Parallel machine scheduling with flexible resources

Parallel machine flexible resource scheduling (PMFRS) problems consider an additional flexible resource (e.g. operators), which can be freely allocated to any jobs and/or any machines and may speed-up the process in proportion to its amount. If job–machine assignment is unspecified, the problem is referred to as unspecified PMFRS (UPMFRS). This paper reviews the mathematical models of both PMFRS and UPMFRS problems in the literature and not only gives some extensions to the model of dynamic PMFRS problem but also presents integer programming (IP) models for static and dynamic UPMFRS problems with the objective of minimizing makespan. To solve large-sized dynamic PMFRS and UPMFRS problems, a relaxed IP based constraint programming (CP) approach is also proposed. All IP models and the proposed IP/CP approach are tested with an extensive computational study. The results of the computational experiments are discussed with respect to the major parameters of the problem and conclusions are drawn.     

Parallel machine match-up scheduling with manufacturing cost considerations

Many scheduling problems in practice involve rescheduling of disrupted schedules. In this study, we show that in contrast to fixed processing times, if we have the flexibility to control the processing times of the jobs, we can generate alternative reactive schedules considering the manufacturing cost implications in response to disruptions. We consider a non-identical parallel machining environment where processing times of the jobs are compressible at a certain manufacturing cost, which is a convex function of the compression on the processing time. In rescheduling it is highly desirable to catch up the original schedule as soon as possible by reassigning the jobs to the machines and compressing their processing times. On the other hand, one must also keep the manufacturing cost due to compression of the jobs low. Thus, one is faced with a tradeoff between match-up time and manufacturing cost criteria. We introduce alternative match-up scheduling problems for finding schedules on the efficient frontier of this time/cost tradeoff. We employ the recent advances in conic mixed-integer programming to model these problems effectively. We further provide a fast heuristic algorithm driven by dual prices of convex subproblems for generating approximate efficient schedules.     

具有树和路约束的平行机排序问题

考虑具有树和路约束的平行机排序问题,其工件集对应于无向图(有向图)的边(弧)集。目标是选取工件集的一个子集使其满足树或路的约束,将其放在平行机上处理,使得机器的最大完工时间(makespan)尽可能地小。通过分析此类问题的组合性质,得到如下结论：在K-树约束下,利用最小支撑K-树的性质可得一个有效多项式时间近似方案;在两固定点间路的约束下,通过构造辅助实例以控制边的权重,分析辅助实例的输出值与目标实例最优值之间的关系,利用最短路的性质可以得到一个2-近似算法;在单源点最短路径树的约束下,根据最短路径树的性质可以得到一个有效多项式时间近似方案;在两固定点间最短路的约束下,在所有的两点间最短路构成的子图基础上,通过构造新的辅助图以控制弧的权重,再利用最短路的性质可以得到一个1.618-近似算法。     

Parallel machine scheduling with precedence constraints and setup times

This paper presents different methods for solving parallel machine scheduling problems with precedence constraints and setup times between the jobs. These problems are strongly NP-hard and it is even conjectured that no list scheduling algorithm can be defined without explicitly considering jointly scheduling and resource allocation. We propose dominance conditions based on the analysis of the problem structure and an extension to setup times of the energetic reasoning constraint propagation algorithm. An exact branch-and-bound procedure and a climbing discrepancy search (CDS) heuristic based on these components are defined. We show how the proposed dominance rules can still be valid in the CDS scheme. The proposed methods are evaluated on a set of randomly generated instances and compared with previous results from the literature and those obtained with an efficient commercial solver. We conclude that our propositions are quite competitive and our results even outperform other approaches in most cases.     

Parallel machine earliness and tardiness scheduling with proportional weights

In this paper we study the problem of scheduling n jobs with a common due date and proportional early and tardy penalties on m identical parallel machines. We show that the problem is NP-hard and propose a dynamic programming algorithm to solve it. We also propose two heuristics to tackle the problem and analyze their worst-case error bounds.Scope and purposeScheduling problems to minimize the total weighted earliness and tardiness (WET) arise in Just-in-time manufacturing systems, where one of the objectives is to complete each job as close to its due date as possible. The earliness and tardiness weights of a job in WET tend to increase with the value of the job. Because processing time is often a good surrogate for the value of a job, it is reasonable to consider weights that are proportional to job processing times. In this paper we study the parallel identical machine WET problem with proportional weights. We propose both exact and approximation algorithms to tackle the problem.     

基于MapReduce模型带任务分割的平行机调度优化

研究一类基于MapReduce模型的两阶段平行机调度问题.该模型中的每个工件包含Map和Reduce两道工序,前一工序的任务可以划分并同步加工,而后一工序不可划分,结合工件的到达时间、交货时间等约束,以最大完工时间和总延迟时间的加权和作为优化目标构建混合整数规划模型,设计采用差分变异策略和逐维角度扰动机制的改进鲸鱼优化算法求解模型.数值仿真实验结果表明,所设计的算法相对于经典的鲸鱼优化算法、粒子群算法的求解效果有显著的提升,验证了模型和所设计算法的有效性.     

