Similarity Measures and Multi-person TOPSIS Method Using m-polar Single-Valued Neutrosophic Sets

Juanyong Wu; Ahmed Mostafa Khalil; Nasruddin Hassan; Florentin Smarandache; A. A. Azzam; Hui Yang

doi:10.2991/ijcis.d.210203.003

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 14, Issue 1, 2021, Pages 869 - 885

Similarity Measures and Multi-person TOPSIS Method Using m-polar Single-Valued Neutrosophic Sets

Authors

Juanyong Wu¹^{, 2}, Ahmed Mostafa Khalil³^,, Nasruddin Hassan⁴^,, Florentin Smarandache⁵^,, A. A. Azzam⁶^{, 7}^{, *}, Hui Yang¹

¹School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, China

²School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, China

³Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt

⁴School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600, Malaysia

⁵Department of Mathematics, University of New Mexico, Gallup, NM, 87301, USA

⁶Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj, 11942, Saudi Arabia

⁷Department of Mathematics, Faculty of Science, New Valley University, Elkharga, 72511, Egypt

^*Corresponding author. Email: nas@ukm.edu.my; azzam0911@yahoo.com

Corresponding Author

A. A. Azzam

Received 25 September 2020, Accepted 22 January 2021, Available Online 19 February 2021.

DOI: 10.2991/ijcis.d.210203.003 How to use a DOI?
Keywords: m-polar single-valued neutrosophic set; Distance measure; Ssimilarity measure; Pattern recognition; Multi-person TOPSIS technique
Abstract: In this paper, we give a new notion of the m-polar single-valued neutrosophic sets (m-PSVNSs) which is a hybrid of the single-valued neutrosophic sets (SVNSs) and the m-polar fuzzy sets (m-PFSs) and study several of the structure operations including subset, equal, union, intersection, and complement. Subsequently, we present the basic definitions, theorems, and examples on m-PSVNSs. Also, we define the certain distance between two m-PSVNSs and a novel similarity measure for m-PSVNSs based on distances. A multi criteria decision-making (MCDM) problem is animated for m-PSVNS data that takes into account the distances for the best alternative (solution) by an application of similarity measure for m-PSVNSs in brand recognition. Finally, we construct a new methodology to extend the TOPSIS to m-PSVNS and illustrate its applicability via a numerical example.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Zadeh [1] defined the concept of fuzzy set (FS) as the generalization of crisp set theory and highlighted the complications of some models (e.g., educational mathematical and crisp set theory, which are unable to handle the difficulty of the statistics containing doubts) with the approach of membership. Smarandache [2,3] defined the neutrosophic set (NS) and gave several operations on NS. Wang et al. [4] defined the single valued neutrosophic set (SVNS) by separately assigning the degree of truth membership, the degree of indeterminacy membership, and the degree of falsity membership. Zhang [5] proposed the concept of bipolar fuzzy sets (BFSs) and relations. Chen et al. [6] presented the concept of m-polar fuzzy set (m-PFS) as a simplified type of BFS. The notions of m-PF graph, characterization of m-PF graph, and metrics in m-PF graph were introduced in [7–10]. A multi-criteria decision-making (MCDM) method on m-PF soft rough sets, hybrid m-PF models with m-PF ELECTRE-I were presented in [11–13].

A similarity measure (SM) for the fuzzy system plays a very considerable role in handling problems that comprehend ambiguous information, but unable to deal with the unclearness and uncertainty of the problems having normal information. The idea of SM of two sets (i.e., fuzzy values and vague sets) were discussed in [14,15], but it had failed to hold in some problems. To resolve this problem, Hong and Kim [16] brought into light some modified measures. Several SM based on SVNSs were proposed in [17–20]. Akram and Waseem [21] implemented a m-PFS and m-PF soft set on SMs to medical diagnosis. Yong [22] gave the new approach of the selection of location for plantation under the terms of the linguistic environment through graded mean reparation based on fuzzy TOPSIS. Adeel et al. [23] animates m-PF through the extension of order preference of MCGDM-TOPSIS for the best alternative. Shih et al. [24] described the technique of order preference through similarity using MADM in a group decision environment based on group preference for TOPSIS. Saeed et al. [25] gave an application in SM on multipolar neutrosophic soft sets structure in decision-making for medical diagnosis. Kang et al. [26] presented an application in BCK/BCI-algebras on multipolar IFS with finite degree. Akram [27,28] defined the notions of single-valued neutrosophic graphs and m-polar fuzzy graphs. There are many published papers on TOPSIS in different fields (for instance, [29–33]).

In normal practice, the problem of SVN information occurs (i.e., which cannot be elaborated well using the existing approaches), while an m-polar single-valued neutrosophic set (m-PSVNS) is used to resolve the uncertain and more variant data, specifically in the SVNS form with m-PFS. Our motivation is to be able to find a solution of many daily life problems enhanced through distance-based SM with the m-PSVNS. It increases the number of applications in various fields, including electronic optimization, industries, and forensic facial portrait. Moreover, a new approach for the best m-PSVNS alternatives based on distance SMs in MCDM is animated.

This article is structured as follows: In Section 2, a brief overview of some fundamental concepts is provided. In Section 3, the concept of m-PSVNS and its basic operations are defined. In Section 4, distance measure formulas are presented on m-PSVNS. Further, the m-PSVNS is used to investigate a problem involving distance-based SMs with an algorithm. In Section 5, the MCDM for m-PSVN data is described with an algorithm for the best solution (alternative). The article is concluded with a review and further work outlook.

2. PRELIMINARIES

We give a short survey of concepts of BFSs, m-PFSs, and SVNSs as indicated below.

2.1. BFSs and m-PFSs

Definition 1.

(cf. [5]). Suppose that Z (i.e., Z={z1,z2,…,zr|r=1,2,…,n},n∈N (natural Numbers)) be a set of elements. A BFS on Z is a pair (α,β), where

α:Z→[0,1] and β:Z→[−1,0]

are two mappings. We note that BFS are an extension of FSs whose membership degree range is [−1, 1].

Definition 2.

(cf. [6]). An m-PFS on Z (i.e., Z={z1,z2,…,zr}) be a set of elements, is a mapping

α:Z→[0,1]m.

Example 1.

Suppose that 3-PFS on Z (i.e., Z={z1,z2,z3}) is defined by

α(z1)=〈0.45,0.42,0.59〉, α(z2)=〈0.51,0.52,0.7〉, and α(z3)=〈0.4,0.53,0.9〉.

2.2. SVNSs

Definition 3.

(cf. [4]). A SVNS Φ=(α,β,γ) on Z (i.e., Z={z1,z2,…,zr}) be a set of elements, is a mapping

(α,β,γ):Z→[0,1]×[0,1]×[0,1],

where α:Z→[0,1] (i.e., the degree of truth membership), β:Z→[0,1] (i.e., the degree of indeterminacy membership), and γ:Z→[0,1] (i.e., the degree of falsity membership) are FSs over Z such that 0≤αΦ(zr)+βΦ(zr)+γΦ(zr)≤3(∀zr∈Z,r=1,2,…,n) or

Φ={(αΦ(zr),βΦ(zr),γΦ(zr))zrzr∈Z}.

Definition 4.

(cf. [4]). Let

Φ={(αΦ(zr),βΦ(zr),γΦ(zr))zrzr∈Z}.

and

Ψ={(ξΨ(zr),ηΨ(zr),εΨ(zr))zrzr∈Z}

are SVNSs. The following five operations are defined by

(Complement) Φc={(γΦ(zr),1−βΦ(zr),αΦ(zr))zrzr∈Z}.
(Inclusion) Φ⊆Ψ⇔αΦ(zr)≤ξΨ(zr),βΦ(zr)≥ηΨ(zr), and γΦ(zr)≥εΨ(zr)(∀zr∈Z).
(Equal) Φ=Ψ⇔Φ⊆Ψ and Ψ⊆Φ.
(Union) Φ∪Ψ={(αΦ(zr)∨ξΨ(zr),βΦ(zr)∧ηΨ(zr),γΦ(zr)∧εΨ(zr))zrzr∈Z}.
(Intersection) Φ∩Ψ={(αΦ(zr)∧ξΨ(zr),βΦ(zr)∨ηΨ(zr),γΦ(zr)∨εΨ(zr))zrzr∈Z}.

