Uniform Machines Research Articles

A better semi-online algorithm for Q3/s1 = s2≤ s3/Cmin with the known largest size

This paper investigates the semi-online machine covering problem on three special uniform machines with the known largest size. Denote by sj the speed of each machine, j = 1, 2, 3. Assume 0 < s1 = s2 = r ≤ t = s3, and let s = t/r be the speed ratio. An algorithm with competitive ratio \(\max \left\{ {2,\tfrac{{3s + 6}} {{s + 6}}} \right\}\) is presented. We also show the lower bound is at least \(\max \left\{ {2,\tfrac{{3s}} {{s + 6}}} \right\}\). For s ≤ 6, the algorithm is an optimal algorithm with the competitive ratio 2. Besides, its overall competitive ratio is 3 which matches the overall lower bound. The algorithm and the lower bound in this paper improve the results of Luo and Sun.

Online scheduling on three uniform machines

This paper investigates the online scheduling on three uniform machines problem. Denote by s j the speed of each machine, j = 1 , 2 , 3 . Assume 0 < s 1 ≤ s 2 ≤ s 3 , and let s = s 2 / s 1 and t = s 3 / s 2 be two speed ratios. We show the greedy algorithm LS is an optimal online algorithm when the speed ratios ( s , t ) ∈ G 1 ∪ G 2 , where G 1 = { ( s , t ) | 1 ≤ t < 1 + 31 6 , s ≥ 3 t 5 + 2 t − 6 t 2 } and G 2 = { ( s , t ) | s ( t − 1 ) t ≥ 1 + s , s ≥ 1 , t ≥ 1 } . The competitive ratio of LS is 1 + s + 2 s t s + s t when ( s , t ) ∈ G 1 and 1 + s s t + 1 when ( s , t ) ∈ G 2 . Moreover, for the general speed ratios, we show the competitive ratio of LS is no more than m i n { 1 + s + 2 s t s + s t , 1 + s s t + 1 , 1 + s + 3 s t 1 + s + s t } and its overall competitive ratio is 2 which matches the overall lower bound of the problem.

Development of machining technology for micropatterns with large surface area

With increasing demand for micropatterns such as V-shaped microgrooves and the trend of large surface areas in developing technologies, precision machining technology for micropatterns with large surface areas is expected to play an increasingly important role in today's manufacturing technology. In large-surface micromachining, machining time is much longer than that in general pattern machining and it is not easy to achieve uniform machining accuracy in the entire machined areas because of various factors. Therefore, systematic machining processes and technical development for achieving precision in each process are essential prerequisites to reduce the errors. In this study, we focused on developing machining technologies, which include a machine vision system for precise tool setting, an on-machine measurement system for large-area measurement, and software for tool path generation and simulation, for the fabrication of large-surface micropatterns in an electroless nickel-plated workpiece with single-crystal diamond tools and a 32-in., 675 × 450-mm mold with tens of V-and pyramid-shaped micropatterns.

Asymptotical optimality of WSEPT for stochastic online scheduling on uniform machines

We study the stochastic online scheduling on m uniform machines with the objective to minimize the expected value of total weighted completion times of a set of jobs that arrive over time. For each job, the processing time is a random variable, and the distribution of processing time is unknown in advance. The actual processing time could be known only when the job is completed. For the problem, we propose a policy which is proved to be asymptotically optimal when the processing times and weights are uniformly bounded, i.e. the relative error of the solution achieved by our policy approaches zero as the number of jobs increases to infinity.

Online scheduling on uniform machines with two hierarchies

In this paper we study online scheduling problem on m parallel uniform machines with two hierarchies. The objective is to minimize the maximum completion time (makespan). Machines are provided with different capability. The machines with speed s can schedule all jobs, while the other machines with speed 1 can only process partial jobs. Online algorithms for any 0<s<? are provided in the paper. For the case of k=1 and m=2, and the case of some values of s, k=1 and m=3, the algorithms are the best possible, where k is the number of machines with hierarchy 1, and m is the number of machines. Lower bounds for some special cases are also presented.

Scheduling job classes on uniform machines

We study a scheduling problem with job classes on parallel uniform machines. All the jobs of a given class share a common due-date. General, non-decreasing and class-dependent earliness and tardiness cost functions are assumed. Two objectives are considered: (i) minmax, where the scheduler is required to minimize the maximum earliness/tardiness cost among all the jobs and (ii) minmax-minsum, where the scheduler minimizes the sum of the maximum earliness/tardiness cost in all job classes. The problem is easily shown to be NP-hard, and we focus here on the introduction of simple heuristics. We introduce LPT (Largest Processing Time first)-based heuristics for the allocation of jobs to machines within each class, followed by a solution of an appropriate non-linear program, which produces for this job allocation an optimal schedule of the classes. We also propose a lower bound, based on balancing the load on the machines. Our numerical tests indicate that the heuristics result in very small optimality gaps.

