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 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.     

Minimizing total completion time in a two-stage hybrid flow shop with dedicated machines at the first stage

This paper considers a two-stage hybrid flow shop scheduling with dedicated machines at stage 1 with the objective of minimizing the total completion time. There exist two machines at stage 1 and one machine at stage 2. Each job must be processed on one of the two dedicated machines at stage 1 depending on the job type; subsequently, the job is processed on the single machine at stage 2.First, we introduce the problem and establish the complexity of the problem. For a special case in which the processing times on the machine at stage 2 are identical, an optimal solution is presented; for three special cases, we show that the decision version is unary NP-complete. For the general case, two simple and intuitive heuristics are introduced, and a worst case bound on the relative error is found for each of the heuristics. Finally, we empirically evaluate the heuristics, including an optimal algorithm for a special case.     

Heuristics for scheduling problems with an unavailability constraint and position-dependent processing times

We study a single machine scheduling problem, where the machine is unavailable for processing for a pre-specified time period. We assume that job processing times are position-dependent. The objective functions considered are minimum makespan, minimum total completion time and minimum number of tardy jobs. All these problems are known to be NP-hard even without position-dependent processing times. For all three cases we introduce simple heuristics which are based on solving the classical assignment problem. Lower bounds, worst case analysis and asymptotic optimality are discussed. All heuristics are shown numerically to perform extremely well.     

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.     

heuristics for parallel machine scheduling with delivery times

A parallel machine scheduling problem is considered in which each job has a processing time and a delivery time. The objective is to find a schedule which minimizes the time by which all jobs are delivered. For a single machine this problem is easily solved in polynomial time, form2 machines it becomes NP-hard. Several heuristics using list scheduling as a subroutine are proposed and a tight worst-case analysis is given. The best one of our heuristics has a worst-case performance guarantee of 2–2/(m+1). For the on-line case we give a heuristic with the (best possible) worst-case performance of two.This research was supported by the Christian Doppler Laboratorium für Diskrete Optimierung.     

Scheduling two-stage hybrid flow shops with parallel batch, release time, and machine eligibility constraints

This paper investigates a difficult scheduling problem on a specialized two-stage hybrid flow shop with multiple processors that appears in semiconductor manufacturing industry, where the first and second stages process serial jobs and parallel batches, respectively. The objective is to seek job-machine, job-batch, and batch-machine assignments such that makespan is minimized, while considering parallel batch, release time, and machine eligibility constraints. We first propose a mixed integer programming (MIP) formulation for this problem, then gives a heuristic approach for solving larger problems. In order to handle real world large-scale scheduling problems, we propose an efficient dispatching rule called BFIFO that assigns jobs or batches to machines based on first-in-first-out principle, and then give several reoptimization techniques using MIP and local search heuristics involving interchange, translocation and transposition among assigned jobs. Computational experiments indicate our proposed re-optimization techniques are efficient. In particular, our approaches can produce good solutions for scheduling up to 160 jobs on 40 machines at both stages within 10?min.     

Flowshop scheduling problems with transportation or deterioration between the batching and single machines

In this paper, we consider two new types of the two-machine flowshop scheduling problems where a batching machine is followed by a single machine. The first type is that normal jobs with transportation between machines are scheduled on the batching and single machines. The second type is that normal jobs are processed on the batching machine while deteriorating jobs are scheduled on the single machine. For the first type, we formulate the problem to minimize the makespan as a mixed integer programming model and prove that it is strongly NP-hard. Furthermore, a heuristic algorithm along with a worst case error bound is derived and the computational experiments are also carried out to verify the effectiveness of the proposed heuristic algorithm. For the second type, the two objectives are considered. For the problem with minimizing the makespan, we find an optimal polynomial algorithm. For the problem with minimizing the sum of completion time, we show that it is strongly NP-hard and propose an optimal polynomial algorithm for its special case.     

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.     

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.     

Minimizing total weighted completion time when scheduling orders in a flexible environment with uniform machines

We consider the scheduling of orders in an environment with m uniform machines in parallel. Each order requests certain amounts of k different product types. Each product type can be produced by any one of the m machines. No setup is required if a machine switches over from one product type to another. Different product types intended for the same order can be produced at the same time (concurrently) on different machines. Each order is released at time zero and has a positive weight. Preemptions are allowed. The completion time of an order is the finish time of the product type that is completed last for that order. The objective is to minimize the total weighted completion time. We propose heuristics for the non-preemptive as well as the preemptive case and obtain worst case bounds that are a function of the number of machines as well as the differences in the speeds of the machines. Even though the worst-case bounds we showed for the two heuristics are not very tight, our experimental results show that they yield solutions that are very close to optimal.     

