optimization of type 2 fuzzy systems based on bio inspired methods a concise review a review of the optimization methods used in the design of type 2 fuzzy systems which are relatively novel models of imprecision has been considered in this work the fundamental focus of the work has been based on the basic reasons of the need for optimizing type 2 fuzzy systems for different areas of application recently bio inspired methods have emerged as powerful optimization algorithms for solving complex problems in the case of designing type 2 fuzzy systems for particular applications the use of bio inspired optimization methods have helped in the complex task of finding the appropriate parameter values and structure of the fuzzy systems in this review we consider the application of genetic algorithms particle swarm optimization and ant colony optimization as three different paradigms that help in the design of optimal type 2 fuzzy systems we also provide a comparison of the different optimization methods for the case of designing type 2 fuzzy systems

a continuum of relative equilibria in the five body problem abstract it is generally believed that the set of relative equilibria equivalence classes in the newtonian n body problem for a given set of positive masses is finite however the result has only been proven for n 3 and remains a difficult open question for n 4 a wintner princeton math ser 5 215 princeton university press princeton nj 1941 s smale mathematical intelligencer 20 1998 7 15 we demonstrate that the condition for the masses being positive is a necessary one by finding a continuum of relative equilibria in the five body problem which unfortunately includes one negative mass this family persists in similar potential functions including the logarithmic potential used to describe the motion of point vortices in a plane of fluid

a comparison of discrete algorithms for fuzzy weighted average fuzzy weighted average fwa which can be applied to various fields such as engineering design decision analysis etc and as function of fuzzy numbers is suitable for the problem of multiple occurrences of fuzzy parameters additional fuzziness may be introduced in the alpha cut arithmetic this paper reviews and compares discrete algorithms for the fwas in both theoretical comparison and numerical comparisons as opposed to the linear programming algorithms that may be efficient but require the help of linear programming software an alternative efficient algorithm is also proposed the algorithm introduces an all candidate criteria ratings weights replaced benchmark adjusting procedure other than a binary dichotomy search in the existing methods the theoretical worst case comparison shows that the algorithms of guu and guh and our alternative algorithm require the same order of calculation complexity o n where n is the number of fwa terms yet the guh algorithm always requires 2 n 1 calculations to be performed the algorithm of guu requires the elemental comparison complexity o n and is superior to the other algorithms in the worst case on the other hand our alternative algorithm and lee and park algorithm have the same comparison complexity o n log n yet our alternative algorithm requires o n calculation complexity which is better than o n log n of the lee and park algorithm the numerical experiments show that the algorithm of guu and our alternative algorithm generally perform and converge faster by requiring fewer calculations than that of lee and park s algorithm the alternative algorithm requires a slightly smaller number of calculations than that of the guu algorithm due to the use of the convergent benchmark adjustment procedure rather than a somewhat fixed binary search conversely in the number of element comparisons lee and park s algorithm is shown numerically generally slightly better than the alternative algorithm due to the simple binary search scheme used the alternative algorithm also requires fewer comparisons than that of the guu algorithm furthermore in comparison of the average cpu time requirements the ranks of the results of these algorithms are consistent with those of the algorithms for the average numbers of calculations in general the alternative algorithm may provide an efficient alternative algorithm to the fwas in the worse case situation the guu algorithm still may be considered as an efficient alternative

a new approach to normalization of interval and fuzzy weights a new approach to normalization of interval and fuzzy weights based on the so called interval extended zero method is proposed the three desirable intuitively obvious properties of the normalization procedure are defined the main of them is based on the assumption that the sum of normalized interval or fuzzy weights should be an interval or a fuzzy value centered around 1 with a minimal width the advantages of a new approach are illustrated with the use of numerical examples it is shown that a new approach performs better than known methods for normalization of interval and fuzzy weights as it provides the results with the properties which are close to the desirable ones

