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

relating different approaches to solve linear programming problems with imprecise costs abstract linear programming problems involving coefficients in the objective function with some lack of precision are usual to solve them several approaches have been proposed in this paper the different methods studied to solve the problem are briefly described next it is shown that the different solutions obtained from them may be obtained by solving a multiobjective linear programming problem

procedure to reduce the effects of modal truncation in eigensolutionreanalysis the influence of structural perturbations on the eigensolutions of a mechanical system is calculated through the use of a discrete model composed of the first normal modes of the base system it consists of modifying the structural perturbations before the application of the classical sensitivity relationships

statistical analysis of music and subspace rotation estimates of sinusoidal frequencies consideration is given to the analysis of the large sample second order properties of multiple signal classification music and subspace rotation sur methods such as esprit for sinusoidal frequency estimation explicit expressions for the covariance elements of the estimation errors associated with either method are derived these expressions of covariances are then used to analyze and compare the statistical performances of the music and sur estimation sure methods both music and sure are based on the eigendecomposition of a sample data covariance matrix the expressions for the estimation error variances derived are used to study the dependence of music and sure performances on the dimension of the data covariance matrix used

newton algorithms for conditional and unconditional maximum likelihood estimation of the parameters of exponential signals in noise the authors present polynomial based newton algorithms for maximum likelihood estimation mle of the parameters of multiple exponential signals in noise this formulation can be used in the estimation for example of the directions of arrival of multiple noise corrupted narrowband plane waves using uniform linear arrays and the frequencies of multiple noise corrupted complex sine waves the algorithms offer rapid convergence and exhibit the computation efficiency associated with the polynomial approach compact closed form expressions are presented for the gradients and hessians various model assumptions concerning the statistics of the underlying signals are considered numerical simulations are presented to demonstrate the algorithms performance

an algorithm for generalized fuzzy binary linear programming problems abstract fuzzy binary linear programming fblp problems are very essential in many fields such as assignment and assembly line balancing problems in operational research multiple projects locations and candidates selection cases in management science as well as representing and reasoning with prepositional knowledge in artificial intelligence although fblp problems play a significant role in human decision environment not very much research has focused on fblp problems this work first proposes a simple means of expressing a triangular fuzzy number as a linear function with an absolute term a method of linearizing absolute terms is also presented the developed goal programming gp model weighted by decision makers dms preference aims to optimize the expected objective function and minimize the sum of possible membership functions deviations after presented a novel way of linearizing product terms the solution algorithm is proposed to generate a crisp trade off promising solution that is also an optimal solution in a certain sense three examples equipment purchasing choice investment project selection and assigning clients to project leaders illustrate that the proposed algorithm can effectively solve generalized fblp problems

two new approaches for assessing the weights of fuzzy opinions in group decision analysis the aggregation of fuzzy opinions is an important component of group decision analysis with fuzzy information this paper proposes two new approaches for the assessment of the weights to be associated with fuzzy opinions these approaches involve respectively the minimization of the sum of squared distances from one weighted fuzzy opinion to another which is called the least squares distance method lsdm and the minimization of the sum of squared differences between the defuzzified values of any two weighted fuzzy opinions which is called the defuzzification based least squares method dlsm the two approaches are developed and numerical examples are presented to illustrate their simplicity and effectiveness in aggregating fuzzy opinions

possibility versus probability the main objective of this note is to try to indicate that a possibility distribution at least in some applications can be interpreted as a family of probabilities possibly subjective an advantage of the latter interpretation is that subsequent derivations can be made by well proven probabilistic arguments rather than relying on an intuition concerning the properties of possibility distributions which is not well developed multistate reliability theory is given as an example of an area of application where the probabilistic approach seems far superior

a fast maximum likelihood algorithm for frequency estimation of a sinusoid based on newton s method a fast algorithm to estimate the frequency of a sinusoid is presented here it is based on newton s method for finding the root of an equation and it is shown that under easily met conditions the root mean square error rmse of the estimator is practically equal to the cramer rao bound after only two iterations of newton s method for all signal to noise ratios snr s above threshold the estimator s probability density function is computed analytically and the rmse is calculated for one and two iterations of newton s method its computational load is shown to be significantly less than other conventional algorithms

