Annals of Operations Research

, Volume 272, Issue 1–2, pp 217–241 | Cite as

Single-machine and parallel-machine serial-batching scheduling problems with position-based learning effect and linear setup time

  • Jun PeiEmail author
  • Bayi Cheng
  • Xinbao LiuEmail author
  • Panos M. Pardalos
  • Min Kong
Advances in Theoretical and Applied Combinatorial Optimization


This paper introduces the serial batching scheduling problems with position-based learning effect, where the actual job processing time is a function of its position. Two scheduling problems respectively for single-machine and parallel-machine are studied, and in each problem the objectives of minimizing maximum earliness and total number of tardy jobs are both considered respectively. In the proposed scheduling models, all jobs are first partitioned into serial batches, and then all batches are processed on the serial-batching machine. We take some practical production features into consideration, i.e., setup time before processing each batch increases with the time, regarded as time-dependent setup time, and we formalize it as a linear function of its starting time. Under the single-machine scheduling setting, structural properties are derived for the problems with the objectives of minimizing maximum earliness and number of tardy jobs respectively, based on which optimization algorithms are developed to solve them. Under the parallel-machine scheduling setting, a hybrid VNS–GSA algorithm combining variable neighborhood search (VNS) and gravitational search algorithm (GSA) is proposed to solve the problems with these two objectives respectively, and the effectiveness and efficiency of the proposed VNS–GSA are demonstrated and compared with the algorithms of GSA, VNS, and simulated annealing (SA). This paper demonstrates that the consideration of different objectives leads to various optimal decisions on jobs assignment, jobs batching, and batches sequencing, which generates a new insight to investigate batching scheduling problems with learning effect under single-machine and parallel-machine settings.


Scheduling Serial-batching Learning effect Single-machine Parallel-machine Setup time 



This work is supported by the National Natural Science Foundation of China (Nos. 71601065, 71231004, 71501058, 71671055, 71690235), and Innovative Research Groups of the National Natural Science Foundation of China (71521001), the Humanities and Social Sciences Foundation of the Chinese Ministry of Education (No. 15YJC630097), Anhui Province Natural Science Foundation (No. 1608085QG167), and the Fundamental Research Funds for the Central Universities (Nos. JZ2016HGTA0709, JZ2016HGTB0727). Panos M. Pardalos is partially supported by the project of “Distinguished International Professor by the Chinese Ministry of Education” (MS2014HFGY026).


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of ManagementHefei University of TechnologyHefeiChina
  2. 2.Center for Applied Optimization, Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.Key Laboratory of Process Optimization and Intelligent Decision-Making of Ministry of EducationHefeiChina

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