monitored heavy fuzzy measures and their role in decision making under uncertainty we discuss the fuzzy measure and the related idea of choquet aggregation we describe a prototypical application of the fuzzy measure in decision making under uncertainty we introduce an extension of the fuzzy measure called the monitored heavy fuzzy measure we give some examples of these new measures using these measures with the choquet integral we obtain a general class of aggregation operators that include both mean type operators and totaling type operators we consider the application of this measure to the problem of decision making under uncertainty in the case in which the uncertain variable can have multiple solutions

handling possibilistic labels in pattern classification using evidential reasoning abstract a category of learning problems in which the class membership of training patterns is assessed by an expert and encoded in the form of a possibility distribution is considered each example i thus consists in a feature vector x i and a possibilistic label u 1 i u c i where u k i denotes the possibility of that example belonging to class k this problem is tackled in the framework of evidence theory the evidential distance based classifier previously introduced by one of the authors is extended to handle possibilistic labeling of training data two approaches are proposed based either on the transformation of each possibility distribution into a consonant belief function or on the use of generalized belief structures with fuzzy focal elements in each case a belief function modeling the expert s beliefs concerning the class membership of each new pattern is obtained this information may then be either interpreted by a human operator to support decision making or automatically processed to yield a final class assignment through the computation of pignistic probabilities experiments with synthetic and real data demonstrate the ability of both classification schemes to make effective use of possibilistic labels as training information

a linear regression model using triangular fuzzy number coefficients fuzzy regression analysis using fuzzy linear models with symmetric triangular fuzzy number coefficients has been formulated earlier the goal of this regression is to find the coefficients of a proposed model for all given input output data sets in this paper we extend the results of a fuzzy linear regression model that uses symmetric triangular coefficient to one with non symmetric fuzzy triangular coefficients this work eradicates the inflexibility of existing fuzzy linear regression models

possibilistic functional dependencies and their relationship to possibility theory this paper introduces possibilistic functional dependencies these dependencies are associated with a particular possibility distribution over possible worlds of a classical database the possibility distribution reflects a layered view of the database the highest layer of the classical database consists of those tuples that certainly belong to it while the other layers add tuples that only possibly belong to the database with different levels of possibility the relation between the confidence levels associated with the tuples and the possibility distribution over possible database worlds is discussed in detail in the setting of possibility theory a possibilistic functional dependency is a classical functional dependency associated with a certainty level that reflects the highest confidence level where the functional dependency no longer holds in the layered database moreover the relationship between possibilistic functional dependencies and possibilistic logic formulas is established related work is reviewed and the intended use of possibilistic functional dependencies is discussed in the conclusion

from fuzzy set theory to fuzzy set theory some critical remarks on existing concepts abstract our aim in this paper is to give some foundational remarks on the nature of fuzziness and their modeling by fuzzy sets we will reveal some fundamental misunderstandings which have pervaded fuzzy set theory just from its very beginnings at the end of the paper we give three examples demonstrating the differences and advantages of our new conception our approach in heavily based on an operational and set theoretical background contrasting hisdal s tee model

a dual ascent procedure for large scale uncapacitated network design the fixed charge network design problem arises in a variety of problem contexts including transportation communication and production scheduling we develop a family of dual ascent algorithms for this problem this approach generalizes known ascent procedures for solving shortest path plant location steiner network and directed spanning tree problems our computational results for several classes of test problems with up to 500 integer and 1 98 million continuous variables and constraints show that the dual ascent procedure and an associated drop add heuristic generate solutions that in almost all cases are guaranteed to be within 1 to 4 of optimality moreover the procedure requires no more than 150 seconds on an ibm 3083 computer the test problems correspond to dense and sparse networks including some models that arise in freight transport

compatibility relations for the representation of associations between variables in knowledge based systems and their use in approximate reasoning abstract compatibility relations are proposed as formal representations of the production if then rules capturing associations between a primary a and a secondary b variable the most general form of the if then rule considered is if a takes on its value in a set then b takes on its value in a set or in different possible sets each with its associated degree of belief conviction moreover with each rule is associated a degree of belief as to its validity we consider an approximate reasoning scheme stated as we know a possibility distribution representing the value of a premise and an association between a and b represented by an if then rule and we seek the induced possibility distribution representing the value of b consequence the approach to approximate reasoning proposed is based on possibility theory and possibility measures induced by multivalued mappings

maximum entropy membership functions for discrete fuzzy variables due to the deficiency of information the membership function of a fuzzy variable cannot be obtained explicitly it is a challenging work to find an appropriate membership function when certain partial information about a fuzzy variable is given such as expected value or moments this paper solves such problems for discrete fuzzy variables via maximum entropy principle and proves some maximum entropy theorems with certain constraints a genetic algorithm is designed to solve the general maximum entropy model for discrete fuzzy variables which is illustrated by some numerical experiments

sampling fuzzy discretization and signal reconstruction abstract the sampling theorem provides the condition under which a continuous time signal can be precisely represented and completely recovered from its instantaneous values equally spaced in time it gives a condition for the signal reconstruction problem the importance of this theorem pertains to the necessity of knowing when a continuous time signal is uniquely represented by its discrete version so that continuous time signals can be properly converted and handled by sampled data systems like digital computers the problem of the equivalence between either continuous time or discrete time signals and their fuzzy descriptions is analyzed here this encompasses the analysis of signal reconstruction when fuzzy systems are used as sampled data systems such as fuzzy filters fuzzy models or fuzzy controllers a necessary and sufficient condition is given for ensuring that the fuzzy system processes the legitimate representations of the signal presented at its interfaces furthermore a sufficient condition suitable for automatic tuning of membership functions is proposed and discussed the discussion encompasses the outline of some requirements that membership functions at fuzzy system interfaces should meet to allow humans to assess their semantic meaning

