Stochastic Production Planning in Manufacturing Systems

We study stochastic production planning in capacity-constrained manufacturing systems, where feasible operating states are restricted to a convex safe-operating region. The objective is to minimize the total cost that combines a quadratic production effort with an inventory holding cost, while automatically halting production when the state leaves the safe region. We derive the associated Hamilton–Jacobi–Bellman (HJB) equation, establish the existence and uniqueness of the value function under broad conditions, and prove a concavity property of the transformed value function that yields a robust gradient-based optimal feedback policy. From an operations perspective, the stopping mechanism encodes hard capacity and safety limits, ensuring bounded risk and finite expected costs. We complement the analysis with numerical methods based on finite differences and illustrate how the resulting policies inform real-time decisions through two application-inspired examples: a single-product case calibrated with typical process-industry parameters and a two-dimensional example motivated by semiconductor fabrication, where interacting production variables must satisfy joint safety constraints. The results bridge rigorous stochastic control with practical production planning and provide actionable guidance for operating under uncertainty and capacity limits.