{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T23:59:24Z","timestamp":1768521564041,"version":"3.49.0"},"reference-count":48,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,21]],"date-time":"2018-10-21T00:00:00Z","timestamp":1540080000000},"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 this paper, the model of imprecise quantity information is an ordered fuzzy number. The purpose of our study is to propose some methods of approximating any ordered fuzzy number using a trapezoidal ordered fuzzy number. The information ambiguity is evaluated by means of an energy measure. The information indistinctness is evaluated by Kosko\u2019s entropy measure. We discuss the problem of approximation of an arbitrary ordered fuzzy number by the nearest trapezoidal ordered fuzzy number. This way, we can simplify arithmetical operations on the linear space of ordered fuzzy numbers. The set of feasible trapezoidal ordered numbers is limited by the combination of the following conditions: invariance of energy measure, invariance of entropy measure, and invariance of information support. Evaluating the influence of individual limits combinations on the utility of given approximations, two combinations of those restraints, recommended for use, were chosen. It was also indicated that one of the recommended approximation problems can be used only for ordered fuzzy numbers characterized by a low level of entropy. The obtained results are currently used in such multi-criterial decision making models as financial portfolio management, evaluation of negotiations offers, the fuzzy TOPSIS model, and the fuzzy SAW model.<\/jats:p>","DOI":"10.3390\/sym10100526","type":"journal-article","created":{"date-parts":[[2018,10,23]],"date-time":"2018-10-23T08:43:36Z","timestamp":1540284216000},"page":"526","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On Approximation of Any Ordered Fuzzy Number by A Trapezoidal Ordered Fuzzy Number"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9629-4167","authenticated-orcid":false,"given":"Krzysztof","family":"Piasecki","sequence":"first","affiliation":[{"name":"Department of Investment and Real Estate, Poznan University of Economics and Business, al. Niepodleglosci 10, 61-875 Pozna\u0144, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6011-7855","authenticated-orcid":false,"given":"Anna","family":"\u0141yczkowska-Han\u0107kowiak","sequence":"additional","affiliation":[{"name":"Department of Finance, WSB University in Pozna\u0144, ul. Powsta\u0144c\u00f3w Wielkopolskich 5, 61-895 Pozna\u0144, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,21]]},"reference":[{"key":"ref_1","first-page":"285","article-title":"Fuzzy numbers and their quotient space with algebraic operations","volume":"41","year":"1993","journal-title":"Bull. Pol. Acad. Sci."},{"key":"ref_2","unstructured":"Klopotek, M., Wierzcho\u0144, S.T., and Michalewicz, M. (2002). Fuzzy reals with algebraic operations: Algorithmic approach. 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