Bipolar Neutrosophic Dombi-Based Heronian Mean Operators and Their Application in Multi-criteria Decision-Making Problems

Abstract

Dombi operations based on the Dombi t-norm (TN) and t-conorm (TCN) have the advantage in terms of operational parameter flexibility in dealing with varying degrees of uncertainty and aggregation requirements. Meanwhile, Heronian mean (HM) operator is an effective technique for capturing the interrelationship between any number of inputs. Bipolar neutrosophic set (BNS) offers the ability to represent both positive and negative information as well as indeterminate information. It is beneficial in cases where there is uncertainty or insufficient information. However, the existing Dombi operator under BNS do not take into account the interrelationship between input arguments. To overcome this limitation, this study incorporates Dombi operator into HM and propose the bipolar neutrosophic Dombi Heronian mean aggregation operator. This paper introduces two type of aggregation operators namely bipolar neutrosophic Dombi-based generalized weighted Heronian mean (BND-GWHM), and bipolar neutrosophic Dombi-based improved generalized weighted Heronian mean (BND-IGWHM). The proposed operators are integrated into MCDM procedure. The influence of different parameter values on decision-making results is discussed. Finally, a comparison analysis with existing methods is also provided.

1 Introduction

Decision-making (DM) processes frequently involve dealing with complex and uncertain data. Traditional crisp sets, characterized by their clear-cut boundaries and binary membership, often struggle to capture the inherent vagueness and ambiguity that real-world decision problems exhibit. In response to this limitation, Zadeh [1] developed the fuzzy set (FS) theory, introducing a more flexible framework where elements can have partial memberships in a set, allowing for a smoother representation of uncertainty. Nevertheless, in some decision-making problems, FSs had limitations in representing intuitionistic attributes of human judgement. As such, Atanassov [2] introduced the concept of an intuitionistic fuzzy set (IFS), which characterized both the membership function and non-membership function between 0 and 1. IFSs can accurately describe and express ambiguous decision information in DM problems. After that, the idea of interval-valued intuitionistic fuzzy set (IVIFS) was proposed by Atanassov and Gargov [3]. However, IFS and IVIFS can only represent contradictory information in real-world DM problems.

The neutrosophic set (NS) is an extension of FSs introduced by Smarandache [4]. It addresses the shortcomings of both crisp and fuzzy sets by incorporating a third parameter, known as "neutrosophy, " which accounts for indeterminacy or inconsistency. This novel approach enables decision-makers to deal with situations in which elements of membership function, non-membership function, and indeterminacy function, thereby creating a more flexible framework for modelling complex decision problems. NS empowers decision-makers to make more informed and robust choices in an increasingly uncertain and complex world by offering a more refined means of handling uncertainty, ambiguity, and indeterminacy. Wang et al. [5] extended the concept of interval neutrosophic set (INS) and Wang et al. [6] introduced the concept of single-valued neutrosophic set (SVNS) to apply NS to practical DM problems.

In real-life situations, some decision-making processes may involve positive and negative aspects in expressing opinions about a specific context. To address this, Zhang [7] introduced the concept of bipolarity into FS theory, giving rise to what is known as a bipolar fuzzy set (BFS). The term "bipolarity" in this context refers to the capacity of the human mind to make decisions based on both positive and negative factors, as highlighted by Bosc and Pivert [8]. Positive information within this framework pertains to possible, satisfactory, or acceptable choices. Conversely, negative information refers to options that are either impossible or forbidden. A positive preference reflects a desire, while a negative preference signifies a constraint in decision-making processes. Recognizing the advantages of BFS and NS, Deli et al. [9] introduced a novel concept called a bipolar neutrosophic set (BNS). Characterized by positive and negative membership degrees, BNS is designed to handle bipolar, inconsistent, and ambiguous data effectively. The research on neutrosophic bipolar set have been integrated with various set theories, including complex bipolar valued-neutrosophic soft set [10] and bipolar fuzzy hypersoft set [11]. This innovative approach has found applications in the domain of multi-criteria decision-making (MCDM). In various scenarios, BNS has been successfully employed to address complex problems such as professional [12], car selection [9], and green supplier evaluation [13]. Through these applications, the utility of bipolar neutrosophic sets has been demonstrated in resolving MCDM challenges associated with diverse decision contexts.

Generally, in the context of MCDM problems, decision-makers commonly evaluate and compare alternatives across multiple criteria. Their preferences, confidence levels, and attitudes toward risk are typically combined by means of aggregation operations. In aggregating these, various aggregation operators (AOs) have been explored by researchers which include Maclaurin symmetric mean (MSM) [14], Choquet integral [15], ordered weighted average (OWA) [16], power averaging [17], ordered weighted geometric [18], Bonferroni mean (BM) [19], Heronian mean (HM) [20], prioritized AO [21], prioritized OWA [22], harmonic mean [23], and logarithmic hybrid [24]. Later, Chen et al. [25] delved into the generation of OWA operators and their implementation in decision-making frameworks. Various methodologies have been devised to precisely determine weight vectors via the AO, enhancing fairness and equity in decision-making under conditions of uncertainty and complexity. Among these methodologies, notable ones include the weight determination model [26], the assignment of weights predicated on bipolar preferences [27], and the innovation of the Reconstructed Weighted Aggregation (RWA) operator [28]. Furthermore, Jin [29] investigated sophisticated techniques to manage uncertainty within probability calculations, incorporating principles such as regular probability intervals and relative proximity. This integration aids in more nuanced modelling and analysis of scenarios where data or outcomes are indeterminate. In the aggregation process, the algebraic operational law, such as algebraic product and algebraic sum are commonly applied for aggregation purposes. Nevertheless, the algebraic operations have some limitations in term of flexibility in handling uncertain information. These methods produce deterministic results that fail to capture the inherent uncertainty in many decision-making scenarios.

The Dombi operations based on Dombi t-norm and Dombi t-conorms, on the other hand offer flexibility in aggregating multiple criteria or inputs [30]. In particular, the Dombi operators are defined with a parameter that assigns the degree of emphasis on the larger of any two operands. This parameter is a real number that can be changed to modify the operator’s behavior. In previous work, Liu et al. [31] introduced a combination of intuitionistic fuzzy and Dombi and applied it to decision-making problems, while He [32] studied Dombi operations under hesitant fuzzy information. The applications of the Dombi operation in MCDM methods under neutrosophic environment had gained attention among researchers. Chen and Ye [33] proposed an aggregation operator of Dombi under SVNS. Khan et al. [34] introduced some Dombi operations for interval neutrosophic sets (INSs). Fan et al. [35] studied Dombi prioritized Bonferroni mean under single-valued triangular neutrosophic sets. Mahmood et al. [36] extended the Dombi operation to aggregate BNS. However, these operators do not consider the interrelationship between input arguments.

An effective aggregation operator should not only have flexible operational parameters but be able to handle interrelationships among input arguments as well. Interrelationship is important in aggregation process to ensure accurate findings. For example, selecting a COVID-19 vaccination location involves a complex interplay of criteria. In this situation, the interrelationship among criteria such as transportation, population, labour, accessibility and community engagement can directly impact the vaccination site efficiency. The location with the highest value is the most suitable for setting up the COVID-19 vaccination location. Dealing with such information requires the use of Heronian mean (HM) operator to aggregate multiple criteria into a single value. HM is a significant AO in handling the interrelationship among input arguments. Some of the extensions of HM have been made to improve its limitations such as generalized HM [37], generalized weighted HM (GWHM) [38], generalized Heronian OWA operator [39]. Further, Liu [40] extended HM to the improved generalized weighted HM (IGWHM). Besides that, the research on HM is expanding in various environments, including but not limited to single valued neutrosophic heronian mean [41], bipolar neutrosophic heronian mean [42], interval neutrosophic vague Heronian mean [43].

Yet, until now, there have been no studies that integrate Dombi and HM in neutrosophic environment. The existing integration of BNS and Dombi mean operator proposed by Mahmood et al. [36] do not consider the interrelationship of input arguments. Aside from that, Fan et al. [42] presented BNS HM, which is limited to algebraic operations and lacks parameter flexibility. Thus, motivated by Mahmood et al. [36], Wu et al. [44], and Ayub et al. [45], this study incorporates Dombi into HM and propose the bipolar neutrosophic Dombi Heronian mean aggregation operator. Furthermore, the GWHM and IGWHM are used because they are a better extended version of HM that can overcome the constraints of classical HM, specifically unsatisfied idempotency and disregard for weight throughout the aggregating process. The motivation behind our research can be encapsulated as follows: despite numerous studies introducing various AOs across different contexts, the current method relying on BNS reveals notable shortcomings regarding their computational procedures and aggregation outcomes. Consequently, this study aims to enhance the approaches established by Mahmood et al. [36] and Fan et al. [42] by devising two novel AOs. These proposed operators merge the Heronian operator with the Dombi operation. Furthermore, the Dombi operation is employed to tackle the constraint of parameter flexibility inherent in algebraic operations.

The objective of this paper is twofold. Firstly, to develop new aggregation operators, known as bipolar neutrosophic Dombi-based generalized weighted Heronian mean (BND-GWHM) and bipolar neutrosophic Dombi-based improved generalized weighted Heronian mean (BND-IGWHM). Secondly, to develop an MCDM algorithm in bipolar neutrosophic setting in which the proposed BND-GWHM and BND-IGWHM operators are employed in the aggregation process. The effectiveness of the proposed algorithm is illustrated and analyzed using numerical example of solving an investment problem.

