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

efficient algorithms for separated continuous linear programs the multicommodity flow problem with holding costs and extensions we give an approximation scheme for separated continuous linear programming problems such problems arise as fluid relaxations of multiclass queueing networks and are used to find approximate solutions to complex job shop scheduling problems in a network with linear flow costs and linear per unit time holding costs our algorithm finds a drainage of the network that for given constants e 0 and i 0 has total cost 1 e opt i where opt is the cost of the minimum cost drainage the complexity of our algorithm is polynomial in the size of the input network 1 e and log 1 i the fluid relaxation is a continuous problem while the problem is known to have a piecewise constant solution it is not known to have a polynomially sized solution we introduce a natural discretization of polynomial size and prove that this discretization produces a solution with low cost this is the first polynomial time algorithm with a provable approximation guarantee for fluid relaxations

a simplex based algorithm to solve separated continuous linear programs we consider the separated continuous linear programming problem with linear data we characterize the form of its optimal solution and present an algorithm which solves it in a finite number of steps using an analog of the simplex method in the space of bounded measurable functions

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

data driven fuzzy preference analysis from an optimization perspective abstract how to leverage massive online data to understand consumer preferences over products and services has accumulated significant attention in business in this paper we analyze consumer preferences by modeling it as a decision making problem of ranking alternatives with consumers online ratings we propose a data driven fuzzy preference analysis d fpa method to obtain the priorities of alternatives we show that the d fpa is tractable and with high computation efficiency in addition we propose a natural indicator to measure the reliability of the derived ranking results and suggest thresholds of this indicator for better control of the method a real world application about online film rating is provided to illustrate the d fpa demonstrating that the derived ranking results converge rapidly and remain stable with the observed empirical data finally we show how to build up an effective recommendation system with empirical data from movielens

convergence of a general class of algorithms for separated continuous linear programs separated continuous linear programs sclp are a type of infinite dimensional linear program which can serve as a useful model for a variety of dynamic network problems where storage is permitted at the nodes this paper proves the convergence of a general class of algorithms for solving sclp under certain restrictions on the problem data this is the first such proof for any nondiscretization algorithm for solving any form of continuous linear programs

strongly damped wave equation with exponential nonlinearities abstract in this paper we study the initial boundary value problem for two dimensional strongly damped wave equation with exponentially growing source and damping terms we first show the well posedness of this problem and then prove the existence of the global attractor in h 0 1 l l 2

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

reducing weight flexibility in fuzzy dea an important outcome of assessing relative efficiencies within a group of decision making units dmus in fuzzy data envelopment analysis is a set of virtual multipliers or weights accorded to each input or output factor taken into account these sets of weights are typically different for each of the participating dmus and in some cases it may be considered unacceptable that the same factor is accorded widely differing weights thus it is important to find a common set of weights csw across the set of dmus in this paper by assessing upper bounds on factor weights and compacting the resulted intervals a csw is determined since resulted efficiencies by the proposed csw are fuzzy numbers rather than crisp values it is more informative for decision maker

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

estimation of a fuzzy regression model using fuzzy distances regression analysis is a powerful statistical tool that has many applications in different areas the problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input output data crisp or fuzzy however we realize that in general most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances minimizing a real error function and the interpretation of fuzzy data as possibility distributions the main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers the new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error the estimation process in general can be considered easier to apply in practice and it is not limited to triangular fuzzy numbers numerical examples are provided to illustrate the usefulness and applicability of these results and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory

a taxonomy and review of the fuzzy data envelopment analysis literature two decades in the making data envelopment analysis dea is a methodology for measuring the relative efficiencies of a set of decision making units dmus that use multiple inputs to produce multiple outputs crisp input and output data are fundamentally indispensable in conventional dea however the observed values of the input and output data in real world problems are sometimes imprecise or vague many researchers have proposed various fuzzy methods for dealing with the imprecise and ambiguous data in dea in this study we provide a taxonomy and review of the fuzzy dea methods we present a classification scheme with four primary categories namely the tolerance approach the a level based approach the fuzzy ranking approach and the possibility approach we discuss each classification scheme and group the fuzzy dea papers published in the literature over the past 20 years to the best of our knowledge this paper appears to be the only review and complete source of references on fuzzy dea

