comparison of fuzzy numbers based on the probability measure of fuzzy events abstract most approaches for ranking fuzzy numbers proposed in the literature are based on fuzzy sets theory only and suffer from lack of discrimination and occasionally conflict with intuition it is true that fuzzy numbers are frequently partial order and cannot be compared however this does not alleviate the need for comparison in practical applications in this paper the order of fuzzy numbers are determined based on the concept of probability measure of fuzzy events due to zadeh it considers both the mean and dispersion of alternatives and gives two groups of indices based on the uniform and the proportional probability distributions the approach is also extended to the comparison of random fuzzy numbers by means of a mean fuzzy number it is shown that several comparison indices in the literature can be obtained based on the probability present measure approach finally some typical examples are used to compare the various different approaches the different interpretations of the dispersion index under different physical situations are emphasized

extended vikor method in comparison with outranking methods the vikor method was developed to solve mcdm problems with conflicting and noncommensurable different units criteria assuming that compromising is acceptable for conflict resolution the decision maker wants a solution that is the closest to the ideal and the alternatives are evaluated according to all established criteria this method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria and on proposing compromise solution one or more the vikor method is extended with a stability analysis determining the weight stability intervals and with trade offs analysis the extended vikor method is compared with three multicriteria decision making methods topsis promethee and electre a numerical example illustrates an application of the vikor method and the results by all four considered methods are compared

generalized defuzzification strategies and their parameter learning procedures defuzzification is a procedure of crucial importance for fuzzy systems because a final crisp output control action is required in many theoretical and practical applications the choice of defuzzification strategy therefore can directly affect the success of such applications among the existing strategies neither the center of area coa nor the mean of maximum mom emerges as the better defuzzification strategy a compromise strategy that combines the two methods may offer a synergetic solution in this paper the authors introduce two new objective defuzzification strategies gaussian distribution transformation based defuzzification gtd and polynomial transformation based defuzzification ptd which are based on a discrete universe of discourse both strategies can perform better than the existing strategies and the ptd strategy offers a generalized defuzzification tool for a wide class of practical problems both strategies include the coa and mom strategies as special cases and both are based on parameter learning processes using the extended kalman filter as their iterative improvement algorithms on sample database containing fuzzy sets and the associate defuzzified values the proposed parameter learning procedures are capable of either off line or on line processing

on the maximum entropy parameterized interval approximation of fuzzy numbers this paper extends the interval valued weighted possibilistic mean of a fuzzy number of fuller and majlender to a general weighting function without the monotonic increasing assumption this weighting function determines a weighted average aggregation of the cuts of the fuzzy number according to the preference of a decision maker some properties of the weighting function are provided and a preference index that qualifies this aggregation and can serve as a parameter for the definition of interval approximation of a fuzzy number is proposed a special class of parameterized weighting functions satisfying the maximal entropy principle is proposed

optimization method for an evolutional type inverse heat conduction problem this paper deals with the determination of a pair q u in the heat conduction equation with initial and boundary conditions from the overspecified data u x t g x t by the time semi discrete scheme the problem is transformed into a sequence of inverse problems in which the unknown coefficients are purely space dependent based on the optimal control framework the existence uniqueness and stability of the solution q u are proved a necessary condition which is a couple system of a parabolic equation and parabolic variational inequality is deduced

fuzzy plane geometry i points and lines abstract we introduce a comprehensive study of fuzzy geometry in this paper by first defining a fuzzy point and a fuzzy line in fuzzy plane geometry we consider the fuzzy distance between fuzzy points and show it is a weak fuzzy metric we study various definitions of a fuzzy line develop their basic properties and investigate parallel fuzzy lines

ranking fuzzy numbers with integral value abstract ranking fuzzy numbers is important in decision making since very often the alternatives are evaluated by fuzzy numbers in a vague environment a comparison between these fuzzy numbers is indeed a comparison between alternatives this paper proposes a method of ranking fuzzy numbers with integral value the method which is independent of the type of membership functions used and the normality of the functions can rank more than two fuzzy numbers simultaneously it is relatively simple in computation especially in ranking triangular and trapezoidal fuzzy numbers further an index of optimism is used to reflect the decision maker s optimistic attitude discussion on comparative advantages is included

numerical inversions for a nonlinear source term in the heat equation by optimal perturbation algorithm this paper deals with an inverse problem of identifying a nonlinear source term g g u in the heat equation u t u x x a x g u by data compatibility analysis the forward problem is proved to have a unique positive solution with a maximum of m 0 with which an optimal perturbation algorithm is applied to determine the source function g u on u 0 m numerical inversions are carried out for g u with functional forms of polynomial trigonometric and index functions the inversion reconstruction sources basically coincide with the true source solution showing that the optimal perturbation algorithm is efficient to the inverse source problem here by the computations we find that the inversion results are better for polynomial sources than those of trigonometric and index sources the inversion algorithm seems to be very sharp if the solution s maximum m of the forward problem is relatively small otherwise the deviations in the source solutions become large especially near the endpoint of u m

