Novel similarity measures and multi-expert TOPSIS method using picture m-polar fuzzy sets

Abstract

In this paper, we give a new notion of the picture m-polar fuzzy sets (Pm-PFSs) (i.e, combination between the picture fuzzy sets (PFSs) and the m-polar fuzzy sets (m-PFSs)) and study several of the structure operations including subset, equal, union, intersection, and complement. After that, the basic definitions, theorems, and examples on Pm-PFSs are explained. Also, the certain distance between two Pm-PFSs and a novel similarity measure for Pm-PFSs based on distances are defined. MCDM is animated for Pm-PFS data that take into account the distances for the best alternative (solution) by proposed an application of similarity measure for Pm-PFSs in decision-making. Finally, we construct a new methodology to extend the TOPSIS to Pm-PFS in which capable of different objects recognizing belonging to the same family and illustrate its applicability via a numerical example.

Keywords

Picture m-polar fuzzy set distance measure similarity measure decision-making multi-expert TOPSIS technique

1 Introduction

Zadeh [1] defines fuzzy set (FS) as the generalization of crisp set theory and highlights 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. Cuong [2] defined an PFS by separately assigning (i.e., positive-membership, neutral-membership, and negative-membership functions). Zhang [3] proposed the concept of bipolar fuzzy sets (BFSs) and relations. Chen et al. [4] presented the concept of m-PFSs as a simplified type of BFS. The notions of m-PF graph, characterization of m-PF graph, and metrics in m-PF graph are introduced in [5 –8]. A multi-criteria decision-making method on m-PF soft rough sets, hybrid m-PF models with m-PF ELECTRE-I are presented in [9 –11].

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) discussed in [12, 13], but it had failed to hold in some problems. To resolve this problem, Hong and Kim [14] brought into light some modified measures. Several SM based on PFSs are proposed in [15 –18]. Akram and Waseem [19] implemented a m-PFS and m-PF soft set on SMs by application of medical diagnosis. Yong [20] 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. [21] animated m-PF through the extension of order preference also MCGDM-TOPSIS for the best alternative. Shih et al. [22] described the technique of order preference through similarity using MADM in a group decision environment based on group preference for TOPSIS. Saeed et al. [23] gave an application in SM on multipolar neutrosophic soft sets structure of in medical diagnosis and also in decision-making.

In normal practice, the problem of PF information occurs (i.e., which cannot be elaborated well using the existing approaches), while an Pm-PFS is used to resolve the uncertain and more variant data, specifically in the PFS 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 Pm-PFS. It increases the number of applications in various fields, including electronic optimization, industries, and forensic facial portrait. Moreover, a new approach for the best Pm-PFS alternatives based on distance similarity measures 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 Pm-PFS and its basic operations are defined. In Section 4, distance measure formulas are presented on Pm-PFS. Further, the Pm-PFS is used to investigate a problem involving distance-based similarity measures with an algorithm. In Section 5, the MCDM for Pm-PFS 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 PFSs as indicated below.

2.1 BFSs and m-PFSs

Definition 2.1 (cf. [3]). Assume that Z (i.e., Z = {z₁, z₂, . . . , z_r|r = 1, 2, . . . , n} , n ∈ N (Natural Numbers)) be a set of elements. A BFS on Z is a pair (α, β) , where $α : Z \to [0, 1] and β : Z \to [- 1, 0]$ are two mappings. We note that BFSs are an extension of FSs whose membership degree range is [-1,1].

Definition 2.2 (cf. [4]). An m-PFS on Z (i.e., Z = {z₁, z₂, . . . , z_r}) be a set of elements, is a mapping $α : Z \to [0, 1]^{m} .$

Example 2.3 Suppose that a 3-picture fuzzy set (briefly, 3-PFS) on Z (i.e., Z = {z₁, z₂, z₃}) is defined by $α (z_{1}) = 〈 0.45, 0.42, 0.59 〉, α (z_{2}) = 〈 0.51, 0.52, 0.7 〉,$ and $α (z_{3}) = 〈 0.4, 0.53, 0.9 〉 .$

2.2 PFSs

Definition 2.4 (cf. [2]). A PFS Φ = (α, β, γ) on Z (i.e., Z = {z₁, z₂, . . . , z_r}) be a set of elements, is a mapping $(α, β, γ) : Z ⟶ [0, 1] \times [0, 1] \times [0, 1],$ where α : Z ⟶ [0, 1] (i.e., the degree of positive-membership), β : Z ⟶ [0, 1] (i.e., the degree of neutral-membership), and γ : Z ⟶ [0, 1] (i.e., the degree of negative-membership) are FSs over Z such that α (z_r) + β (z_r) + γ_Φ (z_r) ≤1 (∀ z_r ∈ Z, r = 1, 2, . . . , n)

or $Φ = {(α_{Φ} (z_{r}), β_{Φ} (z_{r}), γ_{Φ} (z_{r})) / z_{r} | z_{r} \in Z} .$

Definition 2.5 (cf. [2]). Let $Φ = {(α_{Φ} (z_{r}), β_{Φ} (z_{r}), γ_{Φ} (z_{r})) / z_{r} | z_{r} \in Z}$ and $Ψ = {(ξ_{Ψ} (z_{r}), η_{Ψ} (z_{r}), ɛ_{Ψ} (z_{r})) / z_{r} | z_{r} \in Z}$ are PFSs. The following five operations are defined by

(1) (Complement) Φ^c = {(γ_Φ (z_r) , β_Φ (z_r) , α_Φ (z_r))/z_r | z_r ∈ Z} .

(2) (Inclusion) Φ ⊆ Ψ ⇔ α_Φ (z_r) ≤ ξ_Ψ (z_r) , β_Φ (z_r) ≤ η_Ψ (z_r) , and γ_Φ (z_r) ≥ ɛ_Ψ (z_r) (∀ z_r ∈ Z) .

(3) (Equal) Φ = Ψ ⇔ Φ ⊆ Ψ and Ψ ⊆ Φ .

(4) (Union) Φ ∪ Ψ = {(α_Φ (z_r) ∨ ξ_Ψ (z_r) , β_Φ (z_r) ∧ η_Ψ (z_r) , γ_Φ (z_r) ∧ ɛ_Ψ (z_r))/z_r | z_r ∈ Z} .

(5) (Intersection) Φ ∩ Ψ = {(α_Φ (z_r) ∧ ξ_Ψ (z_r) , β_Φ (z_r) ∧ η_Ψ (z_r) , γ_Φ (z_r) ∨ ɛ_Ψ (z_r))/z_r | z_r ∈ Z} .

3 Pm-PFSs

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

Definition 3.1. A Pm-PFS Φ = (α, β, γ) on Z (i.e., Z = {z₁, z₂, . . . , z_r}) be a set of elements, is a mapping $(α, β, γ) : Z ⟶ [0, 1]^{m} \times [0, 1]^{m} \times [0, 1]^{m},$ where α : Z ⟶ [0, 1] ^m, β : Z ⟶ [0, 1] ^m, and γ : Z ⟶ [0, 1] ^m are m-PFSs over Z such that α (z_r) + β (z_r) + γ (z_r) ≤ 1 (∀ z_r ∈ Z, r = 1, 2, . . . , n) , that is, $p_{k} \circ α (z_{r}) + p_{k} \circ β (z_{r}) + p_{k} \circ γ (z_{r}) \leq 1 (\forall z_{r} \in Z, k = 1, 2, . . ., m)$ or

Φ = {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r |z_r ∈ Z,

k = 1, 2, . . . , m} .

Example 3.2. Suppose that a picture 3-polar fuzzy set (briefly, P3-PFS) on three of elements Z = {z₁, z₂, z₃} is defined by

Φ = {(0.6, 0.3, 0.1) , (0.4, 0.2, 0.2) ∖ z₁, (0.1, 0.7, 0.2) ,

(0.4, 0.3, 0.1) ∖ z₂, (0.1, 0.3, 0.5) , (0.2, 0.4, 0.3) ∖ z₃} .

