In order to determine all the required algebraic relationships among the parameters of the grey possibility functions with the given final clustering results, a matrix-based approach for the bidirectional perspective of the grey clustering model based on the grey possibility functions is developed in this work. The definition of the bidirectional problem in a grey clustering model is defined. The matrix representations for the symmetric and asymmetric grey possibility functions and grey clustering coefficients are derived. Two key theorems, one corollary, and two matrix relations are put forward for calculating the parameters in the novel model. Four matrix-based inequalities are proposed to obtain all the available grey possibility functions under the four different situations. The more general grey clustering model with asymmetric grey possibility functions is derived. To explain how it can be utilized in practice, this novel approach is applied to determine the group of the Pacific Rim countries with the partial known results in the Fukushima nuclear wastewater discharge.

为了确定灰色可能性函数的参数与给定的最终聚类结果之间的所有代数关系，本文开发了一种基于矩阵的方法，用于基于灰色可能性函数的灰色聚类模型的双向视角。定义了灰色聚类模型中的双向问题。推导了对称和非对称灰色可能性函数和灰色聚类系数的矩阵表示。为计算新模型中的参数，提出了两个关键定理、一个推论和两个矩阵关系。提出了四种基于矩阵的不等式来获得四种不同情况下所有可用的灰色可能性函数。推导了具有非对称灰色可能性函数的更一般的灰色聚类模型。为了解释如何在实践中使用它，这种新方法被应用于确定在福岛核废水排放中具有部分已知结果的环太平洋国家组。