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Fuzzy measures generalize additive measures and probabilities. Their advantage with respect to additive ones is that they permit to model interactions between objects. Mesiar introduced in 1999 k-order Pan-additive fuzzy measures that generalize k-order additive and k-maxitive ones. They are related to the Möbius transform and related generalizations. In this paper we introduce some other transforms that we call (Max,+) and (Max,⊕) that permit to represent fuzzy measures in a convenient way when we use genetic algorithms in fuzzy measure identification problems. We illustrate its use identifying a measure for a subjective evaluation problem using a Choquet integral and a Sugeno 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 Holder 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.

We outline recent trends in capacity-based aggregation in large universes. Capacities (fuzzy measures) model dependencies among the inputs, and aggregation by the discrete Choquet, Sugeno and other fuzzy integrals accounts for synergies and redundancies. For large number of inputs the exponential complexity of all interactions is a major obstacle. We exemplify the need for aggregation of a large number of dependent inputs on several applications and discuss the challenges and approaches to reducing the complexity of capacity-based aggregation. We also state which mathematical and computational tools are required for large scale capacity modelling.

We consider calculation of the mode of compositional data, the data related to each other through a linear constraint. Compositional data arises in various extensions of the fuzzy sets theory (type-2 interval-valued, intuitionistic, hesitant fuzzy sets), biomedicine (relative abundance, genome sequencing, activity recognition), and data analytics (various wealth indices, interval-valued observations, traffic congestion, etc.). Mode is a pre-aggregation function in the case of single variable, used as a classical estimator robust to outliers, but its multivariate extensions face the challenges of high computational complexity and potential oversmoothing. In this work we present several novel techniques for mode estimation on the unit k-simplex representing compositional, interval-valued, and general vector-valued data. We highlight the re-weighted k-nearest neighbours algorithm based on the Choquet integral with respect to a 2-additive fuzzy measure, compare its performance against other approaches based on spatial partitioning, and illustrate its applications to aggregation of real-world interval-valued data sets.

In this work, we introduce the notion of dG-Choquet integral, which generalizes the discrete Choquet integral replacing, in the first place, the difference between inputs represented by closed subintervals of the unit interval [0,1] by a dissimilarity function; and we also replace the sum by more general appropriate functions. We show that particular cases of dG-Choquet integral are both the discrete Choquet integral and the d-Choquet integral. We define interval-valued fuzzy measures and we show how they can be used with dG-Choquet integrals to define an interval-valued discrete Choquet integral which is monotone with respect to admissible orders. We finally study the validity of this interval-valued Choquet integral by means of an illustrative example in a classification problem. © 2021 This work was supported in part by the Spanish Ministry of Science and Technology, under project PID2019-108392GB-I00 (AEI/10.13039/501100011033), by the project PJUPNA-1926 of the Public University of Navarre and by the project VEGA 1/0267/21 .

Several operators have emerged in the framework of Choquet integral with the purpose of simultaneously generalizing weighted means and ordered weighted averaging (OWA) operators. However, on many occasions, not enough attention has been paid to whether the constructed operators behaved similarly to the weighted means and OWA operators that have been generalized. In this sense, it seems necessary that these new operators preserve the weights assigned to the information sources (which are established through the weighting vector associated with the weighted mean) and that they are able to rule out extreme values (which is an important characteristic of OWA operators). In this paper we analyze a family of operators recently introduced in the literature through the Crescent Method. First, we introduce a broad class of weighting vectors that allow us to guarantee that the games generated with the Crescent Method are capacities. Next we analyze the conjunctive/disjunctive character of the Choquet integrals associated with these capacities and we also give closed-form expressions of some tolerance and importance indices such as k-conjunctiveness/disjunctiveness indices, the veto and favor indices, and the Shapley values. Finally, we give two examples to show the usefulness of the results obtained.

OWA operators were introduced by Yager and their Choquet integral-based representation was shown by Grabisch. Based on bi-capacities and related Choquet integral introduced by Grabisch and Labreuche, a new generalization of OWA operators, namely BIOWA operators are introduced. Our approach is exemplified by several examples. Bi-capacities leading to the standard Yager's OWA operators on real line are completely characterized. We introduce and exemplify quantifiers generating BIOWA operators with an arbitrary arity, and their orness. Finally, orness of BIOWA operators is also introduced and studied.

