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

the revised method of ranking fuzzy numbers with an area between the centroid and original points in 2002 chu and tsao proposed a method to rank fuzzy numbers they employed an area between the centroid and original points to rank fuzzy numbers however there were some problems with the ranking method in this paper we want to indicate these problems of chu and tsao s method and then propose a revised method which can avoid these problems for ranking fuzzy numbers since the revised method is based on the chu and tsao s method it is easy to rank fuzzy numbers in a way similar to the original method

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

higher order constrained hamiltonian systems in this paper we study a hamiltonian formulation of the higher order constrained systems hocss defined previously in lagrangian terms we shall focus on second order constraints in positions i e second order hocss which correspond on the hamiltonian side to first order constraints in momenta we show that for affine second order constraints the existence and uniqueness conditions can be stated for their corresponding equations of motion and that the latter can be written in terms of a leibniz bracket we also show that hamiltonian systems with external forces can be described as suitable affine second order hocss we study some examples as systems with friction and we present an application for the global asymptotic stabilization of servomechanisms

on the geometry of non holonomic lagrangian systems we present a geometric framework for non holonomic lagrangian systems in terms of distributions on the configuration manifold if the constrained system is regular an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics if the constrained system is singular we develop a constraint algorithm which is very similar to that developed by dirac and bergmann and later globalized by gotay and nester special attention to the case of constrained systems given by connections is paid in particular we extend the results of koiller for caplygin systems an application to the so called non holonomic geometry is given

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

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

inconsistent property of lee and li fuzzy ranking method abstract this paper analyzes the lee and li fuzzy ranking method for the normalized trapezoidal fuzzy numbers ntfns every ntfn has associated with it another ntfn which posseses an inconsistent property an inconsistent property is that the results of comparison of these two ntfns are conflicting when two different comparison indices are used based on this property lee and li fuzzy ranking method suffers from lack of discrimination and occasionally conflicts with intuition

population growth model via interactive fuzzy differential equation abstract in this manuscript we study a fuzzy initial value problem fivp which describes an autocorrelated evolutive process for this we use the f correlated derivative f derivative concept on fuzzy number valued functions this approach lets us establish a relationship between the diameter of the fivp solution and its derivative moreover we notice how the derivative influences the interactivity of the process by computing the measure of possibilistic interactivity of the fivp solution we analyze a population growth model namely the logistic model with our approach and compare the results with another approach that uses the generalized derivative of hukuhara gh derivative finally we conclude that in fuzzy differential equations describing locally f correlated fuzzy processes the use of the interactive derivative is more adequate than the gh derivative for a fuzzy number valued function

integrability of implicit differential equations the problem of integrability of differential equations is discussed examples and integrability criteria are given an algorithm for extracting the integrable part of an implicit differential equation is formulated a procedure for generating a class of submanifolds of the cotangent bundle is defined this procedure is then used for generating implicit differential equations in the phase space of a mechanical system integrability criteria of such equations are established and an extraction algorithm is formulated

evaluating weapon systems using ranking fuzzy numbers abstract in this paper we will point out that chen s 1996 paper has some errors a weapon system is large and complex it has multi level multi factor and multi hierarchy features chen s method does not normalize each criterion s scores which will make a wrong decision under the following conditions i there are many levels for rank scores ii a criterion has many factors iii the total scores of systems have a larger difference under this criterion and iv one criterion s weight is larger than the other criteria numerical illustration is shown in section 5 therefore chen s method is not a general method for the evaluation of weapon systems after pointing out some errors in chen s paper to overcome these errors we will revise it and propose a general method for the evaluation of weapon systems our method utilizes fuzzy ratio scales 1 3 5 7 9 the goal is normalizing heterogeneity into homogeneity to indicate the relative strength of the factors in the corresponding criteria then we build a judgement matrix through comparison of the total scores of performance and use 1 3 5 7 9 to represent a weight vector among these criteria we will derive the priority among the alternatives by multiplying the fuzzy judgement matrix with the corresponding fuzzy weight vector the final results become a problem of ranking fuzzy numbers many triangular fuzzy numbers can easily rank its ordering by the intuition ranking method lee and li 1988 pointed out that human intuition would favor a fuzzy number with the following characteristics higher mean value and at the same time lower spread if its ordering cannot rank by figures we can use many other methods of ranking fuzzy numbers therefore the best weapon selection can be obtained by ranking fuzzy numbers

selecting is personnel use fuzzy gdss based on metric distance method abstract in this paper we propose a new approach to rank fuzzy numbers by metric distance for showing our method is a good ranking method we give two examples to compare with other methods the paper also developes a computer based group decision support system fmcgdss to increase the recruiting productivity and to easily compare our method with other fuzzy number ranking methods the fmcgdss includes three ranking methods intuition ranking lee and li s fuzzy mean spread and our metric distance method to help manager make better decision under fuzzy circumstance the result indicates that the new method is coincident with the intuition ranking and the lee and li s fuzzy mean spread method on each type weight

