Uncertainty is an essential factor in any decision-making process. A q-rung orthopair fuzzy set (q-ROFS) is more practical and robust than a fuzzy set (FS), intuitionistic fuzzy set (IFS), and Pythagorean fuzzy set (PFS) to describe uncertainty in numerous decision-making issues. The most attractive aspect of q-ROFSs is that they provide a wider space for membership and non-membership degrees and provide decision-makers more liberty in expressing their legitimate opinions. This study introduces the conception of q-rung orthopair fuzzy relation (q-ROFR), which will help to remove certain limitations associated with intuitionistic fuzzy relation (IFR) and Pythagorean fuzzy relation (PFR). Some basic operations in this regard are given for q-ROFRs. The set of all q-ROFRs leads to several algebraic structures for these operations (semi-group, semi-ring, hemi-ring, and bounded distributive lattice). Additionally, an application of the designed approach is proposed for predicting scores in cricket. Moreover, the comparative analysis of the proposed method with some existing techniques is also presented by giving several examples to authenticate the feasibility and superiority of the proposed approach.

Soft sets ( S_t S) theory provides a general mechanism for handling uncertainty based on the point of view of parameterization tools. The main theme of this manuscript is to extend the notion of Hamacher operators by establishing an interesting connection between two mathematical concepts S_t S theory and q-rung orthopair fuzzy sets (q-ROFS). To be specific, we develop some new Hamacher operations for q-rung orthopair fuzzy soft sets (q-ROF S_t S). In light of these operational laws, we further propose some q-rung orthopair fuzzy soft Hamacher aggregation operators, i.e., q-ROF soft Hamacher averaging and q-ROF soft Hamacher geometric aggregation operators, such as q-ROF soft Hamacher weighted averaging (q-ROF S_t HWA), q-ROF soft Hamacher ordered weighted averaging (q-ROF S_t HOWA) and q-ROF soft Hamacher hybrid averaging (q-ROF S_t HHA) operators. Furthermore, based on Hamacher operator laws, we discuss some geometric aggregation operators such as q-ROF soft Hamacher weighted geometric (q-ROF S_t HWG), q-ROF soft Hamacher ordered weighted geometric (q-ROF S_t HOWG) and q-ROF soft Hamacher hybrid geometric (q-ROF S_t HHG) operators. Meanwhile, the important properties of the developed operators are investigated in detail. Then, a technique for multi-criteria decision making and a stepwise algorithm for decision making are demonstrated by utilizing the proposed approach. Finally, a numerical example for the developed approach is presented and a comparative study of the investigated models with some existing methods is performed. The derived results demonstrate that the investigated models are more effective and useful than the existing approaches.

Information quantification in numerical form for any given data is very useful in decision-making problems. In Atanassov intuitionistic fuzzy sets (A-IFSs), such quantification becomes more important due to uncertainties such as intuitionism and fuzziness. Distribution of these uncertainties is a key to determine knowledge associated with Atanassov intuitionistic values (A-IFVs). In this paper, first distribution of the above-mentioned uncertainties and their relationship are discussed. Then, knowledge measures are defined as a function of entropy and uncertainty index, with certain desired properties. Existence of such knowledge measures has been established. Further, it is shown that how proposed knowledge measures are useful in multi-criteria group decision-making (MCGDM) problems.

