Pythagorean 2-tuple linguistic power aggregation operators in multiple attribute decision making

Abstract

In this paper, we investigate the multiple attribute decision making problems with Pythagorean 2-tuple linguistic information. Then, we utilize power average and power geometric operations to develop some Pythagorean 2-tuple linguistic power aggregation operators: Pythagorean 2-tuple linguistic power weighted average (P2TLPWA) operator, Pythagorean 2-tuple linguistic power weighted geometric (P2TLPWG) operator, Pythagorean 2-tuple linguistic power ordered weighted average (P2TLPOWA) operator, Pythagorean 2-tuple linguistic power ordered weighted geometric (P2TLPOWG) operator, Pythagorean 2-tuple linguistic power hybrid average (P2TLPHA) operator and Pythagorean 2-tuple linguistic power hybrid geometric (P2TLPHG) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the Pythagorean 2-tuple linguistic multiple attribute decision making problems. Finally, a practical example for enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords:

• Multiple attribute decision making
• Pythagorean 2-tuple linguistic set
• Pythagorean 2-tuple linguistic power weighted average (P2TLPWA) operator
• Pythagorean 2-tuple linguistic power weighted geometric (P2TLPWG) operator
• enterprise resource planning (ERP) system selection

JELS CLASSIFICATION:

• C43
• C61
• D81

1. Introduction

Multiple attribute decision making problems under linguistic information processing environment is an interesting research topic having received more and more attention during the last several years. One of the well-known linguistic information processing models are the 2-tuple linguistic computational model (Beg & Rashid, 2015; Dutta & Guha, 2015; Herrera, Herrera-Viedma 2000a, 2000b; Herrera, Martı́nez, & Sánchez, 2005; Herrera and Martínez 2001b; Martínez-López, Rodríguez, & Herrera, 2015; Lin, Wei, Wang, & Zhao, 2014; Wu et al., 2015; Zhang & Liu, 2010; Zavadskas & Turskis, 2011; Zavadskas, Turskis, & Kildienė, 2014; Zhang & Chu, 2009). Herrera and Martínez (2001a) show 2-tuple linguistic information processing manner can effectively avoid the loss and distortion of information. Herrera, Herrera-Viedma (2000a) developed 2-tuple arithmetic average (TAA) operator, 2-tuple weighted average (TWA) operator, 2-tuple ordered weighted average (TOWA) operator and extended 2-tuple weighted average (ET-WA) operator. Herrera et al. (2005) presented the group decision making model for managing non-homogeneous information processing. Herrera-Viedma, Martinez, Mata, and Chiclana (2005) developed the consensus support system with multi-granular linguistic preference relations. Liao, Li, and Lu (2007) used linguistic information processing model for selecting an ERP system. Herrera, Herrera-Viedma, and Martínez (2008) proposed a fuzzy linguistic methodology to deal with unbalanced linguistic term sets. Wang (2009) presented a 2-tuple fuzzy linguistic evaluation model for selecting appropriate agile manufacturing system. Tai and Chen (2009) developed the intellectual capital evaluation model linguistic variable. Fan, Feng, Sun, and Ou (2009) evaluated knowledge management capability of organizations by using a fuzzy linguistic method. Wei extended TOPSIS method to multiple attribute group decision making with 2-tuple linguistic information. Wei proposed ET-WG and ET-OWG operators for multiple attribute group decision making with 2-tuple linguistic information. Fan and Liu (2010) developed the multi-granularity uncertain linguistic group decision making model. Chang and Wen (2010) developed a novel efficient approach for DFMEA combining 2-tuple and the OWA operator. Jiang and Wei (2014) proposed some Bonferroni mean operators with 2-tuple linguistic information. Xu, Ma, Tao, and Wang (2014) developed some methods to deal with unacceptable incomplete 2-tuple fuzzy linguistic preference relations in group decision making. Liu, Lin, and Wu (2014) proposed the dependent interval 2-tuple linguistic aggregation operators for multiple attribute group decision making. Dutta, Guha, and Mesiar (2015) developed a model based on linguistic 2-tuples for dealing with heterogeneous relationship among attributes in multi-expert decision making. Dong & Herrera-Viedma, (2015) proposed the consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets and its use in the linguistic GDM with preference relation. Wang, Wang, Zhang, and Chen (2015) developed the multi-criteria group decision making method based on interval 2-tuple linguistic information and Choquet integral aggregation operators. Qin & Liu, (2016) proposed the 2-tuple linguistic Muirhead mean operators for multiple attribute group decision making and its application to supplier selection. Zhang, Xu, and Wang (2016) developed the consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information.

More recently, Pythagorean fuzzy set (PFS) (Yager, 2013, 2014) has emerged as an effective tool for depicting uncertainty of the MADM problems. The PFS is also characterized by the membership degree and the non-membership degree, whose sum of squares is less than or equal to 1, the PFS is more general than the IFS. In some cases, the PFS can solve the problems that the IFS cannot, for example, if a DM gives the membership degree and the non-membership degree as 0.8 and 0.6, respectively, then it is only valid for the PFS. In other words, all the intuitionistic fuzzy degrees are a part of the Pythagorean fuzzy degrees, which indicates that the PFS is more powerful to handle the uncertain problems. Zhang and Xu (2014) provided the detailed mathematical expression for PFS and introduced the concept of Pythagorean fuzzy number(PFN). Meanwhile, they also developed a Pythagorean fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) for handling the MCDM problem within PFNs. Peng and Yang (2015) proposed the division and subtraction operations for PFNs, and also developed a Pythagorean fuzzy superiority and inferiority ranking method to solve multicriteria group decision making problem with PFNs. Afterwards, Focused on how the notion of “averaging” should be treated in the case of PFNs and how to ensure that the averaging aggregation functions produce outputs consistent with the case of ordinary fuzzy numbers. Reformat & Yager, (2014) applied the PFNs in handling the collaborative-based recommender system. Gou, Xu, and Ren (2016) investigate the Properties of Continuous Pythagorean Fuzzy Information. Ren, Xu, and Gou (2016) proposed the Pythagorean fuzzy TODIM approach to multi-criteria decision making. Garg (2016a) proposed the new generalized Pythagorean fuzzy information aggregation by using Einstein Operations. Zeng, Chen, and Li (2016) developed a hybrid method for Pythagorean fuzzy multiple-criteria decision making. Garg (2016b) studied a novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem.

Although, Pythagorean fuzzy set theory has been successfully applied in some areas, the PFS is also characterized by the membership degree and the non-membership degree, whose sum of squares is less than or equal to 1, the PFS is more general than the IFS. In some cases, the PFS can solve the problems that the IFS cannot, for example, if a DM gives the membership degree and the non-membership degree as 0.8 and 0.6, respectively, then it is only valid for the PFS. In other words, all the intuitionistic fuzzy degrees are a part of the Pythagorean fuzzy degrees, which indicates that the PFS is more powerful to handle the uncertain problems. In order to describe the membership degree and the non-membership degree of an element to a linguistic label, which can reflect the decision maker’s confidence level when they are making an evaluation, Wei et al.(2017) proposed the concept of Pythagorean 2-tuple linguistic sets(P2TLSs) and some Pythagorean 2-tuple linguistic information aggregating operators to solve this problem based on the Pythagorean fuzzy sets (Yager, 2013-2014) and 2-tuple linguistic information processing model.

From above analysis, we can see that most of the existing Pythagorean 2-tuple linguistic aggregation operators are based on the algebraic product and algebraic sum of P2TLSs to carry the aggregation process. However, all these aggregation operators do not take into account the information about the relationship between the values being fused. To overcome this drawback, motivated by the idea of power average (Yager, 2001) and power geometric operations (Xu & Yager, 2010), in this paper, we develop a series of Pythagorean 2-tuple linguistic power aggregation operators, whose weighting vectors depend upon the input arguments and allow values being aggregated to support and reinforce each other, and study their desirable properties. To do so, the remainder of this paper is set out as follows. In the next section, we shall propose the concept of Pythagorean 2-tuple linguistic set on the basis of the Pythagorean fuzzy set and 2-tuple linguistic information processing model. In Section 3, we shall propose some Pythagorean 2-tuple linguistic power aggregation operators. In Section 4, we shall present we shall propose some Pythagorean 2-tuple linguistic power geometric aggregation operators. In Section 5, based on these operators, we shall present some approaches to multiple attribute decision making with Pythagorean 2-tuple linguistic information. In Section 6, we shall present a numerical example for enterprise resource planning (ERP) system selection with Pythagorean 2-tuple linguistic information in order to illustrate the method proposed in this paper. Section 7 concludes the paper with some remarks.

2. Preliminaries

In the following, we introduced some basic concepts related to 2-tuple linguistic term sets and Pythagorean fuzzy sets.

2.1. 2-tuple linguistic term sets

Let $S=\{s_i|i=1,2,\dots ,t\}$ be a linguistic term set with odd cardinality. Any label,$s_i$ represents a possible value for a linguistic variable, and it should satisfy the following characteristics (Herrera & Martínez, 2000a, 2000b; Herrera et al., 2005; Herrera & Martínez, 2001b; Xu, 2004, 2006):

1. The set is ordered:$s_i>s_j,$ if $i>j;$ (2) Max operator:$\max (s_i,s_j)=s_i,$ if $s_i\ge s_j;$ (3) Min operator: $\min (s_i,s_j)=s_i,$ if $s_i\le s_j.$ For example, S can be defined as $$\begin{array}{c}S=\{s_1=\mathit{extremely}\mathit{poor},s_2=\mathit{very}\mathit{poor},s_3=\mathit{poor},s_4=\mathit{medium},\\ s_5=\mathit{good},s_6=\mathit{very}\mathit{good},s_7=\mathit{extremely}\mathit{good}\}\end{array}$$

Herrera and Martínez (2000a, 2000b) developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple $(s_i,{\alpha }_i),$ where $s_i$ is a linguistic label from predefined linguistic term set $S$ and ${\alpha }_i$ is the value of symbolic translation, and ${\alpha }_i\in [-0.5,0.5).$

Definition 1.

Let $\beta$ be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set $S,$ i.e., the result of a symbolic aggregation operation, $\beta \in \left[1,t\right],$ being $t$ the cardinality of $S.$ Let $i=\mathit{round}(\beta )$ and $\alpha =\beta -i$ be two values, such that, $i\in \left[1,t\right]$ and $\alpha \in [-0.5,0.5)$ then $\alpha$ is called a symbolic translation (Herrera & Martínez, 2000a, 2000b).

Definition 2.

