Multigranulation Decision-theoretic Rough Set Research Articles

Feature selection for multi-label learning based on variable-degree multi-granulation decision-theoretic rough sets

Multi-label learning (MLL) suffers from the high-dimensional feature space teeming with irrelevant and redundant features. To tackle this, several multi-label feature selection (MLFS) algorithms have emerged as vital preprocessing steps. Nonetheless, existing MLFS methods have their shortcomings. Primarily, while they excel at harnessing label-feature relationships, they often struggle to leverage inter-feature information effectively. Secondly, numerous MLFS approaches overlook the uncertainty in the boundary domain, despite its critical role in identifying high-quality features. To address these issues, this paper introduces a novel MLFS algorithm, named VMFS. It innovatively integrates multi-granulation rough sets with three-way decision, leveraging multi-granularity decision-theoretic rough sets (MGDRS) with variable degrees for optimal performance. Initially, we construct coarse decision (RDC), fine decision (RDF), and uncertainty decision (RDU) functions for each object based on MGDRS with variable degrees. These decision functions then quantify the dependence of attribute subsets, considering both deterministic and uncertain aspects. Finally, we employ the dependency to assess attribute importance and rank them accordingly. Our proposed method has undergone rigorous evaluation on various standard multi-label datasets, demonstrating its superiority. Experimental results consistently show that VMFS significantly outperforms other algorithms on most datasets, underscoring its effectiveness and reliability in multi-label learning tasks.

Conflict Analysis Triggered by Three-Way Decision and Pythagorean Fuzzy Rough Set

Conflict is ubiquitous in human society and has a profound impact on various fields such as the economy, politics, law, and military. Many scholars have focused on exploring the internal mechanisms and potential solutions to conflicts. Notably, describing agents’ attitudes is an effective way to construct a conflict model. However, in decision-making, agents’ attitudes on issues are often vague and ambiguous. Pythagorean fuzzy set can deal with fuzzy information more accurately than intuitionistic fuzzy set. On the basis of this understanding, we investigate the conflicts from the perspective of Pythagorean fuzzy set. Firstly, we use Pythagorean fuzzy numbers to express the attitudes of agents on issues, and subsequently establish a Pythagorean fuzzy conflict information system. Secondly, we classify agents into three categories by a pair of thresholds to establish a trisected agent set model with risk preference. Thirdly, we construct a three-way conflict analysis model based on multi-granulation Pythagorean fuzzy decision-theoretic rough set and discuss both global and local conflicts by combining conflict analysis with multi-granulation decision-theoretic rough set. Finally, we discuss the relationships and properties of the proposed conflict analysis models.

Three-way decision-making methods with multi-intuitionistic β-neighborhood-based multiattribute group decision-making problems

Multiattribute group decision-making (MAGDM) is a decision analysis method used to address decision-making problems that have multiple attributes and multiple decision alternatives. It is generally believed that multigranulation rough sets (MGRSs) are a twofold scheme to solve MAGDM problems. Thus, in this paper, we aim to generalize MGRSs and decision-theoretic rough sets (DTRSs) to intuitionistic fuzzy β-covering (IFβC) information systems and propose a novel MAGDM method to handle decision-making problems, which not only deepens the cognition of the three-way decision (3WD) model but also expands its application in decision-making problems. The paper first introduces a novel conditional probability using the ideal positive degree of intuitionistic fuzzy numbers (IFNs) and then investigates its properties. Second, in the IFβC environment, we utilize the proposed conditional probabilities to construct two types of multigranulation decision-theoretic rough set (MGDTRS) models, an optimistic strategy and a pessimistic strategy, discuss their theoretical properties and explore the relationship between the two strategies. Third, a novel MAGDM method for multi-intuitionistic β-neighborhoods is established in accordance with the proposed MGDTRS model. We also establish an algorithm for the proposed MAGDM method. Fourth, we apply the proposed method to address several MAGDM problems. By conducting comparative analysis with existing methods, we demonstrate the effectiveness of the proposed method and discuss the advantages and limitations of the proposed method. Furthermore, we conduct extensive parameter experimentation analysis on the proposed method from different perspectives. The experimental results demonstrate the stability and effectiveness of our proposed method.

