Models for multiple attribute decision making with some interval-valued 2-tuple linguistic Pythagorean fuzzy Bonferroni mean operators

Abstract

In this paper, we extend the Bonferroni mean (BM) operator, generalized Bonferroni mean (GBM), and dual generalized Bonferroni mean (DGBM) operator with interval-valued 2-tuple linguistic Pythagorean fuzzy numbers (IV2TLPFNs) to propose interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (IV2TLPFWBM) operator, interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (IV2TLPFWGBM) operator, generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (GIV2TLPFWBM) operator, generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (GIV2TLPFWGBM) operator, dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (DGIV2TLPFWBM) operator, dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (DGIV2TLPFWGBM) operator. Then the MADM methods are proposed with these operators. In the end, we utilize an applicable example for green supplier selection in green supply chain management to prove the proposed methods.

Keywords

Multiple attribute decision making (MADM)neutrosophic numbers interval-valued 2-tuple linguistic Pythagorean fuzzy sets (IV2TLPFSs)IV2TLPFWBM operator GIV2TLPFWBM operator DGIV2TLPFWBM operator green supplier selection green supply chain management

1. Introduction

Pythagorean fuzzy set (PFS) [1, 2] is the generalization of the intuitionistic fuzzy set (IFS) which the square sum of the membership degree (MD) and the non-membership degree (NMD) is equal or less than 1. The PFS is proved to be more flexible than IFS since it has greater space than that of IFS. Hence, PFS can deal with the situations that can’t be dealt with by the IFS. The PFS has received more and more attention, which has been investigated and applied broadly [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Some MADM methods in PFS have been developed. TOPSIS method was extended to solve MADM problems in PFS [13, 14, 15, 16, 17, 18]. Ren et al. [19] developed the Pythagorean fuzzy TODIM approach which consider the DMs’ psychological behaviors. Zhang [20] proposed the hierarchical QUALIFLEX approach in PFS. Chen [21] developed the Pythagorean fuzzy VIKOR models. Peng and Dai [22] studied the Pythagorean fuzzy stochastic MADM with prospect theory. Xue et al. [23] studied the Pythagorean Fuzzy LINMAP model with entropy theory. Wan et al. [24] studied Pythagorean fuzzy mathematical programming mean for MAGDM with Pythagorean fuzzy truth degrees. Garg [25] proposed linear programming for MADM with interval-valued PFS. Liang et al. [26] gave the projection method for MAGDM with PFNs based on GBM. Peng and Yang [27] proposed the MABAC Method for MAGDM with PFNs. Some information measures were investigated by many scholars [28, 29, 30, 31, 32, 33, 34]. Information aggregation operators are of great importance to the application of MADM and decision support [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. The PFS had been generalized to accommodate interval values [52], linguistic arguments [53, 54, 55, 56, 57], hesitant fuzzy value [58, 59], etc.

Deng et al. [60] defined the concept of 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs) and developed some 2-tuple linguistic Pythagorean Hamy Mean operators in MADM. Deng et al. [61] proposed some 2-tuple linguistic Pythagorean Heronian mean operators in MADM. Deng and Gao [62] defined the TODIM method for MADM with 2-tuple linguistic Pythagorean fuzzy information. Furthermore, Wang et al. [63] defined the concept of interval-valued 2-tuple linguistic Pythagorean fuzzy sets (IV2TLPFSs) and developed some interval-valued 2-tuple linguistic Pythagorean Maclaurin symmetric mean (MSM) operators in MADM.

Bonferroni mean (BM) [64, 65, 66, 67, 68] is a famous aggregation operator which can depict interrelationships between any two arguments. Thus, the BM can supply a flexible mode to deal with the information fusion problem to solve MADM problems. Because IV2TLPFSs can easily describe the fuzzy information, and the BM can capture interrelationships between any two arguments, it is necessary to extend the BM operator to deal with the interval-valued 2-tuple linguistic Pythagorean fuzzy numbers (IV2TLPFNs). The paper’s goal is to expend some BM operators with 2TLPFNs, then to study some properties of these operators, and applied them to cope with the MADM with 2TLPFNs.

The structure of our article is organized as follows. Section 2 introduces the IV2TLPFSs. Section 3 combines IV2TLPFNs with WBM operator to develop IV2TLPFWBM and IV2TLPFWGBM operators. Section 4 combines IV2TLPFNs with GWBM operator to develop GIV2TLPFWBM and GIV2TLPFWGBM operators. Section 5 combines IV2TLPFNs with DGWBM operator to develop DGIV2TLPFWBM and DGIV2TLPFWGBM operators. Section 6 briefly introduces an application for safety assessment of construction project. Conclusions are given in Section 7.

2. Preliminaries

In this section, we introduce the concept of interval-valued 2-tuple linguistic Pythagorean fuzzy sets (IV2TLPFSs) based on the IVPFSs [34] and 2TLSs [38, 39].

2.1 2-tuple linguistic sets

Definition 1 [38, 39]. Let $S=\{{s_{i}|{i=0,1,\ldots,t}}\}$ be a linguistic term set with odd cardinality. Any label, $s_{i}$ represents a possible value for a linguistic variable, and $S$ can be defined as:

$\displaystyle S=\begin{Bmatrix}s_{0}=\text{extremely poor},s_{1}=\text{very % poor},s_{2}=\text{poor},s_{3}=\text{medium},\\ s_{4}=\text{good},s_{5}=\text{very good},s_{6}=\text{extremely good}.\end{Bmatrix}$

Herrera and Martinez [38, 39] developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple $({s_{i},\rho_{i}})$ , where $s_{i}$ is a linguistic label for predefined linguistic term set $S$ and $\rho_{i}$ is the value of symbolic translation, and $\rho_{i}\in[-0.5,0.5)$ .

2.2 Pythagorean fuzzy sets

In this section, some basic concepts of the PFSs and IVPFSs have been reviewed over a fix set $X$ .

Definition 2 [4, 5]. A PFS $P$ is defined as

$\displaystyle P=\{{\langle{x,({\mu_{P}(x),\nu_{P}(x)})}\rangle|{x\in X}}\}$ (1)

where the functions $\mu_{P}:X\to[0,1]$ and $\nu_{P}:X\to[0,1]$ defines the degrees of membership and non-membership of the element $x\in X$ to $P$ , such that for each $x$ , the condition $({\mu_{p}(x)})^{2}+({\nu_{p}(x)})^{2}\leqslant 1$ , holds.

Definition 3 [8]. The $p=({\mu,\nu})$ can be called as Pythagorean fuzzy number (PFN) and defined the score and accuracy functions as $S(p)=\mu^{2}-\nu^{2}$ and $H(p)=\mu^{2}+\nu^{2}$ . In order to compare two or more PFNs $p_{1}$ and $p_{2}$ , a comparison law is defined as

(1) (1)

If $S({p_{1}})<S({p_{2}})$ , then $p_{1}<p_{2}$ ;

(2)

If $S({p_{1}})=S({p_{2}})$ , then

(ii) (i)

If $H({p_{1}})=H({p_{2}})$ , then $p_{1}=p_{2}$ ;

(ii)

If $H({p_{1}})<H({p_{2}})$ , then $p_{1}<p_{2}$ .

2.3 Interval-valued pythagorean fuzzy sets

Zhang [10] extended the PFSs to the IVPFSs which is defined as followed over the fix set $X$ .

Definition 4 [10]. An IVPFS $\tilde{P}$ is defined as

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

where $\tilde{\mu}_{\tilde{P}}(x)=[{\mu_{\tilde{P}}^{L}(x),\mu_{\tilde{P}}^{R}(x)}],% \tilde{{\nu}}_{\tilde{P}}(x)=[{\nu_{\tilde{P}}^{L}(x),\nu_{\tilde{P}}^{R}(x)}]$ are the interval numbers of [0, 1] with the condition $0\leqslant(\mu_{\tilde{P}}^{R}(x))^{2}+({\nu_{\tilde{P}}^{R}(x)})^{2}\leqslant 1$ , $\forall x\in X$ . The pair $\tilde{p}=({[{u_{\tilde{p}}^{L},u_{\tilde{p}}^{R}}],[{v_{\tilde{p}}^{L},v_{% \tilde{p}}^{R}}]})$ is called as an IVPF number (IVPFN), where $u_{\tilde{p}},\nu_{\tilde{p}}\subseteq[0,1]$ and $(u_{\tilde{p}}^{R})^{2}+(\nu_{\tilde{p}}^{R})^{2}\leqslant 1$ .

Theorem 1 [14]. For three IVPFNs $\tilde{p}_{1}=({[u_{\tilde{p}_{1}}^{L},u_{\tilde{p}_{1}}^{R}],[v_{\tilde{p}_{1% }}^{L},v_{\tilde{p}_{1}}^{R}]})$ , $\tilde{p}_{2}=({[{u_{\tilde{p}_{2}}^{L},u_{\tilde{p}_{2}}^{R}}],[{v_{\tilde{p}% _{2}}^{L},v_{\tilde{p}_{2}}^{R}}]})$ , and $\tilde{p}=([{u_{\tilde{p}}^{L},u_{\tilde{p}}^{R}}],$ $[v_{\tilde{p}}^{L},v_{\tilde{p}}^{R}])$ , the basic operational laws are defined as follows:

(1) (1)

$\displaystyle\tilde{p}_{1}\oplus\tilde{p}_{2}=\left(\begin{bmatrix}\sqrt{({u_{% \tilde{p}_{1}}^{L}})^{2}+(u_{\tilde{p}_{2}}^{L})^{2}-({u_{\tilde{p}_{1}}^{L}})% ^{2}({u_{\tilde{p}_{2}}^{L}})^{2}},\\ \sqrt{({u_{\tilde{p}_{1}}^{R}})^{2}+(u_{\tilde{p}_{2}}^{R})^{2}-({u_{\tilde{p}% _{1}}^{R}})^{2}({u_{\tilde{p}_{2}}^{R}})^{2}}\end{bmatrix},\left[v_{\tilde{p}_% {1}}^{L}v_{\tilde{p}_{2}}^{L},v_{\tilde{p}_{1}}^{R}v_{\tilde{p}_{2}}^{R}\right% ]\right)$ ;

(2)

$\displaystyle\tilde{p}_{1}\otimes\tilde{p}_{2}=\left(\left[{\mu_{\tilde{p}_{1}% }^{L}v_{\tilde{p}_{2}}^{L},\mu_{\tilde{p}_{1}}^{R}v_{\tilde{p}_{2}}^{R}}\right% ],\begin{bmatrix}\sqrt{({v_{\tilde{p}_{1}}^{L}})^{2}+(v_{\tilde{p}_{2}}^{L})^{% 2}-({v_{\tilde{p}_{1}}^{L}})^{2}({v_{\tilde{p}_{2}}^{L}})^{2}},\\ \sqrt{({v_{\tilde{p}_{1}}^{R}})^{2}+(v_{\tilde{p}_{2}}^{R})^{2}-({v_{\tilde{p}% _{1}}^{R}})^{2}({v_{\tilde{p}_{2}}^{R}})^{2}}\end{bmatrix}\right)$ ;

(3)

$\displaystyle\lambda\tilde{p}=\left(\left[\sqrt{1-(1-(u_{\tilde{p}}^{L})^{2})^% {\lambda}},\sqrt{1-(1-({u_{\tilde{p}}^{R})^{2}})^{\lambda}}\right],\left[{({v_% {\tilde{p}}^{L}})^{\lambda},(v_{\tilde{p}}^{R})^{\lambda}}\right]\right),% \lambda>0$ ;

(4)

$\displaystyle({\tilde{p}})^{\lambda}=(\left[({\mu_{\tilde{p}}^{L}})^{\lambda},% ({\mu_{\tilde{p}}^{R}})^{\lambda}\right],\left[{\sqrt{1-({1-(v_{\tilde{p}}^{L}% )^{2}})^{\lambda}},\sqrt{1-(1-({v_{\tilde{p}}^{R})^{2}})^{\lambda}}}\right]),% \lambda>0$ ;

(5)

$\displaystyle\tilde{p}^{c}=(\left[{v_{\tilde{p}}^{L},v_{\tilde{p}}^{R}}\right]% ,\left[{\mu_{\tilde{p}}^{L},\mu_{\tilde{p}}^{R}}\right])$ .

2.4 Interval-valued 2-tuple linguistic pythagorean fuzzy sets

Then, Wang et al. [63] proposed the concept of interval-valued 2-tuple linguistic Pythagorean fuzzy sets (IV2TLPFSs) based on the IVPFSs [34] and 2TLSs [38, 39].

Definition 5 [63]. Assume that $P=\{{p_{0},p_{1},\ldots,p_{t}}\}$ is a 2TLSs with odd cardinality $t+1$ . If $\tilde{p}=\langle[({s_{\phi}^{L},\varphi^{L}}),$ $({s_{\phi}^{R},\varphi^{R}})],$ $[(s_{\theta}^{L},\vartheta^{L}),({s_{\theta}^{R},\vartheta^{R}})]\rangle$ is defined for $[{(s_{\phi}^{L},\varphi^{L}),({s_{\phi}^{R},\varphi^{R}})}]$ , $[({s_{\theta}^{L},\vartheta^{L}}),$ $({s_{\theta}^{R},\vartheta^{R}})]\in P$ where $[({s_{\phi}^{L},\varphi^{L}}),$ $(s_{\phi}^{R},\varphi^{R})]$ and $[({s_{\theta}^{L},\vartheta^{L}}),$ $({s_{\theta}^{R},\vartheta^{R}})]$ express independently the degree of membership and non-membership by 2TLSs, then IV2TLPFSs is defined as follows:

$\displaystyle\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),({s_% {\phi_{j}}^{R},\varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),% ({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ (3)

where $\Delta^{-1}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\Delta^{-1}({s_{\phi_{j}}^{R},% \varphi_{j}^{R}}),\Delta^{-1}({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),\Delta^{% -1}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})\subseteq[0,t]$ , and $0\leqslant({\Delta^{-1}({s_{\phi_{j}}^{R},\varphi_{j}^{R}})})^{2}+({\Delta^{-1% }({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})})^{2}\leqslant t^{2}.$

Definition 6 [63]. Let $\tilde{p}_{1}=\langle{[{({s_{\phi_{1}}^{L},\varphi_{1}^{L}}),({s_{\phi_{1}}^{R% },\varphi_{1}^{R}})}],[{({s_{\theta_{1}}^{L},\vartheta_{1}^{L}}),({s_{\theta_{% 1}}^{R},\vartheta_{1}^{R}})}]}\rangle$ be an IV2TLPFN in $P$ . Then the score and accuracy functions of $\tilde{p}_{1}$ are defined as follows:

$\displaystyle S({\tilde{p}_{1}})=\Delta\left\{\frac{1}{2t}\sqrt{\begin{pmatrix% }2t^{2}+({\Delta^{-1}({s_{\phi_{j}}^{L},\varphi_{j}^{L}})})^{2}+({\Delta^{-1}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})})^{2}\\ -({\Delta^{-1}({s_{\theta_{j}}^{L},\vartheta_{j}^{L}})})^{2}-({\Delta^{-1}({s_% {\theta_{j}}^{R},\vartheta_{j}^{R}})})^{2}\end{pmatrix}}\right\},S(\tilde{p}_{% 1})\in[0,1]$ (4) $\displaystyle H(\tilde{p}_{1})=\Delta\left\{\frac{1}{2t}\sqrt{\begin{pmatrix}(% {\Delta^{-1}({s_{\phi_{j}}^{L},\varphi_{j}^{L}})})^{2}+({\Delta^{-1}({s_{\phi_% {j}}^{R},\varphi_{j}^{R}})})^{2}\\ +({\Delta^{-1}({s_{\theta_{j}}^{L},\vartheta_{j}^{L}})})^{2}+({\Delta^{-1}({s_% {\theta_{j}}^{R},\vartheta_{j}^{R}})})^{2}\end{pmatrix}}\right\},H({\tilde{p}_% {1}})\in[0,1]$ (5)

Definition 7 [63]. Let $\tilde{p}_{1}=\langle{[{({s_{\phi_{1}}^{L},\varphi_{1}^{L}}),({s_{\phi_{1}}^{R% },\varphi_{1}^{R}})}],[{({s_{\theta_{1}}^{L},\vartheta_{1}^{L}}),({s_{\theta_{% 1}}^{R},\vartheta_{1}^{R}})}]}\rangle$ and $\tilde{p}_{2}=\langle[{({s_{\phi_{2}}^{L},\varphi_{2}^{L}}),({s_{\phi_{2}}^{R}% ,\varphi_{2}^{R}})}],$ $[(s_{\theta_{2}}^{L},$ $\vartheta_{2}^{L}),$ $({s_{\theta_{2}}^{R},\vartheta_{2}^{R}})]\rangle$ be two IV2TLPFNs, then

(1) (1)

If $S({\tilde{p}_{1}})\prec S(\tilde{p}_{2})$ , then $\tilde{p}_{1}\prec\tilde{p}_{2}$ ;

(2)

If $S({\tilde{p}_{1}})\succ S(\tilde{p}_{2})$ , then $\tilde{p}_{1}\succ\tilde{p}_{2}$ ;

(3)

If $S({\tilde{p}_{1}})=S({\tilde{p}_{2}}),H({\tilde{p}_{1}})\prec H({\tilde{p}_{2}})$ , then $\tilde{p}_{1}\prec\tilde{p}_{2}$ ;

(4)

If $S({\tilde{p}_{1}})=S({\tilde{p}_{2}}),H({\tilde{p}_{1}})\succ H({\tilde{p}_{2}})$ , then $\tilde{p}_{1}\succ\tilde{p}_{2}$ ;

(5)

If $S({\tilde{p}_{1}})=S({\tilde{p}_{2}}),H({\tilde{p}_{1}})=H({\tilde{p}_{2}})$ , then $\tilde{p}_{1}=\tilde{p}_{2}$ .

