how to select and how to rank projects the promethee method abstract in this paper we present the promethee methods a new class of outranking methods in multicriteria analysis their main features are simplicity clearness and stability the notion of generalized criterion is used to construct a valued outranking relation all the parameters to be defined have an economic signification so that the decision maker can easily fix them two ways of treatment are proposed it is possible to obtain either a partial preorder promethee i or a complete one promethee ii both on a finite set of feasible actions a comparison is made with the electre iii method the stability of the results given by the two methods is analysed numerical applications are given in order to illustrate the properties of the new methods and some further problems are discussed

a fuzzy gould type integral in this paper we introduce and study a set valued gould type integral of a real function with respect to a fuzzy multimeasure taking values in the family of all nonempty closed subsets of a banach space x some classical properties of the integral are obtained only using the hypothesis of monotonicity for the set valued fuzzy measure

the autocontinuity of set function and the fuzzy integral abstract the autocontinuity and some other concepts of a set function are introduced and sugeno s fuzzy measure with some annexed conditions is studied on a fuzzy measure space egoroffs theorem is proved it is also proved that f mean convergence is equivalent to convergence in measure for a sequence of measurable functions finally some convergence theorems of sequence of fuzzy integrals are proved especially using the concept of autocontinuity a necessary and sufficient condition for the convergence in measure theorem is given

integrals of set valued functions for decomposable measures abstract in this paper based on weber s 1984 integrals for archimedean t conorms decomposable measures integrals of set valued functions for archimedean t conorms decomposable measures are established some results similar to aumann s set valued integrals are given these include convexity closedness convergence theorems etc all these are the generalization of aumann s results

set valued choquet integrals revisited this paper performs a further investigation on the set valued choquet integral given by jang and kwon fuzzy sets and systems 112 2000 233 some incorrect properties are revised it is pointed out that the definition for convergence of sequence of sets or set valued functions is unreasonable by using the usual kuratowski s convergence some new convergence theorems for the set valued choquet integral are shown meanwhile some other properties are obtained

on the representation of choquet integrals of set valued functions and null sets in this paper we consider the definition of set valued choquet integrals we obtain some properties and the representation of set valued choquet integrals and also we discuss some fundamental relations between null sets and set valued choquet integrals

resonant collisions in four dimensional reversible maps a description of scenarios abstract we define a resonant collision of order k 1 in a family of four dimensional 4d reversible maps for any specified k the bifurcation scenario is the collection of the different possible types of bifurcation of a symmetric fixed point that may be encountered through various choices of parameters describing the family of maps under consideration we adopt a perturbative approach coupled with numerical iterations around orbits obtained perturbatively to explore phase space structures in the immediate vicinity of a resonant collision and thereby to obtain a description of the possible scenarios for different values of k the phase space structures typically involve bifurcating periodic orbits families of invariant curves and tori and present interesting possibilities especially around the secondary bifurcations of the periodic orbits see below based on the results of the perturbative and numerical approach we conjecture that three distinct scenarios are involved for the cases k 2 3 4 respectively while there exists a fourth distinctive scenario common to all k 4 and we present what we believe to be a reasonably exhaustive description of these scenarios the case k 1 involves bifurcations of fixed points rather than of periodic orbits and has been investigated numerically in an earlier paper

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

data informed fuzzy measures for fuzzy integration of intervals and fuzzy numbers the fuzzy integral fi with respect to a fuzzy measure fm is a powerful means of aggregating information the most popular fis are the choquet and sugeno and most research focuses on these two variants the arena of the fm is much more populated including numerically derived fms such as the sugeno lambda measure and decomposable measure expert defined fms and data informed fms the drawback of numerically derived and expert defined fms is that one must know something about the relative values of the input sources however there are many problems where this information is unavailable such as crowdsourcing this paper focuses on data informed fms or those fms that are computed by an algorithm that analyzes some property of the input data itself gleaning the importance of each input source by the data they provide the original instantiation of a data informed fm is the agreement fm which assigns high confidence to combinations of sources that numerically agree with one another this paper extends upon our previous work in data informed fms by proposing the uniqueness measure and additive measure of agreement for interval valued evidence we then extend data informed fms to fuzzy number fn valued inputs we demonstrate the proposed fms by aggregating interval and fn evidence with the choquet and sugeno fis for both synthetic and real world data