Multi-granulation rough approximations under normal distribution

Multi-granulation decision-theoretic rough set effectively combines Bayesian decision approaches with multi-granulation rough set theory, and provides an important theoretical framework for studying rough set. In this paper, we explore several extensional models of multi-granulation decision-theoretic rough sets under the normal distribution of the decision loss function. Using the 3σ rule of normal distribution, we transform the decision loss of the multi-granulation decision-theoretic rough set into a set of interval values. We construct the upper and lower approximations from the optimistic, weakly optimistic, pessimistic, weakly pessimistic, optimistic-pessimistic, weakly optimistic-pessimistic, pessimistic-optimistic, and weakly pessimistic-optimistic viewpoints, and provide the decision rules of the proposed rough set models. The work in this paper brings the decision behavior based on a multi-granulation decision-theoretic rough set closer to the actual situation.

Soft ordered double quantitative approximations based three-way decisions and their applications

The classical theory of rough set was established by Pawlak, which mainly focusses on the approximation of sets characterized by a single equivalence relation over the universe. However, most of the current single granulation structure models cannot meet the user demand or the target of solving problems. Multigranulation rough sets approach can better deal with the problems, where data might be spread over various locations. In this article, we present the idea of soft preference and soft dominance relation for the development of soft dominance rough set in an incomplete information system. Subsequently, several important structural properties and results of the proposed model are carefully analyzed. After employing soft dominance based rough set approach to it for any times, we can only get six different sets at most in an incomplete information system. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Based on soft dominance relation, we introduce logical disjunction/conjunction soft dominance optimistic/pessimistic multigranulation decision theoretic rough approximations in an incomplete information. Meanwhile, to measure the uncertainty of soft dominance optimistic/pessimistic multigranulation decision theoretic rough approximation and some of their interesting properties are examined. Thereafter, a novel multi attribute with multi decision making problem approach based on logical disjunction/conjunction soft dominance optimistic/pessimistic multigranulation decision theoretic rough sets approach are developed to solve the selection of medicine to treat the coronavirus disease (COVID-19). The basic principle and the detailed steps of the decision making model (algorithms) are presented in detail. To demonstrate the applicability and potentiality of the proposed model, we present a practical example of a medical diagnosis is given to validate the practicality of the technique.

Rough set-based entropy measure with weighted density outlier detection method

Abstract The rough set theory is a powerful numerical model used to handle the impreciseness and ambiguity of data. Many existing multigranulation rough set models were derived from the multigranulation decision-theoretic rough set framework. The multigranulation rough set theory is very desirable in many practical applications such as high-dimensional knowledge discovery, distributional information systems, and multisource data processing. So far research works were carried out only for multigranulation rough sets in extraction, selection of features, reduction of data, decision rules, and pattern extraction. The proposed approach mainly focuses on anomaly detection in qualitative data with multiple granules. The approximations of the dataset will be derived through multiequivalence relation, and then, the rough set-based entropy measure with weighted density method is applied on every object and attribute. For detecting outliers, threshold value fixation is performed based on the estimated weight. The performance of the algorithm is evaluated and compared with existing outlier detection algorithms. Datasets such as breast cancer, chess, and car evaluation have been taken from the UCI repository to prove its efficiency and performance.