Recently, some improvement has been made in the dominant notion of fuzzy set that is Yager investigated the generalized concept of fuzzy set, Intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS) and called it q-rung orthopair fuzzy (q-ROF) set (q-ROFS). The aim of this manuscript is to present the concept of q-ROF soft (q-ROFSt) set (q-ROFStS) based on the Dombi operations. Since Dombi operational parameter possess natural flexibility with the resilience of variability. Some new operational laws are defined based on hybrid study of soft sets and q-ROFS. The advantage of Dombi operational parameter is very important to express the experts’ attitude in decision making. In this paper, we present q-ROFSt Dombi average (q-ROFSt DA) aggregation operators including q-ROFSt Dombi weighted average (q-ROFSt DWA), q-ROFSt Dombi ordered weighted average (q-ROFSt DOWA) and q-ROFSt Dombi hybrid average (q-ROFSt DHA) operators. Moreover, we investigate q-ROFSt Dombi geometric (q-ROFSt DG) aggregation operators including q-ROFSt Dombi weighted geometric (q-ROFSt DWG), q-ROFSt Dombi ordered weighted geometric (q-ROFSt DOWG), and q-ROFSt Dombi hybrid geometric (q-ROFSt DHG) operators. The basic properties of these operators are presented with detail such us Idempotency, Boundedness, Monotonicity, Shift invariance, and Homogeneity. Thus from the analysis and advantages of proposed model, it is clear that the investigated q-ROFSt DWA operator is the generalized form of IF St DWA, PFSt DWA and q-ROFDWA operators. Similarly, the investigated q-ROFSt DWG operator is the generalized form of IF St DWG, PFSt DWG and q-ROFDWG operators. By applying the develop approach, this manuscript contains the technique and algorithm for multicriteria decision making (MCDM). Further a numerical example is developed to illustrate the flexibility and applicability of the developed operators.

The compromise solution of the multi-criteria decision-making (MCDM) problem by the existing VIKOR method for Pythagorean fuzzy sets (PyFSs) is not closest to the positive ideal solution. This is because the defining function for VIKOR does not obey the axioms for a dissimilarity measure. Thus in this context, the existing notion of dissimilarity measures and the VIKOR method are controversial. This study aims to provide a new dissimilarity measure and refine the VIKOR method for PyFSs accordingly. We define a new dissimilarity measure for PyFSs and refine the existing VIKOR method. We discuss the additional properties of dissimilarity measures and improve the ideas of remoteness and ranking indexes. We provide numerical examples to support the analysis and findings of our study. Finally, we solve the MCDM problems to illustrate the proposed method.

An approximation space plays a vital role in the accuracy of approximations of a subset of the universal set. The main goal of this paper is to develop new kinds of soft rough sets models by using the concept of near open sets, where the accuracy of approximations is enhanced significantly. Firstly, the concepts of near soft rough approximations, denoted by “ $$J_{SR}$$ -approximations” for each $$J\in \{P,S,\gamma ,\alpha ,\beta \}$$ are proposed as a generalization some previously introduced notions. Then, their properties and relationships are disclosed. Comparisons among the proposed methods and the previous one are obtained. An algorithm has been given for decision-making problems. The proposed algorithm is tested on hypothetical data for the purpose of comparison with already existing methods.

Pythagorean fuzzy sets and interval-valued Pythagorean fuzzy sets are more proficient in handling uncertain and imprecise information than intuitionistic fuzzy sets and fuzzy sets. In this article, we put forward a chance-constraint programming method to solve linear programming network problems with interval-valued Pythagorean fuzzy constraints. This practice is developed using score function and upper and lower membership functions of interval-valued Pythagorean fuzzy numbers. The feasibility of the anticipated approach is illustrated by solving an airway network application and shown to be used to solve different types of network problems with objective function having interval-valued Pythagorean fuzzy numbers by employing it on shortest path problem and minimum spanning tree problem. Furthermore, a comparative examination was performed to validate the effectiveness and usefulness of the projected methodology.

A new inclusion called total inclusion relation has improved the existing dissimilarity measure for q-rung orthopair fuzzy sets (qROFSs). For qROFSs, the modified axiomatic definition of dissimilarity measure is proposed. The modified Hamming and Euclidean dissimilarity measures are defined. An algorithmic procedure for a robust VIKOR method based on modified dissimilarity measures is established. The application of the robust VIKOR method in Mass Vaccination Campaigns (MVCs) in the COVID-19 situation is given.

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.

