BULLETIN OF THE POLISH ACADEMY OF SCIENCES MATHEMATICS Vol. 51, No. 3, 2003 COMPUTER SCIENCE Ordered Fuzzy Numbers by Witold KOSIŃSKI, Piotr PROKOPOWICZ and Dominik ŚLĘZAK Presented by Jan RYCHLEWSKI on August 9, 2002 (∗) Summary. Fuzzy counterpart of real numbers is presented. Fuzzy membership functions, which satisfy conditions similar to the quasi-convexity are considered. An extra feature, called the orientation of the curve of fuzzy membership function, is introduced. It leads to a novel concept of an ordered fuzzy number, represented by the ordered pair of real continuous functions. Arithmetic operations defined on ordered fuzzy numbers enable to avoid some drawbacks of the classical approach. 1. Introduction. In real-life problems both parameters and data used in mathematical modeling are vague. The vagueness can be described by fuzzy numbers and fuzzy sets. Fuzzy data analysis requires more than fuzzy logic: it requires fuzzy arithmetic. Its development should be based on a well established theory of fuzzy sets defined on the real axis, in order to build a fuzzy counterpart of real numbers [1], [5], [7], [12], [15]. In the classical approach for numerical handling of fuzzy quantities the so-called extension principle is of fundamental importance. Formulated by Zadeh [17], [18], [19], it provides a formal apparatus to carry over operations (arithmetic or alge- braic) from sets to fuzzy sets. The commonly accepted theory of fuzzy numbers is that set up by Dubois and Prade [3], who proposed a restricted class of membership functions, called (L, R)-numbers. The essence of their representation is that the mem- bership function is of a particular form that is generated by two so-called shape (or spread) functions: L and R. In this context (L, R)-numbers be- came quite popular, because of their good interpretability and relatively easy handling for simple operation, i.e. for the fuzzy addition. 2000 MS Classification: primary: 68Q10; secondary: 6504, 65G30, 12K99, 93C42, 16– 04, 16Z05. Key words: fuzzy numbers, quasi-convexity, orientation, algebraic operations. (∗) Revised version received on December 11, 2002.