comparison of fuzzy numbers based on the probability measure of fuzzy events abstract most approaches for ranking fuzzy numbers proposed in the literature are based on fuzzy sets theory only and suffer from lack of discrimination and occasionally conflict with intuition it is true that fuzzy numbers are frequently partial order and cannot be compared however this does not alleviate the need for comparison in practical applications in this paper the order of fuzzy numbers are determined based on the concept of probability measure of fuzzy events due to zadeh it considers both the mean and dispersion of alternatives and gives two groups of indices based on the uniform and the proportional probability distributions the approach is also extended to the comparison of random fuzzy numbers by means of a mean fuzzy number it is shown that several comparison indices in the literature can be obtained based on the probability present measure approach finally some typical examples are used to compare the various different approaches the different interpretations of the dispersion index under different physical situations are emphasized

strong laws of large numbers for adapted arrays of set valued and fuzzy valued random variables in banach space in this paper we establish some strong laws of large numbers for double adapted arrays of set valued and fuzzy valued random variables in separable banach space our results are obtained with and without conditionally compactly uniformly integrable hypothesis and they are more general than some related results

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

a strong law of large numbers for fuzzy random sets abstract an slln for fuzzy random variables in rp including other convergence theorems has been proved by klement puri and ralescu the present paper is motivated by the definitions and properties developed by puri and ralescu and concerned with an slln for banach spaced valued fuzzy random sets as an easy application of the previous results

ranking fuzzy numbers with integral value abstract ranking fuzzy numbers is important in decision making since very often the alternatives are evaluated by fuzzy numbers in a vague environment a comparison between these fuzzy numbers is indeed a comparison between alternatives this paper proposes a method of ranking fuzzy numbers with integral value the method which is independent of the type of membership functions used and the normality of the functions can rank more than two fuzzy numbers simultaneously it is relatively simple in computation especially in ranking triangular and trapezoidal fuzzy numbers further an index of optimism is used to reflect the decision maker s optimistic attitude discussion on comparative advantages is included

an approach to generalize laws of large numbers for fuzzy random variables the formulation of strong and weak laws of large numbers for fuzzy random variables based on variance is given the concept of the variance of fuzzy random variables introduced by korner fuzzy sets and systems 92 1997 83 and by feng fuzzy sets and systems 120 2001 487 demonstrates its efficiency

a two phase multi attribute decision making approach for new product introduction this study aims at improving the quality and effectiveness of decision making in new product introduction new product development has long been recognized as one of the corporate core functions to be competitive on an increasingly competitive global market however developing new products is a process involving risk and uncertainty in order to solve this stochastic problem companies need to evaluate their new product initiatives carefully and make accurate decisions for this reason a systematic decision process for selecting more rational new product ideas is proposed basically two stages of decision making are described the identification of nondominated new product candidates and the selection of the best new product idea these stages are composed of an integrated approach based on a fuzzy heuristic multi attribute utility method and a hierarchical fuzzy topsis method finally an application is given to demonstrate the potential of the methodology

a probabilistic approach to rank complex fuzzy numbers abstract techniques for ranking simple fuzzy numbers are abundant in the literature however we lack efficient methods for comparing complex fuzzy numbers that are induced by arithmetic operations in this paper a probabilistic approach is taken membership functions of fuzzy numbers are first converted into probability density functions the mellin transform is then introduced to compute the mean and the variance of a complex fuzzy number the fuzzy number with the higher mean is ranked higher if the means are equal the one with the smaller variance is judged higher rank two numerical examples and a fuzzy multiple attribute decision analysis are illustrated

reducing weight flexibility in fuzzy dea an important outcome of assessing relative efficiencies within a group of decision making units dmus in fuzzy data envelopment analysis is a set of virtual multipliers or weights accorded to each input or output factor taken into account these sets of weights are typically different for each of the participating dmus and in some cases it may be considered unacceptable that the same factor is accorded widely differing weights thus it is important to find a common set of weights csw across the set of dmus in this paper by assessing upper bounds on factor weights and compacting the resulted intervals a csw is determined since resulted efficiencies by the proposed csw are fuzzy numbers rather than crisp values it is more informative for decision maker

