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

  • Weitian Tong
  • Eiji Miyano
  • Randy Goebel
  • Guohui Lin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9711)

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 \(\text {NP}\)-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.

Keywords

Multiprocessor scheduling Flow-shop scheduling Makespan Linear program Polynomial-time approximation scheme 

Notes

Acknowledgments

Tong was supported by the FY16 Startup Funding from the Georgia Southern University and an Alberta Innovates Technology Futures (AITF) Graduate Student Scholarship. Miyano is supported by the Grants-in-Aid for Scientific Research of Japan (KAKENHI), Grant Number 26330017. Goebel is supported by the AITF and the Natural Sciences and Engineering Research Council of Canada (NSERC). Lin is partially supported by the NSERC and his work was mostly done during his sabbatical leave at the Kyushu Institute of Technology, Iizuka Campus.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Weitian Tong
    • 1
    • 3
  • Eiji Miyano
    • 2
  • Randy Goebel
    • 3
  • Guohui Lin
    • 3
  1. 1.Department of Computer SciencesGeorgia Southern UniversityGeorgiaUSA
  2. 2.Department of Systems Design and InformaticsKyushu Institute of TechnologyIizuka, FukuokaJapan
  3. 3.Department of Computing ScienceUniversity of AlbertaEdmontonCanada

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