Pythagorean fuzzy soft MCGDM methods based on TOPSIS,VIKOR and aggregation operators

Abstract

We study, in this paper, some notions related to Pythagorean fuzzy soft sets (PFSSs) along with their algebraic structures. We present operations on PFSSs and their peculiar characteristics and elaborate them with real life examples and tabular representation to develop the affluence of linguistic variables based on Pythagorean fuzzy soft (PFS) information. We present an application of PFSSs to the multi-criteria group decision-making (MCGDM) related to site selection, accompanied by Algorithm and flow chart. We develop PFS TOPSIS method and PFS VIKOR method as extensions of the TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) and VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) respectively. Finally, we rendered an application in stock exchange investment problem and tackled it by both PFS TOPSIS and PFS VIKOR methods.

Keywords

Pythagorean fuzzy soft sets linguistic variable aggregation operator TOPSIS VIKOR

1 Introduction

The fuzzy linguistic approach provides favorable outputs in several areas, whose description is relatively qualitative. The encouragement for the utilization of sentences or words instead of numbers is that linguistic characterizations or classifications are usually less absolute than algebraic or arithmetical ones. Problems that are related to uncertain conditions usually exist in decision-making, but are demanding because of the challenging situation of modeling and handling that comes with such uncertainties. To tackle complex real world problems the techniques usually employed in classical mathematics are not always beneficial due to a number of types of uncertainties and vagueness present in these problems. There are numerous techniques, like theory of probability, the interval mathematics, fuzzy set theory, soft and rough set theories which we can be thought of as mathematical models for coping with uncertainties. Unluckily, all these models have their own drawbacks and difficulties. For example, in fuzzy set theory the word beautiful is vague. The criteria for beautiful varies from person to person. To overcome these kinds of deficiencies Zadeh [70] introduced the idea of fuzzy sets as an extension of the traditional crisp set. A fuzzy set is a significant mathematical model to characterize an assembling of objects whose boundary is obscure. Atanassov [8 –10] proposed the idea of intuitionistic fuzzy sets as an extension of fuzzy set by introducing the concepts of membership (denoted by μ (x)) and non-membership grades (denoted by ν (x)) along with the restriction that sum of these two grades must not exceed unity. Atanassov [11] presented geometrical interpretation of the elements of the intuitionistic fuzzy objects. Molodtsov [33] originated the notion of a new kind of sets conventionally known as soft sets, as a mathematical model for sorting out uncertainties. Akram et al. [2 –4] presented certain applications of Pythagorean fuzzy graphs and rough Pythagorean fuzzy bipolar soft information to the decision making problems (DMP). Fuzzy sets, soft set and their generalizations are strong mathematical models for solving many real world problems. The researchers have introduced different mathematical models including soft set, fuzzy soft set, hesitant fuzzy set to deal with real world problems [6 , 59].

Yager [66 –68] introduced Pythagorean fuzzy set as an extension of Atanassov’s intuitionistic fuzzy set and presented Pythagorean membership grades with applications to the multi-criteria decision making (MCDM). Sumera et al. [35] extended the notion of Pythagorean fuzzy sets to Pythagorean fuzzy graphs. Peng and Yang [36] discussed some results for Pythagorean fuzzy sets and gave idea of Pythagorean fuzzy number. Peng et al. [38] introduced some Pythagorean fuzzy information measures and their applications. Peng et al. [38] introduced Pythagorean fuzzy soft set and its application. Peng and Dai [39] studied some approaches to single-valued neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function.

Riaz and Naeem [40, 41] presented some essential ideas of soft sets together with soft sigma algebra. They additionally displayed a few utilizations of soft mappings to the decision making problems (DMP). Riaz and Hashmi [42 –44] investigated certain applications of FPFS-sets, FPFS-topology and FPFS-compact spaces. They investigated fixed point theorems of FNS-mapping with applications to the DMP. Recently, Riaz and Hashmi (2019) [45] introduced novel concept of linear Diophantine fuzzy set and its applications towards multi-attribute decision making problems. Riaz and Hashmi (2019) [46] introduced MAGDM for agribusiness in the environment of various cubic m-polar fuzzy averaging aggregation operators. Riaz et al. [47, 48] introduced soft rough topology with multi-attribute group decision making problems (MAGDM). Riaz et al. [49] introduced N-soft topology and its applications to multi-criteria group decision making (MCGDM). Riaz and Tahrim [50 –52] established the idea of bipolar fuzzy soft topology and cubic bipolar fuzzy ordered weighted geometric aggregation operators and their application using internal and external cubic bipolar fuzzy data. Riaz and Tahrim [53] introduced bipolar fuzzy soft mappings with application to bipolar disorders. Tehrim and Riaz (2019) [54] introduced a novel extension of TOPSIS to MCGDM with bipolar neutrosophic soft topology.

TOPSIS and VIKOR methods for decision making problems have been studied by many researchers: Adeel et al. [1], Akram and Arshad [5], Boran et al. [13], Eraslan and Karaaslan [24], Kumar and Garg [26], Li and Nan [27], Mohd and Abdullah [34], Peng and Dai [39], Selvachandran and Peng [55], Xu and Zhang [65] and Zhang and Xu [72]. Zhan et al. [73 –75] established various properties of soft rough set, soft rough hemirings, Z-soft fuzzy rough set and their applications to the multi-criteria group decision making MCGDM. Zhang et al. [76 –78] introduced fuzzy soft β-covering based fuzzy rough sets, covering-based generalized IF rough sets and novel classes of fuzzy soft with corresponding decision-making applications.

Recently, Tang et al. [58] studied some MADM models employing interval-valued Pythagorean fuzzy Muirhead mean operators and presented their application to green suppliers selection. Wang et al. [30] presented novel concepts of Dice similarity measures along with its characteristics. They also proposed the generalized Dice similarity measures-based MAGDM models with Pythagorean fuzzy information. Lu et al. [29] proposed a bidirectional project method to handle the MADM problem under the dual hesitant Pythagorean fuzzy environment. They presented a fascinating application of proposed method in performance assessment of new rural construction. Wei [61] utilized Hamacher operations and power aggregation operators to develop some new Pythagorean fuzzy Hamacher power aggregation operators along with their prominent properties. Utilizing the proposed operators, he developed some approaches to solve the MADM problems with Pythagorean fuzzy numbers.

Wu et al. [62] studied attitudinal trust recommendation mechanism to balance consensus and harmony in group decision making. Wu et al. [63] presented attitudinal consensus degree to control feedback mechanism in group decision making with different adjustment cost. A harmony degree (HD) is defined to determine the extent of the difference between an original opinion and the corresponding revised opinion after adopting the recommended advices. Wu et al. [64] discussed the idea of trust propagation and collaborative filtering based method for incomplete information in social network group decision making with type-2 linguistic trust. Dong et al. [17] discussed hybrid group decision making framework for achieving agreed solutions based on stable opinions. Zhou et al. [79] studied evidential reasoning approach with multiple kinds of attributes and entropy-based weight assignment.

The goal of this paper is to introduce Pythagorean fuzzy soft sets (PFSSs) as a hybrid structure of soft set and Pythagorean fuzzy set. In order to solve the real world problems that intuitionistic fuzzy soft set ( $ifs$ -set) can not deal with the situation that the sum of membership degree and non-membership degree of the parameter is larger than 1. It makes the MADM limited, and affects the optimum decision. Pythagorean fuzzy soft set (PFSS) provides a large number of applications to the MADM for such real world problems.

The paper is organized as follows: Section 1 provides literature review of Pythagorean fuzzy sets and related fields to solve various real world problems by using different decision making methods. Section 2 introduces basic concepts of fuzzy sets, soft sets, intuitionistic fuzzy sets and Pythagorean fuzzy sets. Section 3 introduces essential operations and fundamental properties of Pythagorean fuzzy soft sets. Section 4 provides an interesting application of multi-criteria group decision making MCGDM by utilizing Pythagorean fuzzy soft matrices. In section 5, we first propose a TOPSIS model and then demonstrated its applicability by applying it on an application from the business world i.e. stock exchange investment problem. In section 6, we propose a VIKOR technique and applied it on the same example of Section 5. Finally, a concrete conclusion is summarized in Section 7.

2 Preliminaries

In this segment, we concisely review some rudiments of different kinds of sets which will be employed in the rest of the paper.

Definition 2.1. [70] Let X be a classical set and μ_A : X → [0, 1] be the membership function. Then a fuzzy set A can be written in the form $A = {(ρ, μ_{A} (ρ)) : ρ \in X}$ The value of the mapping μ_A at ρ ∈ X i.e. μ_A (ρ) denotes the degree of membership of ρ to the fuzzy set A. The aggregate of all fuzzy sets in X is designated as $F (X)$ .

Definition 2.2. [71] A linguistic variable is a variable, whose values are sentences or words in an artificial or natural language. If these words are expressed by fuzzy sets defined over a universal set, then the variable is called a fuzzy linguistic variable.

Definition 2.3. [10] An intuitionistic fuzzy set ( $if$ -set) $A$ over the universe X comprises ordered triplets $(ρ, μ_{A} (ρ), ν_{A} (ρ))$ , where ρ ∈ X. The mappings $μ_{A}$ and $ν_{A}$ map elements of X to the unit closed interval [0, 1], with the restriction that their sum must not exceed unity, are respectively termed as degrees of membership and non-membership of the element ρ to $A$ . This set in set builder form is usually symbolized as $A = {(ρ, μ_{A} (ρ), ν_{A} (ρ)) : ρ \in X}$

Definition 2.4. [33] Let X be a universal set and E a non-void set of attributes. Suppose that 2^X is the power set of X and A is a non-empty sub-collection of E. A doublet (ψ, A), where ψ : A → 2^X is a mapping, is termed as soft set over X. Thus a soft set can be written as $(ψ, A) = {(e, ψ (e)) : e \in A, ψ (e) \in 2^{X}}$ The pair (ψ, A) is also shortly written ψ_A.

Definition 2.5. [16, 31] Let X be a crisp set and E be the family of attributes with A ⊆ E. Suppose that $F (X)$ stands for collection of all fuzzy sets defined over X. A collection of ordered doublets $Γ_{A} = {(e, γ_{A} (e)) : e \in E, γ_{A} \in F (X)}$ is termed as fuzzy soft set (or an fs-set) over X. Here $γ_{A} : E \to F (X)$ such that γ_A (e) = ϕ whenever e ∉ A, is known as fuzzy approximate function of Γ_A. The value γ_A (e) is a set called e-element of the fs-set, for all e ∈ E.

The aggregate of all fuzzy soft sets over X is designated as FS (X).

