Effective constructive heuristics and meta-heuristics for the distributed assembly permutation flowshop scheduling problem

This paper addresses a novel distributed assembly permutation flowshop scheduling problem that has important applications in modern supply chains and manufacturing systems. The problem considers a number of identical factories, each one consisting of a flowshop for part-processing plus an assembly line for product-processing. The objective is to minimize the makespan. To suit the needs of different CPU time and solution quality, we present a mixed integer linear model, three constructive heuristics, two variable neighborhood search methods, and an iterated greedy algorithm. Important problem-specific knowledge is obtained to enhance the effectiveness of the algorithms. Accelerations for evaluating solutions are proposed to save computational efforts. The parameters and operators of the algorithms are calibrated and analyzed using a design of experiments. To prove the algorithms, we present a total of 16 adaptations of other well-known and recent heuristics, variable neighborhood search algorithms, and meta-heuristics for the problem and carry out a comprehensive set of computational and statistical experiments with a total of 810 instances. The results show that the proposed algorithms are very effective and efficient to solve the problem under consideration as they outperform the existing methods by a significant margin.     

置换流水车间调度问题的两阶段分布估计算法

针对置换流水车间调度问题,以最小化总流水时间为目标,提出了一种新颖的两阶段分布估计算法。第一阶段先利用NEH（Nawaz-Enscore-Ham,NEH）启发式构造一个较优的初始个体,然后随机生成初始种群,为保留种群的多样性,提出一种择优机制来选择个体并建立概率模型,同时在当代种群中利用精英机制保留当代种群中的最优解,最后利用概率模型采样并生成下一代种群。第二阶段采用插入、互换操作算子对第一阶段得到的最优解进行邻域搜索,来提高分布估计算法的全局搜索能力,阻止其陷入局部最优解。通过对算例进行实验、对比和分析,证明该算法的可行性和有效性。     

New simple constructive heuristic algorithms for minimizing total flow-time in the permutation flowshop scheduling problem

This paper develops a set of new simple constructive heuristic algorithms to minimize total flow-time for an $n$-jobs×$m$-machines permutation flowshop scheduling problem. We first propose a new iterative algorithm based on the best existing simple heuristic algorithm, and then integrate new indicator variables for weighting jobs into this algorithm. We also propose new decision criteria to select the best partial sequence in each iteration of our algorithm. A comprehensive numerical experiment reveals that our modifications and extensions improve the effectiveness of the best existing simple heuristic without affecting its computational efficiency.     

A linkage mining in block-based evolutionary algorithm for permutation flowshop scheduling problem

A genetic algorithm is a type of heuristic algorithm used to solve permutation flowshop scheduling problems (PFSPs). Producing an optimal offspring with a variety of genes is difficult because of the evolution of the gene selection and a crossover mechanism that leads to local optima. This study proposes a linkage mining in block-based evolutionary algorithm (LMBBEA) for solving the PFSP, in which the association rule extracts various good genes and increases gene diversity. These genes are used to generate various blocks for artificial chromosome combinations. The generated blocks not only improve the chance of finding optimal solutions but also enhance the efficiency of convergence. The proposed LMBBEA is compared with other algorithms through numerical experiments, namely the Taillard and Reeves experiments in the OR-Library. To compare with other algorithms, the solutions produced by the proposed LMBBEA are closest to the optimal solution. The LMBBEA has a high convergence speed and a better solution quality due to an increase in the diversity of solutions.     

A discrete particle swarm optimization algorithm for the no-wait flowshop scheduling problem

In this paper, a discrete particle swarm optimization (DPSO) algorithm is presented to solve the no-wait flowshop scheduling problem with both makespan and total flowtime criteria. The main contribution of this study is due to the fact that particles are represented as discrete job permutations and a new position update method is developed based on the discrete domain. In addition, the DPSO algorithm is hybridized with the variable neighborhood descent (VND) algorithm to further improve the solution quality. Several speed-up methods are proposed for both the swap and insert neighborhood structures. The DPSO algorithm is applied to both 110 benchmark instances of Taillard [Benchmarks for basic scheduling problems. European Journal of Operational Research 1993;64:278–85] by treating them as the no-wait flowshop problem instances with the total flowtime criterion, and to 31 benchmark instances provided by Carlier [Ordonnancements a contraintes disjonctives. RAIRO Recherche operationelle 1978;12:333–51], Heller [Some numerical experiments for an M×J$\mathrm{M}\times \mathrm{J}$ flow shop and its decision-theoretical aspects. Operations Research 1960;8:178–84], and Revees [A genetic algorithm for flowshop sequencing. Computers and Operations Research 1995;22:5–13] for the makespan criterion. For the makespan criterion, the solution quality is evaluated according to the reference makespans generated by Rajendran [A no-wait flowshop scheduling heuristic to minimize makespan. Journal of the Operational Research Society 1994;45:472–8] whereas for the total flowtime criterion, it is evaluated with the optimal solutions, lower bounds and best known solutions provided by Fink and Voß [Solving the continuous flow-shop scheduling problem by metaheuristics. European Journal of Operational Research 2003;151:400–14]. The computational results show that the DPSO algorithm generated either competitive or better results than those reported in the literature. Ultimately, 74 out of 80 best known solutions provided by Fink and Voß [Solving the continuous flow-shop scheduling problem by metaheuristics. European Journal of Operational Research 2003;151:400–14] were improved by the VND version of the DPSO algorithm.     

