Revision of the Kosinski's theory of ordered fuzzy numbers

Abstract

Ordered fuzzy numbers are defined by Kosinski. In this way, he was going to supplement a fuzzy number by orientation. A significant drawback of Kosinski's theory is that there exist such ordered fuzzy numbers which, in fact, are not fuzzy numbers. For this reason, a fully formalized correct definition of ordered fuzzy numbers is proposed here. Also, the arithmetic proposed by Kosinski has a significant disadvantage. The space of ordered fuzzy numbers is not closed under Kosinski's addition. On the other hand, many mathematical applications require the considered space be closed under used arithmetic operations. Therefore, the Kosinski's theory is modified in this way that the space of ordered fuzzy numbers is closed under revised arithmetic operations. In addition, it is shown that the multiple revised sum of finite sequence of ordered fuzzy numbers depends on its summands ordering.