rough prime ideals and rough fuzzy prime ideals in semigroups in this paper we shall introduce the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup and give some properties of such ideals also we will discuss the relations between the upper lower rough prime ideals and the upper lower approximations of their homomorphism images

global stabilization for a class of switched nonlinear feedforward systems abstract the problem of global stabilization for a class of switched nonlinear feedforward systems under arbitrary switchings is investigated in this paper based on the integrator forwarding technique and the common lyapunov function method we design bounded state feedback controllers of individual subsystems to guarantee asymptotic stability of the closed loop system a common coordinate transformation of all subsystems is exploited to avoid individual coordinate transformations for subsystems that are required when applying the forwarding recursive design scheme an example is provided to demonstrate the effectiveness of the proposed design method

granular counting of uncertain data abstract we propose a definition of granular count realized in the presence of uncertain data modeled through possibility distributions we show that the resulting counts are fuzzy intervals in the domain of natural numbers based on this result we devise two algorithms for granular counting an exact counting algorithm with quadratic time complexity and an approximate counting algorithm with linear time complexity we compare the two algorithms on synthetic data and show their application to a bioinformatics scenario concerning the assessment of gene expressions in cells

set approximations in fuzzy formal concept analysis formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems the purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined based on both lattice theoretic and fuzzy set theoretic operators two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined finally properties of the rough fuzzy set approximation operators are presented in detail

a comparison of two kinds of definitions of rough approximations based on a similarity relation in the present paper we investigate the three types of yao s lower and upper approximations of any set with respect to any similarity relation these types based on a right neighborhood also we define and investigate other three types of approximations for any similarity relations these new types based on the intersection of the right neighborhoods moreover we give a comparison between these types lastly the relationship between these definitions is introduced

discernibility matrix simplification for constructing attribute reducts this paper proposes a reduct construction method based on discernibility matrix simplification the method works in a similar way to the classical gaussian elimination method for solving a system of linear equations elementary matrix simplification operations are introduced each operation transforms a matrix into a simpler form by applying these operations a finite number of times one can transform a discernibility matrix into one of its minimum i e the simplest forms elements of a minimum discernibility matrix are either the empty set or singleton subsets in which the union derives a reduct with respect to an ordering of attributes which is either computed based on a certain measure of attributes or directly given by a user two heuristic reduct construction algorithms are presented one algorithm attempts to exclude unimportant attributes from a reduct and the other attempts to include important attributes in a reduct

on the structure of generalized rough sets in this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that 1 any reflexive binary relation determines a topology 2 if q is a reflexive and symmetric relation on a set x then o email protected x q a a is a topology such that a is open if and only if it is closed 3 conversely for every topological space x o satisfying the condition that a is open if and only if it is closed there exists a reflexive and symmetric relation r such that o email protected x r a a 4 let q be an equivalence relation on x for any pseudo w closed subset a of x q a is an w closed set if and only if w x x x email protected a for any email protected x moreover we consider properties of generalized rough sets

a comprehensive study of implicator conjunctor based and noise tolerant fuzzy rough sets definitions properties and robustness analysis abstract both rough and fuzzy set theories offer interesting tools for dealing with imperfect data while the former allows us to work with uncertain and incomplete information the latter provides a formal setting for vague concepts the two theories are highly compatible and since the late 1980s many researchers have studied their hybridization in this paper we critically evaluate most relevant fuzzy rough set models proposed in the literature to this end we establish a formally correct and unified mathematical framework for them both implicator conjunctor based definitions and noise tolerant models are studied we evaluate these models on two different fronts firstly we discuss which properties of the original rough set model can be maintained and secondly we examine how robust they are against both class and attribute noise by highlighting the benefits and drawbacks of the different fuzzy rough set models this study appears a necessary first step to propose and develop new models in future research

a comprehensive study of fuzzy covering based rough set models definitions properties and interrelationships abstract fuzzy covering based rough set models are hybrid models using both rough set and fuzzy set theory the former is often used to deal with uncertain and incomplete information while the latter is used to describe vague concepts the study of fuzzy rough set models has provided very good tools for machine learning algorithms such as feature and instance selection in this article we discuss different types of dual fuzzy rough set models which all consider fuzzy coverings in particular we study two models using non nested level based representation of fuzziness in addition to the study of the theoretical properties for each model interrelationships between the different models are discussed resulting in a hasse diagram of fuzzy covering based rough set models for a finite fuzzy covering an imtl t norm and its residual implicator

on the reducibility of the voigt functions the voigt functions which are employed in such areas as neutron physics and spectroscopy are shown to be capable of representation in terms of circular functions and confluent hypergeometric functions

quadratic stability of a class of switched nonlinear systems quadratic stability of a class of switched nonlinear systems is studied in this note we first transform quadratic stability problem into an equivalent nonlinear programming problem then we derive a necessary and sufficient condition for quadratic stability of this class of switched systems by using karush kuhn tucker condition for nonlinear programming problems the necessary and sufficient condition is given in terms of the strict completeness of a certain set of functions on a subset of the state space which is much easier to check

interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference information systems which contain only crisp data precise and unique attribute values for all objects have been widely investigated due to the fact that in realworld applications imprecise data are abundant uncertainty is inherent in real information systems in this paper information systems are called fuzzy information systems and formalized by objects attributes f in which f is a fuzzy set and expresses some uncertainty between an object and its attribute values to interpret and extract fuzzy decision rules from fuzzy information systems the meta theory based on modal logic proposed by resconi et al is modified the modified meta theory not only expresses uncertainty between objects and their attributes but also uncertainty in the process of recognizing fuzzy information systems in addition according to perception computing proposed by zadeh granules of fuzzy information systems can be represented by fuzzy decision rules so that fuzzy inference methods can be used to obtain the decision attribute of a new object finally a novel way of combining evidences based on the modified meta theory is introduced which extends the concept of combining evidences based on dempster shafer theory

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

robust fuzzy control of nonlinear systems with parametric uncertainties addresses the robust fuzzy control problem for nonlinear systems in the presence of parametric uncertainties the takagi sugeno t s fuzzy model is adopted for fuzzy modeling of the nonlinear system two cases of the t s fuzzy system with parametric uncertainties both continuous time and discrete time cases are considered in both continuous time and discrete time cases sufficient conditions are derived for robust stabilization in the sense of lyapunov asymptotic stability for the t s fuzzy system with parametric uncertainties the sufficient conditions are formulated in the format of linear matrix inequalities the t s fuzzy model of the chaotic lorenz system which has complex nonlinearity is developed as a test bed the effectiveness of the proposed controller design methodology is finally demonstrated through numerical simulations on the chaotic lorenz system

on the relation between rough set reducts and typical testors this paper studies the relations between rough set reducts and typical testors from the so called logical combinatorial approach to pattern recognition definitions comments and observations are formally introduced and supported by illustrative examples furthermore some theorems expressing theoretical relations between reducts and typical testors are enunciated and proved we also discuss several practical applications of these relations that can mutually enrich the development of research and applications in both areas although we focus on the relation between the classical concepts of testor and reduct our study can be expanded to include other types of testors and reducts

the ϑ lower and t upper fuzzy rough approximation operators on a semigroup in this paper we consider a semigroup as a universal set and investigate the properties of q lower and t upper fuzzy rough approximation operators a new fuzzy algebraic structure called a tl fuzzy rough semigroup is introduced and some related properties are studied finally the homomorphism properties of q lower and t upper fuzzy rough approximation operators are investigated

on stability l2 gain and h control for switched systems this paper addresses the issues of stability l 2 gain analysis and h control for switched systems via multiple lyapunov function methods a concept of general lyapunov like functions is presented a necessary and sufficient condition for stability of switched systems is given in terms of multiple generalized lyapunov like functions which enables derivation of improved stability tests an l 2 gain characterization and a design method for stabilizing switching laws a solution to the h control problem for switched systems is also provided

minimum cost attribute reduction in decision theoretic rough set models in classical rough set models attribute reduction generally keeps the positive or non negative regions unchanged as these regions do not decrease with the addition of attributes however the monotonicity property in decision theoretic rough set models does not hold this is partly due to the fact that all regions are determined according to the bayesian decision procedure consequently it is difficult to evaluate and interpret region preservation attribute reduction in decision theoretic rough set models this paper provides a new definition of attribute reduct for decision theoretic rough set models the new attribute reduction is formulated as an optimization problem the objective is to minimize the cost of decisions theoretical analysis shows the meaning of the optimization problem both the problem definition and the objective function have good interpretation a heuristic approach a genetic approach and a simulated annealing approach to the new problem are proposed experimental results on several data sets indicate the efficiency of these approaches

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

rough set theory for the interval valued fuzzy information systems the notion of a rough set was originally proposed by pawlak z pawlak rough sets international journal of computer and information sciences 11 5 1982 341 356 later on dubois and prade d dubois h prade rough fuzzy sets and fuzzy rough sets international journal of general system 17 2 3 1990 191 209 introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets this paper deals with an interval valued fuzzy information system by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory and discusses the basic rough set theory for the interval valued fuzzy information systems in this paper we firstly define the rough approximation of an interval valued fuzzy set on the universe u in the classical pawlak approximation space and the generalized approximation space respectively i e the space on which the interval valued rough fuzzy set model is built secondly several interesting properties of the approximation operators are examined and the interrelationships of the interval valued rough fuzzy set models in the classical pawlak approximation space and the generalized approximation space are investigated thirdly we discuss the attribute reduction of the interval valued fuzzy information systems finally the methods of the knowledge discovery for the interval valued fuzzy information systems are presented with an example

