Robust inventory routing problem considering budget violation under demand uncertainty

This study investigates the inventory routing problem (IRP) under ambiguous demand distribution. Compared with existing research that assumes the demand distribution is either predefined or lies in an ambiguity set, we adopt a globalized distributionally robust approach. We construct both inner and outer ambiguity sets to describe distributional ambiguity, regardless of whether retailer demand follows a discrete distribution or an unspecified type. This approach ensures performance is maintained even when the demand distribution extends beyond the inner ambiguity set. Additionally, we introduce more flexible soft constraints, coupled with a budget violation metric, to manage the gaps between actual holding and penalty costs and the budgets at retailers, particularly when demand distribution lies outside the inner ambiguity set. Analysis of our model demonstrates its ability to generalize distributionally robust, robust, or stochastic IRP models through appropriate input settings, and thereby we establish the bounds for the objective values of these models. Numerical studies, based on a real-case scenario, show that the delivery solutions from our proposed models are more effective in mitigating the impact of retailer demand uncertainty and achieve superior out-of-sample performance compared to other benchmark models.