Choquet 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.

We present a new decision making model with distance measures by using linguistic aggregation operators. We introduce a new aggregation operator called the linguistic ordered weighted averaging distance (LOWAD) operator. This aggregation operator provides a parameterized family of linguistic aggregation operators that includes the maximum distance, the minimum distance, the linguistic normalized Hamming distance and the linguistic weighted Hamming distance, among others. We study some of its main properties and different families of LOWAD operators such as the median-LOWAD, the Olympic-LOWAD, the S-LOWAD and the centered-LOWAD. We also develop an application of the new approach in a decision making problem concerning human resource management.

This article presents an overview of the recent development on mathematical treatment of behavioural finance, primarily in the setting of continuous-time portfolio choice under the cumulative prospect theory. Financial motivations and mathematical challenges of the problem are highlighted. It is demonstrated that the solutions to the problem have in turn led to new financial and mathematical problems and machineries.

Hesitancy is the most common problem in decision making, for which hesitant fuzzy set can be considered as a suitable means allowing several possible degrees for an element to a set. In this paper, we study the aggregation of the hesitancy fuzzy information. Several series of aggregation operators are proposed and the connections of them are discussed. To reflect the correlation of the aggregation arguments, two methods are proposed to determine the aggregation weight vectors. Based on the support degrees among aggregation arguments, the weight vector of decision makers are obtained more objectively. To deal with the correlation of criteria, we apply the Choquet integral to get the weights of criteria. A method is also proposed for group decision making under hesitant fuzzy environment.

Condition-based maintenance is nowadays considered as a key-process in maintenance strategies and prognostics appears to be a very promising activity as it should permit to not engage inopportune spending. Various approaches have been developed and data-driven methods are increasingly applied. The training step of these methods generally requires huge datasets since a lot of methods rely on probability theory and/or on artificial neural networks. This step is thus time-consuming and generally made in batch mode which can be restrictive in practical application when few data are available. A method for prognostics is proposed to face up this problem of lack of information and missing prior knowledge. The approach is based on the integration of three complementary modules and aims at predicting the failure mode early while the system can switch between several functioning modes. The three modules are: 1) observation selection based on information theory and Choquet Integral, 2) prediction relying on an evolving real-time neuro-fuzzy system and 3) classification into one of the possible functioning modes using an evidential Markovian classifier based on Dempster-Shafer theory. Experiments concern the prediction of an engine health based on more than twenty observations.

ABOUT THIS BOOK The theory of value structure concerns the meaning of "better than" and "good," as well as the way in which values serve as a basis for rational decision making. Drawing methodologically from economics and theories of decision making, the aim of serious axiology in metaethics is to do justice to problems that have puzzled philosophers of value for centuries. Can value comparisons be cyclic? Are all values comparable with each other and can decision makers just add up different aspects of an evaluation to determine the best course of action? A Theory of Value Structure: From Values to Decisions starts with a thorough introduction to the modeling of "better than" comparisons from a normative perspective. In the philosophical part of the book, Erich H. Rast argues that aspects of "better than" comparisons can differ qualitatively so much that one aspect may outrank another. Consequently, the classical weighted sum aggregation model fails. Values cannot always be summed up and comparisons may be fundamentally noncompensatory, an indeterminacy that explains problems like the apparent nontransitivity of "better than" and hard cases in decision making. Using a lexicographic method of value comparisons, Rast develops a multidimensional theory of "better than" and shows how and to which extent it can be combined with standard methods of decision making under uncertainty by using rank-dependent utility theory.

ISBN 978-1-7936-1694-4
https://rowman.com/ISBN/9781793616951/Theory-of-Value-Structure-From-Values-to-Decisions

The location of a warehouse is generally one of the most important and strategic decision in the optimization of logistic systems. Warehouse location is a long-term decision and is influenced by many quantitative and qualitative factors. Among the main criteria taken into account in this paper, some are costs, labor characteristics, infrastructure, and markets. This paper also includes some sub-criteria because of the hierarchical structure of the problem, like tax incentives and tax structures, availability of labor force, quality and reliability of modes of transportation, and proximity to customers. The conventional approaches to warehouse location selection problem tend to be less effective in dealing with the imprecise or vague nature of the linguistic assessment. Under many situations, the values of the qualitative criteria are often imprecisely defined for the decision-makers. Choquet integral is a suitable multi-criteria method to capture this imprecise or vague nature. This paper shows a successful application of multi-criteria Choquet integral to a real warehouse location selection problem of a big Turkish logistic firm.