a unified approach to define fuzzy integrals recently the two most general and well known fuzzy integrals sugeno s integral and choquet s integral were characterized it was also shown how both integrals fitted the same formal model and they only differed in the operators used in this paper we try to analyze whether it is possible to improve those characterization theorems and we also study the possibility of defining other fuzzy integrals following the same model but using other operators in a recent paper campos and bolafios 6 characterized two general and well known fuzzy integrals sugeno s integral 11 and choquet s integral 7 the last one also called monotone expectation 2 4 5 6 the two characterization theorems are based on the same formal model of four properties which only differ in the operators used sum and product for choquet s integral maximum and minimum for sugeno s integral two important questions remained open one involved the characterization theorems of these integrals in an attempt to further refine them the second question was to investigate the possibility of defining other fuzzy integrals using different operators but following the same model in this paper we give answers to those problems section 2 is devoted to describing fuzzy integrals defined on sugeno s fuzzy measures particularly sugeno s and choquet s integrals and their characterizations in section 3 we show how the characterization of choquet s integral can be refined whereas the same cannot be made for sugeno s integral finally other fuzzy integrals fitting our formal model are obtained in section 4

efficiencies of two stage systems with fuzzy data two stage dea data envelopment analysis models show the performance of individual processes and thus are more informative than the conventional one stage models for making decisions this paper extends this approach from deterministic to uncertain situations where the observations are represented by fuzzy numbers the extension principle is utilized to develop a pair of two level mathematical programs to calculate the lower and upper bounds of the a cut of the fuzzy efficiency by enumerating various values of a the membership functions of fuzzy efficiencies are constructed numerically it is found that the property of the system efficiency being equal to the product of the two process efficiencies which holds for the deterministic case also holds for the fuzzy case this property can be generalized to series systems with more than two processes an example of non life insurance companies in taiwan is used to explain how to calculate the system and process efficiencies and how to derive their relationship when the data is fuzzy

possibilistic kalman filtering for radar 2d tracking abstract standard kalman filter skf introduced by kalman in the 60s has gained a non estimated importance in control as well as in robotics community its importance arises from the obtained optimal result in the sense of variance minimization under stochastic gaussian and unbiased perturbations and when the state model as well as the measurement model are precisely known however when the last requirement is relaxed such that one or more parameters governing the models are ill defined and rather given in terms of interval evaluations chen et al ieee trans aerospace electr syst 33 1 1997 251 259 have proposed interval kalman filter ikf by extending the arithmetic operations to interval calculus in this paper we rather assume that the uncertainty pervading some parameters of the models are given in terms of possibility distributions 21 this leads to a formulation of possibilistic kalman filtering pkf which agrees with ikf the same example of 2d radar tracking is tackled comparisons with ikf are investigated as well the influence of the modelling process on the performance of the filter besides the proposal permits to capture certainty qualified information which cannot be obtained from ikf

probability of a fuzzy event an axiomatic approach abstract axioms are proposed that could justify the natural definition of the probability of a fuzzy event initially given by zadeh they are based 1 on the postulate that the sum of the conditional probability of a fuzzy event and of its complement given any fuzzy event adds to one or 2 on soft independence for orthogonal sets with independent constitutive elements a general postulate is also required concerning the complement of a fuzzy set the classical definition of the operator representing the complement can also be deduced

possibility and necessity in optimization abstract we employ possibility and necessity in fuzzy programming and in other optimization problems that traditionally use random variables the possibility approach to multiobjective programming produces the same results as fuzzy programming a detailed example is presented to illustrate the use of possibility and necessity in optimization problems that usually employ random variables to describe uncertainty

a lagrangian heuristic based branch and bound approach for the capacitated network design problem the capacitated network design problem is a multicommodity minimal cost network flow problem with fixed charges on the arcs and is well known to be np hard the problem type is very common in the context of transportation networks telecommunication networks etc in this paper we propose an efficient method for this problem based on a lagrangian heuristic within a branch and bound framework the lagrangian heuristic uses a lagrangian relaxation to obtain easily solved subproblems and solves the lagrangian dual by subgradient optimization it also includes techniques for finding primal feasible solutions the lagrangian heuristic is then embedded into a branch and bound scheme that yields further primal improvements special penalty tests and cutting criteria are developed the branch and bound scheme can either be an exact method that guarantees the optimal solution of the problem or be a quicker heuristic the method has been tested on networks of various structures and sizes computational comparisons between this method and a state of the art mixed integer code are presented the method is found to be capable of generating good feasible solutions to large scale problems within reasonable time and data storage limits

belief functions induced by random fuzzy sets a general framework for representing uncertain and fuzzy evidence abstract we revisit zadeh s notion of evidence of the second kind and show that it provides the foundation for a general theory of epistemic random fuzzy sets which generalizes both the dempster shafer theory of belief functions and possibility theory in this perspective dempster shafer theory deals with belief functions generated by random sets while possibility theory deals with belief functions induced by fuzzy sets the more general theory allows us to represent and combine evidence that is both uncertain and fuzzy we demonstrate the application of this formalism to statistical inference and show that it makes it possible to reconcile the possibilistic interpretation of likelihood with bayesian inference

fuzzy integrals on l fuzzy sets abstract the fuzzy measures and fuzzy integrals for zadeh s fuzzy sets were studied by qiao and wang 5 6 7 12 in this paper we consider the real valued fuzzy measures and fuzzy integrals for l fuzzy sets where l is a pseudocomplemented infinitely distributive complete lattice on a fuzzy algebra of l fuzzy sets the concepts of fuzzy measure and fuzzy integral are introduced some equivalent forms of fuzzy integrals on l fuzzy sets are given moreover we discuss several properties of fuzzy integrals on l fuzzy sets and prove some convergence theorems for a sequence of fuzzy integrals on l fuzzy sets

