Flow shop scheduling problems with transportation constraints revisited

This paper investigates two flow shop scheduling problems with two machines A and B, and a single transporter V. In the first problem, the transporter V is located at machine A initially, each job has to be processed first on A, then transported to B for further processing. While in the second problem, the transporter V is located at machine B initially, each job needs to be processed first on A, then on B, finally transported to the destination. In both problems, the transporter V can carry up to c (where c is a constant and c >= 1) jobs at a time and the objective is to minimize the makespan. For the former problem, the best approximation algorithm guarantees a worst-case ratio bound of (53 + E), while for the latter one, the best approximation algorithm provides a worst-case ratio bound of 2. For each problem, we design a polynomial-time approximation scheme (PTAS).