NEH-based heuristics for the permutation flowshop scheduling problem to minimise total tardiness

Since Johnson׳s seminal paper in 1954, scheduling jobs in a permutation flowshop has been receiving the attention of hundreds of practitioners and researchers, being one of the most studied topics in the Operations Research literature. Among the different objectives that can be considered, minimising the total tardiness (i.e. the sum of the surplus of the completion time of each job over its due date) is regarded as a key objective for manufacturing companies, as it entails the fulfilment of the due dates committed to customers. Since this problem is known to be NP-hard, most research has focused on proposing approximate procedures to solve it in reasonable computation times. Particularly, several constructive heuristics have been proposed, with NEHedd being the most efficient one, serving also to provide an initial solution for more elaborate approximate procedures. In this paper, we first analyse in detail the decision problem depending on the generation of the due dates of the jobs, and discuss the similarities with different related decision problems. In addition, for the most characteristic tardiness scenario, the analysis shows that a huge number of ties appear during the construction of the solutions done by the NEHedd heuristic, and that wisely breaking the ties greatly influences the quality of the final solution. Since no tie-breaking mechanism has been designed for this heuristic up to now, we propose several mechanisms that are exhaustively tested. The results show that some of them outperform the original NEHedd by about 25% while keeping the same computational requirements.     

On insertion tie-breaking rules in heuristics for the permutation flowshop scheduling problem

The most efficient approximate procedures so far for the flowshop scheduling problem with makespan objective – i.e. the NEH heuristic and the iterated greedy algorithm – are based on constructing a sequence by iteratively inserting, one by one, the non-scheduled jobs into all positions of an existing subsequence, and then, among the so obtained subsequences, selecting the one yielding the lowest (partial) makespan. This procedure usually causes a high number of ties (different subsequences with the same best partial makespan) that must be broken via a tie-breaking mechanism. The particular tie-breaking mechanism employed is known to have a great influence in the performance of the NEH, therefore different procedures have been proposed in the literature. However, to the best of our knowledge, no tie-breaking mechanism has been proposed for the iterated greedy. In our paper, we present a new tie-breaking mechanism based on an estimation of the idle times of the different subsequences in order to pick the one with the lowest value of the estimation. The computational experiments carried out show that this mechanism outperforms the existing ones both for the NEH and the iterated greedy for different CPU times. Furthermore, embedding the proposed tie-breaking mechanism into the iterated greedy provides the most efficient heuristic for the problem so far.     

Bi-criteria minimization for the permutation flowshop scheduling problem with machine-based learning effects

In traditional scheduling problems, the processing time for the given job is assumed to be a constant regardless of whether the job is scheduled earlier or later. However, the phenomenon named “learning effect” has extensively been studied recently, in which job processing times decline as workers gain more experience. This paper discusses a bi-criteria scheduling problem in an m-machine permutation flowshop environment with varied learning effects on different machines. The objective of this paper is to minimize the weighted sum of the total completion time and the makespan. A dominance criterion and a lower bound are proposed to accelerate the branch-and-bound algorithm for deriving the optimal solution. In addition, the near-optimal solutions are derived by adapting two well-known heuristic algorithms. The computational experiments reveal that the proposed branch-and-bound algorithm can effectively deal with problems with up to 16 jobs, and the proposed heuristic algorithms can yield accurate near-optimal solutions.     

An improved NEH-based heuristic for the permutation flowshop problem

NEH is an effective heuristic for solving the permutation flowshop problem with the objective of makespan. It includes two phases: generate an initial sequence and then construct a solution. The initial sequence is studied and a strategy is proposed to solve job insertion ties which may arise in the construct process. The initial sequence which is generated by combining the average processing time of jobs and their standard deviations shows better performance. The proposed strategy is based on the idea of balancing the utilization among all machines. Experiments show that using this strategy can improve the performance of NEH significantly. Based on the above ideas, a heuristic NEH-D (NEH based on Deviation) is proposed, whose time complexity is O(mn²), the same as that of NEH. Computational results on benchmarks show that the NEH-D is significantly better than the original NEH.     

A grasp algorithm for m-machine flowshop scheduling problem with bicriteria of makespan and maximum tardiness

In this paper we address the problem of minimizing the weighted sum of makespan and maximum tardiness in an m-machine flow shop environment. This is a NP-hard problem in the strong sense. An attempt has been made to solve this problem using a metaheuristic called Greedy Randomized Adaptive Search Procedure (GRASP). GRASP is a competitive algorithm and is a meta-heuristic for solving combinatorial optimization problems. We have customized the basic concepts of GRASP algorithm to solve a bicriteria flow shop problem and a new algorithm named B-GRASP (Bicriteria GRASP algorithm) is proposed. The new proposed algorithm is evaluated using benchmark problems taken from Taillard and compared with the existing simulated annealing based heuristic developed by Chakravarthy and Rajendran. Computational experiments indicate that the proposed algorithm is much better than the existing one in all cases.     

