On-line integrated production and outbound distribution scheduling to minimize the maximum delivery completion time

In this paper, we consider the on-line integrated production and outbound distribution scheduling problem to minimize the maximum delivery completion time. All jobs arrive over time, and each job and its processing time become known at its arrival time. The jobs are first processed on a single machine and then delivered by a vehicle to a single customer. The vehicle can deliver at most c jobs to the customer at a time. When preemption is allowed and c≥2, we can provide an on-line algorithm with the best competitive ratio $\frac{\sqrt{5}+1}{2}\approx1.618$. When preemption is not allowed, we provide an on-line algorithm which has the best competitive ratio $\frac{\sqrt{5}+1}{2}\approx1.618$ for the case c=1 and has an asymptotic competitive ratio $\frac{\sqrt{5}+1}{2}\approx1.618$ for the case c≥2.