The remaining sections of this paper are classified as follows: Sect. 2 presents some basic concept related to BNSs, Dombi operations, and the HM. In Sect. 3, we defined two AOs, which are the bipolar neutrosophic Dombi-based generalized weighted Heronian mean (BND-GWHM) and bipolar neutrosophic Dombi-based improved generalized weighted Heronian mean (BND-IGWHM) aggregation operators, as well as the properties of each of them. In Sect. 4, a MCDM method are developed based on proposed operators. Section 5 contains a numerical illustration on the applicability of the proposed method. In addition, this section discusses the impact of different parameter values on the final ranking results and presents a comparison analysis with existing methods. Section 6 concludes with a conclusion and future research recommendations.

2 Preliminaries

In this section, we present some concepts and definitions related to the bipolar neutrosophic set (BNS), Dombi operations and Heronian mean (HM) operators.

2.1 Bipolar Neutrosophic Set

The following definition of a bipolar neutrosophic set and its arithmetic operations are based on Deli et al. [9].

Definition 1.

[9] Let $X$ be a nonempty set. A bipolar neutrosophic set A is defined as:

$$ A = \left\{ {\left\langle {x, T^{ + } \left( x \right), I^{ + } \left( x \right), F^{ + } \left( x \right), T^{ - } \left( x \right), I^{ - } (x), F^{ - } \left( x \right)} \right\rangle : x \in X} \right\}, $$

(1)

where $T^{ + } , I^{ + } , F^{ + } : X \to \left[ {1, 0} \right]$ and $T^{ - } , I^{ - } , F^{ - } : X \to \left[ { - 1, 0} \right].$

Here, the $T^{ + } \left( x \right), \;I^{ + } \left( x \right), \;F^{ + } \left( x \right)$ represents positive membership degree, while $T^{ - } \left( x \right), \;I^{ - } \left( x \right), \;F^{ - } \left( x \right)$ represents negative membership degree of an element $x \in X.$

Definition 2.

[9] Let $A_{1} = \left\langle {t_{1}^{ + } , i_{1}^{ + } , f_{1}^{ + } , t_{1}^{ - } , i_{1}^{ - } , f_{1}^{ - } } \right\rangle$ and $A_{2} = \left\langle {t_{2}^{ + } , i_{2}^{ + } , f_{2}^{ + } , t_{2}^{ - } , i_{2}^{ - } , f_{2}^{ - } } \right\rangle$ be two BNSs and $\lambda > 0.$ The following are the fundamental operations for BNS are given as:

(i) $A_{1} \oplus A_{2} = \left\langle {t_{1}^{ + } t_{2}^{ + } , i_{1}^{ + } + i_{2}^{ + } - i_{1}^{ + } i_{2}^{ + } , f_{1}^{ + } + f_{2}^{ + } - f_{1}^{ + } f_{2}^{ + } , - \left( { - t_{1}^{ - } - t_{2}^{ - } - t_{1}^{ - } t_{2}^{ - } } \right), - i_{1}^{ - } i_{2}^{ - } , - f_{1}^{ - } f_{2}^{ - } } \right\rangle .$

(ii) $A_{1} \otimes A_{2} = \left\langle {t_{1}^{ + } t_{2}^{ + } , i_{1}^{ + } + i_{2}^{ + } - i_{1}^{ + } i_{2}^{ + } , f_{1}^{ + } + f_{2}^{ + } - f_{1}^{ + } f_{2}^{ + } , - \left( { - t_{1}^{ - } - t_{2}^{ - } - t_{1}^{ - } t_{2}^{ - } } \right), - i_{1}^{ - } i_{2}^{ - } , - f_{1}^{ - } f_{2}^{ - } } \right\rangle .$

(iii) $\lambda A_{1} = \left\langle {1 - \left( {1 - t_{1}^{ + } } \right)^{\lambda } , \left( {i_{1}^{ + } } \right)^{\lambda } , \left( {f_{1}^{ + } } \right)^{\lambda } , - \left( { - t_{1}^{ - } } \right)^{\lambda } , - \left( { - i_{1}^{ - } } \right)^{\lambda } , - \left( {1 - \left( {1 - \left( { - f_{1}^{ - } } \right)} \right)^{\lambda } } \right)} \right\rangle .$

(iv) $A_{1}^{\lambda } = \left\langle {\left( {t_{1}^{ + } } \right)^{\lambda } , 1 - \left( {1 - i_{1}^{ + } } \right)^{\lambda } , 1 - \left( {1 - f_{1}^{ + } } \right)^{\lambda } , - \left( {1 - \left( {1 - \left( { - t_{1}^{ - } } \right)} \right)^{\lambda } } \right), - \left( { - i_{1}^{ - } } \right)^{\lambda } , - \left( { - f_{1}^{ - } } \right)^{\lambda } } \right\rangle .$

In this part, we will propose the score, accuracy and certainty functions of BNS as a generalization or extension of the score, accuracy and certainty functions used for bipolar intuitionistic fuzzy sets [46] in the following way.

Definition 3.

Assuming that $A_{1} = \left\langle {t_{1}^{ + } , i_{1}^{ + } , f_{1}^{ + } , t_{1}^{ - } , i_{1}^{ - } , f_{1}^{ - } } \right\rangle$ is a BNS, the score, accuracy, and certain function of $A_{1}$ are represented by $s\left( {A_{1} } \right),$ $a\left( {A_{1} } \right),$ and $c\left( {A_{1} } \right),$ respectively such that:

$$ s\left( {A_{1} } \right) = \frac{{t_{1}^{ + } + 1 - i_{1}^{ + } + 1 - f_{1}^{ + } + 1 - t_{1}^{ - } + i_{1}^{ - } + f_{1}^{ - } }}{6}\;; $$

(2)

$$ a\left( {A_{1} } \right) = t_{1}^{ + } - f_{1}^{ + } + f_{1}^{ - } - t_{1}^{ - } \;; $$

(3)

$$ c\left( {A_{1} } \right) = t_{1}^{ + } - f_{1}^{ - } . $$

(4)

2.2 Heronian Mean (HM)-Based Operators

The generalized weighted Heronian mean and improved generalized Heronian mean are presented as follows:

Definition 4.

[39] Generalized weighted Heronian mean.

Let $w_{i} = (w_{1} , w_{2} , ..., w_{n} )^{T}$ be the weight vector of a collection of non-negative numbers $x_{i} (i = 1, 2, ..., n),$ and satisfying $p, q \ge 0,$ $w_{i} \in \left[ {0, 1} \right]$ and $\sum_{i = 1}^{n} w_{i} = 1.$ Then,

$$ GWHM^{p, q} \left( {x_{1} , x_{2} , ..., x_{n} } \right) = \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\left( {w_{i} x_{i} } \right)^{p} \otimes \left( {w_{j} x_{j} } \right)^{q} } \right)} \right)^{{\frac{1}{p + q}}} $$

(5)

which is called a generalized weighted Heronian mean (GWHM) operator.

Definition 5.

[40] Improved generalized weighted Heronian mean.

Let $w_{i} = (w_{1} , w_{2} , ..., w_{n} )^{T}$ be the weight vector of a collection of non-negative numbers $x_{i} (i = 1, 2, ..., n),$ and satisfying $p, q \ge 0,$ $w_{i} \in \left[ {0, 1} \right]$ and $\sum_{i = 1}^{n} w_{i} = 1.$ Then,

$$ IGWHM^{p, q} \left( {x_{1} , x_{2} , ..., x_{n} } \right) = \frac{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} x_{i}^{p} \otimes x_{j}^{q} } \right)^{{\frac{1}{p + q}}} }}{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} } \right)^{{\frac{1}{p + q}}} }} $$

(6)

which is called a improved generalized weighted Heronian mean (IGWHM) operator.

2.3 Dombi Operations of BNSs

Definition 6.

[30] Dombi t-norm and t-conorm.

Let $\gamma$ be a positive real number and $x, y \in \left[ {0, 1} \right].$ Then, Dombi t-norm (TN) and t-conorm (TCN) among $x$ and $y$ denoted as $T_{D, \gamma } \left( {x, y} \right)$ and $T_{D, \gamma }^{*} \left( {x, y} \right)$ respectively, such that

$$ T_{D, \gamma } \left( {x, y} \right) = \left( {1 + \left( {\left( {\frac{1 - x}{x}} \right)^{\gamma } + \left( {\frac{1 - y}{y}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

(7)

$$ T_{D, \gamma }^{*} \left( {x, y} \right) = 1 - \left( {1 + \left( {\left( {\frac{x}{1 - x}} \right)^{\gamma } + \left( {\frac{y}{1 - y}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} . $$

(8)

On the basis of the Dombi t-norm and t-conorm, Mahmood et al. [36] defined the following operational rules of BNS.

Definition 7.

[36] Dombi operations of bipolar neutrosophic sets.

