ELSEVIER European Journal of Operational Research 78 (1994) 146-161 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Adjustment of heads and tails for the job-shop problem J. Carlier a,,, E. Pinson b a URA CNRS, HEUDIASYC, Universitg de Technologie de Compi~gne, 60200 Compi~gne Cede.x, France b Institut de Math~matiquesAppliqu~es, Universit~Catholique de l'Ouest, B.P. 808, 49008Angers Cedex 01, France Abstract The efficiency of recent enumerative methods for the job-shop problem crucially depends on immediate selections of disjunctive constraints leading to adjustment of heads and tails. This paper presents new investigations concerning this powerful tool. More efficient algorithms are proposed, and global operations are introduced. We also describe a new lower bound and a new branching scheme which are used to design a branch and bound method. Computational results show that these techniques permit to drastically reduce the size of the search trees. Keywords: Scheduling; Job-shop; Branch and bound method I. Introduction In the job-shop problem, n jobs have to be processed on m machines, subject to both conjunctive and disjunctive constraints, in order to minimize the makespan (Muth and Thompson, 1963). It is an NP-hard problem in the strong sense (Lenstra et al., 1977), which remains probably one of the most computation- ally intractable combinatorial problem to date. This fact justifies the considerable amount of study which this problem has been subject to over the years (see the partial reference list). In a previous paper (Carlier and Pinson, 1989), we proposed an efficient enumerative method which solved for the first time a particular instance with 10 jobs and 10 machines proposed by Muth and Thompson in 1963. In Carlier and Pinson (1990), we introduced the concept of immediate selection and explained how it permits to adjust heads and tails and to select directly some disjunctions. We also designed a polynomial algorithm for optimally adjusting heads and tails and consequently fixing disjunctive constraints, allowing us to efficiently prune the search tree associated with a new branch and bound method. This basic idea has been recently used by Brucker, Jurisch and Sievers (1991) and Brucker, Jurisch and Kramer (1992) for designing new enumerative methods for this problem. With some modifications leading to weaker conditions than the ones we proposed, they obtain an O(max [d,n log2n])-steps algorithm for adjusting heads and tails (d denotes the number of selected disjunctive constraints). They also propose new immediate selections and efficient algorithms to compute them. * Corresponding author. 0377-2217/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0377-2217(94)00077-P