Generalized Orthopair Fuzzy Weighted Power Bonferroni Mean Operator and Its Application in Decision Making

Abstract

The generalized orthopair fuzzy set is more favored by decision-makers and extensively utilized in areas like supply chain management, risk investment, and pattern recognition because it offers a broader decision information boundary than the intuitionistic fuzzy set and Pythagorean fuzzy set. This enables it to express fuzzy information more comprehensively and accurately in multi-attribute decision-making problems. To this end, this paper combines the ability of the power average (PA) operator to eliminate the impact of extreme values and the advantage of the Bonferroni mean ($\mathrm{B}{\mathrm{M}}^{s,t}$) operator in reflecting the relationships between variables, then incorporates weight indicators for different attributes to define the generalized orthopair fuzzy weighted power Bonferroni mean operator. The effectiveness of this operator is demonstrated through aggregation laws for generalized orthopair fuzzy information. Subsequently, the desirable properties of this operator are discussed. Based on these findings, a novel generalized orthopair fuzzy multi-attribute decision-making method, with a correlation between attributes, is proposed. Lastly, an investment decision-making example illustrates the feasibility and superiority of this method.

1. Introduction

In practical decision-making problems, certain attribute values can be represented by definite numerical figures, such as temperature, length, and the speed at which a car travels. However, for some issues, due to the complexity of the problem and the limitations in the decision-maker’s knowledge, it becomes challenging to provide them in a definitive form. Examples include an individual’s perception of cold or the degree of acceptance towards a particular item. Such issues often encompass multifaceted considerations and uncertainties, making it difficult to arrive at an optimal decision using traditional mathematical methods. As a solution, fuzzy set theory, proposed by the cybernetics expert Zadeh, serves as a potent tool for handling fuzzy and uncertain information in decision-making problems. He extended the classical sets to fuzzy sets and the characteristic functions to membership functions. This made it possible to mathematically characterize fuzzy concepts, pioneering a new perspective based on fuzzy sets to study uncertain phenomena [1].

Atanassov first introduced the concept of the intuitionistic fuzzy set (IFS) in 1983 [2]. Intuitionistic fuzzy sets encompass two dimensions, “membership degree $u$” and “non-membership degree $v$”, satisfying the conditions $u\in \left[0,1\right], v\in \left[0,1\right] \mathrm{and} u+v\le 1$. Additionally, intuitionistic fuzzy sets incorporated the notion of hesitancy, defining the hesitancy degree $\pi$ = 1 − $u$ − $v$ to represent the level of uncertainty regarding a particular decision. Subsequently, Yager introduced the concept of the Pythagorean fuzzy set (PFS) [3] and Fermatean fuzzy set (FFS) [4], which broadened the constraints of intuitionistic fuzzy sets and provided a stronger capability in describing fuzzy phenomena.

However, as decision-making conditions have become increasingly complex, the applicability of the Pythagorean fuzzy set and Fermatean fuzzy set has become more restricted. For instance, when experts use 0.9 and 0.7 to represent the “membership degree $u$” and “non-membership degree $v$” of their decision opinion, respectively, since $0.9^2+0.7^2>1$, $0.9^3+0.7^3>1$, the experts’ decision opinion (0.9, 0.7) cannot be represented by the Pythagorean fuzzy set or Fermatean fuzzy set. To address such issues, Yager once again introduced the concept of the generalized orthopair fuzzy set (i.e., $q$-rung orthopair fuzzy set, q-ROF) [5]. The generalized orthopair fuzzy set not only results in less information distortion, it also offers people a broader decision-making scope and provides more possibilities for decision-making.

Information aggregation operators based on intuitionistic fuzzy sets have been extensively studied. For instance, Zhou et al. investigated the power mean operator of intuitionistic triangular fuzzy numbers [6], while Xu and colleagues introduced the intuitionistic fuzzy hybrid average operator [7]. Research on information aggregation operators based on Pythagorean fuzzy set and Fermatean fuzzy set has also made some progress: Akram et al. [8] introduced a series of Pythagorean Dombi fuzzy aggregation operators. Khan et al. [9] extended the prioritized aggregation operators to the Pythagorean fuzzy environment to address decision-making problems where attributes and decision-makers have a hierarchical relationship. Wei et al. [10] proposed a series of Pythagorean fuzzy Hamacher power aggregation operators using Hamacher operation and power aggregation in the Pythagorean fuzzy environment. Senapati et al. [11] studied the Fermatean fuzzy weighted averaging and geometric operators. Senapati and Yager [12] introduced subtraction, division, and Fermatean arithmetic mean operations over the Fermatean fuzzy set. On the other hand, information aggregation operators based on the generalized orthopair fuzzy set have been attracting a significant amount of scholarly attention in recent years. For example, Jun et al. proposed the generalized orthopair fuzzy Maclaurin symmetric mean operator [13], Riaz investigated the generalized orthopair fuzzy geometric aggregation operators [14], Alcantud introduced complemental fuzzy sets and provided semantic justification for generalized orthopair fuzzy sets [15], and so on.

However, most of the current studies tend to aggregate information from a single perspective; to address this limitation and propose operators that are more comprehensive and better suited to increasingly complex multi-variable decision-making problems in reality, this paper selects some of the most representative operators for multi-variable information aggregation: the power average (PA) operator [16] and the Bonferroni mean ($\mathrm{B}{\mathrm{M}}^{s,t}$) operator [17]. The PA operator is leveraged for its ability to mitigate the negative impacts from extreme evaluations made by experts and the $\mathrm{B}{\mathrm{M}}^{s,t}$ operator reflects the correlation among input variables. By merging these operators and incorporating the significance of weight indicators, we introduce the generalized orthopair fuzzy weighted power Bonferroni mean operator and demonstrate its superior properties. Finally, we further illustrate the feasibility and superiority of this operator through practical examples.

