A feedback arc set in a directed graph is a set of edges which removal leaves the graph acyclic.
Clearly, for every directed graph, the minimum length of a circuit is an upper bound for the maximum number of disjoint feedback arc sets. Equality does not always hold, for instance in a random tournament the size of a minimum feedback arc set is almost half the number of edges, thus one cannot find three disjoint, whereas the shortest circuits have length three.