Let $A_{1} = \left\langle {t_{1}^{ + } , i_{1}^{ + } , f_{1}^{ + } , t_{1}^{ - } , i_{1}^{ - } , f_{1}^{ - } } \right\rangle$ and $A_{2} = \left\langle {t_{2}^{ + } , i_{2}^{ + } , f_{2}^{ + } , t_{2}^{ - } , i_{2}^{ - } , f_{2}^{ - } } \right\rangle$ be two BNSs and $\gamma , \delta$ be two positive real numbers. Then, the Dombi sums and product operation are defined as follows:

(a)

$$ \begin{gathered} A_{1} \oplus_{D} A_{2} \hfill \\ = \left\langle {1 - \left( {1 + \left( {\left( {\frac{{t_{1}^{ + } }}{{1 - t_{1}^{ + } }}} \right)^{\gamma } + \left( {\frac{{t_{2}^{ + } }}{{1 - t_{2}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\left( {\frac{{1 - i_{1}^{ + } }}{{i_{1}^{ + } }}} \right)^{\gamma } + \left( {\frac{{1 - i_{2}^{ + } }}{{i_{2}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\left( {\frac{{1 - f_{1}^{ + } }}{{f_{1}^{ + } }}} \right)^{\gamma } + \left( {\frac{{1 - f_{2}^{ + } }}{{f_{2}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right. \hfill \\ \end{gathered} $$

$$ \left. {\quad - \left( {1 + \left( {\left( {\frac{{1 - t_{1}^{ - } }}{{ - t_{1}^{ - } }}} \right)^{\gamma } + \left( {\frac{{1 - t_{2}^{ - } }}{{ - t_{2}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\left( {\frac{{ - i_{1}^{ - } }}{{1 + i_{1}^{ - } }}} \right)^{\gamma } + \left( {\frac{{ - i_{2}^{ - } }}{{1 + i_{2}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, \left( {1 + \left( {\left( {\frac{{ - f_{1}^{ - } }}{{1 + f_{1}^{ - } }}} \right)^{\gamma } + \left( {\frac{{ - f_{2}^{ - } }}{{1 + f_{2}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1} \right\rangle $$

(9)

(b)

$$ \begin{gathered} A_{1} \otimes_{D} A_{2} \hfill \\ = \left\langle {\left( {1 + \left( {\left( {\frac{{1 - t_{1}^{ + } }}{{t_{1}^{ + } }}} \right)^{\gamma } + \left( {\frac{{1 - t_{2}^{ + } }}{{t_{2}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , 1 - \left( {1 + \left( {\left( {\frac{{i_{1}^{ + } }}{{1 - i_{1}^{ + } }}} \right)^{\gamma } + \left( {\frac{{i_{2}^{ + } }}{{1 - i_{2}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , 1 - \left( {1 + \left( {\left( {\frac{{f_{1}^{ + } }}{{1 - f_{1}^{ + } }}} \right)^{\gamma } + \left( {\frac{{f_{2}^{ + } }}{{1 - f_{2}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right. \hfill \\ \end{gathered} $$

(10)

$$ \quad \left. {\left( {1 + \left( {\left( {\frac{{ - t_{1}^{ - } }}{{1 + t_{1}^{ - } }}} \right)^{\gamma } + \left( {\frac{{ - t_{2}^{ - } }}{{1 + t_{2}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, - \left( {1 + \left( {\left( {\frac{{1 + i_{1}^{ - } }}{{ - i_{1}^{ - } }}} \right)^{\gamma } + \left( {\frac{{1 + i_{2}^{ - } }}{{ - i_{2}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , - \left( {1 + \left( {\left( {\frac{{1 + f_{1}^{ - } }}{{ - f_{1}^{ - } }}} \right)^{\gamma } + \left( {\frac{{1 + f_{2}^{ - } }}{{ - f_{2}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} } \right\rangle $$

(c)

$$ \begin{gathered} \delta \cdot A_{1} \hfill \\ = \left\langle {1 - \left( {1 + \left( {\delta \left( {\frac{{t_{1}^{ + } }}{{1 - t_{1}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\delta \left( {\frac{{1 - i_{1}^{ + } }}{{i_{1}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\delta \left( {\frac{{1 - f_{1}^{ + } }}{{f_{1}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right. \hfill \\ \hfill \\ \quad \left. { - \left( {1 + \left( {\delta \left( {\frac{{1 + t_{1}^{ - } }}{{ - t_{1}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\delta \left( {\frac{{ - i_{1}^{ - } }}{{1 + i_{1}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, \left( {1 + \left( {\delta \left( {\frac{{ - f_{1}^{ - } }}{{1 + f_{1}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1} \right\rangle \hfill \\ \end{gathered} $$

(11)

(d)

$$ \begin{gathered} A_{1}^{\delta } \hfill \\ = \left\langle {\left( {1 + \left( {\delta \left( {\frac{{1 - t_{1}^{ + } }}{{t_{1}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , 1 - \left( {1 + \left( {\delta \left( {\frac{{i_{1}^{ + } }}{{1 - i_{1}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , 1 - \left( {1 + \left( {\delta \left( {\frac{{f_{1}^{ + } }}{{1 - f_{1}^{ + } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right. \hfill \\ \hfill \\ \quad \left( {1 + \left( {\delta \left( {\frac{{ - t_{1}^{ - } }}{{1 + t_{1}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, - \left( {1 + \left( {\delta \left( {\frac{{1 + i_{1}^{ - } }}{{ - i_{1}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left. { - \left( {1 + \left( {\delta \left( {\frac{{1 + f_{1}^{ - } }}{{ - f_{1}^{ - } }}} \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} } \right\rangle . \hfill \\ \end{gathered} $$

(12)

3 Proposed Dombi Heronian Mean Operators for BNSs

This section proposes bipolar neutrosophic Dombi-based generalized weighted Heronian mean and bipolar neutrosophic Dombi-based improved generalized weighted Heronian mean operators. Several properties related to both operators are presented, and the proving of some of these properties is elucidated.

3.1 BND-GWHM Operator

Definition 8.

Let $p, q \ge 0$ and $A_{i} = \left\langle {t_{i}^{ + } , i_{i}^{ + } , f_{i}^{ + } , t_{i}^{ - } , i_{i}^{ - } , f_{i}^{ - } } \right\rangle (i = 1, 2, ..., n)$ be a collection of BNSs with weight $w_{i} = (w_{1} , w_{2} , ..., w_{n} )^{T},$ such that $w_{i} \in \left[ {0, 1} \right]$ and $\sum_{i = 1}^{n} w_{i} = 1.$ Then, the bipolar neutrosophic Dombi-based generalized weighted Heronian mean (BND-GWHM) is defined as follows:

$$ BND - GWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) = \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\left( {w_{i} A_{i} } \right)^{p} \otimes \left( {w_{j} A_{j} } \right)^{q} } \right)} \right)^{{\frac{1}{p + q}}} $$

(13)

Theorem 1.

Let $p, q, \gamma \ge 0$ and $A_{i} = \left\langle {t_{i}^{ + } , i_{i}^{ + } , f_{i}^{ + } , t_{i}^{ - } , i_{i}^{ - } , f_{i}^{ - } } \right\rangle (i = 1, 2, ..., n)$ be a collection of BNSs. The fused value produced by the BND-GWHM operators is also a BNS, and let

$$ G_{i} = \frac{{t_{i}^{ + } }}{{1 - t_{i}^{ + } }}, H_{i} = \frac{{1 - i_{i}^{ + } }}{{i_{i}^{ + } }}, I_{i} = \frac{{1 - f_{i}^{ + } }}{{f_{i}^{ + } }}, J_{i} = \frac{{1 + t_{i}^{ - } }}{{ - t_{i}^{ - } }}, K_{i} = \frac{{ - i_{i}^{ - } }}{{1 + i_{i}^{ - } }}, L_{i} = \frac{{ - f_{i}^{ - } }}{{1 + f_{i}^{ - } }}, $$

and

$$ G_{j} = \frac{{t_{j}^{ + } }}{{1 - t_{j}^{ + } }}, H_{j} = \frac{{1 - i_{j}^{ + } }}{{i_{j}^{ + } }}, I_{j} = \frac{{1 - f_{j}^{ + } }}{{f_{j}^{ + } }}, J_{j} = \frac{{1 + t_{j}^{ - } }}{{ - t_{j}^{ - } }}, K_{j} = \frac{{ - i_{j}^{ - } }}{{1 + i_{j}^{ - } }}, L_{j} = \frac{{ - f_{j}^{ - } }}{{1 + f_{j}^{ - } }}. $$

Then, Eq. (13) also yields the following value.

$$ BND - GWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) $$

$$ = \left\langle {\left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} G_{i}^{\gamma } } \right) + q/\left( {w_{j} G_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right.1 - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} H_{i}^{\gamma } } \right) + q/\left( {w_{j} H_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad 1 - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} I_{i}^{\gamma } } \right) + q/\left( {w_{j} I_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} J_{i}^{\gamma } } \right) + q/\left( {w_{j} J_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, $$

$$ \quad - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} K_{i}^{\gamma } } \right) + q/\left( {w_{j} K_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left. { - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} L_{i}^{\gamma } } \right) + q/\left( {w_{j} L_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} } \right\rangle . $$

(14)

Proof

Using Eq. (11), we obtain.

$$ w_{i} A_{i} = \left\langle {1 - \frac{1}{{1 + \left( {w_{i} \left( {G_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {w_{i} \left( {H_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {w_{i} \left( {I_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, } \right.\frac{ - 1}{{1 + \left( {w_{i} \left( {J_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {w_{i} \left( {K_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }} - 1, \left. {\frac{1}{{1 + \left( {w_{i} \left( {L_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }} - 1} \right\rangle $$

and

$$ w_{j} A_{j} = \left\langle {1 - \frac{1}{{1 + \left( {w_{j} \left( {G_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {w_{j} \left( {H_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {w_{j} \left( {I_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, } \right.\frac{ - 1}{{1 + \left( {w_{j} \left( {J_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {w_{j} \left( {K_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }} - 1, \left. {\frac{1}{{1 + \left( {w_{j} \left( {L_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }} - 1} \right\rangle . $$