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

computing 2 d min median and max filters fast algorithms for computing min median max or any other order statistic filter transforms are described the algorithms take constant time per pixel to compute min or max filters and polylog time per pixel in the size of the filter to compute the median filter a logarithmic time per pixel lower bound for the computation of the median filter is shown

higher order discrete variational problems with constraints an interesting family of geometric integrators for lagrangian systems can be defined using discretizations of the hamilton s principle of critical action this family of geometric integrators is called variational integrators in this paper we derive new variational integrators for higher order lagrangian mechanical system subjected to higher order constraints from the discretization of the variational principles we show that our methods are automatically symplectic and in consequence with a very good energy behavior additionally the symmetries of the discrete lagrangian imply that momentum is conserved by the integrator moreover we extend our construction to variational integrators where the lagrangian is explicitly time dependent finally some motivating applications of higher order problems are considered in particular optimal control problems for explicitly time dependent underactuated systems and an interpolation problem on riemannian manifolds

a characterization of a general class of ranking functions on triangular fuzzy numbers this paper aims to the discussion of properties for ranking fuzzy numbers an approach of axiomatic type is followed providing the definition of two groups of requirements which characterize a ranking function through its behaviour on triangular fuzzy numbers particularly the problem of making the evaluations on triangular fuzzy numbers sensitive to their spread has been deeply analyzed

the cournot game under a fuzzy decision environment the conventional precision based decision analysis methodology is invalid for business decision analysis when precise assessment data seldom exist this paper considers the cournot game with fuzzy demand and fuzzy costs that are assumed to be triangular fuzzy numbers our model utilizes the weighted center of gravity wcog method to defuzzify the fuzzy profit function into a crisp value we derive the equilibrium cournot quantity of each firm by simultaneously solving the first order condition of each firm our model explicitly derives the necessary condition to avoid an unreasonable outcome of negative equilibrium quantities and lack of flexibility for modification of the ranking method in addition we examine the standard deviation of the fuzzy profit resulting from the fuzziness of each firm s cost and market demand functions we conduct sensitivity analysis to investigate the effect of parameter perturbations on firms outcomes the results indicate that the center of parameter plays an important role in sensitivity analysis and dominates over variations in equilibrium quantity due to a perturbation of fuzzy parameters

comparison of fuzzy numbers using a fuzzy distance measure a new approach for ranking fuzzy numbers based on a distance measure is introduced a new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first and then it is used to formulate the distance measure for fuzzy numbers the approach is illustrated by numerical examples showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches

generalized nonholonomic mechanics servomechanisms and related brackets it is well known that nonholonomic systems obeying d alembert s principle are described on the hamiltonian side after using the legendre transformation by the so called almost poisson brackets in this paper we define the lagrangian and hamiltonian sides of a class of generalized nonholonomic systems gnhs obeying a generalized version of d alembert s principle such as rubber wheels like some simplified models of pneumatic tires and certain servomechanisms like the controlled inverted pendulum and show that corresponding equations of motion can also be described in terms of a bracket we present essentially all possible brackets in terms of which the mentioned equations can be written down which include the brackets that appear in the literature and point out those if any that are naturally related to each system in particular we show there always exists a leibniz bracket related to a gnhs and conversely that every leibniz system is a gnhs the control of the inverted pendulum on a cart

reduction of constrained systems with symmetries a general model is proposed for constrained dynamical systems on a symplectic manifold which covers among others the description of lagrangian and hamiltonian systems with nonholonomic constraints and the canonical description of mechanical systems with a singular lagrangian the reduction properties of these systems in the presence of symmetry are investigated within this general framework

directional morphological filtering we show that a translation invariant implementation of min max filters along a line segment of slope in the form of an irreducible fraction dy dx can be achieved at the cost of 2 k min max comparisons per image pixel where k max dx dy therefore for a given slope the computation time is constant and independent of the length of the line segment we then present the notion of periodic moving histogram algorithm this allows for a similar performance to be achieved in the more general case of rank filters and rank based morphological filters applications to the filtering of thin nets and computation of both granulometries and orientation fields are detailed finally two extensions are developed the first deals with the decomposition of discrete disks and arbitrarily oriented discrete rectangles while the second concerns min max filters along gray tone periodic line segments

a two phase multi attribute decision making approach for new product introduction this study aims at improving the quality and effectiveness of decision making in new product introduction new product development has long been recognized as one of the corporate core functions to be competitive on an increasingly competitive global market however developing new products is a process involving risk and uncertainty in order to solve this stochastic problem companies need to evaluate their new product initiatives carefully and make accurate decisions for this reason a systematic decision process for selecting more rational new product ideas is proposed basically two stages of decision making are described the identification of nondominated new product candidates and the selection of the best new product idea these stages are composed of an integrated approach based on a fuzzy heuristic multi attribute utility method and a hierarchical fuzzy topsis method finally an application is given to demonstrate the potential of the methodology