Definition 8 [63]. Let $\tilde{p}_{1}=\langle{[({s_{\phi_{1}}^{L},\varphi_{1}^{L}}),({s_{\phi_{1}}^{R}% ,\varphi_{1}^{R}})],[{({s_{\theta_{1}}^{L},\vartheta_{1}^{L}}),({s_{\theta_{1}% }^{R},\vartheta_{1}^{R}})}]}\rangle$ and $\tilde{p}_{2}=\langle[{({s_{\phi_{2}}^{L},\varphi_{2}^{L}}),({s_{\phi_{2}}^{R}% ,\varphi_{2}^{R}})}],$ $[(s_{\theta_{2}}^{L},$ $\vartheta_{2}^{L}),$ $({s_{\theta_{2}}^{R},\vartheta_{2}^{R}})]\rangle$ be two IV2TLPFNs, then

(1) $\tilde{p}_{1}\oplus\tilde{p}_{2}=\resizebox{372.9132pt}{0.0pt}{$\begin{Bmatrix% }\begin{bmatrix}\Delta\left({t\sqrt{\left({{\displaystyle\frac{\Delta^{-1}({s_% {\phi_{1}}^{L},\varphi_{1}^{L}})}{t}}}\right)^{2}+\left({{\displaystyle\frac{% \Delta^{-1}({s_{\phi_{2}}^{L},\varphi_{2}^{L}})}{t}}}\right)^{2}-\left({{% \displaystyle\frac{\Delta^{-1}({s_{\phi_{1}}^{L},\varphi_{1}^{L}})}{t}}}\right% )^{2}\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{2}}^{L},\varphi_{2}^{L})}% {t}}}\right)^{2}}}\right),\\ \\ \Delta\left({t\sqrt{\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{1}}^{R},% \varphi_{1}^{R}})}{t}}}\right)^{2}+\left({{\displaystyle\frac{\Delta^{-1}({s_{% \phi_{2}}^{R},\varphi_{2}^{R}})}{t}}}\right)^{2}-\left({{\displaystyle\frac{% \Delta^{-1}({s_{\phi_{1}}^{R},\varphi_{1}^{R}})}{t}}}\right)^{2}\left({{% \displaystyle\frac{\Delta^{-1}(s_{\phi_{2}}^{R},\varphi_{2}^{R})}{t}}}\right)^% {2}}}\right)\end{bmatrix},\\ \\ \left[{\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{1}}^{L% },\vartheta_{1}^{L}})}{t}}}\right)\left({{\displaystyle\frac{\Delta^{-1}({s_{% \theta_{2}}^{L},\vartheta_{2}^{L}})}{t}}}\right)}\right),\Delta\left({t\left({% {\displaystyle\frac{\Delta^{-1}({s_{\theta_{1}}^{R},\vartheta_{1}^{R}})}{t}}}% \right)\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{2}}^{R},\vartheta_{2% }^{R}})}{t}}}\right)}\right)}\right]\end{Bmatrix}$}$

(2) $\tilde{p}_{1}\otimes\tilde{p}_{2}=\resizebox{372.9132pt}{0.0pt}{$\begin{% Bmatrix}\left[{\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{% 1}}^{L},\varphi_{1}^{L}})}{t}}}\right)\left({{\displaystyle\frac{\Delta^{-1}(s% _{\phi_{2}}^{L},\varphi_{2}^{L})}{t}}}\right)}\right),\Delta\left({t\left({{% \displaystyle\frac{\Delta^{-1}({s_{\phi_{1}}^{R},\varphi_{1}^{R}})}{t}}}\right% )\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{2}}^{R},\varphi_{2}^{R}})}{t% }}}\right)}\right)}\right],\\ \\ \begin{bmatrix}\Delta\left({t\sqrt{\left({{\displaystyle\frac{\Delta^{-1}({s_{% \theta_{1}}^{L},\vartheta_{1}^{L}})}{t}}}\right)^{2}+\left({{\displaystyle% \frac{\Delta^{-1}({s_{\theta_{2}}^{L},\vartheta_{2}^{L}})}{t}}}\right)^{2}-% \left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{1}}^{L},\vartheta_{1}^{L}})% }{t}}}\right)^{2}\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{2}}^{L},% \vartheta_{2}^{L}})}{t}}}\right)^{2}}}\right),\\ \\ \Delta\left({t\sqrt{\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{1}}^{R}% ,\vartheta_{1}^{R}})}{t}}}\right)^{2}+\left({{\displaystyle\frac{\Delta^{-1}({% s_{\theta_{2}}^{R},\vartheta_{2}^{R}})}{t}}}\right)^{2}-\left({{\displaystyle% \frac{\Delta^{-1}({s_{\theta_{1}}^{R},\vartheta_{1}^{R}})}{t}}}\right)^{2}% \left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{2}}^{R},\vartheta_{2}^{R}})% }{t}}}\right)^{2}}}\right)\end{bmatrix}\end{Bmatrix}$}$

(3) $\lambda\tilde{p}_{1}=\resizebox{352.53306pt}{0.0pt}{$\begin{Bmatrix}\left[{% \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {1}}^{L},\varphi_{1}^{L})}{t}}}\right)^{2}}\right)^{\lambda}}}\right),\Delta% \left({t\sqrt{1-\left({1-\left({\displaystyle\frac{\Delta^{-1}(s_{\phi_{1}}^{R% },\varphi_{1}^{R})}{t}}\right)^{2}}\right)^{\lambda}}}\right)}\right],\\ \\ \left[{\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{1}}^{L% },\vartheta_{1}^{L}})}{t}}}\right)^{\lambda}}\right),\Delta\left({t\left({{% \displaystyle\frac{\Delta^{-1}({s_{\theta_{1}}^{R},\vartheta_{1}^{R}})}{t}}}% \right)^{\lambda}}\right)}\right]\end{Bmatrix}$},\lambda>0$ ;

(4) $({\tilde{p}_{1}})^{\lambda}=\resizebox{346.896pt}{0.0pt}{$\begin{Bmatrix}\left% [{\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{1}}^{L},% \varphi_{1}^{L}})}{t}}}\right)^{\lambda}}\right),\Delta\left({t\left({{% \displaystyle\frac{\Delta^{-1}({s_{\phi_{1}}^{R},\varphi_{1}^{R}})}{t}}}\right% )^{\lambda}}\right)}\right],\\ \\ \left[\Delta\left({t\sqrt{1-\left(1-\left({\displaystyle\frac{\Delta^{-1}(s_{% \theta_{1}}^{L},\vartheta_{1}^{L})}{t}}\right)^{2}\right)^{\lambda}}}\right),% \Delta\left({t\sqrt{1-\left({1-\left({\displaystyle\frac{\Delta^{-1}(s_{\theta% _{1}}^{R},\vartheta_{1}^{R})}{t}}\right)^{2}}\right)^{\lambda}}}\right)\right]% \end{Bmatrix}$},\lambda>0.$

2.5 BM operators

Definition 9 [64]. Let $\alpha,\beta>0$ and $b_{i}(i=1,2,\ldots,n)$ be a collection of nonnegative crisp numbers then the Bonferroni mean (BM) is defined as follows:

$\displaystyle{\rm BM}^{\alpha,\beta}({b_{1},b_{2},\ldots,b_{n}})=\left({\frac{% 1}{n({n-1})}\sum\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{b_{i}^{\alpha}}b_{j}^{\beta}}\right)^{\frac{1}{% \alpha+\beta}}$ (5)

Then ${\rm BM}^{\alpha,\beta}$ is called Bonferroni mean (BM) operator.

3. IV2TLPFWBM and IV2TLPFWGBM operators

3.1 IV2TLPFWBM operator

To consider the attribute weights, the weighted Bonferroni mean (WBM) is defined as follows:

Definition 10 [64]. Let $\alpha,\beta>0$ and $b_{i}(i=1,2,\ldots,n)$ be a collection of nonnegative crisp numbers with the weights vector being $\omega=({\omega_{1},\omega_{2},\ldots\omega_{n}})^{T}$ , thereby satisfying $\omega_{i}\in[0,1]$ and $\sum_{i=1}^{n}{\omega_{i}}=1$ . The weighted Bonferroni mean (WBM) is defined as follows:

$\displaystyle{\rm WBM}_{\omega}^{\alpha,\beta}({b_{1},b_{2},\ldots,b_{n}})=% \left({\frac{1}{n({n-1})}\sum\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\omega_{i}\omega_{j}b_{i}^{\alpha}b_{j}^{\beta}}}% \right)^{1/{(\alpha+\beta)}}$ (6)

Then we extend WBM to fuse the IV2TLPFNs and propose the interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (IV2TLPFWBM) aggregation operator.

Definition 11 Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle$ be a set of IV2TLPFNs. The interval-valued 2-tuple linguistic Pythagorean weighted Bonferroni mean (IV2TLPFWBM) operator is:

$\displaystyle\text{IV2TLPFWBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p}% _{2},\ldots,\tilde{p}_{n})=\left({\frac{1}{n(n-1)}\mathop{\oplus}\limits_{{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}({\omega_{i}\omega_{j}(\tilde{p}_{i}^{\alpha}% \otimes\tilde{p}_{j}^{\beta})})}\right)^{1/{({\alpha+\beta})}}$ (7)

Theorem 2. Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle$ be a set of IV2TLPFNs. The aggregated value by using IV2TLPFWBM operators is also an IV2TLPFN where

$\displaystyle\text{IV2TLPFWBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p}% _{2},\ldots,\tilde{p}_{n})=\left({{\displaystyle\frac{1}{n(n-1)}}\mathop{% \oplus}\limits_{{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}({\omega_{i}\omega_{j}(\tilde{p}_{i}^{\alpha}% \otimes\tilde{p}_{j}^{\beta})})}\right)^{1/{({\alpha+\beta})}}$ (8) $\displaystyle=\resizebox{390.258pt}{0.0pt}{$\begin{Bmatrix}\begin{bmatrix}% \Delta\left({t\left({\sqrt{1-\left({\displaystyle\prod\limits_{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_% {\phi_{i}}^{L},\varphi_{i}^{L}})}{t}}}\right)^{2\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{L})}{t}}\right)^{% 2\beta}}\right)^{\omega_{i}\omega_{j}}}}\right)^{\frac{1}{n(n-1)}}}}\right)^{1% /{(\alpha+\beta)}}}\right),\\ \\ \Delta\left({t\left({\sqrt{1-\left({\displaystyle\prod\limits_{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({\displaystyle\frac{\Delta^{-1}({s_{% \phi_{i}}^{R},\varphi_{i}^{R}})}{t}}\right)^{2\alpha}\cdot\left({\displaystyle% \frac{\Delta^{-1}(s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}}\right)^{2\beta}}% \right)^{\omega_{i}\omega_{j}}}}\right)^{\frac{1}{n(n-1)}}}}\right)^{1/{(% \alpha+\beta)}}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\sqrt{1-\left({1-\left({\displaystyle\prod\limits% _{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({1-\left({{\displaystyle\frac{\Delta% ^{-1}(s_{\theta_{i}}^{L},\vartheta_{i}^{L})}{t}}}\right)^{2}}\right)^{\alpha}% \cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{L},% \vartheta_{j}^{L}})}{t}}}\right)^{2}}\right)^{\beta}}\right)^{\omega_{i}\omega% _{j}}}}\right)^{\frac{1}{n(n-1)}}}\right)^{1/{(\alpha+\beta)}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({\displaystyle\prod\limits_{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({1-\left({{\displaystyle\frac{\Delta% ^{-1}(s_{\theta_{i}}^{R},\vartheta_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha}% \cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{R},% \vartheta_{j}^{R}})}{t}}}\right)^{2}}\right)^{\beta}}\right)^{\omega_{i}\omega% _{j}}}}\right)^{\frac{1}{n(n-1)}}}\right)^{1/{(\alpha+\beta)}}}}\right)\end{% bmatrix}\end{Bmatrix}$}$

Proof According to Definition 8, we can obtain

$\displaystyle\tilde{p}_{i}^{\alpha}=\begin{Bmatrix}\left[\Delta\left({t\left({% {\displaystyle\frac{\Delta^{-1}({s_{\phi_{i}}^{L},\varphi_{i}^{L}})}{t}}}% \right)^{\alpha}}\right),∼{}∼{}\Delta\left(t\left({\displaystyle\frac{\Delta^{% -1}(s_{\phi_{i}}^{R},\varphi_{i}^{R})}{t}}\right)^{\alpha}\right)\right],\\ \\ \left[{\Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s% _{\theta_{i}}^{L},\vartheta_{i}^{L})}{t}}}\right)^{2}}\right)^{\alpha}}}\right% ),∼{}∼{}\Delta\left({t\sqrt{1-\left({1-\left({\displaystyle\frac{\Delta^{-1}(s% _{\theta_{i}}^{R},\vartheta_{i}^{R})}{t}}\right)^{2}}\right)^{\alpha}}}\right)% }\right]\\ \end{Bmatrix}$ (9) $\displaystyle\tilde{p}_{j}^{\beta}=\begin{Bmatrix}\left[{\Delta\left({t\left({% {\displaystyle\frac{\Delta^{-1}({s_{\phi_{j}}^{L},\varphi_{j}^{L}})}{t}}}% \right)^{\beta}}\right),∼{}\Delta\left(t\left({\displaystyle\frac{\Delta^{-1}(% s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}}\right)^{\beta}\right)}\right],\\ \\ \left[{\Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s% _{\theta_{j}}^{L},\vartheta_{j}^{L})}{t}}}\right)^{2}}\right)^{\beta}}}\right)% ,∼{}\Delta\left({t\sqrt{1-\left({1-\left({\displaystyle\frac{\Delta^{-1}(s_{% \theta_{j}}^{R},\vartheta_{j}^{R})}{t}}\right)^{2}}\right)^{\beta}}}\right)}% \right]\\ \end{Bmatrix}$

Thus,

$\displaystyle\tilde{p}_{i}^{\alpha}\otimes\tilde{p}_{j}^{\beta}=\begin{Bmatrix% }\begin{bmatrix}\Delta\left({t\left({\left({{\displaystyle\frac{\Delta^{-1}({s% _{\phi_{i}}^{L},\varphi_{i}^{L}})}{t}}}\right)^{\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{L})}{t}}\right)^{% \beta}}\right)}\right),\\ \\ \Delta\left({t\left({\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{i}}^{R},% \varphi_{i}^{R}})}{t}}}\right)^{\alpha}\cdot\left({\displaystyle\frac{\Delta^{% -1}(s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}}\right)^{\beta}}\right)}\right)\end{% bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{% \Delta^{-1}(s_{\theta_{i}}^{L},\vartheta_{i}^{L})}{t}}}\right)^{2}}\right)^{% \alpha}\cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{L% },\vartheta_{j}^{L}})}{t}}}\right)^{2}}\right)^{\beta}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i}}^{R},\vartheta_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha}\cdot\left% ({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{R},\vartheta_{j}^{% R}})}{t}}}\right)^{2}}\right)^{\beta}}}\right)\end{bmatrix}\end{Bmatrix}$ (11)

Thereafter,

$\displaystyle\omega_{i}\omega_{j}({\tilde{p}_{i}^{\alpha}\otimes\tilde{p}_{j}^% {\beta}})=$ (12) $\displaystyle\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\sqrt{1-\left({1-% \left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{i}}^{L},\varphi_{i}^{L})}{t}}}% \right)^{2\alpha}\cdot\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{j}}^{L}% ,\varphi_{j}^{L}})}{t}}}\right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}% \right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {i}}^{R},\varphi_{i}^{R})}{t}}}\right)^{2\alpha}\cdot\left({{\displaystyle% \frac{\Delta^{-1}({s_{\phi_{j}}^{R},\varphi_{j}^{R}})}{t}}}\right)^{2\beta}}% \right)^{\omega_{i}\omega_{j}}}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\left({\sqrt{1-\left({1-\left({{\displaystyle% \frac{\Delta^{-1}(s_{\theta_{i}}^{L},\vartheta_{i}^{L})}{t}}}\right)^{2}}% \right)^{\alpha}\cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{% \theta_{j}}^{L},\vartheta_{j}^{L}})}{t}}}\right)^{2}}\right)^{\beta}}}\right)^% {\omega_{i}\omega_{j}}}\right),\\ \\ \Delta\left({t\left({\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s% _{\theta_{i}}^{R},\vartheta_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha}\cdot% \left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{R},\vartheta_% {j}^{R}})}{t}}}\right)^{2}}\right)^{\beta}}}\right)^{\omega_{i}\omega_{j}}}% \right)\end{bmatrix}\end{Bmatrix}$

Thus,

$\displaystyle\mathop{\oplus}\limits_{{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}({\omega_{i}\omega_{j}(\tilde{p}_{i}^{\alpha}% \otimes\tilde{p}_{j}^{\beta})})$ (13) $\displaystyle=\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\sqrt{1-% \displaystyle\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_% {\phi_{i}}^{L},\varphi_{i}^{L}})}{t}}}\right)^{2\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{L})}{t}}\right)^{% 2\beta}}\right)^{\omega_{i}\omega_{j}}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_% {\phi_{i}}^{R},\varphi_{i}^{R}})}{t}}}\right)^{2\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}}\right)^{% 2\beta}}\right)^{\omega_{i}\omega_{j}}}}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i,j% =1\\ i\neq j\end{subarray}}^{n}{\left({\sqrt{1-\left({1-\left({{\displaystyle\frac{% \Delta^{-1}(s_{\theta_{i}}^{L},\vartheta_{i}^{L})}{t}}}\right)^{2}}\right)^{% \alpha}\cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{L% },\vartheta_{j}^{L}})}{t}}}\right)^{2}}\right)^{\beta}}}\right)^{\omega_{i}% \omega_{j}}}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({\sqrt{1-\left({1-\left({{\displaystyle\frac{% \Delta^{-1}(s_{\theta_{i}}^{R},\vartheta_{i}^{R})}{t}}}\right)^{2}}\right)^{% \alpha}\cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{R% },\vartheta_{j}^{R}})}{t}}}\right)^{2}}\right)^{\beta}}}\right)^{\omega_{i}% \omega_{j}}}}\right)\end{bmatrix}\end{Bmatrix}$

Furthermore,

$\displaystyle\frac{1}{n(n-1)}\mathop{\oplus}\limits_{{\begin{subarray}{c}i,j=1% \\ i\neq j\end{subarray}}}^{n}({\omega_{i}\omega_{j}(\tilde{p}_{i}^{\alpha}% \otimes\tilde{p}_{j}^{\beta})})=$ (14)

Therefore,

(15)

\displaystyle\text{IV2TLPFWBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p}% _{2},\ldots,\tilde{p}_{n})=\left({\frac{1}{n(n-1)}\mathop{\oplus}\limits_{{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}({\omega_{i}\omega_{j}(\tilde{p}_{i}^{\alpha}% \otimes\tilde{p}_{j}^{\beta})})}\right)^{1/{({\alpha+\beta})}}=

Hence, Eq. (8) is kept.

