A multi-start variable neighbourhood descent algorithm for hybrid flowshop rescheduling
Introduction
Hybrid Flowshop (HFS) scheduling is a complicated decision-making process of committing a number of jobs to a set of machines distributed in several stages [1,2]. HFS scheduling is a NP-hard combinatorial optimization problem and arises broadly in flexible manufacturing and process industry, such as electronics, paper, textile and Steelmaking-Continuous Casting (SCC) [1,3,4]. Efficient HFS schedules can improve production productivity greatly and make significant monetary savings. Therefore, the problem has attracted much attention in recent years [1,2,5].
A series of HFS scheduling methods have been reported in the literature, which can be categorized into three groups: mathematical programming methods, heuristics, and metaheuristics. Among the mathematical programming methods, branch and bound, Lagrangian relaxation have been extensively adopted [[6], [7], [8], [9], [10]]. As to heuristics, Gupta et al. [11] designed constructive heuristics for two-stage HFS scheduling where only the first stage contains identical machines. Kurza and Askin [12] proposed cyclic and insertion heuristics for HFS scheduling with Sequence Dependent Setup Times (SDST), while Dios et al. [5] provided a number of heuristics for HFS scheduling with missing operations.
With regard to the metaheuristics, early research mainly concentrated on HFS scheduling with traditional optimization objectives, i.e. minimizing makespan, total flowtime, or weighted completion time. Wang et al. [13] and Pan et al. [14] developed improved Estimation of Distribution Algorithm (EDA) and Artificial Bee Colony (ABC) respectively, where both of them designed novel decoding approaches. Pan and Dong [15] devised an improved Migrating Birds Optimization (MBO) algorithm for HFS. Tang and Wang [16] proposed a modified Particle Swarm Optimization algorithm for realistic HFS scheduling derived from SCC production. Li et al. [17] developed a hybrid Variable Neighbourhood Search (VNS), where a neighbourhood change strategy and EDA are integrated to enhance the diversification. In recent years, HFS scheduling with more realistic and complex characteristics has become increasingly active, since the characteristics such as setup times, due windows, and energy consumption have existed in the production process [[18], [19], [20]]. For example, to solve the HFS scheduling with SDST, Kurza and Askin [18], Ruiz and Maroto [19] presented effective Genetic Algorithms (GA), while Pan et al. [21] proposed three population-based and six trajectory-based methods, where the discrete ABC was demonstrated to be the best. Subsequently, for the HFS scheduling with due windows, Pan et al. [22] devised Iterated Greedy (IG) and Iterated Local Search. Recently, for the HFS scheduling with total/setup energy consumption, Lei et al. [23] and Li et al. [24] designed Teaching-Learning-Based Optimization (TLBO) and multi-objective evolutionary algorithm respectively.
HFS scheduling generally includes a series of assumptions, e.g. machines are always available for processing, jobs are available at time zero [1,25,26]. However, due to complicated production environments, realistic production systems often encounter a wide range of unexpected dynamic events, e.g. machine breakdown, new job arrival, and job release time variation, which would make the original schedule infeasible or ineffective, and have to be re-optimized [25,[27], [28], [29], [30]]. It is essential to respond to the dynamic events immediately and produce new schedules quickly, which corresponds to HFS rescheduling. HFS rescheduling is to rebuild a new optimal HFS schedule under production environments after dynamic events, which is also NP-hard [26]. Since the production environments after dynamic events are different from the original, static HFS scheduling methods may not be suitable for HFS rescheduling [25].