global convergence of karnik mendel algorithms karnik mendel km algorithms are the most commonly used iterative type reduction methods in interval type 2 fuzzy sets and systems as well as new techniques for computing the fuzzy weighted average fwa various extensions and improvements have been proposed however no proof has been provided for the convergence of these extensions it is necessary to provide the proof because many of the iterative algorithms may have divergence cases in the present study we provide a theoretical proof that km algorithms exhibit global convergence different initialization methods and iteration formats can always obtain the same unique optimal solution thus there are no concerns about the possibility of divergence in extensions of km algorithms our proof provides theoretical support for the applications of km algorithms especially the type reduction designs used in type 2 fuzzy systems and fwa computations because of the important roles of km algorithms in these methods

footprint of uncertainty for type 2 fuzzy sets in the paper a new and comprehensible definition is proposed for type 2 fuzzy sets t2 fss and the primary and secondary memberships function are defined respectively by using multi valued mapping a new definition and formula for the footprint of uncertainty fou is presented and based on the new definitions the relation between fou and the original definition of t2 fs is discussed finally the partition method of fou is provided to represent the primary membership grade the fou and the interval type 2 fuzzy sets it2 fs when the upper and lower membership functions umf lmf of fou are given

learning conditional preference networks conditional preference networks cp nets have recently emerged as a popular language capable of representing ordinal preference relations in a compact and structured manner in this paper we investigate the problem of learning cp nets in the well known model of exact identification with equivalence and membership queries the goal is to identify a target preference ordering with a binary valued cp net by interacting with the user through a small number of queries each example supplied by the user or the learner is a preference statement on a pair of outcomes in this model we show that acyclic cp nets are not learnable with equivalence queries alone even if the examples are restricted to swaps for which dominance testing takes linear time by contrast acyclic cp nets are what is called attribute efficiently learnable when both equivalence queries and membership queries are available we indeed provide a learning algorithm whose query complexity is linear in the description size of the target concept but only logarithmic in the total number of attributes interestingly similar properties are derived for tree structured cp nets in the presence of arbitrary examples our learning algorithms are shown to be quasi optimal by deriving lower bounds on the vc dimension of cp nets in a nutshell our results reveal that active queries are required for efficiently learning cp nets in large multi attribute domains

on central configurations of the κn body problem abstract we consider planar central configurations of the newtonian n body problem consisting in groups of regular n gons of equal masses called n crown we derive the equations of central configurations for a general n crown when 2 we prove the existence of a twisted 2 n crown for any value of the mass ratio moreover for n 3 4 and any value of the mass ratio we give the exact number of twisted 2 n crowns and describe their location finally we conjecture that for any value of the mass ratio there exist exactly three 2 n crowns for n 5

on km algorithms for solving type 2 fuzzy set problems computing the centroid and performing type reduction for type 2 fuzzy sets and systems are operations that must be taken into consideration karnik mendel km algorithms are the standard ways to do these operations however because these algorithms are iterative much research has been conducted during the past decade about centroid and type reduction computations this tutorial paper focuses on the research that has been conducted to 1 improve the km algorithms 2 understand the km algorithms leading to further improved algorithms 3 eliminate the need for km algorithms 4 use the km algorithms to solve other nonfuzzy logic system problems and 5 use or not use km algorithms for general type 2 fuzzy sets and fuzzy logic systems

the fuzzy weighted average within a generalized means function the fuzzy weighted average is widely used to solve hierarchical evaluation problems including fuzzy consideration for the operations of scoring weighting and aggregating previous works considered the fuzziness of score and weight and used the additive function to aggregate these weighted scores this study considers the aggregation operator also as a fuzzy variable and uses a generalized means function to fuzzify the aggregation operator within a fuzzy weighted average in practice the proposed model not only considers both the relative important of the criteria and its achieved performance but also conveys the influence of the dm s decision maker s evaluation attitude thus the proposed model can flexibly reflect any dm s evaluation attitude such as open neutral or rigorous thereby the proposed model can make an objective evaluation that approaches a real decision making situation and thus has the potential to be a useful management tool for improved resolution of fuzzy hierarchical evaluation problems