Then we need to prove that Eq. (8) is an IV2TLPFN. We need to prove two conditions as follows:

(1) (1)

$\Delta^{-1}(s_{\phi}^{L},\varphi^{L}),\Delta^{-1}(s_{\phi}^{R},\varphi^{R}),% \Delta^{-1}({s_{\theta}^{L},\vartheta^{L}}),\Delta^{-1}({s_{\theta}^{R},% \vartheta^{L}})\subseteq[0,t]$ ,

(2)

$0\leqslant({\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})})^{2}+({\Delta^{-1}({s_{% \theta}^{R},\vartheta^{R}})})^{2}\leqslant t^{2}.$

Let

\displaystyle\frac{\Delta^{-1}({s_{\phi}^{L},\varphi^{L}})}{t}=\left({\sqrt{1-% \left({\displaystyle\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({\frac{\Delta^{-1}({s_{\phi_{i}}^{L}% ,\varphi_{i}^{L}})}{t}}\right)^{2\alpha}\cdot\left(\frac{\Delta^{-1}(s_{\phi_{% j}}^{L},\varphi_{j}^{L})}{t}\right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}% \right)^{\frac{1}{n(n-1)}}}}\right)^{1/{(\alpha+\beta)}}

\displaystyle\frac{\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})}{t}=\left({\sqrt{1-% \left({\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({\frac{\Delta^{-1}({s_{\phi_{i}}^{R}% ,\varphi_{i}^{R}})}{t}}\right)^{2\alpha}\cdot\left(\frac{\Delta^{-1}(s_{\phi_{% j}}^{R},\varphi_{j}^{R})}{t}\right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}% \right)^{\frac{1}{n(n-1)}}}}\right)^{1/{(\alpha+\beta)}}

\displaystyle\frac{\Delta^{-1}({s_{\theta}^{L},\vartheta^{L}})}{t}=

\displaystyle\sqrt{1-\left({1\!-\!\left({\prod\limits_{\begin{subarray}{c}i,j=% 1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({1-\left({\frac{\Delta^{-1}(s_{% \theta_{i}}^{L},\vartheta_{i}^{L})}{t}}\right)^{2}}\right)^{\alpha}\cdot\left(% {1\!-\!\left({\frac{\Delta^{-1}({s_{\theta_{j}}^{L},\vartheta_{j}^{L}})}{t}}% \right)^{2}}\right)^{\beta}}\right)^{\omega_{i}\omega_{j}}}}\right)^{\frac{1}{% n(n-1)}}}\right)^{1/{(\alpha+\beta)}}}

\displaystyle\frac{\Delta^{-1}({s_{\theta}^{R},\vartheta^{R}})}{t}=

\displaystyle\sqrt{1-\left({1-\left({\displaystyle\prod\limits_{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1\!-\!\left({1\!-\!\left({\frac{\Delta^{-1}(% s_{\theta_{i}}^{R},\vartheta_{i}^{R})}{t}}\right)^{2}}\right)^{\alpha}\cdot% \left({1-\left({\frac{\Delta^{-1}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}{t}}% \right)^{2}}\right)^{\beta}}\right)^{\omega_{i}\omega_{j}}}}\right)^{\frac{1}{% n(n-1)}}}\right)^{1/{(\alpha+\beta)}}}

Proof (1) Since $0\leqslant{\displaystyle\frac{\Delta^{-1}({s_{\phi_{i}}^{L},\varphi_{i}^{L}})}% {t}}\leqslant 1,0\leqslant{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},% \varphi_{j}^{L})}{t}}\leqslant 1$ we get

$\displaystyle 0\leqslant 1-\left({\frac{\Delta^{-1}({s_{\phi_{i}}^{L},\varphi_% {i}^{L}})}{t}}\right)^{2\alpha}\cdot\left({\frac{\Delta^{-1}({s_{\phi_{j}}^{L}% ,\varphi_{j}^{L}})}{t}}\right)^{2\beta}\leqslant 1$ (16)

Then,

$\displaystyle 0\leqslant 1-\left({\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({\frac{\Delta^{-1}({s_{\phi_{i}}^{L}% ,\varphi_{i}^{L}})}{t}}\right)^{2\alpha}\cdot\left(\frac{\Delta^{-1}(s_{\phi_{% j}}^{L},\varphi_{j}^{L})}{t}\right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}% \right)^{\frac{1}{n(n-1)}}\leqslant 1$ (17) $\displaystyle 0\leqslant\left({\sqrt{1-\left({\prod\limits_{\begin{subarray}{c% }i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({\frac{\Delta^{-1}({s_{\phi_{i}}^{L}% ,\varphi_{i}^{L}})}{t}}\right)^{2\alpha}\cdot\left(\frac{\Delta^{-1}(s_{\phi_{% j}}^{L},\varphi_{j}^{L})}{t}\right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}% \right)^{\frac{1}{n(n-1)}}}}\right)^{1/{(\alpha+\beta)}}\leqslant 1$ (18)

That means $0\leqslant\Delta^{-1}({s_{\phi}^{L},\varphi^{L}})\leqslant t$ , similarly, we can get $\Delta^{-1}({s_{\phi}^{R},\varphi^{R}}),\Delta^{-1}({s_{\theta}^{L},\vartheta^% {L}}),\Delta^{-1}({s_{\theta}^{R},\vartheta^{L}})\subseteq[0,t]$ , so

(1) is maintained.

(2) Since $0\leqslant\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{i}}^{R},\varphi_{i}% ^{R}})}{t}}}\right)^{2}+\left({{\displaystyle\frac{\Delta^{-1}(s_{\theta_{i}}^% {R},\vartheta_{i}^{R})}{t}}}\right)^{2}\leqslant 1$ , we get the following inequality

$\displaystyle\left({\frac{\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})}{t}}\right)^% {2}+\left({\frac{\Delta^{-1}({s_{\theta}^{R},\vartheta^{R}})}{t}}\right)^{2}$ $\displaystyle=\left({1-\left({\displaystyle\prod\limits_{\begin{subarray}{c}i,% j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({\frac{\Delta^{-1}({s_{\phi_{i}}^{R}% ,\varphi_{i}^{R}})}{t}}\right)^{2\alpha}\cdot\left(\frac{\Delta^{-1}(s_{\phi_{% j}}^{R},\varphi_{j}^{R})}{t}\right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}% \right)^{\frac{1}{n(n-1)}}}\right)^{1/{(\alpha+\beta)}}$ $\displaystyle+1-\left({1-\left({\displaystyle\prod\limits_{\begin{subarray}{c}% i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1\!-\!\left({1-\left({\frac{\Delta^{-1}(s_{% \theta_{i}}^{R},\vartheta_{i}^{R})}{t}}\right)^{2}}\right)^{\alpha}\cdot\left(% {1-\left({\frac{\Delta^{-1}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}{t}}\right% )^{2}}\right)^{\beta}}\right)^{\omega_{i}\omega_{j}}}}\right)^{\frac{1}{n(n-1)% }}}\right)^{1/{(\alpha+\beta)}}$ $\displaystyle\leqslant\left({1-\left({\displaystyle\prod\limits_{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({1-\left({\frac{\Delta^{-1}(s_{% \theta_{i}}^{R},\vartheta_{i}^{R})}{t}}\right)^{2}}\right)^{\alpha}\cdot\left(% {1-\left({\frac{\Delta^{-1}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}{t}}\right% )^{2}}\right)^{\beta}}\right)^{\omega_{i}\omega_{j}}}}\right)^{\frac{1}{n(n-1)% }}}\right)^{1/{(\alpha+\beta)}}$ $\displaystyle+\!\left({1\!-\!\left({1\!-\!\left({\displaystyle\prod\limits_{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1\!-\!\left({1\!-\!\left({\frac{\Delta^{-1}(% s_{\theta_{i}},\vartheta_{i})}{t}}\right)^{2}}\right)^{\alpha}\cdot\left({1-% \left({\frac{\Delta^{-1}({s_{\theta_{j}},\vartheta_{j}})}{t}}\right)^{2}}% \right)^{\beta}}\right)^{\omega_{i}\omega_{j}}}}\right)^{\frac{1}{n(n-1)}}}% \right)^{1/{(\alpha+\beta)}}}\right)$ $\displaystyle=1$

That means $0\leqslant({\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})})^{2}+({\Delta^{-1}({s_{% \theta}^{R},\vartheta^{R}})})^{2}\leqslant t^{2},$ so Eq. (2) is maintained.

The IV2TLPFWBM operator has the following properties.

Property 1. (Idempotency) If $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),({s_{\phi_{j}}^{R% },\varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{% j}}^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ are equal, then

$\displaystyle\text{IV2TLPFWBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p}% _{2},\ldots,\tilde{p}_{n})=\tilde{p}$ (19)

Property 2. (Monotonicity) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}% ^{L}}),({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})}],[{({s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}}),({s_{\theta_{% \tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})}]}\rangle$ and ${\tilde{p}}^{\prime}_{j}=\langle[(s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde% {p}_{j}}^{L})^{\prime},$ $({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})^{\prime}],[{(s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L})^{\prime},({s_{% \theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})^{\prime}}]\rangle,% j=1,2,\ldots,n$ be two sets of IV2TLPFNs. If $\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}})% \leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}% })^{\prime},\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^% {R}})\leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}% }^{R}})^{\prime}$ and $\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}})% \geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}% ^{L}})^{\prime},\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{% p}_{j}}^{R}})\geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{% \tilde{p}_{j}}^{R}})^{\prime}$ hold for all $j$ , then

$\displaystyle\text{IV2TLPFWBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p}% _{2},\ldots,\tilde{p}_{n})\leqslant{\rm IV}2\text{ TLPFWBM}_{\omega}^{\alpha,% \beta}({{\tilde{p}}^{\prime}_{1},{\tilde{p}}^{\prime}_{2},\ldots,{\tilde{p}}^{% \prime}_{n}})$ (20)

Property 3. (Boundedness) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),({s_{\phi_{j}}^{R% },\varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{% j}}^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ be a set of IV2TLPFNs. If $\tilde{p}^{+}=\langle{[{\max_{j}(s_{\phi_{j}}^{L},\varphi_{j}^{L}),\max_{j}({s% _{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\min_{j}(s_{\theta_{j}}^{L},\vartheta_{j% }^{L}),\min_{j}(s_{\theta_{j}}^{R},\vartheta_{j}^{R})}]}\rangle$ and $\tilde{p}^{-}=\langle{[{\min_{j}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\min_{j}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\max_{j}({s_{\theta_{j}}^{L},\vartheta% _{j}^{L}}),\max_{j}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ then

$\displaystyle\tilde{p}^{-}\leqslant\text{IV2TLPFWBM}_{\omega}^{\alpha,\beta}(% \tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{p}_{n})\leqslant\tilde{p}^{+}$ (21)

3.2 IV2TLPFWGBM operator

Similarly to WBM, to consider the attribute weights, the weighted geometric Bonferroni mean (WGBM) is defined as follows:

Definition 12 [65]. Let $\alpha,\beta>0$ and $\tilde{p}_{i}({i=1,2,\ldots,n})$ be a collection of nonnegative crisp numbers with the weights vector being $\omega=({\omega_{1},\omega_{2},\ldots\omega_{n}})^{T}$ , thereby satisfying $\omega_{i}\in[0,1]$ and $\sum_{i=1}^{n}{\omega_{i}}=1$ . if

$\displaystyle\text{WGBM}_{\omega}^{p,q}({\tilde{p}_{1},\tilde{p}_{2},\ldots,% \tilde{p}_{n}})=\frac{1}{\alpha+\beta}\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{({\alpha\tilde{p}_{i}+\beta\tilde{p}_{j}})^{\omega_% {i}\omega_{j}}}$ (22)

Then we extend WGBM to fuse the IV2TLPFNs and propose the interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (IV2TLPFWGBM) operator.

Definition 13 Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle({j=1,2,\ldots,n})$ be a set of IV2TLPFNs with their weight vector be $w_{i}=(w_{1},w_{2},\ldots,w_{n})^{T}$ , thereby satisfying $w_{i}\in[0,1]$ and $\sum\nolimits_{i=1}^{n}{w_{i}}=1.$ If

$\displaystyle\text{IV2TLPFWGBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p% }_{2},\ldots,\tilde{p}_{n})=\frac{1}{\alpha+\beta}\left({\mathop{\otimes}% \limits_{{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}({\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}})^{% \omega_{i}\omega_{j}}}\right)^{\frac{1}{n(n-1)}}$ (23)

Then we called $\text{IV2TLPFWGBM}_{\omega}^{\alpha,\beta}$ the interval-valued 2-tuple linguistic Pythagorean weighted geometric BM operator.

Theorem 3. Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle({j=1,2,\ldots,n})$ be a set of IV2TLPFNs. The aggregated value by using IV2TLPFWGBM operators is also an IV2TLPFN where

$\displaystyle\text{IV2TLPFWGBM}_{\omega}^{\alpha,\beta}\left(\tilde{p}_{1},% \tilde{p}_{2},\ldots,\tilde{p}_{n}\right)=\frac{1}{\alpha+\beta}\left({\mathop% {\otimes}\limits_{{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}\left({\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}}% \right)^{\omega_{i}\omega_{j}}}\right)^{\frac{1}{n(n-1)}}=$ (24)

Proof From Definition 8, we can obtain,

(25)

\displaystyle\alpha\tilde{p}_{i}=\left\{\begin{bmatrix}\Delta\left({t\sqrt{1-% \left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{i}}^{L},\varphi_{i}^{% L})}{t}}}\right)^{2}}\right)^{\alpha}}}\right),\\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {i}}^{R},\varphi_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha}}}\right)\end{% bmatrix},\begin{bmatrix}\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({% s_{\theta_{i}}^{L},\vartheta_{i}^{L}})}{t}}}\right)^{\alpha}}\right),\\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{i}}^{R},% \vartheta_{i}^{R}})}{t}}}\right)^{\alpha}}\right)\end{bmatrix}\right\}

\displaystyle\beta p_{j}=\left\{\begin{bmatrix}\Delta\left({t\sqrt{1-\left({1-% \left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{L})}{t}}}% \right)^{2}}\right)^{\beta}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {j}}^{R},\varphi_{j}^{R})}{t}}}\right)^{2}}\right)^{\beta}}}\right)\end{% bmatrix},\begin{bmatrix}\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({% s_{\theta_{j}}^{L},\vartheta_{j}^{L}})}{t}}}\right)^{\beta}}\right),\\ \\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{R},% \vartheta_{j}^{R}})}{t}}}\right)^{\beta}}\right)\end{bmatrix}\right\}

(26)

Thus,

$\displaystyle\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}=\begin{Bmatrix}\begin% {bmatrix}\Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}% (s_{\phi_{i}}^{L},\varphi_{i}^{L})}{t}}}\right)^{2}}\right)^{\alpha}\cdot\left% ({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{L})}{% t}}}\right)^{2}}\right)^{\beta}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {i}}^{R},\varphi_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha}\cdot\left({1-\left% ({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}}}% \right)^{2}}\right)^{\beta}}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\left({\left({{\displaystyle\frac{\Delta^{-1}({s_% {\theta_{i}}^{L},\vartheta_{i}^{L}})}{t}}}\right)^{\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\theta_{j}}^{L},\vartheta_{j}^{L})}{t}}% \right)^{\beta}}\right)}\right),\\ \\ \Delta\left({t\left({\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{i}}^{R% },\vartheta_{i}^{R}})}{t}}}\right)^{\alpha}\cdot\left({\displaystyle\frac{% \Delta^{-1}(s_{\theta_{j}}^{R},\vartheta_{j}^{R})}{t}}\right)^{\beta}}\right)}% \right)\end{bmatrix}\end{Bmatrix}$ (27)

