on duality in linear programming under fuzzy environment a pair of linear primal dual problems is introduced under fuzzy environment and appropriate results are proved to establish the duality relationship between them possible extensions are also suggested

fuzzy goal programming complementary slackness conditions and computational schemes abstract goal programming gp is an important category in linear programming and multiobjective versions which minimize the deviation among value of each objective function and its goal with satisfying other constraints of problem in this paper we deal with fuzzy goal programming fgp which is a generalized problem in vague nature where parameters relations and goals are given as fuzzy quantities with respect to imprecise data recently inuiguchi et al m inuiguchi j ramik t tanio m vlach satisficing solutions and duality in interval and fuzzy linear programming fuzzy sets and systems 135 2003 151 177 have proved some important dual theorems on fgp which are very close to the same crisp versions in continuation to their works based on fuzzy extensions of relations we get complementary slackness conditions for fgp which have an essential role in many optimization techniques moreover we obtain the most optimistic and pessimistic satisficing solutions employing these conditions and present a convex combination of them which covers the satisficing solutions for a fixed level of certainty our practical schemes implement an efficient interior point algorithm as sub procedure so they are polynomial time algorithms

structural induction and coinduction in a fibrational setting we present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types our main results provide sufficient criteria for the validity of such principles in the presence of comprehension the induction principle for initial algebras is admissible and dually in the presence of quotient types the coinduction principle for terminal coalgebras is admissible after giving an alternative formulation of induction in terms of binary relations we combine both principles and obtain a mixed induction coinduction principle which allows us to reason about minimal solutionsx x wherexmay occur both positively and negatively in the type constructor we further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the abstract logic we consider is contextually functionally complete all the main results follow from a basic result about adjunctions between categories of algebras inserters

an implementation of the dual affine scaling algorithm for minimum cost flow on bipartite uncapacitated networks this paper describes an implementation of the dual affine scaling algorithm for linear programming specialized to solve minimum cost flow problems on bipartite uncapacitated networks this implementation uses a preconditioned conjugate gradient algorithm to solve the system of linear equations that determines the search direction at each iteration of the interior point algorithm two preconditioners are considered a diagonal preconditioner and a preconditioner based on the incidence matrix of an approximate maximum weighted spanning tree of the network under dual nondegeneracy this spanning tree allows for early identification of the optimal solution by applying an epsilon perturbation to the cost vector an optimal extreme point primal solution is produced in the presence of dual degeneracy the implementation is tested by solving several large instances of randomly generated assignment problems comparing solution times with the network simplex code netflo and the relaxation algorithm code relax

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

canonical fuzzy numbers of dimension two abstract recently several authors have dealt with interactive fuzzy numbers this paper proposes a new approach to interactions between fuzzy numbers by using canonical fuzzy numbers of dimension two

fuzzy n cell numbers and the differential of fuzzy n cell number value mappings in this paper we give the definition of a special kind of n dimension fuzzy numbers fuzzy n cell numbers discuss their operations and representation theorems define a complete metric on the fuzzy n cell number space and prove that the metric is equivalent to the supremum metric derived by the hausdorff metric between the level sets of the n dimension fuzzy numbers and obtain an embedding theorem of the fuzzy n cell number space isometrically embeds it into a concrete banach space we also consider the differential of the fuzzy mappings from an interval into the fuzzy n cell number space by using the embedding theorem

satisficing solutions and duality in interval and fuzzy linear programming in this paper we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions the necessary tools for dealing with such problems in this way we show that the class of crisp classical lp problems can be embedded into the class of flp ones moreover for flp problems we define the concept of duality and prove the weak and strong duality theorems further we define a class of interval linear programming problems as a special subclass of flp problems and apply the previous results to this special case

representation of uncertain multichannel digital signal spaces and study of pattern recognition based on metrics and difference values on fuzzy n cell number spaces in this paper we discuss the problem of characterization for uncertain multichannel digital signal spaces propose using fuzzy n cell number space to represent uncertain n channel digital signal space and put forward a method of constructing such fuzzy n cell numbers we introduce two new metrics and concepts of certain types of difference values on fuzzy n cell number space and study their properties further based on the metrics or difference values appropriately defined we put forward an algorithmic version of pattern recognition in an imprecise or uncertain environment and we also give practical examples to show the application and rationality of the proposed techniques

linear programming with fuzzy variables in practice there are many problems in which all decision parameters are fuzzy numbers and such problems are usually solved by either possibilistic programming or multiobjective programming methods unfortunately all these methods have shortcomings in this note using the concept of comparison of fuzzy numbers we introduce a very effective method for solving these problems then we propose a new method for solving linear programming problems with fuzzy variables