Definition 3.3. Assume that Φ and Ψ are Pm-PFSs on a set of elements Z, where

Φ = {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r |z_r ∈ Z,

k = 1, 2, . . . , m}

and

Ψ = {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) , p_k ∘ ɛ_Ψ (z_r)) ∖ z_r |z_r ∈ Z,

k = 1, 2, . . . , m} .

Then, Φ ⊆ Ψ (i.e., Φ is a Pm-PF subset of Ψ) if for all z_r ∈ Z, p_k ∘ α_Φ (z_r) ≤ p_k ∘ ξ_Ψ (z_r), p_k ∘ β_Φ (z_r) ≤ p_k ∘ η_Ψ (z_r) and p_k ∘ γ_Φ (z_r) ≥ p_k ∘ ɛ_Ψ (z_r).

Example 3.4. (Continued from Example 3.2). Let Ψ be a P2-PFS on three of elements Z = {z₁, z₂, z₃} is defined by

Ψ = {(0.6, 0.3, 0.1) , (0.5, 0.3, 0.2) ∖ z₁, (0.2, 0.7, 0.1) ,

(0.5, 0.4, 0.1) ∖ z₂, (0.6, 0.3, 0.1) , (0.2, 0.4, 0.2) ∖ z₃} .

Hence, Φ ⊆ Ψ (∀ z_r ∈ Z, r = 1, 2, 3) .

Definition 3.5. Assume that Φ and Ψ are Pm-PFSs on a set of elements Z, where

Φ = {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r| z_r ∈ Z,

k = 1, 2, . . . , m}

and

Ψ = {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) , p_k ∘ ɛ_Ψ (z_r)) ∖ z_p| z_r ∈ Z,

k = 1, 2, . . . , m} .

Then, Φ = Ψ (i.e., Φ is a Pm-PF equal of Ψ) if Φ ⊆ Ψ and Φ ⊇ Ψ.

Definition 3.6. Assume that Φ be a Pm-PFS on a set of elements Z, where

Φ = {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r | z_r ∈ Z,

k = 1, 2, . . . , m} .

Then,

(1) Φ is called a Pm-PF null set (denoted by $\hat{\emptyset}),$ is defined by

$\hat{\emptyset} = {(0, 0, 1) ∖ z_{r} | z_{r} \in Z} .$

(2) Φ is called a Pm-PF absolute set (denoted by $\hat{Z}),$ is defined by

$\hat{Z} = {(1, 0, 0) ∖ z_{r} | z_{r} \in Z} .$

Example 3.7 (Continued from Example 3.2). Then, a P2-PFSs $\hat{\emptyset}$ and $\hat{Z}$ are defined by

$\hat{\emptyset} = {(0, 0, 1), (0, 0, 1) ∖ z_{1}, (0, 0, 1), (0, 0, 1) ∖ z_{2},$

(0, 0, 1) , (0, 0, 1) ∖ z₃}

and

$\hat{Z} = {(1, 0, 0), (1, 0, 0) ∖ z_{1}, (1, 0, 0), (1, 0, 0) ∖ z_{2},$

(1, 0, 0) , (1, 0, 0) ∖ z₃} .

Definition 3.8. Assume that Φ and Ψ are Pm-PFSs on a set of elements Z, where

Φ = {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r | z_r ∈ Z,

k = 1, 2, . . . , m}

and

Ψ = {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) , p_k ∘ ɛ_Φ (z_r)) ∖ z_r | z_r ∈ Z,

k = 1, 2, . . . , m} .

(1) The intersection Φ ∩ Ψ, is defined as

Φ ∩ Ψ = {(p_k ∘ α_Φ (z_r) ∧ p_k ∘ ξ_Ψ (z_r) , p_k ∘ β_Φ (z_r)

∧p_k ∘ η_Ψ (z_r) , p_k ∘ γ_Φ (z_r) ∨ p_k ∘ ɛ_Ψ (z_r)) ∖ z_r|

z_r ∈ Z, k = 1, 2, . . . , m} .

(2) The union Φ ∪ Ψ, is defined as

Φ ∪ Ψ = {(p_k ∘ α_Φ (z_r) ∨ p_k ∘ ξ_Ψ (z_r) , p_k ∘ β_Φ (z_r)

∧p_k ∘ η_Ψ (z_r) , p_k ∘ γ_Φ (z_r) ∧ p_k ∘ ɛ_Ψ (z_r)) ∖ z_r|

z_r ∈ Z, k = 1, 2, . . . , m} .

Example 3.9 (Continued from Examples 3.2 and 3.4). Then, the intersection and union of a P2-PFSs, respectively, are given by

Φ ∩ Ψ = {(0.6, 0.3, 0.1) , (0.4, 0.2, 0.2) ∖ z₁, (0.1, 0.7, 0.2) ,

(0.4, 0.3, 0.1) ∖ z₂, (0.1, 0.3, 0.5) , (0.2, 0.4, 0.3) ∖ z₃}

and

Φ ∪ Ψ = {(0.6, 0.3, 0.1) , (0.5, 0.2, 0.2) ∖ z₁, (0.2, 0.7, 0.1) ,

(0.5, 0.3, 0.1) ∖ z₂, (0.6, 0.3, 0.1) , (0.2, 0.4, 0.3) ∖ z₃} .

Proposition 3.10 (Domination law of Pm-PFSs). Assume that Φ be Pm-PFS, $\hat{\emptyset}$ be Pm-PF null set, and $\hat{Z}$ be Pm-PF absolute set. Then the followings hold:

(1) $Φ \cup \hat{Z} = \hat{Z},$

(2) $Φ \cap \hat{\emptyset} = \hat{\emptyset} .$

Proof. (1) $Φ \cup \hat{Z}$

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r | z_r ∈ Z,

k = 1, 2, . . . , m} ∪ {(1, 0, 0) ∖ z_r|z_r ∈ Z}

= {(p_k ∘ α_Φ (z_r) ∨1, p_k ∘ β_Φ (z_r) ∧0, p_k ∘ γ_Φ (z_r) ∧0) ∖ z_r

|z_r ∈ Z, k = 1, 2, . . . , m}

= {(1, 0, 0) ∖ z_r|z_r ∈ Z}

$= \hat{Z} .$

(2) $Φ \cap \hat{\emptyset}$

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(0, 0, 1) ∖ z_r|z_r ∈ Z}

= {(p_k ∘ α_Φ (z_r) ∧0, p_k ∘ β_Φ (z_r) ∧0, p_k ∘ γ_Φ (z_r) ∨1) ∖ z_r

|z_r ∈ Z, k = 1, 2, . . . , m}

= {(0, 0, 1) ∖ z_r|z_r ∈ Z}

$= \hat{\emptyset} .$

Proposition 3.11 (Idempotent law of Pm-PFSs). Assume that Φ be Pm-PFS. Then the followings hold:

(1) Φ ∪ Φ = Φ,

(2) Φ ∩ Φ = Φ .

Proof. Φ ∪ Φ

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∪ {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) ,

p_k ∘ γ_Φ (z_r)) ∖ z_r| z_r ∈ Z, k = 1, 2, . . . , m}

= {(p_k ∘ α_Φ (z_r) ∨ p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) ∧ p_k ∘ β_Φ (z_r) ,

p_k ∘ γ_Φ (z_r) ∧ p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r | z_r ∈ Z,

k = 1, 2, . . . , m}

= Φ .

(1) Φ ∩ Φ

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r| z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) ,

p_k ∘ γ_Φ (z_r)) ∖ z_r | z_p ∈ Z, k = 1, 2, . . . , m}

= {(p_k ∘ α_Φ (z_r) ∧ p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) ∧ p_k ∘ β_Φ (z_r) ,

p_k ∘ γ_Φ (z_r) ∨ p_k ∘ γ_Φ (z_r)) ∖ z_r | z_r ∈ Z, k = 1, 2, . . . , m}

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r | z_r ∈ Z,

k = 1, 2, . . . , m}

= Φ .

Proposition 3.12 (Commutative law of Pm-PFSs). Assume that Φ and Ψ are Pm-PFSs. Then the followings hold:

(1) Φ ∩ Ψ = Ψ ∩ Φ,

(2) Φ ∪ Ψ = Ψ ∪ Φ .