a comparison of multiple criteria analysis techniques for water resource management multiple criteria analysis mca is a framework for evaluating decision options against multiple criteria numerous techniques for solving an mca problem are available this paper applies mca to six water management decision problems the mca methods tested include weighted summation range of value promthee ii evamix and compromise programming we show that different mca methods were in strong agreement with high correlations amongst rankings in the few cases where strong disagreement between mca methods did occur it was due to presence of mixed ordinal cardinal data in the evaluation matrix the results suggest that whilst selection of the mca technique is important more emphasis is needed on the initial structuring of the decision problem which involves choosing criteria and decision options

defuzzification of the discretised generalised type 2 fuzzy set experimental evaluation the work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type 2 fuzzy sets as created by the inference stage of a mamdani fuzzy inferencing system the exhaustive method of defuzzification for type 2 fuzzy sets is extremely slow owing to its enormous computational complexity several approximate methods have been devised in response to this defuzzification bottleneck in this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type 2 fuzzy set 1 vertical slice centroid type reduction 2 the sampling method 3 the elite sampling method and 4 the a planes method we then evaluate the different methods experimentally for accuracy and efficiency for accuracy the exhaustive method is used as the standard the test results are analysed statistically by means of the wilcoxon nonparametric test and the elite sampling method shown to be the most accurate in regards to efficiency vertical slice centroid type reduction is demonstrated to be the fastest technique

composite radiation dose representation using fuzzy set theory composite plans created from different image sets are generated through deformable image registration dir and present a challenge in accurately presenting uncertainties which vary with anatomy our effort focuses on the application of fuzzy set theory to provide an accurate dose representation of such a composite treatment plan the accuracy of the dir is generally verified through geometrical visual checks including the confirmation of the corresponding anatomies with edge features such as bone or organ boundaries however the remaining volume of the image mostly soft tissues has few significant image features and therefore greater uncertainty we fuzzified the deformation vector and derived a fuzzy composite dose the fuzzification was implemented using gaussian functions based on the varying uncertainties in the dir after establishing the theoretical basis for this new approach we present two and three dimensional examples as proof of concept using fuzzy set theory composite dose plans displaying locality based uncertainties were successfully created providing information previously unavailable to clinicians previous to fuzzy set dose presentations clinicians had no measure of confidence in the accuracy of a composite dose plan using fuzzified composite dose presentations clinicians can determine a safe additional dose to previously treated anatomy this will possibly increase the treatment success rate and reduce the rate of complications

trapezoidal approximations of fuzzy numbers the problem of the trapezoidal approximation of fuzzy numbers is discussed a set of criteria for approximation operators is formulated these constraints can be used for direct operator derivation a new nearest trapezoidal approximation operator preserving expected interval is suggested

a fuzzy robustness measure for the scheduling of commissioned product development projects abstract due to a considerable degree of uncertainty the generation of robust schedules for the execution of product development projects is a crucial planning task in the area of robust project scheduling one basic option is to apply surrogate robustness measures as estimates of schedule robustness although project managers typically possess vague information about activity variations most surrogate measures neglect this knowledge and are entirely based on deterministic data in this contribution we therefore present the fuzzy overlap that is to the best of our knowledge the first surrogate robustness measure for schedule stability which accounts for fuzzy activity duration we provide a mathematical formulation and embed the fuzzy overlap in a two stage planning approach for the scheduling of product development projects which aims at balancing the minimization of project costs as the basic scheduling objective with the maximization of schedule robustness in a numerical study we compare the performance of our proposed approach with comparable deterministic surrogate robustness measures the results indicate that the consideration of fuzzy information enhances schedule robustness compared to the application of traditional deterministic surrogate robustness measures

global regularity criterion for the 3d navier stokes equations involving one entry of the velocity gradient tensor in this paper we provide a sufficient condition in terms of only one of the nine entries of the gradient tensor that is the jacobian matrix of the velocity vector field for the global regularity of strong solutions to the three dimensional navier stokes equations in the whole space as well as for the case of periodic boundary conditions

defuzzification criteria and classification in this paper we contribute to the theory and development of defuzzification techniques first we define the core of a fuzzy set then we formulate a set of criteria for defuzzification in arbitrary universes ordered universes and the set of the real numbers finally we classify the most widely used defuzzification techniques into different groups and we examine the prototypes of each group with respect to the defuzzification criteria we show that the maxima methods behave well with respect to the more basic defuzzification criteria and hence are good candidates for fuzzy reasoning systems on the other hand the distribution methods and the area methods do not fulfill the basic criteria but they exhibit the property of continuity that makes them suitable for fuzzy controllers

an inverse problem of identifying the coefficient in a nonlinear parabolic equation abstract this paper deals with the determination of a pair p u in the nonlinear parabolic equation u t u x x p x f u 0 with initial and boundary conditions u x 0 x u x 0 u x 1 0 from the overspecified data u x t g x based on the optimal control framework the problem is transformed into a nonlinear optimization problem and the existence of the minimizer for the control functional is established the necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced since the optimal control problem is nonconvex one may not expect a unique solution in general however the local uniqueness and stability of the solution are proved which is also the main contribution of the paper

existence results for inverse problems associated with a nonlinear parabolic equation this paper deals with the determination of a pair p u in the nonlinear parabolic equation ut u p x f u 0 with initial and boundary conditions u x 0 0 u 0 t g from the overspecified data u t q the problem is reduced to a nonlinear operator equation and solved by taking advantage of a fixed point theory in ordered banach spaces similarly another form of overspecified data is considered

