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.     

Two-agent two-machine flowshop scheduling with learning effects to minimize the total completion time

We study a two-agent scheduling problem in a two-machine permutation flowshop with learning effects. The objective is to minimize the total completion time of the jobs from one agent, given that the maximum tardiness of the jobs from the other agent cannot exceed a bound. We provide a branch-and-bound algorithm for the problem. In addition, we present several genetic algorithms to obtain near-optimal solutions. Computational results indicate that the algorithms perform well in either solving the problem or efficiently generating near-optimal solutions.     

Single-machine and flowshop scheduling with a general learning effect model

Learning effects in scheduling problems have received growing attention recently. Biskup [Biskup, D. (2008). A state-of-the-art review on scheduling with learning effect. European Journal of Operational Research, 188, 315–329] classified the learning effect scheduling models into two diverse approaches. The position-based learning model seems to be a realistic assumption for the case that the actual processing of the job is mainly machine driven, while the sum-of-processing-time-based learning model takes into account the experience the workers gain from producing the jobs. In this paper, we propose a learning model which considers both the machine and human learning effects simultaneously. We first show that the position-based learning and the sum-of-processing-time-based learning models in the literature are special cases of the proposed model. Moreover, we present the solution procedures for some single-machine and some flowshop problems.     

Some single-machine and m-machine flowshop scheduling problems with learning considerations

Scheduling with learning effect has drawn many researchers’ attention since Biskup [D. Biskup, Single-machine scheduling with learning considerations, European Journal of Opterational Research 115 (1999) 173-178] introduced the concept of learning into the scheduling field. Biskup [D. Biskup, A state-of-the-art review on scheduling with learning effect, European Journal of Opterational Research 188 (2008) 315-329] classified the learning approaches in the literature into two main streams. He claimed that the position-based learning seems to be a realistic model for machine learning, while the sum-of-processing-time-based learning is a model for human learning. In some realistic situations, both the machine and human learning might exist simultaneously. For example, robots with neural networks are used in computers, motor vehicles, and many assembly lines. The actions of a robot are constantly modified through self-learning in processing the jobs. On the other hand, the operators in the control center learn how to give the commands efficiently through working experience. In this paper, we propose a new learning model that unifies the two main approaches. We show that some single-machine problems and some specified flowshop problems are polynomially solvable.     

Note on “Single-machine and flowshop scheduling with a general learning effect model” and “Some single-machine and m-machine flowshop scheduling problems with learning considerations”

In this note, we show that the main results in the two papers [C.C. Wu, W.C. Lee, Single-machine and flowshop scheduling with a general learning effect model, Computers and Industrial Engineering 56 (2009) 1553-1558, W.C. Lee, C.C. Wu, Some single-machine and m-machine flowshop scheduling problems with learning considerations, Information Sciences 179 (2009) 3885-3892] are incorrect.     

A Constructive Genetic Algorithm for permutation flowshop scheduling

The general flowshop scheduling problem is a production problem where a set of n jobs have to be processed with identical flow pattern on m machines. In permutation flowshops the sequence of jobs is the same on all machines. A significant research effort has been devoted for sequencing jobs in a flowshop minimizing the makespan. This paper describes the application of a Constructive Genetic Algorithm (CGA) to makespan minimization on flowshop scheduling. The CGA was proposed recently as an alternative to traditional GA approaches, particularly, for evaluating schemata directly. The population initially formed only by schemata, evolves controlled by recombination to a population of well-adapted structures (schemata instantiation). The CGA implemented is based on the NEH classic heuristic and a local search heuristic used to define the fitness functions. The parameters of the CGA are calibrated using a Design of Experiments (DOE) approach. The computational results are compared against some other successful algorithms from the literature on Taillard’s well-known standard benchmark. The computational experience shows that this innovative CGA approach provides competitive results for flowshop scheduling problems.     

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.     

Computational complexity and solution algorithms for flowshop scheduling problems with the learning effect

In this paper, we analyze the two-machine flowshop problem with the makespan minimization and the learning effect, which computational complexity was not determined yet. First, we show that an optimal solution of this problem does not have to be the ‘permutation’ schedule if the learning effect is taken into consideration. Furthermore, it is proved that the permutation and non-permutation versions of this problem are NP-hard even if the learning effect, in a form of a step learning curve, characterizes only one machine. However, if both machines have learning ability and the learning curves are stepwise then the permutation version of this problem is strongly NP-hard. Furthermore, we prove the makespan minimization problem in m-machine permutation proportional flowshop environment remains polynomially solvable with identical job processing times on each machine even if they are described by arbitrary functions (learning curves) dependent on a job position in a sequence. Finally, approximation algorithms for the general problem are proposed and analyzed.     

