Affiliations: [a]
Institute of Operations Research and Cybernetics, Hangzhou Dianzi University, Hangzhou, China
| [b]
College of Automation, Harbin Engineering University, Harbin, China and College of Engineering and Science, Victoria University, Melbourne, Australia
| [c]
Department of Mechanical Engineering, Washington University in St. Louis Campus, St. Louis, MO, USA
| [d]
Graduate School of Science and Technology, Tokai University, Toroku, Kumamoto, Japan
Correspondence:
[*]
Corresponding author. Peng Shi, College of Automation, Harbin Engineering University, Harbin 150001, China and College of Engineering and Science, Victoria University, Melbourne 8001, Australia. Tel.: +61 451 82588175; Fax: +61 451 82588175; E-mail: peng.shi@adelaide.edu.au.
Abstract: In this paper, the operations of fuzzy ellipsoid numbers and the direct relationship between the joint membership function and the edge membership functions are investigated. First, we prove that the general scalar multiplication (defined by Zadeh’s expansion) of fuzzy ellipsoid numbers preserves the closeness of the operation, but the general addition (defined by Zadeh’s expansion) does not preserve the closeness of the operation. Then, a new addition “⊕” of fuzzy ellipsoid numbers is defined, and it is shown that the new addition “⊕” preserves the closeness of the operation, and is the best approximation of the general addition operation in all addition operations which preserves the closeness of the operation for fuzzy ellipsoid numbers. Next, a bijection is established between fuzzy n-ellipsoid number space and n dimension fuzzy vector space. Furthermore a formula is obtained to express the edge membership functions with the joint membership function, and an iterative algorithm is proposed to calculate the joint membership function value as the edge membership functions are known, and the convergence of the iterative algorithm is demonstrated.
