Multiple attribute decision making based on interval-valued Pythagorean uncertain linguistic aggregation operators

Abstract

In this paper, we investigate the multiple attribute decision making problems with interval-valued Pythagorean uncertain linguistic information. Then, we utilize arithmetic and geometric operations to develop some interval-valued Pythagorean uncertain linguistic aggregation operators: interval-valued Pythagorean uncertain linguistic weighted average (IVPULWA) operator, interval-valued Pythagorean uncertain linguistic weighted geometric (IVPULWG) operator, interval-valued Pythagorean uncertain linguistic ordered weighted average (IVPULOWA) operator, interval-valued Pythagorean uncertain linguistic ordered weighted geometric (IVPULOWG) operator, interval-valued Pythagorean uncertain linguistic hybrid average (IVPULHA) operator and interval-valued Pythagorean uncertain linguistic hybrid geometric (IVPULHG) operator, some interval-valued Pythagorean uncertain linguistic correlate aggregation operators, some interval-valued Pythagorean uncertain linguistic induced aggregation operators, some interval-valued Pythagorean uncertain linguistic induced correlate aggregation operators and some interval-valued Pythagorean uncertain linguistic power aggregating operators. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the interval-valued Pythagorean uncertain 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 uncertain linguistic set interval-valued Pythagorean uncertain linguistic set interval-valued Pythagorean uncertain linguistic weighted average (IVPULWA) operator interval-valued pythagorean uncertain linguistic weighted geometric (IVPULWG) operator enterprise resource planning (ERP) system selection

1. Introduction

Multiple attribute decision making (MADM) 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 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Liao et al. [19] used linguistic information processing model for selecting an ERP system. Wang [20] presented a 2-tuple fuzzy linguistic evaluation model for selecting appropriate agile manufacturing system. Tai and Chen [21] developed the intellectual capital evaluation model linguistic variable. Xu et al. [22] developed some methods to deal with unacceptable incomplete 2-tuple fuzzy linguistic preference relations in group decision making. Wang et al. [23] developed the multi-criteria group decision making method based on interval 2-tuple linguistic information and Choquet integral aggregation operators. Qin and Liu [24] proposed the 2-tuple linguistic Muirhead mean operators for multiple attribute group decision making and its application to supplier selection. Zhang et al. [25] developed the consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. Furthermore, Xu [26] developed the uncertain linguistic arithmetic aggregating operators. Xu [27] developed induced uncertain linguistic OWA (IULOWA) operators and applied the IULOWA operators to MADM with uncertain linguistic information. Xu [28] defined the concept of uncertain multiplicative linguistic preference relation and some operational laws of uncertain multiplicative linguistic variables, and proposed some uncertain linguistic geometric aggregation operators. Wei [29] proposed an uncertain linguistic hybrid geometric mean (ULHGM) operator. Zhang and Guo [30] developed a method for multi-granularity uncertain linguistic group decision making with incomplete weight information. Zhou et al. [31] proposed some uncertain linguistic prioritized aggregation operators for MADM problems. Wei et al. [32] developed some uncertain linguistic Bonferroni mean operators for MADM problems. Lin et al. [33] defined the similarity-based approach for group decision making with multi-granularity linguistic information. Meng and Chen [34] proposed an Approach to uncertain linguistic multi-attribute group decision making based on interactive index.

More recently, Pythagorean fuzzy set (PFS) [35, 36] 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 [37] 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 MADM problem within PFNs. Peng and Yang [38] proposed the division and subtraction operations for PFNs, and also developed a Pythagorean fuzzy superiority and inferiority ranking method to solve MAGDM with PFNs. Afterwards, Beliakov and James [39] 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 and Yager [40] applied the PFNs in handling the collaborative-based recommender system. Gou et al. [41] investigate the Properties of Continuous Pythagorean Fuzzy Information. Ren et al. [42] proposed the Pythagorean fuzzy TODIM approach to MADM. Garg [43] proposed the new generalized Pythagorean fuzzy information aggregation by using Einstein operations. Zeng et al. [44] developed a hybrid method for Pythagorean fuzzy multiple-criteria decision making. Garg [45] studied a novel accuracy function under interval-valued Pythagorean fuzzy environment for solving MADM problem. Lu et al. [46] defined the concept of hesitant Pythagorean fuzzy sets and utilized Hamacher operations to develop some hesitant Pythagorean fuzzy aggregation operators: hesitant Pythagorean fuzzy Hamacher weighted average (HPFHWA) operator, hesitant Pythagorean fuzzy Hamacher weighted geometric (HPFHWG) operator, hesitant Pythagorean fuzzy Hamacher ordered weighted average (HPFHOWA) operator, hesitant Pytha- gorean fuzzy Hamacher ordered weighted geometric (HPFHOWG) operator, hesitant Pythagorean fuzzy Hamacher hybrid average (HPFHHA) operator and hesitant Pythagorean fuzzy Hamacher hybrid geometric (HPFHHG) operator. Wei et al. [47] defined the concept of Pythagorean 2-tuple linguistic sets and utilized arithmetic and geometric operations to develop some Pythagorean 2-tuple linguistic aggregation operators: Pythagorean 2-tuple linguistic weighted average (P2TLWA) operator, Pythagorean 2-tuple linguistic weighted geometric (P2TLWG) operator, Pythagorean 2-tuple linguistic ordered weighted average (P2TLOWA) operator, Pythagorean 2-tuple linguistic ordered wei- ghted geometric (P2TLOWG) operator, Pythagorean 2-tuple linguistic hybrid average (P2TLHA) operator and Pythagorean 2-tuple linguistic hybrid geometric (P2TLHG) operator. Lu and Wei [48] defined the concept of Pythagorean uncertain linguistic sets and utilized arithmetic and geometric operations to develop some Pythagorean uncertain linguistic aggregation operators: Pythagorean uncertain linguistic weighted average (PULWA) operator, Pythagorean uncertain linguistic weighted geometric (PULWG) operator, Pythagorean uncertain linguistic ordered weighted average (PULOWA) operator, Pythagorean uncertain linguistic ordered weighted geometric (PULOWG) operator, Pythagorean uncertain linguistic hybrid average (PULHA) operator and Pythagorean uncertain linguistic hybrid geometric (PULHG) operator.

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. However, all the above approaches are unsuitable to describe the interval-valued membership degree and the interval-valued non-membership degree of an element to an uncertain linguistic label, which can reflect the decision makers’ confidence level when they are making an evaluation. In order to overcome this limit, we shall propose the concept of interval-valued Pythagorean uncertain linguistic set to solve this problem based on the interval-valued Pythagorean fuzzy sets [45] and uncertain linguistic information processing model [26, 27]. Thus, how to aggregate these interval-valued Pythagorean uncertain linguistic numbers is an interesting topic. To solve this issue, in this paper, we shall develop some interval-valued Pythagorean uncertain linguistic information aggregation operators on the basis of the traditional arithmetic and geometric operations. In order to do so, the remainder of this paper is set out as follows. In the next section, we shall propose the concept of interval-valued Pythagorean uncertain linguistic set on the basis of the interval-valued Pythagorean fuzzy set and uncertain linguistic variables. In Section 3, we shall propose some interval-valued Pythagorean uncertain linguistic arithmetic aggregation operators. In Section 4, we shall propose some interval-valued Pythagorean uncertain linguistic geometric aggregation operators. In Section 5, we shall propose some interval-valued Pythagorean uncertain linguistic correlate aggregation operators. In Section 6, we shall propose some interval-valued Pythagorean uncertain linguistic induced aggregation operators and some interval-valued Pythagorean uncertain linguistic induced correlate aggregation operators. In Section 7, we shall propose some interval-valued Pythagorean uncertain linguistic prioritized aggregating operators. In Section 8, based on these operators, we shall present some models for MADM problems with interval-valued Pythagorean uncertain linguistic information. In Section 9, we shall present a numerical example for enterprise resource planning (ERP) system selection with interval-valued Pythagorean uncertain linguistic information in order to illustrate the method proposed in this paper. Section 10 concludes the paper with some remarks.

2. Preliminaries

In the following, we introduced some basic concepts related to uncertain linguistic variables and interval-valued Pythagorean fuzzy sets. Then, we shall propose the interval-valued Pythagorean uncertain linguistic sets.

2.1 Uncertain linguistic term sets

Let $S=\{{s_{i}}|{i=1,2,\ldots,t}\}$ 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 [1, 2]:

(1) The set is ordered: $s_{i}>s_{j}$ , if $i>j$ ; (2) Max operator: $\max({s_{i},s_{j}})=s_{i}$ , if $s_{i}\geqslant s_{j}$ ; (3) Min operator: $\min(s_{i},s_{j})=s_{i}$ , if $s_{i}\leqslant s_{j}$ . For example, $S$ can be defined as

$\displaystyle S=\{s_{1}=\textit{extremely poor},s_{2}=\textit{very poor},s_{3}% =\textit{poor},s_{4}=\textit{medium},s_{5}=\textit{good},s_{6}=\textit{very % good},s_{7}=\textit{extremely good}\}$

Let $\tilde{s}=[s_{\alpha},s_{\beta}]$ , where $s_{\alpha},s_{\beta}\in S$ , $s_{\alpha}$ and $s_{\beta}$ are the lower and the upper limits, respectively. We call $\tilde{{s}}$ the uncertain linguistic variable. Let $\tilde{{S}}$ be the set of all the uncertain linguistic variable sets [26, 27].

Consider any three uncertain linguistic variables $\tilde{s}=[s_{\alpha},s_{\beta}]$ , $\tilde{{s}}_{1}=[s_{\alpha_{1}},s_{\beta_{1}}]$ and $\tilde{{s}}_{2}=[s_{\alpha_{2}},s_{\beta_{2}}]$ , $\tilde{{s}},$ $\tilde{{s}}_{1},$ $\tilde{{s}}_{2}\in\tilde{{S}},\lambda\in[0,1]$ , Xu [26, 27] defined their operational laws as follows:

(1) (1)
$\tilde{{s}}_{1}\oplus\tilde{{s}}_{2}=[{s_{\alpha_{1}},s_{\beta_{1}}}]\oplus[{s% _{\alpha_{2}},s_{\beta_{2}}}]=[s_{\alpha_{1}}\oplus s_{\alpha_{2}},$ $s_{\beta_{1}}\oplus s_{\beta_{2}}]=[{s_{\alpha_{1}\div\alpha_{2}},s_{\beta_{1}% \div\beta_{2}}}]$ ;
(2)
$\tilde{{s}}_{1}\otimes\tilde{{s}}_{2}=[{s_{\alpha_{1}},s_{\beta_{1}}}]\otimes[% {s_{\alpha_{2}},s_{\beta_{2}}}]=[s_{\alpha_{1}}\otimes s_{\alpha_{2}},$ $s_{\beta_{1}}\otimes s_{\beta_{2}}]=[{s_{\alpha_{1}\alpha_{2}},s_{\beta_{1}% \beta_{2}}}]$ ;
(3)
$\lambda\tilde{{s}}=\lambda[{s_{\alpha},s_{\beta}}]=[{\lambda s_{\alpha},% \lambda s_{\beta}}]=[{s_{\lambda\alpha},s_{\lambda\beta}}]$ .
(4)
$({\tilde{{s}}})^{\lambda}\mspace{-2.5mu }=\mspace{-2.5mu }({[{s_{\alpha},s_{% \beta}}]})^{\lambda}\mspace{-2.5mu }=\mspace{-2.5mu }[{({s_{\alpha}})^{\lambda% },({s_{\beta}})^{\lambda}}]\mspace{-2.5mu }=\mspace{-2.5mu }[{s_{\alpha^{% \lambda}},s_{\beta^{\lambda}}}]$ .

2.2 Interval-valued Pythagorean fuzzy set

Garg [45] developed the concept of the interval-valued Pythagorean fuzzy sets (IVPFSs).

Definition 1 [45]. Let $X$ be a fix set. An IVPFS is an object having the form

$\displaystyle P=\{{\langle{x,({\tilde{{\mu}}_{P}(x),\tilde{{\nu}}_{P}(x)})}% \rangle|{x\in X}}\}$ (1)

where $\tilde{{\mu}}_{P}(x)\subset[0,1]$ and $\tilde{{\nu}}_{P}(x)\subset[0,1]$ are interval numbers, $\tilde{{\mu}}_{P}(x)=[{\mu_{P}^{L}(x),\mu_{P}^{R}(x)}],\tilde{{\nu}}_{P}(x)=[{% \nu_{P}^{L}(x),\nu_{P}^{R}(x)}]$ with the condition $0\leqslant({\mu_{P}^{R}(x)})^{2}+({\nu_{P}^{R}(x)})^{2}\leqslant 1$ , $\forall x\in X$ . The numbers $\tilde{{\mu}}_{P}(x),\tilde{{\nu}}_{P}(x)$ represent, respectively, the degree of positive membership, degree of negative membership and degree of negative membership of the element $x$ to $P$ . Then for $x\in X$ ,

$\displaystyle\tilde{{\pi}}_{P}(x)=[{\pi_{P}^{L}(x),\pi_{P}^{R}(x)}]=\left[% \sqrt{1-({({\mu_{P}^{R}(x)})^{2}+({\nu_{P}^{R}(x)})^{2}})},\right.\left.\sqrt{% 1-({({\mu_{P}^{L}(x)})^{2}+({\nu_{P}^{L}(x)})^{2}})}\right]$

could be called the degree of refusal membership of the element $x$ to $P$ .

Definition 2 [45]. Let $p_{1}\mspace{-5.0mu }=\mspace{-5.0mu }([{u_{\tilde{{p}}_{1}}^{L},u_{\tilde{{p}% }_{1}}^{R}}],[{v_{\tilde{{p}}_{1}}^{L},v_{\tilde{{p}}_{1}}^{R}}])$ , $p_{2}=$ $([{u_{p_{2}}^{L},u_{p_{2}}^{R}}],[{v_{p_{2}}^{L},v_{p_{2}}^{R}}])$ , and $p=({[{u_{p}^{L},u_{p}^{R}}],[{v_{p}^{L},v_{p}^{R}}]})$ be three interval-valued Pythagorean fuzzy numbers, and some basic operations on them are defined as follows:

(1) (1)

$p_{1}\oplus p_{2}\mspace{-6.0mu }=\mspace{-8.0mu }\left(\mspace{-3.0mu }\begin% {bmatrix}\sqrt{({u_{p_{1}}^{L}})^{2}\mspace{-4.0mu }+\mspace{-4.0mu }({u_{p_{2% }}^{L}})^{2}\mspace{-4.0mu }-\mspace{-4.0mu }({u_{p_{1}}^{L}})^{2}({u_{p_{2}}^% {L}})^{2}},\\ \sqrt{({u_{p_{1}}^{R}})^{2}\mspace{-4.0mu }+\mspace{-4.0mu }({u_{p_{2}}^{R}})^% {2}\mspace{-4.0mu }-\mspace{-4.0mu }({u_{p_{1}}^{R}})^{2}({u_{p_{2}}^{R}})^{2}% }\\ \end{bmatrix}\right.\mspace{-6.0mu },$

$\left.\rule{0.0pt}{20.0pt}[{v_{p_{1}}^{L}v_{p_{2}}^{L},v_{p_{1}}^{R}v_{p_{2}}^% {R}}]\right)$ ;

(2)

$p_{1}\otimes p_{2}=\left([{\mu_{p_{1}}^{L}v_{p_{2}}^{L},\mu_{p_{1}}^{R}v_{p_{2% }}^{R}}],\rule{0.0pt}{20.0pt}\right.$ $\left.\begin{bmatrix}\sqrt{({v_{p_{1}}^{L}})^{2}+({v_{p_{2}}^{L}})^{2}-({v_{p_% {1}}^{L}})^{2}({v_{p_{2}}^{L}})^{2}},\\ \sqrt{({v_{p_{1}}^{R}})^{2}+({v_{p_{2}}^{R}})^{2}-({v_{p_{1}}^{R}})^{2}({v_{p_% {2}}^{R}})^{2}}\\ \end{bmatrix}\right);$

(3)

$\lambda p\mspace{-2.0mu }=\mspace{-3.0mu }\left(\mspace{-2.0mu }\left[\sqrt{1% \mspace{-3.5mu }-\mspace{-3.5mu }(1\mspace{-3.5mu }-\mspace{-3.5mu }({u_{p}^{L% }})^{2})^{\lambda}},\sqrt{1\mspace{-3.5mu }-\mspace{-3.5mu }(1\mspace{-3.5mu }% -\mspace{-3.5mu }({u_{p}^{R}})^{2})^{\lambda}}\right]\right.\mspace{-6.0mu },$ $\left.\rule{0.0pt}{10.0pt}[{({v_{p}^{L}})^{\lambda},({v_{p}^{R}})^{\lambda}}]% \right),\lambda>0;$

(4)

$(p)^{\lambda}\mspace{-2.0mu }=\mspace{-3.0mu }\left(\mspace{-2.0mu }[{({\mu_{p% }^{L}})^{\lambda},({\mu_{p}^{R}})^{\lambda}}],\left[\sqrt{1-({1-({v_{p}^{L}})^% {2}})^{\lambda}}\right.\right.,$ $\left.\left.\sqrt{1-({1-({v_{p}^{R}})^{2}})^{\lambda}}\right]\right),\lambda>0;$

(5)

$p^{c}=(\nu,\mu).$

Based on the Definition 2, Garg [45] derived the following properties easily.

Theorem 1 [45]. Let $p_{1}=({[{u_{\tilde{{p}}_{1}}^{L},u_{\tilde{{p}}_{1}}^{R}}],[{v_{\tilde{{p}}_{% 1}}^{L},v_{\tilde{{p}}_{1}}^{R}}]})$ and $p_{2}=({[{u_{p_{2}}^{L},u_{p_{2}}^{R}}],[{v_{p_{2}}^{L},v_{p_{2}}^{R}}]})$ be two interval-valued Pythagorean fuzzy numbers, $\lambda,\lambda_{1},\lambda_{2}>0$ , then

(1) (1)

$p_{1}\oplus p_{2}=p_{2}\oplus p_{1};$

(2)

$p_{1}\otimes p_{2}=p_{2}\otimes p_{1};$

(3)

$\lambda({p_{1}\oplus p_{2}})=\lambda p_{1}\oplus\lambda p_{2};$

(4)

$({p_{1}\otimes p_{2}})^{\lambda}=({p_{1}})^{\lambda}\otimes({p_{2}})^{\lambda};$

(5)

$\lambda_{1}p_{1}\oplus\lambda_{2}p_{1}=({\lambda_{1}+\lambda_{2}})p_{1};$

(6)

$({p_{1}})^{\lambda_{1}}\otimes({p_{1}})^{\lambda_{2}}=({p_{1}})^{({\lambda_{1}% +\lambda_{2}})};$

(7)

$({({p_{1}})^{\lambda_{1}}})^{\lambda_{2}}=({p_{1}})^{\lambda_{1}\lambda_{2}}.$

2.3 Interval-valued Pythagorean uncertain linguistic sets

In the following, we shall propose the concepts and basic operations of the interval-valued Pythagorean uncertain linguistic sets on the basis of the interval-valued Pythagorean fuzzy sets [45] and uncertain linguistic information processing models [26, 27].

Definition 3. An interval-valued Pythagorean uncertain linguistic sets $\tilde{{P}}$ in $X$ is given

$\displaystyle\tilde{{P}}=\{{\langle{\tilde{{s}}_{\theta(x)},({\tilde{{\mu}}_{% \tilde{{P}}}(x),\tilde{{\nu}}_{P}(x)})}\rangle,x\in X}\}$ (2)

where $\tilde{{s}}_{\theta(x)}({\tilde{{s}}_{\theta(x)}=[{s_{\theta(x)}^{L},s_{\theta% (x)}^{R}}]})s_{\theta(x)}^{L},s_{\theta(x)}^{R}\in S$ , $\tilde{{\mu}}_{\tilde{{P}}}(x)\subset[0,1]$ and $\tilde{{\nu}}_{P}(x)\subset[0,1]$ are interval numbers, $\tilde{{\mu}}_{\tilde{{P}}}(x)=[{\mu_{\tilde{{P}}}^{L}(x),\mu_{\tilde{{P}}}^{R% }(x)}],\tilde{{\nu}}_{\tilde{{P}}}(x)=[\nu_{\tilde{{P}}}^{L}(x),$ $\nu_{\tilde{{P}}}^{R}(x)]$ with the condition $0\leqslant({\mu_{\tilde{{P}}}^{R}(x)})^{2}+({\nu_{\tilde{{P}}}^{R}(x)})^{2}\leqslant 1$ , $\forall x\in X$ . The numbers $\tilde{{\mu}}_{\tilde{{P}}}(x),\tilde{{\nu}}_{\tilde{{P}}}(x)$ represent, respectively, $t$ the degree of membership and degree of non-membership of the element $x$ to uncertain linguistic variable $\tilde{{s}}_{\theta(x)}$ . Then for $x\in X$ , $\tilde{{\pi}}_{\tilde{{P}}}(x)=[\pi{}_{\tilde{{P}}}^{L}(x),\pi_{\tilde{{P}}}^{% R}(x)]=\left[\sqrt{1-({({\mu_{\tilde{{P}}}^{R}(x)})^{2}+({\nu_{\tilde{{P}}}^{R% }(x)})^{2}})},\right.$ $\left.\sqrt{1-({({\mu_{\tilde{{P}}}^{L}(x)})^{2}+({\nu_{\tilde{{P}}}^{L}(x)})^% {2}})}\right]$ could be called the degree of hesitant degree of the element $x$ to uncertain linguistic variable $\tilde{{s}}_{\theta(x)}$ .

