Minimizing the earliness and tardiness cost of a sequence of tasks on a single machine
RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 2, pp. 165-187.

Supposons que n tâches doivent être exécutées par une machine dans un ordre fixé a priori. On connaît pour chaque tâche sa durée, la date souhaitée de son exécution et les coûts unitaires associés d’avance et de retard. On cherche à allouer à chaque tâche une date d’exécution de sorte que le coût total des avances et des retards soit minimum. Un algorithme de complexité O(nlogn) a été proposé par Garey et al. pour le cas particulier de coûts unitaires. Nous proposons d’abord un algorithme de complexité O(n 2 logn) qui étend l’algorithme Garey et al. au cas de coûts asymétriques indépendants des tâches. Pour le cas général de coûts asymétriques dépendants des tâches, nous proposons un algorithme de complexité O(n 3 logn) fondé sur une propriété de dominance permettant de ramener le problème à la recherche d’un chemin de coût minimum dans un graphe valué sans circuit.

Assume that n tasks must be processed by one machine in a fixed sequence. The processing time, the preferred starting time and the earliness and tardiness costs per time unit are known for each task. The problem is to allocate each task a starting time such that the total cost incurred by the early and tardy tasks is minimum. Garey et al. have proposed a nice O(nlogn) algorithm for the special case of symmetric and task-independent costs. In this paper we first extend that algorithm to the case of asymmetric and task-independent cost without increasing its worst-case complexity. For the general case of asymmetric and task-dependent costs, we propose an O(n 3 logn) algorithm based on a strong dominance property that yields to model the scheduling problem as a minimum cost path in a valued directed acyclic graph.

Mots clés : scheduling, algorithm, complexity
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Chrétienne, Philippe. Minimizing the earliness and tardiness cost of a sequence of tasks on a single machine. RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 2, pp. 165-187. http://archive.numdam.org/item/RO_2001__35_2_165_0/

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