AMG-DTRS: Adaptive multi-granulation decision-theoretic rough sets

The Multi-Granulation Decision-Theoretic Rough Set (MG-DTRS) is an effective method for cost-sensitive decision making from multi-view and multi-level. However, the inherent weak point of MG-DTRS model is to compute three regions with a subjectively given pair of probabilistic parameters (i.e., α and β). To overcome this issue, this paper first proposes a generalized MG-DTRS model called Adaptive Multi-Granulation Decision-Theoretic Rough Sets (AMG-DTRS), which can adaptively obtain a pair of probabilistic thresholds by setting a compensation coefficient ζ. Then, three types of mean AMG-DTRS models are investigated, which provide a novel perspective on multi-granulation method for information fusion. Finally, the connections and differences between the proposed AMG-DTRS and the existing MGRS models are analyzed, which show the advantages and generalization of the AMG-DTRS model. In addition, there are numerous existing MGRS models can be derived explicitly by considering various MG-DTRS, MGRS and VP-MGRS based on our model. These results will be conducive to establishing the framework of information fusion for granular computing.

Matrix‐based method for solving decision domains of neighbourhood multigranulation decision‐theoretic rough sets

It is more and more important to analyse and process complex data for gaining more valuable knowledge and making more accurate decisions. The multigranulation decision theory based on conditional probability and cost loss has the advantage of processing decision-making problems from multi-levels and multi-angles, and the neighbourhood rough set model (NRS) can facilitate the analysis and processing of numerical or mixed type data, and can address the limitation of multigranulation decision-theoretic rough sets (MG-DTRS), which is not easy to cope with complex data. Based on the in-depth study of hybrid-valued decision systems and MG-DTRS models, this study analysed neighbourhood MG-DTRS (NMG-DTRS) deeply by fusing MG-DTRS and NRS; a matrix-based approach for approximation sets of NMG-DTRS model was proposed on the basis of the matrix representations of concepts; the positive, boundary and negative domains were constructed from the matrix perspective, and the concept of positive decision recognition rate was introduced. Furthermore, the authors explored the related properties of NMG-DTRS model, and designed and described the corresponding solving algorithms in detail. Finally, some experimental results that were employed not only verified the effectiveness and feasibility of the proposed algorithm, but also showed the relationship between the decision recognition rate and the granularity and threshold.

A new fuzzy multi-attribute group decision-making method with generalized maximal consistent block and its application in emergency management

Decision-making is the most important business activity and becomes more complex in the current big-data situation. Most organizational decision-making is made in a group and has the data analytics function to seek specific answers for specific purposes. Multi-attribute group decision-making (MAGDM) methods provide effective support to decision groups by evaluating and integrating individual group members opining. However, current MAGDM methods often suffer from the problem of opining data and decision environment uncertainty, which is particularly severe in a large decision group or a newly decision problem. As a solution, rough sets and fuzzy sets have been applied in MAGDM to deal with data and decision process uncertainties. Although a lot of efforts have been made in applying rough sets and fuzzy sets to deal with data and decision process uncertainties the area, the disadvantage of rough set models in classification accuracy when similar class as basic knowledge granularity has not been well solved yet. This paper aims to solve this problem by introducing a new concept-maximal consistent block (MCB) and multi-granulation decision-theoretic rough set (MCB-MDTRS) models. Firstly, it establishes a binary tolerance relation on the universe of discourse, defines generalized MCBs, and introduces pessimistic DTRS based on MCBs. Secondly, it extends objective set to a fuzzy environment and proposes four MCB-MDTRFS models with consideration of the weight of each attribute. The steps of the proposed fuzzy MAGDM methods are carefully described in detail. Different from existing fuzzy MAGDM methods, the weight of each attribute is considered in the determination of the positive ideal decision objective and negative ideal decision objective in this paper, named by qualified and unqualified (fuzzy) sets. We solve the vague problem of individual preference evaluation and the uncertainty of setting ideal decision objectives by using the advantages of fuzzy set and rough set theory. Finally, it takes emergency plan selection as a case study to analyze the effectiveness of our methods and compare with other fuzzy MAGDM methods. Our methods outperform the selection of basic knowledge granularity, the determination of ideal decision objective sets and the ranking method, so as to increase the reliability and accuracy of ranking evaluation index.

