Newsvendor Inventory and Pricing Decisions in the Presence of Stochastic Purchase Return

We consider a multi-period Newsvendor problem where the seller decides the initial inventory level and dynamically sets the prices in every periods. A purchase made in a period can be returned at a (random) future period following a return time distribution. Unlike existing works in the literature, which usually assume an exponential return time distribution with memoryless property for tractability, we consider a sufficiently general class of return time distributions that is not Markovian. The stochastic control formulation of this problem is intractable since the inventory dynamics depend on the timing of all past sales that have not been returned. We overcome this challenge by proposing two computationally tractable policies that are designed based on the solution of the deterministic counterpart of the intractable stochastic model. Our first policy (FP) uses a sequence of fixed prices throughout the horizon; our second policy (DBP) dynamically adjusts the prices based on the realized sales and returns. We show that our proposed policies are asymptotically optimal in the setting where the total market size for the whole selling season is large, with the second policy guaranteeing a stronger performance bound than the first one. Our analysis characterizes the dependency of price adjustments across periods. To test the performance of our policies on non-asymptotic setting, we run numerical experiments using relatively small per-period market size. We demonstrate that, in general, ignoring product returns in inventory planning and pricing decisions results in a 20-40% loss and that naive re-optimizations of a formulation that ignores (future) product returns do not help much; in contrast, our FP policy curbs the loss to 4-20% and our DBP policy further shrinks it to 1.5-13%. This highlights both the value of properly taking into account customers' return behavior in inventory and pricing decisions, and the power of properly devised simple policies