Abstract
Intuitionistic fuzzy set (IFS) theory can be applied for multi-aspect systems due to its capability to address uncertainty and incomplete information in terms of membership and non-membership degrees. Unfortunately, classical Γ-structures cannot handle fuzzy and imprecise information in real problems. In fact, there is no rigorous base to practically express the effectiveness of multi-attribute systems in IFS environment. Here, we develop a generalized IFS with the notion of Γ-module called intuitionistic fuzzy Γ-submodule (IFΓM) to establish a novel “Global electronic (e)-Commerce (GeC) Theory”. To simplify the analysis of parameters, (α,β)-cut representation is proposed in terms of comprehensive distribution of fuzzy number for the classification of components. On the other hand, Cartesian product is implemented to correspond the elements. Substantial properties of IFΓM including (α,β)-cut, Cartesian product and t-intuitionistic fuzzy Γ-submodule (t-IFΓM) are characterized with illustrative examples to extend the framework of IFΓM, where (α,β)-cut and support t-IFΓM are verified to be Γ-submodules based on the properties of IFΓM. Through Γ-module homomorphism, image and inverse image, the parametric connections between (α,β)-cuts are systematically investigated. In addition, a mathematical relationship between the Cartesian product and (α,β)-cut is determined. The overlapping intersection of a collection of t-IFΓM is proved to be t-IFΓM, and the image and inverse image are preserved under Γ-module homomorphism. As global e-trades are increasingly expanding after the recent coronavirus disease 2019 (COVID-19) hit, with the growth of 26.7-trillion dollars, businesses are required to transform their traditional functional natures to online (or blended) strategies for cost efficiency and self-survival in the present competitive environment. Therefore, compared to recent studies on IFS in the context of Γ-structures, the main contribution of this study is to provide a theoretical basis for the establishment of a new GeC Theory through the developed IFΓM method and Γ-module M which targets the purchasing rate of customers through e-commerce companies. In the end, the performance of the proposed method in terms of upper and lower cut, t-intuitionistic fuzzy set, support and IFΓM model, is analyzed in the developed GeC Theory. The proposed GeC Theory is validated using real datasets of e-commerce mega companies, i.e., Amazon, Alibaba, eBay, Shopify. They are characterized based on the amount of online shopping by samples (individuals). Compared to the existing methods, the GeC approach is an effective IFS-based method for complex systems with uncertainty.
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