Abstract
It is shown that the job-shop problem with two machines and a fixed number ofk jobs with makespan criterionJ2¦n=k¦C max is polynomially solvable. Sotskov and Shakhlevich (1993) have shown that problemJ3¦n=3¦C maxisNP-hard. Furthermore it is well known that J¦n=2¦C maxin polynomially solvable. Thus, our result settles the remaining open question concerning the complexity status of job-shop problems with fixed numbers of jobs and machines.
Zusammenfassung
Es wird gezeigt, daß das Job-Shop ProblemJ2¦n=k¦C max bei fester Anzahl von Jobs polynomial lösbar ist. Da das ProblemJ3¦n=3¦C max NP-schwierig ist (Sotskov und Shakhlevich 1993) und sichJ¦n=2¦C max ebenfalls polynomial lösen läßt, erhält man durch dieses Ergebnis eine vollständige Antwort auf die Frage nach der Komplexität von Job-Shop Problemen mit fester Anzahl von Maschinen und Jobs.
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References
Kravchenko SA, Sotskov YN (1993) Optimal makespan schedule for three jobs on two machines, Working Paper, Institute of Engineering Cybernetics of the Byelorussian Academy of Sciences, Minsk
Sotskov YN (1991) The complexity of shop-scheduling problems with two or three jobs. Eur J Oper Res 53:326–336
Sotskov YN, Shakhlevich NV (1993) NP-hardness of shopscheduling problems with three jobs. Disc Appl Math (to appear)
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Supported by Deutsche Forschungsgemeinschaft, Project Jop-TAG
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Brucker, P. A polynomial algorithm for the two machine job-shop scheduling problem with a fixed number of jobs. OR Spektrum 16, 5–7 (1994). https://doi.org/10.1007/BF01719698
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DOI: https://doi.org/10.1007/BF01719698