Scheduling a batch-processing machine subject to precedence constraints, release dates and identical processing times
Section snippets
Introduction and problem formulation
Let n jobs J1,J2,…,Jn and a single machine that can handle batch jobs at the same time be given. There are precedence relations ≺ between the jobs. Each job Jj has an integer processing time pj and an integer release date rj. The jobs are processed in batches, where a batch is a subset of jobs and we require that the batches form a partition of the set of all jobs. The processing time of a batch is equal to the longest processing time of all the jobs in the batch.
Suppose that a batch sequence
Release-date modification
Let us consider the problem , where f is any regular objective function. This problem can be solved by a simple algorithm.
We suppose that the job enumeration given here is topological, i.e., for any two jobs Ji and Jj with Ji≺Jj, we must have i<j. According to Brucker [1], a topological job enumeration can be obtained in O(n2) time by the standard “Algorithm Topological Enumeration”.
If Ji and Jj are two jobs such that Ji≺Jj, then the starting time of the batch that
Job system under identical-processing times
DefineIt is clear that there must be an optimal schedule for such that the starting time and the completion time of every batch belong to . Hence, it suffices to consider the schedule with batch starting times in .
Given an instance of the problem , let . We define layers of the jobs in the following way: is called the i-th layer of
Makespan minimization
The problem , denoted by in the sequel, can easily be solved by the following algorithm. Algorithm 4.1.1 Makespan minimization batching rule. At each point, form the next last batch by including all unbatched jobs that have no unbatched successors. Clearly, the complexity of Algorithm 4.1 is O(n2). The correctness of Algorithm 4.1 is implied by the following theorem. Theorem 4.1.2 The makespan of obtained by Algorithm 4.1 is . Proof Suppose that BS=(B1,B2,…,Bk) is the batch sequence obtained
Conclusion
The parallel-batching scheduling problem was studied in this paper. The restriction on jobs with identical processing times largely simplifies the problem, but the presence of the precedence constraints between jobs increases the hardness of the problem. We showed in this paper that the problem can be solved in O(n2) time, can be solved in O(n3) time, and can be solved in O(n7) time. We gave
Acknowledgements
We are grateful for the constructive comments of the referees on an earlier version of this paper. This research was supported in part by The Hong Kong Polytechnic University under a grant from the ASD in China Business Services. The last two authors were also supported in part by the National Natural Science Foundation of China under the grant number 10371112.
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2012, Computers and Industrial EngineeringCitation Excerpt :On a sum-batch machine jobs are completed sequentially and the processing time of a batch is equal to the sum of the processing times of all the jobs in the batch (e.g. Gribkovskaia, Lee, Strusevich, & de Werra, 2006; Schaller, 2007). A max-batch machine treats the jobs simultaneously, as in our problem, and the processing time of a batch is equal to the time of the longest job, again similar to our case (e.g. Cheng, Yuan, & Yang, 2005; Gupta & Sivakumar, 2006; Koh, Koo, Kim, & Hur, 2005). Scheduling problems for systems with different types of batch-processing machines used together or in combination with discrete processing machines have also been treated in literature (e.g. Oulamara, 2007; Oulamara & Finke, 2001; Sung & Kim, 2002).