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In multi-criteria decision-making, the Choquet integral can handle the interaction between criteria; however, the interaction between criteria is not only related to the criteria themselves but may also depend on the performance values of criteria. We present a further extension of the Choquet integral considering this fact. A classic example is used to illustrate the limitations of the Choquet integral, and a performance-dependent assumption is proposed that describes the interactions influenced by the performance of alternatives on criteria. Based on this assumption, we propose an alternative capacity and construct an improved Choquet integral to obtain alternative’s global utility. A preference model based on the improved Choquet integral with a wider range of applicability and more realistic decision results is proposed. Finally, a numerical example is provided to illustrate the feasibility and validity of our reference model.

In this work, we propose a definition of comonotonicity for elements of $B{(H)}_{sa}$, i.e. bounded self-adjoint operators defined over a complex Hilbert space $H$. We show that this notion of comonotonicity coincides with a form of commutativity. Intuitively, comonotonicity is to commutativity as monotonicity is to bounded variation. We also define a notion of Choquet expectation for elements of $B{(H)}_{sa}$ that generalizes quantum expectations. We characterize Choquet expectations as the real-valued functionals over $B{(H)}_{sa}$ which are comonotonic additive, $c$-monotone, and normalized.

A multilinear extension of the n-person cooperative game was introduced by Owen in 1972, and a new extension method is proposed in this paper. For n-person cooperative games, any coalition can equivalently be represented by its characteristic vectors. By means of the Choquet integral, a new fuzzy extension, called the Choquet extension, is developed. Furthermore, a Shapley function in this class of fuzzy cooperative games with the Choquet extension form is defined. Axioms of the Shapley function are proposed, and an explicit formula for the Shapley function is given. Finally, an equivalent definition of this Shapley function is discussed.

We present an approach to automatically extract a pertinent subset of soft output classifiers, and to aggregate them into a global decision rule using the Choquet integral. This approach relies on two key points. The first is a learning algorithm that uses a measure of the confusion between the categories to be recognized. The second is a selection scheme that discards weak or redundant decision rules, keeping only the most relevant subset. An experimental study, based on real world data, is then described. It analyzes the improvements achieved by these points first when used independently, then when combined together.

Due to the radical change in both Chinese and global economic environment, it is essential to develop a practical model to predict financial distress. The support vector machine (SVM), a new outstanding learning machine based on the statistical learning theory, embodying the principle of structural risk minimization instead of empirical risk minimization principle, is a promising method for such financial distress prediction. However, to some extent, the performance of single classifier depends on the sample's pattern characteristics and each single classifier has its own uncertainty. Using the ensemble methods to predict financial distress becomes a rising trend in this field. This research puts forward a SVM ensemble based on the Choquet integral for financial distress prediction in which Bagging algorithm is used to generate new training sets. The proposed ensemble method can be expressed as "Choquet + Bagging + SVMs". With real data from Chinese listed companies, an experiment is carried out to compare the performance of single classifiers with the proposed ensemble method. Empirical results indicate that the proposed ensemble of SVMs based on the Choquet integral for financial distress prediction has higher average accuracy and stability than single SVM classifiers.

Software architects cannot avoid the consideration of quality attributes when designing software architecture. Architectural styles such as Layers and Client-Server are often used by architects to describe the overall structure and behavior of software. Although an architectural style affects the achievement of quality attributes, these quality attributes are directly performed by design decisions called architectural tactics. While the implementation of an architectural tactic supports a specific quality attribute, it often enhances or hurts other quality attributes in the software. In this paper, a framework for selecting the most appropriate architectural tactics according to their best achievement of the required levels of quality attributes when developing transaction processing systems is proposed. The proposed framework is based on fuzzy measures using Choquet Integral approach and takes into account the impact of architectural tactics on quality attributes, the preferences of quality attributes and the interactions between them. It can also be used to compare different potential architectures in terms of their supporting of quality attributes. The abilities and the advantages of the proposed framework are clarified via practical experiments using a case study.

This paper discusses multiattribute preference relations compatible with a value/utility function represented by the Choquet integral with respect to a fuzzy measure, and shows that the additivity of the fuzzy measure is equivalent to each of mutual preferential independence, mutual weak difference independence, mutual difference independence, mutual utility independence, and additive independence.

In this paper, we discuss the defuzzification problem. We first propose a set defuzzification method, (from a fuzzy set to a crisp set) by using the Aumann integral. From the obtained set to a point, we have two methods of defuzzification. One of these uses the mean value method and the other uses a fuzzy measure. In the first case, we compare our mean value method with the method of the center of gravity. In the second case, we compare fuzzy measure method with the Choquet integral method. We also give there a sufficient condition so that the results in the last two methods are equivalent.

