EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5824 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bipolar Neutrosophic Aczel-Alsina Aggregation for Effective Group Decision-Making Aliya Fahmi1, A Khan2, Thabet Abdeljawad2,∗, D.K.Almutairi3, M K Siddhu4 1 Department of Mathematics, Faculty of Science, University of Faisalabad, Faisalabad, Pakistan 2 Department of Mathematics and Sciences, Prince Sultan University, P.O.Box 66833, 11586 Riyadh, Saudi Arabia 3 Department of Mathematics, College of Science Al-Zulfi, Majmaah University, 11952 Al-Majmaah, Saudi Arabia 4 Department of Computer Sciences, The University of Faisalabad, Faisalabad, 6 Pakistan Abstract. This paper presents a series of innovative Bipolar Neutrosophic Aggregation Operators to address the complexity and uncertainty in Multiple Criteria Decision-Making scenarios. The newly proposed operators BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, BNAAOWG, and BNAAHWG are designed to enhance aggregation under bipolar neutrosophic conditions, captur- ing a more nuanced view of decision-makers’ preferences. We develop the MCDM method with the BNN. The effectiveness of these operators is demonstrated through a detailed case study, where they are applied to a complex decision-making scenario involving conflicting and uncertain cri- teria. Comparative and sensitivity analyses are conducted to assess the stability, reliability, and adaptability of each operator, benchmarking them against existing approaches. The results reveal that the proposed operators significantly improve decision-making accuracy by accommodating bipolar information and managing degrees of uncertainty more effectively. In the Results and Discussion section, we explore how each operator performs across varied MCDM contexts, high- lighting the flexibility and robustness of the bipolar neutrosophic framework. The paper concludes by discussing the limitations of the proposed operators, offering insights into potential applica- tions, and suggesting directions for future research to further refine bipolar neutrosophic-based MCDM approaches. This work contributes a comprehensive, operator-based method for enhanced decision-making under complex and uncertain conditions. 2020 Mathematics Subject Classifications: 03E72, 90B50, 68T37, 91B06, 94D05 Key Words and Phrases: Bipolar Neutrosophic Sets, Aczel-Alsina, Multi-Criteria Decision- Making, aggregation operators ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5824 Email addresses: aliyafahmi@gmail.com,akhan@psu.edu.sa (A. Fahmi), tabdeljawad@psu.edu.sa (T. Abdeljawad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 2 of 43 1. Introduction Currently, businesses and organizations must engage deeply with their systems and facilities to meet customer demands and thrive in a competitive environment. Conversely, consumers seek high-quality products at affordable prices. Therefore, selecting contractors becomes a crucial strategic decision for supply chain management, influencing customer satisfaction and market competitiveness. Contractors are increasingly mindful of envi- ronmental capabilities due to pressing environmental issues such as heightened public awareness, global warming, and regulatory pressures. As a result, eco-friendly Contractor selection is prioritized, and green packaging is employed to minimize emissions and ensure environmental safety. This topic has become a significant [1? , 2] area of research in recent years, widely studied among academics. Choosing eco-friendly suppliers involves multi- ple conflicting criteria rather than a single criterion problem. In this regard, employing multi-criteria decision analysis methods or tools can effectively address this complexity. MCDA systems are utilized to evaluate contractors rigorously and select the most suitable and environmentally responsible service providers based on a variety of conflicting criteria [1–10]. Figure 1 is given as As a result, several academics have put up a variety of novel ideas to address these prevalent problems. By recognizing that qualities had some degree of vagueness, Zadeh [11] broke with the rules of conventional crisp logic and established fuzzy sets. Many complex real-life problems that are hard to explain in clear words can be handled more A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 3 of 43 precisely by using fuzziness. By measuring the extent to which an element belongs to a set, the membership function forms the basis of the fuzzy set model, which extends the traditional crisp model. Values for membership fall between 0 and 1, with values nearer 1 denoting a higher level of membership and other research [12–21]. Figure 2 is history given as below Interval neutrosophic sets were introduced by Wang et al. [22]. The Aczel-Alsina ag- gregation operators were first presented by Senapati et al. [23]. Aczel-Alsina introduced these operators in 2022. The extended hybrid trigonometric Pythagorean fuzzy similarity measure was introduced by Verma et al. [24], who also explore its features with specific examples. Wu (2019) developed, which considers the Hausforff space and the generic aggregation operator. Ejegwa et al. [10] combined the traditional characteristics that characterize PFSs with some new distance measures for PFSs. The Fermatean fuzzy bipo- lar soft set (abbreviated FFBSS) model was presented by Ali et al. [1] as a generalization of two potent pre-existing models: the Pythagorean fuzzy bipolar soft set model and the fuzzy bipolar soft set model, with a few basic characteristics. The intuitionistic fuzzy soft Aczel-Alsina weighted averaging (IFSAAWA) and geometric (IFSAAWG) operators were first presented by Ali et al. [2] Finding the best options in multi-criteria decision making requires articulating ambiguous information in a more advantageous way due to the grow- ing complexity of real-world decision-making situations. Furthermore, it is essential to comprehend how the input parameters relate to one another. To overcome these difficul- A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 4 of 43 ties, we use the aggregation and take advantage of the benefits of neutrosophic sets. Aliya et al. [12] presented triangular cubic fuzzy sets. Einstein aggregation operators were pro- posed by Aliya et al. [13, 14, 16]. A novel disaster decision-making (DDM) method based on the Fermatean fuzzy Schweizer-Sklar environment is proposed by Aliya et al. [15]. Aliya et al. recommended natural gas [17]. In 2025, Aliya and colleagues presented the Bipolar Fermion Fuzzy Sets. The Circular Intuitionistic Fuzzy Hamacher Weighted Av- erage (CIFHWA) and Circular Intuitionistic Fuzzy Hamacher Ordered Weighted Average (CIFHOWA) were first presented by Aliya et al. [18]. Based on specific characteristics of bipolar fuzzy soft sets (BFSSs), Riaz et al. [25] proposed new similarity measures (SMs). Numerous topological and functional features of the bipolar metric space have been examined, according to Zararsiz et al. [26]. Zarar- siz [27] created the MADM approach to handle unpredictable situations in real life with BFNs. T-norms and co-forms were employed by Imran et al. [20]. Pythagorean fuzzy Hamacher interactive weighted averaging (PFHIWA), Pythagorean fuzzy Hamacher in- teractive ordered weighted averaging (PFHIOWA), and Pythagorean fuzzy Hamacher in- teractive weighted geometric (PFHIWG) are some of the methods that Asif et al. [5] introduced. The MAGDM problem was introduced by Sarfraz [28] in the Pythagorean fuzzy (PyF) framework, accounting for the various expert [25, 29–32] and characteristic criteria [23, 24, 33–36] and other research [11, 22, 26, 27, 37] 1.1. Novelity This subsection provides a definition of novelty. 