Definition 2.6. [32] Let X be a crisp set and E be the family of attributes with A ⊆ E. Assume that IF^X stands for collection of all $if$ -subsets defined over X. An intuitionistic fuzzy soft set ( $ifs$ -set) over X is designated as (ψ, A) or sometimes as ψ_A, where ψ : A → IF^X is a multi-valued mapping. For any attribute e ∈ A, ψ (e) yields the value in the form of an an $if$ -set. Hence, an $ifs$ -set ψ_A is described by the multi-valued map ψ : A → 2^X.

The set of all $ifs$ -sets over the universe X having set of attributes from E is known as $ifs$ -class and is designated as $ifs (X, E)$ .

Definition 2.7. [66 –68] Let X be the initial universe. A Pythagorean fuzzy set, abbreviated as $pf$ -set, can be written in the form $P = {< ρ, μ_{P} (ρ), ν_{P} (ρ) > : 0 \leq μ_{P}^{2} (ρ) + ν_{P}^{2} (ρ) \leq 1, ρ \in X}$ where μ_P and ν_P are mappings from X to the unit interval [0, 1] with the restriction that sum of their squares should not exceed unity i.e. $0 \leq μ_{P}^{2} (ρ) + ν_{P}^{2} (ρ) \leq 1$ , called respectively the degree of membership and non-membership of ρ ∈ X to the set P.

The space for a $pf$ -set is a unit circular arc in the first quadrant whereas it was a right isosceles triangle having length of the base and altitude each equal to unity in case of an $if$ -set. Hence we have an enlarged space for $pf$ -sets as compared to $if$ -sets as can be seen in Figure 1. In fact we have included $\frac{2 π - 1}{8}$ extra area for $pf$ -sets.

The degree of indeterminacy may be computed employing $I_{P} (ρ) = \sqrt{1 - μ_{P}^{2} (ρ) - ν_{P}^{2} (ρ)}$ . Zhang and Xu, in [72], named the doublet (μ_P (ρ) , ν_P (ρ)) a Pythagorean fuzzy number (PFN) and used the notation p = (μ_P, ν_P) to represent them. They set the distance between p_i and p_j as $d (p_{i}, p_{j}) = \frac{1}{2} {| μ_{p_{i}}^{2} - μ_{p_{j}}^{2} | + | ν_{p_{i}}^{2} - ν_{p_{j}}^{2} | + | I_{p_{i}}^{2} - I_{p_{j}}^{2} |}$ The region for $pf$ & $if$ sets are given by the Figure 1.

Fig. 1

Spaces of $pf$ & $if$ sets.

Definition 2.8. [72] The score function for any PFN p = (μ_p, ν_p) is defined as $s (p) = μ_{p}^{2} - ν_{p}^{2}$ where -1 ≤ s (p) ≤1. If p_i and p_j are two Pythagorean fuzzy numbers, then

If s (p_i) < s (p_j) then p_i precedes p_j i.e. p_i ≺ p_j,

If s (p_i) > s (p_j) then p_i succeeds p_j i.e. p_i ≻ p_j,

If s (p_i) = s (p_j) then p_i ∼ p_j.

Remark. If we choose p₁ = (0.1, 0.1) and p₂ = (0.7, 0.7), then according to Definition 2.8, we have p₁ ∼ p₂ which does not seem to be reasonable. So the idea of accuracy function is floated to amend the comparison rules.

Definition 2.9. [72] The accuracy function for any PFN p = (μ_p, ν_p) is defined as $a (p) = μ_{p}^{2} + ν_{p}^{2}$ where 0 ≤ a (p) ≤1. If p₁ and p₂ are two PFNs, then

s (p_i) = s (p_j) & a (p_i) > a (p_j) ⇒ p_i ≻ p_j,

s (p_i) = s (p_j) & a (p_i) = a (p_j) ⇒ p_i ∼ p_j.

3 Pythagorean fuzzy soft sets

Recently, Peng et al. [38] proposed the idea of Pythagorean fuzzy soft set and rendered some of its applications. Guleria and Bajaj [25] introduced Pythagorean fuzzy soft matrices (PFS matrices), their operations and applications in decision making and medical diagnosis. In this section, we study some primary notions, fundamental properties and algebraic operations on PFS set with the help of examples.

Definition 3.1. Let X be a crisp set and E be the family of attributes. Let A ⊆ E be non-void and PF^X represent the aggregate of all Pythagorean fuzzy subsets over X. A Pythagorean fuzzy soft set (PFS set) on X is denoted as (ψ, A) or ψ_A, where ψ : A → PF^X is a mapping i.e. $\begin{matrix} (ψ, A) = {(e, {ρ, μ_{ψ_{A}} (ρ), ν_{ψ_{A}} (ρ)}) : e \in A, ρ \in X} \\ = {(e, {\frac{ρ}{(μ_{ψ_{A}} (ρ), ν_{ψ_{A}} (ρ))}}) : e \in A, ρ \in X} \\ = {(e, {\frac{(μ_{ψ_{A}} (ρ), ν_{ψ_{A}} (ρ))}{ρ}}) : e \in A, ρ \in X} \end{matrix}$ where μ_{ψ
_A} : X → [0, 1] and ν_{ψ
_A} : X → [0, 1] are mappings along with the property $0 \leq μ_{ψ_{A}}^{2} (ρ) + ν_{ψ_{A}}^{2} (ρ) \leq 1$ The values of these mappings at some ρ ∈ X are termed as Pythagorean fuzzy numbers (PFNs). In particular, μ_{ψ
_A} (ρ) represents the degree of membership and ν_{ψ
_A} (ρ) denotes degree of non-membership of the element ρ ∈ X to the set (ψ, A).

For any attribute e, ψ (e) yields the value of multi-valued mapping at e which is always a Pythagorean fuzzy set. Hence, a PFS set (ψ, A) is described by the multi-valued mapping ψ : A → 2^X.

The set of all PFSSs over X and with set of parameters from E is called PFS class and is designated as PFS (X, E).

If we write a_ij = μ_{ψ
_A} (e_j) (ρ_i) and b_ij = ν_{ψ
_A} (e_j) (ρ_i) where i runs from 1 to m and j runs from 1 to n then the PFS set ψ_A may be represented in tabular form as

ψ _A	e ₁	e ₂	⋯	e _n
ρ ₁	(a₁₁, b₁₁)	(a₁₂, b₁₂)	⋯	(a_1n, b_1n)
ρ ₂	(a₂₁, b₂₁)	(a₂₂, b₂₂)	⋯	(a_2n, b_2n)
⋮	⋮	⋮	⋱	⋮
ρ _m	(a_m1, b_m1)	(a_m2, b_m2)	⋯	(a_mn, b_mn)

and in matrix form as $\begin{matrix} (ψ, A) = [(a_{ij}, b_{ij})]_{m \times n} \\ = (\begin{matrix} (a_{11}, b_{11}) & (a_{12}, b_{12}) & \dots & (a_{1 n}, b_{1 n}) \\ (a_{21}, b_{21}) & (a_{22}, b_{22}) & \dots & (a_{2 n}, b_{2 n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (a_{m 1}, b_{m 1}) & (a_{m 2}, b_{m 2}) & \dots & (a_{mn}, b_{mn}) \end{matrix}) \end{matrix}$ This matrix is called Pythagorean fuzzy soft matrix or shortly PFS matrix.

Example 3.2. Let X = {ρ_i : i = 1, 2, ⋯ , 7} and E = {e_i : i = 1, 2, ⋯ , 5}. Assume that A = {e₁, e₃}. Then, ψ_A = { (e₁, {(ρ₂, 0.3, 0.9) , (ρ₄, 0.5, 0.2) , (ρ₇, 0.7, 0.4)}) , (e₃, {(ρ₁, 0.2, 0.8) , (ρ₃, 0.1, 0.6) , (ρ₅, 0.3, 0.7)}) } is a PFS set over X. The tabular representation of ψ_A is

ψ _A	e ₁	e ₂	e ₃	e ₄	e ₅
ρ ₁	(0,1)	(0,1)	(0.2,0.8)	(0,1)	(0,1
ρ ₂	(0.3,0.9)	(0,1)	(0,1)	(0,1)	(0,1)
ρ ₃	(0,1)	(0,1)	(0.1,0.6)	(0,1)	(0,1)
ρ ₄	(0.5,0.2)	(0,1)	(0,1)	(0,1)	(0,1)
ρ ₅	(0,1)	(0,1)	(0.3,0.7)	(0,1)	(0,1)
ρ ₆	(0,1)	(0,1)	(0,1)	(0,1)	(0,1)
ρ ₇	(0.7,0.4)	(0,1)	(0,1)	(0,1)	(0,1)

The corresponding PFS matrix is $\begin{matrix} (ψ, A) = [a_{ij}, b_{ij}]_{7 \times 5} \\ = (\begin{matrix} (0, 1) & (0, 1) & (0.2, 0.8) & (0, 1) & (0, 1) \\ (0.3, 0.9) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.1, 0.6) & (0, 1) & (0, 1) \\ (0.5, 0.2) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.3, 0.7) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.7, 0.4) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}$

Definition 3.3. Let ψ_A and G_B be PFS set over X. Then ψ_A is PFS subset of G_B i.e. $ψ_{A} \tilde{\subseteq} G_{B}$ , if

A ⊆ B, and

ψ (e) is PFS subset of G (e) for all e ∈ A.

It is worth mentioning that

ψ_{A} \tilde{\subseteq} G_{B}

does not require that every element of ψ_A is also in G_B, as is desired in classical set theory.

Definition 3.4. Let (ψ₁, A₁) and (ψ₂, A₂) be PFSSs defined over X. Then their union is defined as $(ψ, A) = (ψ_{1}, A_{1}) \tilde{\cup} (ψ_{2}, A_{2}),$ where A = A₁ ∪ A₂ and for all e ∈ A, $ψ (e) = {\begin{matrix} ψ_{1} (e), & if e \in A_{1} ∖ A_{2} \\ ψ_{2} (e), & if e \in A_{2} ∖ A_{1} \\ ψ_{1} (e) \cup ψ_{2} (e), & if e \in A_{1} \cap A_{2} \end{matrix}$ where ψ₁ (e) ∪ ψ₂ (e) is the union of two PFS sets.

Definition 3.5. The intersection of two PFS sets (ψ₁, A₁) and (ψ₂, A₂) is a PFS set $(ψ, A) = (ψ_{1}, A_{1}) \tilde{\cap} (ψ_{2}, A_{2})$ , where A = A₁ ∩ A₂ ≠ ϕ and ψ (e) = ψ₁ (e) ∩ ψ₂ (e) for all e ∈ A.

