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

fuzzy optimization models for the design of wdm networks the wavelength division multiplexing network design problem is considered in this paper the literature is replete with prescriptive integer programming models for various versions of wavelength division multiplexing network design problems but in no case they consider the real situation of solving them taking into account their inherent vagueness in this paper this vagueness is assumed and handled by means of fuzzy sets based methodologies in particular two fuzzy integer linear programming models are proposed to be used as auxiliary models for solving the former problem as an original way to effectively solve these models obtaining cost effective network designs we use three metaheuristic procedures that are compared among them

on the sensitivity of membership functions for fuzzy linear programming problems abstract for a fuzzy linear programming problem it is usually assumed that the decision maker exactly knows the shape of the membership functions taking part in it for the sake of simplicity often linear membership functions are supposed but this seems a poor reason here it is shown that in the case of fuzzy linear programming problems whether or not a fuzzy optimal solution has been found by using linear membership functions modeling the constraints possible further changes of those membership functions do not affect the former optimal solution this sensitivity analysis performed for those membership functions and the corresponding solutions shows the convenience of using linear functions instead of other more complicated ones

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

sobolev orlicz inequalities ultracontractivity and spectra of time changed dirichlet forms let mathcal e be a regular dirichlet form on l 2 x m a positive radon measure charging no sets of zero capacity and an n function we prove that the sobolev orlicz inequality soi f 2 l phi x mu leq c mathcal e 1 f for every f in d mathcal e is equivalent to a capacitary type inequality further we show that if d mathcal e is continuously embedded into l 2 x the latter one implies some integrability condition which is nothing else but the classical uniform integrability condition if is finite we also prove that a soi for mathcal e yields a nash type inequality and if further m and is admissible it yields the ultracontractivity of the corresponding semigroup after in the spirit of sois we derive criteria for d mathcal e to be compactly embedded into l 2 provided is finite as an illustration of the theory we shall relate the compactness of the latter embedding to the discreteness of the spectrum of the time changed dirichlet form and shall derive lower bounds for its eigenvalues in term of

post optimality analysis on the membership functions of a fuzzy linear programming problem models of linear programming problems with fuzzy constraints are very well known in the current literature in almost all cases to solve these problems linear membership functions are used because they have very good properties and are very easy to manipulate in some cases however because of the knowledge that the decision maker has such membership functions could be modeled as nonlinear although the complexity of the problem could increase this paper considers the use of nonlinear membership functions in fuzzy linear programming problems to show that the corresponding solution to be obtained can be derived from a parallel linear model moreover it is easier to solve than the nonlinear model making use of a similar procedure to that of post optimal analysis in classical linear programming the case in which these membership functions are defined by means of piecewise linear approximations is also considered and analyzed

fuzzy location problems on networks location problems concern a wide set of fields where it is usually assumed that exact data are known however in real applications the location of the facility considered can be full of linguistic vagueness that can be appropriately modelled using networks with fuzzy values thus fuzzy location problems on networks arise this paper deals with their general formulation and the description of the ways to solve them namely we show the variety of problems that can be considered in this context and for some of them we propose the most operative approaches for their solution

a dual approach to solve the fuzzy linear programming problem abstract a concept of fuzzy objective based on the fuzzification principle is presented in accordance with this concept the fuzzy linear mathematical programming problem is easily solved a relationship of duality among fuzzy constraints and fuzzy objectives is given the dual problem of a fuzzy linear programming problem is also defined

fuzzy programming approach to multi level programming problems multi level programming is characterized as mathematical programming to solve decentralized planning problems we have considered a multi level linear programming problem and applied fuzzy mathematical programming fmp approach to obtain the solution of the system we have suggested fmp method for the minimization of the objectives using linear membership functions fmp is a supervised search procedure supervised by the higher level decision maker dm the higher level dm provides the preferred values of decision variables under his control to enable the lower level dm to search for his optimum in a wider feasible space and the bounds of his objective function to direct the lower level dm to search for his solutions in the right direction

short rational generating functions for solving some families of fuzzy integer programming problems in this paper we present new complexity results for solving single and multiobjective fuzzy integer programs with soft constraints and fuzzy objective function coefficients our method is based on the use of barvinok s short rational generating functions of convenient transformations of the fuzzy problems to crisp ones this analysis allows us to provide new insights on the structure of these families of fuzzy programs

