On the probabilistic meaning of copula-based extensions of fuzzy measures. Applications to target-based utilities and multi-state reliability systems

Aggregations functions, that are obtainable as copula-based extensions of fuzzy measures, may emerge in several applied fields. We concentrate attention on the probabilistic interpretation of such objects. Starting from this issue, we point out the related meaning in the context of multi-attribute target-based utilities and multi-state reliability systems. Analogies and relations between the two fields of decisions under uncertainty and systems’ reliability have been already remarked in the literature. Here, we prove in particular that aggregation functions, specifically emerging in the setting of multi-attribute (target-based) utilities and in the setting of multi-state systems, do exactly share the same analytic structure. We will show that this identification emerges naturally when the multi-state systems are formed with binary components. From a technical viewpoint, the comparison between two different representations for the considered aggregations, and the specific role of the Möbius transform, have a basic relevance in the proofs of our results and in the development of related discussions.