Previous work has shown that the Choquet integral w.r.t. a symmetric k-order additive fuzzy measure is characterized as a weighted sum, with possibly negative weights, of k specific OWA operators, and that the Sugeno integral w.r.t. a symmetric k-order maxitive fuzzy measure is characterized as a weighted maximum of k order statistics. In this paper, these characterizations are extended to the general, non-symmetric case, thereby providing additional insight in the symmetric case. More specifically, a new general decomposition of the Choquet integral w.r.t. a k-order additive fuzzy measure and of the Sugeno integral w.r.t. a k-order maxitive fuzzy measure are established.

The intepretation of aggregation functions in multicriteria decision making is often based on indices such as importance indices that measure the importance of one criterion. For instance, for the Choquet integral, the importance index is the so-called Shapley value. The use of an index must always be limited to the specific context it has been designed for. Here, we are interested in determining on which criteria acts should be improved if we want their global evaluation to increase as much as possible. The Shapley value is not suited for describing this. So, we introduce a new index of importance which represents the mean worth for acts to reach higher scores in a set of criteria. This index is defined for general aggregation functions with the help of several axioms. This importance index is then applied to the Choquet integral. In particular, we computer the worth to reach higher levels in one attribute, and in a couple of attributes. Interestingly, this leads to quantities that are closely related to the Shapley and interaction indices.

In this paper we study the extension of Choquet integrals to ordinal scales. We show that two different integrals can be defined on the basis of two different but equivalent expressions in the numerical case. We prove some properties of the Choquet integrals and show their application to aggregation.

The Choquet and Sugeno (fuzzy) integrals are examples of piecewise linear functions on a finite dimensional space ℝⁿ. In this paper we consider piecewise linear functions as aggregation functions and establish a Max-Min polynomial representation of these functions.

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 are studied, as well as their Choquet integral. Finally, a degree of interaction between the subsets of indifference is defined.

This paper treats the problem of fitting general aggregation operators with unfixed number of arguments to empirical data. We discuss methods applicable to associative operators (t-norms, t-conorms, uninorms and nullnorms), means and Choquet integral based operators with respect to a universal fuzzy measure. Special attention is paid to k-order additive symmetric fuzzy measures.

We provide a survey of recent developments about capacities (or fuzzy measures) and cooperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.

Some theorems concerning representation of fuzzy measure and the Choquet integral are shown independently by Murofushi and Sugeno (J. Math. Anal. Appl. 159, No 2, (1991), 532-549) and by Denneberg (Fuzzy sets and Systems, 92, (1997), 139-156). Using the mediator for representations, it is shown that the theorems are partially equivalent. The difference between them are also stated with some examples.

Linguistic information aggregation has received great attention from researchers, and a variety of operators have been developed for aggregating linguistic information. All the existing linguistic information aggregation operators only consider the situations where all the aggregated linguistic arguments are independent, i.e., they only consider the addition of the importance of individual linguistic arguments, however, in some actual situations, the considered linguistic arguments may be correlative. In this paper, we focus on this issue. Motivated by the idea of the well-known Choquet integrals,¹ we propose two new linguistic information aggregation operators called the linguistic correlated averaging operator and linguistic correlated geometric operator. In the special cases where the aggregated linguistic arguments are independent, the linguistic correlated averaging operator can be reduced to a variety of traditional linguistic averaging aggregation operators; while the linguistic correlated geometric operator can be reduced to a variety of the traditional linguistic geometric aggregation operators. Furthermore, we extend the above results to accommodate uncertain linguistic environments, and illustrate them with a practical problem.

The stability of discrete universal integrals based on copulas is discussed and examined, both with respect to the norms L₁ (Lipschitz stability) and L_∞ (Chebyshev stability). Each of these integrals is shown to be 1-Lipschitz. Exactly the discrete universal integrals based on a copula which is stochastically increasing in its first coordinate turn out to be 1-Chebyshev. A new characterization of stochastically increasing Archimedean copulas is also given.

A new metric is proposed on the space of measurable functions in the setting of non-additive measure theory. The convergence induced from the metric can be used to describe the convergence in measure for sequences of measurable functions. Furthermore, the space of measurable functions is complete under the metric.

Induced quasi-arithmetic aggregation operators are considered to aggregate uncertain linguistic information by using order inducing variables. We introduce the induced correlative uncertain linguistic aggregation operator with Choquet integral and we also present the induced uncertain linguistic aggregation operator by using the Dempster-Shafer theory of evidence. The special cases of the new proposed operators are investigated. Many existing linguistic aggregation operators are special cases of our new operators and more new uncertain linguistic aggregation operators can be derived from them. Decision making methods based on the new aggregation operators are proposed and architecture material supplier selection problems are presented to illustrate the feasibility and efficiency of the new methods.