Stochastic Production Planning: Optimal Control and Analytical Insights
Dragos-Patru Covei
TL;DR
The paper addresses stochastic production planning under uncertainty with a running cost $|p(t)|^{2}+b(|y(t)|)$ and a stopping time $\tau=\inf\{t>0:|y(t)|\ge R\}$, formulating the problem via dynamic programming and the Hamilton–Jacobi–Bellman (HJB) equation. By transforming the nonlinear HJB with $z(x)=-2\sigma^{2}\ln u(x)$, the authors obtain a linear-like elliptic PDE $\Delta u(x)=\frac{1}{\sigma^{4}}b(|x|)u(x)$ under radial symmetry, and prove existence and uniqueness of a radially symmetric, convex, positive solution $u$ with $u(0)=\alpha$ and $u(R)=e^{-Z_{0}/(2\sigma^{2})}$. The optimal feedback control is $p^{*}(x)=-\frac{1}{2}\nabla z(x)$, whose magnitude $|p^{*}(x)|=-\frac{1}{2}h'(r)$ is nondecreasing in the radius, aligning with intuitive inventory-driven production policies. Numerical experiments using an Euler scheme validate the theoretical properties, including inventory boundedness by $R$ and monotone, convex/concave structures in the transformed value function, and an illustrative two-product example. The work advances stochastic production planning by offering a rigorous, radially symmetric framework with a tractable transformed PDE, robust optimal control, and practical numerical validation with real-world relevance.
Abstract
This study investigates a stochastic production planning problem with a running cost composed of quadratic production costs and inventory-dependent costs. The objective is to minimize the expected cost until production stops when inventory reaches a specified level, subject to a boundary condition. Using probability space and Brownian motion, the Hamilton-Jacobi-Bellman (HJB) equation is derived, and optimal feedback control is obtained. The solution demonstrates desirable monotonicity and convexity properties under specific assumptions. An illustrative example further confirms these results with explicit function properties and a practical application.