Single Machine Scheduling with Job Delivery to minimize makespan

In the single machine scheduling problem with job delivery to minimize makespan, jobs are processed on a single machine and delivered by a capacitated vehicle to their respective customers. We first consider the special case with a single customer, that is, all jobs have the same transportation time. Chang and Lee (2004) proved that this case is strongly NP-hard. They also provided a heuristic with the worst-case performance ratio $\frac{5}{3}$, and pointed out that no heuristic can have a worst-case performance ratio less than $\frac{3}{2}$ unless P = NP. In this paper, we provide a new heuristic which has the best possible worst-case performance ratio $\frac{3}{2}$. We also consider an extended version in which the jobs have non-identical transportation times and the transportation time of a delivery batch is defined as the maximum transportation time of the jobs contained in it. We provide a heuristic with the worst-case performance ratio 2 for the extended version, and show that this bound is tight.

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