A Lower Bound Calculationfor the N-Job, M-Machine Job Shop Scheduling Problemto Minimize Lmax

Abstract

An improved lower bound on Lmax is developed for the N-job, M-machine job shop scheduling problem. Improvements occur particularly on problems that are defined by a specific due date range. The procedure allows preemption on all but one machine and then identifies other delays in the processing on that machine. When a delay is found it affects the earliest starts of the remaining operations on that job and the latest finishes of the preceding operations. The delays are found by repeatedly lowering the potential value of Lmax, which, in response, may increase the lower bound. To accommodate the decreasing upper bound, (i.e. Lmax), changes in sequence are made which cause earliest starts and latest finishes to be updated. This then allows for recalculation of the lower bound. The lower bound is still determined using preemption on all but one machine, but now includes more accurate (i.e., tighter) start and finish times.