Therefore,

$\displaystyle({\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}})^{\omega_{i}\omega% _{j}}$ (28) $\displaystyle=\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\left({\sqrt{1-\left% ({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{i}}^{L},\varphi_{i}^{L})}{% t}}}\right)^{2}}\right)^{\alpha}\cdot\left({1-\left({{\displaystyle\frac{% \Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{L})}{t}}}\right)^{2}}\right)^{\beta}% }}\right)^{{}^{\omega_{i}\omega_{j}}}}\right),\\ \\ \Delta\left({t\left({\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s% _{\phi_{i}}^{R},\varphi_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha}\cdot\left({% 1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}% }}\right)^{2}}\right)^{\beta}}}\right)^{{}^{\omega_{i}\omega_{j}}}}\right)\end% {bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{% \Delta^{-1}(s_{\theta_{i}}^{L},\vartheta_{i}^{L})}{t}}}\right)^{2\alpha}\cdot% \left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{L},\vartheta_{j}^{L}})% }{t}}}\right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i}}^{R},\vartheta_{i}^{R})}{t}}}\right)^{2\alpha}\cdot\left({{% \displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}{t}}}% \right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}\right)\end{bmatrix}\end{Bmatrix}$

Then,

$\displaystyle\mathop{\otimes}\limits_{{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}({\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}})^{% \omega_{i}\omega_{j}}$ (29) $\displaystyle=\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\displaystyle\prod% \limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({\sqrt{1-\left({1-\left({{\displaystyle\frac{% \Delta^{-1}(s_{\phi_{i}}^{L},\varphi_{i}^{L})}{t}}}\right)^{2}}\right)^{\alpha% }\cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi% _{j}^{L})}{t}}}\right)^{2}}\right)^{\beta}}}\right)^{{}^{\omega_{i}\omega_{j}}% }}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({\sqrt{1-\left({1-\left({{\displaystyle\frac{% \Delta^{-1}(s_{\phi_{i}}^{R},\varphi_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha% }\cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{R},\varphi% _{j}^{R})}{t}}}\right)^{2}}\right)^{\beta}}}\right)^{{}^{\omega_{i}\omega_{j}}% }}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\sqrt{1-\displaystyle\prod\limits_{% \begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_% {\theta_{i}}^{L},\vartheta_{i}^{L}})}{t}}}\right)^{2\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\theta_{j}}^{L},\vartheta_{j}^{L})}{t}}% \right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}^{n}{\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_% {\theta_{i}}^{R},\vartheta_{i}^{R}})}{t}}}\right)^{2\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\theta_{j}}^{R},\vartheta_{j}^{R})}{t}}% \right)^{2\beta}}\right)^{\omega_{i}\omega_{j}}}}}\right)\end{bmatrix}\end{Bmatrix}$

Thereafter,

$\displaystyle\left({\mathop{\otimes}\limits_{{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}\left({\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}}% \right)^{\omega_{i}\omega_{j}}}\right)^{\frac{1}{n(n-1)}}=$ (30)

Furthermore,

(31)

\displaystyle\text{IV2TLPFWGBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p% }_{2},\ldots,\tilde{p}_{n})=\frac{1}{\alpha+\beta}\left({\mathop{\otimes}% \limits_{{\begin{subarray}{c}i,j=1\\ i\neq j\end{subarray}}}^{n}({\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}})^{% \omega_{i}\omega_{j}}}\right)^{\frac{1}{n(n-1)}}=

Hence, Eq. (24) is kept.

Then we need to prove that Eq. (24) is an IV2TLPFN. We need to prove two conditions as follows:

(1) (1)

$\Delta^{-1}({s_{\phi}^{L},\varphi^{L}}),\Delta^{-1}(s_{\phi}^{R},\varphi^{R}),% \Delta^{-1}({s_{\theta}^{L},\vartheta^{L}}),\Delta^{-1}({s_{\theta}^{R},% \vartheta^{L}})\subseteq[0,t]$ .

(2)

$0\leqslant({\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})})^{2}+({\Delta^{-1}({s_{% \theta}^{R},\vartheta^{R}})})^{2}\leqslant t^{2}.$

For the proof is similar to the IV2TLPFWBM operator, so it’s omitted here to save space.

The IV2TLPFWGBM has the properties as follows.

Property 4(Idempotency). If $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),({s_{\phi_{j}}^{R% },\varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{% j}}^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ are equal, then

(32)

\displaystyle\text{IV2TLPFWGBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p% }_{2},\ldots,\tilde{p}_{n})=\tilde{p}

Property 5 (Monotonicity). Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}% ^{L}}),({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})}],[{({s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}}),({s_{\theta_{% \tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})}]}\rangle$ and ${\tilde{p}}^{\prime}_{j}=\langle[(s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde% {p}_{j}}^{L})^{\prime},$ $({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})^{\prime}],[{(s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L})^{\prime},({s_{% \theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})^{\prime}}]\rangle,% j=1,2,\ldots,n$ be two sets of IV2TLPFNs. If $\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}})% \leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}% })^{\prime},\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^% {R}})\leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}% }^{R}})^{\prime}$ and $\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}})% \geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}% ^{L}})^{\prime},\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{% p}_{j}}^{R}})\geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{% \tilde{p}_{j}}^{R}})^{\prime}$ hold for all $j$ , then

$\displaystyle\text{IV2TLPFWGBM}_{\omega}^{\alpha,\beta}(\tilde{p}_{1},\tilde{p% }_{2},\ldots,\tilde{p}_{n})\leqslant{\rm IV}2\text{ TLPFWGBM}_{\omega}^{\alpha% ,\beta}({{\tilde{p}}^{\prime}_{1},{\tilde{p}}^{\prime}_{2},\ldots,{\tilde{p}}^% {\prime}_{n}})$ (33)

Property 6 (Boundedness). Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),({s_{\phi_{j}}^{R% },\varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{% j}}^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ be a set of IV2TLPFNs. If $\tilde{p}^{+}=\langle[\max_{j}(s_{\phi_{j}}^{L},\varphi_{j}^{L}),$ $\max_{j}({s_{\phi_{j}}^{R},\varphi_{j}^{R}})],$ $[\min_{j}(s_{\theta_{j}}^{L},\vartheta_{j}^{L}),$ $\min_{j}(s_{\theta_{j}}^{R},\vartheta_{j}^{R})]\rangle$ and $\tilde{p}^{-}=\langle[\min_{j}(s_{\phi_{j}}^{L},\varphi_{j}^{L}),$ $\min_{j}(s_{\phi_{j}}^{R},\varphi_{j}^{R})],$ $[\max_{j}(s_{\theta_{j}}^{L},$ $\vartheta_{j}^{L}),$ $\max_{j}(s_{\theta_{j}}^{R},\vartheta_{j}^{R})]\rangle$ then

$\displaystyle\tilde{p}^{-}\leqslant\text{IV2TLPFWGBM}_{\omega}^{\alpha,\beta}(% \tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{p}_{n})\leqslant\tilde{p}^{+}$ (34)

4. GIV2TLPFWBM and GIV2TLPFWGBM operators

4.1 GIV2TLPFWBM operator

The primary advantage of BM is that it can determine the interrelationship between arguments. However, the traditional BM can only consider the correlations of any two aggregated arguments. Thereafter, Beliakov et al. [65] extended the BM and introduced generalized BM (GBM) operator. Zhu et al. [67] introduced generalized weighted BM (GWBM) operator as follows.

Definition 14 [67]. Let $\alpha,\beta,\gamma>0$ and $\tilde{p}_{i}({i=1,2,\ldots,n})$ be a collection of nonnegative crisp numbers with the weights vector being $\omega=(\omega_{1},\omega_{2},\ldots\omega_{n})^{T}$ , thereby satisfying $\omega_{i}\in[0,1]$ and $\sum_{i=1}^{n}{\omega_{i}}=1$ . The generalized weighted Bonferroni mean (GWBM) is defined as follows:

$\displaystyle\text{GWBM}_{\omega}^{\alpha,\beta,\gamma}({\tilde{p}_{1},\tilde{% p}_{2},\ldots,\tilde{p}_{n}})=\left({\sum\limits_{i,j,k=1}^{n}{\omega_{i}% \omega_{j}\omega_{k}\tilde{p}_{i}^{\alpha}\tilde{p}_{j}^{\beta}\tilde{p}_{k}^{% \gamma}}}\right)^{1/{({\alpha+\beta+\gamma})}}$ (35)

Then we extend GWBM to fuse the IV2TLPFNs and propose the generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (GIV2TLPFWBM) aggregation operator.

Definition 15 Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle({j=1,2,\ldots,n})$ be a set of IV2TLPFNs. The generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (G2TLPFWBM) operator is:

$\displaystyle\text{GIV2TLPFWBM}_{\omega}^{\alpha,\beta,\gamma}=\left({\frac{1}% {n(n-1)(n-2)}\mathop{\oplus}\limits_{{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}}^{n}\left({\omega_{i}\omega_{j}\omega_{k}\left({% \tilde{p}_{i}^{\alpha}\otimes\tilde{p}_{j}^{\beta}\otimes\tilde{p}_{k}^{\gamma% }}\right)}\right)}\right)^{1/{({\alpha+\beta+\gamma})}}$ (36)

Theorem 4. Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle({j=1,2,\ldots,n})$ be a set of IV2TLPFNs. The fused value by GIV2TLPFWBM operator is also an IV2TLPFN where

$\displaystyle\text{GIV2TLPFWBM}_{\omega}^{\alpha,\beta,\gamma}(\tilde{p}_{1},% \tilde{p}_{2},\ldots,\tilde{p}_{n})=\left({\displaystyle\frac{1}{n(n\!-\!1)(n% \!-\!2)}}\mathop{\oplus}\limits_{{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}}^{n}\left(\omega_{i}\omega_{j}\omega_{k}(\tilde{p% }_{i}^{\alpha}\otimes\tilde{p}_{j}^{\beta}\otimes\tilde{p}_{k}^{\gamma})\right% )\right)^{1/{\left({\alpha+\beta+\gamma}\right)}}=$ (37)

Proof According to Definition 8, we can obtain

(38)

\displaystyle\tilde{p}_{i}^{\alpha}=\left\{{\begin{bmatrix}\Delta\left({t\left% ({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{i}}^{L},\varphi_{i}^{L}})}{t}}}% \right)^{\alpha}}\right),\\ \\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{i}}^{R},\varphi% _{i}^{R}})}{t}}}\right)^{\alpha}}\right)\end{bmatrix},\begin{bmatrix}\Delta% \left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\theta_{i}}% ^{L},\vartheta_{i}^{L})}{t}}}\right)^{2}}\right)^{\alpha}}}\right)\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i}}^{R},\vartheta_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha}}}\right)% \end{bmatrix}}\right\}

\displaystyle\tilde{p}_{j}^{\beta}=\left\{{\begin{bmatrix}\Delta\left({t\left(% {{\displaystyle\frac{\Delta^{-1}({s_{\phi_{j}}^{L},\varphi_{j}^{L}})}{t}}}% \right)^{\beta}}\right),\\ \\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{j}}^{R},\varphi% _{j}^{R}})}{t}}}\right)^{\beta}}\right)\end{bmatrix},\begin{bmatrix}\Delta% \left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\theta_{j}}% ^{L},\vartheta_{j}^{L})}{t}}}\right)^{2}}\right)^{\beta}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{% \theta_{j}}^{R},\vartheta_{j}^{R})}{t}}}\right)^{2}}\right)^{\beta}}}\right)% \end{bmatrix}}\right\}

(39)

\displaystyle\tilde{p}_{k}^{\gamma}=\left\{{\begin{bmatrix}\Delta\left({t\left% ({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{k}}^{L},\varphi_{k}^{L}})}{t}}}% \right)^{\gamma}}\right),\\ \\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{k}}^{R},\varphi% _{k}^{R}})}{t}}}\right)^{\gamma}}\right)\end{bmatrix},\begin{bmatrix}\Delta% \left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\theta_{k}}% ^{L},\vartheta_{k}^{L})}{t}}}\right)^{2}}\right)^{\gamma}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{% \theta_{k}}^{R},\vartheta_{k}^{R})}{t}}}\right)^{2}}\right)^{\gamma}}}\right)% \end{bmatrix}}\right\}

(40)

Thus,

$\displaystyle\tilde{p}_{i}^{\alpha}\otimes\tilde{p}_{j}^{\beta}\otimes\tilde{p% }_{k}^{\gamma}=$ (41)

Thereafter,

(42)

\displaystyle\omega_{i}\omega_{j}\omega_{k}\left({\tilde{p}_{i}^{\alpha}% \otimes\tilde{p}_{j}^{\beta}\otimes\tilde{p}_{k}^{\gamma}}\right)=

Furthermore,

(43)

\displaystyle\mathop{\oplus}\limits_{{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}}^{n}\left({\omega_{i}\omega_{j}\omega_{k}\left({% \tilde{p}_{i}^{\alpha}\otimes\tilde{p}_{j}^{\beta}\otimes\tilde{p}_{k}^{\gamma% }}\right)}\right)

\displaystyle=\resizebox{407.6028pt}{0.0pt}{$\begin{Bmatrix}\begin{bmatrix}% \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}^{n}{\left({1-\left({{\displaystyle\frac{\Delta^{-% 1}({s_{\phi_{i}}^{L},\varphi_{i}^{L}})}{t}}}\right)^{2\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{L})}{t}}\right)^{% 2\beta}\cdot\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{k}}^{L},\varphi_{k% }^{L})}{t}}}\right)^{2\gamma}}\right)^{\omega_{i}\omega_{j}\omega_{k}}}}}% \right),\\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}^{n}{\left({1-\left({{\displaystyle\frac{\Delta^{-% 1}({s_{\phi_{i}}^{R},\varphi_{i}^{R}})}{t}}}\right)^{2\alpha}\cdot\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}}\right)^{% 2\beta}\cdot\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{k}}^{R},\varphi_{k% }^{R})}{t}}}\right)^{2\gamma}}\right)^{\omega_{i}\omega_{j}\omega_{k}}}}}% \right)\end{bmatrix},\\ \begin{bmatrix}\Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i,j% ,k=1\\ i\neq j\neq k\end{subarray}}^{n}{\left({\sqrt{1-\left({1-\left({{\displaystyle% \frac{\Delta^{-1}(s_{\theta_{i}}^{L},\vartheta_{i}^{L})}{t}}}\right)^{2}}% \right)^{\alpha}\cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{% \theta_{j}}^{L},\vartheta_{j}^{L}})}{t}}}\right)^{2}}\right)^{\beta}\cdot\left% ({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{k}}^{L},\vartheta_{k}^{% L}})}{t}}}\right)^{2}}\right)^{\gamma}}}\right)^{\omega_{i}\omega_{j}\omega_{k% }}}}\right),\\ \Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}^{n}{\left({\sqrt{1-\left({1-\left({{\displaystyle% \frac{\Delta^{-1}(s_{\theta_{i}}^{R},\vartheta_{i}^{R})}{t}}}\right)^{2}}% \right)^{\alpha}\cdot\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{% \theta_{j}}^{R},\vartheta_{j}^{R}})}{t}}}\right)^{2}}\right)^{\beta}\cdot\left% ({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{k}}^{R},\vartheta_{k}^{% R}})}{t}}}\right)^{2}}\right)^{\gamma}}}\right)^{\omega_{i}\omega_{j}\omega_{k% }}}}\right)\end{bmatrix}\end{Bmatrix}$}

Then,

$\displaystyle{\displaystyle\frac{1}{n(n-1)(n-2)}}\mathop{\oplus}\limits_{{% \begin{subarray}{c}i,j,k1\\ i\neq j\neq k\end{subarray}}}^{n}\left({\omega_{i}\omega_{j}\omega_{k}\left({% \tilde{p}_{i}^{\alpha}\otimes\tilde{p}_{j}^{\beta}\otimes\tilde{p}_{k}^{\gamma% }}\right)}\right)=$ (44)

Therefore,

\displaystyle\text{GIV2TLPFWBM}_{\omega}^{\alpha,\beta,\gamma}\left(\tilde{p}_% {1},\tilde{p}_{2},\ldots,\tilde{p}_{n}\right)=\left({{\displaystyle\frac{1}{n(% n-1)(n-2)}}\mathop{\oplus}\limits_{{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}}^{n}\left({\omega_{i}\omega_{j}\omega_{k}\left({% \tilde{p}_{i}^{\alpha}\otimes\tilde{p}_{j}^{\beta}\otimes\tilde{p}_{k}^{\gamma% }}\right)}\right)}\right)^{1/{\left({\alpha+\beta+\gamma}\right)}}=

Hence, Eq. (37) is kept.

Then we need to prove that Eq. (37) is an IV2TLPFN. We need to prove two conditions as follows:

\displaystyle(1)\Delta^{-1}({s_{\phi}^{L},\varphi^{L}}),\Delta^{-1}(s_{\phi}^{% R},\varphi^{R}),\Delta^{-1}({s_{\theta}^{L},\vartheta^{L}}),\Delta^{-1}({s_{% \theta}^{R},\vartheta^{L}})\subseteq[0,t]

\displaystyle(2)0\leqslant({\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})})^{2}+({% \Delta^{-1}({s_{\theta}^{R},\vartheta^{R}})})^{2}\leqslant t^{2}.

For the proof is similar to the IV2TLPFWBM operator, so it’s omitted here to save space.

Then we will discuss some properties of GIV2TLPFWBM operator.

