Problem F2∥Cmax with forbidden jobs in the first or last position is easy

doi:10.1016/j.ejor.2006.01.018

European Journal of Operational Research

Volume 177, Issue 2, 1 March 2007, Pages 1310-1311

https://doi.org/10.1016/j.ejor.2006.01.018 Get rights and content

Abstract

Saadani et al. [N.E.H. Saadani, P. Baptiste, M. Moalla, The simple F2∥C_max with forbidden tasks in first or last position: A problem more complex that it seems, European Journal of Operational Research 161 (2005) 21–31] studied the classical n-job flow shop scheduling problem F2∥C_max with an additional constraint that some jobs cannot be placed in the first or last position. There exists an optimal job sequence for this problem, in which at most one job in the first or last position is deferred from its position in Johnson’s [S.M. Johnson, Optimal two- and three-stage production schedules with setup times included, Naval Research Logistics Quarterly 1 (1954) 61–68] permutation. The problem was solved in O(n²) time by enumerating all candidate job sequences. We suggest a simple O(n) algorithm for this problem provided that Johnson’s permutation is given. Since Johnson’s permutation can be obtained in O(n log n) time, the problem in Saadani et al. (2005) can be solved in O(n log n) time as well.

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Acknowledgement

This research was supported in part by INTAS under grant number 03-51-5501.

References (3)

S.M. Johnson
Optimal two- and three-stage production schedules with setup times included
Naval Research Logistics Quarterly
(1954)

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Cited by (1)

No-idle parallel-machine scheduling of unit-time jobs with a small number of distinct release dates and deadlines
2021, Computers and Operations Research
Citation Excerpt :
An essential loss of profit or efficiency can happen in workforce timetabling if the idle times between successive working periods are useless for the business and for the workers. The bibliography of the scheduling research considering no-idling machines include Kovalyov and Shafransky (1998), Narain and Bagga (2005), El Houda Saadani et al. (2005), Kalczynski and Kamburowski (2007), Kovalyov and Werner (2007), Valente and Alves (2008), Chrétienne (2008, 2014, 2016), Carlier et al. (2010), Jouglet (2012), Quilliot et al. (2013), Kacem and Kellerer (2014), Yazdania and Naderi (2017), Pempera (2017), Wang et al. (2018), Cheng et al. (2019), Nagano et al. (2019), Shao et al. (2018), Billaut et al. (2019), and Antoniadis et al. (2019). Preliminary results of this study are presented at the conference CoDIT’2019 (Brauner et al., 2019).
A problem of scheduling $n$ unit-time jobs with a small number of distinct release dates and deadlines, on identical parallel machines, to minimize the number of active machines is studied. The feature that makes this problem challenging is that no machine can stand idle between its start and completion times. A number of properties of this problem is established, and heuristic and optimal algorithms based on these properties are designed. They include optimal algorithms with running times $O (k n^{k})$ and $O (k n^{2} 3^{n})$ for the general case, where $k$ is the number of distinct release dates and deadlines, optimal $O (n log n)$ algorithms for the cases of a common release date or a common deadline, and an optimal $O (k n^{2})$ algorithm for the case of agreeable release dates and deadlines. Algorithms to find lower and upper bounds on the optimal (minimal) number of active machines are developed, and an integer linear programming (ILP) formulation with $O (k)$ variables and $O (k^{2})$ constraints is provided. Computer experiments demonstrated that the ILP problem is solved by CPLEX to optimality within a few hours for $n \leq 400$ and $k \leq 50$ , and that the quality of the lower and upper bounds is sufficiently good.

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Short CommunicationProblem F2∥Cmax with forbidden jobs in the first or last position is easy

Abstract

Section snippets

Acknowledgement

Optimal two- and three-stage production schedules with setup times included

Naval Research Logistics Quarterly

Short Communication
Problem F2∥C_max with forbidden jobs in the first or last position is easy