applying the fuzzy weighted average approach to evaluate network security systems imprecise or vague importance as well as satisfaction levels of criteria may characterize the evaluation of alternative network security systems and may be treated by the fuzzy set theory as fuzzy numbers are adopted the fuzzy weighted averages fwas may become a suitable computation for the fuzzy criteria ratings and fuzzy weightings in this paper we propose to use the fwa approach that operates on the fuzzy numbers and obtains the fuzzy weighted averages during the evaluations of network security systems the algorithm constructs a benchmark adjustment solution approach for fwa a case study of the evaluation of network security systems is provided and demonstrates the usefulness of this study

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

fractional programming approach to fuzzy weighted average this paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average based on the cut representation of fuzzy sets and the extension principle a pair of fractional programs is formulated to find the cut of fuzzy weighted average owing to the special structure of the fractional programs in most cases the optimal solution can be found analytically consequently the exact form of the membership function can be derived by taking the inverse function of the cut for other cases a discrete but exact solution to fuzzy weighted average is provided via an efficient solution method examples are given for illustration

triple and quadruple nested central configurations for the planar n body problem abstract in this paper we show the existence of families of triple or quadruple nested planar central configurations for the n body problem with n 6 8 9 more precisely for n 6 we nested three 2 collinear central configurations for n 8 we nested four 2 collinear central configurations and for n 9 we nested three equilateral triangular central configurations as far as we know these are the first examples of planar triple and quadruple nested central configurations

fuzzy sets and type 2 under algebraic product and algebraic sum abstract the concept of fuzzy sets of type 2 has been proposed by l a zadeh as an extension of ordinary fuzzy sets a fuzzy set of type 2 can be defined by a fuzzy membership function the grade or fuzzy grade of which is taken to be a fuzzy set in the unit interval 0 1 rather than a point in 0 1 this paper investigates the algebraic properties of fuzzy grades that is fuzzy sets of type 2 under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring moreover the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join meet and algebraic product

central configurations formed by two twisted regular polygons abstract in this paper we study the necessary conditions and sufficient conditions for the central configurations formed by two twisted regular polygons one n regular polygon and one l regular polygon we wish to extend the results of the symmetrical central configurations formed by two twisted n regular polygons however to our own surprise it will be proved that the necessary condition on the central configurations is l n

modeling uncertain variables of the weighted average operation by fuzzy vectors the paper deals with the fuzzy extension of the weighted average operation first we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors we show that in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values next we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights we derive a general formula and we study its special forms the advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation this fact is illustrated by several examples finally we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average

finiteness of central configurations of five bodies in the plane we prove there are finitely many isometry classes of planar central configurations also called relative equilibria in the newtonian 5 body problem except perhaps if the 5 tuple of positive masses belongs to a given codimension 2 subvariety of the mass space

cp and ocf networks a comparison network approaches are used to structure partition and display formalisms in the area of knowledge representation as well as decision making known approaches are for instance ocf networks bayesian style networks where every variable is annotated with a conditional ranking table and cp networks directed acyclic networks with local preferences annotated at each vertex the structures of these networks are similar but their semantics seem to be quite different in this paper we discuss if ocf networks can be used to model the information of cp networks and vice versa to answer this question we investigate which restrictions and conditions have to be presupposed to either of the approaches such that one structure can be used to generate the other

fuzzy weighted average an improved algorithm abstract this paper suggests a modification on the fuzzy weighted average fwa method developed by dong and wong 1987 a steepest descend ascend method is proposed to facilitate steps 3 and 4 of the fwa to find the minimum maximum of f x1 x2 xn w1 w2 wn i 1n wixi i 1n wi an example used by dong and wong is again employed to illustrate the new method the same results are obtained but the new method requires much less evaluations and computations the advantages and disadvantages of the two algorithms are discussed also several relevant theorems are stated and discussed in this paper and the proofs of these theorems are listed in the appendix

