Fuzzy mathematical programming

Fuzzy Linear Programming (FLP) was originally suggested to solve problems which could be formulated as L.P-models, the parameters of which, however, were fuzzy rather than crisp numbers. It has turned out in the meantime that FLP is also well suited to solve LP-problems with several objective functions. FLP belongs to goal programming in the sense that implicitly or explicitly aspiration levels have to be defined at which the membership function of the fuzzy set reach their maximum or minimum. Main advantages of FLP are, that the models are used numerically very efficient and that they can in many ways be well adopted to different decision behaviours and contexts. For less structured problems knowledge based systems have been developed. They used partly fuzzy logic and approximate reasoning to model knowledge and to derive decisions and diagnoses from available knowledge. In real situations often a combination of algorithmic tools and knowledge based systems (so called second generation expert systems) are appropriate. This will be demonstrated using the control of flexible manufacturing systems as an example: the master scheduling or aggregate planning part will be modelled using fuzzy linear programming. Short-term planning or scheduling is too complex a problem to be modelled by even fuzzy linear programming. Therefore aproximate reasoning is used to perform this task. It will be shown that such a system outperforms (dominates) classical approaches with respect to due dates, capacity utilization and waiting time.