Minimising makespan on a single batch processing machine with dynamic job arrivals and non-identical job sizes

Batch scheduling is a prevalent policy in many industries such as burn-in operations in semiconductor manufacturing and heat treatment operations in metalworking. In this paper, we consider the problem of minimising makespan on a single batch processing machine in the presence of dynamic job arrivals and non-identical job sizes. The problem under study is NP-hard. Consequently, we develop a number of efficient construction heuristics. The performance of the proposed heuristics is evaluated by comparing their results to two lower bounds, and other solution approaches published in the literature, namely the first-fit longest processing time-earliest release time (FFLPT-ERT) heuristic, hybrid genetic algorithm (HGA), joint genetic algorithm and dynamic programming (GA+DP) approach and ant colony optimisation (ACO) algorithm. The computational experiments demonstrate the superiority of the proposed heuristics with respect to solution quality, especially for the problems with small size jobs. Moreover, the computational costs of the proposed heuristics are very low.     

Makespan minimisation on parallel batch processing machines with non-identical job sizes and release dates

The problem we study in this paper arises from the washing step of hospital sterilisation services. Washers in the washing step are capable of handling more than one medical device set as long as their capacity is not exceeded. The medical device set sizes and arrival times to the sterilisation service may be different, but they all have the same washing duration. Thus, we model the washing step as a batch scheduling problem where medical device sets are treated as jobs with non-identical sizes and release dates, but equal processing times. The main findings we present in this paper are the following. First, we study two special cases for which polynomial algorithms are presented. We then develop a 2-approximation algorithm for the general problem. Finally, we develop a MILP model and compare it with another MILP model from the literature. Computational results show that our MILP model outperforms the model from the literature.     

Minimising makespan heuristics for scheduling a single batch machine processing machine with non-identical job sizes

In this paper, the problem of minimising maximum completion time on a single batch processing machine is studied. A batch processing is performed on a machine which can simultaneously process several jobs as a batch. The processing time of a batch is determined by the longest processing time of jobs in the batch. The batch processing machine problem is encountered in many manufacturing systems such as burn-in operations in the semiconductor industry and heat treatment operations in the metalworking industries. Heuristics are developed by iterative decomposition of a mixed integer programming model, modified from the successive knapsack problem by Ghazvini and Dupont (1998 Ghazvini, F.J. and Dupont, L. 1998. Minimising mean flow times criteria on a single batch processing machine with non-identical jobs sizes. International Journal of Production Economics, 55: 273–280. [Crossref], [Web of Science ®] , [Google Scholar], Minimising mean flow times criteria on a single batch processing machine with non-identical jobs sizes. International Journal of Production Economics 55: 273–280) and the waste of batch clustering algorithm by Chen, Du, and Huang (2011 Chen, H., Du, B. and Huang, G.Q. 2011. Scheduling a batch processing machine with non-identical job sizes: a clustering perspective. International Journal of Production Research, 49(19): 5755–5778. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Scheduling a batch processing machine with non-identical job sizes: a clustering perspective. International Journal of Production Research 49 (19): 5755–5778). Experimental results show that the suggested heuristics produce high-quality solutions comparable to those of previous heuristics in a reasonable computation time.     

Scheduling a batch processing machine with non-identical job sizes

The problem of scheduling batch processors is important in some industries and, at a more fundamental level, captures an element of complexity common to many practical scheduling problems. We describe a branch and bound procedure applicable to a batch processor model with arbitrary job processing times, job weights and job sizes. The scheduling objective is to minimize total weighted completion time. We find that the procedure returns optimal solutions to problems of up to 25 jobs in reasonable CPU time, and can be adapted for use as a heuristic for larger problems.     

Minimising makespan on parallel batch processing machines with non-identical ready time and arbitrary job sizes

This paper considers the parallel batch processing machine scheduling problem which involves the constraints of unequal ready times, non-identical job sizes, and batch dependent processing times in order to sequence batches on identical parallel batch processing machines with capacity restrictions. This scheduling problem is a practical generalisation of the classical parallel batch processing machine scheduling problem, which has many real-world applications, particularly, in the aging test operation of the module assembly stage in the manufacture of thin film transistor liquid crystal displays (TFT-LCD). The objective of this paper is to seek a schedule with a minimum total completion time for the parallel batch processing machine scheduling problem. A mixed integer linear programming (MILP) model is proposed to optimise the scheduling problem. In addition, to solve the MILP model more efficiently, an effective compound algorithm is proposed to determine the number of batches and to apply this number as one parameter in the MILP model in order to reduce the complexity of the problem. Finally, three efficient heuristic algorithms for solving the large-scale parallel batch processing machine scheduling problem are also provided.     

