2-Distance vertex-distinguishing index of subcubic graphs

This paper addresses the performance of scheduling algorithms for a two-stage no-wait hybrid flowshop environment with inter-stage flexibility, where there exist several parallel machines at each stage. Each job, composed of two operations, must be processed from start to completion without any interruption either on or between the two stages. For each job, the total processing time of its two operations is fixed, and the stage-1 operation is divided into two sub-parts: an obligatory part and an optional part (which is to be determined by a solution), with a constraint that no optional part of a job can be processed in parallel with an idleness of any stage-2 machine. The objective is to minimize the makespan. We prove that even for the special case with only one machine at each stage, this problem is strongly NP-hard. For the case with one machine at stage 1 and m machines at stage 2, we propose two polynomial time approximation algorithms with worst case ratio of \(3-\frac{2}{m+1}\) and \(2-\frac{1}{m+1}\), respectively. For the case with m machines at stage 1 and one machine at stage 2, we propose a polynomial time approximation algorithm with worst case ratio of 2. We also prove that all the worst case ratios are tight.     

Heuristic algorithms for the two-machine flowshop with limited machine availability

The paper studies a flowshop scheduling problem where machines are not available in given time intervals. The objective is to minimize the makespan. The problem is known to be NP-hard for two machines. We analyze constructive and local search based heuristic algorithms for the two-machine case. The algorithms are tested on easy and difficult test problems with up to 100 jobs and 10 intervals of non-availability. Computational results show that the algorithms perform well. For many problems an optimum solution is found.     

Parallel-machine scheduling of deteriorating jobs with potential machine disruptions

We consider parallel-machine scheduling of deteriorating jobs in a disruptive environment in which some of the machines will become unavailable due to potential disruptions. This means that a disruption to some of the machines may occur at a particular time, which will last for a period of time with a certain probability. If a job is disrupted during processing by a disrupted machine and it does not need (needs) to re-start after the machine becomes available again, it is called the resumable (non-resumable) case. By deteriorating jobs, we mean that the actual processing time of a job grows when it is scheduled for processing later because the machine efficiency deteriorates over time due to machine usage and aging. However, a repaired machine will return to its original state of efficiency. We consider two cases, namely performing maintenance immediately on the disrupted machine when a disruption occurs and not performing machine maintenance. In each case, the objective is to determine the optimal schedule to minimize the expected total completion time of the jobs in both non-resumable and resumable cases. We determine the computational complexity status of various cases of the problem, and provide pseudo-polynomial-time solution algorithms and fully polynomial-time approximation schemes for them, if viable.     

Non-resumable scheduling on a single bounded parallel-batch machine with periodic maintenance

We consider the problem of scheduling a set of jobs with different processing times and sizes on a single bounded parallel-batch machine with periodic maintenance. Because the machine is in batch-processing model and the capacity is fixed, several jobs can be processed simultaneously in a batch provided that the total size of the jobs in the batch doesn’t exceed the machine capacity. And the processing time of a batch is the largest processing time of the jobs contained in the batch. Meanwhile, the production of each batch is non-resumable, that is, if a batch cannot be completed processing before some maintenance, that batch needs to be processed anew once the machine returns available. Our goal is to minimize the makespan. We first consider two special cases where the jobs have the same sizes or the same processing times, both of which are strongly NP-hard. We present two different approximation algorithms for them and show that these two algorithms have the same tight worst-case ratio of 2. We then consider the general case where the jobs have the arbitrary processing times and arbitrary sizes, for which we propose a 17/5-approximation algorithm.

     

Scheduling Methods for Efficient Stamping Operations at an Automotive Company

We consider scheduling issues at Beyçelik, a Turkish automotive stamping company that uses presses to give shape to metal sheets in order to produce auto parts. The problem concerns the minimization of the total completion time of job orders (i.e., makespan) during a planning horizon. This problem may be classified as a combined generalized flowshop and flexible flowshop problem with special characteristics. We show that the Stamping Scheduling Problem is NP‐Hard. We develop an integer programming‐based method to build realistic and usable schedules. Our results show that the proposed method is able to find higher quality schedules (i.e., shorter makespan values) than both the company's current process and a model from the literature. However, the proposed method has a relatively long run time, which is not practical for the company in situations when a (new) schedule is needed quickly (e.g., when there is a machine breakdown or a rush order). To improve the solution time, we develop a second method that is inspired by decomposition. We show that the second method provides higher‐quality solutions—and in most cases optimal solutions—in a shorter time. We compare the performance of all three methods with the company's schedules. The second method finds a solution in minutes compared to Beyçelik's current process, which takes 28 hours. Further, the makespan values of the second method are about 6.1% shorter than the company's schedules. We estimate that the company can save over €187,000 annually by using the second method. We believe that the models and methods developed in this study can be used in similar companies and industries.     