Let $S=\left\{s_1,s_2,\dots ,s_t\right\}$ be a linguistic term set and $\beta \in \left[1,t\right]$ is a number value representing the aggregation result of linguistic symbolic. Then the function $\Delta$ used to obtain the 2-tuple linguistic information equivalent to $\beta$ is defined as: (1) $$\Delta :\left[1,t\right]\to S\times [-0.5,0.5),$$(1) (2) $$\Delta \left(\beta \right)=\{\begin{array}{l}{s}_i,i=\mathit{round}\left(\beta \right)\\ \alpha =\beta -i,\alpha \in [-0.5,0.5),\end{array}$$(2) where round(.) is the usual round operation, $s_i$ has the closest index label to $\beta$ and $\alpha$ is the value of the symbolic translation (Herrera & Martínez, 2000a, 2000b).

Definition 3.

Let $S=\left\{s_1,s_2,\dots ,s_t\right\}$ be a linguistic term set and $(s_i,{\alpha }_i)$ be a 2-tuple. There is always a function ${\Delta }^{-1}$ can be defined, such that, from a 2-tuple $(s_i,{\alpha }_i)$ it return its equivalent numerical value $\beta \in \left[1,t\right]\subset R,$ which is (Herrera & Martínez, 2000a, 2000b). (3) $${\Delta }^{-1}:S\times [-0.5,0.5)\to \left[1,t\right],$$(3) (4) $${\Delta }^{-1}\left(s_i,\alpha \right)=i+\alpha =\beta ,$$(4)

From Definitions 1 and 2, we can conclude that the conversion of a linguistic term into a linguistic 2-tuple consists of adding a value 0 as symbolic translation: (5) $$\Delta \left(s_i\right)=\left(s_i,0\right),$$(5)

2.2. Pythagorean fuzzy set

Yager, (2014) developed the concept of the Pythagorean fuzzy sets.

Definition 4.

(Yager, 2014). Let $X$ be a fix set. A PFS is an object having the form (6) $$P=\left\{〈x,\left({\mu }_P\left(x\right),{\nu }_P\left(x\right)\right)〉|x\in X\right\}$$(6) where the function ${\mu }_P:X\to \left[0,1\right]$ defines the degree of membership and the function ${\nu }_P:X\to \left[0,1\right]$ defines the degree of non-membership of the element $x\in X$ to $P,$ respectively, and, for every $x\in X,$ it holds that (7) $${\left({\mu }_p\left(x\right)\right)}^2+{\left({\nu }_p\left(x\right)\right)}^2\le 1.$$(7)

Definition 5.

Let $\tilde{a}=(\mu ,\nu )$ be a Pythagorean fuzzy number, a score function $S$ of a Pythagorean fuzzy number can be represented as follows: (8) $$S\left(\tilde{a}\right)=\frac{1}{2}\left(1+{\mu }^2-{\nu }^2\right),S\left(\tilde{a}\right)\in \left[0,1\right].$$(8)

Definition 6.

Let $\tilde{a}=(\mu ,\nu )$ be a Pythagorean fuzzy number, an accuracy function $H$ of a Pythagorean fuzzy value can be represented as follows: (9) $$H\left(\tilde{a}\right)={\mu }^2+{\nu }^2,H\left(\tilde{a}\right)\in \left[0,1\right].$$(9) to evaluate the degree of accuracy of the Pythagorean fuzzy number $\tilde{a}=(\mu ,\nu ),$ where $H(\tilde{a})\in \left[0,1\right].$ The larger the value of $H(\tilde{a}),$ the more the degree of accuracy of the Pythagorean fuzzy number $\tilde{a}.$

Based on the score function S and the accuracy function $H,$ in the following, we shall give an order relation between two Pythagorean fuzzy numbers, which is defined as follows:

Definition 7.

Let ${\tilde{a}}_1=({\mu }_1,{\nu }_1)$ and ${\tilde{a}}_2=({\mu }_2,{\nu }_2)$ be two Pythagorean fuzzy numbers, $s({\tilde{a}}_1)=\frac{1}{2}(1+{({\mu }_1)}^2-{({\nu }_1)}^2)$ and $s({\tilde{a}}_2)=\frac{1}{2}(1+{({\mu }_2)}^2-{({\nu }_2)}^2)$ be the scores of $\tilde{a}$ and $\tilde{b},$ respectively, and let $H({\tilde{a}}_1)={({\mu }_1)}^2+{({\nu }_1)}^2$ and $H({\tilde{a}}_2)={({\mu }_2)}^2+{({\nu }_2)}^2$ be the accuracy degrees of $\tilde{a}$ and $\tilde{b},$ respectively, then if $S(\tilde{a})<S(\tilde{b}),$ then $\tilde{a}$ is smaller than $\tilde{b},$ denoted by $\tilde{a}<\tilde{b};$ if $S(\tilde{a})=S(\tilde{b}),$ then

1. if $H(\tilde{a})=H(\tilde{b}),$ then $\tilde{a}$ and $\tilde{b}$ represent the same information, denoted by $\tilde{a}=\tilde{b};$ (2)(2) $$\Delta \left(\beta \right)=\{\begin{array}{l}{s}_i,i=\mathit{round}\left(\beta \right)\\ \alpha =\beta -i,\alpha \in [-0.5,0.5),\end{array}$$(2) if $H(\tilde{a})<H(\tilde{b}),$ $\tilde{a}$ is smaller than $\tilde{b},$ denoted by $\tilde{a}<\tilde{b}.$

Definition 8.

(Reformat & Yager, 2014). Let ${\tilde{a}}_1=({\mu }_1,{\nu }_1),$ ${\tilde{a}}_2=({\mu }_2,{\nu }_2),$ and $\tilde{a}=(\mu ,\nu )$ be three Pythagorean fuzzy numbers, and some basic operations on them are defined as follows:

1. ${\tilde{a}}_1\oplus {\tilde{a}}_2=\left(\sqrt{{({\mu }_1)}^2+{({\mu }_2)}^2-{({\mu }_1)}^2{({\mu }_2)}^2},{\nu }_1{\nu }_2\right);$

2. ${\tilde{a}}_1\otimes {\tilde{a}}_2=\left({\mu }_1{\mu }_2,\sqrt{{({\nu }_1)}^2+{({\nu }_2)}^2-{({\nu }_1)}^2{({\nu }_2)}^2}\right);$

3. $\lambda \tilde{a}=(\sqrt{1-{(1-{\mu }^2)}^{\lambda }},{\nu }^{\lambda }),\lambda >0;$

4. ${(\tilde{a})}^{\lambda }=({\mu }^{\lambda },\sqrt{1-{(1-{\nu }^2)}^{\lambda }}),\lambda >0;$

5. ${\tilde{a}}^c=(\nu ,\mu ).$

Based on the Definition 6, we can derive the following properties easily.

Theorem 1.

(Reformat & Yager, 2014). Let ${\tilde{a}}_1=({\mu }_1,{\nu }_1)$ and ${\tilde{a}}_2=({\mu }_2,{\nu }_2)$ be two Pythagorean fuzzy numbers, $\lambda ,{\lambda }_1,{\lambda }_2>0,$ then

1. ${\tilde{a}}_1\oplus {\tilde{a}}_2={\tilde{a}}_2\oplus {\tilde{a}}_1;$

2. ${\tilde{a}}_1\otimes {\tilde{a}}_2={\tilde{a}}_2\otimes {\tilde{a}}_1;$

3. $\lambda ({\tilde{a}}_1\oplus {\tilde{a}}_2)=\lambda {\tilde{a}}_1\oplus \lambda {\tilde{a}}_2;$

4. ${({\tilde{a}}_1\otimes {\tilde{a}}_2)}^{\lambda }={({\tilde{a}}_1)}^{\lambda }\otimes {({\tilde{a}}_2)}^{\lambda };$

5. ${\lambda }_1{\tilde{a}}_1\oplus {\lambda }_2{\tilde{a}}_1=({\lambda }_1+{\lambda }_2){\tilde{a}}_1;$

6. ${({\tilde{a}}_1)}^{{\lambda }_1}\otimes {({\tilde{a}}_1)}^{{\lambda }_2}={({\tilde{a}}_1)}^{({\lambda }_1+{\lambda }_2)};$

7. ${({({\tilde{a}}_1)}^{{\lambda }_1})}^{{\lambda }_2}={({\tilde{a}}_1)}^{{\lambda }_1{\lambda }_2}.$

2.3. Pythagorean 2-tuple linguistic sets

In the following, Wei et al. proposed the concepts and basic operations of the Pythagorean 2-tuple linguistic sets on the basis of the Pythagorean fuzzy sets (Yager, 2013-2014) and 2-tuple linguistic information processing model (Herrera & Martínez, 2000a; Herrera & Martínez, 2000b).

Definition 9.

A Pythagorean 2-tuple linguistic sets $A$ in $X$ is given (10) $$P=\left\{\left(s_{\theta (x)},\rho \right),\left({\mu }_P\left(x\right),{\nu }_P\left(x\right)\right),x\in X\right\}$$(10) where $s_{\theta (a)}\in S,$ $\rho \in [-0.5,0.5),$$u_P(x)\in [0,1]$ and $v_P(x)\in [0,1],$ with the condition $0\le {(u_P(x))}^2+{(v_P(x))}^2\le 1,$$\forall x\in X.$ The numbers ${\mu }_P(x),{\nu }_P(x)$ represent, respectively, the degree of positive membership, degree of negative membership and degree of negative membership of the element $x$ to linguistic variable $(s_{\theta (x)},\rho ).$ Then for $x\in X,$ ${\pi }_P(x)=\sqrt{1-({(u_P(x))}^2+{(v_P(x))}^2)}$ could be called the degree of refusal membership of the element $x$ to linguistic variable $(s_{\theta (x)},\rho ).$

For convenience, we call $\tilde{p}=〈(s_p,\rho ),(u_p,v_p)〉$ a Pythagorean 2-tuple linguistic number (P2TLN), where ${\mu }_p\in \left[0,1\right],{\nu }_p\in \left[0,1\right],$${({\mu }_p)}^2+{({\nu }_p)}^2\le 1,$ $s_{\theta (p)}\in S$ and $\rho \in [-0.5,0.5).$

Definition 10.

Let $\tilde{p}=〈(s_p,\rho ),(u_p,v_p)〉$ be a Pythagorean 2-tuple linguistic number (P2TLN), a score function $\tilde{a}$ of a Pythagorean 2-tuple linguistic number can be represented as follows: (11) $$S\left(\tilde{p}\right)=\Delta \left({\Delta }^{-1}\left(s_{\theta (p)},\rho \right)\cdot \frac{1+{\left({\mu }_p\right)}^2-{\left({\nu }_p\right)}^2}{2}\right),{\Delta }^{-1}\left(S\left(\tilde{p}\right)\right)\in \left[1,t\right].$$(11)

Definition 11.