fast impulsive noise removal a generic n dimensional filter with the primary purpose of eliminating impulsive like noise is presented this recursive nonlinear filter is composed of two conditional rules which are applied independently in any order one after the other it identifies noisy items by inspection of their surrounding neighborhood and afterwards it replaces their values with the most conservative ones out of their neighbors values in this way no new values are introduced and the histogram distribution range is conserved this n dimensional filter can be decomposed recursively to a lower dimensional space each time generating two sets of n n 1 dimensional filters this study which focuses on the case of two dimensional signals gray scale images explores one possible implementation of this new filter and orients the evaluation of its performance toward the median filter as this filter is the basis of many more sophisticated filters for impulsive noise reduction tests were carried out using both real and artificial images we found this new filter to be much faster than the median filter while performing comparably in terms of both image information conservation and noise reduction which suggests that it could replace the median filter for the preliminary processing included in state of the art noise removal filters this new filter should either eliminate or attenuate most noisy pixels in synthetic and natural images not excessively contaminated it has a slight smoothing effect on nonnoisy image regions in addition it is scalable easily implemented and adaptable to specific applications

falscal a fuzzy multidimensional scaling algorithm the conventionally adopted alternating least squares scaling alscal procedure of multidimensional scaling mds is a valuable mathematical scheme for analyzing data in areas where organized concepts and underlying dimensions are inadequately defined or developed fuzzy set theory fst attempts to formulate human reasoning and perceptions therefore targeting problems in areas where human factors significantly impact the result of decision making to our knowledge the fst and alscal approaches have not yet been integrated this study integrates and modifies the fst and alscal procedures fuzzy data collected from fuzzy questionnaires are adopted as the input of the mds ensuring that the uncertainty of input data can be incorporated into the analysis the conventionally adopted alscal procedure is then modified to cope with fuzzy input data by adopting the notion of fuzzy distances fuzzy disparities and fuzzy ranking to represent the similarities between fuzzy data related approximation operations of the triangular fuzzy number are also introduced to facilitate computation in fuzzy alscal

some remarks on distances between fuzzy numbers in this paper we consider different approaches to assigning distances between fuzzy numbers a pseudo metric on the set of fuzzy numbers arising from the idea of the value of a fuzzy number is described and some of its topological properties are noted reducing functions are used to define a family of metrics on the space of fuzzy numbers some convergent properties for these metrics are illustrated finally a fuzzy distance between fuzzy numbers is introduced and its basic properties are studied

d alembert lagrange analytical dynamics for nonholonomic systems the d alembert lagrange principle dlp is designed primarily for dynamical systems under ideal geometric constraints although it can also cover linear velocity constraints its application to nonlinear kinematic constraints has so far remained elusive mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint on recognition that the commutation rule traditionally accepted for velocity displacements in lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements which can be adjoined via lagrange s multipliers to the d alembert lagrange equation to yield the equations of state and also new transpositional relations for nonholonomic systems the equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived in appendix a from the quite different gauss principle the present advance therefore solves a long outstanding problem on the application of dlp to ideal nonholonomic systems and as an aside provides validity to axioms as the chetaev rule previously left theoretically unjustified a more general transpositional form of the boltzmann hamel equation is also obtained

ranking fuzzy numbers with preference weighting function expectations based on the recent research developments of fuzzy number ranking the paper extends the centroid expectation approach and proposes a preference weighting function expectation method to rank fuzzy numbers the weighting function gives different emphasis on the possible values of the fuzzy number which can be used to represent the decision maker s pessimistic neutral or optimistic attitude an optimistic measure to represent this preference information is proposed some properties of the weighting function and optimistic measure are systematically analyzed then the paper proposes a special class weighting function with maximum entropy principle and analyzes some of its properties with this parameterized maximum entropy weighting function we can get preference expectation of any fuzzy number with given optimistic level next the weighting function is extended to represent the confidential attitude of the decision maker by giving different emphasis on the cut level set of the fuzzy number some results about confidential attitude that parallel to the valuation attitude are attained we especially extended the badd basic defuzzification distribution approach with preference expectation of fuzzy numbers and another confidential measure with badd operator is proposed finally the expected valuations of fuzzy numbers are used to ranking the fuzzy numbers compared with other fuzzy number ranking methods the method of this paper is easily interpreted robust flexible and invariant with translating and scaling for real numbers

contribution to application of energy measure of fuzzy sets abstract the works of de luca termini continued by for example knopfmacher loo and gottwald are the most important on the topic of determination of measures of fuzzy sets the matter is to evaluate how fuzzy a fuzzy set is there are two general concepts of measures of fuzzy set i e entropy and energy measures we show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations applications of the use of energy measure discussed in detail include decision making fuzzy process control and prediction in fuzzy systems

