550 A NEW HEURISTIC FOR THE PERMUTATION FLOWSHOP PROBLEM Pawel J. Kalczynski MS#103, College of Business Administration, University of Toledo, Toledo, OH 43606-3390, HPawel.Kalczynski@utoledo.edu H, 419-530-2258 Jerzy Kamburowski MS#103, College of Business Administration, University of Toledo, Toledo, OH 43606-3390, HJerzy.Kamburowski@utoledo.edu H, 419-530-4361 ABSTRACT For over twenty years the NEH heuristic of Nawaz, Enscore, and Ham has been commonly regarded as the best heuristic for minimizing the makespan in permutation flow shops. We present a new heuristic that outperforms NEH. The computational complexity of both heuristics is the same. Keywords: Scheduling, flow shop, makespan, heuristic. INTRODUCTION The ancient flow shop problem is defined as follows. A set of n jobs, J = {1,2,...,n}, available at time zero has to be processed in a shop with m machines MB 1 B, MB 2 B,…,MB m B. Each job is processed first on MB 1 B, next on MB 2 B, and so on, and lastly on MB m B. No machine can process more than one job at a time, no preemption is allowed, all setup times are included into the job processing times, and there is unlimited storage between the machines. The problem, commonly referred to as Fm||CB max B, is to determine a schedule that minimizes the completion time of the last job, also known as the makespan. The schedule with the same job ordering on every machine is called a permutation schedule, and the permutation flow shop problem, Fm|prmu|CB max B, is to find a job sequence that minimizes the makespan. For m=2 and m=3, the search of the optimal schedule can be restricted to permutation schedules, but the optimal schedules may have different job orderings on different machines when m > 3; see e.g. Potts et al. (1991). Both F2||CB max B and F2|prmu|CB max B can be solved in O(nlogn) time by the well known algorithm of Johnson (1954), but F3|prmu|CB max B is strongly NP-hard; see Garey et al. (1976). Moreover, there is no a polynomial time approximation scheme for solving Fm|prmu|CB max B, unless P = NP; see Williamson et al. (1997). Since the pioneer’s work of Johnson (1954) dozens of heuristics have been proposed for solving Fm|prmu|CB max B, but the O(mnP 2 P) NEH heuristic of Nawaz, Enscore & Ham (1983) is known to easily outperform other heuristics, and be even fairly competitive with metaheuristics; see e.g. Watson et al. (2002), Framinan et al. (2004), Ruiz & Maroto (2005), Reza Hejazi & Saghafian (2005), and Kalczynski & Kamburowski (2007). The purpose of this paper is to present a new simple O(mnP 2 P) heuristic that outperforms NEH.