linear optimal control problems with piecewise analytic solutions this paper is concerned with the existence of piecewise analytic optimal solutions for various linear optimal control problems with piecewise analytic problem data some results of this form may be found in the literature but their proofs are generally based on the maximum principle and require certain normality conditions before they can be applied also the control constraints are required to be polyhedral and fixed in time using the recent theory developed for a class of problems known as separated continuous linear programs we are able to remove the normality conditions completely and allow time varying control constraints of both polyhedral and integral form in particular we prove theorems on the existence of piecewise analytic optimal solutions for the linear time optimal control problem and the linear optimal control problem with or without end point constraints

decision risk analysis for an interval topsis method abstract topsis is a multi attribute decision making madm technique for ranking and selection of a number of externally determined alternatives through distance measures when the collected data for each criterion is interval and the risk attitude for a decision maker is unknown we present a new topsis method for normalizing the collected data and ranking the alternatives the results show that the decision maker with different risk attitude ranks the different alternatives

separated continuous conic programming strong duality and an approximation algorithm motivated by recent applications in robust optimization and in sign constrained linear quadratic control we study in this paper a new class of optimization problems called separated continuous conic programming sccp focusing on a symmetric primal dual pair we develop a strong duality theory for the sccp our idea is to use discretization to connect the sccp and its dual to two ordinary conic programs we show if the latter are strongly feasible and with finite optimal values a condition that is readily verifiable then the strong duality holds for the sccp this approach also leads to a polynomial time approximation algorithm that solves the sccp to any required accuracy

on the strongly damped wave equation we prove the existence of the universal attractor for the strongly damped semilinear wave equation in the presence of a quite general nonlinearity of critical growth when the nonlinearity is subcritical we prove the existence of an exponential attractor of optimal regularity having a basin of attraction coinciding with the whole phase space as a byproduct the universal attractor is regular and of finite fractal dimension moreover we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient

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

modeling uncertain activity duration by fuzzy number and discrete event simulation this paper introduces how to incorporate fuzzy set theory and a fuzzy ranking measure with discrete event simulation in order to model uncertain activity duration in simulating a real world system especially when insufficient or no sample data are available fuzzy numbers are used to describe uncertain activity durations reflecting vagueness imprecision and subjectivity in the estimation of them a fuzzy ranking measure is merged with an activity scanning simulation algorithm for performing fuzzy simulation time advancement and event selection for simulation experimentation the uses of the fuzzy activity duration and the probability distribution modeled duration are compared through a series of simulation experiments it is observed that the fuzzy simulation outputs are arrived at through only one cycle of fuzzy discrete event simulation still they contain all the statistical information that are produced through multiple cycles of simulation experiments when the probability distribution approach is adopted

a new algorithm for state constrained separated continuous linear programs during the last few decades significant progress has been made in solving large scale finite dimensional and semi infinite linear programming problems in contrast little progress has been made in solving linear programs in infinite dimensional spaces despite their importance as models in manufacturing and communication systems inspired by the research on separated continuous linear programs we propose a new class of continuous linear programming problems that has a variety of important applications in communications manufacturing and urban traffic control this class of continuous linear programs contains the separated continuous linear programs as a subclass using ideas from quadratic programming we propose an efficient algorithm for solving large scale problems in this new class under mild assumptions on the form of the problem data we prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set we show this class of problems admits piecewise constant optimal control when the optimal solution exists we give conditions for the existence of an optimal solution we also report computational results which illustrate that the new algorithm is effective in solving large scale realistic problems with several hundred continuous variables arising in manufacturing systems