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

generalized differentiability for n dimensional fuzzy number valued functions and fuzzy optimization in this paper we present a characteristic theorem for the generalized difference concept of n dimensional fuzzy numbers by means of support functions the concepts of differentiability for n dimensional fuzzy number valued functions are presented and discussed using the new generalized difference the relationships of g differentiability s differentiability and l differentiability are characterized furthermore optimization problems with s convex and s differentiable fuzzy number valued functions are described

interactive fuzzy programming for multi level linear programming problems with fuzzy parameters this paper presents interactive fuzzy programming for multi level linear programming problems with fuzzy parameters in fuzzy programming for multi level linear programming problems recently developed by lai et al since the fuzzy goals are determined for both an objective function and decision variables at the upper level undesirable solutions are produced when these fuzzy goals are inconsistent in order to overcome such problems after eliminating the fuzzy goals for decision variables interactive fuzzy programming for multi level linear programming problems with fuzzy parameters is presented in our interactive method after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of decision makers with considerations of overall satisfactory balance among all levels illustrative numerical examples for two level and three level linear programming problems are provided to demonstrate the feasibility of the proposed method

dimension free quasiconformal distortion in n space most distortion theorems for k quasiconformal mappings in rn n 2 depend on both n and k in an essential way with bounds that become infinite as n tends to oo the present authors obtain dimension free versions of four well known distortion theorems for quasiconformal mappings namely bounds for the linear dilatation the schwarz lemma the e distortion theorem and the r1 quasisymmetry property of these mappings they show that the upper estimates they have obtained in each of these four main results remain bounded as n tends to oo with k fixed the proofs are based on a dimensioncancellation property of the function t h r 1 r t k t 0 k 0 where r t is the capacity of a teichmiiller extremal ring in rn the authors also prove a dimension free distortion theorem for the absolute cross ratio under k quasiconformal mappings of n from which several other distortion theorems follow as special cases

linear programming with fuzzy coefficients in constraints abstract this paper presents a new method for solving linear programming problems with fuzzy coefficients in constraints it is shown that such problems can be reduced to a linear semi infinite programming problem the relations between optimal solutions and extreme points of the linear semi infinite program are established a cutting plane algorithm is introduced with a convergence proof and a numerical example is included to illustrate the solution procedure

a shortest path problem on a network with fuzzy arc lengths abstract we concentrate on a shortest path problem on a network in which a fuzzy number instead of a real number is assigned to each arc length introducing an order relation between fuzzy numbers based on fuzzy min a nondominated path or pareto optimal path from the specified node to every other node is defined an algorithm for solving the problem is developed on the basis of the multiple labeling method for a multicriteria shortest path as a result a number of nondominated paths can be obtained and is offered to a decision maker however a number of nondominated paths derived from large scale network may be too numerous for him to choose a preferable path due to this situation we propose a method to reduce the number of paths according to a possibility level the proposed algorithm is numerically evaluated on large scale random networks

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

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

joint canonical fuzzy numbers abstract starting from ramik and nakamura s report 1991 we introduce and fully characterize the joint canonical fuzzy numbers of arbitrary dimension the crucial part is to find the least value of the maximum of absolute values of several linear combinations surprisingly there is a formula for it based upon determinants formed by the coefficients of the linear combinations finally we give a full description of joint canonical fuzzy numbers of dimension 2 provided by 5 infinite families