Let $\tilde{p}=〈(s_p,\rho ),(u_p,v_p)〉$ a Pythagorean 2-tuple linguistic number (P2TLN), an accuracy function $H$ of a Pythagorean 2-tuple linguistic number can be represented as follows: (12) $$H\left(\tilde{p}\right)=\Delta \left({\Delta }^{-1}\left(s_{\theta (p)},\rho \right)\cdot \frac{{\left({\mu }_p\right)}^2+{\left({\nu }_p\right)}^2}{2}\right),{\Delta }^{-1}\left(H\left(\tilde{p}\right)\right)\in \left[1,t\right].$$(12) to evaluate the degree of accuracy of the Pythagorean 2-tuple linguistic number $\tilde{p}=〈(s_p,\rho ),(u_p,v_p)〉,$ where ${\Delta }^{-1}(H(\tilde{p}))\in \left[1,t\right].$ The larger the value of $H(\tilde{p}),$ the more the degree of accuracy of the Pythagorean 2-tuple linguistic number $a.$

Based on the score function S and the accuracy function $H,$ in the following, Wei et al. gave an order relation between two Pythagorean 2-tuple linguistic numbers, which is defined as follows:

Definition 12.

Let ${\tilde{p}}_1=〈(s_{p_1},\rho ),(u_{p_1},v_{p_1})〉$ and ${\tilde{p}}_2=〈(s_{p_2},\rho ),(u_{p_2},v_{p_2})〉$ be two Pythagorean 2-tuple linguistic numbers, $S({\tilde{p}}_1)=\Delta ({\Delta }^{-1}(s_{\theta (p_1)},{\rho }_1)\cdot \frac{1+{({\mu }_{p_1})}^2-{({\nu }_{p_1})}^2}{2})$ and $S({\tilde{p}}_2)=\Delta ({\Delta }^{-1}(s_{\theta (p_2)},{\rho }_2)\cdot \frac{1+{({\mu }_{p_2})}^2-{({\nu }_{p_2})}^2}{2})$ be the scores of ${\tilde{a}}_1$ and ${\tilde{a}}_2,$ respectively, and let $H({\tilde{p}}_1)=\Delta ({\Delta }^{-1}(s_{\theta (p_1)},{\rho }_1)\cdot \frac{{({\mu }_{p_1})}^2+{({\nu }_{p_1})}^2}{2})$ and $H({\tilde{p}}_2)=\Delta ({\Delta }^{-1}(s_{\theta (p_2)},{\rho }_2)\cdot \frac{{({\mu }_{p_2})}^2+{({\nu }_{p_2})}^2}{2})$ be the accuracy degrees of ${\tilde{p}}_1$ and ${\tilde{p}}_2,$ respectively, then if $S({\tilde{p}}_1)<S({\tilde{p}}_2),$ then ${\tilde{p}}_1$ is smaller than ${\tilde{p}}_2,$ denoted by ${\tilde{p}}_1<{\tilde{p}}_2;$ if $S({\tilde{p}}_1)=S({\tilde{p}}_2),$ then

1. if $H({\tilde{p}}_1)=H({\tilde{p}}_2),$ then ${\tilde{p}}_1$ and ${\tilde{p}}_2$ represent the same information, denoted by ${\tilde{p}}_1={\tilde{p}}_2;$ (2)(2) $$\Delta \left(\beta \right)=\{\begin{array}{l}{s}_i,i=\mathit{round}\left(\beta \right)\\ \alpha =\beta -i,\alpha \in [-0.5,0.5),\end{array}$$(2) if $H({\tilde{p}}_1)<H({\tilde{p}}_2),$ ${\tilde{p}}_1$ is smaller than ${\tilde{p}}_2,$ denoted by ${\tilde{p}}_1<{\tilde{p}}_2.$

Motivated by the operations of the 2-tuple linguistic information(Herrera & Martínez, 2000a, 2000b) and Definition 5, in the following, Wei et al. defined some operational laws of Pythagorean 2-tuple linguistic numbers.

Definition 13.

Let ${\tilde{p}}_1=〈(s_{p_1},\rho ),(u_{p_1},v_{p_1})〉$ and ${\tilde{p}}_2=〈(s_{p_2},\rho ),(u_{p_2},v_{p_2})〉$ be two Pythagorean 2-tuple linguistic numbers, then $$\begin{array}{c}{\tilde{p}}_1\oplus {\tilde{p}}_2=〈\Delta \left({\Delta }^{-1}\left(s_{\theta (p_1)},{\rho }_1\right)+{\Delta }^{-1}\left(s_{\theta (p_2)},{\rho }_2\right)\right),\\ \left(\sqrt{{\left({\mu }_{p_1}\right)}^2+{\left({\mu }_{p_2}\right)}^2-{\left({\mu }_{p_1}\right)}^2{\left({\mu }_{p_2}\right)}^2},{\nu }_{p_1}{\nu }_{p_2}\right)〉;\\ {\tilde{p}}_1\otimes {\tilde{p}}_2=〈\Delta \left({\Delta }^{-1}\left(s_{\theta (p_1)},{\rho }_1\right)·{\Delta }^{-1}\left(s_{\theta (p_2)},{\rho }_2\right)\right),\\ \left({\mu }_{p_1}{\mu }_{p_2},\sqrt{{\left({\nu }_{p_1}\right)}^2+{\left({\nu }_{p_2}\right)}^2-{\left({\nu }_{p_1}\right)}^2{\left({\nu }_{p_2}\right)}^2}\right)〉;\\ \lambda {\tilde{p}}_1=〈\Delta \left(\lambda {\Delta }^{-1}\left(s_{\theta (p_1)},{\rho }_1\right)\right),\left(\sqrt{1-{\left(1-{\left({\mu }_{p_1}\right)}^2\right)}^{\lambda }},{\left({\nu }_{p_1}\right)}^{\lambda }\right)〉;\\ {\left({\tilde{p}}_1\right)}^{\lambda }=〈\Delta \left({\left({\Delta }^{-1}\left(s_{\theta (p_1)},{\rho }_1\right)\right)}^{\lambda }\right),\left({\left({\mu }_{p_1}\right)}^{\lambda },\sqrt{1-{\left(1-{\left({\nu }_{p_1}\right)}^2\right)}^{\lambda }}\right)〉;\end{array}$$

Based on the Definition 13, Wei et al. derived the following properties easily.

Theorem 2.

For any two Pythagorean 2-tuple linguistic numbers ${\tilde{p}}_1=〈(s_{p_1},\rho ),(u_{p_1},v_{p_1})〉$ and ${\tilde{p}}_2=〈(s_{p_2},\rho ),(u_{p_2},v_{p_2})〉,$ it can be proved the calculation rules shown as follows

1. ${\tilde{p}}_1\oplus {\tilde{p}}_2={\tilde{p}}_2\oplus {\tilde{p}}_1$

2. ${\tilde{p}}_1\otimes {\tilde{p}}_2={\tilde{p}}_2\otimes {\tilde{p}}_1$

3. $\lambda ({\tilde{p}}_1\oplus {\tilde{p}}_2)=\lambda {\tilde{p}}_1\oplus \lambda {\tilde{p}}_2,0\le \lambda \le 1$

4. ${\lambda }_1{\tilde{p}}_1\oplus {\lambda }_2{\tilde{p}}_1=({\lambda }_1\oplus {\lambda }_2){\tilde{p}}_1,0\le {\lambda }_1,{\lambda }_2,{\lambda }_1+{\lambda }_2\le 1$

5. ${\tilde{p}}_1{}^{{\lambda }_1}\otimes {\tilde{p}}_1{}^{{\lambda }_2}={({\tilde{p}}_1)}^{{\lambda }_1+{\lambda }_2},0\le {\lambda }_1,{\lambda }_2,{\lambda }_1+{\lambda }_2\le 1$

6. ${\tilde{p}}_1{}^{{\lambda }_1}\otimes {\tilde{p}}_2{}^{{\lambda }_1}={({\tilde{p}}_1\otimes {\tilde{p}}_2)}^{{\lambda }_1},{\lambda }_1\ge 0.$

7. ${({({\tilde{p}}_1)}^{{\lambda }_1})}^{{\lambda }_2}={({\tilde{p}}_1)}^{{\lambda }_1{\lambda }_2}.$

3. Pythagorean 2-tuple linguistic power aggregation operators

Yager, (2001) developed a nonlinear weighted average aggregation operator called power average (PA) operator, which can be defined as follows: (13) $$\mathit{PA}\left(a_1,a_2,\dots ,a_n\right)=\frac{\sum _{i=1}^n\left(1+T\left(a_i\right)\right)a_i}{\sum _{i=1}^n\left(1+T\left(a_i\right)\right)}$$(13) Where $T(a_i)=\sum _{j=1j\ne i}^n\mathit{Sup}(a_i,a_j),$ and $\mathit{Sup}(a,b)$ is the support for $a$ from $b,$ which satisfies the following three properties: (1)$\mathit{Sup}(a,b)\in \left[0,1\right];$(2)$\mathit{Sup}(a,b)=\mathit{Sup}(b,a);$ (3) $\mathit{Sup}(a,b)\ge \mathit{Sup}(x,y),$ if $\left|a-b\right|<\left|x-y\right|.$ Obvoiusly, the support (Sup) measure is essentially a similarity index. The more similar, the closer two values, and the more they support each other.

In this section, we shall develop some power aggregation operators with Pythagorean 2-tuple linguistic information, such as Pythagorean 2-tuple linguistic power weighted averaging (P2TLPWA) operator, Pythagorean 2-tuple linguistic power ordered weighted averaging (P2TLPOWA) operator and Pythagorean 2-tuple linguistic power hybrid average (P2TLPHA) operator.

Definition 14.