If $s_{\theta(x)}^{L}=s_{\theta(x)}^{R}$ , then the interval-valued Pytha- gorean uncertain linguistic sets reduce to interval-valued Pythagorean linguistic sets; If $\mu_{\tilde{{P}}}^{L}(x)=\mu_{\tilde{{P}}}^{R}(x),$ $\nu_{\tilde{{P}}}^{L}(x)=\nu_{\tilde{{P}}}^{R}(x)$ , then interval-valued Pythagorean uncertain linguistic sets reduce to Pythagorean uncertain linguistic sets; If $s_{\theta(x)}^{L}=s_{\theta(x)}^{R}$ and $\mu_{\tilde{{P}}}^{L}(x)=\mu_{\tilde{{P}}}^{R}(x),$ $\nu_{\tilde{{P}}}^{L}(x)=\nu_{\tilde{{P}}}^{R}(x)$ , then, interval-valued Pythagorean uncertain linguistic sets reduce to Pythagorean linguistic sets.

For convenience, we call $\tilde{{p}}=\langle{\tilde{{s}}_{\theta(p)},({\tilde{{u}}_{\tilde{{p}}},\tilde% {{v}}_{\tilde{{p}}}})}\rangle=\langle[{s_{\theta(p)}^{L},s_{\theta(p)}^{R}}],(% [u_{\tilde{{p}}}^{L},u_{\tilde{{p}}}^{R}],[{v_{\tilde{{p}}}^{L},v_{\tilde{{p}}% }^{R}}])\rangle$ an interval-valued Pythagorean uncertain linguistic number (IVPULN), where $\tilde{{u}}_{\tilde{{p}}}\subset[{0,1}],\tilde{{v}}_{\tilde{{p}}}\subset[{0,1}]$ , $({u_{\tilde{{p}}}^{R}})^{2}+({\nu_{\tilde{{p}}}^{R}})^{2}\leqslant 1$ , $s_{\theta({\tilde{{p}}})}^{L},s_{\theta({\tilde{{p}}})}^{R}\in S$ .

Definition 4. Let $\tilde{{p}}=\langle[s_{\theta({\tilde{{p}}})}^{L},s_{\theta({\tilde{{p}}})}^{R% }],([{u_{\tilde{{p}}}^{L},u_{\tilde{{p}}}^{R}}],$ $[v_{\tilde{{p}}}^{L},$ $v_{\tilde{{p}}}^{R}])\rangle$ be an IVPULN, a score function $\tilde{{p}}$ of an IVPULN can be represented as follows:

$\displaystyle S({\tilde{{p}}})=[{s_{\theta({\tilde{{p}}})}^{L},s_{\theta({% \tilde{{p}}})}^{R}}]\cdot\frac{2+({u_{\tilde{{p}}}^{L}})^{2}+({u_{\tilde{{p}}}% ^{R}})^{2}-({v_{\tilde{{p}}}^{L}})^{2}-({v_{\tilde{{p}}}^{R}})^{2}}{4}=\mspace% {-2.0mu }\left[s_{\theta({\tilde{{p}}})}^{L}{\cdot}\frac{2{+}({u_{\tilde{{p}}}% ^{L}})^{2}{+}({u_{\tilde{{p}}}^{R}})^{2}{-}({v_{\tilde{{p}}}^{L}})^{2}{-}({v_{% \tilde{{p}}}^{R}})^{2}}{4},\right.\mspace{-10.0mu }\mspace{-60.0mu }\left.s_{% \theta({\tilde{{p}}})}^{R}\cdot\frac{2+({u_{\tilde{{p}}}^{L}})^{2}+({u_{\tilde% {{p}}}^{R}})^{2}-({v_{\tilde{{p}}}^{L}})^{2}-({v_{\tilde{{p}}}^{R}})^{2}}{4}\right]$ (3)

Motivated by the operations of the uncertain linguistic variables and Definition 3, in the following, we shall define some operational laws of IVPULNs.

Definition 5. Let $\tilde{{p}}_{1}=\langle[{s_{\theta({\tilde{{p}}_{1}})}^{L},s_{\theta({\tilde{{% p}}_{1}})}^{R}}],([{u_{\tilde{{p}}_{1}}^{L},u_{\tilde{{p}}_{1}}^{R}}],$ $[v_{\tilde{{p}}_{1}}^{L},v_{\tilde{{p}}_{1}}^{R}])\rangle$ and $\tilde{{p}}_{2}=\langle[{s_{\theta({\tilde{{p}}_{2}})}^{L},s_{\theta({\tilde{{% p}}_{2}})}^{R}}],([{u_{\tilde{{p}}_{2}}^{L},u_{\tilde{{p}}_{2}}^{R}}],[v_{% \tilde{{p}}_{2}}^{L},$ $v_{\tilde{{p}}_{2}}^{R}])\rangle$ be two IVPULNs, then

$\displaystyle\tilde{{p}}_{1}\oplus\tilde{{p}}_{2}=\left\langle\left[s_{\theta(% {\tilde{{p}}_{1}})}^{L}\oplus s_{\theta({\tilde{{p}}_{2}})}^{L},s_{\theta({% \tilde{{p}}_{1}})}^{R}\oplus s_{\theta({\tilde{{p}}_{2}})}^{R}\right],\right.% \left(\left[\sqrt{({u_{\tilde{{p}}_{1}}^{L}})^{2}+({u_{\tilde{{p}}_{2}}^{L}})^% {2}-({u_{\tilde{{p}}_{1}}^{L}})^{2}({u_{\tilde{{p}}_{2}}^{L}})^{2}},\right.% \right.\mspace{-10.0mu }\left.\sqrt{({u_{\tilde{{p}}_{1}}^{R}})^{2}+({u_{% \tilde{{p}}_{2}}^{R}})^{2}-({u_{\tilde{{p}}_{1}}^{R}})^{2}({u_{\tilde{{p}}_{2}% }^{R}})^{2}}\right],\left.\left.\rule{0.0pt}{10.0pt}[{v_{\tilde{{p}}_{1}}^{L}v% _{\tilde{{p}}_{2}}^{L},v_{\tilde{{p}}_{1}}^{R}v_{\tilde{{p}}_{2}}^{R}}]\right)% \right\rangle;$ $\displaystyle\tilde{{p}}_{1}\otimes\tilde{{p}}_{2}=\left\langle\left[s_{\theta% ({\tilde{{p}}_{1}})}^{L}\otimes s_{\theta({\tilde{{p}}_{2}})}^{L},s_{\theta({% \tilde{{p}}_{1}})}^{R}\otimes s_{\theta({\tilde{{p}}_{2}})}^{R}\right],\right.% \left(\rule{0.0pt}{10.0pt}[\mu_{\tilde{{p}}_{1}}^{L}v_{\tilde{{p}}_{2}}^{L},% \mu_{\tilde{{p}}_{1}}^{R}v_{\tilde{{p}}_{2}}^{R}],\right.\left[\sqrt{({v_{% \tilde{{p}}_{1}}^{L}})^{2}+({v_{\tilde{{p}}_{2}}^{L}})^{2}-({v_{\tilde{{p}}_{1% }}^{L}})^{2}({v_{\tilde{{p}}_{2}}^{L}})^{2}},\right.\mspace{-20.0mu }\left.% \left.\left.\sqrt{(v_{\tilde{{p}}_{1}}^{R})^{2}{+}({v_{\tilde{{p}}_{2}}^{R}})^% {2}{-}({v_{\tilde{{p}}_{1}}^{R}})^{2}({v_{\tilde{{p}}_{2}}^{R}})^{2}}\right]% \right)\mspace{-3.0mu }\right\rangle;\mspace{-90.0mu }\lambda\tilde{{p}}_{1}=% \left\langle\left[\lambda s_{\theta(\tilde{{p}}_{1})}^{L},\lambda s_{\theta({% \tilde{{p}}_{1}})}^{R}\right],\left(\left[\sqrt{1\mspace{-3.1mu }-\mspace{-3.1% mu }(1\mspace{-3.1mu }-\mspace{-3.1mu }(u_{\tilde{{p}}_{1}}^{L})^{2})^{\lambda% }},\right.\right.\right.\mspace{-45.0mu }\left.\left.\left.\rule{0.0pt}{10.0pt% }\sqrt{1\mspace{-1.0mu }-\mspace{-1.0mu }({1\mspace{-1.0mu }-\mspace{-1.0mu }(% {u_{\tilde{{p}}_{1}}^{R}})^{2}})^{\lambda}}\right],\left[({v_{\tilde{{p}}_{1}}% ^{L}})^{\lambda},(v_{\tilde{{p}}_{1}}^{R})^{\lambda}\right]\right)\right% \rangle;\mspace{-90.0mu }(\tilde{{p}}_{1})^{\lambda}=\left\langle{\left[{\left% ({s_{\theta\left({\tilde{{p}}_{1}}\right)}^{L}}\right)^{\lambda},\left({s_{% \theta\left({\tilde{{p}}_{1}}\right)}^{R}}\right)^{\lambda}}\right]}\right.,% \mspace{-28.0mu }\left(\left[(\mu_{\tilde{{p}}_{1}}^{L})^{\lambda},(\mu_{% \tilde{{p}}_{1}}^{R})^{\lambda}\right],\left[\sqrt{1\mspace{-1.0mu }-\mspace{-% 1.0mu }(1\mspace{-1.0mu }-\mspace{-1.0mu }(v_{\tilde{{p}}_{1}}^{L})^{2})^{% \lambda}},\right.\right.\mspace{-33.0mu }\left.\left.\left.\sqrt{1-({1-({v_{% \tilde{{p}}_{1}}^{R}})^{2}})^{\lambda}}\right]\right)\right\rangle.$

Based on the Definition 5, we can derive the following properties easily.

Theorem 2. For any two IVPULNs $\tilde{{p}}_{1}=\langle[s_{\theta(\tilde{{p}}_{1})}^{L},$ $s_{\theta(\tilde{{p}}_{1})}^{R}],([u_{\tilde{{p}}_{1}}^{L},u_{\tilde{{p}}_{1}}% ^{R}],[v_{\tilde{{p}}_{1}}^{L},v_{\tilde{{p}}_{1}}^{R}])\rangle$ and $\tilde{{p}}_{2}=\langle[s_{\theta(\tilde{{p}}_{2})}^{L},$ $s_{\theta(\tilde{{p}}_{2})}^{R}],$ $([u_{\tilde{{p}}_{2}}^{L},u_{\tilde{{p}}_{2}}^{R}],$ $[v_{\tilde{{p}}_{2}}^{L},$ $v_{\tilde{{p}}_{2}}^{R}])\rangle$ , it can be proved the calculation rules shown as follows

(1) (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)

$\lambda(\tilde{{p}}_{1}\oplus\tilde{{p}}_{2})=\lambda\tilde{{p}}_{1}\oplus% \lambda\tilde{{p}}_{2},0\leqslant\lambda\leqslant 1;$

(4)

$\lambda_{1}\tilde{{p}}_{1}\oplus\lambda_{2}\tilde{{p}}_{1}=(\lambda_{1}\oplus% \lambda_{2})\tilde{{p}}_{1}0\leqslant\lambda_{1},\lambda_{2},\lambda_{1}+% \lambda_{2}\leqslant 1;$

(5)

$\tilde{{p}}_{1}^{\lambda_{1}}\otimes\tilde{{p}}_{1}^{\lambda_{2}}=(\tilde{{p}}% _{1})^{\lambda_{1}+\lambda_{2}}0\leqslant\lambda_{1},\lambda_{2},\lambda_{1}+% \lambda_{2}\leqslant 1;$

(6)

$\tilde{{p}}_{1}^{\lambda_{1}}\otimes\tilde{{p}}_{2}^{\lambda_{1}}=(\tilde{{p}}% _{1}\otimes\tilde{{p}}_{2})^{\lambda_{1}}\lambda_{1}\geqslant 0;$

(7)

$({({\tilde{{p}}_{1}})^{\lambda_{1}}})^{\lambda_{2}}=({\tilde{{p}}_{1}})^{% \lambda_{1}\lambda_{2}}.$

Definition 6 [26, 27]. Let $\tilde{{s}}_{1}=\left[{s_{\alpha_{1}},s_{\beta_{1}}}\right]$ and $\tilde{{s}}_{2}=\left[{s_{\alpha_{2}},s_{\beta_{2}}}\right]$ be two uncertain linguistic variables, and let $len\left({\tilde{{s}}_{1}}\right)=\beta_{1}-\alpha_{1}$ , $len\left({\tilde{{s}}_{2}}\right)=\beta_{2}-\alpha_{2}$ , then the degree of possibility of $\tilde{{s}}_{1}\geqslant\tilde{{s}}_{2}$ is defined as

$\displaystyle p({\tilde{{s}}_{1}\geqslant\tilde{{s}}_{2}})=$ (4) $\displaystyle\frac{\max({0,\textit{len}({\tilde{{s}}_{1}})+\textit{len}({% \tilde{{s}}_{2}})-\max({\beta_{2}-\alpha_{1},0})})}{\textit{len}({\tilde{{s}}_% {1}})+\textit{len}({\tilde{{s}}_{2}})}$

From Definition 6, we can easily get the following results easily:

(1) (1)

$0\leqslant p({\tilde{{s}}_{1}\geqslant\tilde{{s}}_{2}})\leqslant 1,0\leqslant p% ({\tilde{{s}}_{2}\geqslant\tilde{{s}}_{1}})\leqslant 1$ ;

(2)

$p({\tilde{{s}}_{1}\geqslant\tilde{{s}}_{2}})+p({\tilde{{s}}_{2}\geqslant\tilde% {{s}}_{1}})=1$ . Especially, $p({\tilde{{s}}_{1}\geqslant\tilde{{s}}_{1}})=p({\tilde{{s}}_{2}\geqslant\tilde% {{s}}_{2}})=0.5$ .

3. Interval-valued Pythagorean uncertain linguistic arithmetic aggregation operators

In this section, we shall develop some arithmetic aggregation operators with interval-valued Pythagorean uncertain linguistic information, such as interval-valuedPythagorean uncertain linguistic weighted averaging (IVPULWA) operator, interval-valued Pythagorean uncertain linguistic ordered weighted averaging (IVPULOWA) operator and interval-valued Pythagorean uncertain linguistic hybrid average (IVPULHA) operator.

Definition 7. Let $\tilde{{p}}_{j}=\langle[s_{\theta_{j}}^{L},$ $s_{\theta_{j}}^{R}],$ $([\mu_{j}^{L},\mu_{j}^{R}],$ $[\nu_{j}^{L},$ $\nu_{j}^{R}])\rangle$ $(j=1,2,\ldots,n)$ be a collection of IVPULNs. The interval-valued Pythagorean uncertain linguistic weighted averaging (IVPULWA) operator is a mapping $P^{n}\to P$ such that

$\displaystyle\text{IVPULWA}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})=\mathop{\oplus}\limits_{j=1}^{n}({\omega_{j}\tilde{{p}}_{j}})$ (5)

where $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})^{T}$ be the weight vector of $\tilde{{p}}_{j}(j=1,2,\ldots,n)$ , and $\omega_{j}>0$ , $\sum\nolimits_{j=1}^{n}{\omega_{j}}=1$ .

Based on the Definition 7 and Theorem 2, we can get the following result:

Theorem 3. The aggregated value by using IVPULWA operator is also an IVPULN, where

$\displaystyle\text{IVPULWA}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})=\mathop{\oplus}\limits_{j=1}^{n}({\omega_{j}\tilde{{p}}_{j}})$ $\displaystyle=\left\langle\left[{\sum\limits_{j=1}^{n}{\omega_{j}s_{\theta_{j}% }^{L}},\sum\limits_{j=1}^{n}{\omega_{j}s_{\theta_{j}}^{R}}}\right],\right.$ $\displaystyle\quad\left(\begin{bmatrix}\sqrt{1-\prod\limits_{j=1}^{n}{({1-({% \mu_{j}^{L}})^{2}})^{\omega_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{({1-({\mu_{j}^{R}})^{2}})^{\omega_{j}}}}\\ \end{bmatrix},\right.$ $\displaystyle\quad\left.\left.\begin{bmatrix}\prod\limits_{j=1}^{n}{({\nu_{j}^% {L}})^{\omega_{j}}},\\ \prod\limits_{j=1}^{n}{({\nu_{j}^{R}})^{\omega_{j}}}\\ \end{bmatrix}\right)\right\rangle$ (6)

where $\omega=\left({\omega_{1},\omega_{2},\ldots,\omega_{n}}\right)^{T}$ be the weight vector of $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ , and $\omega_{j}>0$ , $\sum_{j=1}^{n}{\omega_{j}}=1$ .

Proof We prove Eq. (3) by mathematical induction on $n$ .

(1)

When $n=2$ , we have

$\displaystyle\text{IVPULWA}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2}})=\omega% _{1}\tilde{{p}}_{1}\oplus\omega_{2}\tilde{{p}}_{2}$

By Theorem 2, we can see that both $\omega_{1}\tilde{{p}}_{1}$ and $\omega_{2}\tilde{{p}}_{2}$ are IVPULNs, and the value of $\omega_{1}\tilde{{p}}_{1}\oplus\omega_{2}\tilde{{p}}_{2}$ is also an IVPULN. From the operational laws of IVPULNs, we have

$\displaystyle\omega_{1}\tilde{{p}}_{1}\mspace{-3.2mu }=\mspace{-3.2mu }\left% \langle[\omega_{1}s_{\theta{}_{1}}^{L},\omega_{1}s_{\theta_{1}}^{R}],\left(% \left[\sqrt{1\mspace{-3.2mu }-\mspace{-3.2mu }({1\mspace{-3.2mu }-\mspace{-3.2% mu }({\mu_{1}^{L}})^{2}})^{\omega_{1}}}\mspace{-1.0mu },\right.\right.\right.$ $\displaystyle∼{}\left.\left.\left.\sqrt{1-({1-({\mu_{1}^{R}})^{2}})^{\omega_{1% }}}\right],[({\nu_{1}^{L}})^{\omega_{1}},({\nu_{1}^{R}})^{\omega_{1}}]\right)% \right\rangle;$

Then

$\displaystyle\text{IVPULWA}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2}})=\omega% _{1}\tilde{{p}}_{1}\oplus\omega_{2}\tilde{{p}}_{2}$ $\displaystyle=\langle{[{\omega_{1}s_{\theta_{1}}^{L}+\omega_{2}s_{\theta_{2}}^% {L},\omega_{1}s_{\theta_{1}}^{R}+\omega_{2}s_{\theta_{2}}^{R}}],}$ $\displaystyle\mspace{-13.0mu }\left.\left(\begin{array}[]{l}\left[\sqrt{\begin% {array}[]{l}2\mspace{-4.9mu }-\mspace{-4.9mu }({1\mspace{-4.9mu }-\mspace{-4.9% mu }({\mu_{1}^{L}})^{2}})^{\omega_{1}}\mspace{-4.9mu }-\mspace{-4.9mu }({1% \mspace{-4.9mu }-\mspace{-4.9mu }({\mu_{2}^{L}})^{2}})^{\omega_{2}}-\\ ({1\mspace{-4.9mu }-\mspace{-4.9mu }({1\mspace{-4.9mu }-\mspace{-4.9mu }({\mu_% {1}^{L}})^{2}})^{\omega_{1}}})({1\mspace{-4.9mu }-\mspace{-4.9mu }({1\mspace{-% 4.9mu }-\mspace{-4.9mu }({\mu_{2}^{L}})^{2}})^{\omega_{2}}})\end{array}}\right% .,\\ \left.\sqrt{\begin{array}[]{l}2\mspace{-4.9mu }-\mspace{-4.9mu }({1\mspace{-4.% 9mu }-\mspace{-4.9mu }({\mu_{1}^{R}})^{2}})^{\omega_{1}}\mspace{-4.9mu }-% \mspace{-4.9mu }({1\mspace{-4.9mu }-\mspace{-4.9mu }({\mu_{2}^{R}})^{2}})^{% \omega_{2}}-\\ ({1\mspace{-4.9mu }-\mspace{-4.9mu }({1\mspace{-4.9mu }-\mspace{-4.9mu }({\mu_% {1}^{R}})^{2}})^{\omega_{1}}})({1\mspace{-4.9mu }-\mspace{-4.9mu }({1\mspace{-% 4.9mu }-\mspace{-4.9mu }({\mu_{2}^{R}})^{2}})^{\omega_{2}}})\end{array}}\right% ],\\ [(\nu_{1}^{L})^{\omega_{1}}(\nu_{2}^{L})^{\omega_{2}},(\nu_{1}^{R})^{\omega_{1% }}({\nu_{2}^{R}})^{\omega_{2}}]\\ \end{array}\right)\right\rangle$ $\displaystyle=\langle[\omega_{1}s_{\theta_{1}}^{L}+\omega_{2}s_{\theta_{2}}^{L% },\omega_{1}s_{\theta_{1}}^{R}+\omega_{2}s_{\theta_{2}}^{R}],$ $\displaystyle\mspace{13.0mu }\left.\left(\begin{array}[]{l}\left[\sqrt{1-({1-(% {\mu_{1}^{L}})^{2}})^{\omega_{1}}({1-({\mu_{2}^{L}})^{2}})^{\omega_{2}}},% \right.\\ \left.\sqrt{1-({1-({\mu_{1}^{R}})^{2}})^{\omega_{1}}({1-({\mu_{2}^{R}})^{2}})^% {\omega_{2}}}\right],\\ [({\nu_{1}^{L}})^{\omega_{1}}({\nu_{2}^{L}})^{\omega_{2}},({\nu_{1}^{R}})^{% \omega_{1}}({\nu_{2}^{R}})^{\omega_{2}}]\\ \end{array}\right)\right\rangle$

