Annals of Operations Research https://doi.org/10.1007/s10479-018-2967-z ORIGINAL RESEARCH Two-machine flowshop scheduling problem with coupled-operations Nadjat Meziani 1,3 · Ammar Oulamara 2 · Mourad Boudhar 3 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract This paper addresses a generalization of the coupled-operations scheduling problem in the context of a flow shop environment. We consider the two-machine scheduling problem with the objective of minimizing the makespan. Each job consists of a coupled-operation to be processed first on the first machine and a single operation to be then processed on the second machine. A coupled-operation contains two operations separated by an exact time delay. The single operation can start on the second machine only when the coupled-operation on the first machine is completed. We prove the NP-completeness of two restricted versions of the general problem, whereas we also exhibit several other well solvable cases. Keywords Flowshop · Coupled-operations · Complexity · Polynomial time algorithms 1 Introduction The coupled-operations scheduling problem was first introduced by Shapiro (1980). It con- sists of a set of n jobs to be scheduled on a single machine. Each job j consists of a coupled-operation. A coupled-operation is made of two operations that have to be processed with an intermediate exact delay L j , i.e., if C 1 j and S 2 j denote the completion time and the start time of the first and the second operation of job j , respectively, then, in a valid schedule, we have S 2 j − C 1 j = L j . Each job j is thus described by a triplet (a j , L j , b j ), where a j and b j denote the processing times of the first and the second operation of job j , respectively, B Ammar Oulamara oulamara@loria.fr Nadjat Meziani ro_nadjet07@yahoo.fr Mourad Boudhar mboudhar@yahoo.fr 1 Abderrahmane Mira University, Bejaia, Algeria 2 LORIA Laboratory, UMR CNRS 75003, University of Lorraine, Campus Scientifique, 615 Rue du Jardin-Botanique, 54506 Vandoeuvre-lès-Nancy, France 3 RECITS Laboratory, Faculty of Mathematics, University of Sciences and Technology Houari Boumediene (USTHB), BP 32, El-Alia, 16111 Bab-Ezzouar, Algiers, Algeria 123