# Pythagoreanm-Polar Fuzzy Weighted Aggregation Operators and Algorithm for the Investment Strategic Decision Making

Authors: Muhammad Riaz; Khalid Naeem; Ronnason Chinram; Aiyared Iampan
Journal: Journal of Mathematics
 (2021)

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## Abstract

The role of multipolar uncertain statistics cannot be unheeded while confronting daily life problems on well-founded basis. Fusion (aggregation) of a number of input values in multipolar form into a sole multipolar output value is an essential tool not merely of physics or mathematics but also of widely held problems of economics, commerce and trade, engineering, social sciences, decision-making problems, life sciences, and many more. The problem of aggregation is very wide-ranging and fascinating, in general. We use, in this article, Pythagorean fuzzy numbers (PFNs) in multipolar form to contrive imprecise information. We introduce Pythagoreanm-polar fuzzy weighted averaging (PmFWA), Pythagorean m-polar fuzzy weighted geometric (PmFWG), symmetric Pythagorean m-polar fuzzy weighted averaging (SPmFWA), and symmetric Pythagorean m-polar fuzzy weighted geometric (SPmFWG) operators for aggregating uncertain data. Finally, we present a practical example to illustrate the application of the proposed operators and to demonstrate its practicality and effectiveness towards investment strategic decision making.


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## Body

## 1. Introduction and Literature Review

The process of MCGDM focuses upon assisting the choice makers in evaluating the most appropriate choice amongst a finite number of options according to some criteria in such a manner that inclination of any member from the group towards a particular choice is diffused. Such knotty problems occur frequently in daily life situations. Due to the presence of uncertain, imprecise, and ever changing information, the decision makers face problems in reaching some unanimous decision. To address the issue of uncertainty, Zadeh [1] founded fuzzy set (FS) theory by annexing membership map to each element of the traditional set. The so-called membership function yields information about level of association of some particular element with the underlying set.Soft set (SS), initiated by Molodtsov [2], is yet another model to handle imprecisions available in data. Zhang [3] suggested bipolar fuzzy sets as a generality of FSs. Lee [4] proposed bipolar-valued fuzzy sets. Ensuing the realization of Zhang and Lee, Chen et al. [5] inaugurated m-polar fuzzy sets as an extension of bipolar fuzzy sets.After the actuation of FSs, the researchers around the globe initiated working on its further expansions in different directions. Atanassov [6] supplemented FSs by including anti-membership map and denominated the resulting family as intuitionistic fuzzy set (IFS). According to Atanassov, the mappings used in an IFS drag members of underlying universe to [0, 1] with the additional restriction that their aggregate should also fall in the same interval.Yager [7] initiated the concept of ordered weighted averaging aggregation operators and information aggregation. Yager [8, 9] adjusted the curtailment imposed on parameters in IFSs so that the sum total of their squared values should lie in [0, 1] and acknowledged the evolved structure as Pythagorean fuzzy set (PFS). Yager [10] further acquainted the notion of q-ROFS as an enlargement of PFS. A short time ago, Pythagorean m-polar fuzzy sets with their practical implementations have been unveiled by Naeem et al. [11, 12]. Well along, Riaz et al. [13] extended the notion of soft sets towards Pythagorean m-polar fuzzy soft sets and prooffered some fascinating utilizations of this model. Riaz et al. [14] unveiled Pythagorean fuzzy multisets with their applications.Peng and Yang [15, 16] proposed some properties of PFSs and interval-valued Pythagorean fuzzy aggregation operators. Peng and Yuan [17] studied fundamental properties of PF aggregation operators. Selvachandran and Peng [18] presented a new approach for the supplier selection problem based on the modified TOPSIS method under vague parameterized vague soft information. Peng and Selvachandran [19] proposed state of the art and future directions for Pythagorean fuzzy set. Peng [20] introduced a new similarity measure and distance measure for Pythagorean fuzzy set. Feng et al. [21] discussed generalized intuitionistic fuzzy soft sets with their practical usage. Feng et al. [22] proposed Minkowski weighted score functions of intuitionistic fuzzy values and developed an algorithm for solving decision-making problems.Aggregation operators are used to fuse a given information as a single resultant from the same structure. Diverse sorts of operators employed on different expansions of FSs along with their practical usage are studied by different researchers. Jose and Kuriaskose [23] studied aggregation operators, score function, and accuracy function for multicriteria decision making in intuitionistic fuzzy context. Kaur and Garg [24] studied cubic intuitionistic fuzzy aggregation operators. Garg and Arora [25] presented t-norm-based generalized intuitionistic fuzzy soft power aggregation operator accompanied by its practical implementation. Garg and Arora [26] proposed scaled prioritized intuitionistic fuzzy soft interaction averaging operator. Garg [27] suggested neutrality operations-based Pythagorean fuzzy aggregation operators. Garg and Kaur [28] introduced a robust correlation coefficient for probabilistic dual hesitant fuzzy sets and its applications. Karaaslan and Hunu [29] introduced type-2 single-valued neutrosophic sets and their applications in multicriteria group decision making based on the TOPSIS method.Liu and Wang [30] discussed some q-rung orthopair fuzzy aggregation operator. Liu et al. [31] extended prioritized weighted aggregation operators. Liu et al. [32] explored the ranking range-based approach to MADM under incomplete context. Li et al. [33] established decision making based on interval-valued complex single-valued neutrosophic hesitant fuzzy generalized hybrid weighted averaging operators. Liu et al. [34] proposed group decision making using complex q-rung orthopair fuzzy Bonferroni mean. Liu and Wang [35] introduced the multiattribute group decision-making method based on intuitionistic fuzzy Einstein interactive operations. Liu et al. [36] introduced the concept of hesitant intuitionistic fuzzy linguistic aggregation operators and their applications to multiattribute decision making.Akram et al. [37] studied Pythagorean Dombi fuzzy aggregation operators. Akram et al. [38, 39] introduced decision-making analysis based on q-rung picture fuzzy graph structures and complex picture fuzzy Hamacher aggregation operators.Lu et al. [40] coined hesitant Pythagorean fuzzy Hamacher aggregation operators. Ma and Xu [41] launched symmetric Pythagorean fuzzy weighted geometric/averaging operators. Akram and Shahzadi [42] established q-rung orthopair fuzzy Yager aggregation operators.Zararsiz and Sengönül [43] introduced certain concepts on the gravity of center of sequence of fuzzy numbers. Zararsiz [44] proposed new similarity measures of sequence of fuzzy numbers and fuzzy risk analysis. Riaz and Hashmi [45] introduced a novel approach to censuses process by using Pythagorean m-polar fuzzy Dombi’s aggregation operators. Riaz et al. [46] introduced a robust q-rung orthopair fuzzy Einstein prioritized aggregation operators with application towards MCGDM. Riaz and Tehrim [47, 48] introduced the concept of cubic bipolar fuzzy set with application to multicriteria group decision making using geometric aggregation operators. They proposed a robust extension of the VIKOR method for bipolar fuzzy sets using connection numbers of SPA theory-based metric spaces.Wei and Lu [49] unveiled PF power aggregation operators. Wei [50] coined PF interaction aggregation operators. Faizi et al. [51] developed Einstein aggregation operational laws for intuitionistic 2-tuple linguistic set and further developed weighted averaging and weighted geometric operators. Xu [52] studied intuitionistic fuzzy aggregation operators. Xu and Cai [53] explored IF information aggregation.The motive behind this article is to study (symmetric) Pythagorean fuzzy weighted averaging and geometric aggregation operators encompassing multipolar information and their characteristics. Contribution of multipolar data cannot be overlooked in coping with daily life problems. Pythagoreanm-polar fuzzy sets have a range of applications in diverse real-life circumstances, and these models boost the management of uncertainty and vagueness by using multipolarity in the membership and nonmembership grades in a broader way. The practical characteristic of PmFSs is that the decision makers (DMs) can be asked to assign multipolar ordered pairs of membership and nonmembership grades with the condition that their sum of squares may not exceed unity. Before reaching a solid decision, we think time and again about the pros and cons of the problem which is indeed a process of manipulating multipolar information.The leftover part of this article is organized as follows. Section2 gives access to preliminary notions mainly including operational laws of Pythagorean m-polar fuzzy numbers. The next segment presents Pythagorean m-polar fuzzy weighted averaging operator in company with its desirable qualities, whereas Section 4 deals with the corresponding geometric operator. Section 5 deals with symmetric Pythagorean m-polar fuzzy weighted averaging operator as well as its worthwhile characteristics, whereas Section 6 is dedicated to deal with the corresponding geometric operator. The four suggested operators are applied on MCGDM problem of capital investment analysis accompanied by an algorithm in Section 7. Comparative analysis and superiority of the proposed work is also rendered in the same segment. We conclude the paper in Section 8 with some further future directions.