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

application of fuzzy sets of type 2 to the solution of fuzzy equation systems abstract the present paper continues the authors investigation on fuzzy equation systems with tolerances we consider the situation that the outputs of an object are modelled by fuzzy sets of type 2 for special membership functions this problem can be attributed to a problem with tolerances moreover this concept enables us to define solutions for fuzzy outputs of type 1 when classical solution concepts may fail

aggregation using the linguistic weighted average and interval type 2 fuzzy sets the focus of this paper is the linguistic weighted average lwa where the weights are always words modeled as interval type 2 fuzzy sets it2 fss and the attributes may also but do not have to be words modeled as it2 fss consequently the output of the lwa is an it2 fs the lwa can be viewed as a generalization of the fuzzy weighted average fwa where the type 1 fuzzy inputs are replaced by it2 fss this paper presents the theory algorithms and an application of the lwa it is shown that finding the lwa can be decomposed into finding two fwas since the lwa can model more uncertainties it should have wide applications in distributed and hierarchical decision making

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

enhanced karnik mendel algorithms the karnik mendel km algorithms are iterative procedures widely used in fuzzy logic theory they are known to converge monotonically and superexponentially fast however several usually two to six iterations are still needed before convergence occurs methods to reduce their computational cost are proposed in this paper extensive simulations show that on average the enhanced km algorithms can save about two iterations which corresponds to more than a 39 reduction in computation time an additional at least 23 computational cost can be saved if no sorting of the inputs is needed

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

computational techniques for a simple theory of conditional preferences a simple logic of conditional preferences is defined with a language that allows the compact representation of certain kinds of conditional preference statements a semantics and a proof theory cp nets and tcp nets can be mapped into this logic and the semantics and proof theory generalise those of cp nets and tcp nets the system can also express preferences of a lexicographic kind the paper derives various sufficient conditions for a set of conditional preferences to be consistent along with algorithmic techniques for checking such conditions and hence confirming consistency these techniques can also be used for totally ordering outcomes in a way that is consistent with the set of preferences and they are further developed to give an approach to the problem of constrained optimisation for conditional preferences

twisted angles for central configurations formed by two twisted regular polygons abstract in this paper we study the necessary conditions and sufficient conditions for the twisted angles of the central configurations formed by two twisted regular polygons in particular we prove that for the 2n body problem the twisted angles must be 0 or n

fuzzy weighted averages revisited the fuzzy weighted average operations are common operations in risk and decision analysis based on zadeh s extension principle dong and wang fuzzy sets and systems 21 1987 183 proposed a fwa method a method based on the cut representation of fuzzy sets and interval analysis the fwa method provides a discrete approximation of the cuts of fuzzy weighted average the fwa method has an exponential complexity subsequently liou and wang fuzzy sets and systems 49 1992 307 proved that indeed the fwa method can be simplified to be a nonconstrainted linear fractional programming problem with lower and upper bounds for each of the variables they proposed an improved algorithm called ifwa which is of o n2 complexity where n is the number of decision variables in ifwa guh et al comput math appl 32 8 1996 115 proposed a max min paired elimination method pfwa they claimed their pfwa was of order n in complexity yet the pfwa is a heuristic method lacking any proof of convergence in 1997 lee and park proposed an algorithm efwa which is of o nlogn complexity most recently kao and liu fuzzy sets and systems 120 2001 435 solved the fwa by transforming the fractional program into a linear program in this note we shall relate the ifwa framework with the nonconstrainted 0 1 integer linear fractional programming problem in the mathematical programming literature several o n time algorithms have been existing for instance robillard naval res logist quart 18 1971 47 and hansen et al math programming 52 1991 255 the algorithm of hansen et al will be presented for easy reference