In this paper, with the aid of fuzzy soft $$\beta $$ -neighborhoods, we introduce fuzzy soft covering-based multi-granulation fuzzy rough set models. We examine some of the relevant properties of fuzzy soft covering based on optimistic, pessimistic, and variable precision multi-granulation fuzzy rough set models. Then, we give fuzzy soft coverings based on $$\psi $$ -optimistic and $${\mathscr {D}}$$ -optimistic ( $$\psi $$ -pessimistic and $${\mathscr {D}}$$ -pessimistic) multi-granulation fuzzy rough sets from fuzzy soft measures. It also discusses the interactions between these forms of fuzzy soft coverings based on multi-granulation fuzzy rough sets. Eventually, we apply the proposed models for solving MAGDM problems. The effectiveness and feasibility of our approach are noted from the introduced comparisons between our method and some methods given in the previous studies.

This paper aims at introducing two types of orthopair soft sets, which might serve as novel approaches to granular computing based on parametrization. These generalized soft sets emerge naturally when linguistic parameters are employed to convey uncertainty attached to elements of certain sets. Concepts of uncertainty measures attached to the parameters and the whole orthopair soft sets are presented as well. The proposed uncertainty measures are useful for classifying elements of the set of parameters. Different types of granularity measures associated with parameters are presented and are extended to orthopair soft sets. Collective wisdom is helpful in decision making based on consensus. A numerical example is given to demonstrate how orthopair soft sets can be employed in this regard.

In intuitionistic fuzzy set and their generalizations such as Pythagorean fuzzy sets and q‐rung orthopair fuzzy sets, ranking is not easy to define. There are several techniques available in literature for ranking values in above mentioned orthopair fuzzy sets. It is interesting to see that almost all the proposed ranking methods produce distinct ranking. Notion of knowledge base is very important to study ranking proposed by different techniques. Aim of this paper is to critically analyze the available ranking techniques for q‐rung orthopair fuzzy values and propose a new graphical ranking method based on hesitancy index and entropy. Several numerical examples are tested with the proposed technique, which shows that the technique is intuitive and convenient for real life applications.

There are many approaches to deal with vagueness and ambiguity including soft sets and rough sets. Feng et al. initiated the concept of possible hybridization of soft sets and rough sets. They introduced the concept of soft rough sets, in which parameterized subsets of a universe set serve as the building blocks for lower and upper approximations of a subset. Topological notions play a vital role in rough sets and soft rough sets. So, the basic objectives of the current work are as follows: first, we find answers to some very important questions, such as how to determine the probability that a subset of the universe is definable. Some more similar questions are answered in rough sets and their extensions. Secondly, we enhance soft rough sets from topological perspective and introduce topological soft rough sets. We explore some of their properties to improve existing techniques. A comparison has been made with some existing studies to show that accuracy measure of proposed technique shows an improvement. Proposed technique has been employed in decision-making problem for diagnosing heart failure. For this two algorithms have been given.

The pioneer paradigm of soft set ( $S_{ft}\text{S}$ ) was investigated by Molodtsov in 1999 by affixing parameterization tools in ordinary sets. $S_{ft}\text{S}$ theory is free from inherit complexity and a nice mathematical tool for handle uncertainties and vagueness. The aim of this paper is to initiate the combine study of $S_{ft}\text{S}$ and q-rung orthopair fuzzy set (q-ROFS) to get the new notion called q-rung orthopair fuzzy soft set (q-ROF $S_{ft}\text{S}$ ). The notion of q-ROF $S_{ft}\text{S}$ is free from those complexities which suffering the contemporary theories because parameterization tool is the most significant character of q-ROF $S_{ft}\text{S}$ . In this manuscript our main contribution to originate the concept of q-ROF soft weighted geometric (q-ROF $S_{ft}$ WG), q-ROF soft ordered weighted geometric (q-ROF $S_{ft}$ OWG) and q-ROF soft hybrid geometric (q-ROF $S_{ft}$ HG) operators in q-ROF $S_{ft}\text{S}$ environment. Moreover, some dominant properties of these developed operators are studied in detail. Based on these proposed approaches, a model is build up for multi-criteria decision making (MCDM) and their step wise algorithm is being presented. Finally, utilizing the developed approach an illustrative example is solved under q-ROF $S_{ft}$ environment. Further a comparative analysis of the investigated models with some existing methods are presented in detail which shows the superiority, competence and ability of the developed model.

