Scheduling of parallel machines with sequence-dependent batches and product incompatibilities in an automotive glass facility

This application is motivated by a complex real-world scheduling problem found in the bottleneck workstation of the production line of an automotive safety glass manufacturing facility. The scheduling problem consists of scheduling jobs (glass parts) on a number of parallel batch processing machines (furnaces), assigning each job to a batch, and sequencing the batches on each machine. The two main objectives are to maximize the utilization of the parallel machines and to minimize the delay in the completion date of each job in relation to a required due date (specific for each job). Aside from the main objectives, the output batches should also produce a balanced workload on the parallel machines, balanced job due dates within each batch, and minimal capacity loss in the batches. The scheduling problem also considers a batch capacity constraint, sequence-dependent processing times, incompatible product families, additional resources, and machine capability. We propose a two-phase heuristic approach that combines exact methods with search heuristics. The first phase comprises a four-stage mixed-integer linear program for building the batches; the second phase is based on a Greedy Randomized Adaptive Search Procedure for sequencing the batches assigned to each machine. We conducted experiments on instances with up to 100 jobs built with real data from the manufacturing facility. The results are encouraging both in terms of computing time—5 min in average—and quality of the solutions—less than 10 % relative gap from the optimal solution in the first phase and less than 5 % in the second phase. Additional experiments were conducted on randomly generated instances of small, medium, and large size.     

Unrelated parallel machine scheduling with setup times using simulated annealing

This paper presents a scheduling problem for unrelated parallel machines with sequence-dependent setup times, using simulated annealing (SA). The problem accounts for allotting work parts of L jobs into M parallel unrelated machines, where a job refers to a lot composed of N items. Some jobs may have different items while every item within each job has an identical processing time with a common due date. Each machine has its own processing times according to the characteristics of the machine as well as job types. Setup times are machine independent but job sequence dependent. SA, a meta-heuristic, is employed in this study to determine a scheduling policy so as to minimize total tardiness. The suggested SA method utilizes six job or item rearranging techniques to generate neighborhood solutions. The experimental analysis shows that the proposed SA method significantly outperforms a neighborhood search method in terms of total tardiness.     

Scheduling Parallel Machines for the Customer Order Problem

This paper considers scheduling problems where jobs are dispatched in batches. The objective is to minimize the sum of the completion times of the batches. While a machine can process only one job at a time, multiple machines can simultaneously process jobs in a batch. This simple environment has a variety of real world applications such as part kitting and customer order scheduling.A heuristic is presented for the parallel machine version of the problem. Also, a tight worst case bound on the relative error is found. For the case of two parallel machines, we examine two heuristics, which are based on simple scheduling rules. We find tight worst case bounds of 6/5 and 9/7 on the relative error and show that neither procedure is superior for all instances. Finally, we empirically evaluate these two heuristics. For large problems, the methods find solutions that are close to optimal.     

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 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.     

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.     

Hybrid genetic algorithms with dispatching rules for unrelated parallel machine scheduling with setup time and production availability

This article considers the unrelated parallel machine scheduling problem with sequence- and machine-dependent setup times and machine-dependent processing times. Furthermore, the machine has a production availability constraint to each job. The objective of this problem is to determine the allocation policy of jobs and the scheduling policy of machines to minimize the total completion time. To solve the problem, a mathematical model for the optimal solution is derived, and hybrid genetic algorithms with three dispatching rules are proposed for large-sized problems. To assess the performance of the algorithms, computational experiments are conducted and evaluated using several randomly generated examples.     

On-line scheduling of two parallel machines with a single server

In this paper, we consider the on-line scheduling of two parallel identical machines sharing a single server with the objective of minimizing the latest completion time of all jobs. Each job has to be setup by the server before being processed on one of the machines. Three special cases: equal length jobs, equal processing times and regular equal setup times are considered and the asymptotic competitive ratios of list scheduling are determined. Also, a lower bound for the equal length job case is given, and two heuristics with tight asymptotic competitive ratios for the other two cases are proposed.     

A dominant class of schedules for malleable jobs in the problem to minimize the total weighted completion time

This paper is about scheduling parallel jobs, i.e. which can be executed on more than one machine at the same time. Malleable jobs is a special class of parallel jobs. The number of machines a malleable job is executed on may change during its execution.In this work, we consider the NP-hard problem of scheduling malleable jobs to minimize the total weighted completion time (or mean weighted flow time). For this problem, we introduce the class of “ascending” schedules in which, for each job, the number of machines assigned to it cannot decrease over time while this job is being processed.We prove that, under a natural assumption on the processing time functions of jobs, the set of ascending schedules is dominant for the problem. This result can be used to reduce the search space while looking for an optimal solution.     

A robust hybrid approach based on particle swarm optimization and genetic algorithm to minimize the total machine load on unrelated parallel machines

This paper dealt with an unrelated parallel machines scheduling problem with past-sequence-dependent setup times, release dates, deteriorating jobs and learning effects, in which the actual processing time of a job on each machine is given as a function of its starting time, release time and position on the corresponding machine. In addition, the setup time of a job on each machine is proportional to the actual processing times of the already processed jobs on the corresponding machine, i.e., the setup times are past-sequence-dependent (p-s-d). The objective is to determine jointly the jobs assigned to each machine and the order of jobs such that the total machine load is minimized. Since the problem is NP-hard, optimal solution for the instances of realistic size cannot be obtained within a reasonable amount of computational time using exact solution approaches. Hence, an efficient method based on the hybrid particle swarm optimization (PSO) and genetic algorithm (GA), denoted by HPSOGA, is proposed to solve the given problem. In view of the fact that efficiency of the meta-heuristic algorithms is significantly depends on the appropriate design of parameters, the Taguchi method is employed to calibrate and select the optimal levels of parameters. The performance of the proposed method is appraised by comparing its results with GA and PSO with and without local search through computational experiments. The computational results for small sized problems show that the mentioned algorithms are fully effective and viable to generate optimal/near optimal solutions, but when the size of the problem is increased, the HPSOGA obtains better results in comparison with other algorithms.     

Search heuristics for a parallel machine scheduling problem with ready times and due dates

We consider a problem of scheduling orders on identical parallel machines. An order can be released after a given ready time and must be completed before its due date. An order is split into multiple jobs (batches) and a job is processed on one of the parallel machines. The objective of the scheduling problem is to minimize the holding costs of orders including work-in-process as well as finished job inventories. We suggest two local search heuristics, simulated annealing and taboo search algorithms, for the problem. Performance of the suggested algorithms is tested through computational experiments on randomly generated test problems.     

多工艺路线车间作业调度遗传算法实现

针对简单遗传算法在解决作业车间调度问题时只适用于简单问题的局限,研究了多工艺路线的批量调度遗传算法实现,论述了3种提高生产效率的调度策略,即采用最小批量原则对零件进行分批调度生产;将批量准备时间和零件加工时间相分离,在工件到达加工机床前做好批量加工准备;在生产加工过程中,将同批加工零件进行多次机床间转移,缩短后续机床的等待时间.同时将工序优先级调度算法加入到简单遗传算法,提出了一种全局优化的多工艺路线批量生产调度混合遗传算法.仿真结果表明,该调度算法能取得较好的效果.     

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.     

A GRASP approach for makespan minimization on parallel batch processing machines

In this paper we consider the problem of scheduling a set of identical batch processing machines arranged in parallel. A Greedy Randomized Adaptive Search Procedure (GRASP) approach is proposed to minimize the makespan under the assumption of non-zero job ready times, arbitrary job sizes and arbitrary processing times. Each machine can process simultaneously several jobs as a batch as long as the machine capacity is not violated. The batch processing time is equal to the largest processing time among those jobs in the batch. Similarly, the batch ready time is equal to the largest ready time among those jobs in the batch. The performance of the proposed GRASP approach was evaluated by comparing its results to a lower bound and heuristics published in the literature. Experimental study suggests that the solution obtained from the GRASP approach is superior compared to other heuristics.     

A Stronger Complexity Result for the Single Machine Multi-Operation Jobs Scheduling Problem to Minimize the Number of Tardy Jobs

We consider the single machine multi-operation jobs scheduling problem to minimize the number of tardy jobs. Each job consists of several operations that belong to different families. In a schedule, each family of job operations may be processed in batches with each batch incurring a setup time. A job completes when all of its operations have been processed. The objective is to minimize the number of tardy jobs. In the literature, this problem has been proved to be strongly NP-hard for arbitrary due-dates. We show in this paper that the problem remains strongly NP-hard even when the due-dates are common and all jobs have the same processing time.     

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.     

极小化加权完工时间和的无界批量机器并行调度问题

考虑无界批量机器并行调度中极小化加权完工时间和问题.设有n个工件和m台批加工同型机.每个工件具有一个正权因子、一个释放时间和一个加工时间.每台机器可以同时加工B≥n个工件.一个批次的加工时间是该批次所包含的所有工件的加工时间的最大者.在同一批次中加工的工件有相同的完工时间,即它们的共同开始时间加上该批次的加工时间.给出了一个多项式时间近似方案(PTAS).     

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.     

A non-delayed relax-and-cut algorithm for scheduling problems with parallel machines,due dates and sequence-dependent setup times

Consider the problem of scheduling a set of jobs to be processed exactly once, on any machine of a set of unrelated parallel machines, without preemption. Each job has a due date, weight, and, for each machine, an associated processing time and sequence-dependent setup time. The objective function considered is to minimize the total weighted tardiness of the jobs.This work proposes a non-delayed relax-and-cut algorithm, based on a Lagrangean relaxation of a time indexed formulation of the problem. A Lagrangean heuristic is also developed to obtain approximate solutions.Using the proposed methods, it is possible to obtain optimal solutions within reasonable time for some instances with up to 180 jobs and six machines. For the solutions for which it is not possible to prove optimality, interesting gaps are obtained.