Gradual Numbers and Fuzzy Solutions to Fuzzy Optimization Problems

This short paper indicates that early examples of fuzzy elements in a fuzzy set, that is, entities that assign elements to membership values, in contrast with fuzzy sets that assign membership values to elements, can be found in papers by Verdegay in the early 1980, following a line of thought opened by Orlovsky. They are so-called fuzzy solutions to fuzzy optimization problems. The notion of fuzzy element, and more specifically gradual number sheds some light on the ambiguous notion of fuzzy number often viewed as generalizing a number while it generalizes intervals. The notion of fuzzy solution is in fact a parameterized solution, in the style of parametric programming. These considerations show the pioneering contributions of Verdegay to the development of fuzzy optimization.

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