on consistency and ranking of alternatives in fuzzy ahp abstract in fuzzy ahp where pairwise comparisons are described by membership values the treatment of consistency has largely been sketchy this paper proposes a fuzzy consistency definition with consideration of a tolerance deviation essentially the fuzzy ratios of relative importance allowing certain tolerance deviation are formulated as constraints on the membership values of the local priorities the fuzzy local and global weights are determined via the extension principle the alternatives are ranked on the basis of the global weights by application of a maximum minimum set ranking method our analysis uses only the data which are considered as consistent the proposed approach is computationally efficient as it consists of solving only linear programs an example is included for illustrative purpose

decision analysis under ambiguity abstract in selecting the preferred course of action decision makers are often uncertain about one or more probabilities of interest the experimental literature has ascertained that this uncertainty ambiguity might affect decision makers preferences then the decision maker might wish to incorporate ambiguity aversion in the analysis we investigate the modeling ambiguity attitudes in the solution of decision analysis problems through functionals well established in the decision theory literature we obtain the multiple event problems for subjective expected utility smooth ambiguity and maximin decision makers this allows us to establish the conditions under which these alternative decision makers face equivalent problems results for certainty equivalents and risk premia in the presence of both risk and ambiguity aversion are obtained a recent generalization of the classical arrow pratt quadratic approximation allows us to quantify the portions of a premium due to risk plxinsert and to ambiguity aversion the numerical implementation of the objective functions is addressed showing that all functionals can be estimated at no additional burden through monte carlo simulation the well known carter racing case study is addressed quantitatively to demonstrate the findings

eliciting dual interval probabilities from interval comparison matrices in recent years dealing with uncertainty using interval probabilities such as combination marginalization condition bayesian inferences has received considerable attention however how to elicit interval probabilities from subjective judgment is still a fundamental problem in applications of interval probabilities in this paper interval valued pairwise comparisons of events are used to characterize the judgment of a person about which one is more likely to occur between each pair of events from the interval comparison matrix we elicit the dual interval probabilities by linear programming problems the ratios of the estimated interval probabilities are used to approximate the interval comparison values from the outside and the inside of them in other words the inclusion relations between the interval comparison values and the ratios of the obtained interval probabilities hold a numerical example concerning a new product development is presented to illustrate how to estimate dual interval probabilities and use interval probabilities to make a decision

fundamental solutions for multivariate difference equations soit b une fonction a support compact definie sur le reseau entier z a on considere l equation aux diferences b a k j zd b k j a j f k k z d

binary nonlinearization of the super akns system under an implicit symmetry constraint for the super akns system an implicit symmetry constraint between the potentials and the eigenfunctions is proposed after introducing some new variables to explicitly express potentials the super akns system is decomposed into two compatible finite dimensional super systems x part and tn part furthermore we show that the obtained super systems are integrable super hamiltonian systems in the supersymmetry manifold

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

a new method of obtaining the priority weights from an interval fuzzy preference relation in analyzing a multiple criteria decision making problem the decision maker may express her his opinions as an interval fuzzy or multiplicative preference relation then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights in this paper a new consistent interval fuzzy preference relation is defined and the corresponding properties are derived the transformation formulae between interval fuzzy and multiplicative preference relations are further given which show that two preference relations consistent interval fuzzy and multiplicative preference relations can be transformed into each other based on the transformation formula the definition of acceptably consistent interval fuzzy preference relation is given furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented finally three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures the numerical results show that the given procedure is feasible effective and not requisite to solve any mathematical programing

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

interpolation by multivariate splines a general interpolation scheme by multivariate splines at regular sample points is introduced this scheme guarantees the local optimal order of approximation to sufficiently smooth data functions a discussion on numerical implementation is included 1 introduction in this paper we introduce a very general interpolation scheme by multivariate splines based on the quasi interpolation formulas devel oped in 3 we show that the interpolating multivariate splines so obtained give the optimal order of approximation to sufficiently smooth functions let q be a nonnegative locally supported piecewise polynomial function sym metric with respect to the origin and s the linear span of all the translates q j j e z8 of q hence s is a multivariate spline space on a certain grid partition a with certain smoothness joining conditions and of certain total degree induced by q denote by q the fourier transform of q we assume that q is normalized that is

