On Probability of Making a Given Decision: A Theoretically Justified Transition from Interval to Fuzzy Uncertainty

In practice, it is often necessary to make a decision under uncertainty. In the case of interval uncertainty, for each alternative i, instead of the exact value vi of the objective function, we only have an interval of possible values. In this case, it is reasonable to assume that each value vi is uniformly distributed on the corresponding interval, and to take the probability that vi is the largest as the probability of selecting the i-th alternative. In some practical situations, we have fuzzy uncertainty, i.e., for every alternative i, we have a fuzzy number describing the value of the objective function. Then, for every degree alpha, we have an interval, the alpha-cut of the corresponding fuzzy number. For each alpha, we can assume the uniform distributions on the corresponding alpha-cuts and get a probability Pi(alpha) that i will be selected for this alpha. From the practical viewpoint, it is desirable to combine these probabilities into a single probability corresponding to fuzzy uncertainty. In deriving the appropriate combination, we use the fact that fuzzy values are not uniquely defined, different procedures can lead to differently scaled values. It turns out that the only scaling-invariant distribution on the set of all the degrees alpha is a uniform distribution. So, we justify the choice of an integral of Pi(alpha) over alpha as the probability that under fuzzy uncertainty, an alternative i will be selected