Stochastic Production Planning in Manufacturing Systems

TL;DR

This paper develops a stochastic production planning model for manufacturing systems with convex capacity constraints, deriving the associated HJB equation, and providing theoretical and numerical insights into optimal control strategies.

Contribution

It extends existing stochastic production planning frameworks to general convex domains, establishing existence, uniqueness, and characterization of optimal controls in this setting.

Findings

Derived the Hamilton--Jacobi--Bellman equation for the model

Proved existence and uniqueness of solutions within the convex domain framework

Numerical experiments demonstrate practical applicability

Abstract

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.