(1)

Suppose that $n=k$ , Eq. (3) holds, i.e.,

$\displaystyle\text{IVPULWA}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{k}})$ $\displaystyle=\omega_{1}\tilde{{p}}_{1}\oplus\omega_{2}\tilde{{p}}_{2}\oplus% \ldots\omega_{k}\tilde{{p}}_{k}$ $\displaystyle=\left\langle\left[{\sum\limits_{j=1}^{k}{\omega_{j}s_{\theta_{j}% }^{L}},\sum\limits_{j=1}^{k}{\omega_{j}s_{\theta_{j}}^{R}}}\right],\right.$ $\displaystyle\mspace{-5.0mu }\left.\left(\mspace{-2.0mu }\left[\begin{array}[]% {l}\sqrt{1\mspace{-3.0mu }-\mspace{-7.0mu }\prod\limits_{j=1}^{k}{(1\mspace{-3% .0mu }-\mspace{-3.0mu }({\mu_{j}^{L}})^{2})^{\omega_{j}}}},\\ \sqrt{1\mspace{-3.0mu }-\mspace{-7.0mu }\prod\limits_{j=1}^{k}{(1\mspace{-3.0% mu }-\mspace{-3.0mu }({\mu_{j}^{R}})^{2})^{\omega_{j}}}}\\ \end{array}\mspace{-2.0mu }\right]\mspace{-3.0mu },\mspace{-3.0mu }\left[{% \begin{array}[]{l}\prod\limits_{j=1}^{k}\mspace{-3.0mu }{(\nu_{j}^{L})^{\omega% _{j}}},\\ \prod\limits_{j=1}^{k}\mspace{-3.0mu }{(\nu_{j}^{R})^{\omega_{j}}}\\ \end{array}}\mspace{-3.0mu }\right]\mspace{-3.0mu }\right)\mspace{-3.0mu }\right\rangle$

And the aggregated value is a IVPULN, Then when $n=k+1$ , by the operational laws of IVPULNs, we have

$\displaystyle\text{IVPULWA}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{k+1}})$ $\displaystyle=\omega_{1}\tilde{{p}}_{1}\oplus\omega_{2}\tilde{{p}}_{2}\oplus% \ldots\omega_{k}\tilde{{p}}_{k}\oplus\omega_{k+1}\tilde{{p}}_{k+1}$ $\displaystyle=\left\langle{\left[{\sum\limits_{j=1}^{k}{\omega_{j}s_{\theta_{j% }}^{L}},\sum\limits_{j=1}^{k}{\omega_{j}s_{\theta_{j}}^{R}}}\right]+[{\omega_{% k}s_{\theta_{k}}^{L},\omega_{k}s_{\theta_{k}}^{R}}]}\right.,$ $\displaystyle\mspace{-41.0mu }\left(\left[\sqrt{\begin{array}[]{l}1\mspace{-5.% 0mu }-\mspace{-5.0mu }\prod\limits_{j=1}^{k}\mspace{-3.0mu }(1\mspace{-5.0mu }% -\mspace{-5.0mu }({\mu_{j}^{L}})^{2})^{\omega_{j}}\mspace{-5.0mu }+\mspace{-5.% 0mu }({1\mspace{-5.0mu }-\mspace{-5.0mu }({1\mspace{-5.0mu }-\mspace{-5.0mu }(% {\mu_{k+1}^{L}})^{2}})^{\omega_{k+1}}})-\\ \left({1\mspace{-5.0mu }-\mspace{-5.0mu }\prod\limits_{j=1}^{k}\mspace{-3.0mu % }(1\mspace{-5.0mu }-\mspace{-5.0mu }({\mu_{j}^{L}})^{2})^{\omega_{j}}}\right)(% {1\mspace{-5.0mu }-\mspace{-5.0mu }({1\mspace{-5.0mu }-\mspace{-5.0mu }({\mu_{% k+1}^{L}})^{2}})^{\omega_{k+1}}})\\ \end{array}},\right.\right.$ $\displaystyle\mspace{-36.0mu }\left.\left.\sqrt{{\begin{array}[]{l}1\mspace{-5% .0mu }-\mspace{-5.0mu }\prod\limits_{j=1}^{k}\mspace{-3.0mu }(1\mspace{-5.0mu % }-\mspace{-5.0mu }({\mu_{j}^{R}})^{2})^{\omega_{j}}\mspace{-5.0mu }+\mspace{-5% .0mu }({1\mspace{-5.0mu }-\mspace{-5.0mu }({1\mspace{-5.0mu }-\mspace{-5.0mu }% ({\mu_{k+1}^{R}})^{2}})^{\omega_{k+1}}})-\\ \left(1\mspace{-5.0mu }-\mspace{-5.0mu }\prod\limits_{j=1}^{k}\mspace{-3.0mu }% (1\mspace{-5.0mu }-\mspace{-5.0mu }({\mu_{j}^{R}})^{2})^{\omega_{j}}\right)({1% \mspace{-5.0mu }-\mspace{-5.0mu }({1\mspace{-5.0mu }-\mspace{-5.0mu }({\mu_{k+% 1}^{R}})^{2}})^{\omega_{k+1}}})\\ \end{array}}}\right],\right.$ $\displaystyle\mspace{12.0mu }\left.\left.\left[{\prod\limits_{j=1}^{k+1}({\nu_% {j}^{L}})^{\omega_{j}},\prod\limits_{j=1}^{k+1}({\nu_{j}^{R}})^{\omega_{j}}}% \right]\right)\right\rangle$ $\displaystyle\mspace{-36.0mu }=\mspace{-2.0mu }\left\langle\mspace{-8.0mu }{% \begin{array}[]{l}\left[{\sum\limits_{j=1}^{k+1}{\omega_{j}s_{\theta_{j}}^{L}}% \mspace{-2.0mu },\mspace{-2.0mu }\sum\limits_{j=1}^{k+1}{\omega_{j}s_{\theta_{% j}}^{R}}}\right]\mspace{-5.0mu },\mspace{-5.0mu }\left[\sqrt{1\mspace{-6.0mu }% -\mspace{-6.0mu }\prod\limits_{j=1}^{k+1}\mspace{-2.0mu }(1\mspace{-6.0mu }-% \mspace{-6.0mu }({\mu_{j}^{L}})^{2})^{\omega_{j}}}\mspace{-2.0mu },\mspace{-2.% 0mu }\right.\\ \left.\sqrt{1\mspace{-6.0mu }-\mspace{-6.0mu }\prod\limits_{j=1}^{k+1}\mspace{% -2.0mu }(1\mspace{-6.0mu }-\mspace{-6.0mu }({\mu_{j}^{R}})^{2})^{\omega_{j}}}% \right]\mspace{-5.0mu },\mspace{-5.0mu }\left[\prod\limits_{j=1}^{k+1}\mspace{% -2.0mu }({\nu_{j}^{L}})^{\omega_{j}}\mspace{-2.0mu },\mspace{-2.0mu }\prod% \limits_{j=1}^{k+1}\mspace{-2.0mu }({\nu_{j}^{R}})^{\omega_{j}}\right]\\ \end{array}}\mspace{-8.0mu }\right\rangle$

By which aggregated value is also a IVPULN, Therefore, when $n=k+1$ , Eq. (3) holds.

Thus, by 1) and 2), we know that Eq. (3) holds for all $n$ . The proof is completed.

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

Property 1 (Idempotency). If all $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULWA}_{\omega}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n})=\tilde{{p}}$ (7)

Property 2 (Boundedness). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant\text{IVPULWA}_{\omega}(\tilde{{p}}_{1},% \tilde{{p}}_{2},\ldots,\tilde{{p}}_{n})\leqslant\tilde{{p}}^{+}$ (8)

Property 3 (Monotonicity). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ and ${\tilde{{p}}}^{\prime}_{j}\left({j=1,2,\ldots,n}\right)$ be two set of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULWA}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle\leqslant\text{IVPULWA}_{\omega}({\tilde{{p}}}^{\prime}_{1},{% \tilde{{p}}}^{\prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n})$ (9)

Further, we give an interval-valued Pythagorean uncertain linguistic ordered weighted averaging (IVPULOWA) operator below:

Definition 8. Let $\tilde{{p}}_{j}\!=\!\langle{[{s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}}],({[{\mu_% {j}^{L},\mu_{j}^{R}}],[{\nu_{j}^{L},\nu_{j}^{R}}]})}\rangle$ $({j=1,2,\ldots,n})$ be a collection of IVPULNs, the interval-valued Pythagorean uncertain linguistic ordered weighted averaging (IVPULOWA) operator of dimension $n$ is a mapping IVPULOWA: $P^{n}\to P$ , that has an associated weight vector $w=(w_{1},w_{2},\ldots,$ $w_{n})^{T}$ such that $w_{j}>0$ and $\sum_{j=1}^{n}{w_{j}}=1$ . Furthermore,

$\displaystyle\text{IVPULOWA}_{w}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}({w_{j}\tilde{{p}}_{\sigma(j)}})$ $\displaystyle=\left\langle\left[\sum\limits_{j=1}^{n}{w_{j}s_{\theta_{\sigma(j% )}}^{L}},\sum\limits_{j=1}^{n}{w_{j}s_{\theta_{\sigma(j)}}^{R}}\right],\right.$ $\displaystyle\mspace{16.0mu }\left(\left[\begin{array}[]{l}\sqrt{1-\prod% \limits_{j=1}^{n}{({1-({\mu_{\sigma(j)}^{L}})^{2}})^{w_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{({1-({\mu_{\sigma(j)}^{R}})^{2}})^{w_{j}}}}\\ \end{array}\right],\right.$ $\displaystyle\mspace{12.0mu }\left.\left.\left[\begin{array}[]{l}\prod\limits_% {j=1}^{n}{({\nu_{\sigma(j)}^{L}})^{w_{j}}},\\ \prod\limits_{j=1}^{n}{({\nu_{\sigma(j)}^{R}})^{w_{j}}}\\ \end{array}\right]\right)\right\rangle$

where $\left({\sigma\left(1\right),\sigma\left(2\right),\ldots,\sigma\left(n\right)}\right)$ is a permutation of $\left({1,2,\ldots,n}\right)$ , such that $\tilde{{p}}_{\sigma\left({j-1}\right)}\geqslant\tilde{{p}}_{\sigma(j)}$ for all $j=2,\ldots,n$ .

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

Property 4 (Idempotency). If all $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULOWA}_{w}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde% {{p}}_{n})=\tilde{{p}}$ (10)

Property 5 (Boundedness). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$ (11)

Then

$\displaystyle\tilde{{p}}^{-}\leqslant\text{IVPULOWA}_{w}(\tilde{{p}}_{1},% \tilde{{p}}_{2},\ldots,\tilde{{p}}_{n})\leqslant\tilde{{p}}^{+}$ (12)

Property 6 (Monotonicity). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ and ${\tilde{{p}}}^{\prime}_{j}\left({j=1,2,\ldots,n}\right)$ be two set of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULOWA}_{w}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde% {{p}}_{n})$ $\displaystyle\leqslant\text{IVPULOWA}_{w}({\tilde{{p}}}^{\prime}_{1},{\tilde{{% p}}}^{\prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n})$ (13)

Property 7 (Commutativity). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ and ${\tilde{{p}}}^{\prime}_{j}\left({j=1,2,\ldots,n}\right)$ be two set of IVPULNs, for all $j$ , then

$\displaystyle\text{IVPULOWA}_{w}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde% {{p}}_{n})$ $\displaystyle=\text{IVPULOWA}_{w}({\tilde{{p}}}^{\prime}_{1},{\tilde{{p}}}^{% \prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n})$ (14)

where ${\tilde{{p}}}^{\prime}_{j}(j=1,2,\ldots,n)$ is any permutation of $\tilde{{p}}_{j}$ $\left({j=1,2,\ldots,n}\right)$ .

From Definitions 10 and 11, we know that the IVPULWA operator only weights the IVPULN itself, while the IVPULOWA operator weights the ordered positions of the IVPULN instead of weighting the arguments itself. Therefore, the weights represent two different aspects in both the IVPULWA and IVPULOWA operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose the interval-valued Pythagorean uncertain linguistic hybrid average (IVPULHA) operator.

Definition 9. Let $\tilde{{p}}_{j}\!=\!\langle{[{s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}}],({[{\mu_% {j}^{L},\mu_{j}^{R}}],[{\nu_{j}^{L},\nu_{j}^{R}}]})}\rangle$ $({j=1,2,\ldots,n})$ be a collection of IVPULNs. An interval-valued Pythagorean uncertain linguistic hybrid average (IVPULHA) operator is a mapping IVPULHA: $P^{n}\to P$ , such that

$\displaystyle\text{IVPULHA}_{w,\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots% ,\tilde{{p}}_{n}})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}({w_{j}\dot{{\tilde{{p}}}}_{% \sigma(j)}})$ $\displaystyle=\left\langle\left[\sum\limits_{j=1}^{n}{w_{j}\dot{{s}}_{\theta_{% \sigma(j)}}^{L}},\sum\limits_{j=1}^{n}{w_{j}\dot{{s}}_{\theta_{\sigma(j)}}^{R}% }\right],\right.$ $\displaystyle\mspace{16.0mu }\left(\left[\begin{array}[]{l}\sqrt{1-\prod% \limits_{j=1}^{n}{({1-({\dot{{\mu}}_{\sigma(j)}^{L}})^{2}})^{w_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{({1-({\dot{{\mu}}_{\sigma(j)}^{R}})^{2}})^{w_{j% }}}}\\ \end{array}\right],\right.$ $\displaystyle\mspace{12.0mu }\left.\left.\left[\begin{array}[]{l}\prod\limits_% {j=1}^{n}{({\dot{{\nu}}_{\sigma(j)}^{L}})^{w_{j}}},\\ \prod\limits_{j=1}^{n}{({\dot{{\nu}}_{\sigma(j)}^{R}})^{w_{j}}}\\ \end{array}\right]\right)\right\rangle$ (15)

where $w=({w_{1},w_{2},\ldots,w_{n}})$ is the associated weighting vector, with $w_{j}\in[{0,1}]$ , $\sum_{j=1}^{n}{w_{j}}=1$ , and $\dot{{\tilde{{p}}}}_{\sigma(j)}$ is the $j$ -th largest element of the interval-valued Pythagorean uncertain linguistic arguments $\dot{{\tilde{{p}}}}_{j}(\dot{{\tilde{{p}}}}_{j}=({n\omega_{j}})\tilde{{p}}_{j}% ,j=1,2,\ldots,n)$ , $\omega=({\omega_{1},\omega_{2},\ldots,\omega_{n}})$ is the weighting vector of interval-valued Pythagorean uncertain linguistic arguments $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ , with $\omega_{j}\in[{0,1}]$ , $\sum_{j=1}^{n}{\omega_{j}}=1$ , and $n$ is the balancing coefficient. Especially, if $w=({1/n,1/n,\ldots,1/n})^{T}$ , then IVPULHA is reduced to the interval-valued Pytha- gorean uncertain linguistic weighted average (IVPULWA) operator; if $\omega=({1/n,1/n,\ldots,1/n})$ , then IVPULHA is reduced to the interval-valued Pythagorean uncertain linguistic ordered weighted average (IVPULOWA) operator.

4. Interval-valued Pythagorean uncertain linguistic geometric aggregation operators

In this section, we shall develop some geometric aggregation operators with interval-valued Pythagorean uncertain linguistic information, such as interval-valued Pythagorean uncertain linguistic weighted geometric (IVPULWG) operator, interval-valued Pytha- gorean uncertain linguistic ordered weighted geometric (IVPULOWG) operator and interval-valued Pythagorean uncertain linguistic hybrid geometric (IVPULHG) operator.

Definition 10. Let $\tilde{{p}}_{j}\!=\!\langle[{s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}}],([\mu_{j}% ^{L},\mu_{j}^{R}],[\nu_{j}^{L},$ $\nu_{j}^{R}])\rangle$ $({j=1,2,\ldots,n})$ be a collection of PULNs. The interval-valued Pythagorean uncertain linguistic weighted geometric (IVPULWG) operator is a mapping $P^{n}\to P$ such that

$\displaystyle\text{IVPULWG}_{\omega}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n})=\mathop{\otimes}\limits_{j=1}^{n}\left({\tilde{{p}}_{j}}% \right)^{\omega_{j}}$ (16)

Based on the Definition 10 and Theorem 2, we can get the following result:

Theorem 4. The aggregated value by using IVPULWG operator is also a IVPULN, where

$\displaystyle\text{IVPULWG}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{{p}}_{j}})^{\omega_{j}}$ $\displaystyle=\left\langle{\begin{array}[]{l}\left[\mathop{\otimes}\limits_{j=% 1}^{n}({s_{\theta_{j}}^{L}})^{\omega_{j}},\mathop{\otimes}\limits_{j=1}^{n}({s% _{\theta_{j}}^{R}})^{\omega_{j}}\right],\\ \left(\begin{array}[]{l}\left[\prod\limits_{j=1}^{n}{({\mu_{j}^{L}})^{\omega_{% j}}},\prod\limits_{j=1}^{n}{({\mu_{j}^{R}})^{\omega_{j}}}\right],\\ \left[\begin{array}[]{l}\sqrt{1-\prod\limits_{j=1}^{n}{({1-({\nu_{j}^{L}})^{2}% })^{\omega_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{({1-({\nu_{j}^{R}})^{2}})^{\omega_{j}}}}\end{% array}\right]\\ \end{array}\right)\\ \end{array}}\right\rangle$ (17)

Proof We prove Eq. (4) by mathematical induction on $n$ .

