, HYBRID HEURISTICS FOR THE PERMUTATION FLOW SHOP PROBLEM M.G. RAVETTI, F.G. NAKAMURA, C.N. MENESES, M.G.C. RESENDE, G.R. MATEUS, AND P.M. PARDALOS ABSTRACT. The Flow Shop Problem (FSP) is known to be NP-hard when more than three machines are considered. Thus, for non-trivial size problem instances, heuristics are needed to find good orderings. We consider the permutation case of this problem. For this case, denoted by F|prmu|Cmax, the sequence of jobs has to remain the same at each machine. We propose and test two hybrid heuristics, combining elements from the standard Greedy Randomized Adaptive Search Procedure (GRASP), Iterated Local Search (ILS), Path Relinking (PR) and Memetic Algorithm (MA). The results obtained are shown to be competitive with existing algorithms. 1. I NTRODUCTION The Flow Shop Problem (FSP) is a scheduling problem in which n jobs have to be pro- cessed by m machines. The problem is to find the sequence of jobs for each machine to minimize completion time, also known as makespan (6). This problem is NP-Hard for m > 3 (8; 13). Several papers in the literature address this problem, proposing models, heuristics, and bounds. Dannenbring (9) tested several heuristics. Nawaz, Enscore, and Ham (16) presented a polynomial time algorithm (NEH), finding interesting results. Until now, NEH is one of the best polynomial time heuristics for this problem. Taillard (26) presented an improvement in the complexity of the NEH algorithm, a heuristic based on tabu search, and a useful characterization of the distribution of the objective function. Tail- lard (27) proposed a series of test problems with strong upper bounds. The running time required by Taillard’s tabu search heuristic was not given and he focused on solution qual- ity. Ben-Daya and Al-Fawzan (7) implemented and tested an improved variant of Taillard’s tabu search, reporting times and comparing their performance with Ogbu and Smith’s sim- ulated annealing algorithm (17). However, they did not match all of Taillard’s results for large instances. St¨ utzle presented and tested an Iterated Local Search (ILS) heuristic ob- taining good results. Ruiz and Maroto (21) compared 25 methods, from very basic ones, such as Johnson’s algorithm (12), to more sophisticated ones, such as tabu search and sim- ulated annealing. The results of their study concluded that NEH was the best polynomial- time heuristic, while St¨ utzle’s ILS (25) and Reeves’s genetic algorithm (18) were the best metaheuristic-based heuristics. Ruiz et al. (22) proposed a new memetic algorithm for this problem, obtaining improved results when compared with the ILS and tabu search. The same authors followed up with another paper in the same direction (23), proposing and test- ing two genetic algorithms and obtaining strong results. Agarwal, Collak, and Eryarsoy (1) implemented a heuristic improvement procedure based on adaptive learning and applied it to the NEH algorithm, leading to additional improvements. However, for larger instances, their results were of poor quality and their algorithm was computationally intensive. These Date: November 5, 2006. Key words and phrases. Heuristics, GRASP, Iterated Local Search, Memetic Algorithm, Flow Shop Problem. AT&T Labs Research Technical Report TD-6V9MEV.. 1