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Mathematics
Volume 13
Issue 24
10.3390/math13243973
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

A Novel Imperialist Competitive Algorithm for Energy-Efficient Permutation Flow Shop Scheduling Problem Considering the Deterioration Effect of Machines

by
Kaiyang Yin
1,
Zhi Li
1,
Ming Li
2,*,
Yaxu Xue
1 and
Yi Chen
3,*
1
School of Electrical and Mechanical Engineering, Pingdingshan University, Pingdingshan 467000, China
2
School of Economics and Management, Anhui Polytechnic University, Wuhu 241000, China
3
School of Intelligent Manufacturing Ecosystem, Xi’an Jiaotong-Liverpool University, Suzhou 215400, China
*
Authors to whom correspondence should be addressed.
Mathematics 2025, 13(24), 3973; https://doi.org/10.3390/math13243973
Submission received: 6 November 2025 / Revised: 5 December 2025 / Accepted: 11 December 2025 / Published: 13 December 2025
(This article belongs to the Special Issue New Advances and Applications of Operational Research and Scheduling Theory)
Versions Notes

Abstract

This study addresses a critical gap in Energy-Efficient Permutation Flow Shop Scheduling (EPFSP) by integrating the often-overlooked time-accumulative equipment degradation inherent in practical manufacturing. This research formalizes and solves the EPFSP with machine deterioration (EPFSP-DEM), aiming to simultaneously minimize the makespan and total energy consumption. To achieve this objective, this study proposes a Diversity-Constrained Imperialist Competitive Algorithm (DCICA) featuring several novel mechanisms. In DCICA, a differentiated assimilation is developed to improve diversity of the population; a knowledge-guided revolution is designed to allocate computing resources efficiently; the convergence and diversity metrics are defined to evaluate the search quality on assimilation and revolution; a novel imperialist competition is also given to enhance information exchanges among empires and strengthen the search for some worse solutions. Finally, extensive experiments are conducted, and the results demonstrate that DCICA outperforms the existing algorithms in solving the investigated problem.
Keywords: permutation flow shop scheduling problem; makespan; total energy consumption; deterioration effect; imperialist competitive algorithm permutation flow shop scheduling problem; makespan; total energy consumption; deterioration effect; imperialist competitive algorithm

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MDPI and ACS Style

Yin, K.; Li, Z.; Li, M.; Xue, Y.; Chen, Y. A Novel Imperialist Competitive Algorithm for Energy-Efficient Permutation Flow Shop Scheduling Problem Considering the Deterioration Effect of Machines. Mathematics 2025, 13, 3973. https://doi.org/10.3390/math13243973

AMA Style

Yin K, Li Z, Li M, Xue Y, Chen Y. A Novel Imperialist Competitive Algorithm for Energy-Efficient Permutation Flow Shop Scheduling Problem Considering the Deterioration Effect of Machines. Mathematics. 2025; 13(24):3973. https://doi.org/10.3390/math13243973

Chicago/Turabian Style

Yin, Kaiyang, Zhi Li, Ming Li, Yaxu Xue, and Yi Chen. 2025. "A Novel Imperialist Competitive Algorithm for Energy-Efficient Permutation Flow Shop Scheduling Problem Considering the Deterioration Effect of Machines" Mathematics 13, no. 24: 3973. https://doi.org/10.3390/math13243973

APA Style

Yin, K., Li, Z., Li, M., Xue, Y., & Chen, Y. (2025). A Novel Imperialist Competitive Algorithm for Energy-Efficient Permutation Flow Shop Scheduling Problem Considering the Deterioration Effect of Machines. Mathematics, 13(24), 3973. https://doi.org/10.3390/math13243973

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