Bounds for the permutation flowshop scheduling problem with exact time lags to minimize the total earliness and tardiness

We consider the problem of n-jobs scheduling in an m-machine permutation flowshop with exact time lags between consecutive operations of each job. The exact time lag is defined as the time elapsed between every couple of successive operations of the same job which is equal to a prescribed value. The aim is to find a feasible schedule that minimizes the total tardiness and earliness. We propose a mathematical formulation, which is then solved by running the commercial software CPLEX to provide an optimal solution for small size problems. As the problem is shown to be strongly NP-hard, we propose two new upper bounds and two lower bounds useful to solve efficiently large size problems. We then evaluate their effectiveness through an extensive computational experiment.