A note on a two-agent scheduling problem related to the total weighted late work
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We revisit a two-agent scheduling problem in which a set of jobs belonging to two agents A and B (without common jobs) will be processed on a single machine for minimizing the total weighted late work of agent A subject to the maximum cost of agent B being bounded. Zhang and Wang (J Comb Optim 33:945–955, 2017) studied three versions of the problem: (i) the A-jobs having a common due date, (ii) the A-jobs having a common processing time, (iii) the general version. The authors presented polynomial-time algorithms for the first two versions and a pseudo-polynomial-time algorithm for the last one. However, their algorithm for the first version is invalid. Then we show the NP-hardness and provide a pseudo-polynomial-time algorithm for the first version with the cost of agent B being makespan, present a polynomial-time algorithm for an extension of the second version, and show that the third version is solvable in pseudo-polynomial-time by a new technique.
KeywordsScheduling Single machine Two-agent Late work
The authors would like to thank the associate editor and two anonymous referees for their constructive comments and kind suggestions. This research was supported by NSFC (11671368) and NSFC (11771406).
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