rough set approach to multi attribute decision analysis abstract we review the methodology of the rough set analysis of multi attribute decision problems rough set theory proved to be a useful tool for analysis of a vague description of decision situations it answers two basic questions related to multi attribute decision problems one about explanation of a decision situation and another about prescription of some decisions basing on analysis of a decision situation we define four classes of multi attribute decision problems depending on the structure of their representation its interpretation and the kind of questions related then we characterize the rough set methodology for each particular class of decision problems we use simple practical examples to illustrate this presentation a review of related literature is made throughout the paper

a class of rough multiple objective programming and its application to solid transportation problem this paper establishes a rough multiple objective programming model for a solid transportation problem furthermore a general model for rough multiple objective programming problem is presented properties of the feasible and efficient solutions of rough multiple objective programming problems are investigated in addition compromise solutions are obtained by using the interactive fuzzy satisfying method to solve the rough multiple objective solid transportation problem the rough simulation based genetic algorithm is proposed in which the rough simulation is embedded into the genetic algorithm finally an application of the solid transportation problem at xiluodu hydropower station is provided as an illustration

approaches to knowledge reduction based on variable precision rough set model this paper deals with approaches to knowledge reduction based on variable precision rough set model the concepts of lower distribution reduct and upper distribution reduct based on variable precision rough sets vprs are first introduced their equivalent definitions are then given and the relationships among lower and upper distribution reducts and alternative types of knowledge reduction in inconsistent systems are investigated it is proved that for some special thresholds lower distribution reduct is equivalent to the maximum distribution reduct whereas upper distribution reduct is equivalent to the possible reduct the judgement theorems and discernibility matrices associated with the lower and upper distribution reducts are also established from which we can obtain the approaches to knowledge reduction in vprs

roughness based on fuzzy ideals the theory of rough set proposed by pawlak and the theory of fuzzy set proposed by zadeh are complementary generalizations of classical set theory many sets are naturally endowed with two binary operations addition and multiplication one concept which does this is a ring this paper concerns a relationship between rough sets fuzzy sets and ring theory it is a continuation of ideas presented by kuroki and wang n kuroki p p wang the lower and upper approximations in a fuzzy group inform sci 90 1996 203 220 we consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal in fact we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring some characterizations of the above approximations are made and some examples are presented

rule acquisition and adjustment based on set valued mapping based on the theory of set valued mapping a new approach on how to acquire fuzzy rule sets from the corresponding relations between the group of data on the two universes of discourse is presented moreover some theories and methods on how to adjust fuzzy rule sets with the change of the group of data are established finally a recurrence theorem of the group of data is also given

stability analysis and state feedback control of continuous time t s fuzzy systems via anew switched fuzzy lyapunov function approach this paper deals with stability analysis and control design problems for continuous time takagi sugeno t s fuzzy systems first a new membership function dependent switching law is proposed and a relaxation parameter is introduced into this switching law to guarantee a minimal dwell time between two consecutive switching compared to the existing methods the most important point is that with the help of the dwell time the discretized lyapunov function dlf technique can be adopted then a new stability criterion of the t s fuzzy system with less conservatism is derived based on a fuzzy discretized lyapunov function fdlf second to estimate the domain of attraction da an algorithm with less iteration steps is proposed based on the proposed switching method sufficient conditions for existence of the state feedback controllers are presented via the switched non parallel distributed compensation non pdc scheme the effectiveness of the proposed method is illustrated through three simulation examples

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

an approach to fuzzy control of nonlinear systems stability and design issues presents a design methodology for stabilization of a class of nonlinear systems first the authors represent a nonlinear plant with a takagi sugeno fuzzy model then a model based fuzzy controller design utilizing the concept of the so called parallel distributed compensation is employed the main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system the design procedure is conceptually simple and natural moreover the stability analysis and control design problems can be reduced to linear matrix inequality lmi problems therefore they can be solved efficiently in practice by convex programming techniques for lmis the design methodology is illustrated by application to the problem of balancing and swing up of an inverted pendulum on a cart

generalized fuzzy rough sets this paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized approximation operators is defined the connections between fuzzy relations and fuzzy rough approximation operators are examined in axiomatic approach various classes of fuzzy rough approximation operators are characterized by different sets of axioms axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators

attribute reduction based on evidence theory in incomplete decision systems attribute reduction is a basic issue in knowledge representation and data mining this paper deals with attribute reduction in incomplete information systems and incomplete decision systems based on dempster shafer theory of evidence the concepts of plausibility reduct and belief reduct in incomplete information systems as well as relative plausibility reduct and relative belief reduct in incomplete decision systems are introduced it is shown that in an incomplete information system an attribute set is a belief reduct if and only if it is a classical reduct and a plausibility consistent set must be a classical consistent set in a consistent incomplete decision system the concepts of relative reduct relative plausibility reduct and relative belief reduct are all equivalent in an inconsistent incomplete decision system an attribute set is a relative plausibility reduct if and only if it is a relative reduct a plausibility consistent set must be a belief consistent set and a belief consistent set is not a plausibility consistent set in general

