canonical fuzzy numbers of dimension two abstract recently several authors have dealt with interactive fuzzy numbers this paper proposes a new approach to interactions between fuzzy numbers by using canonical fuzzy numbers of dimension two

a new framework for the bayesian analysis of single stage decision problems with imprecise utilities a mathematical framework to model the bayesian analysis of single stage decision problems with imprecise utilities is proposed the main advantage of this model with respect to previous models for such a problem is that product measurability of the utility function is not necessary since this model involves iterated expectations instead of an integral over a product space conditions for the equivalence between the extensive and normal forms of the bayesian analysis within the proposed framework are obtained the model is illustrated with an example in which the utility function is not product measurable

on characterization of fuzzy vectors and its applications to fuzzy mathematical programming problems in this paper we consider fuzzy mathematical programming problems which involve fuzzy vectors only in the objective function first we define fuzzy vectors on n dimensional euclidean space r n and investigate their properties especially we extend the concepts of the fuzzy max order and the possibility and the necessity measures given in fuzzy number to those of fuzzy vectors next based on the results we consider fuzzy mathematical programming problems including fuzzy multicriteria optimization problems which involve fuzzy vectors only in the objective function and we give the concepts of optimal solutions and characterize their properties

triangular norm based addition of fuzzy intervals triangular norm based addition of fuzzy intervals on the extended real line is reduced to the addition of shapes on 0 the addition of fuzzy intervals including lr fuzzy intervals lr fuzzy numbers lt and tr fuzzy numbers based on limit t norms is recalled for continuous archimedean t norms special types of incoming summands allowing the exact output computation are introduced several examples are given

associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

fuzzy vectors as a tool for modeling uncertain multidimensional quantities the paper deals with modeling uncertain multidimensional quantities by means of fuzzy vectors in connection with possible interactions among variables the separability of fuzzy vectors is discussed in detail for the case when interconnections among variables are given by a crisp relation a more general form of separability of fuzzy vectors the separability on a given relation e g on a probability simplex is introduced properties of fuzzy vectors separable on a given relation are studied and ways of convenient setting of such fuzzy vectors are proposed finally fuzzy extensions of real valued functions and real vector valued functions to general input fuzzy vectors are investigated

chebyshev type inequalities for interval valued functions abstract we introduce the interval double integral for interval valued functions and give some fundamental properties further we establish some continuous and discrete chebyshev type inequalities and some interesting equalities for interval valued functions also some examples are presented to illustrate our results

cpk index estimation using fuzzy numbers process capacity indices pcis were developed and have been successfully used by companies to compete in and dominate the high profit markets by improving the quality and the productivity since the past two decades there is an essential assumption in the conventional application wherein the output process measurements are precise and distributed as normal random variables since the assumption of normal distribution is untenable errors can occur if the cpk index is computed using non normal data in the present study we address the situation that the output of data from measurement of the quality of a product is insufficiently precise or scarce this is possible when the quality measurement refers to the decision maker s subjective determination in such a situation the linguistic variable that is easier to capture the decision maker s subjective perception is applied to construct the pci cpk the present approach can mitigate the effect when the normal assumption is inappropriate and extends the application of cpk index

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

inferring a possibility distribution from empirical data several transformations from probabilities to possibilities have been proposed in particular dubois and prade s procedure produces the most specific possibility distribution among the ones dominating a given probability distribution in this paper this method is generalized to the case where the probabilities are unknown the only information being a data sample represented by a histogram it is proposed to characterize the probabilities of the different classes by simultaneous confidence intervals with a given confidence level email protected from this imprecise specification a procedure for constructing a possibility distribution is described insuring that the resulting possibility distribution will dominate the true probability distribution in at least 100 email protected of the cases finally a simple efficient algorithm is given which makes the computations tractable even if the number of classes is high

a convergence of geometric mean for t related fuzzy numbers we study a convergence of geometric mean for t related fuzzy numbers where t is an archimedian t norm in particular we show that the class of archimedian t norms can be characterized by the validity of a general convergence of geometric mean for sequences of l r fuzzy numbers

