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Motivated by mail delivery scheduling problems arising in Royal Mail, we study a generalization of the fundamental makespan scheduling problem \(P||C_{\max }\) which we call the Bounded Job Start Scheduling Problem. Given a set of jobs, each one 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 \(g\le m\) on the number of jobs that may simultaneously begin per unit of time. We show that Longest Processing Time First (LPT) algorithm is tightly 2-approximate. After proving that the problem is strongly \(\mathcal {NP}\)-hard even when \(g=1\), we elaborate on improving the 2-approximation ratio for this case. We distinguish the classes of long and short instances satisfying \(p_j\ge m\) and \(p_ju003cm\), 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 solutions with tightly satisfied packing and bounded job start constraints. For a broad family of instances excluding degenerate instances with many very long jobs and instances with few machines, 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 a state-of-the-art lexicographic optimization method for \(P||C_{\max }\) under uncertainty to propose a two-stage robust optimization approach for bounded job start scheduling under uncertainty attaining good trade-offs in terms of makespan and number of used machines. We substantiate this approach numerically using Royal Mail data.

Approximating Bounded Job Start Scheduling with Application in Royal Mail Deliveries Under Uncertainty