Competitive analysis of online machine rental and online parallel machine scheduling problems with workload fence

In this paper, we introduce the concept of “workload fence" into online machine rental and machine scheduling problems. With the knowledge of workload fence, online algorithms acquire the information of a finite number of first released jobs in advance. The concept originates from the frozen time fence in the domain of master scheduling in materials management. The total processing time of the jobs foreseen, corresponding to a finite number of jobs, is called workload fence, which is irrelevant to the job sequence. The remaining jobs in the sequence, however, can only become known on their arrival. This work aims to reveal whether the knowledge of workload fence helps to boost the competitive performance of deterministic online algorithms. For the online machine rental problem, we prove that the competitiveness of online algorithms can be improved with a sufficiently large workload fence. We further propose a best online algorithm for the corresponding scenario. For online parallel machine scheduling with workload fence, we give a positive answer to the above question for the case where the workload fence is equal to the length of the longest job. We also show that the competitiveness of online algorithms may not be improved even with a workload fence strictly larger than the largest length of a job. The results help one manager to make a better decision regarding the tradeoff between the performance improvement of online algorithms and the cost caused to acquire the knowledge of workload fence.

     

Single-machine scheduling problems with general truncated sum-of-actual-processing-time-based learning effect

We study single machine scheduling problems with general truncated sum-of-actual-processing-time-based learning effect. In the general truncated learning model, the actual processing time of a job is affected by the sum of actual processing times of previous jobs and by a job-dependent truncation parameter. We show that the single machine problems to minimize makespan and to minimize the sum of weighted completion times are both at least ordinary NP-hard and the single machine problem to minimize maximum lateness is strongly NP-hard. We then show polynomial solvable cases and approximation algorithms for these problems. Computational experiments are also conducted to show the effectiveness of our approximation algorithms.

     

Evaluation of heuristics for scheduling in a flowshop: a case study

The problem of finding the best permutation schedule for a flowshop has engaged the attention of researchers for almost four decades. In view of its NP-completeness, the problem is not amenable to the development of efficient optimizing algorithms. A number of heuristics have been proposed, most of which have been evaluated by using randomly generated problems for a single measure of performance. However, real-life problems often have more than one objective. This paper discusses a live flowshop problem that has the twin objectives of minimizing the production run-time as well as the total flowtime of jobs. Five heuristics are evaluated in this study and some interesting findings are reported.     

Heuristics for scheduling in a flowshop with setup,processing and removal times separated

The problem of scheduling in a flowshop, where setup, processing and removal times are separable, is considered with the objective of minimizing makespan. Heuristic algorithms are developed by the introduction of simplifying assumptions into the scheduling problem under study. An improvement method is incorporated in the heuristics to enhance the quality of their solutions. The proposed heuristics and an existing heuristic are evaluated by a large number of randomly generated problems. The results of an extensive computational investigation for various values of parameters are presented.     

考虑分时电价的多目标批调度问题蚁群算法求解

对同时优化电力成本和制造跨度的多目标批处理机调度问题进行了研究,设计了两种多目标蚁群算法,基于工件序的多目标蚁群算法(J-PACO,Job-based Pareto Ant Colony Optimization)和基于成批的多目标蚁群算法(B-PACO,Batch-based Pareto Ant Colony Optimization)对问题进行求解分析。由于分时电价中电价是时间的函数,因而在传统批调度进行批排序的基础上,需要进一步确定批加工时间点以测定电力成本。提出的两种蚁群算法分别将工件和批与时间线相结合进行调度对此类问题进行求解。通过仿真实验将两种算法对问题的求解进行了比较,仿真实验表明B-PACO算法通过结合FFLPT(First Fit Longest Processing Time)启发式算法先将工件成批再生成最终方案,提高了算法搜索效率,并且在衡量算法搜索非支配解数量的Q指标和衡量非支配集与Pareto边界接近程度的HV指标上,均优于J-PACO算法。     

TWO‐ AND THREE‐STAGE FLOWSHOP SCHEDULING WITH NO‐WAIT IN PROCESS

In this paper we study the time complexities of some two‐ and three‐stage no‐wait flowshop makespan scheduling problems where, in some stage, all the jobs require a constant processing time and the stage may consist of parallel identical machines. Polynomial time algorithms are presented for certain problems, while several others are proved to be strongly NP‐complete.     

