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

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 large sample confidence region useful in characterizing the stationary point of a quadratic response surface a second order response surface model is often used to approximate the relationship between a response variable and a set of explanatory variables the nature of the stationary point of the surface can be determined by considering the eigenvalues of the matrix of the model s second order terms since the elements of this matrix are estimated from the data however it follows that the eigenvalues are random variables hence the sampling properties of these eigenvalues should be considered in characterizing the nature of the stationary point in this article it is shown how a confidence region around these eigenvalues can be used to aid in the characterization of the stationary point and in an improved ridge analysis of the response surface the delta method is used to construct an approximate confidence region for these eigenvalues box and draper 1987 gave a result that simplifies the calculation of such a confidence region for rotatable or nearly rotatable designs a simulation study was performed to

reasonable properties for the ordering of fuzzy quantities ii as a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches to do so we briefly introduce fuzzy relations on which the third class of approaches is based then we recall some transitivity related concepts and an ordering procedure based on a acyclic fuzzy relation acyclicity is a very weak restriction on a fuzzy relation we prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity so we give a widely applicable formulation to derive a total ranking order from a fuzzy relation with our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part i

theoretical and semantic distinctions of fuzzy possibilistic and mixed fuzzy possibilistic optimization theoretical semantic and algorithmic distinctions among fuzzy possibilistic and mixed fuzzy possibilistic optimization are presented and illustrated the theory underlying fuzzy possibilistic and mixed fuzzy possibilistic optimization is developed and demonstrated and points to the appropriate use of distinct solution methods associated with each type of optimization dependant on the semantics of the problem semantics is key to both the input where one is obtaining the data and constructing the optimization model in the presence of uncertainty and the output where the meaning of the results is necessary for understanding solutions the case in which the optimization model arises from the data that is a combination of fuzzy and possibilistic distributions is also derived lastly examples illustrate the theory this paper is a modification and an amplification of a presentation made at ifsa 05 w a lodwich k d jamison theory and semantics for fuzzy and possibilistic optimization in fuzzy logic soft computing and computational intelligence eleventh internat fuzzy systems association world congress july 28 31 2005 beijing china vol iii pp 1805 1810 26

fuzzy scores of technical efficiency abstract this paper suggests to extend scores of technical efficiency as used in for example data envelopment analysis dea to efficiency scores defined as so called fuzzy intervals fuzzy scores allow the decision maker to use scores of technical efficiency in combination with other sources of available performance information e g expert opinions key figures etc in this sense fuzzy scores become formal images of the evaluation otherwise done implicitly by the decision maker and may therefore enhance a more consistent evaluation it is shown how to rank a set of fuzzy scores

a probability based approach to comparison of fuzzy numbers and applications to target oriented decision making in this paper we introduce a new comparison relation on fuzzy numbers based on their alpha cut representation and comparison probabilities of interval values basically this comparison process combines a widely accepted interpretation of fuzzy sets together with the uncertain characteristics inherent in the representation of fuzzy numbers the proposed comparison relation is then applied to the issue of ranking fuzzy numbers using fuzzy targets in terms of target based evaluations some numerical examples are used to illuminate the proposed ranking technique as well as to compare with previous methods more interestingly according to the interpretation of the new comparison relation on fuzzy numbers we provide a fuzzy target based decision model as a solution to the problem of decision making under uncertainty with which an interesting link between the decision maker s different attitudes about target and different risk attitudes in terms of utility functions can be established moreover an application of the proposed comparison relation to the fuzzy target based decision model for the problem of fuzzy decision making with uncertainty is provided numerical examples are also given for illustration

fuzzy flowsort an integration of the flowsort method and fuzzy set theory for decision making on the basis of inaccurate quantitative data nowadays most real life decision problems can be modeled as sorting problems although it is difficult to quantitatively express the evaluation criteria precisely or precise values are inadequate to model the criteria for these situations multicriteria methods can be extended by using fuzzy theory thereby providing mathematical methods able to deal with decision processes in a fuzzy environment in this context we propose a sorting method called fuzzy flowsort f flowsort based on fuzzy set theory and flowsort which is a promethee based sorting method to demonstrate the usefulness and applicability of this new method an illustrative numerical example is presented the result confirms the potential and the applicability of associating fuzzy logic with the flowsort method for processing imprecise data comparisons are made between the results provided by fuzzy flowsort and flowsort and also between fuzzy flowsort and another algorithm based on flowsort that deals with the imprecision of data when they are defined by intervals in addition the association between the results obtained with the f flowsort and flowsort methods and promethee and f promethee methods is investigated

