A novel approach to censuses process by using Pythagorean m-polar fuzzy Dombi’s aggregation operators

Abstract

This manuscript provides an advanced mathematical model for the censuses process to reduce the drawbacks of the existing methods. From the living ideas of m-polar fuzzy set (MPFS) and Pythagorean fuzzy set (PFS), we establish a novel concept of Pythagorean m-polar fuzzy set (PMPFS). We introduce some fundamental operations on Pythagorean m-polar fuzzy sets and explain these concepts with the help of illustrations. With this novel perspective, we build up modified forms of Dombi’s aggregation operators named as Pythagorean m-polar fuzzy Dombi weighted arithmetic average (PMPFDWAA) operator and Pythagorean m-polar fuzzy Dombi weighted geometric average (PMPFDWGA) operator. We discuss certain properties of the proposed operators based on Pythagorean m-polar fuzzy numbers (PMPFNs). Mathematical modeling on real world problems often implicate multi-factors, multi-attributes and multi-polar information. We discuss a case study for the censuses process to elaborate the proposed algorithm for multi-criteria decision-making (MCDM). We also discuss how the drawbacks of existing methods can be handled by applying this novel perspective. Lastly, we present a comparative analysis, validity of proposed algorithm, influence of operational parameter, convergence and sensitivity analysis to indicate the flexibility and advantages of the proposed method.

Keywords

Pythagorean m-polar fuzzy set (PMPFS)Pythagorean m-polar fuzzy Dombi weighted arithmetic average (PMPFDWAA) operator Pythagorean m-polar fuzzy Dombi weighted geometric average (PMPFDWGA) operator multi-criteria decision-making censuses process

1 Introduction

The genuine ability of fuzzy logic is in its power to treat and manipulate verbally-stated information based on perceptions rather than equations. Zadeh [1] initiated 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. The classical logic has been extended to fuzzy logic, which is characterized by a membership function ranging in [0, 1] and provides a powerful alternative to probability theory to characterize imprecision, uncertainty, and obscureness in various fields. The concept of linguistic variable was also introduced by Zadeh [2] with its applications on approximate reasoning. Atanassov [3] proposed the idea of intuitionistic fuzzy set (IFS) as an extension of fuzzy set by introducing the concepts of membership grades and non-membership grades along with the restriction that sum of both grades must not exceed unity. To eradicate this drawback an extension of IFS was introduced by Yager [5, 6] named as Pythagorean fuzzy set (PFS). Recently, many researchers have been introduced aggregation operators based on PFS for the information fusion. The ideas of score function and accuracy function play an important role in the ranking methodology of the alternatives in the decision analysis. Score function was established by Tversky and Kahneman [8] and further developed by Chen and Tan [7]. Dombi [9] presented the concept of Dombi’s operators with the contribution of t-norm and t-conorm. Ali and Shabir [45] brought out the ideas of logic connectives for soft set and fuzzy soft set (FSS). Feng et al. [10] introduced an adjustable approach to fuzzy soft set based decision-making. Feng et al. [11] presented another view of generalized intuitionistic fuzzy soft sets (GIFSSs) and its application in multi-attribute decision making (MADM). Feng et al. [12] presented Lexicographic orders for intuitionistic fuzzy numbers (IFNs) and their relationships.

Jose and Kuriaskose [13] investigated aggregation operators with the corresponding score function for MCDM based on IFNs. Kaur and Garg [14] established aggregation operators based on cubic intuitionistic fuzzy numbers (CIFNs) and used these operators in decision-making approach for job selection. Lu and Ye [16] established Dombi’s aggregation operators for the linguistic cubic variables and developed its application with the proposed algorithm. Maji et al. [19] introduced some new results on FSSs. Mahmood et al. [20] presented some generalized aggregation operators for cubic hesitant fuzzy numbers (CHFNs) and used them in MCDM. Riaz and Hashmi [21 –25] introduced cubic m-polar fuzzy sets, m-polar neutrosophic sets, soft rough Pythagorean m-polar fuzzy sets, Pythagorean m-polar soft rough fuzzy sets, m-polar neutrosophic topology and clustering analysis. They established different aggregation operators and algorithms for solving MCDM and MAGDM problems. They introduced the novel idea of linear Diophantine fuzzy set as an extension of IFSs and PFSs and presented its application in MADM problems. They developed fixed point theorems based on fuzzy neutrosophic soft (FNS)-mapping. Riaz and Tehrim [26] established MAGDM based on cubic bipolar fuzzy information using averaging aggregation operators. Naeem et al. [27] presented Pythagorean fuzzy soft MCGDM methods based on TOPSIS and VIKOR.

Shi and Ye [30] established some Dombi’s operators for neutrosophic cubic set (NCS) and presented its applications in MADM problems. Xu [31, 32] presented aggregation operations for intuitionistic fuzzy sets and hesitant fuzzy sets and utilized them in the decision analysis. Yager [33] initiated MCDM with ordinal linguistic IFS and used this idea in the mobile application. Ye [35, 36] introduced prioritized aggregation operators in the context of interval-valued hesitant fuzzy set (IVHFS) and presented its application in multi-attribute group decision-making (MAGDM). Lui et al. [17] developed the centroid transformations of intuitionistic fuzzy values with the help of some aggregation operators. Liang and Xu [18] established TOPSIS method for MCDM by using the hesitant Pythagorean fuzzy sets (HPFSs). Yu et al. [34] used unbalanced hesitant fuzzy linguistic term sets and used in the multi-criteria group decision-making (MCGDM) problems. Zhang et al. [40 –42] introduced the idea of deriving priority weights from intuitionistic multiplicative preference relations in the group decision-making. They introduced additive consistency analysis and used this idea for the improvement of hesitant fuzzy preference relations. They established priority weights and consistency for incomplete hesitant fuzzy preference relations. Zhang et al. [43, 44] introduced consensus efficiency in the group decision-making problem and discussed its optimal design with a comprehensive comparative study. They presented consensus-based MAGDM technique on failure mode and effect analysis with the help of linguistic terms. Brown et al. [4] gave his perception in his book named as Envisioning the 2020 census. Killick et al. [15] introduced some results on censuses process as an information source in public policy-making. Many researchers have studied fuzzy set theory, soft set theory, rough set theory and used these set theoretic models in the solving many real world problems (see [11 , 45–47]).

PFS is a mathematical tool to deal with vagueness considering the membership grade $T_{P} (ς)$ and non-membership $F_{P} (ς)$ satisfying the condition $(T_{P} (ς))^{2} + (F_{P} (ς))^{2} \leq 1$ . It can be used to characterize the uncertain information more sufficiently and accurately than IFS. PFS has attracted by many scholars and they have extended it into new types. These extensions have been used in many areas such as decision-making, aggregation operators, and information measures. MPFS is a powerful tool for depicting fuzziness and uncertainty under multi-polar information. From the prevailing concepts of MPFS and PFS, we establish a novel hybrid model namely Pythagorean m-polar fuzzy set (PMPFS). This proposed technique is more flexible and practical for real-world problems, specially when data come from multi-polar information as well as the membership grade $T_{P} (ς)$ and non-membership $F_{P} (ς)$ satisfying the condition $(T_{P} (ς))^{2} + (F_{P} (ς))^{2} \leq 1$ .

The first objective of the proposed model is to provide more flexible and practical technique to deal with uncertainties and vagueness in the field of decision analysis. The proposed technique gives us best ranking methodology by using Pythagorean m-polar fuzzy numbers (PMPFNs).

The second objective of this research is to develop Dombi’s aggregation operators based on PMPFNs with the help of T-norm and T-conorm. Mathematical modeling on real world problems often involve multi-factors, multi-attributes and multi-polar information. The proposed technique is suitable for solving such real world problems. We establish an algorithm for the censuses process based on PMPFNs.

The layout of this paper is systematized as follows. Section 2 briefly recalls some basic concepts of fuzzy sets, IFSs, PFSs and MPFSs. Section 3 introduces the novel concept of PMPFS. Certain properties and operations on PMPFSs are presented. The ideas of score function, accuracy function and certainty functions based on PMPFNs are presented. In section 4 and 5, we establish Dombi’s aggregation operators in the context of PMPFNs. We present Pythagorean m-polar fuzzy Dombi weighted arithmetic average (PMPFDWAA) operator and Pythagorean m-polar fuzzy Dombi weighted geometric average (PMPFDWGA) operator. In section 6, we present a novel approach conducting the census by using proposed operators based on PMPFNs. In section 7, we give a brief comparison of the proposed model. Finally, the conclusion of this research is summarized in section 8.

2 Background

In this section, we review some basic concepts of fuzzy set, IFS, PFS and MPFS. In the whole paper, we use $Q$ as a fixed sample space or universal set. We use $T$ and $F$ as a membership and non-membership grades, respectively. ϖ is the operational parameter, ℑ_℘ are the PMPFNs, where ℘ ∈ Δ (an indexing set). $℧^{N}$ is the collection of all PMPFNs.

Definition 2.1. A fuzzy set (FS) $F$ in $Q$ is a mapping $σ : Q \to [0, 1]$ , where σ (ς), for every $ς \in Q$ , represents the degree of membership related to fuzzy set $F$ . Mathematically, it can be represented as; $F = {(ς, σ (ς)) : ς \in Q}$

Only membership evaluations or truth values of alternatives were not sufficient for the complete scrutiny of decision-making problems. To get rid of this ambiguity, Atanassov [3] suggested that non-membership grades should be taken with some appropriate conditions.

Definition 2.2. [3] An IFS in $Q$ is defined as $I = {〈 ς, σ (ς), ρ (ς) 〉 : ς \in Q}$ where the mappings $σ : Q \to [0, 1]$ and $ϱ : Q \to [0, 1]$ represents the membership and non-membership of object $ς \in Q$ , respectively. It is necessary for an IFS that 0 ≤ σ (ς) + ρ (ς) ≤1 for all $ς \in Q$ . Hesitation or indeterminacy part can be calculated as π (ς) =1 - (σ (ς) + ρ (ς)). Graphically it can be represented as Fig. 1.

Fig. 1

Graphical representation of membership and non-membership of IFS.

Definition 2.3. An abstraction of bipolar fuzzy set was inaugurated by Chen [46] named as MPFS. An MPFS $C$ in a non-empty universal set $Q$ is a function $C : Q \to [0, 1]^{m}$ , symbolized by $C = {〈 ς, P_{i} o Λ (ς) 〉 : ς \in Q; i = 1, 2, 3, . . ., m}$ where P_i : [0, 1] ^m → [0, 1] is the i-th projection of the function (i ∈ m).