Property 7. (Idempotency) If $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),({s_{\phi_{j}}^{R% },\varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{% j}}^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ are equal, then

$\displaystyle\text{GIV2TLPFWBM}_{\omega}^{\alpha,\beta,\gamma}(\tilde{p}_{1},% \tilde{p}_{2},\ldots,\tilde{p}_{n})=\tilde{p}$ (46)

Property 8. (Monotonicity) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}% ^{L}}),({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})}],[{({s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}}),({s_{\theta_{% \tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})}]}\rangle$ and ${\tilde{p}}^{\prime}_{j}=\langle[(s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde% {p}_{j}}^{L})^{\prime},$ $({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})^{\prime}],[{(s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L})^{\prime},({s_{% \theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})^{\prime}}]\rangle,% j=1,2,\ldots,n$ be two sets of IV2TLPFNs. If $\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}})% \leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}% })^{\prime},\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^% {R}})\leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}% }^{R}})^{\prime}$ and $\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}})% \geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}% ^{L}})^{\prime},\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{% p}_{j}}^{R}})\geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{% \tilde{p}_{j}}^{R}})^{\prime}$ hold for all $j$ , then

$\displaystyle\text{GIV2TLPFWBM}_{\omega}^{\alpha,\beta,\gamma}(\tilde{p}_{1},% \tilde{p}_{2},\ldots,\tilde{p}_{n})\leqslant{\rm GIV}2\text{ TLPFWBM}_{\omega}% ^{\alpha,\beta,\gamma}({{\tilde{p}}^{\prime}_{1},{\tilde{p}}^{\prime}_{2},% \ldots,{\tilde{p}}^{\prime}_{n}})$ (47)

Property 9. (Boundedness) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),({s_{\phi_{j}}^{R% },\varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{% j}}^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ be a set of IV2TLPFNs. If $\tilde{p}^{+}=\langle{[{\max_{j}(s_{\phi_{j}}^{L},\varphi_{j}^{L}),\max_{j}({s% _{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\min_{j}(s_{\theta_{j}}^{L},\vartheta_{j% }^{L}),\min_{j}(s_{\theta_{j}}^{R},\vartheta_{j}^{R})}]}\rangle$ and $\tilde{p}^{-}=\langle{[{\min_{j}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\min_{j}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\max_{j}({s_{\theta_{j}}^{L},\vartheta% _{j}^{L}}),\max_{j}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ then

$\displaystyle\tilde{p}^{-}\leqslant\text{GIV2TLPFWBM}_{\omega}^{\alpha,\beta,% \gamma}({\tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{p}_{n}})\leqslant\tilde{p}^% {+}$ (48)

4.2 GIV2TLPFWGBM operator

Similarly to GWBM, to consider the attribute weights, the generalized weighted geometric Bonferroni mean (GWGBM) is defined as follows:

Definition 16 [67]. Let $\alpha,\beta,\gamma>0$ and $\tilde{p}_{i}({i=1,2,\ldots,n})$ be a collection of nonnegative crisp numbers with the weights vector being $\omega=(\omega_{1},\omega_{2},\ldots\omega_{n})^{T}$ , thereby satisfying $\omega_{i}\in[0,1]$ and $\sum\limits_{i=1}^{n}{\omega_{i}}=1$ . if

$\displaystyle{\rm GWGBM}_{\omega}^{\alpha,\beta,\gamma}({\tilde{p}_{1},\tilde{% p}_{2},\ldots,\tilde{p}_{n}})=\frac{1}{\alpha+\beta+\gamma}\prod\limits_{% \begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}^{n}{({\alpha\tilde{p}_{i}+\beta\tilde{p}_{j}+% \gamma\tilde{p}_{k}})^{\omega_{i}\omega_{j}\omega_{k}}}$ (49)

Then we extend GWGBM to fuse the IV2TLPFNs and propose the generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (GIV2TLPFWGBM) aggregation operator.

Definition 17 Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ be a set of IV2TLPFNs with their weight vector be $w_{i}=(w_{1},w_{2},\ldots,w_{n})^{T}$ , thereby satisfying $w_{i}\in[{0,1}]$ and $\sum\nolimits_{i=1}^{n}{w_{i}}=1.$ If

$\displaystyle\text{GIV2TLPFWGBM}_{\omega}^{\alpha,\beta,\gamma}(\tilde{p}_{1},% \tilde{p}_{2},\ldots,\tilde{p}_{n})=\frac{1}{\alpha+\beta+\gamma}\left({% \mathop{\otimes}\limits_{{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}}^{n}({\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}% \oplus\gamma\tilde{p}_{k}})^{\omega_{i}\omega_{j}\omega_{k}}}\right)^{\frac{1}% {n(n-1)({n-2})}}$ (50)

Then we called $\text{GIV2TLPFWGBM}_{\omega}^{\alpha,\beta,\gamma}$ the generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric BM operator.

Theorem 5. Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ be a set of IV2TLPFNs. The fused value by GIV2TLPFWGBM operator is also an IV2TLPFN where

Proof From Definition 8, we can obtain,

(52)

\displaystyle\alpha\tilde{p}_{i}=\left\{{\begin{bmatrix}\Delta\left({t\sqrt{1-% \left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{i}}^{L},\varphi_{i}^{% L})}{t}}}\right)^{2}}\right)^{\alpha}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {i}}^{R},\varphi_{i}^{R})}{t}}}\right)^{2}}\right)^{\alpha}}}\right)\end{% bmatrix},\begin{bmatrix}\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({% s_{\theta_{i}}^{L},\vartheta_{i}^{L}})}{t}}}\right)^{\alpha}}\right),\\ \\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{i}}^{R},% \vartheta_{i}^{R}})}{t}}}\right)^{\alpha}}\right)\end{bmatrix}}\right\}

\displaystyle\beta\tilde{p}_{j}=\left\{{\begin{bmatrix}\Delta\left({t\sqrt{1-% \left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{% L})}{t}}}\right)^{2}}\right)^{\beta}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {j}}^{R},\varphi_{j}^{R})}{t}}}\right)^{2}}\right)^{\beta}}}\right)\end{% bmatrix},\begin{bmatrix}\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({% s_{\theta_{j}}^{L},\vartheta_{j}^{L}})}{t}}}\right)^{\beta}}\right),\\ \\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{j}}^{R},% \vartheta_{j}^{R}})}{t}}}\right)^{\beta}}\right)\end{bmatrix}}\right\}

(53)

\displaystyle\gamma\tilde{p}_{k}=\left\{{\begin{bmatrix}\Delta\left({t\sqrt{1-% \left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{k}}^{L},\varphi_{k}^{% L})}{t}}}\right)^{2}}\right)^{\gamma}}}\right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {k}}^{R},\varphi_{k}^{R})}{t}}}\right)^{2}}\right)^{\gamma}}}\right)\end{% bmatrix},\begin{bmatrix}\Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({% s_{\theta_{k}}^{L},\vartheta_{k}^{L}})}{t}}}\right)^{\gamma}}\right),\\ \\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{k}}^{R},% \vartheta_{k}^{R}})}{t}}}\right)^{\gamma}}\right)\end{bmatrix}}\right\}

(54)

Thus

$\displaystyle\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}\oplus\gamma\tilde{p}_% {k}=$ (55)

Therefore,

(56)

\displaystyle\left({\alpha\tilde{p}_{i}\oplus\beta\tilde{p}_{j}\oplus\gamma% \tilde{p}_{k}}\right)^{\omega_{i}\omega_{j}\omega_{k}}=

Thereafter,

(57)

\displaystyle\mathop{\otimes}\limits_{{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}}^{n}\left({\alpha\tilde{p}_{i}\oplus\beta\tilde{p% }_{j}\oplus\gamma\tilde{p}_{k}}\right)^{\omega_{i}\omega_{j}\omega_{k}}=

Then,

(58)

\displaystyle\left(\mathop{\otimes}\limits_{{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}}^{n}\left({\alpha\tilde{p}_{i}\oplus\beta\tilde{p% }_{j}\oplus\gamma\tilde{p}_{k}}\right)^{\omega_{i}\omega_{j}\omega_{k}}\right)% ^{\frac{1}{n(n-1)(n-2)}}=

Furthermore,

(59)

\displaystyle\text{GIV2TLPFWGBM}_{\omega}^{\alpha,\beta,\gamma}\left(\tilde{p}% _{1},\tilde{p}_{2},\ldots,\tilde{p}_{n}\right)={\displaystyle\frac{1}{\alpha+% \beta+\gamma}}\left({\mathop{\otimes}\limits_{{\begin{subarray}{c}i,j,k=1\\ i\neq j\neq k\end{subarray}}}^{n}\left({\alpha\tilde{p}_{i}\oplus\beta\tilde{p% }_{j}\oplus\gamma\tilde{p}_{k}}\right)^{\omega_{i}\omega_{j}\omega_{k}}}\right% )^{\frac{1}{n(n-1)(n-2)}}=

Hence, Eq. (51) is kept.

Then we need to prove that Eq. (51) is an IV2TLPFN. We need to prove two conditions as follows:

(1) (1)

$\Delta^{-1}({s_{\phi}^{L},\varphi^{L}}),\Delta^{-1}(s_{\phi}^{R},\varphi^{R}),% \Delta^{-1}({s_{\theta}^{L},\vartheta^{L}}),\Delta^{-1}({s_{\theta}^{R},% \vartheta^{L}})\subseteq[0,t]$ .

(2)

$0\leqslant({\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})})^{2}+({\Delta^{-1}({s_{% \theta}^{R},\vartheta^{R}})})^{2}\leqslant t^{2}$ .

For the proof is similar to the IV2TLPFWBM operator, so it’s omitted here to save space.

Then we will discuss some properties of GIV2TLPFWGBM operator.

Property 10. (Idempotency) If $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),(s_{\phi_{j}}^{R}% ,\varphi_{j}^{R})}],[{(s_{\theta_{j}}^{L},\vartheta_{j}^{L}),({s_{\theta_{j}}^% {R},\vartheta_{j}^{R}})}]}\rangle$ $(j=1,2,\ldots,n)$ are equal, then

(60)

\displaystyle\text{GIV2TLPFWGBM}_{\omega}^{\alpha,\beta,\gamma}(\tilde{p}_{1},% \tilde{p}_{2},\ldots,\tilde{p}_{n})=\tilde{p}

Property 11. (Monotonicity) Let $\tilde{p}_{j}\!=\!\langle{[{({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_% {j}}^{L}}),({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})}],[{({% s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}}),({s_{\theta_{% \tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})}]}\rangle$ and ${\tilde{p}}^{\prime}_{j}\!=\!\langle[(s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{% \tilde{p}_{j}}^{L})^{\prime},$ $({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})^{\prime}],[{(s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L})^{\prime},({s_{% \theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})^{\prime}}]\rangle,% j=1,2,\ldots,n$ be two sets of IV2TLPFNs. If $\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}})% \leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}% })^{\prime},\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^% {R}})\leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}% }^{R}})^{\prime}$ and $\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}})% \geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}% ^{L}})^{\prime},\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{% p}_{j}}^{R}})\geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{% \tilde{p}_{j}}^{R}})^{\prime}$ hold for all $j$ , then

$\displaystyle\text{GIV2TLPFWGBM}_{\omega}^{\alpha,\beta,\gamma}(\tilde{p}_{1},% \tilde{p}_{2},\ldots,\tilde{p}_{n})\leqslant{\rm GIV}2\text{ TLPFWGBM}_{\omega% }^{\alpha,\beta,\gamma}({{\tilde{p}}^{\prime}_{1},{\tilde{p}}^{\prime}_{2},% \ldots,{\tilde{p}}^{\prime}_{n}})$ (61)

Property 12.(Boundedness) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),(s_{\phi_{j}}^{R}% ,\varphi_{j}^{R})}],[{(s_{\theta_{j}}^{L},\vartheta_{j}^{L}),({s_{\theta_{j}}^% {R},\vartheta_{j}^{R}})}]}\rangle$ $(j=1,2,\ldots,n)$ be a set of IV2TLPFNs. If $\tilde{p}^{+}=\langle{[{\max_{j}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\max_{j}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\min_{j}({s_{\theta_{j}}^{L},\vartheta% _{j}^{L}}),\min_{j}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ and $\tilde{p}^{-}=\langle{[{\min_{j}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\min_{j}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\max_{j}({s_{\theta_{j}}^{L},\vartheta% _{j}^{L}}),\max_{j}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ then

$\displaystyle\tilde{p}^{-}\leqslant\text{GIV2TLPFWGBM}_{\omega}^{\alpha,\beta,% \gamma}({\tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{p}_{n}})\leqslant\tilde{p}^% {+}$ (62)

5. DGIV2TLPFWBM and DGIV2TLPFWGBM operators

5.1 DGIV2TLPFWBM operator

However, the GBM still has some drawbacks, GBWM and GWGBM can only consider the interrelationship between any three aggregated arguments. So Zhang et al. [68] introduced a new generalization of the traditional BM because the correlations are ubiquitous among all arguments. The new generalization of the traditional BM is called the dual GWBM (DGBM). The DGWBM is defined as follows:

Definition 18 [68]. Let $\tilde{p}_{i}({i=1,2,\ldots,n})$ be a collection of nonnegative crisp numbers with the weights vector being $\omega=({\omega_{1},\omega_{2},\ldots\omega_{n}})^{T}$ , thereby satisfying $\omega_{i}\in[0,1]$ and $\sum\limits_{i=1}^{n}{\omega_{i}}=1$ . The dual generalized weighted Bonferroni mean (DGWBM) is defined as follows:

$\displaystyle\text{DGWBM}_{w}^{r}\left(\tilde{p}_{1},\tilde{p}_{2},\ldots,% \tilde{p}_{n}\right)=\left({\sum\limits_{i_{1},i_{2},\ldots,i_{n}=1}^{n}{\left% ({\displaystyle\prod\limits_{j=1}^{n}{w_{i_{j}}}\tilde{p}_{i_{j}}^{r_{j}}}% \right)}}\right)^{1/{\sum\nolimits_{j=1}^{n}{r_{j}}}}$ (63)

where $r=(r_{1},r_{2},\ldots,r_{n})^{T}$ is the parameter vector with $r_{i}\geqslant 0(i=1,2,\ldots,n)$ .

Then we extend DGWBM to fuse the IV2TLPFNs and propose the dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (DGIV2TLPFWBM) operator.

Definition 19 Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle({j=1,2,\ldots,n})$ be a set of IV2TLPFNs. The dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (DGIV2TLP FWBM) operator is:

$\displaystyle\text{DGIV2TLPFWBM}_{w}^{r}\left(\tilde{p}_{1},\tilde{p}_{2},% \ldots,\tilde{p}_{n}\right)=\left({\frac{1}{n!}\mathop{\otimes}^{n}\limits_{{% \begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}}\left({\mathop{\otimes}^{n}% \limits_{j=1}w_{i_{j}}\tilde{p}_{i_{j}}^{r_{j}}}\right)}\right)^{1/{\sum% \nolimits_{i=1}^{n}{r_{j}}}}$ (64)

Theorem 6. Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle({j=1,2,\ldots,n})$ be a set of IV2TLPFNs. The fused value by DGIV2TLPFWBM operators is also an IV2TLPFN where

$\displaystyle\text{DGIV2TLPFWBM}_{w}^{r}\left(\tilde{p}_{1},\tilde{p}_{2},% \ldots,\tilde{p}_{n}\right)=\left({{\displaystyle\frac{1}{n!}}\mathop{\oplus}^% {n}\limits_{{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}}\left({\mathop{\otimes}^{n}% \limits_{j=1}w_{i_{j}}\tilde{p}_{i_{j}}^{r_{j}}}\right)}\right)^{1/{\sum% \nolimits_{i=1}^{n}{r_{j}}}}=$ (65)

Proof According to Definition 8, we can obtain

(66)

\displaystyle\tilde{p}_{i_{j}}^{r_{j}}=\left\{{\begin{bmatrix}\Delta\left({t% \left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{i_{j}}}^{L},\varphi_{i_{j}}^{% L}})}{t}}}\right)^{r_{j}}}\right),\\ \\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\phi_{i_{j}}}^{R},% \varphi_{i_{j}}^{R}})}{t}}}\right)^{r_{j}}}\right)\end{bmatrix},\begin{bmatrix% }\Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i_{j}}}^{L},\vartheta_{i_{j}}^{L})}{t}}}\right)^{2}}\right)^{r_{j}}}}% \right),\\ \\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i_{j}}}^{R},\vartheta_{i_{j}}^{R})}{t}}}\right)^{2}}\right)^{r_{j}}}}% \right)\end{bmatrix}}\right\}

Thus,

$\displaystyle w_{i_{j}}\tilde{p}_{i_{j}}^{r_{j}}=$ (67)

Thereafter,

(68)

\displaystyle\mathop{\otimes}^{n}\limits_{j=1}w_{i_{j}}\tilde{p}_{i_{j}}^{r_{j% }}=\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\displaystyle\prod\limits_{j=1}% ^{n}{\sqrt{1-\left({1-\left({\displaystyle\frac{\Delta^{-1}(s_{\phi_{i_{j}}}^{% L},\varphi_{i_{j}}^{L})}{t}}\right)^{2r_{j}}}\right)^{w_{i_{j}}}}}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{j=1}^{n}{\sqrt{1-\left({1-\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{i_{j}}}^{R},\varphi_{i_{j}}^{R})}{t}}% \right)^{2r_{j}}}\right)^{w_{i_{j}}}}}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\sqrt{1-\displaystyle\prod\limits_{j=1}^{n}{\left% ({1-\left({1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{i_{j}% }}^{L},\vartheta_{i_{j}}^{L}})}{t}}}\right)^{2}}\right)^{r_{j}}}\right)^{w_{i_% {j}}}}\right)}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({1-% \left({1-\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{i_{j}}}^{R},% \vartheta_{i_{j}}^{R}})}{t}}}\right)^{2}}\right)^{r_{j}}}\right)^{w_{i_{j}}}}% \right)}}}\right)\end{bmatrix}\end{Bmatrix}

