O. , *. }. , O. , and U. Noeud, L'union disjointe de ces chemins donne toutes les arêtes du graphe Chaque noeud v ? O fait partie d'exactement deux chemins : l'un représente la gamme de son ordre de fabrication et l'autre la séquence des opérations sur la machine à laquelle il est affecté. Pour un noeud v ? O, on définit son successeur de gamme par r(v) ? V \ {0} et son successeur de machine par m(v) ? V \ {0}. De la même manière, ses prédécesseurs sont désignés par r ?1 (v) De la même manière, le noeud de fin artificiel * a |J| + |M| arêtes entrantes et pas d'arêtes sortantes. En total, le graphe contient |E| = 2 |O| + |J| + |M| arêtes. Le graphe conjonctif peut être utilisé pour déterminer les dates de début S v des opérations v ? O, Pour chaque ordre de fabrication et pour chaque machine poids l u, v ? N 0 est affecté à chaque arête (u, v) ? E. Ce poids impose une durée minimale entre le début de deux opérations adjacentes : S u + l u, v ? S v pour chaque arrête

?. O. Pour-chaque-opération-v, S. Sa-date-de-début-est-déterminée-par, and =. , on définit le poids des arêtes de la manière suivante Pour les arêtes (0, o 1, j ) ? E qui relient le noeud de départ artificiel 0 avec la première opération o 1, j de l'ordre de fabrication j, le poids de l'arête est fixé à la date de sortie r j de l'ordre de fabrication j. Pour les arêtes (0, o m ) ? E qui relient le noeud de départ artificiel 0 avec la première opération o m prévue sur la machine m ? M, le poids de l'arête est fixé à zéro. Pour les arêtes des gammes (v, r(v)) ? E avec v 0, le poids de l'arête correspond à la durée du traitement p v de l'opération v. Pour les arêtes de séquence machine (v, m(v)) ? E avec v 0, le poids de l'arête correspond à la somme p v + s(? v , ? m(v) ) qui représente la durée du traitement de l'opération v et le temps de réglage qui dépend de l'opération v et de son successeur machine m(v) Ceci représente les machines sans traitement par lot. Ils nous reste à proposer une représentation pour les machines avec regroupement par lot, Ces machines peuvent être modélisées en modifiant la structure du graphe ou en adaptant les poids des arêtes. Les deux paragraphes suivants présentent les deux alternatives. Nous rappelons que chaque regroupement doit respecter la capacité de la machine et doit se faire uniquement par famille. Ces deux contraintes doivent toujours être satisfaites. Nous ne détaillons pas les contrôles connexes pour vérifier la faisabilité d'une solution car nous souhaitons nous concentrer sur les parties essentielles des deux représentations

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