Minimizing total tardiness on a batch processing machine with incompatible job families

Motivated by an application in semiconductor manufacturing, we study the problem of minimizing total tardiness on a batch processing machine with incompatibl8e job families, where all jobs of the same family have identical processing times and jobs of different families cannot be processed together. We present a dynamic programming algorithm which has polynomial time complexity when the number of job families and the batch machine capacity are fixed. We also examine various heuristic solution procedures which can provide near optimal solutions in a reasonable amount of computation time.     

Iterative algorithms for batching and scheduling to minimise the total job tardiness in two-stage hybrid flow shops

This study considers the batching and scheduling problem in two-stage hybrid flow shops in which each job with a distinct due-date is processed through two serial production stages, each of which has identical machines in parallel. Under the fundamental trade-off that large batch sizes with less frequent changeovers may reduce setup costs and hence increase machine utilisation, while small batch sizes may reduce job flow times and hence improve scheduling performance, the problem is to determine the number of batches, the batch compositions, the allocation of batches to the parallel machines at each stage, and the sequence of the batches allocated to each machine for the objective of minimising the total job tardiness. A mixed integer programming model is developed for the reduced problem in which the number of batches is given, and then, three iterative algorithms are proposed in which batching and scheduling are done repeatedly until a good solution is obtained. To show the performance of the algorithms, computational experiments were done on a number of test instances, and the results are reported. In particular, we show that the number of batches decreases as the ratio of the batch setup time to the job processing time increases.     

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.     

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.     

The burn-in test scheduling problem with batch dependent processing time and sequence dependent setup time

The burn-in test scheduling problem (BTSP) is a variation of the complex batch processing machine scheduling problem, which is also a generalisation of the liquid crystal injection scheduling problem with incompatible product families and classical identical parallel machine problem. In the case we investigated on the BTSP, the jobs are clustered by their product families. The product families can be clustered by different product groups. In the same product group, jobs with different product families can be processed as a batch. The batch processing time is dependent on the longest processing time of those jobs in that batch. Setup times between two consecutive batches of different product groups on the same batch machine are sequentially dependent. In addition, the unequal ready times are considered in the BTSP which involves the decisions of batch formation and batch scheduling in order to minimise the total machine workload without violating due dates and the limited machine capacity restrictions. Since the BTSP involves constraints on unequal ready time, batch dependent processing time, and sequence dependent setup times, it is more difficult to solve than the classical parallel batch processing machine scheduling problem with compatible product families or incompatible product families. These restrictions mean that the existing methods cannot be applied into real-world factories directly. Consequently, this paper proposes a mixed integer programming model to solve the BTSP exactly. In addition, two efficient solution procedures which solve the BTSP are also presented.     

The liquid crystal injection scheduling problem (LCISP)

In this paper, the liquid crystal injection scheduling problem (LCISP) involving the constraints on limited maximum waiting times, unequal ready times, and machine setup times is considered to form the batches with incompatible product families and to sequence those batches on identical parallel batch processing machines. The batch scheduling problem, LCISP, has many applications, especially in thin film transistor liquid crystal display (TFT-LCD) factories at the cell assembly stage. In the LCISP, the objective is to minimise the total machine workload without violating the limited maximum waiting time restriction. Furthermore, machine setup times that are sequence dependent for two consecutive batches classified into different product families on the same machine are also considered. Since the LCISP involves constraints on limited maximum waiting times and sequence dependent setup times, it is more difficult to solve than the classical parallel batch processing machine scheduling problem with incompatible product families. These restrictions mean that the existing methods cannot be applied into real-world factories directly. Therefore, this paper proposes a mixed integer programming model to solve the LCISP exactly. In addition, two efficient solution procedures which solve the LCISP are also presented.     