regularity criteria for the 3d mhd equations involving one current density and the gradient of one velocity component abstract in this paper we consider the cauchy problem for the 3 d mhd equations and provide a regularity criterion involving u 3 and j 3 where u 3 and j 3 are the third component of the velocity and the current density respectively as by products we show that assumptions on two vorticity and one current density component are enough to ensure the smoothness of the solution and improve some previous results skalak 2014 zhou and pokorn 2009 2010 for the navier stokes equations

algebraic characteristics of extended fuzzy numbers abstract extended fuzzy numbers previously called fuzzy intervals are discussed by using the resolution identity and the extension principle the regularity and the spread are defined for describing the algebraic properties of extended fuzzy numbers arithmetic operations on level set intervals are suggested instead of general set operations in order to reduce the amount of computation a sufficient and necessary condition for solving a x c is derived the exact solution for a x c is obtained finally a a a fuzzification of the crisp 0 which is a natural extension from the nonfuzzy field is proved

continuity issues of the implicational interpretation of fuzzy rules the implicational interpretation of fuzzy rules has received little attention in real world applications so far this is largely due to the fact that ensuring continuity of the resulting function is not a straightforward task this paper targets this subject departing from consistent linguistic descriptions rule bases we introduce sufficient conditions for the continuity of the implicational interpretation with mean of maximum defuzzification we demonstrate that continuity can be achieved under practically feasible conditions regardless of the dimensionality of the input

only smooth rule bases can generate monotone mamdani assilian models under center of gravity defuzzification at first sight it seems that ordered linguistic values for all input variables and the output variable and a set of rules describing a monotone system are all that is needed for a monotone model however this is not the case in this study we show that the choice of the mathematical operators used when calculating the model output and the properties of the membership functions in the output domain are also of crucial importance to obtain a monotone input output behavior in the mamdani assilian models considered in this study the linguistic values of the input variables as well as the output variable are described by trapezoidal membership functions that form a fuzzy partition the rule base is monotone and the crisp output is obtained by the center of gravity cog defuzzification method it is verified that for each of the three basic t norms i e the minimum tm the product tp and the lukasiewicz t norm tl a monotone input output behavior is obtained for any monotone rule base or at least for any monotone smooth rule base the outcome of this study is a guideline for designers of monotone linguistic fuzzy models for the t norms tm and tl models with a single input variable show a monotone input output behavior for any monotone rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degrees of equal length the latter restriction can easily be bypassed by an auxiliary interpolation procedure for the t norm tp models with a single input variable show a monotone input output behavior for any monotone rule base and any fuzzy output partition when designing a monotone model with more than one input variable one should opt for the t norm tp and use a monotone smooth rule base it is shown that monotonicity of models with two input variables that apply tp is guaranteed for any monotone smooth rule base and any fuzzy output partition finally it is proved that a monotone input output behavior is always obtained for models with three input variables that apply tp and a monotone smooth rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions of identical shape

multi attribute decision making a simulation comparison of select methods abstract several methods have been proposed for solving multi attribute decision making problems madm a major criticism of madm is that different techniques may yield different results when applied to the same problem the problem considered in this study consists of a decision matrix input of n criteria weights and ratings of l alternatives on each criterion the comparative performance of some methods has been investigated in a few mostly field studies in this simulation experiment we investigate the performance of eight methods electre topsis multiplicative exponential weighting mew simple additive weighting saw and four versions of ahp original vs geometric scale and right eigenvector vs mean transformation solution simulation parameters are the number of alternatives criteria and their distribution the solutions are analyzed using twelve measures of similarity of performance similarities and differences in the behavior of these methods are investigated dissimilarities in weights produced by these methods become stronger in problems with few alternatives however the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives although less significant the distribution of criterion weights affects the methods differently in general all ahp versions behave similarly and closer to saw than the other methods electre is the least similar to saw except for closer matching the top ranked alternative followed by mew topsis behaves closer to ahp and differently from electre and mew except for problems with few criteria a similar rank reversal experiment produced the following performance order of methods saw and mew best followed by topsis ahps and electre it should be noted that the electre version used was adapted to the common madm problem and therefore it did not take advantage of the method s capabilities in handling problems with ordinal or imprecise information

continuity focused choice of maxima yet another defuzzification method in this paper we present a new defuzzification method for a mamdani fuzzy controller the method selects an element of the core of the fuzzy output set as defuzzification value in a way that the continuity of the controller can be preserved however in order for this continuity to be guaranteed some constraints on the components of the controller have to be taken into account except for some constraints of continuity convexity and points of intersection for the input and output linguistic terms we will define the property of linguistic continuity that has to be fulfilled by the controller further we show that the new defuzzification method is computationally very efficient and we point out how the fuzzy controller can be tuned by modifying the output linguistic terms finally we also give two variants of the proposed formula

