On Fuzzy Simulations for Expected Values of Functions of Fuzzy Numbers and Intervals

Based on existing fuzzy simulation algorithms, this article presents two innovative techniques for approximating the expected values of fuzzy numbers' monotone functions, which is of utmost importance in fuzzy optimization literature. In this regard, the stochastic discretization algorithm of L...

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Published in:IEEE transactions on fuzzy systems Vol. 29; no. 6; pp. 1446 - 1459
Main Authors: Liu, Yuanyuan, Miao, Yunwen, Pantelous, Athanasios A., Zhou, Jian, Ji, Ping
Format: Journal Article
Language:English
Published: New York IEEE 01.06.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Analytical models
Approximation algorithms
Business
Computational modeling
Discretization
Electronic mail
Expected value
Expected values
fuzzy simulation
Intervals
Mathematical analysis
Monotone functions
Numerical integration
Numerical models
Optimization
regular fuzzy interval
regular fuzzy number
Simulation
Stochastic processes
ISSN:1063-6706, 1941-0034
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Abstract Based on existing fuzzy simulation algorithms, this article presents two innovative techniques for approximating the expected values of fuzzy numbers' monotone functions, which is of utmost importance in fuzzy optimization literature. In this regard, the stochastic discretization algorithm of Liu and Liu (2002) is enhanced by updating the discretization procedure for the simulation of the membership function and the calculation formula for the expected values. This is achieved through initiating a novel uniform sampling process and employing a formula for discrete fuzzy numbers, respectively, as the generated membership function in the stochastic discretization algorithm would adversely affect its accuracy to some extent. What is more, considering that the bisection procedure involved in the numerical integration algorithm of Li (2015) is time-consuming and also, not necessary for the specified types of fuzzy numbers, a special numerical integration algorithm is proposed, which can simplify the simulation procedure by adopting the analytical expressions of <inline-formula><tex-math notation="LaTeX">\alpha</tex-math></inline-formula>-optimistic values. Subsequently, concerning the extensive applications of regular fuzzy intervals, several theorems are introduced and proved as an extended effort to apply the improved stochastic discretization algorithm and the special numerical integration algorithm to the issues of fuzzy intervals. Throughout this article, a series of numerical experiments are conducted from which the superiority of both the two novel techniques over others are conspicuously displayed in aspects of accuracy, stability, and efficiency.
AbstractList Based on existing fuzzy simulation algorithms, this article presents two innovative techniques for approximating the expected values of fuzzy numbers’ monotone functions, which is of utmost importance in fuzzy optimization literature. In this regard, the stochastic discretization algorithm of Liu and Liu (2002) is enhanced by updating the discretization procedure for the simulation of the membership function and the calculation formula for the expected values. This is achieved through initiating a novel uniform sampling process and employing a formula for discrete fuzzy numbers, respectively, as the generated membership function in the stochastic discretization algorithm would adversely affect its accuracy to some extent. What is more, considering that the bisection procedure involved in the numerical integration algorithm of Li (2015) is time-consuming and also, not necessary for the specified types of fuzzy numbers, a special numerical integration algorithm is proposed, which can simplify the simulation procedure by adopting the analytical expressions of [Formula Omitted]-optimistic values. Subsequently, concerning the extensive applications of regular fuzzy intervals, several theorems are introduced and proved as an extended effort to apply the improved stochastic discretization algorithm and the special numerical integration algorithm to the issues of fuzzy intervals. Throughout this article, a series of numerical experiments are conducted from which the superiority of both the two novel techniques over others are conspicuously displayed in aspects of accuracy, stability, and efficiency.
Based on existing fuzzy simulation algorithms, this article presents two innovative techniques for approximating the expected values of fuzzy numbers' monotone functions, which is of utmost importance in fuzzy optimization literature. In this regard, the stochastic discretization algorithm of Liu and Liu (2002) is enhanced by updating the discretization procedure for the simulation of the membership function and the calculation formula for the expected values. This is achieved through initiating a novel uniform sampling process and employing a formula for discrete fuzzy numbers, respectively, as the generated membership function in the stochastic discretization algorithm would adversely affect its accuracy to some extent. What is more, considering that the bisection procedure involved in the numerical integration algorithm of Li (2015) is time-consuming and also, not necessary for the specified types of fuzzy numbers, a special numerical integration algorithm is proposed, which can simplify the simulation procedure by adopting the analytical expressions of <inline-formula><tex-math notation="LaTeX">\alpha</tex-math></inline-formula>-optimistic values. Subsequently, concerning the extensive applications of regular fuzzy intervals, several theorems are introduced and proved as an extended effort to apply the improved stochastic discretization algorithm and the special numerical integration algorithm to the issues of fuzzy intervals. Throughout this article, a series of numerical experiments are conducted from which the superiority of both the two novel techniques over others are conspicuously displayed in aspects of accuracy, stability, and efficiency.
Author Zhou, Jian
Liu, Yuanyuan
Pantelous, Athanasios A.
Ji, Ping
Miao, Yunwen
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SubjectTerms Algorithms
Analytical models
Approximation algorithms
Business
Computational modeling
Discretization
Electronic mail
Expected value
Expected values
fuzzy simulation
Intervals
Mathematical analysis
Monotone functions
Numerical integration
Numerical models
Optimization
regular fuzzy interval
regular fuzzy number
Simulation
Stochastic processes
Title On Fuzzy Simulations for Expected Values of Functions of Fuzzy Numbers and Intervals
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