Proof. (1) Φ ∩ Ψ

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r| z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) ,

p_k ∘ ɛ_Φ (z_r)) ∖ z_r | z_r ∈ Z, k = 1, 2, . . . , m}

= {((p_k ∘ α_Φ (z_r) ∧ p_k ∘ ξ_Ψ (z_r)) , (p_k ∘ β_Φ (z_r)

∧p_k ∘ η_Ψ (z_r)) , (p_k ∘ γ_Φ (z_r) ∨ p_k ∘ ɛ_Ψ (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {((p_k ∘ ξ_Ψ (z_r) ∧ p_k ∘ α_Φ (z_r)) , (p_k ∘ η_Ψ (z_r)

∧p_k ∘ β_Φ (z_r)) , (p_k ∘ ɛ_Ψ (z_r) ∨ p_k ∘ γ_Φ (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) , p_k ∘ ɛ_Ψ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) ,

p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= Ψ ∩ Φ .

(2) Φ ∪ Ψ

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∪ {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) ,

p_k ∘ ɛ_Ψ (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= {((p_k ∘ α_Φ (z_r) ∨ p_k ∘ ξ_Ψ (z_r)) , (p_k ∘ β_Φ (z_r)

∧p_k ∘ η_Ψ (z_r)) , (p_k ∘ γ_Φ (z_r) ∧ p_k ∘ ɛ_Ψ (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {((p_k ∘ ξ_Ψ (z_r) ∨ p_k ∘ α_Φ (z_r)) , (p_k ∘ η_Ψ (z_r)

∧p_k ∘ β_Φ (z_r)) , (p_k ∘ ɛ_Ψ (z_r) ∧ p_k ∘ γ_Φ (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) , p_k ∘ ɛ_Ψ (z_p)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∪ {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) ,

p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= Ψ ∪ Φ .

Proposition 3.13 (Associative law of Pm-PFSs). Assume that Φ, Ψ, and Ω are Pm-PFSs. Then the followings hold:

(1) (Φ ∩ Ψ) ∩ Ω = Φ ∩ (Ψ ∩ Ω) ,

(2) (Φ ∪ Ψ) ∪ Ω = Φ ∪ (Ψ ∪ Ω) .

Proof. (1) (Φ ∩ Ψ) ∩ Ω

= {((p_k ∘ α_Φ (z_r) ∧ p_k ∘ ξ_Ψ (z_r)) , (p_k ∘ β_Φ (z_r)

∧p_k ∘ η_Ψ (z_r)) , (p_k ∘ γ_Φ (z_r) ∨ p_k ∘ ɛ_Ψ (z_r))) ∖ z_r| z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(p_k ∘ ρ_Ω (z_r) , p_k ∘ λ_Ω (z_r) ,

p_k ∘ δ_Ω (z_r)) ∖ z_r| z_r ∈ Z, k = 1, 2, . . . , m}

= {((p_k ∘ α_Φ (z_r) ∧ p_k ∘ ξ_Ψ (z_r) ∧ p_k ∘ ρ_Ω (z_r)) ,

(p_k ∘ β_Φ (z_r) ∧ p_k ∘ η_Ψ (z_r) ∧ p_k ∘ λ_Ω (z_r)) , (p_k ∘ γ_Φ (z_r)

∨p_k ∘ ɛ_Ψ (z_r) ∨ p_k ∘ δ_Ω (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {((p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r))) ∖ z_r| z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(ξ_Ψ (z_r) ∧ p_k ∘ ρ_Ω (z_r) , η_Ψ (z_p)

∧p_k ∘ λ_Ω (z_r) , ɛ_Ψ (z_r) ∨ p_k ∘ δ_Ω (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= Φ ∩ (Ψ ∩ Ω) .

(2) (Φ ∪ Ψ) ∪ Ω

= {((p_k ∘ α_Φ (z_r) ∨ p_k ∘ ξ_Ψ (z_r)) , (p_k ∘ β_Φ (z_r)

∧p_k ∘ η_Ψ (z_r)) , (p_k ∘ γ_Φ (z_r) ∧ p_k ∘ ɛ_Ψ (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∪ {(p_k ∘ ρ_Ω (z_r) , p_k ∘ λ_Ω (z_r) ,

p_k ∘ δ_Ω (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= {((p_k ∘ α_Φ (z_r) ∨ p_k ∘ ξ_Ψ (z_r) ∨ p_k ∘ ρ_Ω (z_r)) , (p_k ∘ β_Φ (z_r)

∧p_k ∘ η_Ψ (z_r) ∧ p_k ∘ λ_Ω (z_r)) , (p_k ∘ γ_Φ (z_r) ∧ p_k ∘ ɛΨ (z_r)

∧p_k ∘ δ_Ω (z_r))) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= {((p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}∪ {(ξ_Ψ (z_r) ∨ p_k ∘ ρ_Ω (z_r) , η_Ψ (z_r) ∧

p_k ∘ λ_Ω (z_r) , ɛ_Ψ (z_r) ∧ p_k ∘ δ_Ω (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= Φ ∪ (Ψ ∪ Ω) .

Proposition 3.14 (Distributive law of Pm-PFSs). Assume that Φ, Ψ, and Ω are Pm-PFSs. Then the followings hold:

(1) Φ ∩ (Ψ ∪ Ω) = (Φ ∩ Ψ) ∪ (Φ ∩ Ω) ,

(2) Φ ∪ (Ψ ∩ Ω) = (Φ ∪ Ψ) ∩ (Φ ∪ Ω) .

Proof. (1) Φ ∩ (Ψ ∪ Ω)

= {((p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(p_k ∘ ξ_Ψ (z_r) ∨ p_k ∘ ρ_Ω (z_r) ,

p_k ∘ η_Ψ (z_r) ∧ p_k ∘ λ_Ω (z_r) , p_k ∘ ɛ_Ψ (z_r)

∧p_k ∘ δ_Ω (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= {(p_k ∘ α_Φ (z_r) ∧ [p_k ∘ ξ_Ψ (z_r) ∨ p_k ∘ ρ_Ω (z_r)]) ,

(p_k ∘ β_Φ (z_r) ∧ [p_k ∘ η_Ψ (z_r) ∧ p_k ∘ λ_Ω (z_r)]) ,

(p_k ∘ γ_Φ (z_r) ∨ [p_k ∘ ɛ_Ψ (z_r) ∧ p_k ∘ δ_Ω (z_r)]) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {([p_k∘ α_Φ (z_r) ∧ p_k ∘ ξ_Ψ (z_r)] ∨ [p_k ∘ α_Φ (z_r) ∧

p_k ∘ ρ_Ω (z_r)]) , ([p_k ∘ β_Φ (z_r) ∧ p_k ∘ η_Ψ (z_r)] ∧ [p_k ∘ β_Φ (z_r)

∧p_k∘ λ_Ω (z_r)]) , ([p_k ∘ γ_Φ (z_r) ∨ p_k ∘ ɛ_Ψ (z_r)] ∧ [p_k ∘

γ_Φ (z_r) ∨ p_k ∘ δ_Ω (z_r)]) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= (Φ ∩ Ψ) ∪ (Φ ∩ Ω) .

(2) Φ ∪ (Ψ ∩ Ω)

= {((p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r))) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}∪ {(p_k ∘ ξ_Ψ (z_r) ∧ p_k ∘ ρ_Ω (z_r) , p_k ∘

η_Ψ (z_r) ∧ p_k ∘ λ_Ω (z_r) , p_k ∘ ɛ_Ψ (z_r) ∨ p_k ∘ δ_Ω (z_r))

∖z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= {(p_k ∘ α_Φ (z_r) ∨ [p_k ∘ ξ_Ψ (z_r) ∧ p_k ∘ ρ_Ω (z_r)]) ,

(p_k ∘ β_Φ (z_r) ∧ [p_k ∘ η_Ψ (z_r) ∧ p_k ∘ λ_Ω (z_r)]) ,

(p_k ∘ γ_Φ (z_r) ∧ [p_k ∘ ɛ_Ψ (z_r) ∨ p_k ∘ δ_Ω (z_r)]) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {([p_k∘ α_Φ (z_r) ∨ p_k ∘ ξ_Ψ (z_r)] ∧ [p_k ∘ α_Φ (z_r) ∨

p_k∘ ρ_Ω (z_r)]) , ([p_k ∘ β_Φ (z_r) ∧ p_k ∘ η_Ψ (z_r)] ∧ [p_k ∘

β_Φ (z_r)∧ p_k ∘ λ_Ω (z_r)]) , ([p_k ∘ γ_Φ (z_r) ∧ p_k ∘ ɛ_Ψ (z_r)] ∨

[p_k ∘ γ_Φ (z_r) ∧ p_k ∘ δ_Ω (z_r)]) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= (Φ ∪ Ψ) ∩ (Φ ∪ Ω) .

