Multiple attribute group decision making approach for selection of robot under induced bipolar neutrosophic aggregation operators

Abstract

In current piece of writing, we bring in the new notion of induced bipolar neutrosophic (BN) AOs by utilizing Einstein operations as the foundation for aggregation operators (AOs), as well as to endow having a real-world problem-related application. The neutrosophic set can rapidly and more efficiently bring out the partial, inconsistent, and ambiguous information. The fundamental definitions and procedures linked to the basic bipolar neutrosophic (BN) set as well as the neutrosophic set (NS), are presented first. Our primary concern is the induced Einstein AOs, like, induced bipolar neutrosophic Einstein weighted average (I-BNEWA), induced bipolar neutrosophic Einstein weighted geometric (I-BNEWG), as well as their different types and required properties. The main advantage of employing the offered methods is that they give decision-makers a more thorough analysis of the problem. These strategies whenever compare to on hand methods, present complete, progressively precise, and accurate result. Finally, utilizing a numerical representation of an example for selection of robot, for a problem involving multi-criteria community decision making, we propose a novel solution. The suitability ratings are then ranked to select the most suitable robot. This demonstrates the practicality as well as usefulness of these novel approaches.

Introduction

In the present-day decision-making era, knowledge being over and over again remains incomplete, undetermined, as well as incompatible. There is no single rule to handle diverse obstacles due to varied uncertainties in real-life circumstances, hence complexity describes the behavior of an object whose components interact in various ways and obey distinct logical principles. It appears that many academics from all around the world have done substantial research on multi-criteria decision-making management strategies. Numerous mathematical techniques have been suggested by scholars to cope with uncertainties. Zadeh [1] was the one who first introduced the fundamental concept of fuzzy set theory, which deal by means of ambiguity and has application in a variety of contemporary domains of the present and the future of society. Fuzzy sets, on the other hand, having the ability to only communicate the values of the function related to membership rather than non-membership is a difficulty. Atanassov [2] created the core idea behind intuitionistic fuzzy sets (IFS) and the theory that underlies how to condense the idea of fuzzy sets in order to address this. Each IFS element is represented as a pair of membership values (truth-membership) $\Im \left( \chi \right)$, along with non-membership values (falsity-membership) $f\left( \chi \right)$ also satisfied the most important condition $\Im \left( \chi \right),f\left( \chi \right) \in \left[ {0,1} \right]$ with $0 \le \Im \left( \chi \right) + f\left( \chi \right) \le 1$. Intuitionistic fuzzy set be able to hold for incomplete information; indeterminate information is not a preference. Florentin Smarandache [3] develops the new notion of neutrosophic set, which include indeterminacy membership values ${I}\left( \chi \right).$ Neutrosophic set is very good at dealing with incomplete, ambiguous, and conflicting information. When $\Im \left( \chi \right) + {I}\left( \chi \right) + f\left( \chi \right) < 1$, it represents that information acquire is indeterminate. When ${\Im }\left( \chi \right) + {I}\left( \chi \right) + f\left( \chi \right) > 1$, it demonstrates so as to this represent inconsistency within a neutrosophic set (NS) devised by Smarandache. Wang and colleagues develop the fundamental idea of using the (SVNS) single valued neutrosophic set to handle real-life situation in science and technology [4], along with condition ${\Im }\left( \chi \right){,}{I}\left( \chi \right){,}f\left( \chi \right) \in \left[ {0,1} \right]$ and $0 \le {\Im }\left( \chi \right) + {I}\left( \chi \right) + f\left( \chi \right) \le 3$. The concept of correlation coefficient for SVNS along with comparison method was developed by Ye [5]. Wang and others devised the range of the values of the truth membership, indeterminacy membership, and false membership as an interval-valued neutrosophic set [6] is from 0 to 1.

Neutrosophic sets can represent uncertainty in a more versatile manner than traditional crisp sets or fuzzy sets. NS use a triadic representation, which includes three components: the truth-membership function, the indeterminacy-membership function, and the falsity-membership function. This triadic structure provides a more nuanced description of the elements' status. They find applications in fields where uncertainty and ambiguity are prevalent, such as decision-making, image processing, medical diagnosis, pattern recognition, and more. By capturing different degrees of truth and falsity, they can provide a more accurate representation of complex systems. Neutrosophic set theory can be combined with other mathematical models like fuzzy sets and interval-valued fuzzy sets to create hybrid approaches that can handle uncertainty in a more sophisticated way.

The aggregation operators (AOs) are attracting the attention of various researchers of modern scientific fields. Since its start, many scientists [7,8,9,10,11] have profoundly influenced the development of IFS theory. Wang and Zhao [12, 13] developed Einstein aggregation operators (AOs) for IFS and applied to decision making. The (BFS) bipolar fuzzy set [14,15,16] has its emergence, along with novel techniques to dealing imprecision in decision making (DM) situation. The range of the membership degree of the bipolar fuzzy set is − 1 to 1. BFS offers both positive and negative membership degrees. BFSs are extremely helpful in several academic subjects, including DM [6, 13, 17,18,19]. Gul [20] defines bipolar averaging as well as geometric AOs for BFS and applied them to different real life decision making situations. Deli et al. [21, 22] introduced the fundamental operations and comparison methods, the bipolar neutrosophic (BN) set was first created. Jamil et al. [23, 24] create bipolar neutrosophic (BN) aggregation operator (AOs) for study of decision making in BN environment. Rahman et al. [25], create aggregation operators (AOs) base on IFS value in addition to apply to multi-criteria decision making. Fan and others created the Heronian mean aggregation operators [26]. In induced aggregation operators, priority of importance is set at the beginning. The induced ordered weighted averaging (IOWA) operator is introduced by Yager [20]. The order inducing value is a variable that guides the argument ordering procedure in certain operators. Xu [27] introduced the idea of I-IFEOWA aggregation operator and developed its application. Jamil and others produce I-GIVIFEHG and apply them to multi-criteria DM problems [28]. The robots are contributing significantly to improving quality and productivity in manufacturing has gained exceptional recognition. Liang et al. use fuzzy multi-criteria decision making technique for selection of robot [29]. Jamil and others introduced [30] the multi-criteria decision-making applications for Einstein aggregation operators in a bipolar neutrosophic environment.

Riaz et al. introduced the concept of cubic bipolar fuzzy set and related AOs [31]. Riaz and others [32] introduced the fundamental concept of bipolar picture fuzzy set and related AOs with multi-criteria decision making approaches. Lin et al. [33] develop state of the art picture fuzzy interactional partitioned Heronian mean aggregation operators with multi-attribute decision making process. Lin and others [34] introduced the linguistic q-rung orthopair interactional partitioned Heronian mean aggregation operators along with its application. Lin et al. develop OWA weights using kernel density estimation [35] and applied to MCDM problems. Lin et al. [36] Linguistic paythagorean fuzzy interaction partitioned Bonferroni mean aggregation operators and develop its DM approaches very effectively.

Our present study main focus on the following important goals:

• (a) Suggestion of the various bipolar neutrosophic (BN) Einstein aggregation operator as well as desire properties.

• (b) A multi-criteria DM technique is also built based on BN values and directed at real-world problems.

• (c) Given a descriptive numerical example was devised for application toward multi-criteria group decision making.

• (d) The SVNSs make dealing by means of uncertain detail easier. The SVNSs is generalization of earlier sets like the fuzzy sets, the IFS sets, as well as the classical sets.

• (e) Because they are capable of handling both positive and negative membership value, BFSs are quite good at handling unpredictable, unexpected situations that arise in real life.

In BNS, each element is characterized by three membership degrees: positive (truth), negative (falsity), and indeterminate (neither truth nor falsity). This dual membership concept enables a richer and more accurate representation of complex situations. Bipolar neutrosophic sets find applications in granular computing, where information is processed at multiple levels of granularity. In decision-making processes, they can help in handling ambiguity and conflicts effectively. Similar to neutrosophic sets, BNS can be combined with other mathematical models like fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets to create hybrid approaches that accommodate various types of uncertainties. BNS has potential applications in various fields, including decision-making, pattern recognition, classification, medical diagnosis, risk assessment, and sentiment analysis, among others.

The study's remainder is organized at the same time as below:

The second section contains various basic definitions along with their connected properties.

In section three, we introduced I-BNEWA aggregation operators and I-BNEWG aggregation operators.

In section four, these novel AOs are apply to multi-criteria decision making in addition to that a numerical example.

In section five, at last, proposed a comparative study along with concluding remarks.

Preliminaries

In current segment, there are provided primary definition associated toward our study of neutrosophic set (NS) theory. Also defined various types of fuzzy sets, BN sets, score functions, accuracy functions, certainty functions, and in addition to the Einstein operation.

Definition 1

[3] Suppose, represents a fixed universal set by P. The (NS) neutrosophic set N be define below;

$$ N = \left\{ { \, \left( { \, \chi , \, {\Im }\left( \chi \right){,} \, {I}\left( \chi \right){, f}\left( \chi \right) \, } \right) \, | \, \chi \in P} \right\}. $$

The mapping for truth's membership functions is ${\Im }:N \to \overline{Q} ,$ the mapping for indeterminacy's membership functions is ${I}:N \to \overline{Q}$, and the mapping for the membership of falsehood functions is, ${f}:N \to \overline{Q}$, where $\overline{Q} = \left] {0^{ - } ,1^{ + } } \right[$ along with condition $0^{ - } \le {\Im }\left( \chi \right) + {I}\left( \chi \right) + {f}\left( \chi \right) \le 3^{ + }$.

Definition 2

[4] Suppose $A_{{{\text{NS}}}}$ be a (SVNS) single valued neutrosophic set and P is a fixed universal set, then $A_{{{\text{NS}}}}$ is stated as;

$$ A_{{{\text{NS}}}} = \left\{ {\left( {\chi ,{\Im }\left( \chi \right){,}{I}\left( \chi \right){,f}\left( \chi \right)} \right)|\chi \in P} \right\} $$

The mapping for truth's membership functions is ${\Im }:A_{{{\text{NS}}}} \to \overline{L}$, the mapping for indeterminacy's membership functions is ${I}:A_{{{\text{NS}}}} \to \overline{L}$, the mapping for the membership of falsehood functions is ${f}:A_{{{\text{NS}}}} \to \overline{L} ,$ where $\overline{L} = \left[ {0,1} \right]$ along with condition $0 \le {\Im }\left( \chi \right) + {I}\left( \chi \right) + {f}\left( \chi \right) \le 3.$

Definition 3

[15] Suppose P to be a fixed universal set. The (BFS) bipolar fuzzy set will then be defined as follows:

$$ F = \left\{ {\left\langle {\chi ,{\Im }_{F}^{ + } \left( \chi \right),{f}_{F}^{ - } \left( \chi \right)} \right\rangle |\chi \in P^{{}} } \right\} $$

where ${\Im }_{{_{{^{F} }} }}^{ + } \left( \chi \right):F \to L^{ + }$ represent positive membership function and ${f}_{{_{F} }}^{ - } \left( \chi \right):F \to K^{ - } ,$ negative membership function, where $K^{ - } = \left[ { - 1,^{{}} 0} \right]$ and $L^{ + } = \left[ {0,^{{}} 1} \right].$

Definition 4

[22] Suppose, represents universal set by P enclosing within it bipolar neutrosophic set (BNS) A is define as;

$$ A = \left\{ {\left( {\chi ,{\Im }^{ + } \left( \chi \right),{I}^{ + } \left( \chi \right),{f}^{ + } \left( \chi \right),{\Im }^{ - } \left( \chi \right),{I}^{ - } \left( \chi \right),{f}^{ - } \left( \chi \right)} \right)|\chi \in P} \right\} $$

Let ${\Im }^{ + } \left( \chi \right),{I}^{ + } \left( \chi \right),{f}^{ + } \left( \chi \right) = {\text{BN}}^{ + }$ as well as ${\Im }^{ - } \left( \chi \right),{I}^{ - } \left( \chi \right),{f}^{ - } \left( \chi \right) = {\text{BN}}^{ - }$.

Here. ${\Im }^{ + } \left( \chi \right),{I}^{ + } \left( \chi \right),{f}^{ + } \left( \chi \right)$ are positive membership functions represent truth, indeterminate as well as false for $\chi \in P$ and ${\Im }^{ - } \left( \chi \right),{I}^{ - } \left( \chi \right),{f}^{ - } \left( \chi \right)$ represent truth, indeterminate as well as false, for negative membership function are used. Now, the mapping for functions with positive memberships are ${\text{BN}}^{ + } :A \to L^{ + }$ as well as ${\text{BN}}^{ - } :A \to L^{ - } ,$ where $L^{ - } = \left[ { - 1,^{{}} 0} \right]$, $L^{ + } = \left[ {0,^{{}} 1} \right]$ with $0 \le {\Im }^{ + } \left( \chi \right) + {I}^{ + } \left( \chi \right) + {f}^{ + } \left( \chi \right) + {\Im }^{ - } \left( \chi \right) + {I}^{ - } \left( \chi \right) + {f}^{ - } \left( \chi \right) \le 6.$

Example

Let $P = \left\{ {\chi_{1} ,\chi_{2} ,\chi_{3} } \right\}$, then

$$ A = \left\{ \begin{gathered} \left( {\chi_{1} ,0.4,0.6,0.2, - 0.5, - 0.6, - 0.5} \right), \hfill \\ \left( {\chi_{2} ,0.3,0.7,0.4, - 0.3, - 0.6, - 0.2} \right), \hfill \\ \left( {\chi_{3} ,0.2,0.6,0.7, - 0.4, - 0.3, - 0.6} \right) \hfill \\ \end{gathered} \right\} $$

represent a BNS subset for the fixed universal set P.