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

the evidential reasoning approach for mada under both probabilistic and fuzzy uncertainties abstract many multiple attribute decision analysis mada problems are characterised by both quantitative and qualitative attributes with various types of uncertainties incompleteness or ignorance and vagueness or fuzziness are among the most common uncertainties in decision analysis the evidential reasoning er approach has been developed in the 1990s and in the recent years to support the solution of mada problems with ignorance a kind of probabilistic uncertainty in this paper the er approach is further developed to deal with mada problems with both probabilistic and fuzzy uncertainties in this newly developed er approach precise data ignorance and fuzziness are all modelled under the unified framework of a distributed fuzzy belief structure leading to a fuzzy belief decision matrix a utility based grade match method is proposed to transform both numerical data and qualitative fuzzy assessment information of various formats into the fuzzy belief structure a new fuzzy er algorithm is developed to aggregate multiple attributes using the information contained in the fuzzy belief matrix resulting in an aggregated fuzzy distributed assessment for each alternative different from the existing er algorithm that is of a recursive nature the new fuzzy er algorithm provides an analytical means for combining all attributes without iteration thus providing scope and flexibility for sensitivity analysis and optimisation a numerical example is provided to illustrate the detailed implementation process of the new er approach and its validity and wide applicability

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

construction of fuzzy σ algebras using triangular norms abstract generalizing the definitions given by the author fuzzy sets and systems4 1980 83 93 we introduce and study t fuzzy algebras t being any triangular norm the main result is that for a large class of triangular norms each t fuzzy algebra is generated i e consists of all functions x 0 1 being measurable with respect to some algebra on x

representation similarity measures and aggregation methods using fuzzy sets for content based recommender systems representation of features of items and user feedback and reasoning about their relationships are major problems in recommender systems this is because item features and user feedback are subjective imprecise and vague the paper presents a fuzzy set theoretic method ftm for recommender systems that handles the non stochastic uncertainty induced from subjectivity vagueness and imprecision in the data and the domain knowledge and the task under consideration the research further advances the application of fuzzy modeling for content based recommender systems initially presented by ronald yager the paper defines a representation method similarity measures and aggregation methods as well as empirically evaluates the methods performance through simulation using a benchmark movie data ftm consist of a representation method for items features and user feedback using fuzzy sets and a content based algorithm based on various fuzzy set theoretic similarity measures the fuzzy set extensions of the jaccard index cosine proximity or correlation similarity measures and aggregation methods for computing recommendation confidence scores the maximum minimum or weighted sum fuzzy set theoretic aggregation methods compared to the baseline crisp set based method csm presented the empirical evaluation of the ftm using the movie data and simulation shows an improvement in precision without loss of recall moreover the paper provides a guideline for recommender systems designers that will help in choosing from a combination of one of the fuzzy set theoretic aggregation methods and similarity measures

fuzzy measure of fuzzy events defined by fuzzy integrals abstract we define the concept of fuzzy measure of a fuzzy event by using a general form of fuzzy integral proposed by murofushi called fuzzy t conorm integral encompassing previous definitions zadeh defined the probability measure of a fuzzy event and later the possibility measure of fuzzy event using a duality property of fuzzy t conorm integral we propose a general definition of fuzzy measure of fuzzy events which is compatible with previous definitions of zadeh and possesses all properties of a fuzzy measure in particular the duality property using our definition we examine the case of decomposable measures and belief functions a comparison with previous works is provided

combining uncertainty and imprecision in models of medical diagnosis the paper presents a unified fuzzy probabilistic framework for modeling processes of medical diagnosis the two basic concepts of the dempster shafer theory i e focal elements and a basic probability assignment correspond to disease symptoms and the significance of an individual symptom in the diagnosis respectively the belief computation is related to diagnostic inference the final conclusion of the inference is the diagnosis with the greatest belief value fuzzy sets are used to describe focal elements it is shown how their membership functions and basic probability assignments are estimated on the basis of experimental data the interpretation of focal elements as fuzzy sets along with individual consideration of evidence imprecision and uncertainty of diagnosis are the essential new aspects of the presented method experimental studies have demonstrated the superiority of the proposed approach over some other modeling alternatives

