X* and *i* = 1, 2, ..., *P*. Therefore, *A*∧˜ *B* is a neutrosophic multi group over *X*, hence the proof.

**Example 3.** *Let us take into consideration the classical group* (*Z*3, +)*. Define the neutrosophic multiset A*, *B on* (*Z*3, +) *as follows:*

*A* = {0;(0.5, 0.3, 0.2, 0.1),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.5), 1;(0.6, 0.4, 0.3, 0.2),(0.1, 0.3, 0.3, 0.4),(0.1, 0.2, 0.4, 0.6), 2;(0.7, 0.5, 0.3, 0.2),(0.2, 0.2, 0.3, 0.4),(0.3, 0.3, 0.4, 0.6), *B* = {0;(0.6, 0.5, 0.4, 0.2),(0.2, 0.2, 0.3, 0.4),(0.3, 0.3, 0.4, 0.6), 1;(0.8, 0.6, 0.4, 0.3),(0.1, 0.1, 0.2, 0.3),(0.2, 0.2, 0.3, 0.4)} *are NM* − *groups*. *A*∧˜ *B* = {(0, 0);(0.5, 0.3, 0.2, 0.1),(0.2, 0.2, 0.2, 0.4),(0.2, 0.3, 0.4, 0.6), (0, 1);(0.5, 0.3, 0.2, 0.1),(0.1, 0.1, 0.2, 0.3),(0.2, 0.2, 0.3, 0.5), (1, 0);(0.6, 0.4, 0.3, 0.2),(0.2, 0.2, 0.3, 0.4),(0.3, 0.3, 0.4, 0.6), (1, 1);(0.6, 0.4, 0.3, 0.2),(0.1, 0.3, 0.3, 0.4),(0.2, 0.2, 0.4, 0.6), (2, 0);(0.6, 0.5, 0.3, 0.2),(0.2, 0.2, 0.3, 0.4),(0.3, 0.3, 0.4, 0.6), (2, 1);(0.7, 0.5, 0.3, 0.2),(0.2, 0.2, 0.3, 0.4),(0.3, 0.3, 0.4, 0.6)}.

*Then, A*∧˜ *B* ∈ *NMG*(*X*).

**Definition 13.** *Let X be a classical group and A*, *B* ∈ *NMS*(*X*). *Then, their "OR" operation is denoted by A*∨˜ *B and is defined by*

$$A \lor B = \{ (\mathbf{x}, \mathbf{y}), T^l{}\_{A \lor B}(\mathbf{x}, \mathbf{y}), I^l{}\_{A \lor B}(\mathbf{x}, \mathbf{y}), F^l{}\_{A \lor B}(\mathbf{x}, \mathbf{y}) : (\mathbf{x}, \mathbf{y}) \in \mathcal{X} \times \mathcal{X} \} \tag{9}$$

*where T<sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>T</sup><sup>i</sup> B*(*y*)*, I<sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>I</sup><sup>i</sup> B*(*y*)*, F<sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>F</sup><sup>i</sup> <sup>B</sup>*(*y*)*.*

**Proposition 4.** *Let <sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *NMG*(*X*). *Then, <sup>T</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*) <sup>≤</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*−1)*, <sup>I</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*) <sup>≥</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*−1)*, Fi <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*) <sup>≥</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*−1)*.*

**Proof.** Let (*x*1, *y*1),(*x*2, *y*2) ∈ *X* × *X*. Then,

$$\begin{array}{ll} T^{i}{}\_{A\odot B}((\mathbf{x}\_{1},y\_{1}),(\mathbf{x}\_{2},y\_{2})^{-1}) &= T^{i}{}\_{A\odot B}(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1},y\_{1}y\_{2}^{-1}) \\ &= T^{i}{}\_{A}(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1}) \vee T^{i}{}\_{B}(y\_{1}y\_{2}^{-1}) \\ &\leq (T^{i}{}\_{A}(\mathbf{x}\_{1}) \vee T^{i}{}\_{A}(\mathbf{x}\_{2})) \vee (T^{i}{}\_{B}(y\_{1}) \vee T^{i}{}\_{B}(y\_{2})) \\ &= (T^{i}{}\_{A}(\mathbf{x}\_{1}) \vee T^{i}{}\_{B}(y\_{1})) \vee (T^{i}{}\_{A}(\mathbf{x}\_{2}) \vee T^{i}{}\_{B}(y\_{2})) \\ &= T^{i}{}\_{A\odot B}(\mathbf{x}\_{1},y\_{1}) \vee T^{i}{}\_{A\odot B}(\mathbf{x}\_{2},y\_{2}) \end{array}$$

$$\begin{array}{ll} \left( \begin{array}{l} \left( \begin{array}{c} (\mathbf{x}\_{1}, y\_{1}), (\mathbf{x}\_{2}, y\_{2}) \right)^{-1} \right) \\ \end{array} \right) & = \left( \begin{array}{l} \mathrm{i}\_{A \lor B} \left( \mathbf{x}\_{1} \mathbf{x}\_{2}^{-1}, y\_{1} y\_{2}^{-1} \right) \\ \end{array} \right) \\ & = \left( \mathrm{i}\_{A}^{\top} \left( \mathbf{x}\_{1} \mathbf{x}\_{2}^{-1} \right) \land \, \mathrm{i}^{\top}\_{B} \left( y\_{1} y\_{2}^{-1} \right) \\ \geq \left( \begin{array}{l} \mathrm{i}\_{A}^{\top} \left( \mathbf{x}\_{1} \right) \land \, \mathrm{i}^{\top}\_{A} \left( \mathbf{x}\_{2} \right) \right) \land \left( \, \mathrm{i}\_{B}^{\top} \left( y\_{1} \right) \land \, \mathrm{i}^{\top}\_{B} \left( y\_{2} \right) \right) \\ = \left( \, \mathrm{i}\_{A}^{\top} \left( \mathbf{x}\_{1} \right) \land \, \mathrm{i}^{\top}\_{B} \left( y\_{1} \right) \right) \land \left( \, \mathrm{i}\_{A}^{\top} \left( \mathbf{x}\_{2} \right) \land \, \mathrm{i}^{\top}\_{B} \left( y\_{2} \right) \right) \\ = \, \mathrm{i}\_{A \mp B}^{\top} \left( \mathbf{x}\_{1}, y\_{1} \right) \land \, \mathrm{i}^{\top}\_{A \mp B} \left( \mathbf{x}\_{2}, y\_{2} \right) \end{array} \end{array}$$

and

$$\begin{array}{ll} F^{i}\_{A \cap B}((\mathbf{x}\_{1}, y\_{1}), (\mathbf{x}\_{2}, y\_{2})^{-1}) &= F^{i}\_{A \cap B}(\mathbf{x}\_{1} \mathbf{x}\_{2}^{-1}, y\_{1} y\_{2}^{-1}) \\ &= F^{i}\_{A}(\mathbf{x}\_{1} \mathbf{x}\_{2}^{-1}) \wedge F^{i}\_{B}(y\_{1} y\_{2}^{-1}) \\ &\geq (F^{i}\_{A}(\mathbf{x}\_{1}) \wedge F^{i}\_{A}(\mathbf{x}\_{2})) \wedge (F^{i}\_{B}(y\_{1}) \wedge F^{i}\_{B}(y\_{2})) \\ &= (F^{i}\_{A}(\mathbf{x}\_{1}) \wedge F^{i}\_{B}(y\_{1})) \wedge (F^{i}\_{A}(\mathbf{x}\_{2}) \wedge F^{i}\_{B}(y\_{2})) \\ &= F^{i}\_{A \cap B}(\mathbf{x}\_{1}, y\_{1}) \wedge F^{i}\_{A \cap B}(\mathbf{x}\_{2}, y\_{2}) \end{array}$$

for all (*x*1, *y*1),(*x*2, *y*2) ∈ *X* and *i* = 1, 2, ..., *P*—hence the proof.

From this, it is clear that, if *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *NMG*(*X*), then *<sup>A</sup>*∨˜ *<sup>B</sup>* <sup>∈</sup> *NMG*(*X*) iff *<sup>T</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) <sup>≥</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*) ∧ *Ti <sup>A</sup>*∨˜ *<sup>B</sup>*(*y*), *<sup>I</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) <sup>≤</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*y*), *<sup>F</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) <sup>≤</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*x*) <sup>∨</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(*y*).

**Corollary 1.** *Let A*, *B* ∈ *NMG*(*X*). *Then, A*∨˜ *B need not be an element of NMG*(*X*).

**Example 4.** *Let us take into consideration the classical group* (*Z*4, +)*. Define the neutrosophic multiset A*, *B on* (*Z*4, +) *as follows:*

*A* = {0;(0.5, 0.3, 0.2, 0.1),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.5), 1;(0.6, 0.4, 0.3, 0.2),(0.1, 0.3, 0.3, 0.4),(0.1, 0.2, 0.4, 0.6), 2;(0.7, 0.5, 0.3, 0.2),(0.2, 0.2, 0.3, 0.4),(0.3, 0.3, 0.4, 0.6), 3;(0.7, 0.6, 0.4, 0.3),(0.2, 0.1, 0.2, 0.3),(0.3, 0.2, 0.1, 0.3)} *B* = {0;(0.6, 0.5, 0.4, 0.2),(0.2, 0.2, 0.3, 0.4),(0.2, 0.3, 0.4, 0.6), 1;(0.8, 0.6, 0.4, 0.3),(0.1, 0.1, 0.2, 0.3),(0.2, 0.2, 0.3, 0.4)} *are NM* − *groups*. *A*∨˜ *B* = {(0, 0);(0.6, 0.5, 0.4, 0.2),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.5), (0, 1);(0.8, 0.6, 0.4, 0.3),(0.1, 0.1, 0.2, 0.3),(0.2, 0.2, 0.3, 0.4), (1, 0);(0.6, 0.5, 0.4, 0.2),(0.1, 0.2, 0.3, 0.4),(0.1, 0.2, 0.4, 0.6), (1, 1);(0.8, 0.6, 0.4, 0.3),(0.1, 0.1, 0.2, 0.3),(0.1, 0.2, 0.3, 0.4), (2, 0);(0.7, 0.5, 0.4, 0.2),(0.1, 0.2, 0.3, 0.4),(0.2, 0.3, 0.4, 0.6), (2, 1);(0.8, 0.6, 0.4, 0.3),(0.1, 0.1, 0.2, 0.3),(0.2, 0.2, 0.3, 0.4), (3, 0);(0.7, 0.6, 0.4, 0.3),(0.2, 0.1, 0.2, 0.3),(0.2, 0.2, 0.1, 0.3), (3, 1);(0.8, 0.6, 0.4, 0.3),(0.1, 0.1, 0.2, 0.3),(0.2, 0.2, 0.1, 0.3)}.

*However, T<sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(3, 0) <sup>≥</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*∨˜ *<sup>B</sup>*(1, 0)*. Then, A*∨˜ *<sup>B</sup>* <sup>∈</sup>/ *NMG*(*X*).

**Theorem 3.** *Let X be a classical group and A* ∈ *NMG*(*X*). *Then, the followings are equivalent:*


**Proof.** 1. (1) ⇒ (2): Let *x*, *y* ∈ *X*. Then,

$$\begin{array}{l} T\_A^i(\mathbf{x}y\mathbf{x}^{-1}) = T\_A^i(\mathbf{x}^{-1}\mathbf{x}y) = T\_A^i(y), \\ I\_A^i(\mathbf{x}y\mathbf{x}^{-1}) = I\_A^i(\mathbf{x}^{-1}\mathbf{x}y) = I\_A^i(y), \\ F\_A^i(\mathbf{x}y\mathbf{x}^{-1}) = F\_A^i(\mathbf{x}^{-1}\mathbf{x}y) = F\_A^i(y). \end{array}$$

2. (2) ⇒ (3): Immediate.

3. (3) ⇒ (4)

$$\begin{array}{l} T\_A^i(\mathbf{x}\mathbf{y}\mathbf{x}^{-1}) \le T\_A^i(\mathbf{x}^{-1}\mathbf{x}\mathbf{y}(\mathbf{x}^{-1})^{-1}) = T\_A^i(\mathbf{y}),\\ I\_A^i(\mathbf{x}\mathbf{y}\mathbf{x}^{-1}) \ge I\_A^i(\mathbf{x}^{-1}\mathbf{x}\mathbf{y}(\mathbf{x}^{-1})^{-1}) = I\_A^i(\mathbf{y}),\\ F\_A^i(\mathbf{x}\mathbf{y}\mathbf{x}^{-1}) \ge F\_A^i(\mathbf{x}^{-1}\mathbf{x}\mathbf{y}(\mathbf{x}^{-1})^{-1}) = F\_A^i(\mathbf{y}).\end{array}$$

4. (4) ⇒ (1): Let *x*, *y* ∈ *X*. Then,

$$\begin{array}{ll} T\_A^i(xy) &= T\_A^i(\mathbf{x}.y\mathbf{x}.\mathbf{x}^{-1}) \\ &\leq T\_A^i(yx) \\ &= T\_A^i(y.xy.y^{-1}) \\ &\leq T\_A^i(xy), \end{array}$$

$$\begin{array}{rcl} \mathrm{I}\_{A}^{i}(\mathbf{x}\mathbf{y}) &=& \mathrm{I}\_{A}^{i}(\mathbf{x},\mathbf{y}\mathbf{x}.\mathbf{x}^{-1}) \\ & \leq \mathrm{I}\_{A}^{i}(\mathbf{y}\mathbf{x}) \\ &=\mathrm{I}\_{A}^{i}(\mathbf{y},\mathbf{x}\mathbf{y}.\mathbf{y}^{-1}) \\ & \leq \mathrm{I}\_{A}^{i}(\mathbf{x}\mathbf{y}), \\ & & \leq \mathrm{I}\_{A}^{i}(\mathbf{x}\mathbf{y}.\mathbf{x}.\mathbf{x}^{-1}) \\ & & \leq \mathrm{I}\_{A}^{i}(\mathbf{y}\mathbf{x}) \\ & & = \mathrm{I}\_{A}^{i}(\mathbf{y}.\mathbf{x}\mathbf{y}.\mathbf{y}^{-1}) \\ & & \leq \mathrm{I}\_{A}^{i}(\mathbf{x}\mathbf{y}) \\ \end{array}$$
  $\text{Hence, } \mathrm{I}\_{A}^{i}(\mathbf{y}\mathbf{x}) = \mathrm{I}\_{A}^{i}(\mathbf{x}\mathbf{y}), \mathrm{I}\_{A}^{i}(\mathbf{y}\mathbf{x}) = \mathrm{I}\_{A}^{i}(\mathbf{x}\mathbf{y}), \mathrm{I}\_{A}^{i}(\mathbf{y}\mathbf{x}) = \mathrm{I}\_{A}^{i}(\mathbf{x}\mathbf{y}).$ 

**Definition 14.** *Let X be a group, A* ∈ *NMS*(*X*) *and B is a nonempty neutrosophic multi subset of A over X*. *Then, B is called an abelian neutrosophic multi subset of A if T<sup>i</sup> <sup>A</sup>*(*yx*) = *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*xy*), *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*yx*) = *Ii <sup>A</sup>*(*xy*) *and F<sup>i</sup> <sup>A</sup>*(*yx*) = *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*xy*) *for all x*, *y* ∈ *X.*

**Example 5.** 1*<sup>X</sup> and* 1*<sup>e</sup> are normal neutrosophic multi subgroup of X*. *If X is a commutative group, every neutrosophic multi subgroup of X is normal.*

**Definition 15.** *Let X be a group, A* ∈ *NMG*(*X*) *and B is a neutrosophic multi subgroup of A over X*. *Then, <sup>B</sup> is called an a normal neutrosophic multi subgroup of A, denoted by <sup>B</sup>*˜ *<sup>A</sup> if it is an abelian neutrosophic multi subset of A over X.*

**Example 6.** *Assume that* (*Z*3, +) *is a classiccal group. Define the neutrosophic multisets A and B on* (*Z*3, +) *as follows:*

> *A* = {0;(0.6, 0.5, 0.4, 0.2),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.6), 1;(0.5, 0.4, 0.4, 0.3),(0.2, 0.1, 0.2, 0.3),(0.3, 0.4, 0.5, 0.6), 2;(0.9, 0.7, 0.6, 0.5),(0.1, 0.1, 0.2, 0.3),(0.2, 0.2, 0.3, 0.5)}

*is a NM-group. If*

$$B\_{\phantom{\phantom{\phantom{0}}}} = \{ \langle 0; (0.6, 0.5, 0.4, 0.2), (0.1, 0.1, 0.2, 0.3), (0.2, 0.3, 0.4, 0.6) \rangle, \\ \langle 1; (0.5, 0.4, 0.4, 0.3), (0.2, 0.1, 0.2, 0.3), (0.3, 0.3, 0.5, 0.4) \rangle \},$$

*then B is a neutrosophic multi subgroup of A over* (*Z*3, +) *and denoted by B*≤˜ *<sup>A</sup>*. *Therefore, B*˜ *<sup>A</sup>*.