$C_{φ} (ς) = (0, 0, . . ., 0)$ is the smallest value in [0, 1] ^m and $C_{\tilde{X}} (ς) = (1, 1, . . ., 1)$ is the greatest value in [0, 1] ^m.

In some real life problems, IFS fails when σ (ς) + ρ (ς) >1. To eradicate this drawback an extension of IFS was familiarized by Yager [5, 6] named as PFS.

Definition 2.4. [5, 6] Let $Q = {ς_{1}, ς_{2}, . . ., ς_{N}}$ be the non-empty fixed reference set, then PFS in $Q$ is an object written as $Q_{P} = {〈 ς, T_{P} (ς), F_{P} (ς) 〉 : ς \in Q}$ where $T_{P} (ς) \in [0, 1]$ and $F_{P} (ς) \in [0, 1]$ are the grades of membership and non-membership of the object $ς \in Q$ respectively and the inequality $T_{P}^{2} (ς) + F_{P}^{2} (ς) \leq 1$ holds for every $ς \in Q$ . The indeterminacy or hesitant part is calculated as $L_{P} (ς) = \sqrt{1 - T_{P}^{2} (ς) - F_{P}^{2} (ς)}$ .

Let $P = (T_{P}, F_{P})$ be the Pythagorean fuzzy number (PFN) satisfying the constraint $T_{P}^{2} + F_{P}^{2} \leq 1$ with $L_{P} = \sqrt{1 - T_{P}^{2} - F_{P}^{2}}$ . Graphically it can be represented as Fig. 2.

Fig. 2

Graphical representation of membership grades and non-membership grades of PFS.

3 Pythagorean m-polar fuzzy set (PMPFS)

In this section, we introduce the novel concept of PMPFS by combining the ideas of PFS and MPFS. The proposed technique is suitable for solving the real world problems that involve multi-polar (paired) information. It gives us membership and non-membership grades of the form $〈 T_{i} (ς), F_{i} (ς) 〉$ , $i = 1, 2, . . ., M$ , for each alternative ς with the condition $T_{i}^{2} (ς) + F_{i}^{2} (ς) \leq 1$ .

Definition 3.1. A PMPFS $P_{M}$ defined on a non-empty fixed sample space can be written as $P_{M} = {(ς, {〈 T_{1} (ς), F_{1} (ς) 〉, 〈 T_{2} (ς), F_{2} (ς) 〉, . . ., 〈 T_{M} (ς), F_{M} (ς) 〉}) : ς \in Q}$ where, $0 \leq T_{i}^{2} (ς) + F_{i}^{2} (ς) \leq 1$ , $i = 1, 2, . . ., M$ and $L_{M} (ς) = \sqrt{1 - T_{M}^{2} (ς) - F_{M}^{2}} (ς), \forall M$ .

The tabular form of PMPFS can be written either by Tables 1 or 2.

Table 1
Tabular representation of PMPFS

$P_{M}$ PMPFS

ς ₁ ${〈 T_{1} (ς_{1}), F_{1} (ς_{1}) 〉, 〈 T_{2} (ς_{1}), F_{2} (ς_{1}) 〉, . . ., 〈 T_{M} (ς_{1}), F_{M} (ς_{1}) 〉}$

ς ₂ ${〈 T_{1} (ς_{2}), F_{1} (ς_{2}) 〉, 〈 T_{2} (ς_{2}), F_{2} (ς_{2}) 〉, . . ., 〈 T_{M} (ς_{2}), F_{M} (ς_{2}) 〉}$

... ... ... ...

$ς_{N}$ ${〈 T_{1} (ς_{N}), F_{1} (ς_{N}) 〉, 〈 T_{2} (ς_{N}), F_{2} (ς_{N}) 〉, . . ., 〈 T_{M} (ς_{N}), F_{M} (ς_{N}) 〉}$

$P_{M}$	PMPFS
ς ₁	${〈 T_{1} (ς_{1}), F_{1} (ς_{1}) 〉, 〈 T_{2} (ς_{1}), F_{2} (ς_{1}) 〉, . . ., 〈 T_{M} (ς_{1}), F_{M} (ς_{1}) 〉}$
ς ₂	${〈 T_{1} (ς_{2}), F_{1} (ς_{2}) 〉, 〈 T_{2} (ς_{2}), F_{2} (ς_{2}) 〉, . . ., 〈 T_{M} (ς_{2}), F_{M} (ς_{2}) 〉}$
...	... ... ...
$ς_{N}$	${〈 T_{1} (ς_{N}), F_{1} (ς_{N}) 〉, 〈 T_{2} (ς_{N}), F_{2} (ς_{N}) 〉, . . ., 〈 T_{M} (ς_{N}), F_{M} (ς_{N}) 〉}$

Table 2

Tabular representation of PMPFS

$P_{M}$	ς ₁	ς ₂	...	$ς_{N}$
1	$〈 T_{1} (ς_{1}), F_{1} (ς_{1}) 〉$	$〈 T_{1} (ς_{2}), F_{1} (ς_{2}) 〉$	...	$〈 T_{1} (ς_{N}), F_{1} (ς_{N}) 〉$
2	$〈 T_{2} (ς_{1}), F_{2} (ς_{1}) 〉$	$〈 T_{2} (ς_{2}), F_{2} (ς_{2}) 〉$	...	$〈 T_{2} (ς_{N}), F_{2} (ς_{N}) 〉$
. . .	. . .	. . .	...	. . .
$M$	$〈 T_{M} (ς_{1}), F_{M} (ς_{1}) 〉$	$〈 T_{M} (ς_{2}), F_{M} (ς_{2}) 〉$	...	$〈 T_{M} (ς_{N}), F_{M} (ς_{N}) 〉$

A PMPFN can be written as $ℑ = (〈 T_{1}, F_{1} 〉, 〈 T_{2}, F_{2} 〉, . . ., 〈 T_{M}, F_{M} 〉)$ , where $0 \leq T_{i}^{2} + F_{i}^{2} \leq 1$ and $L_{i} = \sqrt{1 - T_{i}^{2} - F_{i}^{2}}, \forall i = 1, 2, . . ., M$ .

A PMPFS of the form $^{1} P_{M} = {ς, {〈 1, 0 〉, 〈 1, 0 〉, . . ., 〈 1, 0 〉}}$ is called absolute PMPFS and a PMPFS of the form $^{0} P_{M} = {ς, {〈 0, 1 〉, 〈 0, 1 〉, . . ., 〈 0, 1 〉}}$ is called empty PMPFS.

Definition 3.2. Let Δ be an indexing set. Then we define some operations on PMPFSs. Let $P_{M} = {ς, {〈 T_{1} (ς), F_{1} (ς) 〉, . . ., 〈 T_{M} (ς), F_{M} (ς) 〉} : ς \in Q}$ be PMPFS and $ℑ = (〈 T_{1}, F_{1} 〉, 〈 T_{2}, F_{2} 〉,$ $. . ., 〈 T_{M}, F_{M} 〉)$ be its PMPFN. Let $^{℘} P_{M} = {ς, {〈^{℘} T_{1} (ς),^{℘} F_{1} (ς) 〉, . . ., 〈^{℘} T_{M} (ς),^{℘}$ $F_{M} (ς) 〉} : ς \in Q}$ , ℘ ∈ Δ, be PMPFSs and $F_{p} = (〈^{p} T_{1},^{p} F_{1}〉, 〈^{p} T_{2},^{p} F_{2}〉, \dots, 〈^{p} T_{M},^{p} F_{M}〉 : p \in Δ)$ be their PMPFNs. Then

(i) Complement: $P_{c}^{M} = {ς, {〈 F_{1} (ς), T_{1} (ς) 〉, . . ., 〈 F_{M} (ς), T_{M} (ς) 〉} : ς \in Q}$

(ii) Equality: $^{1} P_{M} =^{2} P_{M}$ if and only if ℑ₁ = ℑ ₂ i.e. and $^{1} F_{1} (ς) =^{2} F_{1} (ς), \dots,^{1} T_{M} (ς) =^{2} T_{M} (ς)$ and $^{1} F_{1} (ς) =^{2} F_{1} (ς)$ .

(iii) Subset: $^{1} P_{M} \subseteq^{2} T_{M}$ if and only if $^{1} T_{1} (ς) \leq^{2} T_{1} (ς),^{1} F_{1} (ς) \geq^{2} F_{1} (ς) \dots,^{1} T_{M} (ς) \leq^{2} T_{M} (ς)$ and $^{1} F_{M} (ς) \geq^{2} F_{M} (ς)$ .

(iv) Union: $⋃_{℘}^{℘} P_{M} = {(ς, {〈 sup_{℘ \in Δ}^{℘} T_{1} (ς), inf_{℘ \in Δ}^{℘} F_{1} (ς) 〉,$ $〈 sup_{℘ \in Δ}^{℘} T_{2} (ς), inf_{℘ \in Δ}^{℘} F_{2} (ς) 〉, . . ., 〈 sup_{℘ \in Δ}^{℘} T_{M} (ς), inf_{℘ \in Δ}^{℘} F_{M} (ς) 〉) : ℘ \in Δ}$ .

(iii) Intersection: $⋂_{℘}^{℘} P_{M} = {(〈 inf_{℘ \in Δ}^{℘} T_{1} (ς), sup_{℘ \in Δ}^{℘} F_{1} (ς) 〉,$ $〈 inf_{℘ \in Δ}^{℘} T_{2} (ς), sup_{℘ \in Δ}^{℘} F_{2} (ς) 〉, . . ., 〈 inf_{℘ \in Δ}^{℘} T_{M} (ς), sup_{℘ \in Δ}^{℘} F_{M} (ς) 〉) : ℘ \in Δ}$ .

Example 3.3. Let us consider three Pythagorean 5-polar fuzzy sets (P5PFSs) namely $^{1} P_{M},^{2} P_{M}$ and $^{3} P_{M}$ for ℘=1, 2, 3 as given in Table 3, then by using Definition 3.2 we calculate union and intersection as given in Table 4.

