Some notes on the pan-integrals of set-valued functions

Article Computer Science, Theory & Methods

Choquet integrals of set-valued functions with respect to set-valued fuzzy measures

Deli Zhang et al.

Summary: As powerful tools for modeling nondeterministic problems, Choquet integrals have attracted much attention in recent years. This study generalizes the existing set-valued Choquet integrals and formulates the theory of Choquet integrals with respect to set -valued fuzzy measures. Various properties, convergence theorems, and set-valued Jensen's inequalities are obtained.

FUZZY SETS AND SYSTEMS (2023)

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Article Computer Science, Theory & Methods

Pan-integrals of set-valued functions based on fuzzy measures

Tong Kang et al.

Summary: This paper proposes the set-valued pan-integral based on number-valued fuzzy measure and explores its properties. The monotonicity of the set-valued pan-integral is demonstrated using a preorder on the class of all nonempty sets. The linearity of the set-valued pan-integral is shown through an equivalence relation based on the preorder. The relationship between the set-valued pan-integral and the set-valued Choquet integral is discussed, and Chebyshev's inequality of set-valued pan-integrals is proven. An open problem regarding the linearity of the set-valued pan-integral is raised.

FUZZY SETS AND SYSTEMS (2023)

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Article Computer Science, Information Systems

Choquet type integrals for single-valued functions with respect to set-functions and set-multifunctions

Deli Zhang et al.

Summary: In this study, two generalization types of Choquet integrals are presented. First, a generalized Choquet type integral of a single-valued function is introduced with respect to a set-function and measure. Several of its properties, such as convergence theorems and Jensen's inequality, are proved. Second, in the spirit of the single-valued Choquet integral, a generalized Choquet type set-valued integral for a single-valued function with respect to a set-multifunction and measure is introduced using Aumann integrals as well as various properties, including convergence theorems.

INFORMATION SCIENCES (2023)

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Article Mathematics