Pythagorean fuzzy prioritized aggregation operators and their application to multi-attribute group decision making

  • Muhammad Sajjad Ali Khan
  • Saleem Abdullah
  • Asad Ali
  • Fazli Amin
Original Paper
  • 45 Downloads

Abstract

Pythagorean fuzzy set is a useful tool to deal with the fuzziness and vagueness. Many aggregation operators have been proposed by many researchers based on Pythagorean fuzzy sets. But the current methods are under the assumption that the decision makers and the attributes are at the same priority level. However, in real group decision-making problems the attribute and decision makers may have different priority level. Therefore, in this paper, we develop multi-attribute group decision-making based on Pythagorean fuzzy sets where there exists a prioritization relationship over the attributes and decision makers. First, we develop Pythagorean fuzzy prioritized weighted average operator and Pythagorean fuzzy prioritized weighted geometric operator. Then we study some of its desirable properties such as idempotency, boundary and monotonicity in detail. Moreover, we propose a multi-attribute group decision-making approach based on the developed operators under Pythagorean fuzzy environment. Finally, a numerical example is provided to illustrate the practicality of the proposed approach.

Keywords

Pythagorean fuzzy set Pythagorean fuzzy prioritized weighted average (PFPWA) operator Pythagorean fuzzy prioritized weighted geometric (PFPWG) operator Multi-attribute group decision making 

1 Introduction

The notion of uncertainty can be integrated into granular computing models via complex type information granules (Pedrycz and Chen 2011) such that, apart from also having the characteristics of information granules, their models also integrate an additional attribute, which is uncertainty management. Granular computing (Pedrycz and Chen 2015a, b) is created on the idea of how the human attention arranges “known information” into certain levels of specificity, and is used as a means for finding enhanced models of known data; and with such, take better decisions. It also modifies and adjusts existing information models so as to improve their overall presentation, as well as improve the interpretability of its information granule representations. In this case, an information granule is a compact, and well-defined, grouping of information which shares similarities.

In 1986, the concept of fuzzy set proposed by Zadeh (1965) was extended by Atanassov and introduced the concept of intuitionistic fuzzy sets (IFS) (Atanassov 1986). The intuitionistic fuzzy set is characterized by membership and non-membership degrees and satisfy the condition that the sum of its membership and non-membership degrees is less than or equal to 1. To deal with multi-attribute decision-making problem Liu and Wang (2018) proposed q-rung orthopair fuzzy weighted averaging operator and the q-rung orthopair fuzzy weighted geometric operator. Several extension of fuzzy set come in to view in the literature of fuzzy set theory but intuitionistic fuzzy set is more precise when dealing with fuzziness and vagueness. Since its appearance many researchers have studied intuitionistic fuzzy aggregation operators in different point of view. Xu and Yager (2006) proposed the intuitionistic fuzzy weighted geometric (IFWG) operator, intuitionistic fuzzy ordered weighted geometric (IFOWG) operator and intuitionistic fuzzy hybrid (IFHG) operator. Xu (2007) proposed intuitionistic fuzzy weighted average (IFWA) operator, intuitionistic fuzzy ordered weighted average (IFOWA) operator and intuitionistic fuzzy hybrid (IFHA) operator and applied to a multi-attribute decision-making problem. Further Zhao et al. (2010) generalized the operators Xu (2007), and developed the generalized intuitionistic fuzzy weighted average (GIFWA) operator, intuitionistic fuzzy ordered weighted average (GIFOWA) operator and intuitionistic fuzzy hybrid (GIFHA) operator. In Liu et al. (2018a, b) the authors developed linguistic intuitionistic fuzzy partitioned Heronian mean (LIFPHM) operator, the linguistic intuitionistic fuzzy weighted partitioned Heronian mean (LIFWPHM) operator, the linguistic intuitionistic fuzzy partitioned geometric Heronian mean (LIFPGHM) operator and linguistic intuitionistic fuzzy weighted partitioned geometric Heronian mean (LIFWPGHM) operator for multi-attribute group decision making (MAGDM) problems. Based on the Archimedean t-conorm and t-norm Liu and Chen (2017) developed intuitionistic fuzzy Archimedean Heronian aggregation (IFAHA) operator and the intuitionistic fuzzy weight Archimedean Heronian aggregation (IFWAHA) operator. Combined the power average operator with Heronian mean operator in (Liu 2017) the author developed interval-valued intuitionistic fuzzy power Heronian aggregation (IVIFPHA) operator, interval-valued intuitionistic fuzzy power weight Heronian aggregation (IVIFPWHA) operator to deal with multi-attribute decision-making problems. Extended the Bonferroni mean (BM) operator on the Dombi operation Liu et al. (2018a, b) developed intuitionistic fuzzy Dombi Bonferroni mean (IFDBM) operator, the intuitionistic fuzzy weighted Dombi Bonferroni mean (IFWDBM) operator, the intuitionistic fuzzy Dombi geometric Bonferroni mean (IFDGBM) operator and the intuitionistic fuzzy weighted Dombi geometric Bonferroni mean (IFWDGBM) operator for dealing with the aggregation of intuitionistic fuzzy numbers (IFNs) and proposed some multi-attribute group decision-making (MAGDM) methods. Liu and Li (2017) developed interval-valued intuitionistic fuzzy power Bonferroni mean (IVIFPBM) operator, the interval-valued intuitionistic fuzzy weighted power Bonferroni mean (IVIFWPBM) operator, the power geometric BM (IVIFPGBM) operator, and the weighted power geometric BM (IVIFWPGBM) operator for IVIFNs are proposed. Liu et al. (2017) extended the portioned Bonferroni mean (PBM) operator and the portioned geometric Bonferroni mean (PGBM) operator and developed intuitionistic fuzzy interaction portioned Bonferroni mean (IFIPBM) operator for intuitionistic fuzzy numbers (IFNs), the intuitionistic fuzzy weighted interaction portioned Bonferroni mean (IFWIPBM) operator for IFNs, the intuitionistic fuzzy interaction portioned geometric Bonferroni mean (IFIPGBM) operator for IFNs and the intuitionistic fuzzy weighted interaction portioned geometric Bonferroni mean (IFWIPGBM) operator for IFNs. The authors applied the proposed operators to MAGDM problems. Based on intuitionistic 2-Tuple Linguistic information (I2LI) Liu and Chen (2018) developed Intuitionistic 2-Tuple Linguistic averaging (I2LA) operator for MAGDM problems.

All of the above operators assuming the priority level of the attributes are of the same type. To deal with the problem of the prioritization of the attributes under intuitionistic fuzzy environment in Yu (2013a), proposed the concept of intuitionistic fuzzy prioritized weighted average (IFPWA) operator and intuitionistic fuzzy prioritized weighted geometric (IFPWG) operator and applied to multi-attribute decision-making problems. Further, Yu (2013b) introduced a new generalized intuitionistic fuzzy prioritized geometric aggregation operator based on Archimedean t-conorm and t-norm. Yu (2012) also proposed two new generalized prioritized aggregation operators such as the generalized intuitionistic fuzzy prioritized weighted average (GIFPWA) operator, generalized intuitionistic fuzzy prioritized weighted geometric (GIFPWG) operator and discussed their applications in multi criteria decision making. Verma and Sharma (2015), proposed the intuitionistic fuzzy Einstein prioritized weighted average (IFEPWA) operator and intuitionistic fuzzy Einstein prioritized weighted geometric (IFEPWG) operator.

Recently Yager (2013), generalized the concept of intuitionistic fuzzy set (IFS) and introduced the concept of Pythagorean fuzzy set (PFS). The Pythagorean fuzzy set (PFS) satisfies the conditions that the square sum of the membership degree and non-membership degree is less than or equal to 1. In Yager and Abbasov (2014), the authors gave an example to state this condition: a decision maker gives his support for membership of an alternative is \(\frac{{\sqrt 3 }}{2}\) and his against membership is \(\frac{1}{2}\). This fullfill the condition that their sum is bigger than 1 and are not presented for IFS, but they are presented for PFS since \({\left( {\frac{{\sqrt 3 }}{2}} \right)^{\text{2}}}+{\left( {\frac{1}{2}} \right)^{\text{2}}} \leq 1\). Obviously PFS is more suitable than IFS to deal with fuzziness and vagueness. Yager (2014) introduced Pythagorean fuzzy weighted average (PFWA) operators, Pythagorean fuzzy weighted geometric (PFWG) operators, Pythagorean fuzzy ordered weighted average (PFOWA) operators and Pythagorean fuzzy ordered weighted geometric (PFOWG) operators under Pythagorean fuzzy environment. Zhang and Xu (2014), developed Pythagorean fuzzy TOPSIS method for multi-criteria decision making (MCDM) under Pythagorean fuzzy environment. In Peng and Yang (2015), discussed relationship between the Pythagorean fuzzy aggregation operators proposed by Yager, and developed a superiority inferiority ranking method for multi-criteria group decision-making (MCDM) problems. Garg (2016), generalized the PFWA operator and PFWG operator and developed Pythagorean fuzzy Einstein weighted average (PFEWA) operator and Pythagorean fuzzy Einstein ordered weighted average (PFEOWA) operator. Rahman et al. (2017a) developed Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator. Rahman et al. (2017b, c, d) discussed the properties of Pythagorean fuzzy hybrid averaging aggregation operator, Pythagorean fuzzy weighted geometric aggregation operator, Pythagorean fuzzy ordered weighted geometric aggregation operator, Pythagorean fuzzy hybrid geometric aggregation operator in detail and applied them to group decision-making. Garg (2017) developed new aggregation operators under Pythagorean fuzzy environment namely, confidence Pythagorean fuzzy weighted averaging (CPFWA) operator, confidence Pythagorean fuzzy weighted ordered averaging (CPFOWA) operator, confidence Pythagorean fuzzy weighted geometric (CPFWG) operator and confidence Pythagorean fuzzy weighted ordered geometric (CPFOWG) operator along with their desired properties. Zeng (2017), proposed Pythagorean fuzzy probabilistic OWA (PFPOWA) operator and applied to MAGDM problem. Rahman et al. (2017e) developed interval-valued Pythagorean fuzzy geometric operator for multi-attribute decision-making problems. Based on Choquet integral in Khan at al. (2018a) the authors developed interval-valued Pythagorean fuzzy TOPSIS method to deal with multi-attribute decision-making problem. Generalized the concept of PFS (Yager 2013) with hesitant fuzzy set (HFS) (Torra 2010), Khan et al. (2017) introduced Pythagorean hesitant fuzzy sets and developed Pythagorean hesitant fuzzy aggregation operators for multi-attribute decision-making problems. Khan et al. (2018b) developed Pythagorean hesitant fuzzy ordered weighted (PHFOW) operators for multi-attribute decision-making problems.