A Self-guided Genetic Algorithm for permutation flowshop scheduling problems

In this paper we develop a Self-guided Genetic Algorithm (Self-guided GA), which belongs to the category of Estimation of Distribution Algorithms (EDAs). Most EDAs explicitly use the probabilistic model to sample new solutions without using traditional genetic operators. EDAs make good use of the global statistical information collected from previous searches but they do not efficiently use the location information about individual solutions. It is recently realized that global statistical information and location information should complement each other during the evolution process. In view of this, we design the Self-guided GA based on a novel strategy to combine these two kinds of information. The Self-guided GA does not sample new solutions from the probabilistic model. Instead, it estimates the quality of a candidate offspring based on the probabilistic model used in its crossover and mutation operations. In such a way, the mutation and crossover operations are able to generate fitter solutions, thus improving the performance of the algorithm. We tested the proposed algorithm by applying it to deal with the NP-complete flowshop scheduling problem to minimize the makespan. The experimental results show that the Self-guided GA is very promising. We also demonstrate that the Self-guided GA can be easily extended to treat other intractable combinatorial problems.     

An iterated local search for the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times

Due to its simplicity yet powerful search ability, iterated local search (ILS) has been widely used to tackle a variety of single-objective combinatorial optimization problems. However, applying ILS to solve multi-objective combinatorial optimization problems is scanty. In this paper we design a multi-objective ILS (MOILS) to solve the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times to minimize the makespan and total weighted tardiness of all jobs. In the MOILS, we design a Pareto-based variable depth search in the multi-objective local search phase. The search depth is dynamically adjusted during the search process of the MOILS to strike a balance between exploration and exploitation. We incorporate an external archive into the MOILS to store the non-dominated solutions and provide initial search points for the MOILS to escape from local optima traps. We compare the MOILS with several multi-objective evolutionary algorithms (MOEAs) shown to be effective for treating the multi-objective permutation flowshop scheduling problem in the literature. The computational results show that the proposed MOILS outperforms the MOEAs.     

改进的粒子群算法求解置换流水车间调度问题

针对置换流水车间调度问题,提出了一种改进的粒子群算法进行求解。改进算法引入了判断粒子群早熟的方法,并在发现粒子群早熟后采用逆转策略对种群最优粒子进行变异,利用模拟退火思想概率接收新的最优粒子。种群最优粒子的改变会引导粒子群跳出局部极值的约束,从而克服粒子群的早熟状态。通过对置换流水车间调度问题中Car系列和Rec系列部分基准数据的测试,证明了该算法的有效性。     

A machine-learning based memetic algorithm for the multi-objective permutation flowshop scheduling problem

In recent years, the historical data during the search process of evolutionary algorithms has received increasing attention from many researchers, and some hybrid evolutionary algorithms with machine-learning have been proposed. However, the majority of the literature is centered on continuous problems with a single optimization objective. There are still a lot of problems to be handled for multi-objective combinatorial optimization problems. Therefore, this paper proposes a machine-learning based multi-objective memetic algorithm (ML-MOMA) for the discrete permutation flowshop scheduling problem. There are two main features in the proposed ML-MOMA. First, each solution is assigned with an individual archive to store the non-dominated solutions found by it and based on these individual archives a new population update method is presented. Second, an adaptive multi-objective local search is developed, in which the analysis of historical data accumulated during the search process is used to adaptively determine which non-dominated solutions should be selected for local search and how the local search should be applied. Computational results based on benchmark problems show that the cooperation of the above two features can help to achieve a balance between evolutionary global search and local search. In addition, many of the best known Pareto fronts for these benchmark problems in the literature can be improved by the proposed ML-MOMA.     

An estimation of distribution algorithm for minimizing the total flowtime in permutation flowshop scheduling problems

In this work we propose an estimation of distribution algorithm (EDA) as a new tool aiming at minimizing the total flowtime in permutation flowshop scheduling problems. A variable neighbourhood search is added to the algorithm as an improvement procedure after creating a new offspring. The experiments show that our approach outperforms all existing techniques employed for the problem and can provide new upper bounds.     

