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.     

On the equivalence of single machine earliness/tardiness problems with job rejection

We present optimal algorithms for single-machine scheduling problems with earliness criteria and job rejection and compare them with the algorithms for the corresponding problems with tardiness objectives. We present an optimal O(n log n) algorithm for minimizing the maximum earliness on a single machine with job rejection. Our algorithm also solves the bi-criteria scheduling problem is which the objective is to simultaneously minimize the maximum earliness of the scheduled jobs and the total rejection cost of the rejected jobs. We also show that the optimal pseudo-polynomial time algorithm for the total tardiness problem with job rejection can be used to solve the corresponding total earliness problem with job rejection.     

A PTAS for parallel batch scheduling with rejection and dynamic job arrivals

In the parallel batch scheduling model, a group of jobs can be scheduled together as a batch while the processing time of this batch is the greatest processing time among its members; in the model of scheduling with rejection, any job can be rejected with a corresponding penalty cost added to the objective value. In this paper, we present a PTAS for the combined model of the above two scheduling models where jobs arrive dynamically. The objective is to minimize the sum of the makespan of the accepted jobs and the total penalty of the rejected ones. Our basic approaches are dynamic programming and roundings.     

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.     

Bounded single-machine parallel-batch scheduling with release dates and rejection

We consider the bounded single-machine parallel-batch scheduling problem with release dates and rejection. A job is either rejected, in which case a certain penalty has to be paid, or accepted and then processed on the machine. The objective is to minimize the sum of the makespan of the accepted jobs and the total penalty of the rejected jobs. When the jobs have identical release dates, we present a polynomial-time algorithm. When the jobs have a constant number of release dates, we give a pseudo-polynomial-time algorithm. For the general problem, we provide a 2-approximation algorithm and a polynomial-time approximation scheme.     

Online scheduling with preemption or non-completion penalties

We consider online preemptive scheduling problems where jobs have deadlines and the objective is to maximize the total weight of jobs completed before their deadlines. In the first problem, preemptions are not free but incur a penalty. In the second problem, a job has to be accepted or rejected immediately upon arrival, and may need to be immediately allocated a fixed scheduling interval as well; if these accepted jobs are not eventually completed, the job is lost, and a penalty is incurred. We give an algorithm with the optimal competitive ratio for the first problem, and new and improved algorithms for the second problem, under different models of preemptions and job weights.     

Real-Time Scheduling with a Budget

Suppose that we are given a set of jobs, where each job has a processing time, a non-negative weight, and a set of possible time intervals in which it can be processed. In addition, each job has a processing cost. Our goal is to schedule a feasible subset of the jobs on a single machine, such that the total weight is maximized, and the cost of the schedule is within a given budget. We refer to this problem as budgeted real-time scheduling (BRS). Indeed, the special case where the budget is unbounded is the well-known real-time scheduling problem. The second problem that we consider is budgeted real-time scheduling with overlaps (BRSO), in which several jobs may be processed simultaneously, and the goal is to maximize the time in which the machine is utilized. Our two variants of this real-time scheduling problem have important applications, in vehicle scheduling, linear combinatorial auctions, and Quality-of-Service management for Internet connections. These problems are the focus of this paper. Both BRS and BRSO are strongly NP-hard, even with unbounded budget. Our main results are (2 + ε)-approximation algorithms for these problems. This ratio coincides with the best known approximation factor for the (unbudgeted) real-time scheduling problem, and is slightly weaker than the best known approximation factor of e/(e - 1) for the (unbudgeted) real-time scheduling with overlaps, presented in this paper. We show that better ratios (or simpler approximation algorithms) can be derived for some special cases, including instances with unit-costs and the budgeted job interval selection problem (JISP). Budgeted JISP is shown to be APX-hard even when overlaps are allowed and with unbounded budget. Finally, our results can be extended to instances with multiple machines.     