There is following list of core operations [22] for BNSs includes;

Let $A_{1} = \left\{ {\left( {\chi ,{\Im }_{1}^{ + } \left( \chi \right),{I}_{1}^{ + } \left( \chi \right),{f}_{1}^{ + } \left( \chi \right),{\Im }_{1}^{ - } \left( \chi \right),{I}_{1}^{ - } \left( \chi \right),{f}_{1}^{ - } \left( \chi \right)} \right)|\chi \in P} \right\}$ and $A_{2} = \left\{ {\left( {\chi ,{\Im }_{2}^{ + } \left( \chi \right),{I}_{2}^{ + } \left( \chi \right),{f}_{2}^{ + } \left( \chi \right),{\Im }_{2}^{ - } \left( \chi \right),{I}_{2}^{ - } \left( \chi \right),{f}_{2}^{ - } \left( \chi \right)} \right)|\chi \in P} \right\}$ be two BNSs.

• i. Then $A_{1} \subseteq A_{2}$ is define as

  $$ {\Im }_{1}^{ + } \left( \chi \right) \le {\Im }_{2}^{ + } \left( \chi \right),{I}_{1}^{ + } \left( \chi \right) \le {I}_{2}^{ + } \left( \chi \right),{f}_{1}^{ + } \left( \chi \right) \ge {f}_{2}^{ + } \left( \chi \right), $$

  and

  $$ {\Im }_{1}^{ - } \left( \chi \right) \ge {\Im }_{2}^{ - } \left( \chi \right),{I}_{1}^{ - } \left( \chi \right) \ge {I}_{2}^{ - } \left( \chi \right),{f}_{1}^{ - } \left( \chi \right) \le {f}_{2}^{ - } \left( \chi \right) $$

• ii. $A_{1} = A_{2}$ if and only if

  $$ {\Im }_{1}^{ + } \left( \chi \right) = {\Im }_{2}^{ + } \left( \chi \right),{I}_{1}^{ + } \left( \chi \right) = {I}_{2}^{ + } \left( \chi \right),{f}_{1}^{ + } \left( \chi \right) = {f}_{2}^{ + } \left( \chi \right), $$

  and

  $$ {\Im }_{1}^{ - } \left( \chi \right) = {\Im }_{2}^{ - } \left( \chi \right),{I}_{1}^{ - } \left( \chi \right) = {I}_{2}^{ - } \left( \chi \right),{f}_{1}^{ - } \left( \chi \right) = {f}_{2}^{ - } \left( \chi \right) $$

• iii. The union is define as below:

  $$ \left( {A_{1} \cup A_{2} } \right) = \left\{ {\left( \begin{gathered} \max^{{}} \left( {_{{}} {\Im }_{1}^{ + } \left( \chi \right),{\Im }_{2}^{ + } \left( \chi \right)} \right),\frac{{{I}_{1}^{ + } \left( \chi \right){ + }{I}_{2}^{ + } \left( \chi \right),}}{2},\min_{{}} \left( {{f}_{1}^{ + } \left( \chi \right){,f}_{2}^{ + } \left( \chi \right)} \right),^{{}} \hfill \\ \min \left( {{\Im }_{1}^{ - } \left( \chi \right){,\Im }_{2}^{ - } \left( \chi \right)} \right),\frac{{{I}_{1}^{ - } \left( \chi \right){ + }{I}_{2}^{ - } \left( \chi \right)}}{2},\max \left( {{f}_{1}^{ - } \left( \chi \right),{f}_{2}^{ - } \left( \chi \right)} \right) \hfill \\ \end{gathered} \right)} \right\} $$

• iv. The intersection is defined as:

  $$ \left( {A_{1} \cap A_{2} } \right) = \left\{ {\left( \begin{gathered} \min \left( {{\Im }_{1}^{ + } \left( \chi \right),{\Im }_{2}^{ + } \left( \chi \right)} \right),\frac{{{I}_{1}^{ + } \left( \chi \right) + {I}_{2}^{ + } \left( \chi \right)}}{2},\max \left( {{f}_{1}^{ + } \left( \chi \right),{f}_{2}^{ + } \left( \chi \right)} \right), \hfill \\ \max \left( {{\Im }_{1}^{ - } \left( \chi \right),{\Im }_{2}^{ - } \left( \chi \right)} \right),\frac{{{I}_{1}^{ - } \left( \chi \right) + {I}_{2}^{ - } \left( \chi \right)}}{2},\min \left( {{f}_{1}^{ - } \left( \chi \right),{f}_{2}^{ - } \left( \chi \right)} \right) \hfill \\ \end{gathered} \right)} \right\}, $$

• v. Let $A = \left\{ {\left( {\chi ,{\Im }^{ + } \left( \chi \right),{I}^{ + } \left( \chi \right),{f}^{ + } \left( \chi \right),{\Im }^{ - } \left( \chi \right),{I}^{ - } \left( \chi \right),{f}^{ - } \left( \chi \right)} \right)|\chi \in P} \right\},$ and be a BNSs. Then, the complement $A^{\prime}$ is define as:

  $$\begin{aligned} {\Im }_{{A^{\prime}}}^{ + } \left( \chi \right) & = \left\{ {1^{ + } } \right\} - {\Im }_{A}^{ + } \left( \chi \right),\\ {I}_{{A^{\prime}}}^{ + } \left( \chi \right) & = \left\{ {1^{ + } } \right\} - {I}_{A}^{ + } \left( \chi \right),{f}_{{A^{\prime}}}^{ + } \left( \chi \right) = \left\{ {1^{ + } } \right\} - {f}_{A}^{ + } \left( \chi \right),\end{aligned} $$

  and

  $$\begin{aligned} {\Im }_{{A^{\prime}}}^{ - } \left( \chi \right) & = \left\{ {1^{ - } } \right\} - {\Im }_{A}^{ - } \left( \chi \right),\\ {I}_{{A^{\prime}}}^{ - } \left( \chi \right) & = \left\{ {1^{ - } } \right\} - {I}_{A}^{ - } \left( \chi \right),{f}_{{A^{\prime}}}^{ - } \left( \chi \right) = \left\{ {1^{ - } } \right\} - {f}_{A}^{ - } \left( \chi \right),\end{aligned} $$

Definition 5

[22] Suppose two (BNNs) bipolar neutrosophic numbers $u_{1} = \left( {{\Im }_{1}^{ + } ,{I}_{1}^{ + } ,{f}_{1}^{ + } ,{\Im }_{1}^{ - } ,{I}_{1}^{ - } ,{f}_{1}^{ - } } \right)$ and $u_{2} = \left( {{\Im }_{2}^{ + } ,{I}_{2}^{ + } ,{f}_{2}^{ + } ,{\Im }_{2}^{ - } ,{I}_{2}^{ - } ,{f}_{2}^{ - } } \right)$. The fundamental operations between two BNNs are as follows:

• (i) $u_{1} + u_{2} = \left( {{\Im }_{1}^{ + } + {\Im }_{2}^{ + } - {\Im }_{1}^{ + } {\Im }_{2}^{ + } ,{I}_{1}^{ + } {I}_{2}^{ + } ,{f}_{1}^{ + } {f}_{2}^{ + } , - {\Im }_{1}^{ - } {\Im }_{2}^{ - } , - \left( { - {I}_{1}^{ - } - {I}_{2}^{ - } - {I}_{1}^{ - } {I}_{2}^{ - } } \right), - \left( { - {f}_{1}^{ - } - {f}_{2}^{ - } - {f}_{1}^{ - } {f}_{2}^{ - } } \right)} \right){;}$.

• (ii) $u_{1} \cdot u_{2} = \left( {{\Im }_{1}^{ + } {\Im }_{2}^{ + } ,{I}_{1}^{ + } + {I}_{2}^{ + } - {I}_{1}^{ + } {I}_{2}^{ + } ,{f}_{1}^{ + } + {f}_{2}^{ + } - {f}_{1}^{ + } {f}_{2}^{ + } , - \left( { - {\Im }_{1}^{ - } - {\Im }_{2}^{ - } - {\Im }_{1}^{ - } {\Im }_{2}^{ - } } \right), - {I}_{1}^{ - } {I}_{2}^{ - } , - {f}_{1}^{ - } {f}_{2}^{ - } } \right){; }$.

• (iii) $\lambda \left( {u_{1} } \right) = \left( {1 - \left( {1 - {\Im }_{1}^{ + } } \right)^{\lambda } ,\left( {{I}_{1}^{ + } } \right)^{\lambda } ,\left( {{f}_{1}^{ + } } \right)^{\lambda } , - \left( { - {\Im }_{1}^{ - } } \right)^{\lambda } , - \left( { - {I}_{1}^{ - } } \right)^{\lambda } , - \left( {1 - \left( {1 - \left( { - {f}_{1}^{ - } } \right)} \right)^{\lambda } } \right)} \right){; }$.

• (iv) $\left( {u_{1} } \right)^{\lambda } = \left( {\left( {{\Im }_{1}^{ + } } \right)^{\lambda } ,1 - \left( {1 - {I}_{1}^{ + } } \right)^{\lambda } ,1 - \left( {1 - {f}_{1}^{ + } } \right)^{\lambda } , - \left( {1 - \left( {1 - \left( { - {\Im }_{1}^{ - } } \right)} \right)^{\lambda } } \right), - \left( { - {I}_{1}^{ - } } \right)^{\lambda } , - \left( { - {f}_{1}^{ - } } \right)^{\lambda } } \right){; }$.

here $\lambda > 0.$

Definition 6

[22] The bipolar neutrosophic number's score function for $u = \left( {{\Im }^{ + } ,{I}^{ + } ,{f}^{ + } ,{\Im }^{ - } ,{I}^{ - } ,{f}^{ - } } \right)$ is denoted by $S\left( u \right)$:

$$ S\left( u \right) = \frac{1}{6}\left( {{\Im }^{ + } + 1 - {I}^{ + } + 1 - {f}^{ + } + 1 + {\Im }^{ - } - {I}^{ - } - {f}^{ - } } \right). $$

Definition 7

[22] The accuracy function be there for a bipolar neutrosophic number $u = \left( {{\Im }^{{ + }{}} ,{I}^{ + }_{{}} ,{f}^{{ + }{}} ,{\Im }^{ - } ,_{{}} {I}^{ - } ,{f}^{ - } } \right)$ denoted by $a\left( u \right)$ is:

$$ a\left( u \right) = {\Im }^{ + } - {f}^{ + } + {\Im }^{ - } - {f}^{ - } {.} $$

Definition 8

[22] Consider $u = \left( {{\Im }^{ + } ,{I}^{ + } ,{f}^{ + } ,{\Im }^{ - } ,{I}^{ - } ,{f}^{ - } } \right)$ be there neutrosophic number, the function for certainty value is $c\left( u \right)$ define as:

$$ c\left( u \right) = {\Im }^{ + } - {f}^{ - } {.} $$

Definition 9

[22] Suppose ${\k{\it u}}_{1} = \left( {\Im_{1}^{ + } ,{I}_{1}^{ + } ,f_{1}^{ + } ,\Im_{1}^{ - } ,{I}_{1}^{ - } ,f_{1}^{ - } } \right)$ and ${\k{\it u}}_{2} = \left( {\Im_{2}^{ + } ,{I}_{2}^{ + } ,f_{2}^{ + } ,\Im_{2}^{ - } ,{I}_{2}^{ - } ,f_{2}^{ - } } \right)$ represent two BN numbers, then evaluation method among BN numbers is:

• (i) If condition $S({\k{\it u}}_{1} ) \succ S({\k{\it u}}_{2} ),$ then ${\k{\it u}}_{1}$ is greater than ${\k{\it u}}_{2}$, as a result ${\k{\it u}}_{1} \succ {\k{\it u}}_{2} ;$

• (ii) If condition $S({\k{\it u}}_{1} ) = S({\k{\it u}}_{2} ),$ $a({\k{\it u}}_{1} ) \succ a({\k{\it u}}_{2} ),$ then ${\k{\it u}}_{1}$ is greater to ${\k{\it u}}_{2}$, as a result ${\k{\it u}}_{1} \succ {\k{\it u}}_{2} ;$

• (iii) If condition $S({\k{\it u}}_{1} ) = S({\k{\it u}}_{2} ),$ $a({\k{\it u}}_{1} ) = a({\k{\it u}}_{2} ),$ as well as $c({\k{\it u}}_{1} ) \succ c({\k{\it u}}_{2} ),$ then ${\k{\it u}}_{1}$ must be greater than ${\k{\it u}}_{2}$, as a result ${\k{\it u}}_{1} \succ {\k{\it u}}_{2} ;$

• (iv) If condition $S({\k{\it u}}_{1} ) = S({\k{\it u}}_{2} ),$ $a({\k{\it u}}_{1} ) = a({\k{\it u}}_{2} ),$ as well as $c({\k{\it u}}_{1} ) = c({\k{\it u}}_{2} ),$ than ${\k{\it u}}_{1}$ must be equal to ${\k{\it u}}_{2}$ as a result ${\k{\it u}}_{1} = {\k{\it u}}_{2} ;$