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

on frequency estimation summary t this paper discusses a least squares procedure and the use of the periodogram for isolating a discrete harmonic of a time series it is shown that the usual asymptotics on estimation of frequency amplitude and phase of such a harmonic have to be used with great caution from a moderate sample perspective computational issues are discussed and some illustrations are provided bolt brillinger 1979 make use of these asymptotic results we consider a time series model of the form x a cos wt f e where e is a stationary noise sequence and one is interested in estimating the amplitude frequency and phase of the harmonic component the asymptotic theory of the least squares estimates of these parameters has a long history whittle 1951 1953 obtained some of the earliest results more recent results are by hasan 1982 hannan 1973 and walker 1971 who formalize and extend whittle s results in these works it is shown that the asymptotic variance of the frequency estimate is of order n 3 and that the asymptotic variances of the other two components are of the more usual order n these results extend when there are several harmonic components the rate for the estimate of w seems almost unbelievably good and our work was motivated by a desire to see how reliable the asymptotic theory is in brief we find that the product of the amplitude and the sample size n must be quite large in order for the asymptotic theory to be meaningful if this product is not large the frequency estimate is much more variable than indicated by the asymptotic theory and the amplitude estimate is severely biased in applications in which the amplitude is small giving rise to a small peak in the periodogram these results suggest that naive application of the asymptotic theory to gauge resolution can be quite misleading section 2 of this paper is devoted to a review and examination of the asymptotic theory we are also concerned with computational issues arising from the least squares problem this problem is nonlinear in the parameters so that some sort of iterative search must be employed typically search methods start from an initial guess and then proceed by a sequence of modified newton raphson steps for this nonlinear least squares problem it turns out that there are many local minima with a separation in frequency about n l which makes the stationary point to which the iterative scheme converges extremely sensitive to the starting values and this problem gets worse as the sample size increases furthermore it follows from the results of 2 that the estimate of the amplitude is very biased unless the frequency is resolved with order o n so that failure to converge to the global minimum may give a very poor estimate of amplitude the problem becomes

a new method for tool steel materials selection under fuzzy environment abstract this paper presents a new method to solve the tool steel materials selection problem under fuzzy environment where the importance weights of different criteria and the ratings of various alternatives under different criteria are assessed in linguistic terms represented by fuzzy numbers because the proposed method uses simple arithmetic operations rather than the complicated arithmetic operations for aggregation and ranking of fuzzy numbers described in wang and chang 1995 its execution is much faster than the one presented in wang and chang 1995

a fuzzy decision making method and its application to a company choice problem abstract we propose a fuzzy decision making method for multiobjective decision problems its main features are the use of fuzzy connectives to represent the decision maker s preference structure and fuzzy probability to express imprecise and uncertain outcomes the preference structure is extracted through interaction between the decision maker and a computer the fuzzy connectives adopted here are zimmermann s operation and its two extensions the extended operations describe the preference structure better than the operation the proposed method consists of the following five steps 1 1 identification of the problem 2 2 a questionnaire process 3 3 parameter identification 4 4 evaluations of alternatives 5 5 aggregated rating and ranking of alternatives finally an application of this method to a company choice problem is presented

comparison of fuzzy values on a continuous domain abstract a comparison measure called the satisfaction function sf is extended in this paper the sf estimates the degree to which arithmetic comparisons between two fuzzy values are satisfied the previously proposed sf was defined on a discrete domain sfd so in order to compare continuous fuzzy values by the sfd the fuzzy values should be converted into discrete ones in order to directly compare continuous fuzzy values the sf is extended to a continuous domain sfc the sfc is defined based on the sfd and its properties are also developed since the sfc is extended from the sfd its properties includes those of the sfd and it has the same interpretation as the sfd thus it can be used in any applications instead of the sfd the sfc requires less computing time than the sfd under some conditions for comparisons between a fuzzy and a crisp value and between two crisp values the sfc are also extended