fuzzy cooperative games arising from linear production programming problems with fuzzy parameters abstract in the present study we consider a production model in which multiple decision makers pool resources to produce finished goods such a production model which is assumed to be linear can be formulated as mathematical programming problems with fuzzy parameters from the viewpoint of experts imprecise or fuzzy understanding of the nature of parameters in a problem formulation process it is shown that a cooperative game with values of coalitions represented by fuzzy numbers arises from the linear production programming problem with fuzzy parameters and such a game is referred to as a fuzzy linear production programming game the values of coalitions of the game can be obtained by employing a parametric approach to solving the linear production programming problem with fuzzy parameters it is proven that the core of the game is not empty and an interval in the core is computed by using the ideas of sensitivity analysis and the duality theory of linear programming problems

solutions based on fuzzy goals in fuzzy linear programming games abstract in this paper we deal with linear programming problems with fuzzy parameters from the viewpoint of experts imprecise or fuzzy understanding of the nature of parameters in a problem formulation process and consider fuzzy linear programming games arising from the linear programming problems constructing fuzzy goals of coalitions with respect to payoff in order to suitably reflect fuzzy environments in which the linear programming games are formulated we define solutions maximizing minimal fuzzy goal and solutions maximizing the sum of fuzzy goals and develop computational methods for obtaining the solutions finally we give a numerical example to illustrate construction of a fuzzy linear programming game and derivation of the solutions

quasiconformal homeomorphisms and the convex hull boundary we investigate the relationship between an open simply connected region s 2 and the boundary y of the hyperbolic convex hull in h 3 of s 2 a counterexample is given to thurston s conjecture that these spaces are related by a 2 quasiconformal homeomorphism which extends to the identity map on their common boundary in the case when the homeomorphism is required to respect any group of mobius transformations which preserves we show that the best possible universal lipschitz constant for the nearest point retraction r y is 2 we find explicit universal constants 0 c 2 c 1 such that no pleating map which bends more than c 1 in some interval of unit length is an embedding and such that any pleating map which bends less than c 2 in each interval of unit length is embedded we show that every k quasiconformal homeomorphism d 2 d 2 is a k k quasi isometry where k is an explicitly computed function the multiplicative constant is best possible and the additive constant k is best possible for some values of k

on the restudy of fuzzy complex analysis part 1 the sequence and series of fuzzy complex numbers and their convergences this paper studies the convergence of the sequence and series of fuzzy complex numbers we define the metric convergence and levelwise convergence of sequences on the metric space of fuzzy complex numbers and discuss the relationship between two convergences all this may be a foundation for researching fuzzy complex analysis

fuzzy complex analysis ii integration abstract this paper continues the author s research in fuzzy complex analysis where we now define and study the basic properties of a fuzzy contour integral for fuzzy mappings which map a rectifiable curve in the complex plane into fuzzy complex numbers

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

approximate reasoning with time abstract the need to make time dependent assumptions is frequently encountered in many systems especially in dynamical systems this paper presents a theoretical study of the representation and manipulation of some vague concepts in the temporally changing world our approach to the modeling is based on zadeh s theory of possibility we introduce a concept of similarity index in reasoning with time we present two different types of widely used models one rule based and another resolution based for reasoning with time we consider simply yet concrete examples to illustrate the usefulness of the proposed models

possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems abstract in this paper several kinds of possibility distributions of fuzzy variables are studied in possibilistic linear programming problems to reflect the inherent fuzziness in fuzzy decision problems interval and triangular possibility distributions are used to express the non interactive cases between the fuzzy decision variables and exponential possibility distributions are used to represent the interrelated cases possibilistic linear programming problems based on exponential possibility distributions become non linear optimization problems in order to solve optimization problems easily algorithms for obtaining center vectors and distribution matrices in sequence are proposed by the proposed algorithms the possibility distribution of fuzzy decision variables can be obtained

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

on polygonal fuzzy sets and numbers in this paper we propose a new mathematical formalization of the concept of polygonal fuzzy numbers and an extension of this notion to fuzzy sets on r n we study the mathematical structure of these families of fuzzy sets and show that each family is a complete and separable metric space when endowed with the generalized hausdorff metric moreover we show that for n 1 the families of polygonal fuzzy numbers are isomorphic to some convex and closed convex cone of a finite dimensional space we obtain generalizations and extensions of some previous results on polygonal fuzzy numbers and simplified proofs of some well known results about approximation of fuzzy n dimensional quantities finally some developments about the approximation of families of fuzzy sets are introduced