acceptable consistency analysis of interval reciprocal comparison matrices when a decision maker expresses his her opinions by means of an interval reciprocal comparison matrix the study of consistency becomes a very important aspect in decision making in order to avoid a misleading solution in the present paper an acceptably consistent interval reciprocal comparison matrix is defined which can be reduced to an acceptably consistent crisp reciprocal comparison matrix when the intervals become exact numbers an interval reciprocal comparison matrix with unacceptable consistency can be easily adjusted such that the revised matrix possesses acceptable consistency utilizing a convex combination method a family of crisp reciprocal comparison matrices with acceptable consistency can be obtained whose weights are further found to exhibit a style of convex combination and aggregated to obtain interval weights from an acceptably consistent interval reciprocal comparison matrix a novel simple yet effective formula of possibility degree is presented to rank interval weights numerical results are calculated to show the quality and quantity of the proposed approaches and compare with other existing procedures

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

a super hierarchy of the vector nonlinear schrodinger equations and hamiltonian structures conservation laws a super hierarchy of the vector nonlinear schrodinger equations associated with the 3 3 matrix spectral problem is proposed by using the lenard recursion equations and the zero curvature equation the corresponding super bi hamiltonian structures are established with the help of the super trace identity furthermore we derive the infinite conservation flows of the super vector nonlinear schrodinger equation and the second equation in the hierarchy by utilizing spectral parameter expansions

an eigenvector method for generating normalized interval and fuzzy weights crisp comparison matrices produce crisp weight estimates it is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate in this paper an eigenvector method em is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix which differs from csutora and buckley s lambda max method in several aspects first the proposed em produces a normalized interval or fuzzy eigenvector weight estimate through the solution of a linear programming model while the lambda max method gives a series of nonnormalized interval eigenvector weight estimate through an iterative solution process next the em directly solves the principal right eigenvector of an interval or fuzzy comparison matrix while the lambda max method needs transforming a fuzzy comparison matrix into a series of interval comparison matrices using a level sets and the extension principle and therefore needs the solution of a series of eigenvalue problems finally the lambda max method needs the help of the principal right eigenvector of a crisp comparison matrix to determine the final interval weights while the em has no such requirement it is also found that not all interval or fuzzy comparison matrices can generate normalized interval or fuzzy eigenvector weights situations where the em is inapplicable are analyzed and the aggregation of local interval or fuzzy weights into global interval or fuzzy weights is also discussed three numerical examples including a hierarchical structure decision making problem are examined using the em to demonstrate its applications

a goal programming method for obtaining interval weights from an interval comparison matrix crisp comparison matrices lead to crisp weight vectors being generated accordingly an interval comparison matrix should give an interval weight estimate in this paper a goal programming gp method is proposed to obtain interval weights from an interval comparison matrix which can be either consistent or inconsistent the interval weights are assumed to be normalized and can be derived from a gp model at a time the proposed gp method is also applicable to crisp comparison matrices comparisons with an interval regression analysis method are also made three numerical examples including a multiple criteria decision making mcdm problem with a hierarchical structure are examined to show the potential applications of the proposed gp method

on the solutions of generalized discrete poisson equation the set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete poisson equation it is shown that its solutions with square summable discrete derivatives are unique up to a constant the proof uses the fourier transform as the main tool the necessary condition for the existence of the solution is provided

consistency analysis of triangular fuzzy reciprocal preference relations abstract in order to simulate the uncertainty associated with impression or vagueness a decision maker may give her his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making the study of their consistency becomes a very important aspect to avoid a misleading solution based on the reciprocity property this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations the new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by buckley 1985 the properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail in addition the shortcomings of the proof procedure of the proposition given by wang and chen 2008 are pointed out and the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations finally using the n 1 restricted comparison ratios a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed and an algorithm is shown to make a consistent decision ranking numerical results are further calculated to illustrate the new definition and the obtained algorithm

shape preserving quasi interpolation and interpolation by box spline surfaces this paper is devoted to the study of shape preserving approximation and interpolation of functions by box spline surfaces on three and four directional meshes the properties of positivity monotonicity and convexity are considered a characterization of the grid spacing is given which guarantees the preservation of these properties for functions in certain lipschitz classes

