#### Title

A PTAS for the Multiple Parallel Identical Multi-Stage Flow-Shops to Minimize the Makespan

#### Document Type

Conference Proceeding

#### Publication Date

2016

#### Publication Title

Proceedings of the International Workshop on Frontiers of Algorithmics

#### DOI

10.1007/978-3-319-39817-4_22

#### ISBN

978-3-319-39817-4

#### ISSN

1611-3349

#### Abstract

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.

#### Recommended Citation

Tong, Weitian, Eiji Miyano, Randy Goebel, Guohui Lin.
2016.
"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

https://digitalcommons.georgiasouthern.edu/compsci-facpubs/71