Definition 10

Consider $u = \left( {{\Im }^{ + } ,{I}^{ + } ,{f}^{ + } ,{\Im }^{ - } ,{I}^{ - } ,{f}^{ - } } \right)$, $u_{1} = \left( {{\Im }_{1}^{ + } ,{I}_{1}^{ + } ,{f}_{1}^{ + } ,{\Im }_{1}^{ - } ,{I}_{1}^{ - } ,{f}_{1}^{ - } } \right)$ and $u_{2} = \left( {{\Im }_{2}^{ + } ,{I}_{2}^{ + } ,{f}_{2}^{ + } ,{\Im }_{2}^{ - } ,{I}_{2}^{ - } ,{f}_{2}^{ - } } \right)$ be BNNs, and $\lambda \succ 0$, basic Einstein operations of BNNs are as follows:

$$\begin{aligned} u_{1} \oplus u_{2} &= \left( \frac{{{\Im }_{1}^{ + } + {\Im }_{2}^{ + } }}{{1 + {\Im }_{1}^{ + } {\Im }_{2}^{ + } }},\frac{{{I}_{1}^{ + } {I}_{2}^{ + } }}{{1 + \left( {1 - {I}_{1}^{ + } } \right)\left( {1 - {I}_{2}^{ + } } \right)}},\right.\\ &\quad \frac{{{f}_{1}^{ + } {f}_{2}^{ + } }}{{1 + \left( {1 - {f}_{1}^{ + } } \right)\left( {1 - {f}_{2}^{ + } } \right)}},\frac{{ - {\Im }_{1}^{ - } {\Im }_{2}^{ - } }}{{1 + \left( {1 + {\Im }_{1}^{ - } } \right)\left( {1 + {\Im }_{2}^{ - } } \right)}},\\ &\quad \left.\frac{{ - \left( { - {I}_{1}^{ - } - {I}_{2}^{ - } } \right)}}{{1 + {I}_{1}^{ - } {I}_{2}^{ - } }},\frac{{ - \left( { - {f}_{1}^{ - } - {f}_{2}^{ - } } \right)}}{{1 + {f}_{1}^{ - } {f}_{2}^{ - } }} \right) \end{aligned}$$

$$ u_{1} \otimes u_{2} = \left( \frac{{{\Im }_{1}^{ + } {\Im }_{2}^{ + } }}{{1 + \left( {1 - {\Im }_{1}^{ + } } \right)\left( {1 - {\Im }_{2}^{ + } } \right)}},\frac{{{I}_{1}^{ + } + {I}_{2}^{ + } }}{{1 + {I}_{1}^{ + } {I}_{2}^{ + } }},\frac{{{f}_{1}^{ + } + {f}_{2}^{ + } }}{{1 + {f}_{1}^{ + } {f}_{2}^{ + } }},\frac{{ - \left( { - {\Im }_{1}^{ - } - {\Im }_{2}^{ - } } \right)}}{{1 + {\Im }_{1}^{ - } {\Im }_{2}^{ - } }},\frac{{ - {I}_{1}^{ - } {I}_{2}^{ - } }}{{1 + \left( {1 + {I}_{1}^{ - } } \right)\left( {1 + {I}_{2}^{ - } } \right)}},\frac{{ - {f}_{1}^{ - } {f}_{2}^{ - } }}{{1 + \left( {1 + {f}_{1}^{ - } } \right)\left( {1 + {f}_{2}^{ - } } \right)}} \right) $$

$$ \lambda \left( u \right) = \left( \frac{{\left( {1 + {\Im }^{ + } } \right)^{\lambda } - \left( {1 - {\Im }^{ + } } \right)^{\lambda } }}{{\left( {1 + {\Im }^{ + } } \right)^{\lambda } + \left( {1 - {\Im }^{ + } } \right)^{\lambda } }},\frac{{2\left( {{I}^{ + } } \right)^{\lambda } }}{{\left( {2 - {I}^{ + } } \right)^{\lambda } + \left( {{I}^{ + } } \right)^{\lambda } }},\frac{{2\left( {{f}^{ + } } \right)^{\lambda } }}{{\left( {2 - {f}^{ + } } \right)^{\lambda } + \left( {{f}^{ + } } \right)^{\lambda } }},\frac{{ - 2\left| {{\Im }^{ - } } \right|^{\lambda } }}{{\left( {2 + {\Im }^{ - } } \right)^{\lambda } + \left| {{\Im }^{ - } } \right|^{\lambda } }},\frac{{ - 2\left| {{I}^{ - } } \right|^{\lambda } }}{{\left( {2 + {I}^{ - } } \right)^{\lambda } + \left| {{I}^{ - } } \right|^{\lambda } }},\frac{{ - \left( {\left( {1 + \left| {{f}^{ - } } \right|} \right)^{\lambda } - \left( {1 + {f}^{ - } } \right)^{\lambda } } \right)}}{{\left( {1 + \left| {{f}^{ - } } \right|} \right)^{\lambda } + \left( {1 + {f}^{ - } } \right)^{\lambda } }} \right) $$

$$ \left( u \right)^{\lambda } = \left( \frac{{2\left( {{\Im }^{ + } } \right)^{\lambda } }}{{\left( {2 - {\Im }^{ + } } \right)^{\lambda } + \left( {{\Im }^{ + } } \right)^{\lambda } }},\frac{{\left( {1 + {I}^{ + } } \right)^{\lambda } - \left( {1 - {I}^{ + } } \right)^{\lambda } }}{{\left( {1 + {I}^{ + } } \right)^{\lambda } + \left( {1 - {I}^{ + } } \right)^{\lambda } }},\frac{{\left( {1 + {f}^{ + } } \right)^{\lambda } - \left( {1 - {f}^{ + } } \right)^{\lambda } }}{{\left( {1 + {f}^{ + } } \right)^{\lambda } + \left( {1 - {f}^{ + } } \right)^{\lambda } }},\frac{{ - \left( {\left( {1 + \left| {{\Im }^{ - } } \right|} \right)^{\lambda } - \left( {1 + {\Im }^{ - } } \right)^{\lambda } } \right)}}{{\left( {1 + \left| {{\Im }^{ - } } \right|} \right)^{\lambda } + \left( {1 + {\Im }^{ - } } \right)^{\lambda } }},\frac{{ - 2\left| {{I}^{ - } } \right|^{\lambda } }}{{\left( {2 + {I}^{ - } } \right)^{\lambda } + \left| {{I}^{ - } } \right|^{\lambda } }},\frac{{ - 2\left| {{f}^{ - } } \right|^{\lambda } }}{{\left( {2 + {f}^{ - } } \right)^{\lambda } + \left| {{f}^{ - } } \right|^{\lambda } }} \right) $$

Induced bipolar neutrosophic Einstein average AOs

Within the following segment of the paper, we offer several fundamental characteristics for bipolar Einstein average AOs, including average I-BNEWA, order average I-BNEOWA, and hybrid I-BNEHA.

Induced bipolar neutrosophic Einstein weighted average AO

Suppose $u_{\ell } = \left( {{\Im }_{\ell }^{ + } ,{I}_{\ell }^{ + } ,{f}_{\ell }^{ + } ,{\Im }_{\ell }^{ - } ,{I}_{\ell }^{ - } ,{f}_{\ell }^{ - } } \right)$ representing a family that contains BN numbers, here $\ell \in \left\{ {1,\ldots,n - 1,n} \right\}$.

Definition 11

Let us consider $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle \; \left( {\ell = 1,2,\ldots,n} \right)$ represent a collection that contains 2-tuple, then (I-BNEWA) AO is define as:

$$\begin{aligned} &{\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u_{\ell } } \right) \\ & \quad = \nu_{1} u_{1} \oplus \nu_{2} u_{2} \oplus \cdots \oplus \nu_{n} u_{n} \end{aligned}$$

(1)

where $\nu = \left( {\nu_{1} ,\nu_{2} , \ldots ,\nu_{n} } \right)^{T}$ represented as weighting vector of $u_{\ell } ,$ $\nu_{\ell } > 0$ as well as $\mathop \sum \nolimits_{\ell = 1}^{n} \nu_{\ell } = 1,$ and $\alpha_{\ell }$ in $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$ known as the order-inducing variable and $u_{\ell }$ for bipolar neutrosophic argument variable.

Theorem 12

Suppose $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,3,\ldots,n - 1,n} \right)$ represent a collection of 2-tuples, then the BN values of (I-BNEWA) gives in returns a BN value with

$$ \begin{aligned} & {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \\ & \quad = \left( {\frac{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{1} - {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{2} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{1} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{2} }},\frac{{{2}\zeta_{1} }}{{\zeta_{2} { + }\zeta_{1} }},\frac{{{2}{\varvec{f}}_{1} }}{{{\varvec{f}}_{2} { + }{\varvec{f}}_{1} }},\frac{{ - {2}{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{3} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{4} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{3} }}, - \frac{{\zeta_{3} - \zeta_{4} }}{{\zeta_{3} { + }\zeta_{4} }}, - \frac{{{\varvec{f}}_{3} - {\varvec{f}}_{4} }}{{{\varvec{f}}_{3} { + }{\varvec{f}}_{4} }}} \right) \\ \end{aligned} $$

(2)

where,

$$\begin{aligned} {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{1} &= \mathop{\varvec{\varPi}}\limits_{{\ell { = 1}}}^{{n}} \left( {1 + {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} ,\zeta_{1} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {{I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} ,\\ & {\varvec{f}}_{1} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {{f}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} , \, {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{2} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {1 - {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} ,\\ & \zeta_{2} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {2 - {I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} , \, {\varvec{f}}_{2} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {2 - {f}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} , \\ & {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{3} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left| {{\Im }_{\ell }^{ - } } \right|^{{\nu_{\ell } }} { ,}\zeta_{3} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {1 + {\ominus }\left| {{I}_{\ell }^{ - } } \right|} \right)^{{\nu_{\ell } }} , \\ & {\varvec{f}}_{3} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {1 + {\ominus }\left| {{f}_{\ell }^{ - } } \right|} \right)^{{\nu_{\ell } }} , \, {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{4} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {2 + {\Im }_{\ell }^{ - } } \right)^{{\nu_{\ell } }} ,\\ & \zeta_{4} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {1 + {I}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} ,{\varvec{f}}_{4} = \mathop \prod \limits_{{\ell { = 1}}}^{{n}} \left( {1 + {f}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} {. } \end{aligned} $$

where $\nu = \left( {\nu_{1} ,\nu_{2} , \ldots ,\nu_{n} } \right)^{T}$ is the weighted vector of $u_{\ell }$ such that $\nu_{\ell } > 0$ and $\mathop \sum \nolimits_{\ell = 1}^{n} \nu_{\ell } = 1.$

In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ as the bipolar neutrosophic arguments variables.

Proof

First we calculate $u_{1}$, we have

$$ u_{1} = \left( \begin{gathered} \frac{{\left( {1 + {\Im }_{1}^{ + } } \right) - \left( {1 - {\Im }_{1}^{ + } } \right)}}{{\left( {1 + {\Im }_{1}^{ + } } \right) + \left( {1 - {\Im }_{1}^{ + } } \right)}},\frac{{2\left( {{I}_{1}^{ + } } \right)}}{{\left( {2 - {I}_{1}^{ + } } \right) + \left( {{I}_{1}^{ + } } \right)}},\frac{{2\left( {{f}_{1}^{ + } } \right)}}{{\left( {2 - {f}_{1}^{ + } } \right) + \left( {{f}_{1}^{ + } } \right)}}, \hfill \\ \frac{{ - 2\left| {{\Im }_{1}^{ - } } \right|}}{{\left( {2 + {\Im }_{1}^{ - } } \right) + \left| {{\Im }_{1}^{ - } } \right|}},\frac{{ - 2\left| {{I}_{1}^{ - } } \right|}}{{\left( {2 + {I}_{1}^{ - } } \right) + \left| {{I}_{1}^{ - } } \right|}},\frac{{ - \left( {\left( {1 + {\ominus }\left| {{f}_{1}^{ - } } \right|} \right) - \left( {1 + {f}_{1}^{ - } } \right)} \right)}}{{\left( {1 + \left| {{f}_{1}^{ - } } \right|} \right) + \left( {1 + {f}_{1}^{ - } } \right)}} \hfill \\ \end{gathered} \right). $$

Now, we calculate $\nu_{1} u_{1} ,$ for this, we have

$$ \nu_{1} u_{1} = \left( \begin{gathered} \frac{{\left( {1 + {\Im }_{1}^{ + } } \right)^{{\nu_{1} }} - \left( {1 - {\Im }_{1}^{ + } } \right)^{{\nu_{1} }} }}{{\left( {1 + {\Im }_{1}^{ + } } \right)^{{\nu_{1} }} + \left( {1 - {\Im }_{1}^{ + } } \right)^{{\nu_{1} }} }},\frac{{2\left( {{I}_{1}^{ + } } \right)^{{\nu_{1} }} }}{{\left( {2 - {I}_{1}^{ + } } \right)^{{\nu_{1} }} + \left( {{I}_{1}^{ + } } \right)^{{\nu_{1} }} }},\frac{{2\left( {{f}_{1}^{ + } } \right)^{{\nu_{1} }} }}{{\left( {2 - {f}_{1}^{ + } } \right)^{{\nu_{1} }} + \left( {{f}_{1}^{ + } } \right)^{{\nu_{1} }} }}, \hfill \\ \frac{{ - 2\left| {{\Im }_{1}^{ - } } \right|^{{\nu_{1} }} }}{{\left( {2 + {\Im }_{1}^{ - } } \right)^{{\nu_{1} }} + \left| {{\Im }_{1}^{ - } } \right|^{{\nu_{1} }} }},\frac{{ - 2\left| {{I}_{1}^{ - } } \right|^{{\nu_{1} }} }}{{\left( {2 + {I}_{1}^{ - } } \right)^{{\nu_{1} }} + \left| {{I}_{1}^{ - } } \right|^{{\nu_{1} }} }},\frac{{ - \left( {\left( {1 + {\ominus }\left| {{f}_{1}^{ - } } \right|} \right)^{{\nu_{1} }} - \left( {1 + {f}_{1}^{ - } } \right)^{{\nu_{1} }} } \right)}}{{\left( {1 + \left| {{f}_{1}^{ - } } \right|} \right)^{{\nu_{1} }} + \left( {1 + {f}_{1}^{ - } } \right)^{{\nu_{1} }} }} \hfill \\ \end{gathered} \right). $$

