Derived fuzzy measures and derived Choquet integrals with some properties

2020 ◽  
pp. 1-1 ◽  
Author(s):  
LeSheng Jin ◽  
Radko Mesiar ◽  
Ronald R. Yager
Keyword(s):  
Fuzzy Measures ◽  
Choquet Integrals

Fuzzy Measures and Choquet Integrals based on Fuzzy Covering Rough Sets

2021 ◽  
pp. 1-1
Author(s):  
Xiaohong Zhang ◽  
Jingqian Wang ◽  
Jianming Zhan ◽  
Jianhua Dai
Keyword(s):  
Rough Sets ◽  
Fuzzy Measures ◽  
Choquet Integrals

scholarly journals Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Chen Li ◽  
Gong Zeng-tai ◽  
Duan Gang
Keyword(s):  
Genetic Algorithm ◽  
Genetic Algorithms ◽  
Real Data ◽  
Difficult Problem ◽  
Particle Swarm Algorithm ◽  
Fuzzy Data ◽  
Algorithm Optimization ◽  
Practical Applications ◽  
Fuzzy Measures ◽  
Choquet Integrals

Fuzzy measures and fuzzy integrals have been successfully used in many real applications. How to determine fuzzy measures is a very difficult problem in these applications. Though there have existed some methodologies for solving this problem, such as genetic algorithms, gradient descent algorithms, neural networks, and particle swarm algorithm, it is hard to say which one is more appropriate and more feasible. Each method has its advantages. Most of the existed works can only deal with the data consisting of classic numbers which may arise limitations in practical applications. It is not reasonable to assume that all data are real data before we elicit them from practical data. Sometimes, fuzzy data may exist, such as in pharmacological, financial and sociological applications. Thus, we make an attempt to determine a more generalized type of general fuzzy measures from fuzzy data by means of genetic algorithms and Choquet integrals. In this paper, we make the first effort to define theσ-λrules. Furthermore we define and characterize the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based onσ-λrules. In addition, we design a special genetic algorithm to determine a type of general fuzzy measures from fuzzy data.

scholarly journals Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zengtai Gong ◽  
Li Chen ◽  
Gang Duan
Keyword(s):  
Integral Equations ◽  
Real Line ◽  
Fuzzy Number ◽  
Laplace Transformation ◽  
Choquet Integral ◽  
Fuzzy Measures ◽  
Nonadditive Measure ◽  
Choquet Integrals ◽  
Lebesgue Measures ◽  
Interval Valued

This paper deals with the Choquet integral of fuzzy-number-valued functions based on the nonnegative real line. We firstly give the definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on the nonadditive measure. Furthermore, the operational schemes of above several classes of integrals on a discrete set are investigated which enable us to calculate Choquet integrals in some applications. Secondly, we give a representation of the Choquet integral of a nonnegative, continuous, and increasing fuzzy-number-valued function with respect to a fuzzy measure. In addition, in order to solve Choquet integral equations of fuzzy-number-valued functions, a concept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations of fuzzy-number-valued functions can be solved by the Laplace transformation. Finally, an example is given to illustrate the main results at the end of the paper.

scholarly journals Forming a Hierarchical Choquet Integral with a GA-Based Heuristic Least Square Method

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1155
Author(s):  
Chen ◽  
Huang
Keyword(s):  
Choquet Integral ◽  
Least Square Method ◽  
Least Square ◽  
Integral Model ◽  
Fuzzy Measures ◽  
Proposed Model ◽  
Choquet Integrals ◽  
The Mean ◽  
Input Variables ◽  
Least Mean Squares Algorithm

: Identifying the fuzzy measures of the Choquet integral model is an important component in resolving complicated multi-criteria decision-making (MCDM) problems. Previous papers solved the above problem by using various mathematical programming models and regression-based methods. However, when considering complicated MCDM problems (e.g., 10 criteria), the presence of too many parameters might result in unavailable or inconsistent solutions. While k-additive or p-symmetric measures are provided to reduce the number of fuzzy measures, they cannot prevent the problem of identifying the fuzzy measures in a high-dimension situation. Therefore, Sugeno and his colleagues proposed a hierarchical Choquet integral model to overcome the problem, but it required the partition information of the criteria, which usually cannot be obtained in practice. In this paper, we proposed a GA-based heuristic least mean-squares algorithm (HLMS) to construct the hierarchical Choquet integral and overcame the above problems. The genetic algorithm (GA) was used here to determine the input variables of the sub-Choquet integrals automatically, according to the objective of the mean square error (MSE), and calculated the fuzzy measures with the HLMS. Then, we summed these sub-Choquet integrals into the final Choquet integral for the purpose of regression or classification. In addition, we tested our method with four datasets and compared these results with the conventional Choquet integral, logit model, and neural network. On the basis of the results, the proposed model was competitive with respect to other models.

On a Choquet-Stieltjes type integral on intervals

2019 ◽  
Vol 69 (4) ◽  
pp. 801-814 ◽  
Author(s):  
Sorin G. Gal
Keyword(s):  
Lebesgue Measure ◽  
Choquet Integral ◽  
Stieltjes Integral ◽  
Line Integral ◽  
Type Integral ◽  
Choquet Integrals ◽  
Lebesgue Measures ◽  
Stieltjes Type

Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

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