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

real time constrained fuzzy arithmetic klir introduced constrained fuzzy arithmetic cfa as a solution to the unnecessary precision loss when dealing with fuzzy quantities that represent linguistic variables since then some attempts have been made to make cfa efficient but none of these solutions is suitable for real time applications in this paper we will propose a new cfa algorithm that can be used in such environments

fuzzy arithmetic with requisite constraints abstract the purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables these constraints referred to as requisite constraints represent additional information when they are ignored as in the standard fuzzy arithmetic the obtained results exhibit in general information deficiency it is thus important as argued in the paper to revise fuzzy arithmetic to take relevant requisite constraints into account some typical requisite constraints are overviewed with a more extensive examination of the requisite equality constraint crisp or fuzzy

falscal a fuzzy multidimensional scaling algorithm the conventionally adopted alternating least squares scaling alscal procedure of multidimensional scaling mds is a valuable mathematical scheme for analyzing data in areas where organized concepts and underlying dimensions are inadequately defined or developed fuzzy set theory fst attempts to formulate human reasoning and perceptions therefore targeting problems in areas where human factors significantly impact the result of decision making to our knowledge the fst and alscal approaches have not yet been integrated this study integrates and modifies the fst and alscal procedures fuzzy data collected from fuzzy questionnaires are adopted as the input of the mds ensuring that the uncertainty of input data can be incorporated into the analysis the conventionally adopted alscal procedure is then modified to cope with fuzzy input data by adopting the notion of fuzzy distances fuzzy disparities and fuzzy ranking to represent the similarities between fuzzy data related approximation operations of the triangular fuzzy number are also introduced to facilitate computation in fuzzy alscal

t sum of l r fuzzy numbers abstract the sum of l r fuzzy numbers based on a given archimedian continuous t norm t is studied a series of improving sufficient conditions of several authors ensuring an analytical form of the output sum is completed by a sufficient and necessary condition i e the best sufficient condition some applications are indicated

constructing the exact significance level for a person fit statistic this paper uses an extension of the network algorithm originally introduced by mehta and patel to construct exact tail probabilities for testing the general hypothesis that item responses are distributed according to the rasch model by assuming that item difficulties are known the algorithm is applicable to the statistical tests either given the maximum likelihood ability estimate or conditioned on the total score a simulation study indicates that the network algorithm is an efficient tool for computing the significance level of a person fit statistic based on test lengths of 30 items or less

on possibilistic correlation in 2004 fuller and majlender introduced the notion of covariance between fuzzy numbers by their joint possibility distribution to measure the degree to which they interact based on this approach in this paper we will present the concept of possibilistic correlation representing an average degree of interaction between marginal distributions of a joint possibility distribution as compared to their respective dispersions moreover we will formulate the classical cauchy schwarz inequality in this possibilistic environment and show that the measure of possibilistic correlation satisfies the same property as its probabilistic counterpart in particular applying the idea of transforming level sets of possibility distributions into uniform probability distributions we will point out a fundamental relationship between our proposed possibilistic approach and the classical probabilistic approach to measuring correlation

fuzzy differential equations with interactive derivative in this paper we introduce and study new concept of differentiability for fuzzy set valued functions this derivative considers possible local interactivity in the process studied several properties of differentiability and integrability are investigated for the new concept and they are compared to similar fuzzy differentiabilities like hukuhara differentiability and generalized hukuhara differentiability furthermore we establish theorems as the fundamental theorem of calculus ultimately we exhibit some results for fuzzy initial value problem and an application

the case for an interval based representation of linguistic truth abstract this paper develops an interval based approach to the concept of linguistic truth a special purpose interval logic is defined and it is argued that for many applications this logic provides a potentially useful alternative to the conventional fuzzy logic the key idea is to interpret the numerical truth value v p of a proposition p as a degree of belief in the logical certainty of p in which case p is regarded as true for example if v p falls within a certain range say the interval 0 7 1 this leads to a logic which although being only a special case of fuzzy logic appears to be no less linguistically correct and at the same time offers definite advantages in terms of mathematical simplicity and computational speed it is also shown that this same interval logic can be generalized to a lattice based logic having the capacity to accommodate propositions p which employ fuzzy predicates of type 2