This paper studies non-cooperative games where players are allowed to play their mixed non-additive strategies. Expected payoffs are expressed by so-called fuzzy integrals: Choquet integral, Sugeno integral and generalizations of Sugeno integral obtained by using triangular norms. We consider the existence problem of Nash equilibrium for such games. Positive results for Sugeno integral and its generalizations are obtained. However we provide some example of a game with Choquet payoffs which have no Nash equilibrium. Such example demonstrates that fuzzy integrals based on the maximum operation are more suitable for possibility capacities than Choquet integral which is based on the addition operation.

The use of the Choquet integral in data fusion processes allows for the effective modelling of interactions and dependencies between data features or criteria. Its application requires identification of the defining capacity (also known as fuzzy measure) values. The main limiting factor is the complexity of the underlying parameter learning problem, which grows exponentially in the number of variables. However, in practice we may have expert knowledge regarding which of the subsets of criteria interact with each other, and which groups are independent. In this paper we study hierarchical aggregation processes, architecturally similar to feed-forward neural networks, but which allow for the simplification of the fitting problem both in terms of the number of variables and monotonicity constraints. We note that the Mobius representation lets us identify a number of relationships between the overall fuzzy measure and the data pipeline structure. Included in our findings are simplified fuzzy measures that generalise both k-intolerant and k-interactive capacities.

We examine sufficient and necessary conditions for the Choquet integral on the support of its integrand to be equal to the integral on a set which includes this support (we consider the problem in the case of a locally compact Hausdorff space).

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.

We introduce a Choquet-Sugeno-like operator generalizing many operators for bounded nonnegative functions and monotone measures from the literature, e.g., the Sugeno-like operator, the Lovasz and Owen measure extensions, the F -decomposition integral with respect to a partition decomposition system, and others. The new operator is based on concepts of dependence relation and conditional aggregation operators, but it does not depend on α -level sets. We also provide conditions under which the Choquet-Sugeno-like operator coincides with some Choquet-like integrals defined on finite spaces and appeared recently in the literature, e.g., the reverse Choquet integral , the d -Choquet integral, the F -based discrete Choquet-like integral, some version of the C F 1 F 2 -integral, the CC -integrals (or Choquet-like Copula-based integral) and the discrete inclusion-exclusion integral. Some basic properties of the Choquet-Sugeno-like operator are studied.

Under the present circumstances, when we are still under the threat of different strains of coronavirus, and since the most widely used method for COVID-19 detection, RT-PCR is a tedious and time-consuming manual procedure with poor precision, the application of Artificial Intelligence (AI) and Computer-Aided Diagnosis (CAD) is inevitable. Though, some vaccines have now been authorized worldwide, it will take huge time to reach everyone, especially in developing countries. In this work, we have analyzed Chest X-ray (CXR) images for the detection of the coronavirus. The primary agenda of this proposed research study is to leverage the classification performance of the deep learning models using ensemble learning. Many papers have proposed different ensemble learning techniques in this field, some methods using aggregation functions like Weighted Arithmetic Mean (WAM) among others. However, none of these methods take into consideration the decisions that subsets of the classifiers take. In this paper, we have applied Choquet integral for ensemble and propose a novel method for the evaluation of fuzzy measures using coalition game theory, information theory, and Lambda fuzzy approximation. Three different sets of fuzzy measures are calculated using three different weighting schemes along with information theory and coalition game theory. Using these three sets of fuzzy measures, three Choquet integrals are calculated and their decisions are finally combined. Besides, we have created a database by combining several image repositories developed recently. Impressive results on the newly developed dataset and the challenging COVIDx dataset support the efficacy and robustness of the proposed method. Our experimental results outperform many recently proposed methods.

