Randić energies in decision making for human trafficking by interval-valued T-spherical fuzzy Hamacher graphs
Ali Hasan Ahmad, Humera Rashid, Hamdan Alshehri, Muhammad Jamil, Haitham Assiri · AIMS Mathematics · 2025
The interval-valued ($ IV $) T-spherical fuzzy set $ (IVTSFS) $ appears to be more effective and practical in dealing with uncertainty and ambiguity while dealing with various decision-making ($ DM $) problems than other fuzzy sets, for example, the q-rung ortho-pair fuzzy set, T-spherical fuzzy set, and picture fuzzy set. In real-life problems, where intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, or interval-valued picture fuzzy sets give unsatisfactory results, the $ IVTSFS $ is a mathematical model that is used to deal with such problems where these method fails, and one can handle it efficiently by using $ IVTSFS. $ If there is a different framework that has a base of four opinions — yes, no, abstains, and refusal — the $ IVTSFS $ looks to be the most beneficial, and it is also proven to be so. In this article, an idea of interval-valued T-spherical fuzzy Hamacher graphs $ (IVTSFHGs) $ is proposed, which is based on the Hamacher t-norm ($ TN $) and Hamacher t-conorm ($ TCN $). It provides enhanced discrimination and flexibility in uncertain environments by capturing a broader spectrum of hesitancy and indeterminacy, which other fuzzy models like interval-valued intuitionistic fuzzy set ($ IVIFS $) or picture fuzzy set ($ PFS $) may overlook. This article aims to examine the energy associated with the splitting of $ IVTSFHGs, $ as well as the energy related to shadow $ IVTSFHGs. $ Moreover, the Randić energy of $ IVTSFHGs $ was presented, and an in-depth analysis of its essential outcomes was conducted. Additionally, this study introduces interval-valued T-spherical fuzzy Hamacher digraphs $ (IVTSFHDGs) $ and explores their diverse outcomes. An algorithm involving $ IVTSFHDGs $ was also studied, which is used to propose energies of $ IVTSFHGs $ in $ DM $, problems, and Hamacher aggregation operators. To validate the proposed results, a comparative study is conducted.