Minimizing the makespan on a batch machine with non-identical job sizes: an exact procedure

A batch processing machine can simultaneously process several jobs forming a batch. This paper considers the problem of scheduling jobs with non-identical capacity requirements, on a single-batch processing machine of a given capacity, to minimize the makespan. The processing time of a batch is equal to the largest processing time of any job in the batch. We present some dominance properties for a general enumeration scheme and for the makespan criterion, and provide a branch and bound method. For large-scale problems, we use this enumeration scheme as a heuristic method.Scope and purposeUsually in classical scheduling problems, a machine can perform only one job at a time. Although, one can find machines that can process several jobs simultaneously as a batch. All jobs of a same batch have common starting and ending times. Batch processing machines are encountered in many different environments, such as burn-in operations in semiconductor industries or heat treatment operations in metalworking industries. In the first case, the capacity of the machine is defined by the number of jobs it can hold. In the second case, each job has a certain capacity requirement and the total size of a batch cannot exceed the capacity of the machine. Hence, the number of jobs contained in each batch may be different. In this paper, we consider this second case (which is more difficult) and we provide an exact method for the makespan criterion (minimizing the last ending time).     

A branch-and-bound algorithm for single-machine scheduling with batch delivery and job release times

This paper addresses scheduling a set of jobs with specified release times on a single machine for delivery in batches to customers or to other machines for further processing. This problem is a natural extension of minimizing the sum of flow times in the presence of release time by considering the possibility of delivering jobs in batches and introducing batch delivery costs. The scheduling objective adopted is that of minimizing the sum of flow times and delivery costs. The extended problem arises in the context of coordination between machine scheduling and a distribution system in a supply chain network. Structural properties of the problem are investigated and used to devise a branch-and-bound solution scheme. Computational experiments show significant improvement over an existing dynamic programming algorithm.     

Minimising makespan for two batch-processing machines with non-identical job sizes in job shop

In this article, the job shop scheduling problem with two batch-processing machines is considered. The machines have limited capacity and the jobs have non-identical job sizes. The jobs are processed in batches and the total size of each batch cannot exceed the machine capacity. The processing times of a job on the two machines are proportional. We show the problem of minimising makespan is NP-hard in the strong sense. Then we provide an approximation algorithm with worst-case ratio no more than 4, and the running time of the algorithm is O(n?log?n). Finally, the performance of the proposed algorithm is tested by different levels of instances. Computational results demonstrate the effectiveness of the algorithm for all the instances.     

Heuristic scheduling of jobs on parallel batch machines with incompatible job families and unequal ready times

This research is motivated by a scheduling problem found in the diffusion and oxidation areas of semiconductor wafer fabrication, where the machines can be modeled as parallel batch processors. We attempt to minimize total weighted tardiness on parallel batch machines with incompatible job families and unequal ready times of the jobs. Given that the problem is NP-hard, we propose two different decomposition approaches. The first approach forms fixed batches, then assigns these batches to the machines using a genetic algorithm (GA), and finally sequences the batches on individual machines. The second approach first assigns jobs to machines using a GA, then forms batches on each machine for the jobs assigned to it, and finally sequences these batches. Dispatching and scheduling rules are used for the batching phase and the sequencing phase of the two approaches. In addition, as part of the second decomposition approach, we develop variations of a time window heuristic based on a decision theory approach for forming and sequencing the batches on a single machine.     

A hybrid scheduling approach for a two-stage flexible flow shop with batch processing machines

In this paper, we discuss a flexible flow shop scheduling problem with batch processing machines at each stage and with jobs that have unequal ready times. Scheduling problems of this type can be found in semiconductor wafer fabrication facilities (wafer fabs). We are interested in minimizing the total weighted tardiness of the jobs. We present a mixed integer programming formulation. The batch scheduling problem is NP-hard. Therefore, an iterative stage-based decomposition approach is proposed that is hybridized with neighborhood search techniques. The decomposition scheme provides internal due dates and ready times for the jobs on the first and second stage, respectively. Each of the resulting parallel machine batch scheduling problems is solved by variable neighborhood search in each iteration. Based on the schedules of the subproblems, the internal due dates and ready times are updated. We present the results of designed computational experiments that also consider the number of machines assigned to each stage as a design factor. It turns out that the proposed hybrid approach outperforms an iterative decomposition scheme where a fairly simple heuristic based on time window decomposition and the apparent tardiness cost dispatching rule is used to solve the subproblems. Recommendations for the design of the two stages with respect to the number of parallel machines on each stage are given.     

