Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times

We revisit the classic problem of preemptive scheduling on m uniformly related machines. In this problem, jobs can be arbitrarily split into parts, under the constraint that every job is processed completely, and that the parts of a job are not assigned to run in parallel on different machines. We study a new objective which is motivated by fairness, where the goal is to minimize the sum of the two maximal job completion times. We design a polynomial time algorithm for computing an optimal solution. The algorithm can act on any set of machine speeds and any set of input jobs. The algorithm has several cases, many of which are very different from algorithms for makespan minimization (algorithms that minimize the maximum completion time of any job), and from algorithms that minimize the total completion time of all jobs.     

Weak-restriction bi-objective optimization algorithm for scheduling with rejection on non-identical batch processing machines

We investigate the problem of scheduling a set of jobs with arbitrary sizes and unequal weights on a set of parallel batch machines with non-identical capacities. The objective is to minimize the makespan of the accepted jobs and the total rejection penalty of the rejected jobs, simultaneously. To address the studied problem, a Pareto-based ant colony optimization algorithm with the first job selection probability (FPACO) is proposed. A weak-restriction selection strategy is proposed to obtain the desirability of candidate jobs. Two objective-oriented heuristic information and pheromone matrices are designed, respectively, to record the experience in different search dimensions. Moreover, a local optimization algorithm is incorporated to improve the solution quality. Finally, the proposed algorithm is compared with four existing algorithms through extensive simulation experiments. The experimental results indicate that the proposed algorithm outperforms all of the compared algorithms within a reasonable time.     

可用性约束资源预留与分配算法

给定一系列作业和只能在有限的时间段可用的资源,如何预留和分配资源以实现作业的最大完成时间最小化的问题是NP难的。本文将其归结为一种新型的尺寸可变装箱问题并给出了作业信息和资源信息完全已知条件下的六种离线算法,理论分析表明所给算法的渐进最坏比为2,在作业相互独立的务件下推广的降序最佳适合(Best Fit Decreasing)算法的平均性能最优,在作业有先后依赖关系的条件下推广的最佳适合(Best Fit)算法的平均性能最优。     

Two-agent two-machine flowshop scheduling with learning effects to minimize the total completion time

We study a two-agent scheduling problem in a two-machine permutation flowshop with learning effects. The objective is to minimize the total completion time of the jobs from one agent, given that the maximum tardiness of the jobs from the other agent cannot exceed a bound. We provide a branch-and-bound algorithm for the problem. In addition, we present several genetic algorithms to obtain near-optimal solutions. Computational results indicate that the algorithms perform well in either solving the problem or efficiently generating near-optimal solutions.     

Preemptive online scheduling with rejection of unit jobs on two uniformly related machines

We consider preemptive online and semi-online scheduling of unit jobs on two uniformly related machines. Jobs are presented one by one to an algorithm, and each job has a rejection penalty associated with it. A new job can either be rejected, in which case the algorithm pays its rejection penalty, or it can be scheduled preemptively on the machines, in which case it may increase the maximum completion time of any machine in the schedule, also known as the makespan of the constructed schedule. The objective is to minimize the sum of the makespan of the schedule of all accepted jobs and the total penalty of all rejected jobs. We study two versions of the problem. The first one is the online problem where the jobs arrive unsorted, and the second variant is the semi-online case, where the jobs arrive sorted by a non-increasing order of penalties. We also show that the variant where the jobs arrive sorted by a non-decreasing order of penalties is equivalent to the unsorted one. We design optimal online algorithms for both cases. These algorithms have smaller competitive ratios than the optimal competitive ratio for the more general problem with arbitrary processing times (except for the case of identical machines), but larger competitive ratios than the optimal competitive ratio for preemptive scheduling of unit jobs without rejection.     

Two-machine flowshop scheduling with intermediate transportation under job physical space consideration

In this paper, we study a coordinated production–transportation scheduling problem in a two-machine flowshop environment where a single transporter may carry several jobs simultaneously as a batch between the machines. Each job has its own physical-space requirement while being loaded into the transporter. The goal is to minimize the makespan. For the jobs with the same size of physical space during transportation, we present a heuristic algorithm with an absolute worst-case ratio of 2 and a polynomial-time optimal algorithm for a special case with given job sequence. For the jobs having different size of physical storage space, a heuristic algorithm is constructed with an absolute worst-case ratio of 7/3 and asymptotic worst-case ratio of 20/9. Computational experiments demonstrate that the two heuristic algorithms developed are capable of generating near-optimal solutions quickly.     

一类线性加工时间单机调度问题

讨论一类线性加工时间单机调度问题.在这类问题中,工件具有相同的基本加工时间, 但每个工件的实际加工时间以其开工时间线性增长.对满足无延迟工件条件下极小化提前惩罚 和问题,满足最大完工时间限制条件下极小化资源消耗总量的问题和满足资源消耗总量限制条 件下极小化最大完工时间的问题,分别给出了最优算法.     

