Stochastic Production Planning in Manufacturing Systems

We extend the stochastic production planning framework to manufacturing systems, where the set of admissible production configurations is described by a general smooth convex domain $\omega $. In our setting, production operations continue as long as the production inventory $y(t)$ remains inside the capacity limits of $\omega $ and are halted once the state exits this region, i.e.,% \begin{equation*} \tau =\inf \{t>0:\Vert y(t)-x_{0}\Vert >\text{dist}(x_{0},\partial \omega )\}. \end{equation*}% The running cost is partitioned into a quadratic production cost $% a(p)=\left\Vert p\right\Vert ^{2}$ and an inventory holding cost modeled by a positive continuous function $b(y)$. We derive the associated Hamilton--Jacobi--Bellman (HJB) equation, verify the supermartingale property of the value function, and characterize the optimal feedback control. Techniques inspired by Lasry, Lions and Alvarez enable us to prove existence and uniqueness within this generalized production planning framework. Numerical experiments and a real-world examples illustrate the practical relevance of our results.