• Title/Summary/Keyword: Choquet integral.

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Structural characterizations of monotone interval-valued set functions defined by the interval-valued Choquet integral (구간치 쇼케이적분에 의해 정의된 단조 구간치 집합함수의 구조적 성질에 관한 연구)

  • Jang, Lee-Chae
    • Journal of the Korean Institute of Intelligent Systems
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    • v.18 no.3
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    • pp.311-315
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    • 2008
  • We introduce nonnegative interval-valued set functions and nonnegative measurable interval-valued Junctions. Then the interval-valued Choquet integral determines a new nonnegative monotone interval-valued set function which is a generalized concept of monotone set function defined by Choquet integral in [17]. We also obtained absolutely continuity of them in [9]. In this paper, we investigate some characterizations of the monotone interval-valued set function defined by the interval-valued Choquet integral, and such as subadditivity, superadditivity, null-additivity, converse-null-additivity.

Choquet integrals and interval-valued necessity measures (쇼케이 적분과 구간치 필요측도)

  • Jang, Lee-Chae;Kim, Tae-Kyun
    • Journal of the Korean Institute of Intelligent Systems
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    • v.19 no.4
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    • pp.499-503
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    • 2009
  • Y. R$\acute{e}$ball$\acute{e}$ [11] discussed the representation of necessity measure through the Choquet integral criterian. He also consider a decision maker who ranks necessity measures related with Choquet integral representation. In this paper, we consider a decision maker have an "ambiguity"(say, interval-valued) necessity measure according to their Choquet's expected utility. Furthermore, we prove two theorems which are weak Choquet integral representation of preferences with a monotone set function for interval-valued necessity measures and strong Choquet integral representation of preferences with an interval-valued utility function for necessity measures.

Interval-valued Choquet Integrals and applications in pricing risks (구간치 쇼케이적분과 위험률 가격 측정에서의 응용)

  • Jang, Lee-Chae
    • Proceedings of the Korean Institute of Intelligent Systems Conference
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    • 2007.04a
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    • pp.209-212
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    • 2007
  • Non-additive measures and their corresponding Choquet integrals are very useful tools which are used in both insurance and financial markets. In both markets, it is important to to update prices to account for additional information. The update price is represented by the Choquet integral with respect to the conditioned non-additive measure. In this paper, we consider a price functional H on interval-valued risks defined by interval-valued Choquet integral with respect to a non-additive measure. In particular, we prove that if an interval-valued pricing functional H satisfies the properties of monotonicity, comonotonic additivity, and continuity, then there exists an two non-additive measures ${\mu}_1,\;{\mu}_2$ such that it is represented by interval-valued choquet integral on interval-valued risks.

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Interval-valued Choquet integrals and applications in pricing risks (구간치 쇼케이적분과 위험률 가격 측정에서의 응용)

  • Jang, Lee-Chae
    • Journal of the Korean Institute of Intelligent Systems
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    • v.17 no.4
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    • pp.451-454
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    • 2007
  • Non-additive measures and their corresponding Choquet integrals are very useful tools which are used in both insurance and financial markets. In both markets, it is important to update prices to account for additional information. The update price is represented by the Choquet integral with respect to the conditioned non-additive measure. In this paper, we consider a price functional H on interval-valued risks defined by interval-valued Choquet integral with respect to a non-additive measure. In particular, we prove that if an interval-valued pricing functional H satisfies the properties of monotonicity, comonotonic additivity, and continuity, then there exists an two non-additive measures ${\mu}1,\;{\mu}2$ such that it is represented by interval-valued choquet integral on interval-valued risks.

A note on Jensen type inequality for Choquet integrals

  • Jang, Lee-Chae
    • International Journal of Fuzzy Logic and Intelligent Systems
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    • v.9 no.2
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    • pp.71-75
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    • 2009
  • The purpose of this paper is to prove a Jensen type inequality for Choquet integrals with respect to a non-additive measure which was introduced by Choquet [1] and Sugeno [20]; $$\Phi((C)\;{\int}\;fd{\mu})\;{\leq}\;(C)\;\int\;\Phi(f)d{\mu},$$ where f is Choquet integrable, ${\Phi}\;:\;[0,\;\infty)\;\rightarrow\;[0,\;\infty)$ is convex, $\Phi(\alpha)\;\leq\;\alpha$ for all $\alpha\;{\in}\;[0,\;{\infty})$ and ${\mu}_f(\alpha)\;{\leq}\;{\mu}_{\Phi(f)}(\alpha)$ for all ${\alpha}\;{\in}\;[0,\;{\infty})$. Furthermore, we give some examples assuring both satisfaction and dissatisfaction of Jensen type inequality for the Choquet integral.

Finding the Mostly Preferred Solution for MADM Problems Using Fuzzy Choquet's Integral (퍼지 Choquet적분을 이용한 다속성 의사결정문제의 최적 선호대안 결정)

  • Cho, Sung-Ku;Lee, Kang-In
    • Journal of Korean Institute of Industrial Engineers
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    • v.23 no.4
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    • pp.635-643
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    • 1997
  • The purpose of this paper is to propose an interactive method, using fuzzy Choquet's integral, which is designed to find out the mostly preferred solutions for deterministic MADM problems with many attributes and alternatives. The basic idea of the paper is essentially the same as that of the one we have published before[1]; subgrouping of attributes and eliminating of inefficient solutions. But the difference between these two methods lies in the fact that the present method evaluates and eliminates alternatives using fuzzy Choquet's integral on the basis of decision-maker's judgements about the relative importance of subgroups of attributes, rather than using mathematical programming on the basis of pair-wise comparisons of alternatives. If such information is obtainable from the decision-maker, the method can be proved to be much easier to understand and more efficient to compute.

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A NOTE ON THE MONOTONE INTERVAL-VALUED SET FUNCTION DEFINED BY THE INTERVAL-VALUED CHOQUET INTEGRAL

  • Jang, Lee-Chae
    • Communications of the Korean Mathematical Society
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    • v.22 no.2
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    • pp.227-234
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    • 2007
  • At first, we consider nonnegative monotone interval-valued set functions and nonnegative measurable interval-valued functions. In this paper we investigate some properties and structural characteristics of the monotone interval-valued set function defined by an interval-valued Choquet integral.

Subsethood Measures Defined by Choquet Integrals

  • Jang, Lee-Chae
    • International Journal of Fuzzy Logic and Intelligent Systems
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    • v.8 no.2
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    • pp.146-150
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    • 2008
  • In this paper, we consider concepts of subsethood measure introduced by Fan et al. [2]. Based on this, we give various subsethood measure defined by Choquet integral with respect to a fuzzy measure on fuzzy sets which is often used in information fusion and data mining as a nonlinear aggregation tool and discuss some properties of them. Furthermore, we introduce simple examples.