3. AN m-PSVNSs

We will introduce the concept of m-PSVNS and study several definitions, theorems, and examples as indicated below.

Definition 5.

Anm-PSVNS Φ on Z (i.e., Z={z1,z2,…,zr} be a set of elements), is a mapping

Φ:Z→[0,1]m×[0,1]m×[0,1]m.

It can be written as

Φ={(〈p1∘αΦ(zr),p1∘βΦ(zr),p1∘γΦ(zr)〉,〈p2∘αΦ(zr),p2∘βΦ(zr),p2∘γΦ(zr)〉,…,〈pk∘αΦ(zr),pk∘βΦ(zr),pk∘γΦ(zr)〉)zzr∈Z},

where (pk∘αΦ,pk∘βΦ,pk∘γΦ):([0,1]×[0,1]×[0,1])→[0,1]×[0,1]×[0,1] is the kth projection mapping. pk∘α is the kth truth membership value, pk∘β is the kth indeterminacy membership value, and pk∘γ is the kth falsity membership value of element in Φ=(〈p1∘αΦ(zr),p1∘βΦ(zr),p1∘γΦ(zr)〉,〈p2∘αΦ(zr),p2∘βΦ(zr),p2∘γΦ(zr)〉,…,〈pk∘αΦ(zr),pk∘βΦ(zr),pk∘γΦ(zr)〉) such that 0≤pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)≤3.

Example 2.

Suppose that an 2-PSVNS on three of elements Z={z1,z2,z3} is defined by

Φ={(〈0.6,0.3,0.7〉,〈0.4,0.2,0.8〉)z1,(〈0.1,0.7,0.9〉,〈0.4,0.3,0.8〉)z2,(〈0.1,0.3,0.7〉,〈0.2,0.4,0.7〉)z3}.

Definition 6.

Assume that Φ and Ψ are m-PSVNS on a set of elements Z, where

Φ={(〈p1∘αΦ(zr),p1∘βΦ(zr),p1∘γΦ(zr)〉,〈p2∘αΦ(zr),p2∘βΦ(zr),p2∘γΦ(zr)〉,…,〈pk∘αΦ(zr),pk∘βΦ(zr),pk∘γΦ(zr)〉)zzr∈Z},

and

Ψ={(〈p1∘ζΨ(zr),p1∘ηΨ(zr),p1∘εΨ(zr)〉,〈p2∘ζΨ(zr),p2∘ηΨ(zr),p2∘εΨ(zr)〉,…,〈pk∘ζΨ(zr),pk∘ηΨ(zr),pk∘εΨ(zr)〉)zzr∈Z}.

Then, Φ⊆Ψ (i.e., Φ is an m-PSVN subset of Ψ) if for all zr∈Z,pk∘αΦ(zr)≤pk∘ζΨ(zr),pk∘βΦ(zr)≥pk∘ηΨ(zr) and pk∘γΦ(zr)≥pk∘εΨ(zr).

Example 3.

(Continued from Example 2). Suppose that Ψ be an 2-PSVNS on three of elements Z={z1,z2,z3} is defined by

Ψ={(〈0.7,0.3,0.4〉,〈0.5,0.1,0.5〉)z1,(〈0.2,0.6,0.4〉,〈0.5,0.2,0.4〉)z2,(〈0.6,0.1,0.6〉,〈0.4,0.4,0.5〉)z3}.

Hence, Φ⊆Ψ(∀zr∈Z,r=1,2,3).

Definition 7.

Assume that Φ and Ψ are m-PSVNSs on a set of elements Z, where

Φ={(〈p1∘αΦ(zr),p1∘βΦ(zr),p1∘γΦ(zr)〉,〈p2∘αΦ(zr),p2∘βΦ(zr),p2∘γΦ(zr)〉,…,〈pk∘αΦ(zr),pk∘βΦ(zr),pk∘γΦ(zr)〉)zzr∈Z},

and

Ψ={(〈p1∘ζΨ(zr),p1∘ηΨ(zr),p1∘εΨ(zr)〉,〈p2∘ζΨ(zr),p2∘ηΨ(zr),p2∘εΨ(zr)〉,…,〈pk∘ζΨ(zr),pk∘ηΨ(zr),pk∘εΨ(zr)〉)zzr∈Z}.

Then, Φ=Ψ (i.e., Φ is an m-PSVN equal of Ψ) if Φ⊆Ψ and Φ⊇Ψ.

Definition 8.

Assume that Φ be an m-PSVNS on a set of elements Z, where

Φ={(〈p1∘αΦ(zr),p1∘βΦ(zr),p1∘γΦ(zr)〉,〈p2∘αΦ(zr),p2∘βΦ(zr),p2∘γΦ(zr)〉,…,〈pk∘αΦ(zr),pk∘βΦ(zr),pk∘γΦ(zr)〉)zzr∈Z}.

Then,

Φ is called an m-PSVN null set (denoted by ∅^), is defined by
∅^={(0,1,1)zrzr∈Z}.
Φ is called an m-PSVN absolute set (denoted by Z^), is defined by
Z^={(1,0,0)zrzr∈Z}.

Example 4.

(Continued from Example 2). Then, an 2-PSVNSs ∅^ and Z^ are defined by

∅^={(〈0,1,1〉,〈0,1,1〉)z1,(〈0,1,1〉,〈0,1,1〉)z2,(〈0,1,1〉,〈0,1,1〉)z3}

and

Z^={(〈1,0,0〉,〈1,0,0〉)z1,(〈1,0,0〉,〈1,0,0〉)z2,(〈1,0,0〉,〈1,0,0〉)z3}.

Definition 9.

Assume that Φ and Ψ are m-PSVNSs on a set of elements Z, where

Φ={(〈p1∘αΦ(zr),p1∘βΦ(zr),p1∘γΦ(zr)〉,〈p2∘αΦ(zr),p2∘βΦ(zr),p2∘γΦ(zr)〉,…,〈pk∘αΦ(zr),pk∘βΦ(zr),pk∘γΦ(zr)〉)zzr∈Z},

and

Ψ={(〈p1∘ζΨ(zr),p1∘ηΨ(zr),p1∘εΨ(zr)〉,〈p2∘ζΨ(zr),p2∘ηΨ(zr),p2∘εΨ(zr)〉,…,〈pk∘ζΨ(zr),pk∘ηΨ(zr),pk∘εΨ(zr)〉)zzr∈Z}.

The intersection Φ∩Ψ, is defined as
Φ∩Ψ={(pk∘αΦ(zr)∧pk∘ζΨ(zr),pk∘βΦ(zr)∨pk∘ηΨ(zr),pk∘γΦ(zr)∨pk∘εΨ(zr))zrzr∈Z,k=1,2,…,m}.
The union Φ∪Ψ, is defined as
Φ∪Ψ={(pk∘αΦ(zr)∨pk∘ζΨ(zr),pk∘βΦ(zr)∧pk∘ηΨ(zr),pk∘γΦ(zr)∧pk∘εΨ(zr))zrzr∈Z,k=1,2,…,m}.

Example 5.