ranking nonnormal p norm trapezoidal fuzzy numbers with integral value this paper considers the ranking fuzzy numbers with integral value proposed by liou and wang for the nonnormal p norm trapezoidal fuzzy numbers two interesting special cases of p norm trapezoidal fuzzy numbers are the well known triangular and trapezoidal fuzzy numbers for the nonnormal fuzzy numbers the differences exist among the membership functions for the triangular trapezoidal and p norm trapezoidal fuzzy numbers these differences can affect their left right and total integral values so we establish the relationship between these three types of fuzzy numbers

multiple resolution texture analysis and classification textures are classified based on the change in their properties with changing resolution the area of the gray level surface is measured at serveral resolutions this area decreases at coarser resolutions since fine details that contribute to the area disappear fractal properties of the picture are computed from the rate of this decrease in area and are used for texture comparison and classification the relation of a texture picture to its negative and directional properties are also discussed

fuzzy job sequencing for a flow shop abstract in job sequencing for a flow shop processing times are frequently not known exactly and only estimated intervals are given fuzzy numbers are ideally suited to represent these intervals in this work the campbell dudek and smith cds job sequencing algorithm is modified to accept trapezoidal fuzzy processing times deterministic sequences result but the sequence performance measurements of makespan and job mean flow time are fuzzy having been calculated using fuzzy arithmetic the use of possibility theory and the fuzzy integral enables the schedular to meaningfully interpret these fuzzy results deterministic approximations to this fuzzy approach are also investigated

new approaches for the comparison of l r fuzzy numbers a theoretical and operational analysis abstract a key issue in operationalizing fuzzy set theory particularly in decision analysis is how to compare fuzzy numbers in this paper the case of l r fuzzy numbers i e the most general form of fuzzy numbers is considered in particular here l r fuzzy numbers represented by continuous convex membership functions allowing also definite integration is taken into consideration normality is not required traditional comparison methods are generally limited to the use of triangular fuzzy numbers and often the shape of the membership function is not taken into account or only a part of it is used leading to a loss of information most of the approaches one can find in the literature are characterised by the use of cuts and credibility levels the use of areas for comparing fuzzy numbers has been proposed only recently in particular in the so called naiade method a new semantic distance able to compare crisp numbers fuzzy numbers and density functions has been developed the basic idea underlying this paper is that if only l r fuzzy numbers are considered other methodologies for comparing fuzzy numbers can be developed three indices based on the use of areas are studied i e the expected value the variance with its decomposition into positive and negative semivariances and the degree of coincidence of two fuzzy numbers a justification of the use of these indices and a first tentative of axiomatisation is given a short discussion on the issue of possible aggregation conventions of these indices is presented and an empirical example is examined too

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

divide and conquer heuristics for minimum weighted euclidean matching we consider the following problem given n points in a unit square in the euclidean plane find a matching of the points such that the cost i e the sum of the lengths of the edges between matched points is minimum in particular we present a class of linear time heuristic algorithms for this problem and analyze their worst case performance the worst case performance of an algorithm is defined as the greatest possible cost as a function of n of the matching produced by the algorithm on a set of n points each of the algorithms studied here divides the unit square into a few smaller regions and then is applied recursively to the points in each of these regions

a generalized fuzzy mathematical morphology and its application in robust 2 d and 3 d object representation in this paper the generalized fuzzy mathematical morphology gfmm is proposed based on a novel definition of the fuzzy inclusion indicator fii fii is a fuzzy set used as a measure of the inclusion of a fuzzy set into another that is proposed to be a fuzzy set it is proven that the fii obeys a set of axioms which are proposed to be extensions of the known axioms that any inclusion indicator should obey and which correspond to the desirable properties of any mathematical morphology operation the gfmm provides a very powerful and flexible tool for morphological operations the binary and grayscale mathematical morphologies can be considered as special cases of the proposed gfmm an application for robust skeletonization and shape decomposition of two dimensional 2 d and three dimensional 3 d objects is presented simulation examples show that the object reconstruction from their skeletal subsets that can be achieved by using the gfmm is better than by using the binary mathematical morphology in most cases furthermore the use of the gfmm for skeletonization and shape decomposition preserves the shape and the location of the skeletal subsets and spines

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

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

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

analysis and simulation of fuzzy queues abstract the aim of this paper is to combine the ability of fuzzy sets to represent practical situations with the well known established traditional queuing approaches which are rigorous but the assumptions are frequently too far from reality it is shown that this approach can utilize the advantages of both the fuzzy and probability approaches to make the model more realistic and less restrictive both cut sets and the use of random variables to represent fuzzy numbers are investigated trapezoidal fuzzy numbers are used and the analytical and simulation results are discussed based on actual examples

