In this work, we consider a recycling facility location-routing planning problem with feedstock volume and composition uncertainty. A salient feature of the problem is that the feedstock condition (volume and composition) exhibits both ambiguity and ‘unobservability’. That is, the distribution of (future) feedstock condition cannot be known at the planning stage, which is regarded as ambiguity or ‘here-and-now uncertainty’. At the same time, the feedstock condition (e.g., the composition) also cannot be perfectly observed even at the implementation stage, which we term the ‘unobservability’ or ‘wait-and-see uncertainty’. This is different from the conventional two-stage adaptive optimization problems with uncertainty. To tackle jointly the unobservability and ambiguity of feedstock condition, we propose a structured paradigm of finitely adaptive distributionally robust optimization, which is developed with a learning machinery integrating clustering analysis and ${\chi }^2$-divergence-based distributional ambiguity set. In particular, the distributionally robust optimization framework is utilized to tackle the ambiguity, while the clustering-based adaptivity over the imperfect feedstock observations is tailored to achieve the unobservability mitigation. Technically, we show that the resulting recycling location optimization problem under the paradigm can be reformulated into a mixed integer second-order cone program which can be handled by off-the-shelf MIP solvers. We also prove that given the feedstock volume uncertainty and feedstock composition uncertainty separately, the optimal value of our model, under some partition condition, is able to converge to that of the fully adaptive distributionally robust optimization model. Finally, sufficient numerical experiments demonstrate the effectiveness of our approach in mitigating the unobservability and ambiguity effects in recycling planning, also the scalability advantage of our approach could motivates its applications in other areas.

在这项工作中，我们考虑了具有原料量和成分不确定性的回收设施选址规划问题。该问题的一个显着特征是原料条件（体积和成分）表现出​​模糊性和“不可观察性”。也就是说，在规划阶段无法知道（未来）原料状况的分布，这被视为模糊或“此时此地的不确定性”。同时，原料条件（例如，组合）即使在实施阶段也无法完美观察，我们称之为“不可观察性”或“观望不确定性”。这不同于传统的具有不确定性的两阶段自适应优化问题。为了共同解决原料条件的不可观察性和模糊性，我们提出了一种有限自适应分布鲁棒优化的结构化范式，该范式是用集成聚类分析和${\chi }^2$-基于散度的分布模糊集。特别是，分布稳健的优化框架被用来解决歧义，而对不完美原料观察的基于聚类的适应性被定制以实现不可观察性的缓解。从技术上讲，我们表明在范式下产生的回收位置优化问题可以重新表述为混合整数二阶锥程序，可以由现成的 MIP 求解器处理。我们还证明，分别给定原料体积的不确定性和原料组成的不确定性，我们模型的最优值在某些分区条件下能够收敛到完全自适应分布鲁棒优化模型的最优值。最后，