大规模集成电路预烧作业中分批排序问题的数学模型

分批排序(Batch Scheduling)是在半导体生产过程的最后阶段提炼出来的一类重要的排序问题。单机分批排序问题就是n个工件在一台机器上加工,要将工件分批,每批最多可以同时加工B个工件,每批的加工时间等于此批工件中的最大的加工时间。Skutella^[8]1998年把平行机排序的P｜｜∑ω_jC_j和R｜｜∑ω_jC_j表述成二次的0-1整数规划,得到一些令人满意的结果;国内罗守成等^[9]、张倩^[10]给出了单机排序问题1｜｜∑ω_jC_j的数学规划表示,对于用数学规划来研究排序问题是一个很有意义的进展。本文首先介绍总完工时间和最小的带权单机分批排序问题1｜B｜∑ω_jC_j,然后将1｜B｜∑ω_jC_j表示成数学规划的形式,并且用数学规划中的对偶理论证明了SPT序是其特殊情况1｜B=1｜∑C_j的最优解。     

A FRAMEWORK FOR MACHINE SCHEDULING PROBLEMS WITH CONTROLLABLE PROCESSING TIMES

This is a study of single and parallel machine scheduling problems with controllable processing time for each job. The processing time for job j depends on the position of the job in the schedule and is a function of the number of resource units allocated to its processing. Processing time functions and processing cost functions are allowed to be nonlinear. The scheduling problems considered here have important applications in industry and include many of the existing scheduling models as special cases. For the single machine problem, the objective is minimization of total compression costs plus a scheduling measure. The scheduling measures include makespan, total flow time, total differences in completion times, total differences in waiting times, and total earliness and tardiness with a common due date for all jobs. Except when the total earliness and tardiness measure is involved, each case the problem is solved efficiently. Under an assumption typically satisfied in just-in-time systems, the problem with total earliness and tardiness measure is also solved efficiently. Finally, for a large class of processing time functions; parallel machine problems with total flow time and total earliness and tardiness measures are solved efficiently. In each case we reduce the problem to a transportation problem.     

Scheduling a single semi-continuous batching machine

This paper addresses a new problem, called semi-continuous batch scheduling, which arises in the heating-operation of tube-billets in the steel industry. Each heating furnace can be regarded as a semi-continuous batching machine, which can handle up to C jobs simultaneously. The jobs in the same batch enter and leave the machine semi-continuously, which differs from the traditional batching machine scheduling where the jobs in same batch have a starting time and a finishing time. In this paper the processing time of a batch depends on the capacity of the semi-continuous batching machine, the longest processing time of jobs in the batch and its size. The objectives are to schedule jobs on the machine so that the makespan and the total completion time are minimized. A schedule for a semi-continuous batching machine consists of a batching and sequencing for the batches. We propose the optimal properties of two different objective functions and present the different dynamic programming algorithms with a running time of O(n²), respectively.     

Minimizing makespan in a blocking flowshop using a revised artificial immune system algorithm

The blocking flowshop scheduling problem has a strong industrial background but is under-represented in the research literature. In this study, a revised artificial immune system (RAIS) algorithm based on the features of artificial immune systems and the annealing process of simulated annealing algorithms was presented to minimize the makespan in a blocking flowshop. To validate the performance of the proposed RAIS algorithm, computational experiments and comparisons were conducted on the well-known benchmark problems of Taillard used in earlier studies. The experimental results show that the proposed RAIS algorithm outperforms the state-of-art algorithms on the same benchmark problem data set.     

