Approximation theorem of the fuzzy transform in fuzzy reasoning and its application to the scheduling problem

In this paper, the theoretical foundation of fuzzy reasoning is analyzed, and the idea that the fuzzy transform given in the fuzzy reasoning method should be continuous with respect to a certain fuzzy distance is proposed. Also, the fuzzy transforms given in the two fuzzy reasoning methods, the Mamdani method and the III method, are proved to be continuous. Based on the continuity of the fuzzy transform, the approximation theorem of the continuous fuzzy number transform is proven. Then, on the basis of the approximation theorem, a simple fuzzy number transform is constructed in order to implement the fuzzy reasoning based on multiple rules. At last, the fuzzy reasoning based on multiple rules implemented by a simple fuzzy number transform is applied to machine scheduling problems, and numerical computational results of different scale scheduling problems with the objective of minimizing the total number of tardy jobs show that it is more effective than usual heuristics based on rules, and in the computational time it has the obvious advantage over the reasoning by fuzzy rules directly.