## 2. Preliminaries

We recall some fundamentals of Pythagoreanm-polar fuzzy sets and their operational laws accompanied by operational laws of corresponding numbers in this segment.Definition 1 (see [11]).
A Pythagoreanm-polar fuzzy set (PmFS) O is characterized by two sets of mappings ϒOi (denoting affiliation degrees) and o¯Oi (meant for dissociation grades) dropping members of X to [0, 1] constrained to obey 0≤ϒOig2+o¯Oig2≤1, for all i. The quantity εOig=1−o¯Oig2−ϒOig2 is known as hesitation margin or indeterminacy degree of g∈X to O. εOi:X↦0,1 are mappings expressing lack of knowledge regarding g∈O or g∉O. The pair ϒi,o¯i is commonly acknowledged as Pythagorean fuzzy number (PFN).
A PmFS is usually expressed as(1)B=gϒOig,o¯Oigi=1m.
IfX=r, then tabulatory array of O is as in Table 1.
The corresponding matrix format is(2)O=ϒO1g1,o¯O1g1ϒO2g1,o¯O2g1⋯ϒOmg1,o¯Omg1ϒO1g2,o¯O1g2ϒO2g2,o¯O2g2⋯ϒOmg2,o¯Omg2⋮⋮⋱⋮ϒO1gr,o¯O1grϒO2gr,o¯O2gr⋯ϒOmgr,o¯Omgr.
This matrix of sizer×m is titled as PmF matrix.Table 1
Tabulatory array ofß.
ßg1ϒß1g1,o¯ß1g1ϒß2g1,o¯ß2g1⋯ϒßmg1,o¯ßmg1g2ϒß1g2,o¯ß1g2ϒß2g2,o¯ß2g2⋯ϒßmg2,o¯ßmg2⋮⋮⋮⋱⋮grϒß1gr,o¯ß1grϒß2gr,o¯ß2gr⋯ϒßmgr,o¯ßmgr


### 2.1. Operational Laws of Pythagoreanm-Polar Fuzzy Sets (PmFSs)

Definition 2.
(see [11]). Let O1=ϒO1iγ,o¯O1iγ and O2=ϒO2iγ,o¯O2iγ be PmFSs on X and λ be a fuzzy number. Then,(1)
O1c=o¯O1iγ,ϒO1iγ.(2)
O1⊑O2 on condition that ϒO1i℘≤ϒO2i℘ and o¯O2i℘≤o¯O1i℘.(3)
O1⊔O2=maxϒO1i℘,ϒO2i℘,mino¯O1i℘,o¯O2i℘.(4)
O1⊓O2=minϒO1i℘,ϒO2i℘,maxo¯O1i℘,o¯O2i℘.(5)
O1⊕O2=ϒO1i℘2+ϒO2i℘2−ϒO1i℘2ϒO2i℘2,o¯O1i℘o¯O2i℘.(6)
O1⊗O2=ϒO1i℘ϒO2i℘,o¯O1i℘2+o¯O2i℘2−o¯O1i℘2o¯O2i℘2.(7)
λO1=1−1−ϒO1i℘2λ,o¯O1i℘λ.(8)
O1λ=ϒO1i℘λ,1−1−o¯O1i℘2λ.


### 2.2. Operational Laws of PythagoreanO1λ=ϒO1i℘λ,1−1−o¯O1i℘2λ-Polar Fuzzy Numbers (PmFNs)

Definition 3.
SupposeO1=ϒ1i,o¯1ii=1m and O2=ϒ2i,o¯2ii=1m are two PmFNs and λ is a fuzzy number. Then,(1)
O1c=o¯1i,ϒ1ii=1m.(2)
O1⊔O2=maxϒ1i,ϒ2i,mino¯1i,o¯2ii=1m.(3)
O1⊓O2=minϒ1i,ϒ2i,maxo¯1i,o¯2ii=1m.(4)
O1⊕O2=ϒ1i2+ϒ2i2−ϒ1i2ϒ2i2,o¯1io¯2ii=1m.(5)
O1⊗O2=ϒ1iϒ2i,o¯1i2+o¯2i2−o¯1i2o¯2i2i=1m.(6)
O1⊞O2=ϒ1iϒ2i/1−ϒ1i21−ϒ2i2+ϒ1iϒ2i2,o¯1io¯2i/1−o¯1i21−o¯2i2+o¯1io¯2i2i=1m.(7)
O1∔O2=1−1−ϒ1i21−ϒ2i2/2−1−ϒ1i21−ϒ2i2−ϒ1iϒ2i2,1−1−o¯1i21−o¯2i2/2−1−o¯1i21−o¯2i2−o¯1io¯2i2i=1m.(8)
λO1=1−1−ϒ1i2λ,o¯1iλi=1m.(9)
λ⊡O1=ϒ1iλ/1−ϒ1i2λ+ϒ1i2λ,o¯1iλ/1−o¯1i2λ+o¯1i2λi=1m.(10)
λ⋅O1=1−1−ϒ1i2λ/2−1−ϒ1i2λ−ϒ1i2λ,1−1−o¯1i2λ/2−1−o¯1i2λ−o¯1i2λi=1m.(11)
O1λ=ϒ1iλ,1−1−o¯1i2λi=1m.Definition 4 (see [12]).
The score function of a PmFN O=ϒi,o¯ii=1m is specified by(3)sO=1m∑i=1mϒOi2−o¯Oi2.
The value of this score function always falls in−1,1.Definition 5 (see [12]).
The accuracy function of a PmFN O=ϒi,o¯ii=1m is determined by(4)aO=1m∑i=1mϒOi2+o¯Oi2.
The value of this accuracy function always falls in0,1.
We get advantage of score and accuracy functions of two PmFNs O1 and O2 in deciding ordering of O1 and O2 as described in Definition 6.Definition 6 (see [12]).
LetO1=ϒ1i,o¯1ii=1m and O2=ϒ2i,o¯2ii=1m be two PmFNs.(1)
IfsO1<sO2, then O1≺O2.(2)
If, however,sO1=sO2 and(i)
aO1<aO2, then O1≺O2.(ii)
aO1=aO2, then O1∼O2.
Note thatO1≺O2 means O1 precedes O2, and O1∼O2 means O1 and O2 are identical (same).