Definition 3.6. The difference (ψ, A) of two PFS sets (ψ₁, A₁) and (ψ₂, A₂) over X is defined by (ψ, e) = (ψ₁, e) \ (ψ₂, e); ∀ e ∈ E and is designated as $(ψ_{1}, A) \tilde{\} (ψ_{2}, A)$ . Hence, $(ψ_{1}, A_{1}) \tilde{\} (ψ_{2}, A_{2})$ = { (e, {ρ, min {μ_ψ₁(e) (ρ) , ν_ψ₂(e) (ρ)} , max {ν_ψ₁(e) (ρ) , μ_ψ₂(e) (ρ)}} ) : ρ ∈ X, e ∈ E}

Definition 3.7. The complement of a PFS set (ψ, A) is a mapping ψ^c : A → PF^X given by ψ^c (e) = [ψ (e)] ^c, for all e ∈ A. It is represented as (ψ, A) ^c or sometimes by (ψ^c, A). Thus, if $ψ (e) = {(ρ, μ_{ψ (e)} (ρ), ν_{ψ (e)} (ρ)) : ρ \in X}$ then $ψ^{c} (e) = {(ρ, ν_{ψ (e)} (ρ), μ_{ψ (e)} (ρ)) : ρ \in X}$ for all e ∈ A.

Definition 3.8. A PFS set (ψ, E) over X is known as null PFS set, symbolized as ψ_ϕ or Φ, if ∀ e ∈ E we have $ψ (e) = \tilde{0}$ , where $\tilde{0}$ denotes null Pythagorean fuzzy set. Hence, $ψ_{ϕ} = {(e, {ρ, 0, 1}) : e \in E, ρ \in X}$

Definition 3.9. A PFS set (ψ, E) over X is called absolute PFS set, denoted by $\overset{˘}{X}$ , if $ψ (e) = \tilde{1}, \forall e \in E$ . Here $\tilde{1}$ denotes absolute Pythagorean fuzzy set. Hence, $\overset{˘}{X} = {(e, {ρ, 1, 0}) : e \in E, ρ \in X}$

We illustrate some of these concepts with the help of following example.

Example 3.10. Let X = {ρ₁, ρ₂, ⋯ , ρ₆} and E = {e₁, e₂, ⋯ , e₆}. Suppose that A₁ = {e₁, e₃} and A₂ = {e₂, e₃, e₄}. Consider the PFS sets (ψ₁, A₁) and (ψ₂, A₂) given by (ψ₁, A₁) = {(e₁, {ρ₁, 0.3, 0.5} , {ρ₃, 0.7, 0.1} , {ρ₆, 0.8, 0.2}) , (e₃, {ρ₂, 0.5, 0.3} , {ρ₆, 0.3, 0})}, and (ψ₂, A₂) = {(e₂, {ρ₃, 0.4, 0.5} , {ρ₄, 0.3, 0.1} , {ρ₅, 0.4, 0.1} , {ρ₆, 0.6, 0.2}) , (e₃, {ρ₁, 0.5, 0.4} , {ρ₆, 0.6, 0.2}) , (e₄, {ρ₂, 0.7, 0.3} , {ρ₃, 0.6, 0.2} , {ρ₆, 0.4, 0.2})}

Their union is $(ψ_{1}, A_{1}) \tilde{\cup} (ψ_{2}, A_{2}) = {(e_{1}, {ρ_{1}, 0.3, 0.5}, {ρ_{3}, 0.7, 0.1}, {ρ_{6}, 0.8, 0.2}), (e_{2}, {ρ_{3}, 0.4, 0.5}, {ρ_{4}, 0.3, 0.1}, {ρ_{5}, 0.4, 0.1}, {ρ_{6}, 0.6, 0.2}), (e_{3}, {ρ_{1}, 0.5, 0.4}, {ρ_{2}, 0.5, 0.3}, {ρ_{6}, 0.6, 0}), (e_{4}, {ρ_{2}, 0.7, 0.3}, {ρ_{3}, 0.6, 0.2}, {ρ_{6}, 0.4, 0.2})}$

Their intersection is $(ψ_{1}, A_{1}) \tilde{\cap} (ψ_{2}, A_{2}) = {(e_{3}, {ρ_{6}, 0.3, 0.2})}$

The complements of (ψ₁, A₁) and (ψ₂, A₂) are (ψ₁, A₁) ^c = {(e₁, {ρ₁, 0.5, 0.3} , {ρ₃, 0.1, 0.7} , {ρ₆, 0.2, 0.8}) , (e₃, {ρ₂, 0.3, 0.5} , {ρ₆, 0, 0.3})} and (ψ₂, A₂) ^c = {(e₂, {ρ₃, 0.5, 0.4} , {ρ₄, 0.1, 0.3} , {ρ₅, 0.1, 0.4} , {ρ₆, 0.2, 0.6}) , (e₃, {ρ₁, 0.4, 0.5} , {ρ₆, 0.2, 0.6}) , (e₄, {ρ₂, 0.3, 0.7} , {ρ₃, 0.2, 0.6} , {ρ₆, 0.2, 0.4})} respectively.

The difference of (ψ₁, A₁) and (ψ₂, A₂) is $(ψ_{1}, A_{1}) \tilde{\} (ψ_{2}, A_{2}) = {(e_{1}, {ρ_{1}, 0.3, 0.5}, {ρ_{3}, 0.7, 0.1}, {ρ_{6}, 0.8, 0.2}), (e_{3}, {ρ_{2}, 0.5, 0.3}, {ρ_{6}, 0.3, 0.2})}$

We can easily comprehend these notions with the help of PFS matrices as given below:(ψ₁, A₁) =

\begin{matrix} (\begin{matrix} (0.3, 0.5) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.5, 0.3) & (0, 1) & (0, 1) & (0, 1) \\ (0.7, 0.1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.8, 0.2) & (0, 1) & (0.3, 0) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}

(ψ₂, A₂) =

\begin{matrix} (\begin{matrix} (0, 1) & (0, 1) & (0.5, 0.4) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0.7, 0.3) & (0, 1) & (0, 1) \\ (0, 1) & (0.4, 0.5) & (0, 1) & (0.6, 0.2) & (0, 1) & (0, 1) \\ (0, 1) & (0.3, 0.1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.4, 0.1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.6, 0.2) & (0.6, 0.2) & (0.4, 0.2) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}

(ψ_{1}, A_{1}) \tilde{\cup} (ψ_{2}, A_{2}) =

\begin{matrix} (\begin{matrix} (0.3, 0.5) & (0, 1) & (0.5, 0.4) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.5, 0.3) & (0.7, 0.3) & (0, 1) & (0, 1) \\ (0.7, 0.1) & (0.4, 0.5) & (0, 1) & (0.6, 0.2) & (0, 1) & (0, 1) \\ (0, 1) & (0.3, 0.1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.4, 0.1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.8, 0.2) & (0.6, 0.2) & (0.6, 0) & (0.4, 0.2) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}

(ψ_{1}, A_{1}) \tilde{\cap} (ψ_{2}, A_{2}) =

\begin{matrix} (\begin{matrix} (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.3, 0.2) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}

(ψ₁, A₁) ^c =

\begin{matrix} (\begin{matrix} (0.5, 0.3) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.3, 0.5) & (0, 1) & (0, 1) & (0, 1) \\ (0.1, 0.7) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.2, 0.8) & (0, 1) & (0, 0.3) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}

(ψ₂, A₂) ^c =

\begin{matrix} (\begin{matrix} (0, 1) & (0, 1) & (0.4, 0.5) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0.3, 0.7) & (0, 1) & (0, 1) \\ (0, 1) & (0.5, 0.4) & (0, 1) & (0.2, 0.6) & (0, 1) & (0, 1) \\ (0, 1) & (0.1, 0.3) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.1, 0.4) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0.2, 0.6) & (0.2, 0.6) & (0.2, 0.4) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}

(ψ_{1}, A_{1}) \tilde{\} (ψ_{2}, A_{2}) =

\begin{matrix} (\begin{matrix} (0.3, 0.5) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0.5, 0.3) & (0, 1) & (0, 1) & (0, 1) \\ (0.7, 0.1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0.8, 0.2) & (0, 1) & (0.3, 0.2) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}

Proposition 3.11. If ψ_A is any PFS set over X, then

$ψ_{ϕ} \tilde{\subseteq} ψ_{A}$ .

$ψ_{A} \tilde{\subseteq} \overset{˘}{X}$ .

Proof.

By definition, degrees of membership and non-membership always fall in [0, 1], so 0 ≤ μ (ρ) and 1 ≥ ν (ρ) for all ρ ∈ X. Thus, it follows from definition of subset and ψ_ϕ that $ψ_{ϕ} \tilde{\subseteq} ψ_{A}$ .

By definition, degrees of membership and non-membership always lie in [0, 1], so μ (ρ) ≤1 and ν (ρ) ≥0 for all ρ ∈ X. Thus, it follows from definition of subset and $\overset{˘}{X}$ that $ψ_{A} \tilde{\subseteq} \overset{˘}{X}$ .□

Proposition 3.12. If ψ_A is any PFS set over X, then

$ψ_{A} \tilde{\cap} ψ_{A} = ψ_{A}$ .

$ψ_{A} \tilde{\cup} ψ_{A} = ψ_{A}$ .

$ψ_{A} \tilde{\cap} ψ_{B} = ψ_{B} \tilde{\cap} ψ_{A}$ .

$ψ_{A} \tilde{\cup} ψ_{B} = ψ_{B} \tilde{\cup} ψ_{A}$ .

$ψ_{A} \tilde{\cap} (ψ_{B} \tilde{\cap} ψ_{C}) = (ψ_{A} \tilde{\cap} ψ_{B}) \tilde{\cap} ψ_{C}$ .

$ψ_{A} \tilde{\cup} (ψ_{B} \tilde{\cup} ψ_{C}) = (ψ_{A} \tilde{\cup} ψ_{B}) \tilde{\cup} ψ_{C}$ .

Proof. The proof is obvious.□

Proposition 3.13. If ψ_A and ψ_B are PFS sets over X, then

$ψ_{A} \tilde{\cap} ψ_{B} \tilde{\subseteq} ψ_{A} \tilde{\subseteq} ψ_{A} \tilde{\cup} ψ_{B}$

$ψ_{A} \tilde{\cap} ψ_{B} \tilde{\subseteq} ψ_{B} \tilde{\subseteq} ψ_{A} \tilde{\cup} ψ_{B}$ .

Proof. We prove only (i). The proof of (ii) may be furnished on parallel track.

