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

non dominance and potential optimality for partial preference relations abstract in this paper we obtain new theoretical results relating the notions of potential optimality and non dominance without an assumption that a value function exists in particular we investigate a decision problem involving the choice of single or multiple best objects our results show that the notions of potential optimality and non dominance are equivalent in a special case of preferences of the decision maker expressed by partial quasi orders

a fuzzy approach to select the location of the distribution center abstract the location selection of distribution center dc is one of the most important decision issues for logistics managers owing to vague concept frequently represented in decision data a new multiple criteria decision making method is proposed to solve the distribution center location selection problem under fuzzy environment in the proposed method the ratings of each alternative and the weight of each criterion are described by linguistic variables which can be expressed in triangular fuzzy numbers the final evaluation value of each dc location is also expressed in a triangular fuzzy number by calculating the difference of final evaluation value between each pair of dc locations a fuzzy preference relation matrix is constructed to represent the intensity of the preferences of one plant location over another and then a stepwise ranking procedure is proposed to determine the ranking order of all candidate locations finally a numerical example is solved to illustrate the procedure of the proposed method at the end of this paper

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

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

ranking 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

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

modeling uncertainty in multi criteria decision analysis abstract his paper provides a review of multiple criteria decision analysis mcda for cases where attribute evaluations are uncertain the main aim is to identify different tools which can be used to represent uncertain evaluations and to broadly survey the available decision models that can be used to support uncertain decision making the review includes models using probabilities or probability like quantities explicit risk measures such as quantiles and variances fuzzy numbers and scenarios the practical assessment of uncertain outcomes and preferences associated with these outcomes is also discussed

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

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

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

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

a fuzzy mcdm method for solving marine transshipment container port selection problems abstract transshipment is a very popular and important issue in the present international trade container transportation market in order to reduce the international trade container transportation operation cost it is very important for shipping companies to choose the best transshipment container port the aim of this paper is to present a new fuzzy multiple criteria decision making method fmcdm for solving the transshipment container port selection problem under fuzzy environment in this paper we present first the canonical representation of multiplication operation on three fuzzy numbers and then this canonical representation is applied to the selection of transshipment container port based on the canonical representation the decision maker of shipping company can determine quickly the ranking order of all candidate transshipment container ports and select easily the best one

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

fuzzy group decision making for facility location selection the selection of a facility location among alternative locations is a multicriteria decision making problem including both quantitative and qualitative criteria the conventional approaches to facility location problem tend to be less effective in dealing with the imprecise or vagueness nature of the linguistic assessment under many situations the values of the qualitative criteria are often imprecisely defined for the decision makers the aim of the paper is to solve facility location problems using different solution approaches of fuzzy multi attribute group decision making the paper includes four different fuzzy multi attribute group decision making approaches the first one is a fuzzy model of group decision proposed by blin the second is the fuzzy synthetic evaluation the third is yager s weighted goals method and the last one is fuzzy analytic hierarchy process although four approaches have the same objective of selecting the best facility location alternative they come from different theoretic backgrounds and relate differently to the discipline of multi attribute group decision making these approaches are extended to select the best facility location alternative by taking into account quantitative and qualitative criteria a short comparative analysis among the approaches and a numeric example to each approach are given

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

fuzzy mcdm based on ideal and anti ideal concepts this paper presents a novel fuzzy multiple criteria decision making mcdm based on the concepts of ideal and anti ideal points the concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix to evaluate the weighted suitability of different alternatives versus various criteria the distance of different alternatives versus positive ideal solution and negative ideal solution are then obtained by using the proposed ranking method finally the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative

robust eigenvalue computation for smoothing operators robust quasi relative galerkin discretization error estimates are derived for the eigenvalue problem associated to a nonnegative compact operator cal k acting in a hilbert space trace discretization error estimates for arbitrarily small positive powers of cal k are obtained as a consequence coupled with bounds on eigenfunction oscillations the results are then applied to the case of an integral operator with piecewise smooth kernel k on a bounded domain and in the context of the h finite element method

an extension to fuzzy mcdm an extension to the fuzzy multiple criteria decision making mcdm model is suggested in this work where the ratings of alternatives versus criteria and the importance weights of all criteria are assessed in linguistic values represented by fuzzy numbers moreover values of alternatives under objective criteria are normalized by a suggested approach meanwhile the membership function of the final fuzzy evaluation value of each alternative can be developed in addition a riemann integral based mean of removals is suggested to rank all the final fuzzy evaluation values for final decision making so that the ranking procedure can be clearly formulated finally a numerical example demonstrates the feasibility of the proposed model