(1) When $n=2$ , we have

$\displaystyle\text{IVPULWG}_{\omega}(\tilde{{p}}_{1},\tilde{{p}}_{2})=({\tilde% {{p}}_{1}})^{\omega_{1}}\otimes({\tilde{{p}}_{2}})^{\omega_{2}}$

By Theorem 2, we can see that both $\left({\tilde{{p}}_{1}}\right)^{\omega_{1}}$ and $\left({\tilde{{p}}_{2}}\right)^{\omega_{2}}$ are IVPULNs, and the value of $\left({\tilde{{p}}_{1}}\right)^{\omega_{1}}\otimes\left({\tilde{{p}}_{2}}% \right)^{\omega_{2}}$ is also a IVPULN. From the operational laws of IVPULN, we have

$\displaystyle({\tilde{{p}}_{1}})^{\omega_{1}}\mspace{-3.0mu }=\mspace{-3.0mu }% \left\langle\rule{0.0pt}{15.0pt}[{({s_{\theta_{1}}^{L}})^{\omega_{1}}\mspace{-% 1.0mu },\mspace{-1.0mu }({s_{\theta_{1}}^{R}})^{\omega_{1}}}]\mspace{-1.0mu },% \mspace{-1.0mu }\left([{({\mu_{1}^{L}})^{\omega_{1}}\mspace{-1.0mu },\mspace{-% 1.0mu }({\mu_{1}^{R}})^{\omega_{1}}}],\right.\right.$ $\displaystyle\left.\left.\left[{\sqrt{1\mspace{-3.5mu }-\mspace{-3.5mu }({1% \mspace{-3.5mu }-\mspace{-3.5mu }({\nu_{1}^{L}})^{2}})^{\omega_{1}}},\sqrt{1% \mspace{-3.5mu }-\mspace{-3.5mu }({1\mspace{-3.5mu }-\mspace{-3.5mu }({\nu_{1}% ^{R}})^{2}})^{\omega_{1}}}}\right]\right)\right\rangle;$ $\displaystyle({\tilde{{p}}_{2}})^{\omega_{2}}\mspace{-3.0mu }=\mspace{-3.0mu }% \left\langle\rule{0.0pt}{15.0pt}[{({s_{\theta_{2}}^{L}})^{\omega_{2}},({s_{% \theta_{2}}^{R}})^{\omega_{2}}}]\mspace{-1.0mu },\mspace{-1.0mu }\left([{({\mu% _{2}^{L}})^{\omega_{1}}\mspace{-1.0mu },\mspace{-1.0mu }({\mu_{2}^{R}})^{% \omega_{1}}}],\right.\right.$ $\displaystyle\left.\left.\left[{\sqrt{1\mspace{-3.5mu }-\mspace{-3.5mu }({1% \mspace{-3.5mu }-\mspace{-3.5mu }({\nu_{2}^{L}})^{2}})^{\omega_{1}}},\sqrt{1% \mspace{-3.5mu }-\mspace{-3.5mu }({1\mspace{-3.5mu }-\mspace{-3.5mu }({\nu_{2}% ^{R}})^{2}})^{\omega_{1}}}}\right]\right)\right\rangle.$

Then

$\displaystyle\text{IVPULWG}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2}})=({% \tilde{{p}}_{1}})^{\omega_{1}}\otimes({\tilde{{p}}_{2}})^{\omega_{2}}$ $\displaystyle=\left\langle\rule{0.0pt}{20.0pt}[{({s_{\theta_{1}}^{L}})^{\omega% _{1}}+({s_{\theta_{2}}^{L}})^{\omega_{2}},({s_{\theta_{1}}^{R}})^{\omega_{1}}+% ({s_{\theta_{2}}^{R}})^{\omega_{2}}}],\right.$ $\displaystyle\mspace{-25.0mu }\left.\left(\begin{array}[]{l}[({\mu_{1}^{L}})^{% \omega_{1}}({\mu_{2}^{L}})^{\omega_{2}},({\mu_{1}^{R}})^{\omega_{1}}({\mu_{2}^% {R}})^{\omega_{2}}],\\ \left[\sqrt{\begin{array}[]{l}2\mspace{-3.0mu }-\mspace{-3.0mu }({1\mspace{-3.% 0mu }-\mspace{-3.0mu }({\nu_{1}^{L}})^{2}})^{\omega_{1}}\mspace{-3.0mu }-% \mspace{-3.0mu }({1\mspace{-3.0mu }-\mspace{-3.0mu }({\nu_{2}^{L}})^{2}})^{% \omega_{2}}-\\ ({1\mspace{-3.0mu }-\mspace{-3.0mu }({1\mspace{-3.0mu }-\mspace{-3.0mu }({\nu_% {1}^{L}})^{2}})^{\omega_{1}}})({1\mspace{-3.0mu }-\mspace{-3.0mu }({1\mspace{-% 3.0mu }-\mspace{-3.0mu }({\nu_{2}^{L}})^{2}})^{\omega_{2}}})\end{array}},% \right.\\ \left.\sqrt{\begin{array}[]{l}2\mspace{-3.0mu }-\mspace{-3.0mu }({1\mspace{-3.% 0mu }-\mspace{-3.0mu }({\nu_{1}^{R}})^{2}})^{\omega_{1}}\mspace{-3.0mu }-% \mspace{-3.0mu }({1\mspace{-3.0mu }-\mspace{-3.0mu }({\nu_{2}^{R}})^{2}})^{% \omega_{2}}-\\ ({1\mspace{-3.0mu }-\mspace{-3.0mu }({1\mspace{-3.0mu }-\mspace{-3.0mu }({\nu_% {1}^{R}})^{2}})^{\omega_{1}}})({1\mspace{-3.0mu }-\mspace{-3.0mu }({1\mspace{-% 3.0mu }-\mspace{-3.0mu }({\nu_{2}^{R}})^{2}})^{\omega_{2}}})\end{array}}\right% ]\\ \end{array}\right)\right\rangle$ $\displaystyle=\left\langle\rule{0.0pt}{20.0pt}[({s_{\theta_{1}}^{L}})^{\omega_% {1}}+({s_{\theta_{2}}^{L}})^{\omega_{2}},({s_{\theta_{1}}^{R}})^{\omega_{1}}+(% {s_{\theta_{2}}^{R}})^{\omega_{2}}],\right.$ $\displaystyle\left.\left(\begin{array}[]{l}[({\mu_{1}^{L}})^{\omega_{1}}({\mu_% {2}^{L}})^{\omega_{2}},({\mu_{1}^{R}})^{\omega_{1}}({\mu_{2}^{R}})^{\omega_{2}% }],\\ \left[\begin{array}[]{l}\sqrt{1-({1-({\nu_{1}^{L}})^{2}})^{\omega_{1}}({1-({% \nu_{2}^{L}})^{2}})^{\omega_{2}}},\\ \sqrt{1-({1-({\nu_{1}^{R}})^{2}})^{\omega_{1}}({1-({\nu_{2}^{R}})^{2}})^{% \omega_{2}}}\\ \end{array}\right]\\ \end{array}\right)\right\rangle$

(2) Suppose that $n=k$ , Eq. (4) holds, i.e.,

$\displaystyle\text{IVPULWG}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{k}})$ $\displaystyle=({\tilde{{p}}_{1}})^{\omega_{1}}\otimes({\tilde{{p}}_{2}})^{% \omega_{2}}\otimes\ldots\otimes({\tilde{{p}}_{k}})^{\omega_{k}}$ $\displaystyle=\left\langle\begin{array}[]{l}\left[\mathop{\otimes}\limits_{j=1% }^{k}({s_{\theta_{j}}^{L}})^{\omega_{j}},\mathop{\otimes}\limits_{j=1}^{k}({s_% {\theta{}_{j}}^{R}})^{\omega_{j}}\right],\\ \left(\begin{array}[]{l}\left[\prod\limits_{j=1}^{k}{({\mu_{j}^{L}})^{\omega_{% j}}},\prod\limits_{j=1}^{k}{({\mu_{j}^{R}})^{\omega_{j}}}\right],\\ \left[\begin{array}[]{l}\sqrt{1-\prod\limits_{j=1}^{k}{({1-({\nu_{j}^{L}})^{2}% })^{\omega_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{k}{({1-({\nu_{j}^{R}})^{2}})^{\omega_{j}}}}\end{% array}\right]\\ \end{array}\right)\\ \end{array}\right\rangle$

and the aggregated value is a IVPULN, Then when $n=k+1$ , by the operational laws of IVPULN, we have

$\displaystyle\text{IVPULWG}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{k}})$ $\displaystyle=\mspace{-2.0mu }({\tilde{{p}}_{1}})^{\omega_{1}}\mspace{-1.0mu }% \otimes\mspace{-1.0mu }({\tilde{{p}}_{2}})^{\omega_{2}}\mspace{-1.0mu }\otimes% \mspace{-1.0mu }\ldots\mspace{-1.0mu }\otimes\mspace{-1.0mu }({\tilde{{p}}_{k}% })^{\omega_{k}}\mspace{-1.0mu }\otimes\mspace{-1.0mu }({\tilde{{p}}_{k+1}})^{% \omega_{k+1}}$ $\displaystyle=\left\langle\left[\mathop{\otimes}\limits_{j=1}^{k}({s_{\theta_{% j}}^{L}})^{\omega_{j}},\mathop{\otimes}\limits_{j=1}^{k}({s_{\theta_{j}}^{R}})% ^{\omega_{j}}\right]\otimes\right.$ $\displaystyle\quad\left[({s_{\theta_{k+1}}^{L}})^{\omega_{k+1}},({s_{\theta_{k% +1}}^{R}})^{\omega_{k+1}}\right],$ $\displaystyle\mspace{-50.0mu }\scalebox{.955}{$\left.\left(\mspace{-3.0mu }% \begin{array}[]{l}\left[\prod\limits_{j=1}^{k+1}{({\mu_{j}^{L}})^{\omega_{j}}}% ,\prod\limits_{j=1}^{k+1}({\mu_{j}^{R}})^{\omega_{j}}\right],\\ \left[\sqrt{\mspace{-4.0mu }\begin{array}[]{l}1\mspace{-4.0mu }-\mspace{-4.0mu% }\prod\limits_{j=1}^{k}\mspace{-2.0mu }({1\mspace{-4.0mu }-\mspace{-4.0mu }({% \nu_{j}^{L}})^{2}})^{\omega_{j}}\mspace{-4.0mu }+\mspace{-4.0mu }({1\mspace{-4% .0mu }-\mspace{-4.0mu }({1\mspace{-4.0mu }-\mspace{-4.0mu }({\nu_{k+1}^{L}})^{% 2}})^{\omega_{k+1}}})-\\ \left(1\mspace{-4.0mu }-\mspace{-4.0mu }\prod\limits_{j=1}^{k}\mspace{-2.0mu }% ({1\mspace{-4.0mu }-\mspace{-4.0mu }({\nu_{j}^{L}})^{2}})^{\omega_{j}}\right)% \mspace{-5.0mu }({1\mspace{-4.0mu }-\mspace{-4.0mu }({1\mspace{-4.0mu }-% \mspace{-4.0mu }({\nu_{k+1}^{L}})^{2}})^{\omega_{k+1}}})\end{array}},\right.\\ \\ \left.\sqrt{\mspace{-4.0mu }\begin{array}[]{l}1\mspace{-4.0mu }-\mspace{-4.0mu% }\prod\limits_{j=1}^{k}\mspace{-2.0mu }({1\mspace{-4.0mu }-\mspace{-4.0mu }({% \nu_{j}^{R}})^{2}})^{\omega_{j}}\mspace{-4.0mu }+\mspace{-4.0mu }(1\mspace{-4.% 0mu }-\mspace{-4.0mu }({1\mspace{-4.0mu }-\mspace{-4.0mu }\mspace{-4.0mu }({% \nu_{k+1}^{R}})^{2}})^{\omega_{k+1}})-\\ \left(1\mspace{-4.0mu }-\mspace{-4.0mu }\prod\limits_{j=1}^{k}\mspace{-2.0mu }% ({1\mspace{-4.0mu }-\mspace{-4.0mu }({\nu_{j}^{R}})^{2}})^{\omega_{j}}\right)% \mspace{-5.0mu }({1\mspace{-4.0mu }-\mspace{-4.0mu }({1\mspace{-4.0mu }-% \mspace{-4.0mu }({\nu_{k+1}^{R}})^{2}})^{\omega_{k+1}}})\\ \end{array}}\right]\end{array}\mspace{-8.0mu }\right)\mspace{-3.0mu }\right% \rangle$}$ $\displaystyle=\left\langle\left[\mathop{\otimes}\limits_{j=1}^{k+1}({s_{\theta% _{j}}^{L}})^{\omega_{j}},\mathop{\otimes}\limits_{j=1}^{k+1}({s_{\theta_{j}}^{% R}})^{\omega_{j}}\right],\right.$ $\displaystyle\mspace{14.0mu }\left.\left(\begin{array}[]{l}\left[\prod\limits_% {j=1}^{k+1}{({\mu_{j}^{L}})^{\omega_{j}}},\prod\limits_{j=1}^{k+1}{({\mu_{j}^{% R}})^{\omega_{j}}}\right],\\ \\ \left[\begin{array}[]{l}\sqrt{1-\prod\limits_{j=1}^{k+1}{({1-({\nu_{j}^{L}})^{% 2}})^{\omega_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{k+1}{({1-({\nu_{j}^{R}})^{2}})^{\omega_{j}}}}\end{% array}\right]\\ \end{array}\right)\right\rangle$

By which aggregated value is also an IVPULN, Therefore, when $n=k+1$ , Eq. (4) holds.

Thus, by Eqs (1) and (2), we know that Eq. (4) holds for all $n$ . The proof is completed.

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

Property 8 (Idempotency). If all $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULWG}_{\omega}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n})=\tilde{{p}}$ (18)

Property 9 (Boundedness). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant\text{IVPULWG}_{\omega}({\tilde{{p}}_{1},% \tilde{{p}}_{2},\ldots,\tilde{{p}}_{n}})\leqslant\tilde{{p}}^{+}$ (19)

Property 10 (Monotonicity). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ and ${\tilde{{p}}}^{\prime}_{j}\left({j=1,2,\ldots,n}\right)$ be two set of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULWG}_{\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle\leqslant\text{IVPULWG}_{\omega}({{\tilde{{p}}}^{\prime}_{1},{% \tilde{{p}}}^{\prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n}})$

Further, we give an interval-valued Pythagorean uncertain linguistic ordered weighted geometric (IVPULOWG) operator below:

Definition 11. Let $\tilde{{p}}_{j}=\langle[{s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}}],([\mu_{j}^{L}% ,\mu_{j}^{R}],$ $[\nu_{j}^{L},$ $\nu_{j}^{R}])\rangle(j=1,2,\ldots,n)$ be a collection of IVPULNs, the interval-valued Pythagorean uncertain linguistic ordered weighted geometric (IVPULOWG) operator of dimension $n$ is a mapping IVPULOWG: $P^{n}\to P$ , that has an associated weight vector $w=(w_{1},w_{2},\ldots,$ $w_{n})^{T}$ such that $w_{j}>0$ and $\sum_{j=1}^{n}{w_{j}}=1$ . Furthermore,

$\displaystyle\text{IVPULOWG}_{w}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{{p}}_{\sigma(j)}})^{w_% {j}}$ $\displaystyle=\left\langle\mspace{-3.0mu }\begin{array}[]{l}\left[\mathop{% \otimes}\limits_{j=1}^{n}\left({s_{\theta_{\sigma(j)}}^{L}}\right)^{\omega_{j}% },\mathop{\otimes}\limits_{j=1}^{n}\left({s_{\theta_{\sigma(j)}}^{R}}\right)^{% \omega_{j}}\right],\\ \left(\begin{array}[]{l}\left[\prod\limits_{j=1}^{n}{\left({\mu_{\sigma(j)}^{L% }}\right)^{w_{j}}},\prod\limits_{j=1}^{n}{\left({\mu_{\sigma(j)}^{R}}\right)^{% w_{j}}}\right],\\ \\ \left[\begin{array}[]{l}\sqrt{1-\prod\limits_{j=1}^{n}{\left({1-\left({\nu_{% \sigma(j)}^{L}}\right)^{2}}\right)^{w_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{\left({1-\left({\nu_{\sigma(j)}^{R}}\right)^{2}% }\right)^{w_{j}}}}\end{array}\right]\\ \end{array}\right)\\ \end{array}\right\rangle$ (20)

where $(\sigma(1),\sigma(2),\ldots,\sigma(n))$ is a permutation of $(1,2,$ $\ldots,n)$ , such that $\tilde{{p}}_{\sigma({j-1})}\geqslant\tilde{{p}}_{\sigma(j)}$ for all $j=2,\ldots,n$ .

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

Property 11 (Idempotency). If all $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULOWG}_{w}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde% {{p}}_{n})=\tilde{{p}}$ (21)

Property 12 (Boundedness). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant\text{IVPULOWG}_{w}(\tilde{{p}}_{1},% \tilde{{p}}_{2},\ldots,\tilde{{p}}_{n})\leqslant\tilde{{p}}^{+}$ (22)

Property 13 (Monotonicity). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ and ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ be two set of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULOWG}_{w}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle\leqslant\text{IVPULOWG}_{w}({{\tilde{{p}}}^{\prime}_{1},{\tilde{% {p}}}^{\prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n}})$

Property 14 (Commutativity). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ and ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ be two set of IVPULNs, for all $j$ , then

$\displaystyle\text{IVPULOWG}_{w}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde% {{p}}_{n})$ $\displaystyle=\text{IVPULOWG}_{w}({{\tilde{{p}}}^{\prime}_{1},{\tilde{{p}}}^{% \prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n}})$ (23)

where ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ is any permutation of $\tilde{{p}}_{j}(j$ $=1,2,\ldots,n)$ .

From Definitions 13 and 14, we know that the IVPULWG operator only weights the IVPULN itself, while the IVPULOWG operator weights the ordered positions of the IVPULN instead of weighting the arguments itself. Therefore, the weights represent two different aspects in both the IVPULWG and IVPULOWG operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose the interval-valued Pythagorean uncertain linguistic hybrid geometric (IVPULHG) operator.

Definition 12. An interval-valued Pythagorean uncertain linguistic hybrid geometric (IVPULHG) operator is a mapping IVPULHG: $P^{n}\to P$ , such that

$\displaystyle\text{IVPULHG}_{w,\omega}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots% ,\tilde{{p}}_{n}})=\mathop{\otimes}\limits_{j=1}^{n}\left(\dot{{\tilde{{p}}}}_% {\sigma(j)}\right)^{w_{j}}$ $\displaystyle=\left\langle\begin{array}[]{l}\left[\mathop{\otimes}\limits_{j=1% }^{n}\left({\dot{{s}}_{\theta_{\sigma(j)}}^{L}}\right)^{\omega_{j}}\mspace{-3.% 0mu },\mspace{-3.0mu }\mathop{\otimes}\limits_{j=1}^{n}\left({\dot{{s}}_{% \theta_{\sigma(j)}}^{R}}\right)^{\omega_{j}}\right],\\ \left({\begin{array}[]{l}\left[\prod\limits_{j=1}^{n}{\left({\dot{{\mu}}_{% \sigma(j)}^{L}}\right)^{w_{j}}}\mspace{-3.0mu },\mspace{-3.0mu }\prod\limits_{% j=1}^{n}{\left({\dot{{\mu}}_{\sigma(j)}^{R}}\right)^{w_{j}}}\right]\mspace{-3.% 0mu },\mspace{-3.0mu }\\ \left[\begin{array}[]{l}\sqrt{1-\prod\limits_{j=1}^{n}{({1-({\dot{{\nu}}_{% \sigma(j)}^{L}})^{2}})^{w_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{({1-({\dot{{\nu}}_{\sigma(j)}^{R}})^{2}})^{w_{j% }}}}\end{array}\right]\\ \end{array}}\right)\\ \end{array}\right\rangle$ (24)

where $w=({w_{1},w_{2},\ldots,w_{n}})$ is the associated weighting vector, with $w_{j}\in[{0,1}]$ , $\sum_{j=1}^{n}{w_{j}}=1$ , and $\dot{{\tilde{{p}}}}_{\sigma(j)}$ is the $j$ -th largest element of the IVPULNs $\dot{{\tilde{{p}}}}_{j}$ $({\dot{{\tilde{{p}}}}_{j}=({\tilde{{p}}_{j}})^{n\omega_{j}},j=1,2,\ldots,n})$ , $\omega=({\omega_{1},\omega_{2},\ldots,\omega_{n}})$ is the weighting vector of IVPULNs $\tilde{{p}}_{j}$ $(j=1,2,\ldots,$ $n)$ , with $\omega_{j}\in[{0,1}]$ , $\sum_{j=1}^{n}{\omega_{j}}=1$ , and $n$ is the balancing coefficient. Especially, if $w=(1/n,1/n,\ldots,$ $1/n)^{T}$ , then IVPULHG is reduced to the interval-valued Pythagorean uncertain linguistic weighted geometric (IVPULWG) operator; if $\omega=(1/n,1/n,\ldots,$ $1/n)$ , then IVPULHG is reduced to the interval-valued Pythagorean uncertain linguistic ordered weighted geometric (IVPULOWG) operator.

5. Some correlated aggregation operators with interval-valued Pythagorean uncertain linguistic information

For real decision making problems, there is always some degree of inter-dependent characteristics between attributes. Usually, there is interaction among attributes of decision makers. However, this assumption is too strong to match decision behaviors in the real world. The independence axiom generally can’t be satisfied. Thus, it is necessary to consider this issue.

Definition 13 [49]. Let $f$ be a positive real-valued function on $X$ , and $\mu$ be a fuzzy measure on $X$ . The discrete Choquet integral of $f$ with respective to $\mu$ is defined by

$\displaystyle C_{\mu}(f)=\sum\limits_{i=1}^{n}{f_{\sigma(i)}}[{\mu({A_{\sigma(% i)}})-\mu({A_{\sigma({i-1})}})}]$ (25)

where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ is a permutation of $(1,2,$ $\ldots,n)$ , such that $f_{\sigma({i-1})}\geqslant f_{\sigma(i)}$ for all $j=2,\ldots,n$ , $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}.}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ .

Based on the aggregation principle for IVPULNs and Choquet integral [50, 51, 52, 53, 54, 55], in the following, we shall develop some correlated aggregation operators with interval-valued Pythagorean uncertain linguistic information.

Definition 14. Let $\tilde{{p}}_{j}=\langle[{s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}}],([{\mu_{j}^{L% },\mu_{j}^{R}}],[\nu_{j}^{L},$ $\nu_{j}^{R}])\rangle$ $({j=1,2,\ldots,n})$ be a collection of IVPULNs on $X$ , and $\mu$ be a fuzzy measure on $X$ , then we call

$\displaystyle\text{IVPULCA}_{\mu}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}({({\mu({A_{\sigma(j)}})-\mu({A_% {\sigma({j-1})}})})\tilde{{p}}_{\sigma(j)}})$ (26)

the interval-valued Pythagorean uncertain linguistic correlated averaging (IVPULCA) operator, where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ is a permutation of $(1,2,\ldots,$ $n)$ , such that $\tilde{{p}}_{\sigma({j-1})}\geqslant\tilde{{p}}_{\sigma(j)}$ for all $j=2,\ldots,n$ , $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ .

With the operation of IVPULNs, the IVPULCA operator can be transformed into the following from by induction on n:

$\displaystyle\text{IVPULCA}_{\mu}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}(({\mu({A_{\sigma(j)}})-\mu({A_{% \sigma({j-1})}})})\tilde{{p}}_{\sigma(j)})$ $\displaystyle=\left\langle\left[\sum\limits_{j=1}^{n}{\mu({A_{\sigma(j)}})-\mu% ({A_{\sigma({j-1})}})s_{\theta_{\sigma(j)}}^{L}},\right.\right.$ $\displaystyle\mspace{19.0mu }\left.\sum\limits_{j=1}^{n}{\mu({A_{\sigma(j)}})-% \mu({A_{\sigma({j-1})}})s_{\theta_{\sigma(j)}}^{R}}\right],$ $\displaystyle\mspace{-8.0mu }\left(\mspace{-3.0mu }\left[\mspace{-3.0mu }\sqrt% {1\mspace{-3.0mu }-\mspace{-3.0mu }\prod\limits_{j=1}^{n}{\mspace{-3.0mu }% \left(1\mspace{-3.0mu }-\mspace{-3.0mu }\left({\mu_{\sigma(j)}^{L}}\right)^{2}% \right)^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}}},\right.\right.% \mspace{-8.0mu }$ $\displaystyle\mspace{-4.0mu }\left.\sqrt{1\mspace{-3.0mu }-\mspace{-3.0mu }% \prod\limits_{j=1}^{n}{\mspace{-3.0mu }\left(1\mspace{-3.0mu }-\mspace{-3.0mu % }\left({\mu_{\sigma(j)}^{R}}\right)^{2}\right)^{\mu(A_{\sigma(j)})-\mu({A_{% \sigma({j-1})}})}}}\mspace{2.0mu }\right],\mspace{-8.0mu }$ $\displaystyle\left[\prod\limits_{j=1}^{n}{\mspace{-3.0mu }\left({\nu_{\sigma(j% )}^{L}}\right)^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}},\right.$ $\displaystyle\left.\left.\left.\prod\limits_{j=1}^{n}{\left({\nu_{\sigma(j)}^{% R}}\right)^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}}\right]\right)\right\rangle$ (27)

whose aggregated value is also an IVPULN.

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

Property 15 (Idempotency). If all $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULCA}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p}}% _{n})=\tilde{{p}}$ (28)

Property 16 (Boundedness). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant\text{IVPULCA}(\tilde{{p}}_{1},\tilde{{p}% }_{2},\ldots,\tilde{{p}}_{n})\leqslant\tilde{{p}}^{+}$ (29)

Property 17 (Monotonicity). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ and ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ be two set of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULCA}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p}% }_{n}})$ $\displaystyle\leqslant\text{IIVPULCA}({\tilde{{p}}}^{\prime}_{1},{\tilde{{p}}}% ^{\prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n})$ (30)

Property 18 (Commutativity). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ and ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ be two set of IVPULNs, for all $j$ , then

$\displaystyle\text{IVPULCA}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p}% }_{n}})$ $\displaystyle=\text{IVPULCA}({{\tilde{{p}}}^{\prime}_{1},{\tilde{{p}}}^{\prime% }_{2},\ldots,{\tilde{{p}}}^{\prime}_{n}})$ (31)

where ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ is any permutation of $\tilde{{p}}_{j}(j$ $=1,2,\ldots,n)$ .

In the following, we shall develop the interval-valued Pythagorean uncertain linguistic correlated geometric (IVPULCG) operator based on the well-known Choquet integral [50, 51, 52, 53, 54, 55].

Definition 15. Let $\tilde{{p}}_{j}=\langle[{s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}}],([{\mu_{j}^{L% },\mu_{j}^{R}}],[\nu_{j}^{L},$ $\nu_{j}^{R}])\rangle({j=1,2,\ldots,n})$ be a collection of IVPULNs on $X$ , then we call

the interval-valued Pythagorean uncertain linguistic correlated geometric (IVPULCG) operator, where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ is a permutation of $(1,2,\ldots,$ $n)$ , such that $\tilde{{p}}_{\sigma({j-1})}\geqslant\tilde{{p}}_{\sigma(j)}$ for all $j=2,\ldots,n$ , $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ .

With the operation of IVPULNs, the IVPULCG operator can be transformed into the following from by induction on $n$ :

$\displaystyle\text{IVPULCG}_{\mu}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}})$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{{p}}_{\sigma(j)}})^{({% \mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}$ $\displaystyle=\left\langle\left[\mathop{\otimes}\limits_{j=1}^{n}\left({s_{% \theta_{\sigma(j)}}^{L}}\right)^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}% })})},\right.\right.$ $\displaystyle\quad\left.\mathop{\otimes}\limits_{j=1}^{n}\left({s_{\theta_{% \sigma(j)}}^{R}}\right)^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}% \right],$ $\displaystyle\quad\left(\left[\prod\limits_{j=1}^{n}{({\mu_{\sigma(j)}^{L}})^{% \left(\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})\right)}},\right.\right.$ $\displaystyle\quad\left.\prod\limits_{j=1}^{n}{({\mu_{\sigma(j)}^{R}})^{\left(% \mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})\right)}}\right],$ $\displaystyle\left[\mspace{-3.0mu }\sqrt{1\mspace{-3.0mu }-\mspace{-3.0mu }% \prod\limits_{j=1}^{n}\mspace{-3.0mu }\left({1\mspace{-3.0mu }-\mspace{-3.0mu % }\left({\nu_{\sigma(j)}^{L}}\right)^{2}}\right)^{({\mu({A_{\sigma(j)}})-\mu({A% _{\sigma(j\mspace{-3.0mu }-\mspace{-3.0mu }1)}})})}},\right.\mspace{-7.0mu }$ $\displaystyle\mspace{-52.0mu }\left.\left.\left.\sqrt{1\mspace{-3.0mu }-% \mspace{-3.0mu }\prod\limits_{j=1}^{n}{\mspace{-3.0mu }\left(\mspace{-2.0mu }1% \mspace{-3.5mu }-\mspace{-3.5mu }\left({\nu_{\sigma(j)}^{R}}\right)^{2}\mspace% {-2.0mu }\right)^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}}}\mspace{% -3.0mu }\right]\mspace{-4.0mu }\right)\mspace{-4.0mu }\right\rangle$ (33)

whose aggregated value is also an IVPULN.

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

Property 19 (Idempotency). If all $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULCG}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p}% }_{n}})=\tilde{{p}}$ (34)

Property 20 (Boundedness). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant\text{IVPULCG}({\tilde{{p}}_{1},\tilde{{p% }}_{2},\ldots,\tilde{{p}}_{n}})\leqslant\tilde{{p}}^{+}$ (35)

Property 21 (Monotonicity). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ and ${\tilde{{p}}}^{\prime}_{j}\left({j=1,2,\ldots,n}\right)$ be two set of IVPULNs, if $\tilde{{p}}_{j}$ $\leqslant$ ${\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULCG}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p}% }_{n}})$ $\displaystyle\leqslant\text{IIVPULCG}({{\tilde{{p}}}^{\prime}_{1},{\tilde{{p}}% }^{\prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n}})$ (36)

Property 22 (Commutativity). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ and ${\tilde{{p}}}^{\prime}_{j}\left({j=1,2,\ldots,n}\right)$ be two set of IVPULNs, for all $j$ , then

$\displaystyle\text{IVPULCG}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p}% }_{n}})$ $\displaystyle=\text{IVPULCG}({{\tilde{{p}}}^{\prime}_{1},{\tilde{{p}}}^{\prime% }_{2},\ldots,{\tilde{{p}}}^{\prime}_{n}})$ (37)

where ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ is any permutation of $\tilde{{p}}_{j}(j$ $=1,2,\ldots,n)$ .

6. Some induced aggregation operators with interval-valued Pythagorean uncertain linguistic information

6.1 Interval-valued Pythagorean uncertain linguistic induced ordered weighted averaging (IVPULIOWA) operator

Yager and Filev [56] developed the induced OWA (IOWA) operator. It is an aggregation operator that uses order inducing variables in the reordering of the arguments. Thus, it is possible to consider more complex reordering processes that can describe the problem in a more complete way. The IOWA operator has been studied by different authors. The induced OWA operator represents an extension of the OWA operator [57], with the difference that the reordering step of the IOWA operator is not defined by the values of the arguments $a_{i}$ , but rather by order inducing variables $u_{i}$ , where the ordered position of the arguments $a_{i}$ depends upon the values of the $u_{i}$ . It can be defined as follows.

Definition 16 [56]. An IOWA operator of dimension $n$ is a mapping IOWA: $R^{n}\to R$ defined by an associated weight vector $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ such that $w_{j}>0$ and $\sum_{j=1}^{n}{w_{j}}=1$ , a set of order-inducing variables $u_{i}$ , according to the following formula:

$\displaystyle\text{IOWA}_{w}({\langle{u_{1},a_{1}}\rangle,\langle{u_{2},a_{2}}% \rangle,\ldots,\langle{u_{n},a_{n}}\rangle})$ $\displaystyle=\sum\limits_{j=1}^{n}{w_{j}a_{\sigma(j)}}$ (38)

$a_{\sigma(j)}$ is the $a_{i}$ value of the OWA pair $\langle{u_{i},a_{i}}\rangle$ having the jth largest $u_{i}({u_{i}\in[{0,1}]})$ , and $u_{i}$ in $\langle{u_{i},a_{i}}\rangle$ is referred to as the order inducing variable and $a_{i}$ are the argument variables.

In the following, we shall develop the interval-valued Pythagorean uncertain linguistic induced ordered weighted averaging (IVPULIOWA) operator which is an extension of induced ordered weighted averaging (IOWA) operator proposed by Yager and Filev [56].

Definition 17. Let $\{{u_{j},\tilde{{p}}_{j}}\}=\{u_{j},\langle[{s_{\theta_{j}}^{L},s_{\theta_{j}}% ^{R}}],([\mu_{j}^{L},$ $\mu_{j}^{R}],[\nu_{j}^{L},\nu_{j}^{R}])\rangle\}({j=1,2,\ldots,n})$ be a collection of 2-tuples, then we define the interval-valued Pythagorean uncertain linguistic induced ordered weighted averaging (IVPULIOWA) operator as follows:

$\displaystyle\text{IVPULIOWA}_{w}(\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{% {p}}_{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}({w_{j}\tilde{{p}}_{\sigma(j)}})$ (39)

where $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ is a weighting vector, such that $w_{j}>0$ , $\sum_{j=1}^{n}{w_{j}}=1$ , $j=1,2,\ldots,n$ , $\tilde{{p}}_{\sigma(j)}$ is the $\tilde{{p}}_{j}$ value of the IVPULIOWA pair $\{{u_{i},\tilde{{p}}_{i}}\}$ having the jth largest $u_{i}({u_{i}\in[{0,1}]})$ , and $u_{i}$ in $\{{u_{i},\tilde{{p}}_{i}}\}$ is referred to as the order inducing variable and $\tilde{{p}}_{i}$ as the interval-valued Pythagorean uncertain linguistic arguments.

Based on sum operations of the IVPULNs described, we can drive the Theorem 5.

Theorem 5. Let $\{{u_{j},\tilde{{p}}_{j}}\}=\{u_{j},\langle[s_{\theta_{j}}^{L},s_{\theta_{j}}^% {R}],([\mu_{j}^{L},\mu_{j}^{R}],$ $[{\nu_{j}^{L},\nu_{j}^{R}}])\rangle\}(j=1,2,\ldots,n)$ be a collection of 2-tuples, then their aggregated value by using the IVPULIOWA operator is also an IVPULN, and

$\displaystyle\text{IVPULIOWA}_{w}(\mspace{-2.0mu }\{{u_{1}\mspace{-1.0mu },% \mspace{-1.0mu }\tilde{{p}}_{1}}\}\mspace{-2.0mu },\mspace{-2.0mu }\{{u_{2}% \mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{2}}\}\mspace{-2.0mu },\ldots,% \mspace{-2.0mu }\{{u_{n}\mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{n}}\}% \mspace{-2.0mu })$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}({w_{j}\tilde{{p}}_{\sigma(j)}})$ $\displaystyle=\left\langle\left[\sum\limits_{j=1}^{n}{w_{j}s_{\theta_{\sigma(j% )}}^{L}},\sum\limits_{j=1}^{n}{w_{j}s_{\theta_{\sigma(j)}}^{R}}\right],\right.$ $\displaystyle\quad\left(\left[{\begin{array}[]{l}\sqrt{1-\prod\limits_{j=1}^{n% }{\left({1-\left({\mu_{\sigma(j)}^{L}}\right)^{2}}\right)^{w_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{\left({1-\left({\mu_{\sigma(j)}^{R}}\right)^{2}% }\right)^{w_{j}}}}\\ \end{array}}\right],\right.$ $\displaystyle\quad∼{}∼{}\left.\left.\left[{\begin{array}[]{l}\prod\limits_{j=1% }^{n}{\left({\nu_{\sigma(j)}^{L}}\right)^{w_{j}}},\\ \prod\limits_{j=1}^{n}{\left({\nu_{\sigma(j)}^{R}}\right)^{w_{j}}}\\ \end{array}}\right]\right)\right\rangle$ (40)

where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ is a permutation of $(1,2,$ $\ldots,n)$ , such that $\tilde{{p}}_{\sigma({j-1})}\geqslant\tilde{{p}}_{\sigma(j)}$ for all $j=2,\ldots,n$ , and $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ is the aggregation-associated weight vector such that $w_{j}\in[{0,1}]$ and $\sum_{j=1}^{n}{w_{j}}=1$ .

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

Property 23 (Idempotency). If all $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULIOWA}_{w}({\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde% {{p}}_{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\}})$ $\displaystyle=\tilde{{p}}$ (41)

Property 24 (Boundedness). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant$ $\displaystyle\text{IVPULIOWA}_{w}(\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{% {p}}_{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\})$ $\displaystyle\leqslant\tilde{{p}}^{+}$ (42)

Property 25 (Monotonicity). Let $\{{u_{j},\tilde{{p}}_{j}}\}(j=1,2,$ $\ldots,n)$ and $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ be two sets of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\mspace{-30.0mu }\text{IVPULIOWA}_{w}(\mspace{-2.0mu }\{{u_{1}% \mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{1}}\}\mspace{-2.0mu },\mspace{-2% .0mu }\{{u_{2}\mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{2}}\}\mspace{-2.0% mu },\ldots,\mspace{-2.0mu }\{{u_{n}\mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p% }}_{n}}\}\mspace{-2.0mu })\mspace{-3.0mu }\leqslant$ $\displaystyle\mspace{-30.0mu }\text{IVPULIOWA}_{w}(\mspace{-2.0mu }\{{{u}^{% \prime}_{1}\mspace{-1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{\prime}_{1}}\}% \mspace{-2.0mu },\mspace{-2.0mu }\{{{u}^{\prime}_{2}\mspace{-1.0mu },\mspace{-% 1.0mu }{\tilde{{p}}}^{\prime}_{2}}\}\mspace{-2.0mu },\ldots,\mspace{-2.0mu }\{% {{u}^{\prime}_{n}\mspace{-1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{\prime}_{n}}% \}\mspace{-2.0mu })$ (43)

Property 26 (Commutativity). Let $\{{u_{j},\tilde{{p}}_{j}}\}(j=1,2,$ $\ldots,n)$ and $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ be two sets of IVPULNs, then

$\displaystyle\mspace{-30.0mu }\text{IVPULIOWA}_{w}(\mspace{-2.0mu }\{{u_{1}% \mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{1}}\}\mspace{-2.0mu },\mspace{-2% .0mu }\{{u_{2}\mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{2}}\}\mspace{-2.0% mu },\ldots,\mspace{-2.0mu }\{{u_{n}\mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p% }}_{n}}\}\mspace{-2.0mu })=$ $\displaystyle\mspace{-30.0mu }\text{IVPULIOWA}_{w}(\mspace{-2.0mu }\{{{u}^{% \prime}_{1}\mspace{-1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{\prime}_{1}}\}% \mspace{-2.0mu },\mspace{-2.0mu }\{{{u}^{\prime}_{2}\mspace{-1.0mu },\mspace{-% 1.0mu }{\tilde{{p}}}^{\prime}_{2}}\}\mspace{-2.0mu },\ldots,\mspace{-2.0mu }\{% {{u}^{\prime}_{n}\mspace{-1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{\prime}_{n}}% \}\mspace{-2.0mu })$ (44)

where $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ is any permutation of $\{{u_{j},\tilde{{p}}_{j}}\}({j=1,2,\ldots,n})$ .

6.2 Interval-valued Pythagorean uncertain linguistic induced ordered weighted geometric (IVPULIOWG) operator

Xu and Da [58] developed the induced OWG (IOWG) operator. It is an aggregation operator that uses order inducing variables in the reordering of the arguments. Thus, it is possible to consider more complex reordering processes that can describe the problem in a more complete way. The IOWG operator has been studied by different authors. The induced OWG operator represents an extension of the OWG operator [59], with the difference that the reordering step of the IOWG operator is not defined by the values of the arguments $a_{i}$ , but rather by order inducing variables $u_{i}$ , where the ordered position of the arguments $a_{i}$ depends upon the values of the $u_{i}$ . It can be defined as follows.

Definition 18 [58]. An IOWG operator of dimension $n$ is a mapping IOWG: $R^{n}\to R$ defined by an associated weight vector $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ such that $w_{j}>0$ and $\sum_{j=1}^{n}{w_{j}}=1$ , a set of order-inducing variables $u_{i}$ , according to the following formula:

$\displaystyle\text{IOWG}_{w}({\langle{u_{1},a_{1}}\rangle,\langle{u_{2},a_{2}}% \rangle,\ldots,\langle{u_{n},a_{n}}\rangle})$ $\displaystyle=\prod\limits_{j=1}^{n}{({a_{\sigma(j)}})^{w_{j}}}$ (45)

$a_{\sigma(j)}$ is the $a_{i}$ value of the OWG pair $\langle{u_{i},a_{i}}\rangle$ having the jth largest $u_{i}({u_{i}\in[{0,1}]})$ , and $u_{i}$ in $\langle{u_{i},a_{i}}\rangle$ is referred to as the order inducing variable and $a_{i}$ are the argument variables.

In the following, we shall develop the interval-valued Pythagorean uncertain linguistic induced ordered weighted geometric (IVPULIOWG) operator which is an extension of induced ordered weighted geometric (IOWG) operator proposed by Xu and Da [58].

Definition 19. Let $\{{u_{j},\tilde{{p}}_{j}}\}=\{u_{j},\langle[s_{\theta_{j}}^{L},s_{\theta_{j}}^% {R}],([\mu_{j}^{L},$ $\mu_{j}^{R}],[\nu_{j}^{L},\nu_{j}^{R}])\rangle\}({j=1,2,\ldots,n})$ be a collection of 2-tuples, then we define the interval-valued Pythagorean uncertain linguistic induced ordered weighted geometric (IVPULIOWG) operator as follows:

where $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ is a weighting vector, such that $w_{j}>0$ , $\sum_{j=1}^{n}{w_{j}}=1$ , $j=1,2,\ldots,n$ , $\tilde{{p}}_{\sigma(j)}$ is the $\tilde{{p}}_{j}$ value of the IVPULIOWG pair $\{{u_{i},\tilde{{p}}_{i}}\}$ having the jth largest $u_{i}({u_{i}\in[{0,1}]})$ , and $u_{i}$ in $\{{u_{i},\tilde{{p}}_{i}}\}$ is referred to as the order inducing variable and $\tilde{{p}}_{i}$ as the interval-valued Pythagorean uncertain linguistic arguments.

Based on sum operations of the IVPULNs described, we can drive the Theorem 6.

