evaluating naval tactical missile systems by fuzzy ahp based on the grade value of membership function abstract many decision making problems which are complicated and fuzzy in nature exist in modern society how to solve them is becoming increasingly important for human society for solving multiple criteria s decision making in a fuzzy environment in this paper we will propose a new algorithm for evaluating naval tactical missile systems by the fuzzy analytical hierarchy process based on grade value of membership function generally we are given scores by experience of experts to represent judgmental objects in this paper from viewpoint of many experts we will build membership functions of judgement criteria for all sub items when the membership function is built we can calculate the grade value by data of missile performance the grade value is called performance score our methods can be summarized in the following 1 1 building membership function of judgement criteria for all sub items it is called fuzzy standard 2 2 calculate the grade of membership function by practical data to represent performance scores 3 3 use fuzzy ahp method and entropy concepts to calculate aggregate weights finally for a simple and efficient computation we have developed a systematic and practical program to calculate all algorithms and apply the new algorithm to a naval tactical missile systems valuation and selection problem

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

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

reasonable properties for the ordering of fuzzy quantities ii as a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches to do so we briefly introduce fuzzy relations on which the third class of approaches is based then we recall some transitivity related concepts and an ordering procedure based on a acyclic fuzzy relation acyclicity is a very weak restriction on a fuzzy relation we prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity so we give a widely applicable formulation to derive a total ranking order from a fuzzy relation with our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part i

on the largest zeroes of orthogonal polynomials for certain weights the asymptotic growth of the largest zero of the orthogonal polynomials for the weights w x i x ih exp k i log i x i i is investigated

multiresolution wavelet analysis of integer scale bessel functions we identify multiresolution subspaces giving rise via hankel transforms to bessel functions they emerge as orthogonal systems derived from geometric hilbert space considerations the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of hilbert spaces we study the theory of representations of the c algebra o 1 arising from this multiresolution analysis a connection with markov chains and representations of o 1 is found projection valued measures arising from the multiresolution analysis give rise to a markov trace for the quantum groups soq

enhancing data consistency in decision matrix adapting hadamard model to mitigate judgment contradiction cardinal and ordinal inconsistencies are important and popular research topics in the study of decision making with pair wise comparison matrices pcms few of the currently employed tactics are capable of simultaneously dealing with both cardinal and ordinal inconsistency issues in one model and most are heavily dependent on the method chosen for weight priorities derivation or the obtained closest matrix by optimization method that may change many of the original values in this paper we propose a hadamard product induced bias matrix model which only requires the use of the data in the original matrix to identify and adjust the cardinally inconsistent element s in a pcm through graph theory and numerical examples we show that the adapted hadamard model is effective in identifying and eliminating the ordinal inconsistencies also for the most inconsistent element identified in the matrix we develop innovative methods to improve the consistency of a pcm the proposed model is only dependent on the original matrix is independent of the methods chosen to derive the priority vectors and preserves most of the original information in matrix a since only the most inconsistent element s need s to be modified our method is much easier to implement than any of the existing models and the values it recommends for replacement outperform those derived from the literature it significantly enhances matrix consistency and improves the reliability of pcm decision making

echocardiogram analysis using fuzzy sets and relations abstract this paper describes part of a larger project whose objective is to obtain completely automated analysis of a set of time sequenced echocardiogram images to yield such heart function parameters as chamber and stroke volumes ejection fraction and indices of regional myocardial function the research reported here is concerned with identifying collections of local regions in different frames which have similar characteristics and hence comprise a global region and those global regions which add together to form other known global regions the local regions are described by a fuzzy vector whose components are fuzzy numbers an equality like relation is defined on the set of local regions and equivalence classes are used to define the global regions a similar technique is used to decide which global regions add to larger global regions the end result is used by a fuzzy expert system which classifies the global regions according to their anatomical significance or as artifacst the results here reported suggest the utility of the fuzzy set techniques employed the possible application of our methods to other pattern recognition problems is discussed