一种差异工件单机批调度问题的蚁群优化算法

由于在利用蚁群算法构建差异工件(即工件有尺寸差异)单机批调度问题的解时,批的加工时间是不确定的.从而不能类似于经典调度问题的蚁群算法把批加工时间的倒数作为蚁群算法中的启发式信息,引入批的利用率和批的负载均衡率作为蚁群算法中的启发式信息,提出了JACO(ant colony optimization based a job sequence)和BACO(ant colony optimization based a batch sequence)两种蚁群优化算法.在算法JACO中,解的编码为工件序列,它对应着用BF(best fit)分批规则生成的调度方案,信息素代表工件间的排列顺序;在算法BACO中,解的编码为批序列,信息素代表工件间的批相关性,由此信息素通过中间信息素量来构造相应的解,并引入特定的局部优化策略,提高了算法的搜索效率.实验表明,与以往文献中的SA(simula-ted annealing)、GA(genetic algorithm)算法以及FFLPT(first-fit longest processing time)、BFLPT (best-fit longest processing time)启发式规则相比,算法JACO和BACO明显优于它们,且BACO算法比JACO算法效果更好.     

Minimizing makespan in an ordered flow shop with?machine-dependent processing times

We consider a generalization of the proportionate flow shop problem with the makespan objective. Each job has a processing requirement and each machine has a characteristic value. In our case, we assume that the time a job occupies a machine is equal to the processing requirement of the job plus a setup time that is equal to the characteristic value of that machine. In this paper, we consider permutation schedules and show that the problem is solvable in polynomial time when the number of machines is fixed.     

A HEURISTIC ALGORITHM FOR THE FLOWSHOP PROBLEM WITH A COMMON JOB SEQUENCE ON ALL MACHINES*

This paper presents a scheduling algorithm to solve flowshop problems with a common job sequence on all machines. This algorithm uses makespan as its criterion. Initially, it chooses a preferred sequence by scanning the processing times matrix and making a few calculations. The makespan time of the preferred job-sequence is further reduced by using an improvement routine that allows interchanges between adjacent jobs. Solutions of 1200 problems are compared with the best solutions previously reported for corresponding size problems in the Campbell-Dudek-Smith (C-D-S) paper [1]. This algorithm offers up to 1% average improvement in reducing the makespan of nearly 50% of the problem sets over the results of the existing algorithms, and its computational time requirements are about one-fifth of that of the C-D-S algorithm.     

A Multi-Exchange Neighborhood for Minimum Makespan Parallel Machine Scheduling Problems

We propose new local search algorithms for minimum makespan parallel machine scheduling problems, which perform multiple exchanges of jobs among machines. Inspired by the work of Thompson and Orlin (1989) on cyclic transfer neighborhood structures, we model multiple exchanges of jobs as special disjoint cycles and paths in a suitably defined improvement graph, by extending definitions and properties introduced in the context of vehicle routing problems (Thompson and Psaraftis, 1993) and of the capacitated minimum spanning tree problem (Ahuja et al., 2001). Several algorithms for searching the neighborhood are suggested.We report the results of a wide computational experimentation, on different families of benchmark instances, performed for the case of identical machines. This problem has been selected as a case study to perform a comparison among the alternative algorithms, and to discover families of instances for which the proposed neighborhood may be promising in practice. Based on the results of the experiments, we can suggest which among the many possible variants of the proposed approaches may be more promising for developing local search algorithms based on multi-exchange moves for related problems. Also, on some families of instances, which are very hard to solve exactly, the most promising multi-exchange algorithms were observed to dominate, in solution quality and in computational time, competitive benchmark heuristics.     

Genetic algorithms with path relinking for the minimum tardiness permutation flowshop problem

In this work three genetic algorithms are presented for the permutation flowshop scheduling problem with total tardiness minimisation criterion. The algorithms include advanced techniques like path relinking, local search and a procedure to control the diversity of the population. We also include a speed up procedure in order to reduce the computational effort needed for the local search technique, which results in large CPU time savings. A complete calibration of the different parameters and operators of the proposed algorithms by means of a design of experiments approach is also given. We carry out a comparative evaluation with the best methods that can be found in the literature for the total tardiness objective, and with adaptations of other state-of-the-art methods originally proposed for other objectives, mainly makespan. All the methods have been implemented with and without the speed up procedure in order to test its effect. The results show that the proposed algorithms are very effective, outperforming the remaining methods of the comparison by a considerable margin.