Abstract
This paper considers multiprocessor task scheduling in a multistage hybrid flow-shop environment. The objective is to minimize the make-span, that is, the completion time of all the tasks in the last stage. This problem is of practical interest in the textile and process industries. A genetic algorithm (GA) is developed to solve the problem. The GA is tested against a lower bound from the literature as well as against heuristic rules on a test bed comprising 400 problems with up to 100 jobs, 10 stages, and with up to five processors on each stage. For small problems, solutions found by the GA are compared to optimal solutions, which are obtained by total enumeration. For larger problems, optimum solutions are estimated by a statistical prediction technique. Computational results show that the GA is both effective and efficient for the current problem. Test problems are provided in a web site at www.benchmark.ibu.edu.tr/mpt-hfsp.
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Appendix: The rationale of the lower bound formulations
Appendix: The rationale of the lower bound formulations
The lower bound given in formula (1) is a stage-based lower bound similar to the lower bound suggested by Santos et al 43 for scheduling ‘single-processor’ jobs in hybrid flowshops. Here, associated with each stage i, i=1,…,m, is a lower bound on the make-span, say LB(i). The overall lower bound LB 1 is the maximum of these bounds, that is, LB 1 = max i∈M LB(i), where LB(i) is defined as follows:
The logic behind the proof for the LB(i) formulation is similar to the one used in the proof provided by Santos et al
43 It is based on the assumption that no idle times occur on the processors throughout the duration of the schedule. Under this assumption, the time needed to start processing on any machine at stage i is at best equal to 
. The minimum time required to finish processing of the jobs at the remaining stages i+1 through to m is at best 
The middle part of the LB(i) formulation pertains to the bound on the duration of the processing of jobs at stage i. The minimum for the duration of the processing of jobs at stage i occurs when the constraints on the simultaneous processing on size[i,j] processors are not respected and the jobs can be preempted as often as required to allow an even distribution of the total work content associated with stage i. The total work content for stage i is given by 
, and when evenly distributed over all the processors, it gives rise to a duration of 
Looking at stage i from a different perspective, it can also be thought that there are size[i,j] replicates of each job j, each with the duration p[i,j], so that there are altogether 
‘single-processor’ jobs in a set J′ to be scheduled on m
i
processors at stage i. The make-span associated with stage i is then bounded from below by 
(see, eg, Syslo et al,44 p 502). But 
and this completes the proof of LB
1.
LB given in formula (2) differs from LB
1 by the refinement associated with the distribution of work content at stage i, i=1, …, m. We can be sure that jobs with size[i,j]>m
i
/2 (ie, jobs j∈A
i
) will have at least one common processor, on which they all will be scheduled, and the best possible way to do this is that they are scheduled one after the other with no inserted idle time. This scheduling will give rise to a duration of magnitude 
on that bottleneck processor.
From the rest of the jobs, jobs with size[i,j]=m
i
/2 (ie, jobs j∈B
i
) can best be scheduled such that their work content is distributed evenly on the processors. This gives rise to a duration of magnitude 
.
Finally, jobs with size[i,j]<m i /2 will, in the best case, be scheduled to fit in to the idle times on the processors not used by the set of jobs in A i so that no idle time and hence no extension on the overall duration occurs.
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şerifoğlu, F., Ulusoy, G. Multiprocessor task scheduling in multistage hybrid flow-shops: a genetic algorithm approach. J Oper Res Soc 55, 504–512 (2004). https://doi.org/10.1057/palgrave.jors.2601716
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DOI: https://doi.org/10.1057/palgrave.jors.2601716