On local multigranulation covering decision-theoretic rough sets

Multi-granulation decision-theoretic rough sets uses the granular structures induced by multiple binary relations to approximate the target concept, which can get a more accurate description of the approximate space. However, Multi-granulation decision-theoretic rough sets is very time-consuming to calculate the approximate value of the target set. Local rough sets not only inherits the advantages of classical rough set in dealing with imprecise, fuzzy and uncertain data, but also breaks through the limitation that classical rough set needs a lot of labeled data. In this paper, in order to make full use of the advantage of computational efficiency of local rough sets and the ability of more accurate approximation space description of multi-granulation decision-theoretic rough sets, we propose to combine the local rough sets and the multigranulation decision-theoretic rough sets in the covering approximation space to obtain the local multigranulation covering decision-theoretic rough sets model. This provides an effective tool for discovering knowledge and making decisions in relation to large data sets. We first propose four types of local multigranulation covering decision-theoretic rough sets models in covering approximation space, where a target concept is approximated by employing the maximal or minimal descriptors of objects. Moreover, some important properties and decision rules are studied. Meanwhile, we explore the reduction among the four types of models. Furthermore, we discuss the relationships of the proposed models and other representative models. Finally, illustrative case of medical diagnosis is given to explain and evaluate the advantage of local multigranulation covering decision-theoretic rough sets model.

Dynamic and Updating Multigranulation Decision-Theoretic Rough Approximations When Adding or Deleting Objects

Along with the development of big data, knowledge updating occurs in various situations, some scholars had studied the dynamic method of updating knowledge in multigranulation decision-theoretic rough sets when adding or deleting granular structures. However, there is no study about the case of adding or deleting objects, which limits the development of multigranulation decision-theoretic rough sets. Based on the matrix method, the dynamic knowledge updating of optimistic multigranulation decision-theoretic rough set and pessimistic multigranulation decision-theoretic rough set were studied in this paper, then given the static algorithm and dynamic algorithms, and the time complexity of three algorithms was analyzed. The theory and experimental results show that two dynamic algorithms are both more effective than the static algorithm.

Updating knowledge in multigranulation decision‐theoretic rough set model based on decision support degree

Based on the majority rules, a multigranulation decision-theoretic rough set model based on the decision support degree is proposed, in which the thresholds can be computed by the decision risk minimisation based on the Bayesian decision-theoretic. In various practical situations, information systems may alter dynamically with time. Incremental learning is an alternative manner for maintaining knowledge by utilising previous computational results under dynamic data. Therefore, the authors investigate dynamic approaches to update the knowledge in the new model when adding or deleting granular structures. Besides, the corresponding dynamic and static algorithms are designed and their time complexities are analysed. Finally, comparative experiments by using six data sets from UCI are carried out; the results illustrate that the proposed dynamic algorithm is effective and is more efficient than the static algorithm.

Multigranulation behavioral three-way group decisions under hesitant fuzzy linguistic environment

Bounded rationality of decision-makers and interaction among criteria plays a key role in the actual multi-criteria group decision-making (MCGDM) process. This paper aims to offer a decision approach to a MCGDM problem under hesitant fuzzy linguistic (HFL) uncertainty, by means of behavioral multigranulation decision-theoretic rough set (DTRS), where criteria interaction is handled by Choquet integral, and decision-makers’ psychological preferences are considered by prospect theory. In specific, we firstly define the behavioral DTRS over two universes, where the threshold parameters are obtained by maximizing expected utility, rather than minimizing expected risk. Then, from the perspective of the difference of alternatives, we develop the calculation methods of loss and benefit functions, and provide the determination of the overall threshold parameters by Choquet integral, in consideration of criteria interaction. With these discussions, the behavioral multigranulation DTRS over two universes in the HFL context is proposed, and the two special cases, i.e., optimistic and pessimistic versions are also discussed. Thus, solutions to a MCGDM issue, green supplier selection, by utilizing the proposed behavioral multigranulation HFL DTRS over two universes, are designed, along with key steps and algorithm. Ultimately, an illustrative example is given to elaborate the practicability and feasibility of our presented model.