Let ${\tilde{p}}_j=〈(r_j,{\alpha }_j),({\mu }_j,{\nu }_j)〉(j=1,2,\dots ,n)$ be a collection of Pythagorean 2-tuple linguistic numbers. The Pythagorean 2-tuple linguistic power averaging (P2TLPA) operator is a mapping $P^n\to P$ such that (14) $$\mathrm{P}2\text{TLPA}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\frac{\underset{j=1}{\overset{n}{\oplus }}\left(\left(1+T\left({\tilde{p}}_j\right)\right){\tilde{p}}_j\right)}{\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}$$(14) where (15) $$T\left({\tilde{p}}_j\right)=\sum _{i=1i\ne j}^n\mathit{Sup}\left({\tilde{p}}_j,{\tilde{p}}_i\right)$$(15) and $\mathit{Sup}({\tilde{p}}_j,{\tilde{p}}_i)$ is the support for ${\tilde{p}}_j$ from ${\tilde{p}}_i,$ with the conditions:

1. $\mathit{Sup}({\tilde{p}}_j,{\tilde{p}}_i)\in \left[0,1\right];$

2. $\mathit{Sup}({\tilde{p}}_j,{\tilde{p}}_i)=\mathit{Sup}({\tilde{p}}_i,{\tilde{p}}_j);$

3. $\mathit{Sup}({\tilde{p}}_j,{\tilde{p}}_i)\ge \mathit{Sup}({\tilde{p}}_s,{\tilde{p}}_t),$ if $d({\tilde{p}}_j,{\tilde{p}}_i)<d({\tilde{p}}_s,{\tilde{p}}_t),$ where $d$ is a distance measure.

Based on the Definition 14 and Theorem 2, we can get the following result:

Theorem 3.

The aggregated value by using P2TLPA operator is also a Pythagorean 2-tuple linguistic numbers, where (16) $$\begin{array}{c}\mathrm{P}2\text{TLPA}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\frac{\underset{j=1}{\overset{n}{\oplus }}\left(\left(1+T\left({\tilde{p}}_j\right)\right){\tilde{p}}_j\right)}{\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}\\ =〈\Delta \left(\sum _{j=1}^n\frac{\left(1+T\left({\tilde{p}}_j\right)\right){\Delta }^{-1}\left(r_j,a_j\right)}{\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}\right),\\ \left(\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\mu }_j\right)}^2\right)}^{\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}},\prod _{j=1}^n{\left({\nu }_j\right)}^{\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}\right)〉\end{array}$$(16) where (17) $$T\left({\tilde{p}}_j\right)=\sum _{i=1i\ne j}^n\mathit{Sup}\left({\tilde{p}}_j,{\tilde{p}}_i\right)$$(17)

Definition 15.

Let ${\tilde{p}}_j=〈(r_j,{\alpha }_j),({\mu }_j,{\nu }_j)〉(j=1,2,\dots ,n)$ be a collection of Pythagorean 2-tuple linguistic numbers, $\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ be the weight vector of ${\tilde{p}}_j(j=1,2,\dots ,n),$ and ${\omega }_j>0,$ ${\sum }_{j=1}^n{\omega }_j=1.$ The Pythagorean 2-tuple linguistic power weighted averaging (P2TLPWA) operator is a mapping $P^n\to P$ such that (18) $${\mathrm{P}2\text{TLPWA}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\frac{\underset{j=1}{\overset{n}{\oplus }}\left({\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right){\tilde{p}}_j\right)}{\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}$$(18) where (19) $$T\left({\tilde{p}}_j\right)=\sum _{j=1j\ne i}^n{\omega }_{i}S\mathit{up}\left({\tilde{p}}_j,{\tilde{p}}_i\right)$$(19)

Based on the Definition 15, Theorem 2 and mathematical induction on $n,$ we can get the following result:

Theorem 4.

The aggregated value by using P2TLPWA operator is also a Pythagorean 2-tuple linguistic numbers, where (20) $$\begin{array}{c}\mathrm{P}2\text{TLPWA}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\\ =\frac{\underset{j=1}{\overset{n}{\oplus }}\left({\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right){\tilde{p}}_j\right)}{\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}\\ =〈\Delta \left(\sum _{j=1}^n\frac{{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right){\Delta }^{-1}\left(r_j,a_j\right)}{\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}\right),\\ \left(\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\mu }_j\right)}^2\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}},\prod _{j=1}^n{\left({\nu }_j\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}\right)〉\end{array}$$(20) where $\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ be the weight vector of ${\tilde{p}}_j(j=1,2,\dots ,n),$ and ${\omega }_j>0,$ ${\sum }_{j=1}^n{\omega }_j=1.$

It can be easily proved that the P2TLPWA operator has the following properties.

Theorem 5.

(Idempotency) If all ${\tilde{p}}_j(j=1,2,\dots ,n)$ are equal, i.e., ${\tilde{p}}_j=\tilde{p}$ for all $j,$ then (21) $${\mathrm{P}2\text{TLPWA}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\tilde{p}$$(21)

Theorem 6.

(Boundedness) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ be a collection of P2TLNs, and let $${\tilde{p}}^{-}=\underset{j}{\min }{\tilde{p}}_j,{\tilde{p}}^{+}=\underset{j}{\max }{\tilde{p}}_j$$

Then (22) $${\tilde{p}}^{-}\le {\mathrm{P}2\text{TLPWA}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\le {\tilde{p}}^{+}$$(22)

Theorem 7.

(Monotonicity) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ and ${{\tilde{p}}^{\prime }}_j(j=1,2,\dots ,n)$ be two set of P2TLNs, if ${\tilde{p}}_j\le {{\tilde{p}}^{\prime }}_j,$ for all $j,$ then (23) $${\mathrm{P}2\text{TLPWA}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\le {\mathrm{P}2\text{TLPWA}}_{\omega }\left({{\tilde{p}}^{\prime }}_1,{{\tilde{p}}^{\prime }}_2,\dots ,{{\tilde{p}}^{\prime }}_n\right)$$(23)

Further, we give a Pythagorean 2-tuple linguistic power ordered weighted averaging (P2TLPOWA) operator below:

Definition 16.

Let ${\tilde{p}}_j=〈(r_j,{\alpha }_j),({\mu }_j,{\nu }_j)〉(j=1,2,\dots ,n)$ be a collection of P2TLNs, the Pythagorean 2-tuple linguistic power ordered weighted averaging (P2TLPOWA) operator of dimension $n$ is a mapping P2TLPOWA:$P^n\to P,$ that has an associated weight vector $w={(w_1,w_2,\dots ,w_n)}^T$ such that $w_j>0$ and ${\sum }_{j=1}^n{w}_j=1.$ Furthermore, (24) $$\begin{array}{c}\mathrm{P}2\text{TLPOWA}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\\ =\frac{\underset{j=1}{\overset{n}{\oplus }}\left(w_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right){\tilde{p}}_{\sigma \left(j\right)}\right)}{\sum _{j=1}^n{w}_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)}\\ =〈\Delta \left(\sum _{j=1}^n\frac{w_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right){\Delta }^{-1}\left(r_j,a_{\sigma \left(j\right)}\right)}{\sum _{j=1}^n{w}_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)}\right),\\ \left(\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\mu }_{\sigma \left(j\right)}\right)}^2\right)}^{w_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)}},\prod _{j=1}^n{\left({\nu }_{\sigma \left(j\right)}\right)}^{w_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)}\right)〉\end{array}$$(24) where $(\sigma (1),\sigma (2),\dots ,\sigma (n))$ is a permutation of $(1,2,\dots ,n),$ such that ${\tilde{p}}_{\sigma (j-1)}\ge {\tilde{p}}_{\sigma (j)}$ for all $j=2,\dots ,n,$$w_j(j=1,2,\dots ,n)$ is collection of weights such that (25) $$w_j=g\left(\frac{R_j}{\mathit{TV}}\right)-g\left(\frac{R_{j-1}}{\mathit{TV}}\right),R_j=\sum _{i=1}^j{V}_{\sigma \left(i\right)},\mathit{TV}=\sum _{i=1}^n{V}_{\sigma \left(i\right)},V_{\sigma \left(i\right)}=1+T\left({\tilde{p}}_{\sigma \left(i\right)}\right)$$(25) and $T({\tilde{p}}_{\sigma (j)})$ denotes the support of the $j\text{th}$ largest Pythagorean 2-tuple linguistic numbers $T({\tilde{p}}_{\sigma (j)})$ by all the other Pythagorean 2-tuple linguistic numbers, i.e., (26) $$T\left({\tilde{p}}_{\sigma \left(j\right)}\right)=\sum _{i=1i\ne j}^n\mathit{Sup}\left({\tilde{p}}_{\sigma \left(j\right)},{\tilde{p}}_{\sigma \left(i\right)}\right)$$(26)

Where $\sum _{i=1i\ne j}^n\mathit{Sup}({\tilde{p}}_{\sigma (j)},{\tilde{p}}_{\sigma (i)})$ indicates the support of $j\text{th}$ largest Pythagorean 2-tuple linguistic number ${\tilde{p}}_{\sigma (j)}$ for the $i\text{th}$ largest Pythagorean 2-tuple linguistic number ${\tilde{p}}_{\sigma (i)},$ and g: $\left[0,1\right]\to \left[0,1\right]$ is a basic unit-interval monotonic(BUM) function, having the properties: g(0)=0, g(1)=1, and $g(x)\ge g(y),\mathit{if}x>y.$

It can be easily proved that the P2TLPOWA operator has the following properties.

Theorem 8.

(Idempotency) If all ${\tilde{p}}_j(j=1,2,\dots ,n)$ are equal, i.e., ${\tilde{p}}_j=\tilde{p}$ for all $j,$ then (27) $$\mathrm{P}2\text{TLPOWA}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\tilde{p}$$(27)

Theorem 9.

(Boundedness) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ be a collection of P2TLNs, and let $${\tilde{p}}^{-}=\underset{j}{\min }{\tilde{p}}_j,{\tilde{p}}^{+}=\underset{j}{\max }{\tilde{p}}_j$$

Then (28) $${\tilde{p}}^{-}\le \mathrm{P}2\text{TLPOWA}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\le {\tilde{p}}^{+}$$(28)

Theorem 10.

(Monotonicity) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ and ${{\tilde{p}}^{\prime }}_j(j=1,2,\dots ,n)$ be two set of P2TLNs, if ${\tilde{p}}_j\le {{\tilde{p}}^{\prime }}_j,$ for all $j,$ then (29) $$\mathrm{P}2\text{TLPOWA}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\le \mathrm{P}2\text{TLPOWA}\left({{\tilde{p}}^{\prime }}_1,{{\tilde{p}}^{\prime }}_2,\dots ,{{\tilde{p}}^{\prime }}_n\right)$$(29)

Theorem 11.

