Abstract
The partially-concurrent open shop scheduling problem is introduced. The standard open shop scheduling problem is generalized by allowing some operations to be processed concurrently. A schedule for the partially-concurrent problem is represented by a digraph. We show that the scheduling problem is equivalent to a problem of orienting a given undirected graph, called a conflict graph. The schedule digraph is then modeled by a matrix, generalizing the rank matrix representation. The problem is shown to be NP-hard. The representation can be used to generalize previously discussed standard open shop issues. It is demonstrated by generalizing the theoretical concept of reducibility and also by using standard open shop heuristic solutions to the partially-concurrent scenario. The presented problem is directly motivated from a real-life timetabling project of assigning technicians to airplanes in an airplane garage.
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Notes
The results for 0 % concurrency are worse than those presented by Bräsel et al. (1993), since the machine lower bound used herein is looser than the (machine and job) lower bound used by Bräsel et al.
The lower bound of a standard OSS is the maximal sum of processing times in a given machine or job. In a unit OSS with \(m>n\) it equals m. For a unit PCOSS the lower bound is \(\max (n, \{\tilde{m_j}\})\), where \(\tilde{m_j}\) is the maximum independent set of the jth job. The problem of finding a maximum independent set is known to be NP-hard.
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Grinshpoun, T., Ilani, H. & Shufan, E. The representation of partially-concurrent open shop problems. Ann Oper Res 252, 455–469 (2017). https://doi.org/10.1007/s10479-015-1934-1
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DOI: https://doi.org/10.1007/s10479-015-1934-1