We consider the problem of minimizing the makespan in a flowshop involving
maximal and minimal time lags. Time lag constraints generalize the classical
precedence constraints between operations. We assume that such constraints are
only defined between operations of the same job. We propose a solution method
and present several extensions.Comment: 2 pages. Also available at http://hal.inria.fr/inria-0000014

Fondrevelle, Julien

Oulamara, Ammar

Portmann, Marie-Claude

English

arXiv

2 pagesWe consider the problem of minimizing the makespan in a flowshop involving maximal and minimal time lags. Time lag constraints generalize the classical precedence constraints between operations. We assume that such constraints are only defined between operations of the same job. We propose a solution method and present several extensions

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HAL Id: inria-00000149https://hal.inria.fr/inria-00000149Submitted on 6 Jul 2005HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Minimizing makespan in flowshop with time lagsJulien Fondrevelle, Ammar Oulamara, Marie-Claude PortmannTo cite this version:Julien Fondrevelle, Ammar Oulamara, Marie-Claude Portmann. Minimizing makespan in flowshopwith time lags. Models and Algorithms for Planning and Scheduling Problems - MAPSP’2005, Jun2005, Sienne/Italie. ￿inria-00000149￿inria-00000149, version 1 - 6 Jul 2005Minimizing makespan in flowshop with time lagsJulien Fondrevelle (Speaker) ∗ Ammar Oulamara † Marie-Claude Portmann ‡1 IntroductionWe consider the problem of minimizing the makespan in a flowshop involving maximal and min-imal time lags, denoted by Fm|θmin, θmax|Cmax. Time lag constraints generalize the classicalprecedence constraints between operations: minimal (respectively maximal) time lags indeedspecify that the time elapsed between two operations must be lower- (respectively upper-)bounded. We assume that such constraints are only defined between operations of the samejob.Time lags may be used to model various industrial situations. In several processes thedelay between some critical operations must not be too long otherwise the products may bedeteriorated. Such examples can be found in food industry with perishable products or inchemistry with unstable chemicals. Minimal time lags may correspond to transportation timesbetween the machines or to communication delays between processors. Detailed case studies ofshop problems involving time lags can be found in Deppner’s thesis [1].From a computational complexity viewpoint, the existence of time lags make the problemNP-hard, even for special cases with two machines: Yu et al. [3] showed that the two-machineflowshop problem with minimal time lags and unit processing times (F2|pi,j = 1, θmin|Cmaxis strongly NP-hard. Besides, the two-machine flowshop with constant maximal time lags(F2|θmax = θ|Cmax) has been shown to be strongly NP-hard as well, even if we restrict theproblem to permutation schedules [2].2 Solution approachIn the remainder of the paper, we consider only permutation schedules, which are not dominanteven for two-machine problems. Since the classical Fmπ||Cmax problem is strongly NP-hardfor m ≥ 3, the corresponding problem with time lags is strongly NP-hard as well. A branch-and-bound procedure has been proposed in [2] to solve this problem, when time lags are onlydefined between successive operations of the jobs. The algorithm makes use of a fundamental∗julien.fondrevelle@loria.fr. MACSI Project LORIA-INRIA Lorraine, Ecole des Mines de Nancy, Parcde Saurupt, 54042 Nancy, France.†ammar.oulamara@loria.fr. MACSI Project LORIA-INRIA Lorraine, Ecole des Mines de Nancy, Parc deSaurupt, 54042 Nancy, France.‡marie-claude.portmann@loria.fr. MACSI Project LORIA-INRIA Lorraine, Ecole des Mines de Nancy,Parc de Saurupt, 54042 Nancy, France.1result which claims that the restricted problem of constructing an optimal schedule for a givenjob sequence is polynomial.We show that this result remains valid when time lags are defined between arbitrary couplesof operations of the same jobs and when an arbitrary regular criterion is considered.We propose also other extensions of the solution method:• It is possible to take into account job release dates by adding a dummy initial machinewith zero processing times and minimal time lags between this machine and the first realone equal to the release date.• By symmetry, tails can be introduced using a dummy final machine. The resulting problemis equivalent to minimizing the maximum lateness (Lmax).Concernig the general flowshop problem, where non-permutation schedules are also takeninto account, we define the restricted problem as constructing an optimal schedule for a givenjob sequence on first machine. We show that this problem is:• polynomial for 2 machines with minimal time lags only• strongly NP-hard for 2 machines with maximal time lags• strongly NP-hard for 3 machines with minimal time lagsTherefore the proposed approach can only be used for the 2-machine flowshop problem withminimal time lags only.References[1] F. Deppner (2004). Ordonnancement d’atelier avec contraintes temporelles entre opra-tions. PhD thesis, Institut National Polytechnique de Lorraine, France (in French).[2] J. Fondrevelle, A. Oulamara and M.-C. Portmann. Permutation flowshop schedul-ing problems with maximal and minimal time lags. Computers and Operations Research,(to appear).[3] W. Yu, H. Hoogeveen and J.K. Lenstra (2004). Minimizing makespan in a two-machine flow shop with delays and unit-time operations is NP-hard. Journal of Scheduling,7, 333-348.2

Minimizing makespan in flowshop with time lags

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Minimizing makespan in flowshop with time lags

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