(Commutativity) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ and ${{\tilde{p}}^{\prime }}_j(j=1,2,\dots ,n)$ be two set of P2TLNs, for all $j,$ then (30) $${\mathrm{P}2\text{TLOWA}}_w\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)={\mathrm{P}2\text{TLOWA}}_w\left({{\tilde{p}}^{\prime }}_1,{{\tilde{p}}^{\prime }}_2,\dots ,{{\tilde{p}}^{\prime }}_n\right)$$(30) where ${{\tilde{p}}^{\prime }}_j(j=1,2,\dots ,n)$ is any permutation of ${\tilde{p}}_j(j=1,2,\dots ,n).$

From Definitions 15-16, we know that the P2TLPWA operators only weights the Pythagorean 2-tuple linguistic number itself, while the P2TLPOWA operators weights the ordered positions of the Pythagorean 2-tuple linguistic number instead of weighting the arguments itself. Therefore, the weights represent two different aspects in both the P2TLPWA and P2TLPOWA operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose the Pythagorean 2-tuple linguistic power hybrid average (P2TLPHA) operator.

Definition 17.

Let ${\tilde{p}}_j=〈(r_j,{\alpha }_j),({\mu }_j,{\nu }_j)〉(j=1,2,\dots ,n)$ be a collection of P2TLNs. A Pythagorean 2-tuple linguistic power hybrid average (P2TLPHA) operator is a mapping P2TLPHA:$P^n\to P,$ such that (31) $$\begin{array}{c}{\mathrm{P}2\text{TLPHA}}_{\omega ,\mathrm{w}}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\\ =\frac{\underset{j=1}{\overset{n}{\oplus }}\left(w_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right){\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)}{\sum _{j=1}^n{w}_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)}\\ =〈\Delta \left(\sum _{j=1}^n\frac{w_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right){\Delta }^{-1}\left({\dot{r}}_j,{\dot{a}}_{\sigma \left(j\right)}\right)}{\sum _{j=1}^n{w}_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)}\right),\\ \left(\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\dot{\mu }}_{\sigma \left(j\right)}\right)}^2\right)}^{w_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)}},\prod _{j=1}^n{({\dot{\nu }}_{\sigma \left(j\right)})}^{w_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)}\right)〉\end{array}$$(31) where $w=(w_1,w_2,\dots ,w_n)$ is the associated weighting vector, with $w_j\in \left[0,1\right],$ ${\sum }_{j=1}^n{w}_j=1,$ and ${\dot{\tilde{p}}}_{\sigma (j)}$ is the j-th largest element of the Pythagorean 2-tuple linguistic arguments ${\dot{\tilde{p}}}_j({\dot{\tilde{p}}}_j=(n{\omega }_j){\tilde{p}}_j,j=1,2,\dots ,n),$$\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the weighting vector of Pythagorean 2-tuple linguistic arguments ${\tilde{p}}_j(j=1,2,\dots ,n),$ with ${\omega }_i\in \left[0,1\right],$${\sum }_{i=1}^n{\omega }_i=1,$ and $n$ is the balancing coefficient. And $w_j(j=1,2,\dots ,n)$ is collection of weights such that (32) $$w_j=g\left(\frac{R_j}{\mathit{TV}}\right)-g\left(\frac{R_{j-1}}{\mathit{TV}}\right),R_j=\sum _{i=1}^j{V}_{\sigma \left(i\right)},\mathit{TV}=\sum _{i=1}^n{V}_{\sigma \left(i\right)},V_{\sigma \left(i\right)}=1+T\left({\dot{\tilde{p}}}_{\sigma \left(i\right)}\right)$$(32) and $T({\dot{\tilde{p}}}_{\sigma (j)})$ denotes the support of the $j\text{th}$ largest Pythagorean 2-tuple linguistic numbers $T({\dot{\tilde{p}}}_{\sigma (j)})$ by all the other Pythagorean 2-tuple linguistic numbers, i.e., (33) $$T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)=\sum _{i=1i\ne j}^n\mathit{Sup}\left({\dot{\tilde{p}}}_{\sigma \left(j\right)},{\dot{\tilde{p}}}_{\sigma \left(i\right)}\right)$$(33) where $\sum _{i=1i\ne j}^n\mathit{Sup}({\dot{\tilde{p}}}_{\sigma (j)},{\dot{\tilde{p}}}_{\sigma (i)})$ indicates the support of $j\text{th}$ largest Pythagorean 2-tuple linguistic number ${\dot{\tilde{p}}}_{\sigma (j)}$ for the $i\text{th}$ largest Pythagorean 2-tuple linguistic number ${\dot{\tilde{p}}}_{\sigma (i)},$ and g: $\left[0,1\right]\to \left[0,1\right]$ is a basic unit-interval monotonic(BUM) function, having the properties: g(0)=0, g(1)=1, and $g(x)\ge g(y),\mathit{if}x>y.$ Especially, if $w={(1/n,1/n,\dots ,1/n)}^T,$ then P2TLPHA is reduced to the Pythagorean 2-tuple linguistic power weighted average (P2TLPWA) operator; if $\omega =(1/n,1/n,\dots ,1/n),$ then P2TLPHA is reduced to the Pythagorean 2-tuple linguistic power ordered weighted average (P2TLPOWA) operator.

4. Pythagorean 2-tuple linguistic power geometric aggregation operators

Based on the PA operator (Yager, 2001) and geometric mean, in the following, Xu & Yager, (2010) further define a power geometric (PG) operator: (34) $$\mathit{PG}\left(a_1,a_2,\dots ,a_n\right)=\prod _{i=1}^n{a}_i{}^{\frac{1+T\left(a_i\right)}{\sum _{i=1}^n\left(1+T\left(a_i\right)\right)}}$$(34)

Obviously, the PA and PG operators are two nonlinear weighted aggregation tools, whose weighting vectors depend upon the input values and allow values being aggregated to support and reinforce each other, that’s to say, the closer $a_i$ and $a_j,$ the more similar they are, and the more they support each other.

In this section, we shall develop some power geometric aggregation operators with Pythagorean 2-tuple linguistic information, such as Pythagorean 2-tuple linguistic power geometric (P2TLPG) operator, Pythagorean 2-tuple linguistic power weighted geometric (P2TLPWG) operator, Pythagorean 2-tuple linguistic power ordered weighted geometric (P2TLPOWG) operator and Pythagorean 2-tuple linguistic power hybrid geometric (P2TLPHG) operator.

Definition 18.

Let ${\tilde{p}}_j=〈(r_j,{\alpha }_j),({\mu }_j,{\nu }_j)〉(j=1,2,\dots ,n)$ be a collection of P2TLNs. The Pythagorean 2-tuple linguistic power geometric (P2TLPG) operator is a mapping $P^n\to P$ such that (35) $$\mathrm{P}2\text{TLPG}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\underset{j=1}{\overset{n}{\otimes }}{\left({\tilde{p}}_j\right)}^{\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}$$(35) where (36) $$T\left({\tilde{p}}_j\right)=\sum _{i=1i\ne j}^n\mathit{Sup}\left({\tilde{p}}_j,{\tilde{p}}_i\right)$$(36) and $\mathit{Sup}({\tilde{p}}_j,{\tilde{p}}_i)$ is the support for ${\tilde{p}}_j$ from ${\tilde{p}}_i,$ with the conditions:

1. $\mathit{Sup}({\tilde{p}}_j,{\tilde{p}}_i)\in \left[0,1\right];$

2. $\mathit{Sup}({\tilde{p}}_j,{\tilde{p}}_i)=\mathit{Sup}({\tilde{p}}_i,{\tilde{p}}_j);$

3. $\mathit{Sup}({\tilde{p}}_j,{\tilde{p}}_i)\ge \mathit{Sup}({\tilde{p}}_s,{\tilde{p}}_t),$ if $d({\tilde{p}}_j,{\tilde{p}}_i)<d({\tilde{p}}_s,{\tilde{p}}_t),$ where $d$ is a distance measure.

Based on the Definition 18 and Theorem 2, we can get the following result:

Theorem 12.

The aggregated value by using P2TLPG operator is also a Pythagorean 2-tuple linguistic numbers, where (37) $$\begin{array}{c}\mathrm{P}2\text{TLPG}\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\\ =\underset{j=1}{\overset{n}{\otimes }}{\left({\tilde{p}}_j\right)}^{\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}\\ =〈\Delta \left(\prod _{j=1}^n{\left({\Delta }^{-1}\left(r_j,a_j\right)\right)}^{\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}\right),\\ \left(\prod _{j=1}^n{\left({\mu }_j\right)}^{\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)},\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\nu }_j\right)}^2\right)}^{\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n\left(1+T\left({\tilde{p}}_j\right)\right)}}\right)〉\end{array}$$(37) where (38) $$T\left({\tilde{p}}_j\right)=\sum _{i=1i\ne j}^n\mathit{Sup}\left({\tilde{p}}_j,{\tilde{p}}_i\right)$$(38)

Definition 19.

Let ${\tilde{p}}_j=〈(r_j,{\alpha }_j),({\mu }_j,{\nu }_j)〉(j=1,2,\dots ,n)$ be a collection of P2TLNs, $\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ be the weight vector of ${\tilde{p}}_j(j=1,2,\dots ,n),$ and ${\omega }_j>0,$ ${\sum }_{j=1}^n{\omega }_j=1.$ The Pythagorean 2-tuple linguistic power weighted geometric (P2TLPWG) operator is a mapping $P^n\to P$ such that (39) $${\mathrm{P}2\text{TLPWG}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\underset{j=1}{\overset{n}{\otimes }}{\left({\tilde{p}}_j\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}$$(39) where (40) $$T\left({\tilde{p}}_j\right)=\sum _{i=1i\ne j}^n{\omega }_{i}S\mathit{up}\left({\tilde{p}}_j,{\tilde{p}}_i\right)$$(40)

Based on the Definition 19, Theorem 2 and mathematical induction on $n,$ we can get the following result:

Theorem 13.

The aggregated value by using P2TLPWG operator is also a Pythagorean 2-tuple linguistic numbers, where (41) $$\begin{array}{c}{\mathrm{P}2\text{TLPWG}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\\ =\underset{j=1}{\overset{n}{\otimes }}{\left({\tilde{p}}_j\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}\\ =〈\Delta \left(\prod _{j=1}^n{\left({\Delta }^{-1}\left(r_j,a_j\right)\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}\right),\\ \left(\prod _{j=1}^n{\left({\mu }_j\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)},\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\nu }_j\right)}^2\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_j\right)\right)}}\right)〉\end{array}$$(41) where $\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ be the weight vector of ${\tilde{p}}_j(j=1,2,\dots ,n),$ and ${\omega }_j>0,$ ${\sum }_{j=1}^n{\omega }_j=1.$

It can be easily proved that the P2TLPWG operator has the following properties.

