Invited ArticleA new iterated greedy algorithm for no-idle permutation flowshop scheduling with the total tardiness criterion
Introduction
Flowshop scheduling has its applicability and practicality in many industrial problems. Developing efficient and effective scheduling techniques along this line of research is therefore highly important. For flowshop scheduling, a number of jobs must be processed on a certain number of machines in the same processing order. The aim is to find permutations of jobs such that a given objective is minimized. Among the different variants of flowshop problems, relatively fewer studies have considered the no-idle case. For the no-idle permutation flowshop scheduling problem (NPFSP), the machines must process all jobs without any interruption. In other words, no idle time is allowed for any machine after the first job in the sequence is started. This constraint is commonly seen in real-world environments such as fiber glass processing (Kalczynski and Kamburowski, 2005) and foundry production (Saadani et al., 2003).
The NPFSP was first studied by Adiri and Pohoryles (1982). Given that exact methods such as branch and bound are able to tackle only small-scale instances of the problem (e.g., see Vachajitpan (1982) and Baptiste and Hguny (1997)), heuristic approaches have been seen as alternatives to solve problem instances of larger sizes. The first heuristic method for the NPFSP can be found in the work of Woollam (1986). By modeling the NPFSP as a traveling salesman problem, a heuristic approach based on the well-known nearest insertion rule was later proposed by Saadani et al. (2005). Another heuristic method was developed by Kalczynski and Kamburowski (2005), and they showed that their algorithm is better than that of Saadani et al. (2005). A greedy algorithm based on an insert move was studied by Baraz and Mosheiov (2008) and shown to be better than the approach of Saadani et al. (2005) too. For a comprehensive review of heuristic algorithms used for the NPFSP, see Goncharov and Sevastyanov (2009).
The ability of metaheuristics to find near-optimal solutions in polynomial time for large-scale problem instances has prompted Pan and Wang to propose two hybrid algorithms based on discrete Particle Swarm Optimization (Pan and Wang, 2008a) and discrete Differential Evolution (DE) (Pan and Wang, 2008b). In their studies, a speed-up method was developed for the insert move. As a result, both their algorithms are greatly dependent on the insert move. After that, an Iterated Greedy Algorithm (IGA) was presented by Ruiz et al. (2009). They tested the proposed algorithm on newly generated benchmark instances and showed that it is better than the algorithms of Pan and Wang. Two improved algorithms based on DE were later proposed by Deng and Gu (2012) and Tasgetiren et al. (2013b) for the NPFSP. Another metaheuristic algorithm – known as Invasive Weed Optimization (IWO) – was presented by Zhou et al. (2014), and their results showed that IWO is able to outperform the algorithm of Pan and Wang (2008a). More recently, Shao et al. (2017) developed a Memetic Algorithm and showed that their algorithm is significantly better than the algorithms of Pan and Wang (2008a), Ruiz et al. (2009), and Tasgetiren et al. (2013b). They updated 89 of the 250 best known solutions. Nagano et al. (2019) studied no-idle flowshops with total flowtime and proposed a highly efficient but simple constructive heuristic as well as a local search algorithm. Cheng et al. (2019) considered mixed no-idle flowshops with the makespan objective in a distributed environment, and proposed an IGA-based algorithm for their problem.
It is worth pointing out that all of the aforementioned studies were focusing on the makespan or flowtime objective. In our work, we focus on minimizing the total tardiness, which is of great importance in manufacturing systems because completing a job after its due date in such an environment would lead to loss of contracts, customers and trust. Tasgetiren et al. considered the total tardiness objective and presented a DE (Tasgetiren et al., 2011) as well as a Discrete Artificial Bee Colony (DABC) algorithm (Tasgetiren et al., 2013a) to address it. For both algorithms, a speed-up method was proposed for the insert move. They compared their DE with other variants of DE (Tasgetiren et al., 2011) and the DABC algorithm to a Genetic Algorithm (GA) (Tasgetiren et al., 2013a). Shen et al. (2015) later proposed a Bi-population Estimation of Distribution Algorithm (BEDA) for the total tardiness-oriented NPFSP. Their results indicated that the BEDA is more efficient and effective than both the GA and DABC of Tasgetiren et al. (2013a). Recently, Shao et al. (2018) proposed a hybrid discrete teaching-learning based meta-heuristic (HDTLM) for the problem, and they demonstrated the effectiveness of their algorithm using some well-known benchmark instances.
