Semi-online scheduling on two identical machines with a common due date to maximize total early work
Introduction
Scheduling with the goal of early work maximization is closely related to late work minimization, which was first proposed by Błażewicz in 1984 [2], and then has been continuously studied for more than 30 years [4], [17], [26]. Since this criterion can depict many production scenarios such as information collection [3], land cultivation [22], production planning [25], burn-in operations [24] and so on, it was widely investigated both from theoretical aspect [6], [23], [30] and practical applications [1], [7], [21].
A job’s late work is equal to the late part executed after its due date, and by contrast, its early work is the part processed before the due date. Therefore, from the viewpoint of optimal offline schedules, early work maximization and late work minimization share the same essence. However, the worst case analysis or the competitive analysis performed for approximation algorithms or online algorithms, respectively, is possible for early work maximization problems only, since late work minimization problems are non-approximable due to the consideration that the optimal criterion value may be zero (from the technical viewpoint, the zero optimal criterion value cannot be used as a denominator in the approximation/competitive ratio calculation, cf. [9]).
In our previous paper [9], we investigated the online version of early work maximization problem with parallel machines and a common due date, and proposed an optimal online algorithm for two identical machines with the competitive ratio . Hence, a natural question arises whether a better bound can be obtained if given partial information on the problem instances, i.e., in the semi-online cases.
In this paper, we study four semi-online versions of scheduling on two identical machines with a common due date to maximize total early work (denoted as in field in the - [14]). Particularly, we assume that,
- (1)
if the total processing time of all jobs () is given in advance, i.e., for problem , we show that its lower bound is equal to , and propose an optimal semi-online algorithm;
- (2)
if the optimal criterion value () is given, we show that the tight bound of this problem (i.e. ) is still ;
- (3)
if is given together with the maximal job processing time (i.e., problem ), the tight bound is reduced to ;
- (4)
if only is given, the lower and upper bounds of problem are and , respectively.
The rest of this paper is organized as follows. The formal definition of the problems and the related work are presented in Section 2. Section 3 is devoted to problems and , where the tight bound for both models is given. Problem is studied in Section 4, where the bound is reduced to and is still tight. In Section 5, we study and show that there is a tiny gap, 0.0144, between the upper and lower bounds. The final conclusions and future work are discussed in Section 6.
Section snippets
Problem definition
The early work of job , denoted as , is always accompanied with its symmetrical parameter — late work . Both parameters can be used to model the scenarios where a perishable commodity is involved [22]. In its seminal paper, Błażewicz [2] got his motivation from data collection in a control system. He considered each job as a set of information with a valid term (i.e. due date, ), and only the parts collected (into the system) before its due date is valuable for the users. Consequently,
Problems and
In this section, we first focus on problem , for which we prove that the lower bound is equal to , and propose an optimal semi-online algorithm. Then we show that the same properties hold for problem as well.
Problem
In this section, we show that if more pieces of information, e.g. both the total and maximal processing time is known before scheduling, the tight bound could be reduced from to .
Problem
Taking into account the results represented in the previous two sections, a natural research direction is to study the case when we know only the maximal processing time (without the knowledge of the total processing time), i.e., problem . In this section, we prove an upper bound 1.1375 and a lower bound 1.1231 for this semi-online problem.
Conclusions
In this paper, we studied four versions of semi-online scheduling on two identical machines to maximize total early work. When the total processing time of all jobs is known in advance, i.e. for problem , we proved that the tight bound is equal to , and this result can be easily transformed to problem . Then, if more information could be given, e.g., for problem , the tight bound was reduced to . However, if only
Acknowledgments
This research was partially supported by Natural Science Foundation of Liaoning, China (No. 2019-MS-170), Overseas Training Foundation of Liaoning, China (No. 2019GJWYB015), and by the statutory funds of Poznan University of Technology, Poland .
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