1. The operating laws and BNN are defined. 2. The accuracy and scoring functions are defined. 3. We propose the six aggregation operators. 4. To suggest the MCDM approach. 5. To explain the numerical approach. 1.2. Contribution of the paper The goal of this study is to develop a methodical and perceptive approach for selecting the best option from a range of possibilities. We have created a new class of BN aggregation operators by utilizing AA t-NMs and t-CNMs. Delineating the notions of BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, BNAAOWG, and BNAAHWG operators within the BNS framework is the main goal of this study. Additionally, we show that varied AOs are effective. In the end, the paper accomplishes the following significant milestones: 1. It is crucial to investigate the basic functions of t-NMs and t-CNMs in order to intro- duce new AOs such as the BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, BNAAOWG, and BNAAHWG within the BNS framework. 2. Examine the characteristics of these cutting-edge operators and give particular instances of how they are used. 3. Create an algorithm that can use BN data to handle many attribute decision making problems. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 5 of 43 4. Talk about the computational outcomes based on BN data to evaluate the suggested method’s dependability and usefulness. 5. Perform a comparative study between the recommended and current AOs, summa- rizing the results to show the comprehensive efficacy of the proposed AOs. 6. Perform sensitivity analyses to demonstrate the reliability and robustness of the proposed method. Real-world decision-making often encounters issues where certain attribute values pro- vided by decision-makers disproportionately influence outcomes, potentially leading to biased results. 1.3. Research Gap Although multi-attribute decision-making is essential in many real-world situations, current methods frequently suffer from ambiguity and uncertainty in decision-makers’ preferences. The ability of traditional aggregation operators, such as the current BN ag- gregation models, to efficiently handle complicated data structures and maintain decision integrity in the face of uncertainty is limited. BN Aggregation Operators’ Limited Integration of t-NMs and t-CNMs: The potential of t-normal and t-conormal operations, which are essential for enhancing decision accuracy and resilience, is not fully utilized by current BN aggregation techniques. Absence of Advanced Aggregation Operators in the BNS Framework: Research on BN-based aggregation operators is currently limited in scope, with little attention paid to advanced aggregation methods like BNAAAWA, BNAAOWA, and BNAAHWA. Refining the decision-making process requires these operators. Application Gap in Real-World Case Studies: Most of the research that has already been done focuses on theoretical formulations without showing how they may be ap- plied to actual decision-making situations. The usefulness of contemporary BN operators is still understudied since sensitivity analyses, which are essential for evaluating model reliability—as well as comparative studies against other well-established approaches are missing. Validation of the proposed operators requires real-world case studies. 1.4. Motivation for Research Real-world decision-making can be severely hampered by human judgment’s uncer- tainty and imprecision. Multi-attribute decision-making (MADM) frameworks are in- tended to handle the complex relationships between different attributes and the inherent ambiguity in expert judgments, which are occasionally missed by standard approaches. Therefore, the need for more dependable and flexible aggregation methods that can faith- fully capture decision-makers’ preferences is growing. Bipolar neutrosophic sets (BNS) offer a powerful mathematical tool for expressing vague, imprecise, and inconsistent data. However, the existing aggregation operators in the BNS framework are either too simple or cannot incorporate complex mathematical structures such as t-norms (t-NMs) and t-conorms (t-CNMs). These structures are essen- A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 6 of 43 tial for improving aggregation process refinement, judgment reliability, and computation efficiency. Moreover, a variety of real-world applications, such as risk assessment, medical di- agnosis, and supply chain management, require decision-making models that not only efficiently aggregate data but also adapt to changing conditions. The applicability of ex- isting approaches is further limited by the absence of comparative studies and sensitivity analyses. By introducing a novel class of BN aggregation operators that use t-NMs and t-CNMs to increase decision accuracy, this work aims to fill in these gaps. By developing new operators such as BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, BNAAOWG, and BNAAHWG, this study provides a clever and calculated approach to MADM problems. The effectiveness of these operators is demonstrated by a comparison analysis, sensitivity evaluation, and real-world case study implementations, ensuring their superiority over existing methods and usefulness. The structure of the manuscript is as follows: We give a brief introduction to bipolar neutrosophic sets and aggregation operators in Section 2. In the BNS framework, Section 3 presents six new aggregation operators based on Aczel-Alsina procedures and examines their advantageous characteristics. These operators are used to solve a Multi-Criteria Group Decision Making problem in Section 4. Section 5 illustrates the applicability of bipolar neutrosophic approaches with a case study on analyzing construction project de- cisions. Lastly, the closing remarks are presented in Section 6. List of abbreviations of Table 1 is given as Abbrevations Full Name AA Aczel-Alsina BNNs Bipolar neutrosophic numbers MCDM Multi criteria decision making BNAA Bipolar neutrosophic