applications of the extent analysis method on fuzzy ahp in this paper a new approach for handling fuzzy ahp is introduced with the use of triangular fuzzy numbers for pairwise comprison scale of fuzzy ahp and the use of the extent analysis method for the synthetic extent value si of the pairwise comparison by applying the principle of the comparison of fuzzy numbers that is v m1 m2 1 iff m1 m2 v m2 m1 hgt m1 m2 m1 d the vectors of weight with respect to each element under a certaine criterion are represented by d ai min v si sk k 1 2 n k i this decision process is demonstrated by an example

fuzzy analytic hierarchy process fallacy of the popular methods abstract the fuzzy analytic hierarchy process fuzzy ahp is a very popular decision making method and literally thousands of papers have been published about it however we find the basic logic of this approach has problems from its methodology the definition and operational rules of fuzzy numbers not only oppose the main logic of fuzzy set theory but also oppose the basic principles of the ahp in dealing with the outcomes fuzzy ahp does not give a generally accepted method to rank fuzzy numbers and a way to check the validity of the results besides we discuss the validity of the analytic hierarchy network process ahp anp in complex and uncertain environments and find that fuzzy anp is a false proposition because there is no fuzzy priority in the super matrix which provides the basis for the anp although fuzzy ahp has been applied in many cases and cited hundreds of times we hoped that those who use fuzzy ahp would understand the problems associated with this method

fuzzy scores of technical efficiency abstract this paper suggests to extend scores of technical efficiency as used in for example data envelopment analysis dea to efficiency scores defined as so called fuzzy intervals fuzzy scores allow the decision maker to use scores of technical efficiency in combination with other sources of available performance information e g expert opinions key figures etc in this sense fuzzy scores become formal images of the evaluation otherwise done implicitly by the decision maker and may therefore enhance a more consistent evaluation it is shown how to rank a set of fuzzy scores

stochastic group preference modelling in the multiplicative ahp a model of group consensus this paper proposes a model for the representation of a consensus seeking group decision making process under the multiplicative variant of the analytic hierarchy process ahp in particular it proposes a group preference model which expresses the group s preference intensity judgements as random variables with associated probability distributions and the structure of the multiplicative ahp is exploited to derive interval judgements for the alternatives final impact scores as perceived by the group these interval judgements can be used to determine the probability that a rank reversal may occur amongst alternatives due to the group preference uncertainty i e to assess the stability of the final impact score vector which in turn offers information about the degree of consensus within the group about the decision

closed form expressions for the finite approximations of first and higher derivatives based on taylor series numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series in this paper we have presented closed form expressions of these approximations of arbitrary order for first and higher derivatives a comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses the comparison reveals that the central difference approximations can be used as digital differentiators because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator it is also observed that central difference approximations are in fact the same as maximally flat digital differentiators in the appendix a computer program written in mathematica is presented which can give the approximation of any order to the derivative of a function at a certain mesh point

algebraic structure of centered m fuzzy numbers abstract in this paper we discuss the algebraic structure of fuzzy numbers assuming that they are generated by a subset of mappings from r to a lattice ordered monoid particularly to a positive or a negative cone and for every fuzzy number there is a center defined using a special ordering on the set of fuzzy numbers and introducing special binary fuzzy algebraic operations we obtain different structural properties of these fuzzy numbers we study these algebraic structures using different class of generator mappings and establish the isomorphism between the real line and the set of the centered fuzzy numbers

the fuzzy mathematics of finance this paper develops fuzzy analogues of the elementary compound interest problems in the mathematics of finance we find the fuzzy present value and fuzzy future value of fuzzy cash amounts using fuzzy interest rates over n periods where n may be crisp or fuzzy this paper also develops a method of comparing fuzzy net cash flows in order to rank fuzzy investment alternatives from best to worst

on the extent analysis method for fuzzy ahp and its applications abstract in a paper by chang d y chang applications of the extent analysis method on fuzzy ahp european journal of operational research 95 1996 649 655 an extent analysis method on fuzzy ahp was proposed to obtain a crisp priority vector from a triangular fuzzy comparison matrix it is found that the extent analysis method cannot estimate the true weights from a fuzzy comparison matrix and has led to quite a number of misapplications in the literature in this paper we show by examples that the priority vectors determined by the extent analysis method do not represent the relative importance of decision criteria or alternatives and that the misapplication of the extent analysis method to fuzzy ahp problems may lead to a wrong decision to be made and some useful decision information such as decision criteria and fuzzy comparison matrices not to be considered we show these problems to avoid any possible misapplications in the future