fuzzy approaches to quality function deployment for new product design for new product development quality function deployment qfd is a useful approach to maximize customer satisfaction the determination of the fulfillment levels of design requirements drs and parts characteristics pcs in phases 1 and 2 is an important issue during qfd processes for new product design unlike the existing literature which mainly focuses on the drs this paper proposes fuzzy nonlinear programming models based on kano s concept to determine the fulfillment levels of pcs with the aim of achieving the determined contribution levels of drs in phase 1 for customer satisfaction in addition to deal with the design risk this study incorporates failure modes and effects analysis fmea into qfd processes and treats it as the constraint factor in the models to cope with the vague nature of product development processes fuzzy approaches are used for both fmea and qfd the applicability of the proposed models in practice is demonstrated with a numerical example

on a regularity criterion for the navier stokes equations involving gradient of one velocity component we improve the regularity criterion for the incompressible navier stokes equations in the full three dimensional space involving the gradient of one velocity component the method is based on recent results of cao and titi see regularity criteria for the three dimensional navier stokes equations indiana univ math j 57 2643 2008 and kukavica and ziane see navier stokes equations with regularity in one direction j math phys 48 065203 2007 in particular for s 2 3 we get that the solution is regular if u3 lt 0 t ls r3 2 t 3 s 2312

classification with nonmetric distances image retrieval and class representation a key problem in appearance based vision is understanding how to use a set of labeled images to classify new images systems that model human performance or that use robust image matching methods often use nonmetric similarity judgments but when the triangle inequality is not obeyed most pattern recognition techniques are not applicable exemplar based nearest neighbor methods can be applied to a wide class of nonmetric similarity functions the key issue however is to find methods for choosing good representatives of a class that accurately characterize it we show that existing condensing techniques are ill suited to deal with nonmetric dataspaces we develop techniques for solving this problem emphasizing two points first we show that the distance between images is not a good measure of how well one image can represent another in nonmetric spaces instead we use the vector correlation between the distances from each image to other previously seen images second we show that in nonmetric spaces boundary points are less significant for capturing the structure of a class than in euclidean spaces we suggest that atypical points may be more important in describing classes we demonstrate the importance of these ideas to learning that generalizes from experience by improving performance we also suggest ways of applying parametric techniques to supervised learning problems that involve a specific nonmetric distance functions showing how to generalize the idea of linear discriminant functions in a way that may be more useful in nonmetric spaces

an algorithmic method to extend topsis for decision making problems with interval data in this paper from among multi criteria models in making complex decisions and multiple attribute models for the most preferable choice technique for order preference by similarity ideal solution topsis approach has been dealt with in some cases determining precisely the exact value of the attributes is difficult and that as a result of this their values are considered as intervals therefore the aim of this paper is to extend the topsis method for decision making problems with interval data by extension of topsis method an algorithm to determine the most preferable choice among all possible choices when data is interval is presented finally an example is shown to highlight the procedure of the proposed algorithm at the end of this paper 2005 elsevier inc all rights reserved

on the regularity of the solutions to the navier stokes equations via the gradient of one velocity component abstract we improve a regularity criterion for the solutions to the navier stokes equations in the full three dimensional space involving the gradient of one velocity component revising the method used in pokorn and zhou 2009 2010 we show that a weak solution u is regular on 0 t provided that u 3 l t 0 t l s where 2 t 3 s 19 10 for s 30 19 10 3 and 2 t 3 s 7 4 1 2 s for s 10 3 it improves the known results for s 30 19 150 77 and s 10 3

a fuzziness measure for fuzzy numbers applications abstract extending the ideas of delgado et al 1995 to include the notion of fuzziness we obtain canonical representations of fuzzy numbers in particular we show that any fuzzy number can be represented by either a trapezoidal or a quasitrapezoidal number having the same basic attributes as the original fuzzy number

fast and accurate center of gravity defuzzification of fuzzy system outputs defined on trapezoidal fuzzy partitions in this article three methods are presented to perform the center of gravity cog defuzzification method in the context of linguistic fuzzy models with t norm based inference one well known method the discretisation method and two new methods the slope based method and the modified transformation function method the methods are worked out for trapezoidal membership functions forming a fuzzy partition in the sense of ruspini experimental results show that the newly introduced methods exhibit excellent accuracy at an extremely low computational cost compared to the widely applied discretisation method

the effects of membership function on fuzzy reasoning abstract fuzzy modus ponens has been considered a generalization of the classic modus ponens logic inference rule it has been expected to provide a more powerful and useful inference capability in the process of fuzzy reasoning the definitions of membership functions of fuzzy sets used in the reasoning process may however adversely affect the reasoning results if the insignificant parts of the membership functions are incompatible this paper explains these undesirable effects and proposes a method to resolve this problem in addition an algorithm for the compositional rule of inference is developed to perform the fuzzy modus ponens syllogism test results shown in this paper indicate that the algorithm and the underlying proposed methods are reasonable approaches to a robust fuzzy logic reasoning system

topological properties of fuzzy numbers fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets topological properties of the resulting metric space are studied the notion of q crisp fuzzy numbers is introduced and in particular it is shown that this class of fuzzy numbers is homeomorphic to the hilbert cube