The above proposed aggregation operators for Pythagorean fuzzy numbers assuming that the attribute is of the same priority level. They are considered by the skill to trade-off between attribute. For example, if \({A_i}\) and \({A_j}\) are two attributes with the weight \({w_i}\) and \({w_j}\) respectively. Based on the above aggregation operator, we can compensate for a decrease of \(\phi\) in satisfaction to attribute \({A_i}\) by gain \(\frac{{{w_j}}}{{{w_i}}}\phi\) in satisfaction to attribute \({A_j}\). But, in many real decision-making problems, this kind of compensation between attribute is not possible. Consider the situation in which a woman is making a decision based on consideration of powdered milk cost and safety for her child. She should not permit a benefit with respect to cost of powdered milk compensate for a loss in safety. This is a typical kind of prioritization of the attribute, i.e., Safety has a higher priority than cost. When making decisions in Hazara University, attribute desired by Vice Chancellor generally has a higher priority than professors and lecturers. Therefore, in this paper we develop the aggregation method for Pythagorean fuzzy numbers (PFNs) which has prioritization relationship among the attributes. Motivated by the paper (Yu 2013a, b) in this paper we developed Pythagorean fuzzy prioritized average (PFPWA) operator and Pythagorean fuzzy prioritized geometric (PFPWG) operator to deal with multi-attribute decision-making problems, where the attribute and decision makers may have different priority level. To do this the remainder of the paper is organized as follows.

In Sect. 2 we briefly review some basic definition and result about Pythagorean fuzzy sets. In Sect. 3 we develop Pythagorean fuzzy prioritized average (PFPWA) operator and Pythagorean fuzzy prioritized geometric (PFPWG) operator. We then discussed some basic properties such as Idempotency, boundary and monotonicity in detail. In Sect. 4 we develop an algorithm for multi-attribute decision making problem under Pythagorean fuzzy environment based on the proposed operators. In Sect. 5 we provide a practical example to show the feasibility and validity of the proposed approach. Conclusion is in Sect. 6.

2 Preliminaries

In this section, we first introduce some basic definition of Pythagorean fuzzy sets (PFSs) and present some of its properties.

2.1 Pythagorean fuzzy sets and their operations

In 2013, Yager introduced the concept of Pythagorean fuzzy set (Yager 2013) as the generalization of intuitionistic fuzzy set (Atanassov 1986). Pythagorean fuzzy set fulfill the condition that the square sum of the membership degree and non-membership degree is less than or equal to 1 and is defined as follows.

Definition 1 (Yager 2013)

If \(X=({x_1},{x_2}, \ldots ,{x_n})\) is a fixed set, then a PFN in \(X\) can be defined as follows:
$$\bar {P}=\left\{ {\left( {{x_i},{\mu _{\bar {P}}}\left( {{x_i}} \right),{\upsilon _{\bar {P}}}\left( {{x_i}} \right)} \right)|{x_i} \in X} \right\},$$
(1)
where \({\mu _{\bar {P}}}({x_i}) \in [0,1]\) and \({\upsilon _{\bar {P}}}({x_i}) \in [0,1]\) are called the degree of membership and the degree of non-membership of the element \({x_i} \in X\) to \(\bar {P}\), and \(0 \leq {\left( {{\mu _{\bar {P}}}({x_i})} \right)^2}+{\left( {{\upsilon _{\bar {P}}}({x_i})} \right)^2} \leq 1\) for all \({x_i} \in X\).

Also, Zhang and Xu (2014) said to be the pair \(({\mu _{\bar {P}}},{\upsilon _{\bar {P}}})\) a Pythagorean fuzzy number (PFN) denoted as \(p\) with the conditions\(0 \leq {\mu _{\bar {P}}} \leq 1\), \(0 \leq {\upsilon _{\bar {P}}} \leq 1\), and \(0 \leq {\mu _{\bar {P}}}^{2}+{\upsilon _{\bar {P}}}^{2} \leq 1\). Throughout this paper, we denote a Pythagorean fuzzy number by \(\bar {\alpha }=({\mu _{\bar {\alpha }}},{\upsilon _{\bar {\alpha }}})\).

For the comparison between PFN Zhang and Xu (2014) defined the score and accuracy functions of PFN\(\bar {\alpha }\), namely, \(S(\bar {\alpha })={\mu _{\bar {\alpha }}}^{2} - {\upsilon _{\bar {\alpha }}}^{2}\) and \(h(\bar {\alpha })={\mu _{\bar {\alpha }}}^{2}+{\upsilon _{\bar {\alpha }}}^{2}\). Zhang and Xu (2014) proposed a total order relation to compare two PFNs \(a\) and \(b\) as follows:

If \(S({\bar {\alpha }_1}) \prec S({\bar {\alpha }_2})\), then \({\bar {\alpha }_1} \prec {\bar {\alpha }_2}\); if \(S({\bar {\alpha }_1})=S({\bar {\alpha }_2})\), then (i) if \(h({\bar {\alpha }_1})=h({\bar {\alpha }_2})\), then \({\bar {\alpha }_2}={\bar {\alpha }_1}\); (ii) if \(h({\bar {\alpha }_1}) \prec h({\bar {\alpha }_2})\), then \({\bar {\alpha }_1} \prec \bar {\alpha }{}_{2}\); (iii) if \(h({\bar {\alpha }_1}) \succ h({\bar {\alpha }_2})\), then \({\bar {\alpha }_1} \succ {\bar {\alpha }_2}\).

Zhang and Xu (2014) further proposed the following operational laws to fuse the PFN information:
  1. 1.
    $${\bar {\alpha }_1} \oplus {\bar {\alpha }_2}=\left( {\sqrt {\mu _{{{{\bar {\alpha }}_1}}}^{2}+\mu _{{{{\bar {\alpha }}_2}}}^{2} - \mu _{{{{\bar {\alpha }}_1}}}^{2}\mu _{{{{\bar {\alpha }}_2}}}^{2}} ,{\upsilon _{{{\bar {\alpha }}_1}}}{\upsilon _{{{\bar {\alpha }}_2}}}} \right);$$
     
  2. 2.
    $${\bar {\alpha }_1} \otimes {\bar {\alpha }_2}=\left( {{\mu _{{{\bar {\alpha }}_1}}}{\mu _{{{\bar {\alpha }}_2}}},\sqrt {\upsilon _{{{{\bar {\alpha }}_1}}}^{2}+\upsilon _{{{{\bar {\alpha }}_2}}}^{2} - \upsilon _{{{{\bar {\alpha }}_1}}}^{2}\upsilon _{{{{\bar {\alpha }}_2}}}^{2}} } \right);$$
     
  3. 3.
    $$\lambda \bar {\alpha }=\left( {\sqrt {1 - {{\left( {1 - \mu _{{\bar {\alpha }}}^{2}} \right)}^\lambda }} ,\upsilon _{{\bar {\alpha }}}^{\lambda }} \right),{\text{ }}\lambda \succ \:\:0,$$
     
  4. 4.
    $${\mkern 1mu} {\bar {\alpha }^\lambda }=\left( {\mu _{{\bar {\alpha }}}^{\lambda },\sqrt {1 - {{\left( {1 - \upsilon _{{\bar {\alpha }}}^{2}} \right)}^\lambda }} } \right),{\text{ }}\lambda \succ \:\:0.$$
     

Obviously we describe the basic definitions and operations of the PFNs containing the score and accuracy functions, the total order relation, and the operational laws, which are the theoretical basis of the following aggregation operators:

2.2 The Pythagorean fuzzy aggregation operators

Since the setting of multi-attribute decision-making with Pythagorean fuzzy information, one of the basic methodologies to construct consensus and achieve the optimal alternative is to fuse all PFN information from the decision maker (DM) on the base of aggregation operators. Therefore, in this section, we presented four conventional Pythagorean fuzzy aggregation operators. On the base these operators we will propose four extended aggregation operators and design the corresponding pessimism and optimism decision-making processes. Moreover, on the base of operational laws, we can obtain the following definitions:

Definition 2 (Yager 2014)

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \((j=1,2, \cdots ,n)\) be a collection of PFNs and \(w{\text{=(}}{w_1}{\text{,}}{w_2}{\text{,}} \ldots ,{w_n}{\text{)}}\) be the weight vector of \({\bar {\alpha }_j}\) \((j=1,2, \ldots ,n)\) with \({w_j} \in [0,1]\) and \(\sum\nolimits_{{j=1}}^{n} {{w_j}} =1\). The Pythagorean fuzzy weighted averaging (PFWA) operator of dimension \(n\) is mapping \({\Omega ^n} \to \Omega\), and is defined as:
$${\text{PFWA(}}{\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n}= \mathop \oplus \limits_{{j=1}}^{n} {w_j}{\bar {\alpha }_j}=\left( {\sqrt {1 - \prod\limits_{{j=1}}^{n} {{{(1 - {{({\mu _{{{\bar {\alpha }}_j}}})}^2})}^{{w_j}}}} } ,\prod\limits_{{i=1}}^{n} {{\upsilon _{{{\bar {\alpha }}_j}}}^{{{w_j}}}} } \right).$$
(2)

Definition 3 (Yager 2014)

Let \({\bar {\alpha }_j}=\left( {{\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}}} \right)\;(j=1,2, \ldots ,n)\) be a collection of PFNs and \(w{\text{=(}}{w_1}{\text{,}}{w_2}{\text{,}} \ldots ,{w_n}{\text{)}}\) be the weight vector of \({\bar {\alpha }_j}\) \((j=1,2, \ldots ,n)\) with \({w_i} \in [0,1]\) and \(\sum\nolimits_{{i=1}}^{n} {{w_i}} =1\). The Pythagorean fuzzy weighted geometric (PFWG) operator of dimension \(n\) is the mapping\({\Omega ^n} \to \Omega\), and is defined as:
$${\text{PFWG(}}{\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n} = \mathop \otimes \limits_{{j=1}}^{n} {\bar {\alpha }_j}^{{{w_j}}}=\left( {\prod\limits_{{j=1}}^{n} {{\mu _{{{\bar {\alpha }}_j}}}^{{{w_j}}}} ,\sqrt {1 - \prod\limits_{{j=1}}^{n} {{{(1 - \upsilon _{{{{\bar {\alpha }}_j}}}^{2})}^{{w_j}}}} } } \right)$$
(3)

2.3 Pythagorean fuzzy prioritized aggregation operators and their properties

In this section, we introduce the concept of Pythagorean fuzzy prioritized weighted average (PFPWA) operator and Pythagorean fuzzy prioritized weighted average (PFPWA) operator. We then discuss some desired properties such as idempotency, boundary and monotonicity in detail.

Definition 4

Let \({\bar {\alpha }_j}=\left( {{\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}}} \right)\;(j=1,2,3, \ldots ,n)\) be the collection of PFNs and let\({\text{PFPWA:}}\;\;{\Omega ^n} \to \Omega\), be a mapping of dimension \(n\). If
$${\text{PFPWA}}\left( {{{\bar {\alpha }}_1},{{\bar {\alpha }}_2}, \ldots ,{{\bar {\alpha }}_n}} \right)=\left( {\frac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_1} \oplus \frac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_2} \oplus \cdots \oplus \frac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_n}} \right),$$
(4)
then the mapping PFPWA is called Pythagorean fuzzy prioritized weighted averaging PFPWA operator where \(T=\mathop \prod \nolimits_{{k=1}}^{{j - 1}} {S^ * }({\bar {\alpha }_k})\), \((j=1,2,3, \ldots ,n)\), \(T=1\) and \({S^ * }({\bar {\alpha }_k})\) is the score of Pythagorean fuzzy numbers \({\bar {\alpha }_k}=({\mu _{{{\bar {\alpha }}_k}}},{\upsilon _{{{\bar {\alpha }}_k}}})\).

Based on the operational law of definition, we have the following result.