canonical fuzzy bags and bag fuzzy measures as a basis for madm with mixed non cardinal data abstract this contribution introduces what will be called sine analysis a new approach to multiple attribute decision making madm with mixed noncardinal data this approach is based upon the notion of canonical fuzzy bags and their scalar evaluations by bag fuzzy measures

ranking of fuzzy sets based on the concept of existence abstract various approaches have been proposed for the comparison or ranking of fuzzy sets however due to the complexity of the problem a general method which can be used for any situation still does not exist this paper formalizes the concept of existence for the ranking of fuzzy sets many of the existing fuzzy ranking methods are shown to be some application of this concept an improved fuzzy ranking method is then introduced based on this concept this newly introduced method is expanded for treating both normal and nonnormal convex and nonconvex fuzzy sets emphasis is placed on the use of the subjectivity of the decision maker such as the optimistic or the pessimistic view points an improved procedure for obtaining linguistic conclusions is also developed finally some numerical examples are given to illustrate the approach

acceptable consistency of aggregated comparison matrices in analytic hierarchy process abstract the analytic hierarchy process is a method for solving multiple criteria decision problems as well as group decision making the weighted geometric mean method is appropriate when aggregation of individual judgements is used this paper presents a new proof which confirms the property that if the comparison matrices of all decision makers are of acceptable consistency then the weighted geometric mean complex judgement matrix wgmcjm also is of acceptable consistency this property was presented and first proved by xu 2000 but lin et al 2008 rejected the proof we also discuss under what conditions the wgmcjm is of acceptable consistency when not all comparison matrices of decision makers are of acceptable consistency for this case we determine the sufficient condition for the wgmcjm to be of acceptable consistency and provide numerical examples for a special case of two decision makers with 3 3 comparison matrices we find out some additional conditions for the wgmcjm to be of acceptable consistency

the zeros of basic bessel functions the functions jv ax x and associated orthogonal polynomials abstract the modified lommel polynomials are generalized and their orthogonality relation is obtained as a byproduct we prove that the non zero roots of jv ax x are real simple and interlace with those of jv ax 1 x the q lommel polynomials are shown to play in the theory of basic bessel function the role played by the lommel polynomials in the bessel function theory the q lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support this is then used to prove that the prositive zeros of x j 2 v x q are real simple and interlace with the zeros of x 1j 2 v 1 x q when 1 we also establish the complete monotonicity of 1x 1 2 j v 1 2 i x q j v 2 i x q

fuzzy eigenvector method for obtaining normalized fuzzy weights from fuzzy pairwise comparison matrices proper formulas for obtaining the fuzzy maximal eigenvalue and the corresponding fuzzy maximal eigenvector of a fuzzy pairwise comparison matrix are proposed in this paper first the formulas for obtaining the fuzzy maximal eigenvalue of a fuzzy pairwise comparison matrix proposed by csutora buckley 2001 and by ishizaka 2014 are reviewed and the flaws in their formulas regarding the violation of the reciprocity of pairwise comparisons are pointed out new formulas for obtaining the fuzzy maximal eigenvalue preserving the reciprocity of pairwise comparisons are then proposed after a fuzzy extension of saaty s consistency index and consistency ratio is introduced in order to verify an acceptable level of inconsistency of a fuzzy pairwise comparison matrix further the methods for obtaining the fuzzy maximal eigenvector corresponding to the fuzzy maximal eigenvalue of a fuzzy pairwise comparison matrix proposed by wang chin 2006 and by ishizaka 2014 are reviewed the flaws in ishizaka s method are pointed out and wang chin s method is studied and modified in order to preserve the reciprocity of pairwise comparisons the fuzzy maximal eigenvalues and the corresponding fuzzy maximal eigenvectors obtained by the new formulas are confronted with those obtained by methods proposed by csutora buckley 2001 wang chin 2006 and ishizaka 2014 and three numerical examples are given for better illustration a proper fuzzy extension of the maximal eigenvector method is proposed the new fuzzy maximal eigenvalue preserves the reciprocity of pairwise comparisons by employing the reciprocity into the method redundant information is eliminated the elements of the fuzzy maximal eigenvector are normalized fuzzy weights

