{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:31:40Z","timestamp":1772253100276,"version":"3.50.1"},"reference-count":16,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,10,14]],"date-time":"2017-10-14T00:00:00Z","timestamp":1507939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In fuzzy decision problems, the ordering of fuzzy numbers is the basic problem. The fuzzy preference relation is the reasonable representation of preference relations by a fuzzy membership function. This paper studies Nakamura\u2019s and Ko\u0142odziejczyk\u2019s preference relations. Eight cases, each representing different levels of overlap between two triangular fuzzy numbers are considered. We analyze the ranking behaviors of all possible combinations of the decomposition and intersection of two fuzzy numbers through eight extensive test cases. The results indicate that decomposition and intersection can affect the fuzzy preference relations, and thereby the final ranking of fuzzy numbers.<\/jats:p>","DOI":"10.3390\/sym9100228","type":"journal-article","created":{"date-parts":[[2017,10,16]],"date-time":"2017-10-16T11:11:09Z","timestamp":1508152269000},"page":"228","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Decomposition and Intersection of Two Fuzzy Numbers for Fuzzy Preference Relations"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9991-6443","authenticated-orcid":false,"given":"Hui-Chin","family":"Tang","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, Kaohsiung 80778, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,10,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Chen, S.J., and Hwang, C.L. 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