fourth order resonant collisions of multipliers in reversible maps period 4 orbits and invariant curves resonant collision of multipliers at i of a symmetric fixed point for a 2 parameter family of 4 dimensional reversible maps is considered bifurcation of period 4 orbits from the fixed point and their linear stability characteristics are briefly reviewed in one of the three possible types of bifurcation see text a small angle secondary collision of the floquet multipliers of the bifurcating periodic orbit takes place leading to the bifurcation of invariant curves from the orbit the invariant curves are calculated in a perturbation scheme in the leading order of perturbation the secondary bifurcation is found to be of superthreshold type an interesting pattern in the vicinity of the resonant collision involving families of invariant curves and 2 tori emerges results of numerical iterations corroborating the picture conjectured on the basis of perturbation calculations are presented corresponding results on resonant collisions at 1 are briefly stated

fuzzy number fuzzy measures and fuzzy integrals i fuzzy integrals of functions with respect to fuzzy number fuzzy measures this paper based on the fuzzy measures and fuzzy integrals given by sugeno first defines interval number fuzzy measures inf measures and fuzzy number fuzzy measures fnf measures and the fuzzy integral of function with respect to inf measures are given then it defines the fuzzy integral of functions with respect to fnf measures and the properties and convergence theorems are obtained all these are generalizations of sugeno s works

note a preference ranking organisation method the promethee method for multiple criteria decision making principles for a new family of outranking methods are given the main aim of the proposed promethee approach is to be as easily understood as possible by the decision maker it is based on extensions of the notion of criterion six possible extensions are considered these extensions can easily be identified by the decision maker because the parameters to be defined at most 2 have an economic significance a valued outranking graph is constructed by using a preference index two possibilities are considered to solve the ranking problem by using this valued graph promethee i provides a partial preorder and promethee ii a total preorder on the set of the possible actions

on the convergence of sequences of fuzzy measures and generalized convergence theorems of fuzzy integrals abstract in this paper we will investigate further fuzzy measures and fuzzy integrals given by sugeno 1974 our purpose is concentrated on the convergence of sequences of fuzzy measures or fuzzy integrals some concepts such as convergence uniform convergence convergence in integral weak convergence are introduced for the sequence of fuzzy measures then the properties and relations of them are discussed furthermore various kinds of convergence theorems of fuzzy integrals are extended these are generalized monotone convergence theorem generalized fatou s lemma etc

the use of domains in multicriteria decision making abstract this paper describes a domain criterion for a multicriteria problem given decision maker it then outlines possibilities for aggregating individual preference structures in the case of multiple decision makers

a note on the interval valued generalized fuzzy integral by means of an interval representable pseudo multiplication and their convergence properties in this paper we consider the generalized fuzzy integral which is a fuzzy integral based on the idempotent pseudo addition supremum and a pseudo multiplication studied by xie and fang in 2006 the purpose of this study is to define the interval valued generalized fuzzy integral with respect to a fuzzy measure by means of an interval representable pseudo multiplication of measurable interval valued functions and to investigate some characterizations and convergence properties of them

some properties of choquet integrals of set valued functions abstract in this paper we will define choquet integrals of set valued functions then we show some properties of choquet integrals of set valued functions

stability of invariant curves in four dimensional reversible mappings near 1 1 resonance recent work on reversible neimark sacker bifurcation in 4d maps is summarized linear stability analysis of families of invariant curves appearing in this bifurcation is presented by a referring to the analogous stability problem in reversible hopt bifurcation in vector fields and b perturbatively calculating a set of quantities termed quasi multipliers for the invariant curves in particular the critical rotation numbers corresponding to transition from elliptic to hyperbolic invariant curves on the subthreshold side in the so called inverted bifurcation are calculated results of numerical iterations corroborating the above analysis are presented the question of exploring the structure of the phase space close to the invariant curves is briefly addressed in the conclusion

fuzzy integrals of fuzzy valued functions abstract this paper based on the fuzzy measures and integrals given by sugeno first defines fuzzy integrals of set valued functions and then the fuzzy integral of fuzzy valued functions is obtained and discussed next it defines the fuzzy integrals of fuzzy mappings then pays much effort to the fuzzy integral of interval valued functions the properties and convergence theorems are obtained all these are generalizations of earlier results

fuzzy integrals of set valued mappings and fuzzy mappings abstract the paper based on sugeno 1974 first defines the concept of fuzzy integrals of set valued mappings by the similar way of aumann 1965 then it is discussed some results similar to aumann integral are obtained these include convexity closedness fatou s lemmas and the lebesgue convergence theorem etc at last the fuzzy integral of fuzzy mappings is defined the extended results corresponding with fuzzy integrals of set valued mappings are showed