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

hierarchical structures of fuzzy ratings in the analysis of strategic goals of enterprises abstract analysis of strategic goals of enterprises involves simultaneous consideration of various goals subgoals and strategies which are changed according to situations and problems and it is a complex unstructured ill defined process incorporating multiattribute comparisons and impreciseness of the data to cope with this analysis a methodology integrating the analytic hierarchy process ahp procedure what if analysis fuzzy sets and fuzzy arithmetic is proposed in this paper

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

new finite difference formulas for numerical differentiation conventional numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series and closed form expressions of these finite difference formulas have already been presented in this paper we present new finite difference formulas which are more accurate than the available ones especially for the oscillating functions having frequency components near the nyquist frequency closed form expressions of the new formulas are given for arbitrary order a comparison of the previously available three types of approximations is given with the presented formulas a computer program written in mathematica based on new formulas is given in the appendix for numerical differentiation of a function at a specified mesh point

remarks on the analytic hierarchy process the analytic hierarchy process ahp is flawed as a procedure for ranking alternatives in that the rankings produced by this procedure are arbitrary this paper provides a brief review of several areas of operational difficulty with the ahp and then focuses on the arbitrary rankings that occur when the principle of hierarchic composition is assumed this principle requires that the weights on the higher levels of a hierarchy can be determined independently of the weights on the lower levels virtually all of the published examples of the use of the ahp to evaluate alternatives relative to a set of criteria have assumed this principle the key to correcting this flaw is the synthesis of the ahp with the concepts of multiattribute utility theory

representations of fuzzy relations with an application to group decision making abstract representations of fuzzy relations of type 1 and type 2 satisfying max and max product transitivities in terms of exact and fuzzy utility functions respectively have been obtained an application to group decision making has been indicated

methods for decision making with cardinal numbers and additive aggregation abstract for multi attribute decision making with cardinal numbers and additive aggregation the methods additive aggregation and outranking for crisp and fuzzy numbers are connected with the concept of linguistic variables preference ranking a generalization of outranking and preference ranking organization methods such as electre and promethee can be divided into three phases where each of them is defined completely by linguistic variables the derived method preference ranking with additive weighting and individual fuzziness has an improved selectivity or discrimination compared to conventional methods moreover the method fulfils four important requirements preference only for sufficiently large weights considering evaluation differences outside of thresholds and regarding aggregational and substitutional properties the simplified method preference ranking with additive weighting and general fuzziness allows a reduction of computational resources

fuzzy hierarchical analysis revisited abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by saaty in the analytic hierarchy process we test our new procedure in two cases where there are formulas for the crisp weights an example is presented where there are five criteria and three alternatives

numerical methods for interval and fuzzy number comparison based on the probabilistic approach and dempster shafer theory there are many different methods for interval and fuzzy number comparison proposed in the literature which provide the results of comparison in the form of a real or boolean value in this paper we use the dempster shafer theory of evidence with its probabilistic interpretation to justify and construct the method which provides the result of comparison in the form of an interval or fuzzy number the complete and consistent set of expressions for inequality and equality relations between intervals and fuzzy numbers has been obtained in the framework of the probabilistic approach these relations make it possible to compare intervals and fuzzy numbers with real values and may be considered as an asymptotic limit of the results we obtain using the dempster shafer theory a natural fuzzy extension of the proposed approach is considered and discussed using some illustrative examples

the r d project selection problem with fuzzy coefficients abstract this paper deals with the multicriteria project selection problem subject to linear constraints with zero one decision variables linguistic rather than numerical values are used in the project appraisals for each criterion with their meanings represented by normal fuzzy sets the grade of membership of a given alternative in the fuzzy set of all nondominated alternatives is computed by an algorithm that converges rapidly on the final solution continuity and normality are the only restrictions imposed on the membership functions an example of the overall procedure and topics for future research are presented

a remainder formula of numerical differentiation for the generalized lagrange interpolation through exploiting the generalized lagrange interpolation for continuous piecewise smooth functions a remainder term of numerical differentiation is given also the error estimates for the numerical differentiation formula are presented