nonholonomic versus vakonomic dynamics on a riemann cartan manifold for the chaplygin s nonholonomic constrained systems the constraint manifold can be endowed with riemann cartan geometric structure by nonholonomic mapping into a riemann manifold the two kinds of existing dynamics nonholonomic dynamics and vakonomic dynamics are compared in the framework of riemann cartan geometry it is proved that the equations of motion for nonholonomic and vakonomic dynamics are described by the equations of autoparallel and geodesic trajectories on the riemann cartan constraint manifold respectively if the metricity condition of riemann cartan connection is satisfied the torsion contorsion of the riemann cartan manifold characterizes the difference between the autoparallel and geodesic trajectories as well as the distinction between the nonholonomic and vakonomic equations

note on some models for deriving the priority weights from interval fuzzy preference relations abstract deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems xu and chen 2008 proposed a number of linear programming models to derive interval weights but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved in this paper a numerical example is given to show how these definitions and models can be improved to increase accuracy a new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation

ranking fuzzy numbers with an area method using radius of gyration ranking fuzzy numbers plays an very important role in linguistic decision making and some other fuzzy application systems many methods have been proposed to deal with ranking fuzzy numbers chu pointed out some shortcomings of the existing distance method and proposed to rank the fuzzy numbers with the area between the centroid point and original point however drawbacks are also found in the area method for example it cannot rank fuzzy numbers when some fuzzy numbers have the same centroid point in this paper we propose a modified area method to rank fuzzy numbers the modified method can effectively rank various fuzzy numbers and their images we also used some comparative examples to illustrate the advantage of the proposed method

area ranking of fuzzy numbers based on positive and negative ideal points the maximizing set and minimizing set method is a popular ranking approach for fuzzy numbers which ranks them based on their left right and total utilities this paper presents an alternative ranking approach for fuzzy numbers called area ranking based on positive and negative ideal points which defines two new alternative indices for the purpose of ranking the two new indices are defined in terms of a decision maker dm s attitude towards risks and the left and the right areas between fuzzy numbers and the two ideal points it is shown that the area ranking approach has strong discrimination power and can rank fuzzy numbers that are unable to be discriminated by the maximizing set and minimizing set method it is also shown that the dm s attitude towards risks may have a significant impact on the ranking of fuzzy numbers as a side product a new defuzzification formula is also developed and discussed

a min max medial axis transformation blum s medial axis transformation mat of the set s of 1 s in a binary picture can be defined by an iterative shrinking and reexpanding process which detects corners on the contours of constant distance from s and thereby yields a skeleton of s for unsegmented gray level pictures one can use an analogous definition in which local min and max operations play the roles of shrinking and expanding to compute a mmmat value at each point of the picture the set of points having high values defines a good skeleton for the set of high gray level points in the given picture

thick pen transformation for time series traditional visualisation of time series data often consists of plotting the time series values against time and connecting the dots we propose an alternative multiscale visualisation technique motivated by the scale space approach in computer vision in brief our method also connects the dots but uses a range of pens of varying thicknesses for this purpose the resulting multiscale map termed the thick pen transform tpt corresponds to viewing the time series from a range of distances we formally prove that the tpt is a discriminatory statistic for two gaussian time series with distinct correlation structures further we show interesting possible applications of the tpt to measuring cross dependence in multivariate time series classifying time series and testing for stationarity in particular we derive the asymptotic distribution of our test statistic and argue that the test is applicable to both linear and nonlinear processes under low moment assumptions various other aspects of the methodology including other possible applications are also discussed

applying fuzzy method for measuring criticality in project network program evaluation and review technique pert is widely used as a tool for managing large scale projects in the traditional pert model the durations of activities tasks are either represented as crisp numbers or drawn from the beta distribution to estimate the task durations such as pessimistic most likely and optimistic times however the operation time for each activity is usually difficult to define and estimate precisely in a real situation the aim of this paper is to present an analytical method for measuring the criticality in a project network with fuzzy activity times triangular fuzzy numbers are used to express the operation times for all activities in a project network a new model that combines fuzzy set theory with the pert technique is proposed to determine the critical degrees of activities tasks and paths in the proposed model a possibility index is defined to identify the likelihood of meeting a specified required time for a project network at the end of the paper an example is presented to compare with those obtained using the proposed method as well as other methods the comparisons reveal that the method proposed in this paper is more effective in determining the activity criticalities and finding the critical path

higher order mechanical systems with constraints a general mathematical theory covering higher order mechanical systems subject to constraints of arbitrary order i e depending on time positions velocities accellerations and higher derivatives is presented including higher order holonomic systems as a particular case within differential geometric setting on higher order jet bundles the concept of a mechanical system not necessarily regular or lagrangian is introduced to be a class of 2 forms equivalent with a dynamical form dynamics are then represented by means of corresponding exterior differential systems higher order constraint structure on a fibered manifold is defined to be a submanifold endowed with a distribution canonical distribution higher order chetaev bundle with help of a constraint structure a constraint force is naturally introduced higher order mechanical systems subject to different kinds of higher order constraints are then geometrically characterized and their dynamics are studied from a geometrical point of view