Now, we calculate $\mathop \oplus \nolimits_{\ell = 1}^{n} \left( {\nu_{\ell } u_{\ell } } \right),$ for this we have

$$ \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u_{\ell } } \right) = \left( \begin{gathered} \frac{{\mathop \prod \nolimits_{{_{\ell = 1} }}^{n} \left( {1 + {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} - \mathop \prod \nolimits_{\ell = 1}^{n} \left( {1 - {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{n} \left( {1 {\textit{+}}{\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} {\textit{+}}\mathop \prod \nolimits_{{^{\ell = 1} }}^{n} \left( {1 - {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }},\frac{{2\mathop \prod \nolimits_{\ell = 1}^{n} \left( {{I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}{{\mathop {^{\prod } }\nolimits_{\ell = 1}^{n} \left( {2 - {I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} + \mathop \Pi \nolimits_{\ell = 1}^{n} \left( {{I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}, \hfill \\ \frac{{2\mathop \Pi \nolimits_{\ell = 1}^{n} \left( {{f}_{\ell }^{ {\textit{+}}} } \right)^{{\nu_{\ell } }} }}{{\mathop \Pi \nolimits_{\ell = 1}^{n} \left( {2 - {f}_{\ell }^{ {\textit{+}}} } \right)^{{\nu_{\ell } }} + \mathop \Pi \nolimits_{\ell = 1}^{n} \left( {{f}_{\ell }^{ {\textit{+}}} } \right)^{{\nu_{\ell } }} }},\frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^{n} \left| {{\Im }_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{n} \left( {2 + {\Im }_{\ell }^{ - } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{\ell = 1}^{n} \left| {{\Im }_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}, \hfill \\ \frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^{n} \left| {{I}_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{n} \left( {2 + {I}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{\ell = 1}^{n} \left| {{I}_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}, - \frac{{\mathop \prod \nolimits_{\ell = 1}^{n} \left( {1 {\textit{+}}{\ominus }\left| {{f}_{\ell }^{ - } } \right|^{{}} } \right)^{{\nu_{\ell } }} - \mathop \prod \nolimits_{\ell = 1}^{n} \left( {1 {\textit{+}}{f}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{n} \left( {1 + \left| {{f}_{\ell }^{ - } } \right|} \right)^{{\nu_{\ell } }} {\textit{+}}\mathop {^{\prod } }\nolimits_{\ell = 1}^{n} \left( {1 + {f}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} }} \hfill \\ \end{gathered} \right), $$

(3)

We have, here using mathematical induction:

when $n = 2$

$$ \nu_{1} u_{1} = \left( \begin{gathered} \frac{{\left( {1 + {\Im }_{1}^{ + } } \right)^{{\nu_{1} }} - \left( {1 - {\Im }_{1}^{ + } } \right)^{{\nu_{1} }} }}{{\left( {1 + {\Im }_{1}^{ + } } \right)^{{\nu_{1} }} + \left( {1 - {\Im }_{1}^{ + } } \right)^{{\nu_{1} }} }},\frac{{2\left( {{I}_{1}^{ + } } \right)^{{\nu_{1} }} }}{{\left( {2 - {I}_{1}^{ + } } \right)^{{\nu_{1} }} + \left( {{I}_{1}^{ + } } \right)^{{\nu_{1} }} }},\frac{{2\left( {{f}_{1}^{ + } } \right)^{{\nu_{1} }} }}{{\left( {2 - {f}_{1}^{ + } } \right)^{{\nu_{1} }} + \left( {{f}_{1}^{ + } } \right)^{{\nu_{1} }} }}, \hfill \\ \frac{{ - 2\left| {{\Im }_{1}^{ - } } \right|^{{\nu_{1} }} }}{{\left( {2 + {\Im }_{1}^{ - } } \right)^{{\nu_{1} }} + \left| {{\Im }_{1}^{ - } } \right|^{{\nu_{1} }} }},\frac{{ - 2\left| {{I}_{1}^{ - } } \right|^{{\nu_{1} }} }}{{\left( {2 + {I}_{1}^{ - } } \right)^{{\nu_{1} }} + \left| {{I}_{1}^{ - } } \right|^{{\nu_{1} }} }},\frac{{ - \left( {\left( {1 + {\ominus }\left| {{f}_{1}^{ - } } \right|} \right)^{{\nu_{1} }} - \left( {1 + {f}_{1}^{ - } } \right)^{{\nu_{1} }} } \right)}}{{\left( {1 + \left| {{f}_{1}^{ - } } \right|} \right)^{{\nu_{1} }} + \left( {1 + {f}_{1}^{ - } } \right)^{{\nu_{1} }} }} \hfill \\ \end{gathered} \right) $$

and

$$ \nu_{2} u_{2} = \left( \begin{gathered} \frac{{\left( {1 + {\Im }_{2}^{ + } } \right)^{{\nu_{2} }} - \left( {1 - {\Im }_{2}^{ + } } \right)^{{\nu_{2} }} }}{{\left( {1 + {\Im }_{2}^{ + } } \right)^{{\nu_{2} }} + \left( {1 - {\Im }_{2}^{ + } } \right)^{{\nu_{2} }} }},\frac{{2\left( {{I}_{2}^{ + } } \right)^{{\nu_{2} }} }}{{\left( {2 - {I}_{2}^{ + } } \right)^{{\nu_{2} }} + \left( {{I}_{2}^{ + } } \right)^{{\nu_{2} }} }},\frac{{2\left( {{f}_{2}^{ + } } \right)^{{\nu_{2} }} }}{{\left( {2 - {f}_{2}^{ + } } \right)^{{\nu_{2} }} + \left( {{f}_{2}^{ + } } \right)^{{\nu_{2} }} }}, \hfill \\ \frac{{ - 2\left| {{\Im }_{2}^{ - } } \right|^{{\nu_{2} }} }}{{\left( {2 + {\Im }_{2}^{ - } } \right)^{{\nu_{2} }} + \left| {{\Im }_{2}^{ - } } \right|^{{\nu_{2} }} }},\frac{{ - 2\left| {{I}_{2}^{ - } } \right|^{{\nu_{2} }} }}{{\left( {2 + {I}_{2}^{ - } } \right)^{{\nu_{2} }} + \left| {{I}_{2}^{ - } } \right|^{{\nu_{2} }} }},\frac{{ - \left( {\left( {1 + \left| {{f}_{2}^{ - } } \right|} \right)^{{\nu_{2} }} - \left( {1 + {f}_{2}^{ - } } \right)^{{\nu_{2} }} } \right)}}{{\left( {1 + \left| {{f}_{2}^{ - } } \right|} \right)^{{\nu_{2} }} + \left( {1 + {f}_{2}^{ - } } \right)^{{\nu_{2} }} }} \hfill \\ \end{gathered} \right) $$

and for

$$\mathop \oplus \limits_{\ell = 1}^2 \left( {{\nu _\ell }{u_\ell }} \right)= \left( \begin{array}{l} \frac{{\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {1 + \Im _\ell ^ + } \right)}^{{\nu _\ell }}} - \mathop \prod \nolimits_{\ell = 1}^2 {{\left( {1 - \Im _\ell ^ + } \right)}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {1 {\textit{+}}\Im _\ell ^ + } \right)}^{{\nu _\ell }}} {\textit{+}}\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {1 - \Im _\ell ^ + } \right)}^{{\nu _\ell }}}}},\frac{{2\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {{I}_\ell ^ + } \right)}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {2 - {I}_\ell ^ + } \right)}^{{\nu _\ell }}} + \mathop \Pi \nolimits_{\ell = 1}^2 {{\left( {{I}_\ell ^ + } \right)}^{{\nu _\ell }}}}},\\ \frac{{2\mathop \Pi \nolimits_{\ell = 1}^2 {{\left( {f_\ell ^{ {\textit{+}}}} \right)}^{{\nu _\ell }}}}}{{\mathop \Pi \nolimits_{\ell = 1}^2 {{\left( {2 - f_\ell ^{ {\textit{+}}}} \right)}^{{\nu _\ell }}} + \mathop \Pi \nolimits_{\ell = 1}^2 {{\left( {f_\ell ^{ {\textit{+}}}} \right)}^{{\nu _\ell }}}}},\frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^2 {{\left| {\Im _\ell ^ - } \right|}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {2 + \Im _\ell ^ - } \right)}^{{\nu _\ell }}} + \mathop \prod \nolimits_{\ell = 1}^2 {{\left| {\Im _\ell ^ - } \right|}^{{\nu _\ell }}}}},\\ \frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^2 {{\left| {{I}_\ell ^ - } \right|}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {2 + {I}_\ell ^ - } \right)}^{{\nu _\ell }}} + \mathop \prod \nolimits_{\ell = 1}^2 {{\left| {{I}_\ell ^ - } \right|}^{{\nu _\ell }}}}}, - \frac{{\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {1 {\textit{+}}\ominus \left| {f_\ell ^ - } \right|} \right)}^{{\nu _\ell }}} - \mathop \prod \nolimits_{\ell = 1}^2 {{\left( {1 {\textit{+}}f_\ell ^ - } \right)}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {1 + \left| {f_\ell ^ - } \right|} \right)}^{{\nu _\ell }}} {\textit{+}}\mathop \prod \nolimits_{\ell = 1}^2 {{\left( {1 + f_\ell ^ - } \right)}^{{\nu _\ell }}}}} \end{array} \right)$$

thus, satisfy for $n^{{}} =_{{}} 2$.

put $n^{{}} =_{{}} r$ for Eq. (3),

$$ \begin{aligned} & {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{r} ,u_{r} } \right\rangle } \right) \, = \left( {\frac{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{1} - {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{2} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{1} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{2} }},\frac{{{2}\zeta_{1} }}{{\zeta_{2} { + }\zeta_{1} }},\frac{{{2}{\varvec{f}}_{1} }}{{{\varvec{f}}_{2} { + }{\varvec{f}}_{1} }},\frac{{ - {2}{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{3} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{4} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{3} }}, - \frac{{\zeta_{3} - \zeta_{4} }}{{\zeta_{3} { + }\zeta_{4} }}, - \frac{{{\varvec{f}}_{3} - {\varvec{f}}_{4} }}{{{\varvec{f}}_{3} { + }{\varvec{f}}_{4} }}} \right) \\ & \quad = \left( \begin{gathered} \frac{{\mathop \prod \nolimits_{{_{\ell = 1} }}^{r} \left( {1 + {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} - \mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 - {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{{r}{}} \left( {_{{}} 1 {\textit{+}}{\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{{^{\ell = 1} }}^{r} \left( {1 - {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }},\frac{{2\mathop \prod \nolimits_{\ell = 1}^{r} \left( {{I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}{{\mathop {^{\prod } }\nolimits_{\ell = 1}^{r} \left( {2 - {I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} + \mathop \Pi \nolimits_{\ell = 1}^{r} \left( {{I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}, \hfill \\ \frac{{2\mathop \Pi \nolimits_{\ell = 1}^{r} \left( {{f}_{\ell }^{ {\textit{+}}} } \right)^{{\nu_{\ell } }} }}{{\mathop \Pi \nolimits_{\ell = 1}^{r} \left( {2 - {f}_{\ell }^{ {\textit{+}}} } \right)^{{\nu_{\ell } }} + \mathop \Pi \nolimits_{\ell = 1}^{r} \left( {{f}_{\ell }^{ {\textit{+}}} } \right)^{{\nu_{\ell } }} }},\frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^{r} \left| {{\Im }_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {2 + {\Im }_{\ell }^{ - } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{\ell = 1}^{r} \left| {{\Im }_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}, \hfill \\ \frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^{r} \left| {{I}_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {2 + {I}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{\ell = 1}^{r} \left| {{I}_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}, - \frac{{\mathop \prod \nolimits_{\ell = 1}^{k} \left( {1 {\textit{+}}{\ominus }\left| {{f}_{\ell }^{ - } } \right|^{{}} } \right)^{{\nu_{\ell } }} - \mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 = *{f}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 + \left| {{f}_{\ell }^{ - } } \right|} \right)^{{\nu_{\ell } }} {\textit{+}}\mathop {^{\prod } }\nolimits_{\ell = 1}^{r} \left( {1 + {f}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} }} \hfill \\ \end{gathered} \right), \, \\ \end{aligned} $$

suppose that it, if $n = r.$

Now, for $n = r + 1,$ thus

$$ \begin{aligned} & {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\left\langle {\alpha_{3} ,u_{3} } \right\rangle , \ldots ,\left\langle {\alpha_{r} ,u_{r} } \right\rangle } \right) \, \\ & \quad = \left( \begin{gathered} \frac{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 {\textit{+}}{\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} - \mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 - {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 {\textit{+}}{\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} {\textit{+}}\mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 - {\Im }_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }},\frac{{2\mathop \prod \nolimits_{\ell = 1}^{r} \left( {{I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {2 - {I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{\ell = 1}^{r} \left( {{I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}, \hfill \\ \frac{{2\mathop \prod \nolimits_{\ell = 1}^{r} \left( {{f}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {2 - {f}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{\ell = 1}^{r} \left( {{f}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} }},\frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^{r} \left| {{\Im }_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {2 + {\Im }_{\ell }^{ - } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{\ell = 1}^{r} \left| {{\Im }_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}, \hfill \\ \frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^{r} \left| {{I}_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {2 + {I}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} + \mathop \prod \nolimits_{\ell = 1}^{r} \left| {{I}_{\ell }^{ - } } \right|^{{\nu_{\ell } }} }}, - \frac{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 + {\ominus }\left| {{f}_{\ell }^{ - } } \right|} \right)^{{\nu_{\ell } }} - \mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 + {f}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} }}{{\mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 + \left| {{f}_{\ell }^{ - } } \right|} \right)^{{\nu_{\ell } }} {\textit{+}}\mathop \prod \nolimits_{\ell = 1}^{r} \left( {1 + {f}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} }} \hfill \\ \end{gathered} \right) \, \\ \end{aligned} $$

