Int J Adv Manuf Technol DOI 10.1007/s00170-016-9557-5 ORIGINAL ARTICLE Solving a capacitated flow-shop problem with minimizing total energy costs Oussama Masmoudi 1 · Alice Yalaoui 1 · Yassine Ouazene 1 · Hicham Chehade 2 Received: 8 July 2016 / Accepted: 3 October 2016 © Springer-Verlag London 2016 Abstract In this paper, a single-item capacitated lot-sizing problem in a flow-shop system with energy consideration is addressed. The planning horizon is split into T periods where each one is characterized by a duration, an electricity cost, a maximum peak power and a demand. This problem is NP-hard, since its simple version is known to be NP- hard. Therefore, to deal with the complexity and to find good quality solutions in a reasonable time, a fix-and-relax heuristic and a genetic algorithm are developed. Computa- tional experiments are performed on different instances to show the efficiency of these proposed heuristics. To eval- uate their performances, problems of different scales have been studied and analyzed. Keywords Capacitated lot-sizing problem · Flow-shop · Energy · Heuristics · Genetic algorithm 1 Introduction Due to the increase of energy price and environmental con- straints, there is an important requirement for manufacturing companies to control their energy consumption. As men- tioned by Cheng and Srai [7], manufacturing systems should take into account contexts of economy, society, environ- ment, and technology. Wang and Li [37] underlined that the  Oussama Masmoudi oussama.masmoudi@utt.fr 1 University of Technology of Troyes, ICD, LOSI (UMR-CNRS 6281), 12 rue Marie Curie, CS 42060, 10004, Troyes, France 2 OPTA LP, Technopole de l’Aube, 2 rue Gustave Eiffel, 10430 Rosires prs Troyes, France industrial sector is the first energy consumer and greenhouse gas emitter in the world. Mouzon and Yildirim [28] and Yildirim and Mouzon [40] proposed a mathematical model that aims to minimize energy consumption and reduce total completion time of a single machine. A generalized case for a set of machines is presented by Liu et al. [21] with the objective to mini- mize total tardiness and total energy consumption. Hait and Artigues [17] studied a foundry scheduling problem with energy and human constraints. A peak power demand con- straint is introduced in the mathematical model proposed by Bruzzone et al. [6] with the objective to minimize total tardiness and makespan. In Xu et al’s. work [38], a schedul- ing problem for a hybrid flow-shop system with minimizing power demand is studied. The peak power load, energy consumption and associated carbon footprint are consid- ered by Fang et al. [10] for a flow-shop scheduling system. Mansouri et al. [23] considered a two machines flow-shop system. A mixed-integer linear multi-objective model is developed to find the Pareto frontier which is a compro- mise between makespan and total energy consumption. Two different problems are proposed by Wang and Li [36]. The first one aims to minimize the energy consumption and the second one aims to minimize the energy cost without affect- ing the throughput. Considering the variation of electricity cost according to periods, a just-for-peak buffer inventory model is presented by Fernandez et al. [11]. The objec- tive is to establish the production planning during ON (with high price of electricity) and OFF (with low price of elec- tricity comparing the ON one) peak periods. Under time of use rates, Zhang et al. [42] developed a mathematical model with the objective to minimize energy cost and car- bon footprint without compromising production throughput. Other research studies taking into account the variation of energy price according to periods are presented by Shrouf