a semidefinite relaxation based algorithm for robust attitude estimation this paper presents a tractable method for solving a robust attitude estimation problem based on a weighted least squares approach with nonlinear constraints attitude estimation requires information of a few vector quantities each obtained from both a sensor and a mathematical model by considering the modeling errors measurement noise sensor biases and offsets as infinity norm bounded uncertainties we formulate a robust optimization problem which is nonconvex with nonlinear cost and constraints the robust min max problem is approximated with a nonconvex minimization problem using an upper bound a new regularization scheme is also proposed to improve the robust performance we then use semidefinite relaxation to convert the suboptimal problem with quadratic cost and constraints into a tractable semidefinite program with a linear objective function and linear matrix inequality constraints we also show how to extract the solution of the suboptimal robust estimation problem from the solution of the semidefinite relaxation further a mathematical proof supported by numerical results is presented stating the gap between the suboptimal problem and its relaxation is zero under a given condition which is mostly true in real life scenarios the usefulness of the proposed algorithm in the presence of uncertainties is evaluated with the help of examples

criteria for evaluating fuzzy ranking methods abstract in this paper four criteria for evaluating fuzzy ranking methods are investigated fuzzy preference representation rationality of fuzzy ordering distinguishability and robustness based on these criteria two existing ranking methods baas and kwakernaak s and nakamura s are evaluated and a new improved ranking method is suggested

a method for ranking fuzzy numbers and its application to decision making since fuzzy numbers represent uncertain numeric values it is difficult to rank them according to their magnitude in the paper a method for ranking fuzzy numbers is proposed the method considers the overall possibility distributions of fuzzy numbers in their evaluations for ranking and provides users with a method of changing viewpoints for evaluations users represent their viewpoints with fuzzy sets the method evaluates fuzzy numbers with a satisfaction function and the viewpoint given by users and then ranks the numbers according to their evaluation values the satisfaction function is a measure of comparisons between fuzzy numbers in order to illustrate the ranking method two numeric examples are shown and for the comparative study our method is compared with four existing ranking methods through eight examples as an example of potential applications the proposed method is applied to a decision making problem a two person game with fuzzy profit and loss the ranking method is used to analyze player choices

on the centroids of fuzzy numbers in a paper by cheng a new approach for ranking fuzzy numbers by distance method fuzzy sets and systems 95 1998 307 317 a centroid based distance method was suggested for ranking fuzzy numbers both normal and non normal where the fuzzy numbers are compared and ranked in terms of their euclidean distances from their centroid points to the origin it is found that the centroid formulae provided by the above paper are incorrect and have led to some misapplications in this paper we present the correct centroid formulae for fuzzy numbers and justify them from the viewpoint of analytical geometry a numerical example demonstrates that cheng s formulae can significantly alter the result of the ranking procedure

on the concept of possibility probability consistency both probability and possibility may be seen as information about an experiment it is conceivable to have at some time these two forms of information about a same experiment and then the question of the relation between them arises at once in this paper some aspects of the concept of possibility probability consistency are studied the consistency is considered as a fuzzy property relative to the coherence between possibilistic and probabilistic information we analyse several measures of the degree of consistency and introduce an axiomatic to characterize them

fuzzy optimization for supply chain planning under supply demand and process uncertainties in today s global marketplace individual firms do not compete as independent entities rather as an integral part of a supply chain this paper proposes a fuzzy mathematical programming model for supply chain planning which considers supply demand and process uncertainties the model has been formulated as a fuzzy mixed integer linear programming model where data are ill known and modelled by triangular fuzzy numbers the fuzzy model provides the decision maker with alternative decision plans for different degrees of satisfaction this proposal is tested by using data from a real automobile supply chain