Since min {μ_{ψ
_A} (ρ) , μ_{ψ
_B} (ρ)} ≤ μ_{ψ
_A} (ρ) ≤ max {μ_{ψ
_A} (ρ) , μ_{ψ
_B} (ρ)} and max {ν_{ψ
_A} (ρ) , ν_{ψ
_B} (ρ)} ≥ ν_{ψ
_A} (ρ) ≥ min {ν_{ψ
_A} (ρ) , ν_{ψ
_B} (ρ)} for all ρ ∈ X. So (i) follows.□

Remark. Let X = {ρ₁, ρ₂, ⋯ , ρ₆}. Take two subsets A₁ = {e₁, e₃} and A₂ = {e₃, e₅} of E = {e₁, e₂, ⋯ , e₆}. Consider the PFS sets (ψ₁, A₁) and (ψ₂, A₂) given by (ψ₁, A₁) = {(e₁, {ρ₁, 0.3, 0.9} , {ρ₃, 0.3, 0.1} , {ρ₆, 0.8, 0.2}) , (e₃, {ρ₂, 0.8, 0.3} , {ρ₆, 0.3, 0}) }, and (ψ₂, A₂) = {(e₃, {ρ₁, 0.7, 0.4} , {ρ₆, 0.6, 0.6}) , (e₅, {ρ₂, 0.7, 0.3} , {ρ₃, 0.6, 0.5} , {ρ₆, 0.4, 0.2}) }

Then, $(ψ_{1}, A_{1}) \tilde{\cup} (ψ_{2}, A_{2}) = {(e_{1}, {ρ_{1}, 0.3, 0.9}, {ρ_{3}, 0.3, 0.1}, {ρ_{6}, 0.8, 0.2}), (e_{3}, {ρ_{1}, 0.7, 0.4}, {ρ_{2}, 0.8, 0.3},$ {ρ₆, 0.6, 0}) , (e₅, {ρ₂, 0.7, 0.3} , {ρ₃, 0.6, 0.5} , {ρ₆, 0.4, 0.2}) } $∴ ((ψ, A_{1}) \tilde{\cup} (ψ_{2}, A_{2}))^{c} = {(e_{1}, {ρ_{1}, 0.9, 0.3}, {ρ_{3}, 0.1, 0.3}, {ρ_{6}, 0.2, 0.8}), (e_{3}, {ρ_{1}, 0.4, 0.7}, {ρ_{2}, 0.3, 0.8}, {ρ_{6}, 0, 0.6}), (e_{5}, {ρ_{2}, 0.3, 0.7}, {ρ_{3}, 0.5, 0.6}, {ρ_{6}, 0.2, 0.4})}$

Also, (ψ₁, A₁) ^c = {(e₁, {ρ₁, 0.9, 0.3} , {ρ₃, 0.1, 0.3} , {ρ₆, 0.2, 0.8}) , (e₃, {ρ₂, 0.3, 0.8} , {ρ₆, 0, 0.3}) }, and (ψ₂, A₂) ^c = {(e₃, {ρ₁, 0.4, 0.7} , {ρ₆, 0.6, 0.6}) , (e₅, {ρ₂, 0.3, 0.7} , {ρ₃, 0.5, 0.6} , {ρ₆, 0.2, 0.4}) } so that $(ψ_{1}, A_{1})^{c} \tilde{\cap} (ψ_{2}, A_{2})^{c} = {(e_{3}, {ρ_{6}, 0, 0.6})}$ Thus, we conclude that De Morgan’s laws do not hold for PFS sets.

Theorem 3.14. If (ψ₁, A₁) and (ψ₂, A₂) are two PFS sets over X, then

$((ψ_{1}, A_{1}) \tilde{\cup} (ψ_{2}, A_{2}))^{c} \neq (ψ_{1}, A_{1})^{c} \tilde{\cap} (ψ_{2}, A_{2})^{c}$ , and

$((ψ_{1}, A_{1}) \tilde{\cap} (ψ_{2}, A_{2}))^{c} \neq (ψ_{1}, A_{1})^{c} \tilde{\cup} (ψ_{2}, A_{2})^{c}$ .

Remark. Let X = {ρ₁, ρ₂, ⋯ , ρ₆} and A = {e₁, e₃, e₆} ⊆ E where E = {e₁, e₂, ⋯ , e₆}. Consider the PFS matrix (ψ, A) = $\begin{matrix} (\begin{matrix} (0.33, 0.52) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.19, 0.98) \\ (0, 1) & (0, 1) & (0.51, 0.43) & (0, 1) & (0, 1) & (0.22, 0.75) \\ (0.72, 0.01) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.37, 0.51) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.07, 0.99) \\ (0.18, 0.52) & (0, 1) & (0.39, 0.88) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}$ so that (ψ, A) ^c = $\begin{matrix} (\begin{matrix} (0.52, 0.33) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.98, 0.19) \\ (1, 0) & (0, 1) & (0.43, 0.51) & (0, 1) & (0, 1) & (0.75, 0.22) \\ (0.01, 0.72) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (1, 0) \\ (1, 0) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.51, 0.37) \\ (1, 0) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.99, 0.07) \\ (0.52, 0.18) & (0, 1) & (0.88, 0.39) & (0, 1) & (0, 1) & (1, 0) \end{matrix}) \end{matrix}$ We observe that $(ψ, A) \tilde{\cup} (ψ, A)^{c} =$ $\begin{matrix} (\begin{matrix} (0.52, 0.33) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.98, 0.19) \\ (1, 0) & (0, 1) & (0.51, 0.43) & (0, 1) & (0, 1) & (0.75, 0.22) \\ (0.72, 0.01) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (1, 0) \\ (1, 0) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.51, 0.37) \\ (1, 0) & (0, 1) & (1, 0) & (0, 1) & (0, 1) & (0.99, 0.07) \\ (0.52, 0.18) & (0, 1) & (0.88, 0.39) & (0, 1) & (0, 1) & (1, 0) \end{matrix}) \end{matrix}$ $\neq \overset{˘}{X}$ and $(ψ, A) \tilde{\cap} (ψ, A)^{c} =$ $\begin{matrix} (\begin{matrix} (0.33, 0.52) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.19, 0.98) \\ (0, 1) & (0, 1) & (0.43, 0.51) & (0, 1) & (0, 1) & (0.22, 0.75) \\ (0.01, 0.72) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.37, 0.51) \\ (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0, 1) & (0.07, 0.99) \\ (0.18, 0.52) & (0, 1) & (0.39, 0.88) & (0, 1) & (0, 1) & (0, 1) \end{matrix}) \end{matrix}$ ≠ψ_ϕ

Hence, we have the following theorem.

Theorem 3.15. Let (ψ, A) be a PFS set over X, then

$(ψ, A) \tilde{\cup} (ψ, A)^{c} \neq \overset{˘}{X}$ , and

$(ψ, A) \tilde{\cap} (ψ, A)^{c} \neq ψ_{ϕ}$ .

Definition 3.16. A PFS set (ψ, A) is called Pythagorean fuzzy soft point (PFS point), denoted as ϑ_ψ, if for the element ϑ ∈ A we have

ψ (ϑ) ≠ ψ_ϕ, and

$ψ (ϑ^{'}) = \overset{˘}{X}$ , for all ϑ′ ∈ A - {ϑ}.

Definition 3.17. A PFS point $ϑ_{ψ} \tilde{\in} (ψ, A)$ is said to be in PFS set (ψ₁, A₁) i.e. $ϑ_{ψ} \tilde{\in} (ψ_{1}, A_{1})$ if ϑ ∈ A₁ ⇒ ψ (ϑ) ⊆ ψ₁ (ϑ).

Example 3.18. Let X = {p, c, k} and E = {ϑ₁, ϑ₂}, then $ϑ_{ψ_{1}} = {(ϑ_{1}, {(p, 0.52, 0.31), (c, 0.47, 0.19)})},$ and $ϑ_{ψ_{2}} = {(ϑ_{2}, {(p, 0.76, 0.35), (k, 0.81, 0.13)})}$ are two distinct $pfs$ -points contained in the PFS set ψ_E = {(ϑ₁, {(p, 0.52, 0.31) , (c, 0.47, 0.19)}) , (ϑ₂, {(p, 0.76, 0.35) , (k, 0.81, 0.13)})}. Note that $ψ_{E} = ϑ_{ψ_{1}} \tilde{\cup} ϑ_{ψ_{2}}$ . i.e. a Pythagorean fuzzy soft set is union of its Pythagorean fuzzy soft points.

Definition 3.19. Two PFS points $ϑ_{ψ}^{1}$ and $ϑ_{ψ}^{2}$ are equal if

$μ_{ϑ_{ψ}^{1}} (e) = μ_{ϑ_{ψ}^{2}} (e)$ , and

$ν_{ϑ_{ψ}^{1}} (e) = ν_{ϑ_{ψ}^{2}} (e)$ .

Definition 3.20. Let ψ_A and ψ_B be two PFS sets, then $ψ_{A} \tilde{\subseteq} ψ_{B}$ if and only if $ϑ_{ψ_{A}} \tilde{\in} ψ_{A} \Rightarrow ϑ_{ψ_{A}} \tilde{\in} ψ_{B}$ .

Definition 3.21. The cardinal set of the PFS set ψ_A over X is again a PFS set over E and is defined as $c ψ_{A} = {\frac{(μ_{c λ_{A}} (e), ν_{c η_{A}} (e))}{e} : e \in E}$ where μ_{cλ
_A} and ν_{cη
_A} are mappings from E to unit interval, respectively given as $μ_{c λ_{A}} (e) = \frac{| λ_{A} (e) |}{| X |}$ and $ν_{c η_{A}} (e) = \frac{| η_{A} (e) |}{| X |}$ where |λ_A (e) | and |η_A (e) | denote the scalar cardinalities of the PFS sets λ_A (e) and η_A (e) respectively, and |X| represents cardinality of the universe X.

The collection of all cardinal sets of PFS sets over X is represented as cPF^X. If A ⊆ E = {e_i : i = 1, 2, ⋯ , n}, then cψ_A ∈ cPF^X may be represented in tabular form as

E	e ₁	e ₂	⋯	e _n
cψ _A	(a₁₁, b₁₁)	(a₁₂, b₁₂)	⋯	(a_1n, b_1n)

and in matrix form as $[(a_{1 j}, b_{1 j})]_{1 \times n} = [\begin{matrix} (a_{11}, b_{11}) & (a_{12}, b_{12}) & \dots & (a_{1 n}, b_{1 n}) \end{matrix}]$ where (a_1j, b_1j) = μ_{cψ
_A} (e_j), for j = 1, 2, ⋯ , n. This matrix is termed as cardinal matrix of cψ_A over E.

