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.     

混合离散差分进化算法在单机批处理调度中的应用

本文研究单机批处理调度问题,批处理机有批次容量限制,批处理时间由每个批次所含作业中的最长作业处理时间决定。每个作业具有不同的大小、处理时间、提前拖期惩罚权重,所有作业具有公共交货期,且交货期无限晚。目标函数为最小化所有作业的加权提前拖期惩罚之和。该问题已被证明为NP难题,本研究找到了其最优解具有的一些性质,在此基础上利用它们提出了一种动态规划(DP)与差分进化(DE)算法相结合的混合离散差分进化(HDDE)算法来求解该问题,通过与传统的遗传算法、模拟退火算法和迭代贪婪算法进行对比,HDDE算法显示了更加强大的全局搜索能力。     

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.     

Scheduling two job families on a single machine 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 single machine. A setup time is needed for a job if it is the first job to be processed on the machine or its processing on the machine follows a job that belongs to another family. Each agent wants to minimize a certain cost function, which depends on the completion times of its jobs only. The aim is to find a schedule for all the jobs of the two agents that minimizes the objective of one agent while keeping the objective of the other agent being bounded by a fixed value \(Q\). Polynomial-time and pseudo-polynomial-time algorithms are designed to solve the problem involving various combinations of regular scheduling objective functions.     

On-Line Scheduling Algorithms for a Batch Machine with Finite Capacity

We study the problem of on-line scheduling jobs with release dates on a batch machine of finite capacity with the objective of minimizing the makespan. We generalize several existing algorithms for the problem to a class of on-line algorithms that are 2-competitive for any arbitrary finite machine capacity. Then, we show that one of these generalized algorithms is in fact 7/4-competitive for machine capacity 2. This is the first on-line algorithm for a finite machine capacity with competitive ratio less than 2.This research is substantially supported by a grant from City Univ. of Hong Kong (Grant No. 7001119). The second author is supported by this grant and by the Natural Science Foundation of China.     

Single machine batch scheduling with release times

Motivated by a high-throughput logging system, we investigate the single machine scheduling problem with batching, where jobs have release times and processing times, and batches require a setup time. Our objective is to minimize the total flow time, in the online setting. For the online problem where all jobs have identical processing times, we propose a 2-competitive algorithm and we prove a corresponding lower bound. Moreover, we show that if jobs with arbitrary processing times can be processed in any order, any online algorithm has a linear competitive ratio in the worst case. A preliminary version of a part of this paper was presented at the 31st International Symposium on Mathematical Foundations of Computer Science (MFCS 2006). We gratefully acknowledge reviewers’ comments that helped to improve the presentation of this work. Supported by the Swiss SBF under contract no. C05.0047 within COST-295 (DYNAMO) of the European Union. Research carried out while B. Weber was affiliated with the Institute of Theoretical Computer Science, ETH Zurich.     

Optimal on-line algorithms for one batch machine with grouped processing times

In this paper, we study on-line scheduling problems on a batch machine with the assumption that all jobs have their processing times in [p, (1+φ)p], where p>0 and \(\phi=(\sqrt{5}-1)/2\). Jobs arrive over time. First, we deal with the on-line problem on a bounded batch machine with the objective to minimize makespan. A class of algorithms with competitive ratio \((\sqrt{5}+1)/2\) are given. Then we consider the scheduling on an unbounded batch machine to minimize the time by which all jobs have been delivered, and provide a class of on-line algorithms with competitive ratio \((\sqrt{5}+1)/2\). The two class of algorithms are optimal for the problems studied here.     

Scheduling in an assembly-type production chain with batch transfer

This paper addresses a three-machine assembly-type flowshop scheduling problem, which frequently arises from manufacturing process management as well as from supply chain management. Machines one and two are arranged in parallel for producing component parts individually, and machine three is an assembly line arranged as the second stage of a flowshop for processing the component parts in batches. Whenever a batch is formed on the second-stage machine, a constant setup time is required. The objective is to minimize the makespan. In this study we establish the strong NP-hardness of the problem for the case where all the jobs have the same processing time on the second-stage machine. We then explore a useful property, based upon which a special case can be optimally solved in polynomial time. We also study several heuristic algorithms to generate quality approximate solutions for the general problem. Computational experiments are conducted to evaluate the effectiveness of the algorithms.     

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.     

MINIMIZING TOTAL WEIGHTED COMPLETION TIME ON A SINGLE BATCH PROCESSING MACHINE

We study the problem of scheduling jobs on a single batch processing machine to minimize the total weighted completion time. A batch processing machine is one that can process a number of jobs simultaneously as a batch. The processing time of a batch is given by the processing time of the longest job in the batch. We present a branch and bound algorithm to obtain optimal solutions and develop lower bounds and dominance conditions. We also develop a number of heuristics and evaluate their performance through extensive computational experiments. Results show that two of the heuristics consistently generate high-quality solutions in modest CPU times.     