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

two new approaches for assessing the weights of fuzzy opinions in group decision analysis the aggregation of fuzzy opinions is an important component of group decision analysis with fuzzy information this paper proposes two new approaches for the assessment of the weights to be associated with fuzzy opinions these approaches involve respectively the minimization of the sum of squared distances from one weighted fuzzy opinion to another which is called the least squares distance method lsdm and the minimization of the sum of squared differences between the defuzzified values of any two weighted fuzzy opinions which is called the defuzzification based least squares method dlsm the two approaches are developed and numerical examples are presented to illustrate their simplicity and effectiveness in aggregating fuzzy opinions

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

duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables linear programming problems with trapezoidal fuzzy variables fvlp have recently attracted some interest some methods have been developed for solving these problems by introducing and solving certain auxiliary problems here we apply a linear ranking function to order trapezoidal fuzzy numbers then we establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results in particular we prove that the auxiliary problem is indeed the dual of the fvlp problem having established the dual problem the results will then follow as natural extensions of duality results for linear programming problems with crisp data finally using the results we develop a new dual algorithm for solving the fvlp problem directly making use of the primal simplex tableau this algorithm will be useful for sensitivity or post optimality analysis when using primal simplex tableaus

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

on wielandt s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix a relatively obscure eigenvalue inequality due to wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix the asymptotic distributions are obtained under a fairly general setting an application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given 1 introduction and summary the derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis although the main idea behind most of the derivations is quite basic i e the expansion of the sample roots about the population roots the derivations themselves are often quite involved these complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem one of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices the method is based upon a relatively obscure eigenvalue inequality

standard errors for the multiple roots in quadratic response surface models in this article we develop the asymptotic distribution for the eigenvalues of a quadratic response surface by using the result of eaton and tyler to extend the results of carter chinchilli and campbell and bisgaard and ankenman to the case including nondistinct eigenvalues in particular we suggest a method for estimating the confidence regions for multiple roots

on the instantiation of possibility distributions we investigate the problem of instantiating a variable whose value is uncertain but known to be constrained by a possibility distribution using a model based upon the dempster shafer belief structure we show that this problem of instantiation is analogous to a compound experiment in which we randomly select a subset and then perform a number of repeated experiments without replacement on this set the probability distribution resulting from this compound experiment leads to a random procedure for instantiating a possibility distribution

the l average value and the fuzzy expectation of a fuzzy random variable abstract the concepts of fuzzy random variable and the associated fuzzy expected value have been introduced by puri and ralescu as an extension of measurable set valued functions random sets and of the aumann integral of these functions respectively on the other hand the average function has been suggested by campos and gonzalez as an appropriate function to rank fuzzy numbers in this paper we are going to analyze some useful properties concerning the average value of the expectation of a fuzzy random variable and some practical implications of these properties are also commented on

ranking fuzzy numbers based on decomposition principle and signed distance by using the decomposition principle and the crisp ranking system on r we construct a new ranking system for fuzzy numbers which is very realistic and also matching our intuition as in r

a multiple criteria linguistic decision model mcldm for human decision making abstract this paper presents a human nature like decision model which makes the decision making process easier for human beings the model which is called the multiple criteria linguistic decision model mcldm can be used to evaluate a set of alternatives over various criteria by using linguistic variables and then produces a linguistic decision output for selecting the best alternative the mcldm uses fuzzy set theory fst to deal with the imprecision measurement of decision makers dms subjective opinions which usually exist in the decision making process the decision model is originally designed for the screening of research projects however it can also be applied to problems such as program evaluation project performance evaluation and many other management decisions

bector chandra type duality in fuzzy linear programming with exponential membership functions in this paper we study a pair of fuzzy primal dual linear programming problems and calculate duality results using an aspiration level approach we use an exponential membership function which is in contrast to the earlier works that relied on a linear membership function as the fuzzy environment causes a duality gap we investigate how choosing the exponential membership function impacts this gap this issue is particularly important for fuzzy linear programming where in general the primal and dual objective values may not be bounded

a fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers the aim of this paper is to develop an effective method for solving matrix games with payoffs of triangular fuzzy numbers tfns which are arbitrary in this method it always assures that players gain floor and loss ceiling have a common tfn type fuzzy value and hereby any matrix game with payoffs of tfns has a tfn type fuzzy value based on duality theorem of linear programming lp and the representation theorem for fuzzy sets the mean and the lower and upper limits of the tfn type fuzzy value are easily computed through solving the derived lp models with data taken from 1 cut set and 0 cut set of fuzzy payoffs hereby the tfn type fuzzy value of any matrix game with payoffs of tfns can be explicitly obtained moreover we can easily compute the upper and lower bounds of any alfa cut set of the tfn type fuzzy value for any matrix game with payoffs of tfns and players optimal mixed strategies through solving the derived lp models at any specified confidence level alfa the proposed method in this paper is demonstrated with a numerical example and compared with other methods to show the validity applicability and superiority

