A polynomial algorithm for special case of the one-machine scheduling problem with time-lags

The standard one-machine scheduling problem consists in scheduling a set of jobs in one machine which can handle only one job at a time, minimizing the maximum lateness. Each job is available for processing at its release date, requires a known processing time and after finishing the processing, it is delivery after a certain time. There also can exists precedence constraints between pairs of jobs, requiring that the first jobs must be completed before the second job can start. An extension of this problem consists in assigning a time interval between the processing of the jobs associated with the precedence constrains, known by finish-start time-lags. In presence of this constraints, the problem is NP-hard even if preemption is allowed. In this work, we consider a special case of the one-machine preemption scheduling problem with time- lags, where the time-lags have a chain form, and propose a polynomial algorithm to solve it. The algorithm consist in a polynomial number of calls of the preemption version of the Longest Tail Heuristic. One of the applicability of the method is to obtain lower bounds for NP-hard one-machine and job-shop scheduling problems. We present some computational results of this application, followed by some conclusions.