**Corollary 2.** *Let A* ∈ *NMG*(*X*) *and B be a neutrosophic multi subgroup of A over X. If X is an abelian group, then B is a normal neutrosophic multi subgroup of A over X.*

#### **4. Applications of Neutrosophic Multi Groups**

In this section, we give some applications to the group theory with respect to neutrosophic multi groups.

**Definition 16.** *Let A be a neutrosophic multiset on X and α* ∈ [0, 1]. *Define the α-level sets of A as follows:*

$$\begin{array}{l}(T^{i}\_{A})\_{\mathfrak{a}} = \{ \mathfrak{x} \in X : T^{i}\_{A}(\mathfrak{x}) \ge \mathfrak{a} \}, \\ (I^{i}\_{A})^{\mathfrak{a}} = \{ \mathfrak{x} \in X : I^{i}\_{A}(\mathfrak{x}) \le \mathfrak{a} \}, \\ (F^{i}\_{A})^{\mathfrak{a}} = \{ \mathfrak{x} \in X : F^{i}\_{A}(\mathfrak{x}) \le \mathfrak{a} \}. \end{array}$$

*It is easy to verify that*

(1) *IfA*⊆˜ *B and <sup>α</sup>* <sup>∈</sup> [0, 1], *then* (*T<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* <sup>⊆</sup> (*T<sup>i</sup> B*)*α*,(*I<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* <sup>⊇</sup> (*I<sup>i</sup> <sup>B</sup>*)*<sup>α</sup> and* (*F<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* <sup>⊇</sup> (*F<sup>i</sup> <sup>B</sup>*)*α*. (2) *<sup>α</sup>* <sup>≤</sup> *<sup>β</sup> implies* (*T<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* <sup>⊇</sup> (*T<sup>i</sup> A*)*β*,(*I<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* <sup>⊆</sup> (*I<sup>i</sup> <sup>A</sup>*)*<sup>β</sup> and* (*F<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* <sup>⊆</sup> (*F<sup>i</sup> <sup>A</sup>*)*β*.

**Proposition 5.** *A is a neutrosophic multi group of a classical group X if and only if for all α* ∈ [0, 1]*, α-level sets of A*,(*T<sup>i</sup> A*)*α*,(*I<sup>i</sup> <sup>A</sup>*)*<sup>α</sup> and* (*F<sup>i</sup> <sup>A</sup>*)*<sup>α</sup> are classical subgroups of X*.

**Proof.** Let *<sup>A</sup>* be a neutrosophic multi subgroup of *<sup>X</sup>*, *<sup>α</sup>* <sup>∈</sup> [0, 1] and *<sup>x</sup>*.*<sup>y</sup>* <sup>∈</sup> (*T<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* (similarly *<sup>x</sup>*.*<sup>y</sup>* <sup>∈</sup> (*I<sup>i</sup> A*)*α*,(*F<sup>i</sup> <sup>A</sup>*)*α*). By the assumption, *T<sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≥</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*y*) ≥ *α* ∧ *α* = *α* (and similarly, *Ii <sup>A</sup>*(*xy*−1) <sup>≤</sup> *<sup>α</sup>* and *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≤</sup> *<sup>α</sup>*). Hence, *xy*−<sup>1</sup> <sup>∈</sup> (*T<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* (and similarly *xy*−<sup>1</sup> <sup>∈</sup> (*I<sup>i</sup> A*)*α*,(*F<sup>i</sup> <sup>A</sup>*)*α*) for each *<sup>α</sup>* <sup>∈</sup> [0, 1]. This means that (*T<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* (and similarly (*I<sup>i</sup> A*)*α*,(*F<sup>i</sup> <sup>A</sup>*)*α*) is a classical subgroup of *X* for each *α* ∈ [0, 1].

Conversely, let (*T<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* be a classical subgroup of *<sup>X</sup>*, for each *<sup>α</sup>* <sup>∈</sup> [0, 1]. Let *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>X</sup>*, *<sup>α</sup>* <sup>=</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) ∧ *Ti <sup>A</sup>*(*y*) and *β* = *T<sup>i</sup> <sup>A</sup>*(*x*). Since (*T<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* and (*T<sup>i</sup> <sup>A</sup>*)*<sup>β</sup>* are classical subgroups of *<sup>X</sup>*, *<sup>x</sup>*.*<sup>y</sup>* <sup>∈</sup> (*T<sup>i</sup> <sup>A</sup>*)*<sup>α</sup>* and *<sup>x</sup>*−<sup>1</sup> <sup>∈</sup> (*T<sup>i</sup> <sup>A</sup>*)*β*. Thus, *T<sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≥</sup> *<sup>α</sup>* <sup>=</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*y*) and *T<sup>i</sup> <sup>A</sup>*(*x*−1) <sup>≥</sup> *<sup>β</sup>* <sup>=</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*). Similarly, *Ii <sup>A</sup>*(*xy*−1) <sup>≤</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*y*) and *F<sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≤</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*y*).

**Theorem 4.** *Let X*1, *X*<sup>2</sup> *be the classical groups and g* : *X*<sup>1</sup> → *X*<sup>2</sup> *be a group homomorphism. If A is a neutrosophic multi subgroup of X*1*, then the image of A*, *g*(*A*) *is a neutrosophic multi subgroup of X*2*.*

**Proof.** Let *<sup>A</sup>* <sup>∈</sup> *NMS*(*X*1) and *<sup>y</sup>*1, *<sup>y</sup>*<sup>2</sup> <sup>∈</sup> *<sup>X</sup>*2. If *<sup>g</sup>*−1(*y*1) = <sup>∅</sup> or *<sup>g</sup>*−1(*y*2) = <sup>∅</sup>, then it is clear that *g*(*A*) ∈ *NMS*(*X*2). Let us assume that there exists *x*1, *x*<sup>2</sup> ∈ *X*<sup>1</sup> such that *g*(*x*1) = *y*<sup>1</sup> and *g*(*x*2) = *y*2. Since *g* is a group homomorphism,

$$\begin{array}{l} \mathcal{g}(T^{i}\_{A})(y\_{1}y\_{2}^{-1}) = \bigvee\_{\mathcal{Y}\_{1}y\_{2}^{-1}=\mathcal{G}(\mathbf{x})} T^{i}\_{A}(\mathbf{x}) \ge T^{i}\_{A}(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1}),\\ \mathcal{g}(I^{i}\_{A})(y\_{1}y\_{2}^{-1}) = \bigwedge\_{y\_{1}y\_{2}^{-1}=\mathcal{G}(\mathbf{x})} I^{i}\_{A}(\mathbf{x}) \le I^{i}\_{A}(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1}),\\ \mathcal{g}(F^{i}\_{A})(y\_{1}y\_{2}^{-1}) = \bigwedge\_{y\_{1}y\_{2}^{-1}=\mathcal{G}(\mathbf{x})} F^{i}\_{A}(\mathbf{x}) \le F^{i}\_{A}(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1}). \end{array}$$

By using the above inequalities, let us prove that *g*(*A*)(*y*1*y*−<sup>1</sup> <sup>2</sup> ) ≥ *g*(*A*)(*y*1) ∧ *g*(*A*)(*y*2) :

$$\begin{array}{ll} g(A)(y\_1y\_2^{-1}) &= g(\bar{T}\_A)(y\_1y\_2^{-1}), g(\bar{I}\_A^i)(y\_1y\_2^{-1}), g(\bar{F}\_A^i)(y\_1y\_2^{-1}) \\ &= \bigvee\_{y\_1y\_2^{-1}=g(\mathbf{x})} T^i\_A(\mathbf{x}), \bigwedge\_{y\_1y\_2^{-1}=g(\mathbf{x})} \bar{I}^i\_A(\mathbf{x}), \bigwedge\_{y\_1y\_2^{-1}=g(\mathbf{x})} F^i\_A(\mathbf{x}) \\ &\geq (\bar{T}\_A^i(\mathbf{x}\_1\mathbf{x}\_2^{-1}), \bar{I}^i\_A(\mathbf{x}\_1\mathbf{x}\_2^{-1}), \bar{F}^i\_A(\mathbf{x}\_1\mathbf{x}\_2^{-1})) \\ &\geq (\bar{T}\_A^i(\mathbf{x}\_1) \wedge \bar{T}^i\_A(\mathbf{x}\_2), \bar{I}^i\_A(\mathbf{x}\_1) \vee \bar{I}^i\_A(\mathbf{x}\_2), \bar{F}^i\_A(\mathbf{x}\_1) \vee \bar{F}^i\_A(\mathbf{x}\_2) \\ &= (\bar{T}\_A^i(\mathbf{x}\_1), \bar{I}^i\_A(\mathbf{x}\_1), \bar{F}^i\_A(\mathbf{x}\_1)) \wedge (\bar{T}^i\_A(\mathbf{x}\_2), \bar{I}^i\_A(\mathbf{x}\_2), \bar{F}^i\_A(\mathbf{x}\_2)). \end{array}$$

This is satisfied for each *x*1, *x*<sup>2</sup> ∈ *X*<sup>1</sup> with *g*(*x*1) = *y*<sup>1</sup> and *g*(*x*2) = *y*2, then it is obvious that

$$\begin{array}{llll} \mathcal{g}(A)(y\_1y\_2^{-1}) & \geq (\vee\_{y\_1=\mathcal{g}(\mathbf{x}\_1)} T^{i}{}\_A(\mathbf{x}\_1), \wedge\_{y\_1=\mathcal{g}(\mathbf{x}\_1)} I^{i}{}\_A(\mathbf{x}\_1), \wedge\_{y\_1=\mathcal{g}(\mathbf{x}\_1)} F^{i}{}\_A(\mathbf{x}\_1)) \\ & \qquad \wedge (\vee\_{y\_2=\mathcal{g}(\mathbf{x}\_2)} T^{i}{}\_A(\mathbf{x}\_2), \wedge\_{y\_2=\mathcal{g}(\mathbf{x}\_2)} I^{i}{}\_A(\mathbf{x}\_2), \wedge\_{y\_2=\mathcal{g}(\mathbf{x}\_2)} F^{i}{}\_A(\mathbf{x}\_2)) \\ & = (\mathcal{g}(T^{i}{}\_A)(y\_1), \mathcal{g}(I^{i}{}\_A)(y\_1), \mathcal{g}(F^{i}{}\_A)(y\_1)) \wedge (\mathcal{g}(T^{i}{}\_A)(y\_2), \mathcal{g}(I^{i}{}\_A)(y\_2), \mathcal{g}(F^{i}{}\_A)(y\_2)) \\ & = \mathcal{g}(A)(y\_1) \wedge \mathcal{g}(A)(y\_2). \end{array}$$

Hence, the image of a neutrosophic multi subgroup is also a neutrosophic multi subgroup.

**Theorem 5.** *Let X*1, *X*<sup>2</sup> *be the classical groups and g* : *X*<sup>1</sup> → *X*<sup>2</sup> *be a group homomorphism. If B is a neutrosophic multi subgroup of X*2*, then the preimage g*−1(*B*) *is a neutrosophic multi subgroup of X*1*.*

**Proof.** Let *B* ∈ *NMS*(*X*2) and *x*1, *x*<sup>2</sup> ∈ *X*1. Since *g* is a group homomorphism, the following inequality is obtained:

$$\begin{array}{llcl} &g^{-1}(B)(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1}) & (T^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1})), I^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1})), F^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1}\mathbf{x}\_{2}^{-1}))) \\ & &= (T^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1})\mathbf{g}(\mathbf{x}\_{2})^{-1}), I^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1})\mathbf{g}(\mathbf{x}\_{2})^{-1}), F^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1})\mathbf{g}(\mathbf{x}\_{2})^{-1})) \\ & & \geq (T^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1})) \wedge T^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{2})), I^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1})) \vee I^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{2})), F^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1})) \vee F^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{2}))) \\ & &= (T^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1})), I^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1})), F^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{1}))) \wedge (T^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{2})), I^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{2}))), F^{i}\_{B}(\mathcal{g}(\mathbf{x}\_{2}))) \\ & &= g^{-1}(B)(\mathbf{x}\_{1}) \wedge g^{-1}(B)(\mathbf{x}\_{2}). \end{array}$$

Therefore, *<sup>g</sup>*−1(*B*) <sup>∈</sup> *NMS*(*X*1).

**Definition 17.** *Let X be a classical group. A* ∈ *NMG*(*X*); *then, the compound function of A and A is defined as*

$$A \exists A(z) = \{ z, T^i{}\_{A \uplus A}(z), I^i{}\_{A \uplus A}(z), F^i{}\_{A \uplus A}(z) : \forall z \in X \}, \tag{10}$$

*where T<sup>i</sup> <sup>A</sup>*◦˜ *<sup>A</sup>*(*z*)=(∨*xy*=*zT<sup>i</sup> <sup>A</sup>*(*y*) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*zy*−1))*, I<sup>i</sup> <sup>A</sup>*◦˜ *<sup>A</sup>*(*z*)=(∧*xy*=*<sup>z</sup> <sup>I</sup><sup>i</sup> <sup>A</sup>*(*y*) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*zy*−1)) *and F<sup>i</sup> <sup>A</sup>*◦˜ *<sup>A</sup>*(*z*) = (∧*xy*=*zF<sup>i</sup> <sup>A</sup>*(*y*) <sup>∨</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*zy*−1)).

**Theorem 6.** *Let A* <sup>∈</sup> *NMS*(*X*). *Then, A* <sup>∈</sup> *NMG*(*X*) *iff A*◦˜ *<sup>A</sup>*⊆˜ *A and A*⊆˜ *<sup>A</sup>*−1.

**Proof.** Let *A* ∈ *NMS*(*X*) and *x*, *y*, *z* ∈ *X*.

$$\begin{array}{l} \Rightarrow \boldsymbol{T}^{i}\_{A}(\boldsymbol{x}\boldsymbol{y}) \geq \boldsymbol{T}^{i}\_{A}(\boldsymbol{x}) \land \boldsymbol{T}^{i}\_{A}(\boldsymbol{y})\\ \Rightarrow \boldsymbol{T}^{i}\_{A}(\boldsymbol{z}) \geq \vee \{\boldsymbol{T}^{i}\_{A}(\boldsymbol{x}) \land \boldsymbol{T}^{i}\_{A}(\boldsymbol{y}); \boldsymbol{x}\boldsymbol{y} = \boldsymbol{z}\} \\ = \boldsymbol{T}^{i}\_{A\circ A}(\boldsymbol{z}) \\\\ \Rightarrow \boldsymbol{I}^{i}\_{A}(\boldsymbol{x}\boldsymbol{y}) \leq \boldsymbol{T}^{i}\_{A}(\boldsymbol{x}) \lor \boldsymbol{I}^{i}\_{A}(\boldsymbol{y}) \\ \Rightarrow \boldsymbol{I}^{i}\_{A}(\boldsymbol{z}) \leq \wedge \{\boldsymbol{I}^{i}\_{A}(\boldsymbol{x}) \lor \boldsymbol{I}^{i}\_{A}(\boldsymbol{y}); \boldsymbol{x}\boldsymbol{y} = \boldsymbol{z}\} \\ = \boldsymbol{I}^{i}\_{A\circ A}(\boldsymbol{x}) \\ \Rightarrow \boldsymbol{F}^{i}\_{A}(\boldsymbol{x}\boldsymbol{y}) \leq \boldsymbol{F}^{i}\_{A}(\boldsymbol{x}) \lor \boldsymbol{T}^{i}\_{A}(\boldsymbol{y}) \\ \Rightarrow \boldsymbol{F}^{i}\_{A}(\boldsymbol{z}) \leq \wedge \{\boldsymbol{F}^{i}\_{A}(\boldsymbol{x}) \lor \boldsymbol{F}^{i}\_{A}(\boldsymbol{y}); \boldsymbol{x}\boldsymbol{y} = \boldsymbol{z}\} \\ = \boldsymbol{F}^{i}\_{A\circ A}(\boldsymbol{z}) \end{array}$$

<sup>⇒</sup> *<sup>A</sup>*◦˜ *<sup>A</sup>*⊆˜ *<sup>A</sup>*.