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

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

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

a study of general dynamic network programs with arc time delays dynamic network flow problems in which there is a delay on the flows in the arcs have been in existence since the early days of modern optimization however most previous work in this area has only considered models which are discrete in the time variable in this paper we present a continuous time model for a very broad class of dynamic network problems with arc time delays the model is a direct extension of the separated continuous linear program sclp to include time delays and is called the separated continuous linear program with time delays sclptd by suitable transformations we are able to rewrite sclptd in a manner which is very close to sclp itself this then allows us to use all the recent theory and algorithms for sclp to derive similar results for sclptd in particular the theory we present includes a characterization of the extreme point solutions an existence theorem for piecewise analytic optimal extreme point solutions and a strong duality theorem we also present a class of convergent algorithms for the solution of sclptd in certain instances

bifurcation with memory a model equation containing a memory integral is posed the extent of the memory the relaxation time lambda controls the bifurcation behavior as the control parameter r is increased small large lambda gives steady periodic bifurcation there is a double eigenvalue at lambda lambda 1 separating purely steady lambda lambda 1 states with t to infty as lambda to lambda 1 analysis leads to the co existence of stable steady periodic states and as r is increased the periodic states give way to the steady states numerical solutions show that this behavior persists away from lambda lambda 1

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

a note on a strongly damped wave equation with fast growing nonlinearities a strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered the main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved the initial boundary value problem for the considered equation is globally well posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space these results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global lyapunov function

development of fuzzy process accuracy index for decision making problems several statistical decision making tools and methods are available to organize evidence evaluate risks and aid in decision making process capability indices are the summary statistics to point out the process performance in this paper these indices are analyzed to obtain a new decision making tool process accuracy index c a measures the degree of process centering and gives alerts when the process mean departures from the target value it focuses on the location of process mean and the distance between mean and target value we modify the traditional process accuracy index to obtain a new tool under fuzziness with the proposed tool specification limits and process mean can be defined as triangular or trapezoidal fuzzy numbers the proposed tool is illustrated to solve a supplier selection problem

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

extended fuzzy linear models and least squares estimates least squares regression of the fuzzy linear model is extended to overcome and interpret the occurrence of negative spreads 1 the idea is to introduce best fit difference models which exploit both the hukuhara difference and the l2 metric distance the fuzzy models use lr fuzzy numbers fitted models are compared by using the coefficient of determination in a similar way to its use in classical statistical least squares fitting the non lr fuzzy case is also considered

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

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

strongly damped wave equations with critical nonlinearities abstract this note deals with the strongly damped nonlinear wave equation u t t u t u f u t g u h with dirichlet boundary conditions where both the nonlinearities f and g exhibit a critical growth while h is a time independent forcing term the existence of an exponential attractor of optimal regularity is proven as a corollary a regular global attractor of finite fractal dimension is obtained

an extended algorithm for separated continuous linear programs separated continuous linear programs sclp are a class of continuous linear programs which among other things can serve as a useful model for dynamic network problems where storage is permitted at the nodes recent work on sclp has produced a detailed duality theory conditions under which an optimal solution exists with a finite number of breakpoints a purification algorithm as well as a convergent algorithm for solving sclp under certain assumptions on the problem data this paper combines much of this work to develop a possible approach for solving a wider range of sclp problems namely those with fairly general costs the techniques required to implement the algorithm are no more than standard finite dimensional linear programming and line searching and the resulting algorithm is simplex like in nature we conclude the paper with the numerical results obtained by using a simple implementation of the algorithm to solve a small problem

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

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

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

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

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

nonsmooth continuous time optimization problems necessary conditions abstract we consider lipschitz continuous time nonlinear optimization problems and provide first order necessary optimality conditions of both fritz john and karush kuhn tucker types

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

multiobjective fuzzy regression with central tendency and possibilistic properties a multiobjective fuzzy regression model mofr is developed this mofr model combines central tendency and possibilistic properties of statistical and fuzzy regressions and overcomes several shortcomings of these two approaches a new class of distance measure for two intervals that takes into account all the points in both intervals is introduced the methodology is illustrated by numerical examples

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

finite dimensional attractors for the quasi linear strongly damped wave equation abstract we present a new method of investigating the so called quasi linear strongly damped wave equations t 2 u t x u x u f u x x u g in bounded 3d domains this method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non linearity is less than 6 and f may have arbitrary polynomial growth rate moreover the existence of a finite dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established in a particular case 0 which corresponds to the so called semi linear strongly damped wave equation our result allows to remove the long standing growth restriction f u c 1 u 5

