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

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

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

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

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

the fuzzy hyperbolic inequality index associated with fuzzy random variables the aim of this paper is focussed on the quantification of the extent of the inequality associated with fuzzy valued random variables in general populations for this purpose the fuzzy hyperbolic inequality index associated with general fuzzy random variables is presented and a detailed discussion of some of the most valuable properties of this index extending those for classical inequality indices is given two examples illustrating the computation of the fuzzy inequality index are also considered some comments and suggestions are finally included

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

dominance relations on fuzzy numbers abstract two dominance relations between fuzzy numbers are defined by means of a ranking function evaluated into r m we show that these different dominances depend on ranking relations between real closed intervals it is also shown both dominances work in a complementary way in the ranking process finally an interpretation of a parameter in the ranking function as an optimism pessimism degree is given the definition of dominance regions depending on this parameter makes the comparison of two fuzzy numbers easy

a note on generalized convexity for fuzzy mappings through a linear ordering in de campos ibanez and gonzalez munoz fuzzy sets syst 29 145 154 1989 6 goestschel and voxman fuzzy sets syst 18 31 43 1986 7 the authors considered a linear ordering on the space of fuzzy intervals for each fuzzy mapping fuzzy interval valued mapping f based on the aforementioned linear ordering they introduced a real valued function t f on the domain of the fuzzy mapping f recently chalco cano et al fuzzy sets syst 231 70 83 2013 4 have studied the relationship between the generalized hukuhara differentiability of a fuzzy mapping f g differentiability for short and the differentiability of t f and some properties of local global minima this paper studies such properties for fuzzy mappings using new concepts which generalize the existing ones

an improved index of interactivity for fuzzy numbers in this paper we will introduce a new index of interactivity between marginal possibility distributions a and b of a joint possibility distribution c the starting point of our approach is to equip each c level set of c with a uniform probability distribution then the probabilistic correlation coefficient between its marginal probability distributions is interpreted as an index of interactivity between the c level sets of a and b then we define the index of interactivity between a and b as the weighted average of these indexes over the set of all membership grades this new index of interactivity is meaningful for the whole family of joint possibility distributions

the mean square error of a random fuzzy vector based on the support function and the steiner point metrics between fuzzy values are a topic with interest for different purposes among them statistics with fuzzy data is growing in modelling and techniques largely through the use of suitable distances between such data this paper introduces a generalized actually parameterized l 2 metric between fuzzy vectors which is based on their representation in terms of their support function and steiner points consequently the metric takes into account the deviation in central location represented by the steiner points and the deviation in shape represented by a deviation defined in terms of the support function and steiner points then sufficient conditions can be given for this representation to characterize fuzzy vectors which is valuable for different aims like optimization studies properties of the metric are analyzed and its application to quantify the mean square error of a fuzzy value in estimating the value of a fuzzy vector valued random element is examined some immediate implications from this mean error are finally described

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

a parameterized l2 metric between fuzzy numbers and its parameter interpretation abstract when handling fuzzy number data it is a common practice to make use of a metric to quantify distances between fuzzy numbers several metrics have been suggested in the literature for this purpose when statistically analyzing fuzzy number valued data l 2 metrics become especially useful this paper introduces a new family of generalized l 2 metrics which take into account key features of the involved fuzzy numbers namely a measure of central location and two measures associated with the shape of the fuzzy numbers are used a crucial property related to these three measures is that necessary and sufficient conditions can be established for them to characterize fuzzy numbers furthermore the family of generalized l 2 metrics depends on one parameter a discussion is provided regarding the interpretation of this parameter which can guide selection of its value in practice

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

statistical self similar properties of complex networks it has been shown that many complex networks shared distinctive features which differ in many ways from the random and the regular networks although these features capture important characteristics of complex networks their applicability depends on the type of networks to unravel ubiquitous characteristics that complex networks may have in common we adopt the clustering coefficient as the probability measure and present a systematic analysis of various types of complex networks from the perspective of statistical self similarity we find that the probability distribution of the clustering coefficient is best characterized by the multifractal moreover the support of the measure had a fractal dimension these two features enable us to describe complex networks in a unified way at the same time offer unforeseen possibilities to comprehend complex networks