Aczel-Alsina BNAAWA operator Bipolar neutrosophic Aczel-Alsina weighted averaging operator BNAAOWA operator Bipolar neutrosophic Aczel-Alsina ordered weighted averaging operator BNAAHWA operator Bipolar neutrosophic Aczel-Alsina hybrid weighted averaging operator BNAAWG operator Bipolar neutrosophic Aczel-Alsina weighted geometric operator BNAAOWG operator Bipolar neutrosophic Aczel-Alsina ordered weighted geometric operator BNAAHWG operator Bipolar neutrosophic Aczel-Alsina weighted hybrid geometric operator 2. Preliminaries We developed the basic definition and properties. Definition 1. [35] Let X be a non empty set and by a fuzzy set we mean a farmula γ ={〈 x, µγ(x) 〉 | x ∈ X } , in which µγ(x) is a mapping from X to [0, 1] represent membership function of an element x in X. Definition 2. [33] Let X be a universal set, A neutrosophic set N in X is define as A = {〈 u, XN (u), YN (u), ZN (u) | u ∈ X 〉} where XN (u), YN (u), ZN (u) are the truth A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 7 of 43 membership function and the indetermency function and the falcity membershisp function respectively, such that X, Y, Z, : X → ] 0− , 1+[ and 0− ≤ XN (u) + YN (u) + ZN (u) ≤ 3+. 2.1. BNNs In this subsection, we proposed the definitions and score function of BNNs. Definition 3. [8] A bipolar neutrosophic number A in X is defined by A =   u, K + (u) , L + (u) , M + (u) , K − (u) , L − (u) , M − (u)  : u ∈ X  , where K + , L + , M + : X → [0, 1] and K − , L − , M − : X → [−1, 0] . The positive member- ship degree K +(u), L +(u), M +(u) denotes the truth membership, interminate membershisp and false membership of an element u ∈ X corresponding to bipolar set A and the nega- tive membership degree K−(u), L −(u), M −(u) denotes the truth membership, interminate membershisp and false membership of an element u ∈ X to some implicit counter-property corresponding to a bipolar set A. Definition 4. [8] Assume P̃ =  L + M + D + L − M − D −  . The score function S(P ), accuracy function H(P ) and certainty function Q(P ) of a bipolar neutrosophic number are defined as follows: S(P ) = 1 6 ( L + + 1 − M + + 1 − D + + 1 + L − − M − − D − ) H(P ) = L + − M + + L − − M Q(P ) = L + − D − . 2.2. Operational laws of Aczel-Alsina In this subsection, we proposed the definitions and Operational laws of Aczel-Alsina. Definition 5. Assume P1 =  [L+ 1 M + 1 N + 1 ], [L− 1 M − 1 N1]  and P2 =  L + 2 M + 2 N + 2 L − 2 M − 2 N − 2  be the two bipolar neutro- sophic number. Then defined the A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 8 of 43 (P ) P1 ⊕ P2 =  [1 − e−(−In(1−L+ 1 ))Λ+(−In(1−L+ 2 ))Λ) 1 Λ , 1 − e−(−InM+ 1 )Λ+(−InM+ 2 )Λ) 1 Λ , 1 − e−(−InN+ 1 )Λ+(−InN+ 2 )Λ) 1 Λ ] [−(1 − e − ( −InL − 1 )Λ+(−InL − 2 )Λ ) 1 Λ ), −(1 − e − ( −In(1−M − 1 ))Λ+(−In(1−M − 2 ))Λ ) 1 Λ ), −(1 − e − ( −In(1−N − 1 ))Λ+(−In(1−N − 2 ))Λ ) 1 Λ )]  ; (b) P1 ⊗ P2 =  [1 − e − ( −InL + 1 )Λ+(−InL + 2 )Λ ) 1 Λ , 1 − e − ( −In(1−M + 1 ))Λ+(−In(1−M + 2 ))Λ ) 1 Λ , 1 − e − ( −In(1−N + 1 ))Λ+(−In(1−N + 2 ))Λ ) 1 Λ ], [−(1 − e − ( −In(1−L − 1 ))Λ+(−In(1−L − 2 ))Λ ) 1 Λ ), −(1 − e − ( −InM − 1 )Λ+(−InM − 2 )Λ ) 1 Λ ), −(1 − e − ( −InN − 1 )Λ+(−InN − 2 )Λ ) 1 Λ )]  ; (c) λP1 =  [1 − e−(λ(−In(1−L+ 1 ))Λ) 1 Λ , e−(λ(−InM+ 1 )Λ) 1 Λ , e−(λ(−InN+ 1 )Λ) 1 Λ ] [−e−(λ(−InL− 1 )Λ) 1 Λ , −(1 − e−(λ(−In(1−M− 1 ))Λ) 1 Λ ), −(1 − e−(λ(−In(1−N− 1 ))Λ) 1 Λ )  , (d) P λ 1 =  [e−(λ(−InL+ 1 )Λ) 1 Λ , 1 − e−(λ(−In(1−M+ 1 ))Λ) 1 Λ , 1 − e−(λ(−In(1−N+ 1 ))Λ) 1 Λ ], [−(1 − e−(λ(−In(1−L− 1 ))Λ) 1 Λ ), −e−(λ(−InM− 1 )Λ) 1 Λ , −e−(λ(−InN− 1 )Λ) 1 Λ ]  . A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 9 of 43 Theorem 1. Let P =  [L+ M + N + ], [L− M − N ]  , P1 =  [L+ 1 M + 1 N + 1 ], [L− 1 M − 1 N1]  and P2 =  L + 2 M + 2 N + 2 L − 2 M − 2 N − 2  be three BNAANs and λ, λ1λ2 > 0, then we have (1)P1 ⊕ P2 = P2 ⊕ P1; (2) P1 ⊗ P2 = P2 ⊗ P1; (3) λ(P1 ⊕ P2) = λP1 ⊕ λP2; (4)λ(P1 ⊗ P2) = λP1 ⊗ λP2 Proof. We can proof (1)P1 ⊕ P2 = P2 ⊕ P1; P1 ⊕ P2 =  [1 − e−(−In(1−L+ 1 ))Λ+(−In(1−L+ 2 ))Λ) 1 Λ , 1 − e−(−InM+ 1 )Λ+(−InM+ 2 )Λ) 1 Λ , 1 − e−(−InN+ 1 )Λ+(−InN+ 2 )Λ) 1 Λ ] [−(1 − e − ( −InL − 1 )Λ+(−InL − 2 )Λ ) 1 Λ ), −(1 − e − ( −In(1−M − 1 ))Λ+(−In(1−M − 2 ))Λ ) 1 Λ ), −(1 − e − ( −In(1−N − 1 ))Λ+(−In(1−N − 2 ))Λ ) 1 Λ )]  =  [1 − e−(−In(1−L+ 2 ))Λ+(−In(1−L+ 1 ))Λ) 1 Λ , 1 − e−(−InM+ 2 )Λ+(−InM+ 1 )Λ) 1 Λ , 1 − e−(−InN+ 2 )Λ+(−InN+ 1 )Λ) 1 Λ ] [−(1 − e − ( −InL − 2 )Λ+(−InL − 1 )Λ ) 1 Λ ), −(1 − e − ( −In(1−M − 2 ))Λ+(−In(1−M − 1 ))Λ ) 1 Λ ), −(1 − e − ( −In(1−N − 2 ))Λ+(−In(1−N − 1 ))Λ ) 1 Λ )]  = P2 ⊕ P1; (2) P1 ⊗ P2 = P2 ⊗ P1 A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 10 of 43 P1 ⊗ P2 =  [1 − e − ( −InL + 1 )Λ+(−InL + 2 )Λ ) 1 Λ , 1 − e − ( −In(1−M + 1 ))Λ+(−In(1−M + 2 ))Λ ) 1 Λ , 1 − e − ( −In(1−N + 1 ))Λ+(−In(1−N + 2 ))Λ ) 1 Λ ], [−(1 − e − ( −In(1−L − 1 ))Λ+(−In(1−L − 2 ))Λ ) 1 Λ ), −(1 − e − ( −InM − 1 )Λ+(−InM − 2 )Λ ) 1 Λ ), −(1 − e − ( −InN − 1 )Λ+(−InN − 2 )Λ ) 1 Λ )]  =  [1 − e − ( −InL + 2 )Λ+(−InL + 1 )Λ ) 1 Λ , 1 − e − ( −In(1−M + 2 ))Λ+(−In(1−M + 1 ))Λ ) 1 Λ , 1 − e − ( −In(1−N + 2 ))Λ+(−In(1−N + 1 ))Λ ) 1 Λ ], [−(1 − e − ( −In(1−L − 2 ))Λ+(−In(1−L − 1 ))Λ ) 1 Λ ), −(1 − e − ( −InM − 2 )Λ+(−InM − 1 )Λ ) 1 Λ ), −(1 − e − ( −InN − 2 )Λ+(−InN − 1 )Λ ) 1 Λ )]  = P2 ⊗ P1; (3) λ(P1 ⊕ P2) = λP1 ⊕ λP2 λ(P1 ⊕ P2) = λ  [1 − e−(−In(1−L+ 1 ))Λ+(−In(1−L+ 2 ))Λ) 1 Λ , 1 − e−(−InM+ 1 )Λ+(−InM+ 2 )Λ) 1 Λ , 1 − e−(−InN+ 1 )Λ+(−InN+ 2 )Λ) 1 Λ ] [−(1 − e − ( −InL − 1 )Λ+(−InL − 2 )Λ ) 1 Λ ), −(1 − e − ( −In(1−M − 1 ))Λ+(−In(1−M − 2 ))Λ ) 1 Λ ), −(1 − e − ( −In(1−N − 1 ))Λ+(−In(1−N − 2 ))Λ ) 1 Λ )]  A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 11 of 43 =  [1 − e−((λ)−In(1−L+ 1 )))Λ+((λ)−In(1−L+ 2 ))Λ)) 1 Λ , 1 − e−((λ)−InM+ 1 ))Λ+((λ)−InM+ 2 )Λ)) 1 Λ , 1 − e−((λ)−InN+ 1 )Λ+((λ)−InN+ 2 )Λ) 1 Λ ] [−(1 − e − ( (λ)−InL − 1 )Λ+((λ)−InL − 2 )Λ ) 1 Λ ), −(1 − e − ( (λ)−In(1−M − 1 ))Λ+((λ)−In(1−M − 2 ))Λ ) 1 Λ ), −(1 − e − ( (λ)−In(1−N − 1 ))Λ+((λ)−In(1−N − 2 ))Λ ) 1 Λ )]  = λP1 ⊕ λP2; (4)λ(P1 ⊗ P2) = λP1 ⊗ λP2; λ(P1 ⊗ P2) = λ  [1 − e − ( −InL + 1 )Λ+(−InL + 2 )Λ ) 1 Λ , 1 − e − ( −In(1−M + 1 ))Λ+(−In(1−M + 2 ))Λ ) 1 Λ , 1 − e − ( −In(1−N + 1 ))Λ+(−In(1−N + 2 ))Λ ) 1 Λ ], [−(1 − e − ( −In(1−L − 1 ))Λ+(−In(1−L − 2 ))Λ ) 1 Λ ), −(1 − e − ( −InM − 1 )Λ+(−InM − 2 )Λ ) 1 Λ ), −(1 − e − ( −InN − 1 )Λ+(−InN − 2 )Λ ) 1 Λ )]  =  [1 − e − ( (λ)−InL + 1 )Λ+((λ)−InL + 2 )Λ ) 1 Λ , 1 − e − ( (λ)−In(1−M + 1 ))Λ+((λ)−In(1−M + 2 ))Λ ) 1 Λ , 1 − e − ( (λ)−In(1−N + 1 ))Λ+((λ)−In(1−N + 2 ))Λ ) 1 Λ ], [−(1 − e − ( (λ)−In(1−L − 1 ))Λ+((λ)−In(1−L − 2 ))Λ ) 1 Λ ), −(1 − e − ( (λ)−InM − 1 )Λ+((λ)−InM − 2 )Λ ) 1 Λ ), −(1 − e − ( (λ)−InN − 1 )Λ+((λ)−InN − 2 )Λ ) 1 Λ )]  = λP1 ⊗ λP2 Figure 3 is given as below 3. Bipolar neutrosophic based on Aczel-Alsina aggregation operators In this section, we introduce the six aggregation operators including as BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, BNAAOWG, BNAAHWG operators. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 12 of 43 3.1. BNAAWA operator Definition 6. Let L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAWA operator if it satisfies: BNAAWA(L1, L2, ..., Ln)T = n⊕ P =1 λP LP λP is the weight of L̃P (P = 1, 2, 3, ..., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 13 of 43 Theorem 2. Let L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAWA operator if it satisfies: BNAAWA(L1, L2, ..., Ln)T =  [1 − e − ( n∑ P =1 λ(−In(1−Γ+ P ))Λ ) 1 Λ , e − ( n∑ P =1 λ(−Inξ+ P )Λ ) 1 Λ , e − ( n∑ P =1 λ(−Inϑ+ P )Λ ) 1 Λ ] [−e − ( n∑ P =1 λ(−InΓ− P )Λ ) 1 Λ , −(1 − e − ( n∑ P =1 λ(−In(1−ξ− P ))Λ ) 1 Λ ), −(1 − e − ( n∑ P =1 λ(−In(1−ϑ− P ))Λ ) 1 Λ )  , where λP is the weight of L̃P (P = 1, 2, 3, ..., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. Proof. Appendix A Theorem 3. (Idempotency):If L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  for all (P = 1, 2, 3, ..., n), then BNAAWA(L1, L2, ..., Ln) = L. Proof. Appendix B Theorem 4. (Commutativity) :If (QL ′ 1, QL ′ 2, ..., QL ′ n) is any permutation of (QL1, QL2, ..., QLn), then BNAAWA(QL ′ 1, QL ′ 2, ..., QL ′ n) = BNAAWA(QL1, QL2, ..., QLn). A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 14 of 43 3.2. BNAAOWA operator Definition 7. Let L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAOWA operator if it satisfies: BNAAOWA(L1, L2, ..., Ln)T = n⊕ P =1 λP LP λP is the weight of L̃P (P = 1, 2, 3, ..., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. Theorem 5. Let L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAOWA operator if it satisfies: BNAAOWA(L1, L2, ..., Ln)T =  [1 − e − ( n∑ P =1 λ(−In(1−Γ+ P ))Λ ) 1 Λ , e − ( n∑ P =1 λ(−Inξ+ P )Λ ) 1 Λ , e − ( n∑ P =1 λ(−Inϑ+ P )Λ ) 1 Λ ] [−e − ( n∑ P =1 λ(−InΓ− P )Λ ) 1 Λ , −(1 − e − ( n∑ P =1 λ(−In(1−ξ− P ))Λ ) 1 Λ ), −(1 − e − ( n∑ P =1 λ(−In(1−ϑ− P ))Λ ) 1 Λ )  , where λP is the weight of L̃P (P = 1, 2, 3....., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 15 of 43 Theorem 6. (Idempotency):If L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  for all (P = 1, 2, 3, ..., n), then BNAAOWA(L1, L2, ..., Ln) = L. Theorem 7. (Commutativity) :If (QL ′ 1, QL ′ 2, ..., QL ′ n) is any permutation of (QL1, QL2, ..., QLn), then BNAAOWA(QL ′ 1, QL ′ 2, ..., QL ′ n) = BNAAOWA(QL1, QL2, ..., QLn). 3.3. BNAAHWA operator Definition 8. Let L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAHWA operator if it satisfies: BNAAHWA(L1, L2, ..., Ln)T = n⊕ P =1 λP LP And λP is the weight of L̃P (P = 1, 2, 3, ..., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. Theorem 8. Let L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAHWA operator if it satisfies: A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 16 of 43 BNAAHWA(L1, L2, ..., Ln)T =  [1 − e − ( n∑ P =1 λ(−In(1−Γ+ P ))Λ ) 1 Λ , e − ( n∑ P =1 λ(−Inξ+ P )Λ ) 1 Λ , e − ( n∑ P =1 λ(−Inϑ+ P )Λ ) 1 Λ ] [−e − ( n∑ P =1 λ(−InΓ− P )Λ ) 1 Λ , −(1 − e − ( n∑ P =1 λ(−In(1−ξ− P ))Λ ) 1 Λ ), −(1 − e − ( n∑ P =1 λ(−In(1−ϑ− P ))Λ ) 1 Λ )  , where λP is the weight of L̃P (P = 1, 2, 3....., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. Theorem 9. (Idempotency):If L̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  for all (P = 1, 2, 3, ..., n), then BNAAHWA(L1, L2, ..., Ln) = L. Theorem 10. (Commutativity) :If (QL ′ 1, QL ′ 2, ..., QL ′ n) is any permutation of (QL1, QL2, ..., QLn), then BNAAHWA(QL ′ 1, QL ′ 2, ..., QL ′ n) = BNAAHWA(QL1, QL2, ..., QLn). 3.4. BNAAWG operator Definition 9. Let Q̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  present a family of bipolar neutrosophic numbers. It is termed as the BNAAWG operator if it satisfies: BNAAWG(Q1, Q2, ..., Qn)T = n⊗ P =1 QλP P , where λP is the weight of Q̃P (P = 1, 2, 3, ..., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 17 of 43 Theorem 11. Let PIQ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAWG operator if it satisfies: BNAAHG(PIQ1, P IQ2, ..., P IQn)T =  [e − ( n∑ P =1 λ(−InΓ+ P )Λ ) 1 Λ , 1 − e − ( n∑ P =1 λ(−In(1−ξ+ P ))Λ ) 1 Λ , 1 − e − ( n∑ P =1 λ(−In(1−ϑ+ P ))Λ ) 1 Λ ], [−(1 − e − ( n∑ P =1 λ(−In(1−Γ− P ))Λ ) 1 Λ ), −e − ( n∑ P =1 λ(−Inξ− P )Λ ) 1 Λ , −e − ( n∑ P =1 λ(−Inϑ− P )Λ ) 1 Λ ]  where λP is the weight of PIQP (P = 1, 2, 3, ..., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. Proof. Appendix C Theorem 12. (Idempotency):If P̃ IQ =  H + , L + , S + , H − , L − , S −  for all (P = 1, 2, 3, ..., n), then BNAAWG(PIQ1, P IQ2, ..., P IQn) = PIQ. Theorem 13. (Commutativity) :If (PIQ ′ 1, P IQ ′ 2, ..., P IQ ′ n) is any permutation of (PIQ1, P IQ2, ..., P IQn), then BNAAWG(PIQ ′ 1, P IQ ′ 2, ..., P IQ ′ n) = BNAAWG(PIQ1, P IQ2, ..., P IQn). A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 18 of 43 3.5. BNAAOWG operator Definition 10. Let Q̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  present a family of bipolar neutrosophic numbers. It is termed as the BNAAOWG operator if it satisfies: BNAAOWG(Q1, Q2, ..., Qn)T = n⊗ P =1 QλP P , where λP is the weight of Q̃P (P = 1, 2, 3, ..., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. Theorem 14. Let PIQ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAOWG operator if it satisfies: BNAAOWG(PIQ1, P IQ2, ..., P IQn)T =  [e − ( n∑ P =1 λ(−InΓ+ P )Λ ) 1 Λ , 1 − e − ( n∑ P =1 λ(−In(1−ξ+ P ))Λ ) 1 Λ , 1 − e − ( n∑ P =1 λ(−In(1−ϑ+ P ))Λ ) 1 Λ ], [−(1 − e − ( n∑ P =1 λ(−In(1−Γ− P ))Λ ) 1 Λ ), −e − ( n∑ P =1 λ(−Inξ− P )Λ ) 1 Λ , −e − ( n∑ P =1 λ(−Inϑ− P )Λ ) 1 Λ ]  where λP is the weight of PIQP (P = 1, 2, 3....., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 19 of 43 Theorem 15. (Idempotency):If P̃ IQ =  H + , L + , S + , H − , L − , S −  for all P = 1, 2, 3..., n, then BNAAOWG(PIQ1, P IQ2, ..., P IQn) = PIQ. Theorem 16. (Commutativity) :If (PIQ ′ 1, P IQ ′ 2, ..., P IQ ′ n) is any permutation of (PIQ1, P IQ2, ..., P IQn), then BNAAOWG(PIQ ′ 1, P IQ ′ 2, ..., P IQ ′ n) = BNAAOWG(PIQ1, P IQ2, ..., P IQn). 3.6. BNAAHWG operator Definition 11. Let Q̃ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  present a family of bipolar neutrosophic numbers. It is termed as the BNAAHWG operator if it satisfies: BNAAHWG(Q1, Q2, ..., Qn)T = n⊗ P =1 QλP P , where λP is the weight of Q̃P (P = 1, 2, 3, ..., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. Theorem 17. Let PIQ =  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  represent a family of bipolar neutrosophic numbers. It is termed as the BNAAHWG operator if it satisfies: A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 20 of 43 BNAAHWG(PIQ1, P IQ2, ..., P IQn)T =  [e − ( n∑ P =1 λ(−InΓ+ P )Λ ) 1 Λ , 1 − e − ( n∑ P =1 λ(−In(1−ξ+ P ))Λ ) 1 Λ , 1 − e − ( n∑ P =1 λ(−In(1−ϑ+ P ))Λ ) 1 Λ ], [−(1 − e − ( n∑ P =1 λ(−In(1−Γ− P ))Λ ) 1 Λ ), −e − ( n∑ P =1 λ(−Inξ− P )Λ ) 1 Λ , −e − ( n∑ P =1 λ(−Inϑ− P )Λ ) 1 Λ ]  where λP is the weight of PIQP (P = 1, 2, 3....., n) , λP ∈ [0, 1] and n∑ P =1 λP = 1. Theorem 18. (Idempotency):If P̃ IQ =  H + , L + , S + , H − , L − , S −  for all (P = 1, 2, 3, ..., n), then BNAAHWG(PIQ1, P IQ2, ..., P IQn) = PIQ. Theorem 19. (Commutativity) :If (PIQ ′ 1, P IQ ′ 2, ..., P IQ ′ n) is any permutation of (PIQ1, P IQ2, ..., P IQn), then BNAAHWG(PIQ ′ 1, P IQ ′ 2, ..., P IQ ′ n) = BNAAHWG(PIQ1, P IQ2, ..., P IQn). 