Table 3

P5PFSs $^{1} P_{M}$ $^{2} P_{M}$ and $^{3} P_{M}$

P5PFSs	$(ς, 〈^{℘} T_{1},^{℘} F_{2} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉〈^{℘} T_{3},^{℘} F_{3} 〉〈^{℘} T_{5},^{℘} F_{5} 〉)$
$^{1} P_{M}$	(ς₁, 〈0.83, 0.18〉, 〈0.65, 0.21〉, 〈0.67, 0.32〉, 〈0.87, 0.21〉, 〈0.88, 0.21〉),
	(ς₂, 〈0.45, 0.23〉, 〈0.45, 0.36〉, 〈0.74, 0.23〉, 〈0.74, 0.22〉, 〈0.63, 0.23〉),
	(ς₃, 〈0.85, 0.36〉, 〈0.84, 0.34〉, 〈0.73, 0.24〉, 〈0.64, 0.24〉, 〈0.85, 0.13〉)
$^{2} P_{M}$	(ς₁, 〈0.74, 0.13〉, 〈0.83, 0.18〉, 〈0.84, 0.32〉, 〈0.83, 0.32〉, 〈0.62, 0.12〉),
	(ς₂, 〈0.28, 0.31〉, 〈0.74, 0.17〉, 〈0.84, 0.15〉, 〈0.55, 0.24〉, 〈0.83, 0.18〉),
	(ς₃, 〈0.73, 0.14〉, 〈0.83, 0.65〉, 〈0.45, 0.13〉, 〈0.63, 0.13〉, 〈0.73, 0.18〉)
$^{3} P_{M}$	(ς₁, 〈0.83, 0.18〉, 〈0.46, 0.11〉, 〈0.73, 0.34〉, 〈0.72, 0.32〉, 〈0.72, 0.23〉),
	(ς₂, 〈0.85, 0.31〉, 〈0.83, 0.18〉, 〈0.73, 0.23〉, 〈0.73, 0.22〉, 〈0.96, 0.18〉),
	(ς₃, 〈0.71, 0.23〉, 〈0.76, 0.13〉, 〈0.83, 0.18〉, 〈0.82, 0.18〉, 〈0.42, 0.11〉)

Table 4

Union and intersection of $^{1} P_{M}$ $^{2} P_{M}$ $^{3} P_{M}$ and

P5PFSs	$(ς, 〈^{℘} T_{1},^{℘} F_{2} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉〈^{℘} T_{3},^{℘} F_{3} 〉〈^{℘} T_{5},^{℘} F_{5} 〉)$
$^{1} P_{M} \cup^{2} P_{M} \cup^{3} P_{M}$	(ς₁, 〈0.83, 0.13〉, 〈0.83, 0.11〉, 〈0.84, 0.32〉, 〈0.87, 0.21〉, 〈0.88, 0.12〉),
	(ς₂, 〈0.85, 0.23〉, 〈0.83, 0.17〉, 〈0.84, 0.15〉, 〈0.74, 0.22〉, 〈0.96, 0.18〉),
	(ς₃, 〈0.85, 0.14〉, 〈0.84, 0.13〉, 〈0.83, 0.13〉, 〈0.82, 0.13〉, 〈0.85, 0.11〉)
$^{1} P_{M} \cap^{2} P_{M} \cap^{3} P_{M}$	(ς₁, 〈0.74, 0.18〉, 〈0.46, 0.21〉, 〈0.67, 0.34〉, 〈0.72, 0.32〉, 〈0.62, 0.23〉),
	(ς₂, 〈0.28, 0.31〉, 〈0.45, 0.36〉, 〈0.73, 0.23〉, 〈0.55, 0.24〉, 〈0.63, 0.23〉),
	(ς₃, 〈0.71, 0.36〉, 〈0.76, 0.65〉, 〈0.45, 0.24〉, 〈0.63, 0.24〉, 〈0.42, 0.18〉)

For the comparative analysis in MCDM of PMPFNs we describe score function, accuracy function and certainty function. The notion of score function was initiated by Chen and Tan [7] for IFSs. Similar concept was proposed by Tversky and Kahneman [8]. That concept can be stretched out for hybrid models of FSs and for PMPFNs defined as:

Definition 3.4. Let $ℑ = (〈 T_{1}, F_{1} 〉, 〈 T_{2}, F_{2} 〉, . . .,$ $〈 T_{M}, F_{M} 〉)$ be a PMPFN, then $£ (ℑ), H (ℑ)$ and $C (ℑ)$ are said to be score function, accuracy function and certainty function of ℑ respectively, and defined as $£ (ℑ) = \frac{1}{2 M} [M + \sum_{i = 1}^{M} (T_{i}^{2} - F_{i}^{2})]$ (3.1) $H (ℑ) = \frac{1}{M} [\sum_{i = 1}^{M} (T_{i}^{2} + F_{i}^{2})]$ (3.2) $C (ℑ) = \frac{1}{M} [\sum_{i = 1}^{M} (T_{i}^{2})]$ (3.3) It is easy to see that $L (^{1} P_{M}) = 1, \forall ς \in^{1} P_{M}$ and $L (^{0} P_{M}) = 0, \forall ς \in^{0} P_{M}$ .

Definition 3.5.Let $F_{1} = (〈^{1} T_{1},^{1} F_{1}〉, 〈^{1} T_{2},^{1} F_{2}〉, ..., 〈^{1} T_{M},^{1} F_{M}〉)$ and $F_{2} = (〈^{2} T_{1},^{2} F_{1}〉, 〈^{2} T_{2},^{2} F_{2}〉, ..., 〈^{2} T_{M},^{2} F_{M}〉)$ be two PMPFNs then,

(i) If £ (ℑ ₁) ≻ £ (ℑ ₂), then ℑ₁ ≻ ℑ ₂.

(ii) If £ (ℑ ₁) = £ (ℑ ₂) and $H (ℑ_{1}) ≻ H (ℑ_{2})$ , then ℑ₁ ≻ ℑ ₂.

(iii) If $£ (ℑ_{1}) = £ (ℑ_{2}), H (ℑ_{1}) = H (ℑ_{2})$ and $C (ℑ_{1}) ≻ C (ℑ_{2})$ , then ℑ₁ ≻ ℑ ₂.

(iv) If $£ (ℑ_{1}) = £ (ℑ_{2}), H (ℑ_{1}) = H (ℑ_{2})$ and $C (ℑ_{1}) = C (ℑ_{2})$ , then ℑ₁ ∼ ℑ ₂.

Example 3.6. Consider two P4PFNs ℑ₁ and ℑ₂ for ℘=1, 2 given in tabular form as Table 5 then by using Definition 3.4 we can calculate scores of ℑ₁ and ℑ₂ given as $£ (ℑ_{1}) = \frac{1}{8} = 0.6811$ Similarly £ (ℑ ₂) =0.7007, so we can compare both PMPFNs by Definition 3.5 as ℑ₂ ≻ ℑ ₁. Graphically it can seen as Fig. 3.

Table 5

Pythagorean 4-polar fuzzy numbers

$Q$	ℑ₁	ℑ₂
$〈^{℘} T_{1},^{℘} F_{1} 〉$	〈0.52, 0.53〉	〈0.78, 0.21〉
$〈^{℘} T_{2},^{℘} F_{2} 〉$	〈0.64, 0.25〉	〈0.71, 0.41〉
$〈^{℘} T_{3},^{℘} F_{3} 〉$	〈0.78, 0.39〉	〈0.93, 0.21〉
$〈^{℘} T_{4},^{℘} F_{4} 〉$	〈0.83, 0.18〉	〈0.47, 0.58〉

Fig. 3

Score values of ℑ₁ and ℑ₂ .

4 Some Dombi operations of PMPFNs

In this section, we introduce the idea of Dombi operators and we discuss some fundamental properties and operations on Dombi operators in the framework of PMPFNs.

Definition 4.1. Dombi T-norm and T-conorm between two real numbers $K_{1}$ and $K_{2}$ are defined as: $D (K_{1}, K_{2}) = \frac{1}{1 + {(\frac{1 - K_{1}}{K_{1}})^{ϖ} + (\frac{1 - K_{2}}{K_{2}})^{ϖ}}^{1 / ϖ}},$ $D^{c} (K_{1}, K_{2}) = 1 - \frac{1}{1 + {(\frac{K_{1}}{1 - K_{1}})^{ϖ} + (\frac{K_{2}}{1 - K_{2}})^{ϖ}}^{1 / ϖ}},$ where $(K_{1}, K_{2}) \in [0, 1] \times [0, 1]$ and ϖ is the operational parameter with the properties given in Table 6.

Table 6
Dombi operator

Operator ϖ Type

$D (K_{1}, K_{2})$ ϖ > 0 Conjunctive

$D^{c} (K_{1}, K_{2})$ ϖ > 0 Disjunctive

$D (K_{1}, K_{2})$ ϖ < 0 Disjunctive

$D^{c} (K_{1}, K_{2})$ ϖ < 0 Conjunctive

Operator	ϖ	Type
$D (K_{1}, K_{2})$	ϖ > 0	Conjunctive
$D^{c} (K_{1}, K_{2})$	ϖ > 0	Disjunctive
$D (K_{1}, K_{2})$	ϖ < 0	Disjunctive
$D^{c} (K_{1}, K_{2})$	ϖ < 0	Conjunctive

Definition 4.2. Let $ℑ = (〈 T_{1}, F_{1} 〉, 〈 T_{2}, F_{2} 〉, . . .,$ $〈 T_{M}, F_{M} 〉)$ be an arbitrary PMPFN and $F_{p} = (〈^{p} T_{1},^{p} F_{1}〉, 〈^{p} T_{2},^{p} F_{2}〉, ..., 〈^{p} T_{M},^{p} F_{M}〉 : p \in Δ)$ be an assembling of PMPFNs, then by using Definition 4.1 we define some Dombi operations on PMPFNs with the arbitrary real numbers ϖ > 0 and η > 0 given as

$ℑ_{1} \oplus ℑ_{2} = (〈 1 - \frac{1}{1 + {{(\frac{^{1} T_{1}}{1 -^{1} T_{1}})}^{ϖ} + {(\frac{^{2} T_{1}}{1 -^{2} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {{(\frac{1 -^{1} F_{1}}{^{1} F_{1}})}^{ϖ} + {(\frac{1 -^{2} F_{1}}{^{2} F_{1}})}^{ϖ}}^{1 / ϖ}} 〉, 〈 1 - \frac{1}{1 + {{(\frac{^{1} T_{2}}{1 -^{1} T_{2}})}^{ϖ} + {(\frac{^{2} T_{2}}{1 -^{2} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {{(\frac{1 -^{1} F_{2}}{^{1} F_{2}})}^{ϖ} + {(\frac{1 -^{2} F_{2}}{^{2} F_{2}})}^{ϖ}}^{1 / ϖ}} 〉, ..., 〈 1 - \frac{1}{1 + {{(\frac{^{1} T_{M}}{1 -^{1} T_{M}})}^{ϖ} + {(\frac{^{2} T_{M}}{1 -^{2} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {{(\frac{1 - 1 F_{M}}{1 F_{M}})}^{ϖ} + {(\frac{1 - 2 F_{M}}{2 F_{M}})}^{ϖ}}^{1 / ϖ}} 〉)$

