Belief-Aware Inventory Control with Deep Mixture Models

This paper introduces a framework that combines deep mixture models with inven- tory optimization under uncertain demand, tackling key challenges at the intersec- tion of machine learning and operations research. We propose deep neural mixture models for demand forecasting that capture multimodal, bounded, and correlated patterns while maintaining computational tractability for downstream optimization. Our approach formulates inventory control as a partially observable Markov deci- sion process (POMDP) where belief states over mixture components evolve via Bayesian updates. In order to enable practical implementation, we develop a belief space clustering approach using medoid clustering that reduces the belief space to a finite set of representative points. We provide theoretical guarantees including contraction properties of belief updates, Lipschitz continuity bounds, and explicit performance bounds under belief discretization. The framework supports diverse neural architectures, including state-of-the-art deep learning time series forecasting models. Experiments on real-world pharmaceutical demand data demonstrate that the method is computationally efficient and can lead to promising performance when the forecasts are well calibrated.