Minimizing maximum tardiness and delivery costs in a batched delivery system

High delivery costs usually urge manufacturers to dispatch their jobs in batches. However, dispatching the jobs in batches can have profound negative effects on important scheduling objective functions such as minimizing maximum tardiness. This paper considers a single machine scheduling problem with the aim of minimizing the maximum tardiness and delivery costs in a single-machine scheduling problem with batched delivery system. A mathematical model is developed for this problem which can serve to solve it with the help of a commercial solver. However, due to the fact that this model happens to be a mixed integer nonlinear programming model the solver cannot guarantee to reach the global solution. For this reason, a branch and bound algorithm (B&B) is presented to obtain the global solution. Besides, a heuristic algorithm for calculation of the initial upper bound is introduced. Computational results show that the algorithm can be beneficial for solving this problem, especially for large size instances.     

A branch-and-bound procedure for a single-machine earliness scheduling problem with two agents

This paper addresses a two-agent scheduling problem on a single machine where the objective is to minimize the total weighted earliness cost of all jobs, while keeping the earliness cost of one agent below or at a fixed level Q. A mixed-integer programming (MIP) model is first formulated to find the optimal solution which is useful for small-size problem instances. To solve medium- to large-size problem instances, a branch-and-bound algorithm incorporating with several dominance properties and a lower bound is then provided to derive the optimal solution. A simulated annealing heuristic algorithm incorporating with a heuristic procedure is developed to derive the near-optimal solutions for the problem. A computational experiment is also conducted to evaluate the performance of the proposed algorithms.     

A variable neighborhood search approach for planning and scheduling of jobs on unrelated parallel machines

In this paper, we study a planning and scheduling problem for unrelated parallel machines. There are n jobs that have to be assigned and sequenced on m unrelated parallel machines. Each job has a weight that represents the priority of the corresponding customer order, a given due date, and a release date. An Automated Guided Vehicle is used to transport at maximum Load _max jobs into a storage space in front of the machines in a given period of time. We consider t _max consecutive periods. We are interested in minimizing the total weighted tardiness of the jobs across the periods. This measure is important when we are interested in a good on-time delivery performance. We present an appropriate mixed integer program. To solve this NP-hard problem, we develop a heuristic methodology based on decomposition and variable neighborhood search (VNS). The proposed approaches are assessed using randomly generated problem instances. We compare them with a simple heuristic based on decomposition and list scheduling using the Apparent Tardiness Cost dispatching rule. The results demonstrate that the heuristic approach based on VNS performs comparably to the mixed integer program while having reasonable solution times and outperforms the simple heuristic and a genetic algorithm (GA) from previous research.     

Minimizing the number of tardy jobs in a single-machine scheduling problem with periodic maintenance

This research focuses on the problem of scheduling jobs on a single machine that requires periodic maintenance with the objective of minimizing the number of tardy jobs. We present a two-phase heuristic algorithm in which an initial solution is obtained first with a method modified from Moore's algorithm for the problem without maintenance and then the solution is improved in the second phase. Performance of the proposed heuristic algorithm is evaluated through computational experiments on randomly generated problem instances and results show that the heuristic gives solutions close to those obtained from a commercial integer programming solver in much shorter time and works better than an existing heuristic algorithm in terms of the solution quality.     

Scheduling on parallel machines with preemption and transportation delays

This paper deals with an identical parallel machines scheduling problem, where independent jobs can be preempted and transported from one machine to another. The transportation of a preempted job requires a time called the transportation delay. The goal is to find a solution that minimizes the total completion time (makespan). We first study the case of equal-size jobs where new complexity results are given. Then, to solve the problem with two identical machines, we present a dynamic programming algorithm and a fully polynomial time approximation scheme (FPTAS). Experimental results show the efficiency of the FPTAS compared to a previously published heuristic.     

