This manuscript aims to analyze the well-known and massive idea of competition graph (CG) in the presence of a new and dominant technique of complex q-rung orthopair fuzzy (CQROF) setting. The mathematical form of the CQROF setting is more flexible and massive consistent for demonstrating the beneficial option from the collection of objectives during the decision-making process. Additionally, the major concept of in-neighbourhood and out-neighbourhood using CQROF diagraph (CQROFDG) are also invented to enhance the quality of the diagnosed approach. The fundamental theory of CQROF k-competition, CQROF p-competition, CQROF neighbourhood and m-step CQROF neighbourhood graphs are also explored. In the availability of the above-described theories, the basic and significant results for the presented work are obtained to show the compatibility and worth of the invented approaches. To show the practicality of the developed approach, we try to verify the proposed work with the help of various examples. Further, to describe the validity and practicality of the invented work, we diagnosed an application using presented approaches based on the CQROF setting is to enhance the major weakness of the existing approaches. Finally, in the availability of the invented ideas, we discussed the sensitivity analysis of the described approaches.

The main influence of this analysis is to derive two different types of aggregation operators under the consideration of algebraic t-norm and t-conorm and Einstein t-norm and t-conorm for CIF set theory. Because these operators are very effective for evaluating the collection of information into a singleton preference. For this, first, we discover the Algebraic and Einstein operational laws for CIF sets. Then, we aim to discover the theory of CCIFWA, CCIFOWA, CCIFWG, CCIFOWG operators and their valuable properties "idempotency, monotonicity and boundedness" and results. Furthermore, we also derive the theory of CCIFEWA, CCIFEOWA, CCIFEWG, CCIFEOWG operators and their valuable properties "idempotency, monotonicity, and boundedness" and results. Some special cases of the derived work are also described in detail. Finally, we illustrate a MADM procedure under the consideration of derived operators to enhance the worth of the presented information. Finally, we compare the presented operators with various existing operators with the help of various suitable examples for showing the reliability and stability of the derived approaches.

The notion of the bipolar complex fuzzy set (BCFS) is a fundamental notion to be considered for tackling tricky and intricate information. Here, in this study, we want to expand the notion of BCFS by giving a general algebraic structure for tackling bipolar complex fuzzy (BCF) data by fusing the conception of BCFS and semigroup. Firstly, we investigate the bipolar complex fuzzy (BCF) sub-semigroups, BCF left ideal (BCFLI), BCF right ideal (BCFRI), BCF two-sided ideal (BCFTSI) over semigroups. We also introduce bipolar complex characteristic function, positive $\left(\omega ,\eta \right)$-cut, negative $\left(\varrho ,\sigma \right)$-cut, positive and $\left(\left(\omega ,\eta \right),\left(\varrho ,\sigma \right)\right)$-cut. Further, we study the algebraic structure of semigroups by employing the most significant concept of BCF set theory. Also, we investigate numerous classes of semigroups such as right regular, left regular, intra-regular, and semi-simple, by the features of the bipolar complex fuzzy ideals. After that, these classes are interpreted concerning BCF left ideals, BCF right ideals, and BCF two-sided ideals. Thus, in this analysis, we portray that for a semigroup $Ş$ and for each BCFLI ${М}_1=\left({\lambda }_{P-{М}_1},{\lambda }_{N-{М}_1}\right)=\left({\lambda }_{RP-{М}_1}+\iota {\lambda }_{IP-{М}_1},{\lambda }_{RN-{М}_1}+\iota {\lambda }_{IN-{М}_1}\right)$ and BCFRI ${М}_2=\left({\lambda }_{P-{М}_2},{\lambda }_{N-{М}_2}\right)=\left({\lambda }_{RP-{М}_2}+\iota {\lambda }_{IP-{М}_2},{\lambda }_{RN-{М}_2}+\iota {\lambda }_{IN-{М}_2}\right)$ over $Ş$, ${М}_1\cap {М}_2={М}_1⊚{М}_2$ if and only if $Ş$ is a regular semigroup. At last, we introduce regular, intra-regular semigroups and show that ${М}_1\cap {М}_2\preccurlyeq {М}_1⊚{М}_2$ for each BCFLI ${М}_1=\left({\lambda }_{P-{М}_1},{\lambda }_{N-{М}_1}\right)=\left({\lambda }_{RP-{М}_1}+\iota {\lambda }_{IP-{М}_1},{\lambda }_{RN-{М}_1}+\iota {\lambda }_{IN-{М}_1}\right)$ and for each BCFRI ${М}_2=\left({\lambda }_{P-{М}_2},{\lambda }_{N-{М}_2}\right)=\left({\lambda }_{RP-{М}_2}+\iota {\lambda }_{IP-{М}_2},{\lambda }_{RN-{М}_2}+\iota {\lambda }_{IN-{М}_2}\right)$ over $Ş$ if and only if a semigroup $Ş$ is regular and intra-regular.