an axiomatic approach of fuzzy rough sets based on residuated lattices rough set theory was developed by pawlak as a formal tool for approximate reasoning about data various fuzzy generalizations of rough approximations have been proposed in the literature as a further generalization of the notion of rough sets l fuzzy rough sets were proposed by radzikowska and kerre in this paper we present an operator oriented characterization of l fuzzy rough sets that is l fuzzy approximation operators are defined by axioms the methods of axiomatization of l fuzzy upper and l fuzzy lower set theoretic operators guarantee the existence of corresponding l fuzzy relations which produce the operators moreover the relationship between l fuzzy rough sets and l topological spaces is obtained the sufficient and necessary condition for the conjecture that an l fuzzy interior closure operator derived from an l fuzzy topological space can associate with an l fuzzy reflexive and transitive relation such that the corresponding l fuzzy lower upper approximation operator is the l fuzzy interior closure operator is examined

roughness in modules the theory of rough sets was proposed by pawlak in 1982 the relations between rough sets and algebraic systems have been already considered by many mathematicians important algebraic structures are groups rings and modules rough groups and rough rings have been investigated by biswas and nanda kuroki and wang and davvaz in this paper we consider an r module as a universal set and we introduce the notion of rough submodule with respect to a submodule of an r module which is an extended notion of a submodule in an r module we also give some properties of the lower and the upper approximations in an r module

data analysis based on discernibility and indiscernibility rough set theory models similarities and difierences of objects based on the notions of indiscernibility and discernibility with respect to any subset of attributes one can deflne two pairs of dual relations the strong indiscernibility and weak discernibility relations and the weak indiscernibility and strong discernibility relations the similarities of objects are examined by the indiscernibility relations and the difierence by the discernibility relations respectively alternatively one can construct an indiscernibility matrix to represent the family of strong indiscernibility or weak discernibility relations one also can construct a discernibility matrix to represent the family of strong discernibility or weak indiscernibility relations the consideration of the matrix counterpart of relations and the relation counterpart of matrices brings more insights into rough set theory based on indiscernibility and discernibility three difierent types of reducts can be constructed keeping the indiscernibility discernibility and indiscernibility anddiscernibility relations respectively although the indiscernibility reducts have been intensively studied in the literature the other two types of reducts are relatively new and require more attention the existing methods for constructing the indiscernibility reducts also can be applied to construct the other two types of reducts an empirical experiment for letter recognition is reported for demonstrating the usefulness of the discussed relations and reducts

generalized cardinal numbers and operations on them abstract in this paper we present a cardinality theory for so called vaguely defined objects which are mild generalizations of fuzzy sets obtained by introducing lower and upper approximations of the membership functions the theory makes use of the sentential calculus in the infinite valued ukasiewicz logic and can be applied to fuzzy sets with arbitrary supports some kinds of twofold fuzzy sets partial sets etc the notion of equipotency of vaguely defined objects is studied in detail the resulting generalized cardinal numbers are convenient tools for describing the powers of vaguely defined objects in the second part of the paper basic operations on the generalized cardinals are defined and carefully investigated similarities and anomalies in comparison with the classical arithmetic of the usual cardinals are indicated

upper and lower images of a fuzzy set induced by a fuzzy relation applications to fuzzy inference and diagnosis in fuzzy set theory the image of a fuzzy set induced by a fuzzy relation is usually obtained by a sup t norm composition this corresponds to the upper image which gathers the elements in relation with at least one element of the fuzzy set however the dual point of view leading to the definition of the lower image as the fuzzy set of elements in relation with all the elements of the fuzzy set whose image is computed is not often considered fuzzy sets and fuzzy relations depending on the situations can be interpreted either in a conjunctive manner or as subsets of mutually exclusive possible values for variables whose precise values are ill known disjunctive view the applications of upper and lower images are investigated in both interpretations the generalized modus ponens used in fuzzy rule based systems corresponds to the disjunctive view the interest of upper and lower images is also emphasized for diagnosis problems where the conjunctive interpretation is encountered

the lower and upper approximations in a quotient hypermodule with respect to fuzzy sets this paper provides a continuation of ideas presented by davvaz and mahdavipour b davvaz m mahdavipour roughness in modules inform sci 176 2006 3658 3674 the notion of hypermodule is a generalization of the notion of module in this paper we consider the quotient hypermodule m a and interpret the lower and upper approximations as subsets of the quotient hypermodule m a then we introduce the concept of quotient rough sub hypermodule also using the concept of fuzzy sets we introduce and discuss the concept of fuzzy rough hypermodules and then we obtain the relation between fuzzy rough sub hypermodules and level rough sets this relation is characterized as a necessary and sufficient condition

consistency measure inclusion degree and fuzzy measure in decision tables classical consistency degree has some limitations for measuring the consistency of a decision table in which the lower approximation of a target decision is only taken into consideration in this paper we focus on how to measure the consistencies of a target concept and a decision table and the fuzziness of a rough set and a rough decision in rough set theory for three types of decision tables complete incomplete and maximal consistent blocks the membership functions of an object are defined through using the equivalence class tolerance class and maximal consistent blocks including itself respectively based on these membership functions we introduce consistency measures to assess the consistencies of a target set and a decision table and define fuzziness measures to compute the fuzziness of a rough set and a rough decision in these three types of decision tables in addition the relationships among the consistency inclusion degree and fuzzy measure are established as well these results will be helpful for understanding the essence of the uncertainty in decision tables and can be applied for rule extraction and rough classification in practical decision issues

generalizing database relational algebra for the treatment of incomplete or uncertain information and vague queries abstract this paper deals with relational databases which are extended in the sense that fuzzily known values are allowed for attributes precise as well as partial imprecise uncertain knowledge concerning the value of the attributes are represented by means of 0 1 valued possibility distributions in zadeh s sense thus we have to manipulate ordinary relations on cartesian products of sets of fuzzy subsets rather than fuzzy relations besides vague queries whose contents are also represented by possibility distributions can be taken into account the basic operations of relational algebra union intersection cartesian product projection and selection are extended in order to deal with partial information and vague queries approximate equalities and inequalities modeled by fuzzy relations can also be taken into account in the selection operation then the main features of a query language based on the extended relational algebra are presented an illustrative example is provided this approach which enables a very general treatment of relational databases with fuzzy attribute values makes an extensive use of dual possibility and necessity measures

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

on a fuzzy bipolar relational algebra this paper presents an extension of relational algebra suitable for the handling of bipolar concepts the type of queries considered involves two parts a first one which expresses a possibly flexible constraint and a second one that corresponds to a possibly flexible wish the framework considered is that of bipolar fuzzy relations where each tuple is associated with a pair of satisfaction degrees

a class of fuzzy rough expected value multi objective decision making model and its application to inventory problems in this paper we concentrate on developing a fuzzy rough multi objective decision making model according to fuzzy rough theory then we prove the existence of a solution of the fuzzy rough multi objective decision making model formulated next we develop an effective algorithm to solve the fuzzy rough expected value multi objective decision making model concerning the inventory problem and prove the convergence of the proposed algorithm finally these are applied to practical inventory problem in which all inventory costs purchasing and selling prices in the objectives and constraints are assumed to be fuzzy rough in nature comparing the results with those from the fuzzy multi objective decision making model we reach that the proposed fuzzy rough multi objective decision making model and algorithm which will be of much use to management and provide significant solutions to construct other inventory models with fuzzy rough variables in real life

the model of fuzzy variable precision rough sets the fuzzy rough set frs model has been introduced to handle databases with real values however frs was sensitive to misclassification and perturbation here misclassification means error or missing values in classification and perturbation means small changes of numerical data the variable precision rough sets vprss model was introduced to handle databases with misclassification however it could not effectively handle the real valued datasets now it is valuable from theoretical and practical aspects to combine frs and vprs so that a powerful tool which not only can handle numerical data but also is less sensitive to misclassification and perturbation can be developed in this paper we set up a model named fuzzy vprss fvprss by combining frs and vprs with the goal of making frs a special case first we study the knowledge representation ways of frs and vprs and then propose the set approximation operators of fvprs second we employ the discernibility matrix approach to investigate the structure of attribute reductions in fvprs and develop an algorithm to find all reductions third in order to overcome the np complete problem of finding all reductions we develop some fast heuristic algorithms to obtain one near optimal attribute reduction finally we compare fvprs with rs frs and several flexible rs based approaches with respect to misclassification and perturbation the experimental comparisons show the feasibility and effectiveness of fvprs

measures for evaluating the decision performance of a decision table in rough set theory as two classical measures approximation accuracy and consistency degree can be employed to evaluate the decision performance of a decision table however these two measures cannot give elaborate depictions of the certainty and consistency of a decision table when their values are equal to zero to overcome this shortcoming we first classify decision tables in rough set theory into three types according to their consistency and introduce three new measures for evaluating the decision performance of a decision rule set extracted from a decision table we then analyze how each of these three measures depends on the condition granulation and decision granulation of each of the three types of decision tables experimental analyses on three practical data sets show that the three new measures appear to be well suited for evaluating the decision performance of a decision rule set and are much better than the two classical measures

reduction methods of type 2 uncertain variables and their applications to solid transportation problem uncertainty theory is a branch of mathematics for dealing with realistic uncertainties arising out of complexity changeability and non decidability of practical environments an uncertain variable is defined as a function from the uncertainty space to the set of real numbers and is characterized by an uncertainty distribution this paper proposes the definition of type 2 uncertain variables within the framework of uncertainty theory through introduction of generalized uncertain measures and focuses on more complex twofold uncertainties some uncertainty reduction methods associated with type 2 uncertain variables are also proposed for convenience of applicability including reduction of optimistic value pessimistic value and expected value moreover four classes of type 2 uncertain variables are reduced to type 1 uncertain variables with specific uncertainty distributions type 2 uncertain optimization methods are applied to solving the fixed charge solid transportation problem with the type 2 uncertain parameters where the solution methods are also provided for the proposed models finally numerical experiments are implemented to demonstrate application and sensitivity analysis of the proposed approaches

