基于正态分布概率区间犹豫模糊集的多属性决策方法及应用

A Multi-Attribute Decision-Making Method and Its Application Based on Normally Distributed Probabilistic Interval-Valued Hesitant Fuzzy Sets

摘要: 针对现有概率区间犹豫模糊集普遍假设区间内隶属度服从均匀分布，与决策者“中间值可能性大、两端可能性小”的实际认知规律不符的问题，本文提出一种基于正态分布假设的概率区间犹豫模糊集，并构建相应的多属性决策模型。首先，将正态分布作为一般分布框架的特例引入概率区间犹豫模糊集，在区间关于均值对称的假设下，利用标准正态分布分位数建立区间端点、置信水平与正态分布参数之间的映射关系。其次，基于全期望公式与全方差公式，推导多区间聚合后的总体均值与总体方差公式。最后，将所提出的正态分布概率区间犹豫模糊集与TOPSIS (Technique for Order Preference by Similarity to Ideal Solution)方法相结合，通过将原始准则扩展为“均值”与“负标准差”两个子准则，引入偏好系数以调节决策者对期望表现与稳定性的权衡，构建多属性群决策模型。汽车选购案例验证了该方法能够有效执行决策流程；控制变量实验表明，当将区间内部分布假设由正态替换为均匀后最优方案发生改变，验证了分布假设对决策结果具有实质性影响。本文方法通过引入更符合认知规律的正态分布假设，并协同考虑均值和稳定性两个维度，为不确定环境下的多属性决策提供了一种逻辑更完善、结果更鲁棒的新工具。

Abstract: To address the issue that existing probabilistic interval-valued hesitant fuzzy sets generally assume a uniform distribution of membership degrees within intervals, which contradicts the actual cognitive pattern of decision-makers that “intermediate values are more likely while values at both ends are less likely”, this paper proposes a probabilistic interval-valued hesitant fuzzy set based on the normal distribution assumption and constructs a corresponding multi-attribute decision-making model. First, the normal distribution is introduced as a special case into the framework of probabilistic interval-valued hesitant fuzzy sets. Under the assumption that the interval is symmetric about the mean, the quantiles of the standard normal distribution are utilized to establish the mapping relationships among interval endpoints, confidence levels, and the parameters of the normal distribution (mean and standard deviation). Second, based on the law of total expectation and the law of total variance, the formulas for the overall mean and overall variance after aggregating multiple intervals are derived. Finally, the proposed normally distributed probabilistic interval-valued hesitant fuzzy set is integrated with the TOPSIS method. By extending the original criteria into two sub-criteria, namely “mean” and “negative standard deviation”, and introducing a preference coefficient to adjust the trade-off between expected performance and stability, a multi-attribute group decision-making model is constructed. A car selection case verifies that the proposed method can effectively execute the decision-making process; a controlled variable experiment shows that when the intra-interval distribution assumption is changed from normal to uniform, the optimal alternative changes, confirming that the distribution assumption has a substantive impact on decision outcomes. By introducing the more cognitively plausible normal distribution assumption and jointly considering both the mean and stability dimensions, this paper provides a new tool with more complete logic and more robust results for multi-attribute decision-making under uncertain environments.