on fuzzy n cell numbers and n dimension fuzzy vectors fuzzy n cell numbers are a kind of special n dimension fuzzy numbers whose sets of cuts are all hyperrectangles n dimension fuzzy vectors are the cartesian products of n one dimension fuzzy numbers in this paper we show that the fuzzy n cell number space and the n dimension fuzzy vector space are identical and obtain the relations of the joint membership function and the edge membership functions of a fuzzy n cell number i e a n dimension fuzzy vector using fuzzy vector form representations of fuzzy n cell numbers we study the metric of two fuzzy n cell numbers the supremum of a set of fuzzy n cell numbers and operational rules of addition scalar multiplication and multiplication of fuzzy n cell numbers and get the fuzzy vector form characterizations of the differentiability measurability and integrability of fuzzy n cell number valued fuzzy mappings

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

some remarks for fuzzy complex analysis abstract this paper gives some results for fuzzy complex sets generalize and improve the corresponding work of fuzzy complex numbers in 1 we also introduce the derivative and integral of functions mapping complex numbers into the set of generalized fuzzy complex numbers and it is different to the definitions in 2 4 all this may be a foundation for researching fuzzy complex analysis

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

numerical solutions of fuzzy differential equations abstract numerical algorithms for solving fuzzy ordinary differential equations fode are considered a scheme based on the classical euler method is discussed in detail and this is followed by a complete error analysis the algorithm is illustrated by solving several linear and nonlinear fuzzy cauchy problems

duality in linear programming with fuzzy parameters and matrix games with fuzzy pay offs abstract a dual for linear programming problems with fuzzy parameters is introduced and it is shown that a two person zero sum matrix game with fuzzy pay offs is equivalent to a primal dual pair of such fuzzy linear programming problems further certain difficulties with similar studies reported in the literature are discussed

solving a special class of large scale fuzzy multiobjective integer linear programming problems we present a method useful in solving a special class of large scale multiobjective integer problems depending on the decomposition algorithm these problems involve fuzzy parameters on the right hand side of the independent constraints the presented solution method is based upon a combination of the decomposition algorithm coupled with the weighting method together with the branch and bound method an illustrative numerical example is given to clarify the theory and the method discussed in this paper

fuzzy complex numbers abstract this paper first defines fuzzy complex numbers and shows closure under the basic arithmetic operations it is shown that arithmetic of fuzzy complex numbers may be performed in terms of cuts next two special types of fuzzy complex numbers based on the forms z x iy and z re i for regular complex numbers are investigated introducing a metric on the space of fuzzy complex numbers then allows us to discuss continuity and differentiability of fuzzy complex functions

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

application of fuzzy minimum cost flow problems to network design under uncertainty this paper deals with fuzzy quantities and relations in multi objective minimum cost flow problem when t norms and t conorms are available the goal programming is applied to minimize the deviation among the multiple costs of fuzzy flows and the given targets when the fuzzy supplies and demands are satisfied to obtain the most optimistic and the most pessimistic satisficing solutions of this problem two polynomial time algorithms are introduced applying some network transformations to demonstrate the performance of this approach in actual substances network design under fuzziness is considered and an efficient scheme is proposed including genetic algorithm together with fuzzy minimum cost flow problem this scheme is applied on a pilot network for more description

canonical fuzzy numbers of dimension two and fuzzy utility difference for understanding preferential judgments abstract a notion of canonical fuzzy numbers of dimension two is introduced to represent flexible joint distributions of two fuzzy numbers whose membership functions are of a common functional type the relationships between fuzzy numbers are given by normalized scalar called relational factors in the joint distributions then the canonical fuzzy numbers are adopted to operate fuzzy utility differences and their aggregations for understanding human uncertain preferential judgments among multiattribute alternatives

possibilistic linear programming with triangular fuzzy numbers abstract we consider a mathematical programming problem where all the parameters may be fuzzy variables specified by their possibility distribution and we define the possibility distribution of the objective function the ideas are illustrated on possibilistic linear programming

