A method for efficiently representing the state equation in a class of max-plus linear systems is proposed. We introduce a construct referred to as 'cell' in which the list of possible longest paths is stored. By imposing interval constraints on the system parameters, we can reduce the complexity of the state equation. The proposed method would be useful in scheduling applications for systems with adjustable system parameters.

This research proposes efficient calculation methods for the transition matrices in discrete event systems, where the adjacency matrices are represented by directed acyclic graphs. The essence of the research focuses on obtaining the Kleene Star of an adjacency matrix. Previous studies have proposed methods for calculating the longest paths focusing on destination nodes. However, in these methods the chosen algorithm depends on whether the adjacency matrix is sparse or dense. In contrast, this research calculates the longest paths focusing on source nodes. The proposed methods are more efficient than the previous ones, and are attractive in that the efficiency is not affected by the density of the adjacency matrix.