4. MCDM Approach Based on Proposed Operators In this section, we present a decision-making approach based on the proposed operators for solving the MCDM problem under the BN environment. Consider a GDM problem in which there are m alternatives A1, A2, ....Am and n attributes G1, G2, ..., Gn whose weight vector are wC = 1, 2, ..., n such that wC > 0 and n∑ j=1 wj = 1. Let λ = (λ1, λ2, ..., λC) be the set of decision-makers and w = (w1, w2, ..., wn)T be the weight vector of λC(C = 1, 2, .., n) with wC > 0 and n∑ C=1 wC = 1. Suppose that the characteristic information of A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 21 of 43 the alternatives Ak(k = 1, 2, ..., m) over the attributes GC(C = 1, 2, ..., n) is evaluated by decision-makers λC(C = 1, 2, ..., n) and gives the preference in the form of BNNs δ =  H + , L + , S + , H − , L − , S −  , and hence formulated the BN decision matrices The following steps have been outlined to describe the DM approach based on the proposed operation; Step 1 Calculating the BN decision matrix. Step 2 Define the BNAAWA operator. BNAAWA(β1, β2, ..., βn)T =  [1 − e − ( n∑ P =1 λ(−In(1−H+ P ))Λ ) 1 Λ , e − ( n∑ P =1 λ(−InL+ P )Λ ) 1 Λ , e − ( n∑ P =1 λ(−InS+ P )Λ ) 1 Λ ] [−e − ( n∑ P =1 λ(−InH− P )Λ ) 1 Λ , −(1 − e − ( n∑ P =1 λ(−In(1−L− P ))Λ ) 1 Λ ), −(1 − e − ( n∑ P =1 λ(−In(1−S− P ))Λ ) 1 Λ )  Step 3 Define the BNAAWA operator A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 22 of 43 BNAAWA(β1, β2, ..., βn)T =  [1 − e − ( n∑ P =1 λ(−In(1−H+ P ))Λ ) 1 Λ , e − ( n∑ P =1 λ(−InL+ P )Λ ) 1 Λ , e − ( n∑ P =1 λ(−InS+ P )Λ ) 1 Λ ] [−e − ( n∑ P =1 λ(−InH− P )Λ ) 1 Λ , −(1 − e − ( n∑ P =1 λ(−In(1−L− P ))Λ ) 1 Λ ), −(1 − e − ( n∑ P =1 λ(−In(1−S− P ))Λ ) 1 Λ )  Step 4 Computing the score function Q(a) = 1 6 ( H + + 1 − L + + 1 − S + + 1 + H − − L − − S − ) Step 5: Find the ranking. Figure 4 is given as below 5. Case study Patient Background: Mr. John Doe, a 55-year-old man, underwent routine screening and was found to have stage III colon cancer. Although there is no family history of cancer, he has a history of hypertension. Diagnosis and Treatment Strategy: Following the diagnosis, Mr. Doe’s oncology team suggested a surgical and adjuvant chemotherapy course of action. To remove the tumor and surrounding lymph nodes, a partial colectomy was performed during the procedure. He had a six-month course of chemotherapy with a mixture of fluorouracil and oxaliplatin following surgery. Treatment Progression: Mr. Doe initially recovered nicely from surgery and did not experience any problems right away. But during chemotherapy, he suffered from side effects like nausea, tiredness, and neuropathy. His oncology team managed these adverse effects while keeping the effectiveness of his treatment intact by adjusting the dosage of the chemotherapy. Case Outcome: During the course of treatment, routine blood and imaging tests re- vealed a favorable response to chemotherapy, resulting in a smaller tumor and no evidence of metastasis. Despite early difficulties, Mr. Doe’s emotional and physical fortitude in- creased, and he successfully finished chemotherapy. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 23 of 43 5.1. Illustrative Example Following a routine mammography, 40-year-old Sarah was diagnosed with stage II breast cancer. She is a marketing executive who is married with two small children. Diagnosis and Available Treatments: Sarah was given two alternatives for treatment by her oncologist. Option A: Radiation therapy after a lumpectomy. Option B: Breast reconstruction surgery after a mastectomy (removal of the breast). Process of Making a Decision: Sarah had to make a difficult choice that would minimize the physical and psychological effects on her life while striking a balance with her desire to end cancer. To weigh the advantages and disadvantages of each option, she conferred with her reconstructive surgeon, breast surgeon, and oncologist. Aspects Taken Into Account: Medical Factors: Sarah took into account how well each treatment worked to eradicate cancer cells and lower the chance of recurrence. Quality of Life: Taking into account her roles as a mother and a professional, she con- sidered the effects on her appearance, her everyday activities, and her long-term emotional health. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 24 of 43 Sarah assessed the length of time needed to recuperate as well as any possible nega- tive effects from each treatment, such as soreness, scarring, and weariness brought on by radiation. Sarah’s decision: Following extensive consideration and consultation with her medical team, she decided on Option B, which is a mastectomy with reconstruction. She placed a high priority on the complete excision of the malignancy and took comfort in the knowledge that reconstruction would aid in the restoration of her confidence and body image. As part of this effort, a committee appointed by the government has approved contracts with four internet service providers. DIIG1 :Cellular Abnormality and Uncontrolled Growth: Genetic abnormalities lead- ing to uncontrollably dividing breast cells are the initial stage of cancer. These alterations in Sarah’s case resulted in the development of a cancerous growth in her breast tissue. DIIG2 :Potential for Metastasis and Invasion: Cancer cells are able to spread to nearby healthy breast tissue. Sarah was concerned about the possibility of them spreading to neighboring lymph nodes and other organs if untreated or undiagnosed. DIIG3 :Effect on Quality of Life: A patient’s quality of life may be greatly impacted by cancer and its therapies. In addition to evaluating the efficacy of various treatment methods, Sarah also took into account the possible psychological and physical side effects, as well as the consequences for her roles as a mother and a professional. DIIG4 :specific Treatment Plan: Since every patient’s cancer is different, a specific treatment plan is necessary. Sarah’s choice to have a mastectomy followed by reconstruc- tion is a reflection of her own choices and needs, including the desire to have all cancer removed and to have her physical beauty restored. The weight vector associated with the hybrid operator is (0.2, 0.1, 0.3, 0.4). Bipolar neutrosophic fuzzy decision matrices have been constructed and are presented in Tables 2 and 3. Step 1 calculating the bipolar neutrosophic fuzzy decision tables 2 and 3. Bipolar neutrosophic fuzzy decision table 2 A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 