on the convergence of t sum series of fuzzy numbers abstract in this paper we study the existence of the limit of the series of fuzzy numbers with different spreads and different shape functions where addition is defined by the sup t norm and we show the uniform continuity of the limit this generalizes the earlier results of hong fuzzy sets and systems 77 1996 253 254

on product sum of triangular fuzzy numbers abstract we study the problem if a i i n are fuzzy numbers of triangular form then what is the membership function of the infinite or finite sum a 1 a 2 defined via the sup product norm convolution

necessary and sufficient conditions for the equality of interactive and non interactive extensions of continuous functions abstract in this contribution we find the class of n dimensional joint possibility distributions with the property that the interactive extension principle coincides with the non interactive extension principle as long as the interactive operations are determined by continuous functions strictly increasing in each argument this result completes recent studies by the authors where the particular case of interactive additions and multiplications versus non interactive additions and multiplications were investigated in addition this time we propose results that also cover the cases when we know the fuzzy numbers only from their membership functions it means that we eliminated the limitations that appear when we cannot pass from membership function representation to parametric representation of fuzzy numbers as important new applications we mention the study on the completely correlated fuzzy numbers also of note is that we propose two simple methods to extend bidimensional joint possibility distributions to n dimensional joint possibility distributions one method is based on an inductive construction while the other one is based on a pairwise construction

a note on fuzzy time series model recently song et al fuzzy sets and systems 73 1995 341 348 proposed a homogeneous fuzzy time series model to model dynamic process whose observations are fuzzy sets or linguistic values by means of defining some new operations on fuzzy numbers in this paper we considered expanding the results to the nonhomogeneous fuzzy time series and the general fuzzy time series by using the weakest t norm based algebraic fuzzy operations and solved the open problem suggested by song et al in their conclusion of the paper

the module structure of a group action on a polynomial ring a finiteness theorem consider a group acting on a polynomial ring over a finite field we study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries for example any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite there are also applications to invariant theory giving a priori bounds on the degrees of the generators

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

modeling uncertain variables of the weighted average operation by fuzzy vectors the paper deals with the fuzzy extension of the weighted average operation first we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors we show that in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values next we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights we derive a general formula and we study its special forms the advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation this fact is illustrated by several examples finally we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average

a refined fuzzy time series model for forecasting abstract fuzzy time series models have been used to model observations where each one of them contains multiple values the formulation of fuzzy relationships and the lengths of intervals are considered to be two of the critical factors that affect forecasting results unfortunately the lengths of the intervals were determined during the early stages of forecasting in these models and they thus often failed to reflect the distribution of observations this study therefore proposes a refined fuzzy time series model to further refine the lengths of intervals this model can refine the lengths of intervals during the formulation of fuzzy relationships and hence capture the fuzzy relationships more appropriately as a result the forecasting results can be improved both the stock index and enrollment are used as the targets in the empirical analysis

defuzzification of fuzzy intervals abstract this paper deals with defuzzification of fuzzy intervals classified as regular or irregular based on the defuzzification method used a fuzzy interval is regular if its membership function is maximized at its defuzzified value otherwise the fuzzy interval is irregular three methods for defuzzification are presented in this paper 1 mean of maxima mom method 2 center of area coa method and 3 fuzzy mean fm method fuzzy intervals with symmetrical membership function are regular and all three defuzzification methods give the same results for any regular fuzzy interval based on the coa method its defuzzified value is equal to the middle point of the corresponding mean value interval for irregular fuzzy intervals two approaches are proposed one is to apply the defuzzification method but relax the reasonable but not necessary rnn requirement another is to apply the rnn requirement but relax the defuzzification method this paper shows that the regularity of a fuzzy interval is preserved after applying an arithmetic operation with a nonzero real number the formulas for calculating the defuzzified value of the arithmetic operations between a regular fuzzy interval and a real number are derived this paper also shows that the regularity of two fuzzy intervals based on the coa method is preserved after applying addition or subtraction operation between them the formulas for calculating the defuzzified value of the sum and the difference of regular fuzzy intervals are derived the properties of fuzzy intervals discussed in this paper can be taken into consideration before or when a defuzzification method is selected