In this paper we define a novel survival function motivated by real life problems, which generalizes the super level measure introduced by Do and Thiele (2015). This concept is based on conditional aggregation operators extending the definition of aggregation operators introduced by Calvo et al. (2002) for all bounded measurable functions and not only for finite sequences. Some basic properties and several examples of conditional aggregation operators are presented. Using the novel survival function, the Choquet-Stieltjes functional is introduced and the conditions are indicated under which this functional can be called an integral. The new functional generalizes several known integrals including the Choquet-Stieltjes integral as well as Choquet integral with respect to level dependent capacity introduced by Greco et al. (2011).

This work concerns solving optimisation problems where the objective function is expressed as a Choquet integral . This objective generalises a linear objective function (with positive weights) and allows for interaction to be modeled between coalitions of the decision variables. We leverage results from optimising the ordered weighted averaging (OWA) operator and propose efficient solution approaches for the asymmetric objectives both for the simplest case of a single constraint and then for multiple comonotone constraints. To solve problems with a large number of variables, we rely on the so-called antibuoyancy property , previously applied to OWA weights, and which we extend to general fuzzy measures. This characterisation not only facilitates a restriction of the domain on which the solution lies but also allows us to relate the Choquet integral’s behavior in such cases to the Pigou-Dalton progressive transfers principle. We characterise the Choquet integrals consistent with the Pigou-Dalton principle. Theoretical results are supported by numerical experiments, which illustrate significant gains in performance. Our results offer opportunities for scalability to a much higher number of variables.

Fuzzy measure-based aggregations allow taking interactions among coalitions of the input sources into account. Their main drawback when applying them in real-world problems, such as combining classifier ensembles, is how to define the fuzzy measure that governs the aggregation and specifies the interactions. However, their usage for combining classifiers has shown its advantage. The learning of the fuzzy measure can be done either in a supervised or unsupervised manner. This paper focuses on supervised approaches. Existing supervised approaches are designed to minimize the mean squared error cost function, even for classification problems. We propose a new fuzzy measure learning algorithm for combining classifiers that can optimize any cost function. To do so, advancements from deep learning frameworks are considered such as automatic gradient computation. Therefore, a gradient-based method is presented together with three new update policies that are required to preserve the monotonicity constraints of the fuzzy measures. The usefulness of the proposal and the optimization of cross-entropy cost are shown in an extensive experimental study with 58 datasets corresponding to both binary and multi-class classification problems. In this framework, the proposed method is compared with other state-of-the-art methods for fuzzy measure learning. Mikel Uriz has been supported by the CDTI and the Spanish Ministry of Science and Innovation under Neotec 2021 (SNEO-20211147). This work was also supported by the Spanish Ministry of Science and Innovation under project PID2019-108392 GB-I00 (AEI/10.13039/501100011033) and by the Public University of Navarre under project PJUPNA25-2022.

In order to obtain high satisfaction from experts, the consensus reaching process (CRP) is an essential requirement for dealing with multi-criteria group decision making (MCGDM) problems. Single-valued neutrosophic number (SVNN) is an effective tool to describe the uncertainty of the expert cognition. Thus, we develop a consensus reaching model for single-valued neutrosophic MCGDM in this paper. First, each expert makes his/her judgment on each alternative with respect to multiple criteria by SVNNs, and the group solution is obtained by the generalized Shapley single-valued neutrosophic Choquet integral (GS-SVNCI) operator to consider the correlations among elements comprehensively. Second, the projection-based consensus measure is proposed to reflect the agreement between the individual and collective opinions. Then, a threshold value is used to determine the CRP whether to be executed based on the expert’s consensus level. If yes, the feedback mechanism provides the experts with personalized adjustment advices based on their psychic utility to group pressure. Finally, we illustrate the feasibility of the proposed consensus model by an example and analyze the superiority by comparing with some existing MCGDM methods and different CRP models. The developed consensus model can consider interrelationships between experts, which is more effective and reasonable to obtain the collective resolution. Further, the consensus measure based on the projection can comprehensively reflect the closeness between the individual and collective opinions. In addition, the personalized adjustment advices considering the experts’ psychic utility to group pressure improve their acceptance of these advices.

