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

Dvir Shabtay, Baruch Mor

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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 Jj has a normal processing time, aj, if it starts to be processed in the shop no later than its deteriorating date, δj. Otherwise, the job’s processing time increases by bj (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, vδ. We show that although the problem is NP-hard when vδ=1, it is fixed parameterized tractable (FPT) for the combined parameters (i) (νδa) and (ii) (νδb), where νa is the number of different normal processing times in the instance, and νb is the number of different deterioration penalties in the instance.

Original languageEnglish
Pages (from-to)239-256
Number of pages18
JournalJournal of Scheduling
Issue number3
StatePublished - 1 Jun 2024


  • Approximation schemes
  • Fixed parameter tractability
  • NP-hardness
  • Proportionate flow shop
  • Step deterioration

ASJC Scopus subject areas

  • Software
  • General Engineering
  • Management Science and Operations Research
  • Artificial Intelligence


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