the binomial gini inequality indices and the binomial decomposition of welfare functions abstract in the context of social welfare and choquet integration we briefly review on the one hand the generalized gini welfare functions and inequality indices for populations of n 2 individuals and on the other hand the mobius representation framework for choquet integration particularly in the case of k additive symmetric capacities we recall the binomial decomposition of owa functions due to calvo and de baets and we examine it in the restricted context of generalized gini welfare functions with the addition of appropriate s concavity conditions we show that the original expression of the binomial decomposition can be formulated in terms of two equivalent functional bases the binomial gini welfare functions and the atkinson kolm sen aks associated binomial gini inequality indices according to blackorby and donaldson s correspondence formula the binomial gini pairs of welfare functions and inequality indices are described by a parameter j 1 n associated with the distributional judgements involved the j th generalized gini pair focuses on the n j 1 n poorest fraction of the population and is insensitive to income transfers within the complementary richest fraction of the population

an axiomatic approach to the measurement of the amount of interaction among criteria or players in the framework of multicriteria decision making based on fuzzy measures or cooperative game theory an axiomatization of the concept of amount of interaction index is given it is based on four axioms a natural property called absence of interaction a boundary condition for which the index reduces to the shapley or banzhaf interaction index a pseudo additivity axiom and a recursivity property the resulting index which can be interpreted as a measure of the average degree of independence among criteria or players takes a particularly simple and natural form for supermodular or submodular measures the approach to the notion of interaction adopted in this paper can be seen as complementary to that recently adopted by grabisch and roubens

on the set of imputations induced by the k additive core an extension to the classical notion of core is the notion of k additive core that is the set of k additive games which dominate a given game where a k additive game has its mobius transform or harsanyi dividends vanishing for subsets of more than k elements therefore the 1 additive core coincides with the classical core the advantages of the k additive core is that it is never empty once k geq 2 and that it preserves the idea of coalitional rationality however it produces k imputations that is imputations on individuals and coalitions of at most k individuals instead of a classical imputation therefore one needs to derive a classical imputation from a k order imputation by a so called sharing rule the paper investigates what set of imputations the k additive core can produce from a given sharing rule

using choquet integral as preference model in interactive evolutionary multiobjective optimization we propose an interactive multiobjective evolutionary algorithm that attempts to discover the most preferred part of the pareto optimal set preference information is elicited by asking the user to compare some solutions pairwise this information is then used to curb the set of compatible user s value functions and the multiobjective evolutionary algorithm is run to simultaneously search for all solutions that could potentially be the most preferred compared to previous similar approaches we implement a much more efficient way of determining potentially preferred solutions that is solutions that are best for at least one value function compatible with the preference information provided by the decision maker for the first time in the context of evolutionary computation we apply the choquet integral as a user s preference model allowing us to capture interactions between objectives as there is a trade off between the flexibility of the value function model and the complexity of learning a faithful model of user s preferences we propose to start the interactive process with a simple linear model but then to switch to the choquet integral as soon as the preference information can no longer be represented using the linear model an experimental analysis demonstrates the effectiveness of the approach

on the f divergence for non additive measures the f divergence evaluates the dissimilarity between two probability distributions defined in terms of the radon nikodym derivative of these two probabilities the f divergence generalizes the hellinger distance and the kullback leibler divergence among other divergence functions in this paper we define an analogous function for non additive measures we discuss them for distorted lebesgue measures and give examples examples focus on the hellinger distance

estimation of the weights of interacting criteria from the set of profiles by means of information theoretic functionals abstract following the recent work of marichal and roubens eur j operat res 124 2000 641 on fuzzy measure identification in multicriteria decision problems we propose an alternative unsupervised identification method based on the estimation of the fuzzy measure coefficients by means of information theoretic functionals

compromise principle based methods of identifying capacities in the framework of multicriteria decision analysis abstract the basic aim of the compromise principle employed in this paper is to seek the capacity by which each alternative has a relatively equal chance to reach as close as possible to its highest reachable overall evaluation according to the compromise principle three types of capacity identification methods the simple arithmetic average based compromise method the least squares based compromise method and the linear programming based compromise method are proposed the input information required for the compromise principle based identification methods consists of the preference information with respect to the decision criteria and the partial evaluations of the alternatives provided by the decision maker an illustrative example is given to show the processes of the proposed methods and a comparison analysis with the maximum entropy principle based identification method is also presented

an axiomatic approach to the definition of the entropy of a discrete choquet capacity to extend the classical shannon entropy to nonadditive measures marichal recently introduced the concept of generalized entropy for discrete choquet capacities we provide a first axiomatization of this new concept on the basis of three axioms the symmetry property a boundary condition for which the entropy reduces to the shannon entropy and a generalized version of the well known recursivity property we also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy like measure lastly we provide an interpretation of it in the framework of aggregation by the discrete choquet integral