Single-machine batch delivery scheduling and common due-date assignment with a rate-modifying activity

We consider batch delivery scheduling on a single machine, where a common due-date is assigned to all the jobs and a rate-modifying activity on the machine may be scheduled, which can change the processing rate of the machine. Thus the actual processing time of a job is variable depending on whether it is processed before or after the rate-modifying activity. The objective is to determine the optimal job sequence, the optimal partition of the job sequence into batches, the optimal assigned common due-date, and the optimal location of the rate-modifying activity simultaneously to minimize the total cost of earliness, job holding, weighted number of tardy jobs, due-date assignment, and batch delivery. We derive some structural properties of the problem, based on which we design polynomial-time algorithms to solve some special cases of the problem.     

A batch splitting method for a job shop scheduling problem in an MRP environment

The job shop scheduling problem has been a major target for many researchers. Unfortunately though, most of the previous research was based on assumptions that are different from the real manufacturing environment. Among those distorted assumptions, two assumptions about set-up time and job composition can greatly influence the performance of a schedule. First, most of the past studies ignored the impact of the before-arrival set-up time. If we know the sequence of operations in advance, we can obtain an improved schedule by preparing the setup before a job arrives. Secondly, most of the past studies assumed that a job consists of only a single part, that is a batch of size one. However, if we assume that a job consists of a batch size greater than one, as in many real manufacturing environments, then we can obtain an improved schedule because we can fill up the idle times of machines with jobs which have smaller processing times by splitting the original batches. However, the number of job orders may then increase due to the split, and the size of the scheduling problem would become too large to be solved in a practical time limit. Consequently, there may be an optimum batch size considering trade-off between better solution and tractability. The current study is the result of an attempt to find an acceptable solution when the production requirement from a MRP system for a planning period exceeds the capacity of a production system. We try to get an improved schedule by splitting the original batch into smaller batches, and consider setting up a machine before the actual arrival of jobs to that machine. Thereby we can meet the due date requirement without resorting to rescheduling of the master production schedule. For the given batch, we disaggregate it according to the algorithm we are proposing. A so-called 'modified shifting bottleneck procedure' is then applied to solve the job shop scheduling problem with a before-arrival family set-up time considering release date, transportation time and due date. The study also shows that we can adapt to unexpected dynamic events more elegantly by allowing the splitting of batches.     

Scheduling of Customized Jobs on a Single Machine under Item Availability

We study a problem of scheduling customized jobs on a single-machine. Each job requires two operations: one standard and one specific. Standard operations are processed in batches under item availability, and each batch requires a set-up time. Based on structural properties of the optimal solution, we introduce a generic dynamic programming scheme that builds an optimal schedule by alternately inserting blocks of operations of two distinct types. Our approach yields efficient algorithms for the sum of completion times problem with agreeable processing times and the maximum lateness problem. The number of late jobs problem is shown to be NP-hard in the ordinary sense, but is pseudo-polynomially solvable. A polynomial algorithm is also given for a special case of this problem. Our results indicate the differences between this problem and its counterpart under batch availability.     

Minimising general setup costs in a two-stage production system

This paper addresses a problem arising in the coordination between two consecutive stages of a production system. Production is organised in batches of identical jobs. Each job is characterised by two distinct attributes, and all jobs sharing the same attributes are processed together as a single batch. Due to the structural and organisational characteristics of the production system, the two stages have to process the same batch sequence. When two consecutive batches with different attributes are processed, at least one stage must pay a setup, in order to reconfigure its own devices. Each stage incurs a setup cost that is a general non-decreasing function of the number of its own setups, and the problem consists of finding a batch sequence minimising the total setup costs of the production system. We present an original solution approach for the considered problem that is shown to be very effective using an extensive experimental campaign.     

A branch and bound algorithm for scheduling unit size jobs on parallel batching machines to minimize makespan

In this paper, we present a branch and bound algorithm for the parallel batch scheduling of jobs having different processing times, release dates and unit sizes. There are identical machines with a fixed capacity and the number of jobs in a batch cannot exceed the machine capacity. All batched jobs are processed together and the processing time of a batch is given by the greatest processing time of jobs in that batch. We compare our method to a mixed integer program as well as a method from the literature that is capable of optimally solving instances with a single machine. Computational experiments show that our method is much more efficient than the other two methods in terms of solution time for finding the optimal solution.     

