generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

rotation and rotation annihilation construction of associative and partially compensatory aggregation operators in this paper it is shown that aggregation operators cannot be associative and strictly compensatory at the same time as a way out the rotation construction and the rotation annihilation construction are applied to construct wide classes of associative aggregation operators that are compensatory on a subdomain only from this point of view the resulting operators are similar to uninorms a class of aggregation operators that is being reported on extensively in the literature the constructions are demonstrated by means of ample examples

transverse vibrations of rectangular plates with edges elastically restrained against translation and rotation abstract the fundamental frequency coefficient for a rectangular plate with edges elastically restrained against both translation and rotation is calculated by using polynomial coordinate functions and the rayleigh ritz method the approach is simple and straightforward and allows the solution of a rather difficult elastodynamics problem complicating factors orthotropic properties in plane forces concentrated masses etc can also be taken into account without formal difficulties

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

aggregation operators ordering and bounds standard ordering of aggregation operators in several distinguished classes is characterized lower and upper bounds of finite systems in those classes of aggregation operators are discussed several examples are given

algorithms for terminal steiner trees the terminal steiner tree problem tst consists of finding a minimum cost steiner tree where each terminal is a leaf we describe a factor 2 r r 3 r 2 approximation algorithm for the tst where r is the approximation factor of a given algorithm for the steiner tree problem considering the current best value of r this improves a previous 3 10 factor to 2 52 for the tst restricted to instances where all edge costs are either 1 or 2 we improve the approximation factor from 1 60 to 1 42

the distributivity property of increasing binary operations abstract this paper is mainly devoted to solving the functional equations of distributivity and conditional distributivity of increasing binary operations with the unit our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory in particular some assumptions namely associativity and commutativity and results about conditional distributivity of uninorms and triangular norms and triangular conorms were simplified moreover some other properties e g componentwise convexity concavity for special operations from the class of quasi copulas is considered

triangular norms position paper i basic analytical and algebraic properties we present the basic analytical and algebraic properties of triangular norms we discuss continuity as well as the important classes of archimedean strict and nilpotent t norms triangular conorms and de morgan triples are also mentioned finally a brief historical survey on triangular norms is given

algebraic representations of the weighted mean abstract we introduce a general framework for the algebraic representation of the weighted mean in the case in which priorities and values refer to a common aggregation domain for instance in hierarchical multicriteria decision models of the ahp type the general framework proposed is based on the semifield structure of open interval domains and provides a natural algebraic description of weighted mean aggregation including the essential mechanism of normalization which transforms priorities into weights such description necessarily involves two operations addition abelian semigroup and multiplication abelian group which generalize the role of addition and multiplication in p 0 in this sense the semifield framework extends recent work by cavallo d apuzzo and squillante on the multiplicative group structure on the basis of the representation of pairwise comparison matrices and their associated priority vectors we consider open interval domains s r whose semifield structures are generated by bijections s r p continuous thus strictly monotonic bijections play a central role in such case continuous strict triangular conorms norms and uninorms emerge naturally in the representation of the semifield structure and in their weighted version they also provide the representation of the weighted mean in both the arithmetic and geometric forms

general fuzzy acceptors for syntactic pattern recognition abstract in this paper the behavior of general fuzzy acceptor over a commutative cl semigroup l is considered and the properties of equivalence relation are investigated from these results we can apply for pattern recognition of deformed images or for enhancement of distorted patterns

pictorial representations of fuzzy connectives part i cases of t norms t conorms and averaging operators part i of this paper shows a number of examples of existing t norms and t conorms which are known as good models of fuzzy set theoretic intersection and union and then the pictorial representations of t norms and t conorms are made with the aid of a computer several new examples of t norms and t conorms are also given averaging operators between min and max are summarized and their pictorial representations are also made

a characterization of a class of uninorms with continuous underlying operators in this paper all uninorms locally internal in the region a e given by the complement in 0 1 2 of 0 e 2 e 1 2 where e is the neutral element of the uninorm having continuous underlying operators are studied and characterized by distinguishing some cases when the underlying t norm and t conorm are not given by ordinal sums it is proved that uninorms locally internal in a e are in fact all possible uninorms with these underlying operators except when both the t norm and the t conorm are strict in which case there is also the class of representable uninorms leading to a finite number of possibilities when at least one of the continuous underlying operators is given by an ordinal sum again there are other possible uninorms than those that are locally internal in a e but all uninorms with this property are also characterized in this case infinitely many possibilities can appear depending on the set of idempotent elements of the uninorm

extended neuro fuzzy models of multilayer perceptrons in this paper the famous neural model the multilayer perceptron is extended to a new neural model that is called the additive takagi sugeno type multilayer perceptron the present study proves that this new model can also act as a universal approximator and thus it can be used in many fields such as system modeling and identification the concept of f duality and the fuzzy operator interactive or are used to prove that the proposed neural model is functionally equal to a kind of fuzzy inference system further this paper presents another new neuro fuzzy model that is called the sigmoid adaptive network based fuzzy inference system simulation studies show that our proposed models both have stronger approximation capability than multilayer perceptrons