Theorem 6. Let $\{{u_{j},\tilde{{p}}_{j}}\}=\{u_{j},\langle[{s_{\theta_{j}}^{L},s_{\theta_{j}}% ^{R}}],([\mu_{j}^{L},\mu_{j}^{R}],$ $[\nu_{j}^{L},\nu_{j}^{R}])\rangle\}$ $({j=1,2,\ldots,n})$ be a collection of 2-tuples, then their aggregated value by using the IVPULIOWG operator is also an IVPULN, and

$\displaystyle\text{IVPULIOWG}_{w}(\mspace{-2.0mu }\{{u_{1}\mspace{-1.2mu },% \mspace{-1.2mu }\tilde{{p}}_{1}}\}\mspace{-2.0mu },\mspace{-2.0mu }\{{u_{2}% \mspace{-1.2mu },\mspace{-1.2mu }\tilde{{p}}_{2}}\}\mspace{-2.0mu },\ldots,% \mspace{-2.0mu }\{{u_{n}\mspace{-1.2mu },\mspace{-1.2mu }\tilde{{p}}_{n}}\}% \mspace{-2.0mu })\mspace{-1.0mu }$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{{p}}_{\sigma(j)}})^{w_% {j}}$ $\displaystyle=\left\langle{\left[{\mathop{\otimes}\limits_{j=1}^{n}({s_{\theta% _{\sigma(j)}}^{L}})^{w_{j}},\mathop{\otimes}\limits_{j=1}^{n}\left({s_{\theta_% {\sigma(j)}}^{R}}\right)^{w_{j}}}\right]}\right.,$ $\displaystyle\quad\left(\left[\prod\limits_{j=1}^{n}{\left({\mu_{\sigma(j)}^{L% }}\right)^{w_{j}}},\prod\limits_{j=1}^{n}{\left({\mu_{\sigma(j)}^{R}}\right)^{% w_{j}}}\right],\right.$ $\displaystyle\quad\left[\sqrt{1-\prod\limits_{j=1}^{n}{\left({1-\left({\nu_{% \sigma(j)}^{L}}\right)^{2}}\right)^{w_{j}}}},\right.$ $\displaystyle\quad\left.\left.\left.\sqrt{1-\prod\limits_{j=1}^{n}{\left({1-% \left({\nu_{\sigma(j)}^{R}}\right)^{2}}\right)^{w_{j}}}}\right]\right)\right\rangle$ (47)

where $w=({w_{1},w_{2},\ldots,w_{n}})^{T}$ is a weighting vector, such that $w_{j}>0$ , $\sum_{j=1}^{n}{w_{j}}=1$ , $j=1,2,\ldots,n$ , $\tilde{{p}}_{\sigma(j)}$ is the $\tilde{{p}}_{j}$ value of the IVPULIOWG pair $\{{u_{i},\tilde{{p}}_{i}}\}$ having the jth largest $u_{i}({u_{i}\in[{0,1}]})$ , and $u_{i}$ in $\{{u_{i},\tilde{{p}}_{i}}\}$ is referred to as the order inducing variable and $\tilde{{p}}_{i}$ as the interval-valued Pythagorean uncertain linguistic arguments.

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

Property 27 (Idempotency). If all $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULIOWG}_{w}(\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{% {p}}_{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\})$ $\displaystyle=\tilde{{p}}$ (48)

Property 28 (Boundedness). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant$ $\displaystyle\text{IVPULIOWG}_{w}(\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{% {p}}_{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\})$ $\displaystyle\leqslant\tilde{{p}}^{+}$ (49)

Property 29 (Monotonicity). Let $\{{u_{j},\tilde{{p}}_{j}}\}(j=1,2,$ $\ldots,n)$ and $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ be two sets of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULIOWG}_{w}(\mspace{-2.0mu }\{{u_{1}\mspace{-1.0mu },% \mspace{-1.0mu }\tilde{{p}}_{1}}\}\mspace{-2.0mu },\mspace{-2.0mu }\{{u_{2}% \mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{2}}\}\mspace{-2.0mu },\ldots,% \mspace{-2.0mu }\{{u_{n}\mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{n}}\}% \mspace{-2.0mu })\mspace{-2.0mu }\leqslant$ $\displaystyle\text{IVPULIOWG}_{w}(\mspace{-2.0mu }\{{{u}^{\prime}_{1}\mspace{-% 1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{\prime}_{1}}\}\mspace{-2.0mu },\mspace{% -2.0mu }\{{{u}^{\prime}_{2}\mspace{-1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{% \prime}_{2}}\}\mspace{-2.0mu },\ldots,\mspace{-2.0mu }\{{{u}^{\prime}_{n}% \mspace{-1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{\prime}_{n}}\}\mspace{-2.0mu })$ (50)

Property 30 (Commutativity). Let $\langle{u_{j},\tilde{{a}}_{j}}\rangle(j=1,2,$ $\ldots,n)$ and $\langle{{u}^{\prime}_{j},{\tilde{{a}}}^{\prime}_{j}}\rangle({j=1,2,\ldots,n})$ be two sets of IVPULNs, then

$\displaystyle\text{IVPULIOWG}_{w}(\mspace{-2.0mu }\{{u_{1}\mspace{-1.0mu },% \mspace{-1.0mu }\tilde{{p}}_{1}}\}\mspace{-2.0mu },\mspace{-2.0mu }\{{u_{2}% \mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{2}}\}\mspace{-2.0mu },\ldots,% \mspace{-2.0mu }\{{u_{n}\mspace{-1.0mu },\mspace{-1.0mu }\tilde{{p}}_{n}}\}% \mspace{-2.0mu })\mspace{-2.0mu }=$ $\displaystyle\text{IVPULIOWG}_{w}(\mspace{-2.0mu }\{{{u}^{\prime}_{1}\mspace{-% 1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{\prime}_{1}}\}\mspace{-2.0mu },\mspace{% -2.0mu }\{{{u}^{\prime}_{2}\mspace{-1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{% \prime}_{2}}\}\mspace{-2.0mu },\ldots,\mspace{-2.0mu }\{{{u}^{\prime}_{n}% \mspace{-1.0mu },\mspace{-1.0mu }{\tilde{{p}}}^{\prime}_{n}}\}\mspace{-2.0mu })$ (51)

where $\langle{{u}^{\prime}_{j},{\tilde{{a}}}^{\prime}_{j}}\rangle({j=1,2,\ldots,n})$ is any permutation of $\langle{u_{j},\tilde{{a}}_{j}}\rangle({j=1,2,\ldots,n})$ .

6.3 Interval-valued Pythagorean uncertain linguistic induced correlated aggregation operators

Definition 20 [60]. Let $f$ be a positive real-valued function on $X$ and $m$ be a fuzzy measure on $X$ . The induced Choquet ordered averaging operator of dimension $n$ is a function I-COA: $({R^{+}\times R^{+}})\to R^{+}$ , which is defined to aggregate the set of second argument of a list of 2-tuples $(\langle{u_{1},f_{1}}\rangle,\langle{u_{2},f_{2}}\rangle,\ldots,\langle u_{n},$ $f_{n}\rangle)$ according to the following expression:

$\displaystyle\text{I-COA}_{m}(\langle{u_{1},f_{1}}\rangle,\langle{u_{2},f_{2}}% \rangle,\ldots,\langle{u_{n},f_{n}}\rangle)$ $\displaystyle=\sum\limits_{j=1}^{n}{f_{\sigma(j)}}[m({A_{\sigma(j)}})-m(A_{% \sigma({j-1})})]$ (52)

where $({\sigma(1),\sigma(2),\ldots,\sigma(n)})$ is a permutation of $(1,2,$ $\ldots,n)$ , such that $u_{\sigma({i-1})}\geqslant u_{\sigma(i)}$ for all $j=2,\ldots,n$ , i.e., $\langle{u_{\sigma(j)},f_{\sigma(j)}}\rangle$ is the 2-tuple with $u_{\sigma(j)}$ the $j$ th largest value in the set $({u_{1},u_{2},\ldots,u_{n}})$ , $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}.}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ .

In the following, we shall develop the interval-valued Pythagorean uncertain linguistic induced correlated averaging (IVPULICA) operator based on the I-COA operator [60].

Definition 21. Let $\{u_{j},\tilde{{p}}_{j}\}=\{u_{j},\langle[s_{\theta_{j}}^{L},s_{\theta_{j}}^{R% }],([\mu_{j}^{L},$ $\mu_{j}^{R}],[{\nu_{j}^{L},\nu_{j}^{R}}])\rangle\}(j=1,2,\ldots,n)$ be a collection of 2-tuples on $X$ , and $\mu$ be a fuzzy measure on $X$ , then we define the interval-valued Pythagorean uncertain linguistic induced correlated averaging (IVPULICA) operator as follows:

where $\tilde{{p}}_{\sigma(j)}$ is the $\tilde{{p}}_{j}$ value of the IVPULICA $\left\{{u_{i},\tilde{{p}}_{i}}\right\}$ having the jth largest $u_{i}\left({u_{i}\in\left[{0,1}\right]}\right)$ , and $u_{i}$ in $\left\{{u_{i},\tilde{{p}}_{i}}\right\}$ is referred to as the order inducing variable and $\tilde{{p}}_{i}$ as the interval-valued Pythagorean uncertain linguistic arguments.

With the operation of IVPULNs, the IVPULICA operator can be transformed into the following from by induction on n:

$\displaystyle\text{IVPULICA}_{\mu}(\mspace{-1.0mu }\{{u_{1},\tilde{{p}}_{1}}\}% \mspace{-1.0mu },\mspace{-1.0mu }\{{u_{2},\tilde{{p}}_{2}}\}\mspace{-1.0mu },% \mspace{-1.0mu }\ldots\mspace{-1.0mu },\mspace{-1.0mu }\{{u_{n},\tilde{{p}}_{n% }}\}\mspace{-1.0mu })$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}({({\mu({A_{\sigma(j)}})-\mu({A_% {\sigma({j-1})}})})\tilde{{p}}_{\sigma(j)}})$ $\displaystyle=\left\langle\left[\sum\limits_{j=1}^{n}{\mu({A_{\sigma(j)}})-\mu% ({A_{\sigma({j-1})}})s_{\theta_{\sigma(j)}}^{L}},\right.\right.$ $\displaystyle\quad\left.\sum\limits_{j=1}^{n}{\mu({A_{\sigma(j)}})-\mu({A_{% \sigma({j-1})}})s_{\theta_{\sigma(j)}}^{R}}\right],$ $\displaystyle\mspace{-14.0mu }\left(\mspace{-2.0mu }\left[\mspace{-2.0mu }% \sqrt{1\mspace{-2.0mu }-\mspace{-2.0mu }\prod\limits_{j=1}^{n}\mspace{-2.0mu }% \left(1\mspace{-2.0mu }-\mspace{-2.0mu }\left({\mu_{\sigma(j)}^{L}}\right)^{2}% \right)^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j\mspace{-2.0mu }-\mspace{-2.0mu% }1})}})}},\right.\right.$ $\displaystyle\mspace{-8.0mu }\left.\sqrt{1\mspace{-2.0mu }-\mspace{-2.0mu }% \prod\limits_{j=1}^{n}{\mspace{-2.0mu }\left(1\mspace{-2.0mu }-\mspace{-2.0mu % }\left({\mu_{\sigma(j)}^{R}}\right)^{2}\right)^{\mu({A_{\sigma(j)}})-\mu({A_{% \sigma({j\mspace{-2.0mu }-\mspace{-2.0mu }1})}})}}}\mspace{1.0mu }\right],$ $\displaystyle\quad\left[\prod\limits_{j=1}^{n}{\left(\nu_{\sigma(j)}^{L}\right% )^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}},\right.$ $\displaystyle\quad\left.\left.\left.\prod\limits_{j=1}^{n}{\left(\nu_{\sigma(j% )}^{R}\right)^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}}\right]\right)\right\rangle$ (54)

whose aggregated value is also an IVPULN.

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

Property 31 (Idempotency). If all $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULICA}({\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{{p}}% _{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\}})$ $\displaystyle=\tilde{{p}}$ (55)

Property 32 (Boundedness). Let $\tilde{{p}}_{j}\left({j=1,2,\ldots,n}\right)$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant$ $\displaystyle\text{IVPULICA}({\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{{p}}% _{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\}})$ $\displaystyle\leqslant\tilde{{p}}^{+}$ (56)

Property 33 (Monotonicity). Let $\{{u_{j},\tilde{{p}}_{j}}\}(j=1,2,$ $\ldots,n)$ and $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ be two sets of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULICA}(\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{{p}}_% {2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\})\leqslant$ $\displaystyle\text{IVPULICA}(\{{{u}^{\prime}_{1},{\tilde{{p}}}^{\prime}_{1}}\}% ,\{{{u}^{\prime}_{2},{\tilde{{p}}}^{\prime}_{2}}\},\ldots,\{{{u}^{\prime}_{n},% {\tilde{{p}}}^{\prime}_{n}}\})$ (57)

Property 34 (Commutativity). Let $\{{u_{j},\tilde{{p}}_{j}}\}(j=1,2,$ $\ldots,n)$ and $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ be two sets of IVPULNs, then

$\displaystyle\text{IVPULICA}(\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{{p}}_% {2}}\},\ldots,\{u_{n},\tilde{{p}}_{n}\})=$ $\displaystyle\text{IVPULICA}(\{{{u}^{\prime}_{1},{\tilde{{p}}}^{\prime}_{1}}\}% ,\{{{u}^{\prime}_{2},{\tilde{{p}}}^{\prime}_{2}}\},\ldots,\{{{u}^{\prime}_{n},% {\tilde{{p}}}^{\prime}_{n}}\})$ (58)

where $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ is any permutation of $\{{u_{j},\tilde{{p}}_{j}}\}({j=1,2,\ldots,n})$ .

In the following, we shall develop the interval-valued Pythagorean uncertain linguistic induced correlated geometric (IVPULICG) operator based on the interval-valued Pythagorean uncertain linguistic induced correlated averaging (IVPULICA) operator and geometric mean [58].

Definition 22. Let $\{{u_{j},\tilde{{p}}_{j}}\}=\{u_{j},\langle[s_{\theta_{j}}^{L},s_{\theta_{j}}^% {R}],([\mu_{j}^{L},$ $\mu_{j}^{R}],$ $[\nu_{j}^{L},\nu_{j}^{R}])\rangle\}({j=1,2,\ldots,n})$ be a collection of 2-tuples on $X$ , and $\mu$ be a fuzzy measure on $X$ , then we define the interval-valued Pythagorean uncertain linguistic induced correlated geometric (IVPULICG) operator as follows:

where $\tilde{{p}}_{\sigma(j)}$ is the $\tilde{{p}}_{j}$ value of the IVPULICG $\{{u_{i},\tilde{{p}}_{i}}\}$ having the jth largest $u_{i}({u_{i}\in[{0,1}]})$ , and $u_{i}$ in $\{{u_{i},\tilde{{p}}_{i}}\}$ is referred to as the order inducing variable and $\tilde{{p}}_{i}$ as the interval-valued Pythagorean uncertain linguistic arguments.

With the operation of IVPULNs, the IVPULICG operator can be transformed into the following from by induction on $n$ :

$\displaystyle\text{IVPULICG}_{\mu}(\mspace{-1.0mu }\{{u_{1},\tilde{{p}}_{1}}\}% \mspace{-1.0mu },\mspace{-1.0mu }\{{u_{2},\tilde{{p}}_{2}}\}\mspace{-1.0mu },% \mspace{-1.0mu }\ldots\mspace{-1.0mu },\mspace{-1.0mu }\{{u_{n},\tilde{{p}}_{n% }}\}\mspace{-1.0mu })$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{{p}}_{\sigma(j)}})^{({% \mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}$ $\displaystyle=\left\langle\left[\mathop{\otimes}\limits_{j=1}^{n}\left({s_{% \theta_{\sigma(j)}}^{L}}\right)^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}% })})},\right.\right.$ $\displaystyle\quad\left.\mathop{\otimes}\limits_{j=1}^{n}\left({s_{\theta_{% \sigma(j)}}^{R}}\right)^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}% \right],$ $\displaystyle\quad\left(\left[\prod\limits_{j=1}^{n}\left({\mu_{\sigma(j)}^{L}% }\right)^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})},\right.\right.$ $\displaystyle\quad\left.\prod\limits_{j=1}^{n}{\left(\mu_{\sigma(j)}^{R}\right% )^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}}\right],$ $\displaystyle\left[\mspace{-2.0mu }\sqrt{\mspace{-2.0mu }1\mspace{-3.0mu }-% \mspace{-3.0mu }\prod\limits_{j=1}^{n}\mspace{-3.0mu }\left({1\mspace{-3.0mu }% -\mspace{-3.0mu }\left({\nu_{\sigma(j)}^{L}}\right)^{2}}\right)^{({\mu({A_{% \sigma(j)}})-\mu({A_{\sigma({j\mspace{-3.0mu }-\mspace{-3.0mu }1})}})})}},% \right.\mspace{-8.0mu }$ $\displaystyle\mspace{-50.0mu }\left.\left.\left.\sqrt{\mspace{-2.0mu }1\mspace% {-4.0mu }-\mspace{-4.0mu }\prod\limits_{j=1}^{n}\mspace{-3.0mu }\left({1% \mspace{-3.0mu }-\mspace{-3.0mu }\left({\nu_{\sigma(j)}^{R}}\right)^{2}}\right% )^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j\mspace{-4.0mu }-\mspace{-4.0mu }1}% )}})})}}\mspace{1.0mu }\right]\mspace{-4.0mu }\right)\mspace{-4.0mu }\right\rangle$ (60)

whose aggregated value is also an IVPULN.

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

Property 35 (Idempotency). If all $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULICG}({\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{{p}}% _{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\}})$ $\displaystyle=\tilde{{p}}$ (61)

Property 36 (Boundedness). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ be a collection of IVPULNs, and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant$ $\displaystyle\text{IVPULICG}({\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{{p}}% _{2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\}})$ $\displaystyle\leqslant\tilde{{p}}^{+}$ (62)

Property 37 (Monotonicity). Let $\{{u_{j},\tilde{{p}}_{j}}\}(j=1,2,$ $\ldots,n)$ and $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ be two sets of IVPULNs, if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULICG}(\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{{p}}_% {2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\})\leqslant$ $\displaystyle\text{IVPULICG}(\{{{u}^{\prime}_{1},{\tilde{{p}}}^{\prime}_{1}}\}% ,\{{{u}^{\prime}_{2},{\tilde{{p}}}^{\prime}_{2}}\},\ldots,\{{{u}^{\prime}_{n},% {\tilde{{p}}}^{\prime}_{n}}\})$ (63)

Property 38 (Commutativity). Let $\{{u_{j},\tilde{{p}}_{j}}\}(j=1,2,$ $\ldots,n)$ and $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ be two sets of IVPULNs, then

$\displaystyle\text{IVPULICG}(\{{u_{1},\tilde{{p}}_{1}}\},\{{u_{2},\tilde{{p}}_% {2}}\},\ldots,\{{u_{n},\tilde{{p}}_{n}}\})=$ $\displaystyle\text{IVPULICG}(\{{{u}^{\prime}_{1},{\tilde{{p}}}^{\prime}_{1}}\}% ,\{{{u}^{\prime}_{2},{\tilde{{p}}}^{\prime}_{2}}\},\ldots,\{{{u}^{\prime}_{n},% {\tilde{{p}}}^{\prime}_{n}}\})$ (64)

where $\{{{u}^{\prime}_{j},{\tilde{{p}}}^{\prime}_{j}}\}({j=1,2,\ldots,n})$ is any permutation of $\{{u_{j},\tilde{{p}}_{j}}\}({j=1,2,\ldots,n})$ .

7. Interval-valued Pythagorean uncertain linguistic prioritized aggregation operators

The Prioritized Average (PA) operator was originally introduced by Yager [61], which was defined as follows:

Definition 23 [61]. Let $G=\{{G_{1},G_{2},\ldots,G_{n}}\}$ be a collection of attribute and that there is a prioritization between the attribute expressed by the linear ordering $G_{1}\succ G_{2}\succ G_{3}\ldots\succ G_{n}$ , indicate attribute $G_{j}$ has a higher priority than $G_{k}$ , if $j<k$ . The value $G_{j}(x)$ is the performance of any alternative $x$ under attribute $G_{j}$ , and satisfies $G_{j}(x)\in[{0,1}]$ . If

$\displaystyle\text{PA}({G_{i}(x)})=\sum\limits_{j=1}^{n}{w_{j}}G_{j}(x)$ (65)

where $w_{j}=\frac{T_{j}}{\sum_{j=1}^{n}{T_{j}}},T_{j}=\prod_{k=1}^{j-1}{G_{k}(x)}(j=% 2,\ldots,$ $n),T=1$ . Then PA is called the prioritized average (PA) operator.

The prioritized average [61] operators, however, have usually been used in situations where the input arguments are the exact values. We shall extend the PA operators [61, 62, 63, 64] to accommodate the situations where the input arguments are interval-valued Pythagorean uncertain linguistic information. In this section, we shall investigate the PA operator under interval-valued Pythagorean uncertain linguistic environments. Based on Definition 23, we give the definition of the interval-valued Pythagorean uncertain linguistic prioritized weighted average (IVPULPWA) operator as follows.

Definition 24. Let $\tilde{{p}}_{j}=\langle[{s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}}],([\mu_{j}^{L}% ,\mu_{j}^{R}],[\nu_{j}^{L},$ $\nu_{j}^{R}])\rangle(j=1,2,\ldots,n)$ be a collection of IVPULNs. The interval-valued Pythagorean uncertain linguistic prioritized weighted average (IVPULPWA) operator is defined as:

$\displaystyle\text{IVPULPWA}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p% }}_{n}})$ $\displaystyle=\frac{T_{1}}{\sum\limits_{j=1}^{n}{T_{j}}}\tilde{{p}}_{1}\oplus% \frac{T_{2}}{\sum\limits_{j=1}^{n}{T_{j}}}\tilde{{p}}_{2}\oplus\ldots\oplus% \frac{T_{n}}{\sum\limits_{j=1}^{n}{T_{j}}}\tilde{{p}}_{n}$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}\left(\frac{T_{j}\tilde{{p}}_{j}% }{\sum\limits_{j=1}^{n}{T_{j}}}\right)$ (66)

where $T_{j}=\prod_{k=1}^{j-1}{s({\tilde{{p}}_{j}})}(j=2,\ldots,n),T_{1}=1$ and $s({\tilde{{p}}_{j}})$ is the expected score values of $\tilde{{p}}_{j}(j=1,2,\ldots,$ $n)$ , that’s to say, $s({\tilde{{p}}_{j}})\!\!=\!\!\frac{\theta_{j}({2+({\mu_{j}^{L}})^{2}+({\mu_{j}% ^{R}})^{2}\!-({v_{j}^{L}})^{2}\!-({v_{j}^{R}})^{2}})}{4}$ .

