{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:01:07Z","timestamp":1760058067107,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,14]],"date-time":"2025-03-14T00:00:00Z","timestamp":1741910400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"],"award-info":[{"award-number":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"publisher","award":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"],"award-info":[{"award-number":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation Project of Heilongjiang Province","award":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"],"award-info":[{"award-number":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"]}]},{"name":"Key Projects of Education Department of Jilin Province","award":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"],"award-info":[{"award-number":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"]}]},{"name":"Jilin Normal University Doctoral Program","award":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"],"award-info":[{"award-number":["61772160","20BZS058","LH2024F045","JJKH20250946KJ","2022014"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the field of manufacturing, complex products with a tree structure often naturally exhibit asymmetric characteristics due to the uniqueness of their constituent elements and hierarchical relationships. In the production scheduling of such small-batch and multi-variety products, due to the influence of process constraints, many pieces of equipment will be idle for a long time in the production process. This increases the makespan of production and reduces the utilization rate of the equipment. Aiming at improving this situation, a resource cooperative scheduling algorithm is proposed to dynamically adjust the idle time of equipment: the integrated scheduling algorithm with dynamic adjustment of machine idle time based on an improved semi-numerical algorithm (ISA_DA). This algorithm adopts the critical path chain and the optimal scheduling strategy. On this basis, an adjustment strategy based on the semi-numerical algorithm considering the backward movement of the process is proposed, which improves the strength of the close connection of the process and the strength of parallel processing. Experimental results show that the effect of the proposed algorithm is better than the comparison algorithms: the hierarchical scheduling order algorithm and the process tree cyclic decomposition algorithm. It can increase the equipment utilization rate by 10% and 11.5%, respectively, and reduce the makespan by 28.6%. These experimental results strongly prove that the proposed algorithm can effectively reduce the total processing time and improve the production efficiency, whether it is for complex products with a symmetric structure or asymmetric structure.<\/jats:p>","DOI":"10.3390\/sym17030434","type":"journal-article","created":{"date-parts":[[2025,3,14]],"date-time":"2025-03-14T07:02:16Z","timestamp":1741935736000},"page":"434","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Integrated Scheduling Algorithm Based on Improved Semi-Numerical Algorithm"],"prefix":"10.3390","volume":"17","author":[{"given":"Xueying","family":"Ding","sequence":"first","affiliation":[{"name":"College of Mathematics and Computer, Jilin Normal University, Siping 136000, China"}]},{"given":"Wei","family":"Zhou","sequence":"additional","affiliation":[{"name":"College of Mathematics and Computer, Jilin Normal University, Siping 136000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7422-1568","authenticated-orcid":false,"given":"Zhiqiang","family":"Xie","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Harbin University of Science and Technology, Harbin 150080, China"}]},{"given":"Ming","family":"Sun","sequence":"additional","affiliation":[{"name":"College of History and Culture, Jilin Normal University, Siping 136000, China"}]},{"given":"Zhenjiang","family":"Tan","sequence":"additional","affiliation":[{"name":"College of Mathematics and Computer, Jilin Normal University, Siping 136000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5666-8331","authenticated-orcid":false,"given":"Wangcheng","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Computer and Information Technology, Mudanjiang Normal University, Mudanjiang 157000, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,14]]},"reference":[{"key":"ref_1","first-page":"36","article-title":"Research on intelligent optimal dispatching of distribution networks based on artificial intelligence","volume":"S1","author":"Pei","year":"2023","journal-title":"Electr. 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