A Hybrid Harmony Search Algorithm for Distributed Permutation Flowshop Scheduling with Multimodal Optimization

• Published in: Mathematics (Basel) Vol. 13; no. 16; p. 2640
• Main Authors: Shen, Hong, Cheng, Yuwei, Li, Yazhi
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 17.08.2025
• Subjects: Algorithms; distributed permutation flowshop; Energy consumption; Evolutionary algorithms; Factories; Greedy algorithms; harmony search; Heuristic; Heuristic methods; iterative optimization; makespan; Manufacturing; Mathematical models; Mathematical optimization; Methods; Military strategy; multimodal optimization; Optimization; Optimization algorithms; Permutations; Schedules; Scheduling; Scheduling (Management); Search algorithms; Search methods; China
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math13162640

Distributed permutation flowshop scheduling is an NP-hard problem that has become a hot research topic in the fields of optimization and manufacturing in recent years. Multimodal optimization finds multiple global and local optimal solutions of a function. This study proposes a harmony search algorithm with iterative optimization operators to solve the NP-hard problem for multimodal optimization with the objective of makespan minimization. First, the initial solution set is constructed by using a distributed NEH operator. Second, after generating new candidate solutions, efficient iterative optimization operations are applied to optimize these solutions, and the worst solutions in the harmony memory (HM) are replaced. Finally, the solutions that satisfy multimodal optimization of the harmony memory are obtained when the stopping condition of the algorithm is met. The constructed algorithm is compared with three meta-heuristics: the iterative greedy meta-heuristic algorithm with a bounded search strategy, the improved Jaya algorithm, and the novel evolutionary algorithm, on 600 newly generated datasets. The results show that the proposed method outperforms the three compared algorithms and is applicable to solving distributed permutation flowshop scheduling problems in practice.