A complete ranking of trapezoidal-valued intuitionistic fuzzy number: an application in evaluating social sustainability

Conventional trapezoidal intuitionistic fuzzy numbers (CTrIFNs) are used in the literature to handle many real-life problems with imprecise information. However, the CTrIFNs are not the real generalization of interval-valued intuitionistic fuzzy numbers (IVIFNs) and triangular intuitionistic fuzzy numbers (TIFNs). This study discusses the non-conventional trapezoidal intuitionistic fuzzy numbers called trapezoidal-valued intuitionistic fuzzy numbers (TrVIFNs) that generalize (contain) all real-valued intuitionistic fuzzy numbers, IVIFNs and TIFNs. Various researchers worldwide are looking for a complete ranking principle on the set of TrVIFNs. However, none of them yields a complete ranking. The paper’s main aim is to propose a new complete ranking principle on the class of TrVIFNs. The complete ranking principle on the set of TrVIFNs makes the decision-making algorithm more robust. To achieve the main aim, firstly, we introduce eight different score functions on the set of TrVIFNs and study their mathematical properties. Secondly, we present a new ranking principle by considering all eight score functions in a linear order. Further, we prove (mathematically) that the introduced ranking principle defines complete ranking on the set of TrVIFNs. Thirdly, we show the significance of the proposed method compared with the existing methods on various classes of fuzzy and intuitionistic fuzzy numbers. Finally, we implement a case application of the proposed complete ranking principle for evaluating the social sustainability performance of Indian firms.