Some Applications of the “Branch-and-Bound” Algorithm to the Machine Scheduling Problem

The "branch-and-bound" algorithm for the exact solution of a three-machine scheduling problem proposed by Lomnicki has been generalized to the case of an arbitrary number of machines (under the assumption of an identical order on all the machines) and it has been adapted for an electronic computer. The flow chart of this algorithm used in the programming of calculations has been given and the efficiency of the method has been illustrated by some experimental evaluations of optimum job sequencing.     

具有学习效应的半导体晶圆制造绿色车间调度问题研究

针对最小化最大完工时间、总碳排放以及总拖期时间的具有学习效应的半导体晶圆制造绿色车间调度问题,构建了双影响因素的新型学习效应模型,提出了改进的多元宇宙优化算法,并对其收敛性进行证明。通过对初始种群进行反向学习、宇宙个体进行莱维飞行扰动和对外部档案中的个体进行邻域搜索变异更新,产生新的父代个体,扩大了种群的多样性,避免算法陷入局部最优。通过对小规模和大规模测试算例的仿真实验,以及利用改进算法求解具有异质性机器的学习型半导体晶圆制造绿色车间调度问题,验证了本文所提出的算法对于求解具有学习效应的半导体晶圆制造绿色车间调度问题的有效性和可行性。     

Scheduling a two-machine robotic cell: A solvable case

The paper deals with the scheduling of a robotic cell in which jobs are processed on two tandem machines. The job transportation between the machines is done by a transportation robot. The robotic cell has limitations on the intermediate space between the machines for storing the work-in-process. What complicates the scheduling problem is that the loading/unloading operation times are non-negligible. Given the total number of operationsn, an optimalO(n logn)-time algorithm is proposed together with the proof of optimality.     

粒子群遗传混合算法求解考虑传输时间的FJSP

在某些生产制造场景中,工件在不同机器间的传输时间对车间调度的总拖期具有重要影响,本文基于此扩展了总拖期最小的柔性作业车间调度模型。针对问题模型的复杂性,采用粒子群优化算法和遗传算法的混合算法进行求解。在初始化过程以一定概率优选加工时间和传输时间短的机器并排除调度频繁的机器,使种群在保持多样性的前提下尽量选择优化结果好的个体;采用线性调整的方式动态改变交叉概率和变异概率的值,使种群在遗传算法的不同阶段具有不同的搜索强度;采用粒子群优化算法进行局部搜索,弥补了遗传算法局部搜索能力的不足。最后采用本文方法和其他方法求解柔性作业车间调度问题实例,并对比不同水平层次传输时间下的总拖期,验证了本文方法的有效性。     

A genetic algorithm for hybrid flowshops with sequence dependent setup times and machine eligibility

After 50 years of research in the field of flowshop scheduling problems the scientific community still observes a noticeable gap between the theory and the practice of scheduling. In this paper we aim to provide a metaheuristic, in the form of a genetic algorithm, to a complex generalized flowshop scheduling problem that results from the addition of unrelated parallel machines at each stage, sequence dependent setup times and machine eligibility. Such a problem is common in the production of textiles and ceramic tiles. The proposed algorithm incorporates new characteristics and four new crossover operators. We show an extensive calibration of the different parameters and operators by means of experimental designs. To evaluate the proposed algorithm we present several adaptations of other well-known and recent metaheuristics to the problem and conduct several experiments with a set of 1320 random instances as well as with real data taken from companies of the ceramic tile manufacturing sector. The results indicate that the proposed algorithm is more effective than all other adaptations.     

An efficient heuristic based on machine workload for the flowshop scheduling problem with setup and removal

We are concerned in this paper with solving ann jobs,M machines flowshop scheduling problem where the objective function is the minimization of the makespan. We take into account setup, processing and removal times separately. After drawing up a synthesis of existing work which addresses this type of problems, we propose a new heuristic algorithm which is based on the machine workload to find an efficient permutation schedule. Computational experiences show that our algorithm yields excellent results, particularly when bottleneck machines are present.     

一种双目标带约束工件成类的平行机器调度启发式规则

提出了一种快速而有效的启发式规则(fam ily slack,简称FSLACK),来求解极小化总延误时间和极小化最大完工时间两个目标,工件按产品类型成组,带模具数量约束的平行机器生产调度问题.本文提出的FSLACK与EDD、LPT及SLACK进行了比较.随机订单的测试结果表明,本文提出的启发式规则在求解双目标带约束工件成类的平行机器调度问题上是有效的.这表明该算法可以应用在成型加工业的现场作业调度.     