the preference order of fuzzy numbers abstract many fuzzy number ranking approaches are developed in the literature for multiattribute decision making problems almost all of the existing approaches focus on quantity measurement of fuzzy numbers for ranking purpose in this paper we consider the ranking process to determine a decision maker s preference order of fuzzy numbers a new ranking index is proposed to not only take quantity measurement but incorporate quality factor into consideration for the need of general decision making problems for measuring quantity several cuts of fuzzy numbers are used a signal noise ratio is defined to evaluate quality of a fuzzy number this ratio considers the middle point and spread of each cut of fuzzy numbers as the signal and noise respectively a fuzzy number with the stronger signal and the weaker noise is considered better moreover the associated levels are treated as the degree of belief about the cut and used as weights in the index for strengthening the influence of cut with higher levels the membership functions of fuzzy numbers are not necessarily to be known beforehand while applying this index only a few left and right boundary values of cuts of fuzzy numbers are required we have proved the feature of the proposed index in a particular case several examples are also used to illustrate the feature and applicability in ranking fuzzy numbers

general explicit difference formulas for numerical differentiation through introducing the generalized vandermonde determinant the linear algebraic system of a kind of vandermonde equations is solved analytically by use of the basic properties of this determinant and then we present general explicit finite difference formulas with arbitrary order accuracy for approximating first and higher derivatives which are applicable to unequally or equally spaced data comparing with other finite difference formulas the new explicit difference formulas have some important advantages basic computer algorithms for the new formulas are given and numerical results show that the new explicit difference formulas are quite effective for estimating first and higher derivatives of equally and unequally spaced data

ranking alternatives using fuzzy numbers a computational approach abstract the method presented by buckley fuzzy sets and systems 15 1985 21 31 for ranking alternatives by using fuzzy numbers is discussed as far as computational aspects are concerned an alternative procedure is presented which allows for the use of any convex and normalized fuzzy set to assess the contribution of each alternative to some overall objective according to a set of criteria the procedure also allows the criteria to be structured in n level hierarchies an example is presented to illustrate the algorithm

eigenvector and logarithmic least squares abstract the eigenvector method deals with two questions simultaneously closeness and order together they belong to the field topology of order the metric idea of closeness is inadequate to judge what is a good approximation to data involving order relations there is usually a condition which relates order in the derived scale to order in the data this is precisely the kind of condition captured by the eigenvector but not by the logarithmic least squares estimate

generalized and extended fuzzy sets with applications abstract this paper introduces generalized fuzzy sets extended fuzzy sets and generalized extended fuzzy sets and shows how these fuzzy sets may arise naturally in some applications the applications presented are chosen from the areas of decision theory hierarchical analysis and expert systems zadeh s method of reducing a level 2 fuzzy set to a regular fuzzy set is generalized to reduce some of these new fuzzy sets to simpler fuzzy sets what is needed then for applications is an algebra of irreducible generalized extended fuzzy sets

on fuzzy number lattice r abstract we give a systematic development of fuzzy number lattice theory many of our results generalize to number lattice over the lattice algebra our first main result is that the real number chain r is a homomorphic image of the fuzzy number lattice r that is r r and the congruence class r is a component of r thus we can be specialized to describe the structure of the fuzzy number lattice r next we study the structure and representation of the congruence class

ranking and defuzzification methods based on area compensation abstract we present some interesting properties related to the area compensation procedure to compare fuzzy numbers it has been proved that this method produces more than a fuzzy interval order it induces a ranking of fuzzy numbers some further results are given about the transitivity property and about computational aspects extensions to non normal fuzzy numbers and fuzzy quantities are also proposed

ranking alternatives using fuzzy numbers we investigate the problem of employing expert opinion to rank alternatives across a set of criteria the experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue s fuzzy ranking this leads to a partition of the alternatives into sets h1 h2 where h1 contains the highest ranked issues h2 has all the second highest ranked alternatives etc the total ranking process is shown to possess a number of important properties an example is presented to illustrate the method