$\begin{array}{l} ℑ_{1} \otimes ℑ_{2} = (〈 \frac{1}{1 + {{(\frac{1 -^{1} T_{1}}{^{1} T_{1}})}^{ϖ} + {(\frac{1 -^{2} T_{1}}{^{2} T_{1}})}^{ϖ}}^{1 / ϖ}}, \\ 1 - \frac{1}{1 + {{(\frac{^{1} F_{1}}{1 -^{1} F_{1}})}^{ϖ} + {(\frac{^{2} F_{1}}{1 -^{2} F_{1}})}^{ϖ}}^{1 / ϖ}} 〉, 〈 \frac{1}{1 + {{(\frac{1 -^{1} T_{2}}{^{1} T_{2}})}^{ϖ} + {(\frac{1 -^{2} T_{2}}{^{2} T_{2}})}^{ϖ}}^{1 / ϖ}}, \\ 1 - \frac{1}{1 + {{(\frac{^{1} F_{2}}{1 -^{1} F_{2}})}^{ϖ} + {(\frac{^{2} F_{2}}{1 -^{2} F_{2}})}^{ϖ}}^{1 / ϖ}} 〉, ..., 〈 \frac{1}{1 + {{(\frac{1 -^{1} T_{M}}{^{1} T_{M}})}^{ϖ} + {(\frac{1 -^{2} T_{M}}{^{2} T_{M}})}^{ϖ}}^{1 / ϖ}}, \\ 1 - \frac{1}{1 + {{(\frac{^{1} F_{M}}{1 -^{1} F_{M}})}^{ϖ} + {(\frac{^{2} F_{M}}{1 -^{2} F_{M}})}^{ϖ}}^{1 / ϖ}} 〉) \end{array}$

$η ℑ = (〈 1 - \frac{1}{1 + {η (\frac{T_{1}}{1 - T_{1}})^{ϖ}}^{1 / ϖ}},$ $\frac{1}{1 + {η (\frac{1 - F_{1}}{F_{1}})^{ϖ}}^{1 / ϖ}} 〉, 〈 1 - \frac{1}{1 + {η (\frac{T_{2}}{1 - T_{2}})^{ϖ}}^{1 / ϖ}},$ $\frac{1}{1 + {η (\frac{1 - F_{2}}{F_{2}})^{ϖ}}^{1 / ϖ}} 〉, . . ., 〈 1 - \frac{1}{1 + {η (\frac{T_{M}}{1 - T_{M}})^{ϖ}}^{1 / ϖ}},$ $\frac{1}{1 + {η (\frac{1 - F_{M}}{F_{M}})^{ϖ}}^{1 / ϖ}} 〉)$

$ℑ^{η} = (〈 \frac{1}{1 + {η (\frac{1 - T_{1}}{T_{1}})^{ϖ}}^{1 / ϖ}},$ $1 - \frac{1}{1 + {η (\frac{F_{1}}{1 - F_{1}})^{ϖ}}^{1 / ϖ}} 〉, 〈 \frac{1}{1 + {η (\frac{1 - T_{2}}{T_{2}})^{ϖ}}^{1 / ϖ}},$ $1 - \frac{1}{1 + {η (\frac{F_{2}}{1 - F_{2}})^{ϖ}}^{1 / ϖ}} 〉, . . ., 〈 \frac{1}{1 + {η (\frac{1 - T_{M}}{T_{M}})^{ϖ}}^{1 / ϖ}},$ $1 - \frac{1}{1 + {η (\frac{F_{M}}{1 - F_{M}})^{ϖ}}^{1 / ϖ}} 〉)$

Remark. For two PMPFNs ℑ₁ and ℑ₂ we observe that ℑ₁⊕ ℑ ₂, ℑ ₁ ⊗ ℑ ₂, η₁ ℑ and $ℑ_{1}^{η}$ are also PMPFNs.

Example 4.3. Consider two P3PFNs ℑ₁ and ℑ₂ for ℘=1, 2 given in tabular form as Table 7.

ℑ₁ ⊕ ℑ ₂ = (〈0.8223, 0.1770〉, 〈0.8086, 0.1838〉, 〈0.9439, 0.1580〉).

ℑ₁ ⊗ ℑ ₂ = (〈0.4535, 0.5821〉, 〈0.5073, 0.5069〉, 〈0.7367, 0.4750〉). Let η = 0.9 and ϖ = 1 then we obtain

η ℑ ₁ = (〈0.4936, 0.5561〉, 〈0.6153, 0.2702〉, 〈0.7613, 0.4153〉).

η ℑ ₁ = (〈0.5462, 0.5036〉, 〈0.6639, 0.2307〉, 〈0.7975, 0.3652〉).

Table 7

Pythagorean 3-polar fuzzy numbers

$Q$	ℑ₁	ℑ₂
$〈^{℘} T_{1},^{℘} F_{1} 〉$	〈0.52, 0.53〉	〈0.78, 0.21〉
$〈^{℘} T_{2},^{℘} F_{2} 〉$	〈0.64, 0.25〉	〈0.71, 0.41〉
$〈^{℘} T_{3},^{℘} F_{3} 〉$	〈0.78, 0.39〉	〈0.93, 0.21〉

5 Dombi weighted aggregation operators for PMPFNs

In this section, we utilize above idea to construct the PMPFDWAA and PMPFDWGA operators, which can be applied to the MCDM problems. We discuss some properties of these operators in the context of PMPFNs.

5.1 PMPFDWAA operator

Definition 5.1. Let ℧ be the collection of all PMPFNs where, $F_{p} = (〈^{p} T_{1},^{p} F_{1}〉, 〈^{p} T_{2},^{p} F_{2}〉, \dots, 〈^{p} T_{M},^{p} F_{M}〉 : p \in Δ)$ for $℘ = 1, 2, . . ., N$ be the corresponding PMPFNs. The associated weight vector is $ζ = (ζ_{1}, ζ_{2}, . . ., ζ_{N})$ , where ζ_℘∈ [0, 1] ∀ ℘ and $\sum_{℘ = 1}^{N} ζ_{N} = 1$ . Then Pythagorean m-polar fuzzy Dombi weighted arithmetic averaging operator $PMPFDWAA : ℧^{N} \to ℧$ is defined as: $\begin{matrix} PMPFDWAA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) \\ = ζ_{1} ℑ_{1} \oplus ζ_{2} ℑ_{2} \oplus . . . \oplus ζ_{N} ℑ_{N} \end{matrix}$

Theorem 5.2. Let $_{p} = (〈^{p} T_{1},^{p} F_{1}〉, 〈^{p} T_{2},^{p} F_{2}〉, \dots, 〈^{p} T_{M},^{p} F_{M}〉 : p \in Δ)$ for $℘ = 1, 2, . . ., N$ be an assembling of PMPFNs with $ζ = (ζ_{1}, ζ_{2}, . . ., ζ_{N})$ , where ζ_℘∈ [0, 1] ∀ ℘ and $\sum_{℘ = 1}^{N} ζ_{N} = 1$ . Then PMPFDWAA operator can be calculated as: $PMPFDWAA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) =$ $(〈 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{^{℘} T_{1}}{1 -^{℘} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{1 -^{℘} F_{1}}{^{℘} F_{1}})}^{ϖ}}^{1 / ϖ}} 〉, 〈 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{^{℘} T_{2}}{1 -^{℘} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{1 -^{℘} F_{2}}{^{℘} F_{2}})}^{ϖ}}^{1 / ϖ}} 〉, ..., 〈 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{^{℘} T_{M}}{1 -^{℘} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{1 -^{℘} F_{M}}{^{℘} F_{M}})}^{ϖ}}^{1 / ϖ}} 〉) .$ (A)

Proof. See Appendix A.□

Remark. The aggregated values calculated by PMPFDWAA operator for PMPFNs is again a PMPFN.

Example 5.3. Consider four P3PFNs ℑ₁, ℑ ₂, ℑ ₃ and ℑ₄ with ζ = (0.3, 0.4, 0.2, 0.1) ^T and $\sum_{℘ = 1}^{4} ζ_{N} = 1$ . In tabular form ℑ₁, ℑ ₂, ℑ ₃ and ℑ₄ can be represented as Table 8. For ϖ = 1 and by using equation (A) for $M = 3$ and $N = 4$ we obtain P3PFDWAA (ℑ ₁, ℑ ₂, ℑ ₃, ℑ ₄) = (〈0.6852, 0.3268〉, 〈0.7308, 0.3214〉, 〈0.8853, 0.3287〉).

Table 8

Pythagorean 3-polar fuzzy numbers

$Q$	ℑ₁	ℑ₂
$〈^{℘} T_{1},^{℘} F_{1} 〉$	〈0.52, 0.53〉	〈0.78, 0.21〉
$〈^{℘} T_{2},^{℘} F_{2} 〉$	〈0.83, 0.18〉	〈0.71, 0.41〉
$〈^{℘} T_{3},^{℘} F_{3} 〉$	〈0.25, 0.71〉	〈0.93, 0.21〉
$Q$	ℑ₃	ℑ₄
$〈^{℘} T_{1},^{℘} F_{1} 〉$	〈0.47, 0.58〉	〈0.72, 0.41〉
$〈^{℘} T_{2},^{℘} F_{2} 〉$	〈0.51, 0.61〉	〈0.39, 0.71〉
$〈^{℘} T_{3},^{℘} F_{3} 〉$	〈0.83, 0.51〉	〈0.93, 0.31〉

Now we will explore some properties of PMPFDWAA operator.

Property 1: (Reducibility) Let $ℑ_{℘} = (〈^{℘} T_{1},^{℘} F_{1} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉, ..., 〈^{℘} T_{M},^{℘} F_{M} 〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ be an assembling of PMPFNs and $ζ = (1 / N, 1 / N, . . ., 1 / N)$ then there exists $PMPFDWAA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) =$ $(〈 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{^{℘} T_{1}}{1 -^{℘} T_{1}})^{ϖ}}^{1 / ϖ}},$ $\frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{1 - ℘ F_{1}}{^{℘} F_{1}})^{ϖ}}^{1 / ϖ}} 〉, 〈 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{^{℘} T_{2}}{1 -^{℘} T_{2}})^{ϖ}}^{1 / ϖ}},$ $\frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{1 -^{℘} F_{2}}{^{℘} F_{2}})^{ϖ}}^{1 / ϖ}} 〉, . . .,$ $〈 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{^{℘} T_{M}}{1 -^{℘} T_{M}})^{ϖ}}^{1 / ϖ}},$ $\frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{1 - ℘ F_{M}}{^{℘} F_{M}})^{ϖ}}^{1 / ϖ}} 〉)$

Proof. We can proof it directly from equation (A).