Heuristic algorithms for two-machine flowshop with availability constraints

The majority of the scheduling studies carry a common assumption that machines are available all the time. However, machines may not always be available in the scheduling period due to breakdown or preventive maintenance. Taking preventive maintenance activity into consideration, we dealt with the two-machine flowshop scheduling problem with makespan objective. The preventive maintenance policy in this paper was dependent on the number of finished jobs. The integer programming model was proposed. We combined two recent constructive heuristics, HI algorithm and H algorithm, with Johnson’s algorithm, and named the combined heuristic H&J algorithm. We also developed a constructive heuristic, HD, with time complexities O(n²). Based on the difference in job processing times on two machines, both H&J and HD showed good performance, and the latter was slightly better. The HD algorithm was able to obtain the optimality in 98.88% of cases. We also employed the branch and bound (B&B) algorithm to obtain the optimum. With a good upper bound and a modified lower bound, the proposed B&B algorithm performed significantly effectively.     

基于柔性维护的堕化单机系统调度研究

针对单机系统,在假设生产系统为堕化系统,且生产过程中作业的加工不可中断的情况下,对考虑柔性时间窗口[[u,v]]下进行长度为[w]的周期预防性维护的调度问题进行了研究。建立了综合考虑生产调度和设备维护的混合整数规划模型,并设计了一套基于贪婪的启发式算法对所研究问题进行优化求解。通过Cplex和启发式算法求解结果的对比证明了算法可以快速、有效地解决此类问题。     

Minimizing the makespan on a single machine with flexible maintenances and jobs’ release dates

This paper addresses the production scheduling problem on a single machine with flexible periodic preventive maintenances (PM), where jobs’ release dates are also considered. Both resumable and non-resumable cases are studied. For the resumable case, it is proved that the problem can be solved in polynomial time with Earliest Release Date (ERD) rule. For the non-resumable case, it is proved to be NP-Hard in strong sense. And, a mixed integer programming (MIP) mathematical model is provided. Then, an effective heuristic ERD-LPT based on the properties of optimal solution is proposed. Meanwhile, a branch-and-bound algorithm (B and B) that utilizes several dominance rules is developed to search the optimal schedule for small-to-medium sized problems. Computational results indicate that the proposed heuristic is highly accurate and the two algorithms are complementary in dealing with different sized problems. Furthermore, the improvement of the integration between production scheduling and PM is significant compared with the First-in-First-out (FIFO) rule which is adopted commonly in industry.     

A heuristic algorithm to minimize total weighted tardiness on a single machine with release times

This paper attempts to solve a single machine‐scheduling problem, in which the objective function is to minimize the total weighted tardiness with different release dates of jobs. To address this scheduling problem, a heuristic scheduling algorithm is presented. A mathematical programming formulation is also formulated to validate the performance of the heuristic scheduling algorithm proposed herein. Experimental results show that the proposed heuristic algorithm can solve this problem rapidly and accurately. Overall, this algorithm can find the optimal solutions for 2200 out of 2400 randomly generated problems (91.67%). For the most complicated 20 job cases, it requires less than 0.0016 s to obtain an ultimate or even optimal solution. This heuristic scheduling algorithm can therefore efficiently solve this kind of problem.     

An efficient heuristic for a two-stage assembly scheduling problem with batch setup times to minimize makespan

This paper considers a two-stage assembly scheduling problem of N products with setup times to minimize the makespan. In this problem, there is a machining machine which produces components in the first stage. When the required components are available, a single assembly machine can assemble these components into products in the second stage. A setup time is needed whenever the machining machine starts processing components, or the item of component is switched on the machine. The problem is formulated as a mixed integer programming model, and several properties for finding optimal solutions are developed. Moreover, an efficient heuristic based on these optimal properties is proposed. A lower bound is derived to evaluate the performance of the proposed heuristic. Computational results show that the proposed heuristic can obtain a near optimal solution in almost zero time and the average percentage deviation is only 0.478.     

