Binary relations are most important in various fields of pure and applied sciences. The concept of linear Diophantine fuzzy sets (LDFSs) proposed by Riaz and Hashmi is a novel mathematical approach to model vagueness and uncertainty in decision-making problems. In LDFS theory, the use of reference or control parameters corresponding to membership and non-membership grades makes it most accommodating towards modeling uncertainties in real-life problems. The main purpose of this paper is to establish a robust fusion of binary relations and LDFSs, and to introduce the concept of linear Diophantine fuzzy relation (LDF-relation) by making the use of reference parameters corresponding to the membership and non-membership fuzzy relations. The novel concept of LDF-relation is more flexible to discuss the symmetry between two or more objects that is superior to the prevailing notion of intuitionistic fuzzy relation (IF-relation). Certain basic operations are defined to investigate some significant results which are very useful in solving real-life problems. Based on these operations and their related results, it is analyzed that the collection of all LDF-relations gives rise to some algebraic structures such as semi-group, semi-ring and hemi-ring. Furthermore, the notion of score function of LDF-relations is introduced to analyze the symmetry of the optimal decision and ranking of feasible alternatives. Additionally, a new algorithm for modeling uncertainty in decision-making problems is proposed based on LDFSs and LDF-relations. A practical application of proposed decision-making approach is illustrated by a numerical example. Proposed LDF-relations, their operations, and related results may serve as a foundation for computational intelligence and modeling uncertainties in decision-making problems.

Aggregation operators are fundamental concept for information fusion in real-life problems. Many researchers developed aggregation operators for multi-criteria decision-making (MCDM) under uncertainty. Unfortunately, the existing operators can be utilized under strict limitations and constraints. In this manuscript, we focused on new prioritized aggregation operators which remove the strict limitations of the existing operators. The addition of reference parameters associated with membership and non-membership grades in the linear Diophantine Fuzzy sets provide a robust modeling for MCDM problems. The primary objective of this manuscript is to introduce new aggregation operators for modeling uncertainty by using linear Diophantine Fuzzy information. For this objective we develop aggregation operators (AO) namely, "linear Diophantine Fuzzy prioritized weighted average" (LDFPWA) operator and "linear Diophantine Fuzzy prioritized weighted geometric" (LDFPWG) operator. Certain essential properties of new prioritized AOs are also proposed. A secondary objective is to discuss a practical application of third party reverse logistic provider (3PRLP) optimization problem. The efficiency, superiority, and rationality of the proposed approach is analyzed by a numerical example to discuss 3PRLP. The symmetry of optimal decision and ranking of feasible alternatives is followed by a comparative analysis.

The idea of bipolar complex fuzzy (BCF) sets, as a genuine modification of both bipolar fuzzy sets and complex fuzzy sets, gives a massive valuable framework for representing and evaluating ambiguous information. In intelligence decision making based on BCF sets, it is a critical dilemma to compare or rank positive and negative membership grades. In this framework, we deliberated various techniques for aggregating the collection of information into a singleton set, called BCF weighted arithmetic averaging (BCFWAA), BCF ordered weighted arithmetic averaging (BCFOWAA), BCF weighted geometric averaging (BCFWGA), and BCF ordered weighted geometric averaging (BCFOWGA) operators for BCF numbers (BCFNs). To illustrate the feasibility and original worth of the diagnosed approaches, we demonstrated various properties of the diagnosed operators, in addition to their capability that the evaluated value of a set of BCF numbers is a unique BCF number. Further, multiattribute decision making (“MADM”) refers to a technique employed to compute a brief and dominant assessment of opinions with multiattributes. The main influence of this theory is implementing the diagnosed theory in the field of the MADM tool using BCF settings. Finally, a benchmark dilemma is used for comparison with various prevailing techniques to justify the cogency and dominancy of the evaluated operators.

In a conventional interpretation of decision-making based on ambiguity, a decision-maker must prefer the best possible opportunity including various feasible possibilities. However, the dilemma of picking the best possible alternative has continued to be a substantial task to resolve. In this manuscript, we improve the existing complex intuitionistic fuzzy soft set (CIFSS), which includes the grade of truth and falsity with the rule that the sum of the real and imaginary parts of both grades is confined to [0, 1]. CIFS is a valuable procedure to determine the authenticity and consistency of the elaborated approaches. The fundamental laws and their related examples are also determined. Moreover, by using these laws, we investigated the complex intuitionistic fuzzy soft prioritized weighted averaging operator (CIFSPWAO), the complex intuitionistic fuzzy soft prioritized ordered weighted averaging operator (CIFSPOWAO), the complex intuitionistic fuzzy soft prioritized weighted geometric operator (CIFSPWGO), complex intuitionistic fuzzy soft prioritized ordered weighted geometric operator (CIFSPOWGO), and their related properties are also developed. Based on the developed operators, the multiattribute decision-making (MADM) tool is developed by using the explored operators based on CIFSS. Some numerical examples are also illustrated by using the investigated operators to determine the feasibility and consistency of the developed approaches. Finally, the comparative analysis and their geometrical manifestations are also determined to enhance the excellence of the performed explorations.

We explore the dynamics of test particles around a deformed anti-de Sitter Schwarzschild black hole and analyze the influence of model parameters on particle motion. Using the effective potential technique, we examine the stability of equatorial circular orbits. We obtain the analytical expressions for the energy and angular momentum of test particles, along with the innermost stable circular orbits and the effective forces acting upon the test particles. We also probe the epicyclic oscillations of test particles close to the equatorial plane and further evaluate analytical expressions for radial, vertical, and orbital frequencies, including the periastron precession frequency. Additionally, we study the thermal characteristics of the black hole, investigating its Gibbs free energy, emission energy, and check the influence of other thermodynamic properties on the black hole model. Our findings indicate that the model parameters M, $$ \Lambda $$ Λ , $$ \alpha $$ α and $$ \beta $$ β considerably affect both the particle motion including epicyclic oscillations and the thermodynamic behavior.