Parallel-batching scheduling of deteriorating jobs with non-identical sizes and rejection on a single machine

  • Min KongEmail author
  • Xinbao Liu
  • Jun PeiEmail author
  • Zhiping Zhou
  • Panos M. Pardalos
Original paper


This paper studies the bounded parallel-batching scheduling problem considering job rejection, deteriorating jobs, setup time, and non-identical job sizes. Each job will be either rejected with a certain penalty cost, or accepted and further processed in batches on a single machine. There is a setup time before processing each batch, and the objective is to minimize the sum of the makespan and the total penalty. Several useful preliminaries for arranging accepted job with identical size are proposed. Based on these preliminaries, we first investigate a special case where all the jobs are considered to have the identical size, and develop a dynamic programming algorithm to solve it. The preliminaries help to reduce the complexity of the dynamic programming algorithm from \( O\left( {n!n^{2} \sum\nolimits_{i = 1}^{n} {w_{j} } } \right) \) to \( O\left( {n^{2} \sum\nolimits_{i = 1}^{n} {w_{j} } } \right) \). For the general problem with non-identical job sizes, we propose a hybrid algorithm combining heuristic with dynamic programming algorithm (H-DP) to obtain satisfactory solutions within reasonable time. Finally, the effectiveness and efficiency of the H-DP algorithm are illustrated by a series of computational experiments.


Parallel-batching scheduling Deteriorating jobs Job rejection Setup time Dynamic programming algorithm Non-identical sizes 



This work is supported by the National Natural Science Foundation of China (Nos. 71871080, 71231004, 71601065, 71690235, 71501058, 71601060), 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), Base of Introducing Talents of Discipline to Universities for Optimization and Decision-making in the Manufacturing Process of Complex Product (111 project), the Project of Key Research Institute of Humanities and Social Science in University of Anhui Province, Open Research Fund Program of Key Laboratory of Process Optimization and Intelligent Decision-making (Hefei University of Technology), Ministry of Education.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ManagementHefei University of TechnologyHefeiPeople’s Republic of China
  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 EducationHefeiPeople’s Republic of China

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