Bao Qing Hu

Three-way decisions space and three-way decisions

Ideas of three-way decisions proposed by Yao come from rough sets. It is well known that there are three basic elements in three-way decisions theory, which are ordered set as to define three regions, object set contained in evaluation function and evaluation function to make three-way decisions. In this paper these three basic elements are called decision measurement, decision condition and evaluation function, respectively. In connection with the three basic elements this paper completes three aspects of work. The first one is to introduce axiomatic definitions for decision measurement, decision condition and evaluation function; the second is to establish three-way decisions space; and the third is to give a variety of three-way decisions on three-way decisions spaces. Existing three-way decisions are the special examples of three-way decisions spaces defined in this paper, such as three-way decisions based on fuzzy sets, random sets and rough sets etc. At the same time, multi-granulation three-way decisions space and its corresponding multi-granulation three-way decisions are also established. Finally this paper introduces novel dynamic two-way decisions and dynamic three-way decisions based on three-way decisions spaces and three-way decisions with a pair of evaluation functions.

Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets

This paper builds the topological and lattice structures of L-fuzzy rough sets by introducing lower and upper sets. In particular, it is shown that when the L-relation is reflexive, the upper (resp. lower) set is equivalent to the lower (resp. upper) L-fuzzy approximation set. Then by the upper (resp. lower) set, it is indicated that an L-preorder is the equivalence condition under which the set of all the lower (resp. upper) L-fuzzy approximation sets and the Alexandrov L-topology are identical. However, associating with an L-preorder, the equivalence condition that L-interior (resp. closure) operator accords with the lower (resp. upper) L-fuzzy approximation operator is investigated. At last, it is proven that the set of all the lower (resp. upper) L-fuzzy approximation sets forms a complete lattice when the L-relation is reflexive.

On type-2 fuzzy sets and their t-norm operations

In this paper, we discuss t-norm extension operations of general binary operation for fuzzy true values on a linearly ordered set, with a unit interval and a real number set as special cases. On the basis of it, t-norm operations of type-2 fuzzy sets and properties of type-2 fuzzy numbers are discussed.

Fuzzy and interval-valued fuzzy decision-theoretic rough set approaches based on fuzzy probability measure

This paper investigates decision-theoretic rough set (DTRS) approach in the frameworks of fuzzy and interval-valued fuzzy (IVF) probabilistic approximation spaces, respectively. It takes fuzzy probability and IVF probability into consideration. Bayesian decision procedure is a basis of DTRS approach. By integrating fuzzy probability measure and IVF probability measure into Bayesian decision procedure, there come fuzzy decision-theoretic rough set (FDTRS) approach and interval-valued fuzzy decision-theoretic rough set (IVF-DTRS) approach. The new approaches have the ability to directly deal with real-valued and interval-valued data. This makes FDTRS and IVF-DTRS more applicable than DTRS. Two methods are presented to compare intervals while constructing the IVF-DTRS approach: one is compatible with DTRS and FDTRS approaches; the other is a total order based on which the decision procedure is much easier to operate. Cases of two different universes of discourse for FDTRS and IVF-DTRS are also taken into account.

Fuzzy rough sets based on generalized residuated lattices

This paper is devoted to propose generalized L-fuzzy rough sets as a further generalization of the notion of L-fuzzy rough sets. A quadruple of approximation operators are defined to suit the situation when generalized residuated lattices are non-commutative. Generalized L-fuzzy rough sets are characterized from both constructive and axiomatic approaches. In the constructive approach, various classes of generalized L-fuzzy rough sets are investigated. Moreover, the relationship between generalized L-fuzzy rough sets and L-topologies on an arbitrary universe is discussed with generalized lower and upper sets. As an application of generalized L-fuzzy rough sets, fuzzy rough sets are proposed and studied on the unit interval [0, 1], which are based on generalized residuated lattices induced by left-continuous pseudo-t-norms.