## 2.1. Operational Laws of Pythagoreanm-Polar Fuzzy Sets (PmFSs)

Definition 2.
(see [11]). Let O1=ϒO1iγ,o¯O1iγ and O2=ϒO2iγ,o¯O2iγ be PmFSs on X and λ be a fuzzy number. Then,(1)
O1c=o¯O1iγ,ϒO1iγ.(2)
O1⊑O2 on condition that ϒO1i℘≤ϒO2i℘ and o¯O2i℘≤o¯O1i℘.(3)
O1⊔O2=maxϒO1i℘,ϒO2i℘,mino¯O1i℘,o¯O2i℘.(4)
O1⊓O2=minϒO1i℘,ϒO2i℘,maxo¯O1i℘,o¯O2i℘.(5)
O1⊕O2=ϒO1i℘2+ϒO2i℘2−ϒO1i℘2ϒO2i℘2,o¯O1i℘o¯O2i℘.(6)
O1⊗O2=ϒO1i℘ϒO2i℘,o¯O1i℘2+o¯O2i℘2−o¯O1i℘2o¯O2i℘2.(7)
λO1=1−1−ϒO1i℘2λ,o¯O1i℘λ.(8)
O1λ=ϒO1i℘λ,1−1−o¯O1i℘2λ.


## 2.2. Operational Laws of PythagoreanO1λ=ϒO1i℘λ,1−1−o¯O1i℘2λ-Polar Fuzzy Numbers (PmFNs)

Definition 3.
SupposeO1=ϒ1i,o¯1ii=1m and O2=ϒ2i,o¯2ii=1m are two PmFNs and λ is a fuzzy number. Then,(1)
O1c=o¯1i,ϒ1ii=1m.(2)
O1⊔O2=maxϒ1i,ϒ2i,mino¯1i,o¯2ii=1m.(3)
O1⊓O2=minϒ1i,ϒ2i,maxo¯1i,o¯2ii=1m.(4)
O1⊕O2=ϒ1i2+ϒ2i2−ϒ1i2ϒ2i2,o¯1io¯2ii=1m.(5)
O1⊗O2=ϒ1iϒ2i,o¯1i2+o¯2i2−o¯1i2o¯2i2i=1m.(6)
O1⊞O2=ϒ1iϒ2i/1−ϒ1i21−ϒ2i2+ϒ1iϒ2i2,o¯1io¯2i/1−o¯1i21−o¯2i2+o¯1io¯2i2i=1m.(7)
O1∔O2=1−1−ϒ1i21−ϒ2i2/2−1−ϒ1i21−ϒ2i2−ϒ1iϒ2i2,1−1−o¯1i21−o¯2i2/2−1−o¯1i21−o¯2i2−o¯1io¯2i2i=1m.(8)
λO1=1−1−ϒ1i2λ,o¯1iλi=1m.(9)
λ⊡O1=ϒ1iλ/1−ϒ1i2λ+ϒ1i2λ,o¯1iλ/1−o¯1i2λ+o¯1i2λi=1m.(10)
λ⋅O1=1−1−ϒ1i2λ/2−1−ϒ1i2λ−ϒ1i2λ,1−1−o¯1i2λ/2−1−o¯1i2λ−o¯1i2λi=1m.(11)
O1λ=ϒ1iλ,1−1−o¯1i2λi=1m.Definition 4 (see [12]).
The score function of a PmFN O=ϒi,o¯ii=1m is specified by(3)sO=1m∑i=1mϒOi2−o¯Oi2.
The value of this score function always falls in−1,1.Definition 5 (see [12]).
The accuracy function of a PmFN O=ϒi,o¯ii=1m is determined by(4)aO=1m∑i=1mϒOi2+o¯Oi2.
The value of this accuracy function always falls in0,1.
We get advantage of score and accuracy functions of two PmFNs O1 and O2 in deciding ordering of O1 and O2 as described in Definition 6.Definition 6 (see [12]).
LetO1=ϒ1i,o¯1ii=1m and O2=ϒ2i,o¯2ii=1m be two PmFNs.(1)
IfsO1<sO2, then O1≺O2.(2)
If, however,sO1=sO2 and(i)
aO1<aO2, then O1≺O2.(ii)
aO1=aO2, then O1∼O2.
Note thatO1≺O2 means O1 precedes O2, and O1∼O2 means O1 and O2 are identical (same).


