Approximating bounded job start scheduling with application in Royal Mail deliveries under uncertainty

Motivated by mail delivery scheduling problems arising in Royal Mail, we study a generalization of the fundamental makespan scheduling P||Cmax problem which we call the "bounded job start scheduling problem". Given a set of jobs, each specified by an integer processing time p_j, that have to be executed non-preemptively by a set of m parallel identical machines, the objective is to compute a minimum makespan schedule subject to an upper bound g=m and p_j<m, respectively, for each job j. We show that LPT is 5/3-approximate for the former and optimal for the latter. Then, we explore the idea of scheduling long jobs in parallel with short jobs to obtain tightly satisfied packing and bounded job start constraints. For a broad family of instances excluding degenerate instances with many very long jobs, we derive a 1.985-approximation ratio. For general instances, we require machine augmentation to obtain better than 2-approximate schedules. Finally, we exploit machine augmentation and lexicographic optimization, which is useful for P||Cmax under uncertainty, to propose a two-stage robust optimization approach for bounded job start scheduling under uncertainty aiming in good trade-offs in terms of makespan and number of used machines. We substantiate this approach numerically using Royal Mail data