The arrangement for the remainder of this paper is as follows. We introduce the preliminaries such as the definition of GOF set, distance measure, and arithmetic laws firstly in Section 2. Secondly, we propose the generalized orthopair fuzzy weighted power Bonferroni mean operator and prove its feasibility, then study some of its desirable properties in Section 3. In Section 4, we introduce a novel multi-attribute decision-making (MADM) model based on the $GOFWPB{M}^{s,t}$ operator. After that, we use a practical example to elucidate the superiority of this novel operator, then conduct some comparative analyses under different parameters and existing methods in Section 5. Finally, we summarize some conclusions and point out the application scenarios of this method in Section 6.

2. Preliminaries

Definition 1

[5]. Let $X$ be a non-empty general set, then the expression for the generalized orthopair fuzzy set $A$ defined on $X$ is given by

$$A=\left\{<x,u_A\left(x\right),v_A\left(x\right)>|x\in X\right\}$$

(1)

where $u_A\left(x\right)$: $X\to \left[0,1\right]$ and $v_A\left(x\right)$: $X\to \left[0,1\right]$ represent the membership function and non-membership function of $A$, respectively; $q$ is a positive integer independent of $x$ and $\forall x\in X, 0\le {u_A\left(x\right)}^q+{v_A\left(x\right)}^q\le 1$. Define the degree of hesitation $\pi \left(x\right)=\sqrt[q]{1-{u_A\left(x\right)}^q-{v_A\left(x\right)}^q}$.

“Orthopair” refers to the simultaneous inclusion of both membership and non-membership dimensions. Consequently, the intuitionistic fuzzy set, Pythagorean fuzzy set, Fermatean fuzzy set, and generalized orthopair fuzzy set all fall within the domain of the orthopair fuzzy set. The constraint condition for the generalized orthopair fuzzy set is that the sum of the $q$th power of membership and the $q$th power of non-membership is less than or equal to 1, that is, $u^q$ + $v^q$ ≤ 1. When $q=1$, the generalized orthopair fuzzy set evolves into an intuitionistic fuzzy set; when $q=2$, it morphs into a Pythagorean fuzzy set; and when $q=3$, it evolves into a Fermatean fuzzy set as illustrated in Figure 1.

The generalized orthopair fuzzy set allows for a broader boundary condition in decision-making information, enabling a more comprehensive and accurate representation of fuzzy information. This aligns more closely with real-life decision-making scenarios, which Alcantud and colleagues have studied [15]. Moreover, it encompasses the special cases of $q=1$ (intuitionistic fuzzy set), $q=2$ (Pythagorean fuzzy set), and $q=3$ (Fermatean fuzzy set), offering greater versatility and a broader range of applications. It serves as an effective tool for depicting phenomena characterized by uncertainty.

It is noteworthy that there are other extensions of intuitionistic fuzzy set and Pythagorean fuzzy set, which do not further extend the “$q$” exponent, but instead hybridize the model through other methods, such as the interval-valued intuitionistic fuzzy set [18], complex Pythagorean fuzzy set [19], Pythagorean fuzzy soft rough set [20], intuitionistic fuzzy soft set [21], and so on.

Definition 2

[22]. Let $a_i=\left(u_i,v_i\right)\left(i=1,2\right), a=\left(u,v\right)$ be three generalized orthopair fuzzy numbers, $\lambda$ is any real number greater than or equal to zero. The operational laws are defined as follows

(1)

$a_1⨁a_2=(\sqrt[q]{u_1^q+u_2^q-u_1^q{u}_2^q},v_1{v}_2)$;

(2)

$a_1⨂a_2=(u_1{u}_2,\sqrt[q]{v_1^q+v_2^q-v_1^q{v}_2^q})$;

(3)

$\lambda a=(\sqrt[q]{1-{\left(1-u^q\right)}^{\lambda }},v^{\lambda })$;

(4)

$a^{\lambda }=(u^{\lambda },\sqrt[q]{1-{\left(1-v^q\right)}^{\lambda }})$.

From the above, we can derive the following conclusions

(1)

$a_1⨁a_2=a_2⨁a_1$;

(2)

$a_1⨂a_2=a_2⨂a_1$;

(3)

$\lambda \left(a_1⨁a_2\right)={\lambda a}_1⨁{\lambda a}_2$;

(4)

$(a_1⨂a_2{)}^{\lambda }=a_1^{\lambda }⨂a_2^{\lambda }$;

(5)

${\lambda }_{1}a⨁{\lambda }_{2}a=({\lambda }_1+{\lambda }_2)a$;

(6)

$a^{{\lambda }_1}⨂a^{{\lambda }_2}=a^{{\lambda }_1+{\lambda }_2}$.

Definition 3

[22]. Let $a=\left(u,v\right)$ be a generalized orthopair fuzzy number, then the score-valued function of $a$ is defined as $S_{(a)}=u^q-v^q$, and the accuracy-valued function of $a$ is defined as $H_{(a)}=u^q+v^q$. Based on the functions $S_{(a)}$ and $H_{(a)}$, for any two generalized orthopair fuzzy numbers $a_1=\left(u_1,v_1\right)$, $a_2=\left(u_2,v_2\right)$, the comparison method is defined as

(1)

if $S_{(a1)}>S_{(a2)}$, then $a_1>a_2$;

(2)

if $S_{(a1)}=S_{(a2)}$ and $H_{\left(a1\right)}<H_{\left(a2\right)}$, then $a_1<a_2$;

(3)

if $S_{(a1)}=S_{(a2)}$ and $H_{\left(a1\right)}=H_{\left(a2\right)}$, then $a_1=a_2$.