□ Property 2: (Idempotency) Let $ℑ_{℘} = (〈^{℘} T_{1},^{℘} F_{1} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉, ..., 〈^{℘} T_{M},^{℘} F_{M} 〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ be a group of PMPFNs. When $ℑ_{℘} = ℑ = (〈 T_{1}, F_{1} 〉, 〈 T_{2}, F_{2} 〉, . . ., 〈 T_{M}, F_{M} 〉)$ ∀ ℘ =1, 2, $. . ., N$ i.e. all ℑ_℘ ’s are equal, then $PMPFDWAA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) = ℑ$ .

Proof. See Appendix B.□

Property 3: (Boundedness) Let $ℑ_{℘} = (〈^{℘} T_{1},^{℘} F_{1} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉, ..., 〈^{℘} T_{M},^{℘} F_{M} 〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ be an assembling of PMPFNs and let $^{℘} ℑ_{\min} = (〈{\min_{℘}}^{℘} T_{1}, {\max_{℘}}^{℘} F_{1}〉,$ $〈 min_{℘}^{℘} T_{2}, min_{℘}^{℘} F_{2} 〉$ , . . . , $〈 min_{℘}^{℘} T_{M}, min_{℘}^{℘} F_{M} 〉 : ℘ \in Δ)$ and $^{℘} ℑ_{\max} = (〈{\max_{℘}}^{℘} T_{1}, {\max_{℘}}^{℘} F_{1}〉,$ $〈 max_{℘}^{℘} T_{2}, max_{℘}^{℘} F_{2} 〉$ , . . . , $〈 max_{℘}^{℘} T_{M}, max_{℘}^{℘} F_{M} 〉 : ℘ \in Δ)$ be two PMPFNs then, $^{℘} ℑ_{\min} \leq P M P F D W G A (ℑ_{1}, ℑ_{2}, ..., ℑ_{N}) \leq^{℘} ℑ_{\max} .$

Proof. See Appendix C.□

5.2 PMPFDWGA operator

Definition 5.4. Let ℧ be the collection of all PMPFNs where, ℑ_℘ = $(〈^{℘} T_{1},^{℘} F_{1} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉, ..., 〈^{℘} T_{M},^{℘} F_{M} 〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ be the corresponding PMPFNs. The associated weight vector is $ζ = (ζ_{1}, ζ_{2}, . . ., ζ_{N})$ , where ζ_℘ ∈ [0, 1] ∀ ℘ and $\sum_{℘ = 1}^{N} ζ_{N} = 1$ . Then Pythagorean m-polar fuzzy Dombi weighted geometric averaging operator $PMPFDWAA : ℧^{N} \to ℧$ is defined as: PMPFDWGA $(ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) =$ $ℑ_{1}^{ζ_{1}} \otimes ℑ_{2}^{ζ_{2}} \otimes . . . \otimes ℑ_{N}^{ζ_{N}}$

Theorem 5.5. Let $ℑ_{℘} = (〈^{℘} T_{1},^{℘} F_{1} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉, ..., 〈^{℘} T_{M},^{℘} F_{M} 〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ be an assembling of PMPFNs with $ζ = (ζ_{1}, ζ_{2}, . . ., ζ_{N})$ , where ζ_℘∈ [0, 1] ∀ ℘ and $\sum_{℘ = 1}^{N} ζ_{N} = 1$ . Then PMPFDWGA operator can be calculated as: $PMPFDWGA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) =$ $\begin{array}{l} (〈 \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{1 -^{℘} T_{1}}{^{℘} T_{1}})}^{ϖ}}^{1 / ϖ}}, 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{^{℘} F_{1}}{1 -^{℘} F_{1}})}^{ϖ}}^{1 / ϖ}} 〉, \\ 〈 \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{1 -^{℘} T_{2}}{^{℘} T_{2}})}^{ϖ}}^{1 / ϖ},} 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{^{℘} F_{2}}{1 -^{℘} F_{2}})}^{ϖ}}^{1 / ϖ}} 〉 \\ , ..., 〈 \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{1 -^{℘} T_{M}}{^{℘} T_{M}})}^{ϖ}}^{1 / ϖ}} \\ , 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} {(\frac{^{℘} F_{M}}{1 -^{℘} F_{M}})}^{ϖ}}^{1 / ϖ}} 〉) . . \end{array}$ (B)

Proof. The proof is similar as Theorem 5.2.□

Remark. The properties of reducibility, idempotency and boundedness also holds for PMPFDWGA operators.

Now we will explore some properties of PMPFDWGA operator.

Property 1: (Reducibility) Let $ℑ_{℘} = (〈^{℘} T_{1},^{℘} F_{1} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉, ..., 〈^{℘} T_{M},^{℘} F_{M} 〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ be an assembling of PMPFNs and $ζ = (1 / N, 1 / N, . . ., 1 / N)$ then there exists $PMPFDWGA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) =$ $(〈 \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{1 -^{℘} T_{1}}{^{℘} T_{1}})^{ϖ}}^{1 / ϖ}},$ $1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{^{℘} F_{1}}{1 -^{℘} F_{1}})^{ϖ}}^{1 / ϖ}} 〉, 〈 \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{1 -^{℘} T_{2}}{^{℘} T_{2}})^{ϖ}}^{1 / ϖ}},$ $1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{^{℘} F_{2}}{1 -^{℘} F_{2}})^{ϖ}}^{1 / ϖ}} 〉, . . .,$ $〈 \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{1 -^{℘} T_{M}}{^{℘} T_{M}})^{ϖ}}^{1 / ϖ}},$ $1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} \frac{1}{N} (\frac{^{℘} F_{M}}{1 -^{℘} F_{M}})^{ϖ}}^{1 / ϖ}} 〉)$ .

Property 2: (Idempotency) Let $ℑ_{℘} = (〈^{℘} T_{1},^{℘} F_{1} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉, ..., 〈^{℘} T_{M},^{℘} F_{M} 〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ be the collection of PMPFNs. When $ℑ_{℘} = ℑ = (〈 T_{1}, F_{1} 〉, 〈 T_{2}, F_{2} 〉, . . ., 〈 T_{M}, F_{M} 〉)$ $\forall ℘ = 1, 2, . . ., N$ i.e. all ℑ_℘ ’s are equal, then $PMPFDWGA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) = ℑ$ .

Property 3: (Boundedness) Let $ℑ_{℘} = (〈^{℘} T_{1},^{℘} F_{1} 〉, 〈^{℘} T_{2},^{℘} F_{2} 〉, ..., 〈^{℘} T_{M},^{℘} F_{M} 〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ be an assembling of PMPFNs and let $^{℘} ℑ_{\min} = (〈{\min_{℘}}^{℘} T_{1}, {\min_{℘}}^{℘} F_{1}〉,$ $〈 min_{℘}^{℘} T_{2}, min_{℘}^{℘} F_{2} 〉$ , . . . , $〈 min_{℘}^{℘} T_{M}, min_{℘}^{℘} F_{M} 〉 : ℘ \in Δ)$ and $〈 max_{℘}^{℘} T_{2}, max_{℘}^{℘} F_{2} 〉$ , . . . , $〈 max_{℘}^{℘} T_{M}, max_{℘}^{℘} F_{M} 〉 : ℘ \in Δ)$ be two PMPFNs then, $^{p} F_{min} ≼ PMPFDWAA(F_{1}, F_{2}, \dots, F_{N}) ≼^{p} F_{max}$ .

Note: The proofs of property1,2 and 3 are similar as we do for PMPFDWAA operator.

6 A novel approach to census process via PMPFS

In this segment, we propose the idea of MCDM under the influence of assembled PMPFD operators. A case study concerning to a census process is presented and can be improved by using the novel methodology under PMPF rules and its associated operators.

6.1 Case study

The word "census" is deduced from a Latin word "cansere" which intends "to estimate". The census is an imaginative count of some country’s population. Fundamentally, this is an authorized count or survey of the population and for property assessment. Generally, a census takes place after every 10 years, but sometimes due to some specific motives and several governmental issues this time duration increases.

The account of conducting census is prehistoric and gigantic. Mostly all the states of the world conduct census on decimal basis. A population census has both virtues and faults. Now, we are going to demonstrate a new approach for census process via PMPFDWAA and PMPFDWGA operators by applying fuzzy logic.

6.2 Significance of census process

Census process of a nation holds many exercises and benefits for the development of that nation. More or less significant tips are listed below:

It detentions a extensive variety of a country’s population data and physiognomies.

It has a broader coverage of a country’s population with the other variables such as housing, salary, hygiene etc.

This worldwide action plan aims to attain anti-poverty goals and utilize this material in many ways for the improvement of the country such as,

For eradicate extreme poverty and hunger.

Achieve general primary education.

Encourage gender equivalence and empower women.

Diminish child mortality.

Improve maternal health.

Guarantee environmental health.

Develop a worldwide corporation of development.

Make country economically strong.

6.3 Disadvantages of existence approach

It is not guaranteed, i.e. It requires time to take the census, usually a solid month or more than that and during this time period the authentic data on the land is constantly fluctuating.

Domestic heads do something give fabricated feedbacks thus compromising the data.

Inaccuracies may result if the persons conducting census are not correctly prepared to avoid double counting or skipping some household.

It receives many other demerits such as,

It is very expensive, •Time consuming.

Labor consuming, •More arithmetical error.

Held after a long time, •training issues.

It has a short set of physiognomies for persons and it is very problematic to summarized and classified all the evaluated data after completing the process.

6.4 Fuzzy logic model for software construction

To reduce disadvantages we are starting to demonstrate a new approach for census process. We construct a mathematical model under fuzzy logic based on PMPFNs which will be helpful to collect the data by using a software instead of manual procedure. A software can be constructed by using fuzzy rules like a biometric systematic technique for census process. This should be an online application and every mortal receives an easy entree to it through their android mobiles or through NADRA (National database and registration authority).