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

adaptive filter design using type 2 fuzzy cerebellar model articulation controller this paper aims to propose an efficient network and applies it as an adaptive filter for the signal processing problems an adaptive filter is proposed using a novel interval type 2 fuzzy cerebellar model articulation controller t2fcmac the t2fcmac realizes an interval type 2 fuzzy logic system based on the structure of the cmac due to the better ability of handling uncertainties type 2 fuzzy sets can solve some complicated problems with outstanding effectiveness than type 1 fuzzy sets in addition the lyapunov function is utilized to derive the conditions of the adaptive learning rates so that the convergence of the filtering error can be guaranteed in order to demonstrate the performance of the proposed adaptive t2fcmac filter it is tested in signal processing applications including a nonlinear channel equalization system a time varying channel equalization system and an adaptive noise cancellation system the advantages of the proposed filter over the other adaptive filters are verified through simulations

solutions of regular polygon with an inner particle for newtonian n 1 body problem abstract in this paper we prove that if the solution of the planar n 1 body problem has n bodies rotating at the vertices of a regular n gon then the n bodies have equal mass and the remaining body must be at the center

robust control of an lusm based x hbox y hbox theta motion control stage using an adaptive interval type 2 fuzzy neural network the robust control of a linear ultrasonic motor based x y thetas motion control stage to track various contours is achieved by using an adaptive interval type 2 fuzzy neural network ait2fnn control system in this study in the proposed ait2fnn control system an it2fnn which combines the merits of an interval type 2 fuzzy logic system and a neural network is developed to approximate an unknown dynamic function moreover adaptive learning algorithms are derived using the lyapunov stability theorem to train the parameters of the it2fnn online furthermore a robust compensator is proposed to confront the uncertainties including the approximation error optimal parameter vectors and higher order terms in taylor series to relax the requirement for the value of lumped uncertainty in the robust compensator an adaptive lumped uncertainty estimation law is also investigated in addition the circle and butterfly contours are planned using a nonuniform rational b spline curve interpolator the experimental results show that the contour tracking performance of the proposed ait2fnn is significantly improved compared with the adaptive type 1 fnn additionally the robustness to parameter variations external disturbances cross coupled interference and frictional force can also be obtained using the proposed ait2fnn

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

fuzzy weighted average the linear programming approach via charnes and cooper s rule abstract to improve the method of dong and wong fuzzy sets and systems 21 1987 183 199 for obtaining the fuzzy weighted average a steepest descent ascent method was proposed by liou and wang fuzzy sets and systems 49 1992 307 315 in this paper we propose to replace the steepest descent ascent method by linear programming which is a much more powerful approach for large problems the reduction of the original nonlinear programming problem into a linear one is achieved by the use of the charnes and cooper s linear transformation method the example used by dong and wong is again used to illustrate the approach

analytical solution methods for the fuzzy weighted average for the fuzzy weighted average fwa despite various discrete solution algorithms and their improvements attempts at analytical solutions are very rare this paper provides an analytical solution method for the fwa based on the conclusions of the karnik mendel km algorithm compared with the two current popular kinds of a cut based computational methods for the fwa mathematical programming transformations and direct iterate computations our method is precise and has a concise structure efficient computation process and sound theoretical proofs we propose two algorithms for computing the analytical solution of the fwa two numerical examples illustrate our proposed approach

on central configurations this paper concerns an old problem which arose in celestial mechanics and has since taken on a life of its own the probem of central configurations of the n body problem