Definition 3.22. Let ψ_A be a PFS set over X and cψ_A ∈ cPF^X. The PFS aggregation operator PFS_agg : cPF^X × PF^X → PFS (X, E) is defined as

\begin{matrix} {PFS}_{agg} (c ψ_{A}, ψ_{A}) & = & {\frac{μ_{ψ_{A}^{*}} (ρ)}{ρ} : ρ \in X} \\ = & {\frac{(μ_{λ_{A}^{*}} (ρ), ν_{η_{A}^{*}} (ρ))}{ρ} : ρ \in X} \end{matrix}

This collection is called aggregate Pythagorean fuzzy set of PFS set ψ_A. The membership function

μ_{λ_{A}^{*}} : X \to [0, 1]

is given by

μ_{λ_{A}^{*}} (ρ) = \frac{1}{| E |} Σ_{e \in E} (μ_{c λ_{A}} (e), μ_{λ_{A}} (e)) (ρ)

and the non-membership function

ν_{η_{A}^{*}} : X \to [0, 1]

is given by

ν_{η_{A}^{*}} (ρ) = \frac{1}{| E |} Σ_{e \in E} (ν_{c η_{A}} (e), ν_{η_{A}} (e)) (ρ)

The set PFS_agg (cψ_A, ψ_A) may be expressed in tabular form as

ψ _A	$μ_{ψ_{A}^{*}}$
ρ ₁	(a₁₁, b₁₁)
ρ ₂	(a₂₁, b₂₁)
⋮	⋮
ρ _m	(a_m1, b_m1)

and in matrix form as $[a_{i 1}, b_{i 1}]_{m \times 1} = [\begin{matrix} (a_{11}, b_{11}) \\ (a_{21}, b_{21}) \\ ⋮ \\ (a_{m 1}, b_{m 1}) \end{matrix}]$ where $[(a_{i 1}, b_{i 1})] = μ_{ψ_{A}^{*}} (ρ_{i})$ , for i = 1, 2, ⋯ , m. This matrix is called PFS aggregate matrix of PFS_agg (cψ_A, ψ_A) over X.

Theorem 3.23. Let ψ_A be a PFS set. Suppose that M_{ψ
_A}, M_{cψ
_A} and $M_{ψ_{A}}^{*}$ are respectively representative matrices of ψ_A, cψ_A and $ψ_{A}^{*}$ , then $M_{ψ_{A}} \times M_{c ψ_{A}}^{t} = M_{ψ_{A}}^{*} \times | E |$

4 Proposed method for MCGDM based on PFS sets

Decision making is a vital part of our daily life. An interviewer asked highly praised graphic designer James Victor what made him so competent? He simply replied: "I make decisions". We make millions of micro-choices daily - from what to focus our energies on, to how to talk to someone, to how to answer back an email, to what to eat which suits our health requirements. One could simply indicate that becoming a better and swifter decision-maker would be the shortest route to improving one’s daily productivity. Every person whether he is a layman or a statesman, whether he is an employee or an employer, whether he is a teacher or a student, whether he is a grown up man or an infant; everyone makes hundreds, if not millions, of decisions in his daily life. When an infant feels hungry, he can’t speak, so he decides to cry to seek attention of his attendant and shows by his body gestures that his stomach is empty.

Often we are played by our emotions to make important decisions in life and later we have to repent upon those decisions. Suppose we are making a hard choice, one that could impact our life significantly. Every time we think we have settled on something, the other option tugs us back to its side. We end up where we started: It’s a draw. Should we make ever-more-detailed lists of pros and cons and seek advice from even more trusted sources? Or should we go with our gut? Deciding how to decide is another core issue. Besides other enormous applications, Mathematics also helps us in making decisions on scientific grounds.

Besides other enormous applications, Mathematics also helps us in making decisions on scientific grounds. In this section, we give a decision making application of PFS sets. Suppose that we have an aggregate PFS set, now it is necessary to choose the best alternative form of this set. We can make a multi-criteria group decision making based on PFS sets by the following algorithm:

Algorithm 1 (Proposed)

Construct PFS set ψ_A over the universe X.

Compute the cardinalities and find the cardinal set cψ_A of ψ_A.

Find aggregate PFS set $ψ_{A}^{*}$ of ψ_A.

Compute the value of score function for each element of X utilizing $s (p) = μ_{p}^{2} - ν_{p}^{2}$ , the element for which s (p) is maximum is the best alternative.

The flow chart of Algorithm 1 is given by Figure 2.

Fig. 2

Flow chart representation of Algorithm 1.

In the forthcoming example, we first elaborate the idea to illustrate the MCGDM approach with PFS set utilizing PFS matrices and then present comparison analysis of the proposed method with two existing techniques in the next subsection.

Example 4.1. A group of a university students plans to visit some recreational site. The list of places worth-visiting is X = {p_i : i = 1, 2, ⋯ , 10}. They want to reach at some decision mathematically about which place should they visit. They agree to focus on the family of traits E = {e_i : i = 1, 2, ⋯ , 5}, where $\begin{matrix} e_{1} & = & Budget friendly \\ e_{2} & = & Close to nature \\ e_{3} & = & Shopping options \\ e_{4} & = & Hospitable locals \\ e_{5} & = & Accommodation options \end{matrix}$ After discussion, each site is evaluated according to a subset of parameters A = {e₁, e₃, e₄} ⊆ E. They apply the above algorithm as follows:

Step-1: They Construct PFS set ψ_A over the universe X as $ψ_{A} = {(e_{1}, {\frac{(0.7, 0.2)}{p_{2}}, \frac{(0.4, 0.4)}{p_{4}}, \frac{(0.2, 0.6)}{p_{7}}, \frac{(0.4, 0.2)}{p_{8}}, \frac{(0.5, 0.3)}{p_{10}}}), (e_{3}, {\frac{(0.8, 0.1)}{p_{1}}, \frac{(0.4, 0.4)}{p_{6}}, \frac{(0.3, 0.6)}{p_{9}}}), (e_{4}, {\frac{(0.6, 0.2)}{p_{1}}, \frac{(0.6, 0.2)}{p_{3}}, \frac{(0.8, 0.1)}{p_{5}}, \frac{(0.2, 0.1)}{p_{7}}})}$ Step-2: The cardinal set of ψ_A is $c ψ_{A} = {\frac{(0.22, 0.17)}{e_{1}}, \frac{(0.15, 0.11)}{e_{3}}, \frac{(0.22, 0.06)}{e_{4}}}$ . Step-3: By virtue of Theorem 3.23, the aggregate PFS set $ψ_{A}^{*}$ of ψ_A is $M_{ψ_{A}}^{*} = \frac{M_{ψ_{A}} \times M_{c ψ_{A}}^{t}}{| E |}$ $\begin{matrix} = \frac{1}{5} {(\begin{matrix} 0 & 0 & 0.8 & 0.6 & 0 \\ 0.7 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.6 & 0 \\ 0.4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.8 & 0 \\ 0 & 0 & 0.4 & 0 & 0 \\ 0.2 & 0 & 0 & 0.2 & 0 \\ 0.4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.3 & 0 & 0 \\ 0.5 & 0 & 0 & 0 & 0 \end{matrix}) (\begin{matrix} 0.22 \\ 0 \\ 0.15 \\ 0.22 \\ 0 \end{matrix}), \\ (\begin{matrix} 0 & 0 & 0.1 & 0.2 & 0 \\ 0.2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.2 & 0 \\ 0.4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.1 & 0 \\ 0 & 0 & 0.4 & 0 & 0 \\ 0.6 & 0 & 0 & 0.1 & 0 \\ 0.2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.6 & 0 & 0 \\ 0.3 & 0 & 0 & 0 & 0 \end{matrix}) (\begin{matrix} 0.17 \\ 0 \\ 0.11 \\ 0.06 \\ 0 \end{matrix})} \end{matrix}$ $\begin{matrix} = {(\begin{matrix} 0.0504 \\ 0.0308 \\ 0.0264 \\ 0.0104 \\ 0.0352 \\ 0.0120 \\ 0.0176 \\ 0.0176 \\ 0.0090 \\ 0.0220 \end{matrix}), (\begin{matrix} 0.0046 \\ 0.0068 \\ 0.0024 \\ 0.0136 \\ 0.0012 \\ 0.0088 \\ 0.0216 \\ 0.0680 \\ 0.0132 \\ 0.0102 \end{matrix})} \end{matrix}$

Hence, $ψ_{A}^{*} = {\frac{(0.0504, 0.0046)}{p_{1}}, \frac{(0.0308, 0.0068)}{p_{2}}, \frac{(0.0264, 0.0024)}{p_{3}}, \frac{(0.0104, 0.0136)}{p_{4}}, \frac{(0.0352, 0.0012)}{p_{5}}, \frac{(0.0120, 0.0088)}{p_{6}}, \frac{(0.0176, 0.0216)}{p_{7}}, \frac{(0.0176, 0.0680)}{p_{8}}, \frac{(0.0090, 0.0132)}{p_{9}}, \frac{(0.0220, 0.0102)}{p_{10}}}$

Step-4: The values of the score function s (p_i) for each element of X are tabulated below:

The rankings are depicted with the aid of bar diagram in the Figure 3.

Fig. 3

Bar chart showing values of score function for each p_i.

Since max {s (p_i)} =0.0025190 which corresponds to p₁, so they must choose the site p₁ for outing.

4.1 Comparison analysis

In this subsection, we represent two other existing algorithms based on Pythagorean fuzzy soft sets for comparison analysis first.

Algorithm 2 (Guleria and Bajaj [25])

Construct the Pythagorean fuzzy soft matrix (PFS matrix) on the basis of the parameters.

Case I (Equal weights): Compute the choice matrix for the membership and non-membership of PFS matrix. Case II (Unequal weights): Compute the weighted choice matrix for the membership and non-membership of PFS matrix.

Choose alternative with highest membership value.

Algorithm 3 (Peng et al. [38])

Obtain the aggregated Pythagorean fuzzy weighted averaging (PFWA) numbers p_i of alternative x_i by using $p_{i} = P_{PFWA} (p_{i 1}, p_{i 2}, . . ., p_{in}) = (\sum_{j = 1}^{n} w_{j} μ_{ij}, \sum_{j = 1}^{n} w_{j} ν_{ij})$

Compute the score function of s (p_i).

Select the optimal alternative by $\max_{i} s (p_{i})$ .

The proposed method is compared with two other methods, Guleria and Bajaj [25] and Peng et al. [38], as indicated in the Table 1 as given below listing the results of the comparison in the final ranking of top 5 alternatives. It can be observed in the comparison Table 1, the best selection made by the proposed method is comparable with the already established methods which is expressive in itself and approves the reliability and validity of the proposed method.

