Increasing evaluation indexes have been involved in the network modeling, and some parameters cannot be described precisely. Fuzzy set theory becomes a promising mathematical method to characterize such uncertain parameters. This study investigates the fuzzy multi-objective path optimization problem (FMOPOP), in which each arc has multiple crisp and fuzzy weights simultaneously. Fuzzy weights are characterized by triangular fuzzy numbers or trapezoidal fuzzy numbers. We adopt two fuzzy number ranking methods based on their fuzzy graded mean values and distances from the fuzzy minimum number. Motivated by the ripple spreading patterns on the natural water surface, we propose a novel ripple-spreading algorithm (RSA) to solve the FMOPOP. Theoretical analyses prove that the RSA can find all Pareto optimal paths from the source node to all other nodes within a single run. Numerical examples and comparative experiments demonstrate the efficiency and robustness of the newly proposed RSA. Moreover, in the first numerical example, the processes of the RSA are illustrated using metaphor-based language and ripple spreading phenomena to be more comprehensible. To the best of our knowledge, the RSA is the first algorithm for the FMOPOP that can adopt various fuzzy numbers and ranking methods while maintaining optimality.

网络建模涉及越来越多的评价指标，有些参数无法准确描述。模糊集理论成为一种很有前途的数学方法来表征这种不确定的参数。本研究研究了模糊多目标路径优化问题（FMOPOP），其中每条弧同时具有多个清晰和模糊的权重。模糊权重的特征是三角模糊数或梯形模糊数。我们采用了两种基于模糊分级平均值和与模糊最小数距离的模糊数排序方法。受天然水面上的波纹扩散模式的启发，我们提出了一种新颖的波纹扩散算法（RSA）来解决 FMOPOP。理论分析证明，RSA 可以在一次运行中找到从源节点到所有其他节点的所有 Pareto 最优路径。数值示例和对比实验证明了新提出的 RSA 的效率和鲁棒性。此外，在第一个数值示例中，使用基于隐喻的语言和波纹传播现象来说明 RSA 的过程，以便更易于理解。据我们所知，RSA 是第一个可以采用各种模糊数和排序方法同时保持最优性的 FMOPOP 算法。使用基于隐喻的语言和涟漪传播现象来说明 RSA 的过程，以便更易于理解。据我们所知，RSA 是第一个可以采用各种模糊数和排序方法同时保持最优性的 FMOPOP 算法。使用基于隐喻的语言和涟漪传播现象来说明 RSA 的过程，以便更易于理解。据我们所知，RSA 是第一个可以采用各种模糊数和排序方法同时保持最优性的 FMOPOP 算法。