A hybrid memetic algorithm for maximizing the weighted number of just-in-time jobs on unrelated parallel machines

This paper presents a hybrid memetic algorithm for the problem of scheduling n jobs on m unrelated parallel machines with the objective of maximizing the weighted number of jobs that are completed exactly at their due dates. For each job, due date, weight, and the processing times on different machines are given. It has been shown that when the numbers of machines are a part of input, this problem is NP-hard in the strong sense. At first, the problem is formulated as an integer linear programming model. This model is practical to solve small-size problems. Afterward, a hybrid memetic algorithm is implemented which uses two heuristic algorithms as constructive algorithms, making initial population set. A data oriented mutation operator is implemented so as to facilitate memetic algorithm search process. Performance of all algorithms including heuristics (H1 and H2), hybrid genetic algorithm and hybrid memetic algorithm are evaluated through computational experiments which showed the capabilities of the proposed hybrid algorithm.     

Two-stage hybrid flowshop scheduling with simultaneous processing machines

Simultaneous processing machines, common in processing industries such as steel and food production, can process several jobs simultaneously in the first-in, first-out manner. However, they are often highly energy-consuming. In this paper, we study a new two-stage hybrid flowshop scheduling problem, with simultaneous processing machines at the first stage and a single no-idle machine with predetermined job sequence at the second stage. A mixed integer programming model is proposed with the objective of minimizing the total processing time to reduce energy consumption and improve production efficiency. We give a sufficient and necessary condition to construct feasible sequencing solutions and present an effective approach to calculate the time variables for a feasible sequencing solution. Based on these results, we design a list scheduling heuristic algorithm and its improvement. Both heuristics can find an optimal solution under certain conditions with complexity O(nlogn), where n is the number of jobs. Our experiments verify the efficiency of these heuristics compared with classical heuristics in the literature and investigate the impacts of problem size and processing times.     

Scheduling independent jobs on uniform parallel machines to minimize tardiness criteria

The problem of scheduling N jobs on M uniform parallel machines is studied. The objective is to minimize the mean tardiness or the weighted sum of tardiness with weights based on jobs, on periods or both. For the mean tardiness criteria in the preemptive case, this problem is NP-hard but good solutions can be calculated with a transportation problem algorithm. In the nonpreemptive case the problem is therefore NP-hard, except for the cases with equal job processing times or with job due dates equal to job processing times. No dominant heuristic is known in the general nonpreemptive case. The author has developed a heuristic to solve the nonpreemptive scheduling problem with unrelated job processing times. Initially, the algorithm calculates a basic solution. Next, it considers the interchanges of job subsets to equal processing time sum interchanging resources (i.e. a machine for a given period). This paper models the scheduling problem. It presents the heuristic and its result quality, solving 576 problems for 18 problem sizes. An application of school timetable scheduling illustrates the use of this heuristic.     

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.     

An integer programming approach to the multimode resource-constrained multiproject scheduling problem

The project scheduling problem (PSP) is the subject of several studies in computer science, mathematics, and operations research because of the hardness of solving it and its practical importance. This work tackles an extended version of the problem known as the multimode resource-constrained multiproject scheduling problem. A solution to this problem consists of a schedule of jobs from various projects, so that the job allocations do not exceed the stipulated limits of renewable and nonrenewable resources. To accomplish this, a set of execution modes for the jobs must be chosen, as the jobs’ duration and amount of needed resources vary depending on the mode selected. Finally, the schedule must also consider precedence constraints between jobs. This work proposes heuristic methods based on integer programming to solve the PSP considered in the Multidisciplinary International Scheduling Conference: Theory and Applications (MISTA) 2013 Challenge. The developed solver was ranked third in the competition, being able to find feasible and competitive solutions for all instances and improving best known solutions for some problems.     