Ant colony optimisation algorithms for two-stage permutation flow shop with batch processing machines and nonidentical job sizes

This paper focuses on minimising the maximum completion time for the two-stage permutation flow shop scheduling problem with batch processing machines and nonidentical job sizes by considering blocking, arbitrary release times, and fixed setup and cleaning times. Two hybrid ant colony optimisation algorithms, one based on job sequencing (JHACO) and the other based on batch sequencing (BHACO), are proposed to solve this problem. First, max-min pheromone restriction rules and a local optimisation rule are embedded into JHACO and BHACO, respectively, to avoid trapping in local optima. Then, an effective lower bound is estimated to evaluate the performances of the different algorithms. Finally, the Taguchi method is adopted to investigate and optimise the parameters for JHACO and BHACO. The performances of the proposed algorithms are compared with that of CPLEX on small-scale instances and those of a hybrid genetic algorithm (HGA) and a hybrid discrete differential evolution (HDDE) algorithm on full-scale instances. The computational results demonstrate that BHACO outperforms JHACO, HDDE and HGA in terms of solution quality. Besides, JHACO strikes a balance between solution quality and run time.     

A method combining rules with genetic algorithm for minimizing makespan on a batch processing machine with preventive maintenance

This paper considers the problem of minimising makespan on a single batch processing machine with flexible periodic preventive maintenance. This problem combines two sub-problems, scheduling on a batch processing machine with jobs’ release dates considered and arranging the preventive maintenance activities on a batch processing machine. The preventive maintenance activities are flexible but the maximum continuous working time of the machine, which is allowed, is determined. A mathematical model for integrating flexible periodic preventive maintenance into batch processing machine problem is proposed, in which the grouping of jobs with incompatible job families, the starting time of batches and the preventive maintenance activities are optimised simultaneously. A method combining rules with the genetic algorithm is proposed to solve this model, in which a batching rule is proposed to group jobs with incompatible job families into batches and a modified genetic algorithm is proposed to schedule batches and arrange preventive maintenance activities. The computational results indicate the method is effective under practical problem sizes. In addition, the influences of jobs’ parameters on the performance of the method are analyzed, such as the number of jobs, the number of job families, jobs’ processing time and jobs’ release time.     

Minimising makespan on a batch processing machine using heuristics improved by an enumeration scheme

Batch processing machines can process several job simultaneously and are encountered in many manufacturing environments. Jobs in a batch are processed together and have the same start and end processing time. Since jobs are non-identical in job sizes and job processing times, they should be reasonably scheduled to improve the machine utilisation and processing efficiency. Two well-known heuristics, first fit longest processing time and best fit longest processing time (BFLPT), are improved in this study by considering identical job sizes and then BFLPT is further improved by an enumeration scheme proposed. Computational experiments are conducted to evaluate the performance of the improvement and the results are compared with the existing heuristics.     

Quantum-inspired ant colony optimisation algorithm for a two-stage permutation flow shop with batch processing machines

This paper studied two-stage permutation flow shop problems with batch processing machines, considering different job sizes and arbitrary arrival times, with the optimisation objective of minimising the makespan. The quantum-inspired ant colony optimisation (QIACO) algorithm was proposed to solve the problem. In the QIACO algorithm, the ants are divided into two groups: one group selects the largest job in terms of job size as the initial job for each batch and the other group selects the smallest job as the initial job for each batch. Each group of ants has its own pheromone matrix. In the computational experiment, our novel algorithm was compared with the hybrid discrete differential evolution (HDDE) algorithm and the batch-based hybrid ant colony optimisation (BHACO) algorithm. Although the HDDE algorithm has a shorter run time, the quality of the solution for large-scale jobs is not good, while the BHACO algorithm always obtains a better solution but requires a longer run time. The computational results show that the QIACO algorithm embedded in the quantum information has advantages in terms of both solution quality and running time.     

