An effective hybrid PSO-based algorithm for flow shop scheduling with limited buffers
Introduction
Production scheduling plays a key role in the manufacturing systems of enterprises for maintaining a competitive position in fast-changing markets, so it is very important to develop effective and efficient advanced manufacturing and scheduling technologies and approaches [1]. Flow shop scheduling problem (FSSP) is a class of widely studied scheduling problems with a strong engineering background, which represents nearly a quarter of manufacturing systems, assembly lines, and information service facilities in use nowadays, and has earned a reputation for being difficult to solve [2], [3], [4], [5], [6]. Whereas, it is often assumed that intermediate buffers between two consecutive machines have infinite capacity and a job can be stored for an unlimited amount of time. With the advent of just-in-time manufacturing and kanban control systems, which maintain a limited in-process inventory, the study on scheduling problem with the limited buffer capacities is of greater significance and hence has attracted much attention [7], [8]. In this paper, we consider the permutation FSSP (PFSSP) with the limited buffers between each two consecutive machines. In such a case, after finishing processing on a machine, a job either directly has to be processed on the next machine or it has to be stored in the buffer between the two machines. If the buffer is completely occupied, the job has to wait on its current machine and this machine is blocked for other jobs. This blocking will be finished if the job, which momentarily is processed on the next machine, leaves that machine. At that time a job from the buffer or the job which blocks its machine may start processing on the next machine. In the literature, the criterion widely used is the minimization of the maximum completion time, i.e., makespan. According to the research work by Papadimitriou and Kanellakis [9], the FSSP with the limited buffers is strongly NP-hard even for two machines only. Thus, due to the significance both in academic and engineering fields, it is important to develop effective and efficient approaches for such a problem.
Compared with a lot of literature on the classical FSSP, the research on FSSP with the limited buffers is not extensive. A good review on blocking and no-wait scheduling problems can be found in [7]. As for the two-machine PFSSP with intermediate buffers, Dutta and Cunningham [10] presented a dynamic programming procedure to generate optimal solutions to minimize the makespan. Smutnicki [11] presented a tabu search (TS) approach by incorporating the reduced neighborhood, search accelerator and technique of back jumps on the search trajectory. Sarper and Henry [12] further considered the dynamic scheduling case operating as a Poisson process, and they found the MDD (Modified Due Date) and the SPTL (Shortest Processing Time Local) performed better than the other four dispatching rules investigated. Soukhal et al. [13] investigated the complexity of two-machine FSSP with both constraints on transportation and buffer capacities, and proved it strongly NP-hard when the transportation capacity is limited to two or three parts with an unlimited buffer at the output of each machine.
As for the multi-machine FSSP with intermediate buffers under the criteria of makespan minimization, Reddi [14] proposed a dynamic programming procedure. Leisten [15] presented systematic comparisons among some well-known heuristics for problems with no buffers, unlimited buffers and limited buffers. It was concluded that the NEH method proposed by Nawaz et al. [2] performs better for the FSSP with the limited buffers. Nowicki [16] proposed a generalization of the notion of a block of jobs suggested by Grabowski et al. [17] and a local search technique controlled by a TS for PFSSP with the limited buffers. And Brucker et al. [18] generalized Nowicki's idea to the case where different job-sequence on machines were allowed and formulated a TS for general FSSP with buffers. McCormick et al. [19] transformed the buffer constrained problem into a blocking one where all machines have no intermediate buffers by modeling the positions of the intermediate buffers as machines with zero processing time. Wang et al. [20] proposed an effective hybrid genetic algorithm (HGA) where both multiple genetic operators and a neighborhood structure based on graph model were employed, and a decision probability was used to control the utilization of mutation operation and local search based on problem-specific information. Norman [21] applied TS for FSSP with sequence-dependent setup time and finite buffers. Sikora [22] presented a modified GA based on new crossover and mutation operators for a lot-sizing FSSP involving limited buffers, sequence-dependent setup time, capacity constraints, and due dates. Wardono and Fathi [23] developed TS for multi-stage parallel machine FSSP with the limited buffer capacities, and a procedure to construct a complete schedule was also introduced.
In addition, batch scheduling in buffer constrained flow shop has also received academic attention. Agnetis et al. [24] proved that a batch sequencing problem in a two-machine FSSP with the limited buffer is NP-hard, and proposed an exact branch-and-bound algorithm by analyzing the batching nature of the problem. Pranzo [25] studied the case in which each batch required separated setup, processing and removal times. It is proved that under some assumptions on the batch size, the problem could be formulated as a Traveling Salesman Problem (TSP) and solved in polynomial time by reducing the TSP cost structure to a two-machine no-wait flow shop. Sawik [26] presented a mixed integer programming formulation for makespan minimization in batch scheduling in a hybrid FSSP with the limited buffers between the stages, which included various cutting constraints exploiting special configurations and some properties of batch processing on parallel machines. In addition, Gourgand et al. [27] addressed a stochastic FSSP with exponentially distributed processing time considering buffers of any capacity (unlimited, limited or null), and proposed a recursive formula based on Markovian analysis and Chapman–Kolomogorov equation to evaluate the performances of schedules and a heuristics based on simulated annealing (SA) with Markovian model was used to obtain the schedules.