designing and learning of adjustable quasi triangular norms with applications to neuro fuzzy systems we introduce a new class of operators called quasi triangular norms they are denoted by h and parameterized by a parameter spl nu h a sub 1 a sub 2 a sub n spl nu from the construction of function h it follows that it becomes a t norm for spl nu 0 and a dual t conorm for spl nu 1 for spl nu close to 0 function h resembles a t norm and for spl nu close to 1 it resembles a t conorm in the paper we also propose adjustable quasi implications and a new class of neuro fuzzy systems most neuro fuzzy systems proposed in the past decade employ engineering implications defined by a t norm as the minimum or product in our proposition a quasi implication i a b spl nu varies from an engineering implication t a b to corresponding s implication as spl nu goes from 0 to 1 consequently the structure of neuro fuzzy systems presented in this work is determined in the process of learning learning procedures are derived and simulation examples are presented

geometric defuzzification revisited abstract in this paper the geometric defuzzification strategy for type 2 fuzzy sets is reappraised for both discretised and geometric fuzzy sets the techniques for type 1 interval type 2 and generalised type 2 defuzzification are presented in turn in the type 2 case the accuracy of geometric defuzzification is assessed through a series of test runs on interval type 2 fuzzy sets using exhaustive defuzzification as the benchmark method these experiments demonstrate the geometric defuzzifier to be wildly inaccurate the test sets take many shapes they are not confined to those type 2 sets with rotational symmetry that have previously been acknowledged by the technique s developers to be problematic as regards accuracy type 2 geometric defuzzification is then examined theoretically the defuzzification strategy is demonstrated to be built upon a fallacious application of the concept of centroid this explains the markedly inaccurate experimental results thus the accuracy issues of type 2 geometric defuzzification are revealed to be inevitable fundamental and significant

evaluations of fuzzy quantities this paper deals with the problem of evaluating a fuzzy quantity we call fuzzy quantity any non normal and non convex fuzzy set defined as the union of two or more non normal fuzzy numbers in order to introduce either ranking or defuzzifing procedures we propose a definition which arises from crisp set theory it is based on a particular fuzzy number evaluation weighted average value wav that works on a cuts levels and depends on two parameters a real number l and an additive measure s l is connected with the optimistic or pessimistic point of view of the decision maker s allows the decision maker to choose evaluations in particular subsets of the fuzzy number support according to his preference

on fuzzy distances and their use in image processing under imprecision abstract this paper proposesa classification of fuzzy distances with respect to the requirements needed for applications in image processing under imprecision we distinguish on the one hand distances that basically compare only the membership functions representing the concerned fuzzy objects and on the other hand distances that combine spatial distance between objects and membership functions to our point of view the second class of methods finds more general applications in image processing since these methods take into account both spatial information and information related to the imprecision attached to the image objects new distances based on mathematical morphology are proposed in this second class

cooperative neighbors in defuzzification abstract defuzzification is a problem of optimized selection of an element from a fuzzy set it is in fact an optimization problem optimization with respect to the whole system under consideration we believe that neighbors can contribute and make a better selection in the process of defuzzification we have proposed here a method that uses a collective decision making of cooperative neighbors natural selection process has been the basis of our model we have also outlined three simple functions to show that adaptiveness can be incorporated in the process of defuzzification without making a possibility probability transformation the method further uses the concept of logical distance and the interaction between the neighbors to select an element the neighbors poll and reach a decision of their most ideal representative

a note on the regularity of the solutions to the navier stokes equations via the gradient of one velocity component we present a regularity criterion for the solutions to the navier stokes equations based on the gradient of one velocity component starting with the method developed by cao and titi global regularity criterion for the 3d navier stokes equations involving one entry of the velocity gradient tensor arch ration mech anal 202 919 932 2011 for the case of one entry of the velocity gradient and using further some inequalities concerning the anisotropic sobolev spaces we show as a main result that a weak solution u is regular on 0 t t 0 provided that u3 lt 0 t ls where 2 t 3 s 3 2 3 4s and s 3 2 2 it improves the known results for s 3 2 15 8

a fuzzy set approach for evaluating and enhancing the mid term competitiveness of a semiconductor factory fuzzy set theory is applied in this study to establish a systematic procedure for evaluating the mid term competitiveness of a semiconductor manufacturing factory subsequently to enhance the overall competitiveness of the semiconductor manufacturing factory a capacity re allocation mechanism is designed to improve the competitiveness of less competitive products without raising the expense of the other products and diminishing their competitiveness for this purpose a fuzzy nonlinear programming fnp model is constructed and is then converted into an equivalent np problem that can be solved with the existing optimization problem solving package to prove the usefulness of the proposed methodology a real case containing the data collected from a dram manufacturing factory was investigated after reallocating its capacity the competitiveness of the whole factory was indeed enhanced besides the capacity plan developed by the capacity re allocation mechanism was shown to be both efficient and practical

several new regularity criteria for the axisymmetric navier stokes equations with swirl abstract in this paper we consider the axisymmetric navier stokes equations with swirl and show that the global regularity is ensured if we add some weighted integrable conditions on z u r r u z r u r z u z or r u

designing a model of fuzzy topsis in multiple criteria decision making decision making is the process of finding the best option among the feasible alternatives in classical multiple attribute decision making madm methods the ratings and the weights of the criteria are known precisely due to vagueness of the decision data the crisp data are inadequate for real life situations since human judgments including preferences are often vague and cannot be expressed by exact numerical values the application of fuzzy concepts in decision making is deemed to be relevant we design a model of topsis for the fuzzy environment with the introduction of appropriate negations for obtaining ideal solutions here we apply a new measurement of fuzzy distance value with a lower bound of alternatives then similarity degree is used for ranking of alternatives examples are shown to demonstrate capabilities of the proposed model

