Minimizing total tardiness of orders with reentrant lots in a hybrid flowshop

This study focuses on a hybrid flowshop scheduling problem, in which there are serial stages, each with identical parallel machines. In the hybrid flowshop, each order is composed of multiple lots with the same due date, and each lot can be processed on any one of parallel machines at each stage. In addition, there are reentrant flows since lots of certain orders have to visit the stages twice. Heuristic algorithms are suggested for the scheduling problem with the objective of minimizing total tardiness of a given set of orders. In these algorithms, the list-scheduling method is employed, and lots are scheduled with priorities determined with a construction method. Computational experiments are performed on randomly generated test problems. Results show that the suggested algorithms perform better than well-known dispatching rules for various scheduling problems and an algorithm that is used in a real system.     

Minimizing makespan in a no-wait flowshop with two batch processing machines using estimation of distribution algorithm

This paper studies the problem of minimising makespan in a no-wait flowshop with two batch processing machines (comprised of a parallel batch processing machine and a serial batch processing machine), non-identical job sizes and unequal ready times. We propose a population-based evolutionary method named estimation of distribution algorithm (EDA). Firstly, the individuals in the population are coded into job sequences. Then, a probabilistic model is built to generate new population and an incremental learning method is developed to update the probabilistic model. Thirdly, the best-fit heuristic is used to group jobs into batches and a least idle/waiting time approach is proposed to sequence the batches on batch processing machines. In addition, some problem-dependent local search heuristics are incorporated into the EDA to further improve the searching quality. Computational simulation and comparisons with some existing algorithms demonstrate the effectiveness and robustness of the proposed algorithm. Furthermore, the effectiveness of embedding the local search method in the EDA is also evaluated.     

Meta-heuristics for the distributed two-stage assembly scheduling problem with bi-criteria of makespan and mean completion time

This paper focuses on the distributed two-stage assembly flowshop scheduling problem for minimising a weighted sum of makespan and mean completion time. This problem involves two inter-dependent decision sub-problems: (1) how to allocate jobs among factories and (2) how to schedule the assigned jobs at each factory. A mathematical model is formulated for solving the small-sized instances of the problem. Since the NP-hardness of the problem, we also proposed a variable neighbourhood search (VNS) algorithm and a hybrid genetic algorithm combined with reduced variable neighbourhood search (GA-RVNS) to solve the distributed two-stage assembly flowshop scheduling problems and approximately optimise makespan and mean completion time simultaneously. Computational experiments have been conducted to compare the performances of the model and proposed algorithms. For a set of small-sized instances, both the model and the proposed algorithms are effective. The proposed algorithms are further evaluated on a set of large-sized instances. The results statistically show that both GA-RVNS and VNS obtain much better performances than the GA without RVNS-based local search step (GA-NOV). For the instances with small numbers of jobs, VNS achieves better performances than GA-RVNS. However, for the instances with large numbers of jobs, GA-RVNS yields better performances than the VNS. It is also shown that the overall performances of VNS are very close to GA-RVNS with different numbers of factories, weights given to makespan and numbers of machines at the first stage.     

A scheduling problem in glass manufacturing

In this paper, a production scheduling problem in glass manufacturing is studied. The production facility consists of multiple identical production lines and each production line includes a number of serially arranged machines. The production is characterized by semi-ordered processing times in each product family, and the last machine in each production line is a bottleneck machine. Significant changeover times are required when products of different families are produced on a production line. The scheduling problem was modeled as a parallel no-delay flowshop scheduling problem (PNDFSP). The PNDFSP combines the parallel machine scheduling problem (PMSP) with the no-delay flowshop scheduling problem (NDFSP). While PMSP and NDFSP have received considerable attention in the literature, PNDFSP has not been well studied. A mixed-integer programming formulation is developed and an efficient heuristic algorithm is proposed. The sequential heuristic algorithm considers simultaneously the line changeover time, no-delay effect, and line utilization in assigning product families to the production lines. The computational results are reported.     

