Simplification Assumptions For Fuzzy Measures
Since fuzzy measures are defined on the power set (or, more formally, on the sigma algebra associated with ), even in discrete cases the number of variables can be quite high . For this reason, in the context of multi-criteria decision analysis and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or -fuzzy measure and k-additive measures, introduced by Sugeno and Grabisch respectively.
Read more about this topic: Fuzzy Measure Theory
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