measuring properties of fuzzy sets a general technique and its use in fuzzy query evaluation abstract a fuzzy set can be viewed as a weighted max combination or as a convex combination of a family of nested ordinary sets the latter point of view allows the interpretation of a fuzzy set as a consonant random set then any numerical set function applied to a fuzzy set leads to a result which is a random number the expectation attached to this random number provides a scalar evaluation of the fuzzy set with respect to the considered set function this approach is very general and can be used for various kinds of measurements cardinality confidence measures attached to the occurrence of events diameter or perimeter of a fuzzy region distance between two sets etc also the fuzziness of the set to evaluate naturally induces a fuzzy set of possible evaluations and expected values of the lower and upper bounds of this fuzzy set of numbers can be computed the weighted max combination view which leads to another type of scalar evaluation is also briefly discussed in this paper the above approach is more particularly applied to the treatment of queries asking for some global evaluation of a set of items specified by means of a fuzzy property this kind of query can be managed by languages like sql in case of a crisp set of items when this set is fuzzily specified a scalar answer can still be provided as well as a fuzzy valued or interval valued one if we want to keep track of the effect of the fuzziness of the query the case where the data base contains precise information but also the more general case of fuzzy information data bases are considered

do inferential processes affect uncertainty frameworks this paper studies the connections among different comparative or numerical degrees of belief in particular we consider in turn a comparative probability or possibility on a given boolean algebra and we prove that their upper extensions to a different boolean algebra are respectively a comparative plausibility or possibility on the other hand in general the upper extension of a comparative necessity is simply a comparative capacity moreover by considering a suitable condition of weak logical independence between the two boolean algebras we prove that the upper ordinal relation is a comparative possibility in all the aforementioned cases we consider specifically also the lower ordinal relations since they may not be the comparative dual relation of the upper ones

generalized probabilities of fuzzy events from fuzzy belief structures abstract we extend shafer s theory of evidence to include the ability to have belief structures involving fuzzy sets we then obtain under the condition of bayesian belief structure a whole family of possible definitions for the probability of fuzzy sets we also suggest a procedure for including belief qualification in pruf

combining forecasts operational adjustments to theoretically optimal rules clemen and winkler 1985 have described the theoretical effectiveness of winkler s 1981 formula for optimally combining forecasts the optimality of winkler s formula is however contingent on actually knowing the forecasters statistical properties i e the variances and covariances of their forecasts in realistic applications of course these properties have to be estimated usually from a set of prior forecasts in this case we show how the operationally optimal combining strategy differs from winkler s theoretically optimal formula specifically we provide figures indicating the operationally optimal strategy for combining two forecasts we then propose a heuristic to choose the best set of parameter estimates in combining any number of forecasters and demonstrate its effectiveness via simulation

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

inferential processes leading to possibility and necessity this paper deals with the upper and lower bounds of a class of uncertainty measures endowed with particular characteristics decomposability monotonicity partial additivity and so on we consider an initial partial assessment consistent with either probability or possibility or necessity then we study the upper and lower envelopes of all possible extensions by resorting to a notion of weak logical independence we get as lower or upper envelope a possibility or a necessity respectively starting either from a probability or from a possibility or from a necessity

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

weighted fuzzy pattern matching the fuzzy pattern matching technique has been developed in the framework of fuzzy set and possibility theory in order to take into account the imprecision and the uncertainty pervading values which have to be compared in a matching process this technique has proved very useful for implementing patterns of approximate reasoning in expert system inference engines and for designing retrieval systems capable of managing incomplete and fuzzy information data bases and vague queries in this paper the fuzzy pattern matching procedure is improved by introducing weights assessing the relative importance of atoms in the pattern matching indices are obtained using weighted versions of the minimum and maximum operations of fuzzy set theory this approach is extended to the case of variable weights and is related to aggregation schemes proposed by yager when modeling soft quantified statements such as most of the criteria must be met a notion of soft partial matching is thus modelled

models and algorithms for network reduction abstract we study models and algorithms for a network reduction nr problem that entails constructing a reduced network in place of an existing large network so that the shortest path lengths between specified node pairs in this network are equal to or only slightly longer than the corresponding shortest path lengths in the original network solving this problem can be very useful both to accelerate shortest path calculations in many practical contexts and to reduce the size of optimization models that contain embedded shortest path problems this work was motivated by a real problem of scheduling and routing resources to perform spatially dispersed jobs but also has other applications we consider two variants of the nr problem a min size nr problem that minimizes the number of arcs in the reduced network while ensuring that the shortest path lengths in this network are close to the original lengths and a min length nr problem that minimizes a weighted sum of shortest path lengths over all the specified node pairs while limiting the number of arcs in the reduced network we model both problems as integer programs with multi commodity flows and propose optimization based heuristic algorithms to solve them these methods include preprocessing a shortest path based procedure with local improvement and a dual ascent algorithm we report on the successful applications to reduce the network for practical infrastructure project planning and to condense a transportation network for distribution planning we also compare these solutions with those obtained using algorithms for minimum length trees

multiple probability assessments by dependent experts abstract when two or more information sources experts provide a decision maker with information on two or more random variables the decision maker using bayes s rule has an opportunity to a update a prior about the random variables and b calibrate the experts calibration is the process of adjusting the decision maker s likelihood about the experts assessments this article presents a model for this two way process and specializes to the case in which the experts assessment errors have a multivariate normal density in general we find that variables which the decision maker and the experts regard as independent a priori will be dependent a posteriori because of dependence in the assessment errors formulas for posterior densities are given for the normal model in this model the posterior density of the random variables depends on only a weighted average of the expert s means with weights that depend on the experts assessments of previously known quantities i also present a special case o