Makespan distribution of permutation flowshop schedules

The makespan distribution of permutation flowshop schedules has been a topic of debate for almost fifty years. Many researchers have confirmed or doubted the famous claim that the makespan distribution of permutation flowshop schedules is asymptotically normal if the number of jobs is sufficiently large. This paper theoretically and empirically investigates the makespan distribution of permutation flowshop schedules and shows that the normality claim is not valid for the job-dominated and machine-dominated flowshops. Errors in the proof of normality of the makespan distribution of permutation flowshop schedules are pointed out. It is shown that the makespan distribution of a permutation flowshop scheduling problem depends on the number of jobs as well as the number of machines.     

A memetic algorithm for the re-entrant permutation flowshop scheduling problem to minimize the makespan

A common assumption in the classical permutation flowshop scheduling model is that each job is processed on each machine at most once. However, this assumption does not hold for a re-entrant flowshop in which a job may be operated by one or more machines many times. Given that the re-entrant permutation flowshop scheduling problem to minimize the makespan is very complex, we adopt the CPLEX solver and develop a memetic algorithm (MA) to tackle the problem. We conduct computational experiments to test the effectiveness of the proposed algorithm and compare it with two existing heuristics. The results show that CPLEX can solve mid-size problem instances in a reasonable computing time, and the proposed MA is effective in treating the problem and outperforms the two existing heuristics.     

变量块内部迭代算法求解零空闲流水车间问题

零空闲流水车间问题（NIFSP）是流水车间问题中带有约束条件的典型NP-hard问题,在大多数现实场景下,零空闲约束是对机器的基本要求。而目前关于NIFSP问题提出的算法对于较大规模算例、综合性能及参数调整的灵活性较差。为此,以最小化最大完工时间为目标,提出了一种可变内部迭代算法VIIA。在VIIA的初始化阶段,使用改进的FRB5产生初始解,提高了FRB5的效率,在保证算法性能的同时极大地缩短了CPU消耗时间。在破坏重建阶段,通过增加对移除工件块数量的内部迭代,从而灵活调整参数值。VIIA增大了邻域搜索,以适应不同规模的算例。为了验证VIIA算法的性能,将该算法与在流水车间调度问题中表现优秀的几种算法进行了比较。实验结果证明了VIIA在NIFSP问题求解上性能的优越性,并且在最优解的搜索上,性能明显优于对比算法。     

A hybrid estimation of distribution algorithm for simulation-based scheduling in a stochastic permutation flowshop

The permutation flowshop scheduling problem (PFSP) is NP-complete and tends to be more complicated when considering stochastic uncertainties in the real-world manufacturing environments. In this paper, a two-stage simulation-based hybrid estimation of distribution algorithm (TSSB-HEDA) is presented to schedule the permutation flowshop under stochastic processing times. To deal with processing time uncertainty, TSSB-HEDA evaluates candidate solutions using a novel two-stage simulation model (TSSM). This model first adopts the regression-based meta-modelling technique to determine a number of promising candidate solutions with less computation cost, and then uses a more accurate but time-consuming simulator to evaluate the performance of these selected ones. In addition, to avoid getting trapped into premature convergence, TSSB-HEDA employs both the probabilistic model of EDA and genetic operators of genetic algorithm (GA) to generate the offspring individuals. Enlightened by the weight training process of neural networks, a self-adaptive learning mechanism (SALM) is employed to dynamically adjust the ratio of offspring individuals generated by the probabilistic model. Computational experiments on Taillard’s benchmarks show that TSSB-HEDA is competitive in terms of both solution quality and computational performance.     

Iterated greedy algorithms for the blocking flowshop scheduling problem with makespan criterion

Recently, iterated greedy algorithms have been successfully applied to solve a variety of combinatorial optimization problems. This paper presents iterated greedy algorithms for solving the blocking flowshop scheduling problem (BFSP) with the makespan criterion. Main contributions of this paper can be summed up as follows. We propose a constructive heuristic to generate an initial solution. The constructive heuristic generates better results than those currently in the literature. We employ and adopt well-known speed-up methods from the literature for both insertion and swap neighborhood structures. In addition, an iteration jumping probability is proposed to change the neighborhood structure from insertion neighborhood to swap neighborhood. Generally speaking, the insertion neighborhood is much more effective than the swap neighborhood for the permutation flowshop scheduling problems. Instead of considering the use of these neighborhood structures in a framework of the variable neighborhood search algorithm, two powerful local search algorithms are designed in such a way that the search process is guided by an iteration jumping probability determining which neighborhood structure will be employed. By doing so, it is shown that some additional enhancements can be achieved by employing the swap neighborhood structure with a speed-up method without jeopardizing the effectiveness of the insertion neighborhood. We also show that the performance of the iterated greedy algorithm significantly depends on the speed-up method employed. The parameters of the proposed iterated greedy algorithms are tuned through a design of experiments on randomly generated benchmark instances. Extensive computational results on Taillard’s well-known benchmark suite show that the iterated greedy algorithms with speed-up methods are equivalent or superior to the best performing algorithms from the literature. Ultimately, 85 out of 120 problem instances are further improved with substantial margins.     