Theorem 14.

(Idempotency) If all ${\tilde{p}}_j(j=1,2,\dots ,n)$ are equal, i.e., ${\tilde{p}}_j=\tilde{p}$ for all $j,$ then (42) $${\mathrm{P}2\text{TLPWG}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\tilde{p}$$(42)

Theorem 15.

(Boundedness) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ be a collection of P2TLNs, and let $${\tilde{p}}^{-}=\underset{j}{\min }{\tilde{p}}_j,{\tilde{p}}^{+}=\underset{j}{\max }{\tilde{p}}_j$$

Then (43) $${\tilde{p}}^{-}\le {\mathrm{P}2\text{TLPWG}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\le {\tilde{p}}^{+}$$(43)

Theorem 16.

(Monotonicity) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ and ${{\tilde{p}}^{\prime }}_j(j=1,2,\dots ,n)$ be two set of P2TLNs, if ${\tilde{p}}_j\le {{\tilde{p}}^{\prime }}_j,$ for all $j,$ then (44) $${\mathrm{P}2\text{TLPWG}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\le {\mathrm{P}2\text{TLPWG}}_{\omega }\left({{\tilde{p}}^{\prime }}_1,{{\tilde{p}}^{\prime }}_2,\dots ,{{\tilde{p}}^{\prime }}_n\right)$$(44)

Further, we give a Pythagorean 2-tuple linguistic power ordered weighted geometric (P2TLPOWG) operator below:

Definition 20.

Let ${\tilde{p}}_j=〈(r_j,{\alpha }_j),({\mu }_j,{\nu }_j)〉(j=1,2,\dots ,n)$ be a collection of P2TLNs, the Pythagorean 2-tuple linguistic power ordered weighted geometric (P2TLPOWG) operator of dimension $n$ is a mapping P2TLPOWG:$P^n\to P,$ that has an associated weight vector $w={(w_1,w_2,\dots ,w_n)}^T$ such that $w_j>0$ and ${\sum }_{j=1}^n{w}_j=1.$ Furthermore, (45) $$\begin{array}{c}{\mathrm{P}2\text{TLPOWG}}_w\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\\ =\underset{j=1}{\overset{n}{\otimes }}{\left({\tilde{p}}_{\sigma \left(j\right)}\right)}^{w_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)}\\ =〈\Delta \left(\prod _{j=1}^n{\left({\Delta }^{-1}\left(r_{\sigma \left(j\right)},a_{\sigma \left(j\right)}\right)\right)}^{w_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)}\right),\\ \left(\prod _{j=1}^n{\left({\mu }_{\sigma \left(j\right)}\right)}^{w_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)},\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\nu }_{\sigma \left(j\right)}\right)}^2\right)}^{w_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\tilde{p}}_{\sigma \left(j\right)}\right)\right)}}\right)〉\end{array}$$(45) where $(\sigma (1),\sigma (2),\dots ,\sigma (n))$ is a permutation of $(1,2,\dots ,n),$ such that ${\tilde{p}}_{\sigma (j-1)}\ge {\tilde{p}}_{\sigma (j)}$ for all $j=2,\dots ,n,$$w_j(j=1,2,\dots ,n)$ is collection of weights such that (46) $$w_j=g\left(\frac{R_j}{\mathit{TV}}\right)-g\left(\frac{R_{j-1}}{\mathit{TV}}\right),R_j=\sum _{i=1}^j{V}_{\sigma \left(i\right)},\mathit{TV}=\sum _{i=1}^n{V}_{\sigma \left(i\right)},V_{\sigma \left(i\right)}=1+T\left({\tilde{p}}_{\sigma \left(i\right)}\right)$$(46) and $T({\tilde{p}}_{\sigma (j)})$ denotes the support of the $j\text{th}$ largest Pythagorean 2-tuple linguistic numbers $T({\tilde{p}}_{\sigma (j)})$ by all the other Pythagorean 2-tuple linguistic numbers, i.e., (47) $$T\left({\tilde{p}}_{\sigma \left(j\right)}\right)=\sum _{i=1i\ne j}^n\mathit{Sup}\left({\tilde{p}}_{\sigma \left(j\right)},{\tilde{p}}_{\sigma \left(i\right)}\right)$$(47)

Where $\sum _{i=1i\ne j}^n\mathit{Sup}({\tilde{p}}_{\sigma (j)},{\tilde{p}}_{\sigma (i)})$ indicates the support of $j\text{th}$ largest Pythagorean 2-tuple linguistic number ${\tilde{p}}_{\sigma (j)}$ for the $i\text{th}$ largest Pythagorean 2-tuple linguistic number ${\tilde{p}}_{\sigma (i)},$ and g: $\left[0,1\right]\to \left[0,1\right]$ is a basic unit-interval monotonic(BUM) function, having the properties: g(0)=0, g(1)=1, and $g(x)\ge g(y),\mathit{if}x>y.$

It can be easily proved that the P2TLPOWG operator has the following properties.

Theorem 17.

(Idempotency) If all ${\tilde{p}}_j(j=1,2,\dots ,n)$ are equal, i.e., ${\tilde{p}}_j=\tilde{p}$ for all $j,$ then (48) $${\mathrm{P}2\text{TLPOWG}}_w\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\tilde{p}$$(48)

Theorem 18.

(Boundedness) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ be a collection of P2TLNs, and let $${\tilde{p}}^{-}=\underset{j}{\min }{\tilde{p}}_j,{\tilde{p}}^{+}=\underset{j}{\max }{\tilde{p}}_j$$

Then (49) $${\tilde{p}}^{-}\le {\mathrm{P}2\text{TLPOWG}}_w\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\le {\tilde{p}}^{+}$$(49)

Theorem 19.

(Monotonicity) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ and ${{\tilde{p}}^{\prime }}_j(j=1,2,\dots ,n)$ be two set of P2TLNs, if ${\tilde{p}}_j\le {{\tilde{p}}^{\prime }}_j,$ for all $j,$ then (50) $${\mathrm{P}2\text{TLPOWG}}_w\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\le {\mathrm{P}2\text{TLPOWG}}_w\left({{\tilde{p}}^{\prime }}_1,{{\tilde{p}}^{\prime }}_2,\dots ,{{\tilde{p}}^{\prime }}_n\right)$$(50)

Theorem 20.

(Commutativity) Let ${\tilde{p}}_j(j=1,2,\dots ,n)$ and ${{\tilde{p}}^{\prime }}_j(j=1,2,\dots ,n)$ be two set of P2TLNs, for all $j,$ then (51) $${\mathrm{P}2\text{TLPOWG}}_w\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)={\mathrm{P}2\text{TLPOWG}}_w\left({{\tilde{p}}^{\prime }}_1,{{\tilde{p}}^{\prime }}_2,\dots ,{{\tilde{p}}^{\prime }}_n\right)$$(51) where ${{\tilde{p}}^{\prime }}_j(j=1,2,\dots ,n)$ is any permutation of ${\tilde{p}}_j(j=1,2,\dots ,n).$

From Definitions 19-20, we know that the P2TLPWG operators only weights the Pythagorean 2-tuple linguistic number itself, while the P2TLPOWG operators weights the ordered positions of the Pythagorean 2-tuple linguistic number instead of weighting the arguments itself. Therefore, the weights represent two different aspects in both the P2TLPWG and P2TLPOWG operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose the Pythagorean 2-tuple linguistic power hybrid geometric (P2TLPHG) operator.

Definition 21. A

Pythagorean 2-tuple linguistic power hybrid geometric (P2TLPHG) operator is a mapping P2TLPHG:$P^n\to P,$ such that (52) $$\begin{array}{c}{\mathrm{P}2\text{TLPHG}}_{w,\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)\\ =\underset{j=1}{\overset{n}{\otimes }}{\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)}^{w_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)}\\ =〈\Delta \left(\prod _{j=1}^n{\left({\Delta }^{-1}\left({\dot{r}}_{\sigma \left(j\right)},{\dot{a}}_{\sigma \left(j\right)}\right)\right)}^{w_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)}\right),\\ (\prod _{j=1}^n{\left({\dot{\mu }}_{\sigma \left(j\right)}\right)}^{w_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)},\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\dot{\nu }}_{\sigma \left(j\right)}\right)}^2\right)}^{w_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)/\sum _{j=1}^n{w}_j\left(1+T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)\right)}})〉\end{array}$$(52) where $w=(w_1,w_2,\dots ,w_n)$ is the associated weighting vector, with $w_j\in \left[0,1\right],$ ${\sum }_{j=1}^n{w}_j=1,$ and ${\dot{\tilde{p}}}_{\sigma (j)}$ is the j-th largest element of the Pythagorean 2-tuple linguistic arguments ${\dot{\tilde{p}}}_j({\dot{\tilde{p}}}_j={({\tilde{p}}_j)}^{n{\omega }_j},j=1,2,\dots ,n),$$\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the weighting vector of Pythagorean 2-tuple linguistic arguments ${\tilde{p}}_j(j=1,2,\dots ,n),$ with ${\omega }_j\in \left[0,1\right],$${\sum }_{j=1}^n{\omega }_j=1,$ and $n$ is the balancing coefficient. And $w_j(j=1,2,\dots ,n)$ is collection of weights such that (53) $$w_j=g\left(\frac{R_j}{\mathit{TV}}\right)-g\left(\frac{R_{j-1}}{\mathit{TV}}\right),R_j=\sum _{i=1}^j{V}_{\sigma \left(i\right)},\mathit{TV}=\sum _{i=1}^n{V}_{\sigma \left(i\right)},V_{\sigma \left(i\right)}=1+T\left({\dot{\tilde{p}}}_{\sigma \left(i\right)}\right)$$(53) and $T({\dot{\tilde{p}}}_{\sigma (j)})$ denotes the support of the $j\text{th}$ largest Pythagorean 2-tuple linguistic numbers $T({\dot{\tilde{p}}}_{\sigma (j)})$ by all the other Pythagorean 2-tuple linguistic numbers, i.e., (54) $$T\left({\dot{\tilde{p}}}_{\sigma \left(j\right)}\right)=\sum _{i=1i\ne j}^n\mathit{Sup}\left({\dot{\tilde{p}}}_{\sigma \left(j\right)},{\dot{\tilde{p}}}_{\sigma \left(i\right)}\right)$$(54) where $\sum _{i=1i\ne j}^n\mathit{Sup}({\dot{\tilde{p}}}_{\sigma (j)},{\dot{\tilde{p}}}_{\sigma (i)})$ indicates the support of $j\text{th}$ largest Pythagorean 2-tuple linguistic number ${\dot{\tilde{p}}}_{\sigma (j)}$ for the $i\text{th}$ largest Pythagorean 2-tuple linguistic number ${\dot{\tilde{p}}}_{\sigma (i)},$ and g: $\left[0,1\right]\to \left[0,1\right]$ is a basic unit-interval monotonic(BUM) function, having the properties: g(0)=0, g(1)=1, and $g(x)\ge g(y),\mathit{if}x>y.$ Especially, if $w={(1/n,1/n,\dots ,1/n)}^T,$ then P2TLPHG is reduced to the Pythagorean 2-tuple linguistic power weighted geometric (P2TLPWG) operator; if $\omega =(1/n,1/n,\dots ,1/n),$ then P2TLPHG is reduced to the Pythagorean 2-tuple linguistic power ordered weighted geometric (P2TLPOWG) operator.