## 3. Pythagoreanm-Polar Fuzzy Weighted Averaging Operator

We dedicate this segment for inauguration of the notion of Pythagoreanm-polar fuzzy weighted averaging operator for Pythagorean m-polar fuzzy numbers along with some of its prime characteristics.Definition 7.
LetOk=ϒki,o¯kii=1mk=1,2,…,n be an assemblage of PmFNs. Define PmFWA: Tn⟶T given by(5)PmFWAO1,O2,…,On=⊕k=1nℏkOk=ℏ1O1⊕ℏ2O2⊕⋯⊕ℏnOn,where Tn is the collection of all PmFNs and ℏk’s are fuzzy weights of O1,O2,…,On, such that addition of all ℏk’s results in unity. Then, PmFWA is called the Pythagorean m-polar fuzzy weighted averaging operator.
If weight vectorW=1/n,1/n,…,1/nt, then PmFWA operator reduces to Pythagorean m-polar fuzzy averaging PmFA operator of dimension n and is given as(6)PmFAO1,O2,…,On=1n⊕k=1nOk=1nO1⊕O2⊕⋯⊕On.
As maintained by operational laws of PmFNs given in Definition 3, the following theorem assists in computing PmFWA for any PmFNs.Theorem 1.
LetOk=ϒki,o¯kii=1mk=1,2,…,n be an assemblage of PmFNs; then,(7)PmFWAO1,O2,…,On=1−∏k=1n1−ϒki2ℏk,∏k=1no¯kiℏki=1m.Proof.
We establish the result by means of induction.
By definition,(8)ℏ1O1=1−1−ϒ1i2ℏ1,o¯1iℏ1i=1m,ℏ2O2=1−1−ϒ2i2ℏ2,o¯2iℏ2i=1m,so that(9)PmFWAO1,O2&=ℏ1O1⊕ℏ2O2&=1−1−ϒ1i2ℏ1,o¯1iℏ1i=1m⊕1−1−ϒ2i2ℏ2,o¯2iℏ2i=1m.
Thex-component of the resultant is(10)1−1−ϒ1i2ℏ12+1−1−ϒ2i2ℏ22−1−1−ϒ1i2ℏ11−1−ϒ2i2ℏ22=1−1−ϒ1i2ℏ1+1−1−ϒ2i2ℏ2−1−1−ϒ1i2ℏ11−1−ϒ2i2ℏ2=1−1−ϒ1i2ℏ1+1−1−ϒ2i2ℏ2−1+1−ϒ1i2ℏ1+1−ϒ2i2ℏ2−1−ϒ1i2ℏ11−ϒ2i2ℏ21/2=1−1−ϒ1i2ℏ11−ϒ2i2ℏ2=1−∏k=121−ϒki2ℏk,and the y-component is o¯1iℏ1o¯2iℏ2. Thus,(11)PmFWAO1,O2=1−∏k=121−ϒki2ℏk,∏k=12o¯kiℏki=1m.
Now assuming that the result is valid forn PmFNs, we exhibit its validity for n+1 PmFNs. By definition,(12)PmFWAO1,O2,…,On+1=⊕k=1nℏkOk⊕ℏn+1On+1=1−∏k=1n1−ϒki2ℏk,∏k=1no¯kiℏki=1m⊕1−1−ϒn+1i2ℏn+1,o¯n+1iℏn+1i=1m.
Thex-component of the resultant would be(13)1−∏k=1n1−ϒki2ℏk2+1−1−ϒn+1i2ℏn+12−1−∏k=1n1−ϒki2ℏk21−1−ϒn+1i2ℏn+121/2=1−∏k=1n1−ϒki2ℏk+1−1−ϒn+1i2ℏn+1−1−∏k=1n1−ϒki2ℏk1−1−ϒn+1i2ℏn+11/2=1−∏k=1n1−ϒki2ℏk+1−1−ϒn+1i2ℏn+1−1+∏k=1n1−ϒki2ℏk+1−ϒn+1i2ℏn+1−∏k=1n+11−ϒki2ℏk1/2=1−∏k=1n+11−ϒki2ℏk,and the y-component is(14)∏k=1no¯iℏk×o¯iℏn+1=∏k=1n+1o¯iℏk.
This concludes the proof.Theorem 2.
IfOk=ϒki,o¯kii=1mk=1,2,…,n is an aggregate of PmFNs, then(15)PmFWAO1,O2,…,On=1−∏k=1n1−ϒki2ℏk,∏k=1no¯kiℏki=1mis also a PmFN.Proof.
Sinceϒki,o¯ki∈0,1, for each k and i,(16)0≤ϒki2≤1⟹0≤1−ϒki2≤1⟹0≤1−ϒki2ℏk≤1⟹0≤∏k=1n1−ϒki2ℏk≤1⟹0≤1−∏k=1n1−ϒki2ℏk≤1⟹0≤1−∏k=1n1−ϒki2ℏk≤1,0≤o¯kiℏk≤1⟹0≤∏k=1no¯kiℏk≤1.
Thus,1−∏k=1n1−ϒki2ℏk,∏k=1no¯kiℏk∈0,1.
Now,(17)ϒki2+o¯ki2≤1⇒o¯ki2≤1−ϒki2⇒o¯ki2ℏk≤1−ϒki2ℏk⇒∏k=1no¯ki2ℏk≤∏k=1n1−ϒki2ℏk⇒∏k=1no¯kiℏk2≤∏k=1n1−ϒki2ℏk,so that(18)0≤1−∏k=1n1−ϒki2ℏk2+∏k=1no¯kiℏk2=1−∏k=1n1−ϒki2ℏk+∏k=1no¯kiℏk2≤1−∏k=1n1−ϒki2ℏk+∏k=1n1−ϒki2ℏk=1,i.e., 0≤1−∏k=1n1−ϒki2ℏk2+∏k=1no¯kiℏk2≤1.Example 1.
Let(19)O1=0.62,0.39,0.55,0.68,0.37,0.26,0.46,0.61,O2=0.41,0.37,0.19,0.73,0.10,0.05,0.37,0.46,O3=0.47,0.68,0.39,0.40,0.84,0.35,0.15,0.92,be three P4FNs with corresponding weights ℏ1=0.4 and ℏ2=ℏ3=0.3. We aggregate the three PmFNs utilizing the result rendered in Theorem 1 as below:(20)P4FWAO1,O2,O3=1−∏k=131−ϒki2ℏk,∏k=13o¯kiℏki=14=0.526,0.454,0.430,0.592,0.590,0.173,0.369,0.634.Theorem 3.
Assume thatOk=ϒki,o¯kii=1mk=1,…,n is an assembly of PmFNs. Then,(1)
(Idempotency) ifOk=O=ϒi,o¯ii=1mk=1,2,…,n for all k, then(21)PmFWAO1,O2,…,On=O.(2)
(Boundedness) ifO−=minϒki,maxo¯ki and O+=maxϒki,mino¯ki, then(22)O−≤PmFWAO1,O2,…,On≤O+.(3)
(Monotonicity) letOk=ϒki,o¯kii=1m and Ok∗=ϒki∗,o¯ki∗i=1mk=1,…,n be two sets of PmFNs such that ϒki≥ϒki∗ and o¯ki≤o¯ki∗ for all k and all permissible value of i; then,(23)PmFWAO1,O2,…,On≥PmFWAO1∗,O2∗,…,On∗.Proof.
For idempotency, consider(24)PmFWAO1,O2,…,On=1−∏k=1n1−ϒi2ℏk,∏k=1no¯kiℏki=1m=1−∏k=1n1−ϒi2ℏk,∏k=1no¯iℏki=1m=1−1−ϒi2∑k=1nℏk,o¯i∑k=1nℏki=1m=1−1−ϒi2,o¯ii=1m=ϒi,o¯ii=1m=O.
Now, we establish boundedness. For membership grades ofPmFWAO1,O2,…,On, we have(25)1−∏k=1n1−minϒki2ℏk≤1−∏k=1n1−ϒki2ℏk≤1−∏k=1n1−maxϒki2ℏk⇒1−1−minϒki2∑k=1nℏk≤1−∏k=1n1−ϒki2ℏk≤1−1−maxϒki2∑k=1nℏk⇒1−1−minϒki2≤1−∏k=1n1−ϒki2ℏk≤1−1−maxϒki2⇒minϒki≤1−∏k=1n1−ϒki2ℏk≤maxϒki,and for the nonmembership grades, we have(26)∏k=1nmino¯kiℏk≤∏k=1no¯kiℏk≤∏k=1nmaxo¯kiℏk⇒mino¯ki∑k=1nℏk≤∏k=1no¯kiℏk≤maxo¯ki∑k=1nℏk⇒mino¯ki≤∏k=1no¯kiℏk≤maxo¯ki.
Now, we prove the monotonicity. Sinceϒki≥ϒki∗ and o¯ki≤o¯ki∗ for all k and all permissible value of i,(27)1−ϒki≤1−ϒki∗⇒1−ϒkiℏk≤1−ϒki∗ℏk⇒∏k=1n1−ϒkiℏk≤∏k=1n1−ϒki∗ℏk⇒1−∏k=1n1−ϒkiℏk≥1−∏k=1n1−ϒki∗ℏk⇒1−∏k=1n1−ϒkiℏk≥1−∏k=1n1−ϒki∗ℏk,o¯kiℏk≤o¯ki∗ℏk⇒∏k=1no¯kiℏk∏k=1no¯ki∗ℏk.
Therefore,(28)1−∏k=1n1−ϒkiℏk2−∏k=1no¯kiℏk2≥1−∏k=1n1−ϒki∗ℏk2−∏k=1no¯ki∗ℏk2.
Assume thatO=PmFWAO1,O2,…,On and O∗=PmFWAO1∗,O2∗,…,On∗; then, sO≥sO∗.(i)
IfsO>sO∗, then O>O∗, i.e., PmFWAO1,O2,…,On>PmFWAO1∗,O2∗,…,On∗.(ii)
IfsO=sO∗, then(29)1−∏k=1n1−ϒkiℏk2−∏k=1no¯kiℏk2=1−∏k=1n1−ϒki∗ℏk2−∏k=1no¯ki∗ℏk2,along with the given conditions ϒki≥ϒki∗ and o¯ki≤o¯ki∗ which yield(30)1−∏k=1n1−ϒkiℏk2=1−∏k=1n1−ϒki∗ℏk2∏k=1no¯kiℏk2=∏k=1no¯ki∗ℏk2,so that(31)aO=1m1−∏k=1n1−ϒkiℏk2+∏k=1no¯kiℏk2=1m1−∏k=1n1−ϒki∗ℏk2+∏k=1no¯ki∗ℏk2=aO∗.
Thus,O=O∗, i.e., PmFWAO1,O2,…,On=PmFWAO1∗,O2∗,…,On∗, and hence(32)PmFWAO1,O2,…,On≥PmFWAO1∗,O2∗,…,On∗.