Now, by Proposition 2, we get the conditions. Conversely, suppose *<sup>A</sup>*◦˜ *<sup>A</sup>*⊆˜ *<sup>A</sup>* and *<sup>A</sup>*⊆˜ *<sup>A</sup>*−<sup>1</sup>

$$\begin{aligned} \Rightarrow & T^{i-1}\_A(\mathbf{x}) \ge T^i\_A(\mathbf{x}) \text{ but } T^{i-1}\_A(\mathbf{x}) = T^i\_A(\mathbf{x}^{-1}) \Rightarrow T^i\_A(\mathbf{x}^{-1}) \ge T^i\_A(\mathbf{x}) \\ \Rightarrow & I^{i-1}\_A(\mathbf{x}) \le T^i\_A(\mathbf{x}) \text{ but } I^{i-1}\_A(\mathbf{x}) = I^i\_A(\mathbf{x}^{-1}) \Rightarrow I^i\_A(\mathbf{x}^{-1}) \le I^i\_A(\mathbf{x}) \\ \Rightarrow & F^{i-1}\_A(\mathbf{x}) \le F^i\_A(\mathbf{x}) \text{ but } F^{i-1}\_A(\mathbf{x}) = F^i\_A(\mathbf{x}^{-1}) \Rightarrow F^i\_A(\mathbf{x}^{-1}) \le F^i\_A(\mathbf{x}) \end{aligned}$$

since *<sup>A</sup>* <sup>∈</sup> *NMS*(*X*); then, to prove *<sup>A</sup>* <sup>∈</sup> *NMG*(*X*), it enough to prove that *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≥</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) ∧ *Ti <sup>A</sup>*(*y*), *I<sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≤</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*y*) and *F<sup>i</sup> <sup>A</sup>*(*xy*−1) <sup>≤</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*y*) ∀ *x*, *y* ∈ *X*.

Now,

$$\begin{array}{ll} T^{i}{}\_{A}(\mathbf{x}\mathbf{y}^{-1}) & \geq T^{i}{}\_{A\odot A}(\mathbf{x}\mathbf{y}^{-1}) \\ & = \vee\_{z\in X} \{ T^{i}{}\_{A}(z) \wedge T^{i}{}\_{A}(z^{-1}\mathbf{x}\mathbf{y}^{-1}) \} \\ & \geq \{ T^{i}{}\_{A}(\mathbf{x}) \wedge T^{i}{}\_{A}(\mathbf{y}^{-1}); z=\mathbf{x} \} \\ & \geq T^{i}{}\_{A}(\mathbf{x}) \wedge T^{i}{}\_{A}(\mathbf{y}) \\ & = \operatorname{I}^{i}{}\_{A\odot A}(\mathbf{x}\mathbf{y}^{-1}) \\ & = \wedge\_{z\in X} \{ \operatorname{I}^{i}{}\_{A}(z) \vee \operatorname{I}^{i}{}\_{A}(z^{-1}\mathbf{x}\mathbf{y}^{-1}) \} \\ & \leq \{ \operatorname{I}^{i}{}\_{A}(\mathbf{x}) \vee \operatorname{I}^{i}{}\_{A}(\mathbf{y}^{-1}); z=\mathbf{x} \} \\ & \leq \operatorname{I}^{i}{}\_{A}(\mathbf{x}) \vee \operatorname{I}^{i}{}\_{A}(\mathbf{y}) \end{array}$$

$$\begin{array}{lcl}F^{i}\_{A}(\boldsymbol{\mathfrak{x}}\boldsymbol{y}^{-1})&\leq F^{i}\_{A\upharpoonright A}(\boldsymbol{\mathfrak{x}}\boldsymbol{y}^{-1})\\&=\wedge\_{\boldsymbol{z}\in X}\{F^{i}\_{A}(\boldsymbol{z})\vee F^{i}\_{A}(\boldsymbol{z}^{-1}\boldsymbol{x}\boldsymbol{y}^{-1})\}\\&\leq \{F^{i}\_{A}(\boldsymbol{\mathfrak{x}})\vee F^{i}\_{A}(\boldsymbol{y}^{-1});\boldsymbol{z}=\boldsymbol{x}\}\\&\leq F^{i}\_{A}(\boldsymbol{\mathfrak{x}})\vee F^{i}\_{A}(\boldsymbol{y}),\end{array}$$

hence the proof.

**Corollary 3.** *Let A* <sup>∈</sup> *NMS*(*X*). *Then, A* <sup>∈</sup> *NMG*(*X*) *iff A*◦˜ *<sup>A</sup>* <sup>=</sup> *A and A*⊆˜ *<sup>A</sup>*−1.

**Proof.** Let *A* ∈ *NMG*(*X*). Then,

$$\begin{array}{lcl}T^{i}\_{A\uplus A}(\mathbf{x}) &= \vee \{T^{i}\_{A}(\mathbf{y}) \wedge T^{i}\_{A}(\mathbf{z}); \mathbf{y}, \mathbf{z} \in \mathbf{X} \text{ and } \mathbf{y}\mathbf{z} = \mathbf{x}\} \\ &\geq \{T^{i}\_{A}(\mathbf{e}) \wedge T^{i}\_{A}(\mathbf{e}^{-1}\mathbf{x})\} \\ &= T^{i}\_{A}(\mathbf{x}) \\\\ I^{i}\_{A\uplus A}(\mathbf{x}) &= \wedge \{I^{i}\_{A}(\mathbf{y}) \vee I^{i}\_{A}(\mathbf{z}); \mathbf{y}, \mathbf{z} \in \mathbf{X} \text{ and } \mathbf{y}\mathbf{z} = \mathbf{x}\} \\ &\leq \{I^{i}\_{A}(\mathbf{e}) \vee I^{i}\_{A}(\mathbf{e}^{-1}\mathbf{x})\} \\ &= I^{i}\_{A}(\mathbf{x}) \\\\ F^{i}\_{A\uplus A}(\mathbf{x}) &= \wedge \{F^{i}\_{A}(\mathbf{y}) \vee F^{i}\_{A}(\mathbf{z}); \mathbf{y}, \mathbf{z} \in \mathbf{X} \text{ and } \mathbf{y}\mathbf{z} = \mathbf{x}\} \\ &\leq \{F^{i}\_{A}(\mathbf{e}) \vee F^{i}\_{A}(\mathbf{e}^{-1}\mathbf{x})\} \\ &= F^{i}\_{A}(\mathbf{x}). \end{array}$$

Therefore, *<sup>A</sup>*⊆˜ *<sup>A</sup>*◦˜ *<sup>A</sup>*.

Hence, by the above theorem, the proof is complete.

**Theorem 7.** *Let X be a classical group and A*, *B* ∈ *NMS*(*X*). *If A*, *B* ∈ *NMG*(*X*)*, then A*∩˜ *B* ∈ *NMG*(*X*).

**Proof.** Let *x*, *y* ∈ *X* be arbitrary:

$$\Rightarrow T^{i}\_{A}(\mathbf{x}y^{-1}) \geq T^{i}\_{A}(\mathbf{x}) \land T^{i}\_{A}(y^{-1}), T^{i}\_{B}(\mathbf{x}y^{-1}) \geq T^{i}\_{B}(\mathbf{x}) \land T^{i}\_{B}(y^{-1})$$

$$I^{i}\_{A}(\mathbf{x}y^{-1}) \leq I^{i}\_{A}(\mathbf{x}) \lor I^{i}\_{A}(y^{-1}), I^{i}\_{B}(\mathbf{x}y^{-1}) \leq I^{i}\_{B}(\mathbf{x}) \lor T^{i}\_{B}(y^{-1})$$

$$F^{i}\_{A}(\mathbf{x}y^{-1}) \leq F^{i}\_{A}(\mathbf{x}) \lor F^{i}\_{A}(y^{-1}), F^{i}\_{B}(\mathbf{x}y^{-1}) \leq F^{i}\_{B}(\mathbf{x}) \lor F^{i}\_{B}(y^{-1}).$$

Now,

$$\begin{array}{rcl} \boldsymbol{T}^{i}\_{A \cap B}(\mathbf{x}\mathbf{y}^{-1}) &= \boldsymbol{T}^{i}\_{A \cap B}(\mathbf{x}) \wedge \boldsymbol{T}^{i}\_{A \cap B}(\mathbf{y}^{-1}) \text{ by definition intersection} \\ &\geq [\boldsymbol{T}^{i}\_{A}(\mathbf{x}) \wedge \boldsymbol{T}^{i}\_{A}(\mathbf{y}^{-1})] \wedge [\boldsymbol{T}^{i}\_{B}(\mathbf{x}) \wedge \boldsymbol{T}^{i}\_{B}(\mathbf{y}^{-1})] \\ &= [\boldsymbol{T}^{i}\_{A}(\mathbf{x}) \wedge \boldsymbol{T}^{i}\_{B}(\mathbf{x})] \wedge [\boldsymbol{T}^{i}\_{A}(\mathbf{y}^{-1}) \wedge \boldsymbol{T}^{i}\_{B}(\mathbf{y}^{-1})] \text{ by commutative property of minimum} \\ &\geq [\boldsymbol{T}^{i}\_{A}(\mathbf{x}) \wedge \boldsymbol{T}^{i}\_{B}(\mathbf{x})] \wedge [\boldsymbol{T}^{i}\_{A}(\mathbf{y}) \wedge \boldsymbol{T}^{i}\_{B}(\mathbf{y})] \text{ since } A, \ B \in \mathrm{MMG}(X) \\ &= \boldsymbol{T}^{i}\_{A \cap B}(\mathbf{x}) \wedge \boldsymbol{T}^{i}\_{A \cap B}(\mathbf{y}) \text{ by definition intersection} \\ &\Rightarrow \boldsymbol{T}^{i}\_{A \cap B}(\mathbf{x}\mathbf{y}^{-1}) \ge \boldsymbol{T}^{i}\_{A \cap B}(\mathbf{x}) \wedge \boldsymbol{T}^{i}\_{A \cap B}(\mathbf{y}) \text{ (1)} \end{array}$$

$$\begin{array}{rcl} \operatorname{i}^{!}\_{A \cap B}(\operatorname{xy}^{-1}) &= \operatorname{i}^{!}\_{A \cap B}(\operatorname{x}) \lor \operatorname{i}^{!}\_{A \cap B}(\operatorname{y}^{-1}) \text{ by definition intersection} \\ &\leq \operatorname{[i}^{!}\_{A}(\operatorname{x}) \lor \operatorname{i}^{!}\_{A}(\operatorname{y}^{-1})] \lor [\operatorname{i}^{!}\_{B}(\operatorname{x}) \lor \operatorname{i}^{!}\_{B}(\operatorname{y}^{-1})] \\ &= [\operatorname{i}^{!}\_{A}(\operatorname{x}) \lor \operatorname{i}^{!}\_{B}(\operatorname{x})] \lor [\operatorname{i}^{!}\_{A}(\operatorname{y}^{-1}) \lor \operatorname{i}^{!}\_{B}(\operatorname{y}^{-1})] \text{ by commutative property of maximum} \\ &\leq [\operatorname{i}^{!}\_{A}(\operatorname{x}) \lor \operatorname{i}^{!}\_{B}(\operatorname{x})] \lor [\operatorname{i}^{!}\_{A}(\operatorname{y}) \lor \operatorname{i}^{!}\_{B}(\operatorname{y})] \text{ since } A, \text{ } B \in NMG(X) \\ &= \operatorname{i}^{!}\_{A \cap B}(\operatorname{x}) \lor \operatorname{i}^{!}\_{A \cap B}(\operatorname{y}) \text{ by definition intersection} \\ &\Rightarrow \operatorname{i}^{!}\_{A \cap B}(\operatorname{x}\operatorname{y}^{-1}) \le \operatorname{i}^{!}\_{A \cap B}(\operatorname{x}) \land \operatorname{i}^{!}\_{A \cap B}(\operatorname{y}) \text{ (2)} \end{array}$$

*Mathematics* **2019**, *7*, 95

$$\begin{array}{rcl} \boldsymbol{F}\_{A \cap B}(\boldsymbol{x}\mathbf{y}^{-1}) &= \boldsymbol{F}\_{A \cap B}^{i}(\boldsymbol{x}) \vee \boldsymbol{F}\_{A \cap B}^{i}(\boldsymbol{y}^{-1}) \text{ by definition intersection} \\ &\leq [\boldsymbol{F}\_{A}^{i}(\boldsymbol{x}) \vee \boldsymbol{F}\_{A}^{i}(\boldsymbol{y}^{-1})] \vee [\boldsymbol{F}\_{B}^{i}(\boldsymbol{x}) \vee \boldsymbol{F}\_{B}^{i}(\boldsymbol{y}^{-1})] \\ &= [\boldsymbol{F}\_{A}^{i}(\boldsymbol{x}) \vee \boldsymbol{F}\_{B}^{i}(\boldsymbol{x})] \vee [\boldsymbol{F}\_{A}^{i}(\boldsymbol{y}^{-1}) \vee \boldsymbol{F}\_{B}^{i}(\boldsymbol{y}^{-1})] \text{ by commutative property of maximum} \\ &\leq [\boldsymbol{F}\_{A}^{i}(\boldsymbol{x}) \vee \boldsymbol{F}\_{B}^{i}(\boldsymbol{x})] \vee [\boldsymbol{F}\_{A}^{i}(\boldsymbol{y}) \vee \boldsymbol{F}\_{B}^{i}(\boldsymbol{y})] \text{ since } A, B \in NMG(X) \\ &= \boldsymbol{F}\_{A \cap B}^{i}(\boldsymbol{x}) \vee \boldsymbol{F}\_{A \cap B}^{i}(\boldsymbol{y}) \text{ by definition intersection} \\ &\Rightarrow \boldsymbol{F}\_{A \cap B}^{i}(\boldsymbol{xy}^{-1}) \leq \boldsymbol{F}\_{A \cap B}^{i}(\boldsymbol{x}) \wedge \boldsymbol{F}\_{A \cap B}^{i}(\boldsymbol{y}) \tag{3} \end{array}$$

From (1), (2) and (3), *A*∩˜ *B* ∈ *NMG*(*X*), hence the proof.

**Remark 1.** *Let X be a classical group and* {*Ai*; *i* ∈ *I*} *be neutrosophic multiset on X. If* {*Ai*; *i* ∈ *I*} *is a family of NMG*(*X*) *over X*, *then their intersection* ˜ ) *<sup>i</sup>*∈*<sup>I</sup> Ai is also a NMG*(*X*) *over X*.

**Proposition 6.** *Let <sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *NMG*(*X*). *Then, <sup>T</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*) <sup>≤</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*−1), *<sup>I</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*) ≥ *Ii <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*−1), *<sup>F</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*) <sup>≥</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*−1).