A hybrid genetic heuristic for scheduling parallel batch processing machines with arbitrary job sizes

This paper investigates the scheduling problem of parallel identical batch processing machines in which each machine can process a group of jobs simultaneously as a batch. Each job is characterized by its size and processing time. The processing time of a batch is given by the longest processing time among all jobs in the batch. Based on developing heuristic approaches, we proposed a hybrid genetic heuristic (HGH) to minimize makespan objective. To verify the performance of our algorithm, comparisons are made through using a simulated annealing (SA) approach addressed in the literature as a comparator algorithm. Computational experiments reveal that affording the knowledge of problem through using heuristic procedures, gives HGH the ability of finding optimal or near optimal solutions in a reasonable time.     

Minimizing makespan on a single burn-in oven with job families and dynamic job arrivals

This paper considers a scheduling problem for a single burn-in oven in the semiconductor manufacturing industry where the oven is a batch processing machine and each batch processing time is represented by the largest processing time among those of all the jobs contained in the batch. Each job belongs to one of the given number of families. Moreover, the release times of the jobs are different from one another. The objective measure of the problem is the maximum completion time (makespan) of all jobs. A dynamic programming algorithm is proposed in the order of polynomial time complexity for a situation where the number of job families is given (fixed). A computational experiment is performed to compare the time complexity of the proposed algorithm with that of another exact algorithm evaluating all possible job sequences based on batching-dynamic programming (BDP). The results of the experiment show that the proposed algorithm is superior to the other.Scope and purposeThis paper considers a scheduling problem on the burn-in operation in a semiconductor manufacturing process. The burn-in operation is a bottleneck process in the final testing process which is one of four major steps including wafer fabrication, wafer probe, assembly, and final testing steps. Thus, its scheduling is very important to improve the productivity of the whole manufacturing line. The objective of this paper is to find a solution technique that will find the optimal schedule that minimizes makespan for problems which are found in the semiconductor manufacturing industry.     

An improved ant colony optimization for scheduling identical parallel batching machines with arbitrary job sizes

In this paper we consider the problem of scheduling parallel batching machines with jobs of arbitrary sizes. The machines have identical capacity of size and processing velocity. The jobs are processed in batches given that the total size of jobs in a batch cannot exceed the machine capacity. Once a batch starts processing, no interruption is allowed until all the jobs are completed. First we present a mixed integer programming model of the problem. We show the computational complexity of the problem and optimality properties. Then we propose a novel ant colony optimization method where the Metropolis Criterion is used to select the paths of ants to overcome the immature convergence. Finally, we generate different scales of instances to test the performance. The computational results show the effectiveness of the algorithm, especially for large-scale instances.     

Using genetic algorithm for lot sizing and scheduling problem with arbitrary job volumes and distinct job due date considerations

This paper considers an integrated lot sizing and scheduling problem for a production–distribution environment with arbitrary job volumes and distinct due dates considerations. In the problem, jobs are firstly batch processed on a batching machine at production stage and then delivered to a pre-specified customer at the subsequent delivery stage by a capacitated vehicle. Each job is associated with a distinct due date and a distinct volume, and has to be delivered to the customer before its due date, i.e. delay is not allowed. The processing time of a batch is a constant independent of the jobs it contains. In production, a constant set-up time as well as a constant set-up cost is required before the first job of this batch is processed. In delivery, a constant delivery time as well as a constant delivery cost is needed for each round-trip delivery between the factory and the customer. Moreover, it is supposed that a job that arrives at the customer before its due date will incur a customer inventory cost. The objective is to find a coordinated lot sizing and scheduling scheme such that the total cost is minimised while guaranteeing a certain customer service level. A mixed integer formulation is proposed for this problem, and then a genetic algorithm is developed to solve it. To evaluate the performance of the proposed genetic algorithm, a lower bound on the objective value is established. Computational experiments show that the proposed genetic algorithm performs well on randomly generated problem instances.     

Unrelated parallel machine scheduling with setup times and a total weighted tardiness objective

This paper presents several search heuristics and their performance in batch scheduling of parallel, unrelated machines. Identical or similar jobs are typically processed in batches in order to decrease setup times and/or processing times. The problem accounts for allotting batched work parts into unrelated parallel machines, where each batch consists of a fixed number of jobs. Some batches may contain different jobs but all jobs within each batch should have an identical processing time and a common due date. Processing time of each job of a batch is determined according to the machine group as well as the batch group to which the job belongs. Major or minor setup times are required between two subsequent batches depending on batch sequence but are independent of machines. The objective of our study is to minimize the total weighted tardiness for the unrelated parallel machine scheduling. Four search heuristics are proposed to address the problem, namely (1) the earliest weighted due date, (2) the shortest weighted processing time, (3) the two-level batch scheduling heuristic, and (4) the simulated annealing method. These proposed local search heuristics are tested through computational experiments with data from dicing operations of a compound semiconductor manufacturing facility.     