A hybrid differential evolution algorithm for a two-stage flow shop on batch processing machines with arbitrary release times and blocking

Batch processing machines (BPMs) have important applications in a variety of industrial systems. This paper considers the problem of scheduling two BPMs in a flow shop with arbitrary release times and blocking such that the makespan is minimised. The problem is formulated as a mixed integer programming model. Subsequently, a hybrid discrete differential evolution (HDDE) algorithm is proposed. In the algorithm, individuals in the population are first represented as discrete job sequences, and mutation and crossover operators are applied based on the representation. Second, after using the first-fit rule to form batches, a novel least idle/blocking time heuristic is developed to schedule the batches in the flow shop. Furthermore, an effective local search technique is embedded in the algorithm to enhance the exploitation ability. The performance of the proposed algorithm is evaluated by comparing its results to a commercial solver (CPLEX), a genetic algorithm and a simulated annealing algorithm. Computational experiments demonstrate the superiority of the HDDE algorithm in terms of solution quality, robustness and run time.     

Modelling and symmetry breaking in scheduling problems on batch processing machines

Problems of scheduling batch-processing machines to minimise the makespan are widely exploited in the literature, mainly motivated by real-world applications, such as burn-in tests in the semiconductor industry. These problems consist of grouping jobs in batches and scheduling them on machines. We consider problems where jobs have non-identical sizes and processing times, and the total size of each batch cannot exceed the machine capacity. The processing time of a batch is defined as the longest processing time among all jobs assigned to it. Jobs can also have non-identical release times, and in this case, a batch can only be processed when all jobs assigned to it are available. This paper discusses four different versions of batch scheduling problems, considering a single processing machine or parallel processing machines and considering jobs with or without release times. New mixed integer linear programming formulations are proposed as enhancements of formulations proposed in the literature, and symmetry breaking constraints are investigated to reduce the size of the feasible sets. Computational results show that the proposed formulations have a better performance than other models in the literature, being able to solve to optimality instances only considered before to be solved by heuristic procedures.     

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.     

Scheduling parallel batch processing machines with arbitrary job sizes and incompatible job families

Motivated by a bottleneck operation in an MLCC (multi-layer ceramic capacitor) production line, we study the scheduling problem of parallel batch processing machines in which a number of jobs can be processed simultaneously in a machine as a batch. Volumes of the jobs are different from each other and each job belongs to the family in which all jobs have the same processing time. In this situation, we analyse three kinds of problems whose performance measures are makespan, total completion time, and total weighted completion time, respectively. Since these problems are known to be NP-hard, we propose a number of heuristics and design genetic algorithms for the problems. Through some computational experiments, we evaluate the performances of the heuristic algorithms proposed, including the genetic algorithms for each of three problems.     

Effective hybrid genetic algorithm for minimizing makespan on a single-batch-processing machine with non-identical job sizes

The paper addresses minimizing makespan by a genetic algorithm (GA) for scheduling jobs with non-identical sizes on a single-batch-processing machine. A batch-processing machine can process up to B jobs simultaneously. The processing time of a batch is equal to the longest processing time among all jobs in the batch. Two different GAs are proposed based on different encoding schemes. The first is a sequence-based GA (SGA) that generates random sequences of jobs using GA operators and applies the batch first fit heuristic to group the jobs. The second is a batch-based hybrid GA (BHGA) that generates random batches of jobs using GA operators and ensures feasibility by using knowledge of the problem based on a heuristic procedure. A greedy local search heuristic based on the problem characteristics is hybridized with a BHGA that has the ability of steering efficiently the search toward the optimal or near-optimal schedules. The performance of proposed GAs is compared with a simulated annealing (SA) approach proposed by Melouk et al. (Melouk, S., Damodaran, P. and Chang, P.Y., Minimizing makespan for single machine batch processing with non-identical job sizes using simulated annealing. Int. J. Prod. Econ., 2004, 87, 141–147) and also against a modified lower bound proposed for the problem. Computational results show that BHGA performs considerably well compared with the modified lower bound and significantly outperforms the SGA and SA in terms of both quality of solutions and required runtimes.     

Minimizing makespan on parallel machines with batch arrivals

Most studies in the scheduling literature assume that jobs arrive at time zero, while some studies assume that jobs arrive individually at non-zero times. However, both assumptions may not be valid in practice because jobs usually arrive in batches. In this article, a scheduling model for an identical parallel machine problem with batch arrivals is formulated. Because of the NP-hardness of the problem, a heuristic based on a simplified version of lexicographical search is proposed. To verify the heuristic, two lower bounding schemes are developed, where one lower bound is tight, and the list scheduling heuristic is compared. Extensive computational experiments demonstrate that the proposed heuristic is quite efficient in obtaining near optimal solution with an average error of less than 1.58%. The percentage improvement (from the lower bound) of the heuristic solution on the solution by the list scheduling is as large as 31.68.     