multicriteria analysis survey and new directions abstract before the advent of multicriteria analysis decision problems often took the form of the optimization of an objective function or a utility function this approach has the advantage of resulting in well defined mathematical problems but is not always representative of reality in fact the comparison of several possible decisions is rarely made according to a single point of view and the preferences on a point of view are in many cases only modelled with difficulty by a function the aim of this paper is to survey the different directions in which multicriteria analysis has developed multiattribute utility theory outranking relations interactive methods the methodology recently proposed by b roy is also presented

efficient multiple kernel classification using feature and decision level fusion kernel methods for classification is a well studied area in which data are implicitly mapped from a lower dimensional space to a higher dimensional space to improve classification accuracy however for most kernel methods one must still choose a kernel to use for the problem since there is in general no way of knowing which kernel is the best multiple kernel learning mkl is a technique used to learn the aggregation of a set of valid kernels into a single ideally superior kernel the aggregation can be done using weighted sums of the precomputed kernels but determining the summation weights is not a trivial task furthermore mkl does not work well with large datasets because of limited storage space and prediction speed in this paper we address all three of these multiple kernel challenges first we introduce a new linear feature level fusion technique and learning algorithm gamklp second we put forth three new algorithms defimkl degamkl and delsmkl for nonlinear fusion of kernels at the decision level to address mkl s storage and speed drawbacks we apply the nystrom approximation to the kernel matrices we compare our methods to a successful and state of the art technique called mkl group lasso mklgl and experiments on several benchmark datasets show that some of our proposed algorithms outperform mklgl when applied to support vector machine svm based classification however to no surprise there does not seem to be a global winner but instead different strategies that a user can employ experiments with our kernel approximation method show that we can routinely discard most of the training data and at least double prediction speed without sacrificing classification accuracy these results suggest that mkl based classification techniques can be applied to big data efficiently which is confirmed by an experiment using a large dataset

fuzzy valued fuzzy measures and generalized fuzzy integrals in this paper the concept of fuzzy valued fuzzy measures is introduced at first and then based on the generalized fuzzy integral given by wu et al fuzzy sets and systems 57 1993 219 the generalized fuzzy integral of fuzzy valued functions with respect to fuzzy valued fuzzy measures is defined and the properties and convergence theorems are given all these are extensions of the sugeno theory of fuzzy integrals and its applications ph d thesis tokyo institute of technology 1974 and wu et al 1993 theory

generalized fuzzy integrals of set valued functions in this paper based on wu s generalized fuzzy integrals of point valued function c wu et al fuzzy sets and systems 70 1995 75 a theory of generalized fuzzy integrals of fuzzy valued functions will be investigated it includes definitions properties and various kinds of convergence theorems these are extensions of results in c wu et al fuzzy sets and systems 70 1995 75 d zhang and z wang fuzzy sets and systems 54 1993 63

preference relations on actions and criteria in multicriteria decision making abstract let a be a finite set of feasible actions which are judged following several criteria an outranking relation is defined on a by considering preference of the decision maker as a weak order on each criterion and the relation among criteria as a semi order on the given set of criteria several ways of constructing outranking relations have been proposed one of the most popular introduced by b roy for instance electre s is based on the use of weights related to criteria in our approach the knowledge of weights is replaced by the existence of a semi order a case study is developed it deals with a computer selection problem

fuzzy number fuzzy measures and fuzzy integrals ii fuzzy integrals of fuzzy valued functions with respect to fuzzy number fuzzy measures on fuzzy sets this is a subsequent paper of 9 by using the concepts of fuzzy number fuzzy measures 9 and fuzzy valued functions 10 a theory of fuzzy integrals of fuzzy valued functions with respect to fuzzy number fuzzy measures is built up so far it is a more general one following sugeno s 5

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

strong integral of multifunctions relative to a fuzzy measure abstract in this paper we define and study an integral for measurable multifunctions taking values in the family of all nonvoid subsets of 0 with respect to a fuzzy measure unlike various ways of defining such an integral e g aumann way using different selections dunford way riemann way we use the sugeno method directly applied to multifunctions the originality of our definition is that the value of the integral is a real scalar and not usually a set some classical integral properties a convergence theorem and relationships with other integrals are presented