p stable exponentially fitted methods for the numerical integration of the schrodinger equation a p stable exponentially fitted method is developed in this paper for the numerical integration of the schrodinger equation an application to the bound states problem we solve the radial schrodinger equation in order to find eigenvalues for which the wavefunction and its derivative are continuous and the boundary conditions are satisfied and the resonance problem the point of a resonance is that phase changes rapidly through of the radial schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially fitted methods of the same kind the method can be applied to any problem of physics and chemistry which can be expressed as system of coupled second order differential equations which have oscillatory or periodic solutions this is because it has the property of the p stability i e the interval of periodic stability of the proposed method is equal to 0 which allow is to integrate successful problems with high oscillatory or periodic solution

taylor series based finite difference approximations of higher degree derivatives a new type of taylor series based finite difference approximations of higher degree derivatives of a function are presented in closed forms with their coefficients given by explicit formulas for arbitrary orders characteristics and accuracies of presented approximations and already presented central difference higher degree approximations are investigated by performing example numerical differentiations it is shown that the presented approximations are more accurate than the central difference approximations especially for odd degrees however for even degrees central difference approximations become attractive as the coefficients of the presented approximations of even degrees do not correspond to equispaced input samples

ranking of fuzzy numbers by sign distance several different strategies have been proposed for ranking of fuzzy numbers these include methods based on the coefficient of variation cv index distance between fuzzy sets centroid point and original point and weighted mean value each of these techniques has been shown to produce non intuitive results in certain cases in this paper we propose a modification of the distance based approach called the sign distance which is both efficient to evaluate and able to overcome the shortcomings of the previous techniques the calculation of the proposed method is far simpler than the other approaches

multi criteria group decision making based on bilateral agreements abstract in multi criteria group decision making a key issue is aggregating individual preference into group one in this study we employ a bilateral agreement to conduct group evaluation of alternatives the bilateral agreement between a pair of individuals could be on the weight of a criterion a criterion evaluation function or willingness to pay any one of three types of bilateral agreements can derive the group utility to express the relationship between the bilateral agreement and individual evaluations the quasi arithmetic mean is used which can ensure the consistency property of the pairwise comparison matrices the minimum requirements are explored to obtain the group preference which shows that n 1 pairs of bilateral agreements are necessary finally several examples are provided to illustrate the proposed methods

an approximate approach for ranking fuzzy numbers based on left and right dominance this study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance the proposed approach only requires a few left and right spreads at some levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other the total dominance is then determined by combining the left and right dominance based on a decision maker s optimistic perspectives such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired the approach proposed herein is relatively simple in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers by using a few left and right spreads two groups of examples demonstrate the accuracy and applicability of the proposed approach

ranking alternatives with fuzzy weights using maximizing set and minimizing set ranking alternatives both qualitative as well as quantitative in a multicriterion environment employing experts opinion preference structure using fuzzy numbers and linguistic variables are presented in this paper fuzzy weights wj i of alternatives a i are computed using standard fuzzy arithmetic concept of maximizing set and minimizing set is introduced to decide total utility or ordering value of each of the alternatives a numerical example is provided at the end to illustrate the method 1999 published by elsevier science b v all rights reserved

an improved rotated staggered grid finite difference method with fourth order temporal accuracy for elastic wave modeling in anisotropic media abstract the rotated staggered grid rsg finite difference method is a powerful tool for elastic wave modeling in 2d anisotropic media where the symmetry axes of anisotropy are not aligned with the coordinate axes we develop an improved rsg scheme with fourth order temporal accuracy to reduce the numerical dispersion associated with prolonged wave propagation or a large temporal step size the high order temporal accuracy is achieved by including high order temporal derivatives which can be converted to high order spatial derivatives to reduce computational cost dispersion analysis and numerical tests show that our method exhibits very low temporal dispersion even with a large temporal step size for elastic wave modeling in complex anisotropic media using the same temporal step size our method is more accurate than the conventional rsg scheme our improved rsg scheme is therefore suitable for prolonged modeling of elastic wave propagation in 2d anisotropic media