Two-agent single-machine scheduling to minimize the weighted sum of the agents’ objective functions

Scheduling with two competing agents on a single machine has become a popular research topic in recent years. Most research focuses on minimizing the objective function of one agent, subject to the objective function of the other agent does not exceed a given limit. In this paper we adopt a weighted combination approach to treat the two-agent single-machine scheduling problem. The objective that we seek to minimize is the weighted sum of the total completion time of the jobs of one agent and the total tardiness of the jobs of the other agent. We provide two branch-and-bound algorithms to solve the problem. In addition, we present a simulated annealing and two genetic algorithms to obtain near-optimal solutions. We report the results of the computational experiments conducted to test the performance of the proposed algorithms.     

Online scheduling with rejection and withdrawal

We study an online scheduling problem with rejection, in which some rearrangement of the solution is allowed. This problem is called scheduling with rejection and withdrawal. Each arriving job has a processing time and a rejection cost associated with it, and it needs to be either assigned to a machine or rejected upon arrival. At termination, it is possible to choose at most a fixed number of scheduled jobs and withdraw them (i.e., decide to reject them). We study the minimization version, where the goal is to minimize the sum of the makespan and the total rejection cost (which corresponds to the penalty), and the maximization problem, where the goal is to maximize the sum of the minimum load and the total rejection cost (which corresponds to profit). We study environments of machines, which are the case of m identical machines and the case of two uniformly related machines, and show a strong relation between these problems and the related classic online scheduling problems which they generalize, in contrast to standard scheduling with rejection, which typically makes the scheduling problems harder.     

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.     

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.     

Common due date assignment and scheduling with ready times

We consider the problem of scheduling a set of nonsimultaneously available jobs on one machine. Each job has a ready time only at or after which the job can be processed. All the jobs have a common due date, which needs to be determined. The problem is to determine a due date and a schedule so as to minimize a total penalty depending on the earliness, tardiness and due date. We show that this problem is strongly NP-hard and give an efficient algorithm that finds an optimal due date and schedule when either the job sequence is predetermined or all jobs have the same processing time. We also propose three approximation algorithms for the general and special cases together with their experimental analysis.

Scope and purpose

We consider the single machine due date assignment problem for scheduling jobs which are ready for processing at different times. The problem under consideration arises in production planning and scheduling concerning the setting of appropriate due dates for a number of customer orders arriving over time. Most of the earlier publications on this subject assumed that the jobs are ready for processing simultaneously. This assumption is too restrictive for real-life production systems where jobs arrive at different times. We show that the problem with unequal ready times is NP-hard and develop fast heuristic algorithms for it, and exact algorithms for two special cases.     

工件可拒绝的有限等待置换流水车间调度算法

有限等待限定了工件在相邻机器间的等待时间上下限,普遍存在于中间产品性质不稳定且存在运输作业的车间环境中.工件可拒绝的有限等待置换流水车间调度是对工件拒绝和工件调度的联合决策,要求确定拒绝工件集合并给出被接受工件的调度方案.针对这一联合决策问题,以最小化总拒绝成本与总拖期成本之和为目标,并为最大完工时间(Makespan)设置上限约束,结合问题特征提出一种协同进化遗传算法.该算法将染色体编码分解为工件拒绝和工件序列两个子集,基于调度规则生成初始种群,引入协同进化策略依次进化子集种群,并提出基于记忆的动态概率参数设计方法以确定遗传算子的执行概率,设计解码规则以保证解的可行性并优化总成本.最后,通过数据实验验证了所提出算法及相关策略的可行性和有效性,并分析了问题参数对算法性能的影响.     

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.     

Single machine scheduling with interfering job sets

We consider two single machine bicriteria scheduling problems in which jobs belong to either of two different disjoint sets, each set having its own performance measure. The problem has been referred to as interfering job sets in the scheduling literature and also been called multi-agent scheduling where each agent's objective function is to be minimized. In the first problem (P1) we look at minimizing total completion time and number of tardy jobs for the two sets of jobs and present a forward SPT-EDD heuristic that attempts to generate the set of non-dominated solutions. The complexity of this specific problem is NP-hard; however some pseudo-polynomial algorithms have been suggested by earlier researchers and they have been used to compare the results from the proposed heuristic. In the second problem (P2) we look at minimizing total weighted completion time and maximum lateness. This is an established NP-hard problem for which we propose a forward WSPT-EDD heuristic that attempts to generate the set of supported points and compare our solution quality with MIP formulations. For both of these problems, we assume that all jobs are available at time zero and the jobs are not allowed to be preempted.     

