带两个服务等级的3台机半在线算法

This paper studies a semi-online hierarchical scheduling problem on three identical machines. In the problem, there is only one machine with hierarchy 1 and two machines with hierarchy 2, and the goal is to minimize the makespan. When the total size of low-hierarchy is known, an online algorithm with the competitive ratio of 5/3 and the lower bound of 3/2 is given. When the total size of high-hierarchy is known, an online algorithm with the competitive ratio of 9/5 and the lower bound of 3/2 is given. When the total size of each hierarchy is known, an online algorithm with the competitive ratio of 3/2 and the lower bound of 4/3 is given. When the total size of jobs is known, a best possible online algorithm with the competitive ratio of 3/2 is given.     

Semi-online two-level supply chain scheduling problems

We consider two-level supply chain scheduling problems where customers release jobs to a manufacturer that has to process the jobs and deliver them to the customers. Processed jobs are grouped into batches, which are delivered to the customers as single shipments. The objective is to minimize the total cost which is the sum of the total flow time and the total delivery cost. Such problems have been considered in the off-line environment where future jobs are known, and in the online environment where at any time there is no information about future jobs. It is known that the best possible competitive ratio for an online algorithm is 2. We consider the problem in the semi-online environment, assuming that a lower bound P for all processing times is available a priori, and present a semi-online algorithm with competitive ratio \(\frac{2D}{D+P}\) where D is the cost of a delivery. Also, for the special case where all processing times are equal, we prove that the algorithm is \(1.045\sqrt{\frac{2-u}{u}}\)-competitive, where u is the density of the instance.     

On-line scheduling with extendable working time on a small number of machines

Speranza and Tuza [Ann. Oper. Res. 86 (1999) 494-506] studied the on-line problem of scheduling jobs on m identical machines with extendable working time. In this problem, each machine is assumed to have an identical regular working time, which can be extended if necessary. The working time of a machine is the larger one between its regular working time and the total processing time of jobs assigned to it. The objective is to minimize the total working time of machines. They presented an on-line algorithm H_x, with a competitive ratio at most 1.228 for any number of machines by choosing an appropriate parameter x. In this paper we consider a small number of machines. The best choices of x are given for m=2,3,4 and the tight bounds, 7/6, 11/9 and 19/16, respectively, are proved. Among them, the algorithm for m=2 is best possible. We then derive a new algorithm for m=3 with a competitive ratio 7/6.     

Semi-online scheduling with machine cost

For most scheduling problems the set of machines is fixed initially and remains unchanged for the duration of the problem.Recently Imreh and Nogaproposed to add the concept of machine cost to scheduling problems and considered the so-called List Model problem.An online algorthm with a competitive ratio 1.618 was given while the lower boud is 4/3.In this paper,two different semi-onlne versions of this problem are studied‘.In the first case,it is assumed that the processing time of the largest job is known a priori.A semi-online algorithm is presented with the competitive ratio at most 1.5309 while the lower bound is 4/3,In the second case,it is assumed that the total processing time of all jobs is known in advance.A semi-online algorithm is presented with the competitive ratio at most 1.414 while the lower bound is 1.161.It is shown that the additional partial available information about the jobs leads to the possibility of constructing a schedule with a smaller competitive ratio than that of online algorithms.     

The complexity of customer order scheduling problems on parallel machines

This paper considers scheduling problems where a set of jobs (customer order) is shipped at the same time. The objective function is associated with the completion time of the orders. While a machine can process only one job at a time, multiple machines can process simultaneously jobs in an order. We first introduce this relatively new class of the customer order scheduling problems on parallel machines. Then, we establish the complexity of several problems with different types of objectives, job restrictions, and machine environments. For some tractable cases, we propose optimal solution procedures.     

Worst-case analysis of LPT scheduling on a small number of non-identical processors

The approximation ratio of the longest processing time (LPT) scheduling algorithm has been investigated in various studies. While the tight approximation ratio is known for the cases when all processors are identical, the ratio is unknown when the processors have different speeds. In this study, we provide a tight approximation ratio for three, four, and five processors. We show that the ratios for these cases are no larger than the lower bound provided by Gonzalez et al. (1977) [14]. The ratios are approximately 1.38, 1.43, and 1.46 for three, four, and five processors, respectively.     

The optimal number of used machines in a two-stage flexible flowshop scheduling problem

We study practical scheduling problems with a major decision referring to the number of machines to be used. We focus on a two-stage flexible flowshop, where each job is processed on the first (critical) machine, and then continues to one of the second-stage parallel machines. Jobs are assumed to have identical processing times, and are processed in batches. A setup time is required when starting a new batch. We consider two objective functions: minimum makespan and minimum flowtime. In both cases, a closed form expression for the optimal number of machines to be used is introduced, and a unique and unusual sequence of decreasing batch sizes is shown to be optimal.     

