A common approach in the multiple criteria decision making is to obtain the overall evaluation by aggregating the partial evaluations. For this, a member of a large family of aggregation operators is used. Many of these operators commonly employed in decision making (weighted average, ordered weighted average, minimum, maximum, ...) can be used only when criteria are independent. On the other hand, the Choquet integral, a generalization of the aforementioned operators, can be used even when some interactions between criteria occur. We present a fuzzified Choquet integral capable of dealing not only with fuzzy partial evaluations (first level fuzzification), but also with fuzzy weights (second level fuzzification). We also provide an effective way to evaluate the fully fuzzified integral, which allows its straightforward application to decision making problems with inherent uncertainty.
The present paper is devoted to the transition from crisp domains of probability to fuzzy domains of probability. First, we start with a simple transportation problem and present its solution. The solution has a probabilistic interpretation and it illustrates the transition from classical random variables to fuzzy random variables in the sense of Gudder and Bugajski. Second, we analyse the process of fuzzification of classical crisp domains of probability within the category ID of D-posets of fuzzy sets and put into perspective our earlier results concerning categorical aspects of fuzzification. For example, we show that (within ID) all nontrivial probability measures have genuine fuzzy quality and we extend the corresponding fuzzification functor to an epireflector. Third, we extend the results to simplex-valued probability domains. In particular, we describe the transition from crisp simplex-valued domains to fuzzy simplex-valued domains via a "simplex'' modification of the fuzzification functor. Both, the fuzzy probability and the simplex-valued fuzzy probability is in a sense minimal extension of the corresponding crisp probability theory which covers some quantum phenomenon.