fuzzy choquet integration of homogeneous possibility and probability distributions the fuzzy integral fi is an extremely flexible and powerful tool for data and information aggregation the fi is parametrized by the fuzzy measure fm a normal and monotone capacity based on the selection of fm the fi produces different aggregation operators in recent years a number of fi extensions have been put forth relative to different types of uncertain information e g real interval and set valued under various constraints herein we study the applicability and behavior of different extensions of the fuzzy choquet integral for fusing homogeneous possibility and probability distributions this analysis is of great utility in terms of understanding what extensions and under what conditions it is possible to aggregate and maintain homogeneity within uncertain information we show that two extensions gfi and ndfi can aggregate both probability and possibility distributions while these extensions do not always maintain homogeneity they do under certain conditions last while we specifically focus on the aggregation of homogeneous uncertain information the propositions put forth also shed light into heterogeneous information aggregation via the gfi and the ndfi

on the dispersion measure of owa operators we describe some basic features of the owa operator we turn to the problem of determining the weights associated with this operator and particularly the maximal dispersion entropy approach we consider the possibility of using minimization of dispersion after discussing concerns with both maximization and minimization of dispersion we investigate the possibility of finding an optimal solution intermediate to these extremes we next consider alternative measures of dispersion we introduce a fundamental requirement for a measure of dispersion called the preference for equal division a number of general classes of dispersion measures are provided notable among these are those based on t norm and t conorm operators

determination of weights of interacting criteria from a reference set abstract in this paper we present a model allowing to determine the weights related to interacting criteria this is done on the basis of the knowledge of a partial ranking over a reference set of alternatives prototypes a partial ranking over the set of criteria and a partial ranking over the set of interactions between pairs of criteria

a multispecies birth death immigration process and its diffusion approximation abstract we consider an extended birth death immigration process defined on a lattice formed by the integers of d semiaxes joined at the origin when the process reaches the origin then it may jump toward any semiaxis with the same rate the dynamics on each ray evolves according to a one dimensional linear birth death process with immigration we investigate the transient and asymptotic behavior of the process via its probability generating function the stationary distribution when existing is a zero modified negative binomial distribution we also study a diffusive approximation of the process which involves a diffusion process with linear drift and infinitesimal variance on each ray it possesses a gamma type transient density admitting a stationary limit as a byproduct of our study we obtain a closed form of the number of permutations with a fixed number of components and a new series form of the polylogarithm function expressed in terms of the gauss hypergeometric function

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra

on the issue of obtaining owa operator weights we first investigate the issue of obtaining the weights associated with the owa aggregation in the situation when we have observed data on the arguments and the aggregated value we next introduce a family of owa operators called exponential owa operators finally we look at a simple procedure for generating the weights given a required degree of orness

fixed points and homology of superelliptic jacobians let eta c f n rightarrow mathbb p 1 be a cyclic cover of mathbb p 1 of degree n which is totally and tamely ramified for all the ramification points we determine the group of fixed points of the cyclic covering group mathrm aut eta simeq mathbb z n mathbb z acting on the jacobian j n mathrm jac c f n for each prime ell distinct from the characteristic of the base field the tate module t ell j n is shown to be a free module over the ring mathbb z ell t sum i 0 n 1 t i we also study the subvarieties of j n and calculate the degree of the induced polarization on the new part j n mathrm new of the jacobian

assessing a set of additive utility functions for multicriteria decision making the uta method abstract the purpose of the method presented in this paper is to assess additive utility functions which aggregate multiple criteria in a composite criterion using the information given by a subjective ranking on a set of stimuli or actions weak order comparison judgments and the multicriteria evaluations of these actions it is an ordinal regression method using linear programming to estimate the parameters of the utility function stability and sensitivity analysis leads to the assessment of a set of utility functions by means of post optimality analysis techniques in linear programming finally a simple illustrative example is presented and some extensions of the method are proposed

endomorphisms of superelliptic jacobians let k be a field of characteristic zero n 5 an integer f x an irreducible polynomial over k of degree n whose galois group contains a doubly transitive simple non abelian group let p be an odd prime mathbb z zeta p the ring of integers in the pth cyclotomic field cf p yp f x the corresponding superelliptic curve and j cf p its jacobian assuming that either n p 1 or p does not divide n n 1 we prove that the ring of all endomorphisms of j cf p coincides with mathbb z zeta p the same is true if n 4 the galois group of f x is the full symmetric group s4 and k contains a primitive pth root of unity

the combination of evidence in the transferable belief model a description of the transferable belief model which is used to quantify degrees of belief based on belief functions is given the impact of open and closed world assumption on conditioning is discussed the nature of the frame of discernment on which a degree of belief will be established is discussed a set of axioms justifying dempster s rule for the combination of belief functions induced by two distinct evidences is presented

on equivalence classes of fuzzy connectives the case of fuzzy integrals an equivalence relation between connectives or operators in the framework of multiple criteria decision making is introduced two operators are said to be equivalent if they lead to the same ranking of alternatives some general results to find equivalence classes of operators are given through the concept of level surfaces in a second part these results are applied to the case of discrete fuzzy integrals which are considered here as n place operators first some general results on the situation of fuzzy integrals among fuzzy operators are given and then the equivalence classes of fuzzy integrals