$$ \oplus \left( \begin{gathered} \frac{{\left( {1 + {\Im }_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r{ + 1}}} }} - \left( {1 - {\Im }_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}{{\left( {1^{{}} {\textit{+}}{\Im }_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} {\textit{+}}\left( {1 - {\Im }_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }},\frac{{2\left( {{I}_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}{{\left( {2 - {I}_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} + \left( {{I}_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}, \hfill \\ \frac{{2\left( {{f}_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}{{\left( {2 - {f}_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} + \left( {{f}_{{{\varvec{r}} {\textit{+}}1}}^{ {\textit{+}}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }},\frac{{ - 2\left| {{\Im }_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right|^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}{{\left( {2 + {\Im }_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} + \left| {{\Im }_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right|^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}, \hfill \\ \frac{{ - 2\left| {{I}_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right|^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}{{\left( {2 + {I}_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} + \left| {_{{}} {I}_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right|^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}, - \frac{{\left( {1 {\textit{+}}{\ominus }\left| {{f}_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right|^{{}} } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} - \left( {1 {\textit{+}}{f}_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }}{{\left( {1 {\textit{+}}\left| {{f}_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right|} \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} {\textit{+}}\left( {1 {\textit{+}}{f}_{{{\varvec{r}} {\textit{+}}1}}^{ - } } \right)^{{{{\varvec{\upnu}}}_{{r {\textit{+}}{\varvec{1}}}} }} }} \hfill \\ \end{gathered} \right) = \left( \begin{array}{l} \frac{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 {\textit{+}}\Im _\ell ^ + } \right)}^{{\nu _\ell }}} - \mathop \prod \nolimits_{\ell = 1}^{^{r + 1}} {{\left( {1 - \Im _\ell ^ + } \right)}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 {\textit{+}}\Im _\ell ^ + } \right)}^{{\nu _\ell }}} {\textit{+}}\mathop \prod \nolimits_{\ell = 1}^{^{r + 1}} {{\left( {1 - \Im _\ell ^ + } \right)}^{{\nu _\ell }}}}},\frac{{2\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {{I}_\ell ^ + } \right)}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {2 - {I}_\ell ^ + } \right)}^{{\nu _\ell }}} + \mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {{I}_\ell ^ + } \right)}^{{\nu _\ell }}}}},\\ \frac{{2\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {f_\ell ^ + } \right)}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {2 - f_\ell ^ + } \right)}^{{\nu _\ell }}} + \mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {f_\ell ^ + } \right)}^{{\nu _\ell }}}}},\frac{{ - 2\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left| {\Im _\ell ^ - } \right|}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {2 + \Im _\ell ^ - } \right)}^{{\nu _\ell }}} + \mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left| {\Im _\ell ^ - } \right|}^{{\nu _\ell }}}}},\\ - \frac{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 + \ominus \left| {{I}_\ell ^ - } \right|} \right)}^{{\nu _\ell }}} - \mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 + {I}_\ell ^ - } \right)}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 + \left| {{I}_\ell ^ - } \right|} \right)}^{{\nu _\ell }}} + \mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 + {I}_\ell ^ - } \right)}^{{\nu _\ell }}}}}, - \frac{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 + \ominus \left| {f_\ell ^ - } \right|} \right)}^{{\nu _\ell }}} - \mathop \prod \nolimits_{\ell = 1}^{r{ + ^{}}1} {{\left( {1 + f_\ell ^ - } \right)}^{{\nu _\ell }}}}}{{\mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 + \left| {f_\ell ^ - } \right|} \right)}^{{\nu _\ell }}} + \mathop \prod \nolimits_{\ell = 1}^{r + 1} {{\left( {1 + f_\ell ^ - } \right)}^{{\nu _\ell }}}}} \end{array} \right)$$

Hence, Eq. (3), all values are satisfied for $n^{{}} =_{{}} r + 1$.

Hence proved.

Theorem 13

(Idempotency) Suppose $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n} \right)$, represent 2-tuples set of equal BN numbers, i.e., $u_{\ell } = u$,

$$ {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = u \, $$

Proof

Since

$$ \begin{gathered} {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1,} u_{1} } \right\rangle ,\left\langle {\alpha_{2,} u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n,} u_{n} } \right\rangle } \right) \hfill \\ = \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u_{\ell } } \right) = \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } } \right)u = \left( u \right)\sum\limits_{\ell = 1}^{n} {\nu_{\ell } } = u \hfill \\ \end{gathered} $$

the proof is complete.

Theorem 14

(Boundedness) Let $_{{}} u^{{ + }{}} =_{{}} \mathop {\max }\nolimits_{\ell }^{{}} u_{\ell }$, $u^{{ - }{}} =_{{}} \mathop {\min }\nolimits_{\ell }^{{}} u_{\ell } ,$ represents minimum, as well as maximum BN value

$$ u^{ - } \le {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \le u^{ + } $$

Proof

Let

$$ {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1,} u_{1} } \right\rangle ,\left\langle {\alpha_{2,} u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n,} u_{n} } \right\rangle } \right) = u\left( {{\Im }_{{}}^{ + } ,{I}_{{}}^{ + } ,{f}_{{}}^{ + } ,{\Im }_{{}}^{ - } ,{I}_{{}}^{ - } ,{f}_{{}}^{ - } } \right). $$

Then,

$s\left( {u^{ - } } \right) \le s\left( {\text{I-BNEWA operator}} \right) \,$

$s\left( {u^{ - } } \right) \le s\left( {\text{I-BNEWA operator}} \right)$

$s\left( {u^{ + } } \right) \ge s\left( {\text{I-BNEWA operator}} \right)$

Combining both equations, we have

$$ u^{ - } \le {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1,} u_{1} } \right\rangle ,\left\langle {\alpha_{2,} u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n,} u_{n} } \right\rangle } \right) \le u^{ + } \, {.} $$

Thus, the proof is complete.

Theorem 15

(Monotonicity) Suppose $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle$, and $\left\langle {\alpha^{\prime}_{\ell ,} u^{\prime}_{\ell } } \right\rangle$, where $\ell \in Z,Z = \left\{ {1,2,\ldots,n} \right\}$ be BN number. Then $u_{\ell } \le u^{\prime}_{\ell }$,

$$ {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \le {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha^{\prime}_{1} ,u^{\prime}_{1} } \right\rangle ,\left\langle {\alpha^{\prime}_{2} ,u^{\prime}_{2} } \right\rangle , \ldots ,\left\langle {\alpha^{\prime}_{n} ,u^{\prime}_{n} } \right\rangle } \right) $$

Proof

Assume that

$$ {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha_{1,} u_{1} } \right\rangle ,\left\langle {\alpha_{2,} u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n,} u_{n} } \right\rangle } \right) = \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u_{\ell } } \right) $$

and

$$ {\text{I-BNEWA}}_{\nu } \left( {\left\langle {\alpha^{\prime}_{1,} u^{\prime}_{1} } \right\rangle ,\left\langle {\alpha^{\prime}_{2,} u^{\prime}_{2} } \right\rangle , \ldots ,\left\langle {\alpha^{\prime}_{n,} u^{\prime}_{n} } \right\rangle } \right) = \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u^{\prime}_{\ell } } \right) $$

Since $u_{\ell } \le u^{\prime}_{\ell }$,then

$$ \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u_{\ell } } \right) \le \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u^{\prime}_{\ell } } \right) $$

Thus, the proof is complete.

Induced bipolar neutrosophic Einstein ordered weighted average AO

Definition 16

Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n - 1,n} \right)$ be there a set of 2-tuples, then (I-BNEOWA) induced bipolar neutrosophic Einstein ordered weighted average aggregation operator is stated as:

$$\begin{aligned} &{\text{I-BNEOWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u_{\rho \left( \ell \right)} } \right) \\ &\quad = \nu_{1} u_{\rho \left( 1 \right)} \oplus \nu_{2} u_{\rho \left( 2 \right)} \oplus \nu_{3} u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma } \left( 3 \right)}} \oplus \cdots \oplus \nu_{n} u_{\rho \left( n \right)} \end{aligned}$$

(4)

here $\left( {\rho \left( 1 \right),\rho \left( 2 \right), \ldots ,\rho \left( n \right)} \right)$ represent a permutation that is $u_{{\rho \left( {\ell - 1} \right)}} \ge u_{\rho \left( \ell \right)}$,$\forall$$\ell \in Z$_, $Z^{{}} =_{{}} \left\{ {1, \ldots ,n - 1,n} \right\}$ as well as $\nu = \left( {\nu_{1} ,\nu_{2} , \ldots ,\nu_{n} } \right)^{T}$ represent associated weighting vector for $u_{\ell }$ with $\nu_{\ell } > 0$, $\mathop \sum \nolimits_{\ell = 1}^{{^{n} }} \nu_{\ell } = 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ as the bipolar neutrosophic arguments variable.

Theorem 17

Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2, \ldots ,n - 1,n} \right)$ be there a set of 2-tuples, then the BN values of (I-BNEOWA) operators give in returns give a BN value as

$$ {\text{I-BNEOWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle , \ldots ,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \, = \quad \left( \frac{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }1}} - {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }2}} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }1}} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{2} }},\frac{{{2}\zeta_{{\boldsymbol{\partial }1}} }}{{\zeta_{{\boldsymbol{\partial }2}} { + }\zeta_{{\boldsymbol{\partial }1}} }},\frac{{{2}{\varvec{f}}_{{\boldsymbol{\partial }1}} }}{{{\varvec{f}}_{{\boldsymbol{\partial 2}}} { + }{\varvec{f}}_{{\boldsymbol{\partial }1}} }},\frac{{ - {{2{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\boldsymbol{\partial }3}} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }4}} {{ + {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\boldsymbol{\partial }3}} }},{ - }\frac{{\zeta_{{\boldsymbol{\partial }3}} { - }\zeta_{{\boldsymbol{\partial }4}} }}{{\zeta_{3} { + }\zeta_{{\boldsymbol{\partial }4}} }},{ - }\frac{{{\varvec{f}}_{{\boldsymbol{\partial }3}} - \user2{f}_{{\boldsymbol{\partial }4}} }}{{{\varvec{f}}_{{\boldsymbol{\partial }3}} { + }{\varvec{f}}_{{\boldsymbol{\partial }4}} }} \right) $$

(5)

here,

$$ \begin{aligned} {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{\partial 1} & = \mathop {^{{}} \prod }\limits_{\ell = 1}^{{{n}}} \left( {1 + {\Im }_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \, {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{\partial 2} =_{{}} \mathop {\prod^{{}} }\limits_{\ell = 1}^{{{n}}} \left( {1 - {\Im }_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \\ \, {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{\partial 3} &= \mathop \prod \limits_{\ell = 1}^{{{n}}} \left| {{\Im }_{\rho \left( \ell \right)}^{ - } } \right|^{{\nu_{\ell } }} { , }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{\partial 4} = \mathop \prod \limits_{\ell = 1}^{{{n}}} \left( {{2 + }{\Im }_{\rho \left( \ell \right)}^{ - } } \right)^{{\nu_{\ell } }} , \\ \zeta_{\partial 1} & = \mathop \prod \limits_{\ell = 1}^{n} \left( {{I}_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \, \zeta_{\partial 2} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 - {I}_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \,\\ \zeta_{\partial 3} &= \mathop{\varvec{\varPi}}\limits_{\ell = 1}^{{{n}}} \left( {1 + {\ominus }\left| {{I}_{\rho \left( \ell \right)}^{ - } } \right|} \right)^{{\nu_{\ell } }} , \, \zeta_{\partial 4} = \mathop{\varvec{\varPi}}\limits_{\ell = 1}^{{{n}}} \left( {{1 + }{I}_{\rho \left( \ell \right)}^{ - } } \right)^{{\nu_{\ell } }} , \\ {\varvec{f}}_{\partial 1} & = \mathop{\varvec{\varPi}}\limits_{\ell = 1}^{{{n}}} \left( {{f}_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \, {\varvec{f}}_{\partial 2} = \mathop{\varvec{\varPi}}\limits_{\ell = 1}^{{{n}}} \left( {2 - {f}_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \\ \, {\varvec{f}}_{\partial 3} &= \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + {\ominus }\left| {{f}_{\rho \left( \ell \right)}^{ - } } \right|} \right)^{{\nu_{\ell } }} , \, {\varvec{f}}_{\partial 4} = \mathop \prod \limits_{\ell = 1}^{n} \left( {{1 + }{f}_{\rho \left( \ell \right)}^{ - } } \right)^{{\nu_{\ell } }} {. } \\ \end{aligned} $$

where $\left( {\rho \left( 1 \right),\rho \left( 2 \right),\rho \left( 3 \right),\ldots,\rho \left( n \right)} \right)$ represent a permutation along with $u_{{\rho \left( {\ell - 1} \right)}} \ge u_{\rho \left( \ell \right)}$, $\forall$$\ell \in Z$_,$Z^{{}} =_{{}} \left\{ {1,2,\ldots,n - 1,n} \right\}$, and ${{\varvec{\upnu}}} = \left( {{{\varvec{\upnu}}}_{1} ,{{\varvec{\upnu}}}_{2} ,{{\varvec{\upnu}}}_{3} ,\ldots,{{\varvec{\upnu}}}_{n - 1} ,{{\varvec{\upnu}}}_{n} } \right)^{T}$ represent associated weighting vector for $u_{\ell }$ with $\nu_{\ell } > 0$ and $\mathop \sum \nolimits_{\ell = 1}^{{n}{}} \nu_{\ell } =_{{}} 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ as the bipolar neutrosophic arguments variable.

Proof

Theorem 12 leads to this proof.

