A projective algorithm for preemptive open shop scheduling with two multiprocessor groups
Section snippets
Preemptive open shop scheduling with multiprocessor groups
In this paper we study a special multiprocessor open shop scheduling problem in which a set of jobs has to be scheduled on a collection of processors. The set of processors is partitioned into two groups, and , and each job consists of single-processor and multiprocessor operations, denoted by () and (), respectively. The operations are dedicated to processors, i.e., requires processor for time units, while requires all
Problem formulation
A feasible schedule can be divided into disjoint slices of the following four types: (a) multiprocessor operations on and single-processor operations or idle time on the processors in , (b) multiprocessor operations on both processor groups, (c) single-processor operations or idle time on processors in and multiprocessor operations on , and (d) single-processor operations or idle time on all processors. By permuting the slices of a feasible schedule we can always obtain a feasible
The polynomial time algorithm
In this section we firstly define and analyze a special integer program in two dimensions and then show how to reduce the open shop scheduling problem to the special integer program. As a by-product, we obtain the desired polynomial time algorithm.
Final remarks
Despite the positive result presented in this paper, an important structural question remained unanswered: Is it true that the makespan of the problem with preemptions at integral times only is always , where are part of an optimal solution for the relaxation of (2)? It has been shown in [3] that when all jobs are binary, this property holds. However, we have neither a proof, nor a counterexample for the general case. Nevertheless, the projective technique
Acknowledgements
The authors are grateful to an anonymous referee for constructive comments that helped to improve the presentation. The research of Tamás Kis was supported by the János Bolyai research grant BO/00380/05. The research of Wieslaw Kubiak was supported by the NSERC research grant OGP0105675.
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