With the operation of IVPULNs, the IVPULPWA operator can be transformed into the following from by induction on n:

$\displaystyle\text{IVPULPWA}(\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p}% }_{n})$ $\displaystyle=\frac{T_{1}}{\sum\limits_{j=1}^{n}{T_{j}}}\tilde{{p}}_{1}\oplus% \frac{T_{2}}{\sum\limits_{j=1}^{n}{T_{j}}}\tilde{{p}}_{2}\oplus\ldots\oplus% \frac{T_{n}}{\sum\limits_{j=1}^{n}{T_{j}}}\tilde{{p}}_{n}$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}\left({\frac{T_{j}\tilde{{p}}_{j% }}{\sum\limits_{j=1}^{n}{T_{j}}}}\right)$ $\displaystyle=\left\langle\left[\sum\limits_{j=1}^{n}{\frac{T_{j}s_{\theta_{j}% }^{L}}{\sum\limits_{j=1}^{n}{T_{j}}}},\sum\limits_{j=1}^{n}{\frac{T_{j}s_{% \theta_{j}}^{R}}{\sum\limits_{j=1}^{n}{T_{j}}}}\right],\right.$ $\displaystyle\left.\left(\begin{array}[]{l}\left[\begin{array}[]{l}\sqrt{% \mspace{-2.0mu }1\mspace{-2.0mu }-\mspace{-2.0mu }\prod\limits_{j=1}^{n}{({1% \mspace{-2.0mu }-\mspace{-2.0mu }({\mu_{j}^{L}})^{2}})^{{T_{j}}/{\sum\limits_{% j=1}^{n}{T_{j}}}}}},\\ \\ \sqrt{\mspace{-2.0mu }1\mspace{-2.0mu }-\mspace{-2.0mu }\prod\limits_{j=1}^{n}% {({1\mspace{-2.0mu }-\mspace{-2.0mu }({\mu_{j}^{R}})^{2}})^{{T_{j}}/{\sum% \limits_{j=1}^{n}{T_{j}}}}}}\end{array}\right],\\ \left[\prod\limits_{j=1}^{n}\mspace{-2.0mu }({\nu_{j}^{L}})^{{T_{j}}/{\sum% \limits_{j=1}^{n}{T_{j}}}}\mspace{-2.0mu },\mspace{-2.0mu }\prod\limits_{j=1}^% {n}\mspace{-2.0mu }({\nu_{j}^{R}})^{{T_{j}}/{\sum\limits_{j=1}^{n}{T_{j}}}}% \right]\\ \end{array}\right)\right\rangle$ (67)

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

Property 39 (Idempotency). Let $\tilde{{p}}_{j}=\langle[s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}],([\mu_{j}^{L},$ $\mu_{j}^{R}],[\nu_{j}^{L},\nu_{j}^{R}])\rangle$ $({j=1,2,\ldots,n})$ be a collection of IVPULNs, where $T_{j}=\prod_{k=1}^{j-1}{s({\tilde{{p}}_{k}})}({j=2,\ldots,n}),$ $T_{1}=1$ and $s({\tilde{{p}}_{k}})$ is the expected score values of $\tilde{{p}}_{k}$ $({j=1,2,\ldots,n})$ . If all $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULPWA}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p% }}_{n}})=\tilde{{p}}$ (68)

Property 40 (Boundedness). Let $\tilde{{p}}_{j}=\langle[s_{\theta_{j}}^{L},$ $s_{\theta_{j}}^{R}],$ $([\mu_{j}^{L},$ $\mu_{j}^{R}],[{\nu_{j}^{L},\nu_{j}^{R}}])\rangle({j=1,2,\ldots,n})$ be a collection of IVPULNs, where $T_{j}=\prod_{k=1}^{j-1}{s({\tilde{{p}}_{k}})}({j=2,\ldots,n}),$ $T_{1}=1$ and $s({\tilde{{p}}_{k}})$ is the expected score values of $\tilde{{p}}_{k}$ $({j=1,2,\ldots,n})$ , and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant\text{IVPULPWA}({\tilde{{p}}_{1},\tilde{{% p}}_{2},\ldots,\tilde{{p}}_{n}})\leqslant\tilde{{p}}^{+}$ (69)

Property 41 (Monotonicity). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ and ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ be two set of IVPULNs, where $T_{j}=\prod_{k=1}^{j-1}{s({\tilde{{p}}_{k}})},{T}^{\prime}_{j}=\prod_{k=1}^{j-% 1}{s({{\tilde{{p}}}^{\prime}_{k}})}({j=2,\ldots,n}),$ $T_{1}={T}^{\prime}_{1}=1$ , $s({\tilde{{p}}_{k}})$ is the is the score values of $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ , $s({{\tilde{{p}}}^{\prime}_{j}})$ is the expected score values of ${\tilde{{p}}}^{\prime}_{j}(j=1,2,\ldots,n)$ , if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULPWA}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p% }}_{n}})$ $\displaystyle\leqslant\text{IVPULPWA}({{\tilde{{p}}}^{\prime}_{1},{\tilde{{p}}% }^{\prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n}})$ (70)

Based on the IVPULPWA operator and the geometric mean, here we define a interval-valued Pythagorean uncertain linguistic prioritized weighted geometric (IVPULPWG) operator.

Definition 25. Let $\tilde{{p}}_{j}=\langle[{s_{\theta_{j}}^{L},s_{\theta_{j}}^{R}}],([\mu_{j}^{L}% ,\mu_{j}^{R}],[\nu_{j}^{L},$ $\nu_{j}^{R}])\rangle(j=1,2,\ldots,n)$ be a collection of IVPULNs. The interval-valued Pythagorean uncertain linguistic prioritized weighted geometric (IVPULPWG) operator is defined as:

$\displaystyle\text{IVPULPWG}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p% }}_{n}})$ $\displaystyle=\tilde{{p}}_{1}^{{T_{1}}/{\sum\limits_{j=1}^{n}{T_{j}}}}\otimes% \tilde{{p}}_{2}^{{T_{2}}/{\sum\limits_{j=1}^{n}{T_{j}}}}\otimes\ldots\otimes% \tilde{{p}}_{n}^{{T_{n}}/{\sum\limits_{j=1}^{n}{T_{j}}}}$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}\tilde{{p}}_{j}^{{T_{j}}/{\sum% \limits_{j=1}^{n}{T_{j}}}}$ (71)

where $T_{j}=\prod_{k=1}^{j-1}{s({\tilde{{p}}_{j}})}$ $({j=2,\ldots,n}),T_{1}=1$ and $s({\tilde{{p}}_{j}})$ is the expected score values of $\tilde{{p}}_{j}(j=1,2,\ldots,$ $n)$ , that’s to say, $s({\tilde{{p}}_{j}})\!\!=\!\!\frac{\theta_{j}({2+({\mu_{j}^{L}})^{2}+({\mu_{j}% ^{R}})^{2}\!-({v_{j}^{L}})^{2}\!-({v_{j}^{R}})^{2}})}{4}$ .

With the operation of IVPULNs, the IVPULPWG operator can be transformed into the following from by induction on $n$ :

$\displaystyle\text{IVPULPWG}\left({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,% \tilde{{p}}_{n}}\right)$ $\displaystyle=\tilde{{p}}_{1}^{{T_{1}}/{\sum\limits_{j=1}^{n}{T_{j}}}}\otimes% \tilde{{p}}_{2}^{{T_{2}}/{\sum\limits_{j=1}^{n}{T_{j}}}}\otimes\ldots\otimes% \tilde{{p}}_{n}^{{T_{n}}/{\sum\limits_{j=1}^{n}{T_{j}}}}$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}\tilde{{p}}_{j}^{{T_{j}}/{\sum% \limits_{j=1}^{n}{T_{j}}}}$ $\displaystyle=\left\langle\left[\mathop{\otimes}\limits_{j=1}^{n}({s_{\theta_{% j}}^{L}})^{{T_{j}}/{\sum\limits_{j=1}^{n}{T_{j}}}},\mathop{\otimes}\limits_{j=% 1}^{n}({s_{\theta_{j}}^{R}})^{{T_{j}}/{\sum\limits_{j=1}^{n}{T_{j}}}}\right],\right.$ $\displaystyle\quad\left(\left[\prod\limits_{j=1}^{n}{({\mu_{j}^{L}})^{{T_{j}}/% {\sum\limits_{j=1}^{n}{T_{j}}}}},\prod\limits_{j=1}^{n}{({\mu_{j}^{R}})^{{T_{j% }}/{\sum\limits_{j=1}^{n}{T_{j}}}}}\right],\right.$ $\displaystyle\quad\left[\sqrt{1-\prod\limits_{j=1}^{n}{({1-({\nu_{j}^{L}})^{2}% })^{{T_{j}}/{\sum\limits_{j=1}^{n}{T_{j}}}}}},\right.$ $\displaystyle\quad\left.\left.\left.\sqrt{1-\prod\limits_{j=1}^{n}{({1-({\nu_{% j}^{R}})^{2}})^{{T_{j}}/{\sum\limits_{j=1}^{n}{T_{j}}}}}}\right]\right)\right\rangle$ (72)

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

Property 42 (Idempotency). Let $\tilde{{p}}_{j}=\langle[s_{\theta_{j}}^{L},$ $s_{\theta_{j}}^{R}],$ $([\mu_{j}^{L},$ $\mu_{j}^{R}],[\nu_{j}^{L},\nu_{j}^{R}])\rangle({j=1,2,\ldots,n})$ be a collection of IVPULNs, where $T_{j}=\prod_{k=1}^{j-1}{s({\tilde{{p}}_{k}})}({j=2,\ldots,n}),$ $T_{1}=1$ and $s({\tilde{{p}}_{k}})$ is the expected score values of $\tilde{{p}}_{k}$ $({j=1,2,\ldots,n})$ . If all $\tilde{{p}}_{j}$ $({j=1,2,\ldots,n})$ are equal, i.e. $\tilde{{p}}_{j}=\tilde{{p}}$ for all $j$ , then

$\displaystyle\text{IVPULPWG}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p% }}_{n}})=\tilde{{p}}$ (73)

Property 43 (Boundedness). Let $\tilde{{p}}_{j}=\langle[s_{\theta_{j}}^{L},$ $s_{\theta_{j}}^{R}],$ $([\mu_{j}^{L},$ $\mu_{j}^{R}],$ $[\nu_{j}^{L},\nu_{j}^{R}])\rangle(j=1,2,\ldots,n)$ be a collection of IVPULNs, where $T_{j}=\prod_{k=1}^{j-1}{s({\tilde{{p}}_{k}})}({j=2,\ldots,n}),$ $T_{1}=1$ and $s({\tilde{{p}}_{k}})$ is the expected score values of $\tilde{{p}}_{k}$ $({j=1,2,\ldots,n})$ , and let

$\displaystyle\tilde{{p}}^{-}=\min\limits_{j}\tilde{{p}}_{j},∼{}\tilde{{p}}^{+}% =\max\limits_{j}\tilde{{p}}_{j}$

Then

$\displaystyle\tilde{{p}}^{-}\leqslant\text{IVPULPWG}({\tilde{{p}}_{1},\tilde{{% p}}_{2},\ldots,\tilde{{p}}_{n}})\leqslant\tilde{{p}}^{+}$ (74)

Property 44 (Monotonicity). Let $\tilde{{p}}_{j}({j=1,2,\ldots,n})$ and ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ be two set of IVPULNs, where $T_{j}=\prod_{k=1}^{j-1}{s({\tilde{{p}}_{k}})},{T}^{\prime}_{j}=\prod_{k=1}^{j-% 1}{s({{\tilde{{p}}}^{\prime}_{k}})}({j=2,\ldots,n}),$ $T_{1}={T}^{\prime}_{1}=1$ , $s({\tilde{{p}}_{k}})$ is the is the score values of $\tilde{{p}}_{j}$ $({j=1,2,\ldots,n})$ , $s({{\tilde{{p}}}^{\prime}_{j}})$ is the expected score values of ${\tilde{{p}}}^{\prime}_{j}({j=1,2,\ldots,n})$ , if $\tilde{{p}}_{j}\leqslant{\tilde{{p}}}^{\prime}_{j}$ , for all $j$ , then

$\displaystyle\text{IVPULPWG}({\tilde{{p}}_{1},\tilde{{p}}_{2},\ldots,\tilde{{p% }}_{n}})$ $\displaystyle\leqslant\text{IVPULPWG}({{\tilde{{p}}}^{\prime}_{1},{\tilde{{p}}% }^{\prime}_{2},\ldots,{\tilde{{p}}}^{\prime}_{n}})$ (75)

8. Models for MADM with interval-valued Pythagorean uncertain linguistic information

Based the IVPULWA (IVPULWG) operators, in this section, we shall propose the model for MADM with interval-valued Pythagorean uncertain linguistic information. Let $A=\{{A_{1},A_{2},\ldots,A_{m}}\}$ be a discrete set of alternatives, and $G=\{{G_{1},G_{2},\ldots,G_{n}}\}$ be the set of attributes, $\omega=({\omega_{1},\omega_{2},\ldots,\omega_{n}})$ is the weighting vector of the attribute $G_{j}({j=1,2,\ldots,n})$ , where $\omega_{j}\in[{0,1}]$ , $\sum_{j=1}^{n}{\omega_{j}}=1$ . Suppose that $\tilde{{R}}=({\tilde{{r}}_{ij}})_{m\times n}=\langle{[{s_{\alpha_{ij}},s_{% \beta_{ij}}}],({[{\mu_{ij}^{L},\mu_{ij}^{R}}],[{\nu_{ij}^{L},\nu_{ij}^{R}}]})}% \rangle_{m\times n}$ is the interval-valued Pythagorean uncertain linguistic decision matrix, where $\tilde{{r}}_{ij}$ take the form of the IVPULNs, where $[{\mu_{ij}^{L},\mu_{ij}^{R}}]$ indicates the degree that the alternative $A_{i}$ satisfies the attribute $G_{j}$ given by the decision maker, $[{\nu_{ij}^{L},\nu_{ij}^{R}}]$ indicates the degree that the alternative $A_{i}$ doesn’t satisfy the attribute $G_{j}$ given by the decision maker, $[{\mu_{ij}^{L},\mu_{ij}^{R}}]\subset[{0,1}]$ , $[{\nu_{ij}^{L},\nu_{ij}^{R}}]\subset[{0,1}]$ , $({\mu_{ij}^{R}})^{2}+({\nu_{ij}^{R}})^{2}\leqslant 1$ , $s_{\alpha_{ij}},s_{\beta_{ij}}\in S$ , $i=1,2,\ldots,m$ , $j=1,2,\ldots,n$ .

Table 1
The Pythagorean uncertain linguistic decision matrix

	G ${}_{1}$	G ${}_{2}$
A ${}_{1}$	$<$ [S ${}_{5}$ , S ${}_{6}$ ], ([0.4, 0.50], [0.30, 0.40]) $>$	$<$ [S ${}_{3}$ , S ${}_{4}$ ], ([0.50, 0.60], [0.20, 0.30]) $>$
A ${}_{2}$	$<$ [S ${}_{1}$ , S ${}_{2}$ ], ([0.60, 0.70], [0.40, 0.50]) $>$	$<$ [S ${}_{4}$ , S ${}_{5}$ ], ([0.60, 0.70], [0.10, 0.20]) $>$
A ${}_{3}$	$<$ [S ${}_{2}$ , S ${}_{4}$ ], ([0.50, 0.60], [0.30, 0.40]) $>$	$<$ [S ${}_{2}$ , S ${}_{3}$ ], ([0.40, 0.50], [0.60, 0.70]) $>$
A ${}_{4}$	$<$ [S ${}_{4}$ , S ${}_{5}$ ], ([0.70, 0.80], [0.10, 0.20]) $>$	$<$ [S ${}_{3}$ , S ${}_{4}$ ], ([0.50, 0.60], [0.20, 0.30]) $>$
A ${}_{5}$	$<$ [S ${}_{3}$ , S ${}_{4}$ ], ([0.50, 0.60], [0.30, 0.40]) $>$	$<$ [S ${}_{4}$ , S ${}_{6}$ ], ([0.30, 0.40], [0.70, 0.80]) $>$
	G ${}_{3}$	G ${}_{4}$
A ${}_{1}$	$<$ [S ${}_{1}$ , S ${}_{2}$ ], ([0.20, 0.30], [0.50, 0.60]) $>$	$<$ [S ${}_{6}$ , S ${}_{7}$ ], ([0.40, 0.50], [0.60, 0.70]) $>$
A ${}_{2}$	$<$ [S ${}_{2}$ , S ${}_{3}$ ], ([0.60, 0.70], [0.10, 0.20]) $>$	$<$ [S ${}_{2}$ , S ${}_{3}$ ], ([0.30, 0.40], [0.40, 0.50]) $>$
A ${}_{3}$	$<$ [S ${}_{3}$ , S ${}_{4}$ ], ([0.40, 0.50], [0.20, 0.30]) $>$	$<$ [S ${}_{4}$ , S ${}_{6}$ ], ([0.50, 0.60], [0.20, 0.30]) $>$
A ${}_{4}$	$<$ [S ${}_{3}$ , S ${}_{5}$ ], ([0.20, 0.30], [0.30, 0.40]) $>$	$<$ [S ${}_{1}$ , S ${}_{3}$ ], ([0.40, 0.50], [0.50, 0.60]) $>$
A ${}_{5}$	$<$ [S ${}_{4}$ , S ${}_{5}$ ], ([0.60, 0.70], [0.50, 0.60]) $>$	$<$ [S ${}_{3}$ , S ${}_{4}$ ], ([0.40, 0.50], [0.70, 0.80]) $>$

Table 2

The aggregating results of the ERP systems by the IVPULWA (IVPULWG) operators

	IVPULWA	IVPULWG
A ${}_{1}$	$<$ [S ${}_{4.00}$ , S ${}_{5.00}$ ], ([0.368, 0.467], [0.443, 0.549]) $>$	$<$ [S ${}_{3.15}$ , S ${}_{4.41}$ ], ([0.332, 0.437], [0.500, 0.601]) $>$
A ${}_{2}$	$<$ [S ${}_{2.00}$ , S ${}_{3.00}$ ], ([0.513, 0.614], [0.230, 0.347]) $>$	$<$ [S ${}_{1.87}$ , S ${}_{2.91}$ ], ([0.455, 0.560], [0.321, 0.415]) $>$
A ${}_{3}$	$<$ [S ${}_{3.10}$ , S ${}_{4.70}$ ], ([0.464, 0.564], [0.242, 0.346]) $>$	$<$ [S ${}_{2.98}$ , S ${}_{4.57}$ ], ([0.457, 0.558], [0.297, 0.393]) $>$
A ${}_{4}$	$<$ [S ${}_{2.40}$ , S ${}_{4.10}$ ], ([0.467, 0.570], [0.284, 0.398]) $>$	$<$ [S ${}_{2.05}$ , S ${}_{3.99}$ ], ([0.372, 0.480], [0.373, 0.469]) $>$
A ${}_{5}$	$<$ [S ${}_{3.40}$ , S ${}_{4.50}$ ], ([0.487, 0.588], [0.534, 0.639]) $>$	$<$ [S ${}_{3.37}$ , S ${}_{4.55}$ ], ([0.459, 0.561], [0.598, 0.702]) $>$

In the following, we apply the IVPULWA (IVPULWG) operator to the MADM problems with interval-valued Pythagorean uncertain linguistic information.

Step 4. Step 1.

