Two-machine flowshop scheduling problem with transportation and blocking features

We consider a coordinated scheduling problem with transportation and blocking features on two-machine flowshop where the first machine processes jobs in batches (i.e., a single batching machine) while the second machine processes jobs individually. The finished jobs on the batching machine are transported immediately to the second machine when the second machine is available, otherwise the jobs stay there. The objective is to minimize the sum of the makespan and total blocking time. We show that this problem is strongly NP-hard by a reduction from 3-partition problem. We further give a greedy heuristic and analyze its asymptotic worst-case ratio is equal to the capacity of the batching machine. This performance analysis established the maximum deviation from optimality that can occur for a given heuristic.