Generalized Choquet Integral Research Articles

Analysis of smooth and enhanced smooth quadrature-inspired generalized Choquet integral

Enhancing the ability to make informed decisions stands as a significant challenge in modern IT. Specifically, there is a growing need to improve the efficiency of classification algorithms. When faced with multiple results derived from various methods, one can select the most probable decision using a robust aggregation operator. A common class of algorithms employed for this purpose is based on extensions of the Choquet integral (CI). In this study, we introduce and extensively analyze a novel aggregation operator concept founded on the generalization of the CI. This approach leverages quadrature formulae to calculate Choquet integral values, but with a unique modification involving a smoothing operation. This refinement results in more precise values, preserving the essential characteristics of the Choquet integral. These refined values can be effectively applied in the aggregation of classifiers and, more broadly, in information fusion processes. A series of numerical experiments demonstrates the efficiency of our approach. Furthermore, we thoroughly discuss and provide mathematical proofs for the properties of the newly constructed operators.

Quadrature-Inspired Generalized Choquet Integral in an Application to Classification Problems

Quadrature-Inspired Generalized Choquet Integral in an Application to Classification Problems

The Choquet integral-based Shapley function for n-person cooperative games with probabilistic hesitant fuzzy coalitions

The use of probabilistic hesitant fuzzy elements (PHFEs) to represent the alliance status in cooperative games can describe the player's participation levels and associated probability. This paper proposes an effective Shapley function for probabilistic hesitant fuzzy cooperative games based on generalized Choquet integral. We first define the probabilistic hesitant fuzzy coalition of cooperative games. After that, a supplementary model of incomplete probability information is established. Without losing any probability information, an adjustment algorithm is developed to derive the consistent probability of coalitions. Then the characteristic function of probabilistic hesitant fuzzy cooperative games is constructed based on generalized Choquet integral. The Choquet integral-based Shapley function is presented according to the proposed characteristic function. Some desirable properties of Shapley function are discussed in detail. The CIG algorithm is established for generating consistent imputation of cooperative games. Finally, the proposed algorithm is applied to the cooperative production of alloy materials.

Jensen's inequality for Choquet integral revisited and a note on Jensen's inequality for generalized Choquet integral

In this paper, it is shown that Jensen's inequality for Choquet integral given by R. Wang ten years ago is incorrect, then it is revisited, the modified Jensen's and reverse Jensen's inequalities for Choquet integral are proved. Then Jensen's inequality for generalized Choquet integral, obtained by the authors in a recent paper, is modified accordingly.

Probabilistic uncertain linguistic TODIM method based on the generalized Choquet integral and its application

PurposeThe purpose of this paper is to develop a probabilistic uncertain linguistic (PUL) TODIM method based on the generalized Choquet integral, with respect to the interdependencies between criteria, for the selection of the best alternate in the context of multiple criteria group decision-making (MCGDM).Design/methodology/approachOwing to decision makers (DMs) do not always show completely rational and may have the preference of bounded rational behavior, this may affect the result of the MCGDM. At the same time, criteria interaction is a focused issue in MCGDM. Hence, a novel TODIM method based on the generalized Choquet integral selects the best alternate using PUL evaluation, where the generalized Choquet integral is used to calculate the weight of criterion. The generalized PUL distance measure between two probabilistic uncertain linguistic elements (PULEs) is calculated and the perceived dominance degree matrices for each alternate relative to other alternates are obtained. Furthermore, the comprehensive perceived dominance degree of each alternate can be calculated to get the ranking.FindingsPotential application of the PUL-TODIM method is demonstrated through an evaluation example with sensitivity and comparative analysis.Originality/valueAs per author's concern, there are no TODIM methods with probabilistic uncertain linguistic sets (PULTSs) to solve MCGDM problems under uncertainty. Compared with the result of existing methods, the final judgment value of alternates using the extended TODIM methodology is highly corroborated, which proves its potential in solving MCGDM problems under qualitative and quantitative environments.

Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral

It is known that the quasi-arithmetic means can be characterized in various ways, with an essential role of a symmetry property. In the expected utility theory, the quasi-arithmetic mean is called the certainty equivalent and it is applied, e.g., in a utility-based insurance contracts pricing. In this paper, we introduce and study the quasi-arithmetic type mean in a more general setting, namely with the expected value being replaced by the generalized Choquet integral. We show that a functional that is defined in this way is a mean. Furthermore, we characterize the equality, positive homogeneity, and translativity in this class of means.

