Calculation Method’s Extension of the Characteristic of Grey Number

Abstract

A grey number plays a key role in a grey system, and the characteristics of the grey number are one of the core researched contents of the grey number. The whitenization weight function, the grey degree, and the kernel are the characteristics of the grey number. Currently, a rectangular whitenization weight function is a foundation to calculate the grey degree and the kernel of the grey number. However, the rectangular whitenization weight function cannot comprehensively represent the distribution of the grey number’s value. In order to expand the application of grey number in a system with the known or empirical information of grey number’s distribution, a multiplying operator is constructed to deal with the inner relationship between the whitenization weight function and the probability density function of the grey number. The concept of sym-probability density function is proposed as the whitenization weight function for the grey number in this paper. Meanwhile, according to the various shapes of whitenization weight function, an integral-form formula is developed to calculate the grey degree, and a centroid-form formula is developed to calculate the kernel of the grey number. These formulas are general formulas for two characteristic values of the grey number. The novel general formulas, which are considered as the extension of traditional calculation formulas, do not only handle the regular whitenization weight function but also cope with an irregular whitenization weight function. Finally, several examples are implemented to demonstrate the performance and efficiency of these opinions.