{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:15:02Z","timestamp":1760235302977,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,4]],"date-time":"2021-08-04T00:00:00Z","timestamp":1628035200000},"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":["71872110"],"award-info":[{"award-number":["71872110"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables\u2019 mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.<\/jats:p>","DOI":"10.3390\/sym13081428","type":"journal-article","created":{"date-parts":[[2021,8,4]],"date-time":"2021-08-04T08:47:52Z","timestamp":1628066872000},"page":"1428","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Scalar Mean Chance and Expected Value of Regular Bifuzzy Variables"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4515-8536","authenticated-orcid":false,"given":"Guang","family":"Wang","sequence":"first","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7560-9326","authenticated-orcid":false,"given":"Yixuan","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4599-024X","authenticated-orcid":false,"given":"Yujiao","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5449-155X","authenticated-orcid":false,"given":"Jiahao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,4]]},"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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