Computational complexity and solution algorithms for flowshop scheduling problems with the learning effect

Abstract

In this paper, we analyze the two-machine flowshop problem with the makespan minimization and the learning effect, which computational complexity was not determined yet. First, we show that an optimal solution of this problem does not have to be the ‘permutation’ schedule if the learning effect is taken into consideration. Furthermore, it is proved that the permutation and non-permutation versions of this problem are NP-hard even if the learning effect, in a form of a step learning curve, characterizes only one machine. However, if both machines have learning ability and the learning curves are stepwise then the permutation version of this problem is strongly NP-hard. Furthermore, we prove the makespan minimization problem in m-machine permutation proportional flowshop environment remains polynomially solvable with identical job processing times on each machine even if they are described by arbitrary functions (learning curves) dependent on a job position in a sequence. Finally, approximation algorithms for the general problem are proposed and analyzed.

Introduction

The classical scheduling theory does not cover problems, where a time required to complete a job decreases together with the number of products already machined (or with the number of repetitions during production of a single item). This phenomenon is called the learning effect and research on its presence in manufacturing systems and on advantages following from that fact are being continuously carried out from 1936 (e.g. Argote, 1996, Jackson, 1998, Keachie and Fontana, 1966, Nadler and Smith, 1963, Yelle, 1979, Young, 1991), when this phenomenon was discovered in aircraft industry by Wright (1936). Nevertheless, the learning effect has attracted particular attention in scheduling just only during the last decade as it became clear that scheduling problems with the learning effect allow to describe many settings that occur inter alia in manufacturing, management, businesses and services sectors (for a survey see Biskup, 2008, Janiak and Rudek, 2009). The learning effect in scheduling is generally modelled by a job processing time formulated as a non-increasing positive function (called the learning curve) of a job position in a sequence (e.g. Bachman and Janiak, 2004, Biskup, 2008, Cheng and Wang, 2000, Ji and Cheng, 2010, Mosheiov, 2001, Okołowski and Gawiejnowicz, 2010, Wang, 2009, Wang and Cheng, 2007b, Wu et al., 2007a, Yang and Yang, 2010).

On the other hand, the classical flowshop scheduling problems, where job processing times are constant, constitute a significant part of the scheduling theory, since they reflect many real-life manufacturing settings (see Brucker, 2001, Gonzalez and Sahni, 1978). Nevertheless, to model more precisely the reality it is worthwhile to take into consideration the learning effect (see Chen et al., 2006, Cheng et al., 2009, Cheng et al., 2010, Cheng et al., 2011, Lee and Wu, 2004, Wang and Liu, 2009, Wu, 2005, Wu and Lee, 2009, Wu et al., 2007b, Wang et al., 2008, Xu et al., 2008, Yang and Kuo, 2009).

Although there is a significant number of papers that focus on the makespan minimization problem in a flowshop environment with the learning effect (e.g., Cheng et al., 2008a, Cheng et al., 2008b, Koulamas and Kyparisis, 2007, Lee and Wu, 2009, Wang and Cheng, 2007a, Wang and Xia, 2005, Yin et al., 2009), the computational complexity status of the two-machine case is still an open issue. Therefore, to fill this gap, we prove that the two-machine flowshop problem to minimize makespan becomes at least NP-hard if the learning effect, expressed by position dependent job processing times, is taken into consideration; it is the main result of this paper.

On the other hand, the makespan minimization in m-machine permutation flowshop environment with an identical job processing time on each machine (i.e., proportional flowshop) remains polynomially solvable if jobs are characterized by a common learning curve (see Yin et al., 2009). However, Mosheiov and Sidney (2003) pointed out that the learning rates differ from job to job, thus, they can be described by different learning curves. Hence, the assumption concerning common learning curves can be too restrictive. Nevertheless, computational complexity of such flowshop problem with distinct arbitrary (free-form) learning curves of jobs was not determined and it also became the motivation for this paper.

To show that such flowshop problem is polynomially solvable, we express it as the problem of the weighted linear combination of sum and bottleneck linear assignment problems. Nevertheless, its computational status also was not determined. Therefore, we design for it an optimal polynomial time algorithm and prove its optimality. In fact, it is even more crucial results than only proving polynomial solvability of the related scheduling problem, since the approach is valid for any problem that can be expressed as the considered combination of assignment problems (see Burkard, 2002). Thus, it is a significant general result beyond the scheduling domain.

Since we prove that the considered general two-machine flowshop problem is at least NP-hard, therefore, to make the paper coherent, we also provide for it some approximation algorithms together with their analysis.

The results presented in this paper have – to the best of our knowledge – never been investigated in the scheduling domain.

This paper is organized as follows. Section 2 contains problems formulation and their computational status is established in Section 3. Polynomial solvable cases are analyzed in Section 4. Approximation algorithms with the analysis of their efficiency are given in Section 5. Final remarks are presented in the last section.