asymptotic analysis of a fast algorithm for efficient multiple frequency estimation based on an asymptotic analysis of the contraction mapping cm method of li and kedem 1993 a bandwidth shrinkage rule is proposed for fast and accurate estimation of the frequencies of multiple sinusoids from noisy measurements the cm frequency estimates are defined as the fixed points of a contractive mapping formed by the lag one autocorrelation coefficient calculated from the output of a parametric filter applied to the observed time series with bandwidth parameters judiciously chosen according to the asymptotic analysis the algorithm is shown to be able to accommodate possibly poor initial values of precision o n sup 1 3 and converge to a final estimate whose accuracy is arbitrarily close to o n sup 3 2 the optimal error rate for frequency estimation under the gaussian assumption the total computational complexity of the algorithm is shown to be o n log n which is comparable to that of n point fast fourier transform fft a novelty in the asymptotic analysis is that it accommodates closely spaced frequencies by allowing not only the filter bandwidth but also the frequency separation to be functions of the sample size n this enables an assessment of the accuracy of the frequency estimates for given bandwidths and initial values in situations where some or all of the frequencies are close to each other

a new fuzzy multiple attributive group decision making methodology and its application to propulsion manoeuvring system selection problem in this paper a new fuzzy multiple attribute decision making fmadm method which is suitable for multiple attributive group decision making gdm problems in fuzzy environment is proposed to deal with the problem of ranking and selection of alternatives since the subjectivity imprecision and vagueness in the estimates of a performance rating enter into multiple attribute decision making madm problems fuzzy set theory provides a mathematical framework for modelling vagueness and imprecision in the proposed approach an attribute based aggregation technique for heterogeneous group of experts is employed and used for dealing with fuzzy opinion aggregation for the subjective attributes of the decision problem the propulsion manoeuvring system selection as a real case study is used to demonstrate the versatility and potential of the proposed method for solving fuzzy multiple attributive group decision making problems the proposed method is a generalised model which can be applied to great variety of practical problems encountered in the naval architecture from propulsion manoeuvring system selection to warship requirements definition

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

maximum likelihood estimation of multiple frequencies with constraints to guarantee unit circle roots an approximate maximum likelihood estimator mle of multiple exponentials converts the frequency estimation problem into a problem of estimating the coefficients of a z polynomial with roots at the desired frequencies theoretically the roots of the estimated polynomial should fall on the unit circle but mle as originally proposed does not guarantee unit circle roots this drawback sometimes causes merged frequency estimates especially at low snr if all the sufficient conditions for the z polynomial to have unit circle roots are incorporated the optimization problem becomes too nonlinear and it loses the desirable weighted quadratic structure of mle in the present paper the exact constraints are imposed on each of the first order factors corresponding to individual frequencies for ensuring unit circle roots the constraints are applied during optimization alternately for each frequency in the absence of any merged frequency estimates the rms values more closely approach the theoretical cramer rao cr bound at low snr levels

a fuzzy programming approach to multiobjective multidimensional 0 1 knapsack problems abstract the objective of this paper is to present an interactive procedure for the multiobjective multidimensional 0 1 knapsack problem that takes into consideration the incorporation of fuzzy goals of the decision maker that is easy to use since it requires from the decision maker to handle only one parameter namely the aspiration level of each objective and that is fast and can treat our problem as a usual 0 1 knapsack problem using already available software namely the primal effective gradient method used primarily to solve the large scale cases to get some statistics on the behavior of the algorithm a number of randomly generated simulation of problems is solved from our numerical experience it is possible to conclude that our proposed method is a worthwhile alternative to existing methods from a practical point of view

