An Ant Colony Optimization algorithm to solve a 2-machine bicriteria flowshop scheduling problem Vincent T’kindt * , Nicolas Monmarch e, Fabrice Tercinet, Daniel La€ ugt Laboratoire d’Informatique, Ecole d’Ing enieurs en Informatique pour l’Industrie, 64 avenue Jean Portalis, 37200 Tours, France Received 1 October 2000; accepted 1 May 2001 Abstract Consider the 2-machine flowshop scheduling problem with the objective of minimizing both the total completion time and the makespan criteria. The latter is assumed to be optimized prior to the former. In view of the NP-hardness of the problem an Ant Colony Optimization approach is proposed to solve it. The heuristic also uses feature of Simulated Annealing search and local search algorithms. Computational experiments show its effectiveness compared to existing heuristics. The extension to the total completion time problem is also studied. Ó 2002 Published by Elsevier Science B.V. Keywords: Scheduling; Multiple objective programming; Ant Colony Optimization 1. Introduction Consider the 2-machine flowshop scheduling problem with n jobs to schedule. It is assumed that the processing of a job cannot be interrupted. Let the processing times of job i be referred to as a i on the first machine and b i on the second machine. We note C i the completion time of job i on ma- chine 2 where i ¼ 1; ... ; n. The makespan crite- rion, noted C max , is defined as the maximum completion time of jobs on machine 2 whilst the total completion time criterion, noted P C i , is defined as the sum of completion time of jobs on machine 2. We assume that the total comple- tion time criterion has to be minimized subject to the condition that the makespan computed is minimum. As these two criteria are regular per- formance measure, the search for an optimal schedule can be restricted to the set of permutation schedules. This problem is referred to as a lexico- graphical minimization and is noted F 2kLexðC max ; P C i Þ [25]. Whilst the F 2kC max problem is poly- nomially solvable by the so-called Johnson’s algorithm [15], the F 2k P C i problem is known to be strongly NP-hard. The lexicographical prob- lem considered is also strongly NP-hard [1]. Multiple criteria scheduling problems have been subject to a growing interest in the last decade. Some earlier surveys [6,8,14,17] consider a basic decomposition of such problems and mainly focus on single machine problems. Multicriteria sched- uling literature has recently been revisited ac- cording to multicriteria optimization concepts [25]. European Journal of Operational Research 142 (2002) 250–257 www.elsevier.com/locate/dsw * Corresponding author. Tel.: +33-2-47-36-14-14; fax: +33-2- 47-36-14-22. E-mail address: tkindt@e3i.univ-tours.fr (V. T’kindt). 0377-2217/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII:S0377-2217(02)00265-5