Recently, a novel evolutionary technique, named particle swarm optimization (PSO), has been proposed [28], whose development was based on observations of the social behavior of animals, such as bird flocking, fish schooling, and swarm theory. PSO is initialized with a population of random solutions. Each individual is assigned with a randomized velocity according to its own and its companions’ flying experiences, and the individuals, called particles, are then flown through the hyperspace. Due to the simple concept, easy implementation, and quick convergence, nowadays, PSO has gained much attention and a wide variety of applications in a variety of fields, mainly for continuous optimization problems [29], [30]. However, most of the published research work about PSO is for continuous optimization problems, while little research work can be found for combinatorial optimization problems, especially for scheduling problems. Obviously, it is a challenge to employ the algorithm in different areas of problems other than those areas that the inventors originally focused on. Recently, Tasgetiren et al. [31] proposed a hybrid PSO (PSOVNS) by incorporating the smallest position value (SPV) encoding rule based on random key [32] and variable neighborhood search (VNS) [33] for the PFSSP. To the best of our knowledge, there is no published work for dealing with FSSP with the limited buffers using PSO. In this paper, we will propose a hybrid PSO-based algorithm (HPSO) for PFSSP with the limited buffers to minimize makespan. The proposed HPSO stresses the balance between global exploration and local exploitation. Not only does HPSO apply the evolutionary searching mechanism of PSO characterized by individual improvement plus population cooperation and competition to effectively perform exploration, but it also utilizes several adaptive local searches to perform exploitation. In particular, first, to apply PSO for scheduling problems, a novel encoding scheme based on random key representation and a ranked-order-value (ROV) rule is developed, which can convert the continuous position values of particles to job permutations. Second, an efficient population initialization based on the famous NEH heuristic [2], as well as on a modification of the NEH heuristic is incorporated into a random initialization of PSO to generate an initial population with certain quality and diversity. Third, a local search strategy based on the generalization of the block elimination properties [16], named block-based local search, is probabilistically applied to some good particles chosen by roulette wheel selection with certain decision probability to balance the exploration and exploitation abilities. Moreover, SA with multiple neighborhoods guided by an adaptive meta-Lamarckian learning strategy [34], [35] is designed and applied to prevent the premature convergence and concentrate computing effort on promising neighbor solutions. Simulation results and comparisons demonstrate the effectiveness of the proposed HPSO.
The remaining contents are organized as follows. In Section 2, the mathematical and graph models of the PFSSP with the limited buffers are provided. In Section 3, the HPSO is proposed after presenting the essential elements used in HPSO. In Section 4, experimental results and comparisons are presented and analyzed, in addition, the effects of some parameters on the optimization performance are discussed. Finally, in Section 5, we end the paper with some conclusions.
Section snippets
PFSSP with the limited buffers
The permutation FSSP (PFSSP) with the limited buffers can be described as follows. There are n jobs and m machines . Each of the n jobs is to be sequentially processed on machine . The processing time of job j on machine i is given. At any time, each machine can process at most one job and each job can be processed on at most one machine. The sequence in which the jobs are to be processed is the same for each machine. Considering the constraints on buffer
Brief introduction to PSO
PSO is an evolutionary computation technique through individual improvement plus population cooperation and competition [28]. A particle's status on the search space is characterized by two factors: its position and velocity. The position and velocity of the ith particle in the d-dimensional search space can be represented as and , respectively. Each particle has its own best position (pbest) corresponding to the personal best
Experimental setup
To test the performance of the proposed HPSO for PFSSP with the limited buffers, computational simulation is carried out with some well-studied benchmarks. In this paper, 29 problems that were contributed to the OR-Library are selected. The first eight problems are called car1, car2 through car8 by Carlier [37]. The other 21 problems are called rec01, rec03 through rec41 by Reeves [38], who used them to compare the performances of SA, GA, and neighborhood search and found these problems to be
Conclusion and future research
By applying special encoding mechanism, PSO-based parallel search and multiple problem-dependent local searches, we proposed a hybrid PSO-based algorithm for the PFSSP with the limited buffers in this paper. To the best of our knowledge, this is the first report to apply PSO-based algorithm to PFSSP with the limited buffers. Due to the hybridization of PSO and local searches, searching behaviors can be enriched, searching ability can be enhanced, and exploitation and exploration are well
Acknowledgments
This research is partially supported by NSFC (Grant No. 60204008, 60374060, and 60574072) as well as the 973 Program (Grant No. 2002CB312200). The authors thank the editors of this special issue and anonymous referees for their constructive comments on the earlier manuscript of this paper.
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2021, Omega (United Kingdom)Citation Excerpt :In this study, we employ PSO as one of the variables selection tools since it leads to the TAN models that achieve a high predictive performance. Perhaps, PSO turns out to be an effective variable selection tool due to its meta-heuristic nature that can manage to obtain a solution for irregular, time-dependent or even noisy functions, thereby having a wide range of uses in various scientific fields [50–52], including machine learning [53], image processing [54], business optimization [55], and operations research [56]. Given that overfitting is a common problem in predictive modeling, it is important to validate the algorithm on an unfamiliar dataset [57].