fuzzy differential games for nonlinear stochastic systems suboptimal approach a fuzzy differential game theory is proposed to solve the n person or n player nonlinear differential noncooperative game and cooperative game team problems which are not easily tackled by the conventional methods in the paper both noncooperative and cooperative quadratic differential games are considered first the nonlinear stochastic system is approximated by a fuzzy model based on the fuzzy model a fuzzy controller is proposed to deal with the noncooperative differential game in the sense of nash equilibrium strategies or with the cooperative game in the sense of pareto optimal strategies using a suboptimal approach the outcomes of the fuzzy differential games for both the noncooperative and the cooperative cases are parameterized in terms of an eigenvalue problem since the state variables are usually unavailable a suboptimal fuzzy observer is also proposed in this study to estimate the states for these differential game problems finally simulation examples are given to illustrate the design procedures and to indicate the performance of the proposed methods

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

monte carlo computation of cluster free energies in the ising model a test for the validity of the capillarity approximation a new monte carlo technique is proposed to compute directly the free energy difference fn 1 fn of two clusters with size n 1 and n the algorithm is applied to the two dimensional square ising model with nearest neighbour interactions results are given for 1 9 but for smaller cluster sizes the obtained energies are larger than predicted by the capillarity model the deviation exhibits no systematic size dependence but a strong temperature dependence the existence of a size independent term in fn is confirmed by the values of the internal energy un obtained as a by product of the calculation

a banzhaf function for a fuzzy game in this paper another important solution for fuzzy games called the banzhaf function is investigated first a sim plified expression for the banzhaf function for games with fuzzy coalition is proposed the relationship between this simplified ex pression and two kinds of particular banzhaf functions are dis cussed in detail furthermore in the framework of games with fuzzy coalitions we give a definition of the potential function of fuzzy games and prove that the marginal values of the potential function are the components of the banzhaf function further more a new axiomatic characterization of the banzhaf function for games with fuzzy coalitions is provided

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

equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals when the game theory is applied to real world problems such as decision making in public and managerial problems on occasions it is difficult to assess payoffs exactly because of inaccuracy of information and fuzzy understanding of situations by experts in such cases games with fuzzy payoffs in which payoffs are represented as fuzzy numbers are often considered in this paper we consider equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs we introduce fuzzy goals for payoffs in order to incorporate ambiguity of a player s judgements and assume that the player tries to maximize degrees of attainment of the fuzzy goals the fuzzy goals for payoffs and the equilibrium solution with respect to the degree of attainment of the fuzzy goal are defined two basic methods one by weighting coefficients and the other by a minimum component are employed to aggregate multiple fuzzy goals when membership functions of fuzzy payoffs and fuzzy goals are all linear and the fuzzy decision in terms of the intersection is employed the necessary conditions that pairs of strategies be the equilibrium solutions is obtained when membership functions of fuzzy payoffs are quadratic functions those of fuzzy goals are linear and the fuzzy decision in terms of the convex combination is employed we also derive the necessary conditions that pairs of strategies be the equilibrium solutions

on the convergence of quadrature formulas for complex weight functions abstract we consider quadrature formulas for complex weights and prove their convergence for a large class of functions

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

orthogonal polynomials and gaussian quadrature rules related to oscillatory weight functions in this paper we consider polynomials orthogonal with respect to an oscillatory weight function w x xeim x on 1 1 where m is an integer the existence of such polynomials as well as several of their properties three term recurrence relation differential equation etc are proved we also consider related quadrature rules and give applications of such quadrature rules to some classes of integrals involving highly oscillatory integrands

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

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

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

fuzzy approaches to the game of chicken game theory deals with decision making processes involving two or more parties also known as players with partly or completely conflicting interests decision makers in a conflict must often make their decisions under risk and under unclear or fuzzy information in this paper two distinct fuzzy approaches are employed to investigate an extensively studied 2 spl times 2 game model the game of chicken the first approach uses a fuzzy multicriteria decision analysis method to obtain optimal strategies for the players it incorporates subjective factors into the decision makers objectives and aggregates objectives using a weight vector the second approach applies the theory of fuzzy moves tfm to the game of chicken the theory of moves tom is designed to bring a dynamic dimension to the classical theory of games by allowing decision makers to look ahead for one or several steps so that they can make a better decision tom is the crisp counterpart of tfm the approach we implement here to deal with games that include fuzzy and uncertain information the application of fuzzy approaches to the game of chicken demonstrates their effectiveness in manipulating subjective uncertain and fuzzy information and provides valuable insights into the strategic aspects of chicken