Definition 4

[22]. Assuming $a_1=\left(u_1,v_1\right), a_2=\left(u_2,v_2\right)$ are any two generalized orthopair fuzzy numbers, the hamming distance $d\left(a_1,a_2\right)$ between $a_1$ and $a_2$ can be defined as

$$d\left(a_1,a_2\right)=\frac{\left|u_1^q-u_2^q\right|+\left|v_1^q-v_2^q\right|+\left|{\pi }_1^q-{\pi }_2^q\right|}{2}$$

(2)

The power average (PA) operator is an aggregation operator that effectively considers the interrelationships among data information. By taking into account the support relationships among input data to calculate attribute weights, it diminishes the adverse impact of outlier data on decision-making results, making the decision information processing more objective and impartial. Accordingly, it has garnered attention from many scholars [23,24,25,26].

Definition 5

[16]. Assuming $a_i\left(i=1,2,\cdots , n\right)$ to be a set of non-negative real numbers, the result aggregated by the power average operator of $a_i\left(i=1,2,\cdots , n\right)$ is given by

$$PA\left(a_1,a_2,\cdots ,a_n\right)={\displaystyle {\sum }_{i=1}^n}\frac{1+T(a_i)}{{\sum }_{i=1}^n(1+T(a_i))}{a}_i$$

(3)

where PA is termed as the power average operator. Here, $T\left(a_i\right)={\sum }_{j=1,j\ne i}^n\mathit{Supp}\left(a_i,a_j\right)\left(i=1,2,\cdots , n\right)$, $\mathit{Supp}\left(a_i,a_j\right)=1-d\left(a_i,a_j\right)$ represents the support degree between $a_i$ and $a_j$.

The ${\mathrm{BM}}^{s,t}$ operator is a particular type of aggregation function. The most significant advantage of this operator is its ability to reflect interrelationships among input variables, akin to the Heronian mean operator. With the continuous advancement of contemporary society, many attributes in real-world decision-making scenarios exhibit strong interrelations. Consequently, due to this characteristic, the ${\mathrm{BM}}^{s,t}$ operator is extensively utilized in information aggregation and multi-attribute decision-making [27,28,29,30,31].

Definition 6

[17]. Let parameters $s, t\ge 0$ and $s+t>0$, $a_i\left(i=1,2,\cdots ,n\right)$ be a series of non-negative real numbers, if

$$B{M}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)=(\frac{1}{n\left(n-1\right)}{\sum }_{i,j=1,i\ne j}^n{a}_i^s{a}_j^t{)}^{\frac{1}{s+t}}$$

(4)

then $B{M}^{s,t}$ is referred to as the Bonferroni mean operator.

3. Generalized Orthopair Fuzzy Weighted Power Bonferroni Mean Operator

3.1. Definition and Demonstration of GOFWPBM Operator

To harness the combined benefits of the $\mathrm{B}{\mathrm{M}}^{s,t}$ operator and the power average (PA) operator, He et al. [32] combined the PA operator with the ${\mathrm{BM}}^{s,t}$ operator, introducing the power Bonferroni mean $\left({\mathrm{PBM}}^{s,t}\right)$ operator. Subsequently, the ${\mathrm{PBM}}^{s,t}$ operator was extended to various fuzzy environments, including hesitant fuzzy sets [32], intuitionistic fuzzy sets [33,34], interval intuitionistic fuzzy sets [35], and linguistic intuitionistic fuzzy sets [36]. Further, Khan et al. [37], employing Dombi operations, expanded the ${\mathrm{PBM}}^{s,t}$ operator into the interval-valued intuitionistic setting to tackle multi-attribute decision-making problems with interval-valued intuitionistic information. However, to date, there has been no research on how to employ the ${\mathrm{PBM}}^{s,t}$ operator to aggregate generalized orthopair fuzzy numbers. Hence, to develop an comprehensive method for generalized orthopair fuzzy numbers and expand the application domain of the ${\mathrm{PBM}}^{s,t}$ operator, we subsequently introduce the generalized orthopair fuzzy power Bonferroni mean operator.

Definition 7

[32]. Let $a_i\left(i=1,2,\cdots , n\right)$ be a set of non-negative real numbers, $s$ and $t$ are non-negative real numbers that are not both 0, then the power Bonferroni mean $\left(PB{M}^{s,t}\right)$ operator can be defined as

$${\mathit{PBMM}}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)=[\frac{1}{n\left(n-1\right)}{\sum }_{i,j=1,i\ne j}^n\left((\frac{n(1+T{(a}_i))}{{\sum }_{k=1}^n(1+T{(a}_k))}{a}_i{)}^s(\frac{n(1+T{(a}_j))}{{\sum }_{k=1}^n(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}$$

(5)

where $T\left(a_i\right)={\sum }_{j=1,j\ne i}^n\mathit{Supp}\left(a_i,a_j\right)\left(i=1,2,\cdots , n\right)$, $\mathit{Supp}\left(a_i,a_j\right)=1-d\left(a_i,a_j\right)$ represents the support degree between $a_i$ and $a_j$.

Definition 8.