analysis and simulation of fuzzy queues abstract the aim of this paper is to combine the ability of fuzzy sets to represent practical situations with the well known established traditional queuing approaches which are rigorous but the assumptions are frequently too far from reality it is shown that this approach can utilize the advantages of both the fuzzy and probability approaches to make the model more realistic and less restrictive both cut sets and the use of random variables to represent fuzzy numbers are investigated trapezoidal fuzzy numbers are used and the analytical and simulation results are discussed based on actual examples

strong laws of large numbers for arrays of rowwise independent random compact sets and fuzzy random sets in this paper we obtain some strong laws of large numbers sllns for arrays of rowwise independent not necessary identically distributed random compact sets and fuzzy random sets whose underlying spaces are separable banach spaces

on the formalization of fuzzy random variables abstract in this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable this discussion is mainly focused on finding an adequate notion of measurability to be coherent with the notions on the space these random elements take values

the revised method of ranking fuzzy numbers with an area between the centroid and original points in 2002 chu and tsao proposed a method to rank fuzzy numbers they employed an area between the centroid and original points to rank fuzzy numbers however there were some problems with the ranking method in this paper we want to indicate these problems of chu and tsao s method and then propose a revised method which can avoid these problems for ranking fuzzy numbers since the revised method is based on the chu and tsao s method it is easy to rank fuzzy numbers in a way similar to the original method

ranking fuzzy numbers with an area method using radius of gyration ranking fuzzy numbers plays an very important role in linguistic decision making and some other fuzzy application systems many methods have been proposed to deal with ranking fuzzy numbers chu pointed out some shortcomings of the existing distance method and proposed to rank the fuzzy numbers with the area between the centroid point and original point however drawbacks are also found in the area method for example it cannot rank fuzzy numbers when some fuzzy numbers have the same centroid point in this paper we propose a modified area method to rank fuzzy numbers the modified method can effectively rank various fuzzy numbers and their images we also used some comparative examples to illustrate the advantage of the proposed method

ranking of fuzzy sets based on the concept of existence abstract various approaches have been proposed for the comparison or ranking of fuzzy sets however due to the complexity of the problem a general method which can be used for any situation still does not exist this paper formalizes the concept of existence for the ranking of fuzzy sets many of the existing fuzzy ranking methods are shown to be some application of this concept an improved fuzzy ranking method is then introduced based on this concept this newly introduced method is expanded for treating both normal and nonnormal convex and nonconvex fuzzy sets emphasis is placed on the use of the subjectivity of the decision maker such as the optimistic or the pessimistic view points an improved procedure for obtaining linguistic conclusions is also developed finally some numerical examples are given to illustrate the approach

topological properties on the space of fuzzy sets abstract in this paper we introduce a new metric on the space f rp of fuzzy sets and prove that f rp is separable and complete also a characterization of compact subsets of f rp is presented and some examples are given

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

a new approach for ranking fuzzy numbers by distance method abstract many ranking methods have been proposed so far however there is yet no method that can always give a satisfactory solution to every situation some are counterintuitive not discriminating some use only the local information of fuzzy values some produce different rankings for the same situation for overcoming the above problems we propose a new method for ranking fuzzy numbers by distance method our method is based on calculating the centroid point where the distance means from original point to the centroid point x 0 y 0 and the x 0 index is the same as murakami et al s x 0 however the y 0 index is integrated from the inverse functions of an lr type fuzzy number thus we use ranking function r a x 2 y 2 distance index as the order quantities in a vague environment our method can rank more than two fuzzy numbers simultaneously and the fuzzy numbers need not be normal furthermore we also propose the coefficient of variation cv index to improve lee and li s method comput math appl 15 1988 887 896 lee and li rank fuzzy numbers based on two different criteria namely the fuzzy mean and the fuzzy spread of the fuzzy numbers and they pointed out that human intuition would favor a fuzzy number with the following characteristics higher mean value and at the same time lower spread however when higher mean value and at the same time higher spread or lower mean value and at the same time lower spread exists it is not easy to compare its orderings clearly our cv index is defined as cv standard error mean which can overcome lee and li s problem efficiently in this way our proposed method can also be easily calculated by the mathematica package to solve problems of ranking fuzzy numbers at last we present three numerical examples to illustrate our proposed method and compare with other ranking methods