Soft dominance based multigranulation decision theoretic rough sets and their applications in conflict problems

The extension of rough set model is a crucial and vast research direction in rough set theory. Meanwhile decision making can be considered as a mental process in which human beings make a choice among several alternatives. However, with the increasing complexity of real decision making problems, the decision makers frequently face the challenge of characterizing their preferences in an uncertain context. In the present paper, we initiate a multi attribute group decision making problem in the presence of multi attribute and multi decision in decision making with preferences. We further present the concept of soft preference relation and soft dominance relation corresponding to decision attribute in the multi criteria and multi decision information system. Further we put forward the idea of two types of optimistic/pessimistic multigranulation (soft dominance based optimistic/pessimistic multigranulation decision theoretic) approximations and their applications in solving a multi agent conflict analysis decision problem. The proposed method addresses the limitations of the Pawlak model and Sun’s conflict analysis model and thus improve these models. Finally, the results on labor management negotiation problems show that the proposed algorithms are more effective and efficient for feasible consensus strategy when compared with other techniques.

Note on “Fuzzy multi-granulation decision-theoretic rough sets based on fuzzy preference relation”

In this note, we point out that the transfer function for computing the fuzzy preference degree in Mandal and Ranadive (Soft Comput, 2018. https://doi.org/10.1007/s00500-018-3411-7 ) for the construction of upward/downward fuzzy relations is not additive consistent. Appropriate counterexample is given. Further, their modified versions are presented.

Incomplete neighbourhood multi‐granulation decision‐theoretic rough set in the hybrid‐valued decision system

It is an important subject to mine valuable knowledge from complex and massive data in the era of big data. Rough set theory is a new mathematical tool for dealing with uncertain and inaccurate data, decision-theoretic rough set model (DTRS), as an extension of classical rough set model, is used to analyze decision information systems and multi-granulation decision-theoretic rough set model (MG-DTRS) can analyze and process target concepts from different angles and levels. However, the classical DTRS model exists some limitations in dealing numerical or hybrid-valued data. Considering the different influence of numerical features and symbolic features on decision-making, the paper proposes an incomplete neighborhood multi-granulation decision-theoretic rough set model in hybrid-valued decision system through integrating MG-DTRS with neighbourhood rough sets, and two types of neighborhood multi-granulation decision-theoretic set models are emphatically analysed. Furthermore, taking pessimistic and optimistic neighborhood multi-granulation decision-theoretic rough sets as examples, the implementation algorithms and related properties of the two type of models are studied. Finally, the relationship between the proposed model and other models is analyzed through formula derivation. The model proposed in this paper can effectively solve the decision-making problem of hybrid-valued incomplete information system through multi-angle and multi-level analysis.

Interval-valued hesitant fuzzy multi-granularity three-way decisions in consensus processes with applications to multi-attribute group decision making

Multi-attribute group decision making (MAGDM) is a common activity for multi-variable complicated decision making situations by integrating collective wisdom. Aiming at fusing granular computing with three-way decisions (3WD) to study scheme synthesis and analysis of solution space, multi-granularity three-way decisions (MG-3WD) provide multi-dimension problem solving methods for MAGDM problems. By using MG-3WD frameworks, this paper intends to study viable strategies of processing consensus and conflicting opinions provided by different decision makers in the interval-valued hesitant fuzzy (IVHF) MAGDM problem. More specifically, after reviewing the relevant literature, four kinds of IVHF multigranulation decision-theoretic rough sets (MG-DTRSs) over two universes are proposed according to different risk appetites of experts firstly. Then, we explore some fundamental propositions of newly proposed models. Afterwards, solutions to MAGDM problems in the context of mergers and acquisitions (M&A) target selections by using the presented IVHF MG-DTRSs over two universes are constructed. At last, a M&A target selection case study, along with a sensitivity analysis and a comparative analysis, is applied to illustrate the established decision making approaches.