Let $s$ and $t$ be non negative real numbers that are not both 0, and let $a_i\left(i=1,2,\cdots , n\right)$ be a set of generalized orthopair fuzzy numbers. Then, the generalized orthopair fuzzy power Bonferroni mean $\left(GOFPB{M}^{s,t}\right)$ operator of $a_i\left(i=1,2,\cdots , n\right)$ can be defined as

$$GOFPB{M}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)=[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left((\frac{n(1+T{(a}_i))}{{\sum }_{k=1}^n(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n(1+T{(a}_j))}{{\sum }_{k=1}^n(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}$$

(6)

where $T\left(a_i\right)={\sum }_{j=1,j\ne i}^n\mathit{Supp}\left(a_i,a_j\right)\left(i=1,2,\cdots , n\right)$, $\mathit{Supp}\left(a_i,a_j\right)=1-d\left(a_i,a_j\right)$ represents the support degree between $a_i$ and $a_j$.

Given the aforementioned formulas, it is evident that while this operator combines the strengths of both the PA and $\mathrm{B}{\mathrm{M}}^{s,t}$ operators, it only contemplates the correlation between the weight vectors derived from the power average operator and the generalized orthopair fuzzy numbers pending aggregation. Notably, the intrinsic significance of the data has not been taken into account, meaning this operator is formulated under the premise that the significance level of the aggregating variables is uniform. However, in many practical decision-making problems, the significance of distinct attributes might differ, thereby leading to non-uniform weights. To incorporate the importance of attribute weights, we will proceed to define the generalized orthopair fuzzy weighted power Bonferroni mean operator.

Definition 9.

Let $s$ and $t$ be non-negative real numbers that are not both 0, and let $a_i\left(i=1,2,\cdots , n\right)$ be a set of generalized orthopair fuzzy numbers. Then, the generalized orthopair fuzzy weighted power Bonferroni mean $\left(GOFWPB{M}^{s,t}\right)$ operator of $a_i\left(i=1,2,\cdots , n\right)$ can be defined as

$$GOFWPB{M}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)=[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left((\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}$$

(7)

where $\omega =({\omega }_1,{\omega }_2,{\omega }_3\dots {\omega }_n{)}^T$ represents a set of weight vectors and $\sum {\omega }_i=1$. $T\left(a_i\right)={\sum }_{j=1,j\ne i}^n{\omega }_j\mathit{Supp}\left(a_i,a_j\right)\left(i=1,2,\cdots , n\right)$.

Theorem 1.

Assume $s$ and $t$ are non-negative real numbers that are not simultaneously 0, and $a_i\left(i=1,2,\cdots , n\right)$ are generalized orthopair fuzzy numbers, then use the $GOFWPB{M}^{s,t}$ operator which aggregates these generalized orthopair fuzzy numbers and the aggregated value is still a generalized orthopair fuzzy number, and

$$GOFWPB{M}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)=$$

$$(\sqrt[q]{{\left(1-[{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n(1-(1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t){]}^{\frac{1}{n\left(n-1\right)}}\right)}^{\frac{1}{s+t}}},$$

$$\sqrt[q]{1-{\left(1-[{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n[1-(1-v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t{]}^{\frac{1}{n\left(n-1\right)}}\right)}^{\frac{1}{s+t}}})$$

(8)

Proof of Theorem 1.

 

Given

$$\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i=(\sqrt[q]{1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}},v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}})$$

And

$$\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j=(\sqrt[q]{1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}},v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}})$$

It follows that

$$(\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s=((\sqrt[q]{1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}}{)}^s,\sqrt[q]{1-(1-v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s}),$$

$$(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t=((\sqrt[q]{1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}}{)}^t,\sqrt[q]{1-(1-v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t})$$

Hence

$$(\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t=$$

$$(\sqrt[q]{1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}}{)}^s(\sqrt[q]{1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}}{)}^t,$$

$$\sqrt[q]{2-(1-v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s-(1-v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t-[1-(1-v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s][1-(1-v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t]})$$

$$\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}[(\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t]=$$

$$(\sqrt[q]{1-{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n[1-(1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t]},$$

$$\sqrt[q]{{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n[1-(1-v_i^{\frac{qn{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{qn{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t]})$$

We use mathematical induction to prove the following theorem

(I)

For $n=2$,

$$\begin{array}{c}2\\ ⨁\\ i,j=1,i\ne j\end{array}[(\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t]=$$

$$[(\frac{2{\omega }_1(1+T{(a}_1))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}{a}_1{)}^s⨂(\frac{2{\omega }_2(1+T{(a}_2))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}{a}_2{)}^t]⨁[(\frac{2{\omega }_2(1+T{(a}_2))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}{a}_2{)}^s⨂(\frac{2{\omega }_1(1+T{(a}_1))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}{a}_1{)}^t]=$$

$$(\sqrt[q]{1-(1-(1-(1-u_1^q{)}^{\frac{2{\omega }_1(1+T{(a}_1))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_2^q{)}^{\frac{2{\omega }_2(1+T{(a}_2))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}}{)}^t)(1-(1-(1-u_2^q{)}^{\frac{2{\omega }_2(1+T{(a}_2))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_1^q{)}^{\frac{2{\omega }_1(1+T{(a}_1))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}}{)}^t},$$

$$\sqrt[q]{(1-(1-v_1^{\frac{2q{\omega }_1(1+T{(a}_1))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_2^{\frac{2q{\omega }_2(1+T{(a}_2))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}}{)}^t)(1-(1-v_2^{\frac{2q{\omega }_2(1+T{(a}_2))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_1^{\frac{2q{\omega }_1(1+T{(a}_1))}{{\sum }_{k=1}^2{\omega }_k(1+T{(a}_k))}}{)}^t)}$$

We can establish that the statement is true for $n=2$.