a new approach for ranking nonnormal p norm trapezoidal fuzzy numbers ranking of fuzzy numbers play an important role in decision making optimization forecasting etc fuzzy numbers must be ranked before an action is taken by a decision maker in this paper with the help of several counter examples it is proved that the results proposed by chen and tang c c chen h c tang ranking of nonnormal p norm trapezoidal fuzzy numbers with integral value computers and mathematics with applications 56 2008 2340 2346 are applicable only for the nonnormal p norm trapezoidal fuzzy numbers with equal heights and a new approach is proposed for the ranking of nonnormal p norm trapezoidal fuzzy numbers with different heights the results proposed by chen and tang are modified and to illustrate the proposed approach the counter examples are solved using the proposed approach it is also shown that the proposed approach and the results obtained by using the proposed approach are valid

area ranking of fuzzy numbers based on positive and negative ideal points the maximizing set and minimizing set method is a popular ranking approach for fuzzy numbers which ranks them based on their left right and total utilities this paper presents an alternative ranking approach for fuzzy numbers called area ranking based on positive and negative ideal points which defines two new alternative indices for the purpose of ranking the two new indices are defined in terms of a decision maker dm s attitude towards risks and the left and the right areas between fuzzy numbers and the two ideal points it is shown that the area ranking approach has strong discrimination power and can rank fuzzy numbers that are unable to be discriminated by the maximizing set and minimizing set method it is also shown that the dm s attitude towards risks may have a significant impact on the ranking of fuzzy numbers as a side product a new defuzzification formula is also developed and discussed

simplifying the search for effective ranking of fuzzy numbers we prove that a ranking index which generates an order on some subset of fuzzy numbers can be reduced to a ranking index with a simpler form and which generates an equivalent order this way the finding of ranking indices generating orders with desirable properties can be essentially simplified such that we need to search only from additive or scale invariant ranking indices we determine exactly making an abstraction of equivalent orders the class of ranking indices on trapezoidal fuzzy numbers which generate orders satisfying all the basic requirements finally ranking indices used to rank trapezoidal fuzzy numbers are extended to ranking indices used to rank arbitrary fuzzy numbers so that the desired properties are preserved

applying fuzzy method for measuring criticality in project network program evaluation and review technique pert is widely used as a tool for managing large scale projects in the traditional pert model the durations of activities tasks are either represented as crisp numbers or drawn from the beta distribution to estimate the task durations such as pessimistic most likely and optimistic times however the operation time for each activity is usually difficult to define and estimate precisely in a real situation the aim of this paper is to present an analytical method for measuring the criticality in a project network with fuzzy activity times triangular fuzzy numbers are used to express the operation times for all activities in a project network a new model that combines fuzzy set theory with the pert technique is proposed to determine the critical degrees of activities tasks and paths in the proposed model a possibility index is defined to identify the likelihood of meeting a specified required time for a project network at the end of the paper an example is presented to compare with those obtained using the proposed method as well as other methods the comparisons reveal that the method proposed in this paper is more effective in determining the activity criticalities and finding the critical path

generalized poya urn for time varying pitman yor processes this article introduces a class of first order stationary time varying pitman yor processes subsuming our construction of time varying dirichlet processes presented in caron et al 2007 these models can be used for time dynamic density estimation and clustering our intuitive and simple construction relies on a generalized polya urn scheme significantly this construction yields marginal distributions at each time point that can be explicitly characterized and easily controlled inference is performed using markov chain monte carlo and sequential monte carlo methods we demonstrate our models and algorithms on epidemiological and video tracking data

on strong laws of large numbers for random upper semicontinuous functions a short proof of the strong law of large numbers for random upper semicontinuous functions fuzzy random variables with compact support is given this approach allows us to deduce many results on convergence of fuzzy random variables from the relevant results on random sets that appear as their level sets

monte carlo maximum likelihood estimation for non gaussian state space models state space models are considered for observations which have non gaussian distributions we obtain accurate approximations to the loglikelihood for such models by monte carlo simulation devices are introduced which improve the accuracy of the approximations and which increase computational efficiency the loglikelihood function is maximised numerically to obtain estimates of the unknown hyperparameters standard errors of the estimates due to simulation are calculated details are given for the important special cases where the observations come from an exponential family distribution and where the observation equation is linear but the observation errors are non gaussian the techniques are illustrated with a series for which the observations have a poisson distribution and a series for which the observation errors have a t distribution

strong convergence for weighted sums of fuzzy random sets in this paper we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of upper semicontinuous fuzzy sets in r p