Definition 3.15 Assume that Φ be Pm-PFS on a set of elements Z, where

Φ = {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r| z_r ∈ Z,

k = 1, 2, . . . , m} .

Then, the complement Φ^c of Φ, is defined by

Φ^c = {(p_k ∘ γ_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ α_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} .

Example 3.16 (Continued from Example 3.2). The complement Φ^c of Φ, is calculated by

Φ^c = {((0.1, 0.3, 0.6) , (0.2, 0.2, 0.4)) ∖ z₁, ((0.2, 0.7, 0.1) ,

(0.1, 0.3, 0.4)) ∖ z₂, ((0.5, 0.3, 0.1) , (0.3, 0.4, 0.2)) ∖ z₃} .

Proposition 3.17 (Complementation and double complementation of Pm-PFSs). Assume that Φ be Pm-PFSs, $\hat{\emptyset}$ be Pm-PF null set, and $\hat{Z}$ be Pm-PF absolute set. Then the followings hold:

(1) ${\hat{\emptyset}}^{c} = \hat{Z},$

(2) ${\hat{Z}}^{c} = \hat{\emptyset},$

(3) (Φ^c) ^c = Φ .

Proof. (1) ${\hat{\emptyset}}^{c} = ({(0, 0, 1) ∖ z_{r} | z_{r} \in Z})^{c}$

$= {(1, 0, 0) ∖ z_{r} | z_{r} \in Z} = \hat{Z} .$

(2) ${\hat{Z}}^{c} = ({(1, 0, 0) ∖ z_{r} | z_{r} \in Z})^{c}$

$= {(0, 0, 1) ∖ z_{r} | z_{r} \in Z} = \hat{\emptyset} .$

(3) (Φ^c) ^c = ({(p_k ∘ γ_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ α_Φ (z_r)) ∖ z_r|

z_r ∈ Z, k = 1, 2, . . . , m}) ^c

= {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= Φ .

Remark 3.18. By the following Example 3.19, we will show the equality of $Φ \cup Φ^{c} = \hat{X}$ and $Φ \cap Φ^{c} = \hat{\emptyset}$ does not hold.

Example 3.19 (Continued from Examples 3.2 and 3.16). Then Φ ∪ Φ^c and Φ ∩ Φ^c are calculated by

Φ ∪ Φ^c = {(0.6, 0.3, 0.1) , (0.4, 0.2, 0.2) ∖ z₁, (0.2, 0.7, 0.1) ,

(0.4, 0.3, 0.1) ∖ z₂, (0.5, 0.3, 0.1) , (0.3, 0.4, 0.2) ∖ z₃}

and

Φ ∩ Φ^c = {(0.1, 0.3, 0.6) , (0.2, 0.2, 0.4) ∖ z₁, (0.1, 0.7, 0.2) ,

(0.1, 0.3, 0.4) ∖ z₂, (0.1, 0.3, 0.5) , (0.2, 0.4, 0.3) ∖ z₃} .

This show $Φ \cup Φ^{c} \neq \hat{X}$ and $Φ \cap Φ^{c} \neq \hat{\emptyset}$ .

Proposition 3.20 (De Morgan’s laws of Pm-PFSs). Assume that Φ and Ψ are Pm-PFSs. Then the followings hold:

(1) (Φ ∪ Ψ) ^c = Φ^c ∩ Ψ^c,

(2) (Φ ∩ Ψ) ^c = Φ^c ∪ Ψ^c .

Proof. (1) (Φ ∪ Ψ) ^c

= ({(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}∪ {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) , p_k ∘

ɛ_Ψ (z_r)) ∖ z_r| z_r ∈ Z, k = 1, 2, . . . , m}) ^c

= ({(p_k ∘ α_Φ (z_r) ∨ p_k ∘ ξ_Ψ (z_r) , p_k ∘ β_Φ (z_r) ∧ p_k ∘ η_Ψ (z_r)

, p_k ∘ γ_Φ (z_r) ∧ p_k ∘ ɛ_Ψ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}) ^c

= {(p_k ∘ γ_Φ (z_r) ∧ p_k ∘ ɛ_Ψ (z_r) , p_k ∘ β_Φ (z_r) ∧ p_k ∘ η_Ψ (z_r) ,

p_k ∘ α_Φ (z_r) ∨ p_k ∘ γ_Ψ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {(p_k ∘ γ_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ α_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(p_k ∘ ɛ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) ,

p_k ∘ ξ_Ψ (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= Φ^c ∩ Ψ^c .

(2) (Φ ∩ Ψ) ^c

= ({(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∩ {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) ,

p_k ∘ ɛ_Ψ (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}) ^c

= ({(p_k ∘ α_Φ (z_r) ∧ p_k ∘ ξ_Ψ (z_r) , p_k ∘ β_Φ (z_r) ∧ p_k ∘ η_Ψ (z_r) ,

p_k ∘ γ_Φ (z_r) ∨ p_k ∘ ɛ_Ψ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}) ^c

= {(p_k ∘ γ_Φ (z_r) ∨ p_k ∘ ɛ_Ψ (z_r) , p_k ∘ β_Φ (z_r) ∧ p_k ∘ η_Ψ (z_r) ,

p_k ∘ α_Φ (z_r) ∧ p_k ∘ ξ_Ψ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

= {(p_k ∘ γ_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ α_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} ∪ {(p_k ∘ ɛ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) ,

p_k ∘ γ_Ψ (z_r)) ∖ z_r|z_r ∈ Z, k = 1, 2, . . . , m}

= Φ^c ∪ Ψ^c .

4 Distance measure and similarity measure for Pm-PFSs

We present a new concepts of distances measure and similarity measure for Pm-PFSs as indicated below.

Definition 4.1. Assume that Φ and Ψ are Pm-PFSs on a set of elements Z (i.e., Z = {z₁, z₂, . . . , z_r|r = 1, 2, . . . , n}) , where

Φ = {(p_k ∘ α_Φ (z_r) , p_k ∘ β_Φ (z_r) , p_k ∘ γ_Φ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m}

and

Ψ = {(p_k ∘ ξ_Ψ (z_r) , p_k ∘ η_Ψ (z_r) , p_k ∘ ɛ_Ψ (z_r)) ∖ z_r|z_r ∈ Z,

k = 1, 2, . . . , m} .