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

polynomial approximations for continuous linear programs continuous linear programs have attracted considerable interest due to their potential for modeling manufacturing scheduling and routing problems while efficient simplex type algorithms have been developed for separated continuous linear programs crude time discretization remains the method of choice for solving general nonseparated problem instances in this paper we propose a more generic approximation scheme for nonseparated continuous linear programs where we approximate the functional decision variables policies by polynomial and piecewise polynomial decision rules this restriction results in an upper bound on the original problem which can be computed efficiently by solving a tractable semidefinite program to estimate the approximation error we also compute a lower bound by solving a dual continuous linear program in piecewise polynomial decision rules we establish the convergence of the primal and dual approximations under slater type constraint qualifications we also highlight the

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

some properties of a class of continuous linear programs this paper discusses a class of linear programs posed in a function space a member of this class is called a separated continuous linear program sclp such problems occur for example in the planning of production and inventory we characterize the l infty extreme point solutions of sclp in a manner analogous to the basic solutions of finite dimensional linear programming and give a sufficient condition for there to exist optimal extreme point solutions with finitely many constant basis intervals sclp is to date the most general continuous linear program for which such strong characterizations have been found

parameter estimation based on interval valued belief structures abstract parameter estimation based on uncertain data represented as belief structures is one of the latest problems in the dempster shafer theory in this paper a novel method is proposed for the parameter estimation in the case where belief structures are uncertain and represented as interval valued belief structures within our proposed method the maximization of likelihood criterion and minimization of estimated parameter s uncertainty are taken into consideration simultaneously as an illustration the proposed method is employed to estimate parameters for deterministic and uncertain belief structures which demonstrates its effectiveness and versatility

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

optimal fuzzy counterparts of scheduling rules the optimality of a fuzzy logic alternative to the usual treatment of uncertainties in a scheduling system using probability theory is examined formally fuzzy scheduling techniques proposed in the literature either fuzzify directly the existing scheduling rules or solve mathematical programming problems to determine the optimal schedules in the former method the fuzzy optimality for the optimal scheduling rules is usually not justified but still assumed in this paper the necessary conditions for fuzzy optimality are defined and fuzzy counterparts of some of the well known scheduling rules such as shortest processing time spt and earliest due date edd are developed

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

nonparametric kernel estimation based on fuzzy random variables this paper extends the classical nonparametric curve fitting methods for fuzzy random variables the study was conducted in two parts 1 fuzzy density estimation and 2 nonparametric regression the classical nonparametric density estimation was first developed based on the kernel method for a given fuzzy random sample at crisp and or fuzzy point based on crisp or fuzzy bandwidth the classical bandwidth selection was also extended when the underlying population was normal with known or unknown fuzzy variance moreover the classical nonparametric kernel based regression model with crisp and or input or output is extended whenever a given bandwidth is a crisp or fuzzy number the cross validation procedure for selecting the optimal value of the crisp smoothing parameter is also extended to fit the proposed nonparametric regression model the large sample properties of the proposed fuzzy estimators were also investigated by some theorems several numerical examples including that of real life data are used to illustrate the proposed methods in curve fitting estimation with crisp and or fuzzy information moreover the proposed methods were examined in comparison with some other existing methods and their effectiveness were clarified via some numerical examples and simulation studies both theatrical and numerical results indicated that the nonparametric curve fitting methods significantly reduced the sum of square errors as well as the spreads of the fuzzy curve fitting estimation

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

computational and theoretical pitfalls in some current performance measurement techniques and a new approach this paper addresses some pitfalls of some fuzzy data envelopment analysis models which have been provided in recent published papers this study deals with these papers from both computational and theoretical points of view moreover a new approach to deal with fuzzy data in dea framework is provided which does not have the mentioned pitfalls

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

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

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

continuous time optimization problems involving invex functions in d h martin the essence of invexity j optim theory appl 47 1985 65 76 martin introduced the notions of kkt invexity and wd invexity for mathematical programming problems these notions are relaxations of invexity in this work we generalize these concepts for continuous time nonlinear optimization problems we prove that the notion of kkt invexity is a necessary and sufficient condition for global optimality of a karush kuhn tucker point and that the notion of wd invexity is a necessary and sufficient condition for weak duality

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

on the existence of a global attractor for the wave equation with nonlinear strong damping perturbed by nonmonotone term abstract the long time behavior of the wave equation with nonmonotone interior damping is considered it is shown that the semigroup generated by this equation possesses a global attractor in h 0 1 l 2