Then, using Eq. (12), we get

$$ \left( {w_{i} A_{i} } \right)^{p} $$

$$ = \left\langle {\frac{1}{{1 + \left( {p/\left( {w_{i} G_{i}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {p/\left( {w_{i} H_{i}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {p/\left( {w_{i} I_{i}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, } \right.\frac{1}{{1 + \left( {p/\left( {w_{i} J_{i}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }} - 1, \left. {\frac{ - 1}{{1 + \left( {p/\left( {w_{i} K_{i}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, \frac{ - 1}{{1 + \left( {p/\left( {w_{i} L_{i}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}} \right\rangle $$

and

$$ \left( {w_{j} A_{j} } \right)^{q} $$

$$ = \left\langle {\frac{1}{{1 + \left( {q/\left( {w_{j} G_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {q/\left( {w_{j} H_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {q/\left( {w_{j} I_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, } \right.\frac{1}{{1 + \left( {q/\left( {w_{j} J_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }} - 1, \left. {\frac{ - 1}{{1 + \left( {q/\left( {w_{j} K_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, \frac{ - 1}{{1 + \left( {q/\left( {w_{j} L_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}} \right\rangle . $$

Thereafter, using Eq. (10) we get

$$ \left( {w_{i} A_{i} } \right)^{p} \otimes \left( {w_{j} A_{j} } \right)^{q} $$

$$ = \left\langle {\frac{1}{{1 + \left( {p/\left( {w_{i} G_{i}^{\gamma } } \right) + q/\left( {w_{j} G_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {p/\left( {w_{i} H_{i}^{\gamma } } \right) + q/\left( {w_{j} H_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, } \right.1 - \frac{1}{{1 + \left( {p/\left( {w_{i} I_{i}^{\gamma } } \right) + q/\left( {w_{j} I_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, $$

$$ \quad \frac{1}{{1 + \left( {p/\left( {w_{i} J_{i}^{\gamma } } \right) + q/\left( {w_{j} J_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }} - 1, \left. {\frac{ - 1}{{1 + \left( {p/\left( {w_{i} K_{i}^{\gamma } } \right) + q/\left( {w_{j} K_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}, \frac{ - 1}{{1 + \left( {p/\left( {w_{i} L_{i}^{\gamma } } \right) + q/\left( {w_{j} L_{j}^{\gamma } } \right)} \right)^{{\frac{1}{\gamma }}} }}} \right\rangle . $$

By employing Eq. (9),

$$ \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\left( {w_{i} A_{i} } \right)^{p} \otimes \left( {w_{j} A_{j} } \right)^{q} } \right) $$

$$ = \left\langle {1 - \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} G_{i}^{\gamma } } \right) + q/\left( {w_{j} G_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} H_{i}^{\gamma } } \right) + q/\left( {w_{j} H_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right. $$

$$ \quad \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} I_{i}^{\gamma } } \right) + q/\left( {w_{j} I_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , - \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} J_{i}^{\gamma } } \right) + q/\left( {w_{j} J_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad \left. {\left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} K_{i}^{\gamma } } \right) + q/\left( {w_{j} K_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} L_{i}^{\gamma } } \right) + q/\left( {w_{j} L_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1} \right\rangle . $$

Using Eq. (11),

$$ \frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\left( {w_{i} A_{i} } \right)^{p} \otimes \left( {w_{j} A_{j} } \right)^{q} } \right) $$

$$ = \left\langle {1 - \left( {1 + \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} G_{i}^{\gamma } } \right) + q/\left( {w_{j} G_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right.\;\left( {1 + \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} H_{i}^{\gamma } } \right) + q/\left( {w_{j} H_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad \left( {1 + \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} I_{i}^{\gamma } } \right) + q/\left( {w_{j} I_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \; - \left( {1 + \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} J_{i}^{\gamma } } \right) + q/\left( {w_{j} J_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad \left( {1 + \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} K_{i}^{\gamma } } \right) + q/\left( {w_{j} K_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, \;\left. {\left( {1 + \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} L_{i}^{\gamma } } \right) + q/\left( {w_{j} L_{j}^{\gamma } } \right)}}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1} \right\rangle . $$

Therefore, using Eq. (12) we get

$$ \left( {\frac{2}{{n\left( {n + 1} \right)}}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\left( {w_{i} A_{i} } \right)^{p} \otimes \left( {w_{j} A_{j} } \right)^{q} } \right)} \right)^{{\frac{1}{p + q}}} $$

$$ = \left\langle {\left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} G_{i}^{\gamma } } \right) + q/\left( {w_{j} G_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right.\;1 - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} H_{i}^{\gamma } } \right) + q/\left( {w_{j} H_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad 1 - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} I_{i}^{\gamma } } \right) + q/\left( {w_{j} I_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \;\left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} J_{i}^{\gamma } } \right) + q/\left( {w_{j} J_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, $$

$$ \quad - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} K_{i}^{\gamma } } \right) + q/\left( {w_{j} K_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \;\left. { - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} L_{i}^{\gamma } } \right) + q/\left( {w_{j} L_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} } \right\rangle . $$

Thus, Eq. (14) is proven.

The following example illustrates the application of the proposed operator BND-GWHM.

Example 1.

Let $A_{{_{1} }} = \left( {0.4, 0.5, 0.3, - 0.6, - 0.4, - 0.5} \right), \;A_{2} = \left( {0.6, 0.1, 0.2, - 0.4, - 0.3, - 0.2} \right),$ and $A_{3} = \left( {0.8, 0.6, 0.5, - 0.3, - 0.2, - 0.1} \right)$ be a collection of BNSs, and $p = 2, q = 1, \gamma = 3,$ $w = \left( {0.2, 0.5, 0.3} \right).$ Then, we employ the BND-GWHM operator to fuse three BNSs.

First, we calculated the values $BND - GWHM^{p, q} \left( {A_{1} , A_{2} , A_{3} } \right)$ using Eq. (14).

The equation for $t^{ + }$ is stated as

$$ t^{ + } = \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{1}{{p/\left( {w_{i} G_{i}^{\gamma } } \right) + q/\left( {w_{j} G_{j}^{\gamma } } \right)}}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} $$

$$ \quad = \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right) \times \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} G_{i}^{\gamma } } \right) + q/\left( {w_{j} G_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} $$

$$ \quad = \left( {1 + \left( {\frac{{3\left( {3 + 1} \right)}}{{2\left( {2 + 1} \right) \times \mathop \sum \limits_{i = 1}^{3} \mathop \sum \limits_{j = i}^{3} \left( {2/\left( {w_{i} G_{i}^{3} } \right) + 1/\left( {w_{j} G_{j}^{3} } \right)} \right)^{ - 1} }}} \right)^{\frac{1}{3}} } \right)^{ - 1} . $$

Note that:$\mathop \sum \nolimits_{i = 1}^{3} \mathop \sum \nolimits_{j = i}^{3} \left( {2/\left( {w_{i} G_{i}^{3} } \right) + 1/\left( {w_{j} G_{j}^{3} } \right)} \right)^{ - 1}.$

which

$$\begin{aligned} & = \left( {2/\left( {w_{1} G_{1}^{3} } \right) + 1/\left( {w_{1} G_{1}^{3} } \right)} \right)^{ - 1} + \left( {2/\left( {w_{1} G_{1}^{3} } \right) + 1/\left( {w_{2} G_{2}^{3} } \right)} \right)^{ - 1} + \left( {2/\left( {w_{1} G_{1}^{3} } \right) + 1/\left( {w_{3} G_{3}^{3} } \right)} \right)^{ - 1} \\ & \quad + \left( {2/\left( {w_{2} G_{2}^{3} } \right) + 1/\left({w_{2} G_{2}^{3} } \right)} \right)^{ - 1} + \left( {2/\left( {w_{2} G_{2}^{3} } \right) + 1/\left( {w_{3} G_{3}^{3} } \right)} \right)^{ - 1} + \left( {2/\left( {w_{3} G_{3}^{3} } \right) + 1/\left( {w_{3} G_{3}^{3} } \right)} \right)^{ - 1}\\ & = \left( {2/\left( {0.2\left( {\frac{0.4}{{1 - 0.4}}} \right)^{3} } \right) + 1/\left( {0.2\left( {\frac{0.4}{{1 - 0.4}}} \right)^{3} } \right)} \right)^{ - 1} + \left( {2/\left( {0.2\left( {\frac{0.4}{{1 - 0.4}}} \right)^{3} } \right) + 1/\left( {0.5\left( {\frac{0.6}{{1 - 0.6}}} \right)^{3} } \right)} \right)^{ - 1} + \left( {2/\left( {0.2\left( {\frac{0.4}{{1 - 0.4}}} \right)^{3} } \right) + 1/\left( {0.3\left( {\frac{0.8}{{1 - 0.8}}} \right)^{3} } \right)} \right)^{ - 1}\\ & \quad + \left( {2/\left( {0.5\left( {\frac{0.6}{{1 - 0.6}}} \right)^{3} } \right) + 1/\left( {0.5\left( {\frac{0.6}{{1 - 0.6}}} \right)^{3} } \right)} \right)^{ - 1} + \left( {2/\left( {0.5\left( {\frac{0.6}{{1 - 0.6}}} \right)^{3} } \right) + 1/\left( {0.3\left( {\frac{0.8}{{1 - 0.8}}} \right)^{3} } \right)} \right)^{ - 1} + \left( {2/\left( {0.3\left( {\frac{0.8}{{1 - 0.8}}} \right)^{3} } \right) + 1/\left( {0.3\left( {\frac{0.8}{{1 - 0.8}}} \right)^{3} } \right)} \right)^{ - 1} . \end{aligned}$$

$$ \quad = 7.8491 $$

Hence,

$$ t^{ + } = \left( {1 + \left( {\frac{{3\left( {3 + 1} \right)}}{{2\left( {2 + 1} \right) \times 7.8491}}} \right)^{\frac{1}{3}} } \right)^{ - 1} $$

$$ \quad = 0.6120. $$

In the same way, we also acquired the values for $i^{ + },$ $f^{ + },$ $t^{ - },$ $i^{ - },$ and $f^{ - }.$

Finally, $BND - GWHM^{2, 1} (A_{1} , A_{2} , A_{3} ) = (0.6120, 0.2026, 0.3468, - 0.4781, - 0.2362, - 0.2528).$

Theorem 2.