(Continued from Examples 2 and 3). Then, the intersection and union of an m-PSVNSs, respectively, given by

Φ∩Ψ={(〈0.6,0.3,0.7〉,〈0.4,0.2,0.8〉)z1,(〈0.1,0.7,0.9〉,〈0.4,0.3,0.8〉)z2,(〈0.1,0.3,0.7〉,〈0.2,0.4,0.7〉)z3}

and

Φ∪Ψ={(〈0.7,0.3,0.4〉,〈0.5,0.1,0.5〉)z1,(〈0.2,0.6,0.4〉,〈0.5,0.2,0.4〉)z2,(〈0.6,0.1,0.6〉,〈0.4,0.4,0.5〉)z3}.

Theorem 1.

(Identity law of m-PSVNSs) Assume that Φ be m-PSVNS, ∅^ be m-PSVN null set, and Z^ be m-PSVN absolute set. Then the followings hold:

Φ∪∅^=Φ,
Φ∩Z^=Φ.

Proof.

Φ∪∅^
={(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zrzr∈Z,k=1,2,…,m}∪{(0,1,1)zrzr∈Z}={(pk∘αΦ(zr)∨0, pk∘βΦ(zr)∧1, pk∘γΦ(zr)∧1)zrzr∈Z,k=1,2,…,m}={(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zrzr∈Z,k=1,2,…,m}=Φ.
Φ∩Z^
={(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zrzr∈Z,k=1,2,…,m}∩{(1,0,0)zrzr∈Z}={(pk∘αΦ(zr)∧1, pk∘βΦ(zr)∨0, pk∘γΦ(zr)∨0)zrzr∈Z,k=1,2,…,m}={(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zrzr∈Z,k=1,2,…,m}=Φ.

Theorem 2.

(Domination law of m-PSVNSs) Assume that Φ be m-PSVNS, ∅^ be m-PSVN null set, and Z^ be m-PSVN absolute set. Then the followings hold:

Φ∪Z^=Z^,
Φ∩∅^=∅^.

Proof.

Φ∪Z^
={(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zrzr∈Z,k=1,2,…,m}∪{(1,0,0)zrzr∈Z}={(pk∘αΦ(zr)∨1, pk∘βΦ(zr)∧0, pk∘γΦ(zr)∧0)zrzr∈Z,k=1,2,…,m}={(1,0,0)zrzr∈Z}=Z^.
Φ∩∅^
={(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zrzr∈Z,k=1,2,…,m}∩{(0,1,1)zrzr∈Z}={(pk∘αΦ(zr)∧0, pk∘βΦ(zr)∨1, pk∘γΦ(zr)∨1)zrzr∈Z,k=1,2,…,m}={(0,1,1)zrzr∈Z}=∅^.

Theorem 3.

(Idempotent law of m-PSVNSs) Assume that Φ be m-PSVNS. Then the followings hold:

Φ∪Φ=Φ,
Φ∩Φ=Φ.

Proof.

Φ ∪ Φ=Φ
=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr zr∈Z, k=1,2,…,m∪(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr zr∈Z, k=1,2,…,m=(pk∘αΦ(zr)∨pk∘αΦ(zr), pk∘βΦ(zr)∧pk∘βΦ(zr), pk∘γΦ(zr)∧pk∘γΦ(zr))zr zr∈Z, k=1,2,…,m=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr zr∈Z, k=1,2,…,m=Φ.
Φ ∩ Φ
=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr zr∈Z, k=1,2,…,m∩(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr zr∈Z, k=1,2,…,m=(pk∘αΦ(zr)∧pk∘αΦ(zr), pk∘βΦ(zr)∨pk∘βΦ(zr), pk∘γΦ(zr)∨pk∘γΦ(zr))zr zr∈Z, k=1,2,…,m=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr zr∈Z, k=1,2,…,m=Φ.

Theorem 4.

(Commutative law of m-PSVNSs) Assume that Φ and Ψ are m-PSVNSs. Then the followings hold:

Φ∩Ψ=Ψ∩Φ,
Φ∪Ψ=Ψ∪Φ.

Proof.

Φ∩Ψ
=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m∩(pk∘ζΨ(zr), pk∘ηΨ(zr), pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m.=(pk∘αΦ(zr)∧pk∘ζΨ(zr), pk∘βΦ(zr)∨pk∘ηΨ(zr), pk∘γΦ(zr)∨pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m=(pk∘ζΨ(zr)∧pk∘αΦ(zr), pk∘ηΨ(zr)∨pk∘βΦ(zr), pk∘εΨ(zr)∨pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m=(pk∘ζΨ(zr), pk∘ηΨ(zr), pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m∩(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m.=Ψ∩Φ.
Φ∪Ψ
=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m∩(pk∘ζΨ(zr), pk∘ηΨ(zr), pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m.=(pk∘αΦ(zr)∨pk∘ζΨ(zr), pk∘βΦ(zr)∧pk∘ηΨ(zr), pk∘γΦ(zr)∧pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m=(pk∘ζΨ(zr)∨pk∘αΦ(zr), pk∘ηΨ(zr)∧pk∘βΦ(zr), pk∘εΨ(zr)∧pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m=(pk∘ζΨ(zr), pk∘ηΨ(zr), pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m∪(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m.=Ψ∪Φ.

Theorem 5.

(Associative law of m-PSVNSs) Assume that Φ,Ψ, and Ω are m-PSVNSs. Then the followings hold:

(Φ∩Ψ)∩Ω=Φ∩(Ψ∩Ω),
(Φ∪Ψ)∪Ω=Φ∪(Ψ∪Ω).

Proof.

(Φ∩Ψ)∩Ω
=(pk∘αΦ(zr)∧pk∘ζΨ(zr), pk∘βΦ(zr)∨pk∘ηΨ(zr), pk∘γΦ(zr)∨pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m∩(pk∘ρΩ(zr), pk∘λΩ(zr), pk∘δΩ(zr))zr | zr∈Z,k=1,2,…,m=(pk∘αΦ(zr)∧pk∘ζΨ(zr)∧pk∘ρΩ(zr), pk∘βΦ(zr)∨pk∘ηΨ(zr)∨pk∘λΩ(zr), pk∘γΦ(zr)∨pk∘εΨ(zr)∨pk∘δΩ(zr))zr| zr∈Z, k=1,2,…,m=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m∩(ζΨ(zr)∧pk∘ρΩ(zr), ηΨ(zr)∨pk∘λΩ(zr), pk∘εΨ(zr)∨pk∘δΩ(zr))zr | zr∈Z,k=1,2,…,m=Φ∩(Ψ∩Ω).
(Φ∪Ψ)∪Ω
=(pk∘αΦ(zr)∨pk∘ζΨ(zr), pk∘βΦ(zr)∧pk∘ηΨ(zr), pk∘γΦ(zr)∧pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m∪(pk∘ρΩ(zr), pk∘λΩ(zr), pk∘δΩ(zr))zr | zr∈Z,k=1,2,…,m=(pk∘αΦ(zr)∨pk∘ζΨ(zr)∨pk∘ρΩ(zr), pk∘βΦ(zr)∧pk∘ηΨ(zr)∧pk∘λΩ(zr), pk∘γΦ(zr)∧pk∘εΨ(zr)∧pk∘δΩ(zr))zr| zr∈Z, k=1,2,…,m=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m∪(ζΨ(zr)∨pk∘ρΩ(zr), ηΨ(zr)∧pk∘λΩ(zr), pk∘εΨ(zr)∧pk∘δΩ(zr))zr | zr∈Z,k=1,2,…,m=Φ∪(Ψ∪Ω).

Theorem 6.

(Distributive law of m-PSVNSs) Assume that Φ,Ψ, and Ω are m-PSVNSs. Then the followings hold:

Φ∩(Ψ∪Ω)=(Φ∩Ψ)∪(Φ∩Ω),
Φ∪(Ψ∩Ω)=(Φ∪Ψ)∩(Φ∪Ω).

Proof.