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。     

New results on two-machine flow-shop scheduling with rejection

In this paper, we consider the off-line and on-line two-machine flow-shop scheduling problems with rejection. The objective is to minimize the sum of the makespan of accepted jobs and the total rejection penalty of rejected jobs. For the off-line version, Shabtay and Gasper (Comput Oper Res 39:1087–1096, 2012) showed that this problem is NP-hard and then provided a pseudo-polynomial-time algorithm, two 2-approximation algorithms and a fully polynomial-time approximation scheme. We further study some special cases in this paper. We show that this problem is still NP-hard even when all jobs have the same processing time on one of the machines or all jobs have the same rejection penalty. Furthermore, we also showed that this problem can be solved in polynomialtime algorithm when all jobs satisfy the agreeable condition on their processing times and rejection penalties. For the on-line version without rejection, Chen and Woeginger [in: Du DZ, Pardalos PM (eds.) Minimax and Applications, 1995] showed that the competitive ratio of any determined on-line algorithm is at least 2. We further show that the competitive ratio of any determined on-line algorithm is at least 2 even when all jobs have the same processing time on the first machine. Finally, for the on-line version with rejection, we present a class of on-line algorithms with the best-possible competitive ratio 2.     

Maximum latency scheduling problem on two-person cooperative games

This paper studies a two-person cooperative game in which a set of jobs has to be processed jointly by two people. Each of them has a single machine and his processing cost is defined as the minimum value of the maximum latency of his negotiably assigned jobs. The objective is to maximize the multiplication of their rational positive cooperative profits. In the case where all jobs have the same processing time, if they have a common due date, the problem is polynomial-time solvable; if due dates can be different, there exits an optimal schedule in which the jobs assigned to each person are scheduled in Earlier Due Date first (EDD) order and a polynomial-time dynamic programming is further proposed. In the case where processing times can be different, the NP-completeness of this problem is proved, and a pseudo-polynomial-time dynamic programming algorithm is developed.     

Single machine scheduling with general positional deterioration and rate-modifying maintenance

We present polynomial-time algorithms for single machine problems with generalized positional deterioration effects and machine maintenance. The decisions should be taken regarding possible sequences of jobs and on the number of maintenance activities to be included into a schedule in order to minimize the overall makespan. We deal with general non-decreasing functions to represent deterioration rates of job processing times. Another novel extension of existing models is our assumption that a maintenance activity does not necessarily fully restore the machine to its original perfect state. In the resulting schedules, the jobs are split into groups, a particular group to be sequenced after a particular maintenance period, and the actual processing time of a job is affected by the group that job is placed into and its position within the group.     

A Model for Resource Constrained Production and Inventory Management

We present a general model for multi-item production and inventory management problems that include a resource restriction. The decision variables in the model can take on a variety of interpretations, but will typically represent cycle times, production batch sizes, number of production runs, or order quantities for each item. We consider environments where item demand rates are approximately constant and performing an activity such as producing a batch of a product or placing an order results in the consumption of a scarceresource that is shared among the items. Some examples of shared resources include limited machine capacity, a restriction on the amount of money that can be tied up in stock, orlimited storage capacity. We focus on the case where the decision variables must be integer valued or selected from a discrete set of choices, such as when an integer number of production runs is desired for each item, or in order quantity problems where the items come in pack sizes containing more than one unit and, therefore, the order quantities must be an integer multiple of the pack sizes. We develop a heuristic and a branch and bound algorithm for solving the problem. The branch and bound algorithm includes reoptimization procedures and the heuristic to improve its performance. Computational testing indicates that the algorithms are effective for solving the general model.     

一类新型批处理机调度问题的理论分析

钢卷在冷轧生产中,为了改进其性能,需要在罩式炉进行退火,退火过程由加热、保温和降温三段组成,而这三段处理时间由于工艺上的要求不能归结为一个时间,这与传统批处理机调度有明显的差别.对新型批处理机的总加权完成时间最小化问题建立了非线性整数规划模型,开发了基于动态规划的启发式算法.通过理论分析,获得该算法的误差性能比为3.对于三段中的某一段板卷的处理时间相同的情况,证明了启发式算法的误差性能比是2,而且证明是紧界.对于三段中的某二段板卷的处理时间相同的情况,证明了启发式算法是最优算法.对启发式算法扩展到带有任意段的加工时间的一般情况进行了性能分析.     

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.

     

Minimizing the Number of Late Jobs under the Group Technology Assumption

We consider the one-machine scheduling problem to minimize the number of late jobs under the group technology assumption, where jobs are classified into groups and all jobs from the same group must be processed contiguously. This problem is shown to be strongly NP-hard, even for the case of unit processing time and zero set-up time. A polynomial time algorithm is developed for the restricted version in which the jobs in each group have the same due date. However, the problem is proved to be ordinarily NP-hard if the jobs in a group have the same processing time as well as the same due date.     

Exact algorithms for scheduling programs with shared tasks

We study a scheduling problem where the jobs we have to perform are composed of one or more tasks. If two jobs sharing a non-empty subset of tasks are scheduled on the same machine, then these shared tasks have to be performed only once. This kind of problem is known in the literature under the names of VM-PACKING or PAGINATION. Our objective is to schedule a set of these objects on two parallel identical machines, with the aim of minimizing the makespan. This problem is NP-complete as an extension of the PARTITION problem. In this paper we present three exact algorithms with worst-case time-complexity guarantees, by exploring different branching techniques. Our first algorithm focuses on the relation between jobs sharing one or more symbols in common, whereas the two other algorithms branches on the shared symbols.

     

On-line machine scheduling with batch setups

We study a class of scheduling problems with batch setups for the online-list and online-time paradigms. Jobs are to be scheduled in batches for processing. All jobs in a batch start and complete together, and a constant setup is prior to each batch. The objective is to minimize the total completion time of all jobs. We primarily consider the special cases of these problems with identical processing times, for which efficient on-line heuristics are proposed and their competitive performance is evaluated.