descent direction algorithm with multicommodity flow problem for signal optimization and traffic assignment jointly network managers wish to optimize control parameters such as signal setting which are very related to the traffic assignment models on the other hand traffic assignment patterns as an important instrument for predicting the amount of flow on network links are dependent to control decisions according to the significance of this concept some important papers about this mutually relation are reviewed in this paper then we implement a nonlinear algorithm on a minimal cost multicommodity flow mcmf problem to optimize some control policies subject to optimal flows although we take signal times into account but this approach has a far more reaching application in urban network control and design we employ a hybrid intelligent algorithm integrating decent direction algorithm and an interior point algorithm in a mutually consistent scheme for obtaining optimal signals and equilibrium flows an example is given to illustrate the effectiveness of our scheme

solving fuzzy transportation problems based on extension principle abstract transportation models play an important role in logistics and supply chain management for reducing cost and improving service this paper develops a procedure to derive the fuzzy objective value of the fuzzy transportation problem in that the cost coefficients and the supply and demand quantities are fuzzy numbers the idea is based on the extension principle a pair of mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level from different values of the membership function of the objective value is constructed two different types of the fuzzy transportation problem are discussed one with inequality constraints and the other with equality constraints it is found that the membership function of the objective value of the equality problem is contained in that of the inequality problem since the objective value is expressed by a membership function rather than by a crisp value more information is provided for making decisions

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters in this paper we present interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters using the level sets of fuzzy parameters the corresponding nonfuzzy two level linear fractional programming problem is introduced in our interactive method after determining fuzzy goals of decision makers at both levels a satisfactory solution is derived efficiently by updating a minimal satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels the satisfactory solution well balanced between both levels is easily computed by combined use of the bisection method the phase one of the simplex method and the variable transformation method by charnes and cooper an illustrative numerical example for two level linear fractional programming problems with fuzzy parameters is provided to demonstrate the feasibility of the proposed method

norm inequalities for vector functions abstract we study vector functions of r n into itself which are of the form x g x x where g 0 0 is a continuous function and call these radial functions in the case when g t t c for some c r we find upper bounds for the distance of image points under such a radial function some of our results refine recent results of l maligranda and s s dragomir in particular we study quasiconformal mappings of this simple type and obtain norm inequalities for such mappings

neural net solutions to fuzzy linear programming we show how a neural net with sign restrictions on its weights can be trained to produce approximate solutions to fuzzy linear programming problems the neural net approximates the joint solution buckley 1995 to the fuzzy linear program

strong categorical datatypes ii a term logic for categorical programming abstract this paper lifts earlier category theoretic results on datatypes to the level of an abstract language suitable for categorical programming language implementation the earlier work built a strongly normalizing categorical combinator reduction system based entirely on functorial strength which allows the distribution of context to the interior of datatypes here we declare inductive initial and coinductive final datatypes in a hagino wraith style to provide an expressive computing environment an inductive resp coinductive datatype consists of a strong type forming functor accompanied by 1 a collection of constructors resp destructors and 2 a fold resp unfold which is parametrized by state transformations to realize the appropriate universal property the high complexity of programming exclusively in categorical notation combinators warrants the development of a friendlier language isomorphic to the distributive categorical setting in this paper such a term logic which is also the programming language of the charity programming system is described this logic is first order and is dictated directly by the underlying categorical semantics it embodies primitive inductive and coinductive principles which reflect the uniqueness properties of the fold and unfold combinators several basic progam equivalences are demonstrated to illustrate how these inductive and coinductive principles can be used

the solution and duality of imprecise network problems duality properties have been investigated by many researchers in the recent literature they are introduced in this paper for a fully fuzzified version of the minimal cost flow problem which is a basic model in network flow theory this model illustrates the least cost of the shipment of a commodity through a capacitated network in terms of the imprecisely known available supplies at certain nodes which should be transmitted to fulfil uncertain demands at other nodes first we review on the most valuable results on fuzzy duality concepts to facilitate the discussion of this paper by applying hukuhara s difference approximated and exact multiplication and wu s scalar production we exhibit the flow in network models then we use combinatorial algorithms on a reduced problem which is derived from fully fuzzified mcfp to acquire fuzzy optimal flows to give duality theorems we utilize a total order on fuzzy numbers due to the level of risk and realize optimality conditions for providing some efficient combinatorial algorithms finally we compare our results with the previous worthwhile works to demonstrate the efficiency and power of our scheme and the reasonability of our solutions in actual decision making problems