ensuring reliability of the weighting vector in the context of pairwise comparison matrices pcms defined over abelian linearly ordered group consistency and transitivity represent a full coherence of the decision maker dm and the minimal logical requirement that dm s preferences should satisfy respectively moreover the mean vector wm is proposed as weighting vector for the decision elements related to the pcm in this paper we investigate the effects of inconsistency of a transitive pcm on wm and in order to ensure its reliability as weighting vector we provide the notion of weak consistency it is weaker than consistency and stronger than transitivity and ensures that vectors associated with a pcm by means of a strictly increasing synthesis functional are reliable for assigning a preference order on the set of related decision elements the mean vector wm is associated with a pcm by means of one of these functionals finally we introduce an order relation on the rows of the pcm that is a simple order if and only if the condition of weak consistency is satisfied the simple order allows us to easily determine the actual ranking on the set of related decision elements

basic concepts for a theory of evaluation hierarchical aggregation via autodistributive connectives in fuzzy set theory abstract the aggregative operator introduced by combi 1982 in his analysis of basic concepts for a theory of evaluation in this journal is based on the axioms of associativity continuity and monotonicity in this paper an alternative approach is given where associativity is replaced by another axiom viz autodistributivity implying a kind of hierarchical aggregation instead of the former horizontal one the new set of properties yields a general class of operators for connecting fuzzy sets which are formally similar to the aggregative operators but are purely compensative in a strong sense moreover this modified approach generalizes a recent approach based on generalized means as compensative connectives it appears to be similar to expected utility approaches in decision making under uncertainty and seems to be a very promising tool to handle representation problems as they grow from other fields of human decision making e g of multicriteria analysis and of welfare theory

representation and construction of self dual aggregation operators abstract two different characterizations of self dual aggregation operators are available in the literature one based on c x y x x 1 y and one based on the arithmetic mean both approaches construct a self dual aggregation operator by combining an aggregation operator with its dual in this paper we fit these approaches into a more general framework and characterize n invariant aggregation operators with n an involutive negator various binary aggregation operators fulfilling some kind of symmetry w r t n and with a sufficiently large range can be used to combine an aggregation operator and its dual into an n invariant aggregation operator moreover using aggregation operators to construct n invariant aggregation operators seems rather restrictive it suffices to consider n ary operators fulfilling some weaker conditions special attention is drawn to the equivalence classes that arise as several of these n ary operators can yield the same n invariant aggregation operator

left and right distributivity equations for semi t operators and uninorms abstract recently the distributivity equation for different kind of operators has become a focus of research as the crucial tool in the numerous applications as utility and optimization theory or integration theory the main purpose of this paper is to study the distributivity equations between two special classes of aggregation operators namely semi t operators and uninorms without the commutativity assumption of semi t operators it is necessary to consider the left and the right distributivity conditions separately in both cases the obtained solutions are significantly different

aggregation functions construction methods conjunctive disjunctive and mixed classes in this second part of our state of the art overview on aggregation theory based again on our recent monograph on aggregation functions we focus on several construction methods for aggregation functions and on special classes of aggregation functions covering the well known conjunctive disjunctive and mixed aggregation functions some fields of applications are included

effect of geometric boundary perturbations on the natural frequencies and buckling loads of vibrating clamped circular plates abstract use is made of conformal mapping and the galerkin method coupled with rayleigh s optimization procedure to determine the fundamental frequency of vibration and the buckling coefficient of clamped circular plates with geometric boundary perturbations

aggregation operators and commuting commuting is an important property in any two step information merging procedure where the results should not depend on the order in which the single steps are performed we investigate the property of commuting for aggregation operators in connection with their relationship to bisymmetry in case of bisymmetric aggregation operators we show a sufficient condition ensuring that two operators commute while for bisymmetric aggregation operators with neutral element we even provide a full characterization of commuting n ary operators by means of unary distributive functions the case of associative operations especially uninorms is considered in detail

social choice axioms for fuzzy set aggregation the problem of aggregating n fuzzy sets f1 f2 fn on a set is viewed as one of merging the opinions of n individuals e g experts that rate objects belonging to this approach contrasts with the pure set theoretic point of view and leads to interpreting already known axioms underlying fuzzy connectives in a way different from that of multiple criteria aggregation various natural properties of a voting procedure including the ones proposed by arrow are expressed in the fuzzy set setting a number of conditions limiting the choice of fuzzy set operations are proposed and classified according to whether they are imperative mainly technical or facultative families of solutions are obtained that include those proposed in earlier works the case of non homogeneous groups is briefly examined lastly the application of the voting paradigm to the management of antagonistic local decision rules in knowledge based systems is outlined