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.     

Online Bi-Objective Scheduling for IaaS Clouds Ensuring Quality of Service

This paper focuses on a bi-objective experimental evaluation of online scheduling in the Infrastructure as a Service model of Cloud computing regarding income and power consumption objectives. In this model, customers have the choice between different service levels. Each service level is associated with a price per unit of job execution time, and a slack factor that determines the maximal time span to deliver the requested amount of computing resources. The system, via the scheduling algorithms, is responsible to guarantee the corresponding quality of service for all accepted jobs. Since we do not consider any optimistic scheduling approach, a job cannot be accepted if its service guarantee will not be observed assuming that all accepted jobs receive the requested resources. In this article, we analyze several scheduling algorithms with different cloud configurations and workloads, considering the maximization of the provider income and minimization of the total power consumption of a schedule. We distinguish algorithms depending on the type and amount of information they require: knowledge free, energy-aware, and speed-aware. First, to provide effective guidance in choosing a good strategy, we present a joint analysis of two conflicting goals based on the degradation in performance. The study addresses the behavior of each strategy under each metric. We assess the performance of different scheduling algorithms by determining a set of non-dominated solutions that approximate the Pareto optimal set. We use a set coverage metric to compare the scheduling algorithms in terms of Pareto dominance. We claim that a rather simple scheduling approach can provide the best energy and income trade-offs. This scheduling algorithm performs well in different scenarios with a variety of workloads and cloud configurations.     

Fast primal-dual distributed algorithms for scheduling and matching problems

In this paper we give efficient distributed algorithms computing approximate solutions to general scheduling and matching problems. All approximation guarantees are within a constant factor of the optimum. By “efficient”, we mean that the number of communication rounds is poly-logarithmic in the size of the input. In the scheduling problem, we have a bipartite graph with computing agents on one side and resources on the other. Agents that share a resource can communicate in one time step. Each agent has a list of jobs, each with its own length and profit, to be executed on a neighbouring resource within a given time-window. Each job is also associated with a rational number in the range between zero and one (width), specifying the amount of resource required by the job. Resources can execute non preemptively multiple jobs whose total width at any given time is at most one. The goal is to maximize the profit of the jobs that are scheduled. We then adapt our algorithm for scheduling, to solve the weighted b-matching problem, which is the generalization of the weighted matching problem where for each vertex v, at most b(v) edges incident to v, can be included in the matching. For this problem we obtain a randomized distributed algorithm with approximation guarantee of \frac16+e{\frac{1}{6+\epsilon}}, for any ${\epsilon >0 }${\epsilon >0 }. For weighted matching, we devise a deterministic distributed algorithm with the same approximation ratio. To our knowledge, we give the first distributed algorithm for the aforementioned scheduling problem as well as the first deterministic distributed algorithm for weighted matching with poly-logaritmic running time. A very interesting feature of our algorithms is that they are all derived in a systematic manner from primal-dual algorithms.     

Multi-objective ACO algorithms to minimise the makespan and the total rejection cost on BPMs with arbitrary job weights

In this paper, we investigate the batch-scheduling problem with rejection on parallel machines with non-identical job sizes and arbitrary job-rejected weights. If a job is rejected, the corresponding penalty has to be paid. Our objective is to minimise the makespan of the processed jobs and the total rejection cost of the rejected jobs. Based on the selected multi-objective optimisation approaches, two problems, P1 and P2, are considered. In P1, the two objectives are linearly combined into one single objective. In P2, the two objectives are simultaneously minimised and the Pareto non-dominated solution set is to be found. Based on the ant colony optimisation (ACO), two algorithms, called LACO and PACO, are proposed to address the two problems, respectively. Two different objective-oriented pheromone matrices and heuristic information are designed. Additionally, a local optimisation algorithm is adopted to improve the solution quality. Finally, simulated experiments are conducted, and the comparative results verify the effectiveness and efficiency of the proposed algorithms, especially on large-scale instances.     

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.