iterative filtering for multiple frequency estimation it is well known that prony s least squares estimator gives inconsistent estimates for multiple frequency estimation in a recent attempt to diminish this problem dragosevic and stankovic 1989 couple the least squares method of autoregressive ar estimation with an iterative filtering scheme discussed by kay 1988 using an all pole filter but the inconsistency still persists this paper attacks the chronic inconsistency with a general approach of parametric filtering that unifies and extends the previous work it is shown that the inconsistency can be eliminated with an appropriately parametrized filter the clue for the correct parametrization comes from a formula for the bias of the least squares ar estimator the fact of the matter is that as long as a filter satisfies the parametrization requirement consistent estimates can be obtained from the least squares ar estimator on the basis of the filtered data in particular the all pole filter considered by dragosevic and stankovic can be easily reparametrized so that it too satisfies the parametrization requirement and thus leads to a consistent estimator experimental results show that the modified method has a higher resolution than the discrete fourier transform and that its overall performance is quite remarkable

on convergence and bias correction of a joint estimation algorithm for multiple sinusoidal frequencies twenty years ago kay proposed an iterative filtering algorithm ifa for jointly estimating the frequencies of multiple complex sinusoids from noisy observations ifa is based on the fact that the noiseless signal is an autoregressive ar process so the frequency estimation problem can be reformulated as the problem of estimating the ar coefficients by iterating the cycle of ar coefficient estimation and ar filtering ifa provides a computationally simple procedure yet capable of accurate frequency estimation especially at low signal to noise ratio snr however the convergence of ifa has not been established beyond simulation and a very special case of a single frequency and infinite sample size this article provides a statistical analysis of the algorithm and makes several important contributions it shows that the poles of the ar filter must be reduced by an extra shrinkage parameter to accommodate poor initial values and avoid being trapped into false solutions it also shows that the ar estima

maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements the problem of estimating the frequencies phases and amplitudes of sinusoidal signals is considered a simplified maximum likelihood gauss newton algorithm which provides asymptotically efficient estimates of these parameters is proposed initial estimates for this algorithm are obtained by a variation of the overdetermined yule walker method and periodogram based procedure use of the maximum likelihood gauss newton algorithm is not however limited to this particular initialization method some other possibilities to get suitable initial estimates are briefly discussed an analytical and numerical study of the shape of the likelihood function associated with the sinusoids in noise process reveals its multimodal structure and clearly sets the importance of the initialization procedure some numerical examples are presented to illustrate the performance of the proposed estimation procedure comparison to the performance corresponding to the cramer rao lower bound is also presented using a simple expression for the asymptotic cramer rao bound covariance matrix derived in the paper

maximum likelihood estimation of superimposed complex sinusoids in white gaussian noise by reduced effort coarse search recs an efficient implementation for the maximum likelihood estimator for the frequencies of superimposed complex sinusoids in white gaussian noise is presented the algorithm uses one dimensional searches and classical optimization techniques and produces data that can be used for model order selection its convergence and statistical properties are verified through simulation the method does not require polynomial rooting frequent manipulation of large matrices or the use of approximate constraints

statistical analysis of pisarenko s method for sinusoidal frequency estimation statistical analysis is performed for pisarenko s method of sinusoidal frequency estimation by using the periodogram technique under the assumption that the data consist of several sinusoids plus white noise we give an asymptotic expression of the error variance of the sinusoidal frequency estimator also a modified method is proposed this is computationally efficient but the statistical property is not improved we also present simulation results and compare them to the theoretical ones lastly we point out that the pisarenko s method can be implemented as an on line bias compensated least squares algorithm thus avoiding the eigenvalue and eigenvector calculations

fuzzy boolean programming problems with fuzzy costs a general study abstract this paper deals with boolean linear programming problems involving coefficients in the objective function as fuzzy numbers in the study of these problems different approaches can be proposed to use ranking fuzzy numbers methods and fuzzy preference relations obtaining auxiliary classical boolean programming problems and to use the representation theorem obtaining a convex set with extreme points defined by the lower or upper bound of the a level sets of the fuzzy coefficients and consequently an auxiliary interval boolean programming problem in this paper we develop and link the different approaches

