Inventory Policies with Returns and Demand Uncertainty: Optimal, Fixed-Quantity, and Order Cancellation Models

This paper develops a continuous-review inventory framework that integrates stochastic demand, stochastic returns, perishability, stochastic lead times, and lost sales. Unlike most inventory studies, which treat returns primarily as a reverse logistics phenomenon, product returns are modeled as an endogenous stochastic replenishment source that directly influences replenishment planning and procurement decisions. The inventory process is formulated as a continuous-time Markov chain, enabling the derivation of exact steady-state performance measures and long-term cost expressions. This paper further examines fixed-quantity replenishment policies under practical ordering constraints. The results show that return flows substantially reduce procurement requirements, inventory levels, and total system costs. The analysis accommodates both classical (r Q) operating regimes and reveals that the latter may emerge naturally when replenishment quantities are exogenously fixed. The model is subsequently extended to a new (Q,r,ψ) policy that allows outstanding replenishment orders to be canceled when inventory is recovered by returning during the lead time. Numerical experiments indicate that cancellation flexibility can generate substantial cost savings, particularly in return-intensive environments with high inventory carrying costs. Overall, the findings demonstrate that returns constitute a strategic replenishment resource and should be explicitly incorporated into inventory planning and control.