New hard benchmark for the 2-stage multi-machine assembly scheduling problem: Design and computational evaluation

Highlights

• Scheduling in two variants of the two stage multi-machine assembly is studied.

• The relationship with the customer order and the parallel machine scheduling problems is studied.

• Two extensive sets of instances are proposed.

• The characteristics of adequacy, empirical hardness, exhaustiveness, and amenity for statistical analysis are satisfied.

• A computational evaluation is performed and the state-of-the-art is clarified.

Abstract

The assembly scheduling problem is a common layout with many applications in real manufacturing scenarios. Despite the high number of studies dealing with this problem, no benchmark has been proposed up-to-now in the literature generating neither hard nor balanced instances. In this paper we present two extensive sets of instances for two variants of the 2-stage assembly scheduling problem. The first set is composed of 240 instances for the variant with one assembly machine in the second stage, while in the second set 960 instances are proposed for the variant with several assembly machines. An exhaustive experimental procedure, generating several preliminary testbeds with different processing times and number of jobs and machines, is carried out in order to identify the most representative instances of the problem under study. A total of 120,000 instances are generated and, among them, 1,200 are selected ensuring that the new benchmarks satisfy the desired characteristics of any benchmark: adequacy, empirical hardness, exhaustiveness, and amenity for statistical analysis. Finally, two computational evaluations are performed comparing and evaluating the existing heuristics in the literature, thus establishing the set of efficient heuristics for this assembly problem.

Introduction

Assembly scheduling problems have many applications in the industry (Sheikh et al., 2018), since many products are made up of different components that need to be manufactured in the earlier stages and then assembled in a later stage. Some examples of applications are: personal computer manufacturing (Potts et al., 1995), fire engine assembly plants (Lee et al., 1993), circuit board production (Cheng and Wang, 1999), food and fertilizer production (Hwang and Lin, 2012), car assembly industry (Fattahi et al., 2013), motor assembly industry (Liao et al., 2015), or plastic industry (Allahverdi and Aydilek, 2015). Other examples can also be found in services/IT, including distributed database systems (Allahverdi and Al-Anzi, 2006, Al-Anzi and Allahverdi, 2006b, Al-Anzi and Allahverdi, 2007), or multi-page invoice printing systems (Zhang et al., 2010).

Among the different assembly scheduling problems, in this paper we focus on the so-called 2-stage assembly scheduling problem. This problem consists of $m_1$ ($m_1>1$, fixed) dedicated parallel machines (${\mathit{DPm}}_1$¹) in the first stage (to manufacture the components) and $m_2$ ($m_2⩾1$, fixed) assembly machines in the second stage. The objective considered is the minimization of the total completion time. The variant with one assembly machine (labelled as SA or Single Assembly in the following) is denoted as ${\mathit{DPm}}_1\to 1\left|\right|\sum C_j$ by Framinan et al. (2019), and is equivalent to the regular two-machine flow-shop scheduling problem if there is only one machine in the first stage. Since the $F2\left|\right|\sum C_j$ problem is strongly NP-hard (Garey et al., 1976), our problem is also NP-hard. The variant with several identical parallel machines in the last stage is denoted as ${\mathit{DPm}}_1\to {\mathit{Pm}}_2\left|\right|\sum C_j$, and is referred as MA (from Multi machine Assembly) in the following. Obviously, this problem is also strongly NP-hard (Garey et al., 1976).

Despite the considerable number of papers published so far proposing constructive and improvement heuristics to solve both variants (for more details see the reviews by Hwang and Lin, 2018, Sheikh et al., 2018, Framinan et al., 2019, the state of the art regarding solution methods for the problem is unclear. One of the causes might be the lack of a standard and representative testbed since:

• Every time that a new approximate method for the problem is proposed, it is tested in different sets of instances with different parameters and processing times. Therefore, it might occur that a method obtains a good performance in a set of instances and a bad performance in another one. This fact may lead to an unclear knowledge about the state-of-the-art algorithms for this problem (see Section 4, where some results are different from those obtained in Talens et al., 2020 where the same heuristics are compared using a different testbed).

• Since the scheduling problems are very sensitive to the input data of the instance (Fernandez-Viagas and Framinan, 2015b), in some cases the methodologies adopted to generate the processing times of the instances do not guarantee that the researchers are solving their specific problems. More specifically, we will show that there is a strong relationship among the variants under study and the customer order and the traditional parallel machine problem, among others (see Section 3.1.1 for more details). In other words, the good performance of some approximate methods might have been established by solving, in fact, instances of a different (albeit related) scheduling problem. As a consequence, to fulfil this requirement, the procedure to design a new benchmark has to follow a methodology which ensures the adequacy of the instances to the scheduling problem under study such as e.g. in Fernandez-Viagas and Framinan (2020).

• The procedures adopted to generate the instances do not ensure that the resulting instances represent the hardest ones. This is an important aspect (see Vallada et al., 2015, Fernandez-Viagas and Framinan, 2020) since, for a given scheduling problem, using a solution procedure that outperforms others in the hardest instances ensures an excellent performance of this procedure when applied to easier instances, whereas the opposite does not have to be true.

This paper is aimed to tackle these issues. More specifically, the contribution is twofold: First, a computational analysis is performed in order to determine the relationship among our variants and the related scheduling problems. Then, using this information, we propose two comprehensive benchmarks for the 2-stage multi-machine scheduling problem with total completion time criterion (one for each variant considered). Secondly, a computational evaluation is performed in order to compare and evaluate the existing heuristics in the literature in both considered variants.

The paper is organised as follows: in Section 2 we define the 2-stage assembly scheduling problem, and review the different sets of instances proposed in the literature; in Section 3 the procedure to generate the proposed instances is detailed; the computational evaluation of heuristics is carried out in Section 4; and, finally, Section 5 presents the conclusions of the paper.