generalized convexity in fuzzy vector optimization through a linear ordering in this article we study efficiency and weakly efficiency in fuzzy vector optimization after formulating the problem we introduce two new concepts of generalized convexity for fuzzy vector mappings based on the generalized hukuhara differentiability pseudoinvexity i and pseudoinvexity ii we prove that pseudoinvexity is the necessary and sufficient condition for a stationary point to be a solution of a fuzzy vector optimization problem we give conditions to insure that a fuzzy vector mapping is invex and pseudoinvex i and ii moreover we present some examples to illustrate the results lastly we use these results to study the class of problems which have uncertainty and inaccuracies in the objective function coefficients of mathematical programming models

ranking function based solutions of fully fuzzified minimal cost flow problem the aim of minimal cost flow problem mcfp is to find the least transportation cost of a single commodity through a capacitated network this paper presents a model to deal with one particular group of such problems in which the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy numbers for easier reference hereafter we refer to this group of problems as fully fuzzified mcfp to represent our model hukuhara s difference and approximated multiplication are used thereafter we sort fuzzy numbers by an order using a ranking function and show that it is a total order i e a reflexive anti symmetric transitive and complete binary relation utilizing the proposed ranking function we transform the fully fuzzified mcfp into three crisp problems solvable in polynomial time from this standpoint combinatorial algorithms are provided to solve the above mentioned problem and find the fuzzy optimal flow furthermore the proposed order is related to the importance weights of the center the left spread and the right spread of each fuzzy number thus this method is capable of handling the decision maker s risk taking by comparing some previous ranking function based works with our method the efficiency of the latter is revealed finally an application of our proposed method to petroleum industry is presented

a d dimensional nucleation and growth model we analyze the relaxation time of a ferromagnetic d dimensional growth model on the lattice the model is characterized by d parameters which represent the activation energies of a site depending on the number of occupied nearest neighbours this model is a natural generalisation of the model studied by dehghanpour and schonmann probab theory relat fields 107 1 123 135 1997 where the activation energy of a site with more than two occupied neighbours is zero

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

influence diagrams with super value nodes involving imprecise information the concept of super value nodes was established to allow dynamic programming to be performed within the theory of influence diagrams and to reduce the computational complexity in solving problems by means of influence diagrams this paper is focused on how influence diagrams with super value nodes are affected by the presence of imprecise information we analyze how to reduce the complexity when evaluating an influence diagram in this framework by modelling these kinds of nodes and random magnitudes in terms of fuzzy random variables finally an applied example of the theoretical results is developed

an estimation model of value at risk portfolio under uncertainty a value at risk portfolio model under uncertainty is discussed in the proposed model randomness and fuzziness are evaluated respectively by the probabilistic expectation and the mean values with evaluation weights and l mean functions the means the variances and the measurements of imprecision for fuzzy numbers fuzzy random variables are evaluated in the possibility case and the necessity case and the rate of return in portfolio is estimated regarding the both random factors and imprecise factors by analytical approach we derive a solution of the value at risk portfolio problem a numerical example is given to illustrate our idea

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

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

a discrete method for studying indifference and order relations between fuzzy numbers abstract this paper proposes a generalization of the ranking function approach to fuzzy numbers permitting the definition of different kinds of dominance conferring a degree on the order relation our generalization enables the subjectivity of the decisionmaker to be taken into account unlike the other methods for ranking fuzzy numbers our model provides an improved indifference relation in comparison with previous models finally some properties of the ranking function are described

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

total orderings defined on the set of all fuzzy numbers abstract in this work a new concept of upper dense sequence in interval 0 1 is introduced there are infinitely many upper dense sequences in interval 0 1 using any upper dense sequence a new decomposition theorem for fuzzy sets is established and proved then using a chosen upper dense sequence as one of the necessary reference systems infinitely many total orderings on the set of all fuzzy numbers can be well defined among them a common upper dense sequence based on the binary numbers is suggested as a natural default option another upper dense sequence based on the rational numbers is also suggested regarding real numbers as special fuzzy numbers all of these total orderings defined by using the suggested upper dense sequences are consistent with the natural ordering of real numbers building total ordering on the set of all fuzzy numbers in such a way is significant for fuzzy data analysis and therefore may be used in decision making with fuzzy information

