Lagrangian relaxation for the permutation flowshop scheduling problem with minimal and maximal time lags

In this research, we are interested in the permutation flowshop scheduling problem with minimal and maximal time lags while minimizing the total tardiness. The processing order of jobs is to be the same for each machine. The time lag is defined as the waiting time between two consecutive operations of each job. It is greater than or equal to a prescribed value called minimal time lag and smaller than or equal to a prescribed value called maximal time lag. A new mathematical formulation is proposed. Then, a new lower bound is derived by applying the Lagrangian relaxation. In order to make this technique a viable approach to the considered problem, an auxiliary formulation is adopted and the Lagrangian multipliers are updated using the sub-gradient algorithm. Then, results of the computational experiments are reported.