Approximate distribution reducts in inconsistent interval-valued ordered decision tables

Abstract Many methods based on the rough set theory to deal with information systems have been proposed in recent decades. In practice, some information systems are based on dominance relations and may be inconsistent because of various factors. Moreover, taking the imprecise evaluations and assignments in the description of objects into account, single-valued information systems have been generalized to interval-valued information systems. In this paper, by introducing a dominance relation to interval-valued ordered information systems, we establish a dominance-based rough set approach, which is mainly based on substitution of the indiscernibility relation by the dominance relation. To extract the minimal decision rules, approximate distribution reducts are proposed in inconsistent interval-valued ordered decision tables. This paper presents a theoretical method based on the discernibility matrix to enumerate all reducts and a practical approach on the basis of significance to find one reduct. And two equivalent definitions of approximate distribution reducts are also introduced. In addition, numerical examples are employed to examine the validity of the approaches proposed in this paper.

A fuzzy covering-based rough set model and its generalization over fuzzy lattice

In this paper, we study a new type of fuzzy covering-based rough set model by introducing the notion of fuzzy β-minimal description. We mainly address the following issues in this paper. First, we present the definition of fuzzy β-minimal description and study its properties. Then, we define a novel type of fuzzy covering-based rough set model and investigate the properties of this model. Furthermore, the axiomizations and matrix representations of the fuzzy covering lower and fuzzy covering upper approximations are the vital problems we investigate in this paper. Moreover, we explore the conditions under which two fuzzy β-coverings generate the same fuzzy covering lower or fuzzy covering upper approximations. Finally, we generalize the model to L-fuzzy covering-based rough set which is defined over fuzzy lattices. Similar to the fuzzy covering-based rough set model, we also address the issues mentioned above to the L-fuzzy covering-based rough set.

Attribute reduction in ordered decision tables via evidence theory

Rough set theory and Dempster-Shafer theory of evidence are two distinct but closely related approaches to modeling and manipulating uncertain information. It is quite natural to set up a hybrid model based on these two theories. In this paper, we investigate the problem of attribute reduction for ordered decision tables based on evidence theory. Belief and plausibility functions, which are strongly connected with lower and upper approximation operators in dominance-based rough set approach, are proposed to define relative belief and plausibility reducts of ordered decision tables. Relationships among various types of relative reducts are thoroughly studied in consistent and inconsistent ordered decision tables. A pair of numeric measures, the inner and outer significance measures of a criterion, is presented to search for a relative belief/plausibility reduct, which is meaningful for practical problems. Some real-world tasks taken from the UCI repository are employed to verify the feasibility and effectiveness of the proposed technique.

Dominance-based rough set approach to incomplete ordered information systems

Dominance-based rough set approach has attracted much attention in practical applications ever since its inception. This theory has greatly promoted the research of multi-criteria decision making problems involving preferential information. This paper mainly deals with approaches to attribute reduction in incomplete ordered information systems in which some attribute values may be lost or absent. By introducing a new kind of dominance relation, named the characteristic-based dominance relation, to incomplete ordered information systems, we expand the potential applications of dominance-based rough set approach. To eliminate information that is not essential, attribute reduction in the sense of reducing attributes is needed. An approach on the basis of the discernibility matrix and the discernibility function to computing all (relative) reducts is investigated in incomplete ordered information systems (consistent incomplete ordered decision tables). To reduce the computational burden, a heuristic algorithm with polynomial time complexity for finding a unique (relative) reduct is designed by using the inner and outer significance measures of each criterion candidate. Moreover, some numerical experiments are employed to verify the feasibility and effectiveness of the proposed algorithms.

Three-way decisions based on semi-three-way decision spaces

Abstract Decision evaluation functions in three-way decision spaces must meet three axioms, the minimum element axiom, the monotonicity axiom and the complement axiom. Maintaining the complement axiom of decision evaluation functions is crucial to three-way decisions to simplify the decision rules based only on the conditional probability and the loss functions. However, some handy functions do not satisfy the complement axiom. This paper introduces the notion of semi-decision evaluation functions not necessarily satisfying the complement axiom but the minimum element axiom and the monotonicity axiom, and presents some transformation methods from semi-decision evaluation functions to decision evaluation functions. Through numerous examples this paper demonstrates the existence of semi-decision evaluation functions and significance of the transformation methods.