a note on the conditional probability of fuzzy subsets of a continuous domain abstract the notion of a mass assignment corresponding to a discrete fuzzy set is generalised to include all fuzzy sets the definition is motivated by considering a possible mechanism by which an intelligent agent could extend set theoretic operations to fuzzy sets this same mechanism also suggests an alternative definition for the conditional probability of fuzzy sets it is noted that this definition is probability possibility consistent and it is shown that in the case where the probability distribution on the domain has a density function then the conditional probability of f given g can be expressed as the probability of f calculated relative to a conditional density function dependent on g this result is used to develop an analytical method for calculating conditional probabilities of fuzzy subsets of the reals where the membership functions of f and g and the density function are continuous piecewise linear functions such a method has applications to the development of a general semantic unification in fril

a hierarchical neural model of matching abstract the issue of matching two fuzzy sets emerges as an essential design aspect in many algorithms of processing fuzzy information including fuzzy controllers pattern classifiers knowledge based systems etc this paper introduces a new model of matching the primordial features of the proposed model involve 1 i matching being carried out with respect to the grades of membership of the discussed fuzzy sets as well as some of their functionals like energy entropy transom 2 ii the concepts of hierarchies of matching in the model leading directly to a straightforward distinction between local and global levels of matching and 3 iii a distributed character of the model that is realized as a logic based neural network with enhanced learning capabilities

fuzzy set connectives as combinations of belief structures abstract consonant belief structures provide a representation for fuzzy sets owing to the fact that their plausibility measures are essentially possibility measures we note that two belief structures are equivalent if their plausibility and belief functions are equal this observation leads us to provide a multiple number of equivalent representations for any belief structure commensurate representations can be induced for two different belief structures by forcing the same number of focal elements with the same weights we show that if we represent two consonant belief structures in a commensurate manner then their aggregations are closed with respect to consonance provided that the additional requirement that the underlying probability distributions satisfy a condition of correlation is imposed the results of this work allow us to use belief structure representations for the manipulation of fuzzy subsets under various logical combinations all basic fuzzy set connectives can thus be interpreted in the framework of the theory of evidence

aggregating constraint satisfaction degrees expressed by possibilistic truth values in information systems one often has to deal with constraints in order to compel the semantics and integrity of the stored information or to express some querying criteria hereby different constraints can be of different importance a method to aggregate the information about the satisfaction of a finite number of constraints for a given data instance is presented central to the proposed method is the use of extended possibilistic truth values to express the degree of satisfaction of a constraint and the use of residual implicators and residual coimplicators to model the impact and relevance of a constraint the proposed method can be applied to any constraint based system a database application is discussed and illustrated

reformulating decision theory using fuzzy set theory and shafer s theory of evidence abstract utilities and probabilities in decision theory are usually assessed by asking individuals to indicate their preferences between various uncertain choices in this paper we argue that 1 the utility of a consequence can be assessed as the membership function of the consequence in the fuzzy set satisfactory 2 the probability of an event instead of being directly assessed should be inferred from the evidence associated with that event the degree of evidence is quantified using shaferian basic probability assignments in addition we use the heisenberg uncertainty principle to argue for a change in one of the technical assumptions underlying decision theory as a result of this change some kinds of evidence will be observable in certain experiments but unobservable in others since probabilities are defined over the potential outcomes of an experiment they will only be defined over some but not all the evidence as a result the probabilities associated with different experiments could be inconsistent this formulation emphasizes the importance of new distinctions and not just new information in updating probabilities we argue that this formulation addresses many of the observed empirical deviations between decision theory and experiment it also addresses the anomalies of quantum physics we close with a brief discussion of directions for further research

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

a behavioral foundation for fuzzy measures in savage 41 a behavioral foundation was given for subjective probabilities to be used in the maximization of expected utility this paper analogously gives a behavioral foundation for fuzzy measures to be used in the maximization of choquet expected utility this opens the way to empirical verification or falsification of fuzzy measures and frees them of their ad hoc character

uncertainty modelling for vague concepts a prototype theory approach an epistemic model of the uncertainty associated with vague concepts is introduced label semantics theory is proposed as a framework for quantifying an agent s uncertainty concerning what labels are appropriate to describe a given example an interpretation of label semantics is then proposed which incorporates prototype theory by introducing uncertain thresholds on the distance between elements and prototypes for description labels this interpretation naturally generates a functional calculus for appropriateness measures a more general model with distinct threshold variables for different labels is discussed and we show how different kinds of semantic dependence can be captured in this model

uncertain robot environment modelling using fuzzy numbers abstract in this paper we present a fuzzy oriented methodology to model an uncertain geometric robot environment and to manipulate geometric uncertainties between robot coordinate frames we describe any geometric primitive of robot environment as a parameter vector in parameter space not only ill known values of the parameterized geometric primitives but the uncertain quantities of coordinate transformations are represented by means of fuzzy numbers restricted to appropriate membership functions for consistent interpretation about geometric primitives between different coordinate frames we manipulate these uncertain quantities using fuzzy arithmetic as an application we deal with a 2 dimensional straight line correspondence problem of a mobile robot vision system this example shows the usefulness of the proposed methodology

fuzzy sets and statistical data abstract specific features of probability and possibility theories are discussed with emphasis on semantical aspects instead of putting forward the acknowledged usefulness of possibility theory for the non statistical modelling of subjective categories we try to figure out how statistical data and possibility theory could be matched as a result procedures for constructing weak possibilistic substitutes of probability measures and for processing imprecise statistical data are outlined they provide new insights on relationship between fuzzy sets and probability theories

