Statistical distributions and reliability functions with type-2 fuzzy parameters

Abstract

Fuzzy sets are useful and effective tools to model the uncertainty problem in real-life applications. The most common fuzzy sets used in these applications are known as type-1 fuzzy sets (T1FSs). Since the membership degrees of T1FSs are crisp numbers, recently, type-2 fuzzy sets (T2FSs) are also preferred by many researchers to express uncertainty in T1FSs. Zadeh [11] introduced T2FS as an extension version of the conventional T1FS. Some important studies about T2FSs can be given as Aisbett et al. [1], Hamrawi [3], Karnik and Mendel [5], Wu and Mendel [9]. Furthermore, some applications of T2FSs can be found in Tao et al. [6], Türkşen [7], Wagenkneckt and Hartmann [8], Wu and Mendel [10]. Also, basic operations on T2FSs were studied by Blewitt et.al. [2], Karnik and Mendel [5]. However, type-2 fuzzy numbers (T2FNs) are required to make theoretical inference about modelling uncertainty. In the literature, limited number of studies can be found related to the operators on T2FNs, e.g., Kardan et al. [4]. Mostly, it is difficult to use these operators on T2FNs due to the computational complexity of T2FSs. This study introduces practical and innovative solutions for arithmetical operations on T2FN using the (α, β)-cut definition. Thus, type-2 fuzzy parameter-based distributions and reliability functions are proposed with regards to their monotonicity. Therefore, we present a novel perspective to perform various arithmetical operators on type-2 fuzzy numbers. The basis of this perspective is structured by an (α, β)-cut definition. This definition is derived from the type-1 operations on three type-1 membership functions (lower, upper and type-1 membership functions) of T2FN. Some operations (such as sum, subtraction, multiplication, division) for T2FNs are determined using the (α, β)-cuts. Then, the membership functions of T2FNs are structured by the (α, β)-cuts. Besides, we utilize this (α, β)-cut definition to form fuzzy function of T2FN under some assumptions. Finally, we give some applications of probability distributions when some parameters of the distributions are the T2FNs. This paper is organized as follows: Section 2 provides mathematical background of type-2 fuzzy sets and numbers, (α, β)-cut definition of T2FN, fuzzy function of T2FN with its monotonicity. The applications based on the statistical distributions and reliability functions of T2FN are given in Section 3. Conclusion is drawn in Section 4.