模糊数及其运算是人工智能研究中的重要课题。不少研究者曾提出了多种模糊数的构造 及其运算的定义方式。 本文提出了一类新的模糊数，称为“高斯概率模糊数”（GPDFN）。其隶属函数是由高斯 分布的概率密度函数与一个非线性映射构造而成。选取不同的非线性映射能够构造出不同形 态的模糊数。我们主要选取了一种特殊的非线性映射，研究由其构成的模糊数。这类模糊数 通过集中度参数 μ ? 与 μ + 来调控形态。集中度参数在无法直接获取的情况下可由控制点坐 标来确定。 对于我们所提出的一类模糊数，还定义了相应的代数运算（包括加、减、数乘）与聚合运 算。常规的聚合运算分为三个层次：中心、支集、集中度。除此以外，还定义了这类模糊数 的带序加权平均值运算，为此先规定了这类模糊数的排序方法。 在以加权平均的方式进行聚合时，聚合权重有时不能直接获得，需要通过参考期望得到 的结果来确定。本文提出了根据期望得到结果来调控各个数据聚合权重的一种方式。 最后演示了我们构造的模糊数及其运算在评分机制中的应用。先通过特制的调查问卷获 取用户的满意度评分数据，再将每个用户的评分表示为一个高斯概率模糊数，然后通过加权 平均或带序加权平均的方式进行聚合。

Fuzzy numbers and their operations are heated topics in the field of artificial intelligence. Various forms of fuzzy numbers and operations has been proposed.

In this article, we put forward a new class of fuzzy number, named Gaussian Probability Density Fuzzy Number, acronymized as GPDFN. It's membership function is made up of probability density function of Gaussian distribution and a nonlinear mapping. Different nonlinear mappings generate different fuzzy numbers. We mainly study on fuzzy numbers generated by a specific nonlinear function. This class of fuzzy number is characterized by parameter μ- and μ+, representing for their concentration degree. In circumstances where concentration degree can't be acquired directly, they can be decided by control points.

On this class of fuzzy number are defined arithmetic operations and aggregation operations. Arithmetic operators include addition, subtraction, and multiplication by real number. Usual aggregation operators are defined in three levels: center, support set, and concentration degree. We also introduce ordered weighted average aggregation operator, which is based on the definition of ordering on our fuzzy number class.

When weighted average operator is used in aggregation, there might be problem that the weight parameter of aggregation is hard to obtain. In this case, we propose a method to decide the weight according to an expected result.

We demonstrate how our fuzzy number and its operations applied in scoring system. In this system, each user show his satisfying score degree via a special questionnaire. Every single score is present by a GPDFN. An intergrated score can be generated by aggregating the GPDFNs mentioned above.