(II)

Assume it holds true for some arbitrary $n=m$, that is

$$\begin{array}{c}m\\ ⨁\\ i,j=1,i\ne j\end{array}[(\frac{m{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^m{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{m{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^m{\omega }_k(1+T{(a}_k))}{a}_j{)}^t]=$$

$$(\sqrt[q]{1-{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^m[1-(1-(1-u_i^q{)}^{\frac{m{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^m{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{m{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^m{\omega }_k(1+T{(a}_k))}}{)}^t]},$$

$$\sqrt[q]{{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^m[1-(1-v_i^{\frac{qm{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^m{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{qm{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^m{\omega }_k(1+T{(a}_k))}}{)}^t]})$$

Then, for $n=m+1$,

$$\begin{array}{c}m+1\\ ⨁\\ i,j=1,i\ne j\end{array}[(\frac{(m+1){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{(m+1){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_j{)}^t]=$$

$$\begin{gathered} (\begin{array}{c}m\\ ⨁\\ i,j=1,i\ne j\end{array}[(\frac{\left(m+1\right){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{\left(m+1\right){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_j{)}^t])⨁(\begin{array}{c}m\\ ⨁\\ i=1\end{array}[(\frac{\left(m+1\right){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{\left(m+1\right){\omega }_{m+1}(1+T{(a}_{m+1}))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_{m+1}{)}^t]) \\ ⨁(\begin{array}{c}m\\ ⨁\\ j=1\end{array}[(\frac{\left(m+1\right){\omega }_{m+1}(1+T{(a}_{m+1}))}{{\sum }_{k=1}^m{\omega }_k(1+T{(a}_k))}{a}_{m+1}{)}^s⨂(\frac{\left(m+1\right){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^m{\omega }_k(1+T{(a}_k))}{a}_j{)}^t]) \end{gathered}$$

And

$$\begin{array}{c}m\\ ⨁\\ i=1\end{array}[(\frac{(m+1){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{(m+1){\omega }_{m+1}(1+T{(a}_{m+1}))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_{m+1}{)}^t]=$$

$$(\sqrt[q]{1-{\prod }_{i=1}^m[1-(1-(1-u_i^q{)}^{\frac{(m+1){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_{m+1}^q{)}^{\frac{(m+1){\omega }_{m+1}(1+T{(a}_{m+1}))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^t]},$$

$$\sqrt[q]{{\prod }_{i=1}^m[1-(1-v_i^{\frac{q(m+1){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_{m+1}^{\frac{q(m+1){\omega }_{m+1}(1+T{(a}_{m+1}))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^t]})$$

$$\begin{array}{c}m\\ ⨁\\ j=1\end{array}[(\frac{(m+1){\omega }_{m+1}(1+T{(a}_{m+1}))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_{m+1}{)}^s⨂(\frac{(m+1){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_j{)}^t]=$$

$$(\sqrt[q]{1-{\prod }_{j=1}^m[1-(1-(1-u_{m+1}^q{)}^{\frac{(m+1){\omega }_{m+1}(1+T{(a}_{m+1}))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{(m+1){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^t]},$$

$$\sqrt[q]{{\prod }_{j=1}^m[1-(1-v_{m+1}^{\frac{q(m+1){\omega }_{m+1}(1+T{(a}_{m+1}))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{q(m+1){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^t]})$$

Considering the above equations, it can be concluded

$$\begin{array}{c}m+1\\ ⨁\\ i,j=1,i\ne j\end{array}[(\frac{(m+1){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{(m+1){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}{a}_j{)}^t]=$$

$$(\sqrt[q]{1-{\prod }_{i,j=1,i\ne j}^{m+1}[1-(1-(1-u_i^q{)}^{\frac{(m+1){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{(m+1){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^t]},$$

$$\sqrt[q]{{\prod }_{i,j=1,i\ne j}^{m+1}[1-(1-v_i^{\frac{q(m+1){\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{q(m+1){\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^{m+1}{\omega }_k(1+T{(a}_k))}}{)}^t]})$$

Consequently, the statement holds true for $n=m+1$. By the principle of mathematical induction, the statement is true for all positive integers $n$.

Thus

$$\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\{(\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t\}{]}^{\frac{1}{s+t}}=$$

$$(\sqrt[q]{1-[{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n(1-(1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t){]}^{\frac{1}{n\left(n-1\right)}}},$$

$$\sqrt[\frac{1}{qn\left(n-1\right)}]{{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n[1-(1-v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t]})$$

$$\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)=[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left((\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}=$$

$$(\frac{1}{n\left(n-1\right)}\left(\sqrt[q]{1-{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n(1-(1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t)}\right.,$$

$$\left.\sqrt[q]{{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n[1-(1-v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t]}\right){)}^{\frac{1}{s+t}}=$$

$$(\sqrt[q]{1-({\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n(1-(1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t){)}^{\frac{1}{n\left(n-1\right)}}},$$

$$\sqrt[q]{({\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n[1-(1-v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t]{)}^{\frac{1}{n\left(n-1\right)}}}{)}^{\frac{1}{s+t}}=$$

$$(\sqrt[q]{{\left(1-[{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n(1-(1-(1-u_i^q{)}^{\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-(1-u_j^q{)}^{\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t{]}^{\frac{1}{n\left(n-1\right)}}\right)}^{\frac{1}{s+t}}},$$

$$\sqrt[q]{{\left(1-[{\prod }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n(1-(1-v_i^{\frac{q{\omega }_{i}n(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^s(1-v_j^{\frac{q{\omega }_{j}n(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}}{)}^t{]}^{\frac{1}{n\left(n-1\right)}}\right)}^{\frac{1}{s+t}}})$$

□

3.2. Properties of GOFWPBM Operator

Some basic properties of the generalized orthopair fuzzy weighted power Bonferroni mean operator are discussed below.

