TY - JOUR

T1 - Exact algorithms and approximation schemes for proportionate flow shop scheduling with step-deteriorating processing times

AU - Shabtay, Dvir

AU - Mor, Baruch

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022.

PY - 2024/6

Y1 - 2024/6

N2 - 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) (νδ,νa) and (ii) (νδ,νb), 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.

AB - 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) (νδ,νa) and (ii) (νδ,νb), 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.

KW - Approximation schemes

KW - Fixed parameter tractability

KW - NP-hardness

KW - Proportionate flow shop

KW - Step deterioration

UR - http://www.scopus.com/inward/record.url?scp=85142431245&partnerID=8YFLogxK

U2 - 10.1007/s10951-022-00766-2

DO - 10.1007/s10951-022-00766-2

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AN - SCOPUS:85142431245

SN - 1094-6136

VL - 27

SP - 239

EP - 256

JO - Journal of Scheduling

JF - Journal of Scheduling

IS - 3

ER -