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

cardinal interpolation by multivariate splines the purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials in particular the notion of a commutator is introduced and its connection with the marsden identity is observed the order of a commutator is shown to be equivalent to the strang and fix conditions that arise in the study of the local approxima tion orders using quasi interpolants we also prove that scaled cardinal interpolants give these local approximation orders introduction cardinal interpolation by bivariate box splines was first studied by de boor hollig and riemenschneider 3 the purpose of our paper is to investigate the cardinal interpolation problem from a different point of view in particular the notion of a commutator is introduced it will be shown that this notion generalizes the marsden identity for univariate splines to the multivariate setting the order of a commutator will be shown to be equivalent to the strang and fix conditions used in the study of the order of controlled approximation by dahmen and micchelli 8 or local approximation by de boor and jia 4 an application to obtain approximation orders through the constructive method of scaled cardinal interpolation will also be studied in this paper 1 preliminaries this section consists of preliminary material for multivariate cardinal interpolation our approach is motivated by the work in 5 and 13 where cardinal interpolation in l2 was connected with certain convolution operators lr let z denote the set of integers and z the nonnegative ones for any given complex sequence by

interval weight generation approaches based on consistency test and interval comparison matrices a simple yet pragmatic method of consistency test is developed to check whether an interval comparison matrix is consistent or not linear programming method is used to derive consistent interval weights from consistent interval comparison matrices and to aggregate local interval weights to generate global interval weights in cases where an interval comparison matrix is inconsistent an eigenvector method based nonlinear programming nlp approach is developed to generate interval weights that can meet pre determined consistency requirements a simple and effective preference ranking method is utilized to compare the interval weights of criteria or rank alternatives three numerical examples including a hierarchical ahp decision problem are provided to illustrate the validity and practicality of the proposed methods

cardinal hermite interpolation with box splines ii in the present work we extent the results in rs on chip i e cardinal hermite interpolation by the span of translates of directional derivatives of a box spline these directional derivatives are that ones which define the type of the hermite interpolation we admit here several linearly independent directions with multiplicities instead of one direction as in rs under the same assumptions on the smoothness of the box spline and its defining matrixt we can prove as in rs chip has a system of fundamental solutions which are inl l 2 together with its directional derivatives mentioned above moreover for data sequences inl p d 1 p 2 there is a spline function inl p 1 p 1 p 1 which solves chip

on the normalization of interval and fuzzy weights the normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis mcda under uncertainty especially in analytic hierarchy process ahp with interval or fuzzy judgements the existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised this paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods

note on on the normalization of interval and fuzzy weights this paper shows by an example that lemma 2 in wang and elhag on the normalization of interval and fuzzy weights fuzzy sets and systems 157 2006 2456 2471 is false and presents the modified lemma and the correct proof of theorem 5

the extended supersymmetrization of the multicomponent kadomtsev petviashvilli hierarchy we describe three different approaches to the extended n 2 supersymmetrization of the multicomponent kp hierarchy in the first one we utilize only superfermions while in the second only superbosons and in the third superbosons as well as superfermions it is shown that many soliton equations can be embedded in the supersymmetry theory by using the first approach even if we do not change these equations in the bosonic limit of the supersymmetry in the second or third approach we obtain a generalization of the soliton equations in the bosonic limit which remains in the class of the usual commuting functions as a byproduct of our analysis we prove that for the first procedure the bosonic part of the one component supersymmetric kp hierarchy coincides with the usual classical two component kp hierarchy

existence and uniqueness of solutions to discrete diffusion equations abstract motivated by the classical one dimensional diffusion equation u t ku xx f x t x r t 0 u x 0 x x r we find the unique solution of the partial difference equation t u x t k 2 x x u x 1 t f x t x z t 0 u x 0 x x z where k 0 1 our approach is to construct the green s function for the appropriate partial difference operator and almost immediately we establish an existence and uniqueness result for the discrete nonhomogenous problem

on various approaches to normalization of interval and fuzzy weights abstract the paper deals with the problem of fuzzification of the procedure of normalization of weights first the existing methods for normalization of interval and fuzzy weights are reviewed second we study the problem of normalization of a fuzzy vector of weights that expresses the joint possibility distribution of initial weights we show that a correct way is to apply the extension principle proposed by zadeh since the result of such normalization is the fuzzy vector of normalized weights that expresses the true joint possibility distribution of normalized weights further we establish some properties of this approach to normalization that are important from the point of view of real applications finally since an n tuple of non interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind we investigate normalization of such kind of fuzzy vectors of weights according to the extension principle we show that from the point of view of the way of modelling uncertain normalized weights the result of this approach can be directly compared only with the result of normalization proposed by wang and elhag 2006 35 we find out that it is not sufficient to express the result of normalization only by an n tuple of normalized interval or fuzzy weights together with the constraint that the sum of the weights is equal to 1 since it can cause a false increase of uncertainty in the model this fact is illustrated by an example