distributivity and conditional distributivity of a uninorm and a continuous t conorm the open problem recalled by klement in the linz2000 closing session related to distributivity and conditional distributivity of a uninorm and a continuous t conorm is solved for the most usual known classes of uninorms from the obtained results it is deduced that distributivity and conditional distributivity are equivalent for these cases it is remarkable that solutions appear involving not only strict t conorms but also ordinal sums of the maximum with a strict t conorm conversely the distributivity of a t conorm over a uninorm is also studied leading only to already known solutions moreover the dual case of distributivity and conditional distributivity involving uninorms and continuous t norms is also solved proving again the equivalence of both kinds of distributivities

cauchy like functional equation based on continuous t conorms and representable uninorms commuting is an important property in any two step information merging procedure where the results should not depend on the order in which the single steps are performed in the case of bisymmetric aggregation operators with the neutral elements saminger mesiar and dubois have already reduced characterization of commuting n ary operators to resolving the unary distributive functional equations but only some sufficient conditions of unary functions distributive over two particular classes of uninorms are given out indeed it is very difficult to get the full characterizations of these equations because they are bound up with the famous cauchy functional equation which has so far not been completely solved along this way of thinking in this paper we will respectively investigate and fully characterize the following two functional equations f u x y s f x f y and f s x y u f x f y where f 0 1 0 1 is an unknown function u is a representable uninorm and s is a continuous t conorm these results are an important step toward obtaining complete characterization of the other unary distributive functional equations previously mentioned our results show that commuting is suitable for only the second equation but not for the first one this is because solutions of the first equation are all piecewise constant functions while those of the second equation are formulas with parameters in addition the two equations both have nonmonotone solutions completely different from those ones obtained

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

different interpretations of triangular norms and related operations triangular norms and triangular conorms and uni norms can be treated as semigroup two place function also as commutative semiring multiplications it can also be derived from the difference operation in a difference poset we discuss here these different interpretations and their consequences

triangular norm based iterative compensatory operators abstract aggregation operators based on a fixed t norm and on suitable transformations of processed data are introduced a special attention is paid to the class of iterative compensatory operators containing also the classical arithmetic geometric and harmonic means several properties of introduced operators are studied e g the symmetry the associativity the idempotency the anihilation etc several examples are given including continuous idempotent bisymmetric iterative compensatory operators with zero anihilator

vibrations of plates of arbitrary shape by a modified galerkin method this paper deals with a modification of galerkin s method which allows for the optimization of the fundamental eigenvalue when a single co ordinate function is used use of the conformal mapping method allows for the consideration of two dimensional domains of complicated boundary shape

the modularity law in some classes of aggregation operators abstract in this study we consider the functional equation of modularity in a special class of aggregation operators with an absorbing element first we focus on the modularity between commutative binary operators and their duals and then we examine the modularity for operators from the class of 2 uninorms which generalize nullnorms by extending their certain conditions in particular for these operators depending on the positions of their neutral elements and possessing the same absorbing element we obtain positive and negative results which are analogous to the results in the general case for operators with different absorbing elements

a general model for deriving preference structures from data in this paper we comment upon the integrated model for valued preferences introduced by fodor ovchinnikov and roubens in particular while on one hand we revise basic assumptions and point out their intuitive meaning on the other hand we propose an alternative mathematical justification of such a model which allows not only a better understanding of the obtained results but also a functional characterization of the whole family of solutions

a single point characterization of representable uninorms we study uninorms of which both the underlying t norm and underlying t conorm are strict such uninorms are the only candidates for being representable by an additive generator we prove that the representability of such a uninorm depends solely on its value at a single arbitrary point in the remaining open part of the unit square more explicitly such a uninorm turns out to be representable if and only if this single value is located strictly between the minimum and the maximum of the corresponding arguments if this single value coincides with one of these bounds then the value of the uninorm at any point in the remaining open part is determined by the same bound

on a certain class of aggregative operators abstract here our starting point is a study of connection with dombi aggregative operators uninorms strict t norms and t conorms we present a new representation theorem of strong negations that explicitly contains the neutral value then the relationships for aggregative operators and strong negations are verified as well as those for the t norm and t conorm using the pan operator concept we introduce the multiplicative pliant concept and give the necessary and sufficient conditions for it we study a certain class of weighted aggregative operators representable uninorms which build a self demorgan class with infinitely many negations we provide the necessary and sufficient conditions for these operators

a spectrum of compromise aggregation operators for multi attribute decision making in many decision making problems a number of independent attributes or criteria are often used to individually rate an alternative from an agent s local perspective and then these individual ratings are combined to produce an overall assessment now in cases where these individual ratings are not in complete agreement the overall rating should be somewhere in between the extremes that have been suggested however there are many possibilities for the aggregated value given this this paper systematically explores the space of possible compromise operators for such multi attribute decision making problems specifically we axiomatically identify the complete spectrum of such operators in terms of the properties they should satisfy and show the main ones that are widely used namely averaging operators uninorms and nullnorms represent only three of the nine types we identify for each type we then go onto analyse their properties and discuss how specific instances can actually be developed finally to illustrate the richness of our framework we show how a wide range of operators are needed to model the various attitudes that a user may have for aggregation in a given scenario bidding in multi attribute auctions