Scheduling with multiple servers

In this paper, we consider the problem of scheduling a set of jobs on a set of identical parallel machines. Before the processing of a job can start, a setup is required which has to be performed by a given set of servers. We consider the complexity of such problems for the minimization of the makespan. For the problem with equal processing times and equal setup times we give a polynomial algorithm. For the problem with unit setup times, m machines and m − 1 servers, we give a pseudopolynomial algorithm. However, the problem with fixed number of machines and servers in the case of minimizing maximum lateness is proven to be unary NP-hard. In addition, recent algorithms for some parallel machine scheduling problems with constant precessing times are generalized to the corresponding server problems for the case of constant setup times. Moreover, we perform a worst case analysis of two list scheduling algorithms for makespan minimization.     

Two-stage hybrid batching flowshop scheduling with blocking and machine availability constraints using genetic algorithm

This research investigates a two-stage hybrid flowshop scheduling problem in a metal-working company. The first stage consists of multiple parallel machines and the second stage has only one machine. Four characteristics of the company have substantiated the complexity of the problem. First, all machines in stage one are able to process multiple jobs simultaneously but the jobs must be sequentially set up one after another. Second, the setup time of each job is separated from its processing time and depends upon its preceding job. Third, a blocking environment exists between two stages with no intermediate buffer storage. Finally, machines are not continuously available due to the preventive maintenance and machine breakdown. Two types of machine unavailability, namely, deterministic case and stochastic case, are identified in this problem. The former occurs on stage-two machine with the start time and the end time known in advance. The latter occurs on one of the parallel machine in stage one and a real-time rescheduling will be triggered. Minimizing the makespan is considered as the objective to develop the optimal scheduling algorithm. A genetic algorithm is used to obtain a near-optimal solution. The computational results with actual data are favorable and superior over the results from existing manual schedules.     

Hybrid tabu search and a truncated branch-and-bound for the unrelated parallel machine scheduling problem

We consider the problem of scheduling a number of jobs on a number of unrelated parallel machines in order to minimize the makespan. We develop three heuristic approaches, i.e., a genetic algorithm, a tabu search algorithm and a hybridization of these heuristics with a truncated branch-and-bound procedure. This hybridization is made in order to accelerate the search process to near-optimal solutions. The branch-and-bound procedure will check whether the solutions obtained by the meta-heuristics can be scheduled within a tight upper bound. We compare the performances of these heuristics on a standard dataset available in the literature. Moreover, the influence of the different heuristic parameters is examined as well. The computational experiments reveal that the hybrid heuristics are able to compete with the best known results from the literature.     

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.     

Completion time variance minimization in single machine and multi-machine systems

In this paper we consider the problem of minimizing the completion time variance of n jobs on a single machine with deterministic processing times. We propose a new heuristic and compare the results with existing popular heuristics for the problem. We also propose a method based on genetic algorithms to solve the problem. We present the worst case performance analysis of the proposed heuristic. We also consider the problem of minimizing the completion time variance of n jobs on m identical parallel machines in both restricted and unrestricted versions. A heuristic method and a method based on genetic algorithms are presented for both the cases and results of computational testing are provided. It is concluded that the proposed methods provide better results compared to existing methods for the single machine case as well as for the multi-machine case.     

针对经典排序问题的一种新算法的近似比分析

给定m台平行机(同型机),n个工件,寻找一种分配方案,使得把这n个工件分配到m台机器后,整体完工时间尽可能短,这个NP-难问题被称为经典排序问题。如果每个工件的加工时间满足一定的条件,则有望能在多项式时间内有效地得到最优的分配方案。Yue等对加工时间满足整除性质的经典排序问题考虑了一种新的算法,该算法总是能得到这种特殊情况的最优分配。该算法在多项式时间内能够得到最优分配,是对于一般的经典排序问题的近似算法。文章在此基础上,考虑该新算法在一般问题上的近似比。文中考虑了这个新算法的两种版本,分别得到了3/2和2-1/2 ^q(q∈Z⁺)的近似比。紧例子表明,文中对算法的两个版本的分析都是最优的。     

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.     

三台机并行工件排序问题的改进的下界

与经典的排序问题不同的是,并行工件排序指的是在加工某些工件时,需要多个机器同时并行工作。竞争比是评价在线算法好坏的一个重要指标,而竞争比的下界则是算法设计的一个重要参考。利用反证法,通过构造一个特殊的反例,分析了由此产生的全部9种可能的情形,建立了它们对应的9种线性规划模型,借助计算软件证明了前8种情形是不可能的,然后详细分析了第9种情形也是不可能的,从而给出了三台机并行工件排序问题的竞争比的一个改进的下界2.07。这个结果优于已知的最好的下界1.999。