## 4. Pythagoreanm-Polar Fuzzy Weighted Geometric Operator

In this segment, we present the notion of Pythagoreanm-polar fuzzy weighted geometric operator for Pythagorean m-polar fuzzy numbers accompanied by some of its prime characteristics.Definition 8.
LetOk=ϒki,o¯kii=1mk=1,2,…,n be an assemblage of PmFNs. Define PmFWG: Tn⟶T given by(33)PmFWGO1,O2,…,On=⊗k=1nOkℏk=O1ℏ1⊗O2ℏ2⊗⋯⊗Onℏn,where Tn is the collection of all PmFNs and ℏk’s are fuzzy weights of O1,O2,…,On, such that their sum equals unity. PmFWG is called Pythagorean m-polar fuzzy weighted geometric operator.
If eachℏk equals 1/n, then PmFWG operator turns down to n-dimensional Pythagorean m-polar fuzzy geometric (PmFG) operator and is given as(34)PmFGO1,O2,…,On=⊗k=1nOk1/n=O1⊗O2⊗⋯⊗On1/n.
In conformity with operational laws of PmFNs given in Definition 3, the following theorem accommodates in aggregating any finite number of PmFNs.Theorem 4.
LetOk=ϒki,o¯kii=1mk=1,2,…,n be an assemblage of PmFNs; then,(35)PmFWGO1,O2,…,On=∏k=1nϒkiℏk,1−∏k=1n1−o¯ki2ℏki=1m.Proof.
The proof may be furnished on the parallel track as proof of Theorem1.Theorem 5.
IfOk=ϒki,o¯kii=1mk=1,2,…,n is an aggregate of PmFNs, then(36)PmFWGO1,O2,…,On=∏k=1nϒkiℏk,1−∏k=1n1−o¯ki2ℏki=1mis also a PmFN.Proof.
The proof may be established in the same manner as the proof of Theorem2.Example 2.
We utilize the input of Example1. The P4FWG, using Theorem 4, is(37)P4FWGO1,O2,O3=∏k=13ϒkiℏk,1−∏k=131−o¯ki2ℏki=14=0.504,0.506,0.361,0.641,0.320,0.256,0.308,0.748.Theorem 6.
Assume thatOk=ϒki,o¯kii=1mk=1,…,n is an assemblage of PmFNs. Then,(1)
(Idempotency) ifOk=O=ϒi,o¯ii=1mk=1,2,…,n for all k, then(38)PmFWGO1,O2,…,On=O.(2)
(Boundedness) ifO−=minϒki,maxo¯ki and O+=maxϒki,mino¯ki, then(39)O−≤PmFWGO1,O2,…,On≤O+.(3)
(Monotonicity) letOk=ϒki,o¯kii=1m and Ok∗=ϒki∗,o¯ki∗i=1mk=1,…,n be two sets of PmFNs such that ϒki≥ϒki∗ and o¯ki≤o¯ki∗ for all k and all permissible value of i; then,(40)PmFWGO1,O2,…,On≥PmFWGO1∗,O2∗,…,On∗.Proof.
The proof may be established in the same fashion as the proof of Theorem3.


