Fuzzy sets and join spaces associated with rough sets

Let μ _X be the rough membership function. One compares μ _A with μ_A∪B and μ_A∪B, by the associated hyperoperations. One finds a condition such that a functionμ ε [0, 1] ^H may be a rough membership function.     

Axiomatic systems for rough sets and fuzzy rough sets

Rough set theory is an important tool for approximate reasoning about data. Axiomatic systems of rough sets are significant for using rough set theory in logical reasoning systems. In this paper, outer product method are used in rough set study for the first time. By this approach, we propose a unified lower approximation axiomatic system for Pawlak’s rough sets and fuzzy rough sets. As the dual of axiomatic systems for lower approximation, a unified upper approximation axiomatic characterization of rough sets and fuzzy rough sets without any restriction on the cardinality of universe is also given. These rough set axiomatic systems will help to understand the structural feature of various approximate operators.     

Fuzzy sets and the modelling of physician decision processes,part II: fuzzy diagnosis decision models

Our concern is with the development of a more accurate mathematical model of physician diagnosis decision processes. Using the complete information net of matrices outputed from Part I, the various diagnoses processes—medical hypothesis, initial preliminary diagnoses, other preliminary diagnosis and final diagnoses, are modelled as fuzzy matrices. A general fuzzy decision model which could be particularized to each of the foregoing cases is then developed. Our general technique is based on fuzzy clustering theory with the criterion function being a modified form of the minimum Minkowski metric.     

An approach to decision making based on intuitionistic fuzzy rough sets over two universes

Rough set theory has been combined with intuitionistic fuzzy sets in dealing with uncertainty decision making. This paper proposes a general decision-making framework based on the intuitionistic fuzzy rough set model over two universes. We first present the intuitionistic fuzzy rough set model over two universes with a constructive approach and discuss the basic properties of this model. We then give a new approach of decision making in uncertainty environment by using the intuitionistic fuzzy rough sets over two universes. Further, the principal steps of the decision method established in this paper are presented in detail. Finally, an example of handling medical diagnosis problem illustrates this approach.     

Triangular fuzzy decision-theoretic rough sets

Based on decision-theoretic rough sets (DTRS), we augment the existing model by introducing into the granular values. More specifically, we generalize a concept of the precise value of loss function to triangular fuzzy decision-theoretic rough sets (TFDTRS). Firstly, ranking the expected loss with triangular fuzzy number is analyzed. In light of Bayesian decision procedure, we calculate three thresholds and derive decision rules. The relationship between the values of the thresholds and the risk attitude index of decision maker presented in the ranking function is analyzed. With the aid of multiple attribute group decision making, we design an algorithm to determine the values of losses used in TFDTRS. It is achieved with the use of particle swarm optimization. Our study provides a solution in the aspect of determining the value of loss function of DTRS and extends its range of applications. Finally, an example is presented to elaborate on the performance of the TFDTRS model.     

Axiomatics for fuzzy rough sets

A fuzzy T-rough set consists of a set X and a T-similarity relation R on X, where T is a lower semi-continuous triangular norm. We generalize the Farinas-Prade definition for the upper approximation operator of a fuzzy T-rough set (X, R); given originally for the special case T = Min, to the case of arbitrary T. We propose a new definition for the lower approximation operator of (X,R). Our definition satisfies the two important identities and , as well as a number of other interesting properties. We provide axiomatics to fully characterize those upper and lower approximations.     

Topology vs generalized rough sets

This paper investigates the relationship between topology and generalized rough sets induced by binary relations. Some known results regarding the relation based rough sets are reviewed, and some new results are given. Particularly, the relationship between different topologies corresponding to the same rough set model is examined. These generalized rough sets are induced by inverse serial relations, reflexive relations and pre-order relations, respectively. We point that inverse serial relations are weakest relations which can induce topological spaces, and that different relation based generalized rough set models will induce different topological spaces. We proved that two known topologies corresponding to reflexive relation based rough set model given recently are different, and gave a condition under which the both are the same topology.     

药片与模糊集

根据概念原理举例说明模糊集取大取小运算的不完备性,即词不达意的问题.     

NMGRS: Neighborhood-based multigranulation rough sets

Recently, a multigranulation rough set (MGRS) has become a new direction in rough set theory, which is based on multiple binary relations on the universe. However, it is worth noticing that the original MGRS can not be used to discover knowledge from information systems with various domains of attributes. In order to extend the theory of MGRS, the objective of this study is to develop a so-called neighborhood-based multigranulation rough set (NMGRS) in the framework of multigranulation rough sets. Furthermore, by using two different approximating strategies, i.e., seeking common reserving difference and seeking common rejecting difference, we first present optimistic and pessimistic 1-type neighborhood-based multigranulation rough sets and optimistic and pessimistic 2-type neighborhood-based multigranulation rough sets, respectively. Through analyzing several important properties of neighborhood-based multigranulation rough sets, we find that the new rough sets degenerate to the original MGRS when the size of neighborhood equals zero. To obtain covering reducts under neighborhood-based multigranulation rough sets, we then propose a new definition of covering reduct to describe the smallest attribute subset that preserves the consistency of the neighborhood decision system, which can be calculated by Chen’s discernibility matrix approach. These results show that the proposed NMGRS largely extends the theory and application of classical MGRS in the context of multiple granulations.     

Self-consistency of decision rules for group decision making

The author treats, in this paper, a group of decision makers, where each of them already has preference on a given set of alternatives but the group as a whole does not have a decision rule to make their group decision, yet. Then, the author examines which decision rules are appropriate. As a criterion of “appropriateness” the author proposes the concepts of self-consistency and universal self-consistency of decision rules. Examining the existence of universally self-consistent decision rules in two cases: (1) decision situations with three decision makers and two alternatives, and (2) those with three decision makers and three alternatives, the author has found that all decision rules are universally self-consistent in the case (1), whereas all universally self-consistent decision rules have one and just one vetoer in the essential cases in (2). The result in the case (2) implies incompatibility of universal self-consistency with symmetry. An example of applications of the concept of self-consistency to a bankruptcy problem is also provided in this paper, where compatibility of self-consistency with symmetry in a particular decision situation is shown.     

Fuzzy sets and Boolean-valued universes

A connection between fuzzy sets and Boolean-valued universes is made by developing a formal axiomatic system for fuzzy sets among whose models are suitably interpreted Boolean-valued universes. The formal system is an independent first-order axiomatization of fuzzy set theory which parallels the Zermelo-Fraenkel development of classical set theory.     

Fuzzy decision trees

We propose here to extend the decision trees method to the case when the involved data (probabilities, cost, profits, losses) appear as words belonging to the common language whose semantic representations are fuzzy sets. First we discuss the reasons why such an extension is to be aimed at. Then in the fuzzy case we carry out a reformalization of the basic concepts of probability and utility theory. Finally we show how these reformalized concepts can be applied to fuzzy decision trees.