Fuzzy integrals (FIs) are powerful aggregation operators that fuse information from multiple sources. The aggregation is parameterized using a fuzzy measure (FM), which encodes the worths of all subsets of sources. Since the FI is defined with respect to an FM, much consideration must be given to defining the FM. However, in practice this is a difficult task—the number of values in an FM scales as $2^n$ , where $n$ is the number of input sources, thus manually specifying an FM quickly becomes tedious. In this article, we review an automatic, data-supported method of learning the FM by minimizing a sum-of-squared error objective function in the context of decision-level fusion of classifiers using the Choquet FI. While this solves the specification problem, we illuminate an issue encountered with many real-world data sets; i.e., if the training data do not contain a significant number of all possible sort orders, many of the FM values are not supported by the data. We propose various regularization strategies to alleviate this issue by pushing the learned FM toward a predefined structure; these regularizers allow the user to encode knowledge of the underlying FM to the learning problem. Furthermore, we propose another regularization strategy that constrains the learned FM's structure to be a linear order statistic. Finally, we perform several experiments using synthetic and real-world data sets and show that our proposed extensions can improve the learned FM behavior and classification accuracy. A previously proposed visualization technique is employed to simultaneously quantitatively illustrate the FM as well as the FI.

The capacities and the Choquet integral are powerful tools to represent decision problems with dependencies and aggregate correlated decision criteria. Random generation of suitable capacities is a vital and challenging task in this decision model because of an exponential number of the involved parameters as well as the associated monotonicity restrictions. In this paper we present various approaches to representing decision makers’ preferences on aggregation through linear constraints , random generation of capacities from the selected polytope , testing the uniformity of the resulting distribution, and constructing the dominance relations between the alternatives, which are subsequently used to get the most credible ranking of the alternatives.

As a new method of granular computing, the three-way decision (3WD) approach has unique advantages in handling uncertain and imprecise problems. Based on decision-theoretic rough sets (DTRSs) and Bayesian minimum risk theory, conditional probability and loss function are the key research issues in 3WD. Many approaches for handling deterministic and complete information have been developed. However, few studies have focused on the construction of an intuitionistic fuzzy three-way decision (IF3WD) model for an intuitionistic fuzzy incomplete information system (IFIIS). In this paper, an IF3WD model based on an IFIIS is proposed to improve the ability to process complex fuzzy incomplete information systems, which extends the application range of the traditional 3WD. Concretely, we first propose a calculation method to measure the degree of information retention of missing data and describe it in two dimensions: coarse-grained and fine-grained. Next, an intuitionistic fuzzy number approximation (IFNA) strategy for missing data is presented. Then, a loss function with three states is given. Furthermore, combined with the Choquet integral, the interaction and influence between acceptance, rejection, and delay decision costs are investigated, and the corresponding IF3WD rules are induced. Finally, the rationality and effectiveness of our proposed model are verified through case analysis and are compared with those of existing methods.

Recently predictive models based on the Choquet integral have been applied successfully in machine learning and multi criteria decision making context. The ability of the Choquet integral to capture non-linear dependencies and its comprehensibility make it a very appealing tool. Yet, its complexity is often a barrier to estimate model parameters. In fact, the number of monotonicity constraints grows exponentially as the number of feature increases. This study addresses a heuristic approach to learn parameters underlying the choquistic regression model. In this regard, this study compares the gain of the proposed approach versus the original formalism of the choquistic regression. In addition, the run-time comparison in the experimental study is presented.

Zadeh's extension principle (ZEP) is a fundamental concept in fuzzy set (FS) theory that enables crisp mathematical operation on FSs. A well-known shortcoming of ZEP is that the height of the output FS is determined by the lowest height of the input FSs. In this article, we introduce a generalized extension principle (GEP) that eliminates this weakness and provides flexibility and control over how membership values are mapped from input to output. Furthermore, we provide a computationally efficient point-based FS representation. In light of our new definition, we discuss two approaches to perform aggregation of FSs using the Choquet integral. The resultant integrals generalize prior work and lay a foundation for future extensions. Last, we demonstrate the extended integrals via a combination of synthetic and real-world examples.