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

convergence analysis of solution sets for fuzzy optimization problems abstract in this paper we first consider fuzzy optimization problem in short fop and sequences of fuzzy optimization problems in short fop n then we establish the concept of d convergence of a sequence of fuzzy mappings and its characterizations finally the painleve kuratowski lower convergence and the painleve kuratowski convergence of approximate solution sets for these problems are proposed using d convergence of a sequence of fuzzy mappings some examples are provided to illustrate the essentialness of the imposed assumptions

alpha cut representation used for defuzzification in rule based systems abstract alpha cut representation of fuzzy sets has been used as a basis for fuzzy numbers ranking in some applications but rarely used for defuzzification of rule based systems or fuzzy controllers moreover such alpha cut defuzzification called acd here is not yet formally linked to the membership function mf or to the common mf based defuzzification methods namely the centroid the acd can be considered as a generalisation of the similar algorithms in fuzzy numbers to any fuzzy set a close form formula for acd is developed that involves both mf and its derivative which shows that acd reflects both static and dynamic aspects of a fuzzy set moreover formal links between acd and some mf based defuzzification methods are shown through two groups of experiments the utility of the new method is compared with centroid defuzzification particularly we examined how the acd significantly outperforms the centroid for noisy time series prediction finally the computation complexity of acd is shown to be about the same as the centroid method for convex fuzzy sets our tests suggest that acd can be considered as a viable alternative defuzzification method for fuzzy system designers

a new evaluation of mean value for fuzzy numbers and its application to american put option under uncertainty this paper discusses two topics on fuzzy random variables in decision making one is a new evaluation method of fuzzy random variables and the other is to present a mathematical model in financial engineering by fuzzy random variables the evaluation method is introduced as mean values defined by fuzzy measures and it is also applicable to fuzzy numbers and fuzzy stochastic process defined by fuzzy random variables the other is to apply the method to an american put option with uncertainty formulated as an optimal stopping problem for fuzzy random variables and the randomness and fuzziness are estimated by the probabilistic expectation and the mean values the optimal expected price of the american put option is given by the mean values with decision maker s subjective parameters numerical examples are given to illustrate our idea

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

on the nearest parametric approximation of a fuzzy number many nearest parametric approximation methods of fuzzy sets are proposed in the literature it is clear that the specific approximations may lead to the loss of information about fuzziness to overcome this problem most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation but these approximations mostly are not flexible to the decision maker s choice hence in this paper we offer a parametric fuzzy approximation method based on the decision maker s strategy as an extension of trapezoidal approximation of a fuzzy number this method comprises the selection of the form of the parametric membership function and its evaluation

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

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

an extension of fubini s theorem for fuzzy random variables in this paper an extension of the classical fubini theorem for fuzzy random variables is given this study requires an analysis of the measurability and integrable boundedness of the expected value of a fuzzy random variable in terms of those of the expected value of a compact convex random set which is first developed

a note on yoshida s optimal stopping model for option pricing abstract we argue that the optimal stopping model which has been used by yoshida to discuss option pricing can many times lead to an overoptimistic evaluation of payoffs put option prices this effect is due to the method used to compare fuzzy payoffs using a sugeno integral it is shown that each fuzzy payoff can be associated to an indifferent non fuzzy payoff which is never smaller than the highest value having membership 1 several examples are given in which this property seems to be incovenient we also show that this inconvenience cannot be avoided by replacing sugeno integrals by any other integral like functional like a t seminormed integral or a choquet integral finally we suggest to use the campos gonzalez ranking criterion instead

calculating insurance claim reserves with fuzzy regression claim provisions are crucial for the financial stability of insurance companies this is why actuarial literature has proposed numerous claim reserving methods which are usually based on statistical concepts however the mutant and uncertain behaviour of insurance environments does not make advisable to use a wide data base when calculating claim reserves on the one hand this involves a considerable loss in reliability of statistical methods but on the other it makes the use of fuzzy set theory very attractive in this paper then we propose a claim reserving method that combines the extension by ishibuchi and nii of tanaka s classical fuzzy regression method with sherman s claim reserving scheme

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 procedure for ranking fuzzy numbers using fuzzy relations abstract this paper presents a method to give fuzzy order relations between fuzzy numbers these are founded on the concept of comparison function defined in the paper and the use of fuzzy measures related with the same numbers the link of such relations with fuzzy operations is investigated too