non additive multi attribute fuzzy target oriented decision analysis as an emerging area considering behavioral aspects of decision making target oriented decision model lies in the philosophical root of bounded rationality as well as represents the s shaped value function this paper deals with multi attribute decision analysis from target oriented viewpoint first the basic random target oriented decision model is extended to involve three types of target preferences benefit target cost target and equal target next since applying fuzzy set theory in decision analysis allows the decision maker to specify imprecise aspiration levels fuzzy target oriented decision analysis is formulated to model three typical types of fuzzy targets fuzzy min fuzzy max and fuzzy equal also different attitudes are used to derive target achievement functions which can be viewed as a support for probability as psychological distance furthermore we have proved that multi attribute target oriented decision analysis has a similar structure with discrete fuzzy measure and choquet integral hence we propose using discrete fuzzy measure and choquet integral to model non additive multi attribute target oriented decision analysis in particular the l measure is applied to reduce the difficulty of collecting information via a designed bisection search algorithm finally a new product development example is used to illustrate the effectiveness and advantages of our model the main advantages of our target oriented decision model are its abilities to model the fuzzy uncertainty of targets as well as capture the non additive behaviors among targets by means of discrete fuzzy measure and choquet integral

fuzzy linear programming models to solve fuzzy matrix games abstract in this paper a zero sum two person game with imprecise values in its matrix of payoffs is considered we propose a method for its solution based on the establishment of a fuzzy linear programming f l p problem for each player the method is shown as a generalization of that conventionally used in the solution of a classical game to solve such f l p problems we propose the auxiliary models resulting from the application of some of the methods for ranking fuzzy numbers hence according to the kind of ranking method that players want to use different solutions to the former game can be obtained we show that these solutions are of the same nature as the parameters defining the game and corresponding to a fuzzy predicate that can be established as the value of the game is around

ranking fuzzy multicriteria alternatives with respect to a decision maker s fuzzy goal abstract a two stage method for ranking fuzzy multicriteria alternatives with respect to a decision maker s fuzzy goal is presented in the first stage the joint membership function of each fuzzy alternative and the membership function of the fuzzy goal are projected orthogonally from the multicriteria space to the one dimensional preference subspace of the decision maker in the second stage a ranking index is used to rank the fuzzy projections in the decision maker s preference subspace a numerical example demonstrates the method

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

the shortest path problem on networks with fuzzy parameters shortest path problem where the costs have vague values is one of the most studied problems in fuzzy sets and systems area however due to its high computational complexity previously published algorithms present peculiarities and problems that need to be addressed e g they find costs without an existing path they determine a fuzzy solution set but do not give any guidelines to help the decision maker choose the best path they can only be applied in graphs with fuzzy non negative parameters etc in this paper one proposes an iterative algorithm that assumes a generic ranking index for comparing the fuzzy numbers involved in the problem in such a way that each time in which the decision maker wants to solve a concrete problem s he can choose or propose the ranking index that best suits that problem this algorithm that solves the above remarked drawbacks is based on the ford moore bellman algorithm for classical graphs and in concrete it can be applied in graphs with negative parameters and it can detect whether there are negative circuits for the sake of illustrating the performance of the algorithm in the paper it has been here developed using only certain order relations but it is not restricted at all to use these comparison relations exclusively the proposed algorithm is easy of understanding as the theoretical base of a decision support system oriented to solving this kind of problems

methods for decision making with cardinal numbers and additive aggregation abstract for multi attribute decision making with cardinal numbers and additive aggregation the methods additive aggregation and outranking for crisp and fuzzy numbers are connected with the concept of linguistic variables preference ranking a generalization of outranking and preference ranking organization methods such as electre and promethee can be divided into three phases where each of them is defined completely by linguistic variables the derived method preference ranking with additive weighting and individual fuzziness has an improved selectivity or discrimination compared to conventional methods moreover the method fulfils four important requirements preference only for sufficiently large weights considering evaluation differences outside of thresholds and regarding aggregational and substitutional properties the simplified method preference ranking with additive weighting and general fuzziness allows a reduction of computational resources

