Single-machine scheduling to minimize the total earliness and tardiness is strongly NP-hard

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Abstract

This paper revisits the classic single-machine scheduling to minimize the total earliness and tardiness. It is known that the problem is NP-hard in the ordinary sense, but the exact complexity of the problem is long-standing open. We show in this paper that this problem 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 n jobs which have to be processed on a single machine with no preemption. Let J={J1,J2,,Jn} be the job set. pj and dj are used to denote the processing time and the due date of job Jj, respectively, j=1,2,,n. Let αj and βj be the earliness penalty and tardiness penalty per unit time of job Jj, respectively, j=1,2,,n. Given a schedule σ for J, let Cj(σ) be the completion time of job Jj in σ,j=1,2,,n. Then the cost of schedule σ is defined to be j=1n{αjEj(σ)+βjTj(σ)}, where Ej(σ)=max{0,djCj(σ)} is the earliness of job Jj in σ and Tj(σ)=max{0,Cj(σ)dj} is the tardiness of job Jj in σ,j=1,2,,n. The objective is to find a schedule with the minimum cost. We follow the three-field notation to represent the problem by 1(αjEj+βjTj). In particular, we call Ej(σ)+Tj(σ)=|Cj(σ)dj| the deviation of job Jj in schedule σ,j=1,2,,n. Then j(Ej(σ)+Tj(σ)) is the total deviation of all jobs in σ.

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 O(nlogn). Garey et al. [4] proved that problem 1(Ej+Tj) 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 1(Ej+Tj) 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 1(Ej+Tj) is strongly NP-hard. Throughout this paper, we use TC(σ)=j(Ej(σ)+Tj(σ)) to denote the total deviation of all jobs in a schedule σ.

Section snippets

The strong NP-hardness proof

We show the strong NP-hardness of problem 1(Ej+Tj) 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 3u+1 positive integers a1,a2,,a3u and B with j=13uaj=uB and B4<aj<B2 for 1j3u, the problem asks whether there exists a partition of A={1,2,,3u} into u disjoint subsets A1,A2,,Au such that |Ai|=3 and jAiaj=B for each i with 1iu.

Without loss of generality, we may assume that aj3 for each j 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).

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