{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:24:45Z","timestamp":1760145885201,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T00:00:00Z","timestamp":1725408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Research Funds for the Central Universities","award":["222201917006"],"award-info":[{"award-number":["222201917006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper presents a novel algorithm named Five-element Cycle Integrated Mutation Optimization (FECOIMO) for solving the Traveling Thief Problem (TTP). The algorithm introduces a five-element cycle structure that integrates various mutation operations to enhance both global exploration and local exploitation capabilities. In experiments, FECOIMO was extensively tested on 39 TTP instances of varying scales and compared with five common metaheuristic algorithms: Enhanced Simulated Annealing (ESA), Improved Grey Wolf Optimization Algorithm (IGWO), Improved Whale Optimization Algorithm (IWOA), Genetic Algorithm (GA), and Profit-Guided Coordination Heuristic (PGCH). The experimental results demonstrate that FECOIMO outperforms the other algorithms across all instances, particularly excelling in large-scale instances. The results of the Friedman test show that FECOIMO significantly outperforms other algorithms in terms of average solution, maximum solution, and solution standard deviation. Additionally, although FECOIMO has a longer execution time, its complexity is comparable to that of other algorithms, and the additional computational overhead in solving complex optimization problems translates into better solutions. Therefore, FECOIMO has proven its effectiveness and robustness in handling complex combinatorial optimization problems.<\/jats:p>","DOI":"10.3390\/sym16091153","type":"journal-article","created":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T05:54:47Z","timestamp":1725429287000},"page":"1153","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Five-Element Cycle Optimization Algorithm Based on an Integrated Mutation Operator for the Traveling Thief Problem"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5279-9307","authenticated-orcid":false,"given":"Yue","family":"Xiang","sequence":"first","affiliation":[{"name":"Key Laboratory of Smart Manufacturing in Energy Chemical Process Ministry of Education, East China University of Science and Technology, Shanghai 200237, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1145-8522","authenticated-orcid":false,"given":"Jingjing","family":"Guo","sequence":"additional","affiliation":[{"name":"Department of Aerospace Science and Technology, Space Engineering University, Beijing 101416, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3280-7322","authenticated-orcid":false,"given":"Zhengyan","family":"Mao","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Computer Software Testing & Evaluating, Shanghai Development Center of Computer Software Technology, Shanghai 201112, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8106-4740","authenticated-orcid":false,"given":"Chao","family":"Jiang","sequence":"additional","affiliation":[{"name":"Faculty of Information Technology, Beijing Key Laboratory of Computational Intelligence and Intelligent System, Engineering Research Center of Digital Community, Ministry of Education, Beijing Artificial Intelligence Institute and Beijing Laboratory for Intelligent Environmental Protection, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5928-4656","authenticated-orcid":false,"given":"Mandan","family":"Liu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Smart Manufacturing in Energy Chemical Process Ministry of Education, East China University of Science and Technology, Shanghai 200237, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bonyadi, M.R., Michalewicz, Z., and Barone, L. 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