coupled fuzzy interval model and method for structural response analysis with non probabilistic hybrid uncertainties abstract in many engineering practices with uncertainty the non probabilistic methods play an increasing important role to overcome the limitation of traditional single uncertainty modeling methods in handling coupled uncertain problems this paper develops a more general hybrid uncertainty analysis framework the non probabilistic hybrid uncertainties are expressed as coupled fuzzy interval variables where the bounds of interval are interpreted as fuzzy sets instead of deterministic values by means of the cut set strategy and decomposition theorem in fuzzy set theory the hybrid uncertain problem is transformed into a series of dual interval problems the conservative and radical extreme value predictions in different variable subspaces are adopted to characterize the coupled uncertainty in output response to further improve the computational efficiency of extreme value prediction an adaptive response surface model using radial basis function is proposed where the potential global optimum points are introduced as the new sample points in the sequential sampling process finally the effectiveness of the proposed method on dealing with non probabilistic hybrid uncertainties is validated by two examples

positive approximation and converse approximation in interval valued fuzzy rough sets methods of fuzzy rule extraction based on rough set theory are rarely reported in incomplete interval valued fuzzy information systems thus this paper deals with such systems instead of obtaining rules by attribute reduction which may have a negative effect on inducting good rules the objective of this paper is to extract rules without computing attribute reducts the data completeness of missing attribute values is first presented positive and converse approximations in interval valued fuzzy rough sets are then defined and their important properties are discussed two algorithms based on positive and converse approximations namely mine rules based on the positive approximation mrbpa and mine rules based on the converse approximation mrbca are proposed for rule extraction the two algorithms are evaluated by several data sets from the uc irvine machine learning repository the experimental results show that mrbpa and mrbca achieve better classification performances than the method based on attribute reduction

observer based robust fuzzy control of nonlinear systems with parametric uncertainties this paper addresses the robust fuzzy control problem for nonlinear systems in the presence of parametric uncertainties and the state variables unavailable for measurement the takagi sugeno t s fuzzy model system with parametric uncertainties is adopted for modeling the nonlinear system and establishing fuzzy state observer sufficient conditions are derived for robust stabilization in the sense of lyapunov asymptotic stability and are formulated in the format of linear matrix inequalities lmis the effectiveness of the proposed fuzzy controller and fuzzy observer design methodology is finally demonstrated through numerical simulations on an inverted pendulum system

some connections between generalized voigt functions with the different parameters the voigt functions play an important role in several diverse fields of physics such as astrophysical spectroscopy and the theory of neutron reactions motivated by the contributions toward the unification and generalization of these functions in this paper we establish some connections between generalized voigt functions with the different parameters involving one and two variable hypergeometric functions further we derive various other interesting results as applications of these connections

remarks on a note by h exton on the reducibility of the voigt functions correct expressions for integrals given in a recent letter by exton ibid vol 14 p l75 1981 are found

generalized fuzzy sets and representation of incomplete knowledge abstract in this paper we focus our attention on a kind of generalized fuzzy sets for such sets the grade of membership of an element in a subset should completely be described from the viewpoint of the obverse as well as the reverse side more precisely this kind of generalized fuzzy subset is defined by a pair of membership functions which are not required to satisfy any additional conditions set operations are given and some examples are briefly discussed

converse approximation and rule extraction from decision tables in rough set theory in this paper the concept of a granulation order is proposed in an information system the converse approximation of a target concept under a granulation order is defined and some of its important properties are obtained which can be used to characterize the structure of a set approximation for a subset of the universe in an information system its converge degree is monotonously increasing under a granulation order this means that a proper family of granulations can be chosen for a target concept approximation according to user requirements as an application of the converse approximation an algorithm based on the converse approximation called rebca is designed for decision rule extraction from a decision table which has a time complexity of o m2 c 2 u log 2 u and its practical applications are illustrated by two examples

fuzzy preference based rough sets preference analysis is an important task in multi criteria decision making the rough set theory has been successfully extended to deal with preference analysis by replacing equivalence relations with dominance relations the existing studies involving preference relations cannot capture the uncertainty presented in numerical and fuzzy criteria in this paper we introduce a method to extract fuzzy preference relations from samples characterized by numerical criteria fuzzy preference relations are incorporated into a fuzzy rough set model which leads to a fuzzy preference based rough set model the measure of attribute dependency of the pawlak s rough set model is generalized to compute the relevance between criteria and decisions the definitions of upward dependency downward dependency and global dependency are introduced algorithms for computing attribute dependency and reducts are proposed and experimentally evaluated by using two publicly available data sets

rough variables with values in measurable spaces an x valued rough variable is introduced as a measurable function from a rough space to a measurable space x based on the concept of x valued rough variables some properties of x valued rough variables are studied furthermore the concept of convergence is presented and convergence of x valued rough variables is discussed