weighted pseudometric approximation of 2 dimensional fuzzy numbers by fuzzy 2 cell prismoid numbers preserving the centroid abstract up to now there have been many research results about approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators however there are few investigations about approximations of n dimensional fuzzy numbers the main reason is that the structure of n dimensional fuzzy numbers are more complex this paper devotes to seek an approximation operator that produces a special kind of 2 dimensional fuzzy number which is the nearest one to the given 2 dimensional fuzzy number along the direction k the concept of four edges approximation for 2 dimensional fuzzy numbers is defined which is a special kind of 2 dimensional fuzzy numbers at the same time the concept of fuzzy 2 cell prismoid numbers is introduced and the nearest fuzzy 2 cell prismoid numbers of 2 dimensional fuzzy numbers which preserve the centroid of the core for the 2 dimensional fuzzy numbers with respect to the weighted pseudometric are discussed finally with respect to the criteria scale invariance identity criterion and nearness criterion some properties of the approximation operator are presented

fuzziness and branch and bound procedures applications to separation sequencing abstract the aim of the paper is to identify optimal and near optimal distillation sequences according to a particular objective function definition the potential solutions are valuated from the point of view of practical considerations rather than purely costing methods four rules of thumb addressing column sequencing expressed in vague and imprecise terms are implemented and quantified by the fuzzy set theory the integer programming problem is solved by a branch and bound procedure where the bounding and branching schemes are carried out from the four heuristic rules with fuzzy quantification furthermore the fuzzy set theory is also implemented for modelling data uncertainties particularly the partial molar flow rates from the results obtained with the fuzzy branch and bound procedure involving fuzzy data we conclude that fuzzy set theory can be considered as a promising tool for modelling uncertainty and vagueness and may have many other applications in process systems engineering

stable fuzzy logic design of point to point control for mechanical systems an approach for the development of fuzzy point to point control laws for second order mechanical systems is presented asymptotic stability of the resulting closed loop system is proved using lyapunov stability theory closed loop performance and robustness are quantified in terms of the parameters of membership functions as opposed to most existing fuzzy control laws the closed loop stability of the proposed controller does not depend on the knowledge of the entire dynamics moreover the approach does not require the plant to be open loop stable the proposed approach is demonstrated on design and simulation study of a fuzzy controller for a two link robotic arm

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

fuzzy complex analysis i differentiation abstract we consider functions mapping real numbers into the set of generalized complex fuzzy numbers c we define two derivatives of such functions which are generalizations of the derivative of real fuzzy mappings presented by dubois and prade 4 5 sufficient conditions on the function are given so that derivative will be a function mapping real numbers into c if the functional value is always a rectangular complex fuzzy number made up to lr real fuzzy numbers then we show that the derivative is easily computed from the derivatives of the lr fuzzy numbers other properties of the derivatives are presented

a fuzzy relational identification algorithm and its application to predict the behaviour of a motor drive system fuzzy relational identification builds a relational model describing a system s behaviour by a nonlinear mapping between its variables in this paper we propose a new fuzzy relational algorithm based on the simplified max min relational equation the algorithm presents an adaptation method applied to the gravity centre of each fuzzy set based on the error integral value between the measured and predicted system s output and uses the concept of time variant universe of discourse the identification algorithm also includes a method to attenuate the noise influence in the extracted system s relational model using a fuzzy filtering mechanism the algorithm is applied to a one step forward prediction of a simulated and experimental motor drive system the identified model has its input output variables stator reference current and motor speed signal treated as fuzzy sets whereas the relations existing between them are described by means of a matrix r defining the relational model extracted by the algorithm the results show the good potentialities of the algorithm in predicting the behaviour of the system and in attenuating through the fuzzy filtering method possible noise distortions in the relational model

fuzzy approximation relations on fuzzy n cell number space and their applications in classification in this paper the problems of fuzzy binary relations on fuzzy n cell number space and their applications are investigated firstly we have defined some fuzzy approximation relations on fuzzy n cell number space and studied their properties secondly as application we have developed an algorithmic version of classification in an imprecise or uncertain environment by using the fuzzy approximation relations practical examples are provided to show the application and rationality of the proposed techniques

