A PTAS for the Multiple Parallel Identical Multi-stage Flow-Shops to Minimize the Makespan
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.
International Workshop on Frontiers in Algorithmics (FAW)
Tong, Weitian, Eiji Miyano, Randy Goebel, Guohui Lin.
"A PTAS for the Multiple Parallel Identical Multi-stage Flow-Shops to Minimize the Makespan."
Computer Science Faculty Presentations.