On-line hierarchical job scheduling on grids with admissible allocation

In this paper, we address non-preemptive online scheduling of parallel jobs on a Grid. Our Grid consists of a large number of identical processors that are divided into several machines. We consider a Grid scheduling model with two stages. At the first stage, jobs are allocated to a suitable machine, while at the second stage, local scheduling is independently applied to each machine. We discuss strategies based on various combinations of allocation strategies and local scheduling algorithms. Finally, we propose and analyze a scheme named adaptive admissible allocation. This includes a competitive analysis for different parameters and constraints. We show that the algorithm is beneficial under certain conditions and allows for an efficient implementation in real systems. Furthermore, a dynamic and adaptive approach is presented which can cope with different workloads and Grid properties.     

Machine scheduling with job class setup and delivery considerations

We study machine scheduling problems in which the jobs belong to different job classes and they need to be delivered to customers after processing. A setup time is required for a job if it is the first job to be processed on a machine or its processing on a machine follows a job that belongs to another class. Processed jobs are delivered in batches to their respective customers. The batch size is limited by the capacity of the delivery vehicles and each shipment incurs a transport cost and takes a fixed amount of time. The objective is to minimize the weighted sum of the last arrival time of jobs to customers and the delivery (transportation) cost. For the problem of processing jobs on a single machine and delivering them to multiple customers, we develop a dynamic programming algorithm to solve the problem optimally. For the problem of processing jobs on parallel machines and delivering them to a single customer, we propose a heuristic and analyze its performance bound.     

Scheduling unrelated parallel batch processing machines with non-identical job sizes and unequal ready times

This research analyzes the problem of scheduling a set of n jobs with arbitrary job sizes and non-zero ready times on a set of m unrelated parallel batch processing machines so as to minimize the makespan. Unrelated parallel machine is a generalization of the identical parallel processing machines and is closer to real-world production systems. Each machine can accommodate and process several jobs simultaneously as a batch as long as the machine capacity is not exceeded. The batch processing time and the batch ready time are respectively equal to the largest processing time and the largest ready time among all the jobs in the batch. Motivated by the computational complexity and the practical relevance of the problem, we present several heuristics based on first-fit and best-fit earliest job ready time rules. We also present a mixed integer programming model for the problem and a lower bound to evaluate the quality of the heuristics. The small computational effort of deterministic heuristics, which is valuable in some practical applications, is also one of the reasons that motivates this study. The results show that the heuristic proposed in this paper has a superior performance compared to the heuristics based on ideas proposed in the literature.     

An Agent-Based Approach for Scheduling Multiple Machines

We present a new agent-based solution approach for the problem of scheduling multiple non-identical machines in the face of sequence dependent setups, job machine restrictions, batch size preferences, fixed costs of assigning jobs to machines and downstream considerations. We consider multiple objectives such as minimizing (weighted) earliness and tardiness, and minimizing job-machine assignment costs. We use an agent-based architecture called Asynchronous Team (A-Team), in which each agent encapsulates a different problem solving strategy and agents cooperate by exchanging results. Computational experiments on large instances of real-world scheduling problems show that the results obtained by this approach are significantly better than any single algorithm or the scheduler alone. This approach has been successfully implemented in an industrial scheduling system.     

双资源生产车间调度问题的研究

将遗传算法(GA)和模拟退火算法(SA)相结合研究了双资源生产车间的调度优化问题,该混合算法将机床设备和工人合理地分配给加工任务,使评价性能指标获得最优。通过与国内外学者的算法进行比较,本算法获得的生产周期最短,机床利用率和工人利用率都较高,并且在某些情况下,平均流动时间也较短。因此可以证明本算法具有一定的优越性。     

Tabu search heuristic for two-machine flowshop with batch processing machines

Batch processing machines are frequently encountered in many industrial environments. A batch processing machine is one which can process several jobs simultaneously as a batch. The processing time of a batch is equal to the largest processing time of any job in the batch. This study deals with the problem of scheduling jobs in a flowshop with two batch processing machines such that the makespan is minimized. A heuristic based on Tabu search (TS) technique is proposed. The proposed heuristic is compared with a heuristic based on mixed integer linear programming (MILP). Because the complexity of the MILP-based heuristic is depended on the number of job batches, the comparison is under up-to-eight batches problem. In order to measure the proposed TS-based heuristic in larger batch problem, the relative error percentage with the lower bound (REP_LB) is used. The results show that the proposed heuristic is efficient and effective for the problems with relative large job sizes.     