Pseudo-integral and generalized Choquet integral

Due to many applications, the Choquet integral as a powerful tool for modeling non-deterministic problems needs to be further extended. Therefore the paper is devoted to a generalization of the Choquet integral. As a basis, the pseudo-integral for bounded integrand is extended to the case for nonnegative integrands at first, and then the generalized Choquet integral is defined. As special cases, pseudo-Choquet Stieltjes integrals, pseudo-fuzzy Stieltjes integrals, g-Choquet integrals, pseudo-(N)fuzzy integrals and pseudo-(S)fuzzy integrals are obtained, and various kinds of properties and convergence theorems are shown, meanwhile Markov, Jensen, Minkowski and Hölder inequalities are proved. In the end, the generalized discrete Choquet integral is discussed. The obtained results for the generalized Choquet integral cover some previous results on different types of nonadditive integrals.

Three types of generalized Choquet integral

In this paper we present three recently obtained important generalizations of the Choquet integral. First generalization is based on different collection of finite systems. Second generalization of the Choquet integral uses sublinear means. Third generalization is based on two fuzzy measures, one of which is pseudo-additive.

Forecasting stock market price by using fuzzified Choquet integral based fuzzy measures with genetic algorithm for parameter optimization

In this paper, fuzzified Choquet integral and fuzzy-valued integrand with respect to separate measures like fuzzy measure, signed fuzzy measure and intuitionistic fuzzy measure are used to develop regression model for forecasting. Fuzzified Choquet integral is used to build a regression model for forecasting time series with multiple attributes as predictor attributes. Linear regression based forecasting models are suffering from low accuracy and unable to approximate the non-linearity in time series. Whereas Choquet integral can be used as a general non-linear regression model with respect to non classical measures. In the Choquet integral based regression model parameters are optimized by using a real coded genetic algorithm (GA). In these forecasting models, fuzzified integrands denote the participation of an individual attribute or a group of attributes to predict the current situation. Here, more generalized Choquet integral,i.e., fuzzified Choquet integral is used in case of non-linear time series forecasting models. Three different real stock exchange data are used to predict the time series forecasting model. It is observed that the accuracy of prediction models highly depends on the non-linearity of the time series.

A REAL PERSONNEL SELECTION PROBLEM USING THE GENERALIZED CHOQUET INTEGRAL METHODOLOGY

The main objective in the selection of personnel is to select the most appropriate candidate for a job. Personnel selection for human resources management is a very important issue.The aim of this paper is to determine the best-performing personnel for promotion using an application of a Multi Criteria Decision Making(MCDM) method, generalized Choquet integral, to a real personnel selection problem of a case study in Turkey and 17 alternatives are ranked according to personnel selection criteria (22 subcriteria are classified under 5 main criteria). The main contribution of this paper is to determine the interdependency among main criteria and subcriteria, the nonlinear relationship among them and the environmental uncertainties while selecting personnel alternatives using the generalized Choquet integral method with the experts’ view. To the authors’ knowledge, this will be the first study which uses the generalized Choquet Integral methodology for human resources.

A REAL PERSONNEL SELECTION PROBLEM USING THE GENERALIZED CHOQUET INTEGRAL METHODOLOGY

The main objective in the selection of personnel is to select the most appropriate candidate for a job. Personnel selection for human resources management is a very important issue.The aim of this paper is to determine the best-performing personnel for promotion using an application of a Multi Criteria Decision Making(MCDM) method, generalized Choquet integral, to a real personnel selection problem of a case study in Turkey and 17 alternatives are ranked according to personnel selection criteria (22 subcriteria are classified under 5 main criteria). The main contribution of this paper is to determine the interdependency among main criteria and subcriteria, the nonlinear relationship among them and the environmental uncertainties while selecting personnel alternatives using the generalized Choquet integral method with the experts’ view. To the authors’ knowledge, this will be the first study which uses the generalized Choquet Integral methodology for human resources.

A new type of fuzzy integrals for decision making based on bivariate symmetric means

We propose a new generalization of the discrete Choquet integral based on an arbitrary bivariate symmetric averaging function (mean). So far only the means with a natural multivariate extension were used for this purpose. In this paper, we use a general method based on a pruned binary tree to extend symmetric means with no obvious multivariate form, such as the logarithmic, identric, Heronian, Lagrangean, and Cauchy means. The generalized Choquet integral is built by computing the extensions of the bivariate means of the ordered inputs, and includes some existing extensions as special cases. Our construction is illustrated with multiple examples.

An extended TODIM method for hesitant fuzzy interactive multicriteria decision making based on generalized Choquet integral

As a kind of behavior decision-making theory, TODIM (an acronym in Portuguese of interactive and multicriteria decision making (MCDM)) method has received more attention, in which the criteria values are in the format of crisp numbers. It can't be used to handle uncertain MCDM problems. In this paper, an extended Choquet-based TODIM method is proposed to solve the hesitant fuzzy MCDM problems, where interactive phenomena among criteria are also considered. First, according to fuzzy measure, the interactive criteria weight is measured by Shapley value. Then the prospect value of criteria for every alternative is calculated through prospect value function of the hesitant fuzzy set. Further, the dominance degree of each alternative over the others is calculated by Choquet integral, the global value of each alternative can be obtained to rank the alternatives. Finally, an illustrate example is given to illustrates the application of the novel method where the influence on decision making for different parameters of value function and different decision-making reference point is discussed.