a t sum bound of lr fuzzy numbers abstract in this paper we provide an upper bound and a lower bound of t sum of lr fuzzy numbers with different spreads where t is archimedean t norm and also show in three examples how close they are to actual membership functions furthermore we study when the membership function of t sum achieves the upper bound

the law of large numbers for fuzzy processes and the estimation problem abstract we study how the law of large numbers can be extended to the case of fuzzy numbers we work with l fuzzy sets where l is a cl commutative monoid with an operation we define the independence of fuzzy numbers we find when is the operation of the lattice that the arithmetical mean of independent fuzzy numbers does not converge we think that this case could be a limit one we show for particular cases of the operation that generally convergence holds we build estimators for fuzzy models we particularly consider the minimum fuzziness estimator we study the convergence of these estimators and compare them by way of the fuzzy subsets they induce we lastly study the case where we have dependent fuzzy numbers and particularly the case of linearly filtered autoindependent fuzzy processes

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

on computation of the compositional rule of inference under triangular norms abstract this paper is devoted to the derivation of exact calculation formulas for the compositional rule of inference under archimedean t norms when both the observation and the relation parts are given by hellendoorn s function 6

equivalent conditions for laws of large numbers for t related l r fuzzy numbers in this paper we give some necessary and sufficient conditions for laws of large numbers for sequence of mutually t related l r fuzzy numbers when t is an continuous archimedean t norm and diameter of support of the fuzzy numbers is not uniformly bounded we also consider some necessary and sufficient conditions for laws of large numbers for l r fuzzy random numbers and generalize earlier results of fuller fuzzy sets and systems 45 1992 299 303 triesch fuzzy sets and systems 58 1993 339 342 and hong and lee fuzzy sets and systems 121 2001 537 543

decomposition and construction of fuzzy sets and their applications to the arithmetic operations on fuzzy quantities abstract the membership function of a fuzzy set can be constructed from a family of subsets of a universal set via the form of resolution identity assuming that the family is nested in this paper we are going to consider the weak case of assuming that only a countable subfamily is nested using this generalized consideration we can overcome the ambiguity of arithmetic operations on fuzzy quantities two well known additions of fuzzy quantities in vector space have been adopted in the literature one is based on the extension principle by directly considering the membership functions of course this addition always exists another one uses the level sets without considering the membership functions however the existence of this addition cannot be guaranteed in this paper we are going to propose two other additions of fuzzy quantities in vector space one generalizes the extension principle and the other one also uses the level sets without considering the membership functions such that this generalized addition always exists we can show that these four additions are equivalent

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

t sum of bell shaped fuzzy intervals the usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of zadeh s extension principle based on a t norm t a t norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation if this arithmetic operation is closed for this class it is important to know which t norms are consistent with a particular type of fuzzy intervals recently dombi and gyorbiro proved that addition is closed if the dombi t norm is used with two bell shaped fuzzy intervals a result proved by mesiar on a strict t norm based shape preserving additions of lr fuzzy intervals with unbounded support is recalled as applications we define a broader class of bell shaped fuzzy intervals then we study t norms which are consistent with these particular types of fuzzy intervals dombi and gyorbiro s results are special cases of the results described in this paper

modular invariants of a vector and a covector a proof of a conjecture of bonnafe and kemper abstract consider a finite dimensional vector space v over a finite field f q we give a minimal generating set for the ring of invariants f q v v gl v and show that this ring is a gorenstein ring but is not a complete intersection these results confirm a conjecture of bonnafe and kemper 3 conjecture 3 1

on the convergence of product sum series of l r fuzzy numbers we study series a1 a2 an of l r fuzzy numbers where addition is defined via the sup product norm convulation and obtain some convergence theorems which generalize earlier results by fuller cf 2