(Idempotency property) Let $A_{i} = \left\langle {t_{i}^{ + } , i_{i}^{ + } , f_{i}^{ + } , t_{i}^{ - } , i_{i}^{ - } , f_{i}^{ - } } \right\rangle (i = 1, 2, ..., n) = A = \left\langle {t_{{}}^{ + } , i_{{}}^{ + } , f_{{}}^{ + } , t_{{}}^{ - } , i_{{}}^{ - } , f_{{}}^{ - } } \right\rangle$ be a collection of BNSs, so $G = G_{i} = G_{j} = \frac{{t_{{}}^{ + } }}{{1 - t_{{}}^{ + } }},$ then $i = j.$

Proof

Suppose $BND - GWHM_{{}}^{p, q} (A_{1} , A_{2} , ..., A_{n} ) = \left\langle {t_{\alpha }^{ + } , i_{\alpha }^{ + } , f_{\alpha }^{ + } , t_{\alpha }^{ - } , i_{\alpha }^{ - } , f_{\alpha }^{ - } } \right\rangle,$ we can obtain that

$$ t_{\alpha }^{ + } = \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {p/\left( {w_{i} G_{i}^{\gamma } } \right) + q/\left( {w_{j} G_{j}^{\gamma } } \right)} \right)^{ - 1} }}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} $$

$$ \quad = \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{j} w_{j} G^{\gamma } }}{{w_{j} p + w_{j} q}}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} $$

$$ \quad = \frac{1}{{1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}}} \right)\left( {\frac{{w_{j} G^{\gamma } }}{p + q}} \right)\left( {\frac{p + q}{{w_{j} }}} \right)}}} \right)^{{\frac{1}{\gamma }}} }} $$

$$ \begin{gathered} \quad = \frac{1}{{1 + \left( {G^{\gamma } } \right)^{{\frac{1}{\gamma }}} }} \hfill \\ \quad = \frac{1}{1 + G}. \hfill \\ \end{gathered} $$

So, $BND - GWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) = A.$ Theorem 2 is proven.

Theorem 3.

(Monotonicity property) Let $A_{i} = \left\langle {t_{{}}^{ + } , i_{{}}^{ + } , f_{{}}^{ + } , t_{{}}^{ - } , i_{{}}^{ - } , f_{{}}^{ - } } \right\rangle$$(i = 1, 2, ..., n)$ and $A_{i} * = \left\langle {t_{{}}^{ + * } , i_{{}}^{ + * } , f_{{}}^{ + * } , t_{{}}^{ - * } , i_{{}}^{ - * } , f_{{}}^{ - * } } \right\rangle (i = 1, 2, ..., n)$ be two collections of BNSs. If $t^{ + } \le t^{ + * } , i^{ + } \ge i^{ + * } , f^{ + } \ge f^{ + * } \quad {\text{and}} \quad t^{ - } \ge t^{ - * } , i^{ - } \le i^{ - * } , f^{ - } \le f^{ - * },$ then

$$ BND - GWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) \le BND - GWHM^{p, q} \left( {A_{1}^{ * } , A_{2}^{ * } , ..., A_{n}^{ * } } \right) $$

Proof

For $t^{ + } \le t^{ + * } , i^{ + } \ge i^{ + * } , f^{ + } \ge f^{ + * } \quad {\text{and}}\quad t^{ - } \ge t^{ - * } , i^{ - } \le i^{ - * } , f^{ - } \le f^{ - * },$ it is obvious that

$$\left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{t_{i}^{ + } }}{{1 - t_{i}^{ + } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{t_{j}^{ + } }}{{1 - t_{j}^{ + } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} \le \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{t_{i}^{ + * } }}{{1 - t_{i}^{ + * } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{t_{j}^{ + * } }}{{1 - t_{j}^{ + * } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

Similarly,

$$ 1 - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{1 - i_{i}^{ + } }}{{i_{i}^{ + } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{1 - i_{j}^{ + } }}{{i_{j}^{ + } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} \ge 1 - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{1 - i_{i}^{ + * } }}{{i_{i}^{ + * } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{1 - i_{j}^{ + * } }}{{i_{j}^{ + * } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ 1 - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{1 - f_{i}^{ + } }}{{f_{i}^{ + } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{1 - f_{j}^{ + } }}{{f_{j}^{ + } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} \ge 1 - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{1 - f_{i}^{ + * } }}{{f_{i}^{ + * } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{1 - f_{j}^{ + * } }}{{f_{j}^{ + * } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{1 + t_{i}^{ - } }}{{ - t_{i}^{ - } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{1 + t_{j}^{ - } }}{{ - t_{j}^{ - } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1 \ge \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{1 + t_{i}^{ - * } }}{{ - t_{i}^{ - * } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{1 + t_{j}^{ - * } }}{{ - t_{j}^{ - * } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, $$

$$ - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{ - i_{i}^{ - } }}{{1 + i_{i}^{ - } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{ - i_{j}^{ - } }}{{1 + i_{j}^{ - } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} \le - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{ - i_{i}^{ - * } }}{{1 + i_{i}^{ - * } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{ - i_{j}^{ - * } }}{{1 + i_{j}^{ - * } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

and

$$- \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{ - f_{i}^{ - } }}{{1 + f_{i}^{ - } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{ - f_{j}^{ - } }}{{1 + f_{j}^{ - } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} \le - \left( {1 + \left( {\frac{{n\left( {n + 1} \right)}}{{2\left( {p + q} \right)}} \times \frac{1}{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} \left( {1/\left( {p/\left( {w_{i} \left( {\frac{{ - f_{i}^{ - * } }}{{1 + f_{i}^{ - * } }}} \right)^{\gamma } } \right) + q/\left( {w_{j} \left( {\frac{{ - f_{j}^{ - * } }}{{1 + f_{j}^{ - * } }}} \right)^{\gamma } } \right)} \right)} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} .$$

This proven Theorem 3.

Theorem 4.

(Boundedness property) Let $A_{i} = \left\langle {t_{i}^{ + } , i_{i}^{ + } , f_{i}^{ + } , t_{i}^{ - } , i_{i}^{ - } , f_{i}^{ - } } \right\rangle \, \left( {i = 1, 2, 3, ..., n} \right)$ be a collection of BNSs and if

$$ \begin{gathered} A^{ - } = \left\langle {min\left( {t_{{_{{}} }}^{ + } } \right), max \left( {i_{{}}^{ + } } \right), max \left( {f_{{}}^{ + } } \right), max \left( {t_{{}}^{ - } } \right), min \left( {i_{{}}^{ - } } \right), min\left( {f_{{}}^{ - } } \right)} \right\rangle \hfill \\ A^{ + } = \left\langle {max\left( {t_{{}}^{ + * } } \right), min \left( {i_{{}}^{ + * } } \right), min \left( {f_{{}}^{ + * } } \right), min \left( {t_{{}}^{ - * } } \right), max \left( {i_{{}}^{ - * } } \right), max\left( {f_{{}}^{ - * } } \right)} \right\rangle \hfill \\ \end{gathered} $$

where $\gamma > 0.$

Proof

Based on Theorems 2 and 3, the following can be obtained:

$$ A^{ + } = BND - GWHM^{p, q} (A^{ + } , A^{ + } , ..., A^{ + } ), \quad A^{ - } = BND - GWHM^{p, q} (A^{ - } , A^{ - } , ..., A^{ - } ). $$

As such, $BND - GWHM^{p, q} (A^{ - } , A^{ - } , ..., A^{ - } ) \le BND - GWHM^{p, q} (A_{1} , A_{2} , ..., A_{n} ) \le BND - GWHM^{p, q}$$(A^{ + } , A^{ + } , ..., A^{ + } ).$

Hence, $A^{ - } \le BND - GWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) \le A^{ + }.$

This proves Theorem 4.

In the following, the proposed BND-IGWHM operator and its related properties is discussed in Sect. 3.2.

3.2 BND-IGWHM Operator

Definition 9.

Let $p, q \ge 0$ and $A_{i} = \left\langle {t_{i}^{ + } , i_{i}^{ + } , f_{i}^{ + } , t_{i}^{ - } , i_{i}^{ - } , f_{i}^{ - } } \right\rangle (i = 1, 2, ..., n)$ be a collection of BNSs with weight $w_{i} = (w_{1} , w_{2} , ..., w_{n} )^{T},$ and satisfying $w_{i} \in \left[ {0, 1} \right]$ and $\sum_{i = 1}^{n} w_{i} = 1.$ Then, the bipolar neutrosophic Dombi-based improved generalized weighted Heronian mean (BND-IGWHM) is defined as follows:

$$ BND - IGWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) = \frac{{\left( {\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} w_{i} w_{j} A_{i}^{p} \otimes A_{j}^{q} } \right)^{{\frac{1}{p + q}}} }}{{\left( {\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = i}^{n} w_{i} w_{j} } \right)^{{\frac{1}{p + q}}} }}. $$

(15)

Theorem 5.