on the extension of pseudo boolean functions for the aggregation of interacting criteria abstract the paper presents an analysis on the use of integrals defined for non additive measures or capacities as the choquet and the sipos integral and the multilinear model all seen as extensions of pseudo boolean functions and used as a means to model interaction between criteria in a multicriteria decision making problem the emphasis is put on the use besides classical comparative information of information about difference of attractiveness between acts and on the existence for each point of view of a neutral level allowing to introduce the absolute notion of attractive or repulsive act it is shown that in this case the sipos integral is a suitable solution although not unique properties of the sipos integral as a new way of aggregating criteria are shown with emphasis on the interaction among criteria

assessing non additive utility for multicriteria decision aid abstract in the framework of multi attribute utility theory maut several methods have been proposed to build a decision maker s dm utility function representing his her preferences among such methods the uta method infers an additive utility function from a set of exemplary decisions using linear programming however the uta method does not guarantee to find a utility function which is coherent with the available information this drawback is due to the underlying utility model of uta viz the additive one which does not allow to include additional information such as an interaction among criteria in this paper we present a methodology for building a non additive utility function in the framework of the so called fuzzy integrals which permits to model preference structures with interaction between criteria like in the uta method we aim at searching a utility function representing the dm s preferences but unlike uta the functional form is a specific fuzzy integral choquet integral as a result we obtain weights which can be interpreted as the importance of coalitions of criteria exploiting the potential interaction between criteria as already proposed by other authors however within the same framework we obtain also the marginal utility functions relative to each one of the considered criteria that are evaluated on a common scale as a consequence of the implemented methodology finally we illustrate our approach with an example

k intolerant capacities and choquet integrals we define an aggregation function to be at most k intolerant if it is bounded from above by its kth lowest input value applying this definition to the discrete choquet integral and its underlying capacity we introduce the concept of k intolerant capacities which when varying k from 1 to n cover all the possible capacities on n objects just as the concepts of k additive capacities and p symmetric capacities have been previously introduced essentially to overcome the problem of computational complexity of capacities k intolerant capacities are proposed here for the same purpose but also for dealing with intolerant or tolerant behaviors of aggregation we also introduce axiomatically indices to appraise the extent to which a given capacity is k intolerant and we apply them on a particular recruiting problem

indicators for the characterization of discrete choquet integrals ordered weighted averaging owa operators and their extensions are powerful tools used in numerous decision making problems this class of operator belongs to a more general family of aggregation operators where each operator of this family is understood as a discrete choquet integral aggregation operators are usually characterized by indicators in this article four indicators usually associated with the owa operator are extended to the discrete choquet integral namely the degree of balance the divergence the variance indicator and renyi entropies all of these indicators are considered from a local and a global perspective linearity of indicators for linear combinations of capacities is investigated and to illustrate the usefulness of results indicators of the probabilistic ordered weighted averaging powa operator are derived finally a numerical example is provided to discuss the results in a specific context

entropy of capacities on lattices and set systems we propose a definition for the entropy of capacities defined on lattices classical capacities are monotone set functions and can be seen as a generalization of probability measures capacities on lattices address the general case where the family of subsets is not necessarily the boolean lattice of all subsets our definition encompasses the classical definition of shannon for probability measures as well as the entropy of marichal defined for classical capacities some properties and examples are given

equivalent representations of set functions this paper introduces four alternative representations of a set function the ma bius transformation the co ma bius transformation and the interactions between elements of any subset of a given set as extensions of shapley and banzhaf values the links between the five equivalent representations of a set function are emphasized in this presentation

laguerre population processes we analyse a number of stochastic processes that give rise to first order number statistics governed by laguerre distributions with properties that lie between poisson and geometric random variables these distributions have hitherto been used to characterize the photon statistics of a coherent mixture of thermal and laser light here we explore a number of discrete population processes that can lead to the same first order statistics and correlation functions and highlight the distinguishing features that may be used to identify the relevant model from data

stochastic multiobjective acceptability analysis for the choquet integral preference model and the scale construction problem abstract the choquet integral preference model is adopted in multiple criteria decision aiding mcda to deal with interactions between criteria while the stochastic multiobjective acceptability analysis smaa is an mcda methodology considered to take into account uncertainty or imprecision on the considered data and preference parameters in this paper we propose to combine the choquet integral preference model with the smaa methodology in order to get robust recommendations taking into account all parameters compatible with the preference information provided by the decision maker dm in case the criteria are on a common scale one has to elicit only a set of non additive weights technically a capacity compatible with the dm s preference information instead if the criteria are on different scales besides the capacity one has to elicit also a common scale compatible with the preferences given by the dm our approach permits to explore the whole space of capacities and common scales compatible with the dm s preference information

entropy of bi capacities in the context of multicriteria decision making whose aggregation process is based on the choquet integral bi capacities can be regarded as a natural extension of capacities when the underlying evaluation scale is bipolar the notion of entropy recently generalized to capacities to measure their uniformity is now extended to bi capacities we show that the resulting entropy measure has a very natural interpretation in terms of the choquet integral and satisfies many natural properties that one would expect from an entropy measure

a method for determining the weights of criteria the centralized weights abstract in multicriteria analysis many methods use weights to represent the relative importance of criteria the noncompensatory aggregation procedure used in outranking methods enables comparisons of subsets of criteria from this point of view an interactive method is presented which requests only ordinal comparisons from the decision maker the relation more important than is assumed to be a semiorder therefore the indifference part of this relation is not necessarily transitive the judgements can be formulated as linear inequalities which constrain the set of feasible weights and threshold values all vertices of this polyhedron are determined the threshold and the unknown real weights are estimated by the centroid some characteristics a possible refinement in the estimation and an illustrative example are also given

