Abstract We present a method for bundling scenarios in a progressive hedging heuristic (PHH) applied to stochastic service network design, where the uncertain demand is represented by a finite number of scenarios. Given the number of scenario bundles, we first calculate a vector of probabilities for every scenario, which measures the association strength of a scenario to each bundle center. This membership score calculation is based on existing soft clustering algorithms such as Fuzzy C-Means (FCM) and Gaussian Mixture Models (GMM). After obtaining the probabilistic membership scores, we propose a strategy to determine the scenario-to-bundle assignment. By contrast, almost all existing scenario bundling methods such as K-Means (KM) assume before the scenario-to-bundle assignment that a scenario belongs to exactly one bundle, which is equivalent to requiring that the membership scores are Boolean values. The probabilistic membership scores bring many advantages over Boolean ones, such as the flexibility to create various degrees of overlap between scenario bundles and the capability to accommodate scenario bundles with different covariance structures. We empirically study the impacts of different degrees of overlap and covariance structures on PHH performance by comparing PHH based on FCM/GMM with that based on KM and the cover method, which represents the state-of-the-art scenario bundling algorithm for stochastic network design. The solution quality is measured against the lower bound provided by CPLEX. The experimental results show that, GMM-based PHH yields the best performance among all methods considered, achieving nearly equivalent solution quality in a fraction of the run-time of the other methods.

中文翻译：

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基于软聚类的场景捆绑，用于随机服务网络设计中的渐进对冲启发式

摘要 我们提出了一种在应用于随机服务网络设计的渐进对冲启发式 (PHH) 中捆绑场景的方法，其中不确定的需求由有限数量的场景表示。给定场景束的数量，我们首先计算每个场景的概率向量，它衡量场景与每个束中心的关联强度。这种成员分数计算基于现有的软聚类算法，例如模糊 C 均值 (FCM) 和高斯混合模型 (GMM)。在获得概率成员分数后，我们提出了一种策略来确定场景到捆绑的分配。相比之下，几乎所有现有的场景捆绑方法，如 K-Means (KM)，在场景到捆绑分配之前都假设一个场景恰好属于一个捆绑，这相当于要求成员分数是布尔值。概率成员分数比布尔分数带来许多优势，例如在场景包之间创建不同程度重叠的灵活性以及适应具有不同协方差结构的场景包的能力。我们通过比较基于 FCM/GMM 的 PHH 与基于 KM 和覆盖方法的 PHH 来实证研究不同程度的重叠和协方差结构对 PHH 性能的影响，该方法代表了随机网络的最先进场景捆绑算法设计。解质量是根据 CPLEX 提供的下限测量的。实验结果表明，在所有考虑的方法中，基于 GMM 的 PHH 产生了最好的性能，