Product batching and batch sequencing for NC punch presses

This paper proposes a two-stage practical product batching and batch sequencing algorithm for NC punch presses used in the production of precision sheet metal electronics components. The first stage of the algorithm batches a given set of products in order to maximize the number of products in a batch as well as the tool magazine utilization. In this stage, tool magazine constraints such as number of available slots, size of slots, tool configurations and tool locations are taken into consideration. The resulting batches are passed to the second stage of the algorithm which sequences these batches. The sequencing is done by using the ‘ nearest neighbour’ heuristic for the ‘ travelling salesman problem’ Comparison of the new method with the procedure previously used by a manufacturer of such components demonstrated a significant reduction in setup times.     

Minimizing makespan on parallel batch processing machines

A simulated annealing approach to minimize makespan for identical parallel batch-processing machines is presented. Each job has a corresponding processing time and size. The machine can process the jobs in batches as long as the total size of all the jobs in a batch does not exceed the machine capacity. The processing time of a batch is equal to the longest processing time among all the jobs in the batch. Random instances were generated to test the approach with respect to solution quality and run time. The results of the simulated annealing approach were compared with CPLEX. The approach outperforms CPLEX on most of the instances.     

Scheduling a no-wait flow shop containing unbounded batching machines

We study the problem of scheduling n jobs in a no-wait flow shop consisting of m batching machines. Each job has to be processed by all the machines. All jobs visit the machines in the same order. A job completed on an upstream machine should be immediately transferred to the downstream machine. Batching machines can process several jobs simultaneously in a batch so that all jobs of the same batch start and complete together. The processing time of a batch is equal to the maximum processing time of the jobs in this batch. We assume that the capacity of any batch is unbounded. The problem is to find an optimal batch schedule such that the maximum job completion time, that is the makespan, is minimized. For m = 2, we prove that there exists an optimal schedule with at most two batches and construct such a schedule in O(n log n) time. For m = 3, we prove that the number of batches can be limited to nine and give an example where all optimal schedules have seven batches. Furthermore, we prove that the best schedules with at most one, two and three batches are 3-, 2- and 3/2-approximate solutions, respectively. The first two bounds are tight for corresponding schedules. Finally, we suggest an assignment method that solves the problem with m machines and at most r batches in O(n^{m(r-2)+1+[m/r]}) time, if m and r are fixed. The method can be generalized to minimize an arbitrary maximum cost or total cost objective function.     

Batching and scheduling to minimize the makespan in the two-machine flowshop

In this paper, we consider a class of batching and scheduling problems in the two-machine flowshop where one of the machines is a discrete processor and the other one is a batch processor. The jobs are processed separately on the discrete processor and processed in batches on the batch processor. The processing time of a batch is equal to the total processing time of the jobs contained in it, and the completion time of a job in a batch is defined as the completion time of the batch containing it. A constant setup time is incurred whenever a batch is formed on the batch processor. The problem is to find the optimal batch compositions and the optimal schedule of the batches so that the makespan is minimized. All problems in this class are shown to be NP-complete in the ordinary sense. We also identify some polynomially solvable cases by introducing their corresponding solution methods.     

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.     

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.     

Integrated production scheduling and batch delivery with fixed departure times and inventory holding costs

In this paper, we address a model for supply chain coordination. There are m manufacturers modelled as single machines, each of which processes a specific set of jobs (products). After processing is completed, jobs are batched, and batches are shipped to a customer by means of vehicles. The problem consists in concurrently finding a production schedule of the jobs, a partition of jobs into delivery batches and an assignment of delivery batches to vehicles, so that jobs are delivered within their deadlines and total costs are minimised. We focus on a scenario characterised by fixed departure times and inventory holding costs. For each departure time there is a given number of vehicles, possibly having limited capacity. Each job incurs a cost proportional to the time from job completion to delivery departure. In this paper, we show that the problem is NP-hard even for a very restricted case, and report various polynomiality results for two scenarios, namely: (i) when the production sequence of each manufacturer is fixed in advance, and (ii) when there is a single manufacturer and processing times are all equal to 1. We also point out several open problems.