Theorem 1

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \((j=1,2,3, \ldots ,n)\) be the collection of PFNs. Then using PFPWA operator the aggregated value of the PFNs is also PFN
$${\text{PFPWA}}\left( {{{\bar {\alpha }}_1},{{\bar {\alpha }}_2}, \ldots ,{{\bar {\alpha }}_n}} \right)=\left( {\sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } ,\:\prod\limits_{{j=1}}^{n} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right)$$
(5)
where \(T=\prod\nolimits_{{k=1}}^{{j - 1}} {{S^ * }({{\bar {\alpha }}_k})}\), \((j=1,2,3, \ldots ,n)\), \(T=1\) and \({S^ * }({\bar {\alpha }_k})\) is the score of Pythagorean fuzzy numbers \({\bar {\alpha }_k}=({\mu _{{{\bar {\alpha }}_k}}},{\upsilon _{{{\bar {\alpha }}_k}}})\).

Proof

First result follows form Definition 4. Next in the following we prove,
$${\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})=\left( {\frac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_1} \oplus \frac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_2} \oplus \cdots \oplus \frac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_n}} \right)=\left( {\sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\frac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \prod\limits_{{j=1}}^{n} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{^{{\frac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} } \right)$$
Using mathematical induction.□
First, we show that Eq. (5) holds for \(n=2\). Since,
$$\frac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{\bar {\alpha }_1}=\left( {\sqrt {1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,{{({\upsilon _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)$$
and
$$\frac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{\bar {\alpha }_2}=\left( {\sqrt {1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,{{({\upsilon _{{{\bar {\alpha }}_2}}})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)$$
thus we have,
$$\begin{aligned} \frac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_1} \oplus \frac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_2} & = \left( {\sqrt {1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,{{({\upsilon _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right) \oplus \left( {\sqrt {1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,{{({\upsilon _{{{\bar {\alpha }}_2}}})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right) \\ & = \left( {\sqrt {1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}+1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} - \left( {1 - {{(1 - \mu _{{{{\bar {\alpha }}_1}}}^{2})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)\left( {1 - {{(1 - \mu _{{{{\bar {\alpha }}_2}}}^{2})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)} ,{{({\upsilon _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}{{({\upsilon _{{{\bar {\alpha }}_2}}})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right) \\ & = \left( {\sqrt {1 - 1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}}+1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}} - \left( {1 - {{(1 - \mu _{{{{\bar {\alpha }}_1}}}^{2})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} - {{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}+{{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}{{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)} ,\;{{({\upsilon _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}{{({\upsilon _{{{\bar {\alpha }}_2}}})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right) \\ & = \left( {\sqrt {1 - 1 - {{(1 - \mu _{{{{\bar {\alpha }}_1}}}^{2})}^{\tfrac{{{T_1}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}}+1 - {{(1 - \mu _{{{{\bar {\alpha }}_2}}}^{2})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} - 1+{{(1 - \mu _{{{{\bar {\alpha }}_1}}}^{2})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}+{{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}}+{{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}{{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,\;{{({\upsilon _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}{{({\upsilon _{{{\bar {\alpha }}_2}}})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right) \\ & = \left( {\sqrt {1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}{{\left( {1 - \mu _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,{{({\upsilon _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}{{({\upsilon _{{{\bar {\alpha }}_2}}})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)& = \left( {\sqrt {1 - \prod\limits_{{j=1}}^{2} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } ,\prod\limits_{{j=1}}^{2} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right) \\ \end{aligned}$$
This shows that Eq. (5) holds for \(n=2\). Assume that Eq. (5) holds for \(n=k\), i.e.,
$${\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_k})=\left( {\frac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_1} \oplus \frac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_2} \oplus \cdots \oplus \frac{{{T_k}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_k}} \right)=\left( {\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}}} } ,\prod\limits_{{j=1}}^{k} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right)$$
then, when \(n=k+1\), by the operational law of Definition 2 we have,
$$\begin{aligned} {\text{PFPWA}}\;({{\bar {\alpha }}_1},{{\bar {\alpha }}_2}, \ldots ,{{\bar {\alpha }}_n}) \oplus {{\bar {\alpha }}_{k+1}}& = \left( {\frac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}{{\bar {\alpha }}_1} \oplus \frac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}{{\bar {\alpha }}_2} \oplus \cdots \oplus \frac{{{T_k}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}{{\bar {\alpha }}_k}} \right) \oplus \left( {{{\bar {\alpha }}_{k+1}}} \right) \\ & = \left( {\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} } ,\prod\limits_{{j=1}}^{k} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} } \right) \oplus & \left( {\sqrt {1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} ,{{({\upsilon _{{{\bar {\alpha }}_{k+1}}}})}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} \right) \\ & = \left( {\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{(1 - \mu _{{{{\bar {\alpha }}_j}}}^{2})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} +1 - {{(1 - \mu _{{{{\bar {\alpha }}_{k+1}}}}^{2})}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}} - \left( {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} } \right)\left( {1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} \right)} ,\prod\limits_{{j=1}}^{k} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} {{({\upsilon _{{{\bar {\alpha }}_{k+1}}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} \right) \\ & = \left( {\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} +1 - {{\left( {1 - \mu _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\limits_{{j=1}}^{{k+1}} {{T_j}} }}}} - 1+\prod\limits_{{j=1}}^{k} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} +{{\left( {1 - \mu _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}} - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} {{\left( {1 - \mu _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} ,{{({\upsilon _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}{{({\upsilon _{{{\bar {\alpha }}_{k+1}}}})}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} \right) \\ & = \left( {\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}{{\left( {1 - \mu _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} ,} {{({\upsilon _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}{{({\upsilon _{{{\bar {\alpha }}_{k+1}}}})}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} \right)=\left( {\sqrt {1 - \prod\limits_{{j=1}}^{{k+1}} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} } ,\prod\limits_{{j=1}}^{{k+1}} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} } \right) \\ \end{aligned}$$

This shows that Eq. (5) holds for \(n=k+1\). Thus Eq. (5) holds for all \(n\). This completes the proof. In the following, we describe some desirable properties of PFPWA operator.

Theorem 2 (Idempotency)

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \((j=1,2,3,\ldots,n)\) be the collection of PFNs. If all \({\bar {\alpha }_j}\), \((j=1,2,3,\ldots,n)\) are equal, i.e,. \({\bar {\alpha }_j}=\bar {\alpha }\) for all \(j\), Then
$$PFPWA({\bar {\alpha }_1},{\bar {\alpha }_2},\ldots,{\bar {\alpha }_n})=\bar {\alpha }$$
(6)
where \(T=\prod\nolimits_{{k=1}}^{{j - 1}} {{S^ * }({{\bar {\alpha }}_k})}\), \((j=2,3,\ldots,n)\), \(T=1\) and \({S^ * }({\bar {\alpha }_k})\) is the score of Pythagorean fuzzy numbers \({\bar {\alpha }_k}\).

Proof

By Definition 4, we have
$$\begin{aligned} {\text{PFPWA}}({{\bar {\alpha }}_1},{{\bar {\alpha }}_2}, \ldots ,{{\bar {\alpha }}_n})& = \left( {\frac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_1} \oplus \frac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_2} \oplus \cdots \oplus \frac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_n}} \right) \\ & = \left( {\sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } ,\prod\limits_{{j=1}}^{n} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right) \\ & = \left( {\sqrt {1 - \prod\limits_{{j=1}}^{n} {{{(1 - \mu _{{\bar {\alpha }}}^{2})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } ,\prod\limits_{{j=1}}^{n} {{{({\upsilon _{\bar {\alpha }}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right) \\ & = \left( {{\mu _{\bar {\alpha }}},{\upsilon _{\bar {\alpha }}}} \right)=\bar {\alpha } \\ \end{aligned}$$
This completes the proof.□

Based on Theorem 2, we have the following corollaries.

Corollary 1

If \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \((j=1,2,3,\ldots,n)\) is a collection of PFNs, that is a for all \(j\) then,
$${\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2} \ldots ,{\bar {\alpha }_n})=({\bar {\alpha }^ * },{\bar {\alpha }^ * }, \ldots ,{\bar {\alpha }^ * })=(1,0),$$
(7)
which is the largest value.

Corollary 2 (Non-compensatory)

If \({\bar {\alpha }_1}=({\mu _{{{\bar {\alpha }}_1}}},{\upsilon _{{{\bar {\alpha }}_1}}})\) is the smallest PFN, i.e., \({\bar {\alpha }_1}={\bar {\alpha }^*}=(0,1)\) for all \(j\), then
$${\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})={\text{PFPWA}}\;({\bar {\alpha }^*},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})=(0,1),$$
(8)
which is the smallest PFN.

Proof

Since, \({\bar {\alpha }_1}=(0,1)\) then by Definition of score function \({S^ * }({\bar {\alpha }_1})=0.\)

Since, \({T_j}=\prod\nolimits_{{k=1}}^{{j - 1}} {{S^ * }({{\bar {\alpha }}_k})}\) \((j=2,3,\ldots,n)\) and \({T_1}=1\)

So we have
$$\begin{aligned} {T_j}=\prod\limits_{{k=1}}^{{j - 1}} {{S^ * }({{\bar {\alpha }}_k})} & = {S^ * }({{\bar {\alpha }}_1}) \times {S^ * }({{\bar {\alpha }}_2}) \times \cdots \times {S^ * }({{\bar {\alpha }}_{j - 1}}) \\ & = 0 \times {S^ * }({{\bar {\alpha }}_1}) \times {S^ * }({{\bar {\alpha }}_2}) \times \cdots \times {S^ * }({{\bar {\alpha }}_{j - 1}}) \\ & = 0\;(j=2,3, \ldots n) \\ \end{aligned}$$
By Definition 4 we have
$$\begin{aligned} {\text{PFPWA}}\left( {{{\bar {\alpha }}_1},{{\bar {\alpha }}_2}, \ldots ,{{\bar {\alpha }}_n}} \right)& =\left( {\frac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_1} \oplus \frac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_2} \oplus \cdots \oplus \frac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}{{\bar {\alpha }}_n}} \right) \\ & =\left( {\frac{1}{1}{{\bar {\alpha }}_1} \oplus \frac{0}{1}{{\bar {\alpha }}_2} \oplus \cdots \oplus \frac{0}{1}{{\bar {\alpha }}_n}} \right)={{\bar {\alpha }}_1}=(0,1). \\ \end{aligned}$$

This completes the proof.□

Theorem 3 (Boundary)

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \((j=1,2,3,\ldots,n)\) be the collection of PFNs and
$${\bar {\alpha }^ - }=(\hbox{min} _{j}^{}({\mu _{{{\bar {\alpha }}_j}}}),\hbox{max} _{j}^{}({\upsilon _{{{\bar {\alpha }}_j}}})),\;{\bar {\alpha }^+}=(\hbox{max} _{j}^{}({\mu _{{{\bar {\alpha }}_j}}}),\hbox{min} _{j}^{}({\upsilon _{{{\bar {\alpha }}_j}}})).$$
Then
$${\bar {\alpha }^ - } \leq PFPWA({\bar {\alpha }_1},{\bar {\alpha }_2},\ldots,{\bar {\alpha }_n}) \leq {\bar {\alpha }^+}$$
(9)
where \(T=\prod\nolimits_{{k=1}}^{{j - 1}} {{S^ * }({{\bar {\alpha }}_k})}\), \((j=2,3,\ldots,n)\), \(T=1\) and \({S^ * }({\bar {\alpha }_k})\) is the score of Pythagorean fuzzy numbers \({\bar {\alpha }_k}\).