fuzzy sets based interpretation of the safety factor abstract safety factor is a universally utilized concept in several branches of engineering on one hand most engineers as it were neglect uncertainty but on the other hand the allowable stress level was introduced long time ago as a ratio of the yield stress to the so called safety factor to provide the region for the safe utilization of the structure thus the uncertainty is introduced into practice by the back door this observation led to a considerable literature dedicated to the probabilistic interpretation of the safety factor the present paper deals with the novel aspect of elucidation of the concept of safety factor through the theory of fuzzy sets apparently for the first time in the literature the aim of the paper is to present the safety factor that is uniformly employed by engineers but in the new light the safety factor in the fuzzy setting is introduced the ideas are illustrated on two strength of material problems simple examples are chosen so as to allow for clearer illustration of ideas

paper evidence measures based on fuzzy information an organized body of results pertaining to the analysis of evidence when the available knowledge is pervaded with imprecision is provided the evidence theory of dempster and shafer is extended to the case of fuzzy observations and fuzzy events upper and lower possibilities of such events are derived by iterating the generation process of upper and lower probabilities as done by dempster both probabilistic and possibilistic models are developed in parallel these evidence measures are used for decision evaluation when the available knowledge is poor the classical model of decision making under uncertainty is thus extended to the case when the consequences of a decision are only roughly described and their probabilities of occurrence modeled by intervals or fuzzy numbers

network of probabilities associated with a capacity of order 2 abstract this paper studies capacities of order 2 from a geometric point of view that is from convex polytope of probabilities defined by them this convex polytope is characterized by means of a network of probabilities representing relationships among vertices defined by the edges interesting properties may be obtained about the capacities from this network

on viewing fuzzy measures as fuzzy subsets we introduce the concept of a fuzzy measure discuss some of their basic properties and look at some notable examples we introduce a correspondence between a fuzzy measure on x and a fuzzy subset over the power set of x these fuzzy subsets are required to have special properties on the space 2x and we refer to these as measure type fuzzy sets we take advantage of this correspondence between a fuzzy measure and a fuzzy set to suggest the formulation of various set operations on fuzzy measures

a stochastic interpretation of membership functions abstract a stochastic interpretation of the so called membership functions of fuzzy logic is presented the membership functions are used to define a probability space and the values of them may be interpreted as probabilities it is believed that the stochastic interpretation will facilitate the analysis and understanding of fuzzy control

using fuzzy sets in operational research abstract during the last four decades a large number of mainly mathematical tools have been developed in and for operational research they are primarily devices to find optimal solutions after a problem has been modelled formally hardly any new tools have been developed for proper modelling of problems which are not well structured and which can not easily be cast into classical mathematical models fuzzy set theory as proposed by l zadeh in 1965 promises to bridge part of this gap it is much more adaptable to different problem structures and better suited to model human evaluation and decision making processes than classical mathematics this paper will primarily focus attention on four questions why fuzzy sets in or what are fuzzy sets where can they be used and how are they being used

a computational approach to approximate and plausible reasoning with applications to expert systems the intended purpose of this paper is twofold proposing a common basis for the modeling of uncertainty and imprecision and discussing various kinds of approximate and plausible reasoning schemes in this framework together with probability different kinds of uncertainty measures credibility and plausibility functions in the sense of shafer possibility measures in the sense of zadeh and the dual measures of necessity sugeno s g fuzzy measures are introduced in a unified way the modeling of imprecision in terms of possibility distribution is then presented and related questions such as the measure of the uncertainty of fuzzy events the probability and possibility qualification of statements the concept of a degree of truth and the truth qualification of propositions are discussed at length deductive inference from premises weighted by different kinds of measures by uncertainty or by truth values in the framework of various multivalued logics is fully investigated then deductive inferences from imprecise or fuzzy premises are dealt with patterns of reasoning where both uncertainty and imprecision are present are also addressed the last section is devoted to the combination of uncertain or imprecise pieces of information given by different sources on the whole this paper is a tentative survey of quantitative approaches in the modeling of uncertainty and imprecision including recent theoretical proposals as well as more empirical techniques such as the ones developed in expert systems such as mycin or prospector the management of uncertainty and imprecision in reasoning patterns being a key issue in artificial intelligence

on differential equations with interactive fuzzy parameter via t norms abstract in this paper we study fuzzy differential equations with uncertain parameters modeled by interactive fuzzy numbers the interactivity is formalized with the help of upper semicontinuous t norms to obtain solutions to fuzzy differential equations we use two different approaches the first uses a family of differential inclusions while the second is the fuzzification of the deterministic problem using the zadeh extension principle we show that the solutions obtained by the two methods are equal interactivity among the parameters and variable of the problem are studies with respect two methods we show that a hierarchy in uncertainties of the solutions arises when we chose one of the basic t norm the minimum the t norm of the product the t norm of lukasiewicz and t norm of the drastic product to model the interaction between parameters finally to illustrate the concepts introduced in the paper we study the malthusian model with interactive parameters

stochastic convergence uniform integrability and convergence in mean on fuzzy measure spaces the paper deals with the convergence of sequences of measurable functions on fuzzy measure spaces a classical result that relates the uniform integrability plus stochastic convergence with convergence in mean is extended to this case where the additivity property of measures is not required we also investigate the sufficient and necessary conditions that fuzzy measures must satisfy in the dominated convergence theorem