vagueness evaluation of the crisp output in a fuzzy inference system fuzzy models generally provide an output characterized by vagueness which is expressed through a solution fuzzy set in many applications the response of the model is transformed in a crisp value through some defuzzification methods for solution fuzzy region thus losing its fuzziness only to preserve a few indications of its vagueness some indices summarizing the spread of the output membership function could be used to associate them with the crisp output such as its standard deviation the quartile deviation the coefficients of skewness and kurtosis the behaviour of such indices is examined in a large number of possible though unlikely output solutions and in an application of a fuzzy inference system for evaluating university teaching activity the results seem to suggest that the 20 80 mid percentile range could be a good measure of the vagueness dispersion while the coefficient of skewness could provide a useful indication about the asymmetry of the solution s shape moreover a rough estimate of dispersion was obtained from a triangle approximating the solution fuzzy region because the results were straightforwardly deduced from formulae involving the abscissae of its vertices the results generally appear to underestimate the true values of the standard deviations the 15 85 mid percentile range of the approximating triangle seemed to be a more suitable rough appraisal of fuzzy output dispersion

why triangular membership functions abstract in this note we look at a certain theoretically sound motivation behind the common use of triangular and trapezoidal membership functions the studies are completed within a conceptual framework of fuzzy modelling whose structure comprises of input and output interfaces linked with a single transformation module aimed at processing linguistic information it is shown that under some additional mild assumptions these triangular fuzzy sets comply with a request for a uniformly excited codebook in the case of the input interfaces and a satisfaction of a zero error reconstruction criterion being formulated for the output interface

adjusted p values for simultaneous inference this paper proposes that results from simultaneous tests be reported as adjusted p values such that if the adjusted p value for an individual hypothesis is less than the chosen significance level of a then the hypothesis is rejected with an experimentwise error rate of no more than ax examples are given of adjusted p values for multiple comparisons in the analysis of variance and of adjusted p values based on the bonferroni procedure and modifications of that procedure by holm 1979 scandinavian journal of statistics 6 65 70 hochberg 1988 biometrika 75 800 802 and hommel 1988 biometrika 75 383 386 the modified bonferroni based procedures are much more powerful than the original bonferroni procedure and they deserve wider use in addition to the above a procedure is outlined for obtaining adjusted p values for any closed test procedure

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

averaging procedures in defuzzification processes defuzzification is the ultimate step in approximate reasoning consisting in the replacement of a fuzzy set with a suitable crisp value this process is decomposed in two steps first replacing a fuzzy set by a crisp set then replacing the obtained crisp set by a single value we investigate the natural conditions the first replacement called averaging procedure should satisfy some interesting examples are given the compatibility of averaging procedures with algebraic and order structures is studied the algebraic structure of the set of averaging procedures is investigated

an experimental study on fuzzy distances for skull face overlay in craniofacial superimposition abstract skull face overlay is the most time consuming and error prone stage in craniofacial superimposition an important skeleton based forensic identification technique this task focuses on achieving the best possible overlay of an unknown skull found and a single ante mortem image of a candidate missing person the process is influenced by some sources of uncertainty since two objects of different nature are involved i e a skull and a face in previous works we have developed a computer aided craniofacial superimposition system aimed to assist forensic anthropologists in obtaining the best possible skull and face overlay the system has successfully allowed us to reduce the processing time simplify the forensic anthropologists work and make the process more objective and reproducible our approach is based on automatically overlaying a skull three dimensional model onto a facial photograph by minimizing the distance between two subsets of corresponding cranial and facial landmarks the proposed method properly deals with the inherent uncertainty sources to the skull face overlay process by considering fuzzy sets to model imprecise landmark location and imprecise cranial and facial landmarks spatial correspondence resulting from the presence of soft tissues in the face accordingly our methodology requires computing two kinds of distance metrics between a point and a fuzzy set and between two fuzzy sets this contribution is devoted to study the performance and influence of the most significant and suitable fuzzy distances proposed in the specialized literature as well as other new ones proposed on our skull face overlay system in particular we have tested the behavior of our automatic method when considering eight different distance measurements the system performance has been objectively evaluated considering 18 case studies resulting from a ground truth dataset following a rigorous statistical experimental setup the fact that the choice of a good distance metric is crucial to our method has been demonstrated since it significantly affects the quality of the final solutions it has been shown that our skull face overlay approach presents the best performance using the weighted mean distance in most of the cases and that the results are both more accurate and robust than the other studied metrics

the regularity criterion for 3d navier stokes equations involving one velocity gradient component abstract in this article we establish sufficient conditions for the regularity of solutions of navier stokes equations based on one of the nine entries of the gradient tensor we improve the recent results of c s cao e s titi c s cao e s titi global regularity criterion for the 3d navier stokes equations involving one entry of the velocity gradient tensor arch ration mech anal 202 2011 919 932 and y zhou m pokorn y zhou m pokorn on the regularity of the solutions of the navier stokes equations via one velocity component nonlinearity 23 2010 1097 1107