Flowshops with processing flexibility across production stages

This paper addresses a scheduling problem for a two stage hybrid flowshop (HFS) environment with significant processing flexibility across stages. A job can be either completely processed at one of the stages or its processing is split in a prespecified fashion between stages. For the latter case, the routing is only in one direction, from the 'upstream' stage to the 'downstream' stage. A stage may consist of one or more parallel identical machines. We first study the special case of a single machine per stage and present a pseudopolynomial dynamic programming algorithm to minimize makespan. Then, we provide a heuristic for the HFS_m1,m2 problem with m_k machines at stage k = 1,2, and prove its worst case and average performance. Also, we extend our algorithms to the case of stage dependent processing times. Finally, we report computational results on the average performance of HFS_m1,m2 with comparable flowshop and parallel machine environments, and provide managerial guidelines for process choice decisions.     

双机无等待流水作业调度算法研究

考虑双机无等待流水作业调度问题,此问题中每台机器都受一个非可用时间的约束,工件都有不同的释放时间。机器的非可用性时间间隔是部分重叠并且已知。目标使Makespan(最大流程时间)最小。通过不同的方式计算上限和下限,完善分支定界法。计算机实验结果显示了所述方法的有效性。     

More MILP models for hybrid flow shop scheduling problem and its extended problems

With the rapid development of computer technology and related softwares for mathematical models, mathematical modelling of scheduling problems is receiving growing attention from researchers. In this work, the hybrid flow shop scheduling problem with unrelated parallel machines (HFSP-UPM) with the objective aimed to minimise the makespan is studied. According to the characteristics of the HFSP-UPM, eight mixed integer linear programming (MILP) models are formulated in order to obtain optimal solutions based on different modelling ideas. Then, these models are extended to solve HFSP-UPM with sequence-dependent setup times (HFSP-UPM-SDST), no-wait HFSP-UPM (HFSP-UPM-NW) and HFSP-UPM with blocking (HFSP-UPM-B). All the proposed models and the existing model are detailedly compared and evaluated under three aspects namely modelling process, size complexity and computational complexity. Numerical experiments show that MILP models dependent on diverse modelling ideas perform very differently. The model developed based on stage precedence is the best one and should be given preference in future applications. In addition, the proposed models of HFSP-UPM-NW and HFSP-UPM-B improve several best known solutions for the test instances in the existing literature.     

Lot streaming and scheduling multiple products in two-machine no-wait flowshops

We consider the problem of minimizing makespan in two-machine no-wait flowshops with multiple products requiring lot streaming. A “product” (or lot) consists of many identical items. Lot streaming (lot sizing) is the process of creating sublots (or transfer batches to move the completed portion of a production )sublot to downstream machines so that operations can be overlapped. The number of sublots for each product is fixed. When the flowshop produces only a single product, we obtain optimal continuous-sized sublots. It is shown that these sublot sizes are also optimal for the problem of simultaneous lot streaming and scheduling of multiple products. The optimal scheduling of products can be accomplished by application of the algorithm due to Gilmore and Gomory [1]. Then, we devise an efficient heuristic for the problem of simultaneous lot streaming (finding optimal integer-sized sublots) and scheduling of multiple products. Computational results indicate that this heuristic can consistently deliver close-to-optimal solutions for the problem. A comparison of this heuristic is also made with a heuristic that first divides items belonging to each product into nearly equal-sized sublots and then constructs a schedule for such sublots. Finally, we extend our solution procedures to a traditional and more general lot streaming model, where the number of sublots for each product is a decision variable.     

Mathematical modelling and optimisation of energy-conscious hybrid flow shop scheduling problem with unrelated parallel machines

This paper investigates an energy-conscious hybrid flow shop scheduling problem with unrelated parallel machines (HFSP-UPM) with the energy-saving strategy of turning off and on. We first analyse the energy consumption of HFSP-UPM and formulate five mixed integer linear programming (MILP) models based on two different modelling ideas namely idle time and idle energy. All the models are compared both in size and computational complexities. The results show that MILP models based on different modelling ideas vary dramatically in both size and computational complexities. HFSP-UPM is NP-Hard, thus, an improved genetic algorithm (IGA) is proposed. Specifically, a new energy-conscious decoding method is designed in IGA. To evaluate the proposed IGA, comparative experiments of different-sized instances are conducted. The results demonstrate that the IGA is more effective than the genetic algorithm (GA), simulating annealing algorithm (SA) and migrating birds optimisation algorithm (MBO). Compared with the best MILP model, the IGA can get the solution that is close to an optimal solution with the gap of no more than 2.17% for small-scale instances. For large-scale instances, the IGA can get a better solution than the best MILP model within no more than 10% of the running time of the best MILP model.     