Furthermore,

$\displaystyle\mathop{\oplus}^{n}\limits_{{\begin{subarray}{c}i_{1},i_{2},% \ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}}\left({\mathop{\otimes}^{n}% \limits_{j=1}w_{i_{j}}\tilde{p}_{i_{j}}^{r_{j}}}\right)$ $\displaystyle=\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\sqrt{1-% \displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({1-\displaystyle% \prod\limits_{j=1}^{n}{\left({1-\left({1-\left({{\displaystyle\frac{\Delta^{-1% }({s_{\phi_{i_{j}}}^{L},\varphi_{i_{j}}^{L}})}{t}}}\right)^{2r_{j}}}\right)^{w% _{i_{j}}}}\right)}}\right)}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2% },\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({1-\displaystyle% \prod\limits_{j=1}^{n}{\left({1-\left({1-\left({{\displaystyle\frac{\Delta^{-1% }({s_{\phi_{i_{j}}}^{R},\varphi_{i_{j}}^{R}})}{t}}}\right)^{2r_{j}}}\right)^{w% _{i_{j}}}}\right)}}\right)}}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i_{% 1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\sqrt{1-\displaystyle% \prod\limits_{j=1}^{n}{\left({1-\left({1-\left({1-\left({{\displaystyle\frac{% \Delta^{-1}({s_{\theta_{i_{j}}}^{L},\vartheta_{i_{j}}^{L}})}{t}}}\right)^{2}}% \right)^{r_{j}}}\right)^{w_{i_{j}}}}\right)}}}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2},\ldots% ,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\sqrt{1-\displaystyle% \prod\limits_{j=1}^{n}{\left({1-\left({1-\left({1-\left({{\displaystyle\frac{% \Delta^{-1}({s_{\theta_{i_{j}}}^{R},\vartheta_{i_{j}}^{R}})}{t}}}\right)^{2}}% \right)^{r_{j}}}\right)^{w_{i_{j}}}}\right)}}}}\right)\end{bmatrix}\end{Bmatrix}$ (69)

Then,

$\displaystyle{\displaystyle\frac{1}{n!}}\mathop{\oplus}^{n}\limits_{{% \begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}}\left({\mathop{\otimes}^{n}% \limits_{j=1}w_{i_{j}}\tilde{p}_{i_{j}}^{r_{j}}}\right)$ (70) $\displaystyle=\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\sqrt{1-% \displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({1-\displaystyle% \prod\limits_{j=1}^{n}{\left({1-\left({1-\left({{\displaystyle\frac{\Delta^{-1% }({s_{\phi_{i_{j}}}^{L},\varphi_{i_{j}}^{L}})}{t}}}\right)^{2r_{j}}}\right)^{w% _{i_{j}}}}\right)}}\right)^{\frac{1}{n!}}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2% },\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({1-\displaystyle% \prod\limits_{j=1}^{n}{\left({1-\left({1-\left({{\displaystyle\frac{\Delta^{-1% }({s_{\phi_{i_{j}}}^{R},\varphi_{i_{j}}^{R}})}{t}}}\right)^{2r_{j}}}\right)^{w% _{i_{j}}}}\right)}}\right)^{\frac{1}{n!}}}}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i_{% 1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\sqrt{1-% \displaystyle\prod\limits_{j=1}^{n}{\left({1-\left(1-\left({1-\left({% \displaystyle\frac{\Delta^{-1}(s_{\theta_{i_{j}}}^{L},\vartheta_{i_{j}}^{L})}{% t}}\right)^{2}}\right)^{r_{j}}\right)^{w_{i_{j}}}}\right)}}}\right)^{\frac{1}{% n!}}}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2},\ldots% ,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\sqrt{1-% \displaystyle\prod\limits_{j=1}^{n}{\left({1-\left(1-\left({1-\left({% \displaystyle\frac{\Delta^{-1}(s_{\theta_{i_{j}}}^{R},\vartheta_{i_{j}}^{R})}{% t}}\right)^{2}}\right)^{r_{j}}\right)^{w_{i_{j}}}}\right)}}}\right)^{\frac{1}{% n!}}}}\right)\end{bmatrix}\end{Bmatrix}$

Therefore,

Hence, Eq. (65) is kept.

Then we need to prove that Eq. (65) is an IV2TLPFN. We need to prove two conditions as follows:

(1) (1)

$\Delta^{-1}({s_{\phi}^{L},\varphi^{L}}),\Delta^{-1}(s_{\phi}^{R},\varphi^{R}),% \Delta^{-1}({s_{\theta}^{L},\vartheta^{L}}),\Delta^{-1}({s_{\theta}^{R},% \vartheta^{L}})\subseteq[0,t]$ .

(2)

$0\leqslant({\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})})^{2}+({\Delta^{-1}({s_{% \theta}^{R},\vartheta^{R}})})^{2}\leqslant t^{2}$ .

For the proof is similar to the IV2TLPFWBM operator, so it’s omitted here to save space.

Then we will discuss some properties of DGIV2TLPFWBM operator.

Property 13. (Monotonicity) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}% ^{L}}),({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})}],[{({s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}}),({s_{\theta_{% \tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})}]}\rangle$ and ${\tilde{p}}^{\prime}_{j}=\langle[(s_{\phi_{\tilde{p}_{j}}}^{L},$ $\varphi_{\tilde{p}_{j}}^{L})^{\prime},({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{% \tilde{p}_{j}}^{R}})^{\prime}],[{(s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{% \tilde{p}_{j}}^{L})^{\prime},({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{% \tilde{p}_{j}}^{R}})^{\prime}}]\rangle,j=1,2,\ldots,n$ be two sets of IV2TLPFNs. If $\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}})% \leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}% })^{\prime},\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^% {R}})\leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}% }^{R}})^{\prime}$ and $\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}})% \geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}% ^{L}})^{\prime},\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{% p}_{j}}^{R}})\geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{% \tilde{p}_{j}}^{R}})^{\prime}$ hold for all $j$ , then

(72)

\displaystyle\text{DGIV2TLPFWBM}_{\omega}^{r}(\tilde{p}_{1},\tilde{p}_{2},% \ldots,\tilde{p}_{n})\leqslant\text{DGIV}2\text{ TLPFWBM}_{\omega}^{r}({{% \tilde{p}}^{\prime}_{1},{\tilde{p}}^{\prime}_{2},\ldots,{\tilde{p}}^{\prime}_{% n}})

Property 14. (Boundedness) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),(s_{\phi_{j}}^{R}% ,\varphi_{j}^{R})}],[{(s_{\theta_{j}}^{L},\vartheta_{j}^{L}),({s_{\theta_{j}}^% {R},\vartheta_{j}^{R}})}]}\rangle$ $(j=1,2,\ldots,n)$ be a set of IV2TLPFNs. If $\tilde{p}^{+}=\langle{[{\max_{j}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\max_{j}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\min_{j}({s_{\theta_{j}}^{L},\vartheta% _{j}^{L}}),\min_{j}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ and $\tilde{p}^{-}=\langle{[{\min_{j}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\min_{j}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\max_{j}({s_{\theta_{j}}^{L},\vartheta% _{j}^{L}}),\max_{j}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ then

$\displaystyle\tilde{p}^{-}\leqslant\text{DGIV2TLPFWBM}_{\omega}^{r}({\tilde{p}% _{1},\tilde{p}_{2},\ldots,\tilde{p}_{n}})\leqslant\tilde{p}^{+}$ (73)

5.2 DGIV2TLPFWGBM operator

Similarly to DGWBM, to consider the attribute weights, the dual generalized weighted geometric Bonferroni mean (DGWGBM) is introduced as follows:

Definition 20 [68]. Let and $\tilde{p}_{i}({i=1,2,\ldots,n})$ be a collection of nonnegative crisp numbers with the weights vector being $\omega=({\omega_{1},\omega_{2},\ldots\omega_{n}})^{T}$ , thereby satisfying $\omega_{i}\in[0,1]$ and $\sum_{i=1}^{n}{\omega_{i}}=1$ . If

$\displaystyle\text{DGWGBM}_{\omega}^{r}\left({\tilde{p}_{1},\tilde{p}_{2},% \ldots,\tilde{p}_{n}}\right)=\frac{1}{\sum_{j=1}^{n}{r_{j}}}\left({% \displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq,\ldots,\neq i_{n}\end{subarray}}^{n}{\left({\sum\limits_{j% =1}^{n}{\left({r_{j}\tilde{p}_{i_{j}}}\right)}}\right)}}\right)^{\prod% \nolimits_{j=1}^{n}{w_{i_{j}}}}$ (74)

where $r=(r_{1},r_{2},\ldots,r_{n})^{T}$ is the parameter vector with $r_{i}\geqslant 0$ $(i=1,2,\ldots,n).$

Then we extend DGWGBM to fuse the IV2TLPFNs and propose dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (DGIV2TLPFWGBM) aggregation operator.

Definition 21 Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ be a set of IV2TLPFNs with their weight vector be $w_{i}=(w_{1},w_{2},\ldots,w_{n})^{T}$ , thereby satisfying $w_{i}\in[{0,1}]$ and $\sum\nolimits_{i=1}^{n}{w_{i}}=1.$ If

$\displaystyle\text{DGIV2TLPFWGBM}_{\omega}^{r}\left(\tilde{p}_{1},\tilde{p}_{2% },\ldots,\tilde{p}_{n}\right)=\frac{1}{\sum\nolimits_{j=1}^{n}{r_{j}}}\left({% \mathop{\oplus}^{n}\limits_{{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}}\left({\mathop{\otimes}^{n}% \limits_{j=1}\left({r_{j}\tilde{p}_{i_{j}}}\right)}\right)^{\prod{{}_{j=1}^{n}% w_{i_{j}}}}}\right)^{\frac{1}{n!}}$ (75)

Then we called $\text{DGIV2TLPFWGBM}_{\omega}^{r}$ the dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric BM operator.

Theorem 7.Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ be a set of IV2TLPFNs. The fused value by DGIV2TLPFWGBM operators is also an IV2TLPFN where

$\displaystyle\text{DGIV2TLPFWGBM}_{w}^{r}\left(\tilde{p}_{1},\tilde{p}_{2},% \ldots,\tilde{p}_{n}\right)={\displaystyle\frac{1}{\sum\nolimits_{j=1}^{n}{r_{% j}}}}\left({\mathop{\oplus}^{n}\limits_{{\begin{subarray}{c}i_{1},i_{2},\ldots% ,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}}\left({\mathop{\otimes}^{n}% \limits_{j=1}\left({r_{j}\tilde{p}_{i_{j}}}\right)}\right)^{\prod{{}_{j=1}^{n}% w_{i_{j}}}}}\right)^{\frac{1}{n!}}$ (76) $\displaystyle=\resizebox{390.258pt}{0.0pt}{$\begin{Bmatrix}\begin{bmatrix}% \Delta\left({t\sqrt{1-\left({1-\displaystyle\prod\limits_{\begin{subarray}{c}i% _{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\left({1-% \displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({\displaystyle\frac{\Delta^% {-1}(s_{\phi_{i_{j}}}^{L},\varphi_{i_{j}}^{L})}{t}}\right)^{2}}\right)^{r_{j}}% }}\right)^{\prod{{}_{j=1}^{n}w_{i_{j}}}}}\right)^{{}^{\frac{1}{n!}}}}}\right)^% {\frac{1}{\sum\nolimits_{j=1}^{n}{r_{j}}}}}}\right),\\ \Delta\left({t\sqrt{1-\left({1-\displaystyle\prod\limits_{\begin{subarray}{c}i% _{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\left({1-% \displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({\displaystyle\frac{\Delta^% {-1}(s_{\phi_{i_{j}}}^{R},\varphi_{i_{j}}^{R})}{t}}\right)^{2}}\right)^{r_{j}}% }}\right)^{\prod{{}_{j=1}^{n}w_{i_{j}}}}}\right)^{{}^{\frac{1}{n!}}}}}\right)^% {\frac{1}{\sum\nolimits_{j=1}^{n}{r_{j}}}}}}\right)\end{bmatrix},\\ \begin{bmatrix}\Delta\left({t\left({\sqrt{1-\displaystyle\prod\limits_{% \begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\left({1-% \displaystyle\prod\limits_{j=1}^{n}{\left({\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i_{j}}}^{L},\vartheta_{i_{j}}^{L})}{t}}\right)^{2r_{j}}}}\right)^{% \prod{{}_{j=1}^{n}w_{i_{j}}}}}\right)^{{}^{\frac{1}{n!}}}}}}\right)^{\frac{1}{% \sum\nolimits_{j=1}^{n}{r_{j}}}}}\right),\\ \Delta\left({t\left({\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i_{% 1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\left({1-% \displaystyle\prod\limits_{j=1}^{n}{\left({\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i_{j}}}^{R},\vartheta_{i_{j}}^{R})}{t}}\right)^{2r_{j}}}}\right)^{% \prod{{}_{j=1}^{n}w_{i_{j}}}}}\right)^{{}^{\frac{1}{n!}}}}}}\right)^{\frac{1}{% \sum\nolimits_{j=1}^{n}{r_{j}}}}}\right)\end{bmatrix}\end{Bmatrix}$}$

Proof From Definition 8, we can obtain,

$\displaystyle r_{j}\tilde{p}_{i_{j}}=\left\{{\begin{bmatrix}\Delta\left({t% \sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{i_{j}}}^{L},% \varphi_{i_{j}}^{L})}{t}}}\right)^{2}}\right)^{r_{j}}}}\right),\\ \Delta\left({t\sqrt{1-\left({1-\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_% {i_{j}}}^{R},\varphi_{i_{j}}^{R})}{t}}}\right)^{2}}\right)^{r_{j}}}}\right)% \end{bmatrix},\begin{bmatrix}\Delta\left({t\left({{\displaystyle\frac{\Delta^{% -1}({s_{\theta_{i_{j}}}^{L},\vartheta_{i_{j}}^{L}})}{t}}}\right)^{r_{j}}}% \right),\\ \Delta\left({t\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{i_{j}}}^{R},% \vartheta_{i_{j}}^{R}})}{t}}}\right)^{r_{j}}}\right)\end{bmatrix}}\right\}$ (77)

Thus,

$\displaystyle\mathop{\otimes}^{n}\limits_{j=1}\left({r_{j}\tilde{p}_{i_{j}}}% \right)=$ (78) $\displaystyle\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\sqrt{1-\displaystyle% \prod\limits_{j=1}^{n}{\left({1-\left({\displaystyle\frac{\Delta^{-1}(s_{\phi_% {i_{j}}}^{L},\varphi_{i_{j}}^{L})}{t}}\right)^{2}}\right)^{r_{j}}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{i_{j}}}^{R},\varphi_{i_{j}}^{R})}{t}}% \right)^{2}}\right)^{r_{j}}}}}\right)\end{bmatrix},\begin{bmatrix}\Delta\left(% {t\displaystyle\prod\limits_{j=1}^{n}{\left({{\displaystyle\frac{\Delta^{-1}(s% _{\theta_{i_{j}}}^{L},\vartheta_{i_{j}}^{L})}{t}}}\right)^{r_{j}}}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{j=1}^{n}{\left({{\displaystyle\frac{% \Delta^{-1}(s_{\theta_{i_{j}}}^{R},\vartheta_{i_{j}}^{R})}{t}}}\right)^{r_{j}}% }}\right)\end{bmatrix}\end{Bmatrix}$

Therefore,

$\displaystyle\left({\mathop{\otimes}^{n}\limits_{j=1}\left({r_{j}\tilde{p}_{i_% {j}}}\right)}\right)^{\prod{{}_{j=1}^{n}w_{i_{j}}}}=$ (79)

Thereafter,

(80)

\displaystyle\mathop{\oplus}^{n}\limits_{{\begin{subarray}{c}i_{1},i_{2},% \ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}}\left({\mathop{\otimes}^{n}% \limits_{j=1}\left({r_{j}\tilde{p}_{i_{j}}}\right)}\right)^{\prod{{}_{j=1}^{n}% w_{i_{j}}}}

\displaystyle=\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\displaystyle\prod% \limits_{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\sqrt{1-% \displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({\displaystyle\frac{\Delta^% {-1}(s_{\phi_{i_{j}}}^{L},\varphi_{i_{j}}^{L})}{t}}\right)^{2}}\right)^{r_{j}}% }}}\right)^{\prod{{}_{j=1}^{n}w_{i_{j}}}}}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2},\ldots% ,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\sqrt{1-% \displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({\displaystyle\frac{\Delta^% {-1}(s_{\phi_{i_{j}}}^{R},\varphi_{i_{j}}^{R})}{t}}\right)^{2}}\right)^{r_{j}}% }}}\right)^{\prod{{}_{j=1}^{n}w_{i_{j}}}}}}\right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\sqrt{1-\displaystyle\prod\limits_{% \begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({1-\displaystyle% \prod\limits_{j=1}^{n}{\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{i_{j% }}}^{L},\vartheta_{i_{j}}^{L}})}{t}}}\right)^{2r_{j}}}}\right)^{\prod{{}_{j=1}% ^{n}w_{i_{j}}}}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2% },\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({1-\displaystyle% \prod\limits_{j=1}^{n}{\left({{\displaystyle\frac{\Delta^{-1}({s_{\theta_{i_{j% }}}^{R},\vartheta_{i_{j}}^{R}})}{t}}}\right)^{2r_{j}}}}\right)^{\prod{{}_{j=1}% ^{n}w_{i_{j}}}}}}}\right)\end{bmatrix}\end{Bmatrix}

Then,

$\displaystyle\left({\mathop{\oplus}^{n}\limits_{{\begin{subarray}{c}i_{1},i_{2% },\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}}\left({\mathop{\otimes}^{n}% \limits_{j=1}\left({r_{j}\tilde{p}_{i_{j}}}\right)}\right)^{\prod{{}_{j=1}^{n}% w_{i_{j}}}}}\right)^{\frac{1}{n!}}$ (81) $\displaystyle=\begin{Bmatrix}\begin{bmatrix}\Delta\left({t\displaystyle\prod% \limits_{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\left({\sqrt{1-% \displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({{\displaystyle\frac{\Delta% ^{-1}({s_{\phi_{i_{j}}}^{L},\varphi_{i_{j}}^{L}})}{t}}}\right)^{2}}\right)^{r_% {j}}}}}\right)^{\prod{{}_{j=1}^{n}w_{i_{j}}}}}\right)^{{}^{\frac{1}{n!}}}}}% \right),\\ \Delta\left({t\displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2},\ldots% ,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\left({\sqrt{1-% \displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({{\displaystyle\frac{\Delta% ^{-1}({s_{\phi_{i_{j}}}^{R},\varphi_{i_{j}}^{R}})}{t}}}\right)^{2}}\right)^{r_% {j}}}}}\right)^{\prod{{}_{j=1}^{n}w_{i_{j}}}}}\right)^{{}^{\frac{1}{n!}}}}}% \right)\end{bmatrix},\\ \\ \begin{bmatrix}\Delta\left({t\sqrt{1-\displaystyle\prod\limits_{% \begin{subarray}{c}i_{1},i_{2},\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\left({1-% \displaystyle\prod\limits_{j=1}^{n}{\left({\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i_{j}}}^{L},\vartheta_{i_{j}}^{L})}{t}}\right)^{2r_{j}}}}\right)^{% \prod{{}_{j=1}^{n}w_{i_{j}}}}}\right)^{{}^{\frac{1}{n!}}}}}}\right),\\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{\begin{subarray}{c}i_{1},i_{2% },\ldots,i_{n}=1\\ i_{1}\neq i_{2}\neq\ldots\neq i_{n}\end{subarray}}^{n}{\left({\left({1-% \displaystyle\prod\limits_{j=1}^{n}{\left({\displaystyle\frac{\Delta^{-1}(s_{% \theta_{i_{j}}}^{R},\vartheta_{i_{j}}^{R})}{t}}\right)^{2r_{j}}}}\right)^{% \prod{{}_{j=1}^{n}w_{i_{j}}}}}\right)^{{}^{\frac{1}{n!}}}}}}\right)\end{% bmatrix}\end{Bmatrix}$

Furthermore,

Hence, Eq. (76) is kept.