Property 1.

When $\frac{(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}=\frac{1}{n}$ then,

$$\begin{array}{cc}\hfill \mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)& =[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left((\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}\hfill \\ & =[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left(({\omega }_i{a}_i{)}^s⨂({\omega }_j{a}_j{)}^t\right){]}^{\frac{1}{s+t}}\hfill \\ & =GOFWB{M}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)\hfill \end{array}$$

At this time, the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$ operator degenerates into the generalized orthopair fuzzy weighted Bonferroni mean $\left(\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{B}{\mathrm{M}}^{s,t}\right)$ operator. This operator does not possess the advantages of the power mean operator, which emphasizes the support among data and excludes outliers, thereby ensuring the overall coherence among data.

Property 2.

When $\omega =({\omega }_1,{\omega }_2,\dots ,{\omega }_n{)}^T=(\frac{1}{n}, \frac{1}{n}, \frac{1}{n}\dots \frac{1}{n}{)}^T$ then,

$$\begin{array}{cc}\hfill \mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)& =[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left((\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}\hfill \\ & =[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left((\frac{n(1+T{(a}_i))}{{\sum }_{k=1}^n(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n(1+T{(a}_j))}{{\sum }_{k=1}^n(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}\hfill \\ & =\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)\hfill \end{array}$$

At this point, the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$ operator degenerates into the generalized orthopair fuzzy power Bonferroni mean $\left(\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}\right)$ operator. Although this operator combines the benefits of the power average operator and the Bonferroni mean operator, it cannot assign weights to each variable indicator.

Property 3.

(Boundedness) Let $a_i\left(i=1,2,\cdots , n\right)$ be a set of generalized orthopair fuzzy numbers. Define $a^{+}=\underset{1\le i\le n}{\mathit{max}}{a}_i, a^{-}=\underset{1\le i\le n}{\mathit{min}}{a}_i$. Then

$$a^{-}\le GOFWPB{M}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)\le a^{+}$$

Proof of Property 3.

$$\begin{array}{cc}\hfill GOFWPB{M}^{s,t}\left(a_1,a_2,a_3,\cdots ,a_n\right)& =[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left((\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}\hfill \\ & \le [\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}(a^{+}{)}^s⨂(a^{+}{)}^t{]}^{\frac{1}{s+t}}=[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}(a^{+}{)}^{s+t}{]}^{\frac{1}{s+t}}=a^{+}\hfill \end{array}$$

$$\begin{array}{cc}\hfill GOFWPB{M}^{s,t}\left(a_1,a_2,a_3,\cdots ,a_n\right)& =[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}\left((\frac{n{\omega }_i(1+T{(a}_i))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_i{)}^s⨂(\frac{n{\omega }_j(1+T{(a}_j))}{{\sum }_{k=1}^n{\omega }_k(1+T{(a}_k))}{a}_j{)}^t\right){]}^{\frac{1}{s+t}}\hfill \\ & \ge [\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}(a^{-}{)}^s⨂(a^{-}{)}^t{]}^{\frac{1}{s+t}}=[\frac{1}{n\left(n-1\right)}\begin{array}{c}n\\ ⨁\\ i,j=1,i\ne j\end{array}(a^{-}{)}^{s+t}{]}^{\frac{1}{s+t}}=a^{-}\hfill \end{array}$$

□

Therefore, by combining the two inequalities, we have

$$a^{-}\le GOFWPB{M}^{s,t}\left(a_1,a_2,\cdots ,a_n\right)\le a^{+}$$

4. The MADM Model Based on the ${\mathit{GOFWPBM}}^{\mathit{s},\mathit{t}}$ Operator

The multi-attribute decision-making model is a current research focus in the field of decision science, and is extensively utilized to address decision-making issues in real-life scenarios. For instance, Garg [38] proposed a multi-attribute decision-making algorithm based on trigonometric generalized orthopair fuzzy numbers. Xu et al. [39] introduced a multi-attribute decision-making method based on fuzzy soft sets. Chen et al. [40] put forward a multi-attribute decision-making model based on the intuitionistic trapezoidal fuzzy generalized Heronian OWA operator, and Li et al. [41] proposed a series of multi-attribute decision-making methods based on the Pythagorean fuzzy power Muirhead mean operators.

From the above analysis, it is evident that the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$ operator combines the advantages of numerous operators in information aggregation. Based on this, we propose a multi-attribute decision-making model based on the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$ operator. This decision-making model will be applied to an investment attraction project in a certain city to verify the feasibility and superiority of the method proposed in this paper.

In the multi-attribute decision-making problem with generalized orthopair fuzzy information, suppose there are $n$ candidate schemes $X=\left(x_1,x_2,\dots x_n\right)$, and $m$ decision attributes $C=$($c_1,c_2,c_3\dots c_m$). The weight vector corresponding to each decision attribute is $\omega =({\omega }_1,{\omega }_2,{\omega }_3\dots {\omega }_m{)}^T$. Experts provide generalized orthopair fuzzy evaluation information. We denote the attribute value of scheme $x_i$ under attribute $c_j$ as $a_{ij}$, where $a_{ij}$ is a generalized orthopair fuzzy number. Thus, a generalized orthopair fuzzy decision matrix $M={\left(a_{ij}\right)}_{mn}$ is obtained, where $a_{ij}=\left(u_{ij},v_{ij}\right)$, $u_{ij}$ represents the membership value and $v_{ij}$ represents the non-membership value of candidate scheme $i$ regarding attribute $j$. The specific steps are as follows.