A branch-and-bound algorithm for a two-stage hybrid flowshop scheduling problem minimizing total tardiness

A two-stage hybrid flowshop-scheduling problem is considered with the objective of minimizing total tardiness of jobs. In the hybrid flowshop, there is one machine at the first stage and multiple identical parallel machines at the second stage. Dominance properties and lower bounds are developed for the problem and a branch-and-bound algorithm is suggested using them. Results of computational experiments show that the suggested algorithm can find optimal solutions for problems with up to 15 jobs in a reasonable amount of central processing unit time.     

No-wait scheduling of a two-machine flow-shop to minimise the makespan under non-availability constraints and different release dates

In this paper, we consider the two-machine no-wait flow-shop scheduling problem, when every machine is subject to one non-availability constraint and jobs have different release dates. The non-availability intervals of the machines overlap and they are known in advance. We aim to find a non-resumable schedule that minimises the makespan. We propose several lower bounds and upper bounds. These bounding procedures are used in a branch-and-bound algorithm. Computational experiments are carried out on a large set of instances and the obtained results show the effectiveness of our method.     

Minimising interference for scheduling two parallel machines with a single server

In this paper, we consider the problem of scheduling a set of jobs on two parallel machines with set-up times. The set-up has to be performed by a single server. The objective is to minimise the forced idle time. The problem of minimising the forced idle time (interference problem) is known to be unary NP-hard for the case of two machines and equal set-up and arbitrary processing times. We propose a mixed integer linear programming model, which describes a special class of schedules where the jobs from a list are scheduled alternatively on the machines, and a heuristic algorithm is tested on instances with up to 100,000 jobs. The computational results indicate that the algorithm has an excellent performance even for very large instances, where mostly an optimal solution is obtained within a very small computational time.     

Minimising makespan in distributed mixed no-idle flowshops

The rapid growth of distributed manufacturing in industry today has recently attracted significant research attention that has focused on distributed scheduling problems. This work studied the distributed mixed no-idle flowshop scheduling problem using makespan as an optimality criterion. To the best of the authors’ knowledge, this is the first paper to study the multi-flowshop extension in which each flowshop has mixed no-idle constraints. A novel cloud theory-based iterated greedy (CTBIG) algorithm was proposed for solving the problem. Computational experiments conducted on a set of test instances revealed that the proposed CTBIG algorithm significantly outperformed classic iterated greedy algorithms.     

A bounded-search iterated greedy algorithm for the distributed permutation flowshop scheduling problem

As the interest of practitioners and researchers in scheduling in a multi-factory environment is growing, there is an increasing need to provide efficient algorithms for this type of decision problems, characterised by simultaneously addressing the assignment of jobs to different factories/workshops and their subsequent scheduling. Here we address the so-called distributed permutation flowshop scheduling problem, in which a set of jobs has to be scheduled over a number of identical factories, each one with its machines arranged as a flowshop. Several heuristics have been designed for this problem, although there is no direct comparison among them. In this paper, we propose a new heuristic which exploits the specific structure of the problem. The computational experience carried out on a well-known testbed shows that the proposed heuristic outperforms existing state-of-the-art heuristics, being able to obtain better upper bounds for more than one quarter of the problems in the testbed.     

A complete immunoglobulin-based artificial immune system algorithm for two-stage assembly flowshop scheduling problem with part splitting and distinct due windows

This paper considers a two-stage assembly flowshop scheduling problem with distinct due windows to minimise the sum of weighted earliness and tardiness. There are several identical parallel machines which produce parts in the first stage. When the required parts are available, a single assembly machine can group these parts into products in the second stage. It is assumed that a part can be split into sub-parts which can be processed independently on the parallel machines in the first stage. Setup is also considered. A mathematical model is established to describe and define the proposed problem. A new decoding method is developed by extending an existing decoding method. Two novel operators, named part splitting (PS) and optimal idle time insertion (ITI), are incorporated into the decoding procedure for improving the quality of the solution. A rule named Priority of Earliness and Tardiness (PET) and a Complete Immunoglobulin-based Artificial Immune System (C-IAIS) algorithm are proposed for solving the problem. To evaluate PET and C-IAIS algorithm, several existing algorithms are used in the experiments. Computational results show that C-IAIS algorithm performs better than other algorithms for solving the proposed problem.     