Proof

Since
$$\mathop {\hbox{min} }\limits_{j}^{} ({\mu _{{{\bar {\alpha }}_j}}}) \leq {\mu _{{{\bar {\alpha }}_j}}} \leq \mathop {\hbox{max} }\limits_{j}^{} ({\mu _{{{\bar {\alpha }}_j}}})$$
(10)
and
$$\mathop {\hbox{min} }\limits_{j}^{} ({\upsilon _{{{\bar {\alpha }}_j}}}) \leq {\upsilon _{{{\bar {\alpha }}_j}}} \leq \mathop {\hbox{max} }\limits_{j}^{} ({\upsilon _{{{\bar {\alpha }}_j}}}),$$
(11)
for all \(j\). From Eq. (10) we have,
$$\begin{aligned} \mathop {\hbox{min} }\limits_{j}^{} ({\mu _{{{\bar {\alpha }}_j}}}) \leq {\mu _{{{\bar {\alpha }}_j}}} \leq \mathop {\hbox{max} }\limits_{j}^{} ({\mu _{{{\bar {\alpha }}_j}}}) \Leftrightarrow & \sqrt {\mathop {\hbox{min} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \leq \sqrt {{{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \leq \sqrt {\mathop {\hbox{max} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \\ \Leftrightarrow & \sqrt {1 - \mathop {\hbox{max} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \leq \sqrt {1 - {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \leq \sqrt {1 - \mathop {\hbox{min} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \\ \Leftrightarrow & \sqrt {{{\left( {1 - \mathop {\hbox{max} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} \leq \sqrt {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} \leq \sqrt {{{\left( {1 - \mathop {\hbox{min} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} \\ \Leftrightarrow & \sqrt {\prod\limits_{{j=1}}^{n} {{{\left( {1 - \mathop {\hbox{max} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} } \leq \sqrt {\prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} } \leq \sqrt {\prod\limits_{{j=1}}^{n} {{{\left( {1 - \mathop {\hbox{min} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} } \\ \Leftrightarrow & \sqrt {1 - \mathop {\hbox{max} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \leq \sqrt {\prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} } \leq \sqrt {1 - \mathop {\hbox{min} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \\ \Leftrightarrow & \sqrt { - 1+\mathop {\hbox{min} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \leq \sqrt {\prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} } \leq \sqrt { - 1+\mathop {\hbox{max} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \\ \Leftrightarrow & \sqrt {1 - 1+\mathop {\hbox{min} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \leq \sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} \leq } \sqrt {1 - 1+\mathop {\hbox{max} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \\ \Leftrightarrow & \sqrt {\mathop {\hbox{min} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \leq \sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} } \leq \sqrt {\mathop {\hbox{max} }\limits_{j} {{({\mu _{{{\bar {\alpha }}_j}}})}^2}} \\ \Leftrightarrow & \mathop {\hbox{min} }\limits_{j} ({\mu _{{{\bar {\alpha }}_j}}}) \leq \sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} } \leq \mathop {\hbox{max} }\limits_{j} ({\mu _{{{\bar {\alpha }}_j}}}). \\ \end{aligned}$$
(12)
Now from Eq. (11), we have, \(\mathop {\hbox{min} }\limits_{j} ({\upsilon _{{{\bar {\alpha }}_j}}}) \leq ({\upsilon _{{{\bar {\alpha }}_j}}}) \leq \mathop {\hbox{max} }\limits_{j} ({\upsilon _{{{\bar {\alpha }}_j}}}).\)
$$\Leftrightarrow \mathop {\hbox{min} }\limits_{j}^{} ({\upsilon _{{{\bar {\alpha }}_j}}}) \leq {\upsilon _{{{\bar {\alpha }}_j}}} \leq \mathop {\hbox{max} }\limits_{j}^{} ({\upsilon _{{{\bar {\alpha }}_j}}}) \Leftrightarrow \mathop {\hbox{min} }\limits_{j}^{} {({\upsilon _{{{\bar {\alpha }}_j}}})^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}} \leq {({\upsilon _{{{\bar {\alpha }}_j}}})^{^{{\tfrac{{{T_j}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}}}} \leq \mathop {\hbox{max} }\limits_{j}^{} {({\upsilon _{{{\bar {\alpha }}_j}}})^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}} \Leftrightarrow \prod\limits_{{j=1}}^{n} {\mathop {\hbox{min} }\limits_{j}^{} {{({\upsilon _{{{\bar {\alpha }}_j}}})}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} \leq \prod\limits_{{j=1}}^{n} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} \leq \prod\limits_{{j=1}}^{n} {\mathop {\hbox{max} }\limits_{j}^{} {{({\upsilon _{{{\bar {\alpha }}_j}}})}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} \Leftrightarrow \mathop {\hbox{min} }\limits_{j}^{} ({\upsilon _{{{\bar {\alpha }}_j}}}) \leq \prod\limits_{{j=1}}^{n} {{{({\upsilon _{{{\bar {\alpha }}_j}}})}^{^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}}}} \leq \mathop {\hbox{max} }\limits_{j}^{} ({\upsilon _{{{\bar {\alpha }}_j}}})$$
(13)
Let
$$PFPWA({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})=\bar {\alpha }=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})$$
Then, \({S^ * }(\bar {\alpha })=\mu _{{{{\bar {\alpha }}_j}}}^{2} - \upsilon _{{{{\bar {\alpha }}_j}}}^{2} \leq \mathop {\hbox{max} }\limits_{j} {({\mu _{{{\bar {\alpha }}_j}}})^2} - \mathop {\hbox{min} }\limits_{j} {({\upsilon _{{{\bar {\alpha }}_j}}})^2}={S^ * }({\bar {\alpha }_{\hbox{max} }})\). Thus, \({S^ * }(\bar {\alpha }) \leq {S^ * }({\bar {\alpha }_{\hbox{max} }})\). Again \({S^ * }(\bar {a})=\mu _{{{{\bar {a}}_j}}}^{2} - \upsilon _{{{{\bar {a}}_j}}}^{2} \geq \mathop {\hbox{min} }\nolimits_{j} {({\mu _{{{\bar {a}}_j}}})^2} - \mathop {\hbox{max} }\limits_{j} {({\upsilon _{{{\bar {a}}_j}}})^2}={S^ * }({\bar {a}_{\hbox{min} }}).\) Thus, \({S^ * }(\bar {\alpha }) \geq {S^ * }({\bar {\alpha }_{\hbox{min} }})\). If \({S^ * }(\bar {\alpha })<{S^ * }({\bar {\alpha }_{\hbox{max} }})\) and \({S^ * }(\bar {\alpha })>{S^ * }({\bar {\alpha }_{\hbox{min} }}).\) Then
$${\bar {\alpha }_{\hbox{min} }}<{\text{PFPWA}}\left( {{{\bar {\alpha }}_1},{{\bar {\alpha }}_2}, \ldots ,{{\bar {\alpha }}_n}} \right)<{\bar {\alpha }_{\hbox{max} }}.$$
(14)
If \({S^ * }(\bar {\alpha })\;\;={S^ * }({\bar {\alpha }_{\hbox{max} }})\). Then \(\mu _{{{{\bar {\alpha }}_j}}}^{2} - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\mathop {\hbox{max} }\limits_{j} {({\mu _{{{\bar {\alpha }}_j}}})^2} - \mathop {\hbox{min} }\limits_{j} {({\upsilon _{{{\bar {\alpha }}_j}}})^2}\)
$$\Leftrightarrow \mu _{{_{{{{\bar {\alpha }}_j}}}}}^{2} - \upsilon _{{_{{{{\bar {\alpha }}_j}}}}}^{2}=\mathop {\hbox{max} }\limits_{j}^{} {({\mu _{{{\bar {\alpha }}_j}}})^2} - \mathop {\hbox{min} }\limits_{j}^{} {({\upsilon _{{{\bar {\alpha }}_j}}})^2} \Leftrightarrow \mu _{{{{\bar {\alpha }}_j}}}^{2}=\mathop {\hbox{max} }\limits_{j}^{} {({\mu _{{{\bar {\alpha }}_j}}})^2},\upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\mathop {\hbox{min} }\limits_{j}^{} {({\upsilon _{{{\bar {\alpha }}_j}}})^2} \Leftrightarrow {\mu _{{{\bar {\alpha }}_j}}}=\mathop {\hbox{max} }\limits_{j}^{} ({\mu _{{{\bar {\alpha }}_j}}}),{\upsilon _{{{\bar {a}}_j}}}=\mathop {\hbox{min} }\limits_{j}^{} ({\upsilon _{{{\bar {\alpha }}_j}}}).$$
Since \({H^ * }(\bar {\alpha })=\mu _{{{{\bar {\alpha }}_j}}}^{2}+\upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\mathop {\hbox{max} }\limits_{j} {({\mu _{{{\bar {\alpha }}_j}}})^2}+\mathop {\hbox{min} }\limits_{j} {({\upsilon _{{{\bar {\alpha }}_j}}})^2}=H({\bar {\alpha }_{\hbox{max} }}).\) Thus,
$${\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})={\bar {\alpha }_{\hbox{max} }}$$
(15)
If \({S^ * }(\bar {\alpha })={S^ * }({\bar {\alpha }^ - })\). Then \(\mu _{{{{\bar {\alpha }}_j}}}^{2} - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\hbox{min} _{j}^{}{({\upsilon _{{{\bar {\alpha }}_j}}})^2} - \hbox{max} _{j}^{}{({\mu _{{{\bar {\alpha }}_j}}})^2}.\)
$$\Leftrightarrow \mu _{{{{\bar {\alpha }}_j}}}^{2} - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\mathop {\hbox{min} }\limits_{j}^{} {({\upsilon _{{{\bar {\alpha }}_j}}})^2} - \mathop {\hbox{max} }\limits_{j}^{} {({\mu _{{{\bar {\alpha }}_j}}})^2} \Leftrightarrow \mu _{{{{\bar {\alpha }}_j}}}^{2}=\mathop {\hbox{min} }\limits_{j}^{} {({\upsilon _{{{\bar {\alpha }}_j}}})^2},\upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\mathop {\hbox{max} }\limits_{j}^{} {({\mu _{{{\bar {\alpha }}_j}}})^2} \Leftrightarrow {\mu _{{{\bar {\alpha }}_j}}}=\mathop {\hbox{min} }\limits_{j}^{} {({\upsilon _{{{\bar {\alpha }}_j}}})^2},{\upsilon _{{{\bar {\alpha }}_j}}}=\mathop {\hbox{max} }\limits_{j}^{} {({\mu _{{{\bar {\alpha }}_j}}})^2}$$
Since \({H^ * }(\bar {\alpha })=\mu _{{{{\bar {\alpha }}_j}}}^{2}+\upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\mathop {\hbox{min} }\limits_{j} {({\upsilon _{{{\bar {\alpha }}_j}}})^2}+\mathop {\hbox{max} }\limits_{j} {({\mu _{{{\bar {\alpha }}_j}}})^2}={H^ * }({\bar {\alpha }_{\hbox{min} }}).\) Thus,
$${\text{PFPWA}}({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})={\bar {\alpha }_{\hbox{min} }}$$
(16)
Thus from Eqs. (14) to (16), we have
$${\bar {\alpha }_{\hbox{min} }} \leq {\text{PFPWA}}({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n}) \leq {\bar {\alpha }_{\hbox{max} }},{\text{ for all }}w.$$

Theorem 4 (Monotonicity)

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \((j=1,2,3, \ldots ,n)\) and \(\bar {\alpha }_{j}^{ * }=({\mu _{\bar {\alpha }_{j}^{ * }}},{\upsilon _{\bar {\alpha }_{j}^{ * }}})\) \((j=1,2,3, \ldots ,n)\) be two collections of PFNs. If \({\mu _{{{\bar {\alpha }}_j}}} \leq {\mu _{\bar {\alpha }_{j}^{ * }}}\) and \({\upsilon _{{{\bar {\alpha }}_j}}} \geq {\upsilon _{\bar {\alpha }_{j}^{ * }}}\), then
$${\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n}) \leq {\text{PFPWA}}\;(\bar {\alpha }_{1}^{ * },\bar {\alpha }_{2}^{ * }, \ldots ,\bar {\alpha }_{n}^{ * })$$
(17)
where \(T=\prod\nolimits_{{k=1}}^{{j - 1}} {{S^ * }({{\bar {\alpha }}_k})}\), \((j=1,2,3, \ldots ,n)\), \({T^ * }=\prod\nolimits_{{k=1}}^{{j - 1}} {{S^ * }(\bar {\alpha }_{k}^{ * })}\), \((j=1,2,3, \ldots ,n)\), \(T={T^ * }=1\) and \({S^ * }({\bar {\alpha }_k})\), \({S^ * }(\bar {\alpha }_{k}^{ * })\) are the scores of Pythagorean fuzzy numbers \({\bar {\alpha }_k}\) and Pythagorean fuzzy numbers \(\bar {\alpha }_{k}^{ * }\) respectively.

