Minimizing resource consumption on uniform parallel machines with a bound on makespan
Introduction
Parallel machine scheduling problems have been studied intensively in the past decades. Pinedo [1] pointed out that a bank of machines in parallel is a setting that is important from both a theoretical and a practical point of view. From a theoretical viewpoint, it is a generalization of the single machine and a special case of the flexible flow shop. From a practical point of view, it is important because the occurrence of resources in parallel is common in the real world. Also, techniques for machines in parallel are often used in decomposition procedures for multistage systems.
However, classical scheduling problems typically strike for increased efficiency in terms of “time”. For instance, minimizing the makespan is achieved by finishing all the jobs in the shortest possible time. In response to the effects of global warming, environmental protection and carbon emission cuts have become more crucial issues. Moreover, the problems of pollution and excessive use of natural resources are prevalent in many industries. For instance, the liquid crystal display industry uses an excessive amount of water, and the petrochemical industry produces severe air pollution. Often, newer machines have faster processing speeds in manufacturing systems. At the same time, the amounts of resource consumption might be different between machines. Motivated by these observations, we introduce a uniform parallel machine scheduling problem where the objective is to minimize the resource consumption given that all the jobs must be finished within a period of time, i.e., the makespan cannot exceed an upper bound.
Cheng and Sin [2] and Mokotoff [3] provided extensive reviews of the research on parallel machine scheduling. The majority of the research used the makespan as the objective function [4], [5], [6], [7]. In addition, some researchers considered the same problem under various environments such as assuming the machines have the maintenance activities [8], [9], or assuming the job processing is shortened due to learning effects [10], or assuming the machines are unrelated [11], [12], [13]. In these articles, the makespan is the primary objective to be minimized. Due to the environmental protection issue, it might be reasonable to sacrifice a little bit time efficiency in exchange for energy savings, reducing the pollutions or excessive usages of natural resources. This motivates us to move the makespan (time efficiency) to the secondary objective as a constraint, and the resource consumption becomes the primary objective to be minimized. To the best of our knowledge, this problem has never been studied before. The remainder of this paper is organized as follows. In Section 2, we formulate the problem. In Section 3, we show that the problem is strongly NP-hard and derive a tight lower bound for this problem. In Section 4, we develop a particle swarm optimization algorithm. The computational experiments are given in Section 5, and the conclusion is presented in Section 6.
Section snippets
Problem formulation
The description of the proposed problem is as follows. There are a set of n independent jobs to be scheduled on a set of uniform parallel machines . All the jobs are available for processing at time 0. Each job has to be processed on either one of the machines. Each machine can process one job at a time, and once a job starts to be processed, it must be completed without interruption. Each job has a processing time , and each machine has a speed . Moreover,
Some results
In this section, we will first prove the problem is strongly NP-hard. The complexity of the problem can be established by a reduction from the 3-partition problem, which is known to be strongly NP-hard [14] in polynomial time. Theorem 1 The problem is strongly NP-hard. Proof At first an instance I of the 3-partition problem is given: Given a set of positive integers , , …, and such that for and , is there a partition of
Heuristic algorithm
In this section, we will develop a heuristic algorithm for this NP-hard problem. Since is the unit cost of processing jobs on machine , it is natural to assign as many jobs as possible to the machine with the lowest rate . If this machine is full or no more jobs can be inserted, we continue to assign jobs to the machine with the second lowest rate. The process is repeated until all jobs have been assigned. In addition, let K be the sufficient number of machines such that all the
Computational experiments
In order to evaluate the performance of the proposed algorithm, a computational experiment is conducted. The algorithm is coded in Fortran 90 and run on a personal computer with 2.66 GHz Intel Core 2 Quad CPU Q9400 and 3.25 GB RAM under Windows XP. The computational experiment consists of three parts.
In the first part of the experiment, the number of jobs (n) is fixed at 11. The processing times () are generated from three discrete uniform distributions, namely U(1, 100), U(100, 200), and U
Conclusion
With the current emphasis on environmental protection, managing natural resources efficiently and/or reducing the amount of carbon emissions has become more important than ever. In this paper, we introduce a uniform parallel machine scheduling problem where the objective is to minimize the total resource consumption given that the maximum completion time of all jobs does not exceed a certain level. We showed that the problem is NP-complete, derived a tight lower bound, and developed a heuristic
Acknowledgements
The authors are grateful to the editor, the associate editor, and the referees, whose constructive comments have led to a substantial improvement in the presentation of the paper. The paper was supported by the National Science Council of Taiwan, ROC, under NSC 100-2221-E-035-029-MY3. Ji was supported in part by the National Natural Science Foundation of China (Grant No. 10801121), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y6100598) and the Contemporary Business and
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