Optimization with soft constraints: case of fuzzy intervals

There exist several method of optimizing a crisp function under fuzzy constraints. These methods require that one knows exactly the values the membership functions. In reality, the values of the membership functions μ(x) can be determined only approximately, so that for every x, one has an interval M(x) of possible values of μ(x). Depending on which values μ(x) from these intervals M(x) one chooses, one may get different solutions to the optimization problem, i.e. different values of x will appear as “best”; in other words, one may have several possibly “best” solutions to the original optimization problem. It is therefore desirable, given interval membership functions, to present all possibly “best” solutions x. One cannot do that by trying all possible membership functions, because there are infinitely many of them. In this paper, the authors describe a simple algorithm that finds all possibly best choices, i.e., all values x that are possibly the solutions to the optimization problem (without going through all possible membership functions)