random generation of k interactive capacities abstract the theory of capacities provides powerful formal methodology to account for criteria dependencies in multiple criteria decision problems the discrete choquet and sugeno integrals aggregate criteria valuations accounting for criteria synergies and redundancies we address an important problem of randomly generating capacities of special classes for simulation studies and for capacity learning through evolutionary algorithms we discuss two efficient methods suitable for k interactive capacities the results are supported by the extensive numerical evidence and provide a useful tool for large scale simulations

minimum variance capacity identification abstract in the framework of multi criteria decision making whose aggregation process is based on the choquet integral we present a maximum entropy like method enabling to determine if it exists the least specific capacity compatible with the initial preferences of the decision maker the proposed approach consists in solving a strictly convex quadratic program whose objective function is equivalently either the opposite of a generalized entropy measure or the variance of the capacity the application of the proposed approach is illustrated on two examples

a note on the sobol indices and interactive criteria the choquet integral and the owen extension or multilinear extension are the most popular tools in multicriteria decision making to take into account the interaction between criteria it is known that the interaction transform and the banzhaf interaction transform arise as the average total variation of the choquet integral and multilinear extension respectively we consider in this note another approach to define interaction by using the sobol indices which are related to the analysis of variance of a multivariate model we prove that the sobol indices of the multilinear extension give the square of the fourier transform a well known concept in computer sciences we also relate the latter to the banzhaf interaction transform and compute the sobol indices for the 2 additive choquet integral

tolerant or intolerant character of interacting criteria in aggregation by the choquet integral abstract in many multi criteria decision making problems the decision criteria present some interaction whose nature may vary from one situation to another for example some criteria may be statistically correlated thus making them somewhat redundant or opposed some others may be somewhat substitutive or complementary depending on the behavior of the decision maker some others may be decisive in the sense that the global score of any alternative obtained by aggregation is bounded by the partial score along one of them in this paper we analyze this latter form of interaction when a criterion bounds the global score from above it is called a blocker or a veto due to its rather intolerant character when it bounds the global score from below it is then called a pusher or a favor we thus investigate the tolerance of criteria or equivalently the tolerance of the weighted aggregation operator here the choquet integral that is used to aggregate criteria more specifically we propose axiomatically indices to appraise the extent to which each criterion behaves like a veto or a favor in the aggregation by the choquet integral previous to this we also propose global tolerance degrees measuring the extent to which the choquet integral is conjunctive or disjunctive

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

an axiomatization of entropy of capacities on set systems we present an axiomatization of the entropy of capacities defined on set systems which are not necessarily the whole power set but satisfy a condition of regularity this entropy encompasses the definition of marichal and roubens for the entropy of capacities its axiomatization is in the spirit of the one of faddeev for the classical shannon entropy in addition we present also an axiomatization of the entropy for capacities proposed by dukhovny

aggregation operators in information retrieval abstract information retrieval is a complex task which deals with both the subjectivity related to the user s needs and the uncertainty and vagueness that characterize the retrieval process for this reason deciding to which extent a document is relevant to a user s needs is not easy and it strongly depends on several dimensions such as topicality novelty user s context and so on one of the most straightforward ways to interpret this activity is as a multi criteria decision making mcdm problem in which the choice of appropriate aggregation operators can play an important role in various tasks related to or characterizing the ir process this article aims to provide a presentation of the main approaches that in the literature have made use of aggregation operators in information retrieval

fuzzy integral based information fusion for classification of highly confusable non speech sounds acoustic event classification may help to describe acoustic scenes and contribute to improve the robustness of speech technologies in this work fusion of different information sources with the fuzzy integral fi and the associated fuzzy measure fm are applied to the problem of classifying a small set of highly confusable human non speech sounds as fi is a meaningful formalism for combining classifier outputs that can capture interactions among the various sources of information it shows in our experiments a significantly better performance than that of any single classifier entering the fi fusion module actually that fi decision level fusion approach shows comparable results to the high performing svm feature level fusion and thus it seems to be a good choice when feature level fusion is not an option we have also observed that the importance and the degree of interaction among the various feature types given by the fm can be used for feature selection and gives a valuable insight into the problem

on the entropy of fuzzy measures fuzzy measures provides a structure for modeling the knowledge available about variables whose values are unknown and uncertain a large class of different types of uncertainty can be represented in this framework in this work we provide a measure of entropy that can be used to calculate the amount of uncertainty associated with a fuzzy measure