A branch-and-price algorithm for scheduling parallel machines with sequence dependent setup times

We consider the problem of scheduling n independent jobs on m unrelated parallel machines with sequence-dependent setup times and availability dates for the machines and release dates for the jobs to minimize a regular additive cost function. In this work, we develop a new branch-and-price optimization algorithm for the solution of this general class of parallel machines scheduling problems. A new column generation accelerating method, termed “primal box”, and a specific branching variable selection rule that significantly reduces the number of explored nodes are proposed. The computational results show that the approach solves problems of large size to optimality within reasonable computational time.     

Multi-objective open shop scheduling by considering human error and preventive maintenance

This study presents an open shop scheduling model by considering human error and preventive maintenance. The proposed mathematical model takes into account conflicting objective functions including makespan, human error and machine availability. In order to find the optimum scheduling, human error, maintenance and production factors are considered, simultaneously. Human error is measured by Human Error Assessment and Reduction Technique (HEART). Three metaheuristic methods including non-dominated sorting genetic algorithm-II (NSGA-II), multi-objective particle swarm optimization (MOPSO) and strength Pareto evolutionary algorithm II (SPEA-II) are developed to find near-optimal solution. The Taguchi method is applied by adjusting parameters of metaheuristic algorithms. Several illustrative examples and a real case study (auto spare parts manufacturer) are applied to show the applicability of the multi-objective mixed integer nonlinear programming model. The proposed approach of this study may be used for similar open shop problems with minor modifications.     

含不相关机的动态可重入柔性流水车间问题的混合DABC-GA算法

为了改善生产线的物流平衡和加强阶段间的时间衔接,扩展一般可重入柔性流水车间调度理论,以最小化总加权完工时间为目标,研究了每阶段含不相关并行机的动态可重入柔性流水车间问题,工件在各阶段的加工时间取决于加工它的机器。鉴于所研究问题为NP-hard问题,首先,建立整数规划模型;其次,设计元胞矩阵编码方案,提出融合离散人工蜂群算法和遗传算法的一种混合算法以获得问题的近优解;最后,为了评估混合算法的性能,将所提出算法和一些元启发式算法进行了不同规模问题的对比测试,实验结果说明了所提算法的有效性。     

Online scheduling with machine cost and rejection

In this paper we define and investigate a new scheduling model. In this new model the number of machines is not fixed; the algorithm has to purchase the used machines, moreover the jobs can be rejected. We show that the simple combinations of the algorithms used in the area of scheduling with rejections and the area of scheduling with machine cost are not constant competitive. We present a 2.618-competitive algorithm called OPTCOPY.     

Parallel machine scheduling with common due windows

With the prevalence of on-time scheduling, timely product submission has become a crucial contributor to customer satisfaction. Studies examining on-time scheduling primarily seek to determine the minimum weighted sum of earliness and tardiness penalties. This study assumes that all machines are identical. Furthermore, this study assumes that jobs are independent and share a common due date window when investigating scheduling problems involving parallel machines with a minimum total number of early and tardy jobs (or maximum number of on-time jobs). This study presents related theorems and a novel simplified algorithm based on the problem. Additionally, rule characteristics are examined, and simulated data are used to verify the effectiveness and timeliness of the proposed algorithm. The theoretical proof and data test results all indicate that the proposed approach obtains the best solution within the shortest time.     

DBR理论求解柔性作业车间调度问题

针对柔性作业车间调度完工时间最小问题,提出一种结合DBR(鼓-缓冲器-绳子)理论和改进遗传算法的方法。在问题初始化时,建立瓶颈机器识别机制改善初始化方法,提高初始解的质量;在运算过程中依据关键路径建立瓶颈机器的识别机制和调度策略。为了更好保留每代中的优良解,采用外部精英库对优良解进行解保留。运用提出的算法求解基准测试问题,实验结果验证了算法的可行性和有效性。     

Routing and scheduling in a flexible job shop by tabu search

A hierarchical algorithm for the flexible job shop scheduling problem is described, based on the tabu search metaheuristic. Hierarchical strategies have been proposed in the literature for complex scheduling problems, and the tabu search metaheuristic, being able to cope with different memory levels, provides a natural background for the development of a hierarchical algorithm. For the case considered, a two level approach has been devised, based on the decomposition in a routing and a job shop scheduling subproblem, which is obtained by assigning each operation of each job to one among the equivalent machines. Both problems are tackled by tabu search. Coordination issues between the two hierarchical levels are considered. Unlike other hierarchical schemes, which are based on a one-way information flow, the one proposed here is based on a two-way information flow. This characteristic, together with the flexibility of local search strategies like tabu search, allows to adapt the same basic algorithm to different objective functions. Preliminary computational experience is reported.     

