Largest Job Size Research Articles

Semi-Online Algorithms for the Hierarchical Extensible Bin-Packing Problem and Early Work Problem

In this paper, we consider two types of semi-online problems with hierarchies. In the extensible bin-packing problem with two hierarchical bins, one bin can pack all items, while the other bin can only pack some items. The initial size of the bin can be expanded, and the goal is to minimize the total size of the two bins. When the largest item size is given in advance, we provide some lower bounds and propose online algorithms. When the total item size is given in advance, we provide some lower bounds and propose online algorithms. In addition, we also consider the relevant early-work-maximization problem on two hierarchical machines; one machine can process any job, while the other machine can only process some jobs. Each job shares a common due date, and the goal is to maximize the total early work. When the largest job size is known, we provide some lower bounds and propose two online algorithms whose competitive ratios are close to the lower bounds.

Single machine multi-criteria scheduling problem: total completion time, maximum lateness, and maximum earliness performance measures

This paper considers the multi-criteria scheduling problem with total completion time, maximum lateness, and maximum earliness as the objectives. The problem is a minimisation problem; the total completion time, a MIN-SUM problem, the maximum earliness, a MIN-MAX problem, and the maximum lateness, a MIN-MAX problem. Though, the problem is NP-hard, shortest processing time (SPT) rule, yields optimal for total completion time while early due date rule (EDD) is the optimal solution for maximum lateness and maximum earliness if each criterion were to be considered separately. Two heuristics, named SOL I and SOL II were proposed and the results for each of the criteria were compared to the optimal of the sub-problems. Results of the computational experiment on small job-sizes (5 ≤ n ≤ 30) and large job-sizes (40 ≤ n ≤ 100) show that the two heuristics results are not significantly different from the optimal at 99% significant level for total completion time and maximum lateness performance measures. However, for maximum earliness, the optimal solutions are significantly better than the two proposed heuristics.

A Two-Stage Three-Machine Flow Shop Assembly Problem Mixed with a Controllable Number and Sum-of-Processing Times-Based Learning Effect by Simulated Annealing Algorithms

The two-stage assembly scheduling problem is widely used in industrial and service industries. This study focuses on the two-stage three-machine flow shop assembly problem mixed with a controllable number and sum-of-processing times-based learning effect, in which the job processing time is considered to be a function of the control of the truncation parameters and learning based on the sum of the processing time. However, the truncation function is very limited in the two-stage flow shop assembly scheduling settings. Thus, this study explores a two-stage three-machine flow shop assembly problem with truncated learning to minimize the makespan criterion. To solve the proposed model, we derive several dominance rules, lemmas, and lower bounds applied in the branch-and-bound method. On the other hand, three simulated annealing algorithms are proposed for finding approximate solutions. In both the small and large size number of job situations, the SA algorithm is better than the JS algorithm in this study. All the experimental results of the proposed algorithm are presented on small and large job sizes, respectively.

Analysis of the Task Assignment based on Guessing Size policy

We study the Task Assignment based on Guessing Size (TAGS) policy in a parallel and homogeneous server system. The policy parameters are the number of servers h and a set of cutoffs s1<s2,ldots,sh−1. In this policy, all the incoming jobs are routed to the first server and jobs are run up to s1 time units. If they complete they leave the system, but jobs that do not complete after s1 time units are killed and moved to the end of the queue of the second server, where service starts from scratch. Likewise, jobs that are executed in server i and complete service before si units of time, leave the system, whereas jobs that do not complete are killed and routed to the next server. We first study the stability of such system and provide a precise utilization threshold for the existence of stable parameters for a given job size distribution. We compute the threshold for several families of distributions and provide bounds for others. We show that TAGS is most stable for bounded Pareto distributions with parameter α=1. Besides, we provide tight bounds on the performance of the TAGS policy where the cutoffs are chosen to minimize average waiting time in the asymptotic regime where the largest job size tends to infinity for the Bounded Pareto distribution and the system load smaller than one. In this case, we show that the performance ratio between TAGS and a version called SITA which does require knowledge of job size is at most 2. We then consider more broadly the same asymptotic regime and consider a bound on the average waiting time for any distribution. This is compatible with having a conservative policy which will work well for any job size distribution. We show rather tight upper and lower bounds which again match those of SITA up to a factor of 2. These bounds are considerably lower than the corresponding bounds of competing policies such as Random or Least Work Remaining. We also show that for all these policies the bounded Pareto distribution with α=1 is close to being the worst possible among all distributions with the same job size range. We show using the stability results that in the same regime, if we increase the load the gap between SITA and TAGS grows dramatically and eventually, the TAGS system cannot be stabilized. The conclusion from all our analysis is that TAGS is best used as a conservative policy with minimal requirements on small systems with relatively low utilization and where the range of job sizes is large.

