A modified fuzzy similarity measure for trapezoidal fuzzy number with their applications

The similarity measure (SM) of fuzzy numbers is vital in decision-making, ranking, and risk analysis, particularly when dealing with qualitative information and fuzzy mathematical models. Traditional set theory struggles with such applications due to the significant presence of uncertainties. Although a wide array of methods are available for SMs applied to fuzzy numbers (FNs), some shortcomings persist. Few recent approaches involving magnitude, distance, and coordinates to measure similarity have demonstrated certain drawbacks when applied to arbitrary trapezoidal fuzzy numbers (TZFNs). These existing measures frequently fail to capture human intuitions, encounter division by zero issues, and account for the unique properties of TZFNs. The proposed work introduces a method to determine the center of gravity (CoG) of TZFNs, which is then utilized along with information on areas, perimeter, weights, and geometric distances of TZFNs to formulate a comprehensive SM. This new approach overcomes the limitations observed in the existing SMs for TZFNs. The mathematical properties of the proposed method are discussed, and the performance of the model is evaluated and compared with recent works. It demonstrates its superior handling of a standard set of 32 TZFNs and for various other special cases, shown numerically in the results section. Additionally, three case studies that involve risk assessment analysis applying the proposed SM are done. A specific scenario concerning service level agreement (SLA) violation in Fog-integrated Cloud is thoroughly explored.