Given a set ofn events (or jobs) which are constrained by a precedence relation, we want to order them into a totally ordered sequence (i. e., one machine schedule). Each event has an integer cost (which may be negative) associated with it, and our objective is to minimize the maximum cumulative cost within a schedule, i. e., to obtain a cumulative cost-optimal schedule. We introduce the concept of “strict optimality” and investigate properties of strictly optimal schedules. The usefulness of these schedules is demonstrated in the special case where the precedence relation is “series-parallel”. For this case we describe an algorithm which finds a cumulative cost-optimal schedule inO (n logn) time.