Proof

Since \({\mu _{{{\bar {\alpha }}_j}}} \leq {\mu _{\bar {\alpha }_{j}^{ * }}}\) and \({\upsilon _{{{\bar {\alpha }}_j}}} \geq {\upsilon _{\bar {\alpha }_{j}^{ * }}}\) If \({\mu _{{{\bar {\alpha }}_j}}} \leq {\mu _{\bar {\alpha }_{j}^{ * }}}\).
$$\Leftrightarrow \mu _{{{{\bar {\alpha }}_j}}}^{2} \leq \mu _{{\bar {\alpha }_{j}^{ * }}}^{2} \Leftrightarrow \sqrt {\mu _{{{{\bar {\alpha }}_j}}}^{2}} \leq \sqrt {\mu _{{\bar {\alpha }_{j}^{ * }}}^{2}} \Leftrightarrow \sqrt {1 - \mu _{{\bar {\alpha }_{j}^{ * }}}^{2}} \leq \sqrt {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \Leftrightarrow \sqrt {{{\left( {1 - \mu _{{\bar {\alpha }_{j}^{ * }}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \leq \sqrt {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \Leftrightarrow \sqrt {\prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{\bar {\alpha }_{j}^{ * }}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \leq \sqrt {\prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \Leftrightarrow \sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \leq \sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \mu _{{\bar {\alpha }_{j}^{ * }}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} }$$
(18)
Now \({\upsilon _{{{\bar {\alpha }}_j}}} \geq {\upsilon _{\bar {\alpha }_{j}^{ * }}}\).
$$\begin{aligned} \Leftrightarrow & \upsilon _{{{{\bar {\alpha }}_j}}}^{{\tfrac{{{T_j}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}} \geq \upsilon _{{\bar {\alpha }_{j}^{ * }}}^{{\tfrac{{{T_j}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}} \\ \Leftrightarrow & \prod\limits_{{j=1}}^{n} {\upsilon _{{{{\bar {\alpha }}_j}}}^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \geq \prod\limits_{{j=1}}^{n} {\upsilon _{{\bar {\alpha }_{j}^{ * }}}^{{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \\ \end{aligned}$$
(19)
Let
$$\bar {\alpha }={\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})$$
and
$${\bar {\alpha }^ * }={\text{PFPWA}}\;(\bar {\alpha }_{1}^{ * },\bar {\alpha }_{2}^{ * }, \ldots ,\bar {\alpha }_{n}^{ * })$$

Thus, from Eqs. (18) and (19) we have.

If
$${S^ * }(\bar {\alpha })\;\;={S^ * }({\bar {\alpha }^ * })$$
Then
$$\Leftrightarrow \mu _{{{{\bar {\alpha }}_j}}}^{2} - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\mu _{{\bar {\alpha }_{j}^{ * }}}^{2} - \upsilon _{{\bar {\alpha }_{j}^{ * }}}^{2} \Leftrightarrow \mu _{{{{\bar {\alpha }}_j}}}^{2}=\mu _{{\bar {\alpha }_{j}^{ * }}}^{2},\upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\upsilon _{{\bar {\alpha }_{j}^{ * }}}^{2} \Leftrightarrow {\mu _{{{\bar {\alpha }}_j}}}={\mu _{\bar {\alpha }_{j}^{ * }}},{\upsilon _{{{\bar {\alpha }}_j}}}={\upsilon _{\bar {\alpha }_{j}^{ * }}}$$
Since \({H^ * }(\bar {\alpha })=\mu _{{{{\bar {\alpha }}_j}}}^{2}+\upsilon _{{{{\bar {\alpha }}_j}}}^{2}=\mu _{{\bar {\alpha }_{j}^{ * }}}^{2}+\upsilon _{{\bar {\alpha }_{j}^{ * }}}^{2}={H^ * }({\bar {\alpha }^ * })\) Thus
$${\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})={\text{PFPWA}}\;(\bar {\alpha }_{1}^{ * },\bar {\alpha }_{2}^{ * }, \ldots ,\bar {\alpha }_{n}^{ * })$$
(20)
Thus from Eqs. (19) and (20), we have
$${\text{PFPWA}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n}) \leq {\text{PFPWA}}\;(\bar {\alpha }_{1}^{ * },\bar {\alpha }_{2}^{ * }, \ldots .,\bar {\alpha }_{n}^{ * })$$

This completes the proof.□

Definition 5

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \((j=1,2,3, \ldots ,n)\) be the collection of PFNs and let\({\text{PFPWG:}}\;{\Omega ^n} \to \Omega\), be a mapping of dimension \(n\). If
$${\text{PFPWG}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})=\left( {\bar {\alpha }_{1}^{{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \bar {\alpha }_{2}^{{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \cdots \otimes \bar {\alpha }_{n}^{{\tfrac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)$$
(21)
then the mapping PFPWG is called Pythagorean fuzzy Prioritized weighted geometric (PFPWG) operator where \(T=\prod\nolimits_{{k=1}}^{{j - 1}} {{S^ * }({{\bar {\alpha }}_k})}\), \((j=1,2,3, \ldots ,n)\), \(T=1\) and \({S^ * }({\bar {\alpha }_k})\) is the score of Pythagorean fuzzy numbers \({\bar {\alpha }_k}=({\mu _{{{\bar {\alpha }}_k}}},{\upsilon _{{{\bar {\alpha }}_k}}})\).

Based on the operational law of defined in Sect. 2, we have the following result.

Theorem 5

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \((j=1,2,3, \ldots ,n)\) be the collection of PFNs. Then using PFPWA operator the aggregated value of the PFNs is also PFN
$${\text{PFPWG}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})=\left( {\bar {\alpha }_{1}^{{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \bar {\alpha }_{2}^{{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \cdots \otimes \bar {\alpha }_{n}^{{\tfrac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)=\left( {\mathop \prod \limits_{{j=1}}^{n} {{({\mu _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}},\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right)$$
(22)
where \(T=\prod\nolimits_{{k=1}}^{{j - 1}} {{S^ * }({{\bar {\alpha }}_k})}\), \((j=1,2,3, \ldots ,n)\), \(T=1\) and \({S^ * }({\bar {\alpha }_k})\) is the score of Pythagorean fuzzy numbers \({\bar {\alpha }_k}=({\mu _{{{\bar {\alpha }}_k}}},{\upsilon _{{{\bar {\alpha }}_k}}})\).

Proof

First result follows from Definition 5. Next in the following we prove,
$${\text{PFPWG}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})=\left( {\bar {\alpha }_{1}^{{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \bar {\alpha }_{2}^{{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \cdots \otimes \bar {\alpha }_{n}^{{\tfrac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)=\left( {\prod\limits_{{j=1}}^{n} {{{({\mu _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,\sqrt {1 - \prod\limits_{{j=1}}^{n} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } } \right)$$

Using mathematical induction.

First, we show that Eq. (22) holds for \(n=2\). Since,
$$\bar {\alpha }_{1}^{{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}=\left( {{{({\mu _{{{\bar {\alpha }}_1}}})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}},\sqrt {1 - {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right)$$
and
$$\bar {\alpha }_{2}^{{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}=\left( {{{({\mu _{{{\bar {\alpha }}_2}}})}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}},\sqrt {1 - {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_2}}}^{2}} \right)}^{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right)$$
thus we have,
$$\begin{aligned} \bar{\alpha }_{1}^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} \otimes \bar{\alpha }_{2}^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}}& = \left( {(\mu _{{\bar{\alpha }_{1} }} )^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} ,\sqrt {1 - \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } } \right) \otimes \left( {(\mu _{{\bar{\alpha }_{2} }} )^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} ,\sqrt {1 - \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } } \right) \\& = \left( {(\mu _{{\bar{\alpha }_{1} }} )^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} (\mu _{{\bar{\alpha }_{2} }} )^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} ,\sqrt {1 - \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} + 1 - \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} - \left( {1 - \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } \right)\left( {1 - \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } \right)} } \right) \\& = \left( {(\mu _{{\bar{\alpha }_{1} }} )^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} (\mu _{{\bar{\alpha }_{2} }} )^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} ,\sqrt {1 - \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} + 1 - \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} - \left( {\left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} + \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } \right)} } \right) \\& = \left( {(\mu _{{\bar{\alpha }_{1} }} )^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} (\mu _{{\bar{\alpha }_{2} }} )^{{\tfrac{{T_{2} }}{{\sum\limits_{{j = 1}}^{n} {T_{j} } }}}} ,\sqrt {1 - \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} + 1 - \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} - 1 + \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} + \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} + \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } } \right) \\& = \left( {(\mu _{{\bar{\alpha }_{1} }} )^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} (\mu _{{\bar{\alpha }_{2} }} )^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} ,\sqrt {1 - \left( {1 - \upsilon _{{\bar{\alpha }_{1} }}^{2} } \right)^{{\tfrac{{T_{1} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} \left( {1 - \upsilon _{{\bar{\alpha }_{2} }}^{2} } \right)^{{\tfrac{{T_{2} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } } \right) \\& = \left( {\prod\limits_{{j = 1}}^{2} {(\mu _{{\bar{\alpha }_{j} }} )^{{\tfrac{{T_{j} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } ,\sqrt {1 - \prod\limits_{{j = 1}}^{2} {\left( {1 - \upsilon _{{\bar{\alpha }_{j} }}^{2} } \right)^{{\tfrac{{T_{j} }}{{\sum\nolimits_{{j = 1}}^{n} {T_{j} } }}}} } } } \right) \\ \end{aligned}$$

This shows that Eq. (22) holds for n = 2.