constructing stationary time series models using auxiliary variables with applications here we present a novel method for modeling stationary time series our approach is to construct the model with a specified marginal family and build the dependence structure around it we show that the resulting time series is linear with a simple autocorrelation structure we construct models that parallel existing structures namely state space models autoregressive conditional heteroscedasticity arch models and generalized arch models we use bayesian techniques to estimate the resulting models we also demonstrate that the models perform well compared with competing methods for the applications considered count models and volatility models

ranking fuzzy numbers with preference weighting function expectations based on the recent research developments of fuzzy number ranking the paper extends the centroid expectation approach and proposes a preference weighting function expectation method to rank fuzzy numbers the weighting function gives different emphasis on the possible values of the fuzzy number which can be used to represent the decision maker s pessimistic neutral or optimistic attitude an optimistic measure to represent this preference information is proposed some properties of the weighting function and optimistic measure are systematically analyzed then the paper proposes a special class weighting function with maximum entropy principle and analyzes some of its properties with this parameterized maximum entropy weighting function we can get preference expectation of any fuzzy number with given optimistic level next the weighting function is extended to represent the confidential attitude of the decision maker by giving different emphasis on the cut level set of the fuzzy number some results about confidential attitude that parallel to the valuation attitude are attained we especially extended the badd basic defuzzification distribution approach with preference expectation of fuzzy numbers and another confidential measure with badd operator is proposed finally the expected valuations of fuzzy numbers are used to ranking the fuzzy numbers compared with other fuzzy number ranking methods the method of this paper is easily interpreted robust flexible and invariant with translating and scaling for real numbers

compactness and convexity on the space of fuzzy sets ii abstract in this paper we establish some characterizations of convex and relatively compact subsets of the space f r p of normal upper semicontinuous and compactly supported fuzzy sets in rp equipped with the skorokhod metric

contribution to application of energy measure of fuzzy sets abstract the works of de luca termini continued by for example knopfmacher loo and gottwald are the most important on the topic of determination of measures of fuzzy sets the matter is to evaluate how fuzzy a fuzzy set is there are two general concepts of measures of fuzzy set i e entropy and energy measures we show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations applications of the use of energy measure discussed in detail include decision making fuzzy process control and prediction in fuzzy systems

application of weighting functions to the ranking of fuzzy numbers this paper proposes a new ranking method for fuzzy numbers which uses a defuzzification of fuzzy numbers and a weighting function following saeidifar and pasha 2008 first we define a weighted distance measure on fuzzy numbers and then by minimizing this distance the weighted interval and point approximations of fuzzy numbers are obtained these indices are applied to rank the fuzzy numbers this method is new and interesting for ranking fuzzy numbers and it can be applied for solving and optimizing engineering and economics problems in a fuzzy environment

inconsistent property of lee and li fuzzy ranking method abstract this paper analyzes the lee and li fuzzy ranking method for the normalized trapezoidal fuzzy numbers ntfns every ntfn has associated with it another ntfn which posseses an inconsistent property an inconsistent property is that the results of comparison of these two ntfns are conflicting when two different comparison indices are used based on this property lee and li fuzzy ranking method suffers from lack of discrimination and occasionally conflicts with intuition

the preference order of fuzzy numbers abstract many fuzzy number ranking approaches are developed in the literature for multiattribute decision making problems almost all of the existing approaches focus on quantity measurement of fuzzy numbers for ranking purpose in this paper we consider the ranking process to determine a decision maker s preference order of fuzzy numbers a new ranking index is proposed to not only take quantity measurement but incorporate quality factor into consideration for the need of general decision making problems for measuring quantity several cuts of fuzzy numbers are used a signal noise ratio is defined to evaluate quality of a fuzzy number this ratio considers the middle point and spread of each cut of fuzzy numbers as the signal and noise respectively a fuzzy number with the stronger signal and the weaker noise is considered better moreover the associated levels are treated as the degree of belief about the cut and used as weights in the index for strengthening the influence of cut with higher levels the membership functions of fuzzy numbers are not necessarily to be known beforehand while applying this index only a few left and right boundary values of cuts of fuzzy numbers are required we have proved the feature of the proposed index in a particular case several examples are also used to illustrate the feature and applicability in ranking fuzzy numbers

ranking and defuzzification methods based on area compensation abstract we present some interesting properties related to the area compensation procedure to compare fuzzy numbers it has been proved that this method produces more than a fuzzy interval order it induces a ranking of fuzzy numbers some further results are given about the transitivity property and about computational aspects extensions to non normal fuzzy numbers and fuzzy quantities are also proposed

a law of large numbers for random sets abstract we show a law of large numbers for random sets taking values in a class of subsets of a banach space which is at least larger than the class of compact subsets of the space if the banach space is of infinite dimension to see this we first define the hausdorff distance on the family of all the non empty subsets of a banach space and give some properties of it motivated by the fact that a study of fuzzy sets is to study of set valued analysis