Step 1. Based on the actual situation, establish the generalized orthopair fuzzy matrix $M={\left(a_{ij}\right)}_{mn}$. Convert each decision attribute value in it to the benefit-type, resulting in the standard matrix [42] $\overline{M}={\left({\overline{a}}_{ij}\right)}_{mn}$, where ${\overline{a}}_{ij}=\left\{\begin{array}{c}{a}_{ij}, \mathrm{if} C_j \mathrm{is\; a\; benefit}-\mathrm{type\; attribute}\\ \left(v_{ij},u_{ij}\right),\mathrm{if} C_j \mathrm{is\; a\; cost}-\mathrm{type\; attribute}\end{array}.\right.$

Step 2. Using the information aggregation ($\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$) operator, compute the attribute values of each solution in the generalized orthopair fuzzy standard matrix $\overline{M}={\left({\overline{a}}_{ij}\right)}_{mn}$. Then, determine the score function for each solution based on these attribute values.

Step 3. Rank the score functions from high to low and select the optimal solution.

5. Case Analysis

5.1. Decision-Making Process

Since the outbreak of the COVID-19 pandemic, the economic development in various regions has been consistently sluggish. Responding to the call for national economic recovery, a particular city government took the lead in allocating funds to stimulate local economic recovery. They planned to invest in some privately-owned enterprise in the city. But, due to reasons such as financial difficulties and policy restrictions, the government aimed to raise fiscal revenue, increase employment opportunities, and reduce investment risks by seeking partners in the market. After rigorous market research, five potential partners were identified: $x_1$ Daily goods manufacturing, $x_2$ Chain catering, $x_3$ Internet company, $x_4$ Civil and building materials, $x_5$ Civil and building materials. Taking into consideration its own situation, the decision-making group composed of government officials and experts decided to inspect the candidates based on the following four factors. C₁: Fiscal Assessment (including company’s financial reserves, investment orientation evaluation, etc.), C₂: Developmental Assessment (including enterprise scale, future development plan, etc.), C₃: Social Influence Assessment (including the company’s social status and public reputation, etc.), C₄: Green Development Assessment (including sustainable development strategies and environment-friendly levels of the enterprise).

Following the assessment by the opinion group, a generalized orthopair fuzzy decision matrix $M={\left(a_{ij}\right)}_{5\times 4}$ was established as shown in

Table 1. Generalized orthopair fuzzy decision matrix $M$.

Schemes	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$
$x_1$	<0.4, 0.5>	<0.5, 0.4>	<0.2, 0.7>	<0.2, 0.5>
$x_2$	<0.6, 0.4>	<0.6, 0.3>	<0.6, 0.3>	<0.3, 0.6>
$x_3$	<0.5, 0.5>	<0.4, 0.5>	<0.4, 0.4>	<0.5, 0.4>
$x_4$	<0.7, 0.2>	<0.5, 0.4>	<0.2, 0.5>	<0.6, 0.7>
$x_5$	<0.5, 0.3>	<0.3, 0.4>	<0.6, 0.2>	<0.4, 0.4>

, where the evaluation value ${ a}_{ij}=\left(u_{ij},v_{ij}\right)$ is a generalized orthopair fuzzy number, and the weight vector $\omega =({\omega }_1,{\omega }_2,{\omega }_3\dots {\omega }_n{)}^T$ is considered equal weight. Based on the decision matrix provided by the opinion group, we will evaluate the five prospective partner companies.

Step 1. Establishing the generalized orthopair fuzzy standard matrix $\overline{M}$ based on Table 1. Since all attributes in this case are of benefit type, the standard matrix $\overline{M}$ is equal to $M$.

Step 2. Using Formula (7), we derived the comprehensive attribute values for each scheme. In this case study, to show the complete results, the value of q is set to be greater than or equal to 2. Here, $q$ = 3. The comprehensive attribute values for each scheme are: $x_1=<0.352,0.536>$, $x_2=<0.549,0.414>$, $x_3=<0.453,0.453>$, $x_4=<0.540,0.483>$, $x_5=<0.464,0.335>$. Based on these comprehensive values, the score function values for each scheme are: $S_{x1}=-0.1103$, $S_{x2}=0.0941$, $S_{x3}=-0.0002$, $S_{x4}=0.0449$, $S_{x5}=0.0621$.

Step 3. According to the aforementioned score function values and ranking them from highest to lowest, we have the order ${ S}_{x2}$ > $S_{x5}$ > $S_{x4}$ > $S_{x3}$ > $S_{x1}$. Thus, the optimal scheme is $x_2$ (Chain catering company).

5.2. Parameters Analysis

The analysis and conclusion above were drawn under the condition of $s,t=1$. To ensure generality and robustness in the decision-making process, it is pertinent to discuss the ranking of the best candidate schemes under different parameter values for $s \mathrm{and} t$, as shown in

Table 2. Composite attribute values of each scheme under different parameters.

$\mathit{s},\mathit{t}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$
1	<0.352, 0.536>	<0.549, 0.414>	<0.453, 0.453>	<0.540, 0.483>	<0.464, 0.335>
2	<0.389, 0.532>	<0.571, 0.411>	<0.458, 0.453>	<0.574, 0.477>	<0.485, 0.335>
3	<0.408, 0.528>	<0.581, 0.408>	<0.464, 0.452>	<0.590, 0.470>	<0.499, 0.334>

.

Table 2 shows the composite attribute values of each scheme under different parameters. Based on this table, the score rankings under different parameters are calculated in

Table 3. Score and ranking of each scheme.

