Scenario-based Distributionally Robust Optimization for the Stochastic Inventory Routing Problem

We consider a class of the inventory routing problem in a discrete and finite time horizon, where the demand for homogeneous products at retail stores is uncertain and varies across different scenarios. The supplier is required to determine the times to visit retailers, the replenishment quantities to each retailer, and the routing of a vehicle so as to minimize the sum of stockout, holding, and transportation costs. We propose a scenario-based distributionally robust optimization framework to tackle this problem. We transform the distributionally robust optimization model into a mixed-integer problem, which can be solved efficiently by our proposed algorithm. We adopt a warm-start procedure that utilizes the solution to the nominal model in our methodological framework. Then we apply a Tabu search algorithm, integrated with column generation, to solve a set-partitioning-like integer linear programming model so that a better route set can be identified. By doing so, a large-scale scenario-based distributionally robust optimization model can be solved. We conduct a case study of a fuel company and construct realistic instances to demonstrate the performance of our proposed method. Computational results suggest that the model taking into account various scenarios is more effective when random demands can be classified; the model with a linear decision rule outperforms a non-adaptive model; and the model with the route set identified by an improved algorithm can deliver a better solution than the original route set.