a general model for fuzzy linear programming abstract in the current literature there are several models of fuzzy linear programming problems the aim of this paper is first to describe the more important problems in fuzzy linear programming and second to give a general model of fuzzy linear programming problems involving all of the above ones finally a resolution method for that general model is proposed

three models of fuzzy integer linear programming abstract in this paper we study some models for dealing with fuzzy integer linear programming problems which have a certain lack of precision of a vague nature in their formulation we present methods to solve them with either fuzzy constraints or fuzzy numbers in the objective function or fuzzy numbers defining the set of constraints these methods are based on the representation theorem and on fuzzy number ranking methods

eigentime identity for transient markov chains abstract an eigentime identity is proved for transient symmetrizable markov chains for general markov chains if the trace of green matrix is finite then the expectation of first leap time is uniformly bounded both of which are proved to be equivalent for single birth processes for birth death processes the explicit formulas are presented as an application we give the bounds of exponential convergence rates of sub markov semigroup p t from l to l

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

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

orlicz sobolev inequalities for sub gaussian measures and ergodicity of markov semi groups abstract we study coercive inequalities in orlicz spaces associated to the probability measures on finite and infinite dimensional spaces which tails decay slower than the gaussian ones we provide necessary and sufficient criteria for such inequalities to hold and discuss relations between various classes of inequalities

functional inequalities for transient birth death processes and their applications abstract we give a new diagram about uniform decay empty essential spectrum and various functional inequalities including poincare inequalities super and weak poincare inequalities for transient birth death processes this diagram is completely opposite to that in ergodic situation and substantially points out the difference between transient birth death processes and recurrent ones the criterion for the empty essential spectrum is achieved some matching sufficient and necessary conditions for weak poincare inequalities and super poincare inequalities are also presented

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

finite state unification automata and relational languages abstract we define the new notion of a finite state unification automaton a device for finite state recognition of relational languages by means of unification transitions words in such a language are formed by composing base relations and have the general form r i 1 x j 1 x k 1 r i n x j n x k n for some n generation of such languages by regarding horn clauses as grammers has been considered before but to the best of our knowledge recognizing such languages by suitably designed automata is a new approach the main result presented is a pumping lemma forming a necessary condition for finite state recognizability some example results about such automata are given

finite memory automata abstract a model of computation dealing with infinite alphabets is proposed this model is based on replacing the equality test by substitution it appears to be a natural generalization of the classical rabin scott finite state automata and possesses many of their closure and decision properties also when restricted to finite alphabets the model is equivalent to finite state automata

algorithms of discrete optimization and their application to problems with fuzzy coefficients an approach to solving optimization problems with fuzzy coefficients in objective functions and constraints is described it consists in formulating and solving one and the same problem within the framework of mutually related models with constructing equivalent analogs with fuzzy coefficients in objective functions alone it enables one to maximally cut off dominated alternatives from below as well as from above since the approach is applied within the context of fuzzy discrete optimization problems several modified algorithms of discrete optimization are discussed these algorithms are associated with the method of normalized functions are based on a combination of formal and heuristic procedures and allow one to obtain quasi optimal solutions after a small number of steps thus overcoming the computational complexity posed the np completeness of discrete optimization problems the subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective decision making in a fuzzy environment with using techniques based on fuzzy preference relations the techniques are also directly applicable to situations in which the decision maker is required to choose alternatives from a set of explicitly available alternatives the results of the paper are of a universal character and can be applied to the design and control of systems and processes of different purposes as well as the enhancement of corresponding cad cam systems and intelligent decision making systems the results of the paper are already being used to solve problems of power engineering

non l r type fuzzy parameters in mathematical programming problems the triangular norm based operations in fuzzy logic usually lead to non l r type fuzzy sets this study considers mathematical programming problems with non l r type fuzzy parameters it shows that the fuzzy solutions to such problems can be obtained by solving an optimization problem on a mixed domain the necessary and sufficient conditions for solving the resulting optimization problems are investigated by employing the theory of convex optimization on mixed domains this is the first attempt to solve the fuzzy optimization problem with non l r type membership functions in view of optimization problems on a mixed domain

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs