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.     

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.     

Scheduling unrelated parallel batch processing machines with non-identical job sizes

Scheduling unrelated parallel batch processing machines to minimize makespan is studied in this paper. Jobs with non-identical sizes are scheduled on batch processing machines that can process several jobs as a batch as long as the machine capacity is not violated. Several heuristics based on best fit longest processing time (BFLPT) in two groups are proposed to solve the problem. A lower bound is also proved to evaluate the quality of the heuristics. Computational experiments were undertaken. These showed that J_SC-BFLPT, considering both load balance of machines and job processing times, was robust and outperformed other heuristics for most of the problem categories.     

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).     

Decomposition heuristics for minimizing earliness–tardiness on parallel burn-in ovens with a common due date

This paper considers a scheduling problem for parallel burn-in ovens in the semiconductor manufacturing industry. An oven is a batch processing machine with restricted capacity. The batch processing time is set by the longest processing time among those of all the jobs contained in the batch. All jobs are assumed to have the same due date. The objective is to minimize the sum of the absolute deviations of completion times from the due date (earliness–tardiness) of all jobs. We suggest three decomposition heuristics. The first heuristic applies the exact algorithm due to Emmons and Hall (for the nonbatching problem) in order to assign the jobs to separate early and tardy job sets for each of the parallel burn-in ovens. Then, we use job sequencing rules and dynamic programming in order to form batches for the early and tardy job sets and sequence them optimally. The second proposed heuristic is based on genetic algorithms. We use a genetic algorithm in order to assign jobs to each single burn-in oven. Then, after forming early and tardy job sets for each oven we apply again sequencing rules and dynamic programming techniques to the early and tardy jobs sets on each single machine in order to form batches. The third heuristic assigns jobs to the m early job sets and m tardy jobs sets in case of m burn-in ovens in parallel via a genetic algorithm and applies again dynamic programming and sequencing rules. We report on computational experiments based on generated test data and compare the results of the heuristics with known exact solution for small size test instances obtained from a branch and bound scheme.     

Scheduling a batching machine with convex resource consumption functions

In various industries jobs undergo a batching, or burn in, process where different tasks are grouped into batches and processed simultaneously. The processing time of each batch is equal to the longest processing time among all jobs contained in the batch. All to date studies dealing with batching machines have considered fixed job processing times. However, in many real life applications job processing times are controllable through the allocation of a limited resource. The most common and realistic model assumes that there exists a non-linear and convex relationship between the amount of resource allocated to a job and its processing time. The scheduler?s task when dealing with controllable processing times is twofold. In addition to solving the sequencing problem, one must establish an optimal resource allocation policy. We combine these two widespread models on a single machine setting, showing that both the makespan and total completion time criteria can be solved in polynomial time. We then show that our proposed approach can be applied to general bi-criteria objective comprising of the makespan and the total completion time.     

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.     

Two-machine flowshop scheduling problem with bounded processing times to minimize total completion time

We consider the two-machine flowshop scheduling problem where jobs have random processing times which are bounded within certain intervals. The objective is to minimize total completion time of all jobs. The decision of finding a solution for the problem has to be made based on the lower and upper bounds on job processing times since this is the only information available. The problem is NP-hard since the special case when the lower and upper bounds are equal, i.e., the deterministic case, is known to be NP-hard. Therefore, a reasonable approach is to come up with well performing heuristics. We propose eleven heuristics which utilize the lower and upper bounds on job processing times based on the Shortest Processing Time (SPT) rule. The proposed heuristics are compared through randomly generated data. The computational analysis has shown that the heuristics using the information on the bounds of job processing times on both machines perform much better than those using the information on one of the two machines. It has also shown that one of the proposed heuristics performs as the best for different distributions with an overall average percentage error of less than one.     

Scheduling a capacitated batch-processing machine to minimize makespan

This paper aims at improving the utilization of a single batch-processing machine. The batch-processing machine can process a batch of jobs, as long as the number of jobs and the total size of all the jobs in a batch do not violate the machine's capacity. The processing time of the job and its size is known. The processing time of a batch is the longest processing time among all the jobs in the batch. The objective is to minimize the makespan. Since the problem under study is NP-hard, a Simulated Annealing (SA) approach is proposed. The effectiveness of our solution procedure in terms of solution quality and run time is evaluated through experiments. The results obtained from the SA approach were compared with a commercial solver called CPLEX. Our computational study demonstrates the effectiveness of our approach in solving problem instances with 20 or more jobs in a shorter run time with better solutions.     

Algorithms for scheduling incompatible job families on single batching machine with limited capacity

Motivated by applications in food processing and semiconductor manufacturing industries, we consider the scheduling problem of a batching machine with jobs of multiple families. The machine has a limited capacity to accommodate jobs. The jobs are in arbitrary sizes and multiple families. Jobs from different families cannot be processed in a batch. We show the problems of minimizing makespan and total batch completion time are both NP-hard in the strong sense. We present a mixed integer programming model for the problems. Then we propose two polynomial time heuristics based on longest processing time first rule and first fit rule. For the special case where a larger job also has a longer processing time, the heuristic for minimizing makespan is optimal. For the general case, we show the performance guarantee of the methods for the two objectives respectively.     

工件具有不同尺寸单台批处理机的调度问题

n个工件要在一台有高度限制的批处理机上分批进行加工,工件j的加工时间和高度分别为Pj和Sj,批的加工时间为批中加工时间最大的工件的加工时间,每批加工时,机器的剩余量为批处理机的高度与批中工件的高度和之差,目标函数最小化机器空余总量和工件总完成时间,该NP-难问题源于钢铁企业的罩式退火炉调度问题.基于部分工件分批性质,提...     

Approximation algorithms for minimizing the total weighted number of late jobs with late deliveries in two-level supply chains

We study a supply chain scheduling problem in which n jobs have to be scheduled on a single machine and delivered to m customers in batches. Each job has a due date, a processing time and a lateness penalty (weight). To save batch-delivery costs, several jobs for the same customer can be delivered together in a batch, including late jobs. The completion time of each job in the same batch coincides with the batch completion time. A batch setup time has to be added before processing the first job in each batch. The objective is to find a schedule which minimizes the sum of the weighted number of late jobs and the delivery costs. We present a pseudo-polynomial algorithm for a restricted case, where late jobs are delivered separately, and show that it becomes polynomial for the special cases when jobs have equal weights and equal delivery costs or equal processing times and equal setup times. We convert the algorithm into an FPTAS and prove that the solution produced by it is near-optimal for the original general problem by performing a parametric analysis of its performance ratio.     

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.     

批处理机上具有两类释放时间的工件集竞争调度问题

研究了两个工件集合竞争在一台批处理机上加工的调度问题, 其中每个集合的工件具有一个共同的释放时间.批处理机可以同时加工多个工件作为一批, 每批的加工时间为该批工件中加工时间的最大值.基于两类释放时间的大小, 针对无界批处理机上最小化一个集合工件的最大完工时间、最大延迟以及总完工时间, 使得另一个集合工件的最大完工时间不超过给定上界问题, 分别给出了最优求解方法.针对有界批处理机上最小化一个集合工件的最大完工时间, 使得另一个集合工件的最大完工时间不超过给定上界问题, 证明为一般意义NP--!难问题, 并给出伪多项式时间最优求解方法.     

A DP algorithm for minimizing makespan and total completion time on a series-batching machine

This paper studies a bicriteria scheduling problem on a series-batching machine with objective of minimizing makespan and total completion time simultaneously. A series-batching machine is a machine that can handle up to b jobs in a batch and the completion time of all jobs in a batch is equal to the finishing time of the last job in the batch and the processing time of a batch is the sum of the processing times of jobs in the batch. In addition, there is a constant setup time s for each batch. For the problem we can find all Pareto optimal solutions in O(n²) time by a dynamic programming algorithm, where n denotes the number of jobs.     

Scheduling jobs with ready times and precedence constraints on parallel batch machines using metaheuristics

In this paper, we discuss a scheduling problem for parallel batch machines where the jobs have ready times. Problems of this type can be found in semiconductor wafer fabrication facilities (wafer fabs). In addition, we consider precedence constraints among the jobs. Such constraints arise, for example, in scheduling subproblems of the shifting bottleneck heuristic when complex job shop scheduling problems are tackled. We use the total weighted tardiness as the performance measure to be optimized. Hence, the problem is NP-hard and we have to rely on heuristic solution approaches. We consider a variable neighborhood search (VNS) scheme and a greedy randomized adaptive search procedure (GRASP) to compute efficient solutions. We assess the performance of the two metaheuristics based on a large set of randomly generated problem instances and based on instances from the literature. The obtained computational results demonstrate that VNS is a very fast heuristic that quickly leads to high-quality solutions, whereas the GRASP is slightly outperformed by the VNS approach. However, the GRASP approach has the advantage that it can be parallelized in a more natural manner compared to VNS.     

An effective hybrid multi-objective genetic algorithm for bi-criteria scheduling on a single batch processing machine with non-identical job sizes

This paper addresses the problem of scheduling jobs with non-identical sizes on a single batch processing machine. A batch processing machine is one which can process multiple jobs simultaneously as a batch as long as the total size of jobs being processed does not exceed the machine capacity. The batch processing time is equal to the longest processing time among all jobs in the batch. For the simultaneous minimization of the bi-criteria of makespan and maximum tardiness, we propose two different multi-objective genetic algorithms based on different representation schemes. While the first algorithm do search via generating sequences of jobs using genetic operators and then batching jobs keeping their order in the sequence, the second algorithm uses the idea of generating batches of jobs directly using genetic operators and ensures feasibility through using heuristic procedures. The type of representation used in the second algorithm allows introducing heuristics with the ability of biasing the search towards each objective and also allows hybridization with a local search heuristic that gives the ability of finding Pareto-optimal or locally efficient Pareto-solutions. Computational results show that the non-dominated solutions obtained by the latter algorithm are very superior in closeness to the true Pareto-optimal solutions and to keep diversity in the obtained Pareto-set, as the problem size increases.     

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.     

Unbounded parallel-batching scheduling with two competitive agents

We consider the scheduling problems arising when two agents, each with a family of jobs, compete to perform their respective jobs on a common unbounded parallel-batching machine. The batching machine can process any number of jobs simultaneously in a batch. The processing time of a batch is equal to the maximum processing time of the jobs in the batch. Two main categories of batch processing based on the compatibility of job families or agents are distinguished. In the case where job families are incompatible, jobs from different families cannot be placed in the same processing batch while all jobs can be placed in the same processing batch when job families are compatible. The goal is to find a schedule for all jobs of the two agents that minimizes the objective of one agent while keeping the objective of the other agent below or at a fixed value Q. Polynomial-time and pseudo-polynomial-time algorithms are provided to solve various combinations of regular objective functions for the scenario in which job families are either incompatible or compatible.     

Single machine scheduling with controllable processing times to minimize total tardiness and earliness

In most deterministic scheduling problems job processing times are considered as invariable and known in advance. Single machine scheduling problem with controllable processing times with no inserted idle time is presented in this study. Job processing times are controllable to some extent that they can be reduced or increased, up to a certain limit, at a cost proportional to the reduction or increase. In this study, our objective is determining a set of compression/expansion of processing times in addition to a sequence of jobs simultaneously, so that total tardiness and earliness are minimized. A mathematical model is proposed firstly and afterward a net benefit compression–net benefit expansion (NBC–NBE) heuristic is presented so as to acquire a set of amounts of compression and expansion of jobs processing times in a given sequence. Three heuristic techniques in small problems and in medium-to-large instances two meta-heuristic approaches, as effective local search methods, as well as these heuristics are employed to solve test examples. The single machine total tardiness problem (SMTTP) is already NP-hard, so the considered problem is NP-hard obviously. The computational experiments demonstrate that our proposed heuristic is efficient approach for such just-in-time (JIT) problem, especially equipped with competent heuristics.