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 $$J_{j}$$ J j has a normal processing time, $$a_{j}$$ a j , if it starts to be processed in the shop no later than its deteriorating date, $$\delta _{j}$$ δ j . Otherwise, the job’s processing time increases by $$b_{j}$$ 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 $$\mathcal{N}\mathcal{P}$$ N P -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 $$ \mathcal{N}\mathcal{P}$$ N P -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, $$v_{\delta }$$ v δ . We show that although the problem is $$\mathcal{N}\mathcal{P}$$ N P -hard when $$v_{\delta }=1$$ v δ = 1 , it is fixed parameterized tractable (FPT) for the combined parameters (i) $$(\nu _{\delta },\nu _{a})$$ ( ν δ , ν a ) and (ii) $$(\nu _{\delta },\nu _{b})$$ ( ν δ , ν b ) , where $$\nu _{a}$$ ν a is the number of different normal processing times in the instance, and $$\nu _{b}$$ ν b is the number of different deterioration penalties in the instance.