Φ∩(Ψ∪Ω)
=(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m∩(pk∘ζΨ(zr)∨pk∘ρΩ(zr), pk∘ηΨ(zr)∧pk∘λΩ(zr), pk∘εΨ(zr)∧pk∘δΩ(zr))zr | zr∈Z,k=1,2,…,m=(pk∘αΦ(zr)∧[pk∘ζΨ(zr)∨pk∘ρΩ(zr)], pk∘βΦ(zr)∨[pk∘ηΨ(zr)∧pk∘λΩ(zr)], pk∘γΦ(zr)∨[pk∘εΨ(zr)∧pk∘δΩ(zr)])zr | zr∈Z,k=1,2,…,m=([pk∘αΦ(zr)∧pk∘ζΨ(zr)]∨[pk∘αΦ(zr)∧pk∘ρΩ(zr)]), ([pk∘βΦ(zr)∨pk∘ηΨ(zr)]∧[pk∘βΦ(zr)∨pk∘λΩ(zr)]), ([pk∘γΦ(zr)∨pk∘εΨ(zr)]∧[pk∘γΦ(zr)∨pk∘δΩ(zr)])zr | zr∈Z,k=1,2,…,m=(Φ∩Ψ)∪(Φ∩Ω).
Φ∪(Ψ∩Ω)
=([pk∘αΦ(zr)∨pk∘ζΨ(zr)]∧[pk∘αΦ(zr)∨pk∘ρΩ(zr)]), ([pk∘βΦ(zr)∧pk∘ηΨ(zr)]∨[pk∘βΦ(zr)∧pk∘λΩ(zr)]), ([pk∘γΦ(zr)∧pk∘εΨ(zr)]∨[pk∘γΦ(zr)∧pk∘δΩ(zr)])zr | zr∈Z,k=1,2,…,m=(pk∘αΦ(zr)∨[pk∘ζΨ(zr)∧pk∘ρΩ(zr)], pk∘βΦ(zr)∧[pk∘ηΨ(zr)∨pk∘λΩ(zr)], pk∘γΦ(zr)∧[pk∘εΨ(zr)∨pk∘δΩ(zr)])zr | zr∈Z,k=1,2,…,m=([pk∘αΦ(zr)∨pk∘ζΨ(zr)]∧[pk∘αΦ(zr)∨pk∘ρΩ(zr)]), ([pk∘βΦ(zr)∧pk∘ηΨ(zr)]∨[pk∘βΦ(zr)∧pk∘λΩ(zr)]), ([pk∘γΦ(zr)∧pk∘εΨ(zr)]∨[pk∘γΦ(zr)∧pk∘δΩ(zr)])zr | zr∈Z,k=1,2,…,m=(Φ∪Ψ)∩(Φ∪Ω).

Definition 10.

Assume that Φ be m-PSVNS on a set of elements Z, where

Φ={(〈p1∘αΦ(zr),p1∘βΦ(zr),p1∘γΦ(zr)〉,〈p2∘αΦ(zr),p2∘βΦ(zr),p2∘γΦ(zr)〉,…,〈pk∘αΦ(zr),pk∘βΦ(zr),pk∘γΦ(zr)〉)z|zr∈Z}

Then, the complement Φc of Φ, is defined by

Φc={(〈p1∘γΦ(zr),1−p1∘βΦ(zr),p1∘αΦ(zr)〉,〈p2∘γΦ(zr),1−p2∘βΦ(zr),p2∘αΦ(zr)〉,…,〈pk∘γΦ(zr),1−pk∘βΦ(zr),pk∘αΦ(zr)〉)z|zr∈Z}

Example 6.

(Continued from Example 2). The complement Φc of Φ, is calculated by

Φc={(〈0.7,0.7,0.6〉,〈0.8,0.8,0.4〉)z1,(〈0.9,0.3,0.1〉,〈0.8,0.7,0.4〉)z2,(〈0.7,0.7,0.1〉,〈0.7,0.6,0.2〉)z3}.

Theorem 7.

(Complementation and double complementation of m-PSVNSs) Assume that Φ be m-PSVNSs, ∅^ be m-PSVN null set, and Z^ be m-PSVN absolute set. Then the followings hold:

∅^c=Z^,
Z^=∅^,
(Φc)c=Φ

Proof.

∅^c=({(0,1,1)zr|zr∈Z})c={(1,0,0)zr|zr∈Z}=Z^.
Z^c=({(1,0,0)zr|zr∈Z})c={(0,1,1)zr|zr∈Z}=∅^.
(Φc)c=({(pk∘γΦ(zr),1−pk∘βΦ(zr),pk∘αΦ(zr))zrzr∈Z, k=1,2,…,m})c={(pk∘αΦ(zr),1−(1−pk∘βΦ(zr)),pk∘γΦ(zr))zrzr∈Z, k=1,2,…,m}={(pk∘αΦ(zr),pk∘βΦ(zr),pk∘γΦ(zr))zrzr∈Z, k=1,2,…,m}=Φ.

Remark 1.

By the following Example 7, we will show the equality of Φ∪Φc=X^ and Φ∩Φc=∅^ does not hold.

Example 7.

(Continued from Examples 2 and 6). Then Φ∪Φc and Φ∩Φc are calculated by

Φ∪Φc={(〈0.7,0.3,0.6〉,〈0.8,0.2,0.4〉)z1,(〈0.9,0.3,0.1〉,〈0.8,0.3,0.4〉)z2,(〈0.7,0.3,0.1〉,〈0.7,0.4,0.2〉)z3}

and

Φ∩Φc={(〈0.6,0.7,0.7〉,〈0.4,0.8,0.8〉)z1,(〈0.7,0.7,0.9〉,〈0.4,0.7,0.8〉)z2,(〈0.1,0.7,0.7〉,〈0.2,0.6,0.7〉)z3}.

This show Φ∪Φc≠X^ and Φ∩Φc≠∅^

Theorem 8.

(De Morgan's laws of m-PSVNSs) Assume that Φ,Ψ, and Ω are m-PSVNSs. Then the followings hold:

(Φ∪Ψ)c=Φc∩Ψc,
(Φ∩Ψ)c=Φc∪Ψc.

Proof.

(Φ∪Ψ)c
=({(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m}∪{(pk∘ζΨ(zr), pk∘ηΨ(zr), pk∘εΨ(zr))zr | zr∈Z,k=1,2,…,m})c=({(pk∘αΦ(zr)∨pk∘ζΨ(zr), pk∘βΦ(zr)∧pk∘ηΨ(zr), pk∘γΦ(zr)∧pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m})c={(pk∘γΦ(zr)∧pk∘εΨ(zr), 1−(pk∘βΦ(zr)∧pk∘ηΨ(zr)), pk∘αΦ(zr)∨pk∘ζΨ(zr))zr | zr∈Z, k=1,2,…,m}={(pk∘γΦ(zr)∧pk∘εΨ(zr),(1−pk∘βΦ(zr))∨(1−pk∘ηΨ(zr)), pk∘αΦ(zr)∨pk∘ζΨ(zr))zr | zr∈Z, k=1,2,…,m}={(pk∘γΦ(zr), 1−pk∘βΦ(zr), pk∘αΦ(zr))zr | zr∈Z, k=1,2,…,m}∩{(pk∘εΨ(zr), 1−pk∘ηΨ(zr), pk∘ζΨ(zr))zr | zr∈Z,k=1,2,…,m}=Φc∩Ψc.
(Φ∩Ψ)c
=({(pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr))zr | zr∈Z, k=1,2,…,m}∩{(pk∘ζΨ(zr), pk∘ηΨ(zr), pk∘εΨ(zr))zr | zr∈Z,k=1,2,…,m})c=({(pk∘αΦ(zr)∧pk∘ζΨ(zr), pk∘βΦ(zr)∨pk∘ηΨ(zr), pk∘γΦ(zr)∨pk∘εΨ(zr))zr | zr∈Z, k=1,2,…,m})c={(pk∘γΦ(zr)∨pk∘εΨ(zr), 1−(pk∘βΦ(zr)∨pk∘ηΨ(zr)), pk∘αΦ(zr)∧pk∘ζΨ(zr))zr | zr∈Z, k=1,2,…,m}={(pk∘γΦ(zr)∨pk∘εΨ(zr), (1−pk∘βΦ(zr))∧(1−pk∘ηΨ(zr)), pk∘αΦ(zr)∧pk∘ζΨ(zr))zr | zr∈Z, k=1,2,…,m}={(pk∘γΦ(zr), 1−pk∘βΦ(zr), pk∘αΦ(zr))zr | zr∈Z, k=1,2,…,m}∪{(pk∘εΨ(zr), 1−pk∘ηΨ(zr), pk∘ζΨ(zr))zr | zr∈Z,k=1,2,…,m}=Φc∪Ψc.

