Distributionally Robust Inventory Management with Advance Purchase Contracts

We propose a distributionally robust inventory model for finding an optimal ordering policy that attains the minimum worst-case expected total cost. In a classical stochastic setting, this problem is typically addressed by dynamic programming and is solved by the famous base-stock policy. This approach however crucially relies on two controversial assumptions: the demands are serially independent and the demand distribution is perfectly known. Aiming to address these issues, inspired by the seminal work of Scarf (1958), we adopt a mean-variance ambiguity set that imposes neither the shape of each marginal demand distribution nor their independence structure, and we focus on the case of advance purchase agreements which are prevalent in the robust inventory literature and have drawn renewed attention because of the Covid-19 vaccine procurement. The proposed distributionally robust inventory model provably reduces to a finite conic optimization problem with however an exponential number of constraints. To gain tractability and to err on the safe side, we propose two conservative approximations. The first approximation is obtained by recognizing the problem as an artificial two-stage robust optimization problem and then by restricting each adaptive decision to a linear decision rule. The second approximation, on the other hand, is obtained by a constraint partitioning and by upper bounding each resultant maximum sum with a sum of maxima. We then present a progressive approximation based on a scenario reduction technique to gauge the quality of the proposed conservative approximations. We prove that this progressive approximation is exact when the inventory problem consists of two periods, and besides we use it to show that our conservative solutions are still close to being optimal when the planning horizon is longer. All of our exact and approximate inventory models are expressed as standard conic programs which allow for the incorporation of additional distributional information. The extensions are readily obtained by deriving a new cone that corresponds to the restricted ambiguity set and embedding it in the original problems. We analytically derive the worst-case demand distribution from the mean-variance ambiguity set and numerically use it to show that our robust inventory policy is more resilient to the misspecification of the demand distribution than the state-of-the-art non-robust policies