Minimizing makespan in flowshop with time lags

We consider the problem of minimizing the makespan in a flowshop involving maximal and minimal time lags, denoted by Fm|θ, θ|Cmax. Time lag constraints generalize the classical precedence constraints between operations: minimal (respectively maximal) time lags indeed specify that the time elapsed between two operations must be lower(respectively upper-) bounded. We assume that such constraints are only defined between operations of the same job. Time lags may be used to model various industrial situations. In several processes the delay between some critical operations must not be too long otherwise the products may be deteriorated. Such examples can be found in food industry with perishable products or in chemistry with unstable chemicals. Minimal time lags may correspond to transportation times between the machines or to communication delays between processors. Detailed case studies of shop problems involving time lags can be found in Deppner’s thesis [1]. From a computational complexity viewpoint, the existence of time lags make the problem NP-hard, even for special cases with two machines: Yu et al. [3] showed that the two-machine flowshop problem with minimal time lags and unit processing times (F2|pi,j = 1, θ |Cmax is strongly NP-hard. Besides, the two-machine flowshop with constant maximal time lags (F2|θ = θ|Cmax) has been shown to be strongly NP-hard as well, even if we restrict the problem to permutation schedules [2].