This paper is based on the investigation of a new property for Choquet integral: -supermodularity. With this definition we want to extend the property of supermodularity for Choquet integral with respect to a supermodular fuzzy measure, in particular by using Frank -norms.

... Krusinska, E., Slowinski, R. and Stefanowski, J., 1992. Discriminant versus rough set approach to vague data analysis. Applied Stochastic Models and Data Analysis 8, pp. ... Lin, T., 1989.Neighborhood systems and approximation in database and knowledge base systems. ...

We extend our approach to the linguistic summarization of (numerical) time series. The main issue boils down to the identification of trends in time series that are characterized by a set of attributes followed by their appropriate aggregation. We propose to use the OWA (ordered weighted averaging) operators for the aggregation of partial trends as an alternative to the use of the classic Zadeh's calculus of linguistically quantified propositions, the Sugeno integral and the Choquet integral. The use of the OWA operators provides a convenient unified aggregation means that can be used to derive diverse types of summaries. The results obtained confirm a high human consistency of linguistic summaries derived.

Evaluating software quality is an important and essential issue in the development process because it helps to deliver a competitive software product. A decision of selecting the best software based on quality attributes is a type of multi-criteria decision-making (MCDM) processes where interactions among criteria should be considered. This paper presents and develops quantitative evaluations by considering  interactions among criteria in the MCDM problems. The aggregator methods such as Arithmetic Mean
(AM) and Weighted Arithmetic Mean (WAM) are introduced, described and compared to Choquet Integral (CI) approach which is a type of fuzzy measure used as a new method for MCDM. The comparisons are shown by evaluating and ranking software alternatives based on six main quality attributes as identified by the ISO 9126-1 standard. The evaluation experiments depend on real data collected from case studies.

We provide a characterization of the consequences of the assumption that a decision maker with a given utility function is Choquet rational : She maximizes expected utility, but possibly with respect to non-additive beliefs, so that her preferences are represented by Choquet expected utility (CEU).

The aim of this paper is to describe the method of induction of generalized annotated programs called IGAP what is a special case of inductive fuzzy logic programming for monotonely classified data. This method is based on the multiple use of two valued ILP and the syntactical equivalence of fuzzy logic programs and a restricted class of generalized annotated programs. Finally we compare our method with several fuzzy ILP methods.

We discuss the equivalence between aggregation of fuzzy sets and integration with respect to a special class of nonadditive set functions. Both fuzzy integral and Choquet integral are considered. First, we study aggregation of a finite family of fuzzy sets and then we extend our results to aggregation of an infinite family. The concepts of comonotonic maxitivity and additivity play a central role. We argue that, for the purpose of aggregating fuzzy sets, comonotonic maxitivity is a more desirable requirement than comonotonic additivity. In the absence of any such requirement we explore a wider class of aggregation procedures.

Ce chapitre d'ouvrage collectif vise à présenter les extensions du modèle EU, basées sur l'intégrale de Choquet, qui permettent de tenir compte des comportements observés comme dans le paradoxe d'Allais dans le risque ou le paradoxe d'Ellsberg dans l'incertain, où il y a violation du modèle d'espérance d'utilité. Sous l'axiome clé d'indépendance comonotone, Schmeidler dans l'incertain, et Quiggin et Yaari dans le rique ont réussi à caractériser des préférences qui généralisent le modèle EU, à l'aide d'une fonctionnelle qui s'avère être une intégrale de Choquet. Ces modèles expliquent la plupart des paradoxes observés et permettent des comportements plus diversi…és à la fois dans l'incertain et dans le risque.

Starting from the concept of 'equiordered functions' we characterize two types of fuzzy integrals that can be defined on any kind of fuzzy measure: Sugeno and Choquet integrals. The characterization theorems that we obtain allow us to carry out a comparative study of these two fuzzy integrals, pointing out their formal similarities and their conceptual differences.

