In this paper, we study four projection-based normalization models and a decision-making method for probabilistic linguistic multi-criteria decision-making problems, in which the assessment information about an alternative with respect to a criterion is incomplete and the criteria weight values are not precisely known but the ranges are available. To apply the projection to the probabilistic linguistic environment, we propose the equivalent expression forms of the probabilistic linguistic term sets, and then the equivalent transformation functions between the probabilistic linguistic term set and its associated vector are presented to realize the conversion between the operations on the probabilistic linguistic term sets and the operations on their associated vectors. Next, the projection formulas of the probabilistic linguistic term sets are introduced to build different normalization models for different types of uncertain probabilistic linguistic multi-criteria decision-making problems. After that, a new deviation degree formula is proposed to account for the rationality and validity of the normalization models from the theoretical perspective. Finally, the probabilistic linguistic two-step method is used to determine the criteria weights values and rank the alternatives, and the validity of these projection-based normalization models and our proposed decision-making method are illustrated by a case about the performance assessment of data hiding techniques.

在本文中，我们研究了四个基于投影的归一化模型和一种针对概率语言多准则决策问题的决策方法，其中关于准则的替代方案的评估信息不完整，准则权重值为尚不清楚，但范围可用。为了将投影应用于概率语言环境，我们提出了概率语言术语集的等价表达形式，然后提出了概率语言术语集及其相关向量之间的等价转换函数，以实现对概率语言项之间的转换。概率语言术语集及其相关向量的运算。下一个，引入概率语言术语集的投影公式，针对不同类型的不确定概率语言多准则决策问题建立不同的归一化模型。之后，提出了一个新的偏差度公式，从理论的角度考虑了归一化模型的合理性和有效性。最后，采用概率语言两步法确定标准权重值并对备选方案进行排序，并以数据性能评估为例说明了这些基于投影的归一化模型和我们提出的决策方法的有效性。隐藏技术。提出了新的偏差度公式，从理论的角度解释了归一化模型的合理性和有效性。最后，采用概率语言两步法确定标准权重值并对备选方案进行排序，并以数据性能评估为例说明了这些基于投影的归一化模型和我们提出的决策方法的有效性。隐藏技术。提出了新的偏差度公式，从理论的角度解释了归一化模型的合理性和有效性。最后，采用概率语言两步法确定标准权重值并对备选方案进行排序，并以数据性能评估为例说明了这些基于投影的归一化模型和我们提出的决策方法的有效性。隐藏技术。