orthogonal laurent polynomials corresponding to certain strong stieltjes distributions with applications to numerical quadratures in this paper we shall be mainly concerned with sequences of orthogonal laurent polynomials associated with a class of strong stieltjes distributions introduced by a s ranga algebraic properties of certain quadratures formulae exactly integrating laurent polynomials along with an application to estimate weighted integrals on 1 1 with nearby singularities are given finally numerical examples involving interpolatory rules whose nodes are zeros of orthogonal laurent polynomials are also presented

fuzzy portfolio selection models with value at risk based on fuzzy value at risk var this paper proposes a new portfolio selection model psm called the var based fuzzy psm var fpsm compared with the existing fpsms the var can directly reflect the greatest loss of a selected case under a given confidence level in this study when the security returns are taken as trapezoidal triangular and gaussian fuzzy numbers several crisp equivalent models of the var fpsm are derived which can be handled by any linear programming solvers in general situations an improved particle swarm optimization algorithm on the basis of fuzzy simulation is designed to search for the approximate optimal solutions to illustrate the proposed model and the behavior of the improved particle swarm optimization algorithm two numerical examples are provided and the results are discussed furthermore the proposed algorithm is compared with some existing approaches to fuzzy portfolio selection such as the genetic algorithm and simulated annealing

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 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

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

droplet theory of correlations in ordered phases the explicit droplet theory of low dimensional phase transitions developed recently is extended to yield a description of the pair correlation function for the universality class of q state potts models in the ordered phase and in zero external field it is shown that in d 1 epsilon dimensions the short distance behaviour of the correlation function and in particular the exponent eta are controlled by nested nearly spherical droplets in contrast it is argued that the spatial dependence of the large distance behaviour in d 2 or above is controlled by highly anisotropic droplets thus illuminating the widom relation sigma xi d 1 approximately constant linking the dimensionless surface tension sigma and the bulk correlation length xi in the particular case of d 2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings in the spirit of recent independent arguments by abraham and by fisher a non ornstein zernike correlation function prefactor and the result of sigma xi 1 2 follow in accord with exact results for the q 2 ising case and with implications for the q to 1 percolation problem

the ising model below tc calculation of nonuniversal amplitudes using a primitive droplet model using recent exact results for the surface free energies of ising droplets on a square lattice and applying a primitive droplet theory due to langer 1967 the author obtains an expression for the principal nonuniversal amplitude appearing in the imaginary part of the analytic continuation of the free energy across the coexistence line for the square lattice ising model below tc the significance of this calculation is the known dependence on this amplitude of the large order terms in the expansion of the physical free energy in powers of h thus allowing a direct numerical comparison with existing series data to be made after carrying out a simple renormalisation of the droplet volume term the author finds excellent numerical agreement

generalized fuzzy games and fuzzy equilibria abstract in this paper we define the concepts of generalized fuzzy game and fuzzy equilibrium next we shall prove new existence theorems of equilibrium in generalized fuzzy games with uncountable number of agents our results generalize the corresponding results due to arrow debreu kim lee and yannelis prabhakar in several aspects

comparison of fuzzy values on a continuous domain abstract a comparison measure called the satisfaction function sf is extended in this paper the sf estimates the degree to which arithmetic comparisons between two fuzzy values are satisfied the previously proposed sf was defined on a discrete domain sfd so in order to compare continuous fuzzy values by the sfd the fuzzy values should be converted into discrete ones in order to directly compare continuous fuzzy values the sf is extended to a continuous domain sfc the sfc is defined based on the sfd and its properties are also developed since the sfc is extended from the sfd its properties includes those of the sfd and it has the same interpretation as the sfd thus it can be used in any applications instead of the sfd the sfc requires less computing time than the sfd under some conditions for comparisons between a fuzzy and a crisp value and between two crisp values the sfc are also extended

a least deviation method to obtain a priority vector of a fuzzy preference relation based on the transfer relationship between fuzzy preference relation and multiplicative preference relation this paper proposes a least deviation method to obtain a priority vector of a fuzzy preference relation and studies its desirable properties a simple convergent iterative algorithm is given and its practicality is shown by two numerical examples

cluster size distribution above the percolation threshold discusses how the cluster size distribution above the percolation threshold is related to the existence of metastable states of the potts model the authors thus find that the average number of clusters per site containing n sites is given by c n approximately n theta exp constant n1 1 d where d is the dimensionality of the lattice the exponent theta is given for a range of values of d