Assume that Eq. (22) holds for \(n=k\), i.e.,
$${\text{PFPWG}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_k})=\left( {\bar {\alpha }_{1}^{{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \bar {\alpha }_{2}^{{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \cdots \otimes \bar {\alpha }_{k}^{{\tfrac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right)=\left( {\prod\limits_{{j=1}}^{k} {{{({\mu _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }_j}}}}}} ,\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } } \right)$$
then, when \(n=k+1\), by the operational law of Definition 2 we have,
$$\begin{aligned} {\text{PFPWG}}\;({{\bar {\alpha }}_1},{{\bar {\alpha }}_2}, \ldots ,{{\bar {\alpha }}_n}) \otimes {{\bar {\alpha }}_{k+1}}& = \left( {\bar {\alpha }_{1}^{{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \bar {\alpha }_{2}^{{\tfrac{{{T_2}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}} \otimes \cdots \otimes \bar {\alpha }_{k}^{{\tfrac{{{T_n}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} \right) \otimes \left( {{{\bar {\alpha }}_{k+1}}} \right) \\ & =\left( {\prod\limits_{{j=1}}^{k} {{{({\mu _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } } \right) \otimes \left( {{{({\mu _{{{\bar {\alpha }}_{k+1}}}})}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}},\sqrt {1 - {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } \right) \\ & =\left( {\prod\limits_{{j=1}}^{k} {{{({\mu _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} {{({\mu _{{{\bar {\alpha }}_{k+1}}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}},\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} +1 - {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}} - \left( {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} } \right)\left( {1 - {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} \right)} } \right) \\ & =\left( {\prod\limits_{{j=1}}^{k} {{{({\mu _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_i}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} {{({\mu _{{{\bar {\alpha }}_{k+1}}}})}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}},\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} +1 - {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}} - 1+\prod\limits_{{j=1}}^{k} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\limits_{{j=1}}^{k} {{T_j}} }}}}} +{{(1 - \upsilon _{{{{\bar {\alpha }}_{k+1}}}}^{2})}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}} - \prod\limits_{{j=1}}^{k} {{{(1 - \upsilon _{{{{\bar {\alpha }}_1}}}^{2})}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} } \right) \\ & =\left( {\prod\limits_{{j=1}}^{k} {{{({\mu _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} {{({\mu _{{{\bar {\alpha }}_{k+1}}}})}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{n+1}} {{T_j}} }}}},\sqrt {1 - \prod\limits_{{j=1}}^{k} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_1}}}^{2}} \right)}^{\tfrac{{{T_1}}}{{\sum\nolimits_{{j=1}}^{k} {{T_j}} }}}}} {{\left( {1 - \upsilon _{{{{\bar {\alpha }}_{k+1}}}}^{2}} \right)}^{\tfrac{{{T_{k+1}}}}{{\sum\nolimits_{{j=1}}^{{k+1}} {{T_j}} }}}}} } \right) \\ & =\left( {\prod\limits_{{j=1}}^{{k+1}} {{{({\mu _{{{\bar {\alpha }}_j}}})}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} ,\sqrt {1 - \prod\limits_{{j=1}}^{{k+1}} {{{\left( {1 - \upsilon _{{{{\bar {\alpha }}_j}}}^{2}} \right)}^{\tfrac{{{T_j}}}{{\sum\nolimits_{{j=1}}^{n} {{T_j}} }}}}} } } \right) \\ \end{aligned}$$
This shows that Eq. (22) holds for \(n=k+1\). Thus, Eq. (22) holds for all \(n\). This completes the Proof.□

Theorem 6 (Idempotency)

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\)\(\left( {j=1,2,3, \ldots ,n} \right)\) be the collection of PFNs. If all \({\bar {\alpha }_j}\),\((j=2,3, \ldots ,n)\) are equal, i.e., \({\bar {\alpha }_j}=\bar {\alpha }\) for all \(j\), Then
$${\text{PFPWG}}\,\,({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n})=\bar {\alpha }$$
(23)
where \(T=\mathop \prod \limits_{{k=1}}^{{j - 1}} {S^ * }({\bar {\alpha }_k})\), \((j=2,3, \ldots ,n)\), \(T=1\) and \({S^ * }({\bar {\alpha }_k})\) is the score of Pythagorean fuzzy numbers \({\bar {\alpha }_k}\).

Proof

Proof of the Theorem is same as Theorem 2.□

Theorem 7 (Boundary)

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \(\left( {j=1,2,3,\ldots,n} \right)\) be the collection of PFNs and
$${\bar {\alpha }^ - }=(\mathop {\hbox{min} }\limits_{j} ({\mu _{{{\bar {\alpha }}_j}}}),\mathop {\hbox{max} }\limits_{j} ({\upsilon _{{{\bar {\alpha }}_j}}})),\;{\bar {\alpha }^+}=(\mathop {\hbox{max} }\limits_{j} ({\mu _{{{\bar {\alpha }}_j}}}),\mathop {\hbox{min} }\limits_{j} ({\upsilon _{{{\bar {\alpha }}_j}}})).$$
Then
$${\bar {\alpha }^ - } \leq {\text{PFPWG}}({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n}) \leq {\bar {\alpha }^+}$$
(24)
where \(T=\mathop \prod \nolimits_{{k=1}}^{{j - 1}} {S^ * }({\bar {\alpha }_k})\), \((j=2,3, \ldots ,n)\), \(T=1\) and \({S^ * }({\bar {\alpha }_k})\) is the score of Pythagorean fuzzy numbers \({\bar {\alpha }_k}\).

Proof

Proof of the Theorem is same as Theorem 3.□

Theorem 8 (Monotonicity)

Let \({\bar {\alpha }_j}=({\mu _{{{\bar {\alpha }}_j}}},{\upsilon _{{{\bar {\alpha }}_j}}})\) \(\left( {j=1,2,3, \ldots ,n} \right)\) and \(\bar {\alpha }_{j}^{ * }=({\mu _{\bar {\alpha }_{j}^{ * }}},{\upsilon _{\bar {\alpha }_{j}^{ * }}})\) \((j=1,2,3, \ldots ,n)\) be two collection of PFNs. If \({\mu _{{{\bar {\alpha }}_j}}} \leq {\mu _{\bar {\alpha }_{j}^{ * }}}\) and \({\upsilon _{{{\bar {\alpha }}_j}}} \geq \upsilon \bar {\alpha }\), then
$${\text{PFPWG}}\;({\bar {\alpha }_1},{\bar {\alpha }_2}, \ldots ,{\bar {\alpha }_n}) \leq {\text{PFPWG}}(\bar {\alpha }_{1}^{ * },\bar {\alpha }_{2}^{ * }, \ldots ,\bar {\alpha }_{n}^{ * })$$
(25)
where \(T=\mathop \prod \nolimits_{{k=1}}^{{j - 1}} {S^ * }({\bar {\alpha }_k})\), \((j=2,3, \ldots ,n)\), \({T^ * }=\mathop \prod \nolimits_{{k=1}}^{{j - 1}} {S^ * }(\bar {\alpha }_{k}^{ * })\), \((j=2,3, \ldots ,n)\), \(T={T^ * }=1\) and \({S^ * }({\bar {\alpha }_k})\), \({S^ * }(\bar {\alpha }_{k}^{ * })\) are the scores of Pythagorean fuzzy numbers \({\bar {\alpha }_k}\) and Pythagorean fuzzy numbers \(\bar {\alpha }_{k}^{ * }\), respectively.

Proof

Proof of the Theorem is same as Theorem 4.□

3 Multi-attribute decision making based on pythagorean fuzzy prioritized aggregation operators

In this section, we utilize the developed aggregation operators namely Pythagorean fuzzy prioritized weighted average (PFPWA) operator and Pythagorean fuzzy prioritized weighted geometric (PFPWG) operator to multiple criteria decision-making problem under Pythagorean fuzzy environment.

In multi-attribute group decision making, suppose that\(X=\{ {X_1},{X_2}, \ldots ,{X_m}\}\) be a set of \(m\) alternatives, and \(A=\{ {A_1},{A_2}, \ldots ,{A_n}\}\) be the set of \(n\) attributes and that there is a prioritization between the attributes expressed by the linear ordering \({A_1} \succ {A_2} \succ \cdots \succ {A_m}\) indicate attribute \({A_j}\) has a higher priority than \({A_i}\) if \(j \prec i\), and \(D=\{ {D_1},{D_2}, \ldots ,{D_l}\}\) be the set of decision makers and that there is a prioritization between decision makers expressed by the linear ordering \({D_1} \succ {D_2} \succ \cdots \succ {D_l}\) indicate the decision maker \({D_\varsigma }\) has a higher priority than \({D_\tau }\) if \(\varsigma \prec \tau\). Let \({M^{\left( k \right)}}={[\bar {\alpha }_{{ij}}^{{(k)}}]_{m \times n}}={[(\mu _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}},\upsilon _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}})]_{m \times n}}\) be a Pythagorean fuzzy decision matrix provided by the decision makers \({D_k} \in D\), which is expressed as a Pythagorean fuzzy number where \(\mu _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}}\) denotes the degree that the alternatives \({X_i} \in X\) satisfies the attribute \({A_j} \in A\) expressed by the decision maker \({D_k}\), and \(\upsilon _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}}\) denotes the degree that the alternatives \({X_i} \in X\) does not satisfies the attribute \({A_j} \in A\) expressed by the decision maker \({D_k}\), such that \(\mu _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}} \in [0,1],\upsilon _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}} \in [0,1],\) \({(\mu _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}})^2}+{(\upsilon _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}})^2} \leq 1,\) \(i=1,2, \ldots ,m;\;j=1,2, \ldots ,n\).

To synchronize the data, first step is to look at the attributes. If all the attributes \({A_j}\left( {j=1,2, \ldots ,n} \right)\) are of same type, then there is no need for normalization. Conversely, if these contain different scales and/or units then there is needed to transform them all to the same scale and/or unit. Let us consider two types of attributes, namely, (1) cost type and the (2) benefit type. Considering their natures, a benefit attribute (the bigger the values better is it) and cost attribute (the smaller the values the better is it) are of rather opposite type. In such cases, we need to first transform the attribute values of cost type into the attribute values of benefit type. Therefore, transform the Pythagorean fuzzy decision matrix \({M^{\left( k \right)}}={[\bar {\alpha }_{{ij}}^{{(k)}}]_{m \times n}}\) into normalized decision matrix \({M^{ * (k)}}={[\bar {\gamma }_{{ij}}^{{(k)}}]_{m \times n}}\), where \(\bar {\gamma }_{{ij}}^{{(k)}}=(\mu _{{{{\bar {\gamma }}_{ij}}}}^{{(k)}},\upsilon _{{{{\bar {\gamma }}_{ij}}}}^{{(k)}})\) and
$$\bar {\gamma }_{{ij}}^{{(k)}}=\left\{ {\begin{array}{*{20}{l}} {\bar {\alpha }_{{ij}}^{{(k)}}{\text{ for benefit attribute, }}i=1,2, \ldots ,m;\;j=1,2, \ldots ,n.} \\ {{{(\bar {\alpha }_{{ij}}^{{(k)}})}^c}{\text{ for cost, }}i=1,2, \ldots ,m;\;j=1,2, \ldots ,n.} \end{array}} \right.$$
where \({(\bar {\alpha }_{{ij}}^{{(k)}})^c}\) is the complement of \(\bar {\alpha }_{{ij}}^{{(k)}}\), such that \({(\bar {\alpha }_{{ij}}^{{(k)}})^c}=(\upsilon _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}},\mu _{{{{\bar {\alpha }}_{ij}}}}^{{(k)}})\).

