Aggregation and signature based comparisons of multi-state systems via decompositions of fuzzy measures

Abstract

In the reliability literature, several results have been presented to compare binary (two states) systems. Often, such results are obtained from copula-based extensions of fuzzy measures, where a fuzzy measure describes the structure of a system and a copula describes the stochastic dependence among the lifetimes of its components. Other similar results have been obtained in terms of the concept of signature. Here, we extend all those results to multi-state systems made up from binary components by suitably constructing corresponding mixed binary systems. For such a construction, we show how any fuzzy measure can be decomposed as a convex combination of {0,1}-valued fuzzy measures and how such a decomposition extends to the corresponding aggregation function. For a mixed system we can furthermore consider its signature and so we can also define a signature for the multi-state system. For mixed systems associated to different multi-state systems, we can thus obtain different comparison results, which can be translated into the corresponding comparisons for the parent multi-state systems. Stochastic comparisons are obtained for the discrete random variables which represent the states of two systems at time t, as well. The arguments in the paper will be illustrated by means of examples and related remarks.