**Proof.** Let *x*, *y* ∈ *X*. Now,

$$\begin{array}{rcl}T^{\bar{i}}\_{A \cap B}(\mathbf{x}^{-1})&=\vee\{T^{\bar{i}}\_{A}(\mathbf{x}^{-1}),T^{\bar{i}}\_{B}(\mathbf{x}^{-1})\} \\ &\geq\vee\{T^{\bar{i}}\_{A}(\mathbf{x}),T^{\bar{i}}\_{B}(\mathbf{x})\}\text{ since }A,B\in NMG(X) \\ &=T^{\bar{i}}\_{A \cap B}(\mathbf{x}) \\\\ \text{where }(\mathbf{x}^{-1})^{\bar{i}}=\lambda\_{A}(\bar{\bar{i}}\_{A}(\mathbf{x}^{-1}),\bar{\bar{i}}\_{B}(\mathbf{x}^{-1})) \end{array}$$

$$\begin{array}{rcl} \boldsymbol{I}\_{A \upharpoonright B}^{i}(\mathbf{x}^{-1}) &=& \land \{ \boldsymbol{I}\_{A}^{i}(\mathbf{x}^{-1}), \boldsymbol{I}\_{B}^{i}(\mathbf{x}^{-1}) \} \\ &\leq& \land \{ \boldsymbol{I}\_{A}^{i}(\mathbf{x}), \boldsymbol{I}\_{B}^{i}(\mathbf{x}) \} \text{ since } A, \ \boldsymbol{B} \in NMG(\mathbf{X}) \\ &=& \boldsymbol{I}\_{A \upharpoonright B}^{i}(\mathbf{x}) \\\\ \Gamma^{i} &=& (\mathbf{x}^{-1})^{\circ} \quad \cdots \quad \{ \Gamma^{i} \quad (\mathbf{x}^{-1}), \Gamma^{i} \quad (\mathbf{x}^{-1}) \} \end{array}$$

$$\begin{array}{rcll}F^{i}\_{A\odot B}(\mathbf{x}^{-1})&=\wedge\{F^{i}\_{A}(\mathbf{x}^{-1}),F^{i}\_{B}(\mathbf{x}^{-1})\}\\&\leq\wedge\{F^{i}\_{A}(\mathbf{x}),F^{i}\_{B}(\mathbf{x})\}\text{ since }A,\ B\in\mathrm{NMG}(X)\\&=F^{i}\_{A\odot B}(\mathbf{x}),\end{array}$$

hence the proof.

From this, it is clear that, if *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *NMG*(*X*), then *<sup>A</sup>*∪˜ *<sup>B</sup>* <sup>∈</sup> *NMG*(*X*) iff *<sup>T</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*xy*) <sup>≥</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*) ∧ *Ti <sup>A</sup>*∪˜ *<sup>B</sup>*(*y*), *<sup>I</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*xy*) <sup>≤</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*y*), *<sup>F</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*xy*) <sup>≤</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*x*) <sup>∨</sup> *<sup>F</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*y*).

**Corollary 4.** *Let A*, *B* ∈ *NMG*(*X*). *Then, A*∪˜ *B need not be an element of NMG*(*X*).

**Example 7.** *Assume that X* = {1, −1, *i*, −*i*} *is a classical group. Then,*

*A* = {1;(0.5, 0.3, 0.2, 0.1),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.5), −1;(0.7, 0.6, 0.4, 0.3),(0.1, 0.2, 0.2, 0.4),(0.2, 0.5, 0.4, 0.3)}, *i*;(0.6, 0.4, 0.3, 0.2),(0.1, 0.3, 0.3, 0.4),(0.1, 0.2, 0.4, 0.6), −*i*;(0.6, 0.4, 0.3, 0.2),(0.1, 0.3, 0.3, 0.4),(0.1, 0.2, 0.4, 0.6)}, *B* = {1;(0.5, 0.6, 0.6, 0.4),(0.1, 0.2, 0.2, 0.3),(0.2, 0.4, 0.4, 0.5), −1;(0.7, 0.6, 0.4, 0.3),(0.2, 0.1, 0.2, 0.3),(0.3, 0.4, 0.5, 0.3)} *are NM* − *groups*. *A* ∪ *B* = {1;(0.5, 0.6, 0.6, 0.4),(0.1, 0.1, 0.2, 0.3),(0.2, 0.3, 0.4, 0.5), −1;(0.7, 0.6, 0.4, 0.3),(0.1, 0.2, 0.2, 0.3),(0.2, 0.4, 0.4, 0.3), *i*;(0.6, 0.4, 0.3, 0.2),(0.1, 0.3, 0.3, 0.4),(0.1, 0.2, 0.4, 0.6), −*i*;(0.6, 0.4, 0.3, 0.2),(0.1, 0.3, 0.3, 0.4),(0.1, 0.2, 0.4, 0.6)}.

*However, T<sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(1) <sup>≥</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(*i*) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*∪˜ *<sup>B</sup>*(−*i*) *as i*.(−*i*) = <sup>1</sup>*. Then, A*∪˜ *<sup>B</sup>* <sup>∈</sup>/ *NMG*(*X*).

**Proposition 7.** *If <sup>A</sup>* <sup>∈</sup> *NMG*(*X*) *and <sup>X</sup>*<sup>1</sup> *is a subgroup of <sup>X</sup>*, *then <sup>A</sup>*|*X*<sup>1</sup> *(i.e., A restricted to X*1) ∈ *NM* − *group*(*X*1) *and is a neutrosophic multi subgroup of A*.

**Proof.** Let *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>X</sup>*1. Then, *xy*−<sup>1</sup> <sup>∈</sup> *<sup>X</sup>*1. Now,

$$T^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{x}\mathbf{y}^{-1}) = T^{i}\_{A}(\mathbf{x}\mathbf{y}^{-1}) \ge T^{i}\_{A}(\mathbf{x}) \wedge T^{i}\_{A}(\mathbf{y}) = T^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{x}) \wedge T^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{y}),$$

$$I^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{x}\mathbf{y}^{-1}) = I^{i}\_{A}(\mathbf{x}\mathbf{y}^{-1}) \le I^{i}\_{A}(\mathbf{x}) \vee I^{i}\_{A}(\mathbf{y}) = I^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{x}) \vee I^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{y}),$$

$$F^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{x}\mathbf{y}^{-1}) = F^{i}\_{A}(\mathbf{x}\mathbf{y}^{-1}) \le F^{i}\_{A}(\mathbf{x}) \vee F^{i}\_{A}(\mathbf{y}) = F^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{x}) \vee F^{i}\_{A\_{|\underline{X}\_{1}}}(\mathbf{y}).$$

The second part is trivial.

**Definition 18.** *Let A* ∈ *NMG*(*X*) *and B* ∈ *NMG*(*Y*) *be two neutrosophic multi groups over the groups X and Y, respectively. Then, the Cartesian product of A and B is defined as* (*A*×˜ *<sup>B</sup>*)(*x*, *<sup>y</sup>*) = *<sup>A</sup>*(*x*)×˜ *<sup>B</sup>*(*y*) *where*

$$A \vec{\times} B = \{ (\mathbf{x}, y), T^i{}\_{A \vec{\times} B} (\mathbf{x}, y), I^i{}\_{A \vec{\times} B} (\mathbf{x}, y), F^i{}\_{A \vec{\times} B} (\mathbf{x}, y) : (\mathbf{x}, y) \in \mathbf{X} \times \mathbf{Y} \},\tag{11}$$

*where T<sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∨</sup> *<sup>T</sup><sup>i</sup> B*(*y*)*, I<sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>I</sup><sup>i</sup> B*(*y*)*, F<sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*, *<sup>y</sup>*) = *<sup>F</sup><sup>i</sup> <sup>A</sup>*(*x*) <sup>∧</sup> *<sup>F</sup><sup>i</sup> <sup>B</sup>*(*y*)*.*

**Example 8.** *Assume that* (*Z*2, +) *and* (*Z*3, +) *are classiccal groups. Define the neutrosophic multi group A on* (*Z*2, +) *and B on* (*Z*3, +) *as follows:*


*Then, A*×˜ *B is a neutrosophic multi group.*

**Theorem 8.** *Let A*, *<sup>B</sup>* ∈ *NMG*(*X*). *The cartesian product of A and B is denoted by A*×˜ *<sup>B</sup>* ∈ *NMG*(*X*).

**Proof.** From the Theorem 1, it is clear that a *NMG*(*X*) is a neutrosophic multi group:

*Ti <sup>A</sup>*×˜ *<sup>B</sup>*((*x*1, *<sup>y</sup>*1),(*x*2, *<sup>y</sup>*2)−1) = *<sup>T</sup><sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*1*x*−<sup>1</sup> <sup>2</sup> , *<sup>y</sup>*1*y*−<sup>1</sup> <sup>2</sup> ) = *T<sup>i</sup> A*(*x*1*x*−<sup>1</sup> <sup>2</sup> ) <sup>∧</sup> *<sup>T</sup><sup>i</sup> B*(*y*1*y*−<sup>1</sup> <sup>2</sup> ) <sup>≥</sup> (*T<sup>i</sup> <sup>A</sup>*(*x*1) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*(*x*2)) <sup>∧</sup> (*T<sup>i</sup> <sup>B</sup>*(*y*1) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>B</sup>*(*y*2)) = (*T<sup>i</sup> <sup>A</sup>*(*x*1) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>B</sup>*(*y*1)) <sup>∧</sup> (*T<sup>i</sup> <sup>A</sup>*(*x*2) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>B</sup>*(*y*2)) = *T<sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*1, *<sup>y</sup>*1) <sup>∧</sup> *<sup>T</sup><sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*2, *<sup>y</sup>*2) *Ii <sup>A</sup>*×˜ *<sup>B</sup>*((*x*1, *<sup>y</sup>*1),(*x*2, *<sup>y</sup>*2)−1) = *<sup>I</sup><sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*1*x*−<sup>1</sup> <sup>2</sup> , *<sup>y</sup>*1*y*−<sup>1</sup> <sup>2</sup> ) = *I<sup>i</sup> A*(*x*1*x*−<sup>1</sup> <sup>2</sup> ) <sup>∨</sup> *<sup>I</sup><sup>i</sup> B*(*y*1*y*−<sup>1</sup> <sup>2</sup> ) <sup>≤</sup> (*I<sup>i</sup> <sup>A</sup>*(*x*1) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*(*x*2)) <sup>∨</sup> (*I<sup>i</sup> <sup>B</sup>*(*y*1) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>B</sup>*(*y*2)) = (*I<sup>i</sup> <sup>A</sup>*(*x*1) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>B</sup>*(*y*1)) <sup>∨</sup> (*I<sup>i</sup> <sup>A</sup>*(*x*2) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>B</sup>*(*y*2)) = *I<sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*1, *<sup>y</sup>*1) <sup>∨</sup> *<sup>I</sup><sup>i</sup> <sup>A</sup>*×˜ *<sup>B</sup>*(*x*2, *<sup>y</sup>*2)

and

$$\begin{array}{ll} F^i{}\_{A \ddot{\times} B}((\mathbf{x}\_1, y\_1), (\mathbf{x}\_2, y\_2)^{-1}) &= F^i{}\_{A \ddot{\times} B}(\mathbf{x}\_1 \mathbf{x}\_2^{-1}, y\_1 y\_2^{-1}) \\ &= F^i{}\_{A}(\mathbf{x}\_1 \mathbf{x}\_2^{-1}) \lor F^i{}\_{B}(y\_1 y\_2^{-1}) \\ &\leq (F^i{}\_{A}(\mathbf{x}\_1) \lor F^i{}\_{A}(\mathbf{x}\_2)) \lor (F^i{}\_{B}(y\_1) \lor F^i{}\_{B}(y\_2)) \\ &= (F^i{}\_{A}(\mathbf{x}\_1) \lor F^i{}\_{B}(y\_1)) \lor (F^i{}\_{A}(\mathbf{x}\_2) \lor F^i{}\_{B}(y\_2)) \\ &= F^i{}\_{A \ddot{\times} B}(\mathbf{x}\_1, y\_1) \lor F^i{}\_{A \ddot{\times} B}(\mathbf{x}\_2, y\_2) \end{array}$$

for all *x*, *y* ∈ *X* and *i* = 1, 2, ..., *P*—hence the proof.

#### **5. Conclusions**

The concept of a group is of fundamental importance in the study of algebra. In this paper, the algebraic structure of neutrosophic multiset is introduced as a neutrosophic multigroup. The neutrosophic multigroup is a generalized case of intuitionistic fuzzy multigroup and fuzzy multigroup. The various basic operations, definitions and theorems related to neutrosophic multigroup have been discussed. The foundations which we made through this paper can be used to get an insight into the higher order structures of group theory.

**Author Contributions:** All authors contributed equally.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:

NMG Neutrosophic Multigroup

NMS Neutrosophic Multiset

IFMS Intuitionistic Fuzzy Multiset

#### **References**


c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Multi-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environment**

#### **Changxing Fan 1,\*, Jun Ye 2, Sheng Feng 1, En Fan <sup>1</sup> and Keli Hu <sup>1</sup>**


Received: 23 November 2018; Accepted: 11 January 2019; Published: 17 January 2019

**Abstract:** In real applications, most decisions are fuzzy decisions, and the decision results mainly depend on the choice of aggregation operators. In order to aggregate information more scientifically and reasonably, the Heronian mean operator was studied in this paper. Considering the advantages and limitations of the Heronian mean (HM) operator, four Heronian mean operators for bipolar neutrosophic number (BNN) are proposed: the BNN generalized weighted HM (BNNGWHM) operator, the BNN improved generalized weighted HM (BNNIGWHM) operator, the BNN generalized weighted geometry HM (BNNGWGHM) operator, and the BNN improved generalized weighted geometry HM (BNNIGWGHM) operator. Then, their propositions were examined. Furthermore, two multi-criteria decision methods based on the proposed BNNIGWHM and BNNIGWGHM operator are introduced under a BNN environment. Lastly, the effectiveness of the new methods was verified with an example.

**Keywords:** bipolar neutrosophic number (BNN); BNN improved generalized weighted HM (BNNIGWHM) operator; BNN improved generalized weighted geometry HM (BNNIGWGHM) operator; decision-making

#### **1. Introduction**

In the real world, there is lots of uncertain information in science, technology, daily life, and so on. Particularly under the background of big data, the uncertainty of information is more complex and diverse. Now, how to make use of mathematical tools to deal with the uncertain information is an urgent problem for researchers. In order to describe uncertain information, Zadeh [1] put forward the concept of fuzzy sets. Considering the complexities and changes of uncertainty in the real environment, there was a certain limit on fuzzy sets to describe complex uncertainty; then, some extension theories [2–4] were put forward. Afterword, the neutrosophic set (NS) containing three neutrosophic components and the single-valued neutrosophic set were proposed by Smarandache [5], and the single-valued neutrosophic set was also mentioned by Wang and Smarandache [6]. Wang and Zhang [7] put forward an interval neutrosophic set (INS) theory. Furthermore, an *n*-value neutrosophic set [8] theory was proposed by Smarandache. The fuzzy set theory changed the binary view of people, but ignored the bipolarity of things. Under the background of big data, the confliction between data became more and more obvious. Traditional fuzzy sets could not do well in analyzing and handing uncertain information with incompatible bipolarity; this phenomenon was identified in 1994. For the first time, Zhang [9] introduced incompatible bipolarity into the fuzzy set theory, and put forward the bipolar fuzzy set (BFS). The founder of the fuzzy set theory, Zadeh, also affirmed

that the bipolar fuzzy set theory was a breakthrough in traditional fuzzy set theory [10]. Then, Zemankova et al. [11] discussed a more generalized multipolar fuzzy problem, and pointed out that the multipolar fuzzy problem can be divided into multiple bipolar fuzzy problems. Chen et al. [12] studied m-polar fuzzy sets. Bosc and Pivert [13] introduced a study on fuzzy bipolar relational algebra. Manemaran and Chellappa [14] gave some applications of bipolar fuzzy groups. Zhou and Li [15] introduced some applications of bipolar fuzzy sets in semiring. Deli et al. [16] put forward a bipolar neutrosophic set (BNS), which can describe bipolar information. Later, some studies about BNS were put forward [17–20]. In this paper, we propose four Heronian mean operators for bipolar neutrosophic number (BNN). Compared with the literature [17–19], the HM operator can embody the interaction between attributes to avoid unreasonable situations in information aggregation. Compared with the literature [20], the Bonferroni mean (BM) aggregation operator not only neglects the relationship between each attribute and itself, but also considers the relationship between each attribute and other attributes repeatedly. However, the BM aggregation operator has large computational complexity, but the Heronian mean (HM) can overcome these two shortcomings.