some algebraic properties of weakly non interactive fuzzy numbers abstract this study focuses on the concept of sup t norm convulution a generalized version of zadeh s extension principle the authors treat the algebraic structure of fuzzy arithmetic based on the sup t norm convolution and especially show some new properties comparing them with those of the conventional fuzzy arithmetic based on sup min convolution

a note on the convergence of t sum series of l r fuzzy numbers abstract we prove the existence of the limit of series a 1 a 2 a n of l r fuzzy numbers where addition is defined via the sup t norm convolution and show uniform continuity of the limit this generalizes the earlier result by triesch 1993

weighted likelihood estimation of ability in item response theory applications of item response theory which depend upon its parameter invariance property require that parameter estimates be unbiased a new method weighted likelihood estimation wle is derived and proved to be less biased than maximum likelihood estimation mle with the same asymptotic variance and normal distribution wle removes the first order bias term from mle two monte carlo studies compare wle with mle and bayesian modal estimation bme of ability in conventional tests and tailored tests assuming the item parameters are known constants the monte carlo studies favor wle over mle and bme on several criteria over a wide range of the ability scale

the many null distributions of person fit indices this paper deals with the situation of an investigator who has collected the scores ofn persons to a set ofk dichotomous items and wants to investigate whether the answers of all respondents are compatible with the one parameter logistic test model of rasch contrary to the standard analysis of the rasch model where all persons are kept in the analysis and badly fittingitems may be removed this paper studies the alternative model in which a small minority ofpersons has an answer strategy not described by the rasch model such persons are called anomalous or aberrant from the response vectors consisting ofk symbols each equal to 0 or 1 it is desired to classify each respondent as either anomalous or as conforming to the model as this model is probabilistic such a classification will possibly involve false positives and false negatives both for the rasch model and for other item response models the literature contains several proposals for a person fit index which expresses for each individual the plausibility that his her behavior follows the model the present paper argues that such indices can only provide a satisfactory solution to the classification problem if their statistical distribution is known under the null hypothesis that all persons answer according to the model this distribution however turns out to be rather different for different values of the person s latent trait value this value will be called ability parameter although our results are equally valid for rasch scales measuring other attributes

a note to the addition of fuzzy numbers based on a continuous archimedean t norm abstract a simple method of computing the t sum of special types of fuzzy numbers is introduced for the lukasiewicz t norm tl the proposed method can be applied to the addition of fuzzy numbers with convex membership functions the classes of fuzzy numbers leading to t idempotents are studied for continuous t norms

saddlepoint approximations of the distribution of the person parameter in the two parameter logistic model large sample theory states the asymptotic normality of the maximum likelihood estimator of the person parameter in the two parameter logistic 2pl model in short tests however the assumption of normality can be grossly wrong as a consequence intended coverage rates may be exceeded and confidence intervals are revealed to be overly conservative methods belonging to the higher order theory more specifically saddlepoint approximations are a convenient way to deal with small sample problems confidence bounds obtained by these means hold the approximate confidence level for a broad range of the person parameter moreover an approximation to the exact distribution permits to compute median unbiased estimates mue that are as likely to overestimate as to underestimate the true person parameter additionally in small samples these mue are less mean biased than the often used maximum likelihood estimator

maximum and minimum between fuzzy symbols in non interactive and weakly non interactive situations abstract we define fuzzy symbols as particular fuzzy sets whose membership functions operate between two linearly ordered spaces and study the operations of maximum and of minimum between two fuzzy symbols we consider the membership functions of the fuzzy symbols as possibility distributions we study those of the maximum and the minimum of two non interactive and weakly non interactive variables