A cooperative dispatching approach for minimizing mean tardiness in a dynamic flowshop

Cooperative Dispatching is a real-time scheduling methodology, which consults downstream machines before making a job dispatching decision on any given machine. This paper proposes such an approach for minimizing the mean tardiness in a dynamic flowshop where new jobs arrive continuously, at random points in time, throughout the production cycle. Cooperative Dispatching is based on the idea that individual machines act self-interestedly, with the objective of optimizing their local performance criteria. A consulted machine attempts to influence upstream dispatching decisions in a manner that promotes its ability to minimize its total local tardiness. A machine's influence in the dispatching decision depends on current congestion and due-date tightness levels in the shop. A multiple regression model is proposed to help determine the weight a consulted machine's preferences will carry in the dispatching decision. Conflicting demands from the different machines are resolved by a minimum regret decision procedure, which aims to minimize the aggregate deviation from the consulted machines' preferences. The winning candidate that ultimately emerges from this procedure is the job that is dispatched. A comparative analysis to evaluate the performance of cooperative dispatching, compared to six other dispatching rules that are commonly favoured for tardiness-based criteria, is performed by means of simulation, using randomly generated test problems. Computational results indicate that Cooperative Dispatching outperforms the other dispatching rules, across a broad range of flowshop congestion and due-date tightness levels.     

A two-machine flowshop makespan scheduling problem with deteriorating jobs

In traditional scheduling problems, the job processing times are assumed to be known and fixed from the first job to be processed to the last job to be completed. However, in many realistic situations, a job will consume more time than it would have consumed if it had begun earlier. This phenomenon is known as deteriorating jobs. In the science literature, the deteriorating job scheduling problems are relatively unexplored in the flowshop settings. In this paper, we study a two-machine flowshop makespan scheduling problem in which job processing times vary as time passes, i.e. they are assumed as increasing functions of their starting times. First, an exact algorithm is established to solve most of the problems of up to 32 jobs in a reasonable amount of time. Then, three heuristic algorithms are provided to derive the near-optimal solutions. A simulation study is conducted to evaluate the performances of the proposed algorithms. In addition, the impact of the deterioration rate is also investigated.     

Efficient constructive and composite heuristics for the Permutation Flowshop to minimise total earliness and tardiness

In this paper we address the problem of scheduling jobs in a permutation flowshop with a just-in-time objective, i.e. the minimisation of the sum of total tardiness and total earliness. Since the problem is NP-hard, there are several approximate procedures available for the problem, although their performance largely depends on the due dates of the specific instance to be solved. After an in-depth analysis of the problem, different cases or sub-problems are identified and, by incorporating this knowledge, four heuristics are proposed: a fast constructive heuristic, and three different local search procedures that use the proposed constructive heuristic as initial solution.The proposed Prod. Type: FLPheuristics have been compared on an extensive set of instances with the best-so-far heuristic for the problem, as well as with adaptations of efficient heuristics for similar scheduling problems. The computational results show the excellent performance of the proposed algorithms. Finally, the positive impact of the efficient heuristics is evaluated by including them as seed sequences for one of the best metaheuristic for the problem.     

Minimizing the bicriteria of makespan and maximum tardiness with an upper bound on maximum tardiness

This paper studies the flowshop scheduling problem with a complex bicriteria objective function. A weighted sum of makespan and maximum tardiness subject to a maximum tardiness threshold value is to be optimized. This problem, with interesting potential applications in practice, has been sparsely studied in the literature. We propose global and local dominance relationships for the three-machine problem and a fast and effective genetic algorithm (GA) for the more general m$m$-machine case. The proposed GA incorporates a novel three-phase fitness assignment mechanism specially targeted at dealing with populations in which both feasible as well as infeasible solutions might coexist. Comprehensive computational and statistical experiments show that the proposed GA outperforms the two most effective existing heuristics by a considerable margin in all scenarios. Furthermore, the proposed GA is also faster and able to find more feasible solutions. It should be noted that when the weight assigned to maximum tardiness is zero, then the problem is reduced to minimizing makespan subject to a maximum tardiness threshold value. Heuristics for both problems have been provided in the literature recently but they have not been compared. Another contribution of this paper is to compare these recent heuristics with each other.     

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.     