Table 1

p s (p_i)

p ₁ 0.0025190

p ₂ 0.0009024

p ₃ 0.0006912

p ₄ -0.0000768

p ₅ 0.0012376

p ₆ 0.00006656

p ₇ -0.0001568

p ₈ -0.00431424

p ₉ -0.00009324

p ₁₀ 0.00037996

p	s (p_i)
p ₁	0.0025190
p ₂	0.0009024
p ₃	0.0006912
p ₄	-0.0000768
p ₅	0.0012376
p ₆	0.00006656
p ₇	-0.0001568
p ₈	-0.00431424
p ₉	-0.00009324
p ₁₀	0.00037996

5 Multiple criteria group decision making using PFS linguistic TOPSIS approach

TOPSIS is employed to decide the superlative alternative from the notions of compromise solution. The solution which is closest to the ideal solution and farthest from negative ideal solution is acknowledged as compromise solution. In this section, we study how PFS-sets may be utilized in multiple criteria group decision making (MCGDM) using TOPSIS. First of all we shall extend TOPSIS to PFS-sets and then shall consider a problem of stock exchange investment. Amongst enormous techniques found in literature, TOPSIS holds a central position in tackling such problems. Depending upon the nature of the problem under consideration, every technique has its own pros and cons.

We make an inception by explaining the proposed technique step by step. The proposed PFS TOPSIS is generalization of fuzzy soft TOPSIS presented by Eraslan and Karaaslan [24].

Algorithm 4 (PFS TOPSIS):

Step 1: Identifying the problem: Assume that $E = {D_{i} : i \in IN}$ is a finite set of decision makers/experts, $V = {{\ddot{v}}_{i} : i \in IN}$ is the finite collection of alternatives and D = {p_i : i ∈ IN} is a finite family of parameters/criterion.

Step 2: By selecting the linguistic terms from Table 2, constructing weighted parameter matrix $P$ as $P = [w_{ij}]_{n \times m} = (\begin{matrix} w_{11} & w_{12} & \dots & w_{1 m} \\ w_{21} & w_{22} & \dots & w_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{i 1} & w_{i 2} & \dots & w_{im} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{n 1} & w_{n 2} & \dots & w_{nm} \end{matrix})$

where w_ij is the weight assigned by the expert $D_{i}$ to the alternative $P_{j}$ by considering linguistic variables as given (for example) in Table 2.

Table 2
Comparison analysis of final ranking with existing methods in given numerical example

Method Ranking of alternatives The optimal alternative

Algorithm 1 (Proposed) p₁ ≻ p₅ ≻ p₂ ≻ p₃ ≻ p₁₀ p ₁

Algorithm 2 Case-I (Guleria and Bajaj [25]) p₁ ≻ p₅ ≻ p₂ ≻ p₃ ≻ p₁₀ p ₁

Algorithm 2 Case-II (Guleria and Bajaj [25]) p₁ ≻ p₂ ≻ p₅ ≻ p₃ ≻ p₁₀ p ₁

Algorithm 3 (Peng et al. [38]) p₁ ≻ p₇ ≻ p₂ ≻ p₈ ≻ p₅ p ₁

Method	Ranking of alternatives	The optimal alternative
Algorithm 1 (Proposed)	p₁ ≻ p₅ ≻ p₂ ≻ p₃ ≻ p₁₀	p ₁
Algorithm 2 Case-I (Guleria and Bajaj [25])	p₁ ≻ p₅ ≻ p₂ ≻ p₃ ≻ p₁₀	p ₁
Algorithm 2 Case-II (Guleria and Bajaj [25])	p₁ ≻ p₂ ≻ p₅ ≻ p₃ ≻ p₁₀	p ₁
Algorithm 3 (Peng et al. [38])	p₁ ≻ p₇ ≻ p₂ ≻ p₈ ≻ p₅	p ₁

Step 3: Construct normalized weighted matrix $\hat{N} = [{\hat{n}}_{ij}]_{n \times m} = (\begin{matrix} {\hat{n}}_{11} & {\hat{n}}_{12} & \dots & {\hat{n}}_{1 m} \\ {\hat{n}}_{21} & {\hat{n}}_{22} & \dots & {\hat{n}}_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\hat{n}}_{i 1} & {\hat{n}}_{i 2} & \dots & {\hat{n}}_{im} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\hat{n}}_{n 1} & {\hat{n}}_{n 2} & \dots & {\hat{n}}_{nm} \end{matrix})$

where ${\hat{n}}_{ij} = \frac{w_{ij}}{\sqrt{Σ_{i = 1}^{n} w_{ij}^{2}}}$ and obtaining the weighted vector $W = (w_{1}, w_{2}, \dots, w_{m})$ , where $w_{i} = \frac{w_{i}}{Σ_{α = 1}^{n} w_{α i}}$ and $w_{j} = \frac{Σ_{i = 1}^{n} {\hat{n}}_{ij}}{n}$ . Notice that $Σ_{α = 1}^{n} w_{α i}$ represents sum of weights of i^th column in the matrix $P$ cited at step 2 above.

Step 4: Construct PFS decision matrix $D_{i} = [v_{jk}^{i}]_{l \times m} = (\begin{matrix} v_{11}^{i} & v_{12}^{i} & \dots & v_{1 m}^{i} \\ v_{21}^{i} & v_{22}^{i} & \dots & v_{2 m}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ v_{j 1}^{i} & v_{j 2}^{i} & \dots & v_{jm}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ v_{l 1}^{i} & v_{l 2}^{i} & \dots & v_{lm}^{i} \end{matrix})$

where $v_{jk}^{i}$ is a PFS-element, for i^th decision maker so that D_i makes PFS-topolgy for each i. Then obtain the aggregating matrix $A = \frac{D_{1} + D_{2} + \dots + D_{n}}{n} = [{\dot{v}}_{jk}]_{l \times m}$

Step 5: Obtain the weighted PFS decision matrix $B = [{\ddot{v}}_{jk}]_{l \times m} = (\begin{matrix} {\ddot{v}}_{11} & {\ddot{v}}_{12} & \dots & {\ddot{v}}_{1 m} \\ {\ddot{v}}_{21} & {\ddot{v}}_{22} & \dots & {\ddot{v}}_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\ddot{v}}_{j 1} & {\ddot{v}}_{j 2} & \dots & {\ddot{v}}_{jm} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\ddot{v}}_{l 1} & {\ddot{v}}_{l 2} & \dots & {\ddot{v}}_{lm} \end{matrix})$

where ${\ddot{v}}_{jk} = w_{k} \times {\dot{v}}_{jk}$ .

Step 6: Obtain PFS-valued positive ideal solution (PFSV-PIS) and PFS-valued negative ideal solution (PFSV-NIS), employing in order $\begin{matrix} PFSV - PIS & = & {{\ddot{v}}_{1}^{+}, {\ddot{v}}_{2}^{+}, \dots, {\ddot{v}}_{l}^{+}} \\ = & {(\lor_{k} {\ddot{v}}_{jk}, \land_{k} {\ddot{v}}_{jk}) : k = 1, 2, \dots, m} \end{matrix}$ and $\begin{matrix} PFSV - NIS & = & {{\ddot{v}}_{1}^{-}, {\ddot{v}}_{2}^{-}, \dots, {\ddot{v}}_{l}^{-}} \\ = & {(\land_{k} {\ddot{v}}_{jk}, \lor_{k} {\ddot{v}}_{jk}) : k = 1, 2, \dots, m} \end{matrix}$ where ∨ stands for PFS union and ∧ represents PFS intersection.

Step 7: Compute PFS-Euclidean distances of each alternative from PFSV-PIS and PFSV-NIS, respectively, utilizing $(d_{j}^{+})^{2} = Σ_{k = 1}^{m} {(μ_{jk}^{+} -^{+} μ_{k}^{+})^{2} + (ν_{jk}^{+} -^{+} ν_{k}^{+})^{2} + (μ_{jk}^{-} -^{+} μ_{k}^{-})^{2} + (ν_{jk}^{-} -^{+} ν_{k}^{-})^{2}}$ and $(d_{j}^{-})^{2} = Σ_{k = 1}^{m} {(μ_{jk}^{+} -^{-} μ_{k}^{+})^{2} + (ν_{jk}^{+} -^{-} ν_{k}^{+})^{2} + (μ_{jk}^{-} -^{-} μ_{k}^{-})^{2} + (ν_{jk}^{-} -^{-} ν_{k}^{-})^{2}}$

Step 8: The closeness coefficient of each alternative with ideal solution by making use of $C^{*} ({\ddot{v}}_{j}) = \frac{d_{j}^{-}}{d_{j}^{+} + d_{j}^{-}} \in [0, 1]$

Step 9: Rank the alternatives in decreasing (or increasing) order to get the preference order of the alternatives.

The flow chart of these procedural steps is given in the Figure 4.

Fig. 4

Flow chart of Algorithm 4.

Example 5.1. Assume that a firm plans to invest some money in stock exchange by purchasing some shares of best four companies. In order to minimize the risk factor, they decide to invest their money in percentage of 40%, 30%, 20% and 10% in accordance with the top ranked four companies.

Step 1: Identifying the problem: Assume that $E = {D_{i} : i = 1, 2, 3, 4}$ is the set comprising financial experts/decision makers, $V = {{\ddot{v}}_{i} : i = 1, 2, \dots, 10}$ is the collection of companies/alternatives under consideration and D = {p_i : i = 1, 2, ⋯ , 6} is a finite family of parameters/criterion, where $\begin{matrix} p_{1} & = & Standingreputationofthecompany \\ p_{2} & = & Prospects \\ p_{3} & = & InvestmentSafeguard \\ p_{4} & = & CollateralValue \\ p_{5} & = & ProductViability \\ p_{6} & = & Impactonsharemarket \end{matrix}$ Step 2: Keeping in view the track record of these companies, the technical team of the firm assigns weights to each parameter using linguistic terms from Table 2 and construct weighted parameter matrix $\begin{matrix} P & = & [w_{ij}]_{4 \times 6} \\ = & (\begin{matrix} VHI & AI & HI & LI & LI & AI \\ HI & AI & VHI & AI & VHI & VHI \\ AI & HI & VHI & AI & HI & HI \\ VHI & HI & VHI & HI & HI & VHI \end{matrix}) \\ = & (\begin{matrix} 0.90 & 0.60 & 0.75 & 0.45 & 0.45 & 0.60 \\ 0.75 & 0.60 & 0.90 & 0.60 & 0.90 & 0.90 \\ 0.60 & 0.75 & 0.90 & 0.60 & 0.75 & 0.75 \\ 0.90 & 0.75 & 0.90 & 0.75 & 0.75 & 0.90 \end{matrix}) \end{matrix}$ where w_ij is the weight provided by the medical specialist D_i (row-wise) to each parameter p_j (column-wise).