4. DISTANCE MEASURE AND SM FOR m-PSVNSs

We present a few new concepts of distances measure and SM for m-PSVNSs as indicated below.

Definition 11.

Assume that Φ and Ψ are m-PSVNSs on a set of elements Z (i.e., Z={z1,z2,…,zr|r=1,2,…,n}), where

Φ={(〈p1∘αΦ(zr), p1∘βΦ(zr),p1∘γΦ(zr)〉,〈p2∘αΦ(zr), p2∘βΦ(zr), p2∘γΦ(zr)〉,…,〈pk∘αΦ(zr), pk∘βΦ(zr), pk∘γΦ(zr)〉)z|zr∈Z},

and

Ψ={(〈p1∘ζΨ(zr), p1∘ηΨ(zr), p1∘εΨ(zr)〉,〈p2∘ζΨ(zr), p2∘ηΨ(zr), p2∘εΨ(zr)〉,…,〈pk∘ζΨ(zr), pk∘ηΨ(zr), pk∘εΨ(zr)〉)z|zr∈Z}

The distances measure between Φ and Ψ are defined by

Hamming distance:
d1(Φ,Ψ)=1m{∑k=1m∑r=1n|pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2|}(1)
Normalized Hamming distance:
d2(Φ,Ψ)=1mn{∑k=1m∑r=1n|pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2|}(2)
Euclidean distance:
d3(Φ,Ψ)=1m{∑k=1m∑r=1n(pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2)2}(3)
Normalized Euclidean distance:
d4(Φ,Ψ)=1mn{∑k=1m∑r=1n(pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2)2}.(4)

Theorem 9.

The distances between Im-PFSs Φ and Ψ satisfy the following four inequalities:

d1(Φ,Ψ)≤n,
d2(Φ,Ψ)≤1,
d3(Φ,Ψ)≤n,
d4(Φ,Ψ)≤1.

Theorem 10.

The distance functions d1,d2,d3, and d4 defined from mPSVNS(Z)→R+, are metric distances.

Proof.

Suppose that Φ,Ψ, and Ω are three m-PSVNSs on Z. Then

d1(Φ,Ψ)≥0.
Let
d1(Φ,Ψ)=0⇔1m{∑k=1m∑r=1npk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2}=0⇔pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2=0⇔pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2=pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2, 1≤k≤m, 1≤r≤n$⇔Φ=Ψ.
d1(Φ,Ψ)=d1(Ψ,Φ).
For any three m-PSVNSs Φ,Ψ, and Ω,
=pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2=pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ρΩ(zr)+pk∘λΩ(zr)+pk∘δΩ(zr)2 +pk∘ρΩ(zr)+pk∘λΩ(zr)+pk∘δΩ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2≤pk∘αΦ(zr)+pk∘βΦ(zr)+pk∘γΦ(zr)2−pk∘ρΩ(zr)+pk∘λΩ(zr)+pk∘δΩ(zr)2+pk∘ρΩ(zr)+pk∘λΩ(zr)+pk∘δΩ(zr)2−pk∘ζΨ(zr)+pk∘ηΨ(zr)+pk∘εΨ(zr)2.

Hence, d1(Φ,Ψ)≤d1(Φ,Ω)+d1(Ω,Ψ).

Definition 12.

The SM of two m-PSVNSs Φ and Ψ, defined by

SM(Φ,Ψ)=11+d(Φ,Ψ)(5)

where d(Φ,Ψ) is any of the above four distances in Definition 11.

Definition 13.

The two m-PSVNSs Φ and Ψ are σ similar if and only if SM(Φ,Ψ)≥σ, i.e.,

Φ≈σΨ⇔SM(Φ,Ψ)≥σ, σ∈(0,1)(6)

Φ and Ψ are are significantly similar if SM(Φ,Ψ)≥12.

Example 8.

Suppose that an 2-PSVNSs on two of elements Z={z1,z2} are defined by

Φ={(〈0.4,0.3,0.9〉,〈0.5,0.2,0.8〉)z1,(〈0.6,0.3,0.7〉,〈0.5,0.5,0.6〉)z2}

and

Ψ={(〈0.6,0.2,0.8〉,〈0.1,0.4,0.7〉)z1,(〈0.3,0.2,0.6〉,〈0.7,0.1,0.8〉)z2}.

Then, the Hamming distance is d1(Φ,Ψ)=0.2 and the SM is SM(Φ,Ψ)=0.833. It shows Φ is significantly similar to Ψ.

Theorem 11.

The SM of two m-PSVNSs Φ and Ψ satisfy the following three properties:

0≤SM(Φ,Ψ)≤1,
SM(Φ,Ψ)=SM(Ψ,Φ),
SM(Φ,Ψ)=1⇔Ψ=Φ.

4.1. An Application of SM for m-PSVNSs in Brand Recognition

Based on the notion of Euclidean distance (i.e., Equation (3)) by SM, we will use m-PSVNSs information to solve brand recognition problem.

In the following, we will build an Algorithm 1 to solve brand recognition problem (i.e., to apply m-PSVNS in brand recognition problem).

Algorithm 1

Determination of how an observer finds similarity to an unknown watch.

First step: Suppose that there are n watches represented by m-PSVNS Te(e=1,2,…,n) in feature space Z (i.e., Z={z1,z2,…,zr}).

Second step: Consider an m-PSVNS ψ is another unknown watch, which is to be recognized.

Third step: Convert the given m-PSVNS data into m-PFS by using the following formulas

pk∘αTe′(zr)=pk∘αTe(zr)+pk∘βTe(zr)+pk∘γTe(zr)3(7)

and

pk∘αψ″(zr)=pk∘αψ(zr)+pk∘βψ(zr)+pk∘γψ(zr)3(8)

Fourth step: Calculate d1,d2,d3, and d4 distance between Te and ψ.

Fifth step: Calculate SM(Te,ψ) between Te and ψ by using the following formula

SM(Te,ψ)=11+d(Te,ψ)(9)

Sixth step: Evaluate the result by choosing the Te, which has the greatest SM with unknown ψ.

In the following example, we will explain and apply the above six steps of Algorithm 1.

Example 9.