Non-additive robust ordinal regression (NAROR), uses primary preferences of decision maker (DM) to define the necessary preference relations (NPR) and possible preference relations (PPR) on alternatives when all compatible fuzzy measures are taken into account and aggregation function is Choquet integral (CI). The question arises as how these NPRs and PPRs can be used in capacity definition problem? This article proposes an algorithm which uses these relations in finding a capacity that is representative to DM and has also evenness property to some extent. The methods based on maximizing evenness leads to results that are not fully representative to DM, so this article improves this drawback by focusing on representativeness of capacity and taking into account capacity evenness by means of decision rules defined in algorithm.

Combining multiple classifiers is a natural way to discover useful information and improve the performances of individual classifiers. It's based on the combination of the outputs of an ensemble of different classifiers. When interactions exist in combining multiple classifiers, fuzzy integral would be a valid method to fuse multiple classifiers. In this fuzzy fusion approach, the fuzzy measure plays an important role. Liu proposed a novel fuzzy measure, L-measure, which is more sensitive than some common measures, like¿- ...

It is a critical issue for the continuity of an organization to efficiently evaluate the performance of its employees. The evaluation process has to guarantee a sufficient level of objectivity and a limited level of subjectivity in order to answer the employees' expectancies. In this study, a multi-criteria approach to this problem has been proposed since the employees' performance depends on various criteria simultaneously. It is also considered that those criteria may have interaction. Hence, Choquet Integral has been used to handle this situation. To illustrate the use of the proposed method, the performance of a set of employees working in a company in medical sector is evaluated.

Risks induced by extreme events are characterized by small or ambiguous probabilities, catastrophic losses, or windfall gains. Through a new functional, that mimics the restricted Bayes-Hurwicz criterion within the Choquet expected utility approach, it is possible to represent the decisionmaker behavior facing both risky (large and reliable probability) and extreme (small or ambiguous probability) events. A new formalization of the precautionary principle (PP) is shown and a new functional, which encompasses both extreme outcomes and expectation of all the possible results for every act, is claimed.

The problem relevant for quality management such as aggregation of many features into one representative is analyzed. Actually, in quality management practice, standard approaches to aggregation are often trivial and as a consequence -inadequate. In this paper, aggregation is treated as a logical and/or pseudo-logical operation that is important from many points of view such as adequacy and interpretations.

A recognition and classification method of multiple moving objects in traffic based on the combination of the Biomimetic Pattern Recognition (BPR) and Choquet Integral (CI) is proposed. The recognition process consists of three stages. At the first stage, vehicles and pedestrians are detected in video images and the area, the shape and the velocity features are obtained by classical methods. At the second stage, BPR is used to classify the Zernike moments extracted at the first stage. At the last stage, CI is then adopted for multi-features fusion based on the output of BPR, and the area and the velocity features obtained at the first stage to improve the recognition accuracy. Experiment results show that this approach is efficient.

We axiomatize a model of decision under objective ambiguity or imprecise risk. The decision maker forms a subjective (non necessarily additive) belief about the likelihood of probability distributions and computes the average expected utility of a given act with respect to this second order ...

A method based on relative entropy is proposed to identify Choquet integral parameters for classification purposes. While the entropy-based method initially developed by Kojadinovic [11] assigns a high importance to the attributes which allow to differentiate many classes, the relative entropybased method proposed focuses on the researched classes. This is done in order to obtain the attribute confidence by the integration of expert knowledge concerning these classes. The values of the relative entropy for different subsets of attributes are considered as the values of Choquet capacity coefficients. The proposed identification method is integrated in an aiding system aiming at interpreting 3D-tomographic images of electrotechnical parts made of composite materials and manufactured by Schneider Electric. Four attributes extracted from the tomographic images are aggregated by the Choquet integral using the parameters identified by the proposed method. The result is a 3D cartography of the sought-after regions within the tomographic image. Quantitative assessments of classification highlight the relevance of the proposed approach.