The remaining sections are organized as follows: some related concepts are reviewed in Section 2. The four operators are defined and their properties are investigated in Section 3; these four operators are BNN generalized weighted HM (BNNGWHM), BNN improved generalized weighted HM (BNNIGWHM), BNN generalized weighted geometry HM (BNNGWGHM), and BNN improved generalized weighted geometry HM (BNNIGWGHM). Multi-criteria decision-making (MCDM) methods based on the BNNIGWHM and BNNIGWGHM operators are established in Section 4. A numerical example is provided and the effects of parameters *p* and *q* are analyzed in Section 5. The conclusion of this paper is given in Section 6.

#### **2. Some Basic Concepts**

#### *2.1. BNN and Its Operational Laws*

**Definition 1 [16].** *Let U* = {*u*1, *u*2,..., *un*} *be a universe; a BNS* Γ *in U is defined as follows:*

$$\Gamma = \{ \langle u, a\_{\Gamma}^+(u), \beta\_{\Gamma}^+(u), \gamma\_{\Gamma}^+(u), a\_{\Gamma}^-(u), \beta\_{\Gamma}^-(u), \gamma\_{\Gamma}^-(u) \rangle | u \in \mathcal{U} \},$$

*in which α*<sup>+</sup> <sup>Γ</sup> (*u*) : *<sup>U</sup>* <sup>→</sup> [0, 1] *means a truth-membership function, <sup>γ</sup>*<sup>+</sup> <sup>Γ</sup> (*u*) : *U* → [0, 1] *means a falsity-membership function and β*<sup>+</sup> <sup>Γ</sup> (*u*) : *U* → [0, 1] *means an indeterminacy-membership function, corresponding to a BNS* Γ *and α*− <sup>Γ</sup> (*u*), *γ*<sup>−</sup> <sup>Γ</sup> (*u*), *β*<sup>−</sup> <sup>Γ</sup> (*u*) : *U* → [−1, 0] mean, respectively, the truth membership, false membership, and indeterminate membership to some implicit counter-property corresponding to a BNS Γ.

**Definition 2 [16].** *Let U be a universe, and* Γ<sup>1</sup> *and* Γ<sup>2</sup> *be two BNSs.*

$$\Gamma\_1 = \{ \langle u, \mathfrak{a}\_{\Gamma\_1}^+(u), \mathfrak{f}\_{\Gamma\_1}^+(u), \gamma\_{\Gamma\_1}^+(u), \mathfrak{a}\_{\Gamma\_1}^-(u), \mathfrak{f}\_{\Gamma\_1}^-(u), \gamma\_{\Gamma\_1}^-(u) \rangle | u \in \mathsf{U} \},$$

$$\Gamma\_2 = \{ \langle u, \mathfrak{a}\_{\Gamma\_2}^+(u), \mathfrak{f}\_{\Gamma\_2}^+(u), \gamma\_{\Gamma\_2}^+(u), \mathfrak{a}\_{\Gamma\_2}^-(u), \mathfrak{f}\_{\Gamma\_2}^-(u), \gamma\_{\Gamma\_2}^-(u) \rangle | u \in \mathsf{U} \}.$$

*Then, the operations of* Γ<sup>1</sup> *and* Γ<sup>2</sup> are defined as follows [16]:

<sup>➀</sup> <sup>Γ</sup><sup>1</sup> <sup>⊆</sup> <sup>Γ</sup>2*, if and only if <sup>α</sup>*<sup>+</sup> Γ1 (*u*) <sup>≤</sup> *<sup>α</sup>*<sup>+</sup> Γ2 (*u*)*, β*<sup>+</sup> Γ1 (*u*) <sup>≥</sup> *<sup>β</sup>*<sup>+</sup> Γ2 (*u*), *γ*<sup>+</sup> Γ1 (*u*) <sup>≥</sup> *<sup>γ</sup>*<sup>+</sup> Γ2 (*u*), *and α*− Γ1 (*u*) ≥ *α*− Γ2 (*u*), *β*− Γ1 (*u*) ≤ *β*<sup>−</sup> Γ2 (*u*), *γ*− Γ1 (*u*) ≤ *γ*<sup>−</sup> Γ2 (*u*)*;* ➁ Γ<sup>1</sup> = Γ2*, if and only if α*<sup>+</sup> (*u*) = *α*<sup>+</sup> (*u*)*, β*<sup>+</sup> (*u*) = *β*<sup>+</sup> (*u*), *γ*<sup>+</sup> (*u*) = *γ*<sup>+</sup> (*u*), *and α*− Γ1 (*u*) =

$$\begin{aligned} \text{w.r.} &= \text{r.r.}\_{\Gamma\_2}(u), \rho\_{\Gamma\_1}^-(u) = \rho\_{\Gamma\_2}^-(u), \rho\_{\Gamma\_1}^-(u) = \rho\_{\Gamma\_2}^-(u), \rho\_{\Gamma\_1}^+(u) = \rho\_{\Gamma\_2}^+(u), \\\ a\_{\Gamma\_2}^-(u), \beta\_{\Gamma\_1}^-(u) = \beta\_{\Gamma\_2}^-(u), \gamma\_{\Gamma\_1}^-(u) &= \gamma\_{\Gamma\_2}^-(u); \\\ \text{r.r.} &= \max\left(a\_{\Gamma\_1}^+(u), a\_{\Gamma\_2}^+(u)\right), \frac{\beta\_{\Gamma\_1}^+(u) + \beta\_{\Gamma\_2}^+(u)}{2}, \min\left(\gamma\_{\Gamma\_1}^+(u), \gamma\_{\Gamma\_2}^+(u)\right), \end{aligned}$$

$$\begin{aligned} \{\otimes \Gamma\_1 \cup \Gamma\_2 = \{ ( \begin{array}{c} u, \max \left( a\_{\Gamma\_1}^+(u), a\_{\Gamma\_2}^+(u) \right), \frac{\beta\_{\Gamma\_1}^+(u) + \beta\_{\Gamma\_2}^+(u)}{2}, \min \left( \gamma\_{\Gamma\_1}^+(u), \gamma\_{\Gamma\_2}^+(u) \right), \\ \min \left( a\_{\Gamma\_1}^-(u), a\_{\Gamma\_2}^-(u) \right), \frac{\beta\_{\Gamma\_1}^-(u) + \beta\_{\Gamma\_2}^-(u)}{2}, \max \left( \gamma\_{\Gamma\_1}^-(u), \gamma\_{\Gamma\_2}^-(u) \right) \end{array} \} \end{aligned}$$

$$\begin{array}{c} \mathbb{I} \otimes \Gamma\_{1} \cap \Gamma\_{2} = \{ \langle \begin{array}{c} u, \min\left(a\_{\Gamma\_{1}}^{+}(u), a\_{\Gamma\_{2}}^{+}(u)\right), \frac{\beta\_{\Gamma\_{1}}^{+}(u) + \beta\_{\Gamma\_{2}}^{+}(u)}{2}, \max\left(\gamma\_{\Gamma\_{1}}^{+}(u), \gamma\_{\Gamma\_{2}}^{+}(u)\right), \\ \max\left(a\_{\Gamma\_{1}}^{-}(u), a\_{\Gamma\_{2}}^{-}(u)\right), \frac{\beta\_{\Gamma\_{1}}^{-}(u) + \beta\_{\Gamma\_{2}}^{-}(u)}{2}, \min\left(\gamma\_{\Gamma\_{1}}^{-}(u), \gamma\_{\Gamma\_{2}}^{-}(u)\right) \end{array} \rangle | u \in \mathsf{U} \}; \end{array}$$

*For convenience, we denote a bipolar neutrosophic number (BNN) by <sup>τ</sup>* <sup>=</sup> *α*<sup>+</sup> *<sup>τ</sup>* , *β*<sup>+</sup> *<sup>τ</sup>* , *γ*<sup>+</sup> *<sup>τ</sup>* , *α*<sup>−</sup> *<sup>τ</sup>* , *β*<sup>−</sup> *<sup>τ</sup>* , *γ*<sup>−</sup> *<sup>τ</sup> .*

**Definition 3 [16].** *Let <sup>τ</sup>*<sup>1</sup> *and <sup>τ</sup>*<sup>2</sup> *be two BNNs, <sup>τ</sup>*<sup>1</sup> <sup>=</sup> *α*<sup>+</sup> *τ*1 , *β*<sup>+</sup> *τ*1 , *γ*<sup>+</sup> *τ*1 , *α*− *τ*1 , *β*− *τ*1 , *γ*− *<sup>τ</sup>*<sup>1</sup> *and τ*2= *α*+ *<sup>τ</sup>*<sup>2</sup> , *<sup>β</sup>*<sup>+</sup> *<sup>τ</sup>*<sup>2</sup> , *<sup>γ</sup>*<sup>+</sup> *<sup>τ</sup>*<sup>2</sup> , *α*<sup>−</sup> *<sup>τ</sup>*<sup>2</sup> , *β*<sup>−</sup> *<sup>τ</sup>*<sup>2</sup> , *γ*<sup>−</sup> *<sup>τ</sup>*<sup>2</sup> *, and δ* > 0*; then, the operations for BNNs are defined as follows [16]:*

$$\tau\_1 \oplus \tau\_2 = \langle a\_{\overline{\tau}\_1}^+ + a\_{\overline{\tau}\_2}^+ - a\_{\overline{\tau}\_1}^+ a\_{\overline{\tau}\_2}^+ \beta\_{\overline{\tau}\_1}^+ \beta\_{\overline{\tau}\_2}^+ \gamma\_{\overline{\tau}\_1}^+ \gamma\_{\overline{\tau}\_2}^+ - a\_{\overline{\tau}\_1}^- a\_{\overline{\tau}\_2}^- - \left( -\beta\_{\overline{\tau}\_1}^- - \beta\_{\overline{\tau}\_2}^- - \beta\_{\overline{\tau}\_1}^- \beta\_{\overline{\tau}\_2}^- \right) , -\left( -\gamma\_{\overline{\tau}\_1}^- - \gamma\_{\overline{\tau}\_2}^- - \gamma\_{\overline{\tau}\_1}^- \gamma\_{\overline{\tau}\_2}^- \right) \rangle ; \tag{1}$$

$$\tau\_1 \odot \ \tau\_2 = \langle a\_{\tau\_1}^+ a\_{\tau\_2}^+, \mathcal{f}\_{\tau\_1}^+ + \mathcal{f}\_{\tau\_2}^+ - \beta\_{\tau\_1}^+ \beta\_{\tau\_2}^+, \gamma\_{\tau\_1}^+ + \gamma\_{\tau\_2}^+ - \gamma\_{\tau\_1}^+ \gamma\_{\tau\_2}^+, - \left(-a\_{\tau\_1}^- - a\_{\tau\_2}^- - a\_{\tau\_1}^- a\_{\tau\_2}^-\right), -\beta\_{\tau\_1}^- \beta\_{\tau\_2}^-, -\gamma\_{\tau\_1}^- \gamma\_{\tau\_2}^-\rangle; \quad \text{(2)}$$

$$\delta\tau\_1 = \langle 1 - \left(1 - \mathfrak{a}\_{\tau\_1}^+\right)^\delta, \left(\mathfrak{k}\_{\tau\_1}^+\right)^\delta, \left(\gamma\_{\tau\_1}^+\right)^\delta, -\left(-\mathfrak{a}\_{\tau\_1}^-\right)^\delta, -\left(1 - \left(1 - \left(-\mathfrak{k}\_{\tau\_1}^-\right)\right)^\delta\right), -\left(1 - \left(1 - \left(-\gamma\_{\tau\_1}^-\right)\right)^\delta\right); \tag{3}$$

$$\tau\_2 \stackrel{\delta}{=} \langle \left(\tau^+\right)^\delta \, 1 - \left(1 - \mathfrak{a}^+\right)^\delta \, 1 - \left(1 - \mathfrak{a}^+\right)^\delta \, -\left(1 - \left(1 - \left(-\mathfrak{a}^-\right)\right)^\delta\right) \, -\left(\mathfrak{a}^-\right)^\delta - \left(\mathfrak{a}^-\right)^\delta \quad \left(\mathfrak{a}^+\right)^\delta\right) \tag{4}$$

$$\tau\_{\rm I}^{\delta} = \langle \left( a\_{\tau \mathbf{I}}^{+} \right)^{\delta}, 1 - \left( 1 - \beta\_{\tau \mathbf{I}}^{+} \right)^{\delta}, 1 - \left( 1 - \gamma\_{\tau \mathbf{I}}^{+} \right)^{\delta}, -\left( 1 - \left( 1 - \left( a\_{\tau \mathbf{I}}^{-} \right) \right)^{\delta} \right), -\left( -\beta\_{\tau \mathbf{I}}^{-} \right)^{\delta}, -\left( -\gamma\_{\tau \mathbf{I}}^{-} \right)^{\delta} \rangle. \tag{4}$$

**Definition 4 [16].** *Let <sup>τ</sup>* <sup>=</sup> *α*<sup>+</sup> *<sup>τ</sup>* , *β*<sup>+</sup> *<sup>τ</sup>* , *γ*<sup>+</sup> *<sup>τ</sup>* , *α*<sup>−</sup> *<sup>τ</sup>* , *β*<sup>−</sup> *<sup>τ</sup>* , *γ*<sup>−</sup> *<sup>τ</sup> be a BNN; then, we define s*(*τ*)*, a*(*τ*)*, and c*(*τ*) *as the score, accuracy, and certain functions, respectively; they are as follows:*

$$\mathbf{s}(\boldsymbol{\tau}) = \frac{1}{6} (\boldsymbol{\alpha}\_{\tau}^{+} + 1 - \boldsymbol{\beta}\_{\tau}^{+} + 1 - \boldsymbol{\gamma}\_{\tau}^{+} + 1 + \boldsymbol{\alpha}\_{\tau}^{-} - \boldsymbol{\beta}\_{\tau}^{-} - \boldsymbol{\gamma}\_{\tau}^{-});\tag{5}$$

$$a(\tau) = a\_{\tau}^{+} - \gamma\_{\tau}^{+} + a\_{\tau}^{-} - \gamma\_{\tau}^{-};\tag{6}$$

$$c(\tau) = a\_{\tau}^{+} - \gamma\_{\tau}^{+}.\tag{7}$$

**Definition 5 [16].** *Let <sup>τ</sup>*<sup>1</sup> *and <sup>τ</sup>*<sup>2</sup> *be two BNNs, <sup>τ</sup>*1<sup>=</sup> *<sup>α</sup>*<sup>+</sup> *τ*1 , *β*<sup>+</sup> *τ*1 , *γ*<sup>+</sup> *τ*1 , *α*− *τ*1 , *β*− *τ*1 , *γ*− *<sup>τ</sup>*<sup>1</sup> *and τ*<sup>2</sup> = *α*<sup>+</sup> *<sup>τ</sup>*<sup>2</sup> , *<sup>β</sup>*<sup>+</sup> *<sup>τ</sup>*<sup>2</sup> , *<sup>γ</sup>*<sup>+</sup> *<sup>τ</sup>*<sup>2</sup> , *α*<sup>−</sup> *<sup>τ</sup>*<sup>2</sup> , *β*<sup>−</sup> *<sup>τ</sup>*<sup>2</sup> , *γ*<sup>−</sup> *<sup>τ</sup>*<sup>2</sup> *; then, we can get Figure 1.*

**Figure 1.** The relationship between *τ*<sup>1</sup> and *τ*2.

*2.2. Generalized Weighted HM (GWHM), Improved Generalized Weighted HM (IGWHM), Generalized Weighted Geometry HM (GWGHM), and Improved Generalized Weighted Geometry HM (IGWGHM) Operators*

**Definition 6 [21].** *Let ε* = (*ε*1,*ε*2, ··· ,*εk*) *be the weight vector of a collection of non-negative real numbers* (*τ*1, *τ*2,..., *τk*)*,* ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1], *and t, s* ≥ *0. Then,*

$$\begin{pmatrix} GWHM^{t,s}(\tau\_1, \ \tau\_2, \dots, \ \tau\_k) \end{pmatrix} = \begin{pmatrix} \frac{2}{k(k+1)} \sum\_{j=1}^k \sum\_{i=j}^k \left(\varepsilon\_j \tau\_j\right)^t \left(\varepsilon\_i \tau\_i\right)^s \end{pmatrix}^{\frac{1}{t+s}},\tag{8}$$