determining the fuzzy measures in multiple criteria decision aiding from the tolerance perspective abstract we consider multiple criteria decision aiding mcda in the case of interactions between criteria in dealing with interactions between criteria fuzzy measures and integrals have demonstrated great advantages nevertheless the determination of fuzzy measures has proven difficult because the capacities of not only single criterion but also all subsets of criteria need to be identified due to the value judgment essence of mcda the attitudes of the decision maker dm are typically modeled to identify fuzzy measures in this paper the tolerance attitudes of the dm which implies a direct requirement instead of partial preference are modeled with regard to the determination of fuzzy measures for the first time with two scales developed in this paper the dm can directly express the tolerance attitudes to certain criteria other than providing partial preference through pairwise comparison as a result it requires less prior knowledge and is more efficient to some extent further the inherent interacting mechanism of criteria under different tolerance attitudes is explored at last the tolerance attitudes are applied to the process of multiple criteria analysis using a choquet integral a classic student evaluation problem is given as an example the evaluation results are compared with additive models this paper not only provides a new inspiration to the determination of fuzzy measures but also improves the descriptive capacity of fuzzy measures to the real world

the midweight method to measure attitudes toward risk and ambiguity this paper introduces a parameter free method for measuring the weighting functions of prospect theory and rank dependent utility these weighting functions capture risk attitudes subjective beliefs and ambiguity attitudes our method called the midweight method is based on a convenient way to obtain midpoints in the weighting function scale it can be used both for risk known probabilities and for uncertainty unknown probabilities the resulting integrated treatment of risk and uncertainty is particularly useful for measuring ambiguity i e the difference between uncertainty and risk compared to existing methods to measure weighting functions and attitudes toward uncertainty and ambiguity our method is more efficient and can accommodate violations of expected utility under risk an experiment demonstrates the tractability of our method yielding plausible results such as ambiguity aversion for moderate and high likelihoods but ambiguity seeking for low likelihoods as predicted by ellsberg this paper was accepted by george wu decision analysis

robust ordinal regression for value functions handling interacting criteria we present a new method called utagms int for ranking a finite set of alternatives evaluated on multiple criteria it belongs to the family of robust ordinal regression ror methods which build a set of preference models compatible with preference information elicited by the decision maker dm the preference model used by utagms int is a general additive value function augmented by two types of components corresponding to bonus or penalty values for positively or negatively interacting pairs of criteria respectively when calculating value of a particular alternative a bonus is added to the additive component of the value function if a given pair of criteria is in a positive synergy for performances of this alternative on the two criteria similarly a penalty is subtracted from the additive component of the value function if a given pair of criteria is in a negative synergy for performances of the considered alternative on the two criteria the preference information elicited by the dm is composed of pairwise comparisons of some reference alternatives as well as of comparisons of some pairs of reference alternatives with respect to intensity of preference either comprehensively or on a particular criterion in utagms int ror starts with identification of pairs of interacting criteria for given preference information by solving a mixed integer linear program once the interacting pairs are validated by the dm ror continues calculations with the whole set of compatible value functions handling the interacting criteria to get necessary and possible preference relations in the considered set of alternatives a single representative value function can be calculated to attribute specific scores to alternatives it also gives values to bonuses and penalties utagms int handles quite general interactions among criteria and provides an interesting alternative to the choquet integral

learning k maxitive fuzzy measures from data by mixed integer programming abstract fuzzy measures model interactions between the inputs in aggregation problems their complexity grows exponentially with the dimensionality of the problem and elicitation of fuzzy measure coefficients either from domain experts or from empirical data poses a significant challenge the notions of k additivity and k maxitivity simplify the fuzzy measures by limiting interactions to subsets of up to k elements learning fuzzy measures from data is an important elicitation technique which relies on solving an optimisation problem a heuristic learning algorithm to identify k maxitive fuzzy measures from the data on the basis of hlms heuristic least mean squares was recently presented in murillo et al 2017 11 we present an alternative formulation of the fitting problem which delivers a globally optimal solution through the solution of a mixed integer programming mip problem to deal with high computational cost of mip in moderate to large dimensions we also propose a simple mip relaxation technique which involves solving two related linear programming problems we also provide a linear programming formulation for fitting k tolerant fuzzy measures we discuss implementations of the fitting methods and present the results of numerical experiments

entropy of discrete choquet capacities we introduce a measure of entropy for any discrete choquet capacity and we interpret it in the setting of aggregation by the choquet integral

on the moments and distribution of discrete choquet integrals from continuous distributions we study the moments and the distribution of the discrete choquet integral when regarded as a real function of a random sample drawn from a continuous distribution since the discrete choquet integral includes weighted arithmetic means ordered weighted averaging functions and lattice polynomial functions as particular cases our results encompass the corresponding results for these aggregation functions after detailing the results obtained in j l marichal i kojadinovic distribution functions of linear combinations of lattice polynomials from the uniform distribution statistics probability letters 78 2008 985 991 in the uniform case we present results for the standard exponential case show how approximations of the moments can be obtained for other continuous distributions such as the standard normal and elaborate on the asymptotic distribution of the choquet integral the results presented in this work can be used to improve the interpretation of discrete choquet integrals when employed as aggregation functions

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

preference learning using the choquet integral the case of multipartite ranking we propose a novel method for preference learning or more specifically learning to rank where the task is to learn a ranking model that takes a subset of alternatives as input and produces a ranking of these alternatives as output just like in the case of conventional classifier learning training information is provided in the form of a set of labeled instances with labels or say preference degrees taken from an ordered categorical scale this setting is known as multipartite ranking in the literature our approach is based on the idea of using the discrete choquet integral as an underlying model for representing ranking functions being an established aggregation function in fields such as multiple criteria decision making and information fusion the choquet integral offers a number of interesting properties that make it attractive from a machine learning perspective too the learning problem itself comes down to properly specifying the fuzzy measure on which the choquet integral is defined this problem is formalized as a margin maximization problem and solved by means of a cutting plane algorithm the performance of our method is tested on a number of benchmark datasets