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

numerical tests of nucleation theories for the ising models the classical nucleation theory cnt is tested systematically by computer simulations of the two dimensional 2d and three dimensional 3d ising models with a glauber type spin flip dynamics while previous studies suggested potential problems with cnt our numerical results show that the fundamental assumption of cnt is correct in particular the becker doring theory accurately predicts the nucleation rate if the correct droplet free energy function is provided as input this validates the coarse graining of the system into a one dimensional markov chain with the largest droplet size as the reaction coordinate furthermore in the 2d ising model the droplet free energy predicted by cnt matches numerical results very well after a logarithmic correction term from langer s field theory and a constant correction term are added but significant discrepancies are found between the numerical results and existing theories on the magnitude of the logarithmic correction term in the 3d ising model our analysis underscores the importance of correctly accounting for the temperature dependence of surface energy when comparing numerical results and nucleation theories

on the convergence of two point pade type approximants for a given function f holomorphic in a domain omega containing the origin and the point at infinity rational interpolants with prescribed poles are constructed and their convergence properties analyzed numerical results are also given

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 characterization of equilibrium strategy of two person zero sum games with fuzzy payoffs in this paper we consider fuzzy matrix games namely two person zero sum games with fuzzy payoffs based on fuzzy max order for such games we define three kinds of concepts of minimax equilibrium strategies and investigate their properties first we shall show that these equilibrium strategies are characterized as nash equilibrium strategies of a family of parametric bi matrix games with crisp payoffs second we investigate properties of values of fuzzy matrix games by means of possibility and necessity measures in addition we give a numerical example to illustrate utility of our approaches

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

droplet theory in low dimensions ising systems in zero field the authors develop a theory of the universal configurational physics underlying critical point phenomena in the ising universality class the theory is formally justifiable in d 1 epsilon dimensions and may be regarded as the natural continuation of the kink based theory of one dimension which it incorporates as a limiting case in d 1 epsilon the configurational building block is the droplet the typical droplet is not spherical and the many droplet assembly is not dilute the implied problems are handled with renormalisation group methods it is found that droplet shape fluctuation effects control the correlation length exponent nu while the nesting of droplets within droplets control the order parameter exponent beta the exponents nu and beta thus effectively define respectively the fractal dimensions of the droplet surface and the droplet volume the theory is used to determine and illuminate the critical behaviour of further quantities including the free energy the susceptibility the droplet number distribution and the distribution of the intra droplet order

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

positive interpolatory quadrature formulas and para orthogonal polynomials we establish a relation between quadrature formulas on the interval 1 1 that approximate integrals of the form j f 11 f x x dx and szego quadrature formulas on the unit circle that approximate integrals of the form i f f ei d the functions x and are assumed to be weight functions on 1 1 and respectively and are related by cos sin it is well known that the nodes of szego formulas are the zeros of the so called para orthogonal polynomials bn z n z n z 1 n z and n z being the orthogonal and reciprocal polynomials respectively with respect to the weight function furthermore for 1 we have recently obtained gauss type quadrature formulas on 1 1 see bultheel et al j comput appl math 132 1 2000 1 in this paper making use of the para orthogonal polynomials with 1 a one parameter family of interpolatory quadrature formulas with positive coefficients for j f is obtained along with error expressions for analytic integrands finally some illustrative numerical examples are also included