Step 3: The normalized weighted matrix is $\begin{matrix} \hat{N} & = & [{\hat{n}}_{ij}]_{4 \times 6} \\ = & (\begin{matrix} 0.564 & 0.442 & 0.434 & 0.369 & 0.308 & 0.376 \\ 0.470 & 0.442 & 0.520 & 0.492 & 0.616 & 0.564 \\ 0.376 & 0.552 & 0.520 & 0.492 & 0.513 & 0.470 \\ 0.564 & 0.552 & 0.520 & 0.615 & 0.513 & 0.564 \end{matrix}) \end{matrix}$ and hence the weighted vector is $W = (0.15, 0.19, 0.14, 0.20, 0.17, 0.15)$ .

Step 4: Keeping in view the track record of the companies under consideration, the PFS decision matrix D_i of each specialist is provided in which alternatives are represented row-wise whereas parameters are represented column-wise. Hence, the aggregated decision matrix comes out to be $A = \frac{D_{1} + D_{2} + D_{3} + D_{4}}{4} =$ $\begin{matrix} (\begin{matrix} (0.47, 0.63) & (0.59, 0.34) & (0.82, 0.21) & (0.64, 0.63) & (0.28, 0.36) & (0.43, 0.59) \\ (0.23, 0.19) & (0, 1) & (0.79, 0.18) & (0.89, 0.32) & (0.26, 0.17) & (0, 1) \\ (0.26, 0.35) & (0.78, 0.15) & (0, 1) & (0.54, 0.41) & (0.27, 0.28) & (0.76, 0.64) \\ (0.85, 0.42) & (0.19, 0.12) & (0.33, 0.47) & (0.36, 0.64) & (0.54, 0.87) & (0.81, 0.25) \\ (0.26, 0.18) & (0.33, 0.67) & (0.45, 0.61) & (0.85, 0.28) & (0, 1) & (0.16, 0.16) \\ (0.72, 0.12) & (0.63, 0.26) & (0, 1) & (0.27, 0.34) & (0.88, 0.43) & (0.17, 0.26) \\ (0.27, 0.31) & (0.81, 0.18) & (0.11, 0.22) & (0.45, 0.44) & (0.17, 0.12) & (0.55, 0.56) \\ (0.81, 0.29) & (0.91, 0.12) & (0.21, 0.19) & (0, 1) & (0.45, 0.45) & (0.28, 0.29) \\ (0.64, 0.37) & (0.45, 0.61) & (0.23, 0.31) & (0.28, 0.29) & (0.17, 0.15) & (0.61, 0.18) \\ (0.37, 0.71) & (0.72, 0.19) & (0.19, 0.47) & (0.14, 0.08) & (0.03, 0.16) & (0.07, 0.12) \end{matrix}) \end{matrix}$ $= [{\dot{v}}_{jk}]_{10 \times 6}$

Step 5: The weighted PFS decision matrix $B = [{\ddot{v}}_{jk}]_{10 \times 6} =$ $\begin{matrix} (\begin{matrix} (0.07, 0.09) & (0.11, 0.06) & (0.11, 0.03) & (0.13, 0.13) & (0.05, 0.06) & (0.06, 0.09) \\ (0.03, 0.03) & (0.00, 0.19) & (0.11, 0.03) & (0.18, 0.06) & (0.04, 0.03) & (0.00, 0.15) \\ (0.04, 0.05) & (0.15, 0.03) & (0.00, 0.14) & (0.11, 0.08) & (0.05, 0.05) & (0.11, 0.10) \\ (0.13, 0.06) & (0.04, 0.02) & (0.05, 0.07) & (0.07, 0.13) & (0.09, 0.15) & (0.12, 0.04) \\ (0.04, 0.03) & (0.06, 0.13) & (0.06, 0.09) & (0.17, 0.06) & (0.00, 0.17) & (0.02, 0.02) \\ (0.11, 0.02) & (0.12, 0.05) & (0.00, 0.14) & (0.05, 0.07) & (0.15, 0.07) & (0.03, 0.04) \\ (0.04, 0.05) & (0.15, 0.03) & (0.02, 0.03) & (0.09, 0.09) & (0.03, 0.02) & (0.08, 0.08) \\ (0.12, 0.04) & (0.17, 0.02) & (0.03, 0.03) & (0.00, 0.20) & (0.08, 0.08) & (0.04, 0.04) \\ (0.10, 0.06) & (0.09, 0.12) & (0.03, 0.04) & (0.06, 0.06) & (0.03, 0.03) & (0.09, 0.03) \\ (0.06, 0.11) & (0.14, 0.04) & (0.03, 0.07) & (0.03, 0.02) & (0.01, 0.03) & (0.01, 0.02) \end{matrix}) \end{matrix}$ where ${\ddot{v}}_{jk} = w_{k} \times {\dot{v}}_{jk}$ .

Step 6: Now, we locate PFS-valued positive ideal solution (PFSV-PIS) and PFS-valued negative ideal solution (PFSV-NIS) and are listed, respectively, as $PFSV - PIS = {{\ddot{v}}_{1}^{+}, {\ddot{v}}_{2}^{+}, {\ddot{v}}_{3}^{+}, {\ddot{v}}_{4}^{+}, {\ddot{v}}_{5}^{+}, {\ddot{v}}_{6}^{+}, {\ddot{v}}_{7}^{+}, {\ddot{v}}_{8}^{+}, {\ddot{v}}_{9}^{+}, {\ddot{v}}_{10}^{+}} = {(0.13, 0.03), (0.18, 0.03), (0.15, 0.03), (0.13, 0.02), (0.17, 0.02), (0.15, 0.02), (0.15, 0.02), (0.17, 0.02), (0.10, 0.03), (0.14, 0.02)}$ and $PFSV - NIS = {{\ddot{v}}_{1}^{-}, {\ddot{v}}_{2}^{-}, {\ddot{v}}_{3}^{-}, {\ddot{v}}_{4}^{-}, {\ddot{v}}_{5}^{-}, {\ddot{v}}_{6}^{-}, {\ddot{v}}_{7}^{-}, {\ddot{v}}_{8}^{-}, {\ddot{v}}_{9}^{-}, {\ddot{v}}_{10}^{-}}$ = {(0.05, 0.13) , (0.00, 0.19) , (0.00, 0.14) , (0.04, 0.15) , (0.00, 0.17) , (0.00, 0.14) , (0.02, 0.09) , (0.00, 0.20) , (0.03, 0.12) , (0.01, 0.11)}

Step 7, 8: The PFS-Euclidean distances of each alternative from PFSV-PIS and PFSV-NIS along with closeness coefficients are given in Table 3 below:

Table 3

Linguistic terms for judging alternatives

Linguistic Terms	Fuzzy Weights
Very High Importance (VHI)	0.90
High Importance (HI)	0.75
Average Importance (AI)	0.60
Low Importance (LI)	0.45

Step 9: The preference order of the alternatives, therefore, is ${\ddot{v}}_{9} ≻ {\ddot{v}}_{8} ≻ {\ddot{v}}_{6} ≻ {\ddot{v}}_{3} ≻ {\ddot{v}}_{2} ≻ {\ddot{v}}_{4} ≻ {\ddot{v}}_{1} ≻ {\ddot{v}}_{5} ≻ {\ddot{v}}_{10} ≻ {\ddot{v}}_{7}$ These rankings are depicted in Figure 5 cited below:

Fig. 5

Line chart of rankings.

Keeping in view the above ranking, it may be concluded that the firm should invest 40% of the money on ${\ddot{v}}_{9}$ , 30% on ${\ddot{v}}_{8}$ , 20% on ${\ddot{v}}_{6}$ and the rest 10% on ${\ddot{v}}_{3}$ .

6 Multiple criteria group decision making using PFS VIKOR method

VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) is a Serbian word used to mean multiple criteria optimization/analysis and compromise solution. This method was initially established by Serafim Opricovic to tackle decision making problems with disagreeing and non-commensurable criteria, supposing that compromise is suitable for conflict determination, the decision maker looks for a solution that is closest to the ideal, and the alternatives are assessed according to all recognized principles. This method has emerged as a quite popular multi-criteria decision making technique due to its computational ease and solution exactitude. It emphases on deciding on and ranking from a collection of viable choices, and decides compromise solution for a problem with inconsistent standards to comfort the decision makers in reaching a final conclusion. It determines the compromise grading list established for the specific degree of closeness to the ideal solution.

We begin by explaining the proposed technique step by step as follows:

Algorithm 5 (PFS VIKOR):

where w_ij is the weight assigned by the expert $D_{i}$ to the alternative $P_{j}$ by considering linguistic variables as given (for example) in Table 2.

where ${\ddot{v}}_{jk} = w_{k} \times {\dot{v}}_{jk}$ .

Step 6: Obtain PFS-valued positive ideal solution (PFSV-PIS) and PFS-valued negative ideal solution (PFSV-NIS), employing in order $\begin{matrix} PFSV - PIS & = & {{\ddot{v}}_{1}^{+}, {\ddot{v}}_{2}^{+}, \dots, {\ddot{v}}_{l}^{+}} \\ = & {(\lor_{j} {\ddot{v}}_{jk}, \land_{k} {\ddot{v}}_{jk}) : j = 1, 2, \dots, l} \end{matrix}$ and $\begin{matrix} PFSV - NIS & = & {{\ddot{v}}_{1}^{-}, {\ddot{v}}_{2}^{-}, \dots, {\ddot{v}}_{l}^{-}} \\ = & {(\land_{j} {\ddot{v}}_{jk}, \lor_{k} {\ddot{v}}_{jk}) : j = 1, 2, \dots, l} \end{matrix}$ where ∨ stands for PFS union and ∧ represents PFS intersection.

Step 7: Compute the key values of VIKOR method for each alternative i.e. the group utility value S_i, the individual regret value R_i, and compromise value Q_i by making use of $\begin{matrix} S_{i} & = & Σ_{j = 1}^{m} w_{j} (\frac{d ({\ddot{v}}_{j}^{+}, {\ddot{v}}_{ij})}{d ({\ddot{v}}_{j}^{+}, {\ddot{v}}_{j}^{-})}) \\ R_{i} & = & {max}_{j = 1}^{m} w_{j} (\frac{d ({\ddot{v}}_{j}^{+}, {\ddot{v}}_{ij})}{d ({\ddot{v}}_{j}^{+}, {\ddot{v}}_{j}^{-})}) \\ Q_{i} & = & κ (\frac{S_{i} - S^{-}}{S^{+} - S^{-}}) + (1 - κ) (\frac{R_{i} - R^{-}}{R^{+} - R^{-}}) \end{matrix}$ where $S^{+} = max_{i} S_{i}$ , $S^{-} = min_{i} S_{i}$ , $R^{+} = max_{i} R_{i}$ , and $R^{-} = min_{i} R_{i}$ . The real number κ is termed as coefficient of decision mechanism. The role of the coefficient κ is that if compromise solution is to be selected by majority, we choose κ > 0.5; for consensus we use κ = 0.5, and κ < 0.5 represents veto. $w_{j}$ represents the weight of the j^th criteria, which expresses its relative importance.