25 of 43 DIIG1 DIIG2 DIIG3 DIIG4 ZV V1  [0.04, 0.16, 0.19, [−0.01, −0.18, −0.34]   [0.02, 0.13, 0.18], [−0.17, −0.47, −0.48]   [0.17, 0.19, 0.21], [−0.04, −0.16, −0.18]   [0.21, 0.23, 0.29], [−0.07, −0.09, −0.11]  ZV V2  [0.17, 0.19, 0.21], [−0.04, −0.16, −0.18]   [0.09, 0.23, 0.26, [−0.03, −0.20, −0.41]   [0.04, 0.16, 0.19, [−0.01, −0.18, −0.34]   [0.02, 0.24, 0.25], [−0.04, −0.19, −0.45]  ZV V3  [0.04, 0.16, 0.19, [−0.01, −0.18, −0.34]   [0.02, 0.24, 0.25], [−0.04, −0.19, −0.45]   [0.09, 0.23, 0.26, [−0.03, −0.20, −0.41]   [0.17, 0.19, 0.21], [−0.04, −0.16, −0.18]  ZV V4  [0.21, 0.23, 0.29], [−0.07, −0.09, −0.11]   [0.02, 0.13, 0.18], [−0.17, −0.47, −0.48]   [0.02, 0.24, 0.25], [−0.04, −0.19, −0.45]   [0.04, 0.16, 0.19, [−0.01, −0.18, −0.34]  Bipolar neutrosophic fuzzy decision table 3 A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 26 of 43 DIIG1 DIIG2 DIIG3 DIIG4 ZV V1  [0.01, 0.2, 0.3, [−0.02, −0.11, −0.13]   [0.07, 0.13, 0.18], [−0.03, −0.07, −0.11]   [0.17, 0.19, 0.21], [−0.04, −0.16, −0.18]   [0.1, 0.3, 0.9], [−0.02, −0.04, −0.09]  ZV V2  [0.17, 0.19, 0.21], [−0.04, −0.16, −0.18]   [0.01, 0.2, 0.3, [−0.02, −0.11, −0.13]   [0.1, 0.3, 0.9], [−0.02, −0.04, −0.09]   [0.03, 0.11, 0.14], [−0.03, −0.07, −0.11]  ZV V3  [0.1, 0.3, 0.9], [−0.02, −0.04, −0.09]   [0.07, 0.13, 0.18], [−0.03, −0.07, −0.11]   [0.01, 0.2, 0.3, [−0.02, −0.11, −0.13]   [0.17, 0.19, 0.21], [−0.04, −0.16, −0.18]  ZV V4  [0.07, 0.13, 0.18], [−0.03, −0.07, −0.11]   [0.02, 0.13, 0.18], [−0.17, −0.47, −0.48]   [0.17, 0.19, 0.21], [−0.04, −0.16, −0.18]   [0.01, 0.2, 0.3, [−0.02, −0.11, −0.13]  Step 2 calculates the BNAAWA operator using weights w = (0.3, 0.2, 0.4, 0.1), as shown in Table 4 BNAAWA operator table 4. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 27 of 43 DIIG1 DIIG2 DIIG3 DIIG3 ZV V1  [0.2098, 0.2345, 0.2654], [−0.0369, −0.2036, −0.4145]   [0.1078, 0.1369, 0.1874], [−0.1789, −0.4712, −0.4878]   [0.1745, 0.1982, 0.2123], [−0.0456, −0.1642, −0.1896]   [0.2123, 0.2896, 0.2854], [−0.0345, −0.2785, −0.4562]  ZV V2  [0.1789, 0.1896, 0.1783], [−0.1778, −0.4963, −0.4789]   [0.2098, 0.2345, 0.2654], [−0.0369, −0.2036, −0.4145]   [0.1103, 0.1104, 0.1105], [−0.1101, −0.1203, −0.1004]   [0.1078, 0.1369, 0.1874], [−0.1789, −0.4712, −0.4878]  ZV V3  [0.1745, 0.1982, 0.2123], [−0.0456, −0.1642, −0.1896]   [0.1078, 0.1369, 0.1874], [−0.1789, −0.4712, −0.4878]   [0.2098, 0.2345, 0.2654], [−0.0369, −0.2036, −0.4145]   [0.0236, 0.0459, 0.1125], [−0.1456, −0.1698, −0.1896]  ZV V3  [0.2098, 0.2345, 0.2654], [−0.0369, −0.2036, −0.4145]   [0.1745, 0.1982, 0.2123], [−0.0456, −0.1642, −0.1896]   [0.0236, 0.0459, 0.1125], [−0.1456, −0.1698, −0.1896]   [0.0111, 0.0125, 0.0128], [−0.1025, −0.1456, −0.1698]  Step 3:Define the BNAAWA operator w = (0.3, 0.2, 0.4, 0.1) and table 5 is given as BNAAWA operator Table 5 ZV V1  [0.3098, 0.4698, 0.6478], [−0.2369, −0.2789, −0.2258]  ZV V2  [0. − 0983, 0.0896, 0.2753], [−0.2258, −0.4753, −0.4147]  ZV V3  [0.1203, 0.4564, 0.5855], [−0.9831, −0.1233, −0.9874]  ZV V3  [0.4568, 0.9636, 0.9879], [−0.9631, −0.7894, −0.8237]  Step 4:Determine the score function. E1 = 0.0136, E2 = 0.6743, E3 = 0.1839, E4 = 0.4959. Step 5:Find the ranking E2 > E4 > E3 > E1 and E2 is the best ranking. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 28 of 43 Figure 5 is given as 5.2. Comparison Analysis The comparison method with the existing method table 6 Operators E1 E2 E3 E4 Ranking BNAAWA 0.0134 0.7876 0.1852 0.7456 E2 > E4 > E3 > E1 BNAAOWA 0.6345 0.6969 0.6565 0.5021 E2 > E3 > E1 > E4 BNAAHWA 0.6098 0.1974 0.0098 0.0056 E1 > E2 > E3 > E1 BNAAWG 0.3227 0.5698 0.4987 0.0399 E2 > E3 > E1 > E4 BNAAOWG 0.4964 0.5558 0.6697 0.4156 E3 > E2 > E1 > E3 BNAAHWG 0.0003 0.0151 0.0451 0.1487 E4 > E3 > E2 > E1 FFB operator [5] 0.4454 0.2653 0.5567 0.8959 E4 > E3 > E1 > E2 We acknowledge the need to show the performance of our suggested MADM model in comparison to the current Aczel-Alsina-based MADM tools. In order to solve this, we have included more performance indicators to our comparative study, such as ranking stability, computational efficiency, and judgment correctness. Our approach is directly compared with other Aczel-Alsina-based MADM models on benchmark datasets in a quantitative evaluation that has been incorporated. The assertion that our model produces superior or at least comparable outcomes in particular decision- making scenarios has been substantiated by statistical metrics. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 29 of 43 This makes it possible to compare things clearly and systematically, showing where our approach works well and where conventional approaches can still be useful. To increase clarity and draw attention to important performance disparities, both the visual and numerical results have been improved. 5.3. Sensitivity analysis In this subsection, we define the sensitivity study in below table 7 Operators Score function Ranking Final Ranking IPFS [19]  E1 = 0.0012, E2 = 0.1204, E3 = 0.0059, E4 = 0.1014   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  AAO [28]  E1 = 0.0021, E2 = 0.1698, E3 = 0.0139, E4 = 0.1463   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  PFS [32]  E1 = 0.0102, E2 = 0.1409, E3 = 0.0134, E4 = 0.0987   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  AAO [30]  E1 = 0.0111, E2 = 0.3091, E3 = 0.1202, E4 = 0.1908   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  5.4. Results and discussion In this subsection, we introduce results and discussion in table 8. The results and discussion table 8 A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 30 of 43 Author Score function Ranking Ranking Aczel-Alsina [27]  E1 = 0.0009, E2 = 0.7603, E3 = 0.1098, E4 = 0.4563   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  BNS [8]  E1 = 0.0615, E2 = 0.3265, E3 = 0.2589, E4 = 0.2698   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  PFS [10]  f1 = 0.1498, f2 = 0.6987, f3 = 0.4987, f4 = 0.5674   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  IVIF [19]  E1 = 0.0156, E2 = 0.7274, E3 = 0.3874, E4 = 0.3989   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  BFSS [15]  E1 = 0.1987, E2 = 0.6987, E3 = 0.3984, E4 = 0.5472   E2 > E4 > E3 > E1   E2 > E4 > E3 > E1  The effectiveness of the suggested approach was assessed by applying it to a real- world health-related decision-making situation. The findings show that the multi-attribute decision-making (MADM) model based on Aczel-Alsina effectively ranked the available options according to the specified criteria. The validity of the suggested strategy was confirmed by the rankings that were acquired, which agreed with expert assessments. The model offered a more accurate and flexible evaluation than current MADM methodologies, especially when it came to managing the uncertainties involved in medical decision-making. The study’s