单台批处理机总加权完成时间最小化的启发式算法

批处理机总加权完成时间最小化问题的复杂性目前还没有确定，因此有必要研究该问题的启发式算法．基于对该问题最优解性质的分析，提出了工件分批的最优性质．分别基于WSPT规则和SPT规则对工件进行总排序，利用工件最优分批性质进行分批，提出了两种启发式算法（简称WSPTS和SPTS）．为了检验算法的性能．将提出的算法与此问题的基准算法和常规算法进行了比较，结果表明，启发式算法WSPTS要优于其他的算法，而SPTS算法的性能最优．     

Non-permutation flowshop scheduling with dual resources

Typically, in order to process jobs in a flowshop both machines and labor are required. However, in traditional scheduling problems, labor is assumed to be plentiful and only machine is considered to be a constraint. This assumption could be due to the lower cost of labor compared to machines or the complexity of dual-resource constrained problems. In this paper a mathematical model is developed to minimize the work-in-process inventory while maximizing the service level in a flowshop with dual resources. The model focuses on optimizing a non-permutation flowshop. There are different skill levels considered for labor and the setup times on machines are sequence-dependent. Jobs are allowed to skip one or more stages in the flowshop. Job release and machine availability times are considered to be dynamic. The problem is solved in two layers. The outer layer is a search algorithm to find the schedule of jobs on the machine (traditional flowshop scheduling problem) and the inner layer is a three-step heuristic to find a schedule of jobs on labor in accordance to the machine schedule. Three different search algorithms are developed to solve the proposed NP-hard problem. First algorithm can solve a permutation flowshop while the other two are developed to solve a non-permutation flowshop. The comparison between the optimal solution and the search algorithms in small examples shows a good performance of the algorithms with an average deviation of only 2.00%. An experimental design analyzes the effectiveness and efficiency of the algorithms statistically. The results show that non-permutation algorithms perform better than the permutation algorithm, although the former are less efficient. The effectiveness and efficiency in all three algorithms have an inverse relation. To the best of our knowledge, this research is the first of its kind to provide a comprehensive mathematical model for dual resource flowshop scheduling problem.     

Minimizing makespan subject to minimum flowtime on two identical parallel machines

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem.Scope and purposeMultiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.     

A two-stage Ant Colony Optimization algorithm to minimize the makespan on unrelated parallel machines with sequence-dependent setup times

This paper addresses the non-preemptive unrelated parallel machine scheduling problem with machine-dependent and sequence-dependent setup times. All jobs are available at time zero, all times are deterministic, and the objective is to minimize the makespan. An Ant Colony Optimization (ACO) algorithm is introduced in this paper and is applied to this NP-hard problem; in particular, the proposed ACO tackles a special structure of the problem, where the ratio of the number of jobs to the number of machines is large (i.e., for a highly utilized set of machines). Its performance is evaluated by comparing its solutions to solutions obtained using Tabu Search and other existing heuristics for the same problem, namely the Partitioning Heuristic and Meta-RaPS. The results show that ACO outperformed the other algorithms.     

Minimizing weighted earliness–tardiness on parallel machines using hybrid metaheuristics

We consider the problem of scheduling a set of jobs on a set of identical parallel machines where the objective is to minimize the total weighted earliness and tardiness penalties with respect to a common due date. We propose a hybrid heuristic algorithm for constructing good solutions, combining priority rules for assigning jobs to machines and a local search with exact procedures for solving the one-machine subproblems. These solutions are then used in two metaheuristic frameworks, Path Relinking and Scatter Search, to obtain high quality solutions for the problem.The algorithms are tested on a large number of test instances to assess the efficiency of the proposed strategies.The results show that our algorithms consistently outperform the best reported results for this problem.     

Ant colony optimization algorithm for scheduling jobs with fuzzy processing time on parallel batch machines with different capacities

We study the problem of scheduling on parallel batch processing machines with different capacities under a fuzzy environment to minimize the makespan. The jobs have non-identical sizes and fuzzy processing times. After constructing a mathematical model of the problem, we propose a fuzzy ant colony optimization (FACO) algorithm. Based on the machine capacity constraint, two candidate job lists are adopted to select the jobs for building the batches. Moreover, based on the unoccupied space of the solution, heuristic information is designed for each candidate list to guide the ants. In addition, a fuzzy local optimization algorithm is incorporated to improve the solution quality. Finally, the proposed algorithm is compared with several state-of-the-art algorithms through extensive simulated experiments and statistical tests. The comparative results indicate that the proposed algorithm can find better solutions within reasonable time than all the other compared algorithms.     

Exact and parallel metaheuristic algorithms for the single processor total weighted completion time scheduling problem with the sum-of-processing-time based models

In this paper, the single processor scheduling problem to minimize the total weighted completion times is analysed, where the processing times of jobs are described by functions dependent on the sum of the normal processing times of previously processed jobs, which can model learning or aging (deteriorating) effects. We construct the exact pseudopolynomial time algorithm based on the dynamic programming, which solves the problem, where the processing time of each job is described by an arbitrary stepwise function. Moreover, the parallel metaheuristic algorithms are provided for the general version of the problem with arbitrary sum-of-processing time based models. The efficiency of the proposed algorithms is evaluated during numerical analysis.     