Heuristic algorithms for scheduling on uniform parallel machines with heads and tails

This paper considers the uniform parallel machine scheduling problem with unequal release dates and delivery times to minimize the maximum completion time. For this NP-hard problem, the largest sum of release date, processing time and delivery time first rule is designed to determine a certain machine for each job, and the largest difference between delivery time and release date first rule is designed to sequence the jobs scheduled on the same machine, and then a novel algorithm for the scheduling problem is built. To evaluate the performance of the proposed algorithm, a lower bound for the problem is proposed. The accuracy of the proposed algorithm is tested based on the data with problem size varying from 200 jobs to 600 jobs. The computational results indicate that the average relative error between the proposed algorithm and the lower bound is only 0.667%, therefore the solutions obtained by the proposed algorithm are very accurate.

Batch scheduling on uniform machines to minimize total flow-time

The solution of the classical batch scheduling problem with identical jobs and setup times to minimize flowtime is known for twenty five years. In this paper we extend this result to a setting of two uniform machines with machine-dependent setup times. We introduce an O(n) solution for the relaxed version (allowing non-integer batch sizes), followed by a simple rounding procedure to obtain integer batch sizes.

On the optimality of list scheduling for online uniform machines scheduling

This paper considers classical online scheduling problems on uniform machines. We show the tight competitive ratio of LS for any combinations of speeds of three machines. We prove that LS is optimal when s3 ≥ s2 ≥ s1 = 1 and \({s_3^2\geq s_2^2+s_2s_3+s_2}\), where s1, s2, s3 are the speeds of three machines. On the other hand, LS can not be optimal for all combinations of machine speeds, even restricted to the case of 1 = s1 = s2 < s3. For m ≥ 4 machines, LS remains optimal when one machine has very large speed, and the remaining machines have the same speed.

Scheduling of uniform parallel machines with s-precedence constraints

This paper considers a problem of scheduling s-precedence constrained jobs on machines in parallel which have different speeds. The objective is to minimize the weighted total completion time. The s-precedence relation between two jobs i and j represents the situation where job j is constrained from processing until job i starts processing, which is different from the standard definition of a precedence relation where j cannot start until i completes. An LP-based heuristic procedure is derived for the problem. Numerical experiments are conducted to show that the derived heuristic finds effective solutions.

Non-clairvoyant Scheduling Games

In a scheduling game, each player owns a job and chooses a machine to execute it. While the social cost is the maximal load over all machines (makespan), the cost (disutility) of each player is the completion time of its own job. In the game, players may follow selfish strategies to optimize their cost and therefore their behaviors do not necessarily lead the game to an equilibrium. Even in the case there is an equilibrium, its makespan might be much larger than the social optimum, and this inefficiency is measured by the price of anarchy—the worst ratio between the makespan of an equilibrium and the optimum. Coordination mechanisms aim to reduce the price of anarchy by designing scheduling policies that specify how jobs assigned to a same machine are to be scheduled. Typically these policies define the schedule according to the processing times as announced by the jobs. One could wonder if there are policies that do not require this knowledge, and still provide a good price of anarchy. This would make the processing times be private information and avoid the problem of truthfulness. In this paper we study these so-called non-clairvoyant policies. In particular, we study the RANDOM policy that schedules the jobs in a random order without preemption, and the EQUI policy that schedules the jobs in parallel using time-multiplexing, assigning each job an equal fraction of CPU time. For these models we study two important questions, the existence of Nash equilibria and the price of anarchy. We show that the game under RANDOM policy is a potential game for uniform machines or for two unrelated machines. However, it is not a potential game for three or more unrelated machines. Moreover, we prove that the game under the EQUI policy is a potential game. Next, we analyze the inefficiency of EQUI policy. Interestingly, the (strong) price of anarchy of EQUI, a non-clairvoyant policy, is asymptotically the same as that of the best strongly local policy—policies in which a machine may look at the processing time of jobs assigned to it. The result also indicates that knowledge of jobs’ characteristics is not necessarily needed.

Simulated Annealing Algorithm to Minimize Makespanin Single Machine Scheduling Problem withUniform Parallel Machines

This paper presents a simulated annealing algorithm to minimize makespan of single machine scheduling problem with uniform parallel machines. The single machine scheduling problem with uniform parallel machines consists of n jobs, each with single operation, which are to be scheduled on m parallel machines with different speeds. Since, this scheduling problem is a combinatorial problem; usage of a heuristic is inevitable to obtain the solution in polynomial time. In this paper, simulated annealing algorithm is presented. In the first phase, a seed generation algorithm is given. Then, it is followed by three variations of the simulated annealing algorithms and their comparison using ANOVA in terms of their solutions on makespan.

GA Based Heuristic to Minimize Makespan in Single Machine Scheduling Problem with Uniform Parallel Machines

This paper considers the single machine scheduling problem with uniform parallel machines in which the objective is to minimize the makespan. Four different GA based heuristics are designed by taking different combinations of crossover methods, viz. single point crossover method and two point crossover method, and job allocation methods while generating initial population, viz. equal number of jobs allocation to machines and proportionate number of jobs allocation to machines based on machine speeds. A detailed experiment has been conducted by assuming three factors, viz. Problem size, crossover method and job allocation method on 135 problem sizes each with two replications generated randomly. Finally, it is suggested to use the GA based heuristic with single point crossover method, in which the proportionate number of jobs allocated to machines based on machine speeds.

