Minimizing Makespan in Permutation Flow Shop Scheduling with Proportional Deterioration

The [Formula: see text]-job and [Formula: see text]-machine permutation flow shop scheduling problem with a proportional deterioration is considered in which all machines process the jobs in the same order, i.e., a permutation schedule. A proportional deterioration means that the job deterioration as an increasing function that is proportional to a linear function of time. The objective is to minimize the makespan, i.e., the maximum completion time. When some dominant relationships between [Formula: see text] machines can be satisfied, we show that some special cases of the problem can be polynomial solvable. For the general case, we also propose a heuristic algorithm and give the computational experiments.