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.     

Parallel machine scheduling problems using memetic algorithms

In this paper, we investigate how to apply the hybrid genetic algorithms (the memetic algorithms) to solve the parallel machine scheduling problem. There are two essential issues to be dealt with for all kinds of parallel machine scheduling problems: job partition among machines and job sequence within each machine. The basic idea of the proposed method is that (a) use the genetic algorithms to evolve the job partition and then (b) apply a local optimizer to adjust the job permutation to push each chromosome climb to his local optima. Preliminary computational experiments demonstrate that the hybrid genetic algorithm outperforms the genetic algorithms and the conventional heuristics.     

Parallel machine scheduling problem with preemptive jobs and transportation delay

In this research the parallel machine scheduling problem with preemption by considering a constant transportation delay for migrated jobs and minimization of makespan as the criterion i.e., P_m|pmtn(delay)|C_max is investigated. It is assumed that when a preempted job resumes on another machine, it is required a delay between the processing time of the job on these two machines. This delay is called transportation delay. We criticize an existing mathematical model for the research problem in the literature [Boudhar M, Haned A. Preemptive scheduling in the presence of transportation times. Computers & Operations Research 2009; 36(8):2387–93]. Then, we prove that there exists an optimal schedule with at most (m−1) preemptions with transportation among machines for the problem. We also propose a linear programming formulation and an exact algorithm for the research problem in case of equal transportation delay. The experiments show that the proposed exact algorithm performs better than the proposed mathematical model.     

Single machine scheduling with job delivery to multiple customers

We investigate a single machine scheduling problem with job delivery to multiple customers. In this problem, each job needs to be processed on the single machine, and then delivered by a single vehicle to its customer, where the job has a physical size representing the fraction of space it occupies on the vehicle. The vehicle delivers a shipment from the machine to a customer and has to return back to the machine for delivering the next shipment. It takes different constant time for the round trips between the machine and the different customers. The goal is to minimize the makespan, by that time all the jobs are processed and delivered to their respective customers, and the vehicle returns back to the machine. We propose a 2-approximation algorithm for the general case; when there are only two customers, we present an improved 5/3-approximation algorithm. The design and performance analysis of these two algorithms integrate several known results and techniques for the single machine scheduling problem, the bin-packing problem, and the knapsack problem.     

Parallel machine scheduling problems in green manufacturing industry

Manufacturing companies are now more conscious about the environment. As such, there are more concerns in reducing the consumption of energy and the production of pollutants. Reduced consumption of energy will save cost, while reduction of pollutants will decrease the cost of cleaning up the environment. This paper considers scheduling problems that arise in green manufacturing companies. Suppose the manufacturing company has a set of parallel machines. Each machine has a cost per unit time that differs from machine to machine. The cost here is the sum of the energy cost and the clean up cost. A set of jobs is to be processed by these machines. Our goal is to find a schedule that minimizes the makespan (schedule length) or the total completion time, subject to the constraint that the total cost is not more than a given threshold value. We propose efficient heuristics and show, by computational experiments, that they perform very well in practice.     

Parallel machine total tardiness scheduling with a new hybrid metaheuristic approach

This work proposes a hybrid metaheuristic (HMH) approach which integrates several features from tabu search (TS), simulated annealing (SA) and variable neighbourhood search (VNS) in a new configurable scheduling algorithm. In particular, either a deterministic or a random candidate list strategy can be used to generate the neighbourhood of a solution, both a tabu list mechanism and the SA probabilistic rule can be adopted to accept solutions, and the dimension of the explored neighbourhood can be dynamically modified. The considered class of scheduling problems is characterized by a set of independent jobs to be executed on a set of parallel machines with non-zero ready times and sequence dependent setups. In particular, the NP-hard generalized parallel machine total tardiness problem (GPMTP) recently defined by Bilge et al. [A tabu search algorithm for parallel machine total tardiness problem. Computers & Operations Research 2004;31:397–414], is faced. Several alternative configurations of the HMH have been tested on the same benchmark set used by Bilge et al. The results obtained highlight the appropriateness of the proposed approach.     

Energy-efficient scheduling in multi-core servers

In this paper we develop techniques for analyzing and optimizing energy management in multi-core servers with speed scaling capabilities. Our framework incorporates the processor’s dynamic power, but it also accounts for other intricate and relevant power features such as the static (leakage) power and switching overhead between speed levels. Using stochastic fluid models to capture traffic burst dynamics, we propose and study different strategies for adapting the multi-core processor speeds based on the observable buffer content, so as to optimize objective functions that balance energy consumption and performance. The strategies can be non-hysteretic (i.e., the processor speed depends on current buffer level relative to the buffer thresholds) or hysteretic (i.e., it matters in which direction the buffer thresholds are crossed). It is shown that, under rather general conditions, strategies which use more threshold levels are more efficient with respect to power consumption; however, most of the efficiency gain is achieved with 1 or 2 thresholds only. In addition, the optimal power consumptions of the different strategies are only very mildly sensitive to perturbations in the input parameters, implying the highly advantageous property that the performance is robust to estimation errors in the system’s input traffic parameters.     