## 5. Symmetric Pythagoreanm-Polar Fuzzy Weighted Averaging Operator

In this portion, we render the notion of symmetric Pythagoreanm-polar fuzzy weighted averaging operator for Pythagorean m-polar fuzzy numbers accompanied by some of its prime characteristics.Definition 9.
LetOk=ϒki,o¯kii=1mk=1,2,…,n be an assemblage of PmFNs. Define SPmFWA: Tn⟶T given by(41)SPmFWAO1,O2,…,On=∔k=1nℏk⋅Ok=ℏ1⋅O1∔ℏ2⋅O2∔⋯∔ℏn⋅On,where Tn is the collection of all PmFNs and ℏk’s are fuzzy weights of O1,O2,…,On, such that their addition yields unity. SPmFWA is called symmetric Pythagorean m-polar fuzzy weighted averaging operator.
IfW=w,w,…,wt=1/n,1/n,…,1/nt, then SPmFWA operator shrinks to symmetric Pythagorean m-polar fuzzy averaging (SPmFA) operator of dimension n and is given as(42)SPmFAO1,O2,…,On=1n⋅∔k=1nOk=1n⋅O1∔O2∔⋯∔On.
In keeping with operational laws of PmFNs given in Definition 3, the forthcoming theorem benefits in aggregating PmFNs.Theorem 7.
LetOk=ϒki,o¯kii=1mk=1,…,n be a family of PmFNs; then,(43)SPmFWAO1,O2,…,On=1−∏k=1n1−ϒki2ℏk2−∏k=1n1−ϒki2ℏk−∏k=1nϒki2ℏk,1−∏k=1n1−o¯ki2ℏk2−∏k=1n1−o¯ki2ℏk−∏k=1no¯ki2ℏki=1m.Proof.
The proof may be furnished by means of induction.Theorem 8.
IfOk=ϒki,o¯kii=1mk=1,2,…,n is an aggregate of PmFNs, then(44)SPmFWAO1,O2,…,On=1−∏k=1n1−ϒki2ℏk2−∏k=1n1−ϒki2ℏk−∏k=1nϒki2ℏk,1−∏k=1n1−o¯ki2ℏk2−∏k=1n1−o¯ki2ℏk−∏k=1no¯ki2ℏki=1mis also a PmFN.Proof.
Straight forward.Example 3.
Consider the data of Example1. The SP4FWA, using Theorem 7, is(45)SPmFWAO1,O2,O3=1−∏k=131−ϒki2ℏk2−∏k=131−ϒki3ℏk−∏k=13ϒki2ℏk,1−∏k=131−o¯ki2ℏk2−∏k=131−o¯ki2ℏk−∏k=13o¯ki2ℏki=14=0.520,0.494,0.419,0.623,0.529,0.252,0.361,0.695.Theorem 9.
Assume thatOk=ϒki,o¯kii=1mk=1,…,n is a setting of PmFNs. Then,(1)
(Idempotency) ifOk=O=ϒi,o¯ii=1mk=1,2,…,n for all k, then(46)SPmFWAO1,O2,…,On=O.(2)
(Boundedness) ifO−=minϒki,maxo¯ki and O+=maxϒki,mino¯ki, then(47)O−≤SPmFWAO1,O2,…,On≤O+.(3)
(Monotonicity) letOk=ϒki,o¯kii=1m and Ok∗=ϒki∗,o¯ki∗i=1mk=1,…,n be two collections of PmFNs such that ϒki≥ϒki∗ and o¯ki≤o¯ki∗ for all k and all permissible value of i; then,(48)SPmFWAO1,O2,…,On≥SPmFWAO1∗,O2∗,…,On∗.Proof.
Straight forward.


## 6. Symmetric Pythagoreanm-Polar Fuzzy Weighted Geometric Operator

We assign this unit to render the notion of symmetric Pythagoreanm-polar fuzzy weighted geometric operator for Pythagorean m-polar fuzzy numbers in company with some of its prime characteristics.Definition 10.
LetOk=ϒki,o¯kii=1mk=1,2,…,n be an assemblage of PmFNs. Define SPmFWG: Tn⟶T given by(49)SPmFWGO1,O2,…,On=⊞k=1nℏk⊡Ok=ℏ1⊡O1⊞ℏ2⊡O2⊞⋯⊡ℏn⊡On,where Tn is the collection of all PmFNs and ℏk’s are fuzzy weights of O1,O2,…,On, bearing the constraint that they add up to unity. SPmFWG is called symmetric Pythagorean m-polar fuzzy weighted geometric operator.
IfW=w,w,…,wt=1/n,1/n,…,1/nt, then SPmFWG operator diminishes to symmetric Pythagorean m-polar fuzzy geometric (SPmFG) operator of dimension n and is specified as(50)SPmFGO1,O2,…,On=1n⊡⊞k=1nOk=1n⊡O1⊞O2⊞⋯⊞On.
Relying upon the operational laws of PmFNs given in Definition 3, the approaching theorem suggests mathematical formulation of SPmFWG operator meant for aggregating PmFNs.Theorem 10.
LetOk=ϒki,o¯kii=1mk=1,2,…,n be an assemblage of PmFNs; then,(51)SPmFWGO1,O2,…,On=∏k=1nϒkiℏk∏k=1n1−ϒki2ℏk+∏k=1nϒki2ℏk,∏k=1no¯kiℏk∏k=1n1−o¯ki2ℏk+∏k=1no¯ki2ℏki=1m.Proof.
The proof may be furnished by means of induction.Theorem 11.
IfOk=ϒki,o¯kii=1mk=1,2,…,n is an aggregate of PmFNs, then(52)SPmFWGO1,O2,…,On=∏k=1nϒkiℏk∏k=1n1−ϒki2ℏk+∏k=1nϒki2ℏk,∏k=1no¯kiℏk∏k=1n1−o¯ki2ℏk+∏k=1no¯ki2ℏki=1mis also a PmFN.Proof.
Straight forward.Example 4.
Consider the data of Example1. The SP4FWA, using Theorem 10, is(53)SPmFWGO1,O2,O3=∏k=13ϒkiℏk∏k=131−ϒki2ℏk+∏k=13ϒki2ℏk,∏k=13o¯kiℏk∏k=131−o¯ki2ℏk+∏k=13o¯ki2ℏki=14=0.510,0.466,0.134,0.362,0.118,0.177,0.314,0.691.Theorem 12.
Assume thatOk=ϒki,o¯kii=1mk=1,…,n is a family of PmFNs. Then,(1)
(Idempotency) ifOk=O=ϒi,o¯ii=1mk=1,2,…,n for all k, then(54)SPmFWGO1,O2,…,On=O.(2)
(Boundedness) ifO−=minϒki,maxo¯ki and O+=maxϒki,mino¯ki, then(55)O−≤SPmFWGO1,O2,…,On≤O+.(3)
(Monotonicity) letOk=ϒki,o¯kii=1m and Ok∗=ϒki∗,o¯ki∗i=1mk=1,…,n be two sets of PmFNs such that ϒki≥ϒki∗ and o¯ki≤o¯ki∗ for all k and all permissible value of i; then,(56)SPmFWGO1,O2,…,On≥SPmFWGO1∗,O2∗,…,On∗.Proof.
Straight forward.


## 7. Robust Decision Making through Pythagoreanm-Polar Fuzzy Weighted Aggregation Operators

In the wake of investment, a venture capitalist usually faces manifold challenges in deciding about pros and cons of the trade and commerce industry. Companies entice the investor by cutting down the prices of their commodities, despite the fact that they have evaluated that consumer consummation is one of the most significant and fundamental features to stay alive and subsist in the market. A view of capital market is shown in Figure1.Figure 1
Capital market (source:cushmanwakefield.com).To take a clearer, rewarding, and intelligent decision, a financier will definitely want to have awareness about which market is suitable for investment and then consult a team of experts to get benefitted from their experience to have better safeguard for his investment. So, subsequent upon their prefatory scrutinization, a commission has been instituted to act as aide in investing the finances in the paramount markets where there is least chance of loss, according to the following major criteria:Safeguard of principal: protection of funds financed is one of the indispensable components of any worthy investment program. Security of principal indicates fortification against any probable forfeiture under fluctuating environments. Protection of principal may be accomplished over and done with a watchful analysis of fiscal and industrial inclinations afore deciding on nature of investment. Obviously, no one can predict the yet to come commercial conditions with ultimate exactitude. To defend against definite slips which may sneak in while taking a decision on investment, far-reaching diversification is recommended.Liquidity and collateral value: an investment that may be transformed into cash instantly without having any financial loss is known as liquid investment. Liquid investments comfort financiers to meet crises and disasters. Stocks are with no trouble merchantable only when they make available satisfactory profit through dividends and funds appreciation. Assortment of liquid investments empowers the financiers to raise funds through sale of liquid securities or borrowing by proposing them as collateral security. The venture capitalist finances in top ranked and readily profit-making investments for ensuring their liquidity and collateral value.Tax implications: associated tax implications should be earnestly and well thought out before scheduling an investment plan. Singularly, the amount of revenue that investment offers and the liability of income tax levied on that revenue must be pondered well. Financiers in small revenue brackets go on to make best use of cash earnings on their monies and hence are diffident to take extreme jeopardies. On the contrary, venture capitalists who are not specific about cash returns do not cogitate tax implications earnestly.Steady revenue: financiers endow their treasuries in such assets that offer steady revenue. Monotony of revenue is consistent with a good investment program. Investors are attracted towards those programs that generate revenue not only stably but also adequately.Permanency of buying power: investment is utilization of money with the aim of receiving capital appreciation or profits. Stated differently, current assets are surrendered with the object of getting loftier volumes of future funds. Thus, the financier must deliberate the buying power of future funds. For maintaining the constancy of buying power, the financier must scrutinize the projected price level inflation and the likelihoods of additions and sufferings in the investment accessible to them.Capital growth: capital appreciation is one of the essential main beliefs of investment. The firmness of an industry warranties its allied companies to flourish and progress. So, by identifying the association flanked by industry evolution and assets appreciation, the financiers should capitalize in growth stocks. In brief, right matter in the suitable business must be taken on board at the appropriate stage.Lawfulness: the financier must capitalize only in those assets which are legitimate and sanctioned by law. Illegitimate securities land the financier in misfortune. In addition to being mollified with the rightfulness of investment, the financier ought to be at liberty from administration of securities.We develop an algorithm (Algorithm1) first to intelligibly decipher a decision-making problem.Algorithm 1:
Input:(1)
Analyze the problem: LetX=l1,l2,…,lp be the collection of choices and the set of criteria be E=c1,c2,…,cq. Suppose that ℏ1,ℏ2,…,ℏn are respective weights of n decision experts.Computations:(2)
Present the information given by experts in the form of PF matrices asM1,M2,…,Mn.(3)
Present the matricesM1,M2,…,Mn in Pythagorean m-polar fuzzy array.(4)
Use PmFWA, PmFWG, SPmFWA, or SPmFWG operator to aggregate the PFNs for each choice.(5)
Compute value of score functions for each choice.Output:(6)
The alternative with highest value ofs is the desired alternative.The flowchart of the algorithm is portrayed in Figure2.Figure 2
Flowchart of the algorithm.Example 5.
Consider the decision-making problem of capital investment comprising three experts. Assume that there are five choicesl1,l2,l3,l4,l5 that are to be assessed by the financial experts keeping in view four attributes c1, c2, c3, and c4, where(57)c1=analysis of permanency of buying power,c2=analysis of liquidity and collateral value,c3=principal safeguard analysis,c4=analysis of capital growth.
The weights of three experts, in order, are assigned asℏ1=0.40 and ℏ2=ℏ3=0.30 according to their expertise and importance of their opinion. The three experts provide the information in the form of PFNs which are transformed in the form of PF matrices in which rows represent choices and the columns are meant for criteria.(58)M1=0.37,0.490.76,0.360.72,0.410.61,0.080.77,0.480.81,0.390.32,0.890.21,0.560.42,0.710.56,0.540.37,0.800.11,0.390.56,0.210.42,0.060.58,0.600.45,0.820.54,0.210.31,0.730.50,0.590.62,0.15,M2=0.76,0.290.54,0.090.11,0.230.37,0.520.48,0.610.58,0.630.67,0.360.91,0.400.52,0.530.48,0.210.34,0.790.54,0.210.28,0.090.48,0.190.21,0.860.40,0.900.33,0.760.79,0.210.67,0.710.49,0.36,M3=0.54,0.110.28,0.560.38,0.210.06,0.820.47,0.540.39,0.460.41,0.430.35,0.110.37,0.240.54,0.110.48,0.420.47,0.180.36,0.290.58,0.160.39,0.220.41,0.320.41,0.490.54,0.410.37,0.180.46,0.33.
We present these PF matrices in lamellar formation in Table2.
The values ofPmFWAO1,O2,O3 for each choice are given in Table 3.
The values of score function against each choice are given in Table4.
Hence, the rank of choices is(59)l1≻l2≻l5≻l4≻l3.
Let us experience what happens if we proceed with PmFWG operator instead of PmFWA operator. The values of PmFWAO1,O2,O3 for each choice are given in Table 5.
The score function values against each choice are given in Table6.
Hence, the rank of choices is(60)l1≻l5≻l2≻l3≻l4.
Let us employ SPmFWA operator to experience if there is any change in the optimal choice. The values of SPmFWAO1,O2,O3 for each choice are given in Table 7.
The values of score function against each choice are given in Table8.
Hence, the rank of choices is(61)l1≻l2≻l5≻l4≻l3
Finally, we wield SPmFWG operator to discuss whether this operator brings any revision in the choice of optimal option. The values of SPmFWAO1,O2,O3 for each choice are given in Table 9.
The values of score function against each choice are given in Table10.
Hence, the rank of choices is(62)l1≻l5≻l2≻l4≻l3.
From these four ranking indices, we observe that the optimal choice, which isl1, stays unaltered. We exhibit the four ranking catalogues through the medium of horizontal bar chart cited in Figure 3.Table 2
PmFNs for each choice.
ChoicePFNsl1ß1=0.37,0.49,0.76,0.36,0.72,0.41,0.61,0.08ß2=0.76,0.29,0.54,0.09,0.11,0.23,0.37,0.52ß3=0.54,0.11,0.28,0.56,0.38,0.21,0.06,0.82l2ß1=0.77,0.48,0.81,0.39,0.32,0.89,0.21,0.56ß2=0.48,0.61,0.58,0.63,0.67,0.36,0.91,0.40ß3=0.47,0.54,0.39,0.46,0.41,0.43,0.35,0.11l3ß1=0.42,0.71,0.56,0.54,0.37,0.80,0.11,0.39ß2=0.52,0.53,0.48,0.21,0.34,0.79,0.54,0.21ß3=0.37,0.24,0.54,0.11,0.48,0.42,0.47,0.18l4ß1=0.56,0.21,0.42,0.06,0.58,0.60,0.45,0.82ß2=0.28,0.09,0.48,0.19,0.21,0.86,0.40,0.90ß3=0.36,0.29,0.58,0.16,0.39,0.22,0.41,0.32l5ß1=0.54,0.21,0.31,0.73,0.50,0.59,0.62,0.15ß2=0.33,0.76,0.79,0.21,0.67,0.71,0.49,0.36ß3=0.41,0.49,0.54,0.41,0.37,0.18,0.46,0.33Table 3
Values ofPmFWAß1,ß2,ß3 for each choice.
ChoicePmFWAß1,ß2,ß3l10.586,0.267,0.614,0.271,0.539,0.282,0.455,0.282l20.633,0.534,0.671,0.473,0.492,0.545,0.666,0.311l30.441,0.470,0.532,0.252,0.400,0.657,0.409,0.257l40.441,0.179,0.494,0.114,0.451,0.495,0.424,0.636l50.451,0.398,0.595,0.423,0.536,0.437,0.542,0.247Table 4
Values of score function against each choice.
Choicesl10.229l20.158l30.006l40.032l50.137Table 5
Values ofPmFWAß1,ß2,ß3 for each choice.
ChoicePmFWAß1,ß2,ß3l10.514,0.361,0.508,0.396,0.338,0.313,0.262,0.592l20.576,0.542,0.588,0.501,0.430,0.720,0.380,0.432l30.431,0.573,0.529,0.379,0.390,0.730,0.274,0.292l40.398,0.214,0.482,0.142,0.380,0.671,0.422,0.790l50.429,0.549,0.485,0.558,0.499,0.569,0.528,0.286Table 6
Score function values against each choice.
Choicesl1−0.008l2−0.061l3−0.099l4−0.107l5−0.018Table 7
Values ofSPmFWAß1,ß2,ß3 for each choice.
ChoiceSPmFWAß1,ß2,ß3l10.564,0.351,0.580,0.381,0.497,0.311,0.426,0.525l20.613,0.540,0.639,0.494,0.479,0.652,0.584,0.414l30.439,0.544,0.531,0.365,0.398,0.695,0.391,0.290l40.434,0.212,0.491,0.141,0.438,0.611,0.424,0.715l50.447,0.514,0.563,0.524,0.526,0.535,0.538,0.283Table 8
Values of score function against each choice.
Choicesl10.111l20.056l3−0.053l4−0.037l50.044Table 9
Values ofSPmFWAß1,ß2,ß3 for each choice.
ChoiceSPmFWAß1,ß2,ß3l10.536,0.276,0.541,0.283,0.373,0.285,0.282,0.093l20.344,0.537,0.622,0.480,0.443,0.618,0.454,0.326l30.433,0.497,0.530,0.263,0.392,0.693,0.288,0.259l40.406,0.181,0.484,0.114,0.391,0.555,0.423,0.720l50.433,0.430,0.517,0.454,0.509,0.469,0.532,0.247Table 10
Values of score function against each choice.
Choicesl10.138l2−0.025l3−0.040l4−0.035l50.081Figure 3
Horizontal bar chart of the two rankings.