on various approaches to normalization of interval and fuzzy weights abstract the paper deals with the problem of fuzzification of the procedure of normalization of weights first the existing methods for normalization of interval and fuzzy weights are reviewed second we study the problem of normalization of a fuzzy vector of weights that expresses the joint possibility distribution of initial weights we show that a correct way is to apply the extension principle proposed by zadeh since the result of such normalization is the fuzzy vector of normalized weights that expresses the true joint possibility distribution of normalized weights further we establish some properties of this approach to normalization that are important from the point of view of real applications finally since an n tuple of non interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind we investigate normalization of such kind of fuzzy vectors of weights according to the extension principle we show that from the point of view of the way of modelling uncertain normalized weights the result of this approach can be directly compared only with the result of normalization proposed by wang and elhag 2006 35 we find out that it is not sufficient to express the result of normalization only by an n tuple of normalized interval or fuzzy weights together with the constraint that the sum of the weights is equal to 1 since it can cause a false increase of uncertainty in the model this fact is illustrated by an example

fuzzy sets and their operations ii among the basic operations which can be performed on fuzzy sets are the operations of union intersection complement algebraic product and algebraic sum in addition to these operations new operations called bounded sum and bounded difference were defined by l a zadeh to investigate the fuzzy reasoning which provides a way of dealing with the reasoning problems which are too complex for precise solution this paper investigates the algebraic properties of fuzzy sets under these nex operations of bounded sum and bounded difference and the properties of fuzzy sets in the case where these new operations are combined with the well known operations of union intersection algebraic product and algebraic sum

fuzzy probability vectors abstract several recent papers have discussed an extension of decision analysis both the rewards and probabilities are modelled using fuzzy sets however arbitrary specification of the membership functions of the fuzzy probabilities in these models may lead to internal inconsistencies to avoid this the fuzzy beta possibility distribution is introduced

regranulation a granular algorithm enabling communication between granular worlds abstract in this paper we describe a granular algorithm for translating information between two granular worlds represented as fuzzy rulebases these granular worlds are defined on the same universe of discourse but employ different granulations of this universe in order to translate information from one granular world to the other we must regranulate the information so that it matches the information granularity of the target world this is accomplished through the use of a first order interpolation algorithm implemented using linguistic arithmetic a set of elementary granular computing operations we first demonstrate this algorithm by studying the common fuzzy pd rulebase at several different granularities and conclude that the 3 3 granulation may be too coarse for this objective we then examine the question of what the natural granularity of a system might be this is studied through a 10 fold cross validation experiment involving three different granulations of the same underlying mapping for the problem under consideration we find that a 7 7 granulation appears to be the minimum necessary precision

a two stage logarithmic goal programming method for generating weights from interval comparison matrices a two stage logarithmic goal programming tlgp method is proposed to generate weights from interval comparison matrices which can be either consistent or inconsistent the first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency the weights are assumed to be multiplicative rather than additive in the case of hierarchical structures a nonlinear programming method is used to aggregate local interval weights into global interval weights a simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process the proposed tlgp is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using a level sets and the extension principle six numerical examples including a group decision analysis problem with a group of comparison matrices a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods comparisons with other existing procedures are made whenever possible

the ump exact test and the confidence interval for person parameters in irt models in educational and psychological measurement when short test forms are used the asymptotic normality of the maximum likelihood estimator of the person parameter of item response models does not hold as a result hypothesis tests or confidence intervals of the person parameter based on the normal distribution are likely to be problematic inferences based on the exact distribution on the other hand do not suffer from this limitation however the computation involved for the exact distribution approach is often prohibitively expensive in this paper we propose a general framework for constructing hypothesis tests and confidence intervals for irt models within the exponential family based on exact distribution in addition an efficient branch and bound algorithm for calculating the exact p value is introduced the type i error rate and statistical power of the proposed exact test as well as the coverage rate and the lengths of the associated confidence interval are examined through a simulation we also demonstrate its practical use by analyzing three real data sets

on the algebra of interactive fuzzy numbers the continuous case abstract in 1 we discussed default methods involving an approximate joint distribution or the extension principle to calculated m n m n and evaluate m n m n etc for positively and negatively associated discrete fuzzy numbers m and n this paper continues that research where now m and n are continuous fuzzy numbers we also discuss default procedures for extended max and min