Single facility scheduling with nonlinear processing times

This paper considers the static single facility scheduling problem where the processing times of jobs are a monotonically increasing function of their starting (waiting) times and the objective is to minimize the total elapsed time (called the makespan) in which all jobs complete their processing. Based on the combinatorial analysis of the problem, an exact optimization algorithm is developed for the general processing time function which is then specialized for the linear case. In view of the excessive computational burden of the exact optimization algorithm for the nonlinear processing time functions, heuristic algorithms are proposed. The effectiveness of these proposed alogrithms is empirically evaluated and found to indicate that these heuristic algorithms yield optimal or near optimal schedules in many cases.     

Speed Scaling on Parallel Processors

In this paper we investigate dynamic speed scaling, a technique to reduce energy consumption in variable-speed microprocessors. While prior research has focused mostly on single processor environments, in this paper we investigate multiprocessor settings. We study the basic problem of scheduling a set of jobs, each specified by a release date, a deadline and a processing volume, on variable-speed processors so as to minimize the total energy consumption. We first settle the problem complexity if unit size jobs have to be scheduled. More specifically, we devise a polynomial time algorithm for jobs with agreeable deadlines and prove NP-hardness results if jobs have arbitrary deadlines. For the latter setting we also develop a polynomial time algorithm achieving a constant factor approximation guarantee. Additionally, we study problem settings where jobs have arbitrary processing requirements and, again, develop constant factor approximation algorithms. We finally transform our offline algorithms into constant competitive online strategies.     

Genetic algorithms for a two-agent single-machine problem with release time

Scheduling with two competing agents has drawn a lot of attention lately. However, it is assumed that all the jobs are available in the beginning in most of the research. In this paper, we study a single-machine problem in which jobs have different release times. The objective is to minimize the total tardiness of jobs from the first agent given that the maximum tardiness of jobs from the second agent does not exceed an upper bound. Three genetic algorithms are proposed to obtain the near-optimal solutions. Computational results show that the branch-and-bound algorithm could solve most of the problems with 16 jobs within a reasonable amount of time. In addition, it shows that the performance of the combined genetic algorithm is very good with mean error percentages of less than 0.2% for all the cases.     

A tabu method for a two-agent single-machine scheduling with deterioration jobs

In recent 10 years, the multi-agent idea applied in scheduling issues has received continuing attention. However, the study of the multi-agent scheduling with deteriorating jobs is relatively limited. In light of this, this paper deliberates upon a two-agent single-machine scheduling problem with deteriorating jobs. Taking the proposed model, the actual processing time of a job from both the first agent and the second agent is modeled as a linearly increasing function of its starting time. The goal of this paper is to minimize the total weighted number of tardy jobs of the first agent subject to the condition that the maximum lateness of the second agent is allowed to have an upper bound. The complexity of the model concerned in the paper is claimed as an NP-hard one. Following that, several dominance rules and a lower bound are proposed to be applied in a branch-and-bound algorithm for the optimal solution, and a tabu algorithm is applied to find near-optimal solutions for the problem. The simulation results obtained from all the proposed algorithms are also reported.     

Minimizing total weighted completion time under makespan constraint for two-agent scheduling with job-dependent aging effects

In this paper, we consider a two-agent single-machine scheduling problem with linear position-based aging effects and job-dependent aging ratios. The objective is to minimize the total weighted completion time of all jobs for two agents, where the makespan for one agent is constrained under an upper bound. After showing that this problem is at least NP-hard, we develop two solution algorithms: First, we devise a branch-and-bound algorithm to find an optimal solution through the establishment of several dominance and feasibility properties, and a lower bound. Second, we propose efficient simulated annealing algorithms, using three different methods to generate an initial solution. Through a numerical experiment, we demonstrate that the suggested algorithms can be applied to efficiently find near-optimal solutions within a reasonable amount of CPU time. In particular, we show that the initial solution method (arranging the jobs for one agent in non-increasing order of aging ratio, and scheduling the jobs for the other in the weighted shortest normal processing time order) is superior to others. Moreover, through scalability testing, we verify its consistent and relatively outstanding performance for larger systems with many processing jobs.     

Two-agent singe-machine scheduling with release times to minimize the total weighted completion time

In many management situations multiple agents pursuing different objectives compete on the usage of common processing resources. In this paper we study a two-agent single-machine scheduling problem with release times where the objective is to minimize the total weighted completion time of the jobs of one agent with the constraint that the maximum lateness of the jobs of the other agent does not exceed a given limit. We propose a branch-and-bound algorithm to solve the problem, and a primary and a secondary simulated annealing algorithm to find near-optimal solutions. We conduct computational experiments to test the effectiveness of the algorithms. The computational results show that the branch-and-bound algorithm can solve most of the problem instances with up to 24 jobs in a reasonable amount of time and the primary simulated annealing algorithm performs well with an average percentage error of less than 0.5% for all the tested cases.