In this paper, we present a new IGA to solve the NPFSP considering the total tardiness criterion. The IGA is a simple and powerful algorithm with just a few parameters to tune (Ruiz and Stützle, 2007). It has been successfully applied to different variants of the flowshop scheduling problems, such as permutation flowshop scheduling (Dubois-Lacoste, Pagnozzi, Stützle, 2017, Karabulut, 2016, Ruiz, Stützle, 2007), blocking flowshop scheduling (Ding, Song, Gupta, Wang, Zhang, Wu, 2016, Riahi, Newton, Su, Sattar, 2019, Ribas, Companys, Tort-Martorell, 2011, Tasgetiren, Kizilay, Pan, Suganthan, 2017), no-idle flowshop scheduling (Pan, Ruiz, 2014, Tasgetiren, Pan, Suganthan, Buyukdagli, 2013), distributed no-idle flowshop scheduling (Ying et al., 2017), no-wait flowshop scheduling (Ding, Song, Gupta, Zhang, Chiong, Wu, 2015, Pan, Wang, 2008), and distributed flowshop scheduling (Fernandez-Viagas, Framinan, 2015, Ruiz, Pan, Naderi, 2019). Motivated by its simplicity and efficiency, we incorporate the following novel/improved elements to our proposed IGA:
I) Initial solutions: A new heuristic based on the NEH algorithm (Nawaz et al., 1983) is designed for the total tardiness-oriented NPFSP. Here, a new initial sequence is generated by taking both the total tardiness objective and no-idle constraint into account.
II) Local search: A variable local search based on an iterative insert move with two possible job selection mechanisms is developed. The two selection mechanisms are random job selection and greedy job selection. For random job selection, jobs are selected randomly and inserted in any possible positions of the sequence. For the greedy selection, jobs are selected based on their positions in a reference permutation. The iterative nature of our local search means the list of jobs will be updated when an improvement is observed.
III) Destruction-construction phases: Unlike the original IGA’s destruction-construction phases (Ruiz and Stützle, 2007), where jobs are randomly selected and removed from the current solution, and then reinserted in the best positions of a partial sequence, we introduce a new job-dependent parameter such that its value differs depending on the problem size. Additionally, instead of random selection a tardiness-guided procedure is proposed to select jobs. The idea behind this procedure is that jobs with higher tardiness values would have priority over jobs with lower tardiness values (i.e., to fix the more problematic parts of a solution). We also apply a swap move alongside the insert move during the construction phase to increase solution diversity.
IV) Acceptance probability: To accept new or non-improving solutions, most existing studies used a Simulated Annealing (SA)-based acceptance criterion as per Ruiz and Stützle (2007), where the probability remains the same at all time as the algorithm progresses. We propose a new time-dependent acceptance criterion that gives higher probabilities to non-improving solutions in early iterations. Then, when final iterations approach, this acceptance probability becomes close to zero. The basic idea is to explore the search space more in early iterations, and focus on exploitation in the later iterations. This time-dependent acceptance function is not only computationally simpler, its value would also reduce directly as the algorithm progresses.
To the best of our knowledge, our work represents one of the few studies in this area. Systematic computational experiments based on more than 100 benchmark problem instances from the literature with three different due date scenarios ( > 300 instances in total) clearly show that our proposed algorithm is able to outperform the best-performing algorithms in the literature. As a result, over 50% of the existing best solutions for the benchmark instances used have been updated.
The rest of this paper is structured as follows. In Section 2, we present the problem description. Details of the proposed algorithm are then given in Section 3. In Section 4, we discuss the setup of our computational evaluation as well as results obtained. Finally, we conclude the paper in Section 5, with possible future research directions highlighted.
Section snippets
Problem description
The NPFSP consists of n jobs that should be processed on m machines with the same sequence. p(j, i) denotes the time of processing job j on machine i (including the setup time), while dj denotes the due date of job j. At any time, each machine can process only one job, and each job can be processed on only one machine. To satisfy the no-idle constraint, machines must process jobs without any interruption from the start of processing the first job to the completion of the
The proposed IGA
The IGA is a simple and effective algorithm proposed by Ruiz and Stützle (2007) for flowshop scheduling. It has a number of main steps. At first, the algorithm begins with an initial solution, generated either randomly or through a constructive heuristic, and then it goes through a loop until the stopping criterion is reached. The loop consists of two main phases: destruction and construction. In the destruction phase, some jobs are removed from the current sequence. In the construction phase,
Computational experiments and results
In this section, we report on comprehensive computational experiments conducted to validate the effectiveness of the proposed EIGA. Experimental settings are first given in Section 4.1, and parameter tuning is discussed in Section 4.2. A comparison between our NNEH and the original NEH (Nawaz et al., 1983) is then presented in Section 4.3.
After that, we compare the EIGA with existing IGAs in the literature, including IGARS1 (Ruiz and Stützle, 2007), IGARS2 (Ruiz and Stützle, 2008), IGAPTL (
Conclusion
In this paper, we proposed a new, improved IGA for no-idle permutation flowshop scheduling with the total tardiness criterion. No-idle flowshop scheduling is widely seen in practical applications related to processing technology, e.g., fiber glass processing where glass batches are reduced to molten glass (Kalczynski and Kamburowski, 2005), or casting/assembly lines in which maximizing throughputs is of utmost importance, e.g., foundry production (Saadani et al., 2003). Core production machines
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