ranking l r fuzzy number based on deviation degree this paper proposed a novel approach to ranking fuzzy numbers based on the left and right deviation degree l r deviation degree in the approach the maximal and minimal reference sets are defined to measure l r deviation degree of fuzzy number and then the transfer coefficient is defined to measure the relative variation of l r deviation degree of fuzzy number furthermore the ranking index value is obtained based on the l r deviation degree and relative variation of fuzzy numbers additionally to compare the proposed approach with the existing approaches five numerical examples are used the comparative results illustrate that the approach proposed in this paper is simpler and better

an investigation into relations between some transitivity related concepts abstract there exists a variety of transitivity notions in the literature to eliminate the possible inconsistency adherent to a fuzzy preference relation in ranking fuzzy quantities or alternatives the relationships among max min transitivity restricted max min transitivity quasitransitivity weak transitivity consistency and acyclicity are investigated we point out that max min transitivity and transitivity are very strong restrictions on a fuzzy preference relation and acyclicity is the weakest one with a certain transitivity condition an ordering procedure is suggested to obtain a total order relation among fuzzy quantities or alternatives based on a fuzzy preference relation

derivative kernels numerics and applications a generalized framework for numerical differentiation nd is proposed for constructing a finite impulse response fir filter in closed form the framework regulates the frequency response of nd filters for arbitrary derivative order and cutoff frequency selected parameters relying on interpolating power polynomials and maximally flat design techniques compared with the state of the art solutions such as gaussian kernels the proposed nd filter is sharply localized in the fourier domain with ripple free artifacts here we construct 2d maxflat kernels for image directional differentiation to calculate image differentials for arbitrary derivative order cutoff level and steering angle the resulted kernel library renders a new solution capable of delivering discrete approximation of gradients hessian and higher order tensors in numerous applications we tested the utility of this library on three different imaging applications with main focus on the unsharp masking the reported results highlight the high efficiency of the 2d maxflat kernel and its versatility with respect to robustness and parameter control accuracy

a birth and death process related to the rogers ramanujan continued fraction abstract time dependent system size probabilities of a birth and death process related to the rogers ramanujan continued fraction are obtained the range for the parameter in this continued fraction is obtained to ensure the positivity of the recursively defined birth and death rates the general behavior of the birth and death rates is described and the asymptotic behavior of the transition probabilities and the quasi stationary distribution is determined for the transient case the birth and death process can be seen as a model in queuing theory where the length of the queue encourages customers to join the queue the more the merrier

a simple method to improve the consistency ratio of the pair wise comparison matrix in anp tests of consistency for the pair wise comparison matrices have been studied extensively since ahp was introduced by saaty in 1970s however existing methods are either too complicated to be applied in the revising process of the inconsistent comparison matrix or are difficult to preserve most of the original comparison information due to the use of a new pair wise comparison matrix those methods might work for ahp but not for anp as the comparison matrix of anp needs to be strictly consistent to improve the consistency ratio this paper proposes a simple method which combines the theorem of matrix multiplication vectors dot product and the definition of consistent pair wise comparison matrix to identify the inconsistent elements the correctness of the proposed method is proved mathematically the experimental studies have also shown that the proposed method is accurate and efficient in decision maker s revising process to satisfy the consistency requirements of ahp anp

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

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

ranking fuzzy numbers based on the areas on the left and the right sides of fuzzy number in this paper a novel method based on the areas on the left and the right sides of fuzzy numbers is proposed for ranking fuzzy numbers the merits of the results given here is to overcome certain shortcomings in the recent literature that mostly does not end in the right ordering of fuzzy numbers the method also has very easy and simple calculations compared to other methods moreover numerical examples are given to compare the proposed method with other existing ones

fuzzy method in group decision making abstract this paper investigates the problems of decision making with multiple judge multiple criteria in a fuzzy environment where the performance of alternatives and the importance of criteria are imprecisely defined and represented by fuzzy sets a fuzzy model associated with the solution algorithm is proposed on the basis of an level weighted fuzzy preference relation a numerical example is solved for illustration

the multiple judge multiple criteria ranking problem a fuzzy set approach abstract this paper investigates the problem of selecting from a set of issues those which best satisfy a collection of criteria a group of judges have fuzzy sets defined over the issues for each criterion whose values lie in a finite linearly ordered set l these judges also have fuzzy sets defined over the set of criteria the paper discusses methods of aggregating all the fuzzy sets into one fuzzy set defined on the issues so that a l gives the final ranking for issue a

a modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process this paper revisits the fuzzy logarithmic least squares method llsm in the analytic hierarchy process and points out its incorrectness in the normalization of local fuzzy weights infeasibility in deriving the local fuzzy weights of a fuzzy comparison matrix when the lower bound value of a non normalized fuzzy weight turns out to be greater than its upper bound value uncertainty of local fuzzy weights for incomplete fuzzy comparison matrices and unreality of global fuzzy weights a modified fuzzy llsm which is formulated as a constrained nonlinear optimization model is therefore suggested to tackle all these problems a numerical example is examined to show the applicability of the modified fuzzy llsm and its advantages

strong transitivity rationality and weak monotonicity in fuzzy pairwise comparisons strong transitivity and weak monotonicity are two important axioms in the theory of value and utility measurement in this paper we define strong transitivity and weak monotonicity for fuzzy pairwise comparisons matrices using the requirements of strong transitivity we develop a metric for measuring the degree of strong transitivity in fuzzy pairwise comparisons we further demonstrate that a matrix which satisfies the metric of strong transitivity will satisfy the weak monotonicity axiom as well

fuzzy capital budgeting fuzzy equivalents of all the classical capital budgeting investment choice methods are generalised or proposed these equivalents can be used to evaluate and compare projects in which the cash ows duration time and required rate of return cost of capital are given imprecisely in the form of a fuzzy number c 2000 elsevier science b v all rights reserved

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

tool steel materials selection under fuzzy environment a fuzzy set multiple criteria decision making approach is proposed to help selecting the most suitable tool steel materials for a specific manufacturing application such as die design jig and fixture design the important weights of different criteria and the material suitability ratings of various alternatives under different criteria are given in linguistic terms subsequently they are represented in fuzzy linguistic values and fuzzy numbers through aggregation and ranking of fuzzy numbers the most suitable tool steel material is determined a hypothetical example is illustrated further by computerizing the proposed method and incorporating appropriate interface design together with a data base of tool steel materials assessment information from the handbook a decision support system can be constructed

the ratio of q like orthogonal polynomials abstract it is shown that the ratio of orthogonal polynomials with exponentially increasing recurrence coefficients is closely related to some new orthogonal q polynomials of which the recurrence coefficients converge exponentially fast to zero these q polynomials are investigated in detail

an algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy in this paper we introduce an algorithm and a computer code for numerical differentiation of discrete functions the algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points the algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas in particular when the derivatives are required with high approximation accuracy moreover the given matlab computer code can be implemented to solve boundary value ordinary and partial differential equations with high numerical accuracy the numerical technique is based on the undetermined coefficient method in conjunction with taylor s expansion to avoid the difficulty of solving a system of linear equations an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions this is done by using an explicit closed formula for the vandermonde matrix inverse moreover the code is designed to give a unified approximation order throughout the given domain a numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code it is found that the method and the code work properly for any degree of derivative and any order of accuracy

representation for the lagrangian numerical differentiation formula involving elementary symmetric functions by using elementary symmetric functions this paper presents an explicit representation for the lagrangian numerical differentiation formula as well as the error estimate for local approximation and we also point out that the numerical differentiation formula constructed by li j p li general explicit difference formulas for numerical differentiation j comput appl math 183 2005 29 52 is actually a special case of the lagrangian numerical differentiation formula to approximate the values of the derivatives at the nodes

an efficient approach to solving fuzzy madm problems abstract an approach using defuzzifying methods is proposed for the fuzzy multiple attribute decision making madm problem the computational effectiveness and efficiency of six defuzzifying methods combined with the simple additive weighting saw method are evaluated based on a comparison to the improved fuzzy weighted average ifwa followed by four fuzzy ranking methods numerical examples are discussed to demonstrate the implementation and effectiveness of the methods

on the interval approximation of a fuzzy number the notion of an approximation interval of a fuzzy number is introduced it is the interval which fulfills two conditions in the first its width is equal to the width of a fuzzy number being approximated in the second the hamming distance between this interval and the approximated number is minimal the introduced notion is compared with the notion of expected interval of a fuzzy number known in literature