oblique fuzzy vectors and their use in possibilistic linear programming in this paper we propose oblique fuzzy vectors to treat the interactivity among fuzzy numbers oblique fuzzy vectors are extensions of fuzzy numbers and vectors of non interactive fuzzy numbers the interactivity among fuzzy numbers can be treated by a non singular matrix in an oblique fuzzy vector we discuss characterization of an oblique fuzzy vector and the tractability of manipulation of oblique fuzzy vectors in fuzzy linear functions moreover we discuss possibilistic linear programming problems with oblique fuzzy vectors it is shown that the possibilistic linear programming problems are reduced to linear programming problems with a special structure

novel weighted averages versus normalized sums in computing with words in this paper some properties of novel weighted averages that are related to the concepts of possibility theory are examined it is shown that novel weighted averages have certain interpretations in terms of addition of interactive interval or fuzzy constraints to do this alternative forms of novel weighted averages are provided in particular an alternative form of novel weighted averages represented by the extension principle is determined it is shown that when fuzzy set models of words are obtained by collecting data from subjects in a computing with words setting interactive addition of fuzzy sets is not a well defined method and the optimization problems related to it may have no solutions although interactive addition is recommended in the literature for solving multicriteria decision making problems and for dealing with uncertain probabilities on the other hand novel weighted averages perform a specific normalization that guarantees that they always exist

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

on shape preserving additions of fuzzy intervals abstract we prove an open problem suggested by mesiar 1997 fuzzy sets and systems86 73 78 concerning continuous t norm based additions preserving the lr fuzzy intervals

optimal and most exact confidence intervals for person parameters in item response theory models the common way to calculate confidence intervals for item response theory models is to assume that the standardized maximum likelihood estimator for the person parameter is normally distributed however this approximation is often inadequate for short and medium test lengths as a result the coverage probabilities fall below the given level of significance in many cases and therefore the corresponding intervals are no longer confidence intervals in terms of the actual definition in the present work confidence intervals are defined more precisely by utilizing the relationship between confidence intervals and hypothesis testing two approaches to confidence interval construction are explored that are optimal with respect to criteria of smallness and consistency with the standard approach

a convexity problem and a new proof for linearity preserving additions of fuzzy intervals in this paper we propose a new sufficient condition for convexity of function this result provides a simple new proof for linearity preserving additions of fuzzy intervals obtained by the author in 2002 which was a conjecture suggested by mesiar in 1997

unbiased estimators of ability parameters of their variance and of their parallel forms reliability given known item parameters unbiased estimators are derived i for an examinee s ability parameter and for his proportion correct true score ii for the variances of and across examinees in the group tested and iii for the parallel forms reliability of the maximum likelihood estimator hat theta

power divergence family of statistics for person parameters in irt models we generalize the power divergence pd family of statistics to the two parameter logistic irt model for the purpose of constructing hypothesis tests and confidence intervals of the person parameter the well known score test statistic is a special case of the proposed pd family we also prove the proposed pd statistics are asymptotically equivalent and converge in distribution to chi 1 2 in addition a moment matching method is introduced to compare statistics and choose the optimal one within the pd family simulation results suggest that the coverage rate of the associated confidence interval is well controlled even under small sample sizes for some pd statistics compared to some other approaches the associated confidence intervals exhibit smaller lengths while maintaining adequate coverage rates the utilities of the proposed method are demonstrated by analyzing a real data set

two families of fuzzy integrals abstract in this paper we present two definitions of fuzzy integrals arising from the t seminorm and the t semiconorm concepts we show that sugeno s fuzzy integral is a particular case of them and we study their principal properties finally we suggest some applications of these results

the use of fuzzy variables in pert abstract in this paper we present a method called fpert for estimating a project completion time in the situation when activity duration times in the project network model are given in the form of fuzzy variables fuzzy sets on time space theoretical foundations of the method as well as results of calculations derived from a simple example are included

