摘要

This study was devoted to investigate the differentiation and necessity of the cardinal GRA models. From a localized GRA viewpoint, we focused on Wu's, Wen's, Hsia's, and Nagai's cardinal GRA models. We proved that the grey relational order of series by using these cardinal GRA models can be exactly determined by the order of certain function of x(j)* (k), i.e. scores after data pre-processing. In Wen's and Hsia's models this function is 1/n Sigma(n)(k=1) x(i)(*)(k); in Nagai's (when rho = 1) models this function is Sigma(n)(k=1) x(i)* (k); and in Wu's and Nagai's (when p = 2) models this function is Sigma(k=1) (n) [x(i)* (k)(2) - 2x(i)*(k)]. In other words, form the viewpoint of ranking grey relational order of series, the Wen's, Hsia's and Nagai's (when p = 1) models are functional equivalent; and the Wu's1 and Nagai's (when p = 2) models are functional equivalent. Moreover, if conjoined with certain data pre-processing method, the computing of (sic) Gamma(0i) (i.e. grey relational grade) in Hsia's and Nagai's (when p = 1) models can be omitted because the value of roi is equal to the value of 1/n Sigma(k=1) (n) x(i)* (k) and [Sigma(k=1) (n) x(i)* (k)] - n + 1, respectively. In conclusion, for multiple criteria decision making (MCDM), it seems not necessary to use these cardinal GRA models if they are used as a localized GRA model. Besides, by using a GRA model, we should consider data pre-processing holistically.

  • 出版日期2017