Lower bounds for online makespan minimization on a small number of related machines

In online makespan minimization, the jobs characterized by their processing time arrive one-by-one and each has to be assigned to one of the m uniformly related machines. The goal is to minimize the length of the schedule. We prove new combinatorial lower bounds for m=4 and m=5, and computer-assisted lower bounds for m≤11.     

Optimal algorithms for semi-online preemptive scheduling problems on two uniform machines

This paper investigates two preemptive semi-online scheduling problems to minimize makespan on two uniform machines. In the first semi-online problem, we know in advance that all jobs have their processing times in between p and rp . In the second semi-online problem, we know the size of the largest job in advance. We design an optimal semi-online algorithm which is optimal for every combination of machine speed ratio and job processing time ratio for the first problem, and an optimal semi-online algorithm for every machine speed ratio for the second problem.Received: 2 December 2003, Published online: 16 January 2004This research is supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, China, and National Natural Science Foundation of China (10271110).     

三台等级机器上带重排的半在线问题

研究了3台机上带2种等级的重排问题,当所有工件都被分配之后,在等级约束下,可以重排一台机器上的最后一个工件,目标是最小化最大完工时间。3台机上带2种等级分为2种情形：第1种是有1台机器的等级为1,另2台机器的等级为2;第2种是2台机器的等级为1,另1台机器的等级为2。针对第1种情形给出了一个竞争比下界为3/2,并提出了一个竞争比至多为5/3的在线算法;针对第2种情形给出了一个竞争比下界为3/2,并提出了一个竞争比至多为12/7的在线算法。     

Optimal semi-online algorithms for scheduling problems with reassignment on two identical machines

In this paper, we consider semi-online minimum makespan scheduling problem with reassignment on two identical machines. Two versions are discussed. In the first version, one can reassign the last job of one machine that is based on the problem proposed by Tan and Yu (2008) [1], in which case the last job of each machine is allowed to be reassigned. An optimal algorithm which has the same competitive ratio is presented. In the second version we consider the combination of the next two conditions: the total size of all jobs is known in advance and one can reassign the last job of one machine. For this problem an optimal algorithm with competitive ratio is also given.     

Using quadratic programming to solve high multiplicity scheduling problems on parallel machines

We introduce and analyze several models of schedulingn different types (groups) of jobs onm parallel machines, where in each group all jobs are identical. Our main goal is to exhibit the usefulness of quadratic programming approaches to solve these classes of high multiplicity scheduling problems, with the total weighted completion time as the minimization criterion. We develop polynomial algorithms for some models, and strongly polynomial algorithms for certain special cases. In particular, the model in which the weights are job independent, as well as the generally weighted model in which processing requirements are job independent, can be formulated as an integer convex separable quadratic cost flow problem, and therefore solved in polynomial time. When we specialize further, strongly polynomial bounds are achievable. Specifically, for the weighted model with job-independent processing requirements if we restrict the weights to be machine independent (while still assuming different machine speeds), anO(mn+n logn) algorithm is developed. If it is also assumed that all the machines have the same speed, the complexity of the algorithm can be improved toO(m logm+n logn). These results can be extended to related unweighted models with variable processing requirements in which all the machines are available at time zero. The research of Frieda Granot was partially supported by Natural Sciences and Engineering Research Council of Canada Grant 5-83998. The research of Jadranka Skorin-Kapov was partially supported by National Science Foundation Grant DDM-8909206.     

Online scheduling on semi-related machines

We study the problem of minimizing the makespan on related machines in the following setting: jobs arrive over time, and the machines may become available or unavailable. In either case, advance warning is given, i.e., the next point of time where a job is released or a machine changes state is revealed a little time ahead. We present an optimal online algorithm for this problem.     

Flow shop scheduling problems with decreasing linear deterioration under dominant machines

This paper considers the general, no-wait and no-idle flow shop scheduling problems with deteriorating jobs, respectively. By a deteriorating job, we mean that the processing time is a decreasing function of its execution starting time. The normal processing time proportional to its decreasing rate is assumed and some dominant relationships between machines can be satisfied. It is shown that for the problems to minimize the makespan or the weighted sum of completion time, polynomial algorithms still exist, although these problems are more complicated than the classical ones. When the objective is to minimize the maximum lateness, the solutions of a classical version may not hold.     

Resource constrained scheduling on multiple machines

This paper is devoted to Resource Constrained Scheduling. An instance for this problem is given by a set of n jobs with lengths and weights and a set of m machines with capacities. At every time each machine can run arbitrary many jobs in parallel but the total weight of these jobs must not exceed the capacity of the machine. Furthermore the m machines work in parallel and we wish to find a schedule that minimizes the makespan or the sum of completion times. Thus load balancing on a cluster of servers is a typical application for scheduling under resource constraints. For both measures the problem is -complete even for m=1. We show that the problem is approximable within factors of 2 (makespan) and 14.85 (sum of completion times) for arbitrary m.