Improved algorithms for two single machine scheduling problems

In this paper, we investigate two single machine scheduling problems. The first problem addresses a class of the two-stage scheduling problems in which the first stage is job production and the second stage is job delivery. For the case that jobs are processed on a single machine and delivered by a single vehicle to one customer area, with the objective of minimizing the time when all jobs are completed and delivered to the customer area and the vehicle returns to the machine, an approximation algorithm with a worst-case ratio of is known and no approximation can have a worst-case of unless . We present an improved approximation algorithm with a worst-case ratio of , which only leaves a gap of . The second problem is a single machine scheduling problem subject to a period of maintenance. The objective is to minimize the total completion time. The best known approximation algorithm has a worst-case ratio of . We present a polynomial time approximation scheme.