Branch-and-bound Methods for the Multi-processor Job Shop and Flow Shop Scheduling Problems

The Multi-Processor Job Shop and Flow Shop scheduling problems are variants on the standard Job Shop and Flow Shop problems in which each machine has the capacity to process more than one job simultaneously. This class of problems has attracted the attention of researchers in recent years and several lower bounds has been proposed for a Branch-and-Bound solution method. But only a few attempts has been made to develop a complete Branch-and-Bound scheme in which these lower bounds can be used. In this paper, which is based on 16], we present three diierent Branch-and-Bound methods, where the rst two are similar to existing methods proposed in 7] and 4] and the third is our new proposal. To completely test the Branch-and-Bound methods, some heuristic methods to build initial solutions (based on the Shifting Bottleneck procedure 1] and Simulated Annealing), diierent branching strategies and the known lower bounds has been implemented. These lower bounds include the O(n 2) SubSet Bound and the Adjusted SubSet Bound proposed by A. Vandevelde et al. 18] and an improved O(mnlogn) Jack-son's Pseudo Preemptive Schedule proposed by J. Carlier and E. Pin-son. In addition a test suite of 252 problems has been generated for the purpose of testing the methods. The results show that, within 60 seconds of computation, the two rst Branch-and-Bound methods based on existing, published methods 1 are able to solve resp. 140 and 159 of the 252 test problems, whereas our proposed Branch-and-Bound method is able to solve 209 of the 252 test problems.