numerical methods for interval and fuzzy number comparison based on the probabilistic approach and dempster shafer theory there are many different methods for interval and fuzzy number comparison proposed in the literature which provide the results of comparison in the form of a real or boolean value in this paper we use the dempster shafer theory of evidence with its probabilistic interpretation to justify and construct the method which provides the result of comparison in the form of an interval or fuzzy number the complete and consistent set of expressions for inequality and equality relations between intervals and fuzzy numbers has been obtained in the framework of the probabilistic approach these relations make it possible to compare intervals and fuzzy numbers with real values and may be considered as an asymptotic limit of the results we obtain using the dempster shafer theory a natural fuzzy extension of the proposed approach is considered and discussed using some illustrative examples

on the gap between the quadratic integer programming problem and its semidefinite relaxation consider the semidefinite relaxation sdr of the quadratic integer program qip where q is a given symmetric matrix and d is diagonal we consider the sdr gap we establish the uniqueness of the sdr solution and prove that if and only if r n 1max xtvvtx x 1 1 n 1 where v is an orthogonal matrix whose columns span the r dimensional null space of d q and where d is the unique sdr solution we also give a test for establishing whether that involves 2r 1 function evaluations in the case that r 1 we derive an upper bound on which is tighter than thus we show that breaching the sdr gap for the qip problem is as difficult as the solution of a qip with the rank of the cost function matrix equal to the dimension of the null space of d q this reduced rank qip problem has been recently shown to be solvable in polynomial time for fixed r

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

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

confidence intervals and an improved ridge analysis of response surfaces the user of rsm techniques gains considerable insight into the nature of the stationary point and the nature of the underlying response surface from estimates of the eigenvalues of the matrix of pure and mixed quadratic regression coefftcients b this article presents and illustrates methodology for constructing conservative confidence limits on the eigenvalues of this matrix as well as the mean response at a constrained optimum when a confidence interval about an eigenvalue of b includes zero a change in the strategy of analysis is required one is suggested and illustrated

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

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

on the use of the ranking function approach to solve fuzzy matrix games in a direct way abstract a general method of solving fuzzy matrix games is presented this method is suitable when players choose their fuzzy number ranking procedures in a wide class i e that which may be represented by means of a linear ranking functions in the first case described both players use the same criterion to rank fuzzy numbers and in the second case each player uses different criteria several situations of players knowledge are analyzed in the second case and game solutions are provided in all of them

modeling uncertainty using partial information abstract we are concerned here with the issue of representing the available information which is often only partial about the value of a variable we consider the dempster shafer belief structure and describe a number of different semantics that can be associated with it among these is a probabilistic interpretation in which some of the information is not available we do not have all the distributions this probabilistic interpretation leads us to consider a view involving selecting balls from an urn using this urn framework as a representation of our partial knowledge about the value of a variable leads to number of different uncertainty representations depending upon what information is assumed known

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

ranking of fuzzy numbers by sign distance several different strategies have been proposed for ranking of fuzzy numbers these include methods based on the coefficient of variation cv index distance between fuzzy sets centroid point and original point and weighted mean value each of these techniques has been shown to produce non intuitive results in certain cases in this paper we propose a modification of the distance based approach called the sign distance which is both efficient to evaluate and able to overcome the shortcomings of the previous techniques the calculation of the proposed method is far simpler than the other approaches

a subjective approach for ranking fuzzy numbers abstract starting from a subjective assignation of weights related to the relative importance of the different level sets we define a new method of comparison of fuzzy numbers this procedure is an extension of some well known indices adamo tsumura et al yager some properties of our index are studied in this paper as well as its behaviour on several particular cases

a context dependent method for ordering fuzzy numbers using probabilites in this work we suggest an approach to comparing fuzzy numbers motivated by a probabilistic view of the underlying uncertainty an important aspect of the method suggested is its context dependency the numbers being compared affect the process we discuss the effect of decision attitude and show that this approach is particularly useful for aiding decision makers having a temperate decision attitude optimism in the face of adversity and conservatism in the face plenty