Online and semi-online hierarchical scheduling for load balancing on uniform machines

In this paper, we consider online and semi-online hierarchical scheduling for load balancing on m parallel uniform machines with two hierarchies. There are k machines with speed s and hierarchy 1 which can process all the jobs, while the remaining m − k machines with speed 1 and hierarchy 2 can only process jobs with hierarchy 2. For the model with the objective to maximize the minimum machine completion time, we show that no online algorithm can achieve bounded competitive ratio, i.e., there exists no competitive algorithm. We overcome this barrier by way of fractional assignment, where each job can be arbitrarily split between the machines. We design a best possible algorithm with competitive ratio 2 k s + m − k k s + m − k for any 0 < s < ∞ . For the model with the objective of minimizing makespan, we give a best possible algorithm with competitive ratio ( k s + m − k ) 2 k 2 s 2 + k ( m − k ) s + ( m − k ) 2 for any 0 < s < ∞ . For the semi-online model with the objective to minimize makespan, where the total job size ( s u m ) is known in advance, we present an optimal algorithm (i.e., an algorithm of competitive ratio 1).

An FPTAS for uniform machine scheduling to minimize makespan with linear deterioration

This paper consider m uniform (parallel) machine scheduling with linear deterioration to minimize the makespan. In an uniform machine environment, all machines have different processing speeds. Linear deterioration means that job's actual processing time is a linear increasing function on its execution starting time. We propose a fully polynomial-time approximation scheme (FPTAS) to show the problem is NP-hard in the ordinary sense.

Minimizing total tardiness on parallel machines with preemptions

The basic scheduling problem we are dealing with in this paper is the following one. A set of jobs has to be scheduled on a set of parallel uniform machines. Each machine can handle at most one job at a time. Each job becomes available for processing at its release date. All jobs have the same execution requirement and arbitrary due dates. Each machine has a known speed. The processing of any job may be interrupted arbitrarily often and resumed later on any machine. The goal is to find a schedule that minimizes the sum of tardiness, i.e., we consider problem Q?r j ,p j =p, pmtn??T j whose complexity status was open. Recently, Tian et al. (J. Sched. 9:343---364, 2006) proposed a polynomial algorithm for problem 1?r j ,p j =p, pmtn??T j . We show that both the problem P? pmtn??T j of minimizing total tardiness on a set of parallel machines with allowed preemptions and the problem P?r j ,p j =p, pmtn??T j of minimizing total tardiness on a set of parallel machines with release dates, equal processing times and allowed preemptions are NP-hard. Moreover, we give a polynomial algorithm for the case of uniform machines without release dates, i.e., for problem Q?p j =p, pmtn??T j .

Hybrid meta-heuristics for minimizing the total weighted completion time on uniform parallel machines

Abstract This paper considers the problem of scheduling n independent jobs on m uniform parallel machines, with the aim of minimizing the total weighted completion time. We present two hybrid meta-heuristics to solve this problem. Based on a large set of instances, an experimental study has been carried out to evaluate our methods.

Competitive ratio of List Scheduling on uniform machines and randomized heuristics

We study online scheduling on m uniform machines, where m−1 of them have a reference speed 1 and the last one a speed s with 0≤s≤1. The competitive ratio of the well-known List Scheduling (LS) algorithm is determined. For some values of s and m=3, LS is proven to be the best deterministic algorithm. We describe a randomized heuristic for m machines. Finally, for the case m=3, we develop and analyze the competitive ratio of a randomized algorithm which outperforms LS for any s.

Online scheduling with reassignment on two uniform machines

In this paper, we investigate the online scheduling problem on two uniform machines, where the last job of each machine can be reassigned after all jobs have been assigned. The objective is to minimize the makespan. We prove that the classical List Scheduling algorithm with the competitive ratio s + 1 s is optimal for s ≥ 1 + 5 2 , where s is the speed ratio between the two machines. Also, we prove the lower bound s + 1 for 1 ≤ s < 1 + 5 2 and design an algorithm that matches the bound.

Optimal semi-online algorithm for scheduling with rejection on two uniform machines

In this paper we consider a semi-online scheduling problem with rejection on two uniform machines with speed 1 and s?1, respectively. A sequence of independent jobs are given and each job is characterized by its size (processing time) and its penalty, in the sense that, jobs arrive one by one and can be either rejected by paying a certain penalty or assigned to some machine. No preemption is allowed. The objective is to minimize the sum of the makespan of schedule, which is yielded by all accepted jobs and the total penalties of all rejected ones. Further, two rejection strategies are permitted thus an algorithm can propose two different schemes, from which the better solution is chosen. For the above version, we present an optimal semi-online algorithm H that achieves a competitive ratio ? H (s) as a piecewise function in terms of the speed ratio s.