5. Models for multiple attribute decision making with pythagorean 2-tuple linguistic information

Based the P2TLPWA (P2TLPWG) operators, in this section, we shall propose the model for multiple attribute decision making with Pythagorean 2-tuple linguistic information. Let $A=\left\{A_1,A_2,\dots ,A_m\right\}$ be a discrete set of alternatives, and $G=\left\{G_1,G_2,\dots ,G_n\right\}$ be the set of attributes, $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n)$ is the weighting vector of the attribute $G_j(j=1,2,\dots ,n),$ where ${\omega }_j\in \left[0,1\right],$ ${\sum }_{j=1}^n{\omega }_j=1.$ Suppose that $\tilde{R}={({\tilde{r}}_{\mathit{ij}})}_{m\times n}={〈(s_{\mathit{ij}},{\rho }_{\mathit{ij}}),({\mu }_{\mathit{ij}},{\nu }_{\mathit{ij}})〉}_{m\times n}$ is the Pythagorean 2-tuple linguistic decision matrix, where ${\tilde{r}}_{\mathit{ij}}$ take the form of the Pythagorean 2-tuple linguistic numbers, where ${\mu }_{\mathit{ij}}$ indicates the degree that the alternative $A_i$ satisfies the attribute $G_j$ given by the decision maker, ${\nu }_{\mathit{ij}}$ indicates the degree that the alternative $A_i$ doesn’t satisfy the attribute $G_j$ given by the decision maker,${\mu }_{\mathit{ij}}\in \left[0,1\right],$ ${\eta }_{\mathit{ij}}\in \left[0,1\right]$ ${\nu }_{\mathit{ij}}\in \left[0,1\right],$ ${({\mu }_{\mathit{ij}})}^2+{({\nu }_{\mathit{ij}})}^2\le 1,$ ${\pi }_{\mathit{ij}}=\sqrt{1-({({\mu }_{\mathit{ij}})}^2+{({\nu }_{\mathit{ij}})}^2)},$ $s_{\mathit{ij}}\in S,$ ${\rho }_{\mathit{ij}}\in [-0.5,0.5),$ $i=1,2,\dots ,m,$ $j=1,2,\dots ,n.$

In the following, we apply the P2TLPWA (P2TLPWG) operator to the MADM problems with hesitant fuzzy information.

• Step 1. Calculate the supports:

(55) $$\mathit{Sup}\left({\tilde{p}}_{\mathit{ij}},{\tilde{p}}_{\mathit{ik}}\right)=1-d\left({\tilde{p}}_{\mathit{ij}},{\tilde{p}}_{\mathit{ik}}\right),j,k=1,2,\dots ,n.$$(55)

which satisfies the support conditions (1)–(3) in Section 3. Here, without loss of generality, we calculate $d({\tilde{p}}_{\mathit{ij}},{\tilde{p}}_{\mathit{ik}})$ with the normalized Hamming distance [9]: $$d\left({\tilde{p}}_{\mathit{ij}},{\tilde{p}}_{\mathit{ik}}\right)=\frac{\left|{\Delta }^{-1}\left(s_{\mathit{ij}},{\rho }_{\mathit{ij}}\right)-{\Delta }^{-1}\left(s_{\mathit{ik}},{\rho }_{\mathit{ik}}\right)\right|}{t}\cdot \frac{\left(\left|{\mu }_{\mathit{ij}}-{\mu }_{\mathit{ik}}\right|+\left|{\nu }_{\mathit{ij}}-{\nu }_{\mathit{ik}}\right|\right)}{2}$$ (56) $$j,k=1,2,\dots ,n.$$(56)

• Step 2. Utilize the weights ${\omega }_j(j=1,2,\dots ,n)$ of the attribute $G_j(j=1,2,\dots ,n)$ to calculate the weighted support $T({\tilde{p}}_{\mathit{ij}})$ of the P2TLN ${\tilde{p}}_{\mathit{ij}}$ by the other P2TLN ${\tilde{p}}_{\mathit{ik}}(k=1,2,\dots ,n,k\ne j):$

(57) $$T\left({\tilde{p}}_{\mathit{ij}}\right)=\sum _{k=1k\ne j}^n{\omega }_{j}S\mathit{up}\left({\tilde{p}}_{\mathit{ij}},{\tilde{p}}_{\mathit{ik}}\right)$$(57)

and calculate the weight ${\xi }_{\mathit{ij}}(j=1,2,\dots ,n)$ associated with the P2TLN ${\tilde{p}}_{\mathit{ij}}(i=1,2,\dots ,m,j=1,2,\dots ,n):$ (58) $${\xi }_{\mathit{ij}}=\frac{{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)}{\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)},i=1,2,\dots ,m,j=1,2,\dots ,n$$(58) where ${\xi }_{\mathit{ij}}\ge 0,i=1,2,\dots ,m,j=1,2,\dots ,n,$ and ${\sum }_{j=1}^n{\xi }_{\mathit{ij}}=1,i=1,2,\dots ,m.$

• Step 3. We utilize the decision information given in matrix $\tilde{P},$ and the P2TLPWA operator

(59) $$\begin{array}{c}{\tilde{p}}_i={\mathrm{P}2\text{TLPWA}}_{\omega }\left({\tilde{p}}_{i1},{\tilde{p}}_{i2},\dots ,{\tilde{p}}_{\mathit{in}}\right)=\frac{\underset{j=1}{\overset{n}{\oplus }}\left({\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right){\tilde{p}}_{\mathit{ij}}\right)}{\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)}\\ =〈\Delta \left(\sum _{j=1}^n\frac{{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right){\Delta }^{-1}\left(s_{\mathit{ij}},{\rho }_{\mathit{ij}}\right)}{\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)}\right),\\ \left(\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\mu }_{\mathit{ij}}\right)}^2\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)}},\prod _{j=1}^n{\left({\nu }_{\mathit{ij}}\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)}\right)〉\end{array}$$(59)

Or (60) $$\begin{array}{c}{\tilde{p}}_i={\mathrm{P}2\text{TLPWG}}_{\omega }\left({\tilde{p}}_{i1},{\tilde{p}}_{i2},\dots ,{\tilde{p}}_{\mathit{in}}\right)=\underset{j=1}{\overset{n}{\otimes }}{\left({\tilde{p}}_{\mathit{ij}}\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)}\\ =〈\Delta \left(\prod _{j=1}^n{\left({\Delta }^{-1}\left(s_{\mathit{ij}},{\rho }_{\mathit{ij}}\right)\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)}\right),\\ \left(\prod _{j=1}^n{\left({\mu }_{\mathit{ij}}\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)},\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\nu }_{\mathit{ij}}\right)}^2\right)}^{{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)/\sum _{j=1}^n{\omega }_j\left(1+T\left({\tilde{p}}_{\mathit{ij}}\right)\right)}}\right)〉\end{array}$$(60) to derive the overall preference values ${\tilde{p}}_i(i=1,2,\dots ,m)$ of the alternative $A_i.$

• Step 4. Calculate the scores $S({\tilde{p}}_i)(i=1,2,\dots ,m)$ of the overall Pythagorean 2-tuple linguistic numbers ${\tilde{p}}_i(i=1,2,\dots ,m)$ to rank all the alternatives $A_i(i=1,2,\dots ,m)$ and then to select the best one(s). If there is no difference between two scores $S({\tilde{p}}_i)$ and $S({\tilde{p}}_j),$ then we need to calculate the accuracy degrees $H({\tilde{p}}_i)$ and $H({\tilde{p}}_j)$ of the overall Pythagorean 2-tuple linguistic numbers ${\tilde{p}}_i$ and ${\tilde{p}}_j,$ respectively, and then rank the alternatives $A_i$ and $A_j$ in accordance with the accuracy degrees $H({\tilde{p}}_i)$ and $H({\tilde{p}}_j).$

• Step 5. Rank all the alternatives $A_i(i=1,2,\dots ,m)$ and select the best one(s) in accordance with $S({\tilde{p}}_i)(i=1,2,\dots ,m).$

• Step 6. End.

6. Numerical example and comparative analysis

6.1. Numerical example

In this section, we utilize a practical multiple attribute decision making problems to illustrate the application of the developed approaches. Suppose an organization plans to implement enterprise resource planning (ERP) system (adapted from Liao et al., 2007). The first step is to form a project team that consists of CIO and two senior representatives from user departments. By collecting all possible information about ERP vendors and systems, project term choose five potential ERP systems $A_i(i=1,2,\dots ,5)$ as candidates. The company employs some external professional organizations (or experts) to aid this decision-making. The project team selects four attributes to evaluate the alternatives: (1) function and technology G₁, (2) strategic fitness G₂, (3) vendor’s ability G₃; (4) vendor’s reputation G₄. The five possible ERP systems $A_i(i=1,2,\dots ,5)$ are to be evaluated using the Pythagorean 2-tuple linguistic numbers by the decision makers under the above four attributes (whose weighting vector is $\omega =(0.2,0.1,0.3,0.4)$), and construct the following matrix $\tilde{R}={({\tilde{r}}_{\mathit{ij}})}_{5\times 4}$ is shown in Table 1.

Table 1. The Pythagorean 2-tuple linguistic decision matrix.

