Abstract
Metric functions are introduced for various classes of single-machine scheduling problems. It is shown how approximate solutions of NP-hard problems can be found using these functions. The metric value is determined by solving a linear programming problem with constraints being systems of linear inequalities for polynomial or pseudopolynomial solvable instances of the problem under study. In fact, the initial instance is projected onto the subspace of solvable problem instances in the introduced metric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. Brucker and S. Knust, Complex Scheduling (Springer-Verlag, Berlin, 2011).
H. M. Wagner, “An integer linear-programming model for machine scheduling,” Naval Res. Logist. Q. 6 (2), 131–140 (1959).
M. H. Rothkopf, “Scheduling independent tasks on parallel processors,” Manage. Sci. 12 (5), 437–447 (1966).
E. Ignall and L. Schrage, “Application of the branch and bound technique to some flow-shop scheduling problems,” Oper. Res 13 (3), 400–412 (1965).
D. Lemtyuzhnikova and V. Leonov, “Large-scale problems with quasi-block matrices,” J. Comput. Syst. Sci. Int. 58 (4), 571–578 (2019).
F. Werner, “A survey of genetic algorithms for shop scheduling problems,” in P. Siarry, Heuristics: Theory and Applications (Nova Sci., 2013), pp. 161–222.
L. Zuo, L. Shu, S. Dong, C. Zhu, and T. Hara, “A multi-objective optimization scheduling method based on the ant colony algorithm in cloud computing,” IEEE Access 3, 2687–2699 (2015).
Q. Pan, “An effective co-evolutionary artificial bee colony algorithm for steelmaking-continuous casting scheduling,” Eur. J. Oper. Res. 250 (3), 702–714 (2016).
V. Sels, J. Coelho, A. Dias, and M. Vanhoucke, “Hybrid tabu search and a truncated branch-and-bound for the unrelated parallel machine scheduling problem,” Comput. Oper. Res. 53, 107–117 (2015).
A. Lazarev, “Estimation of absolute error in scheduling problems of minimizing the maximum lateness,” Dokl. Math. 76, 572–574 (2007).
A. A. Lazarev, Scheduling Theory: Methods and Algorithms (Inst. Probl. Upr. Ross. Akad. Nauk, Moscow, 2019) [in Russian].
A. A. Lazarev and D. I. Arkhipov, “Estimation of the absolute error and polynomial solvability for a classical NP-hard scheduling problem,” Dokl. Math. 97 (3), 262–265 (2018).
A. A. Lazarev, P. S. Korenev, and A. A. Sologub, “Metric for minimum total tardiness problem,” Upr. Bol’shimi Sist. 57, 123–137 (2015).
A. A. Lazarev and A. G. Kvaratskheliya, “Metrics in scheduling problems,” Dokl. Math. 81 (3), 497–499 (2010).
J. Du and J. Y.-T. Leung, “Minimizing total tardiness on one processor is \(NP\)-hard,” Math. Oper. Res. 15, 483–495 (1990).
E. L. Lawler, “A pseudopolynomial algorithm for sequencing jobs to minimize total tardiness,” Ann. Discrete Math. 1, 331–342 (1977).
R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, “Optimization and approximation in deterministic sequencing and scheduling: A survey,” Ann. Discrete Math. 5, 287–326 (1979).
E. L. Lawler, “A fully polynomial approximation scheme for the total tardiness problem,” Oper. Res. Lett. 1 (6), 207–208 (1982).
M. Y. Kovalyov, “Improving the complexities of approximation algorithms for optimization problems,” Oper. Res. Lett. 17, 85–87 (1995).
A. A. Lazarev and F. Werner, “Algorithms for special single machine total tardiness problem and an application to the even-odd partition problem,” Math. Comput. Model. 49, 2078–2089 (2009).
Ph. Baptiste, “Scheduling equal-length jobs on identical parallel machines,” Discret. Appl. Math. 103, 21–32 (2000).
A. A. Lazarev, R. R. Sadykov, and S. V. Sevast’yanov, “A scheme for the approximate solution of the problem \(1\,{\text{|}}\,{{r}_{j}}\,{\text{|}}\,{{L}_{{\max}}}\),” J. Appl. Ind. Math. 1, (4), 468–480 (2007).
A. A. Lazarev and E. R. Gafarov, Scheduling Theory: Minimization of Total Tardiness for a Single Machine (Vychisl. Tsentr im. A.A. Dorodnitsyna Ross. Akad. Nauk, Moscow, 2006) [in Russian].
Funding
This work was supported in part by the Russian Foundation for Basic Research, project nos. 18-31-00458, 20-58-S52006.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Lazarev, A.A., Lemtyuzhnikova, D.V. & Pravdivets, N.A. Metric Approach for Finding Approximate Solutions of Scheduling Problems. Comput. Math. and Math. Phys. 61, 1169–1180 (2021). https://doi.org/10.1134/S0965542521070125
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542521070125