On fuzzy ellipsoid numbers and membership functions

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.