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

2 additive capacity identification methods from multicriteria correlation preference information the essential role of the particular families of capacities and the capacity identification methods is to help the decision maker to deal with the exponential complexity inherent in the construction process of the capacity the 2 additive capacities appear to be the most popular among the particular families of capacities since they permit to model interactions between criteria while preserving simplicity besides the preference with respect to the decision criteria most of the capacity identification methods also need to provide the desired overall evaluations of the decision alternatives in the learning set which is a time consuming task for the decision maker in this paper we propose some models to identify 2 additive capacities only from a kind of refined preference information with respect to the decision criteria called the multicriteria correlation preference information mccpi the mccpi is a group of 2 d preference information which can be described and obtained by the refined diamond diagram the common principle of the proposed models is to minimize the different kinds of deviations between the mccpi and the most desired 2 additive capacity ies a multicriteria decision making example is presented to show the feasibility of the proposed methods and a 2 d scale of the mccpi is also introduced in the further discussion of the illustrative example

an analysis of the fixation probability of a mutant on special classes of non directed graphs there is a growing interest in the study of evolutionary dynamics on populations with some non homogeneous structure in this paper we follow the model of lieberman et al lieberman et al 2005 na

k additivity and c decomposability of bi capacities and its integral k additivity is a convenient way to have less complex bi capacities this paper gives a new characterization of k additivity introduced by grabisch and labreuche of bi capacities and contrasts between the existing characterization and the new one that differs from the one of saminger and mesiar besides in the same way for capacities a concept of c decomposability distinct from the proposal of saminger and mesiar but closely linked to k additivity is introduced for bi capacities moreover the concept of c decomposability applies to the choquet integral with respect to bi capacities

on policy capturing with fuzzy measures policy capturing methods generally apply linear regression analysis to model human judgment in this paper we examine the application of fuzzy set and fuzzy measure theories to obtain subjective descriptions of cue importance for policy capturing at the heart of the approach is a method of learning fuzzy measures the shapley values associated with the fuzzy measures provide a basis for comparison with the results of linear regression however the fuzzy measure theoretical approach provides additional insight into interaction effects corresponding to the nonlinear noncompensatory nature of the underlying decision model to illustrate the methodology we estimated the importance of factors and the interactions among them that influence decisions related to strategic investments in telecommunications infrastructure and compared the results from the fuzzy approach to those obtained from traditional statistical methods

elicitation of multiattribute value functions through high dimensional model representations monotonicity and interactions abstract this work addresses the early phases of the elicitation of multiattribute value functions proposing a practical method for assessing interactions and monotonicity we exploit the link between multiattribute value functions and the theory of high dimensional model representations the resulting elicitation method does not state any a priori assumption on an individual s preference structure we test the approach via an experiment in a riskless context in which subjects are asked to evaluate mobile phone packages that differ on three attributes

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

a review of methods for capacity identification in choquet integral based multi attribute utility theory applications of the kappalab r package the application of multi attribute utility theory whose aggregation process is based on the choquet integral requires the prior identification of a capacity the main approaches to capacity identification proposed in the literature are reviewed and their advantages and inconveniences are discussed all the reviewed methods have been implemented within the kappalab r package their application is illustrated on a detailed example

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

learning choquet integral based metrics for semisupervised clustering we consider an application of fuzzy measures to the problem of metric learning in semisupervised clustering we investigate the necessary and sufficient conditions on the underlying fuzzy measure that make the discrete choquet integral suitable for defining a metric as a byproduct we can obtain the analogous conditions for the ordered weighted averaging owa operators which constitute a special case we then generalize these results for power based choquet and owa operators we show that this metric learning problem can be formulated as a linear programming problem and specify the required sets of linear constraints we present the results of numerical experiments on artificial and real world datasets which illustrate the potential usefulness and limitations of this construction

non additive robust ordinal regression a multiple criteria decision model based on the choquet integral abstract within the multicriteria aggregation disaggregation framework ordinal regression aims at inducing the parameters of a decision model for example those of a utility function which have to represent some holistic preference comparisons of a decision maker dm usually among the many utility functions representing the dm s preference information only one utility function is selected since such a choice is arbitrary to some extent recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the dm s preference information until now robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence in this paper we propose a non additive robust ordinal regression on a set of alternatives a whose utility is evaluated in terms of the choquet integral which permits to represent the interaction among criteria modelled by the fuzzy measures parameterizing our approach in our methodology besides holistic pairwise preference comparisons of alternatives from a subset of reference alternatives a a the dm is also requested to express the intensity of preference on pairs of alternatives from a and to supply pairwise comparisons on the importance of criteria and the sign and intensity of interaction among pairs of criteria the output of our approach defines a set of fuzzy measures capacities such that the corresponding choquet integral is compatible with the dm s preference information moreover using linear programming our decision model establishes two preference relations for any a b a the necessary weak preference relation if for all compatible fuzzy measures the utility of a is not smaller than the utility of b and the possible weak preference relation if for at least one compatible fuzzy measure the utility of a is not smaller than the utility of b