Energy efficient scheduler of aperiodic jobs for real-time embedded systems

Energy consumption has become a key metric for evaluating how good an embedded system is, alongside more performance metrics like respecting operation deadlines and speed of execution. Schedulability improvement is no longer the only metric by which optimality is judged. In fact, energy efficiency is becoming a preferred choice with a fundamental objective to optimize the system’s lifetime. In this work, we propose an optimal energy efficient scheduling algorithm for aperiodic real-time jobs to reduce CPU energy consumption. Specifically, we apply the concept of real-time process scheduling to a dynamic voltage and frequency scaling (DVFS) technique. We address a variant of earliest deadline first (EDF) scheduling algorithm called energy saving-dynamic voltage and frequency scaling (ES-DVFS) algorithm that is suited to unpredictable future energy production and irregular job arrivals. We prove that ES-DVFS cannot attain a total value greater than C/000000, where 000000 is the minimum speed of any job and C is the available energy capacity. We also investigate the implications of having in advance, information about the largest job size and the minimum speed used for the competitive factor of ES-DVFS. We show that such advance knowledge makes possible the design of semi-on-line algorithm, ES-DVFS**, that achieved a constant competitive factor of 0.5 which is proved as an optimal competitive factor. The experimental study demonstrates that substantial energy savings and highest percentage of feasible job sets can be obtained through our solution that combines EDF and DVFS optimally under the given aperiodic jobs and energy models.

Semi-online scheduling problems on two uniform machines under a grade of service provision

This paper studies the semi-online scheduling on two uniform machines under a grade of service (GoS) provision, where one machine is available for all jobs and the other one is only available for partial jobs. The objective is to minimize the makespan. We consider three variants, where the optimal makespan, the total size of jobs, and the largest job size are known in advance respectively, and design optimal algorithms for all of them.

Competitive analysis of online real-time scheduling algorithms under hard energy constraint

In this paper, we undertake the competitive analysis of the online real-time scheduling problems under a given hard energy constraint. Specifically, we derive worst-case performance bounds that apply to any online algorithm, when compared to an optimal algorithm that has the knowledge of the input sequence in advance. First, by focusing on uniform value-density settings, we prove that no online algorithm can achieve a competitive factor greater than \(1-\frac{e_{\max}}{E}\), where e max is the upper bound on the size of any job and E is the available energy budget. Then we propose a variant of EDF algorithm, EC-EDF, that is able to achieve this upper bound. We show that a priori information about the largest job size in the actual input sequence makes possible the design of a semi-online algorithm EC-EDF ∗ which achieves a constant competitive factor of 0.5. This turns out to be the best achievable competitive factor in these settings. In non-uniform value density settings, we derive an upper bound on the competitive factor achievable by any online algorithm, as well as the competitive factors of EC-EDF and EC-EDF ∗ algorithms. We also investigate the implications of having additional constraints on the online scheduling algorithm, including non-idling and non-preemptive execution constraints. Moreover, we analyze the case where the processor can adjust its speed at run-time through Dynamic Voltage Scaling (DVS) capability.