*which is called a GWHM operator.*

**Definition 7 [22].** *Let ε* = (*ε*1,*ε*2, ··· ,*εk*) *be the weight vector of a collection of non-negative real numbers* (*τ*1, *τ*2,..., *τk*)*,* ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1], *and t, s* ≥ *0. Then,*

$$\mathbf{G}\mathbf{W}\mathbf{H}\mathbf{M}^{t,s}\begin{pmatrix}\tau\_{1} & \tau\_{2},\dots, & \tau\_{k}\end{pmatrix} = \left(\begin{array}{cc} \frac{1}{\mathcal{X}} & \stackrel{k}{\oplus} & \stackrel{k}{\oplus} \left(\varepsilon\_{f}^{t}\varepsilon\_{i}^{s}\tau\_{f}^{t}\otimes\tau\_{i}^{s}\right) \end{array}\right)^{\frac{1}{1+s}},\tag{9}$$

*where λ* = ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>k</sup> <sup>i</sup>*=*<sup>j</sup> ε<sup>j</sup> t εi <sup>s</sup> is called an IGWHM operator.*

**Definition 8 [21].** *Let ε* = (*ε*1,*ε*2, ··· ,*εk*) *be the weight vector of a collection of non-negative real numbers* (*τ*1, *τ*2,..., *τk*)*,* ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1], *and t, s* ≥ *0. Then,*

$$\text{GWG}HM^{t,s}(\tau\_1, \tau\_2, \dots, \tau\_k) = \frac{1}{t+s} \underset{j=1}{\stackrel{k}{\underset{j=1}{\odot}}} \underset{i=j}{\stackrel{k}{\underset{j=1}{\odot}}} \left( (t\tau\_j)^{\varepsilon\_j} \oplus (s\tau\_i)^{\varepsilon\_i} \right)^{\frac{2}{k(k+1)}},\tag{10}$$

*which is called a GWGHM operator.*

**Definition 9 [22].** *Let ε* = (*ε*1,*ε*2, ··· ,*εk*) *be the weight vector of a collection of non-negative real numbers* (*τ*1, *τ*2,..., *τk*)*,* ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1], *and t, s* ≥ *0. Then,*

$$IGWGHM^{t,s}(\tau\_1, \ \tau\_2, \dots, \ \tau\_k) = \frac{1}{t+s} \left( \bigotimes\_{j=1}^k \bigotimes\_{i=j}^k \left( t\tau\_j \oplus s\tau\_i \right)^{\frac{2(k+1-j)}{k(k+1)} \frac{\tau\_i}{\sum\_{m=j}^k s\alpha}} \right),\tag{11}$$

*which is called an IGWGHM operator.*

#### **3. Some BNN Aggregation Operators**

*3.1. GWHM Operators for BNNs*

**Definition 10.** *Let t, s* <sup>≥</sup> *0, and <sup>t</sup>* <sup>+</sup> *<sup>s</sup>* <sup>=</sup> <sup>0</sup>*, a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNN; then, we define the BNNGWHM operator as follows:*

$$BNNGWM^{t\cdot s}(\tau\_1,\ \tau\_2,\ldots,\ \tau\_k) = \left(\frac{2}{k(k+1)}\sum\_{j=1}^k \sum\_{\substack{i=1 \ i\neq j}}^k (\varepsilon\_j \tau\_j)^t (\varepsilon\_i \tau\_i)^s\right)^{\frac{1}{t+s}},\tag{12}$$

*where* ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1]*.*

According to Definitions 3 and 10, the following theorem can be attained:

**Theorem 1.** *Set a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNNs, using the BNNGWHM operator; then, the aggregation result is still a BNN, which is given by the following form:*

*BNNGWHMt*, *<sup>s</sup>* (*τ*1, *τ*2,..., *τk*) = ⎛ ⎜⎝ <sup>2</sup> *k*(*k*+1) *k* ∑ *j*=1 *k* ∑ *i* = *j* (*εjτj*) *t* (*εiτi*) *s* ⎞ ⎟⎠ 1 *t*+*s* = <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τj εj t* 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ,1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *β*+ *τj εj t* 1 − *β*+ *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* , 1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *γ*<sup>+</sup> *τj εj t* 1 − *γ*<sup>+</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ,− ⎛ ⎜⎝<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − −*α*<sup>−</sup> *τj εj t* 1 − −*α*<sup>−</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ⎞ ⎟⎠, − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − −*β*<sup>−</sup> *τj εj t* 1 − 1 − −*β*<sup>−</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ,− <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − −*γ*<sup>−</sup> *τj εj t* 1 − 1 − −*γ*<sup>−</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* (13)

*where* <sup>1</sup> *<sup>λ</sup>* = <sup>2</sup> *<sup>k</sup>*(*k*+1), <sup>∑</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1].

**Proof.**

(1) *εjτ<sup>j</sup>* = 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τj εj* , *β*+ *τj εj* , *γ*<sup>+</sup> *τj εj* , − −*α*<sup>−</sup> *τj εj* , − 1 − 1 − −*β*<sup>−</sup> *τj εj* , − 1 − 1 − −*γ*<sup>−</sup> *τj εj* ; (2) *εiτ<sup>i</sup>* = 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τi εi* , *β*+ *τi εi* , *γ*<sup>+</sup> *τi εi* , − −*α*<sup>−</sup> *τi εi* , − 1 − 1 − −*β*<sup>−</sup> *τi εi* , − 1 − 1 − −*γ*<sup>−</sup> *τi εi* ; (3)(*εjτj*) *<sup>t</sup>* <sup>=</sup> 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τj εj t* , 1 − 1 − *β*+ *τj εj t* , 1 − 1 − *γ*<sup>+</sup> *τj εj t* , − 1 − 1 − −*α*<sup>−</sup> *τj εj t* , − 1 − 1 − −*β*<sup>−</sup> *τj εj t* , − 1 − 1 − −*γ*<sup>−</sup> *τj εj t* ; (4) (*εiτi*) *<sup>s</sup>* <sup>=</sup> 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τi εi s* , 1 − 1 − *β*+ *τi εi s* , 1 − 1 − *γ*<sup>+</sup> *τi εi s* , − 1 − 1 − −*α*<sup>−</sup> *τi εi s* , − 1 − 1 − −*β*<sup>−</sup> *τi εi s* , − 1 − 1 − −*γ*<sup>−</sup> *τi εi s* ; (5) *εjτ<sup>j</sup> t* (*εiτi*) *<sup>s</sup>* <sup>=</sup> 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τj εj t* 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τi εi s* , 1 − 1 − *β*+ *τj εj t* + 1 − 1 − *β*+ *τi εi s* − 1 − 1 − *β*+ *τj εj t* <sup>1</sup> <sup>−</sup> 1 − *β*+ *τi εi s* , 1 − 1 − *γ*<sup>+</sup> *τj εj t* + 1 − 1 − *γ*<sup>+</sup> *τi εi s* − 1 − 1 − *γ*<sup>+</sup> *τj εj t* <sup>1</sup> <sup>−</sup> 1 − *γ*<sup>+</sup> *τi εi s* , − <sup>1</sup> <sup>−</sup> 1 − −*α*<sup>−</sup> *τj εj t* + 1 − 1 − −*α*<sup>−</sup> *τi εi s* − 1 − 1 − −*α*<sup>−</sup> *τj εj t* <sup>1</sup> <sup>−</sup> 1 − −*α*<sup>−</sup> *τi εi s* , − 1 − 1 − −*β*<sup>−</sup> *τj εj t* 1 − 1 − −*β*<sup>−</sup> *τi εi s* , − 1 − 1 − −*γ*<sup>−</sup> *τj εj t* 1 − 1 − −*γ*<sup>−</sup> *τi εi s* (6) ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>k</sup> i* = *j εjτ<sup>j</sup> t* (*εiτi*) *<sup>s</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τj εj t* 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τi εi s* , ∏*k <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − *β*+ *τj εj t* 1 − *β*+ *τi εi s* , ∏*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>n</sup> i*=*j* 1 − 1 − *γ*<sup>+</sup> *τj εj t* 1 − *γ*<sup>+</sup> *τi εi s* , <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − −*α*<sup>−</sup> *τj εj t* 1 − −*α*<sup>−</sup> *τi εi s* , − <sup>1</sup> <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − 1 − −*β*<sup>−</sup> *τj εj t* 1 − 1 − −*β*<sup>−</sup> *τi εi s* , − <sup>1</sup> <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − 1 − −*γ*<sup>−</sup> *τj εj t* 1 − 1 − −*γ*<sup>−</sup> *τi εi s* ;

(7) <sup>2</sup> *<sup>k</sup>*(*k*+1) <sup>∑</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>k</sup> i* = *j εjτ<sup>j</sup> t* (*εiτi*) *<sup>s</sup>* = <sup>1</sup> *<sup>λ</sup>* <sup>∑</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>k</sup> i* = *j εjτ<sup>j</sup> t* (*εiτi*) *<sup>s</sup>* = <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τj εj t* 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τi εi s* 1 *λ* , *k* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *β*+ *τj εj t* 1 − *β*+ *τi εi s* 1 *λ* , *k* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *γ*<sup>+</sup> *τj εj t* 1 − *γ*<sup>+</sup> *τi εi s* 1 *λ* , − *k* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − −*α*<sup>−</sup> *τj εj t* 1 − −*α*<sup>−</sup> *τi εi s* 1 *λ* , − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − −*β*<sup>−</sup> *τj εj t* 1 − 1 − −*β*<sup>−</sup> *τi εi s* 1 *λ* , − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − −*γ*<sup>−</sup> *τj εj t* 1 − 1 − −*γ*<sup>−</sup> *τi εi s* 1 *λ* ; (8) 1 *<sup>λ</sup>* <sup>∑</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>k</sup> i* = *j εjτ<sup>j</sup> t* (*εiτi*) *s* <sup>1</sup> *t*+*s* = <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τj εj t* 1 − <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>+</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* , 1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *β*+ *τj εj t* 1 − *β*+ *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* , 1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *γ*<sup>+</sup> *τj εj t* 1 − *γ*<sup>+</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* , − ⎛ ⎜⎝ 1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − −*α*<sup>−</sup> *τj εj t* 1 − −*α*<sup>−</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ⎞ ⎟⎠, − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − −*β*<sup>−</sup> *τj εj t* 1 − 1 − −*β*<sup>−</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* , − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − −*γ*<sup>−</sup> *τj εj t* 1 − 1 − −*γ*<sup>−</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* .

This proves Theorem 1. -

**Theorem 2.** *(Monotonicity). Set <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *and σ<sup>j</sup>* = *α*<sup>+</sup> *σj* , *β*<sup>+</sup> *σj* , *γ*<sup>+</sup> *σj* , *α*− *σj* , *β*− *σj* , *γ*− *σj* (*<sup>j</sup>* <sup>=</sup> 1, 2, ··· , *<sup>k</sup>*) *as two collections of BNNs; if <sup>α</sup>*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>+</sup> *σj* , *β*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≥</sup> *<sup>β</sup>*<sup>+</sup> *σj* , *γ*<sup>+</sup> *τ<sup>j</sup>* ≥ *γ*<sup>+</sup> *<sup>σ</sup><sup>j</sup> and α*<sup>−</sup> *<sup>τ</sup><sup>j</sup>* ≥ *α*<sup>−</sup> *σj* , *β*− *<sup>τ</sup><sup>j</sup>* ≤ *β*<sup>−</sup> *σj* , *γ*− *<sup>τ</sup><sup>j</sup>* ≤ *γ*<sup>−</sup> *<sup>σ</sup><sup>j</sup>* , *then*

$$BNNIGGWHM^{t,s}(\tau\_1,\ \tau\_2,\ldots,\ \tau\_k) \le BNNGWHM^{t,s}(\sigma\_1,\ \sigma\_2,\ldots,\ \sigma\_k).$$

**Proof.** For *α*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>+</sup> *σj* , *β*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≥</sup> *<sup>β</sup>*<sup>+</sup> *σj* , *γ*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≥</sup> *<sup>γ</sup>*<sup>+</sup> *<sup>σ</sup><sup>j</sup>* and *α*<sup>−</sup> *<sup>τ</sup><sup>j</sup>* ≥ *α*<sup>−</sup> *σj* , *β*− *<sup>τ</sup><sup>j</sup>* ≤ *β*<sup>−</sup> *σj* , *γ*− *<sup>τ</sup><sup>j</sup>* ≤ *γ*<sup>−</sup> *σj* , it is obvious that

$$\left(1-\left(1-a\_{\tau\_{j}}^{+}\right)^{\varepsilon\_{j}}\right)^{t}\left(1-\left(1-a\_{\tau\_{i}}^{+}\right)^{\varepsilon\_{i}}\right)^{s}\leq\left(1-\left(1-a\_{\sigma\_{j}}^{+}\right)^{\varepsilon\_{j}}\right)^{t}\left(1-\left(1-a\_{\sigma\_{i}}^{+}\right)^{\varepsilon\_{i}}\right)^{s}$$

$$\begin{split} & \left(1 - \prod\_{j=1}^{k} \prod\_{i=j}^{k} \left(1 - \left(1 - \left(1 - a\_{\tau\_{j}^{+}}^{+}\right)^{\varepsilon\_{j}}\right)^{t} \left(1 - \left(1 - a\_{\tau\_{i}^{-}}^{+}\right)^{\varepsilon\_{i}}\right)^{s}\right)^{\frac{1}{\lambda}}\right)^{\frac{1}{1+s}} \\ & \leq \left(1 - \prod\_{j=1}^{k} \prod\_{i=j}^{k} \left(1 - \left(1 - \left(1 - a\_{\sigma\_{j}^{+}}^{+}\right)^{\varepsilon\_{j}}\right)^{t} \left(1 - \left(1 - a\_{\sigma\_{i}^{-}}^{+}\right)^{\varepsilon\_{i}}\right)^{s}\right)^{\frac{1}{1+s}}\right)^{\frac{1}{1+s}}.\end{split}$$

Similarly

1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *β*+ *τj εj t* 1 − *β*+ *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ≥ 1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *β*+ σ*j εj t* 1 − *β*+ σ*i εi s* 1 *λ* <sup>1</sup> *t*+*s* , 1 − <sup>1</sup> <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − *γ*<sup>+</sup> *τj εj t* 1 − *γ*<sup>+</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ≥ 1 − <sup>1</sup> <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − *γ*<sup>+</sup> *σj εj t* 1 − *γ*<sup>+</sup> *σi εi s* 1 *λ* <sup>1</sup> *t*+*s* , − ⎛ ⎜⎝ 1 − <sup>1</sup> <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − −*α*<sup>−</sup> *τj εj t* 1 − −*α*<sup>−</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ⎞ ⎟⎠ ≥ − ⎛ ⎜⎝ 1 − <sup>1</sup> <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − −*α*<sup>−</sup> *σj εj t* 1 − −*α*<sup>−</sup> *σi εi s* 1 *λ* <sup>1</sup> *t*+*s* ⎞ ⎟⎠, − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − −*β*<sup>−</sup> *τj εj t* 1 − 1 − −*β*<sup>−</sup> *τi εi s* 1 *λ* <sup>1</sup> *t*+*s* ≤ − <sup>1</sup> <sup>−</sup> <sup>∏</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∏</sup>*<sup>k</sup> i*=*j* 1 − 1 − 1 − −*β*<sup>−</sup> *σj εj t* 1 − 1 − −*β*<sup>−</sup> *σi εi s* 1 *λ* <sup>1</sup> *t*+*s*

and

$$\begin{split} & -\left(1-\prod\_{j=1}^{k}\prod\_{i=j}^{k}\left(1-\left(1-\left(1-\left(-\gamma\_{\overline{\tau}\_{j}}^{-}\right)\right)^{\varepsilon\_{j}}\right)^{t}\left(1-\left(1-\left(-\gamma\_{\overline{\tau}\_{i}}^{-}\right)\right)^{\varepsilon\_{j}}\right)^{\frac{1}{\lambda}}\right)^{\frac{1}{\lambda+t}}\leq \\ & -\left(1-\prod\_{j=1}^{k}\prod\_{i=j}^{k}\left(1-\left(1-\left(1-\left(-\gamma\_{\overline{\tau}\_{j}}^{-}\right)\right)^{\varepsilon\_{j}}\right)^{t}\left(1-\left(1-\left(-\gamma\_{\overline{\tau}\_{i}}^{-}\right)\right)^{\varepsilon\_{j}}\right)^{\frac{1}{\lambda}}\right)^{\frac{1}{\lambda}}\right)^{\frac{1}{\lambda+t}}. \end{split}$$