Then we need to prove that Eq. (76) is an IV2TLPFN. We need to prove two conditions as follows:

(1) (1)

$\Delta^{-1}({s_{\phi}^{L},\varphi^{L}}),\Delta^{-1}(s_{\phi}^{R},\varphi^{R}),% \Delta^{-1}({s_{\theta}^{L},\vartheta^{L}}),\Delta^{-1}({s_{\theta}^{R},% \vartheta^{L}})\subseteq[0,t]$ .

(2)

$0\leqslant({\Delta^{-1}({s_{\phi}^{R},\varphi^{R}})})^{2}+({\Delta^{-1}({s_{% \theta}^{R},\vartheta^{R}})})^{2}\leqslant t^{2}.$

For the proof is similar to the IV2TLPFWBM operator, so it’s omitted here to save space.

Then we will discuss some properties of DGIV2TLPFWBM operator.

Property 15. (Monotonicity) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}% ^{L}}),({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^{R}})}],[{({s_{% \theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}}),({s_{\theta_{% \tilde{p}_{j}}}^{R},\vartheta_{\tilde{p}_{j}}^{R}})}]}\rangle$ and ${\tilde{p}}^{\prime}_{j}=\langle[(s_{\phi_{\tilde{p}_{j}}}^{L},$ $\varphi_{\tilde{p}_{j}}^{L})^{\prime},({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{% \tilde{p}_{j}}^{R}})^{\prime}],[{(s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{% \tilde{p}_{j}}^{L})^{\prime},({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{% \tilde{p}_{j}}^{R}})^{\prime}}]\rangle,j=1,2,\ldots,n$ be two sets of IV2TLPFNs. If $\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}})% \leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{L},\varphi_{\tilde{p}_{j}}^{L}% })^{\prime},\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}}^% {R}})\leqslant\Delta^{-1}({s_{\phi_{\tilde{p}_{j}}}^{R},\varphi_{\tilde{p}_{j}% }^{R}})^{\prime}$ and $\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}^{L}})% \geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{L},\vartheta_{\tilde{p}_{j}}% ^{L}})^{\prime},\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{\tilde{% p}_{j}}^{R}})\geqslant\Delta^{-1}({s_{\theta_{\tilde{p}_{j}}}^{R},\vartheta_{% \tilde{p}_{j}}^{R}})^{\prime}$ hold for all $j$ , then

(83)

\displaystyle\text{DGIV2TLPFWGBM}_{\omega}^{r}(\tilde{p}_{1},\tilde{p}_{2},% \ldots,\tilde{p}_{n})\leqslant\text{DGIV}2\text{ TLPFWGBM}_{\omega}^{r}({{% \tilde{p}}^{\prime}_{1},{\tilde{p}}^{\prime}_{2},\ldots,{\tilde{p}}^{\prime}_{% n}})

Property 16. (Boundedness) Let $\tilde{p}_{j}=\langle{[{({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),(s_{\phi_{j}}^{R}% ,\varphi_{j}^{R})}],[{(s_{\theta_{j}}^{L},\vartheta_{j}^{L}),({s_{\theta_{j}}^% {R},\vartheta_{j}^{R}})}]}\rangle(j=1,2,\ldots,n)$ be a set of IV2TLPFNs. If $\tilde{p}^{+}=\langle{[{\max_{j}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\max_{j}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\min_{j}({s_{\theta_{j}}^{L},\vartheta% _{j}^{L}}),\min_{j}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ and $\tilde{p}^{-}=\langle{[{\min_{j}({s_{\phi_{j}}^{L},\varphi_{j}^{L}}),\min_{j}(% {s_{\phi_{j}}^{R},\varphi_{j}^{R}})}],[{\max_{j}({s_{\theta_{j}}^{L},\vartheta% _{j}^{L}}),\max_{j}({s_{\theta_{j}}^{R},\vartheta_{j}^{R}})}]}\rangle$ then

$\displaystyle\tilde{p}^{-}\leqslant\text{DGIV2TLPFWGBM}_{\omega}^{r}(\tilde{p}% _{1},\tilde{p}_{2},\ldots,\tilde{p}_{n})\leqslant\tilde{p}^{+}$ (84)

6. Numerical example and comparative analysis

6.1 Numerical example

In this section we shall present a numerical example to select green suppliers in green supply chain management with IV2TLPFNs in order to illustrate the method proposed in this paper. There is a panel with five possible green suppliers in green supply chain management $\tilde{Q}_{i}({i=1,2,3,4,5})$ to select. The experts selects four attribute to evaluate the five possible green suppliers: (1) $C_{1}$ is the product quality factor; (2) $C_{2}$ is the environmental factors; (3) $C_{3}$ is the delivery factor; (4) $C_{4}$ is the price factor. The five possible green suppliers $\tilde{Q}_{i}({i=1,2,3,4,5})$ are to be evaluated using the IV2TLPFNs by the decision maker under the above four attributes (whose weighting vector $\omega=(0.12,0.24,0.36,0.28)$ ).

In the following, we utilize the approach developed to select green suppliers in green supply chain management.

Step 1. According to Table 1, we can aggregate all IV2TLPFNs $\tilde{q}_{ij}$ by using the DGIV2TLPFWBM (DGIV2TLP FWGBM) operator to get the overall IV2TLPFNs $\tilde{Q}_{i}({i=1,2,3,4,5})$ of the green suppliers $\tilde{Q}_{i}$ . Suppose that $r=(3,3,3,3)$ , then the aggregating results are shown in Table 2.

Table 1
IV2TLPFN decision matrix $R$

	C ${}_{1}$	C ${}_{2}$
$\tilde{Q}_{1}$	$\{{[{({s_{4},0}),({s_{5},0})}],[{({s_{1},0}),({s_{2},0})}]}\}$	$\{{[{({s_{3},0}),({s_{5},0})}],[{({s_{2},0}),({s_{3},0})}]}\}$
$\tilde{Q}_{2}$	$\{{[{({s_{3},0}),({s_{4},0})}],[{({s_{2},0}),({s_{3},0})}]}\}$	$\{{[{({s_{2},0}),({s_{4},0})}],[{({s_{3},0}),({s_{5},0})}]}\}$
$\tilde{Q}_{3}$	$\{{[{({s_{2},0}),({s_{3},0})}],[{({s_{3},0}),({s_{4},0})}]}\}$	$\{{[{({s_{2},0}),({s_{3},0})}],[{({s_{1},0}),({s_{4},0})}]}\}$
$\tilde{Q}_{4}$	$\{{[{({s_{1},0}),({s_{2},0})}],[{({s_{2},0}),({s_{4},0})}]}\}$	$\{{[{({s_{3},0}),({s_{4},0})}],[{({s_{2},0}),({s_{3},0})}]}\}$
$\tilde{Q}_{5}$	$\{{[{({s_{3},0}),({s_{4},0})}],[{({s_{2},0}),({s_{4},0})}]}\}$	$\{{[{({s_{2},0}),({s_{5},0})}],[{({s_{1},0}),({s_{4},0})}]}\}$
	C ${}_{3}$	C ${}_{4}$
$\tilde{Q}_{1}$	$\{{[{({s_{3},0}),({s_{4},0})}],[{({s_{1},0}),({s_{4},0})}]}\}$	$\{{[{({s_{3},0}),({s_{4},0})}],[{({s_{2},0}),({s_{5},0})}]}\}$
$\tilde{Q}_{2}$	$\{{[{({s_{2},0}),({s_{3},0})}],[{({s_{2},0}),({s_{4},0})}]}\}$	$\{{[{({s_{2},0}),({s_{3},0})}],[{({s_{4},0}),({s_{5},0})}]}\}$
$\tilde{Q}_{3}$	$\{{[{({s_{3},0}),({s_{5},0})}],[{({s_{2},0}),({s_{5},0})}]}\}$	$\{{[{({s_{3},0}),({s_{4},0})}],[{({s_{1},0}),({s_{3},0})}]}\}$
$\tilde{Q}_{4}$	$\{{[{({s_{1},0}),({s_{2},0})}],[{({s_{2},0}),({s_{3},0})}]}\}$	$\{{[{({s_{2},0}),({s_{5},0})}],[{({s_{3},0}),({s_{5},0})}]}\}$
$\tilde{Q}_{5}$	$\{{[{({s_{2},0}),({s_{4},0})}],[{({s_{2},0}),({s_{3},0})}]}\}$	$\{{[{({s_{3},0}),({s_{4},0})}],[{({s_{2},0}),({s_{4},0})}]}\}$

Step 3. According to the aggregating results shown in Table 2 and the score functions of the green suppliers are shown in Table 3.

Table 2

The aggregating results by the DGIV2TLPFWBM (DGIV2TLPFWGBM) operator

	DGIV2TLPFWBM	DGIV2TLPFWGBM
$\tilde{Q}_{1}$	$\begin{Bmatrix}[{({s_{0},0.0119}),({s_{0},0.0197})}],\\ \end{Bmatrix}$	$\begin{Bmatrix}[{({s_{4},0.0409}),({s_{5},-0.2644})}],\\ \end{Bmatrix}$
$\tilde{Q}_{2}$	$\begin{Bmatrix}[{({s_{0},0.0526}),({s_{0},0.0668})}],\\ {}[{({s_{4},-0.0860}),({s_{5},-0.3213})}]\end{Bmatrix}$	$\begin{Bmatrix}[{({s_{4},-0.3778}),({s_{4},0.1836})}],\\ {}[{({s_{0},0.0071}),({s_{0},0.1623})}]\end{Bmatrix}$
$\tilde{Q}_{3}$	$\begin{Bmatrix}[{({s_{0},0.0036}),({s_{0},0.0056})}],\\ {}[{({s_{3},0.4505}),({s_{5},-0.4905})}]\end{Bmatrix}$	$\begin{Bmatrix}[{({s_{4},-0.2167}),({s_{4},0.4012})}],\\ {}[{({s_{0},0.0003}),({s_{0},0.0832})}]\end{Bmatrix}$
$\tilde{Q}_{4}$	$\begin{Bmatrix}[{({s_{0},0.0002}),({s_{0},0.0004})}],\\ {}[{({s_{4},-0.3350}),({s_{4},0.3608})}]\end{Bmatrix}$	$\begin{Bmatrix}[{({s_{4},-0.4832}),({s_{4},0.2060})}],\\ {}[{({s_{0},0.0017}),({s_{0},0.0501})}]\end{Bmatrix}$
$\tilde{Q}_{5}$	$\begin{Bmatrix}[{({s_{0},0.0077}),({s_{0},0.0142})}],\\ {}[{({s_{3},0.4647}),({s_{4},0.3199})}]\end{Bmatrix}$	$\begin{Bmatrix}[{({s_{4},-0.2558}),({s_{5},-0.4008})}],\\ {}[{({s_{0},0.0004}),({s_{0},0.0494})}]\end{Bmatrix}$

Table 3

The score functions of the green suppliers

	DGIV2TLPFWBM	DGIV2TLPFWGBM
$\tilde{Q}_{1}$	0.5401	0.8770
$\tilde{Q}_{2}$	0.4916	0.8441
$\tilde{Q}_{3}$	0.5255	0.8567
$\tilde{Q}_{4}$	0.5241	0.8419
$\tilde{Q}_{5}$	0.5358	0.8627

Step 4. According to the score functions shown in Table 3 and the comparison formula of score functions, the ordering of the green suppliers are shown in Table 4.

Table 4

Ordering of the green suppliers

	Ordering
DGIV2TLPFWBM	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$
DGIV2TLPFWGBM	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$

6.2 Influence of the parameter on the final result

In order to show the effects on the ranking results by changing parameters of $r$ in the DGIV2TLPFWBM (DGIV2 TLPFWGBM) operators, all the results are shown in Tables 5 and 6.

Table 5
Ranking results for different operational parameters of the DGIV2TLPFWBM operator

$r$	$\tilde{Q}_{1}$	$\tilde{Q}_{2}$	$\tilde{Q}_{3}$	$\tilde{Q}_{4}$	$\tilde{Q}_{5}$	Ordering
(1, 1, 1, 1)	0.4580	0.4571	0.4417	0.4098	0.4564	$\tilde{Q}_{1}>\tilde{Q}_{2}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}$
(2, 2, 2, 2)	0.4972	0.4533	0.4837	0.4803	0.4946	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$
(3, 3, 3, 3)	0.5401	0.4916	0.5255	0.5241	0.5358	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$
(4, 4, 4, 4)	0.5651	0.5156	0.5492	0.5488	0.5587	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$
(5, 5, 5, 5)	0.5814	0.5312	0.5648	0.5650	0.5731	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$
(6, 6, 6, 6)	0.5929	0.5422	0.5759	0.5764	0.5831	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{4}>\tilde{Q}_{3}>\tilde{Q}_{2}$
(7, 7, 7, 7)	0.6014	0.5504	0.5842	0.5849	0.5904	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{4}>\tilde{Q}_{3}>\tilde{Q}_{2}$
(8, 8, 8, 8)	0.6080	0.5569	0.5908	0.5915	0.5960	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{4}>\tilde{Q}_{3}>\tilde{Q}_{2}$
(9, 9, 9, 9)	0.6132	0.5620	0.5960	0.5967	0.6004	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{4}>\tilde{Q}_{3}>\tilde{Q}_{2}$
(10, 10, 10, 10)	0.6174	0.5662	0.6002	0.6010	0.6040	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{4}>\tilde{Q}_{3}>\tilde{Q}_{2}$

Table 6

Ranking results for different operational parameters of the DGIV2TLPFWGBM operator

$r$	$\tilde{Q}_{1}$	$\tilde{Q}_{2}$	$\tilde{Q}_{3}$	$\tilde{Q}_{4}$	$\tilde{Q}_{5}$	Ordering
(1, 1, 1, 1)	0.9128	0.8766	0.8977	0.8873	0.9020	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$
(2, 2, 2, 2)	0.8974	0.8672	0.8803	0.8681	0.8851	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$
(3, 3, 3, 3)	0.8770	0.8441	0.8567	0.8419	0.8627	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$
(4, 4, 4, 4)	0.8643	0.8287	0.8410	0.8248	0.8484	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$
(5, 5, 5, 5)	0.8557	0.8183	0.8301	0.8130	0.8387	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$
(6, 6, 6, 6)	0.8496	0.8109	0.8222	0.8043	0.8317	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$
(7, 7, 7, 7)	0.8450	0.8053	0.8161	0.7977	0.8263	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$
(8, 8, 8, 8)	0.8414	0.8009	0.8113	0.7924	0.8220	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$
(9, 9, 9, 9)	0.8386	0.7974	0.8074	0.7882	0.8186	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$
(10, 10, 10, 10)	0.8362	0.7945	0.8042	0.7846	0.8158	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{2}>\tilde{Q}_{4}$

6.3 Comparative analysis

Then, we compare our proposed method with other methods including IV2TLPFWA operator and IV2TLPFWG operator.