Let $p, q \ge 0$ and $A_{i} = \left\langle {t_{i}^{ + } , i_{i}^{ + } , f_{i}^{ + } , t_{i}^{ - } , i_{i}^{ - } , f_{i}^{ - } } \right\rangle (i = 1, 2, ..., n)$ be a collection of BNSs. The fused value produced by the BND-IGWHM operators is also a BNS, and let

$$ A_{{_{i} }} = \frac{{1 - t_{i}^{ + } }}{{t_{i}^{ + } }}, B_{i} = \frac{{i_{i}^{ + } }}{{1 - i_{i}^{ + } }}, C_{i} = \frac{{f_{i}^{ + } }}{{1 - f_{i}^{ + } }}, D_{i} = \frac{{ - t_{i}^{ - } }}{{1 + t_{i}^{ - } }}, E_{i} = \frac{{1 + i_{i}^{ - } }}{{ - i_{i}^{ - } }}, F_{i} = \frac{{1 + f_{i}^{ - } }}{{ - f_{i}^{ - } }}, $$

and

$$ A_{{_{j} }} = \frac{{1 - t_{j}^{ + } }}{{t_{j}^{ + } }}, B_{j} = \frac{{i_{j}^{ + } }}{{1 - i_{j}^{ + } }}, C_{j} = \frac{{f_{j}^{ + } }}{{1 - f_{j}^{ + } }}, D_{j} = \frac{{ - t_{j}^{ - } }}{{1 + t_{j}^{ - } }}, E_{j} = \frac{{1 + i_{j}^{ - } }}{{ - i_{j}^{ - } }}, F_{j} = \frac{{1 + f_{j}^{ - } }}{{ - f_{j}^{ - } }}. $$

Then, the value derived from Eq. (15) is represented as:

$$\begin{aligned} BND - IGWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) \\= \left\langle {\left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {w_{i} w_{j} /\left( {pA_{i}^{\gamma } } \right) + \left( {qA_{j}^{\gamma } } \right)} \right)} \right)^{ - 1} } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right. \end{aligned} $$

$$ \quad 1 - \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {w_{i} w_{j} /\left( {\left( {pB_{i}^{\gamma } } \right) + \left( {qB_{j}^{\gamma } } \right)} \right)} \right)} \right)^{ - 1} } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad 1 - \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {w_{i} w_{j} /\left( {\left( {pC_{i}^{\gamma } } \right) + \left( {qC_{j}^{\gamma } } \right)} \right)} \right)} \right)^{ - 1} } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {w_{i} w_{j} /\left( {\left( {pD_{i}^{\gamma } } \right) + \left( {qD_{j}^{\gamma } } \right)} \right)} \right)} \right)^{ - 1} } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, $$

$$ \quad - \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {w_{i} w_{j} /\left( {\left( {pE_{i}^{\gamma } } \right) + \left( {qE_{j}^{\gamma } } \right)} \right)} \right)} \right)^{ - 1} } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \left. {\quad - \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {w_{i} w_{j} /\left( {\left( {pF_{i}^{\gamma } } \right) + \left( {qF_{j}^{\gamma } } \right)} \right)} \right)} \right)^{ - 1} } \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} } \right\rangle . $$

(16)

Proof

$$ A_{{_{i} }}^{p} = \left\langle {\frac{1}{{1 + \left( {p\left( {A_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {p\left( {B_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {p\left( {C_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}} \right., \frac{1}{{1 + \left( {p\left( {D_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }} - 1, \frac{ - 1}{{1 + \left( {p\left( {E_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \left. {\frac{ - 1}{{1 + \left( {p\left( {F_{i} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}} \right\rangle $$

and

$$ A_{{_{j} }}^{q} = \left\langle {\frac{1}{{1 + \left( {q\left( {A_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {q\left( {B_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {q\left( {C_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}} \right., \frac{1}{{1 + \left( {q\left( {D_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }} - 1, \frac{ - 1}{{1 + \left( {q\left( {E_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \left. {\frac{ - 1}{{1 + \left( {q\left( {F_{j} } \right)^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}} \right\rangle . $$

Using Eq. (10), we get

$$ A_{{_{i} }}^{p} \otimes A_{{_{j} }}^{q} $$

$$ = \left\langle {\frac{1}{{1 + \left( {pA_{i}^{\gamma } + qA_{j}^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {pB_{i}^{\gamma } + qB_{j}^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, 1 - \frac{1}{{1 + \left( {pC_{i}^{\gamma } + qC_{j}^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, } \right.\frac{1}{{1 + \left( {pD_{i}^{\gamma } + qD_{j}^{\gamma } } \right)^{{\frac{1}{\gamma }}} }} - 1, \left. {\frac{ - 1}{{1 + \left( {pE_{i}^{\gamma } + qE_{j}^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}, \frac{ - 1}{{1 + \left( {pF_{i}^{\gamma } + qF_{j}^{\gamma } } \right)^{{\frac{1}{\gamma }}} }}} \right\rangle . $$

Next, using Eq. (11) we get

$$ w_{i} w_{j} A_{i}^{p} A_{j}^{q} $$

$$ = \left\langle {1 - \frac{1}{{1 + \left( {\frac{{w_{i} w_{j} }}{{pA_{i}^{\gamma } + qA_{j}^{\gamma } }}} \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {\frac{{w_{i} w_{j} }}{{pB_{i}^{\gamma } + qB_{j}^{\gamma } }}} \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {\frac{{w_{i} w_{j} }}{{pC_{i}^{\gamma } + qC_{j}^{\gamma } }}} \right)^{{\frac{1}{\gamma }}} }}, } \right.\left. {\frac{ - 1}{{1 + \left( {\frac{{w_{i} w_{j} }}{{pD_{i}^{\gamma } + qD_{j}^{\gamma } }}} \right)^{{\frac{1}{\gamma }}} }}, \frac{1}{{1 + \left( {\frac{{w_{i} w_{j} }}{{pE_{i}^{\gamma } + qE_{j}^{\gamma } }}} \right)^{{\frac{1}{\gamma }}} }} - 1, \frac{1}{{1 + \left( {\frac{{w_{i} w_{j} }}{{pF_{i}^{\gamma } + qF_{j}^{\gamma } }}} \right)^{{\frac{1}{\gamma }}} }} - 1} \right\rangle . $$

Afterwards, using Eq. (9) we get

$$ \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} A_{i}^{p} A_{j}^{q} $$

$$ = \left\langle {1 - \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pA_{i}^{\gamma } + qA_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pB_{i}^{\gamma } + qB_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right.\left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pC_{i}^{\gamma } + qC_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad - \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pD_{i}^{\gamma } + qD_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left. {\left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pE_{i}^{\gamma } + qE_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, \left( {1 + \left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pF_{i}^{\gamma } + qF_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1} \right\rangle . $$

Thereafter, based on Eq. (11),

$$ \frac{1}{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} } \right)}}\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} A_{i}^{p} A_{j}^{q} } \right) $$

$$ = \left\langle {1 - \left( {1 + \left( {\frac{1}{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} } \right)}} \times \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pA_{i}^{\gamma } + qA_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right.\left( {1 + \left( {\frac{1}{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} } \right)}} \times \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pB_{i}^{\gamma } + qB_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad \left( {1 + \left( {\frac{1}{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} } \right)}} \times \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pC_{i}^{\gamma } + qC_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , - \left( {1 + \left( {\frac{1}{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} } \right)}} \times \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pD_{i}^{\gamma } + qD_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad \left( {1 + \left( {\frac{1}{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} } \right)}} \times \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pE_{i}^{\gamma } + qE_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, \left. {\left( {1 + \left( {\frac{1}{{\left( {\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} } \right)}} \times \mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pF_{i}^{\gamma } + qF_{j}^{\gamma } }}} \right)} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1} \right\rangle . $$

Therefore, based on Eq. (12) we get

$$ \left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} A_{i}^{p} A_{j}^{q} }}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}} \right)^{{\frac{1}{p + q}}} $$

$$ = \left\langle {\left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pA_{i}^{\gamma } + qA_{j}^{\gamma } }}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , } \right.1 - \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pB_{i}^{\gamma } + qB_{j}^{\gamma } }}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , $$

$$ \quad 1 - \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pC_{i}^{\gamma } + qC_{j}^{\gamma } }}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pD_{i}^{\gamma } + qD_{j}^{\gamma } }}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} - 1, $$

$$ \quad - \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pE_{i}^{\gamma } + qE_{j}^{\gamma } }}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} , \left. { - \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pF_{i}^{\gamma } + qF_{j}^{\gamma } }}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} } \right\rangle . $$

Thus, Eq. (16) is proven.

Example 2 illustrates the implementation of the operator BND-IGWHM on BNSs.

Example 2.

Assume $A_{{_{1} }} = \left( {0.4, 0.5, 0.3, - 0.6, - 0.4, - 0.5} \right),$ $A_{2} = \left( {0.6, 0.1, 0.2, - 0.4, - 0.3, - 0.2} \right)$ and $A_{3} = \left( {0.8, 0.6, 0.5, - 0.3, - 0.2, - 0.1} \right)$ be collection of BNSs, and $p = 2,$ $q = 1, \, \, \gamma = 3,$$w = \left( {w_{1} , w_{2} , w_{3} } \right)$ $= \left( {0.2, 0.5, 0.3} \right).$ Then we employ the BND-IGWHM operator to fuse three BNSs.