application of weighting functions to the ranking of fuzzy numbers this paper proposes a new ranking method for fuzzy numbers which uses a defuzzification of fuzzy numbers and a weighting function following saeidifar and pasha 2008 first we define a weighted distance measure on fuzzy numbers and then by minimizing this distance the weighted interval and point approximations of fuzzy numbers are obtained these indices are applied to rank the fuzzy numbers this method is new and interesting for ranking fuzzy numbers and it can be applied for solving and optimizing engineering and economics problems in a fuzzy environment

min max operators in texture analysis a signature is generated for a given picture by operating on it with different masks the operations are gray level generalizations of shrink and expand for binary pictures using serra s morphological methods 10 the signature is a set of numbers each corresponding to an application of an operator at a certain scale and direction and can be used to analyze the discriminate textures it is shown that this family of signatures includes as special cases several currently used texture descriptors

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

fuzzy credibility relation method for multiple criteria decision making problems combining the concepts of fuzzy set theory entropy ideal and grey relation analysis a fuzzy grey relation method for multiple criteria decision making problems is proposed first triangular fuzzy numbers and linguistic values characterized by triangular fuzzy numbers are used to evaluate the importance weights of all criteria and the superiority of all alternatives versus various criteria above the alternative level then the concept of entropy is utilized to solve the adjusted integration weight of all objective criteria above the alternative level furthermore using the concepts of ideal the grey ration grades of various alternatives versus ideal solution are ranked to determine the best alternative finally a numerical example of selecting most appropriate company to build a new highway is used to demonstrate the applicability of proposed method the study results show that this method is an effective means for tackling mcdm problems in fuzzy and grey environments

a new approach for ranking fuzzy numbers based on fuzzy simulation analysis method a new approach for ranking fuzzy numbers based on a fuzzy simulation analysis fsa method is proposed a combination method including computer and math application is developed first the method is intuitive and can be used to order fuzzy numbers easily the approach is illustrated by numerical examples showing that it overcomes the shortcoming existing fuzzy ranking approaches especially when it is difficult to use other methods to solve ranking fuzzy problems

applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy ahp the lack of consistency in decision making can lead to inconsistent conclusions in fuzzy analytic hierarchy process fuzzy ahp method it is difficult to ensure a consistent pairwise comparison furthermore establishing a pairwise comparison matrix requires nx n 1 2 judgments for a level with n criteria alternatives the number of comparisons increases as the number of criteria increases therefore the decision makers judgments will most likely be inconsistent to alleviate inconsistencies this study applies fuzzy linguistic preference relations fuzzy linprera to construct a pairwise comparison matrix with additive reciprocal property and consistency in this study the fuzzy ahp method is reviewed and then the fuzzy linprera method is proposed finally the presented method is applied to the example addressed by kahraman et al c kahraman d ruan i dogan fuzzy group decision making for facility location selection information sciences 157 2003 135 153 this study reveals that the proposed method yields consistent decision rankings from only n 1 pairwise comparisons which is the same result as in kahraman et al research the presented fuzzy linguistic preference relations method is an easy and practical way to provide a mechanism for improving consistency in fuzzy ahp method

particle swarm optimization combined with genetic operators for job shop scheduling problem with fuzzy processing time job shop scheduling problem is a np hard problem the processing time for each job is often imprecise in many real world applications and the imprecision in the data is critical for the scheduling procedures therefore job shop scheduling problem with fuzzy processing time is addressed in the paper the processing time is described by triangular fuzzy numbers the objective is to find a job sequence that minimizes the makespan and the uncertainty of the makespan by using an approach for ranking fuzzy numbers the particle swarm optimization pso is a randomized population based optimization method that was inspired by the flocking behavior of birds and human social interactions pso has been successfully applied to various real world applications but there is a little literature reported regarding application to scheduling problems as it was unsuitable for them in this paper pso is redefined and modified by introducing genetic operators such as crossover and mutation operator to update the particles we call this particle swarm optimization combined with genetic operators gpso this is successfully employed to solve the formulated problem ten benchmarks with fuzzy processing time are used to test gpso the feasibility as well as the efficiency of the proposed method is assessed in comparison with genetic algorithm ga

lagrangian systems with higher order constraints a class of mechanical systems subject to higher order constraints i e constraints involving higher order derivatives of the position of the system are studied we call them higher order constrained systems hocss they include simplified models of elastic rolling bodies and also the so called generalized nonholonomic systems gnhss whose constraints only involve the velocities of the system i e first order derivatives in the position of the system one of the features of this kind of systems is that d alembert s principle or its nonlinear higher order generalization the chetaev s principle is not necessarily satisfied we present here as another interesting example of hocs systems subjected to friction forces showing that those forces can be encoded in a second order kinematic constraint the main aim of the paper is to show that every hocs is equivalent to a gnhs with linear constraints in a canonical way that is to say systems with higher order constraints can be described in terms of one with linear constraints in velocities we illustrate this fact with a system with friction and with rocard s model dynamique generale des vibrations 1949 chap xv p 246 and l instabilite en mecanique automobiles avions ponts suspendus 1954 of a pneumatic tire as a by product we introduce some applications on higher order tangent bundles which we expect to be useful for the study of intrinsic aspects of the geometry of such bundles