some numerical aspects of center of area defuzzification method this paper deals with center of area or gravity defuzzification method via a computationally attractive method called transformation function method it is shown that the proposed method enormously reduces number of mathematical operations and amount of memory space required to compute defuzzified value moreover the computational superiority of transformation function method over the conventional numerical technique is also shown

simultaneous determination of source terms in a linear parabolic problem from the final overdetermination weak solution approach the problem of determining the pair w f x t t0 t of source terms in the parabolic equation ut k x ux x f x t and robin boundary condition k l ux l t v u l t t0 t from the measured final data t x u x t is formulated it is proved that both components of the frechet gradient of the cost functional j w t x u x t w 02 can be found via the same solution of the adjoint parabolic problem lipschitz continuity of the gradient is derived the obtained results permit one to prove existence of a quasi solution of the considered inverse problem as well as to construct a monotone iteration scheme based on a gradient method

compromise solution by mcdm methods a comparative analysis of vikor and topsis the multiple criteria decision making mcdm methods vikor and topsis are based on an aggregating function representing closeness to the ideal which originated in the compromise programming method in vikor linear normalization and in topsis vector normalization is used to eliminate the units of criterion functions the vikor method of compromise ranking determines a compromise solution providing a maximum group utility for the majority and a minimum of an individual regret for the opponent the topsis method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative ideal solution but it does not consider the relative importance of these distances a comparative analysis of these two methods is illustrated with a numerical example showing their similarity and some differences 2003 elsevier b v all rights reserved

distances between fuzzy sets representing grey level images the authors wish to present a class of mathematical functions that can be used to calculate the distance between fuzzy sets essential is that this distance is given by a semi pseudometric i e the triangle inequality is not fulfilled the relation to the ordinary pseudometrics is explained also some theoretical examples are provided

adaptive learning defuzzification techniques and applications abstract in this paper adaptive learning defuzzification techniques are studied under the consideration of system performance indices by treating defuzzification processes as continuous mappings from space 0 1 n to the real line the concept of the optimal defuzzification mapping can be developed since all the continuous defuzzification mappings considered in this paper form a banach space approximation to the optimal mapping with some known functions can be expressed as the parametric optimization problem to find the optimal parameters adaptive learning of the optimal defuzzification mapping is investigated learning laws for the parameters in the defuzzification mapping are derived in one case numerical results indicate that the adaptive learning defuzzification method can give superior defuzzification results to some popular defuzzification methods

on the existence of an inverse problem in this paper we consider the existence of the solution to the inverse problem to identify q u satisfying tu i aij x t ju bi x t iu q x c x t u f x t vu 0 u t 0 g x u t t z x we use the galerkin method to reduce the above mentioned problem to a finite dimensional problem and then use the degree theory to obtain the existence of the inverse problem when the unknown parameter q is of definite signs

extensions of the topsis for group decision making under fuzzy environment the aim of this paper is to extend the topsis to the fuzzy environment owing to vague concepts frequently represented in decision data the crisp value are inadequate to model real life situations in this paper the rating of each alternative and the weight of each criterion are described by linguistic terms which can be expressed in triangular fuzzy numbers then a vertex method is proposed to calculate the distance between two triangular fuzzy numbers according to the concept of the topsis a closeness coefficient is defined to determine the ranking order of all alternatives by calculating the distances to both the fuzzy positive ideal solution fpis and fuzzy negative ideal solution fnis simultaneously finally an example is shown to highlight the procedure of the proposed method at the end of this paper

fuzzy mcdm based on ideal and anti ideal concepts this paper presents a novel fuzzy multiple criteria decision making mcdm based on the concepts of ideal and anti ideal points the concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix to evaluate the weighted suitability of different alternatives versus various criteria the distance of different alternatives versus positive ideal solution and negative ideal solution are then obtained by using the proposed ranking method finally the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative

parameter identification for a class of abstract nonlinear parabolic distributed parameter systems this paper studies the parameter identification problem of nonlinear abstract parabolic distributed parameter systems via variational method 1 based on the fundamental optimal control theory and the transposition method studied in 2 the existence of optimal parameter is proved and the necessary condition for the optimal parameter is established

solving the inverse problem of an sis epidemic reaction diffusion model by optimal control methods in this contribution a novel inverse problem on an sis epidemic reaction diffusion model is investigated by employing the optimal control methods by setting a proper regular cost functional and choosing the coefficients to be identified as control variables we embed the original inverse problem which is very difficult to solve into an optimal control problem the existence of the optimal controls and the first order necessary optimality condition satisfied by the optimal controls are established respectively in particular the local uniqueness and stability of the identification of the coefficients are proved by using analytical techniques

ranking fuzzy numbers with maximizing set and minimizing set abstract up to now these are five methods of ranking n fuzzy numbers in order but these methods contain some confusions and occasionally conflict with intuition this paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers in addition we give a method for calculating the ordering value of each fuzzy number with triangular trapezoidal and two sided drum like shaped membership functions