It is a critical issue for the continuity of an organization to efficiently evaluate the performance of its employees. The evaluation process has to guarantee a sufficient level of objectivity and a limited level of subjectivity in order to answer the employees' expectancies. In this study, a multi-criteria approach to this problem has been proposed since the employees' performance depends on various criteria simultaneously. It is also considered that those criteria may have interaction. Hence, Choquet Integral has been used to handle this situation. To illustrate the use of the proposed method, the performance of a set of employees working in a company in medical sector is evaluated.

This paper investigates the statistical properties of the Choquet and Sugeno integrals, used as multiattribute models. The investigation is done on an empirical basis, and focuses on two topics: the distribution of the output of these integrals when the input is corrupted with noise, and the robustness of these models, when they are identified using some set of learning data through some learning procedure.

We consider future cash flows that are contingent both on dates in time and on uncertain states. The decision maker (DM) values the cash flows according to its 2 decision criterion: Here the payoffs' expectation with respect to a capacity measure. The subjective measure grasps the DM's behaviour in front of the future, in the spirit of and of Yaari's (1987) Dual Theory in the case of risk. Decomposition of the criterion into two criteria that represent the DM's preferences on uncertain payoffs and time contingent payoffs are derived from results. Conditional Choquet integrals are defined by dynamic consistency requirements and conditional capacities are derived, under some conditions on information. In contrast with other models referring to dynamic consistency, ours doesn't collapse into a linear one because it violates a weak version of consequentialism.

The paper presents and discusses a method for rank-ordering alternatives on the basis of con- straints induced by generic principles (expressing for instance the relative importance of criteria), or by examples of orderings between particular alter- natives, without resorting to the use of an aggrega- tion operation for evaluating the alternatives. The approach, which remains qualitative, is based on the

The two-parts state-of-art overview of aggregation theory summarizes the essential information concerning aggregation issues. Overview of aggregation properties is given, including the basic classification of aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with open arity (extended means).

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, modeled by the fuzzy measures, parameterizing our approach.

The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.

In this paper we propose a generalization of the concept of symmetric fuzzy measure based in a decomposition of the universal set in what we have called subsets of indifference. Some properties of these measures are studied, as well as their Choquet integral. Finally, a degree of interaction between the subsets of indifference is defined.

Variety of decision models have been proposed in contemporary literature to tackle the problem of screening product innovations. Although linear models have gained considerable attention and recommendation, contemporary literature contains strong evidence in support of nonlinear noncompensatory models. In this paper, the authors first demonstrate how fuzzy measures, which are defined on subsets of decision attributes, and their Choquet-integral formulation, which exhibits both compensatory and noncompensatory properties, have meaningful behavioral interpretations within the context of new-product screening. Then, they show how to address the complex problem of building such measures by applying a learning algorithm that relies on methods of judgment analysis. An accompanying case study demonstrates how organizations may customize a new product decision aid and fine tune their business strategy as actual results accrue.

This paper formulates and studies a general continuous-time behavioral portfolio selection model under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility (value) functions and probability distortions. Unlike the conventional expected utility maximization model, such a behavioral model could be easily misformulated (a.k.a. ill-posed) if its different components do not coordinate well with each other. Certain classes of an ill-posed model are identified. A systematic approach, which is fundamentally different from the ones employed for the utility model, is developed to solve a well-posed model, assuming a complete market and general Itô processes for asset prices. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure reminiscent of a gambling policy betting on a good state of the world while accepting a fixed, known loss in case of a bad one. An example with a two-piece CRRA utility is presented to illustrate the general results obtained, and is solved completely for all admissible parameters. The effect of the behavioral criterion on the risky allocations is finally discussed.

We develop a vegetation mapping method using long-wave hyperspectral imagery and apply it to landmine detection. The novel aspect of the method is that it makes use of emissivity skewness. The main purpose of vegetation detection for mine detection is to minimize false alarms. Vegetation, such as round bushes, may be mistaken as mines by mine detection algorithms, particularly in synthetic aperture radar (SAR) imagery. We employ an unsupervised vegetation detection algorithm that exploits statistics of emissivity spectra of vegetation in the long-wave infrared spectrum for identification. This information is incorporated into a Choquet integral-based fusion structure, which fuses detector outputs from hyperspectral imagery and SAR imagery. Vegetation mapping is shown to improve mine detection results over a variety of images and fusion models.