possible and necessary optimality tests in possibilistic linear programming problems in this paper two kinds of optimality concepts are introduced into fuzzy linear programming problems in order to discuss the rationality of an obtained solution one is possible optimality evaluating the possibility degree to which the solution is optimal the other the dual concept is necessary optimality evaluating the necessity degree to which the solution is optimal when possibility distributions are defined by closed intervals degrees of possible and necessary optimalities become binary necessary and sufficient conditions for possible and necessary optimalities are induced in this case extending the results methods of calculating the degrees of possible and necessary optimalities are proposed some numerical examples are given to demonstrate the proposed methods

evolutionary algorithm solution to fuzzy problems fuzzy linear programming in this paper we wish to find solutions to the fully fuzzified linear program where all the parameters and variables are fuzzy numbers we first change the problem of maximizing a fuzzy number the value of the objective function into a multi objective fuzzy linear programming problem we prove that fuzzy flexible programming can be used to explore the whole undominated set to the multi objective fuzzy linear program an evolutionary algorithm is designed to solve the fuzzy flexible program and we apply this program to two applications to generate good solutions

a co induction principle for recursively defined domains abstract this paper establishes a new property of predomains recursively defined using the cartesian product disjoint union partial function space and convex powerdomain constructors we prove that the partial order on such a recursive predomain d is the greatest fixed point of a certain monotone operator associated to d this provides a structurally defined family of proof principles for these recursive predomains to show that one element of d approximates another it suffices to find a binary relation containing the two elements that is a post fixed point for the associated monotone operator the statement of the proof principles is independent of any of the various methods available for explicit construction of recursive predomains following milner and tofte 1991 the method of proof is called co induction it closely resembles the way bisimulations are used in concurrent process calculi milner 1989 two specific instances of the co induction principle already occur in the work of abramsky 1990 1991 in the form of internal full abstraction theorems for denotational semantics of sccs and lazy lambda calculus in the first case post fixed binary relations are precisely abramsky s partial bisimulations whereas in the second case they are his applicative bisimulations the co induction principle also provides an apparently useful tool for reasoning about equality of elements of recursively defined datatypes in strict or lazy higher order functional programming languages

ranking of multidimensional uncertain information based on metrics on the fuzzy ellipsoid number space on the usual high dimensional fuzzy number space the usual supremum metric d derived by the hausdorff metric between the level sets of usual n dimension fuzzy numbers and the p metric rho p are the most common metrics however due to the complexity of the level sets of usual n dimensional fuzzy numbers the two kinds of metrics not only have a tendency to be rougher but also are difficult to give concrete expression formulas this affects their theory and application research in this paper some new metrics on fuzzy ellipsoid number space are introduced which not only can better reveal the difference between two different fuzzy ellipsoid numbers but also have concrete expression formulas expressed with the level set functions of fuzzy ellipsoid numbers and the properties of the new introduced metrics and the relationships between the new metrics and the usual metrics d and rho p are studied and some results are obtained then we give the concept of supremum infimum of bounded subsets of fuzzy ellipsoid number space and obtain its concrete calculation formula and then by using the obtained results we propose a method to rank multidimensional uncertain information and give a practical example to show the application and the rationality of the proposed techniques

some properties of extended fuzzy preference relations using modalities abstract a fuzzy preference relation is extended to six kinds of relations between fuzzy sets the properties of the six extended relations are investigated especially a sufficient condition for reducing the extended relations to four kinds is discussed when fuzzy sets are bounded and normal and some examples are given to illustrate it further a sufficient condition for the extended negative relations to coincide with the extended inverse relations is described

on mori s theorem for quasiconformal maps in the n space r fehlmann and m vuorinen proved in 1988 that mori s constant m n k for k quasiconformal maps of the unit ball in r n onto itself keeping the origin fixed satisfies m n k 1 when k 1 we give here an alternative proof of this fact with a quantitative upper bound for the constant in terms of elementary functions our proof is based on a refinement of a method due to g d anderson and m k vamanamurthy we also give an explicit version of the schwarz lemma for quasiconformal self maps of the unit disk some experimental results are provided to compare the various bounds for the mori constant when n 2