### 7.1. Comparison Analysis and Superiority of the Proposed Work

We have observed that the optimal solution remains the same by use of either of the four proposed operators in this article. Further, the optimal choice attained through our suggested techniques does not alter by use of other methods. No computationally easy to use aggregation operator for PmFSs has yet been introduced so far, according to our best knowledge. Our suggested technique is simple to apply and yields definitive outputs. It can handle the data given at repeated spans of times or by different decision experts efficiently. The comparison of presented aggregation operators with some existing operators is given in Table 11.Table 11
Comparison of proposed operators with some existing operators.
Aggregation operatorsOptimal choiceOrdered weighted averaging aggregation operators (Yager [7])l1Pythagorean fuzzy aggregation operators (Peng and Yuan [17])l1Generalized intuitionistic fuzzy soft power aggregation operator (Garg and Arora [25])l1q-Rung orthopair fuzzy aggregation operator (Liu and Wang [30])l1Prioritized weighted aggregation operators (Liu et al. [31])l1Pythagoreanm-polar fuzzy Dombi’s aggregation operators (Hashmi and Riaz [45])l1Intuitionistic fuzzy aggregation operators (Xu [52])l1Pythagoreanm-polar fuzzy weighted averaging operator (proposed)l1Pythagoreanm-polar fuzzy weighted geometric operator (proposed)l1Symmetric Pythagoreanm-polar fuzzy weighted averaging operator (proposed)l1Symmetric Pythagoreanm-polar fuzzy weighted geometric operator (proposed)l1


## 7.1. Comparison Analysis and Superiority of the Proposed Work

We have observed that the optimal solution remains the same by use of either of the four proposed operators in this article. Further, the optimal choice attained through our suggested techniques does not alter by use of other methods. No computationally easy to use aggregation operator for PmFSs has yet been introduced so far, according to our best knowledge. Our suggested technique is simple to apply and yields definitive outputs. It can handle the data given at repeated spans of times or by different decision experts efficiently. The comparison of presented aggregation operators with some existing operators is given in Table 11.Table 11
Comparison of proposed operators with some existing operators.
Aggregation operatorsOptimal choiceOrdered weighted averaging aggregation operators (Yager [7])l1Pythagorean fuzzy aggregation operators (Peng and Yuan [17])l1Generalized intuitionistic fuzzy soft power aggregation operator (Garg and Arora [25])l1q-Rung orthopair fuzzy aggregation operator (Liu and Wang [30])l1Prioritized weighted aggregation operators (Liu et al. [31])l1Pythagoreanm-polar fuzzy Dombi’s aggregation operators (Hashmi and Riaz [45])l1Intuitionistic fuzzy aggregation operators (Xu [52])l1Pythagoreanm-polar fuzzy weighted averaging operator (proposed)l1Pythagoreanm-polar fuzzy weighted geometric operator (proposed)l1Symmetric Pythagoreanm-polar fuzzy weighted averaging operator (proposed)l1Symmetric Pythagoreanm-polar fuzzy weighted geometric operator (proposed)l1


## 8. Conclusion

Pythagoreanm-polar fuzzy set is a mighty model for examining the information given in multipolar form. We suggested four operators, namely, Pythagorean m-polar fuzzy weighted averaging operator, Pythagorean m-polar fuzzy weighted geometric operator, symmetric Pythagorean m-polar fuzzy weighted averaging operator, and symmetric Pythagorean m-polar fuzzy weighted geometric operator for the sake of aggregating the statistics given in multipolar form. The aggregated resultant falling in the same structure has also been manifested. We established the desirable qualities of idempotency, monotonicity, and boundedness for the proposed operators.The results presented in this article are also valid for intuitionisticm-polar fuzzy sets and have potential to be generalized to q-rung orthopair m-polar fuzzy sets and many other models. We rendered an algorithm for capital investment analysis problem as practical usage of the suggested operators in daily life situations and found that our computed results are compatible with the existing techniques. The suggested algorithm may be used in human resource management problems, life sciences, economics analysis, business and trade analysis, pattern recognition, water management problems, agribusiness, and many other areas. We anticipate that this article will attract the attention of vibrant researchers working in this field.



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*Source: 6644994-2021-02-25.xml*