In this manuscript, we generalized the notions of three-way decisions (3WD) and decision theoretic rough sets (DTRS) in the framework of Complex q-rung orthopair 2-tuple linguistic variables (CQRO2-TLV) and then deliberated some of its important properties. Moreover, we considered some very useful and prominent aggregation operators in the framework of CQRO2-TLV, while further observing the importance of the generalized Maclurin symmetric mean (GMSM) due to its applications in symmetry analysis, interpolation techniques, analyzing inequalities, measuring central tendency, mathematical analysis and many other real life problems. We initiated complex q-rung orthopair 2-tuple linguistic (CQRO2-TL) information and GMSM to introduce the CQRO2-TL GMSM (CQRO2-TLGMSM) operator and the weighted CQRO2-TL GMSM (WCQRO2-TLGMSM) operator, and then demonstrated their properties such as idempotency, commutativity, monotonicity and boundedness. We also investigated a CQRO2-TL DTRS model. In the end, a comparative study is given to prove the authenticity, supremacy, and effectiveness of our proposed notions.

Aczel-Alsina t-norm and t-conorm are great substitutes for sum and product and recently various scholars developed notions based on the Aczel-Alsina t-norm and t-conorm. The theory of bipolar complex fuzzy set that deals with ambiguous and complex data that contains positive and negative aspects along with a second dimension. So, based on Aczel-Alsina operational laws and the dominant structure of the bipolar complex fuzzy set, we develop the notion of bipolar complex fuzzy Aczel-Alsina weighted geometric, bipolar complex fuzzy Aczel Alsina ordered weighted geometric and bipolar complex fuzzy Aczel Alsina hybrid geometric operators. Moreover, multi-attribute border approximation area comparison technique is a valuable technique that can cover many decision-making situations and have dominant results. So, based on bipolar complex fuzzy Aczel-Alsina aggregation operators, we demonstrate the notion of a multi-attribute border approximation area comparison approach for coping with bipolar complex fuzzy information. After that, we take a numerical example by taking artificial data for various types of operating systems and determining the finest operating system for a computer. In the end, we compare the deduced multi-attribute border approximation area comparison approach and deduced aggregation operators with numerous prevailing works.

In our daily life we have to make many decisions and sometimes in a single day we met the situations when correct decision is very compulsory to handle some complicated situations. However, in a professional environment, we need decision-making (DM) techniques to determine the finest alternative from the given alternatives. In this manuscript, we develop one of the finest DM techniques by employing interpreted aggregation operators (AOs). Furthermore, to aggregate the collection of a finite number of information into a singleton set, the Bonferroni mean (BM) operator plays a very beneficial and dominant role. The BM operator is massively powerful than the averaging/geometric operators because they are the specific cases of the BM operator. Based on the above advantages-we initiate the notion of bipolar complex fuzzy BM (BCFBM) operator, bipolar complex fuzzy normalized weighted BM (BCFNWBM) operator and bipolar complex fuzzy ordered weighted BM (BCFOWBM) operator. Furthermore, some well-known and useful properties and results of the initiated operators will be established. We will also apply the described AOs, and evaluate a DM technique, called multi-attribute DM (MADM) to prove the trustworthiness and practicality of the evaluated theory. Finally, to compare the presented work with some prevailing operators, we illustrate some examples and try to evaluate the graphical interpretation of the established work to improve the worth of the proposed theory.