Theorem 18

(Idempotency) Suppose $\left\langle {{{\varvec{\upalpha}}}_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,3,\ldots,n} \right)$, represent a set containing equal BN numbers, i.e., $u_{\ell } = u$,

$$ {\text{I-BNEOWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = u $$

Theorem 19

(Bounded) Suppose $u^{ + } = \mathop {\max }\nolimits_{\ell } u_{\ell }$, $u^{ - } = \mathop {\min }\nolimits_{\ell } u_{\ell } ,$ represent minimum and maximum BNN.

$$ u^{ - } \le {\text{I-BNEOWA}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \le u^{ + } $$

Theorem 20

(Monotonicity) Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,3,\ldots,n} \right)$, and $\left\langle {\alpha^{\prime}_{\ell ,} u^{\prime}_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n} \right)$, where $\ell \in Z$ be BN numbers. If there is $u_{\ell } \le u^{\prime}_{\ell }$,

$$ {\text{I-BNEOWA}}_{\varpi } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \le {\text{I-BNEOWA}}_{\varpi } \left( {\left\langle {\alpha^{\prime}_{1} ,u^{\prime}_{1} } \right\rangle ,\ldots,\left\langle {\alpha^{\prime}_{n - 1} ,u^{\prime}_{n - 1} } \right\rangle ,\left\langle {\alpha^{\prime}_{n} ,u^{\prime}_{n} } \right\rangle } \right) $$

Induced bipolar neutrosophic Einstein hybrid average AO

Definition 21

Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n - 1,n} \right)$ be there a set of 2-tuples, then (I-BNEHA) induced bipolar neutrosophic Einstein hybrid average aggregation operator is defined below:

$$ \begin{aligned} & {\text{I-BNEHA}}_{w,\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = \mathop \oplus \limits_{\ell = 1}^{n} \left( {\nu_{\ell } \dot{u}_{\rho \left( \ell \right)} } \right) \\ & \quad = \nu_{1} \dot{u}_{\rho \left( 1 \right)} \oplus \nu_{2} \dot{u}_{\rho \left( 2 \right)} \oplus \nu_{3} \dot{u}_{\rho \left( 3 \right)} \oplus \cdots \oplus \nu_{n} \dot{u}_{\rho \left( n \right)} , \\ \end{aligned} $$

(6)

here $w = \left( {w_{1} ,w_{2} ,w_{3} ,\ldots,w_{n} } \right)$ represents weighted vector for $u_{\ell }$, $w_{\ell } \in \left[ {0,1} \right],$$\mathop \sum \nolimits_{\ell = 1}^{n} w_{\ell } = 1.$ Here $\dot{u}_{\rho \left( \ell \right)}$ be a $\ell$-th biggest element from BN numbers $\dot{u}_{\ell } ,\left( {\dot{u}_{\ell } = \left( {n\nu_{\ell } } \right)u_{\ell } ,\ell = 1,2,\ldots,n} \right)$, also $\nu = \left( {\nu_{1} ,\nu_{2} ,\ldots,\nu_{n} } \right)$ be a weighted vector for BN argument $u_{\ell }$, with $\nu_{\ell } \in \left[ {0,1} \right],$$\mathop \sum \nolimits_{\ell = 1}^{n} \nu_{\ell } = 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ as the bipolar neutrosophic arguments variable.

Remarks

I-BNEHA reduce to I-BNEWA if $w = \left( {\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}} \right)^{T}$ as well as I-BNEOWA operator if $\nu = \left( {\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}} \right).$

Theorem 22

Suppose $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,\ldots,n - 1,n} \right)$ be there represents a collection that contains 2-tuple, then (I-BNEHA) operators proceed a bipolar neutrosophic number with.

$$ \begin{aligned} & {\text{I-BNEHA}}_{w,\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \\ & = \quad \left( {\frac{{\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{1} - \mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{2} }}{{\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{1} { + }\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{2} }},\frac{{{2}\mathop \zeta \limits^{ \bullet }_{1} }}{{\mathop \zeta \limits^{ \bullet }_{2} { + }\mathop \zeta \limits^{ \bullet }_{1} }},\frac{{{2}\mathop {\varvec{f}}\limits^{ \bullet }_{1} }}{{\mathop {\varvec{f}}\limits^{ \bullet }_{2} { + }\mathop {\varvec{f}}\limits^{ \bullet }_{1} }},\frac{{ - {2}\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{3} }}{{\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{4} { + }\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{3} }},{ - }\frac{{\mathop \zeta \limits^{ \bullet }_{3} { - }\mathop \zeta \limits^{ \bullet }_{4} }}{{\mathop \zeta \limits^{ \bullet }_{3} { + }\mathop \zeta \limits^{ \bullet }_{4} }},{ - }\frac{{\mathop {\varvec{f}}\limits^{ \bullet }_{3} { - }\mathop {\varvec{f}}\limits^{ \bullet }_{4} }}{{\mathop {\varvec{f}}\limits^{ \bullet }_{3} { + }\mathop {\varvec{f}}\limits^{ \bullet }_{4} }}} \right) \\ \end{aligned} $$

(7)

where,

$$ \begin{aligned} &\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{1} }\limits^{ \bullet } = \mathop \prod \limits_{{_{\ell = 1} }}^{n} \left( {1 + {\dot{\Im }}_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \, \mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{2} }\limits^{ \bullet } = \mathop \prod \limits_{{_{\ell = 1} }}^{n} \left( {1 - {\dot{\Im }}_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \,\\ &\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{3} = \mathop \prod \limits_{\ell = 1}^{n} \left| {{\dot{\Im }}_{\rho \left( \ell \right)}^{ - } } \right|^{{\nu_{\ell } }} { , }\mathop {{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}\limits^{ \bullet }_{4} = \mathop \prod \limits_{{\ell =_{{}} 1}}^{n} \left( {2 + {\dot{\Im }}_{\rho \left( \ell \right)}^{ - } } \right)^{{\nu_{\ell } }} , \\ &\mathop \zeta \limits^{ \bullet }_{1} = \mathop \prod \limits_{\ell = 1}^{n} \left( {{\dot{I}}_{{_{\rho \left( \ell \right)} }}^{ + } } \right)^{{\nu_{\ell } }} , \, \mathop \zeta \limits^{ \bullet }_{2} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 - {\dot{I}}_{{_{\rho \left( \ell \right)} }}^{ + } } \right)^{{\nu_{\ell } }} , \\ &\, \mathop \zeta \limits^{ \bullet }_{3} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + {\ominus }\left| {{\dot{I}}_{{_{\rho \left( \ell \right)} }}^{ - } } \right|} \right)^{{\nu_{\ell } }} , \, \mathop \zeta \limits^{ \bullet }_{4} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + {\dot{I}}_{{_{\rho \left( \ell \right)} }}^{ - } } \right)^{{\nu_{\ell } }} , \\ &\mathop {\varvec{f}}\limits^{ \bullet }_{1} = \mathop \prod \limits_{\ell = 1}^{n} \left( {{\dot{f}}_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \, \mathop {\varvec{f}}\limits^{ \bullet }_{2} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 - {\dot{f}}_{\rho \left( \ell \right)}^{ + } } \right)^{{\nu_{\ell } }} , \\ &\, \mathop {\varvec{f}}\limits^{ \bullet }_{3} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + {\ominus }\left| {{\dot{f}}_{\rho \left( \ell \right)}^{ - } } \right|} \right)^{{\nu_{\ell } }} , \, \mathop {\varvec{f}}\limits^{ \bullet }_{4} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + {\dot{f}}_{\rho \left( \ell \right)}^{ - } } \right)^{{\nu_{\ell } }} {. } \end{aligned}$$

here $w = \left( {w_{1} ,\ldots,w_{n - 1} ,w_{n} } \right)$ be a weighted vector of $u_{\ell }$, $w_{\ell } \in \left[ {0,^{{}} 1} \right],$$\mathop \sum \nolimits_{\ell = 1}^{n} w_{\ell } = 1$ as well as $\dot{u}_{\rho \left( \ell \right)}$ be a $\ell$-th largest element of BN numbers $\dot{u}_{\ell } \left( {\dot{u}_{\ell } = \left( {n\nu_{\ell } } \right)u_{\ell } ,\ell = 1,2,\ldots,n} \right)$, also $\nu = \left( {\nu_{1} ,\nu_{2} ,\ldots,\nu_{n} } \right)$ represent weighted vector of BN numbers $u_{\ell }$, with $\nu_{\ell } \in \left[ {0,^{{}} 1} \right],$$\mathop {\sum^{{}} }\nolimits_{\ell = 1}^{n} \nu_{\ell } = 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ as the bipolar neutrosophic arguments variable.

Proof

Theorem is straight forward.

Induced bipolar neutrosophic Einstein geometric aggregation operators

The next segment broadens our research to consider the conditions necessary for each of the induced bipolar neutrosophic Einstein geometric aggregation operator, like geometric I-BNEWG, ordered geometric I-BNEOWG, as well as hybrid geometric I-BNHG.

Induced bipolar neutrosophic Einstein weighted geometric AOs

Consider $u_{\ell } = \left( {\Im_{\ell }^{ + } ,{I}_{\ell }^{ + } ,{f}_{\ell }^{ + } ,\Im_{\ell }^{ - } ,{I}_{\ell }^{ - } ,{f}_{\ell }^{ - } } \right)\left( {\ell = 1,2,\ldots,n} \right)$ stand for family of BN values.

Definition 23

Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n - 1,n} \right)$ represent a set of 2-tuples, then (I-BNEWG) induced BN Einstein weighted geometric operator define as:

$$\begin{aligned} &{\text{I-BNEWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \\ &\quad = \mathop \otimes \limits_{\ell = 1}^{n} \left( {\nu_{\ell } u_{\ell } } \right) =^{{}} \nu_{1} u_{1} \otimes \nu_{2} u_{2} \otimes \cdots \otimes \nu_{n} u_{n} \end{aligned}$$

(8)

here $\nu = \left( {_{{}} \nu_{1} ,\nu_{2} ,\nu_{3} ,\ldots,\nu_{n} } \right)^{T}$ represents a weighting vector for $u_{\ell }$, like $\nu_{\ell } > 0$ and $\mathop \sum \nolimits_{\ell = 1}^{n} \nu_{\ell } = 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ for the bipolar neutrosophic arguments variables.

Theorem 24

Suppose $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n - 1,n} \right)$ be there a set that contains 2-tuples, then (I-BNEWG) operators returns a BNV with

$$ \begin{aligned} & {\text{I-BNEWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \\ & \quad = \left( {\frac{{{2}{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{1}}} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{2}}} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{1}}} }},\frac{{\zeta_{{\tilde{1}}} - \zeta_{{\tilde{2}}} }}{{\zeta_{{\tilde{1}}} { + }\zeta_{{\tilde{2}}} }},\frac{{{\varvec{f}}_{{\tilde{1}}} - {\varvec{f}}_{{\tilde{2}}} }}{{{\varvec{f}}_{{\tilde{1}}} { + }{\varvec{f}}_{{\tilde{2}}} }}, - \frac{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{3}}} - {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{4}}} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{3}}} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{4}}} }},\frac{{ - {2}\zeta_{{\tilde{3}}} }}{{\zeta_{{\tilde{4}}} { + }\zeta_{{\tilde{3}}} }},\frac{{ - {2}{\varvec{f}}_{{\tilde{3}}} }}{{{\varvec{f}}_{{\tilde{4}}} { + }{\varvec{f}}_{{\tilde{3}}} }}} \right) \hfill\\ \end{aligned} $$

(9)

where,

$$ \begin{aligned} &{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{1}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {\Im_{\ell }^{ + } } \right)^{{\nu_{\ell } }} ,\zeta_{{\tilde{1}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1{ + }{I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} , \\ &\, {\varvec{f}}_{{\tilde{1}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1{ + }{f}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} ,{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{2}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 - \Im_{\ell }^{ + } } \right)^{{\nu_{\ell } }} ,\\ &\zeta_{{\tilde{2}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 - {I}_{\ell }^{ + } } \right)^{{\nu_{\ell } }} ,{\varvec{f}}_{{\tilde{2}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 - {f}_{\ell }^{ + } } \right)^{{^{{\nu_{\ell } }} }} , \, \hfill \\ &{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{3}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + {\ominus }\left| {\Im_{\ell }^{ - } } \right|} \right)^{{\nu_{\ell } }} { ,}\zeta_{{\tilde{3}}} = \mathop \prod \limits_{\ell = 1}^{n} \left| {{I}_{\ell }^{ - } } \right|^{{\nu_{\ell } }} ,\\ &{\varvec{f}}_{{\tilde{3}}} = \mathop \prod \limits_{\ell = 1}^{n} \left| {{f}_{\ell }^{ - } } \right|^{{\nu_{\ell } }} , \, {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{4}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + \Im_{\ell }^{ - } } \right)^{{\nu_{\ell } }} ,\\ &\zeta_{{\tilde{4}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 + {I}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} , \, {\varvec{f}}_{{\tilde{4}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 + {f}_{\ell }^{ - } } \right)^{{\nu_{\ell } }} {.} \end{aligned} $$

here $\nu = \left( {_{{}} \nu_{1} ,\nu_{2} ,\ldots,\nu_{n} } \right)^{T}$ be a weighting vector for $u_{\ell }$, like $\nu_{\ell } > 0$ as well as $\mathop \sum \nolimits_{\ell = 1}^{n} \nu_{\ell } = 1.$ Where $\alpha_{\ell }$ in $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$ known as the order-inducing variable and $u_{\ell }$ the bipolar neutrosophic argument variables.

Proof

Theorem 1 leads to this proof.