exact and best confidence intervals for the ability parameter of the rasch model a commonly used method to evaluate the accuracy of a measurement is to provide a confidence interval that contains the parameter of interest with a given high probability smallest exact confidence intervals for the ability parameter of the rasch model are derived and compared to the traditional asymptotically valid intervals based on the fisher information tables of the exact confidence intervals termed clopper pearson intervals can be routinely drawn up by applying a computer program designed by and obtainable from the author these tables are particularly useful for tests of only moderate lengths where the asymptotic method does not provide valid confidence intervals

some results on the addition of fuzzy intervals abstract a simple new method of computing t sum of fuzzy intervals having the same results as the sum of fuzzy intervals based on the weakest t norm t w is introduced this work extends that of mesiar fuzzy sets and systems 91 1997 231 237 and markova fuzzy sets and systems 91 1997 253 258

parameter estimations of mutually t related fuzzy variables abstract cai fuzzy sets and systems 55 1993 179 185 discussed parameter estimation methods for normal fuzzy variables in nahmias sense in this paper we investigate more the parameter estimations of general t related not necessarily normal fuzzy variables

on a canonical representation of fuzzy numbers abstract fuzzy numbers and more generally linguistic values are approximate assessments given by experts and accepted by decision makers when obtaining more accurate values is impossible or unnecessary to simplify the task of representing and handling fuzzy numbers several authors have introduced real indices in order to capture the information contained in a fuzzy number in this paper we propose two parameters value and ambiguity for this purpose we use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision making problems several examples illustrate these ideas

fuzzy point estimation and its application on fuzzy supply chain analysis this work describes fuzzy point estimation that reflects the subjective measurement of a real number or an interval of real numbers and a new set of definitions for arithmetic operations on fuzzy points in today s changing business world there is a growing need to manage the supply chains of products to allow businesses to offer competitive prices to their customers managing the supply chains is becoming more complex because of market uncertainties this paper provides a new approach to model the uncertainties involved in the supply chain management using the fuzzy point estimation this work provides a clearer description and analysis framework mathematically for supply chain management based on the stationary strategy established by the author the order up to levels may be calculated by means of formulae directly under a certain stationary condition and then the optimal solution may be gotten for a stationary supply chain this paper is intended to open a window toward the complete results for the analysis of non stationary supply chains

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

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

a behavioural model for vague probability assessments i present an hierarchical uncertainty model that is able to represent vague probability assessments and to make inferences based on them this model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty and is based on walley s theory of imprecise probabilities it is formally closely related to zadeh s fuzzy probabilities but it has a different interpretation and a different calculus through rationality coherence arguments the hierarchical model is shown to lead to an imprecise first order uncertainty model that can be used in decision making and as a prior in statistical reasoning

the ring of upper triangular invariants as a module over the dickson invariants in this paper we study the ring of invariants hn of un fp the upper triangular group with l s on the main diagonal as a module over the ring of invariants d of gin fp the full general linear group over the finite field fp for p a prime both groups acting as algebra automorphisms of the polynomial algebra fp xl xn both of these rings of invariants are known to be polynomial algebras hn fp hl hn and dn fp dl d see sect 3 the latter ring of invariants is known as the ring of dickson invariants now d serves as a homogeneous system of parameters for h in other words h is a finitely generated module over d and since am is a polynomial algebra and hence cohen macaulay h is in fact a free dn module see sect 2 in this situation we have a lot of information concerning this module structure in particular there is a poincar polynomial which gives the number and dimension of the elements in a free basis this polynomial admits a factorization sect 5 which suggests a free basis of a certain form and we are able to show that the suggested basis is indeed a basis see theorem 6 3 once we have a free basis we may define a natural epimorphism of d modules hn dn as follows any free basis contains 1 since dn c h and so any polynomial h c h admits a unique expression h d 9 1 f where d e d and f is a dn linear combination of the non trivial basis elements we define h d given our particular choice of basis we think of as rewriting since the value of on h is calculated from the unique expression for h in terms of our basis on the other hand it is well known that u fp is a p sylow subgroup of gl fp since this is the case we may define another natural epimorphism of d modules p hn d by averaging h hn over a set of coset