We utilize the decision information given in matrix $\tilde{{R}}$ , and the IVPULWA operator $\displaystyle\tilde{{p}}_{i}=\text{IVPULWA}(\tilde{{r}}_{i1},\tilde{{r}}_{i2},% \ldots,\tilde{{r}}_{in})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}(\omega_{j}\tilde{{r}}_{ij})$ $\displaystyle=\left\langle\begin{array}[]{l}\left[\mathop{\oplus}\limits_{j=1}% ^{n}\omega_{j}s_{\alpha_{ij}},\mathop{\oplus}\limits_{j=1}^{n}\omega_{j}s_{% \beta_{ij}}\right],\\ \left(\begin{array}[]{l}\left[\begin{array}[]{l}\sqrt{1-\prod\limits_{j=1}^{n}% {({1-({\mu_{ij}^{L}})^{2}})^{\omega_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{({1-({\mu_{ij}^{R}})^{2}})^{\omega_{j}}}}\end{% array}\right],\\ \left[\prod\limits_{j=1}^{n}{({\nu_{ij}^{L}})^{\omega_{j}}},\prod\limits_{j=1}% ^{n}{({\nu_{ij}^{R}})^{\omega_{j}}}\right]\\ \end{array}\right)\\ \end{array}\right\rangle,$ $\displaystyle i=1,2,\ldots,m.$ (76)

$\displaystyle\tilde{{p}}_{i}=\text{IVPULWG}({\tilde{{r}}_{i1},\tilde{{r}}_{i2}% ,\ldots,\tilde{{r}}_{in}})$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{{r}}_{ij}})^{\omega_{j}}$

$\displaystyle=\left\langle\begin{array}[]{l}\left[\mathop{\otimes}\limits_{j=1% }^{n}({s_{\alpha_{ij}}})^{\omega_{j}},\mathop{\otimes}\limits_{j=1}^{n}({s_{% \beta_{ij}}})^{\omega_{j}}\right],\\ \left(\begin{array}[]{l}\left[\prod\limits_{j=1}^{n}{({\mu_{ij}^{L}})^{\omega_% {j}}},\prod\limits_{j=1}^{n}{({\mu_{ij}^{R}})^{\omega_{j}}}\right],\\ \\ \left[\begin{array}[]{l}\sqrt{1-\prod\limits_{j=1}^{n}{({1-({\nu_{ij}^{L}})^{2% }})^{\omega_{j}}}},\\ \sqrt{1-\prod\limits_{j=1}^{n}{({1-({\nu_{ij}^{R}})^{2}})^{\omega_{j}}}}\end{% array}\right]\\ \end{array}\right)\\ \end{array}\right\rangle,$ $\displaystyle\quad i=1,2,\ldots,m.$ (77)

to derive the overall preference values $\tilde{{p}}_{i}(i$ $=1,2,\ldots,m)$ of the alternative $A_{i}$ .

Step 2.

Calculate the scores $S({\tilde{{p}}_{i}})$ $(i=1,2,\ldots,m)$ of the overall IVPULNs $\tilde{{p}}_{i}(i=1,2,\ldots,$ $m)$ .

Step 3.

To rank these the scores $S({\tilde{{p}}_{i}})$ $(i=1,2,$ $\ldots,m)$ of the overall IVPULNs $\tilde{{p}}_{i}$ $(i=1,$ $2,$ $\ldots,$ $m)$ , we first compare each $S({\tilde{{p}}_{i}})$ with all the $S({\tilde{{p}}_{j}})$ $({j=1,2,\ldots,m})$ by using Eq. (7). For simplicity, we let $c_{ij}=p({S({\tilde{{p}}_{i}})\geqslant S({\tilde{{p}}_{j}})})$ , then we develop a complementary matrix as $C=({c_{ij}})_{m\times m}$ , where $c_{ij}\geqslant 0$ , $c_{ij}+c_{ji}=1$ , $c_{ii}=0.5$ , $i,$ $j$ $=1,2,$ $\ldots,n$ .

Summing all the elements in each line of matrix $C$ , we have

$c_{i}=\sum\limits_{j=1}^{m}{c_{ij}},i=1,2,\ldots,m.$

Step 4.

Rank all the alternatives $A_{i}(i=1,2,\ldots,$ $m)$ and select the best one(s) in accordance with $c_{i}({i=1,2,\ldots,m})$ .

Step 5.

End.

9. Numerical example

In this section, we utilize a practical MADM problem to illustrate the application of the developed approaches. Suppose an organization plans to implement enterprise resource planning (ERP) system (adapted from Liao et al. [19]). 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 $A_{i}({i=1,2,\ldots,5})$ 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 G ${}_{1}$ , (2) strategic fitness G ${}_{2}$ , (3) vendor’s ability G ${}_{3}$ ; (4) vendor’s reputation G ${}_{4}$ . The five possible ERP systems $A_{i}({i=1,2,\ldots,5})$ are to be evaluated using the Pythagorean uncertain 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 $\tilde{{R}}=({\tilde{{r}}_{ij}})_{5\times 4}$ is shown in Table 1.

In the following, in order to select the most desirable ERP systems, we utilize the IVPULWA (IVPULWG) operator to develop an approach to MADM problems with interval-valued Pythagorean uncertain linguistic information, which can be described as following.

Step 1. Step 1.
According to Table 1, aggregate all IVPULNs $\tilde{{r}}_{ij}({j=1,2,\ldots,n})$ by using the IVPULWA (IVPULWG) operator to derive the overall IVPULNs $\tilde{{p}}_{i}({i=1,2,3,4,5})$ of the alternative $A_{i}$ . The aggregating results are shown in Table 2.
Step 2.
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

IVPULWA IVPULWG

A ${}_{1}$ [S ${}_{1.71}$ , S ${}_{2.14}$ ] [S ${}_{1.09}$ , S ${}_{1.52}$ ]

A ${}_{2}$ [S ${}_{1.47}$ , S ${}_{2.20}$ ] [S ${}_{1.16}$ , S ${}_{1.81}$ ]

A ${}_{3}$ [S ${}_{2.10}$ , S ${}_{3.18}$ ] [S ${}_{1.90}$ , S ${}_{2.92}$ ]

A ${}_{4}$ [S ${}_{1.56}$ , S ${}_{2.67}$ ] [S ${}_{1.03}$ , S ${}_{2.01}$ ]

A ${}_{5}$ [S ${}_{1.51}$ , S ${}_{2.00}$ ] [S ${}_{1.14}$ , S ${}_{1.50}$ ]

Step 3.
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 same, but the best ERP system is A ${}_{3}$ .

Table 4
Ordering of the ERP systems

Ordering

IVPULWA A ${}_{3}$ $>$ A ${}_{4}$ $>$ A ${}_{1}$ $>$ A ${}_{2}$ $>$ A ${}_{5}$

IVPULWG A ${}_{3}$ $>$ A ${}_{4}$ $>$ A ${}_{2}$ $>$ A ${}_{5}$ $>$ A ${}_{1}$

10. Conclusion

	Ordering
IVPULWA	A ${}_{3}$ $>$ A ${}_{4}$ $>$ A ${}_{1}$ $>$ A ${}_{2}$ $>$ A ${}_{5}$
IVPULWG	A ${}_{3}$ $>$ A ${}_{4}$ $>$ A ${}_{2}$ $>$ A ${}_{5}$ $>$ A ${}_{1}$

In this paper, we investigate the multiple attribute decision making problems with interval-valued Pytha- gorean uncertain linguistic information. Then, we utilize arithmetic and geometric operations to develop some interval-valued Pythagorean uncertain linguistic aggregation operators: interval-valued Pythagorean uncertain linguistic weighted average (IVPULWA) operator, interval-valued Pythagorean uncertain linguistic weighted geometric (IVPULWG) operator, interval-valued Pythagorean uncertain linguistic ordered weighted average (IVPULOWA) operator, interval-valued Pythagorean uncertain linguistic ordered weighted geometric (IVPULOWG) operator, interval-valued Pythagorean uncertain linguistic hybrid average (IVPULHA) operator and interval-valued Pythagorean uncertain linguistic hybrid geometric (IVPULHG) operator, some interval-valued Pythagorean uncertain linguistic correlate aggregation operators, some interval-valued Pythagorean uncertain linguistic induced aggregation operators, some interval-valued Pythagorean uncertain linguistic induced correlate aggregation operators and some interval-valued Pythagorean uncertain linguistic power aggregating operators. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the interval-valued Pythagorean uncertain linguistic MADM 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 IVPULSs needs to be explored in the decision making, risk analysis and many other fields under uncertain environment [65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86].

Footnotes

Acknowledgments

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 (17XJA630003) and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004).

References

Herrera

and Martinez

, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 8 (2000), 746–752.

Herrera

and Martinez

, An approach for combining linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 8 (2000), 539–562.

Wei

G.W.

, Interval-valued dual hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(3) (2017), 1881–1893.

Wei

G.W.

and Zhao

X.F.

, Methods for probabilistic decision making with linguistic information, Technological and Economic Development of Economy 20(2) (2014), 193–209.

Wei

G.W.

, Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making, International Journal of Machine Learning and Cybernetics 7(6) (2016), 1093–1114.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making, Iranian Journal of Fuzzy Systems 13(4) (2016), 1–16.

Lin

Wei

G.W.

Wang

H.J.

and Zhao

X.F.

, Choquet integrals of weighted triangular fuzzy linguistic information and their applications to multiple attribute decision making, Journal of Business Economics and Management 15(5) (2014), 795–809.

Jiang

X.P.

and Wei

G.W.

, Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 27 (2014), 2153–2162.

Wei

G.W.

, Extension of TOPSIS method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Knowledge and Information Systems 25 (2010), 623–634.

10.

Wei

G.W.

, Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Expert Systems with Applications 38 (2011), 4824–4828.

11.

Q.X.

Zhao

X.F.

and Wei

G.W.

, Model for software quality evaluation with hesitant fuzzy uncertain linguistic information, Journal of Intelligent and Fuzzy Systems 26(6) (2014), 2639–2647.

12.

Wei

G.W.

, A method for multiple attribute group decision making based on the ET-WG and ET-OWG operators with 2-tuple linguistic information, Expert Systems with Applications 37(12) (2010), 7895–7900.

13.

Wei

G.W.

, Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making, Computers & Industrial Engineering 61 (2011), 32–38.

14.

Wei

G.W.

, Picture 2-tuple linguistic Bonferroni mean operators and their application to multiple attribute decision making, International Journal of Fuzzy System 19(4) (2017), 997–1010.

15.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, Bipolar 2-tuple linguistic aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 1197–1207.

16.

Wei

G.W.

and Lu

, Dual hesitant Pythagorean fuzzy Hamacher aggregation operators in multiple attribute decision making, Archives of Control Sciences 27(3) (2017), 365–395.

17.

Merigo

J.M.

Gil-Lafuente

A.M.

Zhou

L.G.

and Chen

H.Y.

, Generalization of the linguistic aggregation operator and its application in decision making, Journal of Systems Engineering and Electronics 22 (2011), 593–603.

18.

Wei

G.W.

, Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making, Informatica 28(3) (2017), 547–564.

19.

Liao

X.W.

and Lu

, A model for selecting an ERP system based on linguistic information processing, Information Systems 32(7) (2007), 1005–1017.

20.

Wang

W.P.

, Evaluating new product development performance by fuzzy linguistic computing, Expert Systems with Applications 36(6) (2009), 9759–9766.

21.

Tai

W.S.

and Chen

C.T.

, A new evaluation model for intellectual capital based on computing with linguistic variable, Expert Systems with Applications 36(2) (2009), 3483–3488.

22.

Y.J.

Tao

F.F.

and Wang

H.M.

, Some methods to deal with unacceptable incomplete 2-tuple fuzzy linguistic preference relations in group decision making, Knowledge-Based Systems 56 (2014), 179–190.

23.

Wang

J.Q.

Wang

D.D.

Zhang

H.Y.

and Chen

X.H.

, Multi-criteria group decision making method based on interval 2-tuple linguistic information and Choquet integral aggregation operators, Soft Computing 19(2) (2015), 389–405.

24.

Qin

J.D.

and Liu

X.W.

, 2-tuple linguistic Muirhead mean operators for multiple attribute group decision making and its application to supplier selection, Kybernetes 45(1) (2016), 22–29.

25.

Zhang

W.C.

Y.J.

and Wang

H.M.

, A consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Int J Systems Science 47(2) (2016), 389–405.

26.

Z.S.

, Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment, Information Science 168 (2004), 171–184.

27.

Z.S.

, Induced uncertain linguistic OWA operators applied to group decision making, Information Fusion 7 (2006), 231–238.

28.

Z.S.

, An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations, Decision Support Systems 41 (2006), 488–499.

29.

Wei

G.W.

, Uncertain linguistic hybrid geometric mean operator and its application to group decision making under uncertain linguistic environment, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17(2) (2009), 251–267.

30.

Zhang

and Guo

C.H.

, A method for multi-granularity uncertain linguistic group decision making with incomplete weight information, Knowledge-based Systems 26 (2012), 111–119.

31.

Zhou

L.Y.

Lin

Zhao

X.F.

and Wei

G.W.

, Uncertain linguistic prioritized aggregation operators and their application to multiple attribute group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21(4) (2013), 603–627.

32.

Wei

G.W.

Zhao

X.F.

Lin

and Wang

H.J.

, Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making, Applied Mathematical Modelling 37 (2013), 5277–5285.

33.

Lin

Chen

R.Q.

and Zhang

, Similarity-based approach for group decision making with multi-granularity linguistic information, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 24(6) (2016), 873–900.

34.

Meng

F.Y.

and Chen

X.H.

, An approach to uncertain linguistic multi-attribute group decision making based on interactive index, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 23(3) (2015), 319–344.

35.

Yager

R.R.

, Pythagorean fuzzy subsets, in: Proceeding of The Joint IFSA Wprld Congress and NAFIPS Annual Meeting, Edmonton, Canada (2013), 57–61.

36.

Yager

R.R.

, Pythagorean membership grades in multicriteria decision making, IEEE Trans Fuzzy Syst 22 (2014), 958–965.

37.

Zhang

X.L.

and Xu

Z.S.

, Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets, Int J Intell Syst 29 (2014), 1061–1078.

38.

Peng

and Yang

, Some results for pythagorean fuzzy sets, International Journal of Intelligent Systems 30 (2015), 133–1160.

39.

Beliakov

and James

, Averaging aggregation functions for preferences expressed as Pythagorean membership grades and fuzzy orthopairs, FUZZ-IEEE (2014), 298–305.

40.

Reformat

and Yager

R.R.

, Suggesting recommendations using pythagorean fuzzy sets illustrated using netflix movie data, IPMU (1) (2014), 546–556.

41.

Gou

and Ren

, The properties of continuous pythagorean fuzzy information, International Journal of Intelligent Systems 31(5) (2016), 401–424.

42.

Ren

and Gou

, Pythagorean fuzzy TODIM approach to multi-criteria decision making, Applied Soft Computing 42 (2016), 246–259.

43.

Garg

, A new generalized pythagorean fuzzy information aggregation using einstein operations and its application to decision making, International Journal of Intelligent Systems 31(9) (2016), 886–920.

44.

Zeng

Chen

and Li

, A hybrid method for pythagorean fuzzy multiple-criteria decision making, International Journal of Information Technology and Decision Making 15(2) (2016), 403–422.

45.

Garg

, A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem, Journal of Intelligent and Fuzzy Systems 31(1) (2016), 529–540.

46.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, Hesitant pythagorean fuzzy hamacher aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 1105–1117.

47.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, Pythagorean 2-tuple linguistic aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 1129–1142.

48.

and Wei

G.W.

, Pythagorean uncertain linguistic aggregation operators for multiple attribute decision making, International Journal of Knowledge-based and Intelligent Engineering Systems 21(3) (2017), 165–179.

49.

Grabisch

Murofushi

and Sugeno

, Fuzzy Measure and Integrals, New York: Physica-Verlag, 2000.

50.

Choquet

, Theory of capacities, Ann Inst Fourier 5 (1953), 131–295.

51.

Wei

G.W.

Lin

Zhao

X.F.

and Wang

H.J.

, An approach to multiple attribute decision making based on the induced Choquet integral with fuzzy number intuitionistic fuzzy information, Journal of Business Economics and Management 15(2) (2014), 277–298.

52.

Wei

G.W.

Zhao

X.F.

and Lin

, Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making, Knowledge-Based Systems 46 (2013), 43–53.

53.

Wei

G.W.

and Zhao

X.F.

, Induced hesitant interval-valued fuzzy einstein aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 24 (2013), 789–803.

54.

Wei

G.W.

Zhao

X.F.

Lin

and Wang

H.J.

, Generalized triangular fuzzy correlated averaging operator and their application to multiple attribute decision making, Applied Mathematical Modelling 36(7) (2012), 2975–2982.

55.

Wei

G.W.

Zhao

X.F.

Wang

H.J.

and Lin

, Hesitant fuzzy choquet integral aggregation operators and their applications to multiple attribute decision making, Information: An International Interdisciplinary Journal 15(2) (2012), 441–448.

56.

Yager

R.R.

and Filev

D.P.

, Induced ordered weighted averaging operators, IEEE Transactions on Systems, Man, and Cybernetics-Part B 29 (1999), 141–150.

57.

Yager

R.R.

, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Transactions on Systems Man and Cybernetics 18 (1988), 183–190.

58.

Z.S.

and Da

Q.L.

, An overview of operators for aggregating information, International Journal of Intelligent System 18 (2003), 953–969.

59.

Chiclana

Herrera

and Herrera-Viedma

, The ordered weighted geometric operator: Properties and application, in: Proc of 8th Int Conf on Information Processing and Management of Uncertainty in Knowledge-based Systems, Madrid, (2000), 985–991.

60.

Yager

R.R.

, Induced aggregation operators, Fuzzy Sets and Systems 137 (2003), 59–69.

61.

Yager

R.R.

, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008), 263–274.

62.

Ran

L.G.

and Wei

G.W.

, Uncertain prioritized operators and their application to multiple attribute group decision making, Technological and Economic Development of Economy 21(1) (2015), 118–139.

63.

Zhao

X.F.

and Wei

G.W.

, Some prioritized aggregating operators with linguistic information and their application to multiple attribute group decision making, Journal of Intelligent and Fuzzy Systems 26(4) (2014), 1619–1630.

64.

Lin

Zhao

X.F.

and Wei

G.W.

, Fuzzy number intuitionistic fuzzy prioritized operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 24 (2013), 879–888.

65.

Wei

G.W.

Alsaadi

F.E.

Hayat

Alsaedi

Bipolar

, Fuzzy aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 1119–1128.

66.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, Hesitant pythagorean fuzzy hamacher aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 1105–1117.

67.

Wei

G.W.

, Picture fuzzy aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 713–724.

68.

Wei

G.W.

, Some similarity measures for picture fuzzy sets and their applications, Iranian Journal of Fuzzy Systems 15(1) (2018), 77–89.

69.

Wei

G.W.

and Wei

, Similarity measures of Pythagorean fuzzy sets based on cosine function and their applications, International Journal of Intelligent Systems 33(3) (2018), 634–652.

70.

Wang

H.J.

Zhao

X.F.

and Wei

G.W.

, Dual hesitant fuzzy aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2281–2290.

71.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, A linear assignment method for multiple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure, International Journal of Fuzzy Systems 19(3) (2017), 607–614.

72.

Wei

G.W.

and Wang

J.M.

, A comparative study of robust efficiency analysis and data envelopment analysis with imprecise data, Expert Systems with Applications 81 (2017), 28–38.

73.

Wei

G.W.

, Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(4) (2017), 2119–2132.

74.

X.R.

and Wei

G.W.

, Dual hesitant bipolar fuzzy aggregation operators in multiple attribute decision making, International Journal of Knowledge-based and Intelligent Engineering Systems 21(3) (2017), 155–164.

75.

Wei

G.W.

, Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management 17(4) (2016), 491–502.

76.

Wei

G.W.

, Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making, Fundamenta Informaticae 157(3) (2018), 271–320. doi: 10.3233/FI-2018-1628.

77.

Wei

G.W.

Alsaadi

F.E.

Hayat

and Alsaedi

, Picture 2-tuple linguistic aggregation operators in multiple attribute decision making, Soft Computing 22(3) (2018), 989–1002.

78.

Jiang

Han

Liu

G.R.

and Liu

G.P.

, A nonlinear interval number programming method for uncertain optimization problems, European Journal of Operational Research 188(1) (2008), 1–13.

79.

Wei

G.W.

X.R.

and Deng

D.X.

, Interval-valued dual hesitant fuzzy linguistic geometric aggregation operators in multiple attribute decision making, International Journal of Knowledge-based and Intelligent Engineering Systems 20(4) (2016), 189–196.

80.

Wei

G.W.

, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information, International Journal of Fuzzy Systems 17(3) (2015), 484–489.

81.

Wei

G.W.

, Picture uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making, Kybernetes 46(10) (2017), 1777–1800.

82.

Lin

Chen

R.Q.

and Zhang

83.

Wei

G.W.

and Lu

, Pythagorean fuzzy power aggregation operators in multiple attribute decision making, International Journal of Intelligent Systems 33(1) (2018), 169–186.

84.

Zhao

X.F.

Lin

and Wei

G.W.

, Hesitant triangular fuzzy information aggregation based on Einstein operations and their application to multiple attribute decision making, Expert Systems with Applications 41(4) (2014), 1086–1094.

85.

Zhao

X.F.

and Wei

G.W.

, Some intuitionistic fuzzy einstein hybrid aggregation operators and their application to multiple attribute decision making, Knowledge-Based Systems 37 (2013), 472–479.

86.

Wei

G.W.

Zhao

X.F.

and Lin

, Some hybrid aggregating operators in linguistic decision making with Dempster-Shafer belief structure, Computers & Industrial Engineering 65 (2013), 646–651.

Multiple attribute decision making based on interval-valued Pythagorean uncertain linguistic aggregation operators

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 Uncertain linguistic term sets

6.1 Interval-valued Pythagorean uncertain linguistic induced ordered weighted averaging (IVPULIOWA) operator

Table 1 The Pythagorean uncertain linguistic decision matrix

Footnotes

Acknowledgments

References

Table 1
The Pythagorean uncertain linguistic decision matrix