a comparative study of the numerical scales and the prioritization methods in ahp abstract in the analytic hierarchy process ahp a decision maker first gives linguistic pairwise comparisons then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them and finally derives a priority vector from the numerical pairwise comparisons in particular the validity of this decision making tool relies on the choice of numerical scale and the design of prioritization method by introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices lpcms and by defining the deviation measures of lpcms we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods using these performance measure algorithms we compare the most common numerical scales the saaty scale the geometrical scale the ma zheng scale and the salo hamalainen scale and the prioritization methods the eigenvalue method and the logarithmic least squares method in addition we also discuss the parameter of the geometrical scale develop a new prioritization method and construct an optimization model to select the appropriate numerical scales for the ahp decision makers the findings in this paper can help the ahp decision makers select suitable numerical scales and prioritization methods

interval priorities in ahp by interval regression analysis abstract this paper proposes an interval approach for obtaining interval weights of priorities in the analytic hierarchy process ahp reflecting inconsistency of pairwise comparison ratios given by a decision maker even if pairwise comparison values with respect to preference are given as crisp values the priority weights should be estimated as intervals because of a decision maker s uncertainty of judgments we deal with two types of comparison matrices represented as crisp and interval elements this allows some uncertainty of estimations made by a decision maker the proposed approach can obtain the estimated interval priorities since interval regression analysis is based on the concept of possibility the interval weights obtained by the proposed method can be interpreted as the possible ranges estimated from the given data the sum of widths of the obtained intervals can be regarded as the degree of inconsistency included in the given data it is emphasized that interval data of preference can obtain the lower and upper models for the estimated weight intervals

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

a two stage logarithmic goal programming method for generating weights from interval comparison matrices a two stage logarithmic goal programming tlgp method is proposed to generate weights from interval comparison matrices which can be either consistent or inconsistent the first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency the weights are assumed to be multiplicative rather than additive in the case of hierarchical structures a nonlinear programming method is used to aggregate local interval weights into global interval weights a simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process the proposed tlgp is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using a level sets and the extension principle six numerical examples including a group decision analysis problem with a group of comparison matrices a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods comparisons with other existing procedures are made whenever possible

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

a characterization of multivariate quasi interpolation formulas and its applications let be a compactly supported function on s ands the linear space withgenerator that is s is the linear span of the multiinteger translates of it is well known that corresponding to a generator there are infinitely many quasi interpolation formulas a characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications and in addition provides a clear formulation of the difficult problem of minimally supported quasi interpolants we introduce a generalization of interpolation called interpolation and a notion of higher order quasi interpolation called approximation a characterization of approximants similar to that of quasi interpolants is obtained with similar applications among these applications are estimating least squares approximants without matrix inversion surface fitting to incomplete or semi scattered discrete data and constructing generators with one point quasi interpolation formulas it will be seen that the exact values of the generator at the multi integers s facilitates the above study an algorithm to yield this information for box splines is discussed

the fully supersymmetric akns equations the general fully supersymmetric approach to the akns method is presented the supersymmetric kdv and mkdv equations are considered as examples the supersymmetric backlund transformation is constructed for the supersymmetric kdv equation

acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices existing research on acceptability of pairwise interval comparison matrices focuses on acceptable consistency by controlling their inconsistency levels to within a certain threshold however a perfectly consistent but highly indeterminate interval comparison matrix can be unacceptable as it contains little sometimes no useful decision information this paper first analyzes the current definition of acceptable consistency for interval multiplicative comparison matrices imcms and shows its technical deficiencies we then introduce a new notion of acceptable imcms considering both inconsistency and indeterminacy levels in imcms a geometric mean based index is proposed to measure the indeterminacy ratio of an imcm and useful properties are derived for consistent imcms and acceptable imcms an indeterminacy ratio and geometric mean based transformation equation is subsequently put forward to convert normalized acceptable interval multiplicative weights into an acceptable imcm with consistency by introducing an auxiliary constraint a logarithmic least square model is established to generate interval multiplicative weights from acceptable imcms a geometric mean based possibility degree formula is designed to compare and rank normalized interval multiplicative weights two numerical examples are presented to illustrate how to utilize the proposed framework