Distributed assembly permutation flow shop problem; Single seekers society algorithm

The distributed manufacturing and assembly systems have an important role at the point of overcoming the difficulties faced by today's mass-production industry. By using both of these systems together in the same production system, the advantages of this integration can make industries more flexible and stronger. Besides, optimizing these systems is more complicated since the multiple production systems can undoubtfully affect the production system’s performance. In this paper, two new mixed-integer linear programming (MILP) models are proposed for the distributed assembly permutation flow shop problem (DAPFSP), inspiring by the multiple-travelling salesman structure. Moreover, a single seekers society (SSS) algorithm is proposed for solving the DAPFSP to minimize the maximum completion time of all products. The performance of the proposed MILP models is evaluated using 900 small-sized benchmark instances. The proposed MILP models were effective and were able to find more optimal solutions or improve the best-found solutions for the small-sized DAPFSP benchmark instances. Similarly, the SSS algorithm is statistically compared with the best-known algorithms developed for solving the DAPFSP on 900 small and 810 large-sized benchmark instances. The proposed SSS algorithm shows superior performance compared to other algorithms in solving the small-sized DAPFSP instances in terms of finding better solutions. In addition, it is as effective as the best performing algorithms developed to solve the large-sized DAPFSP instances. Furthermore, the best-found solutions for 40 numbers of test problems reported to be improved in this paper.     

Polynomial time algorithms for the UET permutation flowshop problem with time delays

This paper addresses the problem of scheduling a set of n unit execution time (UET) jobs on an m-permutation flowshop with arbitrary time delays, so as to minimize the makespan criterion. A polynomial time algorithm is exhibited for the three-machine and four-machine cases, respectively.     

An effective hybrid teaching–learning-based optimization algorithm for permutation flow shop scheduling problem

Permutation flow shop scheduling (PFSP) is among the most studied scheduling settings. In this paper, a hybrid Teaching–Learning-Based Optimization algorithm (HTLBO), which combines a novel teaching–learning-based optimization algorithm for solution evolution and a variable neighborhood search (VNS) for fast solution improvement, is proposed for PFSP to determine the job sequence with minimization of makespan criterion and minimization of maximum lateness criterion, respectively. To convert the individual to the job permutation, a largest order value (LOV) rule is utilized. Furthermore, a simulated annealing (SA) is adopted as the local search method of VNS after the shaking procedure. Experimental comparisons over public PFSP test instances with other competitive algorithms show the effectiveness of the proposed algorithm. For the DMU problems, 19 new upper bounds are obtained for the instances with makespan criterion and 88 new upper bounds are obtained for the instances with maximum lateness criterion.     

基于TSP方法求解等待时间受限的置换流水车间调度

等待时间受限的置换流水车间调度问题要求工件在连续两个机器间的等待时间满足上限值约束.对此,分析了工件序列中相邻工件的加工持续时间及其上下界关系,并且提出一种启发式方法.首先,建立旅行商间题（TSP）以生成初始调度;然后,采用扩展插入方法优化调度解.为了衡量算法性能,给出问题下界的计算方法和相关评价指标,并通过数据实验验证了该启发式和下界计算方法的可行性和有效性.     

Real-time scheduling for distributed permutation flowshops with dynamic job arrivals using deep reinforcement learning

Distributed manufacturing plays an important role for large-scale companies to reduce production and transportation costs for globalized orders. However, how to real-timely and properly assign dynamic orders to distributed workshops is a challenging problem. To provide real-time and intelligent decision-making of scheduling for distributed flowshops, we studied the distributed permutation flowshop scheduling problem (DPFSP) with dynamic job arrivals using deep reinforcement learning (DRL). The objective is to minimize the total tardiness cost of all jobs. We provided the training and execution procedures of intelligent scheduling based on DRL for the dynamic DPFSP. In addition, we established a DRL-based scheduling model for distributed flowshops by designing suitable reward function, scheduling actions, and state features. A novel reward function is designed to directly relate to the objective. Various problem-specific dispatching rules are introduced to provide efficient actions for different production states. Furthermore, four efficient DRL algorithms, including deep Q-network (DQN), double DQN (DbDQN), dueling DQN (DlDQN), and advantage actor-critic (A2C), are adapted to train the scheduling agent. The training curves show that the agent learned to generate better solutions effectively and validate that the system design is reasonable. After training, all DRL algorithms outperform traditional meta-heuristics and well-known priority dispatching rules (PDRs) by a large margin in terms of solution quality and computation efficiency. This work shows the effectiveness of DRL for the real-time scheduling of dynamic DPFSP.