First, we used Eq. (16) to determined the valued of $BND - IGWHM^{p, q} \left( {A_{1} , A_{2} , A_{3} } \right)$$= \left( {t^{ + } , i^{ + } , f^{ + } , t^{ - } , i^{ - } , f^{ - } } \right).$

For $t^{ + },$ the equation is defined as

$$ t^{ + } = \left( {1 + \left( {\left( {\frac{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} w_{i} w_{j} }}{p + q}} \right) \times \frac{1}{{\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{j = i}^{n} \left( {\frac{{w_{i} w_{j} }}{{pA_{i}^{\gamma } + qA_{j}^{\gamma } }}} \right)}}} \right)^{{\frac{1}{\gamma }}} } \right)^{ - 1} $$

$$ \quad = \left( {1 + \left( {\frac{{\mathop \sum \limits_{i = 1}^{3} \mathop \sum \limits_{j = i}^{3} w_{i} w_{j} }}{{\left( {2 + 1} \right) \times \mathop \sum \limits_{i = 1}^{3} \mathop \sum \limits_{j = i}^{3} \left( {\frac{{w_{i} w_{j} }}{{(2)A_{i}^{3} + (1)A_{j}^{3} }}} \right)}}} \right)^{\frac{1}{3}} } \right)^{ - 1} $$

$$ \quad = \left( {1 + \left( {\frac{{w_{1} w_{1} + w_{1} w_{2} + w_{1} w_{3} + w_{2} w_{2} + w_{2} w_{3} + w_{3} w_{3} }}{{\left( {2 + 1} \right) \times \left( {\frac{{w_{1} w_{1} }}{{(2)A_{1}^{3} + (1)A_{1}^{3} }} + \frac{{w_{1} w_{2} }}{{(2)A_{1}^{3} + (1)A_{2}^{3} }} + \frac{{w_{1} w_{3} }}{{(2)A_{1}^{3} + (1)A_{3}^{3} }} + \frac{{w_{2} w_{2} }}{{(2)A_{2}^{3} + (1)A_{2}^{3} }} + \frac{{w_{2} w_{3} }}{{(2)A_{2}^{3} + (1)A_{3}^{3} }} + \frac{{w_{3} w_{3} }}{{(2)A_{3}^{3} + (1)A_{3}^{3} }}} \right)}}} \right)^{\frac{1}{3}} } \right)^{ - 1} $$

$$ \quad = \left( {1 + \left( {\frac{{\left( {0.2 \times 0.2} \right) + \left( {0.2 \times 0.5} \right) + \left( {0.2 \times 0.3} \right) + \left( {0.5 \times 0.5} \right) + \left( {0.5 \times 0.3} \right) + \left( {0.3 \times 0.3} \right)}}{{\left( {2 + 1} \right) \times \left( {\frac{{\left( {0.2 \times 0.2} \right)}}{{(2)A_{1}^{3} + (1)A_{1}^{3} }} + \frac{{\left( {0.2 \times 0.5} \right)}}{{(2)A_{1}^{3} + (1)A_{2}^{3} }} + \frac{{\left( {0.2 \times 0.3} \right)}}{{(2)A_{1}^{3} + (1)A_{3}^{3} }} + \frac{{\left( {0.5 \times 0.5} \right)}}{{(2)A_{2}^{3} + (1)A_{2}^{3} }} + \frac{{\left( {0.5 \times 0.3} \right)}}{{(2)A_{2}^{3} + (1)A_{3}^{3} }} + \frac{{\left( {0.3 \times 0.3} \right)}}{{(2)A_{3}^{3} + (1)A_{3}^{3} }}} \right)}}} \right)^{\frac{1}{3}} } \right)^{ - 1} . $$

Afterwards, substitute the value of $A_{1},$ $A_{2},$ and $A_{3}$ which written as:

$$ \quad A_{1} = \frac{1 - 0.4}{{0.4}}, \quad A_{2} = \frac{1 - 0.6}{{0.6}}, \quad A_{3} = \frac{1 - 0.8}{{0.8}}. $$

Then,

$$ t^{ + } = \left( {1 + \left( {\frac{0.69}{{\left( {2 + 1} \right) \times \left( {0.0040 + 0.0142 + 0.0089 + 0.2813 + 0.2466 + 1.9200} \right)}}} \right)^{\frac{1}{3}} } \right)^{ - 1} . $$

$$ \quad = 0.6883 $$

Also, we obtained the values for $i^{ + },$ $f^{ + },$ $t^{ - },$ $i^{ - },$ and $f^{ - }$ in the same manner.

Therefore, $BND - IGWHM^{2, 1} (A_{1} , A_{2} , A_{3} ) = \left( {0.6883, 0.1348, 0.2490, } \right. - 0.3871, - 0.3021, \left. { - 0.2933} \right).$

Theorem 6.

(Idempotency property). Let $A_{i} = \left\langle {t_{i}^{ + } , i_{i}^{ + } , f_{i}^{ + } , t_{i}^{ - } , i_{i}^{ - } , f_{i}^{ - } } \right\rangle (i = 1, 2, ..., n)$ be a collection of BNSs, if all BNSs are equal, that is $A_{i} = A = \left\langle {t_{{}}^{ + } , i_{{}}^{ + } , f_{{}}^{ + } , t_{{}}^{ - } , i_{{}}^{ - } , f_{{}}^{ - } } \right\rangle,$ then

$$ BND - IGWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) = A. $$

The proof of Theorem 6 is similar to Theorem 2.

Theorem 7.

(Monotonicity property). Let $A_{i} = \left\langle {t_{{}}^{ + } , i_{{}}^{ + } , f_{{}}^{ + } , t_{{}}^{ - } , i_{{}}^{ - } , f_{{}}^{ - } } \right\rangle$$(i = 1, 2, ..., n)$ and $A_{i} * = \left\langle {t_{{}}^{ + * } , i_{{}}^{ + * } , f_{{}}^{ + * } , t_{{}}^{ - * } , i_{{}}^{ - * } , f_{{}}^{ - * } } \right\rangle$$(i = 1, 2, ..., n)$ be two collections of BNSs. If $t^{ + } \le t^{ + * } , i^{ + } \ge i^{ + * } , f^{ + } \ge f^{ + * }$ and $t^{ - } \ge t^{ - * } , i^{ - } \le i^{ - * } , f^{ - } \le f^{ - * }$ then

$BND - IGWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) \le BND - IGWHM^{p, q} \left( {A_{1}^{ * } , A_{2}^{ * } , ..., A_{n}^{ * } } \right).$

This theorem is proved similarly to Theorem 3.

Theorem 8.

(Boundedness property). Let $A_{i} = \left\langle {t_{i}^{ + } , i_{i}^{ + } , f_{i}^{ + } , t_{i}^{ - } , i_{i}^{ - } , f_{i}^{ - } } \right\rangle \, \left( {i = 1, 2, 3, ..., n} \right)$ where $\gamma > 0,$ be a collection of BNSs and

$$ \begin{gathered} A^{ - } = \left\langle {min\left( {t_{{}}^{ + } } \right), max \left( {i_{{}}^{ + } } \right), max \left( {f_{{}}^{ + } } \right), max \left( {t_{{}}^{ - } } \right), min \left( {i_{{}}^{ - } } \right), min\left( {f_{{}}^{ - } } \right)} \right\rangle \hfill \\ A^{ + } = \left\langle {max\left( {t_{{}}^{ + * } } \right), min \left( {i_{{}}^{ + * } } \right), min \left( {f_{{}}^{ + * } } \right), min \left( {t_{{}}^{ - * } } \right), max \left( {i_{{}}^{ - * } } \right), max\left( {f_{{}}^{ - * } } \right)} \right\rangle \hfill \\ \end{gathered} $$

Then, $A^{ - } \le BND - IGWHM^{p, q} \left( {A_{1} , A_{2} , ..., A_{n} } \right) \le A^{ + }.$

Also, Theorem 8 has the same proof as Theorem 4.

Clearly, the BND-IGWHM operator holds these properties. The proof is similar to BND-GWHM. Hence, we omit them.

4 Proposed MCDM Procedure Based on BND-GWHM and BND-IGWHM Operators

The following is an example of MCDM problem with bipolar neutrosophic information. Let $A = \left\{ {\tilde{A}_{1} , \tilde{A}_{2} , ..., \tilde{A}_{m} } \right\}$ be a set of alternatives and $C = \left\{ {C_{1} , C_{2} , ..., C_{n} } \right\}$ be a set of criteria with the weight vector given as $w = (w_{1} , w_{2} , ..., w_{n} )^{T}$ satisfying $w_{i} \in \left[ {0, 1} \right]$ and $\mathop \sum \nolimits_{i}^{n} w_{i} = 1.$ Based on the BND-GWHM and BND-IGWHM operators, we develop an algorithm of the MCDM process. The algorithm of the MCDM process is shown in Fig. 1.