Then, the distance measure between Φ and Ψ is defined by

(1) Hamming distance: $d_{1} (Φ, Ψ) = \frac{1}{m} {\sum_{k = 1}^{m} \sum_{r = 1}^{n} | (p_{k} \circ α_{Φ} (z_{r}) + p_{k} \circ β_{Φ} (z_{r}) + p_{k} \circ γ_{Φ} (z_{r})) ∖ 3 - (p_{k} \circ ξ_{Ψ} (z_{r}) + p_{k} \circ η_{Ψ} (z_{r}) + p_{k} \circ ɛ_{Ψ} (z_{r})) ∖ 3 |} (1)$ (2) Normalized Hamming distance: $d_{2} (Φ, Ψ) = \frac{1}{mn} {\sum_{k = 1}^{m} \sum_{r = 1}^{n} | (p_{k} \circ α_{Φ} (z_{r}) + p_{k} \circ β_{Φ} (z_{r}) + p_{k} \circ γ_{Φ} (z_{r})) ∖ 3 - (p_{k} \circ ξ_{Ψ} (z_{r}) + p_{k} \circ η_{Ψ} (z_{r}) + p_{k} \circ ɛ_{Ψ} (z_{r})) ∖ 3 |} (2)$ (3) Euclidean distance: $d_{3} (Φ, Ψ) = \sqrt{\frac{1}{m} {\sum_{k = 1}^{m} \sum_{r = 1}^{n} (p_{k} \circ α_{Φ} (z_{r}) + p_{k} \circ β_{Φ} (z_{r}) + p_{k} \circ γ_{Φ} (z_{r})) ∖ 3 - (p_{k} \circ ξ_{Ψ} (z_{r}) + p_{k} \circ η_{Ψ} (z_{r}) + p_{k} \circ ɛ_{Ψ} (z_{r})) ∖ 3)^{2}}} (3)$ }} (4) Normalized Euclidean distance: $d_{4} (Φ, Ψ) = \sqrt{\frac{1}{mn} {\sum_{k = 1}^{m} \sum_{r = 1}^{n} (p_{k} \circ α_{Φ} (z_{r}) + p_{k} \circ β_{Φ} (z_{r}) + p_{k} \circ γ_{Φ} (z_{r})) ∖ 3 - (p_{k} \circ ξ_{Ψ} (z_{r}) + p_{k} \circ η_{Ψ} (z_{r}) + p_{k} \circ ɛ_{Ψ} (z_{r})) ∖ 3)^{2}}} (4) .$

Proposition 4.2. The distances between Pm-PFSs Φ and Ψ satisfy the following four inequalities.

(1) d₁ (Φ, Ψ) ≤ n,

(2) d₂ (Φ, Ψ) ≤1,

(3) $d_{3} (Φ, Ψ) \leq \sqrt{n},$

(4) d₄ (Φ, Ψ) ≤1 .

Proposition 4.3. The distance functions d₁, d₂, d₃, and d₄ defined from Pm - PF (Z) ⟶ R⁺, are metric distances.

Proof. Suppose that Φ, Ψ, and Ω are three Pm-PFSs on Z . Then

(1) d₁ (Φ, Ψ) ≥0 .

(2) Let d₁ (Φ, Ψ) =0

$\Leftrightarrow \frac{1}{m} {\sum_{k = 1}^{m} \sum_{r = 1}^{n} | (p_{k} \circ α_{Φ} (z_{r}) + p_{k} \circ β_{Φ} (z_{r})$

+p_k ∘ γ_Φ (z_r)) ∖3 - (p_k ∘ ξ_Ψ (z_r) + p_k ∘ η_Ψ (z_r)

+p_k ∘ ɛ_Ψ (z_r)) ∖3|} = 0

⇔| (p_k ∘ α_Φ (z_r) + p_k ∘ β_Φ (z_r) + p_k ∘ γ_Φ (z_r)) ∖3

- (p_k ∘ ξ_Ψ (z_r) + p_k ∘ η_Ψ (z_r) + p_k ∘ ɛ_Ψ (z_r)) ∖3| = 0

⇔ (p_k ∘ α_Φ (z_r) + p_k ∘ β_Φ (z_r) + p_k ∘ γ_Φ (z_r)) ∖3

= (p_k ∘ ξ_Ψ (z_r) + p_k ∘ η_Ψ (z_r) + p_k ∘ ɛ_Ψ (z_r)) ∖3,

1 ≤ k ≤ m, 1 ≤ r ≤ n

⇔Φ = Ψ .

(3) d₁ (Φ, Ψ) = d₁ (Ψ, Φ) .

(4) For any three Pm-PFSs Φ, Ψ, and Ω,

| (p_k ∘ α_Φ (z_r) + p_k ∘ β_Φ (z_r) + p_k ∘ γ_Φ (z_r)) ∖3 - (p_k ∘ ξ_Ψ (z_r)

+p_k ∘ η_Ψ (z_r) + p_k ∘ ɛ_Ψ (z_r)) ∖3|

= | (p_k ∘ α_Φ (z_r) + p_k ∘ β_Φ (z_r) + p_k ∘ γ_Φ (z_r)) ∖3 - (p_k ∘ ρ_Ω (z_r)

+p_k∘ λ_Ω (z_r) + p_k ∘ δ_Ω (z_r)) ∖3 + (p_k ∘ ρ_Ω (z_r) + p_k ∘

λ_Ω (z_r) + p_k ∘ δ_Ω (z_r)) ∖3 - (p_k ∘ ξ_Ψ (z_r) + p_k ∘ η_Ψ (z_r)

+p_k ∘ ɛ_Ψ (z_r)) ∖3|

≤| (p_k ∘ α_Φ (z_r) + p_k ∘ β_Φ (z_r) + p_k ∘ γ_Φ (z_r)) ∖3 - (p_k ∘ ρ_Ω (z_r)

+p_k∘ λ_Ω (z_r) + p_k ∘ δ_Ω (z_r)) ∖3| + | (p_k ∘ ρ_Ω (z_r) + p_k ∘

λ_Ω (z_r) + p_k ∘ δ_Ω (z_r)) ∖3 - (p_k ∘ ξ_Ψ (z_r) + p_k ∘ η_Ψ (z_r)

+p_k ∘ ɛ_Ψ (z_r)) ∖3| .

Hence, d₁ (Φ, Ψ) ≤ d₁ (Φ, Ω) + d₁ (Ω, Ψ) .

Definition 4.4. The SM of two Pm-PFSs Φ and Ψ, defined by $SM (Φ, Ψ) = 1 ∖ (1 + d (Φ, Ψ)) (5)$ where d (Φ, Ψ) is any of the above four distances in Definition 4.1.

Definition 4.5. The two Pm-PFSs Φ and Ψ are σ similar if and only if SM (Φ, Ψ) ≥ σ, i.e., $Φ \approx^{σ} Ψ \Leftrightarrow SM (Φ, Ψ) \geq σ, σ \in (0, 1) (6)$ Φ and Ψ are significantly similar if SM (Φ, Ψ) ≥1 ∖ 2 .

Example 4.6. Suppose that an P2-PFSs on two of elements Z = {z₁, z₂} are defined by

Φ = {(0.4, 0.3, 0.2) , (0.5, 0.2, 0.1) ∖ z₁, (0.6, 0.3, 0.1) ,

(0.5, 0.4, 0.1) ∖ z₂}

and

Ψ = {(0.6, 0.2, 0.1) , (0.1, 0.4, 0.4) ∖ z₁, (0.3, 0.2, 0.4) ,

(0.7, 0.1, 0.1) ∖ z₂} .

Then, the Hamming distance is d₁ (Φ, Ψ) =0.07 and the similarity measure is SM (Φ, Ψ) =0.93 . It shows Φ is significantly similar to Ψ.

Proposition 4.7. The SM of two Pm-PFSs Φ and Ψ satisfy the following three properties:

(1) 0 ≤ SM (Φ, Ψ) ≤1,

(2) SM (Φ, Ψ) = SM (Ψ, Φ) ,

(3) SM (Φ, Ψ) =1 ⇔ Ψ = Φ .

4.1 An application of SM for Pm-PFSs in decision-making

Based on the notion of Euclidean distance (i.e., Equation (3)) by similarity measure, we will use Pm-PFSs information to solve decision-making problem.

In the following, we will build an Algorithm I to solve decision-making problem (i.e., for apply of Pm-PFS in decision-making problem).

Algorithm I Determination how an observer finds much similar to an unknown watch.

First step: Suppose that there are n watches are represented by Pm-PFS T_e (e = 1, 2, . . . , n) in feature space Z (i.e., Z = {z₁, z₂, . . . , z_r}) .

