Note on “Unrelated parallel-machine scheduling with rate-modifying activities to minimize the total completion time”
Introduction
The problem of joint scheduling and rate-modifying activities has received the attention by many researchers in recent years. Lee and Leon [3] were the pioneers in this area. They introduced a class of scheduling problems in which a rate-modifying activity may be performed on a single-machine setting. They assumed that the processing time of job i is δipi if the job is scheduled after the rate-modifying activity, where pi is the normal processing time of job i and δi > 0 is its modifying rate. They provided polynomial algorithms for solving problems of minimizing the makespan and the total completion time and proposed pseudo-polynomial algorithms for solving problems of minimizing the total weighted completion time and the maximum lateness.
In the succeeding study, Zhao et al. [6] extended some of the objectives studied by Lee and Leon [3] to the identical parallel-machine setting. They assumed that each machine may require a rate-modifying activity during the scheduling horizon. They further assumed that the processing time of job i on machine j is δijpij if the job is scheduled after the rate-modifying activity, where pij is the normal processing time of job i on machine j and δij > 0 is its modifying rate. They developed an O(n2m+3) time algorithm for solving the total completion time minimization problem and a pseudo-polynomial algorithm for solving the total weighted completion time minimization problem, where n is the number of jobs and m is the number of machines. Later, Hsu et al. [2] extended one of the objectives studied by Zhao et al. [6] to the unrelated parallel-machine setting. They showed that the total completion time minimization problem can be solved in O(nm+3) time if 0 < δij ⩽ 1 and O(n2m+2) time if δij > 0.
In this note, we study the same model proposed by Hsu et al. [2] and show that it can be solved in O(nm+3) time even though δij > 1. The rest of this paper is organized as follows: We formulate the problem in Section 2. In Section 3, we show that the time complexity of the model studied by Hsu et al. [2] can be reduced. We conclude the paper and suggest some topics for future research in the last section.
Section snippets
Problem formulation
We follow the notation and terminology used by Hsu et al. [2] as possible throughout the paper and will introduce additional notation when needed. Following Hsu et al. [2], the problem considered can be formally described as follows: There are n independent jobs J = {J1, J2, … , Jn} simultaneously available at time zero which have to be processed on m (m < n) unrelated parallel machines M = {M1, M2, … , Mm}. Preemption is not allowed and each machine is only able to process one job at a time. We denote by nj
Minimization of the total completion time
In this section, we show that the Rm|rma| ∑ Ci problem can be solved in O(nm+3) time no matter what 0 < δij ⩽ 1 or δij > 0.
First, the following two lemmas are applicable for solving the problem. Lemma 1 The number of nonnegative integer solutions to x1 + x2 + ⋯ + xm = n is . Lemma 2 The number C(n + m, m) is bounded from above by .[4]
[2]
Let (l1, l2, … , lm) and Ci(l1, l2, … , lm) respectively denote the rate-modifying activity position vector and the completion time of job Ji that is processed on one of the
Conclusions
In this note, we show that the problem proposed by Hsu et al. [2] can be optimally solved in O(nm+3) time even though δij > 1. Further research might be to consider the problem with resource allocations or optimizing other performance measures.
Acknowledgments
The authors thank the Editor and two anonymous reviewers for their helpful comments and suggestions on an earlier version of the paper. This research was supported in part by the National Science Council of Taiwan, Republic of China, under Grant numbers NSC 102-2221-E-150-042 and NSC 102-2221-E-252-010-MY2.
References (6)
- et al.
Unrelated parallel-machine scheduling with rate-modifying activities to minimize the total completion time
Information Sciences
(2011) - et al.
Machine scheduling with a rate-modifying activity
European Journal of Operational Research
(2001) - et al.
Two-parallel machines scheduling with rate-modifying activities to minimize total completion time
European Journal of Operational Research
(2009)