Two MILP models for the N × M SDST flowshop sequencing problem

This paper presents the derivation and evaluation of two new MILP models for the SDST flowshop sequencing problem. The first model, TS1, was derived directly from the assignment-problem-based MILP model for the regular flowshop that Stafford modified from the original Wagner three-machine all-integer model. The second model, TS2, combined the properties of model TS1 with the looser constraints approach of Srikar and Ghosh, as modified by Stafford and Tseng (model SG*). Three experiments were used to compare both new models to the SG* model. Both new models were found to use significantly less computation time than the SG* model, especially for problem sizes of 6 or more jobs and 5 or more machines. The TS1 model used significantly less computation time than the TS2 model, making it the current best MILP model in the SDST flowshop literature.     

考虑任务拆分特性与簇准备时间的并行机调度

在印制电路板钻孔任务调度等工程实际中,普遍存在一类具有任务拆分特性与簇准备时间的并行机调度问题,尚缺乏高效的优化模型和方法。针对该问题,首先建立以总拖期最小为目标的数学模型,以约束的形式将两个现有优势定理嵌入其中。为了高效求解实际规模问题,进一步提出嵌入优势定理的模拟退火算法。最后,基于随机生成的算例构造计算实验,以验证所建模型和算法的有效性。实验结果表明,嵌入优势定理的数学模型在问题求解规模和计算效率方面均优于现有数学模型,嵌入优势定理的模拟退火算法同样优于现有模拟退火算法。     

Makespan minimisation for a parallel machine scheduling problem with preemption and job incompatibility

In this paper, an extension of the graph colouring problem is introduced to model a parallel machine scheduling problem with job incompatibility. To get closer to real-world applications, where the number of machines is limited and jobs have different processing times, each vertex of the graph requires multiple colours and the number of vertices with the same colour is bounded. In addition, several objectives related to scheduling are considered: makespan, number of pre-emptions and summation over the jobs’ throughput times. Different solution methods are proposed, namely, two greedy heuristics, two tabu search methods and an adaptive memory algorithm. The latter uses multiple recombination operators, each one being designed for optimising a subset of objectives. The most appropriate operator is selected dynamically at each iteration, depending on its past performance. Experiments show that the proposed algorithm is effective and robust, while providing high-quality solutions on benchmark instances for the graph multi-colouring problem, a simplification of the considered problem.     

A bi-objective parallel machine problem with eligibility,release dates and delivery times of the jobs

The scheduling of parallel machines is a well-known problem in many companies. Nevertheless, not always all the jobs can be manufactured in any machine and the eligibility appears. Based on a real-life problem, we present a model which has m parallel machines with different level of quality from the highest level for the first machine till the lowest level for the last machine. The set of jobs to be scheduled on these m parallel machines are also distributed among these m levels: one job from a level can be manufactured in a machine of the same or higher level but a penalty, depending on the level, appears when a job is manufactured in a machine different from the highest level i.e. different from the first machine. Besides, there are release dates and delivery times associated to each job. The tackled problem is bi-objective with the criteria: minimisation of the final date – i.e. the maximum for all the jobs of their completion time plus the delivery time – and the minimisation of the total penalty generated by the jobs. In a first step, we analyse the sub-problem of minimisation of the final date on a single machine for jobs with release dates and delivery times. Four heuristics and an improvement algorithm are proposed and compared on didactic examples and on a large set of instances. In a second step an algorithm is proposed to approximate the set of efficient solutions and the Pareto front of the bi-objective problem. This algorithm contains two phases: the first is a depth search phase and the second is a backtracking phase. The procedure is illustrated in detail on an instance with 20 jobs and 3 machines. Then extensive numerical experiments are realised on two different sets of instances, with 20, 30 and 50 jobs, 3 or 4 machines and various values of penalties. Except for the case of 50 jobs, the results are compared with the exact Pareto front.     

Minimizing makespan for a no-wait flowshop using genetic algorithm

This paper explains minimization of makespan or total completion time for n-jobs, m-machine, no-wait flowshop problem (NW-FSSP). A spread sheet based general purpose genetic algorithm is proposed for the NW-FSSP. The example analysis shows that the proposed approach produces results are comparable to the previous approaches cited in the literature. Additionally, it is demonstrated that the current application is a general purpose approach whereby the objective function can be tailored without any change in the logic of the GA routine.