Unfortunately, however, compared with extensive studies on HFS scheduling, less attention has been paid to HFS rescheduling. Meanwhile, much of the existing work are based on metaheuristics. In the early studies, the traditional efficiency optimization objectives such as makespan are adopted, and machine breakdown was generally considered as the dynamic event. Mirabi et al. [30] devised a heuristic for the two-stage HFS rescheduling. Zandieh and Gholami [31] adopted an immune algorithm to solve HFS scheduling with SDST. Tang et al. [25] constructed a neural network model for HFS rescheduling with dynamic job arrivals, where average tardy time, the percentage of tardy jobs and average flow time were utilized as optimization objectives. Thereafter, Tang et al. [32] proposed a modified differential evolution algorithm to solve the SCC rescheduling. Tang et al. [33] developed a modified PSO for energy-efficient HFS rescheduling considering new job arrivals and machine breakdown, where makespan and energy consumption were used as optimization objectives. Moreover, differ from most of the existing research, Wang and Choi [34] and Wang et al. [35] devised decomposition-based approaches, the main idea is to decompose a rescheduling problem into several cluster scheduling problems first, and then solve them by different approaches. In the above researches, system instability is not considered usually, i.e. the difference in performance between new and original schedule. This is not consistent with the industrial reality. Recently, system instability has been given more attention and also been considered as one of the optimization objectives. Rahmani and Ramezanian [36] adopted VNS to address general HFS rescheduling considering new job arrival, where instability and total weighted tardiness were minimized. Li et al. [26] and Peng et al. [4] studied SCC rescheduling with machine breakdown, and devised a hybrid fruit fly optimization algorithm and an improved ABC, respectively.
Besides HFS rescheduling, other kinds of production rescheduling has also been studied in the literature, where Permutation Flow Shop (PFS) rescheduling and Flexible Job Shop (FJS) rescheduling have received more attention. For the PFS rescheduling, Katragjini et al. [29] devised a rescheduling framework under three dynamic events, i.e. machine breakdown, new job arrival, and job ready time variation. Lower and upper bounds for makespan and instability were provided, and an IG rescheduling algorithm was also developed. Later, in their extension work [37], rescheduling under five more dynamic events was further studied. Subsequently, Li et al. [38] proposed an effective TLBO when five different dynamic events were considered simultaneously, where an IG is integrated as local search. Liu et al. [39] provided four hybrid algorithms for rescheduling with the stochastic and dynamic events. Lu et al. [40] presented a hybrid multi-objective grey wolf optimization algorithm for rescheduling with controllable processing times., while Liu et al. [41] designed heuristics for rescheduling with new job arrival. For the PJS rescheduling considering new job arrival and fuzzy processing time, Gao et al. [42] developed an improved ABC, where probability-based solution generation operators were designed. Very recently, Gao et al. [43] developed a discrete Jaya algorithm for the FJS rescheduling with new job arrival, where five objective-oriented local search strategies were devised. It is worth noting that, although these PFS and PJS rescheduling methods cannot be applied for the HFS rescheduling directly, their main idea and schemes, e.g. objective-oriented local search, can be borrowed to devise the HFS rescheduling methods.
To summarize, there has been limited work dealing with HFS rescheduling in the literature. These researches mainly concentrate on two kinds of dynamic events: machine breakdown and new job arrival. Moreover, majority of the literature considered one and only one dynamic event. This is different from industrial standards since realistic production systems may suffer from several types of dynamic events. Therefore, HFS rescheduling simultaneously considering several dynamic events is essential to meet the demands of modern industry. However, to the best of our knowledge, the problem has seldom been investigated to this date. Thus, it is urgent to study the problem and devise effective HFS rescheduling methods.
Variable Neighbourhood Descent (VND) is an excellent trajectory-based metaheuristic for combinational optimization problems [[44], [45], [46], [47], [48]]. Due to the advantages of a few control parameters and powerful performance, the VND has been successfully utilized to solve a number of production scheduling problems, e.g. flowshop scheduling, block flowshop scheduling, HFS scheduling and distributed PFS scheduling [22,[49], [50], [51]]. However, to the best of our knowledge, there is little research on the application of the VND to deal with the HFS rescheduling.