In view of the environment competencies, selecting the optimal green supplier is one of the crucial issues for enterprises, and multi-criteria decision-making (MCDM) methodologies can more easily solve this green supplier selection (GSS) problem. In addition, prioritized aggregation (PA) operator can focus on the prioritization relationship over the criteria, Choquet integral (CI) operator can fully take account of the importance of criteria and the interactions among them, and Bonferroni mean (BM) operator can capture the interrelationships of criteria. However, most existing researches cannot simultaneously consider the interactions, interrelationships and prioritizations over the criteria, which are involved in the GSS process. Moreover, the interval type-2 fuzzy set (IT2FS) is a more effective tool to represent the fuzziness. Therefore, based on the advantages of PA, CI, BM and IT2FS, in this paper, the interval type-2 fuzzy prioritized Choquet normalized weighted BM operators with $\lambda$ fuzzy measure and generalized prioritized measure are proposed, and some properties are discussed. Then, a novel MCDM approach for GSS based upon the presented operators is developed, and detailed decision steps are given. Finally, the applicability and practicability of the proposed methodology are demonstrated by its application in the shared-bike GSS and by comparisons with other methods. The advantages of the proposed method are that it can consider interactions, interrelationships and prioritizations over the criteria simultaneously.

The modern era of machine learning is focused on data-driven solutions. While this has resulted in astonishing leaps in numerous applications, explainability has not witnessed the same growth. The reality is, most machine learning solutions are black boxes. Herein, we focus on data/information fusion in machine learning. Specifically, we explore four eXplainable Artificial Intelligence (XAI) questions relative to Choquet integral; (i) what is the quality of our inputs and their interactions, (ii) how is the information being combined, (iii) what is the quality of our training data (and thus our learned models), and (iv) what trust do we place in an output? Previously, we derived an initial set of indices for (i)–(iv) on the premise of perfect knowledge. Herein, we make XAI more accurate by taking into consideration what the machine learned. A combination of synthetic data and real-world experiments from remote sensing for fusing deep learners in the context of classification are explored. Our approach leads to performance gain, insights into what was learned, and it helps us realize better future solutions.

The problem of hotel selection involves lots of uncertain information and multiple factors and can be identified as a multi-criteria decision-making problem. Aiming at the information description, measure, and fusion of hotel selection problems, this study develops multi-criteria decision-making methods based on Z-numbers and their projection measure and fusion techniques. To manage the three-dimensional structure of Z-numbers effectively, an optimization model is introduced to determine the potential probability distributions involved in Z-numbers. Then, the module, inner product, and cosine of Z-numbers are defined, and the projection measure of Z-numbers is presented by dealing with the three-dimensional structure of Z-numbers directly. Moreover, some Z-number Choquet integral projection operators are proposed for fusing Z-number evaluation information. Subsequently, an improved Multi-Objective Optimization by Ratio Analysis plus the Full Multiplicative Form method and an innovative pairwise evaluation-based multi-criteria decision-making method are developed. Based on the ideas of content-based recommendation and collaborative filtering recommendation, the above two methods are applied to solve hotel selection problems. And the sensitivity analysis and comparison discussion are conducted to demonstrate the applicability and validity of the two methods.

Entropy is the most important concept used in information theory and measuring uncertainty. In Choquet calculus, Sugeno (2013) [10] and Torra and Narukawa (2016) [2] studied Choquet integral and derivative with respect to monotone measures on the real line. Then as a very challenging problem, the definition of entropy and relative entropy on monotone measures for infinite sets based on Choquet integral was proposed by Torra (2017) [1] and Agahi (2019) [12] . These results show that based on the submodularity condition on monotone measures, entropy and relative entropy for Choquet integral are non-negative. In this paper, we first introduce the concept of Lin divergence (Lin, 1991, [8] ), including Choquet integral and derivative with respect to monotone measures. Then some fundamental properties of this concept in information theory are given. In special case, we show that we can omit the submodularity condition in previous results on entropy and relative entropy for Choquet integral.

