Abstract
The routing open shop problem with preemption allowed is a natural combination of the metric TSP problem and the classical preemptive open shop scheduling problem. While metric TSP is strongly NP-hard, the preemptive open shop is polynomially solvable for any (even unbounded) number of machines. The previous research on the preemptive routing open shop is mostly focused on the case with just two nodes of the transportation network (problem on a link). It is known to be strongly NP-hard in the case of an unbounded number of machines and polynomially solvable for the two-machine case. The algorithmic complexity of both two-machine problem on a triangular network and a three-machine problem with two nodes are still unknown. The problem with a general transportation network is a generalization of the metric TSP and therefore is strongly NP-hard.
We describe a wide polynomially solvable subclass of the preemptive routing open shop on a tree. This class allows building an optimal schedule with at most one preemption in linear time. For any instance from that class optimal makespan coincides with the standard lower bound. Therefore, the result, previously known for the problem on a link, is generalized on a special case on an arbitrary tree. The algorithmic complexity of the general case of the two-machine problem on a tree remains unknown.
This research was supported by the program of fundamental scientific researches of the SB RAS No I.5.1., project No 0314-2016-0014, and by the Russian Foundation for Basic Research, projects 17-01-00170, 17-07-00513 and 18-01-00747.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Averbakh, I., Berman, O., Chernykh, I.: A 6/5-approximation algorithm for the two-machine routing open-shop problem on a two-node network. Eur. J. Oper. Res. 166(1), 3–24 (2005). https://doi.org/10.1016/j.ejor.2003.06.050
Averbakh, I., Berman, O., Chernykh, I.: The routing open-shop problem on a network: complexity and approximation. Eur. J. Oper. Res. 173(2), 531–539 (2006). https://doi.org/10.1016/j.ejor.2005.01.034
Brucker, P., Knust, S., Edwin Cheng, T., Shakhlevich, N.: Complexity results for flow-shop and open-shop scheduling problems with transportation delays. Ann. Oper. Res. 129, 81–106 (2004). https://doi.org/10.1023/b:anor.0000030683.64615.c8
Chernykh, I.: Routing open shop with unrelated travel times. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 272–283. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_22
Chernykh, I., Kuzevanov, M.: Sufficient condition of polynomial solvability of two-machine routing open shop with preemption allowed. Intellektual’nye sistemy 17(1–4), 552–556 (2013). (in Russian)
Chernykh, I., Lgotina, E.: The 2-machine routing open shop on a triangular transportation network. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 284–297. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_23
Chernykh, I., Pyatkin, A.: Refinement of the optima localization for the two-machine routing open shop. In: Proceedings of the 8th International Conference on Optimization and Applications (OPTIMA 2017). CEUR Workshop Proceedings (1987), vol. 1987, pp. 131–138 (2017)
Gonzalez, T.F., Sahni, S.: Open shop scheduling to minimize finish time. J. ACM 23(4), 665–679 (1976). https://doi.org/10.1145/321978.321985
Kononov, A.: On the routing open shop problem with two machines on a two-vertex network. J. Appl. Ind. Math. 6(3), 318–331 (2012). https://doi.org/10.1134/s1990478912030064
Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, G.B.: Sequencing and scheduling: algorithms and complexity. In: Logistics of Production and Inventory. Elsevier (1993)
Lushchakova, I., Soper, A., Strusevich, V.: Transporting jobs through a two-machine open shop. Nav. Res. Logist. 56, 1–18 (2009). https://doi.org/10.1002/nav.20323
Pyatkin, A., Chernykh, I.: The open shop problem with routing at a two-node network and allowed preemption. J. Appl. Ind. Math. 6(3), 346–354 (2012). https://doi.org/10.1134/s199047891203009x
Strusevich, V.: A heuristic for the two-machine open-shop scheduling problem with transportation times. Discret. Appl. Math. 93(2), 287–304 (1999). https://doi.org/10.1016/S0166-218X(99)00115-8
Williamson, D.P., Hall, L.A., Hoogeveen, J.A., Hurkens, C.A.J., Lenstra, J.K., Sevast’janov, S.V., Shmoys, D.B.: Short shop schedules. Oper. Res. 45(2), 288–294 (1997). https://doi.org/10.1287/opre.45.2.288
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Chernykh, I. (2019). Sufficient Conditions of Polynomial Solvability of the Two-Machine Preemptive Routing Open Shop on a Tree. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-10934-9_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-10933-2
Online ISBN: 978-3-030-10934-9
eBook Packages: Computer ScienceComputer Science (R0)