In this paper, the HFS rescheduling problem considering simultaneously machine breakdown, new job arrival, and job release time variation is studied. Moreover, a Multi-Start Variable Neighbourhood Descent (MSVND) algorithm is developed to deal with the problem. More specifically, the main contributions are described as follows:
- (1)
The HFS rescheduling problem considering simultaneously three types of dynamic events is studied. The mathematical model with objectives of minimizing makespan and system instability is established.
- (2)
Approaches for determining lower and upper bounds of the optimization objectives are presented.
- (3)
A MSVND algorithm is devised to solve the HFS rescheduling problem.
- (4)
A hybrid decoding considering the problem characteristics is devised.
- (5)
Intensification and diversification strategies are devised to balance the exploitative and explorative tendencies of the search algorithm.
Extensive experimental comparisons demonstrate the efficiency of the specially-devised strategies. Additionally, comprehensive comparisons against seven state-of-the-art algorithms further verify the effectiveness of the MSVND. In these experiments, three combinations of objective weight coefficients and three termination conditions are tested.
The remainder of this paper is structured as follows. Section 2 introduces the HFS rescheduling problem in detail. Section 3 gives lower and upper bounds calculation approaches for the makespan and system instability. Section 4 proposes the devised MSVND for the HFS rescheduling. Section 5 provides the extensive computational experiments. Section 6 concludes the paper and unearths some avenues for future work.
Section snippets
Problem description
The HFS rescheduling problem studied in this paper consists of m stages (M = {1,2, …,m}). Every stage k ∈ M has nk identical parallel machines, where nk ≥ 2 for at least one stage. n jobs (J = {1,2, …,n}) are to be processed in the m stages. Each job j ∈ J must be processed sequentially from the 1st stage to the mth stage. Accordingly, each job j ∈ J has m operations: O1j,O2j, …,Omj. In each stage k ∈ M, each job j ∈ J must be processed on a machine i∈{1,2, …,nk} in this stage [14,15]. At time
Lower and upper bounds of optimization objectives
As the makespan and instability are in different units and have different orders of magnitude, normalization is utilized to scale their values into the range of [0,1] in this paper. Note that normalization has been used for the PFS rescheduling in Katragjini et al. [29] and Li et al. [38]. By applying the normalization, the objective function is obtained as follows:whereanddenote the normalization values of the makespan and system
The proposed MSVND for HFS rescheduling
In this section, we discuss the proposed MSVND method in sufficient details. First, we introduce the basic VND. Subsequently, the encoding and hybrid decoding approach are elaborated. Then, we sequentially describe the initialization, three neighbourhood structures, neighbourhood changing strategy, intensification strategies, and diversification strategies. Finally, we provide a complete framework of the MSVND.
Experimental results
To demonstrate the effectiveness of the proposed MSVND, we compare it to seven effective algorithms in the literature, which are listed as follows:
- (1)
three high-performance flow shop rescheduling algorithms, i.e. discrete Teaching-Learning-Based Optimization algorithm (TLBO) in Li et al. [38], Iterated Greedy (IG) in Katragjini et al. [29], Iterated Greedy with Variable Neighbourhood Descent (IGVND), where the framework is same as that of the IG of Katragjini et al. [29], and the VND serves as the
Conclusions
In this paper, the HFS rescheduling problem considering simultaneously machine breakdown, new job arrival and job release time variation, is investigated. By hybridizing the VND with the specially-designed hybrid decoding, intensification and diversification strategies, an effective MSVND algorithm is developed for the problem. Seven effective algorithms have been adopted for comparison. Extensive computational comparisons on a total of 150 instances have confirmed the efficiency and
Acknowledgments
The authors sincerely thank the editor and anonymous reviewers for their valuable comments and helpful suggestions. This project is supported by National Natural Science Foundation of China (Grant No. 51705177, 51575212, 51825502, 51775216), Natural Science Foundation of Hubei Province (Grant No. 2018CFA078), and Program for HUST Academic Frontier Youth Team.
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