Theorem 25

(Idempotency) Suppose $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n} \right)$ is collection that contains equal BN values, that is, $u_{\ell } = u$,

$$ {\text{I-BNEWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = u $$

Theorem 26

(Bounded) Suppose $u^{ + } = \mathop {\max }\nolimits_{\ell } u_{\ell } ,$ $u^{ - } = \mathop {\min }\nolimits_{\ell } u_{\ell } ,$ the

$$ u^{ - } \le {\text{I-BNEWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \le u^{ + } $$

Theorem 27

(Monotonicity) Suppose $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n} \right)$, as well as $\left\langle {\alpha^{\prime}_{\ell ,} u^{\prime}_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n} \right)$ two set that contains bipolar neutrosophic values. If, we have $u_{\ell } \le u^{\prime}_{\ell }$, the

$$ {\text{I-BNEWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \le {\text{I-BNEWG}}_{\nu } \left( \left\langle {\alpha^{\prime}_{1} ,u^{\prime}_{1} } \right\rangle ,\ldots,\left\langle {\alpha^{\prime}_{n} ,u^{\prime}_{n} } \right\rangle \right) $$

Induced bipolar neutrosophic Einstein ordered weighted geometric AO

Definition 28

Suppose $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle$ be there a set of 2-tuples, then (I-BNEOWG) induced bipolar neutrosophic Einstein ordered weighted geometric AOs is defined below:

$$\begin{aligned} &{\text{I-BNEOWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = \mathop \otimes \limits_{\ell = 1}^{n} \\ &\quad \left( {\nu_{\ell } u_{\sigma \left( \ell \right)} } \right) =^{{}} \nu_{1} u_{\sigma \left( 1 \right)} \otimes \nu_{2} u_{\sigma \left( 2 \right)} \otimes \nu_{3} u_{\sigma \left( 3 \right)} \otimes \cdots \otimes \nu_{\ell } u_{\sigma \left( \ell \right)} , \end{aligned}$$

(10)

where $\left( {\sigma \left( 1 \right),\ldots,\sigma \left( {n - 1} \right),\sigma \left( n \right)} \right)$ represent permutation of $(1,2,3,\ldots,n)$ with $u_{{\sigma \left( {\ell - 1} \right)}} \ge^{{}} u_{\sigma \left( \ell \right)}$, $(\ell = 2,3,\ldots,n)$ and $\nu = \left( {_{{}} \nu_{1} ,\nu_{2} ,\ldots,\nu_{n} } \right)^{T}$ represent weighting vector for $u_{\ell }$, like $\nu_{\ell } > 0$ as well as $\mathop \sum \nolimits_{\ell = 1}^{n} \nu_{\ell } = 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ the bipolar neutrosophic arguments variables.

Theorem 29

Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle$ be there a set of 2-tuples, then (I-BNEOWG) operators returns a BN number

$$ {\text{I-BNEOWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = \left( \frac{{{2}{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\partial \tilde{1}}} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }\tilde{2}}} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\partial \tilde{1}}} }},\frac{{\zeta_{{\partial \tilde{1}}} - \zeta_{{\partial \tilde{2}}} }}{{\zeta_{{\partial \tilde{1}}} { + }\zeta_{{\partial \tilde{2}}} }},\frac{{{\varvec{f}}_{{\partial \tilde{1}}} - {\varvec{f}}_{{\partial \tilde{2}}} }}{{{\varvec{f}}_{{\partial \tilde{1}}} { + }{\varvec{f}}_{{\partial \tilde{2}}} }}, - \frac{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }\tilde{3}}} - {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }\tilde{4}}} }}{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }\tilde{3}}} { + }{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\boldsymbol{\partial }\tilde{4}}} }},\frac{{ - {2}\zeta_{{\partial \tilde{3}}} }}{{\zeta_{{\partial \tilde{4}}} { + }\zeta_{{\partial \tilde{3}}} }},\frac{{ - {2}{\varvec{f}}_{{\partial \tilde{3}}} }}{{{\varvec{f}}_{{\partial \tilde{4}}} { + }{\varvec{f}}_{{\partial \tilde{3}}} }} \right) \\ $$

(11)

where,

$$ \begin{aligned} & {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\partial \tilde{1}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {\Im_{\sigma \left( \ell \right)}^{ + } } \right)^{{\omega_{j} }} ,\zeta_{{\partial \tilde{1}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1{ + }{I}_{\sigma \left( \ell \right)}^{ + } } \right)^{{\omega_{j} }} ,\\ &{\varvec{f}}_{{\partial \tilde{1}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1{ + }{f}_{\sigma \left( j \right)}^{ + } } \right)^{{\omega_{j} }} , \, {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\partial \tilde{2}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 - \Im_{\sigma \left( \ell \right)}^{ + } } \right)^{{\omega_{j} }} ,\\ &\zeta_{{\partial \tilde{2}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 - {I}_{\sigma \left( \ell \right)}^{ + } } \right)^{{\omega_{j} }} ,{\varvec{f}}_{{\partial \tilde{2}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 - {f}_{\sigma \left( \ell \right)}^{ + } } \right)^{{\omega_{j} }} , \, \\ & {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\partial \tilde{3}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + {\ominus }\left| {\Im_{\sigma \left( \ell \right)}^{ - } } \right|} \right)^{{\omega_{j} }} { ,}\zeta_{{\partial \tilde{3}}} = \mathop \prod \limits_{\ell = 1}^{n} \left| {{I}_{\sigma \left( \ell \right)}^{ - } } \right|^{{\omega_{j} }} ,\\ &{\varvec{f}}_{{\partial \tilde{3}}} = \mathop \prod \limits_{\ell = 1}^{n} \left| {{f}_{\sigma \left( \ell \right)}^{ - } } \right|^{{\omega_{j} }} , \, {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\partial \tilde{4}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {1 + \Im_{\sigma \left( \ell \right)}^{ - } } \right)^{{\omega_{j} }} ,\\ &\zeta_{{\partial \tilde{4}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 + {I}_{\sigma \left( \ell \right)}^{ - } } \right)^{{\omega_{j} }} ,{\varvec{f}}_{{\partial \tilde{4}}} = \mathop \prod \limits_{\ell = 1}^{n} \left( {2 + {f}_{\sigma \left( j \right)}^{ - } } \right)^{{\omega_{j} }} {. } \\ \end{aligned} $$

where $\left( {\sigma \left( 1 \right),\ldots,\sigma \left( {n - 1} \right),\sigma \left( n \right)} \right)$ represent permutation of $(1,2,3,\ldots,n)$ with $u_{{\sigma \left( {\ell - 1} \right)}} \ge^{{}} u_{\sigma \left( \ell \right)}$, $(\ell = 2,3,\ldots,n)$ and $\nu = \left( {_{{}} \nu_{1} ,\nu_{2} ,\ldots,\nu_{n} } \right)^{T}$ represent weighting vector for $u_{\ell }$, like $\nu_{\ell } > 0$ as well as $\mathop \sum \nolimits_{\ell = 1}^{n} \nu_{\ell } = 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ the bipolar neutrosophic arguments variables.

Proof

Theorem is easily understood.

Theorem 29

(Idempotency) Consider $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n} \right)$ representing set that contains equal BN values, i.e., $u_{\ell } = u$,

$$ {\text{I-BNEOWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) = u $$

Theorem 30

(Bounded) Suppose $u^{ + } = \mathop {\max }\nolimits_{\ell } u_{\ell }$, $u^{ - } = \mathop {\min }\nolimits_{\ell } u_{\ell } ,$ so

$$ u^{ - } \le {\text{I-BNEOWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \le u^{ + } $$

Theorem 31

(Monotonicity) Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n - 1,n} \right)$, and $\left\langle {\alpha^{\prime}_{\ell ,} u^{\prime}_{\ell } } \right\rangle\; \left( {\ell = 1,2,3,\ldots,n} \right)$ set that contains two BN number. If we have the condition $u_{\ell } \le u^{\prime}_{\ell }$, then

$$\begin{aligned} &{\text{I-BNEOWG}}_{\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \\ &\quad \le {\text{I-BNEOWG}}_{\nu } \left( {\left\langle {\alpha^{\prime}_{1} ,u^{\prime}_{1} } \right\rangle ,\ldots,\left\langle {\alpha^{\prime}_{n} ,u^{\prime}_{n} } \right\rangle } \right)\end{aligned} $$

Induced bipolar neutrosophic Einstein hybrid geometric AO

Definition 32

Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle\; \left( {\ell = 1,2,\ldots,n} \right)$ represent a collection of 2-tuples, then induced bipolar neutrosophic Einstein hybrid geometric (I-BNEHG) operators defined as fellow:

$$ \begin{aligned} & {\text{I-BNEHG}}_{w,\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \\ & \quad = \mathop \otimes \limits_{\ell = 1}^{n} \left( {\nu_{\ell } \dot{u}_{\sigma \left( \ell \right)} } \right) =^{{}} \nu_{1} \dot{u}_{\sigma \left( 1 \right)} \otimes \nu_{2} \dot{u}_{\sigma \left( 2 \right)} \otimes \nu_{3} \dot{u}_{\sigma \left( 3 \right)} \otimes \cdots \otimes \nu_{\ell } \dot{u}_{\sigma \left( \ell \right)} , \\ \end{aligned} $$

(12)

here $w = \left( {w_{1} ,w_{2} ,w_{3} ,\ldots,w_{n} } \right)$ represent a weighting vector for $u_{\ell }$, by means of $w_{\ell } \in \left[ {0,1} \right]$, as well as $\mathop \sum \nolimits_{\ell = 1}^{n} w_{\ell } = 1,$$\dot{u}_{\sigma \left( \ell \right)}$ be a $\ell$-th biggest element from BN numbers $\dot{u}_{\ell } \left( {\dot{u}_{\ell } = \left( {n\nu_{\ell } } \right)u_{\ell } } \right)$, $\nu = \left( {\nu_{1} ,\nu_{2} ,\ldots,\nu_{n} } \right)$ represent weighting vector of BN argument $u_{\ell }$ with $\omega_{\ell } \in \left[ {0,1} \right],$$\mathop \sum \nolimits_{\ell = 1}^{n} \omega_{\ell } = 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ as the bipolar neutrosophic arguments variables.

Remarks

I-BNEHG operator reduce to I-BNEWG operator if $w = \left( {\frac{1}{n},\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}} \right)^{T}$, I-BNEOWG operator when $\nu = \left( {\frac{1}{n}^{{}} ,\frac{1}{n}_{{}} ,\ldots,\frac{1}{n}} \right).$

Theorem 33

Let $\left\langle {\alpha_{\ell ,} u_{\ell } } \right\rangle$ be there a collection of 2-tuple, then (I-BNEHG) aggregation operators give in returns a BN values

$$ \begin{aligned} & {\text{I-BNEHG}}_{w,\nu } \left( {\left\langle {\alpha_{1} ,u_{1} } \right\rangle ,\left\langle {\alpha_{2} ,u_{2} } \right\rangle ,\ldots,\left\langle {\alpha_{n} ,u_{n} } \right\rangle } \right) \\ &\quad = \left( {\frac{{{{2\dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{1}}} }}{{{{\dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{2}}} {{ + \dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{1}}} }},\frac{{\dot{\zeta }_{{\tilde{1}}} - \dot{\zeta }_{{\tilde{2}}} }}{{\dot{\zeta }_{{\tilde{1}}} { + }\dot{\zeta }_{{\tilde{2}}} }},\frac{{\dot{\user2{f}}_{{\tilde{1}}} - \dot{\user2{f}}_{{\tilde{2}}} }}{{\dot{\user2{f}}_{{\tilde{1}}} { + }\dot{\user2{f}}_{{\tilde{2}}} }}, - \frac{{{{\dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{3}}} - {{\dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{4}}} }}{{{{\dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{3}}} {{ + \dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{4}}} }},\frac{{ - {2}\dot{\zeta }_{{\tilde{3}}} }}{{\dot{\zeta }_{{\tilde{4}}} { + }\dot{\zeta }_{{\tilde{3}}} }},\frac{{ - {2}\dot{\user2{f}}_{{\tilde{3}}} }}{{\dot{\user2{f}}_{{\tilde{4}}} { + }\dot{\user2{f}}_{{\tilde{3}}} }}} \right) \end{aligned}$$