The paper introduces a new class of functions from [0,1]n to [0,n] called d-Choquet integrals. These functions are a generalization of the 'standard' Choquet integral obtained by replacing the difference in the definition of the usual Choquet integral by a dissimilarity function. In particular, the class of all d-Choquet integrals encompasses the class of all 'standard' Choquet integrals but the use of dissimilarities provides higher flexibility and generality. We show that some d-Choquet integrals are aggregation/pre-aggregation/averaging/functions and some of them are not. The conditions under which this happens are stated and other properties of the d-Choquet integrals are studied. This work was supported in part by the Spanish Ministry of Science and Technology under project TIN2016-77356-P (AEI/FEDER, UE), by the Public University of Navarra under project PJUPNA13 and by grant VEGA 1/0614/18 . Z. Takáč was supported by the project VEGA 1/0545/20. R. Mesiar was supported by the project of Grant Agency of the Czech Republic (GACR) no. 18-06915S and by the Slovak grant APVV-17-0066 . G.P. Dimuro was supported by Brazilian agency CNPq under the grant 301618/2019-4 and FAPERGS (Proc. 19/2551-0001660 ). B. Bedregal was supported by Brazilian agency CNPq under the grant 307781/2016-0 and Caixa y Fundación Caja Navarra.

The concept of sponge city receives more and more attention by Chinese government, and the evaluation of sponge city construction is an important aspect. To cope with the complexity and uncertainty of the evaluation process, this paper adopts interval type-2 trapezoidal fuzzy numbers (IT2TFNs) to express decision-making information and develops an approach for evaluating sponge city construction. To do these, two prioritized-guided interval type-2 trapezoidal fuzzy Hamacher operators are first defined to infuse IT2TFNs offered by experts, which can cope with the situation where there is prioritization among experts/attributes. In order to further consider the interactions among experts/attributes, two generalized-Shapley interval type-2 trapezoidal fuzzy prioritized Hamacher Choquet integral operators are presented. To measure the discrimination degree between IT2TFNs, a new interval type-2 trapezoidal fuzzy cross-entropy is defined. After that, cross-entropy based models for obtaining the optimal fuzzy measure on the expert/attribute set are constructed to handle the situation where the weighting information is interactive and partly known. Furthermore, an interval type-2 trapezoidal fuzzy multi-attribute group decision-making approach is developed. Finally, a practical example about the evaluation of residential land design plans in sponge city is provided to illustrate the utilization of the new method, and comparison analysis is provided.

Action recognition based on skeleton key joints has gained popularity due to its cost effectiveness and low complexity. Existing Convolutional Neural Network (CNN) based models mostly fail to capture various aspects of the skeleton sequence. To this end, four feature representations, which capture complementary characteristics of the sequence of key joints, are extracted with novel contribution of features estimated from angular information, and kinematics of the human actions. Single channel grayscale images are used to encode these features for classification using four CNNs, with the complementary nature verified through Kullback-Leibler (KL) and Jensen-Shannon (JS) divergences. As opposed to straightforward classifier combination generally used in existing literature, fuzzy fusion through the Choquet integral leverages the degree of uncertainty of decision scores obtained from four CNNs. Experimental results support the efficacy of fuzzy combination of CNNs to adaptively generate final decision score based upon confidence of each information source. Impressive results on the challenging UTD-MHAD, HDM05, G3D, and NTU RGB+D 60 and 120 datasets demonstrate the effectiveness of the proposed method. The source code for our method is available at https://github.com/theavicaster/fuzzy-integral-cnn-fusion-3d-har

In this paper we discuss integral inequalities for collection integrals that are a special subclass of decomposition integrals introduced as a general framework for many non-linear integrals, including the Choquet integral, the Shilkret integral, the PAN integral, and the concave integral. We give a full characterization of collection integrals that are comonotone additive and for which Chebyshev's, Jensen's, Cauchy's, and Holder's integral inequalities hold. Interestingly, all these classes of collection integrals coincide and thus we introduce a special subclass of collection integrals, called PCC integrals. The paper is complemented with some examples and remarks for collection and decomposition integrals.

We further investigate (M)-property, an important structure characteristic of monotone measures. Some necessary and/or sufficient conditions of (M)-property are shown and some characteristics of (M)-property are described. It is shown that the restriction of a monotone measure to an atom of its own possesses (M)-property on this atom. By means of the (M)-property we characterize the equivalence among the Choquet integral, the pan-integral, the concave integral, and the Shilkret integral on atoms of monotone measures and obtain some special results. We also show a necessary and sufficient condition for the concave integral and the pan-integral to coincide on atoms of monotone measures.