	G1	G2
A1	<(S4,0),(0.50,0.80)>	<(S2,0), (0.60,0.30)>
A2	<(S1,0), (0.70,0.50)>	<(S4,0), (0.70,0.20)>
A3	<(S5,0), (0.60,0.40)>	<(S1,0), (0.50,0.70)>
A4	<(S5,0), (0.80,0.10)>	<(S6,0), (0.60,0.30)>
A5	<(S3,0), (0.60,0.40)>	<(S1,0), (0.40,0.80)>
	G3	G4
A1	<(S1,0), (0.30,0.60)>	<(S3,0), (0.50,0.70)>
A2	<(S2,0), (0.70,0.20)>	<(S4,0), (0.40,0.50)>
A3	<(S4,0), (0.50,0.30)>	<(S2,0), (0.60,0.30)>
A4	<(S7,0), (0.30,0.40)>	<(S1,0), (0.50,0.60)>
A5	<(S3,0), (0.70,0.60)>	<(S1,0), (0.50,0.80)>

Source: Author calculation.

In the following, in order to select the most desirable ERP systems, we utilize the P2TLPWA (P2TLPWG) operator to develop an approach to multiple attribute decision making problems with Pythagorean 2-tuple linguistic information, which can be described as following.

• Step 1. Utilize (53)–(56) to calculate the weight ${\xi }_{\mathit{ij}}(i=1,2,3,4,5,j=1,2,3,4)$ associated with the P2TLN ${\tilde{p}}_{\mathit{ij}}(i=1,2,3,4,5,j=1,2,3,4),$ which are contained in the matrix $\tilde{R}={({\tilde{r}}_{\mathit{ij}})}_{5\times 4}$ which is shown in Table 1.

$$\xi =\left[\begin{array}{c}0.19860.09960.30040.4014\\ 0.20100.10050.30310.3954\\ 0.20010.09800.30090.4010\\ 0.20310.10300.30690.3869\\ 0.20150.09880.30250.3972\end{array}\right]$$

• Step 2. According to $\xi$ and Table 1, aggregate all Pythagorean 2-tuple linguistic numbers ${\tilde{r}}_{\mathit{ij}}(j=1,2,\dots ,n)$ by using the P2TLPWA (P2TLPWG) operator to derive the overall Pythagorean 2-tuple linguistic numbers ${\tilde{p}}_i(i=1,2,3,4,5)$ of the alternative $A_i.$ The aggregating results are shown in Table 2.

  Table 2. The aggregating results of the ERP systems by the P2TLPWA (P2TLPWG) operators.

  	P2TLPWA	P2TLPWG
  A1	<(s3,-0.50),(0.46,0.63)>	<(s2,0.19), (0.44,0.67)>
  A2	<(s3,-0.21), (0.61,0.35)>	<(s2,0.45), (0.56,0.40)>
  A3	<(s3,0.10), (0.56,0.35)>	<(s3,-0.33), (0.56,0.38)>
  A4	<(s4,0.17), (0.55,0.34)>	<(s3,0.03), (0.48,0.43)>
  A5	<(s2,0.01), (0.58,0. 64)>	<(s2,-0.26), (0.56,0.69)>

  Source: Author calculation.

• Step 3. According to the aggregating results shown in Table 2 and the score functions of the ERP systems are shown in Table 3.

  Table 3. The score functions of the ERP systems.

  	P2TLPWA	P2TLPWG
  A1	(s1,0.02)	(s1,-0.19)
  A2	(s2,-0.24)	(s1,0.42)
  A3	(s2,-0.14)	(s2,-0.38)
  A4	(s3,-0.47)	(s2,-0.42)
  A5	(s1,-0.06)	(s1,-0.27)

  Source: Author calculation.

• Step 4. According to the score functions shown in Table 3 and the comparison formula of score functions, the ordering of the ERP systems are shown in Table 4. Note that “﹥” means “preferred to”. As we can see, depending on the aggregation operators used, the ordering of the ERP systems is slightly different, and the best ERP system is A₄ or A₃.

  Table 4. Ordering of the ERP systems.

  	Ordering
  P2TLPWA	A4﹥A3﹥A2﹥A1﹥A5
  P2TLPWG	A3﹥A4﹥A2﹥A1﹥A5

  Source: Author calculation.

• From Tables 4, we can easily find that these two operators may generate slightly different ranking results. The main reason causing this ranking result difference is that the P2TLPWA operator emphasize the group influences, however the P2TLPWG operator emphasize the individual influences.

6.2. Comparative analysis

In what follows, we compare our proposed method with other existing methods including the Pythagorean 2-tuple linguistic weighted average (P2TLWA) operator and Pythagorean 2-tuple linguistic weighted geometric (P2TLWG) operator which are proposed as follows:

Definition 22.

Let ${\tilde{p}}_j=〈(r_j,{\alpha }_j),({\mu }_j,{\nu }_j)〉(j=1,2,\dots ,n)$ be a collection of Pythagorean 2-tuple linguistic numbers, $\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ be the weight vector of ${\tilde{a}}_j(j=1,2,\dots ,n),$ and ${\omega }_j>0,$ ${\sum }_{j=1}^n{\omega }_j=1.$ Then (61) $$\begin{array}{c}{\mathrm{P}2\text{TLWA}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\underset{j=1}{\overset{n}{\oplus }}\left({\omega }_j{\tilde{p}}_j\right)\\ =〈\Delta \left(\sum _{j=1}^n{\omega }_j{\Delta }^{-1}\left(r_j,a_j\right)\right),\left(\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\mu }_j\right)}^2\right)}^{w_j}},\prod _{j=1}^n{\left({\nu }_j\right)}^{{\omega }_j}\right)〉\end{array}$$(61) (62) $$\begin{array}{c}{\mathrm{P}2\text{TLWG}}_{\omega }\left({\tilde{p}}_1,{\tilde{p}}_2,\dots ,{\tilde{p}}_n\right)=\underset{j=1}{\overset{n}{\otimes }}{\left({\tilde{p}}_j\right)}^{{\omega }_j}\\ =〈\Delta \left(\prod _{j=1}^n\left({\Delta }^{-1}{\left(r_j,a_j\right)}^{{\omega }_j}\right)\right),\left(\prod _{j=1}^n{\left({\mu }_j\right)}^{{\omega }_j},\sqrt{1-\prod _{j=1}^n{\left(1-{\left({\nu }_j\right)}^2\right)}^{{\omega }_j}}\right)〉\end{array}$$(62)

By utilizing the decision information given in matrix $\tilde{R},$ and the P2TLWA and P2TLWG operators, the aggregating results are shown in Table 5.

Table 5. The aggregating results of the ERP systems by the P2TLWA (P2TLWG) operators.

	P2TLWA	P2TLWG
A1	<(s3,-0.50),(0.47,0.46)>	<(s2,0.19), (0.44,0.68)>
A2	<(s3,-0.20), (0.61,0.35)>	<(s2,0.46), (0.56,0.42)>
A3	<(s3,0.10), (0.56,0.35)>	<(s3,-0.24), (0.56,0.39)>
A4	<(s4,0.10), (0.57,0.35)>	<(s3,-0.04), (0.48,0.46)>
A5	<(s2,0.00), (0.59,0.64)>	<(s2,-0.27), (0.56,0.70)>

Source: Author calculation.

According to the aggregating results shown in Table 5 and the score functions of the ERP systems are shown in Table 6.

Table 6. The score functions of the ERP systems.

	P2TLWA	P2TLWG
A1	(s2,-0.48)	(s1,0.31)
A2	(s2,-0.07)	(s2,-0.38)
A3	(s2,0.04)	(s2,-0.19)
A4	(s3,-0.28)	(s2,-0.18)
A5	(s1,0.35)	(s1,0.14)

Source: Author calculation.

According to the score functions shown in Table 6 and the comparison formula of score functions, the ordering of the e ERP systems are shown in Table 7. Note that “﹥” means “preferred to”. As we can see, depending on the aggregation operators used, the ordering of the ERP systems is same, and the best ERP systems is A₄.

Table 7. Ordering of the ERP systems.

	Ordering
P2TLWA	A4﹥A3﹥A2﹥A1﹥A5
P2TLWG	A4﹥A3﹥A2﹥A1﹥A5

Source: Author calculation.

From Tables 4 and 7, we can easily find that these two above mentioned models may generate slightly different ranking results. The main reason causing this ranking result difference is that the P2TLPWA (P2TLPWG) operator can more accurately model the relationships between attributes by introducing the relationship structure of power operations. However, the P2TLWA (P2TLWG) operators which are proposed does not consider such actual situations that some of arguments may be related to all other input arguments.

7. Conclusion

In this paper, we investigate the multiple attribute decision making problems with Pythagorean 2-tuple linguistic information. Then, we utilize power average(Yager, 2001) and power geometric operations(Xu & Yager, 2010) to develop some Pythagorean 2-tuple linguistic power aggregation operators: Pythagorean 2-tuple linguistic power weighted average (P2TLPWA) operator, Pythagorean 2-tuple linguistic power weighted geometric (P2TLPWG) operator, Pythagorean 2-tuple linguistic power ordered weighted average (P2TLPOWA) operator, Pythagorean 2-tuple linguistic power ordered weighted geometric (P2TLPOWG) operator, Pythagorean 2-tuple linguistic power hybrid average (P2TLPHA) operator and Pythagorean 2-tuple linguistic power hybrid geometric (P2TLPHG) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the Pythagorean 2-tuple linguistic multiple attribute decision making problems. Finally, a practical example for enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness. In the future, the application of the proposed aggregating operators of P2TLSs needs to be explored in the decision making, risk analysis and many other fields under uncertain environment (Deng & Gao, 2019; Deng, Wang, & Wei, 2019; Gao, Lu, Wei, & Wei, 2018; Han & Liu, 2011; Li & Lu, 2019; Li, Wei, & Lu, 2018; Lin, Zhao, & Wei, 2013; Liu, 2009; Liu, Jin, Zhang, Su, & Wang, 2011; Liu & Zhang, 2010; Liu, Liu, Liu, & Pang, 2017; Lu, Tang, Wei, Wei, & Wei, 2019; Lu & Wei, 2019; Mardani et al. 2015; Merigó, 2008, 2009a, 2009b, 2010; Merigó, Casanovas, & Martínez, 2010; Merigó & Casanovas, 2009; Ngan, 2011; Tang et al., 2019; Wang, Wang, & Wei, 2019; Wang, Gao, & Lu, 2019; Wang, Wang, et al., 2019; Wang, Wang, et al., 2019a,b; Wang, Gao, et al., 2019; Wang, Lu, et al., 2019; Wang, Wei, et al., 2019; Wei, 2018, 2019a, 2019b, 2019c; Wei, Wang, Wei, Wei, & Zhang, 2019a, 2019b; Wei & Wei, 2018; Wu, Gao, & Wei, 2019; Wu, Gao, et al., 2019; Ye, 2009a,b).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Natural Science Foundation of China under Grant No. 61174149 and 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (No. 17XJA630003) and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004).