a generalized multiple lyapunov functions method based non fragile guaranteed cost control for uncertain switched fuzzy systems abstract this paper is concerned with a non fragile guaranteed cost control problem for a class of uncertain switched fuzzy systems which are described by switched takagi sugeno t s fuzzy models by employing the generalized multiple lyapunov functions method we present a sufficient condition to guarantee the closed loop system asymptotically stable and possess an upper bound of the non fragile guaranteed cost function meanwhile a non fragile state feedback control law and a switching law are designed moreover an optimization problem in which the minimal non fragile guaranteed cost can be obtained is solved finally a simulation example is given to verify the feasibility and effectiveness of the proposed method

fuzzy rough set theory for the interval valued fuzzy information systems the concept of the rough set was originally proposed by pawlak as a formal tool for modelling and processing incomplete information in information systems then in 1990 dubois and prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets the aim of this paper is to present a new extension of the rough set theory by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory i e the interval valued fuzzy rough set model is presented based on the interval valued fuzzy information systems which is defined in this paper by a binary interval valued fuzzy relations r f i uxu on the universe u several properties of the rough set model are given and the relationships of this model and the others rough set models are also examined furthermore we also discuss the knowledge reduction of the classical pawlak information systems and the interval valued fuzzy information systems respectively finally the knowledge reduction theorems of the interval valued fuzzy information systems are built

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

a systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

knowledge reduction in random information systems via dempster shafer theory of evidence knowledge reduction is one of the main problems in the study of rough set theory this paper deals with knowledge reduction in random information systems based on dempster shafer theory of evidence the concepts of belief and plausibility reducts in random information systems are first introduced it is proved that both of belief reduct and plausibility reduct are equivalent to classical reduct in random information systems the relative belief and plausibility reducts in consistent and inconsistent random decision systems are then proposed and compared to the relative reduct and relationships between the new reducts and some existing ones are examined

on multiindices and multivariables presentation of the voigt functions this paper aims at presenting multiindices and multivariables study of the unified or generalized voigt functions which play an important role in the several diverse field of physics such as astrophysical spectroscopy and the theory of neutron reactions some expressions representations of these functionl are given in terms of familiar special functions of multivariables further representations and series expansions involving multidimensional classical polynomials laguerre and hermite of mathematical physics are established

rough sets analysis of diagnostic capacity of vibroacoustic symptoms abstract the paper refers to the problem of diagnostic classification of mechanical objects using vibroacoustic symptoms a new approach based on the rough sets theory is applied to evaluate the symptoms from the point of view of their diagnostic capacity i e the quality of estimation of a technical state of a mechanical object the approach enables reduction of the set of symptoms to a minimal subset ensuring a satisfactory estimation the minimal subset is then used to create a classifier of a technical state particular attention is paid to a comparison of different methods of calculation of symptom limit values which divide domains of symptoms into intervals corresponding to classes of technical states the analysed set of data concerns the technical state of rolling bearings installed in a laboratory stand they are described by a set of symptoms which result from measurements of noise and vibration of bearing housings the bearings are in good or bad technical states the paper presents particular steps of the rough sets methodology and gives as a final result a classifier of a technical state of bearings based on a minimal subset of symptoms with the greatest diagnostic capacity

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

incomplete conjunctive information abstract many information systems capable of handling incomplete or fuzzy information manipulate objects with single valued attributes information is then said to be disjunctive information is said to be conjunctive when pertaining to many valued attributes while a piece of incomplete disjunctive information is easily represented by means of a set of mutually exclusive possible values modeling incomplete conjunctive information theoretically leads to consider families of sets since attributes are then set valued under complete information some proposals are made in order to efficiently and rigorously represent incomplete conjunctive information and deal with query evaluation especially in the case where only upper and or lower bounds of the set of values of a many valued attribute are known applications of this approach can be expected for the processing of time intervals as well as spatial reasoning among other topics in knowledge base management

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

the necessary and sufficient conditions for a fuzzy relation being τ euclidean the connections between special types of fuzzy relations and properties of fuzzy rough approximation operators have been established in recent years but euclidean fuzzy relation has not been considered yet in the present paper the necessary and sufficient conditions for a fuzzy relation being euclidean are investigated in three different fuzzy rough approximation spaces meanwhile it is proved that in i fuzzy rough approximation space where i is an r implication the properties the euclidean i fuzzy rough approximation operators possess are just the same as those in rough fuzzy approximation space

twofold fuzzy sets and rough sets some issues in knowledge representation abstract this paper deals with the representation of sets where the membership of some elements may be ill known rather than just a matter of degree as in a fuzzy set the notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete a twofold fuzzy set is made of a nested pair of fuzzy sets the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by pawlak set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined twofold fuzzy relations are also introduced finally various applications of twofold fuzzy sets in knowledge representation are briefly discussed

