Exact algorithms and approximation schemes for proportionate flow shop scheduling with step-deteriorating processing times

We study two scheduling problems in a proportionate flow shop environment, where job processing times are machine independent. In contrast to classical proportionate flow shop models, we assume (in both problems) that processing times are step-deteriorating. Accordingly, each job J_j has a normal processing time, a_j, if it starts to be processed in the shop no later than its deteriorating date, δ_j. Otherwise, the job’s processing time increases by b_j (the job’s deterioration penalty). Our aim is to find a job schedule that minimizes either the makespan or the total load. These two problems are known to be NP-hard for the special case of a single machine, even when all jobs have the same deteriorating date. In this paper, we derive several positive results in relation to the two problems. We first show that the two problems can be represented in a unified way. We then prove that the unified problem is only ordinary NP-hard by providing a pseudo-polynomial time algorithm for its solution. We also show that the pseudo-polynomial time algorithm can be converted into a fully polynomial time approximation scheme (FPTAS). Finally, we analyze the parameterized complexity of the problem with respect to the number of different deteriorating dates in the instance, v_δ. We show that although the problem is NP-hard when v_δ=1, it is fixed parameterized tractable (FPT) for the combined parameters (i) (ν_δ,ν_a) and (ii) (ν_δ,ν_b), where ν_a is the number of different normal processing times in the instance, and ν_b is the number of different deterioration penalties in the instance.