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

a new approach for ranking fuzzy numbers based on fuzzy simulation analysis method a new approach for ranking fuzzy numbers based on a fuzzy simulation analysis fsa method is proposed a combination method including computer and math application is developed first the method is intuitive and can be used to order fuzzy numbers easily the approach is illustrated by numerical examples showing that it overcomes the shortcoming existing fuzzy ranking approaches especially when it is difficult to use other methods to solve ranking fuzzy problems

ranking nonnormal p norm trapezoidal fuzzy numbers with integral value this paper considers the ranking fuzzy numbers with integral value proposed by liou and wang for the nonnormal p norm trapezoidal fuzzy numbers two interesting special cases of p norm trapezoidal fuzzy numbers are the well known triangular and trapezoidal fuzzy numbers for the nonnormal fuzzy numbers the differences exist among the membership functions for the triangular trapezoidal and p norm trapezoidal fuzzy numbers these differences can affect their left right and total integral values so we establish the relationship between these three types of fuzzy numbers

evaluating weapon systems using ranking fuzzy numbers abstract in this paper we will point out that chen s 1996 paper has some errors a weapon system is large and complex it has multi level multi factor and multi hierarchy features chen s method does not normalize each criterion s scores which will make a wrong decision under the following conditions i there are many levels for rank scores ii a criterion has many factors iii the total scores of systems have a larger difference under this criterion and iv one criterion s weight is larger than the other criteria numerical illustration is shown in section 5 therefore chen s method is not a general method for the evaluation of weapon systems after pointing out some errors in chen s paper to overcome these errors we will revise it and propose a general method for the evaluation of weapon systems our method utilizes fuzzy ratio scales 1 3 5 7 9 the goal is normalizing heterogeneity into homogeneity to indicate the relative strength of the factors in the corresponding criteria then we build a judgement matrix through comparison of the total scores of performance and use 1 3 5 7 9 to represent a weight vector among these criteria we will derive the priority among the alternatives by multiplying the fuzzy judgement matrix with the corresponding fuzzy weight vector the final results become a problem of ranking fuzzy numbers many triangular fuzzy numbers can easily rank its ordering by the intuition ranking method lee and li 1988 pointed out that human intuition would favor a fuzzy number with the following characteristics higher mean value and at the same time lower spread if its ordering cannot rank by figures we can use many other methods of ranking fuzzy numbers therefore the best weapon selection can be obtained by ranking fuzzy numbers

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

an approximate approach for ranking fuzzy numbers based on left and right dominance this study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance the proposed approach only requires a few left and right spreads at some levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other the total dominance is then determined by combining the left and right dominance based on a decision maker s optimistic perspectives such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired the approach proposed herein is relatively simple in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers by using a few left and right spreads two groups of examples demonstrate the accuracy and applicability of the proposed approach

selecting is personnel use fuzzy gdss based on metric distance method abstract in this paper we propose a new approach to rank fuzzy numbers by metric distance for showing our method is a good ranking method we give two examples to compare with other methods the paper also developes a computer based group decision support system fmcgdss to increase the recruiting productivity and to easily compare our method with other fuzzy number ranking methods the fmcgdss includes three ranking methods intuition ranking lee and li s fuzzy mean spread and our metric distance method to help manager make better decision under fuzzy circumstance the result indicates that the new method is coincident with the intuition ranking and the lee and li s fuzzy mean spread method on each type weight