interval multiplicative pairwise comparison matrix consistency indeterminacy and normality abstract to manifest human judgments a long established method called pairwise comparison pc has been successfully applied in the analytic hierarchy process ahp in practice human judgments are often made with uncertainty and can be characterized by an interval multiplicative pairwise comparison matrix impcm since consistency is a key issue that has plagued decision makers and researchers for a long time it is useful to propose a transformation that can effectively convert an inconsistent impcm into a consistent one especially in group decision making however a consistent impcm is not sufficient to be acceptable indeterminacy should also be considered moreover the interval priority weights should be normalized to consider consistency indeterminacy and normality simultaneously we put forward a new definition of acceptable impcm to obtain such an acceptable impcm we propose a theorem of consistency a consistent transformation and a normalized prioritization scheme as a result the proposed methods guarantee an inconsistent impcm can be directly converted into an acceptable impcm five theorems are proved to corroborate the proposed methods a numerical example is presented to illustrate the validity and superiority of the proposed methods finally discussion and conclusions are given

a new approach for ranking of trapezoidal fuzzy numbers ranking fuzzy numbers plays an very important role in linguistic decision making and some other fuzzy application systems several strategies have been proposed for ranking of fuzzy numbers each of these techniques have been shown to produce non intuitive results in certain cases in this paper we will introduce a new approach for ranking of trapezoidal fuzzy numbers based on the left and the right spreads at some a levels of trapezoidal fuzzy numbers the calculation of the proposed method is far simpler and easier finally some comparative examples are used to illustrate the advantage of the proposed method

consistency analysis and priority derivation of triangular fuzzy preference relations based on modal value and geometric mean existing consistency notions of triangular fuzzy preference relations tfprs are illustrated to be non robust new consistency and acceptable consistency definitions are put forward for tfprs we propose a notion of normalized triangular fuzzy multiplicative weights ntfmws transformation formulae are provided to convert ntfmws into consistent tfprs a logarithmic least square model is developed to derive a ntfmw vector from a tfpr triangular fuzzy preference relation tfpr is an effective framework to model pairwise estimations with imprecision and vagueness in order to obtain a reliable and rational decision result it is important to investigate consistency and priority derivation of tfprs the paper analyzes existing definitions and properties of consistent tfprs and illustrates that they have no invariance with respect to permutations of decision alternatives a new triangular fuzzy arithmetic based transitivity equation is introduced to define consistent tfprs the new transitivity equation reflects multiplicative consistency of modal values and multiplicative consistency of geometric means of triangular fuzzy estimations some properties are presented for consistent tfprs and a notion of acceptable consistency is put forward for tfprs geometric mean and uncertainty ratio based transformation formulae are devised to convert normalized triangular fuzzy multiplicative weights into consistent tfprs a logarithmic least square model is further established for deriving a normalized triangular fuzzy multiplicative weight vector from a tfpr with acceptable consistency a geometric mean based method is developed to compare and rank triangular fuzzy multiplicative weights three numerical examples including a group decision making problem are examined to demonstrate validity and advantages of the proposed models

a faber series approach to cardinal interpolation for a compactly supported function 9 in rd we study quasiinterpolants based on point evaluations at the integer lattice we restrict ourselves to the case where the coefficient sequence af for given data f is computed by applying a univariate polynomial q to the sequence i zd and then convolving with the data flzd such operators appear in the well known neumann series formulation of quasi interpolation a criterion for the polynomial q is given such that the corresponding operator defines a quasi interpolant since our main application is cardinal interpolation which is well defined if the symbol of p does not vanish we choose q as the partial sum of a certain faber series this series can be computed recursively by this approach we avoid the restriction that the range of the symbol of p must be contained in a disk of the complex plane excluding the origin which is necessary for convergence of the neumann series furthermore for symmetric a we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the neumann series

some issues on consistency of fuzzy preference relations in decision making in order to avoid misleading solutions the study of consistency when the decision makers express their opinions by means of preference relations becomes a very important aspect in order to avoid misleading solutions in decision making problems based on fuzzy preference relations the study of consistency is associated with the study of the transitivity property in this paper a new characterization of the consistency property defined by the additive transitivity property of the fuzzy preference relations is presented using this new characterization a method for constructing consistent fuzzy preference relations from a set of n 1 preference data is proposed applying this method it is possible to assure better consistency of the fuzzy preference relations provided by the decision makers and in such a way to avoid the inconsistent solutions in the decision making processes additionally a similar study of consistency is developed for the case of multiplicative preference relations 2002 elsevier b v all rights reserved