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

ranking alternatives with fuzzy weights using maximizing set and minimizing set ranking alternatives both qualitative as well as quantitative in a multicriterion environment employing experts opinion preference structure using fuzzy numbers and linguistic variables are presented in this paper fuzzy weights wj i of alternatives a i are computed using standard fuzzy arithmetic concept of maximizing set and minimizing set is introduced to decide total utility or ordering value of each of the alternatives a numerical example is provided at the end to illustrate the method 1999 published by elsevier science b v all rights reserved

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

standard errors for the eigenvalues in second order response surface models in the interpretation of the geometry of second order response surface models standard errors and confidence intervals for the eigenvalues of the second order coefficient matrix play an important role in this article we propose a new method for estimating the standard errors and hence approximate confidence intervals of these eigenvalues the method is simple in both concept and execution it involves the refitting of a full quadratic model after rotating the coordinate system to coincide with the canonical axes the estimated standard errors of the pure quadratic terms from this refitting are then used as approximate standard errors of the eigenvalues because this approach is based on the canonical form it is geometrically intuitive and easily taught our method is intended as a way for practitioners to get quick estimates of the standard errors of the eigenvalues in our justification of the approach we show that it is equivalent to using the delta method proposed by carter chinchilli and camp

linear programming problems and ranking of fuzzy numbers abstract in this paper linear programming problems with fuzzy constraints and fuzzy coefficients in both matrix and right hand side of the constraint set are considered because of fuzzy coefficients in both members of each constraint ranking methods for fuzzy numbers must be considered the diversity of such methods provides a lot of different models of conventional linear programming problems from which fuzzy solutions to the former problem can be obtained

an index for ordering fuzzy numbers abstract a new index that is useful for ordering ranking of fuzzy numbers is proposed to use this index the membership functions of fuzzy numbers need not be normal or convex properties of this index and its relationship to the 1981 yager s index are explained a numerical comparison between the proposed and four other ordering procedures is presented

fuzzy nash equilibriums in crisp and fuzzy games in this paper we introduce fuzzy nash equilibrium to determine a graded representation of nash equilibriums in crisp and fuzzy games this interpretation shows the distribution of equilibriums in the matrix form of a game and handles uncertainties in payoffs in addition a new method to rank fuzzy values with the user s viewpoint is investigated by this means the definition of satisfaction function which provides the result of comparison in the form of real value is developed when users have preferences regarding the payoffs

the transferable belief model abstract we describe the transferable belief model a model for representing quantified beliefs based on belief functions beliefs can be held at two levels 1 a credal level where beliefs are entertained and quantified by belief functions 2 a pignistic level where beliefs can be used to make decisions and are quantified by probability functions the relation between the belief function and the probability function when decisions must be made is derived and justified four paradigms are analyzed in order to compare bayesian upper and lower probability and the transferable belief approaches

a study of the ranking function approach through mean values abstract the average value was introduced to help in the ordering of fuzzy numbers and it was defined by means of an integrating process of a parametric function representing the position of every cut in the real line by considering the relation between a particular case of the average value and the notion of expectation for fuzzy numbers we interpret the general ranking function approach as a ranking process using a mean value and an interval relation some well known indices are included in this schema using the general average value a more general mean value for a fuzzy number is defined finally some properties of the general mean value are studied as well as its behaviour on several particular cases

fuzzy quadratic minimum spanning tree problem in this paper a fuzzy quadratic minimum spanning tree problem is formulated as expected value model chance constrained programming and dependent chance programming according to different decision criteria then the crisp equivalents are derived when the fuzzy costs are characterized by trapezoidal fuzzy numbers furthermore a simulation based genetic algorithm using prufer number representation is designed for solving the proposed fuzzy programming models as well as their crisp equivalents finally a numerical example is provided for illustrating the effectiveness of the genetic algorithm

approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval the problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well known metric which preserves the expected interval of the fuzzy number is completely solved the previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number properties of this new trapezoidal approximation operator are studied

