Computational performances of a simple interchange heuristic for a scheduling problem with an availability constraint

This paper deals with a scheduling problem on a single machine with an availability constraint. The problem is known to be NP-complete and admits several approximation algorithms. In this paper we study the approximation scheme described in He et al. [Y. He, W. Zhong, H. Gu, Improved algorithms for two single machine scheduling problems, Theoretical Computer Science 363 (2006) 257–265]. We provide the computation of an improved relative error of this heuristic, as well as a proof that this new bound is tight. We also present some computational experiments to test this heuristic on random instances. These experiments include an implementation of the fully-polynomial time approximation scheme given in Kacem and Ridha Mahjoub [I. Kacem, A. Ridha Mahjoub, Fully polynomial time approximation scheme for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers and Industrial Engineering 56 (2009) 1708–1712]. 2013 Elsevier Ltd. All rights reserved. 1. A single machine total completion time scheduling problem with an availability constraint Scheduling jobs under maintenance constraints is an important issue in many real-life situations. For instance, Seristö (1995) claims that maintenance represents between 10% and 15% of the total operating expenses of airlines. Recall that, for an airline like Lufthansa Group or Air France–KLM, the yearly total operating expenses lied around 30 billions US dollars in 2009 (World Airline Report, 2010). More generally, efficiently scheduling jobs under availability constraints (due to, e.g., maintenance) is a challenge which is often motivated by consequent financial stakes. Besides, scheduling problems with availability constraints are widely studied in the literature (see for instance (Lee, 2004; Sanlaville & Schmidt, 1998; Schmidt, 2000) for surveys), and is an active area of research. In this paper, we consider the problem of scheduling jobs on a single machine having one period of maintenance. This period of maintenance is known in advance, and is such that no job can be done during it. In other words, preemption is not allowed, and the machine is not available for processing jobs during the maintenance. We wish to minimize the total completion time of the jobs. Since the period of maintenance is known in advance, this problems models also other situations where the machine is unavailable besides maintenance. This particular problem is usually denoted 1; h1= P Ci. Adiri, Bruno, Frostig, and Rinnooy Kan (1989) and Lee and Liman (1992) showed that this problem is NP-hard. Lee and Liman also showed that the SPT heuristic, which consists in sorting the jobs in non-decreasing order of their processing times, leads to a heuristic of relative error 7. Sadfi, Penz, Rapine, Bła _ zewicz, and Formanowicz (2005) proposed an improved heuristic for this problem, having a relative error of 3 17. Their heuristic is a post-optimization of SPT using a 2-OPT procedure. More precisely, let us denote A and B the sets of jobs scheduled respectively after and before the maintenance by the SPT algorithm. The heuristic consists in exchanging one job of A with one job of B in order to improve the total completion time. They call their procedure MSPT, for Modified SPT. In He, Zhong, and Gu (2006) the authors study a generalization of MSPT, that we call here MSPT-k. This heuristic consists in exchanging at most k jobs of A with at most k jobs of B, with k a fixed positive constant. In He et al. (2006) they prove that for all k P 2, MSPT-k has a relative error bounded by 0360-8352/$ see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.08.017 q This manuscript was processed by Area Editor T.C. Edwin Cheng. ⇑ Corresponding author at: CNRS – LAAS Université de Toulouse UPS, INSA, INP, ISAE; UT1, UTM, LAAS 7 avenue du Colonel Roche 31077 Toulouse Cedex 4, France. E-mail addresses: [email protected] (J. Moncel), [email protected] (J. Thiery), [email protected] (A. Waserhole). Computers & Industrial Engineering 67 (2014) 216–222