Bi-objective dynamic control of batch processor with non-identical jobs in mould manufacturing

The heat-treatment operation in dynamic mould manufacturing often involves non-identical jobs, which allow for simultaneous processing yet with different weights and due dates. Effective production control of this operation is essential to improve the on-time delivery and decrease the manufacturing cost of the mould. This paper considers the dynamic control of a batch processor for dealing with such non-identical jobs. A new look-ahead batching strategy called MLAB (mould: look-ahead batching) has been proposed. In MLAB, the control decisions are made by the joint use of both near-future arrival information of upstream operations and workload level information of downstream operations. MLAB strategy is used to control two kinds of conflicting objectives related to the delivery and utilisation performances and finally achieve trade-off based on compromise programming method. Computational experiments are conducted to verify the effectiveness of the MLAB strategy and show that the results are promising as compared to benchmark control strategies.     

Heuristic control of multiple batch processors with incompatible job families and future job arrivals

We analyse the problem of minimising the mean cycle time of a batch processing stage containing K?>?1 batch processors in parallel with incompatible job families and future job arrivals. We provide an integer linear programming formulation and a dynamic program formulation for small problem instances. For larger problem instances, we propose an online heuristic policy MPC_REPEAT. At each instance a decision has to be made, MPC_REPEAT decomposes the problem of simultaneously assigning multiple batches to multiple processors into sequentially assigning multiple batches to multiple processors. When job families are uncorrelated, we show via simulation experiments that MPC_REPEAT has significantly lower mean cycle time than a previously proposed look-ahead method except when: (MPC_REPEAT ignores some job families AND the traffic intensity is high.) Our experiments also reveal that increasing the job family correlation of consecutive job arrivals results, with a few exceptions, in a mean cycle-time reduction, for both policies evaluated. This reduction in cycle time generally increases with: increasing number of job families, decreasing number of processors, and increasing time between job arrivals. Our findings imply that controlling the upstream processors, such that job families of consecutive job arrivals are correlated, can reduce the cycle time at the batch processing stage. Furthermore, the expected mean cycle time reduction due to this strategy can be substantially larger than that expected from switching to a more complex batch processing stage policy, under less stringent conditions.     

Single machine batch scheduling with sequential job processing

The problem of scheduling n jobs on a single machine in batches to minimize some regular cost functions is studied. Jobs within each batch are processed sequentially so that the processing time of a batch is equal to the sum of the processing times of the jobs contained in it. Jobs in the same batch are completed at the same time when the last job of the batch has finished its processing. A constant set-up time precedes the processing of each batch. The number of jobs in each batch is bounded by some value b. If b < n, then the problem is called bounded. Otherwise, it is unbounded. For both the bounded and unbounded problems, dynamic programming algorithms are presented for minimizing the maximum lateness, the number of late jobs, the total tardiness, the total weighted completion time, and the total weighted tardiness when all due dates are equal, which are polynomial if there is a fixed number of distinct due dates or processing times. More efficient algorithms are derived for some special cases of both the bounded and unbounded problems in which all due dates and/or processing times are equal. Several special cases of the bounded problem are shown to be NP-hard. Thus, a comprehensive classification of the computational complexities of the special cases is provided.     

Single machine scheduling problem with batch setups involving positional deterioration effects and multiple rate-modifying activities

This article considers a single machine scheduling problem with batch setups, positional deterioration effects, and multiple optional rate-modifying activities to minimize the total completion time. This problem is formulated as an integer quadratic programming problem. In view of the complexity of optimally solving this problem, a two-phase heuristic algorithm is proposed where an optimal but non-integer solution is obtained in the first phase by solving a continuous relaxed version of the problem. This solution serves as a lower bound for the optimal value of the total completion time. The second phase of the algorithm generates an integer solution using a simple rounding scheme that is optimum or very close to optimum for this problem. Empirical evaluation and comparison with an existing heuristic algorithm show that the proposed heuristic algorithm is substantially more effective in solving large-size problem instances.