## 1 Introduction  
In the modern age of managerial decision-making (DM), knowledge frequently remains incomplete, undetermined and incompatible. [PERSON] was the first to propose the fuzzy set theory [1], which deals with uncertainty and has applications in a wide range of modern fields of present and future society. Yet, fuzzy set has a problem in that it can simply convey the value of membership button non-membership. To address this, [PERSON] [2] developed the intuitionistic fuzzy set (IFS), as well as associated theory, in order to summarize concept of fuzzy set.A pair of membership values is used to represent each IFS element, truth-membership \(\mathfrak{I}(\chi)\) and a non-membership value (falsity-membership) \(f(\chi)\), as well as satisfy the condition \(\mathfrak{I}(\chi),f(\chi)\in[0,1]\) with \(0\leq\mathfrak{I}(\chi)+f(\chi)\leq 1\). IFS can only handle incomplete data, meaningindeterminate data are not an option. [PERSON] [3] developed the novel concept of NS, which adds an indeterminacy membership value \(I(\chi)\) to IFS. NS has a strong ability to deal with knowledge that is imperfect, uncertain and contradicting. When \(\Im(\chi)+I(\chi)+f(\chi)<1\), the information acquired is indeterminate. When \(\Im(\chi)+I(\chi)+f(\chi)>1\), this represents inconsistency in an NS. [PERSON] and others devised the single-valued neutrosophic set (SVNS) to deal with real-world situations [4], with the conditions \(\Im(\chi),I(\chi),f(\chi)\in[0,1]\) and \(0\leq\Im(\chi)+I(\chi)+f(\chi)\leq 3\). [PERSON] [5] proposed a method for comparing SVNS, and defined the correlation coefficient.Wang et al. formulated an interval-valued neutrosophic set [6] and broadened the values of truth, indeterminacy and false memberships, which are all between 0 and 1. For researchers, aggregation operators (AOs) are extremely important.Since its introduction, several scientists [7; 8; 9; 10; 11; 12; 13; 14] have contributed significantly to the theory development of IFS. [PERSON] et al. [15] introduced the concept of distinct IF aggregation operators based on IFS (AOs). [PERSON] et al. and [PERSON] et al. [16; 17] created Einstein aggregation operators (AOs). The Einstein t-norm typically gives the same smooth approximations as the product and sum of algebra. The algebraic t-norm (product \(\otimes\)) and t-conorm (sum \(\oplus\)), respectively, are as below:  
\[ \begin{array}{l}T(a,b)=a\otimes b=ab\\ T^{*}(a,b)=a\oplus b=a+b-ab\end{array} \]  
The Einstein operations t-norm (product \(\otimes\)) and t-conorm (sum \(\oplus\)), respectively, are as below:  
\[ \begin{array}{l}T(a,b)=a\otimes b=\frac{ab}{1+1-a)(1-b)}\\ T^{*}(a,b)=a\oplus b=\frac{\oplus b}{1+ab}\end{array} \]  
The bipolar fuzzy set (BFS) [18; 19; 20] has emerged as a new technique to deal with vagueness in DM situations.The bipolar fuzzy set's membership degree varies between \(-1\) and \(1\). Positive as well as negative membership degrees are available in BFS. BFSs are incredibly beneficial for several fields of study, as well as DM [21; 22]. [PERSON] [23] defined bipolar averaging and geometric aggregations operators (AOs). The bipolar neutrosophic set was created by [PERSON] and others [24; 25] by using fundamental operations and a comparison mechanism [26; 27]. [PERSON] and others created aggregation operators (AOs) based on BN values and applied them to DM problems. Heronian mean aggregation operators were developed by [PERSON] and others [28]. [PERSON] et al. introduced the idea of a bipolar soft set and its application to decision-making [29]. [PERSON] and other developed a bipolar neutrosophic soft set and applied itto decision-making [30]. [PERSON] et al. introduced complex fuzzy set and its properties [31]. [PERSON] et al. introduced bipolar complex fuzzy set and its aggregation operators [32]. [PERSON] and others applieda multi-criteria decision-making approach to the bipolar neutrosophic set [33]. Regardless of the available information, there is lot of literature related to the subject.