,

Thus, *BNNGWHMt*, *<sup>s</sup>* (*τ*1, *<sup>τ</sup>*2,..., *<sup>τ</sup>k*) <sup>≤</sup> *BNNGWHMt*, *<sup>s</sup>* (*σ*1, *σ*2,..., *σk*); this proves Theorem 2. -

#### *3.2. Improved Generalized Weighted HM Operators for BNNs*

**Definition 11.** *Let t, s* <sup>≥</sup> *0, and <sup>t</sup>* <sup>+</sup> *<sup>s</sup>* <sup>=</sup> <sup>0</sup>*, a collection <sup>τ</sup>j*<sup>=</sup> *<sup>α</sup>*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *<sup>τ</sup><sup>j</sup>* (*j* = 1, 2, ··· , *k*) *of BNN; then, we define the BNNIGWHM operator as follows:*

$$BNNIGWHM^{t,s}(\tau\_1, \ \tau\_2, \dots, \ \tau\_k) = \left(\begin{array}{c} \frac{1}{\sum\_{j=1}^k \sum\_{i=j}^k \varepsilon\_{j\bar{t}i}} \stackrel{\text{\rightarrow}}{\leftrightarrow} \frac{1}{i} \big(\varepsilon\_j \varepsilon\_{\bar{t}} \tau\_{\bar{j}}^t \otimes \tau\_{\bar{t}}^s\right) \end{array}\right)^{\frac{1}{\bar{t}+s}},\tag{14}$$

*where* ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1]. *Mathematics* **2019**, *7*, 97

According to Definitions 3 and 11, the following theorem can be attained:

**Theorem 3.** *Set a collection <sup>τ</sup>j*<sup>=</sup> *<sup>α</sup>*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *<sup>τ</sup><sup>j</sup>* (*j* = 1, 2, ··· , *k*) *of BNNs, using BNNIGWHM operator; then, the aggregation result is still a BNN, which is given by the following form:*

*BNNIGWHMt*, *<sup>s</sup>* (*τ*1, *τ*2,..., *τk*) = <sup>1</sup> ∑*k <sup>j</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>k</sup> <sup>i</sup>*=*<sup>j</sup> εjε<sup>i</sup> k* ⊕ *j*=1 *k* ⊕ *i*=*j εjεiτ<sup>j</sup> <sup>t</sup>* <sup>⊗</sup> *<sup>τ</sup><sup>i</sup> s* <sup>1</sup> *t*+*s* = ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − *α*+ *τj t α*+ *τi s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* 1 − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − <sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>+</sup> *τj t* <sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>+</sup> *τi s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* , 1 − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − <sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>+</sup> *τj t* <sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>+</sup> *τi s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* , − ⎛ ⎜⎜⎝ 1 − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − −*α*<sup>−</sup> *τj t* 1 − −*α*<sup>−</sup> *τi s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* ⎞ ⎟⎟⎠, − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − −*β*<sup>−</sup> *τj t* −*β*<sup>−</sup> *τi s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* , − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − −*γ*<sup>−</sup> *τj t* −*γ*<sup>−</sup> *τi s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* , (15)

*where λ* = ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>k</sup> <sup>i</sup>*=*<sup>j</sup> <sup>ε</sup>jεi*, *<sup>k</sup>* ∑ *j*=1 *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1].

The proof of Theorem 3 can be achieved according to the proof of Theorem 1; thus, we omit it here.

**Theorem 4.** *(Idempotency). Set a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNNs; if τ<sup>j</sup>* = *τ, then*

> *BNN IGWHM <sup>t</sup>*, *<sup>s</sup>* (*τ*1, *τ*2,..., *τk*) = *BNN IGWHM <sup>t</sup>*, *<sup>s</sup>* (*τ*, *τ*,... *τ*) = *τ*.

**Proof.** For *τ<sup>j</sup>* = *τ*(*j* = 1, 2, . . . , *k*), the following result can be easily attained:

*BNN IGWHM <sup>t</sup>*, *<sup>s</sup>* (*τ*1, *τ*2,..., *τk*) = *BNN IGWHM <sup>t</sup>*, *<sup>s</sup>* (*τ*, *τ*,... *τ*) = = ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* <sup>1</sup> <sup>−</sup> (*α*<sup>+</sup> *τ* ) *t* (*α*<sup>+</sup> *τ* ) *s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* 1 − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* <sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>+</sup> *τ* ) *t* (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>+</sup> *τ* ) *s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* , 1 − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* <sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>+</sup> *τ* ) *t* (<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>+</sup> *τ* ) *s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* , − ⎛ ⎜⎜⎝ 1 − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − (1 − (−*α*<sup>−</sup> *<sup>τ</sup>* ))*<sup>t</sup>* (1 − (−*α*<sup>−</sup> *<sup>τ</sup>* ))*<sup>s</sup> εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* ⎞ ⎟⎟⎠, − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − (−*β*<sup>−</sup> *τ* ) *t* (−*β*<sup>−</sup> *τ* ) *s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* , − ⎛ ⎝1 − *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − (−*γ*<sup>−</sup> *τ* ) *t* (−*γ*<sup>−</sup> *τ* ) *s εjε<sup>i</sup>* 1 *λ* ⎞ ⎠ 1 *t*+*s* (*α*<sup>+</sup> *τ* ) *t*+*s* 1 *t*+*s* , 1 − (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*<sup>+</sup> *τ* ) *t*+*s* 1 *t*+*s* , = 1 − (<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>+</sup> *τ* ) *t*+*s* 1 *t*+*s* , − 1 − (1 − (−*α*<sup>−</sup> *<sup>τ</sup>* ))*t*+*<sup>s</sup>* 1 *t*+*s* ,<sup>=</sup> *α*<sup>+</sup> *<sup>τ</sup>* , *β*<sup>+</sup> *<sup>τ</sup>* , *γ*<sup>+</sup> *<sup>τ</sup>* , *α*<sup>−</sup> *<sup>τ</sup>* , *β*<sup>−</sup> *<sup>τ</sup>* , *γ*<sup>−</sup> *<sup>τ</sup>* = *τ* − (−*β*<sup>−</sup> *τ* ) *t*+*s* 1 *t*+*s* , − (−*γ*<sup>−</sup> *τ* ) *t*+*s* 1 *t*+*s* .

This proves Theorem 4. -

**Theorem 5.** *(Monotonicity). Set <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *and σ<sup>j</sup>* = *α*<sup>+</sup> *σj* , *β*<sup>+</sup> *σj* , *γ*<sup>+</sup> *σj* , *α*− *σj* , *β*− *σj* , *γ*− *σj* (*<sup>j</sup>* <sup>=</sup> 1, 2, ··· , *<sup>k</sup>*) *as two collections of BNNs; if <sup>α</sup>*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>+</sup> *σj* , *β*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≥</sup> *<sup>β</sup>*<sup>+</sup> *σj* , *γ*<sup>+</sup> *τ<sup>j</sup>* ≥ *γ*<sup>+</sup> *<sup>σ</sup><sup>j</sup> and α*<sup>−</sup> *<sup>τ</sup><sup>j</sup>* ≥ *α*<sup>−</sup> *σj* , *β*− *<sup>τ</sup><sup>j</sup>* ≤ *β*<sup>−</sup> *σj* , *γ*− *<sup>τ</sup><sup>j</sup>* ≤ *γ*<sup>−</sup> *<sup>σ</sup><sup>j</sup>* , *then,*

$$BNNIGGWHM^{t,s}(\tau\_1, \tau\_2, \dots, \tau\_k) \le BNNIGWMHM^{t,s}(\sigma\_1, \sigma\_2, \dots, \sigma\_k).$$

*The proof of Theorem 5 is similar to Theorem 2; thus, we omit it.*

$$\begin{array}{lcl} \textbf{Theorem 6.} (Boundaryness). \textbf{ Set } a \text{ collection } \tau\_{i} = \langle a\_{\tau\_{i}}^{+}, \beta\_{\tau\_{i}}^{+}, \gamma\_{\tau\_{i}}^{+}, a\_{\tau\_{i}}^{-}, \delta\_{\tau\_{i}}^{-}, \gamma\_{\tau\_{i}}^{-} \rangle \text{ } (j = 1, 2, \cdots, k) \text{ of BNNs, and} \\\ \textbf{let } \tau^{-} = \langle \max\big(a\_{\tau\_{i}}^{+} \big), \max\big(\beta\_{\tau\_{i}}^{+} \big), \max\big(\gamma\_{\tau\_{i}}^{-} \big) \big( \\\max\big(a\_{\tau\_{i}}^{-} \big), \min\big(\beta\_{\tau\_{i}}^{-} \big), \min\big(\gamma\_{\tau\_{i}}^{-} \big) \big( \\\end{array} \text{ and } \tau^{+} = \langle \max\big(a\_{\tau\_{i}}^{+} \big), \min\big(\beta\_{\tau\_{i}}^{-} \big) \big( \\\max\big(\gamma\_{\tau\_{i}}^{-} \big) \rangle; \text{ then,} \\\ \textbf{return } \tau^{-} \leq \textit{BNNN} \textit{WHM}^{4 \cdot s} (\tau\_{1}, \ \tau\_{2}, \dots, \tau\_{k}) \leq \tau^{+}. \end{array}$$

*Based on Theorems 4 and 5, the following can be obtained:*

$$
\tau^- = BNNIGWHM^{t,s} \ (\tau^-, \tau^-, \dots, \tau^-) \\
\text{and } \tau^+ = BNNIGWHM^{t,s} \ (\tau^+, \tau^+, \dots, \tau^+).
$$

$$
\begin{array}{l}
BNNIGWHM^{t,s}(\tau^-, \tau^-, \dots, \tau^-) \leq BNNIGWHM^{t,s} \ (\tau\_1, \tau\_2, \dots, \tau\_k) \\
\leq BNNIGWHM^{t,s} \ (\tau^+, \tau^+, \dots, \tau^+).
\end{array}
$$

*Mathematics* **2019**, *7*, 97

$$\text{Then, } \mathfrak{r}^- \le BNNIGWM^{t,s} \text{ ( $\mathfrak{r}\_1$ ,  $\mathfrak{r}\_2$ ,  $\dots$ ,  $\mathfrak{r}\_k$ ) \le \mathfrak{r}^+.$$

*This proves Theorem 6.*

#### *3.3. GWGHM Operators of BNNs*

**Definition 12.** *Let t, s* <sup>≥</sup> *0, <sup>t</sup>* <sup>+</sup> *<sup>s</sup>* <sup>=</sup> <sup>0</sup>*, a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNNs; then, we define the BNNGWGHM operator as follows:*

$$BNNGWGHM^{t,s}(\mathbf{r}\_1, \mathbf{r}\_2, \dots, \mathbf{r}\_k) = \begin{array}{c} \mathbf{1} \\ \mathbf{t} + \mathbf{s} \end{array} \stackrel{k}{\underset{j=1}{\otimes}} \stackrel{k}{\otimes} \begin{array}{c} ((t\mathbf{r}\_j)^{\varepsilon\_j} \oplus (s\mathbf{r}\_l)^{\varepsilon\_l})^{\frac{2}{\mathbf{r}(k+1)}}, \end{array} \tag{16}$$

*where* ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1].

According to Definitions 3 and 12, the following theorem can be attained:

**Theorem 7.** *Set a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNNs, using the BNNGWGHM operator; then, the aggregation result is still a BNN, which is given by the following form:*

*BNNGWGHMt*, *<sup>s</sup>* (*τ*1, *τ*2,..., *τk*) = <sup>1</sup> *t*+*s k* ⊗ *j*=1 *k* ⊗ *i*=*j* (*tτj*) *<sup>ε</sup><sup>j</sup>* <sup>⊕</sup> (*sτi*) *εi* <sup>2</sup> *<sup>k</sup>*(*k*+1) = 1 − ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − 1 − *α*+ *τj t ε<sup>j</sup>* <sup>1</sup> <sup>−</sup> 1 − 1 − *α*+ *τi s εi* <sup>1</sup> *λ* ⎞ ⎠ 1 *t*+*s* , ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − *β*+ *τj t ε<sup>j</sup>* <sup>1</sup> <sup>−</sup> 1 − *β*+ *τi s εi* <sup>1</sup> *λ* ⎞ ⎠ 1 *t*+*s* , ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − *γ*<sup>+</sup> *τj t ε<sup>j</sup>* <sup>1</sup> <sup>−</sup> 1 − *γ*<sup>+</sup> *τi s εi* <sup>1</sup> *λ* ⎞ ⎠ 1 *t*+*s* , − ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − −*α*<sup>−</sup> *τj t ε<sup>j</sup>* <sup>1</sup> <sup>−</sup> 1 − −*α*<sup>−</sup> *τi s εi* <sup>1</sup> *λ* ⎞ ⎠ 1 *t*+*s* , − ⎛ ⎜⎝ 1 − ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − 1 − −*β*<sup>−</sup> *τj t ε<sup>j</sup>* <sup>1</sup> <sup>−</sup> 1 − 1 − −*β*<sup>−</sup> *τi s εi* <sup>1</sup> *λ* ⎞ ⎠ 1 *t*+*s* ⎞ ⎟⎠, − ⎛ ⎜⎝ 1 − ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − 1 − 1 − −*γ*<sup>−</sup> *τj t ε<sup>j</sup>* <sup>1</sup> <sup>−</sup> 1 − 1 − −*γ*<sup>−</sup> *τi s εi* <sup>1</sup> *λ* ⎞ ⎠ 1 *t*+*s* ⎞ ⎟⎠ , (17)

*where* <sup>1</sup> *<sup>λ</sup>* = <sup>2</sup> *<sup>k</sup>*(*k*+1), <sup>∑</sup>*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1].

**Theorem 8.** *(Monotonicity). Set <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *and σ<sup>j</sup>* = *α*<sup>+</sup> *σj* , *β*<sup>+</sup> *σj* , *γ*<sup>+</sup> *σj* , *α*− *σj* , *β*− *σj* , *γ*− *σj* (*<sup>j</sup>* <sup>=</sup> 1, 2, ··· , *<sup>k</sup>*) *as two collections of BNNs; if <sup>α</sup>*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>+</sup> *σj* , *β*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≥</sup> *<sup>β</sup>*<sup>+</sup> *σj* , *γ*<sup>+</sup> *τ<sup>j</sup>* ≥ *γ*<sup>+</sup> *<sup>σ</sup><sup>j</sup> and α*<sup>−</sup> *<sup>τ</sup><sup>j</sup>* ≥ *α*<sup>−</sup> *σj* , *β*− *<sup>τ</sup><sup>j</sup>* ≤ *β*<sup>−</sup> *σj* , *γ*− *<sup>τ</sup><sup>j</sup>* ≤ *γ*<sup>−</sup> *<sup>σ</sup><sup>j</sup>* , *then,*

$$BNNNGNGHM^{t,s}(\tau\_1, \ \tau\_2, \dots, \ \tau\_k) \le BNNGNGHM^{t,s}(\sigma\_1, \ \sigma\_2, \dots, \ \sigma\_k).$$

*The proofs of theorems about BNNGWGHM are similar to those about BNNGWHM; thus, we omit them.*

*Mathematics* **2019**, *7*, 97

#### *3.4. IGWGHM Operators of BNNs*

**Definition 13.** *Let t, s* <sup>≥</sup> *0, and <sup>t</sup>* <sup>+</sup> *<sup>s</sup>* <sup>=</sup> <sup>0</sup>*, a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNNs; then, we define the BNNIGWGHM operator as follows:*

$$BNNIGWGHM^{t.s.}\left(\mathfrak{r}\_{1},\ \mathfrak{r}\_{2},\ldots,\ \mathfrak{r}\_{k}\right) = \ \ \ \frac{1}{t+s} \begin{pmatrix} \stackrel{k}{\underset{\rightarrow}{\otimes}} & \stackrel{k}{\underset{\rightarrow}{\otimes}} \begin{pmatrix} \text{tr}\_{\hat{\mathbb{P}}} \oplus s\mathfrak{r}\_{i} \end{pmatrix} \stackrel{\frac{2(k+1-j)}{k(k+1)}\stackrel{\mathfrak{r}\_{i}}{\sum\_{m=j}^{k}m} \\\ \text{tr}\_{\hat{\mathbb{P}}} & \end{pmatrix}}{} \end{pmatrix},\tag{18}$$

*where* ∑*<sup>k</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1].

