Single Machine Scheduling and Job Delivery to Minimize Makespan with the Processing Times of Jobs Being Proportional to Their Sizes

In the single machine scheduling and job delivery problem to minimize makespan, jobs are processed on a single machine and delivered by a capacitated vehicle to a single customer. When the processing times of jobs are independent on their sizes, Chang and Lee [4] proved that this problem is strongly NP-hard. They also provided a heuristic with the worst-case performance ratio of 5/3, and this bound is tight. In this paper, we consider that the restricted version in which the processing times of jobs are proportional to their sizes. We show that Chang-Lee's heuristic has a better worst-case performance ratio of 53/35. We also provide a new heuristic which has the worst-case performance ratio of 3/2, and this bound is best.