slide a simple adaptive defuzzification method we introduce a parameterized family of defuznfica tion operators called the semi linear defuzzification slide method this method is based upon a simple transformation of the fuzzy output set of the controller we suggest an algorithm for the learning of the parameter from a data set in an attempt to simplify the parameter learning we suggest a modified version of the slide method which results in a simple learning algorithm the development of the learning algorithm is based upon the use of the kalman filter n fuzzy logic control systems l 2 the defuzzification i step involves the selection of one value as the output of the controller more specifically starting with a fuzzy subset possibility distribution f over the output space x of the controller the defuzzification step uses this fuzzy subset to select a representative element x the two most often used methods of defuzzification found in the literature are the center of area coa and mean of maxima mom 3 6 methods we recall that the mom method takes as its defuzzified value the mean of the elements that attain the maximum membership grade in f the coa method takes as its defuzzified value d ci ui xi where ui f xi cj f xj in 7 and 8 we formulated a general defuzzification method via basic defuzzification distribution badd transformations the main idea of this badd approach is to transform the possibility distribution f into a probability distribution p and then the defuzzified value is the expected value of the probability distribution the process of transforming f into p was seen as a two step process the first step was to transform f into a new possibility distribution e and then to normalize e to obtain p the step of obtaining e from f involved the use of a badd transformation in which e xi f z where 6 e o co it was shown in 7 that the generalized badd method implies the coa and mom methods as special cases in particular the coa method is obtained for a 1 and the mom is obtained for 6 00 when s 0 we get a defuzzified value that is the unweighted mean of the output space d xi xi the badd transformation has the advantage of being based on a single parameter which allows us the potential of adaptively learning the best method of defuzzification for a given controller while the family of all defuzzified values that can be obtained by the generalized badd defuzzification method was parameterized an undesirable property of the method was the nonlinear

a decision support system for multiple criteria alternative ranking using topsis and vikor in fuzzy and nonfuzzy environments abstract topsis and vikor are two well known and widely used multiple attribute decision making methods many extensions of these methods have been proposed that either use different techniques to rank alternatives or utilize fuzzy logic to handle alternatives and criteria that are unquantifiable and or incomplete in this paper we present the implementation of a web based decision support system that incorporates topsis and vikor to solve multicriteria decision making problems either in a nonfuzzy or in a fuzzy environment the aim of this paper is to present a tool that will be used by decision makers to compare various alternative solutions and understand how robust a decision will be the proposed system can be used both in single and in group decision making problems in addition we review several variations for each step of these methods and implement different techniques when applicable hence decision makers can experiment with different techniques in each method we implement ten normalization techniques three methods for the calculation of the ideal and anti ideal solutions fifteen distance metrics five fuzzy distance metrics and eight defuzzification techniques an illustrative example is presented to highlight the key features of the implemented system and the different scenarios that can be built using the proposed dss

navier stokes equations with regularity in one direction we consider sufficient conditions for the regularity of leray hopf solutions of the navier stokes equations we prove that if the third derivative of the velocity u x3 belongs to the space lts0lxr0 where 2 s0 3 r0 2 and 9 4 r0 3 then the solution is regular this extends a result of beirao da veiga chin ann math ser b 16 407 412 1995 c r acad sci ser i math 321 405 408 1995 by making a requirement only on one direction of the velocity instead of on the full gradient the derivative u x3 can be substituted with any directional derivative of u

an approach to generalization of fuzzy topsis method the topsis method is a technique for establishing order preference by similarity to the ideal solution and was primarily developed for dealing with real valued data this technique is currently one of most popular methods for multiple criteria decision making mcdm in many cases it is hard to present precisely exact ratings of alternatives with respect to local criteria and as a result these ratings are seen as fuzzy values a number of papers have been devoted to fuzzy extensions of the topsis method in the literature but these extensions are not complete since the ideal solutions are usually presented as real values not by fuzzy values or as fuzzy values which are not attainable in the decision matrix in most of these papers a defuzzification of elements of the fuzzy decision matrix is used which leads inevitably to a loss of important information and may even produce the wrong results in this paper we propose a new direct approach to the fuzzy extension of the topsis method which is free of the limitations of other known approaches we show that the distances of the alternatives from the ideal solutions may be treated in some sense as modified weighted sums of local criteria it is known that using weighted sums is not the best approach to the aggregation of local criteria in many real world situations therefore here we propose the use in addition to weighted sums some other types of local criteria aggregation in the topsis method and we develop a method for the generalization of different aggregation modes providing compromised final results

nonlinear modelling and classification based on the micog defuzzification abstract in the paper we study the micog modified indexed center of gravity defuzzification applied to neuro fuzzy architectures with various inference methods it is proven that the micog defuzzification can be used in a combination with the mamdani type reasoning and selected s and q fuzzy implications it is also shown that the micog defuzzification cannot be used with r fuzzy implications if we require interpretable rules the experiments confirm our theoretical results

on the regularity criterion for the navier stokes equations involving the diagonal entry of the velocity gradient abstract in this paper we provide a regularity criterion for the navier stokes equations involving the diagonal entry of the velocity gradient which says that if 3 u 3 l 0 t l 2 r 3 then the weak solution is smooth on 0 t this is an important case in the sense that the strong solution is in this class moreover we verify the limiting case of one regularity criterion in cao and titi 2011