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

Third step: Convert the given Pm-PF data into (m+1)-PFS by using the following formulas (k = 1, 2): $α_{T_{e}}^{'} (z_{r}) = (p_{1} \circ α_{T_{e}} (z_{r}) + p_{2} \circ α_{T_{e}} (z_{r})) ∖ 2$ $β_{T_{e}}^{'} (z_{r}) = (p_{1} \circ β_{T_{e}} (z_{r}) + p_{2} \circ β_{T_{e}} (z_{r})) ∖ 2$ $γ_{T_{e}}^{'} (z_{r}) = (p_{1} \circ γ_{T_{e}} (z_{r}) + p_{2} \circ γ_{T_{e}} (z_{r})) ∖ 2 (7)$ and $α_{ψ}^{'} (z_{r}) = (p_{1} \circ α_{ψ} (z_{r}) + p_{2} \circ α_{ψ} (z_{r})) ∖ 2$ $β_{ψ}^{'} (z_{r}) = (p_{1} \circ β_{ψ} (z_{r}) + p_{2} \circ β_{ψ} (z_{r})) ∖ 2$ $γ_{ψ}^{'} (z_{r}) = (p_{1} \circ γ_{ψ} (z_{r}) + p_{2} \circ γ_{ψ} (z_{r})) ∖ 2 (8) .$ Fourth step: Calculate d₁, d₂, d₃, and d₄ distance between T_e and ψ .

Fifth step: Calculate SM (T_e, ψ) between T_e and ψ by using the following formula $SM (T_{e}, ψ) = 1 ∖ (1 + d (T_{e}, ψ)) (9) .$

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

In the following example, we will explain and apply the above six steps of Algorithm I as:

Example 4.8. 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 P2-PFSs of four brands in first step of the Algorithm I are given in the following Table 1:

Table 1
Data of P2-PFSs of four brands

Brands z ₁ z ₂ z ₃ z ₄

T ₁ (0.4, 0.5,0.1),(0.7, 0.2,0.1) (0.2, 0.6,0.2),(0.3, 0.1,0.3) (0.4, 0.2,0.1),(0.3, 0.3,0.1) (0.3, 0.2,0.4),(0.4, 0.1,0.1)

T ₂ (0.3, 0.1,0.2),(0.5, 0.3,0.1) (0.1,0.7, 0.1),(0.3,0.4, 0.2) (0.2,0.6, 0.1),(0.4,0.3, 0.1) (0.4, 0.3,0.2),(0.2, 0.2,0.4)

T ₃ (0.6, 0.1,0.2),(0.4, 0.2,0.1) (0.5, 0.2,0.1),(0.4, 0.1,0.3) (0.6, 0.2,0.2),(0.5, 0.3,0.2) (0.4, 0.1,0.2),(0.4, 0.2,0.3)

T ₄ (0.2, 0.6,0.2),(0.1, 0.4,0.5) (0.3, 0.4,0.1),(0.4, 0.3,0.2) (0.2, 0.4,0.3),(0.1, 0.5,0.2) (0.7, 0.2,0.1),(0.2, 0.1,0.4)

Brands	z ₁	z ₂	z ₃	z ₄
T ₁	(0.4, 0.5,0.1),(0.7, 0.2,0.1)	(0.2, 0.6,0.2),(0.3, 0.1,0.3)	(0.4, 0.2,0.1),(0.3, 0.3,0.1)	(0.3, 0.2,0.4),(0.4, 0.1,0.1)
T ₂	(0.3, 0.1,0.2),(0.5, 0.3,0.1)	(0.1,0.7, 0.1),(0.3,0.4, 0.2)	(0.2,0.6, 0.1),(0.4,0.3, 0.1)	(0.4, 0.3,0.2),(0.2, 0.2,0.4)
T ₃	(0.6, 0.1,0.2),(0.4, 0.2,0.1)	(0.5, 0.2,0.1),(0.4, 0.1,0.3)	(0.6, 0.2,0.2),(0.5, 0.3,0.2)	(0.4, 0.1,0.2),(0.4, 0.2,0.3)
T ₄	(0.2, 0.6,0.2),(0.1, 0.4,0.5)	(0.3, 0.4,0.1),(0.4, 0.3,0.2)	(0.2, 0.4,0.3),(0.1, 0.5,0.2)	(0.7, 0.2,0.1),(0.2, 0.1,0.4)

Also, the P2-PFS of unknown watch in second step of the Algorithm I is given by

ψ = {(0.2, 0.3, 0.1) , (0.3, 0.2, 0.2) ∖ z₁, (0.1, 0.6, 0.2) ,

(0.4, 0.3, 0.3) ∖ z₂, (0.1, 0.3, 0.3) , (0.2, 0.4, 0.2) ∖ z₃,

(0.5, 0.1, 0.3) , (0.3, 0.2, 0.1) ∖ z₄} .

By third step of the Algorithm I and by Equation (7), we convert P2-PFSs of four brands (i.e., T₁, T₂, T₃, and T₄) into 3-PFSs sets as shown in the following Table 2:

Table 2

3-PFSs of four brands

Brands	z ₁	z ₂	z ₃	z ₄
T ₁	(0.55,0.35,0.1)	(0.25, 0.35,0.25)	(0.35,0.25,0.1)	(0.35,0.15,0.25)
T ₂	(0.4,0.2,0.15)	(0.2, 0.55,0.15)	(0.3,0.45,0.1)	(0.3,0.25,0.3)
T ₃	(0.5,0.15,0.15)	(0.45, 0.15,0.2)	(0.55,0.25,0.2)	(0.4,0.15,0.25)
T ₄	(0.15,0.5,0.35)	(0.35, 0.35,0.15)	(0.15,0.45,0.25)	(0.45,0.15,0.25)

By Equation (8) (i.e., the P2-PFS ψ of unknown watch), we obtain 3-PFSs as follows:

ψ = {(0.25, 0.25, 0.15) ∖ z₁, (0.25, 0.45, 0.25) ∖ z₂,

(0.15, 0.35, 0.25) ∖ z₃, (0.4, 0.15, 0.2) ∖ z₄} .

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

d₃ (T₁, ψ) =0.086, d₃ (T₂, ψ) =0.042, d₃ (T₃, ψ) =0.078, d₃ (T₄, ψ) =0.092 .

By Equation (9) (i.e., SM), we compute the SM of T_e (e = 1, 2, 3, 4) and ψ in fifth step of the Algorithm I as follows: $SM (T_{1}, ψ) = 1 ∖ (1 + d_{3} (T_{1}, ψ)) = 1 ∖ (1 + 0.086) = 0.921,$ $SM (T_{2}, ψ) = 1 ∖ (1 + d_{3} (T_{2}, ψ)) = 1 ∖ (1 + 0.042) = 0.959,$ $SM (T_{3}, ψ) = 1 ∖ (1 + d_{3} (T_{3}, ψ)) = 1 ∖ (1 + 0.078) = 0.928,$ $SM (T_{4}, ψ) = 1 ∖ (1 + d_{3} (T_{4}, ψ)) = 1 ∖ (1 + 0.092) = 0.916 .$ Finally, although for every brand SM is greater than 0.5 but is T₂ much greater than the rest of all, that’s why the unknown watch ψ is too much closer to brand T₂.

5 A novel methodology to extend the TOPSIS to Pm-PFSs

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

Now, we propose an Algorithm II of the multi-decision maker multi-criteria decision-making of Pm-PFSs as follows:

Algorithm II Determination the optimal decision based on picture m-polar fuzzy sets.

First step: Assume that there is a group of the decision-maker of n persons, they evaluate the ratings of alternatives Z_r (r = 1, 2, . . . , p) concerning criteria C_s (s = 1, 2, . . . , q) in picture value form and the information about criterion weights (i.e., the concern weight of each criterion can be found by either directly assign or indirectly using pairwise comparisons) is known.