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

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

quadrature and orthogonal rational functions abstract classical interpolatory or gaussian quadrature formulas are exact on sets of polynomials the szeg quadrature formulas are the analogs for quadrature on the complex unit circle here the formulas are exact on sets of laurent polynomials in this paper we consider generalizations of these ideas where the laurent polynomials are replaced by rational functions that have prescribed poles these quadrature formulas are closely related to certain multipoint rational approximants of cauchy or riesz herglotz transforms of a positive or general complex measure we consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence

two person zero sum game approach for fuzzy multiple attribute decision making problems the multiple attribute decision making madm problem deals with candidate priority alternatives with respect to various attributes madm techniques are popularly used in diverse fields such as engineering economics management science transportation planning etc several approaches have been developed for assessing the weights of madm problems e g the eigenvector method electre topsis etc in this paper we obtained weights of a madm problem with a fuzzy decision matrix by formulating it as a two person zero sum game with an uncertain payoff matrix moreover the equilibrium solution and the resolution method for the madm game have also been developed these results are validated by a product development example of nano materials our study is an innovative approach for applying the game theory on madm problems with management of technology mot

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

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

droplet theory in low dimensions ising systems in an ordering field the authors extend the microscopic droplet theory of ising systems developed recently to incorporate the effects of an ordering field the theory yields a free energy which is the solution of a renormalisation group equation describing droplet nesting and which has a full scaling form the behaviour near the coexistence curve is investigated and found to display the essential singularity suggested by primitive droplet models but with parameters renormalised in a physically intelligible way the droplet population function implied by the theory is parametrised in a simple way in space dimension d 2 to yield predictions for various thermodynamic properties including the equation of state which are found to be in fair accord with series based results the structure of the field dependent droplet number distributions is investigated and contrasted with the forms assumed in phenomenological droplet theories the domain of validity of the theory is assessed it is concluded that the theory fails to describe sufficiently large droplets in the presence of a finite field and is thus in principle trustworthy only in a region close to the coexistence curve

validity of classical nucleation theory for ising models while the classical nucleation theory cnt is widely used to predict the rate of first order phase transitions its validity has been questioned due to discrepancies with experiments we systematically test the individual components of cnt by computer simulations of the ising models and confirm its fundamental assumptions under a wide range of conditions h 0 01 text char21 0 13j t 0 44 text char21 0 84 t c in two dimensions and h 0 30 text char21 0 60j t 0 48 text char21 0 62 t c in three dimensions first cnt accurately predicts the nucleation rate if the correct droplet free energy is provided furthermore theoretical prediction of droplet free energy matches numerical results very well in the two dimensional 2d ising model if appropriate correction terms are added this establishes the 2d ising model as an important reference point where existing theories can predict nucleation rate accurately with no adjustable parameters

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

orthogonality interpolation and quadratures on the unit circle and the interval 1 1 given a weight function s x on 1 1 or more generally a positive borel measure the erdos turan theorem assures the convergence in l 2 s norm to a function f of its sequence of interpolating polynomials at the zeros of the orthogonal polynomials or equivalently at the nodes of the gauss christoffel quadrature formulas associated with s in this paper we will extend this result to the nodes of the gauss radau and gauss lobatto quadrature formulas by passing to the unit circle and taking advantage of the results on interpolation by means of laurent polynomials at the zeros of certain para orthogonal polynomials with respect to the weight function w q s cos q sin q on p p as a consequence an application to the construction of certain product integration rules on finite intervals of the real line will be given several numerical experiments are finally carried out

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

linear quadratic uncertain differential game with application to resource extraction problem uncertain differential game investigates interactive decision making of players over time and the system dynamics is described by an uncertain differential equation this paper goes further to study the two player zero sum uncertain differential game in order to guarantee the saddle point nash equilibrium a max min theorem is provided furthermore when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic the existence of saddle point nash equilibrium is obtained via the solvability of a corresponding riccati equation finally a resource extraction problem is analyzed by using the theory proposed in this paper

a fuzzy approach to strategic games a game is a decision making situation with many players each having objectives that conflict with each other the players involved in the game usually make their decisions under conditions of risk or uncertainty in the paper a fuzzy approach is proposed to solve the strategic game problem in which the pure strategy set for each player is already defined based on the concepts of fuzzy set theory the approach uses a multicriteria decision making method to obtain the optimal strategy in the game a method which shows more advantages than the classical game methods moreover with this approach some useful conclusions are reached concerning the famous prisoner s dilemma problem in game theory