an axiomatic approach of the discrete choquet integral as a tool to aggregate interacting criteria the most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean however in many problems the criteria considered interact and a substitute to the weighted arithmetic mean has to be adopted we show that under rather natural conditions the discrete choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by taking into consideration of the interaction among criteria the axiomatic that supports the choquet integral is presented and an intuitive approach is proposed as well

an axiomatic approach of the discrete sugeno integral as a tool to aggregate interacting criteria in a qualitative framework we present a model allowing us to aggregate decision criteria when the available information is of a qualitative nature the use of the sugeno integral as an aggregation function is justified by an axiomatic approach it is also shown that the mutual preferential independence of criteria reduces the sugeno integral to a dictatorial aggregation

the choquet integral for the aggregation of interval scales in multicriteria decision making this paper addresses the question of which models fit with information concerning the preferences of the decision maker over each attribute and his preferences about aggregation of criteria interacting criteria we show that the conditions induced by these information plus some intuitive conditions lead to a unique possible aggregation operator the choquet integral

decision aid and decision making the objective of multiple criteria approaches is to help us to make better decisions but what is the meaning of better this field has developed considerably in the past twenty years the development makes us fully aware of the limitations on objectivity encountered in the field of decision aid and consequently of the virtual impossibility of providing a truly scientific foundation for an optimal decision work carried out under the rubric of mcdm multiple criteria decision making bases its claims to legitimacy on a framework in which these limitations are left aside multiple criteria decision aid mcda must be seen from a different perspective its aim is above all to enable us to enhance the degree of conformity and coherence between the evolution of a decision making process and the value systems and objectives of those involved in this process the purpose of decision aid is therefore to help us make our way in the presence of ambiguity uncertainty and an abundance of bifurcations we shall analyse multiple criteria concepts models and procedures from both these viewpoints

robust optimization of the choquet integral we study the problem of the choquet integral maximization for the case when preferences of the decision maker do not define a unique capacity but rather some convex set we introduce a robust version of the problem based on the minimax regret criterion and construct a global optimization algorithm it is shown that the decision rule based on regret minimization over a set of capacities is equivalent to maximization of the choquet integral with respect to a certain capacity belonging to this set we compare our method to various capacity identification approaches developed in the literature and provide a semantic interpretation in terms of information value

evidential weights of multiple preferences for competitiveness the alternative selections or feasible sets in multi attribute utility theory usually face problems of inaccuracy or inconsistency in weighting evidential weight and weight of evidence still have not resolved these problems in an efficient manner this research proposes evidential weights of multiple preferences by using evidence and inference based on the rough set theory an evidential probability is constructed to resolve this weight technically the evidential probability is required to satisfy the following conditions uniqueness independence and consistency to identify various evidence and noise and to provide quantities of certain evidence to resolve weights empirically the resolved weights are fused into competitiveness utility for verification a practical case successfully classifies dominating nations of world competitiveness yearbook another case analyzes the competitiveness trends of china during 1997 to 2012 predicting china will continue rising in near future

data driven compression and efficient learning of the choquet integral the choquet integral chi is a parametric nonlinear aggregation function defined with respect to the fuzzy measure fm to date application of the chi has sadly been restricted to problems with relatively few numbers of inputs primarily as the fm has 2 n variables for n inputs and n 2 n 1 1 monotonicity constraints in return the community has turned to density based imputation e g sugeno lambda fm or the number of interactions fm variables are restricted e g k additivity herein we propose a new scalable data driven way to represent and learn the chi making learning computationally manageable for larger n first data supported variables are identified and used in optimization identification of these variables also allows us recognize future ill posed fusion scenarios chis involving variable subsets not supported by data second we outline an imputation function framework to address data unsupported variables third we present a lossless way to compress redundant variables and associated monotonicity constraints finally we outline a lossy approximation method to further compress the chi if when desired computational complexity analysis and experiments conducted on synthetic datasets with known fms demonstrate the effectiveness and efficiency of the proposed theory

applying young diagrams to 2 symmetric fuzzy measures with an application to general fuzzy measures abstract in this paper we apply young diagrams a well known object appearing in combinatorics and group representation theory to study some properties of the polytope of 2 symmetric fuzzy measures with respect to a given partition a subpolytope of the set of fuzzy measures the main result in the paper allows to build a simple and fast algorithm for generating points on this polytope in a random fashion besides we also study some other properties of this polytope as for example its volume in the last section we give an application of this result to the problem of identification of general fuzzy measures

