The primitive notions of rough sets and intuitionistic fuzzy set (IFS) are general mathematical tools having the ability to handle the uncertain and imprecise knowledge easily. EDA $\mathcal {S}$ (Evaluation based on distance from average solution) method has a significant role in decision making problems, especially when more conflicting criteria exist in multicriteria group decision making (MCGDM) problems. The aim of this manuscript is to present intuitionistic fuzzy rough- EDA $\mathcal {S}$ (IFR- EDA $\mathcal {S}$ ) method based on IF rough averaging and geometric aggregation operators. In addition, we put forward the concept of IF rough weighted averaging (IFRWA), IF rough ordered weighted averaging (IFROWA) and IF rough hybrid averaging (IFRHA) aggregation operators. Furthermore, the concepts of IF rough weighted geometric (IFRWG), IF rough ordered weighted geometric (IFROWG) and IF rough hybrid geometric (IFRHG) aggregation operators are investigated. The basic desirable characteristics of the investigated operator are given in detail. A new score and accuracy functions are defined for the proposed operators. Next, IFR-EDA $\mathcal {S}$ model for MCGDM and their stepwise algorithm are demonstrated by utilizing the proposed approach. Finally, a numerical example for the developed model is presented and a comparative study of the investigated models with some existing methods are expressed broadly which show that the investigated models are more effective and useful than the existing approaches.

This article introduces the notion of bipolar complex fuzzy soft set as a generalization of bipolar complex fuzzy set and soft set. Furthermore, this article contains elementary operations for bipolar complex fuzzy soft sets such as complement, union, intersection, extended intersection, and related properties. The OR and AND operations for bipolar complex fuzzy soft set are also initiated in this study. Moreover, this study contains the decision-making algorithm and real-life examples to display the success and usability of bipolar complex fuzzy soft sets. Finally, the comparative study of initiated notions with some prevailing ideas are also interpreted in this study.

The linear Diophantine fuzzy set (LDFS) has been proved to be an efficient tool in expressing decision maker (DM) evaluation values in multicriteria decision-making (MCDM) procedure. To more effectively represent DMs’ evaluation information in complicated MCDM process, this paper proposes a MCDM method based on proposed novel aggregation operators (AOs) under linear Diophantine fuzzy set (LDFS). A q -Rung orthopair fuzzy set ( q -ROFS), Pythagorean fuzzy set (PFS), and intuitionistic fuzzy set (IFS) are rudimentary concepts in computational intelligence, which have diverse applications in modeling uncertainty and MCDM. Unfortunately, these theories have their own limitations related to the membership and nonmembership grades. The linear Diophantine fuzzy set (LDFS) is a new approach towards uncertainty which has the ability to relax the strict constraints of IFS, PFS, and q –ROFS by considering reference/control parameters. LDFS provides an appropriate way to the decision experts (DEs) in order to deal with vague and uncertain information in a comprehensive way. Under these environments, we introduce several AOs named as linear Diophantine fuzzy Einstein weighted averaging (LDFEWA) operator, linear Diophantine fuzzy Einstein ordered weighted averaging (LDFEOWA) operator, linear Diophantine fuzzy Einstein weighted geometric (LDFEWG) operator, and linear Diophantine fuzzy Einstein ordered weighted geometric (LDFEOWG) operator. We investigate certain characteristics and operational laws with some illustrations. Ultimately, an innovative approach for MCDM under the linear Diophantine fuzzy information is examined by implementing suggested aggregation operators. A useful example related to a country’s national health administration (NHA) to create a fully developed postacute care (PAC) model network for the health recovery of patients suffering from cerebrovascular diseases (CVDs) is exhibited to specify the practicability and efficacy of the intended approach.

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.