Fig. 1

Flowchart of the MCDM algorithm

Algorithm

Step 1. Construct a decision matrix in terms of BNSs. If BNS is denoted as $A_{ij} = \left\langle {t_{ij}^{ + } , i_{ij}^{ + } , f_{ij}^{ + } , t_{ij}^{ - } , i_{ij}^{ - } , f_{ij}^{ - } } \right\rangle,$ $i = 1, 2, ..., m,$ $j = 1, 2, ..., n$ then general form of the decision matrix, where every column refers to criteria and a row refers to alternatives, is as follows:

$$ D = \left[ {\begin{array}{*{20}c} {A_{12} } & {A_{12} } & \cdots & {A_{1m} } \\ {A_{21} } & {A_{22} } & \cdots & {A_{2m} } \\ \vdots & \vdots & \ddots & \vdots \\ {A_{n1} } & {A_{n2} } & \cdots & {A_{nm} } \\ \end{array} } \right]. $$

(17)

Step 2. Determine the aggregated values for each alternative $A_{i} (i = 1, 2, 3, 4)$ using the BND-GWHM and BND-IGWHM operators. Each alternative for every aggregation operator can be expressed in the following manner:

$$ \begin{array}{*{20}c} {} \\ {\tilde{A}_{1} } \\ {\tilde{A}_{2} } \\ \vdots \\ {\tilde{A}_{m} } \\ \end{array} \begin{array}{*{20}c} {C_{j} } \\ {\left[ {\begin{array}{*{20}c} {A_{1} } \\ {A_{2} } \\ \vdots \\ {A_{m} } \\ \end{array} } \right]} \\ \end{array} . $$

(18)

Step 3. Determine the score value of $s\left( {A_{i} } \right)$ for $A_{i} (i = 1, 2, ..., n)$ using Eq. (2) and restated as follows:

$$ s\left( {A_{1} } \right) = \frac{{t_{1}^{ + } + 1 - i_{1}^{ + } + 1 - f_{1}^{ + } + 1 - t_{1}^{ - } + i_{1}^{ - } + f_{1}^{ - } }}{6}\;; $$

Then, the score values for each alternative can be written as:

$$ \begin{array}{*{20}c} {} \\ {\tilde{A}_{1} } \\ {\tilde{A}_{2} } \\ \vdots \\ {\tilde{A}_{m} } \\ \end{array} \begin{array}{*{20}c} {C_{j} } \\ {\left[ {\begin{array}{*{20}c} {s\left( {A_{1} } \right)} \\ {s\left( {A_{2} } \right)} \\ \vdots \\ {s\left( {A_{m} } \right)} \\ \end{array} } \right]} \\ \end{array} . $$

(19)

Step 4. Rank all the alternatives that correspond to $s\left( {A_{i} } \right).$ In general, the alternatives with the highest score value will be the most preferable.

In the following, a numerical example is presented to illustrate using the proposed MCDM approach.

5 Illustrative Example

In this section, a numerical example is solved using MCDM method by utilizing the proposed BND-GWHM and BND-IGWHM for selection of the best alternative.

Suppose an investment company seeks to invest its funds in the most suitable option. As alternative options, an automobile company ($\tilde{A}_{1}$), a green energy company ($\tilde{A}_{2}$), a computer company ($\tilde{A}_{3}$) and a biotech company ($\tilde{A}_{4}$) may be considered. Alternatives are evaluated based on a set of criteria, including market conditions ($C_{1}$), growth factor ($C_{2}$), return on investment (ROI) ($C_{3}$), and environmental impact ($C_{4}$). To solve this problem, the weight vector of the four criteria is defined as $w = (0.30, 0.20, 0.30, 0.20)^{T}.$

Step 1. The BNSs decision matrix depend on criteria, $C$ for each alternative, $\tilde{A}$ are shown in Table 1.

Table 1 BNSs decision matrix

Step 2. Determine the aggregated values from BND-GWHM and BND-IGWHM operators of alternatives $\tilde{A}_{i} (i = 1, 2, 3, 4)$(assume $p = q = 1, \;\gamma = 1$). The aggregated values are displayed in Table 2.

Table 2 The aggregation values obtained from BND-GWHM and BND-IGWHM

Step 3. From Table 2, the score function can be obtained, as presented in Table 3 (Table 4).

Table 3 The score values of alternatives

Step 4. Rank all the alternative according score to score in Table 3. As the results, we can get:

Table 4 Ranking results

Thus, the most optimal option using both operators is $\tilde{A}_{2}.$

5.1 Effect of the Parameter ${\varvec{\gamma}},$ ${\varvec{p}}$ and ${\varvec{q}}$ on the Final Ranking Results

In this section, we compute the alternatives $\tilde{A}_{i} (i = 1, 2, 3, 4)$ with different operational roles for the parameter $\gamma$ and the parameters $p$ and $q.$ The parameters $\gamma$ in Dombi TN and Dombi TCN play crucial roles in controlling the shape and behavior of the aggregation process. It also provides the ability to fine-tune the behavior of these operators, making them responsive to various situations and preferences in BNS aggregation. Tables 5 and 6 show the score values and ranking orders for different variables $\gamma$ based on the BND-GWHM and BND-IGWHM operators, respectively (suppose that the parameter value remains constant, $p = q = 1$).

Table 5 The ranking orders of different $\gamma$ for BND-GWHM operator

Table 6 The ranking orders of different $\gamma$ for BND-IGWHM operator

Based on Tables 5 and 6, when the value of $\gamma$ for both the BND-GWHM and BND-IGWHM operators changes, the ranking orders are consistent. Thus, the most suitable plan is $\tilde{A}_{3}.$ In the following, Tables 7 and 8 show the ranking orders with different parameters $p$ and $q$ based on BND-GWHM and BND-IGWHM operator, respectively. In this analysis, assume the operational roles for parameter $\gamma$ remain constant where $\gamma = 1.$

Table 7 The ranking orders with different $p$ and $q$ based on BND-GWHM operator

Table 8 The ranking orders with different $p$ and $q$ based on BND-IGWHM operator

Based on Tables 7 and 8, it can be seen that ranking results for both operators are almost consistent where $\tilde{A}_{2}$ or $\tilde{A}_{3}$ the best option and $\tilde{A}_{1}$ or $\tilde{A}_{4}$ the lowest option. Except for parameter $p = 0, \;q = 1,$ where $\tilde{A}_{1}$ is best option meanwhile $\tilde{A}_{4}$ is the lowest option based on BND-GWHM operator. In the following, the comparison analysis is presented to show the validity of the proposed operator.

5.2 Comparison Analysis

In this section, the comparison analysis uses a bipolar neutrosophic environment with different types of aggregation operators proposed by Fan et al. [42] and Mahmood et al. [36]. The ranking results are shown in Table 9. Figure 2 portrays the ranking behavior of the four alternatives with respect to the four aggregation methods from presented in Table 9.

Table 9 The ranking orders obtained using different method

Fig. 2

Comparison of ranking of alternatives

Note that the ranking order of the alternative with respect to the BND-GWHM method and existing method [42] remains the same. While, the results from [36] and proposed methods yield different results, with $\tilde{A}_{2}$ being the best and $\tilde{A}_{1},$ $\tilde{A}_{3}$ or $\tilde{A}_{4}$ the least options. The slight difference in ranking order is related to the choice of AO. The existing method by Fan et al. [42] using IGWHM, meanwhile the operator proposed by Mahmood et al. [36] using weighted average (WA). Nevertheless, the proposed method under BNS information is more practical compared to existing methods as it can consider the interrelationship of input argument with flexible $\gamma$ parameter. The characteristics of some existing aggregation operators and the proposed approach are summarized in Table 10.

Table 10 Characteristics of the aggregation operators

Based on Table 10, the existing methods proposed by Chen and Ye [33] and Mahmood et al. [36] do not capture the interrelationship of the input argument since they use traditional operators (WA). Furthermore, Li et al. [41] and Fan et al. [42] presented an approach based on the HM AO to consider the interrelationship between arguments. However, these approaches used algebraic product and algebraic sum in the operation. In comparison to these existing approaches, the proposed approach has the advantage of reflecting the interrelationships of all input arguments with flexible parameters in Dombi operation.

6 Conclusion

This study integrates bipolar neutrosophic Dombi Heronian mean aggregation operator by incorporating Dombi into HM operator. In particular, we propose two aggregation operators, namely bipolar neutrosophic Dombi-based generalized weighted Heronian mean (BND-GWHM) and bipolar neutrosophic Dombi-based improved generalized weighted Heronian mean (BND-IGWHM) operators. These proposed AOs offer significance for dealing with the interrelationships between input arguments with flexible operational parameters to aggregate BNS information. Besides that, certain properties are demonstrated to verify that the suggested operators are valid operators. Following that, an MCDM approach is developed to solve the bipolar neutrosophic problems. An illustrative example involving an investment choice is provided to demonstrate the practicality of the proposed operators. The reliability and adaptability of the proposed methods were examined further using parameter analyses. Finally, a comparison analysis is conducted with different aggregation operators, emphasizing the superiority of the proposed approach by showcasing its ability to effectively capture the interrelationships among all input arguments with flexible parameters in Dombi operation.

The advantages of the research can be seen from several perspectives. Firstly, based on Dombi operation for BNS proposed by Mahmood et al. [36], two AOs are constructed by incorporating Heronian operator. BND-GWHM and BND-IGWHM operators can take into account the interrelationships of input arguments with flexible parameters in Dombi operation. Secondly, this study used improved version of HM, namely GWHM and IGWHM. This improved version overcomes the limitations of classical HM, particularly unfulfilled idempotency and a disregard for weight throughout the aggregating process. Thirdly, a MCDM procedure is constructed by incorporating BND-GWHM and BND-IGWHM operators. The proposed MCDM method outperforms previous methods when dealing with situations involving interrelationships of input arguments and parameter flexibility in the Dombi operation.

The proposed operator's limitations can be summarized in two points. First, BNS may not be suitable for all decision-making problems. They are primarily useful in situations where the degrees of truth, indeterminacy and falsity are essential for capturing bipolarity information. Second, the weight of criteria utilized is determined by the assumption approach. The weight of criteria should be calculated using the weighting method. Nevertheless, this limitation can be transformed into an opportunity for future research by delving into a real-life case study.

In the future, the method proposed can be extensible to other neutrosophic and fuzzy environments, including but not limited to Pythagorean neutrosophic set (PNS) [47], farmatean neutrosophic set [48] and novel information theory like basic uncertain information [49]. Besides that, methods like Shapley fuzzy measure and the entropy approach should be utilized to provide precise weight of criteria. Proposed AO also can be extended using Archimedean t-norms and t-conorms operations [50, 51]. In addition, the proposed methods also can be applied to a variety of applications in the science and technology areas, such as pattern recognition and ambiguous programming.