Optimising retailer-supplier inventory systems with defective returns: a continuous-time Markov chain and neural network-based approach

This study investigates a retailer-supplier inventory system by integrating demand forecasting through an LSTM neural network within a Continuous-Time Markov Chain framework. The proposed model optimises reorder points and order quantities under demand uncertainty, uncertain delivery time, and product returns. By combining predictive analytics with stochastic modelling, the framework captures nonlinear demand patterns influenced by price, discounts, and seasonality. Stationary inventory distributions are derived, and a profit-maximising nonlinear integer programming model is developed to determine optimal policies. Numerical results confirm the concavity of the profit function and support effective policy identification. The framework incorporates defective returns and price-dependent demand, enhancing its practical relevance. Managerial insights show that a 50% increase in supplier delivery rate reduces optimal reorder levels by approximately 20% while increasing expected profit by 2%, reflecting an unexpected profit gain despite reduced inventory.