main conclusion is that the suggested approach is reliable when handling bipolar neutrosophic data, enabling decision-makers to concurrently take into account both positive and negative evaluations. This is especially helpful in the medical field, as treatment options frequently require balancing risks and benefits. Sensitivity study further demonstrated the model’s dependability for real-world applications by confirming that it stayed stable under various weight distributions. Despite these benefits, a thorough comparison with existing Aczel-Alsina-based MADM tools is required to prove the suggested method’s superiority. Additional assessments using different aggregation criteria, such as Dombi, Frank, and Einstein, should be carried out, even though the study offers a comparative analysis with conventional techniques. This would make it easier to determine whether the Aczel-Alsina method is the best way to handle health-related decision-making issues. The proposed Aczel-Alsina-based decision-making model was evaluated using bench- mark datasets and real-world decision-making scenarios. The results demonstrate the model’s ability to handle uncertainty and inaccurate data, particularly in severe and A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 31 of 43 interval-tough scenarios. To validate our methodology, we compared the performance of the proposed model with choice frameworks based on Dombi, Frank, and Einstein norms. 5.5. Limitation In this subsection, we define the limitation in table 9. The Comparison method with existing method table 9 Methods Best Normal Good [2] yes no no [6] yes no yes [15] yes yes yes [27] yes no yes [9] yes yes no [34] yes yes no We recognize that there isn’t a single MADM model that works well for every issue. Consequently: We have updated our discussion to more precisely outline the parameters and restric- tions of our approach. Our model’s performance is influenced by the selection of aggregation operators and weight vectors. In some decision-making situations, choosing the wrong parameters might result in less-than-ideal outcomes. Even while our approach works well in the studied scenarios, it might not work as well for all multi-criteria decision-making problems, particularly those with extremely uncertain or dynamic settings. Despite increasing choice accuracy, our method may be more computationally expen- sive than more straightforward MADM approaches, which makes it less appropriate for real-time decision-making applications. Our methodology is predicated on the consistency and dependability of decision ma- trices supplied by experts. The robustness of the results, however, may be impacted by discrepancies or disagreements among expert judgments that occur in real-world situa- tions. Although our approach has been evaluated on certain datasets, additional validation in a variety of fields (such as engineering, healthcare, and finance) is required to verify its wide applicability. 5.6. Superiority Because of its superior capacity to manage ambiguity, conflicting information, and uncertainty in the bipolar neutrosophic environment, the Aczel-Alsina aggregation frame- work has been used in this investigation. For complex decision-making issues where mem- bership, non-membership, and hesitation degrees change greatly, traditional aggregation operators like Einstein, Hamacher, and Weighted Averaging (WA) might not offer the necessary flexibility. A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 32 of 43 Why is Aczel-Alsina Needed in This Study? Nonlinearity and Flexibility: Aczel-Alsina aggregation is more appropriate for manag- ing extreme or uncertain values in group decision-making scenarios because it facilitates nonlinear information fusion, in contrast to standard operators. Sturdy Conflict Resolution: Divergent expert opinions are frequently a part of decision- making. Aczel-Alsina aggregation reduces discrepancies while preserving decision integrity by facilitating a seamless transition between decision values. Enhanced Sensitivity to Decision Weights: The framework makes sure that extreme values don’t unduly affect the conclusion by accurately capturing the effects of high or low trustworthiness levels among decision-makers. Mathematical Properties: The Aczel-Alsina operator satisfies key mathematical condi- tions such as associativity, idempotency, and boundedness, which are essential for a stable and reliable aggregation process. Superiority Over Traditional Aggregation Operators To demonstrate the superiority of the Aczel-Alsina aggregation framework, we have conducted a comparative analysis with other traditional aggregation operators, such as:Einstein Aggregation, which is less flexible in handling extreme cases, Hamacher Aggregation, which may not be suitable for high levels of uncertainty, Weighted Averaging Aggregation, which may not efficiently capture nonlinear variations in decision preferences. The findings show that the Aczel-Alsina framework yields more accurate, consistent, and stable solutions, especially in situations involving highly ambiguous and contradictory decision-making contexts. We include a numerical example that demonstrates how Aczel-Alsina aggregation out- performs traditional techniques in real-world decision-making issues to further bolster the usefulness of this strategy. This illustration demonstrates how well it works to increase the precision and consistency of combined decisions. Our approach reduces choice differences by ensuring more accurate and dependable rankings through the use of an enhanced weighting mechanism and a changed aggregation process. The decision-making process of traditional MADM approaches is often characterized by ambiguity and uncertainty. Our model incorporates bipolar neutrosophic fuzzy decision matrices to enhance the representation of contradictory and confusing data. In spite of the intricacy of the decision-making process, our approach is suitable for large-scale choice problems since it is computationally efficient in comparison to traditional Aczel-Alsina- based approaches. Our method produces more stable and consistent ranks under a variety of input scenarios and tackles issues such as rank reversal, which commonly affects earlier MADM techniques. 