Intuitionistic fuzzy sets (IFSs) have advantage over fuzzy sets and made it possible to describe imprecise information considering its positive and negative aspects simultaneously. In an information system mass assignment and possibility theory are very useful to assign membership grades to elements in a fuzzy set. Unfortunately the situation differs for IFSs in assigning membership function (MF) and nonmembership function (NMF). In this paper, it is shown that the above-mentioned theories fail to produce the MF and NMF for IFSs. Aim of this paper is to present an alternate algorithm to generate these grading functions based on q-rung orthopair fuzzy set. Consequently, it will be extremely convenient to model imprecise and vague information using this approach.

As a generalization of fuzzy rough sets, the concept of generalized hesitant fuzzy rough sets (GHFRS) is presented in this paper. It is an endeavor to define rough approximations of a collection of hesitant fuzzy sets over a given universe. To this end, elements of the universe are initially clustered using a set-valued map, and then, hesitant fuzzy sets are aggregated by using lower and upper approximation operators. These operators produce hesitant fuzzy sets which aggregate hesitant fuzzy elements. Structural and topological properties associated with GHFRS have been examined. The model is further employed to design a three-way decision analysis technique which preserves many properties of classical techniques but needs less effort and computation. Unlike the existing approaches, the alternatives can be clustered and selected jointly by using a set-valued mapping. This feature makes its application area broader. Moreover, this method is applied to an example, where risk analysis issue is discussed for the selection of energy projects.

Z-soft rough covering models introduced by zhan et al are important generalizations of classical rough set theory to deal with more complex problems of real world. So far, the existing studies mainly focus on constructing various forms of approximation operators and their related properties by means of neighborhoods. In this paper, we introduce different kinds of uncertainty measures related to Z-soft rough covering sets and discuss their limitations. An axiomatic definition of knowledge granulation for soft covering approximations space is introduced. Some main theoretical results are obtained and investigated with the help of examples. Finally, a fully developed example describing the application of the proposed theory in multicriteria decision making is constructed.

Initially some concepts related to soft vector spaces have been studied. It is shown that there is a soft matrix associated with a soft linear transform. Optimization of a process as a whole can be studied with the help of soft matrices. In this regard notion of a convex soft set is investigated. Therefore idea of soft linear programming emerges as a natural consequence. Thus soft optimality theorem is presented.

Recently, Yager presented the new concept of q-rung orthopair fuzzy (q-ROF) set (q-ROFS) which emerged as the most significant generalization of Pythagorean fuzzy set (PFS). From the analysis of q-ROFS, it is clear that the rung q is the most significant feature of this notion. When the rung q increases, the orthopair adjusts in the boundary range which is needed. Thus, the input range of q-ROFS is more flexible, resilient, and suitable than the intuitionistic fuzzy set (IFS) and PFS. The aim of this manuscript is to investigate the hybrid concept of soft set ( S t S) and rough set with the notion of q-ROFS to obtain the new notion of q-ROF soft rough (q-ROF S t R) set (q-ROF S t RS). In addition, some averaging aggregation operators such as q-ROF S t R weighted averaging (q-ROF S t RWA), q-ROF S t R ordered weighted averaging (q-ROF S t ROWA), and q-ROF S t R hybrid averaging (q-ROF S t RHA) operators are presented. Then, the basic desirable properties of these investigated averaging operators are discussed in detail. Moreover, we investigated the geometric aggregation operators, such as q-ROF S t R weighted geometric (q-ROF S t RWG), q-ROF S t R ordered weighted geometric (q-ROF S t ROWG), and q-ROF S t R hybrid geometric (q-ROF S t RHG) operators, and proposed the basic desirable characteristics of the investigated geometric operators. The technique for multicriteria decision-making (MCDM) and the stepwise algorithm for decision-making by utilizing the proposed approaches are demonstrated clearly. Finally, a numerical example for the developed approach is presented and a comparative study of the investigated models with some existing methods is brought to light in detail which shows that the initiated models are more effective and useful than the existing methodologies.