求解置换流水车间调度问题的混合鸟群算法

针对置换流水车间调度问题（PFSP）,提出了一种混合鸟群算法（HBSA）以更加有效地最小化最大完工时间。首先,为了改善初始种群的质量和多样性,结合一种基于NEH（Nawaz-Enscore-Ham）的启发式算法和混沌映射提出了一种新的种群初始化方法;其次,为了使算法能够处理离散的调度问题,采用最大排序值（LRV）规则将连续的位置值转换为离散的工件排序;最后,为了强化算法对解空间的探索能力,借鉴变邻域搜索（VNS）和迭代贪婪（IG）算法的思想针对个体最佳工件排序和种群最佳工件排序分别提出了局部搜索方法。针对广泛使用的Rec标准测试集进行了仿真测试,并与目前有效的元启发式算法——刘等提出的混合差分进化算法（L-HDE）、混合共生生物搜索算法（HSOS）、离散狼群算法（DWPA）、多班级教学优化算法（MCTLBO）相比较,结果表明,HBSA取得的最佳相对误差（BRE）、平均相对误差（ARE）的平均值比上述四种算法至少下降了73.3%、76.8%,从而证明HBSA具有更强的寻优能力和更好的稳定性。尤其是针对测试算例Rec25和Rec27,仅HBSA的求解结果达到了目前已知最优解,进一步证明了其优越性。     

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.     

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.     

Minimising total tardiness in the m-machine flowshop problem: A review and evaluation of heuristics and metaheuristics

In this work, a review and comprehensive evaluation of heuristics and metaheuristics for the m-machine flowshop scheduling problem with the objective of minimising total tardiness is presented. Published reviews about this objective usually deal with a single machine or parallel machines and no recent methods are compared. Moreover, the existing reviews do not use the same benchmark of instances and the results are difficult to reproduce and generalise. We have implemented a total of 40 different heuristics and metaheuristics and we have analysed their performance under the same benchmark of instances in order to make a global and fair comparison. In this comparison, we study from the classical priority rules to the most recent tabu search, simulated annealing and genetic algorithms. In the evaluations we use the experimental design approach and careful statistical analyses to validate the effectiveness of the different methods tested. The results allow us to clearly identify the state-of-the-art methods.     

Dispatching heuristics for the single machine weighted quadratic tardiness scheduling problem

In this paper, we consider the single machine scheduling problem with weighted quadratic tardiness costs. Several efficient dispatching rules are proposed. These include existing heuristics for the linear problem, as well as procedures suitably adapted to the quadratic objective function. Also, both forward and backward scheduling procedures are considered.The computational results show that the heuristics that specifically take into account the quadratic objective significantly outperform their linear counterparts. Also, the backward scheduling approach proves to be superior, and the difference in performance is even more noticeable for the harder instances.The best of the backward scheduling heuristics is both quite efficient and effective. Indeed, this procedure can quickly generate a schedule even for large instances. Also, its relative deviation from the optimum is usually rather low, and it performs adequately even for the more difficult instances.     

Heuristic algorithms for preemptive scheduling in a two-stage hybrid flowshop with additional renewable resources at each stage

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines and renewable resources at both the stages. The resource requirements are of a 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Four heuristic algorithms using linear programming are proposed for solving this problem. Performance of the algorithms is analyzed experimentally by comparing heuristic solutions with the lower bound on the optimal makespan. Statistical comparative analysis of the developed algorithms is carried out. The results of a computational experiment show that the proposed algorithms are able to produce good quality solutions in a small amount of computation time.     

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。     

Scheduling in a two-machine flowshop for the minimization of the mean absolute deviation from a common due date

This paper addresses the minimization of the mean absolute deviation from a common due date in a two-machine flowshop scheduling problem. We present heuristics that use an algorithm, based on proposed properties, which obtains an optimal schedule for a given job sequence. A new set of benchmark problems is presented with the purpose of evaluating the heuristics. Computational experiments show that the developed heuristics outperform results found in the literature for problems up to 500 jobs.     

A multi-objective electromagnetism algorithm for a bi-objective flowshop scheduling problem

This paper deals with a bi-objective flowshop scheduling problem minimizing the makespan and total weighted tardiness, in which all jobs may not be processed by all machines. Furthermore, we consider transportation times between machines. Obtaining an optimal solution for this type of complex, large-sized problem in reasonable computational time by using traditional approaches and optimization tools is extremely difficult. This paper presents a new multi-objective electromagnetism algorithm (MOEM). The motivation behind this algorithm has risen from the attraction–repulsion mechanism of electromagnetic theories. Along with MOEA, we apply simulated annealing to solve the given problem. A set of experimental instances are carried out to evaluate the algorithm by advanced multi-objective performance measures. The related results show that a variant of our proposed MOEM provides sound performance comparing with other algorithms.     

Heuristics for a two-stage assembly flowshop with bicriteria of maximum lateness and makespan

We consider a two-stage assembly flowshop scheduling problem with the objective of minimizing a weighted sum of makespan and maximum lateness. The problem is known to be NP-hard, and therefore, we propose heuristics to solve the problem. The proposed heuristics are Tabu search (Tabu), particle swarm optimization (PSO), and self-adaptive differential evolution (SDE). An extensive computational experiment is conducted to compare performances of the proposed heuristics. The computational experiment reveals that both PSO and SDE are much superior to Tabu. Moreover, it is statistically shown that PSO performs better than SDE. The computation times of both PSO and SDE are close to each other and they are less than 40 and 45 s, respectively, for the largest size problem considered.