a practical approach for estimating weights of interacting criteria from profile sets aggregating multi criteria data is an important problem with many applications commonly used additive aggregation methods such as the weighted arithmetic mean cannot account for the criteria interactions encountered in many practical multi criteria decision problems the choquet integral is a suitable aggregation operator in the presence of interacting criteria it replaces the weight vector with a fuzzy measure that models the importance of each subset or coalition of criteria rather than just the importance of individual criteria however estimating the fuzzy measures in practice has been problematic conventional approaches are cognitively challenging for decision makers while more recent approaches suffer from prohibitive data requirements in this paper we present a formulation for the weights of the choquet integral that uses principal component analysis to account for criteria interaction this novel unsupervised approach to estimating the required fuzzy measures overcomes the limitations of other methods the approach is applied to two case studies in environmental and sustainability analysis and the results are compared with those of the weighted arithmetic mean the first case is a triple bottom line analysis of 135 australian industry sectors evaluated against 11 criteria while the second case is an environmental life cycle assessment of 8 alternative biosolids management options evaluated against 5 criteria these examples demonstrate the ability of the proposed approach to account for criteria interaction in these and other decision contexts

an interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy measure abstract in this paper the non additivity of sugeno s fuzzy measures is interpreted in terms of addition and the rationality of the choquet integral is discussed it is pointed out that a fuzzy measure on a set x expresses the interaction between the subsets of x and can be represented by an additive measure it is shown through concrete examples that the choquet integral is reasonable as an integral with respect to a fuzzy measure it is also found that the choquet integral is closely related with the representation of a fuzzy measure

inconsistency and non additive capacities the analytic hierarchy process in the framework of choquet integration we examine the ahp in the framework of choquet integration and we propose an extension of the standard ahp aggregation scheme on the basis of the shapley values associated with the criteria in our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the ahp the measure of dominance inconsistency is then used to construct a non additive capacity whose associated shapley values reduce to the standard ahp priority weights in the consistency case in the general inconsistency case however the extended aggregation scheme based on the shapley weighted mean tends to attenuate resp emphasize the priority weights of the criteria with higher resp lower average dominance inconsistency with respect to the other criteria

functional anova ultramodularity and monotonicity applications in multiattribute utility theory abstract utility function properties as monotonicity and concavity play a fundamental role in reflecting a decision maker s preference structure these properties are usually characterized via partial derivatives however elicitation methods do not necessarily lead to twice differentiable utility functions furthermore while in a single attribute context concavity fully reflects risk aversion in multiattribute problems such correspondence is not one to one we show that tsetlin and winkler s multivariate risk attitudes imply ultramodularity of the utility function we demonstrate that geometric properties of a multivariate utility function can be successfully studied by utilizing an integral function expansion functional anova the necessary and sufficient conditions under which monotonicity and or ultramodularity of single attribute functions imply the monotonicity and or ultramodularity of the corresponding multiattribute function under additive preferential and mutual utility independence are then established without reliance on the utility function differentiability we also investigate the relationship between the presence of interactions among the attributes of a multiattribute utility function and the decision maker s multivariate risk attitudes

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

effects of imprecise weighting in hierarchical preference programming abstract preference programming is a general term for multi criteria decision analytical approaches allowing incomplete preference information in the pairs method interval judgments are assigned to weight ratios between attributes to model imprecision in multi attribute value trees this paper studies the effects of a hierarchical model structure on the overall imprecision as the form of the hierarchy also affects the form of imprecision that can be assigned to the model the aim is to find out good procedural practices for reducing overall imprecision descending inherently from the model structure the study provides simulation results about the ability of various weighting schemes to identify dominated alternatives which are discussed with respect to other issues related to the weighting process according to the results a hierarchical model is structurally somewhat more unable to identify dominances than a corresponding nonhierarchical model but its cognitive advantages often cancel out this the results also suggest paying reasonable attention to the precision of the lower level judgments and to identifying possible correlations between the criteria

the problem of collective identity in a fuzzy environment we define the problem of group identification in a fuzzy environment we concentrate on the case where the society is required to self determine the belongingness of each member to a specific group characterized by a single attribute in general terms this case consists of a collective identity issue that can be regarded as an aggregation problem of individual assessments within a group here we introduce the possibility that both the original assessments and the resulting output attach partial memberships to the collectivity for each potential member we propose relevant classes of rules and some are axiomatically characterized our new approach provides a way to circumvent classical impossibility results like kasher and rubinstein s

the choquet integral with respect to a level dependent capacity we present a generalization of choquet integral in which the capacity depends also on the value of the aggregated variables we show that as particular cases of our generalization of choquet integral there are the sugeno integral the sipos integral and the cumulative prospect theory functional we also show that many concepts such as mobius transform importance index interaction index k order capacities and owa operators introduced in the research about choquet integral can be generalized in the considered context

set characterization selection towards classification based on interaction index in many real world datasets both the individual and coordinated action of features may be relevant for class identification in this paper a computational strategy for relevant feature selection based on the characterization of redundant or complementary features is proposed the characterization is achieved using fuzzy measures and an interaction index computed from fuzzy measure coefficients fuzzy measure identification requires raw data to be turned into confidence degrees this key step is carried out considering the distributions of feature values across all the classes fuzzy measure coefficients are then estimated with an improved version of the heuristic least mean squares algorithm that includes an efficient management of untouched coefficients then a generalization of the shapley index for an arbitrary number of features is used simulations experiments on synthetic datasets are performed to study the behavior of this generalized interaction index for extreme datasets containing either redundant or complementary features as well as noise the index value is defined by mathematical formula this result is used to motivate feature selection guidelines that take into account feature interactions experimental results on benchmark datasets show that the proposal allows for the design of compact interpretable and competitive classification models