Definition 22 Let $\tilde{p}_{j}=\langle{[{(s_{\phi_{j}}^{L},\varphi_{j}^{L}),({s_{\phi_{j}}^{R},% \varphi_{j}^{R}})}],[{({s_{\theta_{j}}^{L},\vartheta_{j}^{L}}),({s_{\theta_{j}% }^{R},\vartheta_{j}^{R}})}]}\rangle({j=1,2,\ldots,n})$ be a set of IV2TLPFNs with their weight vector be $w_{i}=(w_{1},w_{2},\ldots,w_{n})^{T}$ , thereby satisfying $w_{i}\in[0,1]$ and $\sum\nolimits_{i=1}^{n}{w_{i}}=1$ , then we can obtain

$\displaystyle\text{IV2TLPFWA}\left({\tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{% p}_{n}}\right)=\sum\limits_{j=1}^{n}{w_{j}\tilde{p}_{j}}$ (85) $\displaystyle=\left\{{\begin{bmatrix}\Delta\left({t\sqrt{1-\displaystyle\prod% \limits_{j=1}^{n}{\left({1-\left({\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^% {L},\varphi_{j}^{L})}{t}}\right)^{2}}\right)^{w_{j}}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({% \displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}}\right)^{% 2}}\right)^{w_{j}}}}}\right)\end{bmatrix},\begin{bmatrix}\Delta\left({t% \displaystyle\prod\limits_{j=1}^{n}{\left({{\displaystyle\frac{\Delta^{-1}(s_{% \theta_{j}}^{L},\vartheta_{j}^{L})}{t}}}\right)^{w_{j}}}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{j=1}^{n}{\left({{\displaystyle\frac{% \Delta^{-1}(s_{\theta_{j}}^{R},\vartheta_{j}^{R})}{t}}}\right)^{w_{j}}}}\right% )\end{bmatrix}}\right\}$ $\displaystyle\text{IV2TLPFWG}\left({\tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{% p}_{n}}\right)=\prod\limits_{j=1}^{n}{\left({\tilde{p}_{j}}\right)^{w_{j}}}$ (86) $\displaystyle=\left\{{\begin{bmatrix}\Delta\left({t\displaystyle\prod\limits_{% j=1}^{n}{\left({{\displaystyle\frac{\Delta^{-1}(s_{\phi_{j}}^{L},\varphi_{j}^{% L})}{t}}}\right)^{w_{j}}}}\right),\\ \\ \Delta\left({t\displaystyle\prod\limits_{j=1}^{n}{\left({{\displaystyle\frac{% \Delta^{-1}(s_{\phi_{j}}^{R},\varphi_{j}^{R})}{t}}}\right)^{w_{j}}}}\right)% \end{bmatrix},\begin{bmatrix}\Delta\left({t\sqrt{1-\displaystyle\prod\limits_{% j=1}^{n}{\left({1-\left({\displaystyle\frac{\Delta^{-1}(s_{\theta_{j}}^{L},% \vartheta_{j}^{L})}{t}}\right)^{2}}\right)^{w_{j}}}}}\right),\\ \\ \Delta\left({t\sqrt{1-\displaystyle\prod\limits_{j=1}^{n}{\left({1-\left({% \displaystyle\frac{\Delta^{-1}(s_{\theta_{j}}^{R},\vartheta_{j}^{R})}{t}}% \right)^{2}}\right)^{w_{j}}}}}\right)\end{bmatrix}}\right\}$

Compare with the computing results of Table 4, the results are slightly different to show the practicality and effectiveness of the proposed approaches. However, such as IV2TLPFWA operator and IV2TLPFWG operator, do not consider the information about the relationship between arguments being aggregated, and thus cannot eliminate the influence of unfair arguments on decision result. Our proposed DGIV2TLPFWBM and DGIV2TLPFWGBM operators consider the information about the relationship between arguments being aggregated.

Table 7

Ordering of the green suppliers

	Ordering
IV2TLPFWA	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$
IV2TLPFWG	$\tilde{Q}_{1}>\tilde{Q}_{5}>\tilde{Q}_{3}>\tilde{Q}_{4}>\tilde{Q}_{2}$

7. Conclusion

In this paper, we investigate the MADM problems with IV2TLPFNs. Then, we utilize the Bonferroni mean (BM) operator, generalized Bonferroni mean (GBM) operator and dual generalized Bonferroni mean (DGBM) operator to develop some Bonferroni mean aggregation operators with IV2TLPFNs: interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (IV2TLPFWBM) operator, interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (IV2TLPFWGBM) operator, generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (GIV2TLPFWBM) operator, generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (GIV2TLPFWGBM) operator, dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted Bonferroni mean (DGIV2TLPFW BM) operator, dual generalized interval-valued 2-tuple linguistic Pythagorean fuzzy weighted geometric Bonferroni mean (DGIV2TLPFWGBM) operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the MADM problems with IV2TLPFNs. Finally, a practical example for green suppliers selection in green supply chain management 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 IV2TLPFNs needs to be explored in the decision making, risk analysis and many other fields under uncertain environments [69, 70, 71, 72, 73, 74, 75, 76].

Footnotes

Acknowledgments

The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China under Grant No. 16YJA630033.

References

Yager

R.R.

, Pythagorean fuzzy subsets, In: Proceeding of The Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada, 2013, pp. 57–61.

Yager

R.R.

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

Peng

and Yang

, Some results for Pythagorean Fuzzy Sets, International Journal of Intelligent systems 30 (2015), 1133–1160.

Beliakov

and James

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

Peng

X.D.

and Yang

, Some results for Pythagorean fuzzy sets, Int J Intell Syst 30(11) (2015), 1133–1160.

Wei

G.W.

et al., Dual hesitant Pythagorean fuzzy hamy mean operators in multiple attribute decision making, IEEE Access 7(1) (2019), 86697–86716.

J.P.

et al., Bidirectional project method for dual hesitant Pythagorean fuzzy multiple attribute decision-making and their application to performance assessment of new rural construction, International Journal of Intelligent Systems 34(8) (2019), 1920–1934.

Tang

X.Y.

and Wei

G.W.

, Dual hesitant Pythagorean fuzzy Bonferroni mean operators in multi-attribute decision making, Archives of Control Sciences 29(2) (2019), 339–386.

Gou

X.J.

Z.S.

and Ren

P.J.

, The properties of continuous Pythagorean fuzzy information, Int J Intell Syst 31(5) (2016), 401–424.

10.

Tang

X.Y.

Wei

G.W.

and Gao

, Models for multiple attribute decision making with interval-valued Pythagorean fuzzy muirhead mean operators and their application to green suppliers selection, Informatica 30(1) (2019), 153–186.

11.

Tang

X.Y.

and Wei

G.W.

, Multiple attribute decision-making with dual hesitant Pythagorean fuzzy information, Cognitive Computation 11(2) (2019), 193–211.

12.

Tang

X.Y.

Wei

G.W.

and Gao

, Pythagorean fuzzy muirhead mean operators in multiple attribute decision making for evaluating of emerging technology commercialization, Economic Research-Ekonomska Istraživanja 32(1) (2019), 1667–1696.

13.

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.

14.

Yang

et al., A note on extension of topsis to multiple criteria decision making with Pythagorean fuzzy sets, Int J Intell Syst 31(1) (2016), 68–72.

15.

Liang

D.C.

and Xu

Z.S.

, The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets, Appl Soft Comput 60 (2017), 167–179.

16.

Liang

D.C.

et al., Method for three-way decisions using ideal TOPSIS solutions at Pythagorean fuzzy information, Inf Sci 435 (2018), 282–295.

17.

Khan

M.S.A.

et al., Extension of TOPSIS method base on Choquet integral under interval-valued Pythagorean fuzzy environment, Journal of Intelligent and Fuzzy Systems 34(1) (2018), 267–282.

18.

Onar

S.Ç.

Öztaysi

and Kahraman

, Multicriteria evaluation of cloud service providers using Pythagorean fuzzy TOPSIS, Multiple-Valued Logic and Soft Computing 30(2–3) (2018), 263–283.

19.

Ren

P.J.

Z.S.

and Gou

X.J.

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

20.

Zhang

X.L.

, Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods, Inf Sci 330 (2016), 104–124.

21.

Chen

T.Y.

, Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis, Information Fusion 41 (2018), 129–150.

22.

Peng

X.D.

and Dai

J.G.

, Approaches to Pythagorean fuzzy stochastic multi-criteria decision making based on prospect theory and regret theory with new distance measure and score function, Int J Intell Syst 32(11) (2017), 1187–1214.

23.

Xue

W.T.

et al., Pythagorean fuzzy LINMAP method based on the entropy theory for railway project investment decision making, Int J Intell Syst 33(1) (2018), 93–125.

24.

Wan

S.P.

Jin

and Dong

J.Y.

, Pythagorean fuzzy mathematical programming method for multi-attribute group decision making with Pythagorean fuzzy truth degrees, Knowl Inf Syst 55(2) (2018), 437–466.

25.

Garg

, A linear programming method based on an improved score function for interval-valued Pythagorean fuzzy numbers and its application to decision-making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 26(1) (2018), 67–80.

26.

Liang

D.C.

Z.S.

and Darko

A.P.

, Projection model for fusing the information of Pythagorean fuzzy multicriteria group decision making based on geometric bonferroni mean, Int J Intell Syst 32(9) (2017), 966–987.

27.

Peng

X.D.

and Yang

, Pythagorean fuzzy choquet integral based MABAC Method for multiple attribute group decision making, Int J Intell Syst 31(10) (2016), 989–1020.

28.

Peng

X.D.

Yuan

H.Y.

and Yang

, Pythagorean fuzzy information measures and their applications, Int J Intell Syst 32(10) (2017), 991–1029.

29.

Peng

X.D.

and Dai

J.G.

30.

Qin

Liu

and Hong

Z.Y.

, Multicriteria decision making method based on generalized Pythagorean fuzzy ordered weighted distance measures, Journal of Intelligent and Fuzzy Systems 33(6) (2017), 3665–3675.

31.

Wei

G.W.

and Wei

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

32.

Wang

Gao

and Wei

G.W.

, The generalized Dice similarity measures for Pythagorean fuzzy multiple attribute group decision making, International Journal of Intelligent Systems 34(6) (2019), 1158–1183.

33.

Zhang

X.L.

, A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making, International Journal of Intelligent Systems 31(6) (2016), 593–611.

34.

Garg

, A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes, International Journal of Intelligent Systems 31(12) (2016), 1234–1252.

35.

Z.M.

and Xu

Z.S.

, Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision-making problems, International Journal of Intelligent Systems 31(12) (2016), 1198–1219.

36.

Peng

X.D.

and Yang

, Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators, Int J Intell Syst 31(5) (2016), 444–487.

37.

Peng

X.D.

and Yuan

H.Y.

, Fundamental properties of Pythagorean fuzzy aggregation operators, Fundam Inform 147(4) (2016), 415–446.

38.

Rahman

et al., Pythagorean fuzzy Einstein weighted geometric aggregation operator and their application to multiple attribute group decision making, Journal of Intelligent and Fuzzy Systems 33(1) (2017), 635–647.

39.

Wei

G.W.

, Pythagorean fuzzy hamacher power aggregation operators in multiple attribute decision making, Fundamenta Informaticae 166(1) (2019), 57–85.

40.

Wei

G.W.

and Lu

, Pythagorean fuzzy maclaurin symmetric mean operators in multiple attribute decision making, International Journal of Intelligent Systems 33(5) (2018), 1043–1070.

41.

Wang

et al., Dual hesitant q-Rung orthopair fuzzy muirhead mean operators in multiple attribute decision making, IEEE Access 7(1) (2019), 67139–67166.

42.

Zeng

S.Z.

Z.M.

and Balezentis

, A novel aggregation method for Pythagorean fuzzy multiple attribute group decision making, Int J Intell Syst 33(3) (2018), 573–585.

43.

Wang

et al., Methods for multiple-attribute group decision making with q-rung interval-valued orthopair fuzzy information and their applications to the selection of green suppliers, Symmetry 11(1) (2019), 56.

44.

Mohagheghi

Mousavi

S.M.

and Vahdani

, Enhancing decision-making flexibility by introducing a new last aggregation evaluating approach based on multi-criteria group decision making and Pythagorean fuzzy sets, Appl Soft Comput 61 (2017), 527–535.

45.

Garg

, Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process, Computational & Mathematical Organization Theory 23(4) (2017), 546–571.

46.

Garg

, Generalized Pythagorean fuzzy geometric aggregation operators using einstein t-Norm and t-Conorm for multicriteria decision-making process, Int J Intell Syst 32(6) (2017), 597–630.

47.

Tang

et al., Dual hesitant pythagorean fuzzy heronian mean operators in multiple attribute decision making, Mathematics 7(4) (2019), 344.

48.

Liang

D.C.

et al., Pythagorean fuzzy Bonferroni mean aggregation operator and its accelerative calculating algorithm with the multithreading, Int J Intell Syst 33(3) (2018), 615–633.

49.

Wei

and Lu

, Pythagorean fuzzy hamy mean operators in multiple attribute group decision making and their application to supplier selection, Symmetry 10(10) (2018), 505.

50.

Garg

, New exponential operational laws and their aggregation operators for interval-valued Pythagorean fuzzy multicriteria decision-making, Int J Intell Syst 33(3) (2018), 653–683.

51.

Garg

, Some methods for strategic decision-making problems with immediate probabilities in Pythagorean fuzzy environment, Int J Intell Syst 33(4) (2018), 687–712.

52.

Garg

, A novel improved accuracy function for interval valued Pythagorean fuzzy sets and its applications in the decision-making process, Int J Intell Syst 32(12) (2017), 1247–1260.

53.

Tang

X.Y.

Huang

Y.H.

and Wei

G.W.

, Approaches to multiple-attribute decision-making based on Pythagorean 2-tuple linguistic bonferroni mean operators, Algorithms 11(1) (2018),5.

54.

Tang

X.Y.

and Wei

G.W.

, Models for green supplier selection in green supply chain management with Pythagorean 2-tuple linguistic information, IEEE Access 6 (2018), 18042–18060.

55.

Gao

and Wei

G.W.

, Multiple attribute decision making based on interval-valued Pythagorean uncertain linguistic aggregation operators, International Journal of Knowledge-based and Intelligent Engineering Systems 22(1) (2018), 59–81.

56.

Wei

G.W.

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

57.

Wei

G.W.

Gao

and Wei

, Some q-Rung orthopair fuzzy heronian mean operators in multiple attribute decision making, International Journal of Intelligent Systems 33(7) (2018), 1426–1458.

58.

et al., Ahmed 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.

59.

Wei

G.W.

et al., Pythagorean Hesitant Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making, International Journal of Intelligent Systems 33(6) (2018), 1197–1233.

60.

Deng

et al., Models for multiple attribute decision making with some 2-tuple linguistic Pythagorean fuzzy hamy mean operators, Mathematics 6(11) (2018), 236.

61.

Deng

X.M.

Wang

and Wei

G.W.

, Some 2-tuple linguistic Pythagorean Heronian mean operators and their application to multiple attribute decision making, Journal of Experimental & Theoretical Artificial Intelligence 31(4) (2019), 555–574.

62.

Deng

and Gao

, TODIM method for multiple attribute decision making with 2-tuple linguistic Pythagorean fuzzy information, Journal of Intelligent and Fuzzy Systems (2019), 1–12.

63.

Wang

Wei

and Gao

, Approaches to multiple attribute decision making with interval-valued 2-tuple linguistic Pythagorean fuzzy information, Mathematics 6(10) (2018), 201.

64.

Bonferroni

, Sulle medie multiple di potenze, Bolletino Matematica Italiana 5 (1950), 267–270.

65.

Beliakov

et al., Generalized Bonferroni mean operators in multicriteria aggregation, Fuzzy Sets and Systems 161 (2010), 2227–2242.

66.

Wei

G.W.

, The generalized dice similarity measures for multiple attribute decision making with hesitant fuzzy linguistic information, Economic Research-Ekonomska Istraživanja 32(1) (2019), 1498–1520.

67.

Zhu

Z.S.

and Xia

M.M.

, Hesitant fuzzy geometric Bonferroni means, Information sciences 205 (2012), 72–85.

68.

Zhang

R.T.

et al., Some generalized Pythagorean fuzzy bonferroni mean aggregation operators with their application to multiattribute group decision-making, Complexity 2017 (2017), 1–16.

69.

Mohagheghi

Mousavi

S.M.

and Vahdani

70.

L.P.

Wang

and Gao

, Models for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators, Journal of Intelligent and Fuzzy Systems 36(6) (2019), 5693–5709.

71.

Hao

Y.H.

and Chen

X.G.

, Study on the ranking problems in multiple attribute decision making based on interval-valued intuitionistic fuzzy numbers, Int J Intell Syst 33(3) (2018), 560–572.

72.

Gao

, Pythagorean fuzzy hamacher prioritized aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 35(2) (2018), 2229–2245.

73.

Baloglu

U.B.

and Demir

, An agent-based Pythagorean fuzzy approach for demand analysis with incomplete information, Int J Intell Syst 33(5) (2018), 983–997.

74.

Wei

et al., Oil price fluctuation, stock market and macroeconomic fundamentals: Evidence from China before and after the financial crisis, Finance Research Letters 30 (2019), 23–29.

75.

Wei

et al., Hot money and China’s stock market volatility: Further evidence using the GARCH-MIDAS model, Physica A: Statistical Mechanics and its Applications 492 (2018), 923–930.

76.

et al., Some interval-valued intuitionistic fuzzy dombi hamy mean operators and their application for evaluating the elderly tourism service quality in tourism destination, Mathematics 6(12) (2018), 294.

Models for multiple attribute decision making with some interval-valued 2-tuple linguistic Pythagorean fuzzy Bonferroni mean operators

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 2-tuple linguistic sets

2.2 Pythagorean fuzzy sets

3.1 IV2TLPFWBM operator

4.1 GIV2TLPFWBM operator

5.1 DGIV2TLPFWBM operator

6.1 Numerical example

Table 1 IV2TLPFN decision matrix R

Table 5 Ranking results for different operational parameters of the DGIV2TLPFWBM operator

Footnotes

Acknowledgments

References

Table 1
IV2TLPFN decision matrix $R$

Table 5
Ranking results for different operational parameters of the DGIV2TLPFWBM operator