A PTAS for the Multiple Parallel Identical Multi-Stage Flow-Shops to Minimize the Makespan
Proceedings of the International Workshop on Frontiers of Algorithmics
In the parallel k-stage flow-shops problem, we are given m identical k-stage flow-shops and a set of jobs. Each job can be processed by any one of the flow-shops but switching between flow-shops is not allowed. The objective is to minimize the makespan, which is the finishing time of the last job. This problem generalizes the classical parallel identical machine scheduling (where k=1) and the classical flow-shop scheduling (where m=1) problems, and thus it is NP-hard. We present a polynomial-time approximation scheme for the problem, when m and k are fixed constants. The key technique is to enumerate over schedules for big jobs, solve a linear programming for small jobs, and add the fractional small jobs at the end. Such a technique has been used in the design of similar approximation schemes.
Tong, Weitian, Eiji Miyano, Randy Goebel, Guohui Lin.
"A PTAS for the Multiple Parallel Identical Multi-Stage Flow-Shops to Minimize the Makespan."
Proceedings of the International Workshop on Frontiers of Algorithmics, Daming Zhu and Sergey Bereg (Ed.), 9711: 277-237 Cham, Switzerland: Springer International Publishing.
doi: 10.1007/978-3-319-39817-4_22 isbn: 978-3-319-39817-4