Efficient non-population-based algorithms for the permutation flowshop scheduling problem with makespan minimisation subject to a maximum tardiness

This paper focuses on the problem of scheduling jobs in a permutation flowshop with the objective of makespan minimisation subject to a maximum allowed tardiness for the jobs, a problem that combines two desirable manufacturing objectives related to machine utilisation and to customer satisfaction. Although several approximate algorithms have been proposed for this NP-hard problem, none of them can use the excellent speed-up method by Taillard (1990) [22] for makespan minimisation due to the special structure of the problem under consideration. In this paper, several properties of the problem are defined in order to be able to partly apply Taillard׳s acceleration. This mechanism, together with a novel feasible tabu local search method, allows us to further exploit the structure of solutions of the problem, and are incorporated in two proposed algorithms: a bounded-insertion-based constructive heuristic and an advanced non-population-based algorithm. These algorithms are compared with state-of-the-art algorithms under the same computer conditions. The results show that both algorithms improve existing ones and therefore, constitute the new state-of-art approximate solution procedures for the problem.     

Some scheduling problems with sum-of-processing-times-based and job-position-based learning effects

In this paper we introduce a new scheduling model with learning effects in which the actual processing time of a job is a function of the total normal processing times of the jobs already processed and of the job’s scheduled position. We show that the single-machine problems to minimize makespan and total completion time are polynomially solvable. In addition, we show that the problems to minimize total weighted completion time and maximum lateness are polynomially solvable under certain agreeable conditions. Finally, we present polynomial-time optimal solutions for some special cases of the m-machine flowshop problems to minimize makespan and total completion time.     

A heuristic-based genetic algorithm for the two-machine flowshop scheduling with learning consideration

In scheduling problems, the learning phenomenon is often seen in some practical applications such as in the processing of certain chemicals in oil refineries and in the steel plates or bars produced by a foundry. A review of the literature reveals that most researchers paid more attention to the scheduling with both the single-machine settings and the learning without a bound. This is at odds with reality and thereby highlights the importance of addressing the issue by different approaches. This paper tackles the issue by considering a two-machine flowshop problem with a truncated learning consideration where the objective function is to minimize the makespan. In order to solve the proposed model, a branch-and-bound algorithm is first developed for the optimal solution. Then four genetic heuristic-based algorithms are proposed for the near-optimal solution. In addition, the experimental results of all proposed algorithms are also provided.     

A single-machine scheduling problem with learning effects in intermittent batch production

This paper studies a single-machine scheduling problem with three models of learning and forgetting effects in intermittent batch production. They are the models of no transmission, partial transmission and total transmission of learning from batch to batch. The phenomena exist in many realistic production systems. The objective is to minimize the makespan. We show that the problems with the models of no transmission and partial transmission of learning from batch to batch are polynomially solvable. We also provide two polynomial time algorithms for two special cases in the problem with the total transmission model.     

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.     

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.     

The single-machine total weighted tardiness scheduling problem with position-based learning effects

This study addresses the problem of minimizing the total weighted tardiness on a single-machine with a position-based learning effect. Several dominance properties are established to develop branch and bound algorithm and a lower bound is provided to derive the optimal solution. In addition, three heuristic procedures are developed for near-optimal solutions. Computational results are also presented to evaluate the performance of the proposed algorithms.     

Two-machine flow shop problem with effects of deterioration and learning

In this paper we consider a two-machine flow shop scheduling problem with effects of deterioration and learning. By the effects of deterioration and learning, we mean that the processing time of a job is a function of its execution starting time and its position in a sequence. The objective is to find a sequence that minimizes the total completion time. Optimal solutions are obtained for some special cases. For the general case, several dominance properties and some lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm. A heuristic algorithm is also proposed, which is shown by computational experiments to perform effectively and efficiently in obtaining near-optimal solutions.     

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。     

Some scheduling problems with deteriorating jobs and learning effects

Although scheduling with deteriorating jobs and learning effect has been widely investigated, scheduling research has seldom considered the two phenomena simultaneously. However, job deterioration and learning co-exist in many realistic scheduling situations. In this paper, we introduce a new scheduling model in which both job deterioration and learning exist simultaneously. The actual processing time of a job depends not only on the processing times of the jobs already processed but also on its scheduled position. For the single-machine case, we derive polynomial-time optimal solutions for the problems to minimize makespan and total completion time. In addition, we show that the problems to minimize total weighted completion time and maximum lateness are polynomially solvable under certain agreeable conditions. For the case of an m-machine permutation flowshop, we present polynomial-time optimal solutions for some special cases of the problems to minimize makespan and total completion time.     

A two-machine flowshop problem with two agents

The multiple-agent scheduling problems have received increasing attention recently. However, most of the research focuses on deriving feasible/optimal solutions or examining the computational complexity of the intractable cases in a single machine. Often a number of operations have to be done on every job in many manufacturing and assembly facilities (Pinedo, 2002 [1]). In this paper, we consider a two-machine flowshop problem where the objective is to minimize the total completion time of the first agent with no tardy jobs for the second agent. We develop a branch-and-bound algorithm and simulated annealing heuristic algorithms to search for the optimal solution and near-optimal solutions for the problem, respectively.     

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.