M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. SusiloThe 3rd International Conference on Mathematics and Statistics (ICoMS-3) Institut Pertanian Bogor, Indonesia, 5-6 August 2008
Abstract. The activity times in a network is seldom precisely known, and then could be represented into the fuzzy numbers. With max-plus algebra approach, the periodical properties of the network dynamic could be analyzed through the eigenvalues and eigenvectors of matrices over max-plus algebra in the its modelling. This paper aims to determine the eigenvalues and eigenvectors of matrices overfuzzy number max-plus algebra. The result of this paper can be used to analyze the periodical properties of the network dynamic with its activity times which is represented using the fuzzy numbers.
This paper is a theoretical investigation based on literature and computation using MATLAB program. The maximum and addition operations of the fuzzy number is defined through its alpha-cuts which are the closed intervals. The eigenvalues and eigenvectors of matrices overmax-plus algebra is extended into eigenvalues and eigenvectors of matrices overfuzzy number max-plus algebra, through eigenvalues and eigenvectors of matrices over interval max-plus algebra.
The finding shows that eigenvalues and eigenvectors of matrices over fuzzy number max-plus algebra could be determined the eigenvalues and eigenvector of every its alpha-cuts matrices firstly. Based on the Decomposition Theorem, we can determine the membership function of the eigenvalues and membership functions of the elements of eigenvectors corresponding to the eigenvalues. Moreover, the eigenvalue is unique if the matrices is irreducible.
Keywords: max-plus algebra, eigenvalues, eigenvectors, fuzzy number
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