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

aggregation using the fuzzy weighted average as computed by the karnik mendel algorithms by connecting work from two different problems the fuzzy weighted average fwa and the generalized centroid of an interval type 2 fuzzy set a new alpha cut algorithm for solving the fwa problem has been obtained one that is monotonically and superexponentially convergent this new algorithm uses the karnik mendel km algorithms to compute the fwa cut end points it appears that the km cut algorithms approach for computing the fwa requires the fewest iterations to date and may therefore be the fastest available fwa algorithm to date

an efficient centroid type reduction strategy for general type 2 fuzzy logic system in this paper an efficient centroid type reduction strategy for general type 2 fuzzy set is introduced this strategy makes use of the result of a plane representation and performs the centroid type reduction on each a plane simulations show that it usually needs only several resolution of a value such that the defuzzified value converges to a real value consequently comparing with the exhaustive computation approach this approach can tremendously decrease the computation complexity from exponential into linear

an efficient algorithm for fuzzy weighted average in multisensor intelligent systems the information fusion plays an important role several algorithms have been proposed for the purpose of aggregating imprecise sensory information represented by fuzzy numbers this paper proposes an efficient algorithm to compute fuzzy weighted average which turned out to be superior to the previous works by reducing the number of comparisons and arithmetic operations to o n log n

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

study on enhanced karnik mendel algorithms initialization explanations and computation improvements computing the centroid of an interval type 2 fuzzy set is an important operation in a type 2 fuzzy logic system and is usually implemented by karnik mendel km iterative algorithms by connecting km algorithms and continuous km algorithms together this paper gives theoretical explanations on the initialization methods of km and enhanced karnik mendel ekm algorithms proposes exact methods for centroid computation of an interval type 2 fuzzy set and extends the enhanced karnik mendel ekm algorithms to three different forms of weighted ekm wekm algorithms it shows that ekm algorithms become a special case of the wekm algorithms when the weights of the latter are constant value it also shows that in general the weighted ekm algorithms have smaller absolute error and faster convergence speed than the ekm algorithms which make them very attractive for real time applications of fuzzy logic system four numerical examples are used to illustrate and analyze the performance of wekm algorithms

stacked central configurations for newtonian n 2p body problems abstract we show the existence of spatial central configurations for the n 2 p body problems in such a configuration n bodies are at the vertices of a regular n gon t 2 p bodies are symmetric with respect to the center of t and located on the straight line which is perpendicular to the regular n gon t and passes through the center of t the masses located on the vertices of the regular n gon are assumed to be equal the masses located on the same line and symmetric with respect to the center of t are equal

fuzzy weighted average a max min paired elimination method by the use of max min paired elimination an efficient method for obtaining the fuzzy weighted average fwa was developed this approach is much more efficient than either the approach developed by dong and wong 1 or the improved approach developed by liou and wang 2 the computational requirements for the method of dong and wong are 22n and the largest computational requirements for the improved method of liou and wang are 2 n n 1 the computational requirements for the proposed approach are proportional to the problem size n and the total computational requirements are 2 n 1 the same example used by dong and wong is used to illustrate the approach in addition a more complicated problem is also solved

four body central configurations with some equal masses we prove firstly that any convex non collinear central configuration of the planar 4 body problem with equal opposite masses 0 such that the diagonal corresponding to the mass is not shorter than that corresponding to the mass must possess a symmetry and therefore must be a kite then by a recent result of bernat llibre and perez chavela this kite is actually a rhombus secondly we prove that a convex non collinear planar 4 body central configuration with three equal masses must be a kite too we also prove that the concave central configuration with three equal masses forming a triangle and the fourth one with any given mass in the interior must be either an equilateral triangle with the fourth mass at its geometric center or an isosceles triangle with the fourth mass on the symmetry axis

an extended type reduction method for general type 2 fuzzy sets a centroid type reduction strategy for computing the centroids of type 2 fuzzy sets based on decomposed alpha planes was proposed by liu however it cannot be applied to type 2 fuzzy sets with concave secondary membership functions in this paper we extend the liu s method so that the centroids of type 2 fuzzy sets with concave secondary membership functions can be derived for each decomposed alpha plane we convert it into a group of interval type 2 fuzzy sets the union of the centroids of its member interval type 2 fuzzy sets constitutes the centroid of the alpha plane then the weighted union of the centroids of the decomposed alpha planes becomes the centroid type reduced set of the original type 2 fuzzy set when dealing with type 2 fuzzy sets with convex secondary membership functions our proposed method is reduced to the liu s method

