Ranking fuzzy quantities based on the angle of the reference functions

• Published in: Applied mathematical modelling Vol. 37; no. 22; pp. 9230 - 9241
• Main Authors: Nasseri, S.H., Zadeh, M.M., Kardoost, M., Behmanesh, E.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.11.2013
• Subjects: Algorithms; Angle of fuzzy number; Fuzzy; Fuzzy logic; Fuzzy quantities; Fuzzy ranking; Fuzzy set theory; Fuzzy systems; Mathematical analysis; Mathematical models; Ranking; α-Cut; Fuzzy quantities; Fuzzy ranking; α-Cut; Angle of fuzzy number
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2013.04.002

Ordering fuzzy quantities and their comparison play a key tool in many applied models in the world and in particular decision-making procedures. However a huge number of researches is attracted to this filed but until now there is any unique accepted method to rank the fuzzy quantities. In fact, each proposed method may has some shortcoming. So we are going to present a novel method based on the angle of the reference functions to cover a wide range of fuzzy quantities by over coming the draw backs of some existing methods. In the mentioned firstly, the angle between the left and right membership functions (the reference functions) of every fuzzy set is called Angle of Fuzzy Set (AFS), and then in order to extend ranking of two fuzzy sets the angle of fuzzy sets and α-cuts is used. The method is illustrated by some numerical examples and in particular the results of ranking by the proposed method and some common and existing methods for ranking fuzzy sets is compared to verify the advantage of the new approach. In particular, based on the results of comparison of our method with well known methods which are exist in the literature, we will see that against of most existing ranking approaches, our proposed approach can rank fuzzy numbers that have the same mode and symmetric spreads. In fact, the proposed method in this paper can effectively rank symmetric fuzzy numbers as well as the effective methods which are appeared in the literature. Moreover, unlike of most existing ranking approaches, our proposed approach can rank non-normal fuzzy sets. Finally, we emphasize that the concept of fuzzy ordering is one of key role in establishing the numerical algorithms in operations research such as fuzzy primal simplex algorithms, fuzzy dual simplex algorithms and as well as discussed in the works of Ebrahimnejad and Nasseri and coworkers [1–7].