a generalization of chetaev s principle for a class of higher order nonholonomic constraints the constraint distribution in nonholonomic mechanics has a double role on the one hand it is a kinematic constraint that is it is a restriction on the motion itself on the other hand it is also a restriction on the allowed variations when using d alembert s principle to derive the equations of motion we will show that many systems of physical interest where d alembert s principle does not apply can be conveniently modeled within the general idea of the principle of virtual work by the introduction of both kinematic constraints and variational constraints as being independent entities this includes for example elastic rolling bodies and pneumatic tires also d alembert s principle and chetaev s principle fall into this scheme we emphasize the geometric point of view avoiding the use of local coordinates which is the appropriate setting for dealing with questions of global nature like reduction

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

ranking fuzzy numbers with an area between the centroid point and original point abstract to improve the ranking method of lee and li 1 cheng 2 proposed the coefficient of variation cv index shortcomings are also found in the cv index cheng 2 also proposed the distance method to improve the ranking method of murakami et al however the distance method is not sound either moreover the cv index contradicts the distance method in ranking some fuzzy numbers therefore to overcome the above shortcomings we propose ranking fuzzy numbers with the area between the centroid point and original point

ranking fuzzy numbers by preference ratio we propose a ranking method for fuzzy numbers in this method a preference function is defined by which fuzzy numbers are measured point by point and at each point the most preferred number is identified then these numbers are ranked on the basis of their preference ratio therefore fuzzy numbers are compared relatively and not necessarily one is preferred absolutely over the others this method is especially designed to evaluate alternatives in multi criteria or multi attribute decision making the method is intuitive and can be used to discriminate between numbers easily the method is specially tailored for triangular fuzzy numbers tfn and an algorithm is presented to determine the preference ratio of a pair of tfns we show that at most two feasible turning points exist this fact is very helpful to make the algorithm much easier

a multi scale morphologic edge detector in this paper we present a morphologic edge detection methods using multi scale approach for detecting edges of various fineness under noisy condition it is shown that the proposed edge detector has the desirable properties that a good edge detector should have comparative study reveals its superiority over other morphologic edge detectors

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

a probabilistic approach to rank complex fuzzy numbers abstract techniques for ranking simple fuzzy numbers are abundant in the literature however we lack efficient methods for comparing complex fuzzy numbers that are induced by arithmetic operations in this paper a probabilistic approach is taken membership functions of fuzzy numbers are first converted into probability density functions the mellin transform is then introduced to compute the mean and the variance of a complex fuzzy number the fuzzy number with the higher mean is ranked higher if the means are equal the one with the smaller variance is judged higher rank two numerical examples and a fuzzy multiple attribute decision analysis are illustrated

improving gil werman algorithm for running min and max filters the current best bound on the number of comparison operations needed to compute the running maximum or minimum over a p element sliding data window is approximately three comparisons per output sample this bound is probabilistic for some algorithms and is derived for their complexities on the average for independent identically distributed i i d input signals the worst case complexities of these algorithms are o p the worst case complexity c sub r 3 4 p comparisons per output sample for 1d signals is achieved in the gil werman algorithm 1993 in this correspondence we propose a modification of the gil werman algorithm with the same worst case complexity but with a lower average complexity a theoretical analysis shows that using the proposed modification the complexities of sliding max or min 1d and 2d filters over i i d signals are reduced to c sub 1 2 5 3 5 p 1 p sup 2 and c sub 2 5 7 p 2 p sup 2 comparisons per output sample on the average respectively simulations confirm the theoretical results moreover experiments show that even for highly correlated data namely for real images the behavior of the algorithm remains the same as for i i d signals

a new approach for ranking fuzzy numbers by distance method abstract many ranking methods have been proposed so far however there is yet no method that can always give a satisfactory solution to every situation some are counterintuitive not discriminating some use only the local information of fuzzy values some produce different rankings for the same situation for overcoming the above problems we propose a new method for ranking fuzzy numbers by distance method our method is based on calculating the centroid point where the distance means from original point to the centroid point x 0 y 0 and the x 0 index is the same as murakami et al s x 0 however the y 0 index is integrated from the inverse functions of an lr type fuzzy number thus we use ranking function r a x 2 y 2 distance index as the order quantities in a vague environment our method can rank more than two fuzzy numbers simultaneously and the fuzzy numbers need not be normal furthermore we also propose the coefficient of variation cv index to improve lee and li s method comput math appl 15 1988 887 896 lee and li rank fuzzy numbers based on two different criteria namely the fuzzy mean and the fuzzy spread of the fuzzy numbers and they pointed out that human intuition would favor a fuzzy number with the following characteristics higher mean value and at the same time lower spread however when higher mean value and at the same time higher spread or lower mean value and at the same time lower spread exists it is not easy to compare its orderings clearly our cv index is defined as cv standard error mean which can overcome lee and li s problem efficiently in this way our proposed method can also be easily calculated by the mathematica package to solve problems of ranking fuzzy numbers at last we present three numerical examples to illustrate our proposed method and compare with other ranking methods