To deal with multicriteria decision making (MCDM) problems with interaction criteria, the Choquet integral (CI) is one of effective tools. This paper first proposes the reverse Choquet integral (RCI), which defines the importance of the ordered elements in an opposite principle to the CI. To show the principle of the RCI, we offer its concrete expression in view of the Mobius representation by which one can clearly see the difference and the relationship between the CI and the RCI. Then, we propose the “bi-direction Choquet integral” (BDCI), which is a convex combination of the CI and the RCI. To get the interactions of ordered coalitions comprehensively, this paper further proposes the generalized Shapley bi-direction Choquet integral (GSBDCI). Furthermore, the hybrid generalized Shapley bi-direction Choquet integral (HGSBDCI) is proposed, which defines the importance of ordered positions and the criteria with interactions simultaneously. With respect to these types of CIs, their exponent forms are also discussed. Finally, we use an application case to show the utilization of the proposed new CIs for MCDM. The proposed new Choquet integrals provide us a very useful way to deal with MCDM problems.

Recently, we have presented a new family of kernels on the basis of the discrete Choquet integral. While a naive computation of this kernel has an exponential complexity in the number of features, we have proposed an efficient approach with computational complexity of O ( m 2 log ( m ) ) . 1 This kernel family is able to recognize dependencies between features and moreover it can be regularized through a proper selection of q-additivity. In fact, to reduce the effect of over-fitting there is an opportunity to restrict the flexibility of kernel to a lower degree. A key feature of the Choquet integral in a data-driven way is its monotonicity, however, this representation does not consider any monotonicity constraint; hence it is versatile for other applications, too. This issue is highlighted in the experimental study. In this regard, we apply the Choquet kernel for regression task and compare the performance of the proposed kernel versus state-of-the-art support kernel-based regression methods as well as random forest.

The construction of functions that simultaneously generalize weighted means and OWA operators is an interesting topic that has received special attention in recent years. Due to the properties they satisfy, one of the most interesting generalization are SUOWA operators, which have been widely studied in the literature. In a recent paper, a new generalization has been introduced, the Semi-SUOWA operators, which have a close relationship with SUOWA operators. The main aim of this paper is to analyze the games associated with Semi-SUOWA operators. In this respect, we give conditions under which we can guarantee the monotonicity of these games. Moreover, we establish some relationships between some games associated with SUOWA and Semi-SUOWA operators and show the pointwise convergence of certain games.

Yager has proposed the decision making under measure-based granular uncertainty, which can make decision with the aid of Choquet integral, measure and representative payoffs. The decision making under measure-based granular uncertainty is an effective tool to deal with uncertain issues. The intuitionistic fuzzy environment is the more real environment. Since the decision making under measure-based granular uncertainty is not based on intuitionistic fuzzy environment, it cannot effectively solve the decision issues in the intuitionistic fuzzy environment. Then, when the issues of decision making are under intuitionistic fuzzy environment, what is the decision making under measure-based granular uncertainty with intuitionistic fuzzy sets is still an open issue. To deal with this kind of issues, this paper proposes the decision making under measure-based granular uncertainty with intuitionistic fuzzy sets. The decision making under measure-based granular uncertainty with intuitionistic fuzzy sets can effectively solve the decision making issues in the intuitionistic fuzzy environment, in other words, it can extend the decision making under measure-based granular uncertainty to the intuitionistic fuzzy environment. Numerical examples are applied to verify the validity of the decision making under measure-based granular uncertainty with intuitionistic fuzzy sets. The experimental results demonstrate that the decision making under measure-based granular uncertainty with intuitionistic fuzzy sets can represent the objects successfully and make decision effectively. In addition, a practical application of applied intelligence is used to compare the performance between the proposed model and the decision making under measure-based granular uncertainty. The experimental results show that the proposed model can solve some decision problems that the decision making under measure-based granular uncertainty cannot solve.