(13)

where,

$$ \begin{aligned} &\dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{1}}} = \mathop \prod \limits_{j = 1}^{n} \left( {\dot{\Im }_{\sigma \left( j \right)}^{ + } } \right)^{{\omega_{j} }} ,\dot{\zeta }_{{\tilde{1}}} = \mathop \prod \limits_{j = 1}^{n} \left( {1^{{}} { + }\dot{I}_{\sigma \left( j \right)}^{ + } } \right)^{{\omega_{j} }} ,\\ &\dot{\user2{f}}_{{\tilde{1}}} = \mathop \prod \limits_{j = 1}^{n} \left( {1{ + }{\dot{f}}_{\sigma \left( j \right)}^{ + } } \right)^{{\omega_{j} }} , \, \dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{2}}} = \mathop \prod \limits_{j = 1}^{n} \left( {2 - \dot{\Im }_{\sigma \left( j \right)}^{ + } } \right)^{{\omega_{j} }} ,\\ &\dot{\zeta }_{{\tilde{2}}} = \mathop \prod \limits_{j = 1}^{n} \left( {1^{{}} - \dot{I}_{\sigma \left( j \right)}^{ + } } \right)^{{\omega_{j} }} ,\dot{\user2{f}}_{{\tilde{2}}} = \mathop \prod \limits_{j = 1}^{n} \left( {1 -_{{}} {\dot{f}}_{\sigma \left( j \right)}^{ + } } \right)^{{\omega_{j} }} , \, \\ &\dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{3}}} = \mathop \prod \limits_{j = 1}^{n} \left( {1{ + }_{{}} \left| {\dot{\Im }_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{\omega_{j} }} { , }\dot{\zeta }_{{\tilde{3}}} = \mathop \prod \limits_{j = 1}^{n} \left| {\dot{I}_{\sigma \left( j \right)}^{ - } } \right|^{{\omega_{j} }} , \\ &\, \dot{\user2{f}}_{{\tilde{3}}} = \mathop \prod \limits_{j = 1}^{{n}{}} \left| {{\dot{f}}_{\sigma \left( j \right)}^{ - } } \right|^{{\omega_{j} }} , \, \dot{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}_{{\tilde{4}}} = \mathop \prod \limits_{j = 1}^{n} \left( {1{ + }^{{}} \dot{\Im }_{\sigma \left( j \right)}^{ - } } \right)^{{\omega_{j} }} , \\ &\, \dot{\zeta }_{{\tilde{4}}} = \mathop \prod \limits_{j = 1}^{n} \left( {2 + \dot{I}_{\sigma \left( j \right)}^{ - } } \right)^{{\omega_{j} }} , \, {\varvec{f}}_{{\tilde{4}}} = \mathop \prod \limits_{j = 1}^{n} \left( {2 + {\dot{f}}_{\sigma \left( j \right)}^{ - } } \right)^{{\omega_{j} }} {.} \\ \end{aligned} $$

here $w = \left( {w_{1} ,w_{2} ,w_{3} ,\ldots,w_{n} } \right)$ be a weighting vector of $u_{\ell }$, with $w_{\ell } \in \left[ {0,1} \right]$, $\mathop \sum \nolimits_{\ell = 1}^{n} w_{\ell } = 1,$ and $\dot{u}_{\sigma \left( \ell \right)}$ be a $\ell$-th biggest element from BN numbers $\dot{u}_{\ell } \left( {\dot{u}_{\ell } = \left( {n\nu_{\ell } } \right)u_{\ell } } \right)$, $\nu = \left( {\nu_{1} ,\nu_{2} ,\nu_{3} ,\ldots,\nu_{n - 1} ,\nu_{n} } \right)$ represent weighting vector of BN argument $u_{\ell }$ with $\omega_{\ell } \in \left[ {0,1} \right],$$\mathop \sum \nolimits_{\ell = 1}^{n} \omega_{\ell } = 1.$ In 2-tuples $\left\langle {\alpha_{\ell } ,u_{\ell } } \right\rangle$, $\alpha_{\ell }$ represented as an order-inducing variable and $u_{\ell }$ as the bipolar neutrosophic arguments variables. Here, n represent balancing coefficient.

Proof

Theorem is straightforward.

Note that the I-BNEHG operator reduce to I-BNEWG operator if $w = \left( {\frac{1}{n},^{{}} \frac{1}{n},\frac{1}{n},_{{}} \ldots,\frac{1}{n}} \right)^{T}$, I-BNEOWG operator when $\nu = \left( {\frac{1}{n}^{{}} ,\frac{1}{n}_{{}} ,\ldots,\frac{1}{n}} \right).$

Multi-criteria DM based on induced bipolar neutrosophic Einstein AOs for robot selection

The present portion has a multi-criteria group decision-making application that uses induced bipolar neutrosophic Einstein AOs (Fig. 1), together with crisp numbers that serve as attribute with weight and bipolar neutrosophic numbers that serve as attribute values.

Fig. 1

Flowchart for MAGDM

Algorithm

Suppose $R = \left\{ {R_{1} ,R_{2} ,\ldots,R_{m} } \right\}$ be there collection of m alternative, $C = \left\{ {C_{1} ,C_{2} ,C_{3} ,\ldots,C_{n - 1} ,C_{n} } \right\}$ correspond to a set of fixed n attribute in addition to $D^{{}} =_{{}} \left\{ {D_{1} ,D_{2} ,\ldots,D_{k - 1} ,D_{k} } \right\}$ represent a fixed numbers of k decision-maker. Consider $\nu = \left( {\nu_{1} ,\nu_{2} ,\ldots,\nu_{n} } \right)^{T}$ represent the weighting vectors for representing decision-makers $\overline{{D^{s} }} \left( {s = 1,2,\ldots,k} \right),$ along with $\nu_{\ell } \in \left[ {0,^{{}} 1} \right]$ and $\mathop \sum \nolimits_{\ell = 1}^{n} \nu_{\ell } = 1.$ Suppose $w = \left( {w_{1} ,w_{2} ,\ldots,w_{n} } \right)^{T}$ represent weighting vectors for the collection of attributes $C = \left\{ {C_{1} ,C_{2} ,C_{3} ,\ldots,C_{n - 1} ,C_{n} } \right\}$ with $w_{\ell } \in \left[ {0,1} \right]$ as well as $\mathop \sum \nolimits_{\ell = 1}^{n} w_{\ell } = 1.$ The decision-maker evaluates a potential course of action using a set of finite criteria, the value of these are determined by BN values. Suppose $u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{\left( s \right)} = \left[ {\left( {\Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ + } ,{I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ + } ,{f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ + } ,\Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ - } ,{I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ - } ,{f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ - } } \right)} \right]_{m \times n}$ represent the decision matrices constructed for decision by specialist decision-makers, $u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{\left( s \right)}$ represent a BN number for $C_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}$ attributes that associated with alternative. We have $\Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ + } ,{I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ + } ,{f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ + } ,\Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ - } ,{I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ - }$ and ${f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ - } \in \left[ {0,^{{}} 1} \right]$ along with $0 \le \Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ + } + {I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{{ + }{}} +_{{}} {f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{{ + }{}} +_{{}} \Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ - } + {I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{ - } + {f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{{ - }{}} \le_{{}} 6$ here $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i}^{{}} =_{{}} 1,2,\ldots,m$ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j}^{{}} =_{{}} 1,2,\ldots,n$.

Step 1: Matrices (Tables 1, 2, 3) constructions for decision-making $\overline{{D^{s} }} = \left[ {u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{j} }}^{\left( s \right)} } \right]_{m \times n} \left( {s = 1,2,3,\ldots,k - 1,k} \right)$.

Table 1 BN decision matrix $D_{{\tilde{1}}}$

Table 2 BN decision matrix $D_{{\tilde{2}}}$

Table 3 BN decision matrix $D_{{\tilde{3}}}$

Step 2: Computing (Table 4) ${\text{I-BNEWG}}_{\nu } \left( {u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 1}} ,u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 2}} ,\ldots,u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} n}} } \right)$ where $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i}^{{}} =_{{}} 1,2,3,\ldots,m - 1,m$.

$$ \begin{aligned} & r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }} = \left( {\Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ + } ,{I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ + } ,{f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ + } ,\Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ - } ,{I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ - } ,{f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ - } } \right) \\ & \quad = {\text{I-BNEWG}}_{\nu } \left( {r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 1}} ,r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 2}} , \ldots ,r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} n}} } \right) = \mathop \otimes \limits_{\ell = 1}^{n} \left( {\nu_{\ell } r_{i\ell } } \right) \\ & \quad = \left( {\frac{{{{2{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{1}}} }}{{{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{2}}} {{ + {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{1}}} }},\frac{{{\varvec{f}}_{{\tilde{1}}} - {\varvec{f}}_{{\tilde{2}}} }}{{{\varvec{f}}_{{\tilde{1}}} { + }{\varvec{f}}_{{\tilde{2}}} }},\frac{{\zeta_{{\tilde{1}}} - \zeta_{{\tilde{2}}} }}{{\zeta_{{\tilde{1}}} { + }\zeta_{{\tilde{2}}} }}, - \frac{{{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{3}}} - {{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{4}}} }}{{{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{3}}} {{ + {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{4}}} }},\frac{{ - {2}\zeta_{{\tilde{3}}} }}{{\zeta_{{\tilde{4}}} { + }\zeta_{{\tilde{3}}} }},\frac{{ - {2}{\varvec{f}}_{{\tilde{3}}} }}{{{\varvec{f}}_{{\tilde{4}}} { + }{\varvec{f}}_{{\tilde{3}}} }}} \right) \, \\ \end{aligned} $$

Table 4 Decision matrix R for a collective BN

Step 3: Compute the score function for BN values $S\left( {r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }} } \right)$, $\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} = 1,2,\ldots,m} \right)$.

Step 4: Compute the software system, which are ranked for ${\text{I-BNEWG}}_{\nu } \left( {u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 1}} ,u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 2}} ,\ldots,u_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} n}} } \right)$ as per their score.

Step 5: Decide on the most excellent option available (s).

Illustrative example

A company desires to buy few robots (R) as alternatives. For this reason, the organization forms three decision-makers in a working group with weighting vectors $\nu = (0.3,0.3,0.6)^{T} .$ When choosing the most adorable robot suitable to their business, there are various factors to evaluate. The committee of decision-makers just analysis the three criterions mention, with weighting vector $\nu = (0.2,0.3,0.5)^{T} .$ Following the initial screening by decision makers, three robots $R_{j} \left( {j = 1,2,3} \right)$ will be move forward to the next set of round of the process based on the three qualities given below, the committee must reach a decision.

• (1) C₁: man–machine interface

• (2) C₂: programming flexibility

• (3) C₃: purchase cost

Step 1: Construct decision matrix for decision.

Step 2: Compute ${\text{I-BNEWG}}_{\nu } \left( {r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 1}} ,r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 2}} ,\ldots,r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} n}} } \right)$

$$ \begin{aligned} & r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }} = \left( {\Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ + } ,{I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ + } ,{f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ + } ,\Im_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ - } ,{I}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ - } ,{f}_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }}^{ - } } \right) \\ & \quad = {\text{I-BNEWG}}_{\nu } \left( {r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 1}} ,r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} 2}} ,\ldots,r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} n}} } \right) = \mathop \otimes \limits_{\ell = 1}^{n} \left( {\nu_{\ell } r_{i\ell } } \right) \\ & \quad = \left( {\frac{{{{2{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{1}}} }}{{{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{2}}} {{ + {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{1}}} }},\frac{{\zeta_{{\tilde{1}}} - \zeta_{{\tilde{2}}} }}{{\zeta_{{\tilde{1}}} { + }\zeta_{{\tilde{2}}} }},\frac{{{\varvec{f}}_{{\tilde{1}}} - {\varvec{f}}_{{\tilde{2}}} }}{{{\varvec{f}}_{{\tilde{1}}} { + }{\varvec{f}}_{{\tilde{2}}} }}, - \frac{{{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{3}}} - {{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{4}}} }}{{{{{{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{3}}} {{ + {{T}\hspace*{-4pt} \raisebox{4pt}{\rotatebox{-175}{{\sf c}}}}}}_{{\tilde{4}}} }},\frac{{ - {2}\zeta_{{\tilde{3}}} }}{{\zeta_{{\tilde{4}}} { + }\zeta_{{\tilde{3}}} }},\frac{{ - {2}{\varvec{f}}_{{\tilde{3}}} }}{{{\varvec{f}}_{{\tilde{4}}} { + }{\varvec{f}}_{{\tilde{3}}} }}} \right) \, \\ \end{aligned} $$

Step 3: Calculate now the scoring function of

$$ \begin{gathered} r_{1} = \left( {{0}{{.3269, 0}}{.6146, 0}{{.6045, - 0}}{.7310, - 0}{{.4127, - 0}}{.4341}} \right), \hfill \\ r_{2} = \left( {{0}{{.3082, 0}}{.5477, 0}{{.7406, - 0}}{.6510, - 0}{{.4490, - 0}}{.4688}} \right), \hfill \\ r_{3} = \left( {{0}{{.3637, 0}}{.5514, 0}{{.7183, - 0}}{.6387, - 0}{{.4105, - 0}}{.5640}} \right) \hfill \\ \end{gathered} $$

$$ S\left( {r_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{i} }} } \right) = \frac{1}{6}\left( {\Im^{ + } + 1 - {I}^{ + } + 1 - {f}^{ + } + 1 + \Im^{ - } - {I}^{ - } - {f}^{ - } } \right){, } $$

$$ S\left( {r_{1} } \right) = {.3706},S\left( {r_{2} } \right) = {.3811},S\left( {r_{3} } \right) = {.4050,} $$

Step 4: We calculated the scores and came to a conclusion.

$$ R_{3} \succ R_{2} \succ R_{1} $$

Step 5: The most excellent choice is robot R_3.

Comparison analysis

In this section, we will explore the characteristics of the proposed model and compare them to the existing methods used in multi-criteria decision making. Numerous researchers have employed various decision-making techniques thus far, contributing to a diverse landscape in this field. Chen and others [26] used fuzzy set, and Atanassov [2] used intuitionistic fuzzy set, Zavadskas and others [17] used neutrosophic sets, Dubois and others [5] utilized the bipolar fuzzy set, and Irfan and others [22] utilized the bipolar neutrosophic set, for decision-making problems. We employ induced Einstein operations to bipolarity of neutrosophic set in current study.

This hybrid model's construction aims to close the research hole left by earlier approaches. In decision analysis, the bipolar fuzzy set and the neutrosophic set can be combined with induced Einstein operations. We can use its counter characteristics to deal with the alternatives' grades for truth, indeterminacy, falsehood.

The induced aggregation operators (AOs) in present piece study are more broad and flexible, which is an improvement of our anticipated methods. R₃ is, however, the most excellent robot for the company (Table 5).

Table 5 Comparison analysis of the proposed operators and existing operators

Conclusion

This study's objective is to investigate various induced bipolar neutrosophic aggregation operators (AOs) that use BN values as a criterion in multi-criteria community DM built by means of the aid of Einstein t-norms and t-conorms. We proposed induced bipolar neutrosophic Einstein aggregation operators, which were inspired by Einstein operations. We started by looking at induced BN Einstein aggregation operators and the requirements for them. These AOs' characteristics include (I-BNEWA) and (I-BNEWG). Finally, we demonstrated how to use a framework to make multi-criteria decisions. The best Robot assortment has been explored using a descriptive instance. The findings of this research demonstrate that when used, the suggested procedures are both more accurate and useful. We plan to adopt the suggested methodology moving forward for research projects like pattern identification and risk assessment. It will also be used with other fields.