Suppose that there are four brands of watches (i.e., T1,T2,T3, and T4) and let Z={z1,z2,z3,z4}, where z1 is “Material,” z2 is “Glass kind,” z3 is “Water Resistance,” and z4 is “Beautiful Finishing” be feature space of watches. The data of 2-PSVNSs of four brands in the first step of the Algorithm 1 are given in Table 1:

Brands	z₁	z₂	z₃	z₄
T₁	(〈0.4, 0.5, 0.7〉, 〈0.7, 0.2, 0.8〉)	(〈0.2, 0.6, 0.4〉, 〈0.3, 0.1, 0.8〉)	(〈0.8, 0.2, 0.7〉, 〈0.5, 0.5, 0.6〉)	(〈0.3, 0.2, 0.7〉, 〈0.4, 0.1, 0.8〉)
T₂	(〈0.3, 0.1, 0.7〉, 〈0.5, 0.3, 0.6〉)	(〈0.7, 0.1, 0.6〉, 〈0.4, 0.2, 0.9〉)	(〈0.9, 0.1, 0.3〉, 〈0.6, 0.1, 0.4〉)	(〈0.4, 0.3, 0.5〉, 〈0.2, 0.2, 0.7〉)
T₃	(〈0.6, 0.1, 0.5〉, 〈0.4, 0.2, 0.6〉)	(〈0.5, 0.2, 0.6〉, 〈0.4, 0.1, 0.8〉)	(〈0.6, 0.2, 0.7〉, 〈0.5, 0.3, 0.6〉)	(〈0.4, 0.1, 0.9〉, 〈0.4, 0.2, 0.8〉)
T₄	(〈0.2, 0.6, 0.7〉, 〈0.1, 0.4, 0.6〉)	(〈0.3, 0.4, 0.5〉, 〈0.4, 0.6, 0.5〉)	(〈0.2, 0.4, 0.6〉, 〈0.1, 0.5, 0.5〉)	(〈0.7, 0.2, 0.4〉, 〈0.2, 0.1, 0.7〉)

Table 1

Data of 2-PSVNSs of four brands.

Also, the 2-PSVNS of unknown watch in the second step of the Algorithm 1 is given by

ψ={(〈0.6,0.3,0.4〉,〈0.4,0.2,0.5〉)z1,(〈0.1,0.7,0.8〉,〈0.4,0.3,0.6〉)z2,(〈0.1,0.3,0.9〉,〈0.2,0.4,0.8〉)z3,(〈0.5,0.1,0.5〉,〈0.3,0.2,0.6〉)z4}.

By the third step of the Algorithm 1 and by Equation (7), we convert 2-PSVNSs of four brands (i.e., T1,T2,T3, and T4) into 2-PFSs as shown in the following Table 2:

Brands	z₁	z₂	z₃	z₄
T₁	(0.53, 0.56)	(0.4, 0.4)	(0.56, 0.53)	(0.4, 0.43)
T₂	(0.36, 0.46)	(0.46, 0.5)	(0.43, 0.36)	(0.4, 0.36)
T₃	(0.4, 0.4)	(0.43, 0.43)	(0.5, 0.46)	(0.46, 0.46)
T₄	(0.5, 0.36)	(0.4, 0.5)	(0.4, 0.36)	(0.43, 0.33)

Table 2

2-PFSs of four brands.

By Equation (8) (i.e., the 2-PSVNS ψ of unknown watch), we obtain 2-PFSs as follows:

ψ={(0.43,0.36)z1,(0.53,0.43)z2,(0.43,0.46)z3,(0.36,0.36)z4}.

Then, by Equation (3) (i.e., Euclidean distance), we compute the Euclidean distance measure of Te(e=1,2,3,4) and ψ in the fourth step of the Algorithm 1 as follows:

d3(T1,ψ)=0.145,d3(T2,ψ)=0.039,d3(T3,ψ)=0.083,d3(T4,ψ)=0.058.

By Equation (9) (i.e., SM), we compute the SM of Te(e=1,2,3,4) and ψ in the fifth step of the Algorithm 1 as follows:

SM(T1,ψ)=11+d3(T1,ψ)=11+0.145=0.873,

SM(T2,ψ)=11+d3(T2,ψ)=11+0.039=0.962,

SM(T3,ψ)=11+d3(T3,ψ)=11+0.083=0.923,

SM(T4,ψ)=11+d3(T4,ψ)=11+0.058=0.945.

Finally, although for every brand SM is greater than 0.5, SM of T2 is the highest. Hence the unknown watch ψ is closest in similarity to brand T2.

5. A NOVEL METHODOLOGY TO EXTEND THE TOPSIS TO m-PSVNSs

We construct a new methodology to extend the TOPSIS to m-PSVNSs (i.e., this process is very applicable to deal with the group decision-making problem under m-PSVNS system).

Now, we propose an Algorithm 2 of the multi-decision maker multi-criteria decision-making of m-PSVNSs as follows:

Algorithm 2

Determination of the optimal decision based on m-PSVNSs.

First step: Assume that there is a group of n persons decision-makers to evaluate the ratings of alternatives Zr(r=1,2,…,p) concerning criteria Cs(s=1,2,…,q) in single valued neutrosophic value form where the information about criterion weights is known.

Second step: Create an m-PSVNS decision (i.e., a multi-criteria group decision-making problem) which can be represented in matrix form as

ℳp×q=(ϑ11ϑ12⋯,ϑ1qϑ21ϑ22⋯,ϑ2q⋮⋮⋱⋮ϑp1ϑp2⋯,ϑpq).

where p (i.e., the number of alternatives), q (i.e., the number of criteria), and ϑrs∀r(r=1,2,…,p),∀s(s=1,2,…,q); represents the ratings of r-alternatives concerning the s-criteria in the single valued neutrosophic value. The multiple data of n persons of decision-maker group for rating ϑrs can be expressed as

ϑrs=(pk∘αrs,pk∘βrs,pk∘γrs)(k=1,2,..,m).

Third step: Convert the m-PSVNS value data of alternatives into m-PF n-single-value data by using the following formula

pk∘γrs=pk∘αrs+pk∘βrs+pk∘γrs2(10)

and we obtain rating of alternatives as

γ^rs=(p1∘γrs,p2∘γrs,…,pm∘γrs)(k=1,2,…,m)

and the weights of criterion is expressed in information as

w^s=(p1∘ws,p2∘ws,…,pm∘ws)(k=1,2,…,m;s=1,2,..,q).

Fourth step: By normalization (i.e., the normalization process is to preserve the property of m-PSVNS that the ranges of membership of elements is [0, 1]) we get the normalized m-PSVNS to m-PF decision matrix (i.e., fuzzy numbers belong to [0, 1]), denoted by E^, is defined by

E^=(ℰrs)p×q,

where

ℰrs=(pk∘ℰ^rs=pk∘γrs∑r=1p(pk∘γrs)2) (k=1,2,…,m;s=1,2,‥,q)(11)

Fifth step: Construct the weighted normalized m-PSVNS decision matrix, denoted by F^, is defined by

F^=(frs)p×q,

such that

frs=ℰrs(.)w^s=(pk°ℰrs(.)pk°w^s) (k=1,2,…,m;s=1,2,‥,q)(12)

where the elements frs(r=1,2,…,p;s=1,2,…,q) are normalized the given m values to belong their ranges to from closed interval [0, 1].

Sixth step: Determine the m-fuzzy positive ideal solution (denoted by A∗) and m-fuzzy negative ideal solution (denoted by A−), are defined by

A∗=(f1∗,f2∗,…,fq∗) and A−=(f1−,f2−,…,fq−),

where

fs∗=pk∘f^s∗={(1,1,…,1),s∈ℬ(0,0,…,0),s∈C,fs−=pk∘f^s−={(0,0,…,0),s∈ℬ(1,1,…,1),s∈C

for all k=1,2,…,m;s=1,2,…,q and ℬ denotes the benefit criteria and C denotes the cost criteria.

Seventh step: Compute the separation measure (i.e., Euclidean distance) of each alternative from A∗ and A−, respectively, as follows:

Sr∗=d(frs,fs∗)=1m{∑k=1m∑s=1q(pk∘f^rs−pk∘f^s∗)2}(13)

and

Sr−=d(frs,fs−)=1m{∑k=1m∑s=1q(pk∘f^rs−pk∘f^s−)2}(14)

Eighth step: Compute the closeness coefficient (denoted by, Cfr¯) of each alternative Zr(r=1,2,…,p) (i.e., as alternative Zr is near to A∗ and far from A− as closeness coefficient Cfr¯ go closer to 1), is defined by

Cfr¯=Sr−Sr−+Sr∗(15)

Ninth step: According to the Cfr¯, we can give the rank to all alternatives and select the best one of them that nearer to 1.

