Closed Forms of the Interval Type 2 Fuzzy Sets Additions Based on Archimedean T-norms with Application in Decision Making Aggregation

Abstract

Fuzzy addition is one of the operations widely used in the application of fuzzy systems. The analytical and closed form expressions of the fuzzy additions of the generalized fuzzy numbers and the interval type 2 fuzzy numbers (IT2FS) are developed. The generalized fuzzy numbers is a non-flat and subnormal fuzzy number with different left and right shape functions. Classical fuzzy operations based on the minimum t-norm could lead to uncontrollable fuzziness. Fuzzy addition based on generalized t-norm can better control ambiguity. In order to control the ambiguity of fuzzy addition, this paper is to develop a general type 1 fuzzy set and interval type 2 fuzzy set (IT2FS) additions based on t-norm. The closed forms of the generalized IT2FS additions based on some popular t-norms are formulated. The specific analytical and closed form expressions based on Archimedean t-norms: e.g. drastic, Łukasiewicz, product and the families of Schweizer-Sklar and Yager t-norms are derived. The IT2FS are widely used to characterize the natural words, e.g. the decision making linguistic judgments, the fuzzy real time control systems with linguistic rules. Furthermore, based on the analytical T1FS additions results, the application to the linguistic multiple criteria decision making aggregation of the IT2FS additions based on t-norms is introduced. A comparative MCDM numerical example is demonstrated for the verification of the developed closed form expressions.