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One of the CO problems is the Wireless Sensor Network (WSN) coverage optimization problem, which plays an important role in Internet of Things (IoT). A novel hybrid algorithm is proposed, named hybrid particle swarm butterfly algorithm (HPSBA), by combining their strengths of particle swarm optimization (PSO) and butterfly optimization algorithm (BOA), for solving this problem. Significantly, the value of individual scent intensity should be non-negative without consideration of the basic BOA, which is calculated with absolute value of the proposed HPSBA. Moreover, the performance of the HPSBA is comprehensively compared with the fundamental BOA, numerous potential BOA variants, and tried-and-true algorithms, for solving the twenty-six commonly used benchmark functions. The results show that HPSBA has a competitive overall performance. Finally, when compared to PSO, BOA, and MBOA, HPSBA is used to solve the node coverage optimization problem in WSN. The experimental results demonstrate that the HPSBA optimized coverage has a higher coverage rate, which effectively reduces node redundancy and extends WSN survival time.<\/jats:p>","DOI":"10.3390\/axioms11120675","type":"journal-article","created":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T06:04:36Z","timestamp":1669615476000},"page":"675","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["HPSBA: A Modified Hybrid Framework with Convergence Analysis for Solving Wireless Sensor Network Coverage Optimization Problem"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8546-9972","authenticated-orcid":false,"given":"Mengjian","family":"Zhang","sequence":"first","affiliation":[{"name":"Electrical Engineering College, Guizhou University, Guiyang 550025, China"},{"name":"School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4936-8773","authenticated-orcid":false,"given":"Deguang","family":"Wang","sequence":"additional","affiliation":[{"name":"Electrical Engineering College, Guizhou University, Guiyang 550025, China"},{"name":"Guizhou Provincial Key Laboratory of Internet + Intelligent Manufacturing, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4470-3467","authenticated-orcid":false,"given":"Ming","family":"Yang","sequence":"additional","affiliation":[{"name":"Electrical Engineering College, Guizhou University, Guiyang 550025, China"},{"name":"Guizhou Provincial Key Laboratory of Internet + Intelligent Manufacturing, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei","family":"Tan","sequence":"additional","affiliation":[{"name":"College of Forestry, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6407-1276","authenticated-orcid":false,"given":"Jing","family":"Yang","sequence":"additional","affiliation":[{"name":"Electrical Engineering College, Guizhou University, Guiyang 550025, China"},{"name":"Guizhou Provincial Key Laboratory of Internet + Intelligent Manufacturing, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,27]]},"reference":[{"key":"ref_1","unstructured":"Kennedy, J., and Eberhart, R. 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