In the following example, we will explain and apply the above nine steps of Algorithm 2.

Example 10.

Let us consider an investment company, which wants to invest some money in the best alternative. The company forms a committee of three members for the selection of best alternative (i.e., Zr(r=1,2,3,4)) among a panel of four possible alternatives to invest the money, where Z1 is “a medicine company,” Z2 is “an electronics company,” Z3 is “a food company,” and Z4 is “a telecommunication company.” The committee shall make a decision concerning the following four criteria (i.e., Cs(s=1,2,3,4)): C1 is “the environmental impact analysis,” C2 is “the risk analysis,” C3 is “the social-political impact analysis,” and C4 is “the growth analysis.” The data of 3-PSVN decision matrix (in the range (0,10)) and 3-polar weights by the committee of three decision-makers pk(k=1,2,3) evaluate all four possible alternatives (i.e, Z1,Z2,Z3,Z4) in the first and the second steps of Algorithm 2 are given by

M=(C1C2C3C4Z1((3,4,5),(2,4,5),(2,3,4))((1,1,1),(4,5,4),(2,4,5))((1,1,1),(2,5,2),(3,2,6))((2,4,6),(4,4,1),(3,4,4))Z2((2,1,5),(3,2,7),(1,1,3))((1,2,2),(3,5,6),(1,3,5))((4,6,6),(2,1,3),(1,5,5))((2,4,6),(1,3,3),(1,3,5))Z3((1,2,2),(1,3,5),(2,3,6))((2,4,4),(2,2,7),(1,2,4))((3,4,4),(2,4,4),(5,4,5))((1,3,3),(2,2,4),(1,3,4))Z4((1,3,4),(2,2,4),(4,4,4))((2,2,2),(2,3,3),(3,4,6))((1,5,5),(3,3,6),(3,6,7))((2,2,4),(1,3,3),(3,4,4))w^(0.35,0.26,0.38)(0.42,0.31,0.25)(0.26,0.45,0.15)(0.20,0.36,0.24)).

By Equation (10) in the third step of Algorithm 2, we obtain the following 3-polar decision matrix as follows:

M′=(C1C2C3C4Z1(6,5.5,4.5)(2.5,6.5,5.5)(2.5,4.5,5.5)(6,4.5,5.5)Z2(4,6,2.5)(2.5,7,4.5)(8,3,5.5)(6,3.5,4.5)Z3(2.5,4.5,5.5)(5,5.5,3.5)(5.5,5,7)(3.5,4,4)Z4(4,4,6)(3,4,6.5)(5.5,6,8)(4,3.5,5.5)).

By Equation (11) in the fourth step of Algorithm 2, we normalized the 3-polar decision matrix to get the normalized 3-PF decision matrix as follows:

M″=(C1C2C3C4Z1(0.69,0.54,0.47)(0.36,0.55,0.53)(0.21,0.47,0.41)(0.59,0.57,0.55)Z2(0.46,0.59,0.26)(0.36,0.59,0.43)(0.69,0.31,0.41)(0.59,0.44,0.45)Z3(0.29,0.44,0.57)(0.73,0.46,0.34)(0.48,0.32,0.53)(0.34,0.51,0.40)Z4(0.46,0.39,0.62)(0.43,0.34,0.63)(0.48,0.63,0.60)(0.39,0.44,0.55)).

By Equation (12) in the fifth step of the Algorithm 2, we obtain the weighted normalized 3-PSVNS decision matrix as follows:

M″′=(C1C2C3C4Z1(0.24,0.14,0.17)(0.15,0.17,0.13)(0.05,0.21,0.06)(0.11,0.20,0.14)Z2(0.16,0.15,0.09)(0.15,0.18,0.10)(0.17,0.13,0.06)(0.11,0.15,0.10)Z3(0.10,0.11,0.21)(0.30,0.14,0.08)(0.12,0.14,0.07)(0.06,0.18,0.09)Z4(0.16,0.10,0.23)(0.12,0.10,0.15)(0.12,0.28,0.09)(0.07,0.15,0.13)).

As C1 and C2 (resp., C3 and C4) are cost criteria (resp., are benefit criteria). Then the 3-fuzzy positive ideal solution A∗ and 3-fuzzy negative ideal solution A− in sixth step of the Algorithm 2 are given as follows:

A∗=((0,0,0),(0,0,0),(1,1,1),(1,1,1))

A−=((1,1,1),(1,1,1),(0,0,0),(0,0,0)).

In the seventh step of the Algorithm 2, by computing separation measure (Sr∗,Sr−) between each attribute with A∗ and A− by using Equations (13) and (14), respectively, we get the following results as follows (r=1,2,3,4):

S1∗(Z1,A∗)=1.25,S2∗(Z2,A∗)=1.26,S2∗(Z3,A∗)=1.28,S2∗(Z4,A∗)=1.23

and

S1−(Z1,A−)=1.19,S2−(Z2,A−)=1.23,S2−(Z3,A−)=1.20,S2−(Z4,A−)=1.23.

In the eighth step of the Algorithm 2, by computing closeness coefficient Cfr¯ of each alternative Zr(r=1,2,3,4) and by Equation (15), we get the following results as follows:

Cfr¯(Z1)=0.48,Cfr¯(Z2)=0.49,Cfr¯(Z3)=0.48,Cfr¯(Z4)=0.5.

Finally, as Cfr¯(Z4)=0.5 is the highest among others, Z4 is the best ideal solution for this multi-person TOPSIS method. Consequently, the outcome of the decision-making committee of the company is to invest money in a medicine company to obtain the best return of investment.

6. CONCLUSION

We presented the concept of the m-PSVNS as a new m-PFS model. We studied the several structure operations of the m-PSVNS and also discussed the basic properties of the m-PSVNS. Then, we introduced novel concepts of distances measure and SM for m-PSVNSs. Further, we constructed an Algorithm 1 to solve brand recognition problem by applying the m-PSVNS. Moreover, a new approach for the best m-PSVNS alternatives based on distance SMs in MCDM is animated. Finally, a new methodology to extend the TOPSIS to m-PSVNSs is proposed and its applicability was illustrated through a numerical example. In future, an m-PSVNS will definitely open new ways to apply with or without restriction existing results of m-F soft set, m-F soft rough set. Also, by combining the m-PFS and the other sets (e.g., interval-valued FSs [34], picture FSs [35], and spherical FSs [36]) we can extend our work to obtain more novel m-PFS models.

CONFLICTS OF INTEREST

The authors declare no conflicts of interest.

AUTHORS' CONTRIBUTIONS

Author 1, author 5 and author 6 sourced the funding; author 2 drafted the manuscript; author 3 revised the manuscript; author 4 supervised the work.

Funding Statement

The work is the introduced talent foundation of Guizhou University (No. 201504, 201811) and the natural science foundation of Guizhou Province (QKH[2019]1123; QKHKY[2021]088).

ACKNOWLEDGMENTS

Authors thank the editors and the anonymous reviewers for their insightful comments which improved the quality of the paper.

REFERENCES

1.L.A. Zadeh, Fuzzy sets, Inf. Control, Vol. 8, 1965, pp. 338-353.