an alternative approach to the handling of subnormal possibility distributions abstract in this short note a definition of the necessity measure in case of a subnormal possibility distribution already proposed by the authors is recalled this definition both satisfies the axiomatics of necessity measures and preserves the desirable inequality between possibility and necessity measures the handling of subnormal basic probability assignments in the more general framework of shafer s belief functions is also discussed

the well calibrated bayesian abstract suppose that a forecaster sequentially assigns probabilities to events he is well calibrated if for example of those events to which he assigns a probability 30 percent the long run proportion that actually occurs turns out to be 30 percent we prove a theorem to the effect that a coherent bayesian expects to be well calibrated and consider its destructive implications for the theory of coherence

reliability estimation and design with insufficient data based on possibility theory early in the engineering design cycle it is difficult to quantify product reliability or compliance to performance targets because of insufficient data or information for modeling the uncertainties design decisions are therefore based on fuzzy information that is vague imprecise qualitative linguistic or incomplete the uncertain information is usually available as intervals with lower and upper limits in this work the possibility theory is used to assess design reliability with incomplete information the possibility theory can be viewed as a variant of fuzzy set theory the formal theories to handle uncertainty are first introduced using the theoretical fundamentals of fuzzy measures a computationally efficient and accurate hybrid global local optimization approach is subsequently described for calculating the confidence level of fuzzy response the method combines the advantages of the commonly used vertex and discretization methods subsequently the possibility theory is used in design a possibility based design optimization method is proposed where all design constraints are expressed in a possibilistic way it is shown that the method gives a conservative solution compared with all conventional reliability based designs obtained with different probability distributions finally a general possibility based design optimization method which handles a combination of random and possibilistic design variables is presented several numerical examples demonstrate the application of possibility theory in design

combining probability distributions from dependent information sources inferences or decisions in the face of uncertainty should be based on all available information thus when probability distributions for an uncertain quantity are obtained from experts models or

belief functions generated by fuzzy and randomized compatibility relations the standard combinatoric model of belief functions can be generalized in two ways 1 the crisp compatibility relation is replaced by a fuzzy relation and 2 the values of crisp compatibility relation are observed through a binary information channel with noise and perhaps subjected to a deformation in other terms the original compatibility relation is replaced by its randomized modification both the generalizations are investigated separately in more detail

fuzzy set theory observer bias and probability theory abstract it is known that given certain seemingly implausible assumption probability functions act like fuzzy set membership functions this paper shows that these seemingly implausible assumptions are satisfied in probabilistic situations where the sole uncertainty arises from unknown but systematic observer bias now such situations could be considered as on the borderline between the domain of applicability of probability functions and the domain of fuzzy set membership functions hence the convergence between the rules for combining probabilistic functions and those for combining fuzzy set membership functions in these borderline instances is intuitively quite satisfying this convergence also suggests a number of other ways in which possibility theory and probability theory are interrelated

modeling vague beliefs using fuzzy valued belief structures abstract this paper presents a rational approach to the representation and manipulation of imprecise degrees of belief in the framework of evidence theory we adopt as a starting point the non probabilistic interpretation of belief functions provided by smets transferable belief model as well as previous generalizations of evidence theory allowing to deal with fuzzy propositions we then introduce the concepts of interval valued and fuzzy valued belief structures defined respectively as crisp and fuzzy sets of belief structures verifying hard or elastic constraints we then proceed with a generalization of various concepts of dempster shafer theory including those of belief and plausibility functions combination rules and normalization procedures most calculations implied by the manipulation of these concepts are based on simple forms of linear programming problems for which analytical solutions exist making the whole scheme computationally tractable we discuss the application of this framework in the areas of decision making under uncertainty and classification of fuzzy data

probabilities from fuzzy observations abstract we introduce a new definition for getting the probability of a fuzzy event from a collection of fuzzy observations this definition is based upon the use of the compatibility of the target event with the observation we show that this new definition satisfies the komolgoroff conditions we show how it naturally extends all the properties of the classical probability theory

decision making problems in a general environment abstract a general model for decision problems is presented based on the consideration of an evidence given by a basic probability assignment which is the information about states of nature decision rules are defined in two steps by means of a composition of functions firstly the lower and upper expected values provide limits to the expected utility value for each decision the function of the second step reflects the decision maker s subjectivity and lets us model it between these two boundaries by means of a convex combination the relationships between evidences and fuzzy measures are shown in section 2 and some relevants properties of lower and upper expected values are proved in section 3 in section 4 a definition of inclusion relation on the set of evidence is used to study existing relations among results which can be obtained from different basic probability assignments and finally we show the way in which our model includes the most classical criteria such as min max or mathematical expectation criteria in uncertainty or risk environments

a capacity improvement lower bound for fixed charge network design problems network design problems concern flows over networks in which a fixed charge must be incurred before an arc becomes available for use the uncapacitated multicommodity network design problem is modeled with aggregate and disaggregate forcing constraints forcing constraints ensure logical relationships between the fixed charge related and the flow related decision variables a new lower bound for this problem referred to as the capacity improvement ci bound is presented and an efficient implementation scheme using shortest path and linearized knapsack programs is described a key feature of the ci lower bound is that it is based on the lp relaxation of the aggregate version of the problem a numerical example illustrates that the ci lower bound can converge to the optimal objective function value of the ip formulation