Step 8: Rank the choices and derive compromise solution. Arrange S_i, R_i, and Q_i in increasing order to make three ranking lists S_[.], R_[.], and Q_[.]. The alternative ${\ddot{v}}_{α}$ will be declared compromise solution if it ranks the best in Q_[.] (having least value) and satisfies the following two requirements simultaneously: [C - 1] Acceptable advantage: If ${\ddot{v}}_{α}$ and ${\ddot{v}}_{β}$ represent top two alternatives in Q_i, then $Q ({\ddot{v}}_{β}) - Q ({\ddot{v}}_{α}) \geq \frac{1}{n - 1}$ where n is the number of parameters. [C - 2] Acceptable stability: The alternative ${\ddot{v}}_{α}$ should be best ranked by S_i and/or R_i. If above two conditions are not met simultaneously, then there exist multiple compromise solutions: [(i)] If only condition C-1 is satisfied then both alternatives ${\ddot{v}}_{α}$ and ${\ddot{v}}_{β}$ will be the compromise solutions. [(ii)] If condition C-1 is not fulfilled then the alternatives ${\ddot{v}}_{α}, {\ddot{v}}_{β}, \dots, {\ddot{v}}_{η}$ would be the compromise solutions, where ${\ddot{v}}_{η}$ may be determined using $Q ({\ddot{v}}_{η}) - Q ({\ddot{v}}_{α}) \geq \frac{1}{n - 1}$ for the maximum.

The flow chart of these procedural steps is given in the Figure 6.

Fig. 6

Flow chart of Algorithm 5.

Example 6.1. We re-solve Example 5.1 using VIKOR method employing the technique given in Algorithm 5. The first five steps are the same as in Example 5.1. So we begin with step 6.

Step 7: Taking κ = 0.5, the group utility value S_i, the individual regret value R_i, and compromise value Q_i for each alternative ${\ddot{v}}_{i}$ are computed by making use of $\begin{matrix} S_{i} & = & Σ_{j = 1}^{m} w_{j} (\frac{d ({\ddot{v}}_{j}^{+}, {\ddot{v}}_{ij})}{d ({\ddot{v}}_{j}^{+}, {\ddot{v}}_{j}^{-})}) \\ R_{i} & = & {max}_{j = 1}^{m} w_{j} (\frac{d ({\ddot{v}}_{j}^{+}, {\ddot{v}}_{ij})}{d ({\ddot{v}}_{j}^{+}, {\ddot{v}}_{j}^{-})}) \\ Q_{i} & = & κ (\frac{S_{i} - S^{-}}{S^{+} - S^{-}}) + (1 - κ) (\frac{R_{i} - R^{-}}{R^{+} - R^{-}}) \end{matrix}$ and are given in Table 4 below:

Table 4

Distance measures & closeness coefficient of each alternative

Alternative $({\ddot{v}}_{i})$	$d_{i}^{+} = d^{+} ({\ddot{v}}_{i}, PFSV - PIS)$	$d_{i}^{-} = d^{+} ({\ddot{v}}_{i}, PFSV - NIS)$	$C_{i}^{*}$
${\ddot{v}}_{1}$	0.1863	0.1926	0.5083
${\ddot{v}}_{2}$	0.3615	0.3772	0.5106
${\ddot{v}}_{3}$	0.2610	0.2903	0.5266
${\ddot{v}}_{4}$	0.2309	0.2406	0.5103
${\ddot{v}}_{5}$	0.3657	0.3167	0.4641
${\ddot{v}}_{6}$	0.2648	0.3077	0.5375
${\ddot{v}}_{7}$	0.2480	0.1992	0.4454
${\ddot{v}}_{8}$	0.3360	0.4232	0.5574
${\ddot{v}}_{9}$	0.1470	0.2069	0.5846
${\ddot{v}}_{10}$	0.2748	0.2220	0.4469

Table 5

Values of S_i, R_i and Q_i for all alternatives

Alternative $({\ddot{v}}_{i})$	S _i	R _i	Q _i
${\ddot{v}}_{1}$	0.419	0.102	0.005
${\ddot{v}}_{2}$	0.570	0.190	0.913
${\ddot{v}}_{3}$	0.487	0.140	0.406
${\ddot{v}}_{4}$	0.466	0.122	0.250
${\ddot{v}}_{5}$	0.582	0.170	0.848
${\ddot{v}}_{6}$	0.435	0.140	0.246
${\ddot{v}}_{7}$	0.450	0.106	0.120
${\ddot{v}}_{8}$	0.439	0.200	0.561
${\ddot{v}}_{9}$	0.452	0.101	0.101
${\ddot{v}}_{10}$	0.559	0.127	0.561

Step 8: The rank of choices is as under: $By Q_{i} : {\ddot{v}}_{1} ≺ {\ddot{v}}_{9} ≺ {\ddot{v}}_{7} ≺ {\ddot{v}}_{6} ≺ {\ddot{v}}_{4} ≺ {\ddot{v}}_{3} ≺ {\ddot{v}}_{8} = {\ddot{v}}_{10} ≺ {\ddot{v}}_{5} ≺ {\ddot{v}}_{2}$ $By S_{i} : {\ddot{v}}_{1} ≺ {\ddot{v}}_{6} ≺ {\ddot{v}}_{8} ≺ {\ddot{v}}_{7} ≺ {\ddot{v}}_{9} ≺ {\ddot{v}}_{4} ≺ {\ddot{v}}_{3} ≺ {\ddot{v}}_{10} ≺ {\ddot{v}}_{2} ≺ {\ddot{v}}_{5}$ $By R_{i} : {\ddot{v}}_{9} ≺ {\ddot{v}}_{1} ≺ {\ddot{v}}_{7} ≺ {\ddot{v}}_{4} ≺ {\ddot{v}}_{10} ≺ {\ddot{v}}_{6} = {\ddot{v}}_{3} ≺ {\ddot{v}}_{5} ≺ {\ddot{v}}_{2} ≺ {\ddot{v}}_{8}$ Since $Q ({\ddot{v}}_{9}) - ({\ddot{v}}_{1}) = 0.096 ≱ \frac{1}{5}$ so the condition C-1 is not satisfied. Further, $Q ({\ddot{v}}_{6}) - ({\ddot{v}}_{1}) = 0.241 \geq \frac{1}{5}$ Thus, we conclude that ${\ddot{v}}_{1}, {\ddot{v}}_{9}, {\ddot{v}}_{7}$ and ${\ddot{v}}_{6}$ are multiple compromise solutions. These rankings are depicted in Figure 7.

Fig. 7

Bar chart of rankings.

Hence, the firm should invest 40% on ${\ddot{v}}_{1}$ , 30% on ${\ddot{v}}_{9}$ , 30% on ${\ddot{v}}_{7}$ , and the remaining 10% on ${\ddot{v}}_{6}$ .

6.1 Comparison analysis of PFS linguistic TOPSIS approach and PFS VIKOR

Both models assume a scale factor for each criterion. This scale have need of eliminating the different units of all criteria values. For ranking the values calculated by these methods are defined with the help of an aggregating function. The major difference between two methods appears in the aggregation approaches. The VIKOR method provides an aggregating function representing the distances from ideal solution. In addition to TOPSIS, VIKOR method provides a compromise solution with an advantage rate.

The normalization procedures are different in the two methods. VIKOR method uses linear normalization whereas in TOPSIS vector normalization is used. TOPSIS method introduces the ranking index including the distances from the ideal point and from the negative-ideal point. These distances in TOPSIS are simply summed without paying heed to their relative importance.

The TOPSIS method uses n-dimensional Euclidean distance that by itself could represent some balance between total and individual satisfaction, but uses it in a different way than VIKOR, where weight κ is introduced. Both methods provide a ranking list. The highest ranked alternative by VIKOR is closest to the ideal solution. However, the highest ranked alternative by TOPSIS is the best in terms of the ranking index, which does not mean that it is always closest to the ideal solution. In addition to ranking, the VIKOR method proposes a compromise solution with an advantage rate.

Both methods are mostly used MCDM methods in decision making projects. Ability to adapt methods to Geographical Information Systems increase the applicability in the field of environmental, business, stock exchange, trade, commerce, meteorological, medical diagnosis, armed forces decisions and engineering problems. Different solutions enhance the decision process and increase the preference of decision makers. Thus, right decision giving can be possible with two probabilities.

7 Conclusion

We studied various properties of Pythagorean fuzzy soft sets along with some of their particular characteristics, in this paper. We used tables and matrices to conceive the notions effectively. We also studied some set theoretic properties of these sets. We proposed three algorithms accompanied by flow charts in this paper, for MCGDM under PFS environment. The first one is by making use of PFS aggregation operator and score function values based technique. We applied this method on site selection problem. Two other proposed algorithms are based on PFS linguistic TOPSIS and PFS linguistic VIKOR approaches. We rendered an application of stock exchange investment problem by these two techniques. We made use of different sorts of statistical charts to visualize the rankings of different alternatives under consideration. We hope this study will open new doors of enquiry for the knowledge seekers and researchers in various fields including real world problems, artificial intelligence, pattern recognition, image processing and forecasting.

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Pythagorean fuzzy soft MCGDM methods based on TOPSIS,VIKOR and aggregation operators

Abstract

Keywords

1 Introduction

2 Preliminaries

4 Proposed method for MCGDM based on PFS sets

Table 1 p s (p i ) p 1 0.0025190 p 2 0.0009024 p 3 0.0006912 p 4 -0.0000768 p 5 0.0012376 p 6 0.00006656 p 7 -0.0001568 p 8 -0.00431424 p 9 -0.00009324 p 10 0.00037996

7 Conclusion

References

Table 1

p s (p_i)

p ₁ 0.0025190

p ₂ 0.0009024

p ₃ 0.0006912

p ₄ -0.0000768

p ₅ 0.0012376

p ₆ 0.00006656

p ₇ -0.0001568

p ₈ -0.00431424

p ₉ -0.00009324

p ₁₀ 0.00037996