According to Definitions 3 and 13, the following theorem can be attained:

**Theorem 9.** *Set a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNNs, using the BNNIGWGHM operator; then, the aggregation result is still a BNN, which is given by the following form:*

*BNN IGWGHMt*, *<sup>s</sup>* (*τ*1, *τ*2,..., *τk*) = <sup>1</sup> *t*+*s* ⎛ ⎝ *k* ⊕ *j*=1 *k* ⊗ *i*=*j tτ<sup>j</sup>* ⊕ *sτ<sup>i</sup>* 2(*k*+1−*j*) *k*(*k*+1) *εi* ∑*k <sup>m</sup>*=*<sup>j</sup> <sup>ε</sup><sup>m</sup>* ⎞ ⎠ = 1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − *α*+ *τj t* 1 − *α*+ *τi s* 1 *λ* <sup>1</sup> *t*+*s* , <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − *β*+ *τj t β*+ *τi s* 1 *λ* <sup>1</sup> *t*+*s* , <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − *γ*<sup>+</sup> *τj t γ*<sup>+</sup> *τi s* 1 *λ* <sup>1</sup> *t*+*s* − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − −*α*<sup>−</sup> *τj t* −*α*<sup>−</sup> *τi s* 1 *λ* <sup>1</sup> *t*+*s* , − ⎛ ⎜⎝ 1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − −*β*<sup>−</sup> *τj t* 1 − −*β*<sup>−</sup> *τi s* 1 *λ* <sup>1</sup> *t*+*s* ⎞ ⎟⎠, − ⎛ ⎜⎝ 1 − <sup>1</sup> <sup>−</sup> *<sup>k</sup>* ∏ *j*=1 *k* ∏ *i*=*j* 1 − 1 − −*γ*<sup>−</sup> *τj t* 1 − −*γ*<sup>−</sup> *τi s* 1 *λ* <sup>1</sup> *t*+*s* ⎞ ⎟⎠ , (19)

*where* <sup>1</sup> *<sup>λ</sup>* <sup>=</sup> <sup>2</sup>(*k*+1−*j*) *k*(*k*+1) *εi* ∑*k <sup>m</sup>*=*<sup>j</sup> ε<sup>m</sup>* , *k* ∑ *j*=1 *ε<sup>j</sup>* = 1 *and ε<sup>j</sup>* ∈ [0, 1].

**Theorem 10.** *(Monotonicity). Set <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *and σ<sup>j</sup>* = *α*+ *σj* , *β*<sup>+</sup> *σj* , *γ*<sup>+</sup> *σj* , *α*− *σj* , *β*− *σj* , *γ*− *<sup>σ</sup><sup>j</sup>* (*<sup>j</sup>* <sup>=</sup> 1, 2, ··· , *<sup>k</sup>*) *as two collections of BNNs; if <sup>α</sup>*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>+</sup> *σj* , *β*<sup>+</sup> *<sup>τ</sup><sup>j</sup>* <sup>≥</sup> *<sup>β</sup>*<sup>+</sup> *σj* , *γ*<sup>+</sup> *τ<sup>j</sup>* ≥ *γ*<sup>+</sup> *<sup>σ</sup><sup>j</sup> and α*<sup>−</sup> *<sup>τ</sup><sup>j</sup>* ≥ *α*<sup>−</sup> *σj* , *β*− *<sup>τ</sup><sup>j</sup>* ≤ *β*<sup>−</sup> *σj* , *γ*− *<sup>τ</sup><sup>j</sup>* ≤ *γ*<sup>−</sup> *σj* , *then,*

$$BNNIGGWGHM^{t\_1,s} \begin{pmatrix} \tau\_1, & \tau\_2, \dots, & \tau\_k \end{pmatrix} \le BNNIGGWGHM^{t\_1,s} \begin{pmatrix} \sigma\_1, & \sigma\_2, \dots, & \sigma\_k \end{pmatrix}.$$

**Theorem 11.** *(Idempotency). Set a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNNs; if τj= τ, then,*

$$BNNIGGWGHM^{t,s} \left(\pi\_1, \ \pi\_2, \dots, \ \pi\_k\right) = BNNIGGWGHM^{t,s} \left(\pi, \pi, \dots, \pi\right) = \tau.$$

**Theorem 12.** *(Boundedness). Set a collection <sup>τ</sup><sup>j</sup>* <sup>=</sup> *α*<sup>+</sup> *τj* , *β*<sup>+</sup> *τj* , *γ*<sup>+</sup> *τj* , *α*− *τj* , *β*− *τj* , *γ*− *τj* (*j* = 1, 2, ··· , *k*) *of BNN, and let <sup>τ</sup>*<sup>−</sup> <sup>=</sup> min *α*+ *τj* , max *β*+ *τj* , max *γ*<sup>+</sup> *τj* , *max α*− *τj* , *min β*− *τj* , *min γ*− *τj* , *and <sup>τ</sup>*<sup>+</sup> <sup>=</sup> *max α*+ *τj* , *min β*+ *τj* , *min*(*γ*<sup>+</sup> *τj* ), *min α*− *τj* , *max β*− *τj* , *max γ*− *τj* ; *then,*

*<sup>τ</sup>*<sup>−</sup> <sup>≤</sup> *BNN IGWHMt*, *<sup>s</sup>* (*τ*1, *<sup>τ</sup>*2,..., *<sup>τ</sup>k*) <sup>≤</sup> *<sup>τ</sup>*+.

The proofs of theorems about BNNIGWGHM are similar to those about BNNIGWHM; thus, we omit them.

#### **4. MCDM Methods Based on the BNNIGWHM and BNNIGWGHM Operator**

We applied the BNNIGWHM and BNNIGWGHM operator to manage MCDM problems within BNN information in this section.

Suppose that a set Γ = {Γ1, Γ2, ... , Γ*n*} of alternatives and a set Φ = {Φ1, Φ2, ... , Φ*m*} of attributes, with the weight vector *ε* = (*ε*1,*ε*2, ...,*εm*) of Φ*j*(*j* = 1, 2, . . . , *m*), in which ∑*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> *ε<sup>j</sup>* = 1 and *ε<sup>j</sup>* ∈ [0, 1]. Decision-makers use BNNs to evaluate the alternatives. The evaluation values *τij* for Γ*<sup>i</sup>* associated with the attribute Φ*<sup>j</sup>* are represented by the form of BNNs. Assume that *τij <sup>n</sup>*×*<sup>m</sup>* <sup>=</sup> *α*<sup>+</sup> *<sup>τ</sup>ij* , *<sup>β</sup>*<sup>+</sup> *<sup>τ</sup>ij* , *<sup>γ</sup>*<sup>+</sup> *<sup>τ</sup>ij* , *α*<sup>−</sup> *<sup>τ</sup>ij* , *β*<sup>−</sup> *<sup>τ</sup>ij* , *γ*<sup>−</sup> *τij <sup>n</sup>*×*<sup>m</sup>* is the BNN decision matrix.

Now, based on the BNNIGWHM and BNNIGWGHM operator, we can develop some decision algorithms:

Step 1: Construct the decision matrix:

$$\left(\left(\tau\_{ij}\right)\_{n\times m} = \left(\left<\alpha^+\_{\tau\_{ij}\prime}\beta^+\_{\tau\_{ij}\prime}\gamma^+\_{\tau\_{ij}\prime}\alpha^-\_{\tau\_{ij}\prime}\beta^-\_{\tau\_{ij}\prime}\gamma^-\_{\tau\_{ij}}\right\rangle\right)\_{n\times m}.$$

Step 2: According to Definition 11 or Definition 13, calculate *τi*.

Step 3: According to the Equation (5), calculate the score value of *s*(*τi*) for *τi*(*i* = 1, 2, . . . , *n*).

Step 4: According to Definition 5, rank all the alternatives corresponding to the values of *s*(*τi*).

#### **5. Illustrative Example**

In this section, we used a numerical example adapted from the literature [16]. A woman wants to buy a car. Now, four kinds of cars Γ1, Γ2, Γ3, and Γ<sup>4</sup> are taken into account according to gasoline consumption (Φ1), aerodynamics (Φ2), comfort (Φ3), and safety performances (Φ4). The importance of these four attributes is given as *ε* = (0.5, 0.25, 0.125, 0.125) *<sup>T</sup>*. Then, she evaluates four alternatives under the above four attributes in the form of BNNs.

#### *5.1. The Decision-Making Process Based on the BNNIGWHM Operator or BNNIGWGHM Operator*

Step 1: Establish the BNN decision matrix (*τij*) <sup>4</sup>×<sup>4</sup> provided by customer, as shown in Table 1.

**Table 1.** The decision matrix (*τij*) 4×4 .


Step 2: According to Definition 11 (suppose *p* = *q* = 1) and *ε* of attributes, calculate *τi*(*i* = 1, 2, 3, 4):

*τ*<sup>1</sup> = 0.4656, 0.5984, 0.3248, −0.6874, −0.4906, −0.5832,

*τ*<sup>2</sup> = 0.8362, 0.5751, 0.5918, −0.5868, −0.6108, −0.2872,

$$\pi\_3 = \langle 0.4212, 0.3684, 0.2341, -0.5268, -0.4254, -0.5540 \rangle\_{\prime} \rangle$$

$$
\pi\_4 = \langle 0.7456, 0.5504, 0.2669, -0.5838, -0.5793, -0.2006 \rangle \rangle.
$$

Step 3: According to Equation (5), calculate thscore value of *s*(*τi*) for *τi*(*i* = 1, 2, 3, 4):

$$s(\tau\_1) = 0.4881; s(\tau\_2) = 0.4968; \ s(\tau\_3) = 0.5458; \ s(\tau\_4) = 0.5207.$$

Step 4: According to Definition 5, rank Γ<sup>3</sup> Γ<sup>4</sup> Γ<sup>2</sup> Γ<sup>1</sup> corresponding to *s*(*τi*); thus, Γ<sup>3</sup> is the best choice among all the alternatives.

Now, we use the BNNIGWGHM operator (set *p* =1, *q* = 1) to deal with this problem.

Step 1': Just as described in step 1.

Step 2': According to Definition 13 (suppose *p* = *q* = 1) and *ε* of attributes, calculate *τi*(*i* = 1, 2, 3, 4):

> *τ*<sup>1</sup> = 0.3834, 0.5909, 0.4846, −0.6881, −0.4467, −0.5722, *τ*<sup>2</sup> = 0.7371, 0.5369, 0.6627, −0.5747, −0.4484, −0.2381, *τ*<sup>3</sup> = 0.4112, 0.3994, 0.2991, −0.5106, −0.3982, −0.3551, *τ*<sup>4</sup> = 0.4922, 0.5086, 0.4579, −0.5674, −0.5684, −0.2139.

Step 3': According to Equation (5), calculate the score value of *s*(*τi*). for *τi*(*i* = 1, 2, 3, 4):

$$s(\tau\_1) = 0.4398; \; s(\tau\_2) = 0.4416; \; s(\tau\_3) = 0.4926; \; s(\tau\_4) = 0.4568.$$

Step 4': According to Definition 5, rank Γ<sup>3</sup> Γ<sup>4</sup> Γ<sup>2</sup> Γ<sup>1</sup> corresponding to *s*(*τi*); thus, Γ<sup>3</sup> is the best choice among all the alternatives.

#### *5.2. Analyzing the Effects of the Parameters p and q*

In this section, we took different parameters *p* and *q* for calculating *τi*(*i* = 1, 2, 3, 4) for the alternative Γ*i*, and then we analyzed the influence of the parameters *p* and *q* for the ranking result. Tables 2 and 3 show the values of *s*(*τ*1) to *s*(*τ*4) and the ranking results.

**Table 2.** Ranking results with different values of *p* and *q* based on bipolar neutrosophic number improved generalized weighted Heronian mean (BNNIGWHM) operator.


**Table 3.** The ranking with different *p* and *q* based on BNN improved generalized weighted geometry HM (BNNIGWGHM) operator.


#### *Mathematics* **2019**, *7*, 97

From the decision results based on BNNIGWHM in Table 2, we can see that all the ranking orders are Γ<sup>3</sup> Γ<sup>4</sup> Γ<sup>1</sup> Γ<sup>2</sup> in No. 1–2 and all the ranking orders are Γ<sup>3</sup> Γ<sup>4</sup> Γ<sup>2</sup> Γ<sup>1</sup> in No. 3–7; thus, the best choice is Γ3. From the decision results based on BNNIGWGHM in Table 3, we can see that the ranking order is Γ<sup>3</sup> Γ<sup>4</sup> Γ<sup>1</sup> Γ<sup>2</sup> in No. 6 and the others are Γ<sup>3</sup> Γ<sup>4</sup> Γ<sup>2</sup> Γ1; thus, the best choice is also Γ3.

IGWHM and IGWGHM aggregation operators can take into account the correlation between attribute values and can better reflect the preferences of decision-makers and make the decision results more reasonable and reliable. A BNS has two fully independent parts, one part has three independent positive membership functions and the other has three independent negative membership functions, which can deal with uncertain information containing incompatible polarity. Here, we used the BNNIGWHM and BNNIGWGHM operators to solve real problems and analyze the influences of parameters *p* and *q* on the results of decisions, using different parameter values for sorting and comparing the corresponding results. Then, it could be found that the influences of parameters *p* and *q* on the results of decisions were small in these both methods. Comparing the results of the two methods, it can be found that their results were consistent; therefore, the proposed methods in this paper have feasibility and generality.

#### *5.3. Comparison with Related Methods*

In this section, we compared the methods proposed in this paper with other related methods proposed in the literature [16,19]. Table 4 lists the ranking results.


**Table 4.** Decision results based on four aggregation operators.

In Table 4, we can see that the ranking results were different; Γ<sup>3</sup> was obtained as the optimal alternative except the method in Reference [19] with *λ* = 0.9. Compared with these related methods, the BNNIGWHM and BNNIGWGHM operators considered the correlation between attribute values and could better reflect the preferences of decision-makers and make the decision results more reasonable and reliable while dealing with uncertain information containing incompatible polarity. Thus, we think the proposed methods in this paper are more suitable to handle these decision-making problems.

#### **6. Conclusions**

This paper firstly proposed the BNNGWHM, BNNIGWHM, BNNGWGHM, and BNNIGWGHM operators for BNNs and discussed the related properties of these four operators. Furthermore, we developed two methods of MCDM in a BNN environment based on the BNNIGWHM and BNNIGWGHM operators. Finally, these two methods were used for a numerical example to establish their effectiveness and application. Dealing with the calculation, we took different values for *p* and

*q* to observe the sorting results and found that both parameters had little influence on the decision results. Furthermore, we compared the proposed methods with related methods and discovered that the selection result using the proposed methods was the same as the majority of existing methods. In the future, we will make further research bipolar neutrosophic sets, using, e.g., the technique for order preference by similarity to an ideal solution (TOPSIS) and VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje, that means: multicriteria optimization and compromise solution, with pronunciation: vikor) methods with BNS [23], the weighted aggregated sum product assessment (WASPAS) method with BNS [24], the Multi-Attribute Market Value Assessment ( MAMVA) method with BNS [25], and so on [26–28].

**Author Contributions:** C.F. proposed the BNNIGWHM and BNNIGWGHM operators and investigated their properties, C.F., S.F. and K.H. presented the organization and decision making method of this paper, J.Y. and E.F. provided the calculation and analysis of the illustrative example; All authors wrote the paper together.

**Funding:** This research was funded by [the National Natural Science Foundation of China] grant number [61703280, 61603258], [the Science and Technology Planning Project of Shaoxing City of China] grant number [2017B70056, 2018C10013], [the General Research Project of Zhejiang Provincial Department of Education] grant number [Y201839944], and [the Public Welfare Technology Research Project of Zhejiang Province] grant number [LGG19F020007].

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

#### *Article*