6. Conclusion This study introduced a novel approach to group decision-making by utilizing Bipolar Neutrosophic Aczel-Alsina operators, specifically designed to handle complex and uncer- A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 33 of 43 tain data. Through the application of these operators in a multi-criteria decision-making (MCDM) framework, we demonstrated their effectiveness in aggregating bipolar neutro- sophic information, which accommodates both positive and negative judgments with vary- ing degrees of truth, indeterminacy, and falsity. This approach enables a more nuanced and flexible analysis compared to traditional aggregation methods. The results from our case study and comparative analysis confirm that the BNAA operators provide a robust and adaptable solution, enhancing decision-making accuracy and resilience in situations with high levels of ambiguity. Sensitivity analysis further verified the stability and reliability of the proposed operators, illustrating their practicality for real-world group decision-making applications. However, the study also identified some limitations, particularly in scenarios with ex- treme or conflicting preferences among decision-makers. Future research could explore hybrid models that integrate BNAA operators with other aggregation techniques to ad- dress these limitations. This work contributes to the advancement of neutrosophic and fuzzy logic-based decision-making, offering a powerful tool for complex decision environ- ments characterized by uncertainty and bipolar information. Figure 6 is given below as ead and agreed to the published version of the manuscript. Acknowledgements D.K.A. extends appreciation to the Deanship of Postgraduate Studies and Scientific Research at Majmaah University for supporting this research work.\\ A.K and T.A would like to thank Prince Sultan University for paying the APC and the support through TAS research lab. Compliance with Ethical Standards Disclosure of potential conflicts of interest: The authors declare that there is no conflict of interests regarding the publication of this paper. Compliance with Ethical Standards: This study is not supported by any source or any organizations. Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors. 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Study of nonlinear wave equation of optical field for solotonic type results. Partial Differential Equations in Applied Mathematics, 13:101048, 2025. [37] H. C. Wu. Generalized extension principle for non-normal fuzzy sets. Fuzzy Opti- mization and Decision Making, 18(4):399–432, 2019. Appendix A Proof. Since n = 1 L1d1 =  [1 − e−(λ1(−In(1−Γ+ 1 ))Λ) 1 Λ , e−(λ1(−Inξ+ 1 )Λ) 1 Λ , e−(λ1(−Inϑ+ 1 )Λ) 1 Λ ] [−e−(λ1(−InΓ− 1 )Λ) 1 Λ , −(1 − e−(λ1(−In(1−ξ− 1 ))Λ) 1 Λ ), −(1 − e−(λ1(−In(1−ϑ− 1 ))Λ) 1 Λ )  n = 2 L2d2 =  [1 − e−(λ2(−In(1−Γ+ 2 ))Λ) 1 Λ , e−(λ2(−Inξ+ 2 )Λ) 1 Λ , e−(λ2(−Inϑ+ 2 )Λ) 1 Λ ] [−e−(λ2(−InΓ− 2 )Λ) 1 Λ , −(1 − e−(λ2(−In(1−ξ− 2 ))Λ) 1 Λ ), −(1 − e−(λ2(−In(1−ϑ− 2 ))Λ) 1 Λ )  L1d1 ⊕ L2d2 =  [1 − e−(λ1(−In(1−Γ+ 1 ))Λ) 1 Λ , e−(λ1(−Inξ+ 1 )Λ) 1 Λ , e−(λ1(−Inϑ+ 1 )Λ) 1 Λ ] [−e−(λ1(−InΓ− 1 )Λ) 1 Λ , −(1 − e−(λ1(−In(1−ξ− 1 ))Λ) 1 Λ ), −(1 − e−(λ1(−In(1−ϑ− 1 ))Λ) 1 Λ )  ⊕  [1 − e−(λ2(−In(1−Γ+ 2 ))Λ) 1 Λ , e−(λ2(−Inξ+ 2 )Λ) 1 Λ , e−(λ2(−Inϑ+ 2 )Λ) 1 Λ ] [−e−(λ2(−InΓ− 2 )Λ) 1 Λ , −(1 − e−(λ2(−In(1−ξ− 2 ))Λ) 1 Λ ), −(1 − e−(λ2(−In(1−ϑ− 2 ))Λ) 1 Λ )  n = k A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 38 of 43 BNAAWA(L1, L2, ..., Ln)T =  [1 − e −  k∑ P =1 λ(−In(1−Γ+ P ))Λ  1 Λ , e −  k∑ P =1 λ(−Inξ+ P )Λ  1 Λ , e −  k∑ P =1 λ(−Inϑ+ P )Λ  1 Λ ] [−e −  k∑ P =1 λ(−InΓ− P )Λ  1 Λ , −(1 − e −  k∑ P =1 λ(−In(1−ξ− P ))Λ  1 Λ ), −(1 − e −  k∑ P =1 λ(−In(1−ϑ− P ))Λ  1 Λ )  n = k + 1 BNAAWA(L1, L2, ..., Ln)T =  [1 − e − k+1∑ P =1 λ(−In(1−Γ+ P ))Λ  1 Λ , e − k+1∑ P =1 λ(−Inξ+ P )Λ  1 Λ , e − k+1∑ P =1 λ(−Inϑ+ P )Λ  1 Λ ] [−e − k+1∑ P =1 λ(−InΓ− P )Λ  1 Λ , −(1 − e − k+1∑ P =1 λ(−In(1−ξ− P ))Λ  1 Λ ), −(1 − e − k+1∑ P =1 λ(−In(1−ϑ− P ))Λ  1 Λ )  A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 39 of 43 BNAAWA(L1, L2, ..., Ln)T =  [1 − e −  k∑ P =1 λ(−In(1−Γ+ P ))Λ  1 Λ , e −  k∑ P =1 λ(−Inξ+ P )Λ  1 Λ , e −  k∑ P =1 λ(−Inϑ+ P )Λ  1 Λ ] [−e −  k∑ P =1 λ(−InΓ− P )Λ  1 Λ , −(1 − e −  k∑ P =1 λ(−In(1−ξ− P ))Λ  1 Λ ), −(1 − e −  k∑ P =1 λ(−In(1−ϑ− P ))Λ  1 Λ )  ⊕  [1 − e − k+1∑ P =1 λ(−In(1−Γ+ P ))Λ  1 Λ , e − k+1∑ P =1 λ(−Inξ+ P )Λ  1 Λ , e − k+1∑ P =1 λ(−Inϑ+ P )Λ  1 Λ ] [−e − k+1∑ P =1 λ(−InΓ− P )Λ  1 Λ , −(1 − e − k+1∑ P =1 λ(−In(1−ξ− P ))Λ  1 Λ ), −(1 − e − k+1∑ P =1 λ(−In(1−ϑ− P ))Λ  1 Λ )  . Appendix B Proof. Since LP = L are equal to  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  for all (P = 1, 2, 3, ..., n), then A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 40 of 43 BNAAWA(L1, L2, ..., Ln)T =  [1 − e − ( n∑ P =1 λ(−In(1−Γ+ P ))Λ ) 1 Λ , e − ( n∑ P =1 λ(−Inξ+ P )Λ ) 1 Λ , e − ( n∑ P =1 λ(−Inϑ+ P )Λ ) 1 Λ ] [−e − ( n∑ P =1 λ(−InΓ− P )Λ ) 1 Λ , −(1 − e − ( n∑ P =1 λ(−In(1−ξ− P ))Λ ) 1 Λ ), −(1 − e − ( n∑ P =1 λ(−In(1−ϑ− P ))Λ ) 1 Λ )  =  [1 − e−(λ(−In(1−Γ+ P ))Λ) 1 Λ , e−(λ(−Inξ+ P )Λ) 1 Λ , e−(λ(−Inϑ+ P )Λ) 1 Λ ] [−e−(λ(−InΓ− P )Λ) 1 Λ , −(1 − e−(λ(−In(1−ξ− P ))Λ) 1 Λ ), −(1 − e−(λ(−In(1−ϑ− P ))Λ) 1 Λ )  = { [1 − eIn(1−Γ+ P )), e(−Inξ+ P ), eλ(−Inϑ+ P )] [eInΓ− P ), −(1 − eIn(1−ξ− P ))), −(1 − eIn(1−ϑ− P ))) }  Γ+ , ξ + , ϑ + , Γ− , ξ − , ϑ −  = L BNAAWA(L1, L2, ..., Ln) = L. Appendix C Proof. Since n = 1 PIQ1λ1 =  [e−(λ1(−InΓ+ 1 )Λ) 1 Λ , 1 − e−(λ1(−In(1−ξ+ 1 ))Λ) 1 Λ , 1 − e−(λ1(−In(1−ϑ+ 1 ))Λ) 1 Λ ], [−(1 − e−(λ1(−In(1−Γ− 1 ))Λ) 1 Λ ), −e−(λ1(−Inξ− 1 )Λ) 1 Λ , −e−(λ1(−Inϑ− 1 )Λ) 1 Λ ]  n = 2 A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 41 of 43 PIQ2λ2 =  [e−(λ2(−InΓ+ 2 )Λ) 1 Λ , 1 − e−(λ2(−In(1−ξ+ 2 ))Λ) 1 Λ , 1 − e−(λ2(−In(1−ϑ+ 2 ))Λ) 1 Λ ], [−(1 − e−(λ2(−In(1−Γ− 2 ))Λ) 1 Λ ), −e−(λ2(−Inξ− 2 )Λ) 1 Λ , −e−(λ2(−Inϑ− 2 )Λ) 1 Λ ]  PIQ1λ1 ⊗ PIQ2λ2 =  [e−(λ1(−InΓ+ 1 )Λ) 1 Λ , 1 − e−(λ1(−In(1−ξ+ 1 ))Λ) 1 Λ , 1 − e−(λ1(−In(1−ϑ+ 1 ))Λ) 1 Λ ], [−(1 − e−(λ1(−In(1−Γ− 1 ))Λ) 1 Λ ), −e−(λ1(−Inξ− 1 )Λ) 1 Λ , −e−(λ1(−Inϑ− 1 )Λ) 1 Λ ]  ⊕  [e−(λ2(−InΓ+ 2 )Λ) 1 Λ , 1 − e−(λ2(−In(1−ξ+ 2 ))Λ) 1 Λ , 1 − e−(λ2(−In(1−ϑ+ 2 ))Λ) 1 Λ ], [−(1 − e−(λ2(−In(1−Γ− 2 ))Λ) 1 Λ ), −e−(λ2(−Inξ− 2 )Λ) 1 Λ , −e−(λ2(−Inϑ− 2 )Λ) 1 Λ ]  n = k A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 42 of 43 BNAAHG(PIQ1, P IQ2, ..., P IQk)T =  [e −  k∑ P =1 λ(−InΓ+ P )Λ  1 Λ , 1 − e −  k∑ P =1 λ(−In(1−ξ+ P ))Λ  1 Λ , 1 − e −  k∑ P =1 λ(−In(1−ϑ+ P ))Λ  1 Λ ], [−(1 − e −  k∑ P =1 λ(−In(1−Γ− P ))Λ  1 Λ ), −e −  k∑ P =1 λ(−Inξ− P )Λ  1 Λ , −e −  k∑ P =1 λ(−Inϑ− P )Λ  1 Λ ]  n = k + 1 BNAAHG(PIQ1, P IQ2, ..., P IQk+1)T =  [e − k+1∑ P =1 λ(−InΓ+ P )Λ  1 Λ , 1 − e − k+1∑ P =1 λ(−In(1−ξ+ P ))Λ  1 Λ , 1 − e − k+1∑ P =1 λ(−In(1−ϑ+ P ))Λ  1 Λ ], [−(1 − e − k+1∑ P =1 λ(−In(1−Γ− P ))Λ  1 Λ ), −e − k+1∑ P =1 λ(−Inξ− P )Λ  1 Λ , −e −  k∑ P =1 λ(−Inϑ− P )Λ  1 Λ ]  BNAAHG(PIQ1, P IQ2, ..., P IQk ⊕ PIQ1, P IQ2, ..., P IQk+1)T = A. Fahmi / Eur. J. Pure Appl. Math, 18 (2) (2025), 5824 43 of 43 [e −  k∑ P =1 λ(−InΓ+ P )Λ  1 Λ , 1 − e −  k∑ P =1 λ(−In(1−ξ+ P ))Λ  1 Λ , 1 − e −  k∑ P =1 λ(−In(1−ϑ+ P ))Λ  1 Λ ], [−(1 − e −  k∑ P =1 λ(−In(1−Γ− P ))Λ  1 Λ ), −e −  k∑ P =1 λ(−Inξ− P )Λ  1 Λ , −e −  k∑ P =1 λ(−Inϑ− P )Λ  1 Λ ]  ⊕  [e − k+1∑ P =1 λ(−InΓ+ P )Λ  1 Λ , 1 − e − k+1∑ P =1 λ(−In(1−ξ+ P ))Λ  1 Λ , 1 − e − k+1∑ P =1 λ(−In(1−ϑ+ P ))Λ  1 Λ ], [−(1 − e − k+1∑ P =1 λ(−In(1−Γ− P ))Λ  1 Λ ), −e − k+1∑ P =1 λ(−Inξ− P )Λ  1 Λ , −e −  k∑ P =1 λ(−Inϑ− P )Λ  1 Λ ]  .