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

In the parallelk-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. Th...

Full description

Saved in:
Bibliographic Details
Published inFrontiers in Algorithmics Vol. 9711; pp. 227 - 237
Main Authors Tong, Weitian, Miyano, Eiji, Goebel, Randy, Lin, Guohui
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 01.01.2016
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Algorithms & data structures
Flow-shop scheduling
Linear program
Makespan
Multiprocessor scheduling
Polynomial-time approximation scheme
Online AccessGet full text
ISBN3319398164
9783319398167
ISSN0302-9743
1611-3349
DOI10.1007/978-3-319-39817-4_22

Cover

More Information
Summary:In the parallelk-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.
Bibliography:Original Abstract: In the parallelk-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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 1$$\end{document}) and the classical flow-shop scheduling (where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = 1$$\end{document}) problems, and thus it is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {NP}$$\end{document}-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.
ISBN:3319398164
9783319398167
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-319-39817-4_22