In multiattribute decision-making (MADM), more and more attention is paid to the interaction between attributes when considering the actual decision environment. As a result, interactive MADM has become an emerging and challenging area of research whose success will greatly facilitate the development of decision-making. This paper models the interactive MADM problem and its contribution is multifaceted. First, the concept of probabilistic linguistic term set (PLTS) is extended by Dempster–Shafer theory (DST), which helps to express more uncertain information, followed by some basic operations, such as score function and aggregation operator. In virtue of evidential best-worst method and the principle of maximum entropy, then a novel nonadditive measure determination method is developed based on the 2-order additive measure to better model the interaction between attributes. Further, the generalized PLTS-based Choquet integral is defined by which generalized PLTSs on a nonadditive measure can be reasonably aggregated. Finally, an interactive MADM model is constructed and the technical details are described. The proposed approach is implemented to select the supplier for medical devices, and its effectiveness is emphasized by comparison with other methods.

In group decision-making problems, decision makers prefer to use several linguistic terms to describe their own perception and knowledge, and give their preference intensity of each possible linguistic terms based on their own understanding and interpretation. Due to the nonlinearity of decision maker’s cognition, the gaps between adjacent linguistic terms are unbalanced. The unbalanced probabilistic linguistic term set (UPLTS) is proposed to present such situation. To this phenomenon, a resolution framework is constructed to analyze multiple attribute group decision-making problems under unbalanced probabilistic linguistic environment. Firstly, the integration model based on evidential reasoning theory is proposed to aggregate UPLTSs from different groups in view of incomplete probabilistic distributions in UPLTS. Secondly, the transformation function based on proportional 2 tuple is developed to transform UPLTS into probabilistic linguistic term set, making it easier for subsequent analysis and processing. Thirdly, Based on the multiple types of partitioned structure relationship among attributes, partitioned fuzzy measure is developed to globally capture these interactions among attributes. Then the probabilistic linguistic Choquet integral operator with partitioned fuzzy measure is proposed to obtain the comprehensive performances of alternatives. Lastly, the effectiveness and practicability of the proposed method is demonstrated using three numerical examples and comparing with other methods.

The uncertain database retrieval with measure-based belief function attribute values can resolve concerns of database retrieval and decision making, particularly in the circumstances of classical fuzzy set, which has an assuring perspective. However, how to implement the design to explore the things based on the intuitionistic fuzzy set (IFS) in an ambiguous database is yet an open problem. This paper addresses the issues of database retrieval based on the uncertain database with the things related to IFS. In the design, a query associated with each attribute of the objects is located in terms of IFS. Based on the gathered data, the “plausibilities (Pl)” and “beliefs (Bel)” measures are measured for each attribute and then aggregated with the cooperation of the Choquet integral (CI) operator. The “satisfaction degree” of the query in the form of an interval is circumscribed as [Bel, Pl]. The defuzzified value of this interval is affirmed with the aid of golden rule representative value, to link these satisfaction degrees. Hence, the appearance of the stated algorithm is more trustworthy than the others under an unpredictable environment. The description of the asserted algorithm is illustrated with an application correlated to the library database.

Existing winner determination models tend to overlook the sustainable attributes of third party logistics (3PL) providers. This paper investigates a novel green winner determination problem that has several features, including (i) the sustainable attributes with conflicting and interactive properties of potential 3PLs, and (ii) the loss-averse behavior with an internal reference point of a fourth party logistics (4PL) provider. For the attributes with a combination of crisp data, interval numbers and intuitionistic 2-tuple linguistic terms, we integrate the prospect theory (PT) and Choquet integral with the “benefits, opportunities, costs and risks (BOCR)” framework to propose a novel PTC-BOCR solution method. Numerical experiments are conducted to illustrate the effectiveness and applicability of PTC-BOCR by comparing it with some known methods. Comparison analysis indicates that PTC-BOCR is robust with respect to the variance of 3PLs’ attribute values, while behavioral parameter analysis reveals that the loss-averse behavior of the 4PL is intensified as the difference of 3PLs varies. Managerial insights are also drawn for green 3PLs to win the auction. This study is a significant extension of traditional decision-making methods, which could benefit the realization of a sustainable logistics system in a cost-effective way for 4PLs.