Robust recycling facility location with clustering

Abstract

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 χ2-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.