树型网格计算环境下的独立任务调度

任务调度是实现高性能网格计算的一个基本问题,然而,设计和实现高效的调度算法是非常具有挑战性的.讨论了在网格资源计算能力和网络通信速度异构的树型计算网格环境下,独立任务的调度问题.与实现最小化任务总的执行时间不同(该问题已被证明是NP难题),为该任务调度问题建立了整数线性规划模型,并从该线性规划模型中得到最优任务分配方案??各计算节点最优任务分配数.然后,基于最优任务分配方案,构造了两种动态的需求驱动的任务分配启发式算法:OPCHATA(optimization-based priority-computation heuristic algorithm for task allocation)和OPBHATA(optimization-basedpriority-bandwidth heuristic algorithm for task allocation).实验结果表明:在异构的树型计算网格环境下实现大量独立任务调度时,该算法的性能明显优于其他算法.     

一种基于QoS的自适应网格失效检测器

失效检测器是构建可靠的网格计算环境所必需的基础组件之一.由于网格中存在大量对失效检测有着不同QoS需求的分布式应用,对于一个网格失效检测器来说,为保持其有效性和可扩展性,应该既能够准确提供应用程序所需的失效检测QoS,又能够避免为满足不同QoS而设计多套失效检测器所产生的多余负载.基于QoS基本评价指标,采用PULL模式主动检测策略实现了一种新的失效检测器--GA-FD(adaptive failure detector for grid),可以同时支持多个应用程序定量描述的QoS需求,不需要关于消息行为和时钟同步的任何假设.同时,证明了GA-FD在部分同步模型下可实现一个◇P类的失效检测器,并给出了相应的实验及数据.     

A network-based approach to the multi-activity combined timetabling and crew scheduling problem: Workforce scheduling for public health policy implementation

This paper introduces the multi-activity combined timetabling and crew scheduling problem. The goal of this problem is to schedule the minimum number of workers required in order to successfully visit a set of customers characterized by services needed matched against schedule availability. Two solution strategies are proposed. The first is based on mathematical programming whilst the second uses a heuristic procedure in order to reduce computational time. The proposed model combines timetabling with crew scheduling decisions in one mixed integer programming model which considers multiple activities. The algorithms are tested on randomly generated and real instances provided by the Health to School Initiative, a program based at Bogotá’s local Health Department. The results show that the Initiative can increase its coverage by up to 68% using the proposed heuristic approach as a planning process tool.     

阶段感知的跨域数据分析作业保障机制

为缓解边缘计算的不稳定性,设计了适用于跨域规模的任务复制与分配机制以保障作业各阶段的完成（Geo-distributed job Insurance Mechanism,GIM）。为满足并行任务执行特征和资源限制,先将问题形式化为Min-Max多项式整数规划问题,再放松整数约束并采用feasible-point算法快速地收敛到最优解,最后基于高危任务优先原则对最优解进行取整,得到满足原问题约束的高质量任务复制与分配方案。模拟实验表明,在不同的系统负载下,相比于当前的冗余执行策略,GIM至少能够减少22%的作业平均完成时间。     

A new Lagrangian relaxation algorithm for hybrid flowshop scheduling to minimize total weighted completion time

We investigate the problem of scheduling n jobs in s-stage hybrid flowshops with parallel identical machines at each stage. The objective is to find a schedule that minimizes the sum of weighted completion times of the jobs. This problem has been proven to be NP-hard. In this paper, an integer programming formulation is constructed for the problem. A new Lagrangian relaxation algorithm is presented in which precedence constraints are relaxed to the objective function by introducing Lagrangian multipliers, unlike the commonly used method of relaxing capacity constraints. In this way the relaxed problem can be decomposed into machine type subproblems, each of which corresponds to a specific stage. A dynamic programming algorithm is designed for solving parallel identical machine subproblems where jobs may have negative weights. The multipliers are then iteratively updated along a subgradient direction. The new algorithm is computationally compared with the commonly used Lagrangian relaxation algorithms which, after capacity constraints are relaxed, decompose the relaxed problem into job level subproblems and solve the subproblems by using the regular and speed-up dynamic programming algorithms, respectively. Numerical results show that the new Lagrangian relaxation method produces better schedules in much shorter computation time, especially for large-scale problems.