modelling and residual analysis of nonlinear autoregressive time series in exponential variables abstract an approach to modelling and residual analysis of nonlinear autoregressive time series in exponential variables is presented the approach is illustrated by analysis of a long series of wind velocity data which has first been detrended and then transformed into a stationary series with an exponential marginal distribution the stationary series is modelled with a newly developed type of second order autoregressive process with random coefficients called the near 2 model it has a second order autoregressive correlation structure but is nonlinear because its coefficients are random the exponential distributional assumptions involved in this model highlight a very broad four parameter structure which combines five exponential random variables into a sixth exponential random variable other applications of this structure are briefly considered dependency in the near 2 process not accounted for by standard autocorrelations is explored by developing a residual analysis for time series having autoregressive correlation structure this involves defining linear uncorrelated residuals which are dependent and then assessing this higher order dependence by standard time series computations application of this residual analysis to the wind velocity data illustrates both the utility and difficulty of nonlinear time series modelling

ranking l r fuzzy number based on deviation degree this paper proposed a novel approach to ranking fuzzy numbers based on the left and right deviation degree l r deviation degree in the approach the maximal and minimal reference sets are defined to measure l r deviation degree of fuzzy number and then the transfer coefficient is defined to measure the relative variation of l r deviation degree of fuzzy number furthermore the ranking index value is obtained based on the l r deviation degree and relative variation of fuzzy numbers additionally to compare the proposed approach with the existing approaches five numerical examples are used the comparative results illustrate that the approach proposed in this paper is simpler and better

strong limit theorems for random sets and fuzzy random sets with slowly varying weights theories of random sets and fuzzy random sets are useful concepts which are frequently applied in scientific areas including information science probability and statistics in this paper strong limit theorems are derived for random sets and fuzzy random sets with slowly varying weights in separable banach spaces both independent and dependent cases are covered to provide a wide range of applications

fuzzy job sequencing for a flow shop abstract in job sequencing for a flow shop processing times are frequently not known exactly and only estimated intervals are given fuzzy numbers are ideally suited to represent these intervals in this work the campbell dudek and smith cds job sequencing algorithm is modified to accept trapezoidal fuzzy processing times deterministic sequences result but the sequence performance measurements of makespan and job mean flow time are fuzzy having been calculated using fuzzy arithmetic the use of possibility theory and the fuzzy integral enables the schedular to meaningfully interpret these fuzzy results deterministic approximations to this fuzzy approach are also investigated

granular risk based design optimization optimization considering uncertainty is an increasingly important and continuously developing uncertainty mitigation technique for modern design compared with its well established branches i e reliability based design optimization rbdo and robust design optimization risk based design optimization rdo is just regarded as an extension of rbdo by incorporating future cost hence it has not received much deep theoretical study based on the generalized theory of uncertainty we introduce different levels of probability granulation into rdo and propose a granular risk based design optimization grdo methodology the risks are modeled as granular probabilities their mean values and standard deviations two multiobjective optimization mo formulations of grdo are proposed and solved by multiobjective evolutionary algorithm based on decomposition aided with solution filtering criterion based on the results of a structural design example the capability of grdo on uncertainty management is validated by comparing the performances of different mo formulations while uncertainty mitigation using grdo is achieved by controlling the risks associated with fixed level of uncertainty this way grdo is approved as a general design frame rather than just an uncertainty mitigation technique like conventional rdo

marginal likelihood from the gibbs output abstract in the context of bayes estimation via gibbs sampling with or without data augmentation a simple approach is developed for computing the marginal density of the sample data marginal likelihood given parameter draws from the posterior distribution consequently bayes factors for model comparisons can be routinely computed as a by product of the simulation hitherto this calculation has proved extremely challenging our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density this simple identity holds for any parameter value an estimate of the posterior density is shown to be available if all complete conditional densities used in the gibbs sampler have closed form expressions to improve accuracy the posterior density is estimated at a high density point and the numerical standard error of resulting estimate is derived the ideas are applied to probit regression and finite mixture models