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

job sequencing with fuzzy processing times abstract in practice processing times can be more accurately represented as intervals with the most probable completion time somewhere near the middle of the interval a fuzzy number which is essentially a generalized interval can represent this processing time interval exactly and naturally in this work triangular and trapezoidal fuzzy numbers are used to represent those vague job processing times in job shop production systems the job sequencing algorithms of johnson and ignall and schrage are modified to accept fuzzy job processing times fuzzy makespans and fuzzy mean flow times are then calculated for greater decision making information numerous examples are used to illustrate the approach

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

simplifying the search for effective ranking of fuzzy numbers we prove that a ranking index which generates an order on some subset of fuzzy numbers can be reduced to a ranking index with a simpler form and which generates an equivalent order this way the finding of ranking indices generating orders with desirable properties can be essentially simplified such that we need to search only from additive or scale invariant ranking indices we determine exactly making an abstraction of equivalent orders the class of ranking indices on trapezoidal fuzzy numbers which generate orders satisfying all the basic requirements finally ranking indices used to rank trapezoidal fuzzy numbers are extended to ranking indices used to rank arbitrary fuzzy numbers so that the desired properties are preserved

fuzzy decision making based on likelihood based comparison relations in this paper we present a new fuzzy decision making method which is based on likelihood based comparison relations first we introduce the concepts of likelihood based comparison relations for intervals then we propose the concept of likelihood based comparison relations for type 1 fuzzy sets and interval type 2 fuzzy sets then we present a new method to rank fuzzy sets by using fuzzy targets based on the proposed likelihood based comparison relations for fuzzy sets finally we present a new fuzzy decision making method based on the proposed likelihood based comparison relations for fuzzy sets and the proposed fuzzy ranking method the proposed fuzzy decision making method has the advantage that the evaluated values can either be represented by crisp values intervals type 1 fuzzy sets or interval type 2 fuzzy sets it can overcome the drawbacks of huynh et al s method due to the fact that huynh et al s method cannot deal with the ranking of interval type 2 fuzzy sets for fuzzy decision making and cannot distinguish the ranking order between the alternatives in some situations

desingularization of implicit analytic differential equations the question of finding solutions to a given implicit differential equation ide is an important one in part because it appears very naturally in several problems in physics engineering and many other fields in this work we show how to reduce a given analytic ide to an analytic ide of locally constant rank this can be done by using some fundamental results on subanalytic subsets and desingularization of closed subanalytic subsets an example from nonholonomic mechanics is studied using these methods

ordering fuzzy sets over the real line an approach based on decision making under uncertainty abstract the ordering of fuzzy sets over the real line is approached from the point of view of ordering intervals rather than ordering numbers first the maximax and maximin criteria which are commonly used for ordering intervals are expressed in terms of characteristic functions these criteria and the hurwicz criterion for decision making under complete ignorance are then reformulated in a manner that allows for their generalization to the fuzzy case transitivity is established for these ordering rules a criterion based on the principle of diminishing marginal utility is also presented

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

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

granular risk based design optimization optimization considering uncertainty is an increasingly important and continuously developing uncertainty mitigation technique for modern design compared with its well established branches i e reliability based design optimization rbdo and robust design optimization risk based design optimization rdo is just regarded as an extension of rbdo by incorporating future cost hence it has not received much deep theoretical study based on the generalized theory of uncertainty we introduce different levels of probability granulation into rdo and propose a granular risk based design optimization grdo methodology the risks are modeled as granular probabilities their mean values and standard deviations two multiobjective optimization mo formulations of grdo are proposed and solved by multiobjective evolutionary algorithm based on decomposition aided with solution filtering criterion based on the results of a structural design example the capability of grdo on uncertainty management is validated by comparing the performances of different mo formulations while uncertainty mitigation using grdo is achieved by controlling the risks associated with fixed level of uncertainty this way grdo is approved as a general design frame rather than just an uncertainty mitigation technique like conventional rdo

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

an efficient approach for large scale project planning based on fuzzy delphi method abstract the goal of this paper is to replace probabilistic or deterministic considerations in the project network analysis by possibilistic ones and to reduce the difficulty arising from the inexact and insufficient information of activity times the activity times are considered as fuzzy numbers fuzzy intervals or time intervals and the fuzzy delphi method is used to estimate a reliable time interval of each activity based on these time estimates we then propose an efficient methodology for calculating the fuzzy project completion time and the degree of criticality for each path in a project