A branch and price algorithm to minimize makespan on a single batch processing machine with non-identical job sizes

In this paper, we consider the scheduling problem on a single batch processing machine with non-identical job sizes; in which the machine has a limited capacity and can process a group of jobs simultaneously as a batch. The processing time of a batch is the longest processing time of all jobs in the batch. The objective is to minimize the makespan. We formulate the problem using Dantzig–Wolfe decomposition as a set partitioning problem. Based on the set partitioning formulation, we present a tight lower bound using column generation method. A heuristic algorithm is also developed to generate the basic solution in the column generation method. A branch and price algorithm which combines the column generation technique with branch and bound method is then presented to obtain the optimal solution of the problem. The efficiency of the proposed branch and price algorithm is ultimately compared to the branch and bound algorithm from the literature, based on the generated sample problems.     

Dynamic scheduling of a batch of parallel task jobs on heterogeneous clusters

This paper addresses the problem of minimizing the scheduling length (make-span) of a batch of jobs with different arrival times. A job is described by a direct acyclic graph (DAG) of parallel tasks. The paper proposes a dynamic scheduling method that adapts the schedule when new jobs are submitted and that may change the processors assigned to a job during its execution. The scheduling method is divided into a scheduling strategy and a scheduling algorithm. We also propose an adaptation of the Heterogeneous Earliest-Finish-Time (HEFT) algorithm, called here P-HEFT, to handle parallel tasks in heterogeneous clusters with good efficiency without compromising the makespan. The results of a comparison of this algorithm with another DAG scheduler using a simulation of several machine configurations and job types shows that P-HEFT gives a shorter makespan for a single DAG but scores worse for multiple DAGs. Finally, the results of the dynamic scheduling of a batch of jobs using the proposed scheduler method showed significant improvements for more heavily loaded machines when compared to the alternative resource reservation approach.     

An Enhanced tabu search algorithm to minimize a bi-criteria objective in batching and scheduling problems on unrelated-parallel machines with desired lower bounds on batch sizes

This paper addresses a sequence- and machine-dependent batch scheduling problem on a set of unrelated-parallel machines where the objective is to minimize a linear combination of total weighted completion time and total weighted tardiness. In particular, batch scheduling disregards the group technology assumptions by allowing for the possibility of splitting pre-determined groups of jobs into batches with respect to desired lower bounds on batch sizes. With regard to bounds on batch sizes, the MILP model is developed as two integrated batching and scheduling phases to present the problem. A benchmark of small-size instances on group scheduling shows the superior performance of batch scheduling up to 37% reduction in the objective function value. An efficient meta-heuristic algorithm based on tabu search with multi-level diversification and multi-tabu structure is developed at three levels, which moves back and forth between batching and scheduling phases. To eliminate searching in ineffective neighborhoods and thus enhance computational efficiency of search algorithms, several lemmas are proposed and proven. The results of applying lemmas reflect up to 40% reduction in computational times. Comparing the optimal solutions found by CPLEX and tabu search shows that the tabu search algorithm could find solutions, at least as good as CPLEX but in incredibly shorter computational time. In order to trigger the search algorithm, two different initial solution finding mechanisms have been developed and implemented. Also, due to lack of a qualified benchmark for unrelated-parallel machines, a comprehensive data generation mechanism has been developed in a way that it fairly reflects the real world situations encountered in practice. The machine availability times and job release times are considered to be dynamic and the run time of each job differs on different machines based upon the capability of the machines.     

An exact algorithm for parallel machine scheduling with conflicts

We consider an extension of classic parallel machine scheduling where a set of jobs is scheduled on identical parallel machines and an undirected conflict graph is part of the input. Each node in the graph represents a job, and an edge implies that its two jobs are conflicting, meaning that they cannot be scheduled on the same machine. The goal is to find an assignment of the jobs to the machines such that the maximum completion time (makespan) is minimized. We present an exact algorithm based on branch and price that combines methods from bin packing, scheduling, and graph coloring, with appropriate modifications. The algorithm has a good computational performance even for parallel machine scheduling without conflicting jobs.