Branch Less, Cut More and Schedule Jobs with Release and Delivery Times on Uniform Machines

Mathematics

Nodari Vakhania¹  Frank Werner² 

[1] Centro de Investigacion en Ciences, Universidad Autónoma del Estado de Morelos, Cuernavaca 62209, Mexico;Faculty of Mathemaics, Otto-von-Guericke University Magdeburg, PSF 4120, 39016 Magdeburg, Germany;

关键词: scheduling;    uniform machines;    release time;    delivery time;    time complexity;    algorithm;   

DOI  :  10.3390/math9060633

来源: DOAJ

【 摘 要 】

We consider the problem of scheduling n jobs with identical processing times and given release as well as delivery times on m uniform machines. The goal is to minimize the makespan, i.e., the maximum full completion time of any job. This problem is well-known to have an open complexity status even if the number of jobs is fixed. We present a polynomial-time algorithm for the problem which is based on the earlier introduced algorithmic framework blesscmore (“branch less and cut more”). We extend the analysis of the so-called behavior alternatives developed earlier for the version of the problem with identical parallel machines and show how the earlier used technique for identical machines can be extended to the uniform machine environment if a special condition on the job parameters is imposed. The time complexity of the proposed algorithm is $O(\gamma m^{2}nlogn)$, where $\gamma$ can be either n or the maximum job delivery time $q_{max}$. This complexity can even be reduced further by using a smaller number $\kappa <n$ in the estimation describing the number of jobs of particular types. However, this number $\kappa$ becomes only known when the algorithm has terminated.

【 授权许可】

Unknown