AN ITERATIVITY CONDITION FOR THE MEAN-VALUE PRINCIPLE UNDER CUMULATIVE PROSPECT THEORY

AbstractIn this paper, we present the full characterization of the iterativity condition for the mean-value principle under the cumulative prospect theory. It turns out that the premium principle is iterative for exactly six pairs of probability distortion functions. Some of the corresponding premium principles are the classical mean-value principle, essential infimum or essential supremum of the random loss. Moreover, from the proof of the main theorem of this paper, it follows that the iterativity of the mean-value principle is equivalent to the iterativity of the generalized Choquet integral.

Uncertain generalized aggregation operators

The aim of this paper is to extend the generalized ordered weighted averaging operator and provide a new class of operators called the uncertain generalized ordered weighted averaging (UGOWA) operator. It provides a very general formulation that includes as special cases a wide range of aggregation operators and aggregates the input arguments taking the form of intervals rather than exact numbers. We further generalize the UGOWA operator to obtain the uncertain generalized hybrid averaging operator, the quasi uncertain ordered weighted averaging operator and the uncertain generalized Choquet integral aggregation operator. In the meanwhile, a new approach to determining the UGOWA weights is proposed based on the relative deviation measure. Finally, a numerical example is presented to illustrate the proposed approach to group decision making with UGOWA operator.

AXIOMATIZATIONS OF SIGNED DISCRETE CHOQUET INTEGRALS

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovász extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory.

Induced and Linguistic Generalized Aggregation Operators and Their Application in Linguistic Group Decision Making

We introduce a wide range of induced and linguistic generalized aggregation operators. First, we present the induced linguistic generalized ordered weighted averaging (ILGOWA) operator. It is a generalization of the OWA operator that uses linguistic variables, order inducing variables and generalized means in order to provide a more general formulation. One of its main results is that it includes a wide range of linguistic aggregation operators such as the induced linguistic OWA (ILOWA), the induced linguistic OWG (ILOWG) and the linguistic generalized OWA (LGOWA) operator. We further generalize the ILGOWA operator by using quasi-arithmetic means obtaining the induced linguistic quasi-arithmetic OWA (Quasi-ILOWA) operator and by using hybrid averages forming the induced linguistic generalized hybrid average (ILGHA) operator. We also present a further extension with Choquet integrals. We call it the induced linguistic generalized Choquet integral aggregation (ILGCIA). We end the paper with an application of the new approach in a linguistic group decision making problem.

Continuous generalized OWA operator and its application to decision making

In this paper we present a new class of operators called the continuous generalized ordered weighted averaging (C-GOWA) operators, which extends the continuous ordered weighted averaging operator. To adapt to uncertain and complex situations in decision making, we use differentiated aggregation method and produce special forms of C-GOWA operator. In addition, we apply the C-GOWA operator to the aggregation of multiple interval arguments and obtain a wide range of aggregation operators. We especially discuss the controlling parameter and the selection of basic unit-interval monotonic function, introduced in the information aggregation, to make it more feasible. We further generalize the previous approaches by using Choquet integral and quasi-arithmetic means, obtaining the combined continuous generalized Choquet integral aggregation operator and the combined continuous quasi-arithmetic Choquet integral aggregation operator. We also present some further extension by using hierarchical aggregation rules, obtaining the ordered hierarchies of the combined C-GOWA operator. Meanwhile, some desirable properties of these operators are investigated. Finally, we give a numerical example to illustrate the application of these operators to group decision making with interval arguments.

Learning Weights in the Generalized OWA Operators

This paper discusses identification of parameters of generalized ordered weighted averaging (GOWA) operators from empirical data. Similarly to ordinary OWA operators, GOWA are characterized by a vector of weights, as well as the power to which the arguments are raised. We develop optimization techniques which allow one to fit such operators to the observed data. We also generalize these methods for functional defined GOWA and generalized Choquet integral based aggregation operators.

The evaluation of service quality using generalized Choquet integral

In view of the compatibility between psychology and linguistic terms, this study generalizes the standard Choquet integral, whose measurable evidence and fuzzy measures are real numbers. The proposed generalization can deal with fuzzy-number types of measurable evidence and fuzzy measures. In comparison with the previous research on service quality and fuzzy sets, this study integrates the three-column format SERVQUAL into the generalized Choquet integral. A numerical example evaluating the overall service quality of e-stores is provided to illustrate how the generalized Choquet integral evaluates service quality in the three-column format SERVQUAL. This integral is also compared with the possibility/necessity model when measuring service quality.