delay dependent and decay rate dependent conditions for exponential mean stability and non fragile controller design of positive markov jump linear systems with time delay abstract this paper proposes some new sufficient conditions and necessary conditions for exponential mean stability with prescribed decay rate and sufficient conditions for non fragile controller design of both continuous time and discrete time positive markov jump linear systems with time delay first sufficient stability conditions are derived by constructing novel linear stochastic co positive lyapunov krasovskii functionals second the corresponding necessary conditions are established by applying a model transformation technique and analyzing the relationship between stochastic stability of the transformed systems and exponential mean stability with given decay rate of the original systems compared with the existing conditions the proposed stability conditions are not only delay dependent but also decay rate dependent and the sufficient conditions plus the necessary conditions can be used to judge the system stability more precisely third based on the new sufficient conditions non fragile controllers are designed by solving linear programming problems such that the closed loop systems are positive and exponentially mean stable with an expected decay rate finally by numerical examples the effects of both time delay and decay rate on exponential mean stability are exploited and the validity of the non fragile controller design conditions is demonstrated

learning fuzzy rules from fuzzy samples based on rough set technique although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness its performance in dealing with initial fuzzy data is usually poor this paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment the extension is twofold one is knowledge representation and the other is knowledge reduction first we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects second we extend a number of underlying concepts of knowledge reduction such as the reduct and core to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data finally we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm one of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension

rough fuzzy functions in classification this paper generalizes the concept of rough membership functions in pattern classification tasks to rough fuzzy membership functions and rough fuzzy ownership functions unlike the rough membership value of a pattern which is sensitive only towards the rough uncertainty associated with the pattern the rough fuzzy membership or ownership value of the pattern signifies the rough uncertainty as well as the fuzzy uncertainty associated with the pattern in this paper various set theoretic properties of the rough fuzzy functions are exploited to characterize the concept of rough fuzzy sets these properties are also used to measure the rough fuzzy uncertainty associated with the given output class finally a few possible applications of the rough fuzzy functions are mentioned

using fsbt technique with rough set theory for personal investment portfolio analysis this study proposes a novel forward search and backward trace fsbt technique based on rough set theory to improve data analysis and extend the scope of observations made from sample data to solve personal investment portfolio problems rough set theory mathematically classifies data into class sets the class set with the most objects may generate one decision rule the rules generated from rst are rough and fragmented that are very difficult to interpret the information an empirical case is used to generate more than 85 rules by the rst method in comparison with fsbt method which only generated 14 rules this result can show our proposed method is better than traditional rst method based on class sets that contain the most objects much of human knowledge is described in natural language it is a very important thing to convert information from computer databases into normal human language sample data taken from features with the same backgrounds are used to compile different portfolios that investment companies and investment advisors can employ to satisfy the investor needs the method not only can provide decision making rules but also can offer alternative strategies for better data analysis we believe that the fsbt technique can be fully applied in research on investment marketing

fuzzy relation equations and causal reasoning abstract fuzzy relation based models for handling uncertainty in a non probabilistic way in diagnosis problems have been introduced by sanchez about 20 years ago and since then applied and developed by several researchers the paper first reviews the existing proposals also including the association based abductive model proposed by reggia and his co workers which uses non fuzzy relations a new model is presented in order to have a more expensive representation capable of handling uncertainty and also of distinguishing i between manifestations which are certainly absent from those which are not yet observed and ii between manifestations which cannot be caused by a given disorder and manifestations for which we do not know if they can or cannot be caused by this disorder this new model is based on possibility theory and the so called twofold fuzzy sets previously introduced by the authors

a multi objective decision making model with fuzzy rough coefficients and its application to the inventory problem in this paper we concentrate on developing a fuzzy rough multi objective decision making model according to uncertainty theory we present some equivalent models and a traditional algorithm based on an interactive fuzzy satisfying method which is similar to the interactive fuzzy rough satisfying method in order to obtain a satisfying solution for the decision maker in addition the technique of fuzzy rough simulation is applied to deal with general fuzzy rough objective functions and fuzzy rough constraints which are usually difficult to convert into their equivalents furthermore combined with the techniques of fuzzy rough simulation a genetic algorithm using the compromise approach is designed for solving a fuzzy rough multi objective programming problem finally a model is applied to an inventory problem to illustrate the usefulness of the proposed model and algorithm and then a sensitivity analysis is made

a comparative study of rough sets for hybrid data to discover knowledge from hybrid data using rough sets researchers have developed several fuzzy rough set models and a neighborhood rough set model these models have been applied to many hybrid data processing applications for a particular purpose thus neglecting the issue of selecting an appropriate model to address this issue this paper mainly concerns the relationships among these rough set models investigating fuzzy and neighborhood hybrid granules reveals an important relationship between these two granules analyzing the relationships among rough approximations of these models shows that hu s fuzzy rough approximations are special cases of neighborhood and wang s fuzzy rough approximations respectively furthermore one to one correspondence relationships exist between wang s fuzzy and neighborhood rough approximations this study also finds that wang s fuzzy and neighborhood rough approximations are cut sets of dubois fuzzy rough approximations and radzikowska and kerre s fuzzy rough approximations respectively