交叉销售下基于支持向量聚类的数据驱动多产品库存鲁棒优化模型

针对单周期环境下考虑交叉销售的多产品库存决策问题，在市场需求不确定条件下，建立了带有预算约束的交叉销售多产品库存鲁棒优化模型。针对不确定市场需求，采用支持向量聚类(SVC)方法构建了满足一定置信水平的数据驱动不确定集。进一步，运用拉格朗日对偶方法将所建模型等价转化为易于求解的线性规划问题。最后，通过数值计算对比分析了SVC不确定集下及传统不确定集下的零售商利润绩效，并评估了SVC数据驱动鲁棒优化方法导致的绩效损失，进而分析了预算及交叉销售系数对零售商利润绩效的影响。结果表明，SVC数据驱动鲁棒优化方法具有良好的鲁棒性，能够有效抑制需求不确定性对从事多产品销售的零售商利润绩效的影响。特别地，需求分布信息的缺失虽然会给零售商带来一定的绩效损失，但损失值很小，表明文中提出的基于SVC的数据驱动鲁棒优化方法可以为管理者在需求不确定性环境下制定库存策略提供有效决策借鉴。

Support Vector Clustering-based Data-driven Robust Optimization Model for Multi-product Inventory Problem with Cross-selling

The data-driven robust optimization model for a single-period multi-product inventory problem with cross-selling effect and budget constraint is established under the uncertain market demand. Cross-selling, as an important mutual relationship between items, is rarely introduced into the problem of inventory management. Cross-selling means that when a particular product has been purchased, other related products may also be purchased together due to their unknown interior associations. At the same time, if one of the set of jointly demanded items is out of stock, the sales pattern for the associated items will be altered and demands for the other items in this set may decrease as a result. The key to data-driven robust optimization is the choice of an appropriate uncertainty set which should flexibly adapt to the intrinsic structure of the data, thereby well characterizes the demand probability distributions and ameliorates the suboptimality of solutions. In this study, a support vector clustering (SVC) approach is used to construct the data-driven uncertainty set to which the uncertain demands belong with a certain confidence level. Furthermore, the proposed robust optimization model is transformed into a tractable linear programming model by Lagrange dual method. At last, numerical studies are conducted to compare and analyze the retailer's profit under the SVC-based uncertainty set and the traditional uncertainty sets. Besides, the performance loss caused by the SVC-based data-driven robust optimization method is evaluated, and then the impacts of budget and cross-selling coefficient on the retailer’s profit performance are analyzed. The results show that the SVC-based data-driven robust optimization approach is robust and can effectively restrain the impact of demand uncertainties on the retailer’s profit performance. In particular, although the lack of demand distribution information can incur a certain loss of inventory performance, the loss is very small, which indicates that the proposed SVC-based data-driven robust optimization method can provide effective support for managers to make inventory strategy under demand uncertainties. Some recommendations for performance improvement are proposed. For example, the inventory managers should adjust the regularization parameter appropriately to control the conservatism as well as the complexity of the induced robust optimization problems and should keep a good record of the daily sales data to improve the effectiveness of inventory decisions.