With attributes normalized and using the PFPWA/PFPWG operator, we now develop an algorithm to solve multiple attribute group decision-making problems under Pythagorean fuzzy environment:

Step 1. Calculate the value of \(T_{{ij}}^{{(k)}}\;(k=1,2, \ldots ,l)\) as follows:
$$T_{{ij}}^{{(k)}}=\mathop \prod \limits_{{\theta =1}}^{{k - 1}} {S^ * }(\bar {\gamma }_{{ij}}^{{(k)}})(k=2, \ldots ,l)$$
(26)
$$T_{{ij}}^{{(1)}}=1$$
(27)
Step 2. Utilize the PFPWA/PFPWG aggregation operator
$${\bar {\gamma }_{ij}}=({\mu _{{{\bar {\gamma }}_{ij}}}},{\upsilon _{{{\bar {\gamma }}_{ij}}}})={\text{PFPWA}}\left( {\bar {\gamma }_{{ij}}^{{(1)}},\bar {\gamma }_{{ij}}^{{(2)}}, \ldots ,\bar {\gamma }_{{ij}}^{{(l)}}} \right)=\left( {\sqrt {1 - \prod\limits_{{k=1}}^{l} {{{\left( {1 - {{(\mu _{{{{\bar {\gamma }}_{ij}}}}^{{(k)}})}^2}} \right)}^{\tfrac{{T_{{ij}}^{{(k)}}}}{{\sum\nolimits_{{k=1}}^{l} {T_{{ij}}^{{(k)}}} }}}}} } ,\prod\limits_{{k=1}}^{l} {{{(\upsilon _{{{{\bar {\gamma }}_{ij}}}}^{{(k)}})}^{\tfrac{{T_{{ij}}^{{(k)}}}}{{\sum\nolimits_{{k=1}}^{l} {T_{{ij}}^{{(k)}}} }}}}} } \right)$$
(28)
or
$${\bar {\gamma }_{ij}}=({\mu _{{{\bar {\gamma }}_{ij}}}},{\upsilon _{{{\bar {\gamma }}_{ij}}}})={\text{PFPWG}}(\bar {\gamma }_{{ij}}^{{(1)}},\bar {\gamma }_{{ij}}^{{(2)}}, \ldots ,\bar {\gamma }_{{ij}}^{{(l)}})=\left( {\prod\limits_{{k=1}}^{l} {{{(\mu _{{{{\bar {\gamma }}_{ij}}}}^{{(k)}})}^{\tfrac{{T_{{ij}}^{{(k)}}}}{{\sum\nolimits_{{k=1}}^{l} {T_{{ij}}^{{(k)}}} }}}}} ,\sqrt {1 - \prod\limits_{{k=1}}^{l} {{{\left( {1 - {{(\upsilon _{{{{\bar {\gamma }}_{ij}}}}^{{(k)}})}^2}} \right)}^{\tfrac{{T_{{ij}}^{{(k)}}}}{{\sum\nolimits_{{k=1}}^{l} {T_{{ij}}^{{(k)}}} }}}}} } } \right)$$
(29)
to aggregate all the individual Pythagorean fuzzy decision matrices \({M^ * }={[\bar {\gamma }_{{ij}}^{{(k)}}]_{m \times n}}(k=1,2,\ldots,l)\) provided by the decision makers into a collective Pythagorean fuzzy decision matrix \(M={[{\bar {\gamma }_{ij}}]_{m \times n}}\) \(i=1,2,\ldots,m;\, j=1,2,\ldots,n\).
Step 3. Calculate the value of \({T_{ij}}\), \(i=1,2,\ldots,m;\ ,j=1,2,\ldots,n\), as follows:
$${T_{ij}}=\prod\limits_{{q=1}}^{{j - 1}} {{S^ * }({{\bar {\gamma }}_{iq}})} \;(i=1,2, \ldots ,m;\,j=1,2, \ldots ,n)$$
(30)
$${T_{i1}}=1.$$
(31)
Step 4. Aggregate all the Pythagorean fuzzy numbers \({\bar {\gamma }_{ij}}\) for each alternative \({X_i}\) by utilizing PFPWA/PFPWG operator.
$${\bar {\gamma }_i}=({\mu _{{{\bar {\gamma }}_i}}},{\upsilon _{{{\bar {\gamma }}_i}}})={\text{PFPWA}}\left( {{{\bar {\gamma }}_{i1}},{{\bar {\gamma }}_{i2}}, \ldots ,{{\bar {\gamma }}_{in}}} \right)=\left( {\sqrt {1 - \prod\limits_{{k=1}}^{n} {{{\left( {1 - \mu _{{{{\bar {\gamma }}_{ij}}}}^{2}} \right)}^{\tfrac{{{T_{ij}}}}{{\sum\nolimits_{{k=1}}^{n} {{T_{ij}}} }}}}} } ,\prod\limits_{{k=1}}^{n} {{{({\upsilon _{{{\bar {\gamma }}_{ij}}}})}^{\tfrac{{{T_{ij}}}}{{\sum\nolimits_{{k=1}}^{n} {{T_{ij}}} }}}}} } \right)$$
(32)
or
$${\bar {\gamma }_i}=({\mu _{{{\bar {\gamma }}_i}}},{\upsilon _{{{\bar {\gamma }}_i}}})={\text{PFPWG}}({\bar {\gamma }_{i1}},{\bar {\gamma }_{i2}}, \ldots ,{\bar {\gamma }_{in}})=\left( {\prod\limits_{{j=1}}^{n} {{{({\mu _{{{\bar {\gamma }}_{ij}}}})}^{\tfrac{{{T_{ij}}}}{{\sum\nolimits_{{k=1}}^{n} {{T_{ij}}} }}}}} ,\sqrt {1 - \prod\limits_{{k=1}}^{n} {{{(1 - \upsilon _{{{{\bar {\gamma }}_{ij}}}}^{2})}^{\tfrac{{{T_{ij}}}}{{\sum\nolimits_{{k=1}}^{n} {{T_{ij}}} }}}}} } } \right)$$
(33)
to derive the overall Pythagorean fuzzy preference numbers \({\bar {\gamma }_i}\), \(i=1,2,\ldots,m\), of the alternatives \({X_i}\) (\(i=1,2, \ldots ,m\)).
Step 5. Calculate the score value as follows:
$${S^ * }({\bar {\gamma }_i})=\frac{{1+\mu _{{{{\bar {\gamma }}_i}}}^{2} - \upsilon _{{{{\bar {\gamma }}_i}}}^{2}}}{2},\;(i=1,2, \ldots ,m)$$
(34)

Step 6. Rank all the alternative \({X_i}\) (\(i=1,2, \ldots ,m\)), according to the score values \({S^ * }({\bar {\gamma }_i})\) (\(i=1,2, \ldots ,m\)), in descending order. The greater\({X_i}\), with the highest value of \({S^ * }({\bar {\gamma }_i})\), is the best alternative.

4 Illustrative example

The building of teacher’s body can be promoted by hard work for strengthening academic education. It is the program of Hazara University Mathematics Department to introduce excellent and remarkable oversea teachers. This introduction has outstretched great attention from the Department and a panel of three decision markers, Vice Chancellor of the University \({D_1}\), dean of sciences \({D_2}\), and human resource officer \({D_3}\) sets that will take the whole responsibility for this introduction. They made strict evaluation for four candidates (alternatives) \({X_i}\) (\(i=1,2,3,4\)) from four aspects (attributes), namely morality \({A_1}\) research capability \({A_2}\) teaching skill \({A_3}\) and educational background \({A_4}\). The Vice Chancellor of University has the absolute priority for decision making; dean of sciences comes next. Besides, this introduction will be in strict accordance with the principle of combined ability with political integrity. The prioritization relationship for the criteria is as; \({A_1} \succ {A_2} \succ {A_3} \succ {A_4}\). Three decision makers evaluated the candidates \({X_i}\) (\(i=1,2,3,4\)) with respect to the attributes \({A_j}\) (\(i=1,2,3,4\)) and construct the following three Pythagorean fuzzy decision matrix \({M^{\left( k \right)}}={[\bar {\alpha }_{{ij}}^{{(k)}}]_{m \times n}}\) (see Tables 1, 2, 3). Since all the attributes are of benefit types, therefore, no need for normalization and \({M^{\left( k \right)}}={M^{ * \left( k \right)}}={[\bar {\alpha }_{{ij}}^{{(k)}}]_{m \times n}}={[\bar {\gamma }_{{ij}}^{{(k)}}]_{m \times n}}\).

Table 1

Pythagorean fuzzy decision matrix \({M^{(1)}}={[{\bar {\gamma }^{(1)}}_{{ij}}]_{4 \times 4}}\)

 

\({A_1}\)

\({A_2}\)

\({A_3}\)

\({A_4}\)

\({X_1}\)

\((0.9,0.4)\)

\((0.7,0.6)\)

\(\,(0.8,0.5)\)

\((0.4,0.7)\)

\({X_2}\)

\(\,\,(0.6,0.5)\)

\((0.3,0.9)\)

\((0.7,0.4)\)

\((0.8,0.4)\)

\({X_3}\)

\((0.5,0.8)\)

\((0.8,0.4)\)

\((0.9,0.3)\)

\((0.8,0.5)\)

\({X_4}\)

\((0.7,0.6)\)

\((0.6,0.5)\)

\(\,(0.6,0.8)\)

\((0.9,0.4)\)

Table 2

Pythagorean fuzzy decision matrix \({M^{(2)}}={[{\bar {\gamma }^{(2)}}_{{ij}}]_{4 \times 4}}\)

 

\({A_1}\)

\({A_2}\)

\({A_3}\)

\({A_4}\)

\({X_1}\)

\((0.7,0.6)\)

\(\,(0.4,0.9)\)

\((0.8,0.4)\)

\((0.6,0.5)\)

\({X_2}\)

\((0.5,0.8)\)

\((0.7,0.6)\)

\((0.9,0.3)\)

\((0.8,0.6)\)

\({X_3}\)

\((0.9,0.4)\)

\(\,(0.8,0.5)\)

\((0.4,0.7)\)

\(\,(0.7,0.5)\)

\({X_4}\)

\((0.8,0.3)\)

\((0.7,0.4)\)

\(\,(0.8,0.5)\)

\((0.3,0.9)\)

Table 3

Pythagorean fuzzy decision matrix \({M^{(3)}}={[{\bar {\gamma }^{(3)}}_{{ij}}]_{4 \times 4}}\)

 

\({A_1}\)

\({A_2}\)

\({A_3}\)

\({A_4}\)

\({X_1}\)

\((0.9,0.2)\)

\((0.8,0.5)\)

\((0.7,0.6)\)

\((0.3,0.9)\)

\({X_2}\)

\((0.8,0.6)\)

\((0.9,0.4)\)

\((0.5,0.8)\)

\((0.6,0.5)\)

\({X_3}\)

\((0.7,0.5)\)

\((0.4,0.7)\)

\((0.9,0.3)\)

\((0.8,0.4)\)

\({X_4}\)

\((0.4,0.9)\)

\((0.7,0.6)\)

\((0.8,0.4)\)

\((0.9,0.2)\)

Based on PFWPWA operator, the main steps are as follows.