Second step: Create an Pm-PF decision (i.e., a multi-criteria group decision-making problem) which can be represented in matrix form as $M_{p \times q} = (\begin{matrix} ϑ_{11} & ϑ_{12} & \dots, & ϑ_{1 q} \\ ϑ_{21} & ϑ_{22} & \dots, & ϑ_{2 q} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ϑ_{p 1} & ϑ_{p 2} & \dots, & ϑ_{pq} \end{matrix}) .$ where p (i.e., the number of alternatives), q (i.e., the number of criteria), and ϑ_rs ∀r (r = 1, 2, . . . , p) , ∀ s (s = 1, 2, . . . , q); represents the ratings of r-alternatives concerning the s-criteria in the picture value. The multiple data of n persons of decision-maker group for rating ϑ_rs can be expressed as $ϑ_{rs} = (p_{k} \circ α_{rs}, p_{k} \circ β_{rs}, p_{k} \circ γ_{rs}) (k = 1, 2, . ., m) .$

Third step: Convert the Pm-PF value data of alternatives into m-PFS value data by using the following formula $p_{k} \circ ζ_{rs} = (p_{k} \circ α_{rs} + p_{k} \circ β_{rs} + p_{k} \circ γ_{rs}) ∖ 2 (10)$ and we obtain rating of alternatives as ${\hat{ζ}}_{rs} = (p_{1} \circ ζ_{rs}, p_{2} \circ ζ_{rs}, . . ., p_{m} \circ ζ_{rs}) (k = 1, 2, . . ., m)$ and the weights of criterion is expressed in information as ${\hat{w}}_{s} = (p_{1} \circ w_{s}, p_{2} \circ w_{s}, . . ., p_{m} \circ w_{s}) (k = 1, 2, . . ., m; s = 1, 2, . ., q) .$

Fourth step: By normalization (i.e., the normalization process is to preserve the property of Pm-PFS that the ranges of membership of elements is [0, 1]) we get the normalized Pm-PF to m-PF decision matrix (i.e., fuzzy numbers belong to [0,1]), denoted by $\hat{E},$ is defined as follows (∀ k = 1, 2, . . . , m ; ∀ s = 1, 2, . . , q) : $\hat{E} = (E_{rs})_{p \times q},$ where

$E_{rs} = (p_{k} \circ {\hat{E}}_{rs} = \frac{(p_{k} \circ ζ_{rs})}{\sqrt{\sum_{r = 1}^{p} (p_{k} \circ ζ_{rs})^{2}}}) (11)$

Fifth step: Construct the weighted normalized Pm-PFS decision matrix, denoted by $\hat{F},$ is defined as follows (∀ k = 1, 2, . . . , m ; ∀ s = 1, 2, . . , q): $\hat{F} = (f_{rs})_{p \times q},$ such that $f_{rs} = E_{rs} (.) {\hat{w}}_{s} = (p_{k} \circ E_{rs} (.) p_{k} \circ {\hat{w}}_{s}) (12)$ where the elements f_rs (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 $A^{*} = (f_{1}^{*}, f_{2}^{*}, . . ., f_{q}^{*}) and A^{-} = (f_{1}^{-}, f_{2}^{-}, . . ., f_{q}^{-}),$ where $f_{s}^{*} = p_{k} \circ {\hat{f}}_{s}^{*} = {\begin{matrix} (1, 1, . . ., 1), & s \in B \\ (0, 0, . . ., 0), & s \in C \end{matrix},$ $f_{s}^{-} = p_{k} \circ {\hat{f}}_{s}^{-} = {\begin{matrix} (0, 0, . . ., 0), & s \in B \\ (1, 1, . . ., 1), & s \in C \end{matrix}$ 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: $S_{r}^{*} = d (f_{rs}, f_{s}^{*}) = \sqrt{\frac{1}{m} {\sum_{k = 1}^{m} \sum_{s = 1}^{q} (p_{k} \circ {\hat{f}}_{rs} - p_{k} \circ {\hat{f}}_{s}^{*})^{2}}} ∥ (13)$ and $S_{r}^{-} = d (f_{rs}, f_{s}^{-}) = \sqrt{\frac{1}{m} {\sum_{k = 1}^{m} \sum_{s = 1}^{q} (p_{k} \circ {\hat{f}}_{rs} - p_{k} \circ {\hat{f}}_{s}^{-})^{2}}} (14) .$

Eighth step: Compute the closeness coefficient (denoted by, $\bar{{Cf}_{r}}$ ) of each alternative Z_r (r = 1, 2, . . . , p) (i.e., as alternative Z_r is near to $A^{*}$ and far from $A^{-}$ as closeness coefficient $\bar{{Cf}_{r}}$ go closer to 1), is defined by $\bar{{Cf}_{r}} = \frac{S_{r}^{-}}{(S_{r}^{-} + S_{r}^{*})} (15) .$

Ninth step: According to the $\bar{{Cf}_{r}},$ 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 II as:

Example 5.1 Let us consider there is an investment company, which wants to invest some money in the best alternative. The company made a committee of three members for the selection of best alternative (i.e., Z_r (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 should make a decision concerning the following four criteria (i.e., C_s (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 P3-PFS decision matrix and 3-polar weights by the committee of three decision-makers p_k (k = 1, 2, 3) evaluate all four possible alternatives (i.e, Z₁, Z₂, Z₃, Z₄) in first and second steps of Algorithm II are given as shown at the top of next page.

By Equation (10) in third step of Algorithm II, we obtain the following 3-polar decision matrix.

$M^{'} = (\begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} \\ Z_{1} & (6, 5.5, 4.5) & (2.5, 6.5, 5.5) & (2.5, 4.5, 5.5) & (6, 4.5, 5.5) \\ Z_{2} & (4, 6, 2.5) & (2.5, 7, 4.5) & (8, 3, 5.5) & (6, 3.5, 4.5) \\ Z_{3} & (2.5, 4.5, 5.5) & (5, 5.5, 3.5) & (5.5, 5, 7) & (3.5, 4, 4) \\ Z_{4} & (4, 4, 6) & (3, 4, 6.5) & (5.5, 6, 8) & (4, 3.5, 5.5) \end{matrix}) .$

By Equation (11) in fourth step of Algorithm II, we normalized the 3-polar decision matrix to get the normalized 3-polar fuzzy decision matrix.

$M^{''} = (\begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} \\ Z_{1} & (0.69, 0.54, 0.47) & (0.36, 0.55, 0.53) & (0.21, 0.47, 0.41) & (0.59, 0.57, 0.55) \\ Z_{2} & (0.46, 0.59, 0.26) & (0.36, 0.59, 0.43) & (0.69, 0.31, 0.41) & (0.59, 0.44, 0.45) \\ Z_{3} & (0.29, 0.44, 0.57) & (0.73, 0.46, 0.34) & (0.48, 0.32, 0.53) & (0.34, 0.51, 0.40) \\ Z_{4} & (0.46, 0.39, 0.62) & (0.43, 0.34, 0.63) & (0.48, 0.63, 0.60) & (0.39, 0.44, 0.55) \end{matrix}) .$

By Equation (12) in fifth step of the Algorithm II, we obtain the weighted normalized P3-PFS decision matrix.

$M^{'''} = (\begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} \\ Z_{1} & (0.24, 0.14, 0.17) & (0.15, 0.17, 0.13) & (0.05, 0.21, 0.06) & (0.11, 0.20, 0.14) \\ Z_{2} & (0.16, 0.15, 0.09) & (0.15, 0.18, 0.10) & (0.17, 0.13, 0.06) & (0.11, 0.15, 0.10) \\ Z_{3} & (0.10, 0.11, 0.21) & (0.30, 0.14, 0.08) & (0.12, 0.14, 0.07) & (0.06, 0.18, 0.09) \\ Z_{4} & (0.16, 0.10, 0.23) & (0.12, 0.10, 0.15) & (0.12, 0.28, 0.09) & (0.07, 0.15, 0.13) \end{matrix}) .$

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 II 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 seventh step of the Algorithm II, by computing separation measure $(S_{r}^{*}, S_{r}^{-})$ 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):

As $\sum_{k = 1}^{3} \sum_{s = 1}^{4} (Z_{1}, A^{*})^{2} = (0.24 - 0)^{2} + (0.14 - 0)^{2} + (0.17 - 0)^{2} + (0.15 - 0)^{2} + (0.17 - 0)^{2} + (0.13 - 0)^{2} + (0.05 - 1)^{2} + (0.21 - 1)^{2} + (0.06 - 1)^{2} + (0.11 - 1)^{2} + (0.20 - 1)^{2} + (0.14 - 1)^{2} =$