$\mathit{s},\mathit{t}$	Score	Ranking
1	$S=(-0.1103, 0.0941,-0.0002, 0.0449, 0.0621)$	$S_{x2}>S_{x5}>S_{x4}>S_{x3}>S_{x1}$
2	$S=(-0.0916, 0.1164, 0.0035, 0.0805, 0.0765)$	$S_{x2}>S_{x4}>S_{x5}>S_{x3}>S_{x1}$
3	$S=(-0.0790, 0.1283, 0.0072, 0.1013, 0.0871)$	$S_{x2}>S_{x4}>S_{x5}>S_{x3}>S_{x1}$

.

Table 3 presents the ranking results under different parameters using the generalized orthopair fuzzy weighted power Bonferroni mean ($\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$) operator. Due to the novel operator having two parameters, s and t, it allows for greater flexibility and convenience in information aggregation. Therefore, during the decision-making process, the expert group can select appropriate parameter values based on the complexity of the problem combined with their risk preferences. Moreover, the comparison results from the Table 3 above show that the scores of each candidate will increase as the values of the parameters s and t increase. However, due to the stability of the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$ operator, it does not distort information with excessively large values of s and t like other operators. This avoids influencing the final judgment results, making it more suitable for today’s increasingly complex decision-making environment.

5.3. Comparative Analysis

To further demonstrate the superiority of the operator proposed in this paper, the following text will compare this operator with existing operators. Specifically, we selected the generalized orthopair fuzzy power average $\left(\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{P}\mathrm{A}\right)$ operator and the generalized orthopair fuzzy Bonferroni mean $\left({\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{B}\mathrm{M}}^{s,t}\right)$ operator for comparison. For convenience of calculation, parameter values in various information aggregation operators are uniformly taken as one. The comparison results are shown in

Table 4. Best solution ranking under different operators.

Operators	Score	Ranking
$GOFPA$	$S=(0.2251, 0.5930, 0.3767, 0.6091, 0.4259)$	$S_{x4}$$> S_{x2}$$> S_{x5}$$> S_{x3}$$> S_{x1}$
${GOFBM}^{s,t}$	$S=(-0.4886, -0.2530, -0.3233, -0.3823, -0.1440)$	$S_{x5}$$> S_{x2}$$> S_{x3}$$> S_{x4}$$> S_{x1}$
$GOFWPB{M}^{s,t }$	$S=(-0.1103, 0.0941,-0.0002, 0.0449, 0.0621)$	$S_{x2}$$> S_{x5}$$> S_{x4}$$> S_{x3}$$> S_{x1}$

.

Based on the comparative outcomes presented in the Table 4, it is discernible that score values and the resultant rankings exhibit slight differences across various methodologies. The underlying reason for these divergences can be attributed to the distinct information aggregation methodologies employed by each of the emblematic information aggregation operators. Even though all these operators are grounded in the average-based paradigm, their focal points during the information aggregation process vary notably. It can be observed that the candidate scheme $x_4$ consistently exhibits extreme values across various attributes. As a result, using the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{P}\mathrm{A}$ operator, the outcome points to $x_4$. This is primarily because the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{P}\mathrm{A}$ operator can negate the adverse impact brought about by extreme values encountered in expert evaluations. Additionally, it emphasizes the support degree between data, ensuring cohesion among them. However, it does not take into consideration potential inter-relationships between attributes. In this particular case, it is evident that the attribute $C_2$ influences the development of $C_3$. On the other hand, the ${\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{B}\mathrm{M}}^{s,t}$ operator primarily captures the interrelationships between pieces of information. Yet, its vulnerability to extreme values can lead to distorted outcomes. Consequently, in this instance, after being influenced by the extreme value, the result derived from ${\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{B}\mathrm{M}}^{s,t}$ operator is $x_5$.

The newly defined $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$ operator in this study combines the advantages of both: it not only takes into account the overall balance among the data but also considers the weights of each attribute and the potential correlations that may exist between different attributes. This helps prevent biased decision-makers from skewing the results with outlier preference values (i.e., exceptionally high or low values in the original data), ensuring a more fair and objective decision-making process. Furthermore, the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$ operator retains the parameters $s$ and $t$, allowing decision-makers to adjust the parameters based on their risk preferences, making the decision-making process more flexible. Additionally, the $\mathrm{G}\mathrm{O}\mathrm{F}\mathrm{W}\mathrm{P}\mathrm{B}{\mathrm{M}}^{s,t}$ operator introduces the concept of weights, providing an avenue to account for the intrinsic importance of the data, thereby rendering the decision-making process more rational and in line with real-world multi-attribute decision-making problems.

6. Conclusions

In response to the growing scholarly interest in generalized orthopair fuzzy multi-attribute decision-making problems, this study delves into the theory of fuzzy sets and information aggregation operators, introducing a novel generalized orthopair fuzzy aggregation operator, termed the generalized orthopair fuzzy weighted power Bonferroni mean operator. This novel operator combines the commendable attributes of both the power average operator and the Bonferroni mean operator. It not only takes into account the overall balance among various data but also considers the weights of individual attributes and potential correlations that might exist between different attributes. Moreover, the paper explores some admirable properties and specific scenarios pertaining to this novel operator. Building on this, a multi-attribute decision-making method based on the generalized orthopair fuzzy weighted power Bonferroni mean operator is elucidated, with its effectiveness and superiority validated through empirical instances and comparative analysis. The decision-making methodology proposed herein holds potential for further application in areas such as supplier selection assessment, product scheme evaluation, and talent recruitment recommendations in human resource departments, showcasing both theoretical and practical significance.