Local multigranulation decision-theoretic rough set in ordered information systems

As a generalized extension of Pawlak’s rough set model, the multigranulation decision-theoretic rough set model in ordered information systems utilizes the basic set assignment function to construct probability measure spaces through dominance relations. It is an effective tool to deal with uncertain problems and widely used in practical decision problems. However, when the scale of dataset is large, it takes a lot of time to characterize the approximations of the target concept, as well as complicated calculation processes. In this paper, we develop a novel model called local multigranulation decision-theoretic rough set in an ordered information system to overcome the above-mentioned limitation. Firstly, to reduce the computing time of the information granule independent of the target concept, we only use the characterization of the elements in the target concept to approximate this target concept. Moreover, the corresponding local multigranulation decision-theoretic rough set in an ordered information system is addressed according to the established local model, and the comparisons are made between the proposed local algorithm and the algorithm of original multigranulation decision-theoretic rough set in ordered information systems. Finally, the validity of the local approximation operators is verified through the experimental evaluation using six datasets coming from the University of California-Irvine (UCI) repository.

Multi-granularity three-way decisions with adjustable hesitant fuzzy linguistic multigranulation decision-theoretic rough sets over two universes

The notion of hesitant fuzzy linguistic term sets (HFLTSs), which enables experts to utilize a few possible linguistic terms to evaluate varieties of common qualitative information, plays a significant role in handling situations in cases where these experts are hesitant in offering linguistic expressions. For addressing the challenges of information analysis and information fusion in hesitant fuzzy linguistic (HFL) group decision making, in accordance with the multi-granularity three-way decisions paradigm, the primary purpose of this study is to develop the notion of multigranulation decision-theoretic rough sets (MG-DTRSs) into the HFL background within the two-universe framework. Having revisited the relevant literature, we first propose a hybrid model named adjustable HFL MG-DTRSs over two universes by introducing an adjustable parameter for the expected risk appetite of experts, in which both optimistic and pessimistic versions of HFL MG-DTRSs over two universes are special cases of the adjustable version. Second, some of the fundamental properties of the proposed model are discussed. Then, on the basis of the presented hybrid model, a group decision making approach within the HFL context is further constructed. Finally, a practical example, a comparative analysis, and a validity test concerning person-job fit problems are explored to reveal the rationality and practicability of the constructed decision making rule.

Multi-granulation Pythagorean fuzzy decision-theoretic rough sets based on inclusion measure and their application in incomplete multi-source information systems

Multi-granulation rough sets (MGRSs) and decision-theoretic rough sets (DTRSs) are two important and popular generalizations of classical rough sets. The combination of two generalized rough sets have been investigated by numerous researchers in different extensions of fuzzy settings such as interval-valued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFSs), bipolar-valued fuzzy sets (BVFSs), etc. Pythagorean fuzzy (PF) set is another extension of fuzzy set, which is more capable in comparison to IFS handle vagueness in real world. However, few studies have focused on the combination of the two rough sets in PF settings. In this study, we combine the two generalized rough sets in PF settings. First, we introduce a type of PF subset (of subset of the given universe) of the PF Set (of the given universe). Then we establish two basic models of multi-granulation PF DTRS (MG-PF-DTRS) of PF subset of the PF set based on PF inclusion measure within the framework of multi-granulation PF approximation space. One model is based on a combination of PF relations (PFRs) and the construction of approximations with respect to the combined PFR. By combining PFRs through intersection and union, respectively, we construct two models. The other model is based on the construction of approximations from PFRs and a combination of the approximations. By using intersection and union to combine the approximations, respectively, we again get two models. As a result, we have total four models. Further for different constraints on parameters, we obtain three kinds of each model of the MG-PF-DTRSs. Then, their principal structure, basic properties and uncertainty measure methods are investigated as well. Second, we give a way to compute PF similarity degrees between two objects and also give a way to compute PF decision-making objects from incomplete multi-source information systems (IMSISs). Then we design an algorithm for decision-making to IMSISs using MG-PFDTRSs and their uncertainty measure methods. Finally, an example about the mutual funds investment is included to show the feasibility and potential of the theoretic results obtained.