4.7563 and thus $S_{1}^{*} (Z_{1}, A^{*}) = \sqrt{\frac{1}{3} (4.7563)} = 1.25 .$ Similarly, $S_{2}^{*} (Z_{2}, A^{*}) = 1.26, S_{2}^{*} (Z_{3}, A^{*}) = 1.28, S_{2}^{*} (Z_{4}, A^{*}) = 1.23$

and

$S_{1}^{-} (Z_{1}, A^{-}) = 1.19, S_{2}^{-} (Z_{2}, A^{-}) = 1.23, S_{2}^{-} (Z_{3}, A^{-}) = 1.20, S_{2}^{-} (Z_{4}, A^{-}) = 1.23 .$

In eighth step of the Algorithm II, by computing closeness coefficient $\bar{{Cf}_{r}}$ of each alternative Z_r (r = 1, 2, 3, 4) and by Equation (15), we get the following results as follows:

$\bar{{Cf}_{r}} (Z_{1}) = 0.48, \bar{{Cf}_{r}} (Z_{2}) = 0.49, \bar{{Cf}_{r}} (Z_{3}) = 0.48,$

$\bar{{Cf}_{r}} (Z_{4}) = 0.5 .$

Finally, as $\bar{{Cf}_{r}} (Z_{4}) = 0.5$ is highest among others, Z₄ is the best ideal solution the best ideal solution we got using multi-person TOPSIS method.

6 Conclusions

We presented the concept of the Pm-PFS as a new m-PFS model. We stuided the several structure operations of the Pm-PFS and also discussed the basic properties of the Pm-PFS. Then, we introduced a novel concepts of distances measure and similarity measure for Pm-PFSs. Further, we constructed an Algorithm I to solve brand recognition problem (i.e., for apply of Pm-PFS in brand recognition problem). Moreover, a new approach for the best Pm-PFS alternatives based on distance similarity measures in MCDM is animated. Finally, a new methodology to extend the TOPSIS to Pm-PFSs is proposed and illustrate its applicability through a numerical example. In the future, an Pm-PFS has definitely will open the 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., intuitionistic fuzzy sets [24], interval-valued fuzzy sets [25], and spherical fuzzy sets [26]) we can extent our work to obtain a novel m-PFS models. In addition to, we will construct comparison analysis among our work and other related work.

$M = (\begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} \\ Z_{1} & ((3, 4, 5), (2, 4, 5), (2, 3, 4)) & ((1, 1, 1), (4, 5, 4), (2, 4, 5)) & ((1, 1, 1), (2, 5, 2), (3, 2, 6)) & ((2, 4, 6), (4, 4, 1), (3, 4, 4)) \\ Z_{2} & ((2, 1, 5), (3, 2, 7), (1, 1, 3)) & ((1, 2, 2), (3, 5, 6), (1, 3, 5)) & ((4, 6, 6), (2, 1, 3), (1, 5, 5)) & ((2, 4, 6), (1, 3, 3), (1, 3, 5)) \\ Z_{3} & ((1, 2, 2), (1, 3, 5), (2, 3, 6)) & ((2, 4, 4), (2, 2, 7), (1, 2, 4)) & ((3, 4, 4), (2, 4, 4), (5, 4, 5)) & ((1, 3, 3), (2, 2, 4), (1, 3, 4)) \\ Z_{4} & ((1, 3, 4), (2, 2, 4), (4, 4, 4)) & ((2, 2, 2), (2, 3, 3), (3, 4, 6)) & ((1, 5, 5), (3, 3, 6), (3, 6, 7)) & ((2, 2, 4), (1, 3, 3), (3, 4, 4)) \\ \hat{w} & (0.35, 0.26, 0.38) & (0.42, 0.31, 0.25) & (0.26, 0.45, 0.15) & (0.20, 0.36, 0.24) \end{matrix}) .$

Footnotes

Acknowledgments

Authors thank the editors and the anonymous reviewers for their insightful comments which improved the quality of the paper. This research is supported by the National Natural Science Foundation of China (No. 11871475).

References

Zadeh

L.A.

, Fuzzy sets,, Information and Control 8 (1965), 338–353.

Cuong

B.C.

, Picture fuzzy sets, Journal of Computer Science and Cybernetics 30 (2014), 409–420.

Zhang

W.R.

, 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 Intelligence, (1994), 1305–1309.

Chen

, Li

, Ma

and Wang

, m-polar fuzzy sets: An extension of bipolar fuzzy sets, The scientific world journal 2014 (2014), 416530.

Akram

and Sarwar

, New applications of m-polar fuzzy competition graphs, New Mathematics and Natural Computation 14 (2018), 249–276.

Akram

and Adeel

, m-polar fuzzy graphs and m-polar fuzzy line graphs, Journal of Discrete Mathematical Sciences and Cryptography 20 (2017), 1597–1617.

Akram

, Waseem

and Davvaz

, Certain types of domination in m-polar fuzzy graphs, Journal of Multiple-Valued Logic & Soft Computing 29 (2017), 619–646.

Sarwar

and Akram

, Representation of graphs using m-polar fuzzy environment, Italian Journal of Pure and Applied Mathematics 38 (2017), 291–312.

Akram

, Ali

and Alshehri

N.O.

, A new multi-attribute decision-making method based on m-polar fuzzy soft rough sets, Symmetry 9 (2017), 271.

10.

Akram

, Waseem

and Liu

, Novel approach in decision making with m-polar fuzzy ELECTRE-I, International Journal of Fuzzy Systems 21 (2019), 1117–1129.

11.

Adeel

, Akram

, Ahmed

and Nazar

, Novel m-polar fuzzy linguistic ELECTRE-I method for group decision-making, Symmetry 11 (2019), 471.

12.

Chen

S.M.

, Yeh

M.S.

and Hsiao

P.Y.

, A comparison of similarity measures of fuzzy values, Fuzzy Sets and Systems 72 (1995), 79–89.

13.

Chen

S.M.

, A comparison of similarity measures of vague sets, Fuzzy Sets and Systems 74 (1995), 217–223.

14.

Hong

D.H.

and Kim

, A note on similarity measures between vague sets and between elements, Information Sciences 115(1-4) (1999), 83–96.

15.

Thao

N.X.

, Similarity measures of picture fuzzy sets based on entropy and their application in MCDM, Pattern Analysis and Applications 23 (2020), 1203–1213. Doi: 10.1007/s10044-019-00861-9.

16.

Wei

G.W.

, Some similarity measures for picture fuzzy sets and their applications, Iranian Journal of Fuzzy Systems 15(1) (2018), 77–89.

17.

Son

L.H.

, Generalized picture distance measure and applications to picture fuzzy clustering, Applied Soft Computing 46 (2016), 284–295.

18.

Peng

X.D.

and Dai

J.G.

, Algorithm for picture fuzzy multiple attribute decision-making based on new distance measure, International Journal for Uncertainty Quantification 7(2) (2017), 177–187.

19.

Akram

and Waseem

, Similarity measures for new hybrid models: mF sets and mF soft sets, Punjab University Journal of Mathematics 51(5) (2019), 115–130.

20.

Yong

, Plant location selection based on fuzzy TOPSIS, The International Journal of Advanced Manufacturing Technology 28 (2006), 839–844.

21.

Adeel

, Akram

and Koam

A.N.A.

, Group decision-making based on m-polar fuzzy linguistic TOPSIS method, Symmetry 11 (2019), 735.

22.

Shih

H.S.

, Shyur

H.J.

and Lee

E.S.

, An extension of TOPSIS for group decision making, Mathematical and computer modelling 45(7-8) (2007), 801–813.

23.

Saeed

, Saqlain

, Asad

, Khushbakht

and Sonia

, Multi-polar neutrosophic soft sets with application in medical diagnosis and decision-making, Neutrosophic Sets and Systems 33 (2020), 183–207.

24.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

25.

Gorzałzany

M.B.

, A method of inference in approximate reasoning based on interval valued fuzzy sets, Fuzzy Sets and Systems 21 (1987), 1–17.

26.

Ashraf

, Abdullah

, Mahmood

, Ghani

and Mahmood

, Spherical fuzzy sets and their applications in multi-attribute decision making problems, Journal of Intelligent & Fuzzy Systems 36(3) (2019), 2829–2844.