Step 1. Compute the values of \(T_{{ij}}^{{(1)}},T_{{ij}}^{{(2)}},T_{{ij}}^{{(3)}}\) by utilizing Eqs. (26) and (27).
$$T_{{ij}}^{{(1)}}=\left[ {\begin{array}{*{20}{c}} 1&1&1&1 \\ 1&1&1&1 \\ 1&1&1&1 \\ 1&1&1&1 \end{array}} \right]\;\;T_{{ij}}^{{(2)}}=\left[ {\begin{array}{*{20}{c}} {0.825}&{0.565}&{0.695}&{0.335} \\ {0.555}&{0.140}&{0.665}&{0.740} \\ {0.305}&{0.740}&{0.860}&{0.695} \\ {0.565}&{0.555}&{0.360}&{0.825} \end{array}} \right]\;T_{{ij}}^{{(3)}}=\left[ {\begin{array}{*{20}{c}} {0.466}&{0.099}&{0.514}&{0.186} \\ {0.169}&{0.079}&{0.572}&{0.474} \\ {0.252}&{0.514}&{0.288}&{0.431} \\ {0.438}&{0.369}&{0.250}&{0.115} \end{array}} \right]$$

Step 2. Utilize the PFPWA operator (Eq. (28)) to aggregate all the individual matrices \({M^{(k)}}={[\bar {\gamma }_{{ij}}^{{(k)}}]_{m \times n}}\) (k = 1, 2, 3) into collective Pythagorean fuzzy decision matrix \(M={[{\bar {\gamma }_{ij}}]_{m \times n}}\) (see Table 4).

Table 4

Collective PFPWA decision matrix \(M={[{\bar {\gamma }_{ij}}]_{4 \times 4}}\)

 

\({A_1}\)

\({A_2}\)

\({A_3}\)

\({A_4}\)

\({X_1}\)

\((0.8537,0.4020)\)

\((0.6389,0.7408)\)

\((0.7808,0.4863)\)

\((0.4484,0.6703)\)

\({X_2}\)

\((0.6028,0.5922)\)

\((0.4804,0.8151)\)

\((0.7618,0.4384)\)

\((0.7699,0.4805)\)

\({X_3}\)

\((0.6794,0.6473)\)

\((0.7505,0.4890)\)

\((0.8096,0.4212)\)

\((0.7724,0.4779)\)

\({X_4}\)

\((0.6960,0.5392)\)

\((0.6528,0.4855)\)

\((0.6967,0.6467)\)

\((0.7938,0.5420)\)

Step 3. Compute the value of \({T_{ij}}(i=1,2,3,\ldots,m;j=1,2,3,\ldots,n)\) based on Eqs. (30) and (31)
$${T_{ij}}=\left[ {\begin{array}{*{20}{c}} 1&{0.7836}&{0.3367}&{0.2312} \\ 1&{0.4722}&{0.1337}&{0.0928} \\ 1&{0.5213}&{0.3452}&{0.2552} \\ 1&{0.5968}&{0.3552}&{0.1895} \end{array}} \right]$$
Step 4. Utilize the PFPWA operator (Eq. 32) to aggregate all the preference values \({\bar {\gamma }_{ij}}(i=1,2,3,4)\) in the ith line of M and get the overall preference values \({\bar {\gamma }_i}\).
$${\bar {\gamma }_1}=(0.7668,0.5326),\;{\bar {\gamma }_2}=(0.6052,0.6349),\;{\bar {\gamma }_3}=(0.7353,0.5432),\;{\bar {\gamma }_4}=(0.6960,0.5400).$$
Step 5. Compute the score of \({\bar {\gamma }_i} (i=1,2,3,4)\), respectively:
$${S^ * }({\bar {\gamma }_1})=0.6522,\;{S^ * }({\bar {\gamma }_2})=0.4816,\;{S^ * }({\bar {\gamma }_3})=0.6228,\;{S^ * }({\bar {\gamma }_4})=0.5964.$$

Since, \({S^ * }({\bar {\gamma }_1}) \succ {S^ * }({\bar {\gamma }_3}) \succ {S^ * }({\bar {\gamma }_4}) \succ {S^ * }({\bar {\gamma }_2})\).

Therefore, \({X_1} \succ {X_3} \succ {X_4} \succ {X_2}\). Thus, optimal alternative is \({X_1}\).

Based on the PFPWG operator, the main steps are as follows.

Step 1′. See Step 1.

Step 2′. Utilize the PFPWG operator (Eq. 29) to aggregate all the individual matrices \({M^{(k)}}={[\bar {\gamma }_{{ij}}^{{(k)}}]_{m \times n}}\) (k = 1, 2, 3) into collective Pythagorean fuzzy decision matrix \(M={[{\bar {\gamma }_{ij}}]_{m \times n}}\) (see Table 5).

Table 5

Collective PFPWG decision matrix \(M={[{\bar {\gamma }_{ij}}]_{4 \times 4}}\)

 

\({A_1}\)

\({A_2}\)

\({A_3}\)

\({A_4}\)

\({X_1}\)

\(\,(0.8221,0.4663)\)

\((0.5835,0.7346)\)

\((0.7755,0.5009)\)

\((0.4223,0.7127)\)

\({X_2}\)

\((0.5820,0.6457)\)

\((0.3551,0.8715)\)

\((0.6921,0.5544)\)

\((0.7522,0.5014)\)

\({X_3}\)

\((0.5924,0.7221)\)

\((0.6830,0.5270)\)

\((0.6505,0.5275)\)

\((0.7658,0.4823)\)

\({X_4}\)

\((0.6432,0.6768)\)

\((0.6461,0.4983)\)

\((0.6691,0.7184)\)

\((0.5641,0.7416)\)

Step 3′. Compute the value of\({T_{ij}}\)(\(i=1,2,3,\ldots,m;\,\, j=1,2,3,\ldots,n\)) based on Eqs. (30) and (31)
$${T_{ij}}=\left[ {\begin{array}{*{20}{c}} 1&{0.7292}&{0.2920}&{0.1972} \\ 1&{0.4609}&{0.0845}&{0.0495} \\ 1&{0.4148}&{0.2466}&{0.1412} \\ 1&{0.4778}&{0.2793}&{0.1301} \end{array}} \right]$$
Step 4′. Utilize the PFPWG operator (Eq. 33) to aggregate all the preference values \({\bar {\gamma }_{ij}}\)(\(i=1,2,3,4\)) in the ith line of M and get the overall preference values \({\bar {\gamma }_i}\).
$${\bar {\gamma }_1}=(0.6870,0.6106),\;{\bar {\gamma }_2}=(0.5133,0.7355),\;{\bar {\gamma }_3}=(0.6326,0.6500),\;{\bar {\gamma }_4}=(0.6419,0.6545).$$
Step 5′. Compute the score of \({\bar {\gamma }_i}\) (\(i=1,2,3,4\)) respectively:
$${S^ * }({\bar {\gamma }_1})=0.5496,\;{S^ * }({\bar {\gamma }_2})=0.3613,\;{S^ * }({\bar {\gamma }_1})=0.4888,\;{S^ * }({\bar {\gamma }_4})=0.4918.$$

Since, \({S^ * }({\bar {\gamma }_1}) \succ {S^ * }({\bar {\gamma }_4}) \succ {S^ * }({\bar {\gamma }_3}) \succ {S^ * }({\bar {\gamma }_2})\).

Therefore, \({X_1} \succ {X_4} \succ {X_3} \succ {X_2}\). Thus, optimal alternative is \({X_1}\).

The optimum decision has changed the sort result by the PFPWG operator is different at \({X_3}\) and \({X_4}\) from that by the PFPWA operator. The main reason is that PFPWA operator focuses on the impact of overall data while the PFPWG operator highlights the role of individual data.

4.1 Comparison analysis

If the attributes or the decision makers are at the same priority level, then the developed Pythagorean fuzzy prioritized weighted average (PFPWA) operator and Pythagorean fuzzy prioritized weighted geometric (PFPWG) operator reduced into the Pythagorean fuzzy weighted average (PFWA) operator and Pythagorean fuzzy weighted geometric (PFWG) operator. If we consider the weight of attributes as \(w={\left( {0.25,0.3,0.28,0.17} \right)^{\text{T}}}\), and apply the PFWA/ PFWG operator (Yager 2014) we get the score values of the alternatives as:

\(S({\bar {\gamma }_1})={\text{0.2323,}}\,\,\,S({\bar {\gamma }_2})={\text{0.0161,}}\,\,\,S({\bar {\gamma }_3})={\text{0.3704,}}\,S({\bar {\gamma }_4})={\text{0.2438}}\), and \(S({\bar {\gamma }_1})={\text{0.0071,}}\,\,\,S({\bar {\gamma }_2})= - 0.2471,\,\,\,S({\bar {\gamma }_3})={\text{0.1563,}}\,S({\bar {\gamma }_4})={\text{0.0108}}\), respectively.

Since, \(S({\bar {\gamma }_3}) \succ S({\bar {\gamma }_4}) \succ S({\bar {\gamma }_1}) \succ S({\bar {\gamma }_2})\) and \(\,S({\bar {\gamma }_3}) \succ S({\bar {\gamma }_4}) \succ S({\bar {\gamma }_1}) \succ S({\bar {\gamma }_2})\).

Therefore, in both the cases: \({X_3} \succ {X_4} \succ {X_1} \succ {X_2}\).

However, the priority levels among the decision makers and the attributes are different. For example, the candidate when received poor evaluation form the Vice Chancellor of the University is very hard to select. From another point of view, it is impossible for the candidate to be selected, who has bad moral character, no matter how good performance he has received on research capabilities, teaching skill and education background.

Thus, we must consider the prioritization among the attributes and the decision makers. To deal with the situations, the PFPWA operator and the PFPWG operator are very useful tools.

From the above analysis, the main advantages of our developed Pythagorean fuzzy prioritized aggregation operators is that, it not only accommodate the Pythagorean fuzzy environment but also more feasible and practical when considering the prioritization among the attributes and the decision makers as compared with the traditional Pythagorean fuzzy aggregation operators. Also our proposed Pythagorean fuzzy prioritized aggregation operators are more flexible than the intuitionistic fuzzy prioritized aggregation operators as in the Pythagorean fuzzy environment the decision makers deals with the situations where the degree of membership and non-membership of particular attributes are such that its sum is greater than 1.

5 Conclusions

Since in decision-making aggregation operators play a vital role, therefore, in this paper we introduced some aggregation operators based on the idea of prioritized average, namely the Pythagorean fuzzy prioritized weighted average (PFPWA) operator and Pythagorean fuzzy prioritized weighted geometric (PFPWG) operator. The important achievement of the developed Pythagorean fuzzy prioritized aggregation operators is that, it considers prioritization among the attributes or the decision makers. We discussed some basic properties of the developed operators namely idempotency, boundary and monotonicity. Furthermore, we applied the proposed aggregation operators to multi-attribute decision making under Pythagorean fuzzy environment. Finally, to verify and demonstrate the practicality and effectiveness of the developed operators an illustrative example was given. The developed multi-criteria group decision making process studies prioritization relationship among these attributes and decision makers, which allows our method to have wider practical application possibilities.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Muhammad Sajjad Ali Khan
    • 1
  • Saleem Abdullah
    • 2
  • Asad Ali
    • 1
  • Fazli Amin
    • 1
  1. 1.Department of MathematicsHazara University MansehraMansehraPakistan
  2. 2.Department of MathematicsAbdul Wali Khan University MadranMadranPakistan

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