A multilevel machine and vehicle scheduling in a flexible manufacturing system

The paper presents a multilevel decision model for simultaneous machine and vehicle scheduling in a flexible manufacturing system. The system is composed of various machine types and a set of automated guided vehicles that permit each part to move between any pair of machines. The upper level of the decision model involves machine loading and part routing for which a bicriterion integer formulation is presented with the objective of balancing machine workloads and intermachine flows of parts. The lower level involves simultaneous scheduling of machines and vehicles for which a period-by-period heuristic is proposed based on a family of complex dispatching rules. The scheduling objective is to meet all part type requirements in a minimum time. Computational examples are included to illustrate the approach proposed.     

A hybrid genetic algorithm for an identical parallel-machine problem with maintenance activity

The scheduling of maintenance activities has been extensively studied, with most studies focusing on single-machine problems. In real-world applications, however, multiple machines or assembly lines process numerous jobs simultaneously. In this paper, we study a parallel-machine scheduling problem in which the objective is to minimize the total tardiness given that there is a maintenance activity on each machine. We develop a branch-and-bound algorithm to solve the problem with a small problem size. In addition, we propose a hybrid genetic algorithm to obtain the approximate solutions when the number of jobs is large. The performance of the proposed algorithms is evaluated based mainly on computational results.     

An Almost Optimal Heuristic for Preemptive C max Scheduling of Dependent Tasks on Parallel Identical Machines

We consider the problem of scheduling preemptable, dependent tasks on parallel, identical machines to minimize the makespan. The computational complexity of this problem remains open if the number of machines is fixed and larger than 2. The aim of this paper is to compare two heuristic algorithms on a basis of a computational experiment. The solutions generated by the heuristics are compared with optimal solutions obtained by a branch-and-bound algorithm. Computational results show that the heuristic based on node ordering finds optimal schedules for 99.9% of instances with the maximum relative deviation from optimum of 4.8%.     

Exploiting process plan flexibility in production scheduling: A multi-objective approach

When solving a product/process design problem, we must exploit the available degrees of freedom to cope with a variety of issues. Alternative process plans can be generated for a given product, and choosing one of them has implications on manufacturing functions downstream, including planning/scheduling. Flexible process plans can be exploited in real time to react to machine failures, but they are also relevant for off-line scheduling. On the one hand, we should select a process plan in order to avoid creating bottleneck machines, which would deteriorate the schedule quality; on the other one we should aim at minimizing costs. Assessing the tradeoff between these possibly conflicting objectives is difficult; actually, it is a multi-objective problem, for which available scheduling packages offer little support. Since coping with a multi-objective scheduling problem with flexible process plans by an exact optimization algorithm is out of the question, we propose a hierarchical approach, based on a decomposition into a machine loading and a scheduling sub-problem. The aim of machine loading is to generate a set of efficient (non-dominated) solutions with respect to the load balancing and cost objectives, leaving to the user the task of selecting a compromise solution. Solving the machine loading sub-problem essentially amounts to selecting a process plan for each job and to routing jobs to the machines; then a schedule must be determined. In this paper we deal only with the machine loading sub-problem, as many scheduling methods are already available for the problem with fixed process plans. The machine loading problem is formulated as a bicriterion integer programming model, and two different heuristics are proposed, one based on surrogate duality theory and one based on a genetic descent algorithm. The heuristics are tested on a set of benchmark problems.     

Minimizing makespan and total completion time for parallel batch processing machines with non-identical job sizes

This paper considers the scheduling problem of parallel batch processing machines with non-identical job sizes. The jobs are processed in batches and the machines have the same capacity. The models of minimizing makespan and total completion time are given using mixed integer programming method and the computational complexity is analyzed. The bound on the number of feasible solutions is given and the properties of the optimal solutions are presented. Then a polynomial time algorithm is proposed and the worst case ratios for minimizing total completion time and makespan is proved to be 2 and (8/3–2/3 m) respectively. To test the proposed algorithm, we generate different levels of random instances. The computational results demonstrate the effectiveness of the algorithm for minimizing the two objectives.     

需要安装时间的平行多功能机排序问题的启发式算法

考虑需要安装时间的平行多功能机排序问题。在该模型中,每个工件对应机器集合的一个子集,其只能在这个子集中的任一台机器上加工,称这个子集为该工件的加工集合;工件分组,同组工件具有相同的加工时间和加工集合,不同组中的工件在同一台机器上连续加工需要安装时间,目标函数为极小化最大完工时间。对该问题NP-难的一般情况设计启发式算法:首先按照特定的规则将所有工件组都整组地安排到各台机器上,然后通过在各机器间转移工件不断改进当前最大完工时间。通过与下界的比较检验算法的性能,大量的计算实验表明,算法是实用而有效的。