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

on generalized fuzzy belief functions in infinite spaces determined by a fuzzy implication operator a general type of fuzzy belief structure and its induced dual pair of fuzzy belief and plausibility functions in infinite universes of discourse are first defined relationship between the belief structure based and the belief space based fuzzy dempster shafer models is then established it is shown that the lower and upper fuzzy probabilities induced by the fuzzy belief space yield a dual pair of fuzzy belief and plausibility functions for any fuzzy belief structure there must exist a fuzzy belief space such that the fuzzy belief and plausibility functions defined by the given fuzzy belief structure are just the lower and upper fuzzy probabilities induced by the fuzzy belief space respectively essential properties of the fuzzy belief and plausibility functions are also examined the fuzzy belief and plausibility functions are respectively a fuzzy monotone choquet capacity and a fuzzy alternating choquet capacity of infinite order

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

on some optimization problems under uncertainty abstract the last ten years show a diversification of the approaches for handling uncertainties in optimization problems by certain relaxations or on the contrary seeking a more conservative solution in this paper we present some new results and points of view in nonlinear inexact and fuzzy mathematical programming possible applications to capital budgeting of interrelated projects and to production planning are also discussed

medical diagnosis fuzzy sets and degrees of belief abstract a model simulating the medical diagnostic process is presented diagnostic groups are considered as fuzzy sets partial knowledge is quantified by belief functions the process is considered as the evaluation by the clinician of his degree of belief concerning the belonging of his patient to a fuzzy set given fuzzy and partial informations

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

characterization and comparison of sugeno and choquet integrals abstract starting from the concept of equiordered functions we characterize two types of fuzzy integrals that can be defined on any kind of fuzzy measure sugeno and choquet integrals the characterization theorems that we obtain allow us to carry out a comparative study of these two fuzzy integrals pointing out their formal similarities and their conceptual differences

railroad blocking a network design application in this study we formulate the railroad blocking problems as a network design problem with maximum degree and flow constraints on the nodes and propose a heuristic lagrangian relaxation approach to solve the problem the newapproach decomposes the complicated mixed integer programming problem into two simple subproblems so that the storage requirement and computational effort are greatly reduced a set of inequalities are added to one subproblem to tighten the lower bounds and facilitate generating feasible solutions subgradient optimization is used to solve the lagrangian dual an advanced dual feasible solution is generated to speed up the convergence of the subgradient method the model is tested on blocking problems from a major railroad and the results show that the blocking plans generated have the potential to reduce the railroad s operating costs by millions of dollars annually

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

advanced fuzzy inference engines in situation aware computing we focus on the very important family of context aware applications context aware computing relies on tasks like capturing sensing environmental parameters e g lightness location classifying context and inferring further knowledge about that context determine the situation the user is currently in however the relevant applications have to deal with the inherent imperfection of context sensing for decision making we propose an extension of context representation and inference for situation aware applications our model relies on fuzzy set theory to accommodate the imperfection of sensed context based on this model we have developed three fuzzy inference engines which rely on advanced semantics specialization mereological and compatibility relations we have evaluated the proposed engines through a series of experiments involving real users our findings indicate the strong points of the proposed context classification and inference processes

possibilistic information fusion using maximal coherent subsets when multiple sources provide information about the same unknown quantity their fusion into a synthetic interpretable message is often a tricky problem especially when sources are conflicting in this paper we propose to use possibility theory and the notion of maximal coherent subsets mcss often used in logic based representations to build a fuzzy belief structure that will be instrumental both for extracting useful insight about various features of the information conveyed by the sources and for compressing this information into a unique possibility distribution extentions and properties of the basic fusion rule are also studied

user preferences discovery using fuzzy models user preferences discovery aims to learn the patterns of user preferences for various services or items such as movies preferences discovery is essential to the development of intelligent personalization applications based on decision and utility theories traditional approaches to preferences discovery explicitly query users about the behavior of value function or utility of every outcome with respect to each decision criterion consequently these approaches are generally error prone and labor intensive although implicit elicitation approaches have been proposed to address the above limitations extent approaches largely ignore multi valued nature of item features and uncertainty associated with item features and user preferences to address uncertainty due to vagueness and imprecision this research proposed a general framework for preferences discovery based on fuzzy set theories in addition new fuzzy models were created for preferences discovery and representation further an algorithm was developed to predict user preferences with uncertainty and visualization of item features user feedback and the discovered preferences helped improve the interpretation of the discovered knowledge the results of the simulation evaluation using a benchmark movie dataset revealed that the proposed preference discovery method 1 doubled the accuracy of preference discovery as compared to random prediction and 2 outperformed conventional techniques in making movie recommendation these findings suggest that fuzzy models are effective for preferences patterns discovery and personalized recommendation application

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

a set of practical experiments to validate the fuzzy kalman filter abstract the fuzzy kalman filter fkf was recently reformulated with a number of basic examples included in order to demonstrate its applicability we now look ahead to its practical implementation introducing deep analysis which includes very detailed simulations and complete real applications this will allow us to not only validate the fkf but also to draw important conclusions which will help users in the adjustment of the fuzzy filter parameters and the choosing of possibilistic distributions

fuzzy inference relation based on the theory of falling shadows abstract in this paper we establish a theoretical approach to define a fuzzy inference relation based on the theory of falling shadows the main characteristic of our definition of fuzzy inference relation is that it is semantically dependent in the sense that the formula of our definition will vary according to the correlation of the antecedent and the consequence of the given implication we shall show that the formulae of fuzzy inference relation given by ukasiewicz zadeh and the probability formula are consequences of our definition under three different correlations of the propositions