A new mathematical framework called complex probabilistic hesitant fuzzy sets (CPHFS) has emerged, aiming to improve multi-attribute decision making (MADM) by combining two-dimensional fuzzy information with randomness. Traditional fuzzy sets often fall short in expressing the complexity and uncertainty inherent to real-world decision-making scenarios. CPHFS stands out as it addresses these limitations, providing richer alternatives for evaluating choices and preferences.
Researchers have developed CPHFS as extensions of existing fuzzy sets, particularly focusing on incorporating probabilities to handle the hesitancy present when decision-makers choose among multiple attributes. This innovation is pivotal as it bridges the gap between subjective preferences and objective outcomes.
Through the creation of various operations on CPHFS, including aggregation operators like complex probabilistic hesitant fuzzy weighted averaging (CPHFWA) and complex probabilistic hesitant fuzzy weighted geometric operators (CPHFWG), the framework allows users to synthesize multiple evaluation criteria efficiently. For example, these operators can capture the interrelationships among different decision factors, enabling clearer analysis and prioritization.
To validate the proposed methodology, researchers applied these new operators to practical MADM situations, including the selection of vehicles based on several attributes such as quality, cost, and appearance. This practical demonstration illustrated the effectiveness of CPHFS in providing comprehensive evaluations, leading to more informed decisions.
The study finds significant strengths of CPHFS; particularly, it reflects interactions among attributes more accurately than previous models. This becomes evident during comparative analyses where CPHFS consistently outperformed traditional fuzzy decision-making frameworks across various metrics.
Conclusively, the innovative CPHFS framework not only fills gaps left by existing methods but also sets the groundwork for future research avenues, particularly concerning integrating dimensions of randomness and uncertainty within other mathematical and decision-making models.
Overall, the findings indicate CPHFS represent substantial progress toward enhancing the robustness and precision of decision-making methodologies, addressing the increasing complexity and variability of real-world problems.