joint propagation and exploitation of probabilistic and possibilistic information in risk assessment random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk while random variability can be represented by probability distribution functions imprecision or partial ignorance is better accounted for by possibility distributions or families of probability distributions because practical situations of risk computation often involve both types of uncertainty methods are needed to combine these two modes of uncertainty representation in the propagation step a hybrid method is presented here which jointly propagates probabilistic and possibilistic uncertainty it produces results in the form of a random fuzzy interval this paper focuses on how to properly summarize this kind of information and how to address questions pertaining to the potential violation of some tolerance threshold while exploitation procedures proposed previously entertain a confusion between variability and imprecision thus yielding overly conservative results a new approach is proposed based on the theory of evidence and is illustrated using synthetic examples

a robust lot sizing problem with ill known demands the paper deals with a lot sizing problem with ill known demands modeled by fuzzy intervals whose membership functions are possibility distributions for the values of the uncertain demands optimization criteria in the setting of possibility theory that lead to choose robust production plans under fuzzy demands are given some algorithms for determining optimal robust production plans with respect to the proposed criteria and for evaluating production plans are provided some computational experiments are presented

a single period inventory model with fuzzy demand abstract the single period inventory models have wide applications in the real world in assisting the decision maker to determine the optimal quantity to order due to lack of historical data the demand has to be subjectively determined in many cases in this paper a single period inventory model for cases of fuzzy demand is constructed the costs considered include the procurement cost shortage cost and holding cost for different fuzzy total cost resulted from different order quantity a method for ranking fuzzy numbers is adopted to find the optimal order quantity in terms of the cost when the profit gained from selling one item is less greater than the loss incurred due to one unsold item the optimal order quantity lies in the range defined for the left shape right shape function of the fuzzy demand if the unit profit is equal to the unit loss then all quantities with a membership grade 1 are optimal to be ordered the methodology of this paper can be applied to construct other inventory models with fuzzy demand

robust mean squared error estimation in the presence of model uncertainties we consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties where the vector x is known to satisfy a weighted norm constraint we first assume that the model is known exactly and seek the linear estimator that minimizes the worst case mean squared error mse across all possible values of x we show that for an arbitrary choice of weighting the optimal minimax mse estimator can be formulated as a solution to a semidefinite programming problem sdp which can be solved very efficiently we then develop a closed form expression for the minimax mse estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of mayer and willke with a specific choice of shrinkage factor that explicitly takes the prior information into account next we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst case mse across all possible values of x and all possible values of the model matrix as we show the robust minimax mse estimator can also be formulated as a solution to an sdp finally we demonstrate through several examples that the minimax mse estimator can significantly increase the performance over the conventional least squares estimator and when the model matrix is subject to uncertainties the robust minimax mse estimator can lead to a considerable improvement in performance over the minimax mse estimator

comparison of clusters from fuzzy numbers fuzzy numbers are used for representation of numerical quantities in a vague environment their comparison or ranking is important for application purposes a new index for comparing of fuzzy numbers based on their geometrical properties is suggested in this paper this geometrical index is tested on a group of selected examples and compared with the other well known indexes a method for comparison of m tuples of fuzzy numbers and an algorithm for comparison of subsets clusters of similar closed m tuples of trapezoidal fuzzy numbers are presented

structured least squares problems and robust estimators a novel approach is proposed to provide robust and accurate estimates for linear regression problems when both the measurement vector and the coefficient matrix are structured and subject to errors or uncertainty a new analytic formulation is developed in terms of the gradient flow of the residual norm to analyze and provide estimates to the regression the presented analysis enables us to establish theoretical performance guarantees to compare with existing methods and also offers a criterion to choose the regularization parameter autonomously theoretical results and simulations in applications such as blind identification multiple frequency estimation and deconvolution show that the proposed technique outperforms alternative methods in mean squared error for a significant range of signal to noise ratio values

a possibilistic multiple objective pricing and lot sizing model with multiple demand classes we address an inventory marketing system to determine the production lot size marketing expenditure and selling prices where a firm faces demand from two or more market segments in which the firm can set different prices considering pricing marketing and lot sizing decisions simultaneously the model maximizes the profit and return on inventory investment under multiple time varying demand classes the model is formulated as a fuzzy non linear multi objective one where some parameters are ill known and modeled by fuzzy numbers a hybrid possibilistic flexible programming approach is proposed to handle imprecise data and soft constraints concurrently after transforming the original model into an equivalent multi objective crisp model it is then converted to a classical mono objective one by a fuzzy goal programming method an efficient solution procedure using particle swarm optimization pso is also provided to solve the resulting non linear problem