The updated census form issued by the government should be available for this application and having all the necessary features for data aggregation. At that place must be a process of a sign in (login) account for every head of a family. All the people who have easy access to this application through their internet sources can fill the contour for their family with identity card numbers of all the members of the household. The masses who cannot afford all this, they can visit to NADRA to register their family’s information free of monetary value. Government should give notice to the public that they should finish this operation within ten to fifteen days, otherwise after the deadline the people who do not fill their data for census should be fined by the government. After finishing the process government can be found the unregistered identity cards easily by comparing the accumulated data and already existing data from NADRA. Due to this caution every individual gets census process seriously and register themselves before the deadline. The constructed software works on fuzzy logics as given in the Fig. 4.

Fig. 4

Fuzzy logic model for software construction.

6.5 Mathematical modeling

We use PMPFNs for data input and PMPFDWAA and PMPFDWGA operators for fuzzification and assortment of input data. We choose some particular alternatives and some specific criteria to show the modeling and this algorithm can be extendable to $N$ number of alternatives and $M$ number of measures. The pseudocode of proposed algorithm 1 can be seen in Fig. 5.

Fig. 5

Pseudocode of proposed algorithm 1 for census process by using PMPFDWAA operator.

Algorithm 1 Algorithm for census process

Input:

Step 1: Input PMPF-data for suitable number of alternatives $ℑ_{℘^{'}}; (℘^{'} = 1, 2, 3, . . ., N)$ which are related to different criteria $Y_{℘}; (℘ = 1, 2, 3, . . ., N^{'})$ .

Step 2: Normalize the input PMPF-data: It is necessary to normalize the input information before further calculations to obtain the best and precise solutions. Therefore the PMPF evaluation can be normalized by ${\tilde{ℑ}}_{℘^{'}}^{℘} = {\begin{matrix} 〈^{℘} T_{M},^{℘} F_{M} 〉; for same type \\ 〈^{℘} F_{M},^{℘} T_{M} 〉; for different type \end{matrix}$ If the type is same for all attributes, then there is no need to normalize the information. In our given application all the alternatives are of same types and then we do not normalize our input and instantly use it for our deliberations.

Step 3: According to the experts obtain associated weighted vector ζ = (ζ₁, ζ₂, . . . , ζ_m) for further calculations. Calculations:

Step 4: Compute the aggregated values of alternatives $O_{℘^{'}}; (℘^{'} = 1, 2, 3, . . ., N^{'})$ corresponding to the different criteria $Y_{℘}; (℘ = 1, 2, 3, . . ., N)$ by using PMPFDWAA and PMPFDWGA operators given in equations (A) and (B) respectively, and hence the evaluated aggregated values are given by $O_{℘^{'}}; (℘^{'} = 1, 2, 3, . . ., N^{'})$ .

Step 5: We variate the values of operational parameter ϖ for the proposed operators and repeat all above steps for calculations to observe the sensitivity of operators according to the operational parameter ϖ.

Step 6: Using $O_{℘^{'}}; (℘^{'} = 1, 2, 3, . . ., N^{'})$ for every operator separately calculate score values by using Definition 3.4.

Output:

Step 7: We rank the alternative on the basis of score values according to the Definition 3.5.

Step 8: Alternative with the higher score has the maximum rank according to the given numerical example.

Suppose that a country desires to start census process in the broad spectrum of its population and other characteristics for the evolution. Initially, we choose only the economic measure for three cities $Q = {ℑ_{1}, ℑ_{2}, ℑ_{3}}$ for the standards $Y_{1}, Y_{2}$ and $Y_{3}$ , where $Y_{1} =$ Women, $Y_{2} =$ Men, $Y_{3} =$ Transgender.

The characteristic chosen here is the economic measure of the people related to the corresponding cities. This contains the information about educational attributes, literacy, occupation, place of work, employment related information and labor force participation. The weighted vector is given as ζ = (0.6, 0.3, 0.1) such that $\sum_{℘ = 1}^{3} ζ_{℘} = 1$ . The associated ζ indicates the current condition of every city ℑ₁, ℑ ₂ and ℑ₃ and can be constructed through experts by analyzing all the factors of the cities. The input data can be written in tabular form as Table 9. In Table 9 we write P3PFNs for each attribute and criteria, where each pair $〈^{℘} T_{M},^{℘} F_{M} 〉$ represent the economical measure of people of the corresponding city. We take three economical measures,

Table 9

Input for census based on PMPFNs

Economical measure	ℑ₁	ℑ₂	ℑ₃
$〈^{℘} T_{1},^{℘} F_{1} 〉$	$Y_{1}$	〈0.83, 0.18〉	〈0.78, 0.21〉	〈0.64, 0.25〉
$〈^{℘} T_{2},^{℘} F_{2} 〉$		〈0.25, 0.71〉	〈0.41, 0.71〉	〈0.83, 0.18〉
$〈^{℘} T_{3},^{℘} F_{3} 〉$		〈0.31, 0.93〉	〈0.21, 0.93〉	〈0.25, 0.71〉
$〈^{℘} T_{1},^{℘} F_{1} 〉$	$Y_{2}$	〈0.71, 0.39〉	〈0.39, 0.78〉	〈0.39, 0.71〉
$〈^{℘} T_{2},^{℘} F_{2} 〉$		〈0.41, 0.72〉	〈0.64, 0.25〉	〈0.29, 0.11〉
$〈^{℘} T_{3},^{℘} F_{3} 〉$		〈0.21, 0.78〉	〈0.93, 0.31〉	〈0.93, 0.31〉
$〈^{℘} T_{1},^{℘} F_{1} 〉$	$Y_{3}$	〈0.83, 0.18〉	〈0.52, 0.53〉	〈0.48, 0.71〉
$〈^{℘} T_{2},^{℘} F_{2} 〉$		〈0.51, 0.61〉	〈0.83, 0.18〉	〈0.64, 0.25〉
$〈^{℘} T_{3},^{℘} F_{3} 〉$		〈0.25, 0.71〉	〈0.25, 0.71〉	〈0.78, 0.39〉

•Rich

•Merely adequate

•Poor. Measure evaluation form for census consists different type of data like,

•Economical measure

•Literacy,

•Marital status

•Languages,

•Religion

•Age factor,

•Occupation

•Educational attributes.

•Information about buildings,

•Geographic and migration information,

•Information about births and deaths,

•Living quarters and related facilities,

•Employment related information,

On the same pattern for different characteristics different input tables can be constructed for the data collection and this data can be saved easily by using constructed online application for census process.

■ Calculations for PMPFDWAA operator:

•For ℑ₁: We take the operational parameter ϖ = 1. Then by using equation (A) we get $O_{1} = (〈 0.8059, 0.2146 〉, 〈 0.3388, 0.7014 〉,$ 〈0.2767, 0.8542〉).

Similarly we can calculate $O_{2}$ and $O_{3}$ for ℑ₂ and ℑ₃ given as $O_{2} = (〈 0.7082, 0.2915 〉, 〈 0.5899, 0.3845 〉,$ 〈0.8068, 0.5702〉). $O_{3} = (〈 0.5745, 0.3374 〉, 〈 0.7635, 0.1547 〉,$ 〈0.8195, 0.4832〉). The score values of above aggregated P3PFNs can be obtained by using Definition 3.4 given as, $£ (O_{1}) = 0.4288, £ (O_{2}) = 0.6571, £ (O_{3}) = 0.7022$ The aggregated values $O_{1}, O_{2}$ and $O_{3}$ shows the economical strength of every city ℑ₁, ℑ ₂ and ℑ₃ respectively and the calculated score values shows the population measure of cities ℑ₁, ℑ ₂ and ℑ₃. These score values shows that ℑ₃ ≻ ℑ ₂ ≻ ℑ ₁ (See Fig. 6). Hence city ℑ₃ has the maximum population as compared to other cities.

Fig. 6

Score values for PMPFDWAA operator.

■ Calculations for PMPFDWGA operator: We take the operational parameter ϖ = 1. By using equation (B) on PMPF input data we get the aggregated values $O_{1}, O_{2}$ and $O_{3}$ for cities ℑ₁, ℑ ₂ and ℑ₃ given as $O_{1} = (〈 0.7900, 0.2567 〉, 〈 0.3005, 0.7055 〉,$ 〈0.2656, 0.9027〉). $O_{2} = (〈 0.6758, 0.5718 〉, 〈 0.4871, 0.6140 〉,$ 〈0.2793, 0.8930〉). $O_{3} = (〈 0.5221, 0.5411 〉, 〈 0.5226, 0.1680 〉,$ 〈0.3507, 0.6251〉). The score values of above aggregated P3PFNs can be obtained by using Definition 3.4 given as, $£ (O_{1}) = 0.4010, £ (O_{2}) = 0.3579, £ (O_{3}) = 0.4982$ The aggregated values $O_{1}, O_{2}$ and $O_{3}$ shows the economical strength of every city ℑ₁, ℑ ₂ and ℑ₃ respectively and the calculated score values shows the population measure of cities ℑ₁, ℑ ₂ and ℑ₃. These score values shows that ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂. Hence city ℑ₃ has the maximum population as compared to other cities. (See Fig. 7). The flow chart diagram of above application with the proposed algorithm and all the calculations for all defined operators for ϖ = 1 is given as Fig. 8.

Fig. 7

Score values for PMPFDWGA operator.

Fig. 8

Flow chart diagram of proposed algorithm.

7 Discussion and comparison analysis

In this segment, we present a brief comparison between the suggested operators and see the influence of operational parameter in the aggregated outcomes. We will observe that by the convergence of the proposed approach the arithmetical error can be minimized by the given algorithm. We discuss and observe that how the staged approach is more flexible and efficient than the old method. The comparability of the previous and proposed attack is presented in tabular form as Table 10.

7.1 Validity of the method

The suggested method is valid and desirable for the census process because this approach covers many disadvantages of old method. In this approach, we can discuss about various properties and alternatives with the corresponding criteria at the same time. The data aggregation can be summed as a specific PMPF data input and this hybrid model can be employed to compile the information on a large scale. It can be used to define multivalued relations and multivalued social networks.

7.2 Comparison analysis

The proposed technique is more flexible and practical technique to deal with uncertainties and vagueness in the field of decision analysis. The proposed technique gives us best ranking methodology by using PMPFNs. Table 10 shows that the existing methodologies have some drawbacks and limitations, which can be handled by using the proposed technique for the census process. From Table 11, it is clear that PMPFS is suitable technique when the input data is available in the form of multi-polar (paired) information.

