Hong-Ying Zhang

Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure

This article proposes a new axiomatic definition of entropy of interval-valued fuzzy sets (IVFSs) and discusses its relation with similarity measure. First, we propose an axiomatic definition of entropy for IVFS based on distance which is consistent with the axiomatic definition of entropy of a fuzzy set introduced by De Luca, Termini and Liu. Next, some formulae are derived to calculate this kind of entropy. Furthermore we investigate the relationship between entropy and similarity measure of IVFSs and prove that similarity measure can be transformed by entropy. Finally, a numerical example is given to show that the proposed entropy measures are more reasonable and reliable for representing the degree of fuzziness of an IVFS.

Variable-precision-dominance-based rough set approach to interval-valued information systems

Abstract This paper proposes a general framework for the study of interval-valued information systems by integrating the variable-precision-dominance-based rough set theory with inclusion measure theory. By introducing a α -dominance relation based on inclusion measures between two interval numbers, we propose a variable-precision-dominance-based rough set approach based on the substitution of indiscernibility relation by the α -dominance relation. The knowledge discovery framework is formulated for interval-valued information systems. Furthermore, knowledge reduction of interval-valued decision systems based on the variable-precision-dominance-based rough set model is postulated. Relationships between these reducts and discernibility matrices are also established to substantiate knowledge reduction in the variable-precision-dominance-based rough set model.

Hybrid monotonic inclusion measure and its use in measuring similarity and distance between fuzzy sets

In fuzzy set theory, an inclusion measure indicates the degree to which a given fuzzy set is contained in another fuzzy set. In this paper, we introduce a new definition of inclusion measure: hybrid monotonic inclusion measure (S inclusion measure for short). Some inclusion measures defined by other authors and us are proved to be S inclusion measures. In addition, as the rational generalization of crisp inclusion relation, we prove most of the S inclusion measures possess some kind of T-transitivity and some properties postulated by Sinha and Dougherty for a reasonable inclusion measure. Similarity measure and distance between two fuzzy sets derived from S inclusion measure possess properties of normal similarity measure and distance. Furthermore, some similarity measures possess T-transitivity and distributivity.

Ranking interval sets based on inclusion measures and applications to three-way decisions

Three-way decisions provide an approach to obtain a ternary classification of the universe as acceptance region, rejection region and uncertainty region respectively. Interval set theory is a new tool for representing partially known concepts, especially it corresponds to a three-way decision. This paper proposes a framework for comparing two interval sets by inclusion measures. Firstly, we review the basic notations, interpretation and operation of interval sets and classify the orders on interval sets into partial order, preorder and quasi-order. Secondly, we define inclusion measure which indicates the degree to which one interval set is less than another one and construct different inclusion measures to present the quantitative ranking of interval sets. Furthermore, we present similarity measures and distances of interval sets and investigate their relationship with inclusion measures. In addition, we propose the fuzziness measure and ambiguity measure to show the uncertainty embedded in an interval set. Lastly, we study the application of inclusion measures, similarity measures and uncertainty measures of interval sets by a special case of three-way decisions: rough set model and the results show that these measures are efficient to three-way decision processing.

Feature selection and approximate reasoning of large-scale set-valued decision tables based on α-dominance-based quantitative rough sets

Set-valued data are a common type of data for characterizing uncertain and missing information. Traditional dominance-based rough sets can not efficiently deal with large-scale set-valued decision tables and usually neglect the disjunctive semantics of sets. In this paper, we propose a general framework of feature selection and approximate reasoning for large-scale set-valued information tables by integrating quantitative rough sets and dominance-based rough sets. Firstly, we define two new partial orders for set-valued data via the conjunctive and disjunctive semantics of a set. Secondly, based on α-disjunctive dominance relation and α-conjunctive dominance relation defined by the inclusion measure, we present α-dominance-based quantitative rough set models for these two types of set-valued decision tables. Furthermore, we study the issue of feature selection in set-valued decision tables by employing α-dominance-based quantitative rough set models and discuss the relationships between the relative reductions and discernibility matrices. We also present approximate reasoning models based on α-dominance-based quantitative rough sets. Finally, the application of the approach is illustrated by some real-world data sets.

Inclusion measure for typical hesitant fuzzy sets, the relative similarity measure and fuzzy entropy

Typical hesitant fuzzy sets (THFSs), possessing a finite-set-valued fuzzy membership degrees called typical hesitant fuzzy elements (THFEs), is a special kind of hesitant fuzzy sets. Fuzzy inclusion relationship, as the order structure in fuzzy mathematics, plays an elementary role in the theoretical research and practical applications of fuzzy sets. In this paper, a new partial order for THFEs is defined via the disjunctive semantic meaning of a set, based on which fuzzy inclusion relationship is defined for THFSs. Furthermore, inclusion measures are defined to present the quantitative ranking of every two THFEs and THFSs and different inclusion measures are constructed. The related similarity measure, distance and fuzzy entropy of THFSs are presented and their relationship with inclusion measures are investigated. Finally, an example is given to show that the inclusion measure can be applied effectively in hesitant fuzzy multi-attribute decision making.

Methods and Practices of Three-Way Decisions for Complex Problem Solving

A theory of three-way decisions is formulated based on the notions of three regions and associated actions for processing the three regions. Three-way decisions play a key role in everyday decision-making and have been widely used in many fields and disciplines. A group of Chinese researchers further investigated the theory of three-way decision and applied it in different domains. Their research results are highlighted in an edited Chinese book entitled “Three-way Decisions: Methods and Practices for Complex Problem Solving.” Based on the contributed chapters of the edited book, this paper introduces and reviews most recent studies on three-way decisions.

On inclusion measures of intuitionistic and interval-valued intuitionistic fuzzy values and their applications to group decision making

AbstractRanking intuitionistic fuzzy values (IFVs) and interval-valued intuitionistic fuzzy values (IVIFVs) is an important and necessary work in intuitionistic fuzzy group decision making. Since the set of all IFVs is a poset and inclusion measure indicates the degree to which a given element of a poset is contained in another one. This paper studies hybrid monotonic (HM) inclusion measures of IFVs and IVIFVs respectively and discuss their applications to group decision making. Firstly, HM inclusion measure is defined on the posets of all IFVs and IVIFVs respectively. Then HM inclusion measures are studied by constructive approach. Furthermore, the HM inclusion measures are employed to make intuitionistic and interval-valued intuitionistic fuzzy group decisions. Lastly, practical examples are provided to illustrate the developed approaches respectively.

A Ranking Approach with Inclusion Measure in Multiple-Attribute Interval-Valued Decision Making

This paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers and then, on the basis of this, gives the inclusion measure approach to compare any two interval numbers. The monotonic inclusion measure is defined over the strict partial order set proposed by Moore and illustrate that the possibility degrees in the literature are monotonic inclusion measures defined in this paper; Then a series of monotonic inclusion measures are constructed based on t-norms. Finally, we give illustrations by using the monotonic inclusion measures and gain good results.

Inclusion measure and its use in measuring similarity and distance measure between hesitant fuzzy sets

In this paper, we propose a variety of inclusion measures for hesitant fuzzy sets (HFSs), based on which the corresponding similarity and distance measures can be obtained. First, we investigate the hybrid monotonic inclusion measures between two hesitant fuzzy elements (HFEs) and present their properties. Second, the hybrid monotonic inclusion measure between two HFSs are proposed. Furthermore, similarity measures and distance are postulated by inclusion measures. Last, numerical examples are provided to illustrate the inclusion measures, similarity measures and distance.