Publication Date



Technical Report: UTEP-CS-18-13

To appear in Proceedings of the World Conference on Soft Computing, Baku, Azerbaijan, May 29-31, 2018.


In 1970, Richard Bellman and Lotfi Zadeh proposed a method for finding the maximum of a function under fuzzy constraints. The problem with this method is that it requires the knowledge of the minimum and the maximum of the objective function over the corresponding crisp set, and minor changes in this crisp set can lead to a drastic change in the resulting maximum. It is known that if we use a product "and"-operation (t-norm), the dependence on the maximum disappears. Natural questions are: what if we use other t-norms? Can we eliminate the dependence on the minimum? What if we use a different scaling in our derivation of the Bellman-Zadeh formula? In this paper, we provide answers to all these questions. It turns out that the product is the only t-norm for which there is no dependence on maximum, that it is impossible to eliminate the dependence on the minimum, and we also provide t-norms corresponding to the use of general scaling functions.