estimation of the parameters of sinusoidal signals in non gaussian noise accurate estimation of the amplitude and frequency parameters of sinusoidal signals from noisy observations is an important problem in many signal processing applications in this paper the problem is investigated under the assumption of non gaussian noise in general and laplace noise in particular it is proven mathematically that the maximum likelihood estimator derived under the condition of laplace white noise is able to attain an asymptotic cramer rao lower bound which is one half of that achieved by periodogram maximization and nonlinear least squares it is also proven that when applied to non laplace situations the laplace maximum likelihood estimator which may also be referred to as the nonlinear least absolute deviations estimator can achieve an even higher statistical efficiency especially when the noise distribution has heavy tails a computational procedure is proposed to overcome the difficulty of local extrema in the likelihood function simulation results are provided to validate the analytical findings

optimal structural design with damping constraint limitations the algorithm optimization presented uses the optimality criteria method so as to minimize the total weight of the structure in the presence of a constraint of the damping type the goal of this study is to increase the structural damping by optimizing the viscoelastic and elastic material distribution over the structure the types of damping considered are hysteristic the gradient of the constraint function in relation to the design variables is calculated with the aid of a complex variable sensitivity method several numerical results are presented to validate the algorithm proposed

optimization with frequency constraints limitations abstract after outlining an efficient algorithm for designing minimum weight structures for a specified frequency of vibration the limitations of designing such structures are stressed it is shown that the amount of material that would be necessary for an optimum bar or a beam of a fixed configuration length and boundary conditions to attain an abitrarily prescribed frequency can be disproportionately high yielding a design that is completely impractical in some cases there is an upper bound on the maximum frequency that can be attained irrespective of the amount of material used the effectiveness of configuration changes in conjunction with optimization is emphasized to circumvent this difficulty

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

an iterative method for exact maximum likelihood estimation of the parameters of a harmonic series a procedure is described for determining the exact maximum likelihood ml estimates of the parameters of a harmonic series i e the fundamental frequency and the amplitude and phase of each harmonic existing ml methods are only approximate in the sense that terms present due to mixing between the harmonics are ignored these terms asymptotically reduce to zero as the sample size increases to infinity it is argued that these terms can be significant for short signal lengths the application of the expectation maximization algorithm results in an iterative procedure that converges to a stationary point on the true parameter likelihood surface if global convergence results this point yields the exact ml estimates simulation studies illustrate the advantages of the method when short data lengths are used

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

evaluating weapon systems using fuzzy arithmetic operations in this paper we present a new method to deal with the performance evaluation of weapon systems using fuzzy arithmetic operations an example of tactical missile systems selection is used to illustrate the performance evaluation process of weapon systems because the proposed methods uses simplified fuzzy arithmetic operations of fuzzy numbers rather than the complicated entropy weight calculations mentioned in 7 its execution is much faster than the one presented in 7

boolean programming problems with fuzzy constraints abstract boolean linear programming problems for which there exist some lack of precision of vague nature in the constraints are considered an cut based approach is considered to solve them then an algorithm providing a fuzzy solution is proposed and analyzed the linking with other known solution methods is also studied

decisions and evaluations by hierarchical aggregation of information usually in preparing a decision a hierarchy of criteria for evaluating alternatives can he agreed upon by the decision makers the problem addressed in this paper is the appropriate modelling of such evaluations and their aggregation in order to arrive at a proper overall evaluation function by which the decision alternatives can be ordered easily representing the criteria subjective categories by fuzzy sets it is possible to use aggregation operators as models for the amalgamation of those criteria in the framework of an empirical research project a hierarchy of criteria was developed for the evaluation of credit worthiness of customers 50 credit applications were evaluated by a number of credit managers of banks and their judgement could be appropriately modelled by a four level hierarchy of criteria

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