Table 10
Comparison analysis

Census Previous approaches [4, 15]

process

1 Time consuming

2 Inconvenient

3 Labor consuming

4 More statistical error

5 Costly

6 Short set of characteristics

7 Held after a long time

8 Training issues to labor

9 Difficult to assorted the data

Census Proposed approach

process

1 Reduce time period

2 Convenient

3 No labor required

4 Statistical error reduced

5 Cheap

6 Set of characteristics increses

7 Conducted after a short time period

8 No training required

9 Data assorted easily by software

Census	Previous approaches [4, 15]
1	Time consuming
2	Inconvenient
3	Labor consuming
4	More statistical error
5	Costly
6	Short set of characteristics
7	Held after a long time
8	Training issues to labor
9	Difficult to assorted the data
Census	Proposed approach
process
1	Reduce time period
2	Convenient
3	No labor required
4	Statistical error reduced
5	Cheap
6	Set of characteristics increses
7	Conducted after a short time period
8	No training required
9	Data assorted easily by software

Table 11

Comparison of PMPFS with existing structures

Set	Membership grade	Non-membership grade	Multiple membership grades
Fuzzy set [1]	✓	×	×
Intuitionistic fuzzy set [3]	✓	✓	×
Pythagorean fuzzy set [5, 6]	✓	✓	×
m-polar fuzzy set [46]	✓	×	✓
Pythagorean m-polar fuzzy set (proposed)	✓	✓	✓

7.3 The Influence of parameter ϖ

We calculate the aggregated PMPFNs for different values of operational parameter ϖ from the input PMPF-data and will see the behavior of the operator by analyzing their score values. The values are given in tabular form as Tables 12 and 13. The parameter ϖ has no consequence on the ranking results of PMPFDWGA. This signifies that the obtained ranking results from PMPFDWGA operator are not sensitive to the parameter ϖ. While, the ranking results are sensitive to ϖ for the PMPFDWAA operator. When the value of ϖ increases, then ranking of the city changes and after that for a very large value all the ranking becomes similar. The answers show that PMPFDWGA is more elastic and desirable for the MCDM problems. The combined ranking results of both operators for different values of parameter ϖ can be seen from Figs. 11 and 12.

Table 12
Ranking order of PMPFDWAA for ϖ

ϖ $£ (O_{1}), £ (O_{2}), £ (O_{3})$ Ranking Order Result

1 0.4288, 0.6571, 0.7022 ℑ₃ ≻ ℑ ₂ ≻ ℑ ₁ ℑ₃

2 0.4424, 0.7302, 0.7436 ℑ₃ ≻ ℑ ₂ ≻ ℑ ₁ ℑ₃

5 0.4682, 0.7899, 0.7760 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

10 0.4893, 0.8114, 0.7876 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

20 0.5053, 0.8218, 0.7932 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

50 0.5157, 0.8277, 0.7966 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

100 0.5192, 0.8296, 0.7976 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

500 0.5220, 0.8312, 0.7985 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

1000 0.5223, 0.8314, 0.7986 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

ϖ	$£ (O_{1}), £ (O_{2}), £ (O_{3})$	Ranking Order	Result
1	0.4288, 0.6571, 0.7022	ℑ₃ ≻ ℑ ₂ ≻ ℑ ₁	ℑ₃
2	0.4424, 0.7302, 0.7436	ℑ₃ ≻ ℑ ₂ ≻ ℑ ₁	ℑ₃
5	0.4682, 0.7899, 0.7760	ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁	ℑ₂
10	0.4893, 0.8114, 0.7876	ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁	ℑ₂
20	0.5053, 0.8218, 0.7932	ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁	ℑ₂
50	0.5157, 0.8277, 0.7966	ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁	ℑ₂
100	0.5192, 0.8296, 0.7976	ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁	ℑ₂
500	0.5220, 0.8312, 0.7985	ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁	ℑ₂
1000	0.5223, 0.8314, 0.7986	ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁	ℑ₂

Table 13

Ranking order of PMPFDWGA for ϖ

ϖ	$£ (O_{1}), £ (O_{2}), £ (O_{3})$	Ranking Order	Result
1	0.4010, 0.3579, 0.4928	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃
2	0.3908, 0.3066, 0.4463	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃
5	0.3722, 0.2620, 0.4045	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃
10	0.3605, 0.2463, 0.3886	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃
20	0.3538, 0.2386, 0.3800	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃
50	0.3494, 0.2341, 0.3748	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃
100	0.3477, 0.2326, 0.3730	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃
500	0.3462, 0.2314, 0.3716	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃
1000	0.3460, 0.2313, 0.3715	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃

7.4 Convergence

It is observed from the Table 13 that when the value of the operational parameter ϖ increases, then score values converges to some specific point. This demonstrates that due to increase in ϖ our approximations converges to the actual values and to a very large value of ϖ all approximated values approaches to real values and statistical error approaches to zero. At some time for very large values of ϖ we get no effect on score values and all the obtained values remains constant. Thus by applying this novel technique we can manage and solve the disadvantages and uncertainties of manual data aggregation. (Actualvalue - Approximatedvalue) →0 when $ϖ \to Z$ , where $Z$ is very very large positive integer. We can see the convergence behavior of operators graphically as Figs. 9 and 10.

Fig. 9

Ranking results for PMPFDWAA operator.

Fig. 10

Ranking results for PMPFDWGA operator.

Fig. 11

Ranking results for PMPFDWGA operator.

Fig. 12

Ranking results for PMPFDWAA operator.

7.5 Sensitivity analysis

Finally, we will discuss about the sensitivity of the proposed operators and can be seen from Table 14. The PMPFDWAA operator is more flexible to its related attributes and specially for operational parameter ϖ for the information fusion. We observe that the previous calculations that changing the values of ϖ affects the aggregated result of PMPFDWAA operator. On the other side, PMPFDWGA operator is not sensitive to ϖ because by varying the values of ϖ results remains same from ϖ = (1 - 1000). This operator only converge to the actual conditions and remove ambiguities. The graphs are broken to ensure the changing in aggregated score values of all the attributes for different ϖ.

Table 14
Sensitivity of operators to parametric values

Operators ϖ Ranking results Higher rank

PMPFDWAA ϖ = 1, 2 ℑ₃ ≻ ℑ ₂ ≻ ℑ ₁ ℑ₃

PMPFDWAA ϖ = 5 -1000 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

PMPFDWGA ϖ = 1 -1000 ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂ ℑ₃

Operators	ϖ	Ranking results	Higher rank
PMPFDWAA	ϖ = 1, 2	ℑ₃ ≻ ℑ ₂ ≻ ℑ ₁	ℑ₃
PMPFDWAA	ϖ = 5 -1000	ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁	ℑ₂
PMPFDWGA	ϖ = 1 -1000	ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂	ℑ₃

8 Conclusion

From the prevailing ideas of MPFS and PFS, we have introduced a novel concept of PMPFS. We have introduced some fundamental operations on PMPFSs and explained these concepts with the help of illustrations. With this novel perspective, we have developed modified forms of Dombi’s aggregation operators named as Pythagorean m-polar fuzzy Dombi weighted arithmetic average (PMPFDWAA) operator and Pythagorean m-polar fuzzy Dombi weighted geometric average (PMPFDWGA) operator. We have discussed certain properties of the proposed operators based on PMPFSs. For the best ranking of alternatives in the multi-criteria decision analysis based PMPFNs, we have introduced score function, accuracy function and certainty function. A case study concerning to a census process have been presented with the help of proposed PMPFDWAA and PMPFDWGA operators. We have presented a brief comparison analysis between the existing and proposed technique. We have explained the influence of operational parameter to obtain aggregated outcomes. In future, this work can be gone for different type of hybrid structures and decision-making problems.

Footnotes

Appendix A:

Proof. We have to proof that equation (A) holds for PMPFDWAA operator. For this we will use the methodology of mathematical induction technique. As we know that $F_{℘} = (〈^{℘} T_{1},^{℘} F_{1}〉, 〈^{℘} T_{2},^{℘} F_{2}〉, \dots, 〈^{℘} T_{M},^{℘} F_{M}〉 : ℘ \in Δ)$ for $℘ = 1, 2, . . ., N$ is an assembling of PMPFNs, then we will follow these three steps given below: Step 1: For $N = 2$ we have $F_{1} = (〈^{1} T_{1},^{1} F_{1}〉, 〈^{1} T_{2},^{1} F_{2}〉, \dots, 〈^{1} T_{M},^{1} F_{M}〉)$ and $F_{2} = (〈^{2} T_{1},^{2} F_{1}〉, 〈^{2} T_{2},^{2} F_{2}〉, \dots, 〈^{2} T_{M},^{2} F_{M}〉)$ and for ζ₁ and ζ₂ we can write that $η_{1} F_{1} = (\begin{array}{l} 〈1 - \frac{1}{1 + {ζ_{1} {(\frac{^{1} T_{1}}{1 -^{1} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{1} {(\frac{^{1} F_{1}}{1 -^{1} F_{1}})}^{ϖ}}^{1 / ϖ}}〉, \\ 〈1 - \frac{1}{1 + {ζ_{1} {(\frac{^{1} T_{2}}{1 -^{1} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{1} {(\frac{^{1} F_{2}}{1 -^{1} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {ζ_{1} {(\frac{^{1} T_{M}}{1 -^{1} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{1} {(\frac{^{1} F_{M}}{1 -^{1} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array})$ and $η_{2} F_{2} = (\begin{array}{l} 〈1 - \frac{1}{1 + {ζ_{1} {(\frac{^{2} T_{1}}{1 -^{2} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{1} {(\frac{^{2} F_{1}}{1 -^{2} F_{1}})}^{ϖ}}^{1 / ϖ}}〉, \\ 〈1 - \frac{1}{1 + {ζ_{1} {(\frac{^{2} T_{2}}{1 -^{2} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{1} {(\frac{^{2} F_{2}}{1 -^{2} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {ζ_{1} {(\frac{^{2} T_{M}}{1 -^{2} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{1} {(\frac{^{2} F_{M}}{1 -^{2} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array})$ . Now PMPFDWAA (ℑ ₁, ℑ ₂) = ζ₁ ℑ ₁ ⊕ ζ₂ ℑ ₂ PMPFDWAA (ℑ ₁, ℑ ₂) = $\begin{array}{l} (\begin{array}{l} 〈\begin{array}{l} 1 - \frac{1}{1 + {ζ_{1} {(\frac{^{1} T_{1}}{1 -^{1} T_{1}})}^{ϖ} + ζ_{2} {(\frac{^{2} T_{1}}{1 -^{2} T_{1}})}^{ϖ}}^{1 / ϖ}}, \\ \frac{1}{1 + {ζ_{1} {(\frac{1 -^{1} F_{1}}{^{1} F_{1}})}^{ϖ} + ζ_{2} {(\frac{1 -^{2} F_{1}}{^{2} F_{1}})}^{ϖ}}^{1 / ϖ}} \end{array}〉, \\ 〈1 - \frac{1}{1 + {ζ_{1} {(\frac{^{1} T_{2}}{1 -^{1} T_{2}})}^{ϖ} + ζ_{2} {(\frac{^{2} T_{2}}{1 -^{2} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{1} {(\frac{1 -^{1} F_{2}}{^{1} F_{2}})}^{ϖ} + ζ_{2} {(\frac{1 -^{2} F_{2}}{^{2} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {ζ_{1} {(\frac{^{1} T_{M}}{1 -^{1} T_{M}})}^{ϖ} + ζ_{2} {(\frac{^{2} T_{M}}{1 -^{2} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{1} {(\frac{1 -^{2} F_{M}}{^{2} F_{M}})}^{ϖ} + ζ_{2} {(\frac{1 -^{2} F_{M}}{^{2} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array}) \end{array}$ PMPFDWAA (ℑ ₁, ℑ ₂) = $(\begin{array}{l} 〈1 - \frac{1}{1 + {\sum_{p = 1}^{2} ζ_{p} {(\frac{^{p} T_{1}}{^{p} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{2} ζ_{p} {(\frac{1 -^{p} F_{1}}{^{p} F_{1}})}^{ϖ}}^{1 / ϖ}}〉, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{2} ζ_{p} {(\frac{^{p} T_{2}}{^{p} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{2} ζ_{p} {(\frac{1 -^{p} F_{2}}{^{p} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{2} ζ_{p} {(\frac{^{p} T_{M}}{^{p} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{2} ζ_{p} {(\frac{1 -^{p} F_{M}}{^{p} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array}) .$ This shows that equation (A) holds for $N = 2$ . Step 2: Suppose that (A) holds for $N = k$ , given as PMPFDWAA (ℑ ₁, ℑ ₂, . . . , ℑ _k) = $\begin{array}{l} 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{^{p} T_{1}}{^{p} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{1 -^{p} F_{1}}{^{p} F_{1}})}^{ϖ}}^{1 / ϖ}}〉, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{^{p} T_{2}}{^{p} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{1 -^{p} F_{2}}{^{p} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{^{p} T_{M}}{^{p} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{1 -^{p} F_{M}}{^{p} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array}$ . Step 3: Now we will show that (A) holds for $N = k + 1$ . Consider PMPFDWAA (ℑ ₁, ℑ ₂, . . . , ℑ _k+1) = $(\oplus_{℘ = 1}^{k} ζ_{℘} ℑ_{℘}) \oplus ζ_{k + 1} ℑ_{k + 1}$ . PMPFDWAA (ℑ ₁, ℑ ₂, . . . , ℑ _k+1) = $(\begin{array}{l} 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{^{p} T_{1}}{^{p} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{1 -^{p} F_{1}}{^{p} F_{1}})}^{ϖ}}^{1 / ϖ}}〉, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{^{p} T_{2}}{^{p} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{1 -^{p} F_{2}}{^{p} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{^{p} T_{M}}{^{p} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k} ζ_{p} {(\frac{1 -^{p} F_{M}}{^{p} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array}) \oplus (\begin{array}{l} 〈1 - \frac{1}{1 + {ζ_{k + 1} {(\frac{^{k + 1} T_{1}}{1 -^{k + 1} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{k + 1} {(\frac{1 -^{k + 1} F_{1}}{^{k + 1} F_{1}})}^{ϖ}}^{1 / ϖ}}〉, \\ 〈1 - \frac{1}{1 + {ζ_{k + 1} {(\frac{^{k + 1} T_{2}}{1 -^{k + 1} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{k + 1} {(\frac{1 -^{k + 1} F_{2}}{^{k + 1} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {ζ_{k + 1} {(\frac{^{k + 1} T_{M}}{1 -^{k + 1} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {ζ_{k + 1} {(\frac{1 -^{k + 1} F_{M}}{^{k + 1} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array})$ . PMPFDWAA (ℑ ₁, ℑ ₂, . . . , ℑ _k+1) = $(\begin{array}{l} 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k + 1} ζ_{p} {(\frac{^{p} T_{1}}{1 -^{p} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k + 1} ζ_{p} {(\frac{1 -^{p} F_{1}}{^{p} F_{1}})}^{ϖ}}^{1 / ϖ}}〉, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k + 1} ζ_{p} {(\frac{^{p} T_{2}}{1 -^{p} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k + 1} ζ_{p} {(\frac{1 -^{p} F_{2}}{^{p} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{k + 1} ζ_{p} {(\frac{^{p} T_{M}}{1 -^{p} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{k + 1} ζ_{p} {(\frac{1 -^{p} F_{M}}{^{p} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array})$ . Hence (A) is true for all $N$ .□

Appendix B:

Proof. As we know that from equation (A), $PMPFDWAA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) =$ $(\begin{array}{l} 〈1 - \frac{1}{1 + {\sum_{p = 1}^{N} ζ_{p} {(\frac{^{p} T_{1}}{1 -^{p} T_{1}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{N} ζ_{p} {(\frac{1 -^{p} F_{1}}{^{p} F_{1}})}^{ϖ}}^{1 / ϖ}}〉, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{N} ζ_{p} {(\frac{^{p} T_{2}}{1 -^{p} T_{2}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{N} ζ_{p} {(\frac{1 -^{p} F_{2}}{^{p} F_{2}})}^{ϖ}}^{1 / ϖ}}〉, \dots, \\ 〈1 - \frac{1}{1 + {\sum_{p = 1}^{N} ζ_{p} {(\frac{^{p} T_{M}}{1 -^{p} T_{M}})}^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {\sum_{p = 1}^{N} ζ_{p} {(\frac{1 -^{p} F_{M}}{^{p} F_{M}})}^{ϖ}}^{1 / ϖ}}〉 \end{array})$

. We substitute ℑ_℘ =ℑ $\forall ℘ = 1, 2, . . ., N$ and we get $(〈 1 - \frac{1}{1 + {(\frac{T_{1}}{1 - T_{1}})^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {(\frac{1 - F_{1}}{F_{1}})^{ϖ}}^{1 / ϖ}} 〉,$ $〈 1 - \frac{1}{1 + {(\frac{T_{2}}{1 - T_{2}})^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {(\frac{1 - F_{2}}{F_{2}})^{ϖ}}^{1 / ϖ}} 〉, . . .,$ $〈 1 - \frac{1}{1 + {(\frac{T_{M}}{1 - T_{M}})^{ϖ}}^{1 / ϖ}}, \frac{1}{1 + {(\frac{1 - F_{M}}{F_{M}})^{ϖ}}^{1 / ϖ}} 〉)$ . By cancelation of ϖ and 1/ϖ we get the PMPFN ℑ and thus, $PMPFDWAA (ℑ_{1}, ℑ_{2}, . . ., ℑ_{N}) = ℑ$ □

Appendix C:

Proof. The proof is obvious by using equation (A). Since $min_{℘}^{℘} T_{M} ⪯^{℘} T_{M} ⪯ max_{℘}^{℘} T_{M}$ and $min_{℘}^{℘} F_{M} ⪯^{℘} F_{M} ⪯ max_{℘}^{℘} F_{M}$ is true for all $M$ . Then we have the following inequalities, $1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} (\frac{{min}_{℘}^{℘} T_{M}}{1 - {min}_{℘}^{℘} T_{M}})^{ϖ}}^{1 / ϖ}}$ $⪯ 1 - \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} (\frac{{max}_{℘}^{℘} T_{M}}{1 - {max}_{℘}^{℘} T_{M}})^{ϖ}}^{1 / ϖ}}$ , for all $M$ . and $\frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} (\frac{1 - {min}_{℘}^{℘} F_{M}}{{min}_{℘}^{℘} F_{M}})^{ϖ}}^{1 / ϖ}}$ $⪯ \frac{1}{1 + {\sum_{℘ = 1}^{N} ζ_{℘} (\frac{1 - {max}_{℘}^{℘} F_{M}}{{max}_{℘}^{℘} F_{M}})^{ϖ}}^{1 / ϖ}}$ , for all $M$ . Hence for equation (A) above inequalities shows that $^{p} F_{min} ≼ PMPFDWAA(F_{1}, F_{2}, \dots, F_{N}) ≼^{p} F_{max}$ .□

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A novel approach to censuses process by using Pythagorean m-polar fuzzy Dombi’s aggregation operators

Abstract

Keywords

1 Introduction

2 Background

Table 6 Dombi operator Operator ϖ Type D ( K 1 , K 2 ) ϖ > 0 Conjunctive D c ( K 1 , K 2 ) ϖ > 0 Disjunctive D ( K 1 , K 2 ) ϖ < 0 Disjunctive D c ( K 1 , K 2 ) ϖ < 0 Conjunctive

5.1 PMPFDWAA operator

6.1 Case study

6.2 Significance of census process

6.3 Disadvantages of existence approach

6.4 Fuzzy logic model for software construction

7.1 Validity of the method

7.2 Comparison analysis

Table 14 Sensitivity of operators to parametric values Operators ϖ Ranking results Higher rank PMPFDWAA ϖ = 1, 2 ℑ3 ≻ ℑ 2 ≻ ℑ 1 ℑ3 PMPFDWAA ϖ = 5 -1000 ℑ2 ≻ ℑ 3 ≻ ℑ 1 ℑ2 PMPFDWGA ϖ = 1 -1000 ℑ3 ≻ ℑ 1 ≻ ℑ 2 ℑ3

Footnotes

Appendix A:

Appendix B:

Appendix C:

References

Table 6
Dombi operator

Operator ϖ Type

$D (K_{1}, K_{2})$ ϖ > 0 Conjunctive

$D^{c} (K_{1}, K_{2})$ ϖ > 0 Disjunctive

$D (K_{1}, K_{2})$ ϖ < 0 Disjunctive

$D^{c} (K_{1}, K_{2})$ ϖ < 0 Conjunctive

Table 14
Sensitivity of operators to parametric values

Operators ϖ Ranking results Higher rank

PMPFDWAA ϖ = 1, 2 ℑ₃ ≻ ℑ ₂ ≻ ℑ ₁ ℑ₃

PMPFDWAA ϖ = 5 -1000 ℑ₂ ≻ ℑ ₃ ≻ ℑ ₁ ℑ₂

PMPFDWGA ϖ = 1 -1000 ℑ₃ ≻ ℑ ₁ ≻ ℑ ₂ ℑ₃