{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T10:52:25Z","timestamp":1781175145964,"version":"3.54.1"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T00:00:00Z","timestamp":1610409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"The National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61773019"],"award-info":[{"award-number":["61773019"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The picture fuzzy set is a generation of an intuitionistic fuzzy set. The aggregation operators are important tools in the process of information aggregation. Some aggregation operators for picture fuzzy sets have been proposed in previous papers, but some of them are defective for picture fuzzy multi-attribute decision making. In this paper, we introduce a transformation method for a picture fuzzy number and trapezoidal fuzzy number. Based on this method, we proposed a picture fuzzy multiplication operation and a picture fuzzy power operation. Moreover, we develop the picture fuzzy weighted geometric (PFWG) aggregation operator, the picture fuzzy ordered weighted geometric (PFOWG) aggregation operator and the picture fuzzy hybrid geometric (PFHG) aggregation operator. The related properties are also studied. Finally, we apply the proposed aggregation operators to multi-attribute decision making and pattern recognition.<\/jats:p>","DOI":"10.3390\/sym13010119","type":"journal-article","created":{"date-parts":[[2021,1,12]],"date-time":"2021-01-12T20:11:31Z","timestamp":1610482291000},"page":"119","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Picture Fuzzy Geometric Aggregation Operators Based on a Trapezoidal Fuzzy Number and Its Application"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9400-3533","authenticated-orcid":false,"given":"Minxia","family":"Luo","sequence":"first","affiliation":[{"name":"Department of Information and Computing Science, China Jiliang University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huifeng","family":"Long","sequence":"additional","affiliation":[{"name":"Department of Information and Computing Science, China Jiliang University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. 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