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

properties of moments of a family of garch processes this paper considers the moments of a family of first order garch processes first a general condition for the existence of any integer moment of the absolute values of the observations is given second a general expression for this moment as a function of lower order moments is derived third the kurtosis and the autocorrelation function of the squared and absolute valued observations are derived the results apply to a number of different garch parameterizations finally the existence or lack thereof of the theoretical counterpart to the so called taylor effect in some members of this garch family is discussed possibilities of extending the results to higher order garch processes are indicated and potential applications of the statistical theory proposed

a characterization of a general class of ranking functions on triangular fuzzy numbers this paper aims to the discussion of properties for ranking fuzzy numbers an approach of axiomatic type is followed providing the definition of two groups of requirements which characterize a ranking function through its behaviour on triangular fuzzy numbers particularly the problem of making the evaluations on triangular fuzzy numbers sensitive to their spread has been deeply analyzed

the cournot game under a fuzzy decision environment the conventional precision based decision analysis methodology is invalid for business decision analysis when precise assessment data seldom exist this paper considers the cournot game with fuzzy demand and fuzzy costs that are assumed to be triangular fuzzy numbers our model utilizes the weighted center of gravity wcog method to defuzzify the fuzzy profit function into a crisp value we derive the equilibrium cournot quantity of each firm by simultaneously solving the first order condition of each firm our model explicitly derives the necessary condition to avoid an unreasonable outcome of negative equilibrium quantities and lack of flexibility for modification of the ranking method in addition we examine the standard deviation of the fuzzy profit resulting from the fuzziness of each firm s cost and market demand functions we conduct sensitivity analysis to investigate the effect of parameter perturbations on firms outcomes the results indicate that the center of parameter plays an important role in sensitivity analysis and dominates over variations in equilibrium quantity due to a perturbation of fuzzy parameters

measurability for fuzzy valued functions in this paper we characterize the borel field of the space of fuzzy sets endowed with the skorokhod metric as a result the measurability for a fuzzy valued function is discussed

a new approach for ranking of trapezoidal fuzzy numbers ranking fuzzy numbers plays an very important role in linguistic decision making and some other fuzzy application systems several strategies have been proposed for ranking of fuzzy numbers each of these techniques have been shown to produce non intuitive results in certain cases in this paper we will introduce a new approach for ranking of trapezoidal fuzzy numbers based on the left and the right spreads at some a levels of trapezoidal fuzzy numbers the calculation of the proposed method is far simpler and easier finally some comparative examples are used to illustrate the advantage of the proposed method

comparison of fuzzy numbers using a fuzzy distance measure a new approach for ranking fuzzy numbers based on a distance measure is introduced a new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first and then it is used to formulate the distance measure for fuzzy numbers the approach is illustrated by numerical examples showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches

ranking fuzzy numbers with an area between the centroid point and original point abstract to improve the ranking method of lee and li 1 cheng 2 proposed the coefficient of variation cv index shortcomings are also found in the cv index cheng 2 also proposed the distance method to improve the ranking method of murakami et al however the distance method is not sound either moreover the cv index contradicts the distance method in ranking some fuzzy numbers therefore to overcome the above shortcomings we propose ranking fuzzy numbers with the area between the centroid point and original point

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 decision support system for multiple criteria alternative ranking using topsis and vikor in fuzzy and nonfuzzy environments abstract topsis and vikor are two well known and widely used multiple attribute decision making methods many extensions of these methods have been proposed that either use different techniques to rank alternatives or utilize fuzzy logic to handle alternatives and criteria that are unquantifiable and or incomplete in this paper we present the implementation of a web based decision support system that incorporates topsis and vikor to solve multicriteria decision making problems either in a nonfuzzy or in a fuzzy environment the aim of this paper is to present a tool that will be used by decision makers to compare various alternative solutions and understand how robust a decision will be the proposed system can be used both in single and in group decision making problems in addition we review several variations for each step of these methods and implement different techniques when applicable hence decision makers can experiment with different techniques in each method we implement ten normalization techniques three methods for the calculation of the ideal and anti ideal solutions fifteen distance metrics five fuzzy distance metrics and eight defuzzification techniques an illustrative example is presented to highlight the key features of the implemented system and the different scenarios that can be built using the proposed dss

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

ranking fuzzy numbers by preference ratio we propose a ranking method for fuzzy numbers in this method a preference function is defined by which fuzzy numbers are measured point by point and at each point the most preferred number is identified then these numbers are ranked on the basis of their preference ratio therefore fuzzy numbers are compared relatively and not necessarily one is preferred absolutely over the others this method is especially designed to evaluate alternatives in multi criteria or multi attribute decision making the method is intuitive and can be used to discriminate between numbers easily the method is specially tailored for triangular fuzzy numbers tfn and an algorithm is presented to determine the preference ratio of a pair of tfns we show that at most two feasible turning points exist this fact is very helpful to make the algorithm much easier