2.F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, ProQuest Information & Learning, Ann Arbor, MI, USA, 2007, pp. 158. http://fs.unm.edu/eBook-Neutrosophics6.pdf

3.F. Smarandache, Neutrosophic set is a generalization of intuitionistic fuzzy set, inconsistent intuitionistic fuzzy set (picture fuzzy set, ternary fuzzy set), pythagorean fuzzy set, spherical fuzzy set, and q-rung orthopair fuzzy set, while neutrosophication is a generalization of regret theory, grey system theory, and three-ways decision (revisited), J. New Theory, Vol. 29, 2019, pp. 1-31. https://dergipark.org.tr/en/pub/jnt/issue/51172/666629

4.H. Wang, F. Smarandache, Y. Zhang, and R. Sunderraman, Single valued neutrosophic sets, Multispace and Multistructure, Vol. 4, 2010, pp. 410-413. https://vixra.org/pdf/1004.0113v1.pdf

5.W.R. Zhang, Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, in NAFIPS/IFIS/NASA-94 Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, The Industrial Fuzzy Control and Intellige (San Antonio, TX, USA), 1994, pp. 305-309.

6.J. Chen, S. Li, S. Ma, and X. Wang, m-polar fuzzy sets: an extension of bipolar fuzzy sets, Sci. World J., Vol. 2014, 2014, pp. 416530.

7.M. Akram and M. Sarwar, New applications of m-polar fuzzy competition graphs, New Math. Nat. Comput., Vol. 14, 2018, pp. 249-276.

8.M. Akram and A. Adeel, m-polar fuzzy graphs and m-polar fuzzy line graphs, J. Discret. Math. Sci. Cryptogr., Vol. 20, 2017, pp. 1597-1617.

9.M. Akram, N. Waseem, and B. Davvaz, Certain types of domination in m-polar fuzzy graphs, J. Mult.-Valued Logic Soft Comput., Vol. 29, 2017, pp. 619-646. http://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-29-number-6-2017/mvlsc-29-6-p-619-646/

10.M. Sarwar and M. Akram, Representation of graphs using m-polar fuzzy environment, Ital. J. Pure Appl. Math., Vol. 38, 2017, pp. 291-312. https://ijpam.uniud.it/online_issue/201738/28-Akram-Sarwar.pdf

11.M. Akram, G. Ali, and N.O. Alshehri, A new multi-attribute decision-making method based on m-polar fuzzy soft rough sets, Symmetry, Vol. 9, 2017, pp. 271.

12.M. Akram, N. Waseem, and P. Liu, Novel approach in decision making with m-polar fuzzy ELECTRE-I, Int. J. Fuzzy Syst., Vol. 21, 2019, pp. 1117-1129.

13.A. Adeel, M. Akram, I. Ahmed, and K. Nazar, Novel m-polar fuzzy linguistic ELECTRE-I method for group decision-making, Symmetry, Vol. 11, 2019, pp. 471.

14.S.M. Chen, M.S. Yeh, and P.Y. Hsiao, A comparison of similarity measures of fuzzy values, Fuzzy Sets Syst., Vol. 72, 1995, pp. 79-89.

15.S.M. Chen, A comparison of similarity measures of vague sets, Fuzzy Sets Syst., Vol. 74, 1995, pp. 217-223.

16.D.H. Hong and C. Kim, A note on similarity measures between vague sets and between elements, Inf. Sci., Vol. 115, 1999, pp. 83-96.

17.J. Ye, Clustering methods using distance-based similarity measures of single-valued neutrosophic sets, J. Intell. Syst., Vol. 23, 2014, pp. 379-389.

18.L.H. Han, New distance measure of single-valued neutrosophic sets and its application, Int. J. Intell. Syst., Vol. 31, 2016, pp. 1021-1032.

19.X. Peng and J. Dai, Approaches to single-valued neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function, Neural Comput. Appl., Vol. 29, 2018, pp. 939-954.

20.H. Garg and Nancy, Some new biparametric distance measures on single-valued neutrosophic sets with applications to Pattern recognition and medical diagnosis, Information., Vol. 8, 2017, pp. 162.

21.M. Akram and N. Waseem, Similarity measures for new hybrid models: mF sets and mF soft sets, Punjab Univ. J. Math., Vol. 51, 2019, pp. 115-130. http://pu.edu.pk/images/journal/maths/PDF/Paper-8_51_6_2019.pdf

22.D. Yong, Plant location selection based on fuzzy TOPSIS, Int. J. Adv. Manuf. Tech., Vol. 28, 2006, pp. 839-844.

23.A. Adeel, M. Akram, and A.N.A. Koam, Group decision-making based on m-polar fuzzy linguistic TOPSIS method, Symmetry., Vol. 11, 2019, pp. 735.

24.H.S. Shih, H.J. Shyur, and E.S. Lee, An extension of TOPSIS for group decision making, Math. Comput. Model., Vol. 45, 2007, pp. 801-813.

25.M. Saeed, M. Saqlain, M. Asad, N. Khushbakht, and Y. Sonia, Multi-polar neutrosophic soft sets with application in medical diagnosis and decision-making, Neutros. Sets Syst., Vol. 33, 2020, pp. 183-207.

26.K.T. Kang, S.-Z. Song, and Y.B. Jun, Multipolar intuitionistic fuzzy set with finite degree and its application in BCK/BCI-algebras, Mathematics., Vol. 8, 2020, pp. 177.

27.M. Akram, Infosys Science Foundation Series in Mathematical Sciences, Springer, Singapore, 2018.

28.M. Akram, m-Polar Fuzzy Graphs, Studies in Fuzziness and Soft Computing, Springer, Cham, Switzerland, Vol. 371, 2019.

29.M. Akram, S. Alsulami, A. Khan, and F. Karaaslan, Multi-criteria group decision-making using spherical fuzzy prioritized weighted aggregation operators, Int. J. Comput. Intell. Syst., Vol. 13, 2020, pp. 1429-1446.

30.Y. Liu, J. Liu, and Z. Hong, A multiple attribute decision making approach based on new similarity measures of interval-valued hesitant fuzzy sets, Int. J. Comput. Intell. Syst., Vol. 11, 2018, pp. 15-32.

31.Y. Ou, L. Yi, B. Zou, and Z. Pei, The linguistic intuitionistic fuzzy set TOPSIS method for linguistic multi-criteria decision makings, Int. J. Comput. Intell. Syst., Vol. 11, 2018, pp. 120-132.

32.H. Liu, L. Jiang, and Z. Xu, Entropy measures of probabilistic linguistic term sets, Int. J. Comput. Intell. Syst., Vol. 11, 2018, pp. 45-57.

33.M. Cai, Y. Wang, Z. Gong, and G. Wei, A novel comparative linguistic distance measure based on hesitant fuzzy linguistic term sets and its application in group decision-making, Int. J. Comput. Intell. Syst., Vol. 12, 2018, pp. 227-237.

34.M.B. Gorzałzany, A method of inference in approximate reasoning based on interval valued fuzzy sets, Fuzzy Sets Syst., Vol. 21, 1987, pp. 1-17.

35.B.C. Cuong, Picture fuzzy sets, J. Comput. Sci. Cybern., Vol. 30, 2014, pp. 409-420.

36.S. Ashraf, S. Abdullah, T. Mahmood, F. Ghani, and T. Mahmood, Spherical fuzzy sets and their applications in multi-attribute decision making problems, J. Intell. Fuzzy Syst., Vol. 36, 2019, pp. 2829-2844.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 869 - 885
Publication Date: 2021/02/19
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.210203.003 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Juanyong Wu
AU  - Ahmed Mostafa Khalil
AU  - Nasruddin Hassan
AU  - Florentin Smarandache
AU  - A. A. Azzam
AU  - Hui Yang
PY  - 2021
DA  - 2021/02/19
TI  - Similarity Measures and Multi-person TOPSIS Method Using m-polar Single-Valued Neutrosophic Sets
JO  - International Journal of Computational Intelligence Systems
SP  - 869
EP  - 885
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210203.003
DO  - 10.2991/ijcis.d.210203.003
ID  - Wu2021
ER  -

download .riscopy to clipboard