Parallel machine scheduling with splitting jobs by a hybrid differential evolution algorithm

The problem of parallel machine scheduling for minimizing the makespan is an open scheduling problem with extensive practical relevance. It has been proved to be non-deterministic polynomial hard. Considering a job’s batch size greater than one in the real manufacturing environment, this paper investigates into the parallel machine scheduling with splitting jobs. Differential evolution is employed as a solution approach due to its distinctive feature, and a new crossover method and a new mutation method are brought forward in the global search procedure, according to the job splitting constraint. A specific local search method is further designed to gain a better performance, based on the analytical result from the single product problem. Numerical experiments on the performance of the proposed hybrid DE on parallel machine scheduling problems with splitting jobs covering identical and unrelated machine kinds and a realistic problem are performed, and the results indicate that the algorithm is feasible and efficient.     

Parallel machine scheduling to minimize the makespan with sequence dependent deteriorating effects

A new unrelated parallel machine scheduling problem with deteriorating effect and the objective of makespan minimization is presented in this paper. The deterioration of each machine (and therefore of the job processing times) is a function of the sequence of jobs that have been processed by the machine and not (as considered in the literature) by the time at which each job is assigned to the machine or by the number of jobs already processed by the machine. It is showed that for a single machine the problem can be solved in polynomial time, whereas the problem is NP-hard when the number of machines is greater or equal than two. For the last case, a set of list scheduling algorithms and simulated annealing meta-heuristics are designed and the effectiveness of these approaches is evaluated by solving a large number of benchmark instances.     

Parallel machine selection and job scheduling to minimize machine cost and job tardiness

This paper discusses the problem of simultaneously selecting and scheduling parallel machines to minimize the sum of machine holding cost and job tardiness cost. A combinatorial optimization model is developed for this purpose. Solving the developed model is NP-hard. A heuristic algorithm is developed to locate the optimal or near optimal solutions based on a Tabu search mechanism specially designed to control the search process in the solution neighborhood for jobs scheduled on specific machines. Numerical examples show that the solutions of the model lead to compromises between the system cost related to machine selection and the operational cost related to job tardiness penalties. The examples also show that the developed algorithm is effective and computationally efficient.     

Multi-stage resource-aware scheduling for data centers with heterogeneous servers

This paper presents a three-stage algorithm for resource-aware scheduling of computational jobs in a large-scale heterogeneous data center. The algorithm aims to allocate job classes to machine configurations to attain an efficient mapping between job resource request profiles and machine resource capacity profiles. The first stage uses a queueing model that treats the system in an aggregated manner with pooled machines and jobs represented as a fluid flow. The latter two stages use combinatorial optimization techniques to solve a shorter-term, more accurate representation of the problem using the first-stage, long-term solution for heuristic guidance. In the second stage, jobs and machines are discretized. A linear programming model is used to obtain a solution to the discrete problem that maximizes the system capacity given a restriction on the job class and machine configuration pairings based on the solution of the first stage. The final stage is a scheduling policy that uses the solution from the second stage to guide the dispatching of arriving jobs to machines. We present experimental results of our algorithm on both Google workload trace data and generated data and show that it outperforms existing schedulers. These results illustrate the importance of considering heterogeneity of both job and machine configuration profiles in making effective scheduling decisions.     

解不同交货期并行机调度问题的并行遗传算法

为有效地解决不同交货期窗口下的非等同并行多机提前/拖后调度问题,设计了一种分段编码的混合遗传算法。此编码方式能反映工件的分配序列,并利用调度优先级规则和最好适应值规则相结合的启发式算法对其顺序进行了调整,加快了收敛速度。同时为了更好地适应调度实时性和解大规模此类问题的需要,基于遗传算法自然并行性特点的基础上,实现了主从式控制网络模式下并行混合遗传算法。计算结果表明,此算法是有效的,优于遗传算法,有着较高的并行性,并能适用于大规模不同交货期窗口下非等同并行多机提前/拖后调度问题。     

Aperiodic servers in a deadline scheduling environment

A real-time system may have tasks with soft deadlines, as well as hard deadlines. While earliest-deadline-first scheduling is effective for hard-deadline tasks, applying it to soft-deadline tasks may waste schedulable processor capacity or sacrifice average response time. Better average response time may be obtained, while still guaranteeing hard deadlines, with an aperiodic server. Three scheduling algorithms for aperiodic servers are described, and schedulability tests are derived for them. A simulation provides performance data for these three algorithms on random aperiodic tasks. The performances of the deadline aperiodic servers are compared with those of several alternatives, including background service, a deadline polling server, and rate-monotonic servers, and with estimates based on the M/M/1 queueing model. This adds to the evidence in support of deadline scheduling,versus fixed priority scheduling.