Single-machine scheduling to minimize the total earliness and tardiness is strongly NP-hard
Introduction
In the scheduling research, the problems related to earliness and tardiness are widely studied in the past decades. The topic is motivated by the Just-In-Time production which emphasizes the notion of earliness as well as tardiness. In a Just-In-Time scheduling environment, jobs that complete either earlier or later than due dates will be punished. Therefore, an ideal schedule is the one in which all jobs complete exactly at their due dates. There are many measures of performance among which the most common one is to minimize the deviation of job completion times around their due dates. The problem considered in this paper can be stated as follows. There are jobs which have to be processed on a single machine with no preemption. Let be the job set. and are used to denote the processing time and the due date of job , respectively, . Let and be the earliness penalty and tardiness penalty per unit time of job , respectively, . Given a schedule for , let be the completion time of job in . Then the cost of schedule is defined to be , where is the earliness of job in and is the tardiness of job in . The objective is to find a schedule with the minimum cost. We follow the three-field notation to represent the problem by . In particular, we call the deviation of job in schedule . Then is the total deviation of all jobs in .
Baker and Scudder [1] provided a review of the single-machine scheduling problems with earliness and tardiness penalties. The earliest papers focused on the objective of minimizing the maximum earliness or tardiness penalty on a single machine. Sidney [9] gave an efficient algorithm to get an optimal schedule. Lakshminarayan et al. [7] developed an improved algorithm with complexity of . Garey et al. [4] proved that problem is NP-hard in the ordinary sense by a reduction from Even-Odd Partition. Hall et al. [5] further proved that the problem is NP-hard in the ordinary sense even if all jobs have a common due date. But the exact complexity (strongly NP-hard or pseudo-polynomially solvable) of problem is long-standing open. More related papers are referred to Hall and Posner [6], Fry et al. [2], Lee and Choi [8] and Ventura and Radhakrishnan [10], among many others.
In this paper, we show that problem is strongly NP-hard. Throughout this paper, we use to denote the total deviation of all jobs in a schedule .
Section snippets
The strong NP-hardness proof
We show the strong NP-hardness of problem by using the strongly NP-complete 3-Partition Problem (Garey and Johnson, [3]) for the reduction. 3-Partition Problem 3PP in Short Given a set of positive integers and with and for , the problem asks whether there exists a partition of into disjoint subsets such that and for each with .
Without loss of generality, we may assume that for each in the instance of 3PP. Given an
Acknowledgments
The authors would like to thank the associate editor and an anonymous referee for their constructive comments and kind suggestions. This research was supported by NSFC (11271338) and NSFC (11171313).
References (10)
- et al.
A genetic algorithm for job sequencing problems with distinct due dates and general early-tardy penalty weights
Computers and Operations Research
(1995) - et al.
Single machine scheduling with symmetric earliness and tardiness penalties
European Journal of Operational Research
(2003) - et al.
Sequencing with earliness and tardiness penalities: a review
Operations Research
(1990) - et al.
Minimizing weighted absolute deviation in single machine scheduling
IIE Transactions
(1990) - et al.
Computers and Intractability: A Guide to the Theory of NP-Completeness
(1979)
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