Tabu search heuristic for two-machine flowshop with batch processing machines

Batch processing machines are frequently encountered in many industrial environments. A batch processing machine is one which can process several jobs simultaneously as a batch. The processing time of a batch is equal to the largest processing time of any job in the batch. This study deals with the problem of scheduling jobs in a flowshop with two batch processing machines such that the makespan is minimized. A heuristic based on Tabu search (TS) technique is proposed. The proposed heuristic is compared with a heuristic based on mixed integer linear programming (MILP). Because the complexity of the MILP-based heuristic is depended on the number of job batches, the comparison is under up-to-eight batches problem. In order to measure the proposed TS-based heuristic in larger batch problem, the relative error percentage with the lower bound (REP LB) is used. The results show that the proposed heuristic is efficient and effective for the problems with relative large job sizes.

Optimal semi-online algorithms for preemptive scheduling problems with inexact partial information

In semi-online scheduling problems, we always assume that some partial additional information is exactly known in advance. This may not be true in some application. This paper considers semi-online problems on identical machines with inexact partial information. Three problems are considered, where we know in advance that the optimal value, or the largest job size are in given intervals, respectively, while their exact values are unknown. We give both lower bounds of the problems and competitive ratios of algorithms as functions of a so-called disturbance parameter r ∈[1, ∞). We establish for which r the inexact partial information is useful to improve the performance of a semi-online algorithm with respect to its pure online problem. Optimal preemptive semi-online algorithms are then obtained.

Semi-online scheduling problems on two identical machines with inexact partial information

In semi-online scheduling problems, we always assume that some partial additional information is exactly known in advance. This may not be true in some applications. This paper considers semi-online scheduling problems on two identical machines with inexact partial information. Three versions are considered, where we know in advance that the total size of all jobs, the optimal value, and the largest job size are in given intervals, respectively, while their exact values are unknown. We give both lower bounds of the problems and competitive ratios of algorithms as functions of a so-called disturbance parameter r ∈ [ 1 , ∞ ) . We establish for which r the inexact partial information is useful to improve the performance of a semi-online algorithm with respect to its pure online problem. Optimal or near optimal algorithms are then obtained.

A two-machine flowshop maximum tardiness scheduling problem with a learning effect

The primary objective of this paper is to study a two-machine flowshop scheduling problem with a learning effect where the goal is to find a sequence that minimizes the maximum tardiness. We employ a branch-and-bound method and a simulated annealing (SA) method to search for the optimal solution and a near-optimal solution, respectively. Computational results, using Fisher’s (Math Program 11:229–251 1971) framework, show that the mean and maximum number of nodes for the branch-and-bound algorithm decrease when the learning effect is stronger, the value of the tardiness factor is smaller, or the value of the due date range is larger. In addition, comparisons between the SA method and the earliest due date first (EDD) rule are provided for large-job sizes. Results indicate that the percentage of time that the SA solution outperforms the EDD solution decreases as the job size increases and the learning effect becomes greater. Additionally, the SA solution is never worse than the EDD solution.

A makespan study of the two-machine flowshop scheduling problem with a learning effect

This paper studies the problem of minimizing the makespan in a two-machine flowshop scheduling problem with learning considerations. Johnson’s rule might not provide the optimal solution for this problem. Thus, this paper not only develops several dominance relations and proposes a branch-and-bound algorithm based on them but also establishes two heuristic algorithms. Computational experiments show the accuracy of Johnson’s rule and both proposed heuristics, and a comparison of the results to the optimal solution obtained by a branch-and-bound technique is provided for small job sizes. In addition, the performances of the three heuristics, including Johnson’s rule, are also reported for large job sizes.

Classifying scheduling policies with respect to unfairness in an M/GI/1

It is common to evaluate scheduling policies based on their mean response times. Another important, but sometimes opposing, performance metric is a scheduling policy's fairness. For example, a policy that biases towards small job sizes so as to minimize mean response time may end up being unfair to large job sizes. In this paper we define three types of unfairness and demonstrate large classes of scheduling policies that fall into each type. We end with a discussion on which jobs are the ones being treated unfairly.