correspondence note on the lack of equality between fuzzy weighted average and fuzzy convex sum the paper shows that the equality of the interval valued weighted average based on interval weights and the convex sum based on normalized interval weights presented in dubois the role of fuzzy sets in decision sciences old techniques and new directions fuzzy sets syst 184 2011 3 28 is generally false for n 3 and discusses its consequences

new results about the centroid of an interval type 2 fuzzy set including the centroid of a fuzzy granule the centroid of an interval type 2 fuzzy set it2 fs provides a measure of the uncertainty of such a fs its calculation is very widely used in interval type 2 fuzzy logic systems in this paper we present properties about the centroid of an it2 fs we also illustrate many of the general results for a t2 fuzzy granule fg in order to develop some understanding about the uncertainty of the fg in terms of its vertical and horizontal dimensions at present the t2 fg is the only it2 fs for which it is possible to obtain closed form formulas for the centroid and those formulas are in this paper

fuzzy multiple attributes and multiple hierarchical decision making abstract a procedure is proposed to solve the multiple attributes and multiple hierarchical system under fuzzy environment the approach is based on 1 1 fuzzy representation 2 2 hierarchical performance evaluation structure 3 3 gradient eigenvector method for rating the fuzzy criteria weighting and 4 4 using the max min paired elimination method for aggregation to illustrate the approach an example on the evaluation of teaching performance in higher education is solved

possibilistic preference networks this paper studies the use of product based possibilistic networks for representing preferences in multidimensional decision problems this approach uses symbolic possibility weights and defines a partial preference order among solutions to a set of conditional preference statements on the domains of discrete decision variables in the case of boolean decision variables this partial ordering is shown to be consistent with the preference ordering induced by the ceteris paribus assumption adopted in cp nets namely by completing the possibilistic net ordering with suitable constraints between products of symbolic weights all cp net preferences can be recovered computing procedures for comparing solutions are provided the flexibility and representational power of the approach is stressed

gradual numbers and their application to fuzzy interval analysis in this paper we introduce a new way of looking at fuzzy intervals instead of considering them as fuzzy sets we see them as crisp sets of entities we call gradual real numbers they are a gradual extension of real numbers not of intervals such a concept is apparently missing in fuzzy set theory gradual numbers basically have the same algebraic properties as real numbers but they are functions a fuzzy interval is then viewed as a pair of fuzzy thresholds which are monotonic gradual real numbers this view enables interval analysis to be directly extended to fuzzy intervals without resorting to alpha cuts in agreement with zadeh s extension principle several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end points of intervals are changed into left and right fuzzy bounds our approach is illustrated on two known problems computing fuzzy weighted averages and determining fuzzy floats and latest starting times in activity network scheduling

spl alpha cut fuzzy arithmetic simplifying rules and a fuzzy function optimization with a decision variable the problems of alpha cut fuzzy arithmetic have been shown like in interval arithmetic that distinct states of fuzzy parameters or fuzzy variable values may be chosen and produce an overestimated fuzziness meanwhile local extrema of a function may exist inside the support of fuzzy parameters and cause an underestimation of fuzziness and an illegal fuzzy number s result previous approaches to overcoming these problems have appeared in literature yet the computational burden of these approaches became even heavier thus this paper is based on the vertex method in literature and extensively proposes newly devised rules observed greatly useful for simplifying the vertex method these rules are devised through a function partitioned into subfunctions distinguishing the types of fuzzy parameter variable occurrences and types of subfunctions or functions with the various observations the improved efficiency has been found able to significantly reduce the combination vertex test of the vertex method for the fuzzy parameters alpha cut endpoints possibly to only a few fuzzy parameters endpoint combinations also as related a procedure for the fuzzy optimization of fuzzy functions with a fuzzy blurred argument a single variable is examined with the vertex method as well a proper and useful preliminary algorithm is proposed numerical examples with results are provided