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Stochastic Production Planning in Manufacturing Systems

Dragos-Patru Covei

TL;DR

This work generalizes stochastic production planning to smooth convex domains $\omega$, introducing a stopping time when the inventory exits the safe region. The authors derive a generalized HJB equation, transform it to a linear PDE via $z=-\!2\sigma^{2}\log u$, and prove existence, uniqueness, and concavity properties of the value function, with a martingale-based verification yielding the optimal feedback $p^{*}(y)=-\tfrac{1}{2}\nabla z(y)$. They develop sub- and supersolution techniques to establish a well-posed, convex solution for $u$ and obtain asymptotics of the optimal control near the boundary. Numerically, they demonstrate 1D and 2D finite-difference implementations (Gauss–Seidel and monotone iterations), recover $z$ and $p^{*}$, and validate the approach with simulations of the inventory process, including a real-world flavor through parameterized examples. The results offer a rigorous, geometry-aware framework for robust, optimal production planning under uncertainty with practical stopping-time safety considerations.

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 $ω$. In our setting, production operations continue as long as the production inventory $y(t)$ remains inside the capacity limits of $ω$ and are halted once the state exits this region, i.e.,% \begin{equation*} τ=\inf \{t>0:\Vert y(t)-x_{0}\Vert >\text{dist}(x_{0},\partial ω)\}. \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.

Stochastic Production Planning in Manufacturing Systems

TL;DR

This work generalizes stochastic production planning to smooth convex domains , introducing a stopping time when the inventory exits the safe region. The authors derive a generalized HJB equation, transform it to a linear PDE via , and prove existence, uniqueness, and concavity properties of the value function, with a martingale-based verification yielding the optimal feedback . They develop sub- and supersolution techniques to establish a well-posed, convex solution for and obtain asymptotics of the optimal control near the boundary. Numerically, they demonstrate 1D and 2D finite-difference implementations (Gauss–Seidel and monotone iterations), recover and , and validate the approach with simulations of the inventory process, including a real-world flavor through parameterized examples. The results offer a rigorous, geometry-aware framework for robust, optimal production planning under uncertainty with practical stopping-time safety considerations.

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 . In our setting, production operations continue as long as the production inventory remains inside the capacity limits of and are halted once the state exits this region, i.e.,% \begin{equation*} τ=\inf \{t>0:\Vert y(t)-x_{0}\Vert >\text{dist}(x_{0},\partial ω)\}. \end{equation*}% The running cost is partitioned into a quadratic production cost and an inventory holding cost modeled by a positive continuous function . 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.

Paper Structure

This paper contains 19 sections, 2 theorems, 91 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Generalized Hamilton--Jacobi--Bellman Equation
  3. Results on the Value Function
  4. Preliminaries
  5. Existence and Uniqueness
  6. Verification via Martingale Property
  7. Definition of the Process
  8. Application of Itô's Lemma and Verification
  9. Optimal Feedback Control
  10. Two Examples of Numerical Implementation by Finite Difference Method
  11. Solve 1D PDE on an open interval
  12. Solve 2D PDE on Elliptical Domain
  13. Conclusion and Future Directions
  14. Declarations
  15. Conflict of interest
  16. ...and 4 more sections

Key Result

Theorem 2.1

Suppose $\omega \subset \mathbb{R}^{N}$ is a convex $C^{2}$, open bounded domain representing the admissible set of production inventory states, with smooth boundary $\partial \omega$, and that $b:\overline{\omega }\rightarrow \lbrack 0,\infty )$ is continuous with its zero set exactly the closure o Moreover, the asymptotic behavior of the magnitude of the optimal control is characterized by where

Figures (5)

  • Figure 1: Caption describing the two figures.
  • Figure 2: Caption describing the two figures.
  • Figure 3: Caption describing the two figures.
  • Figure 4: Caption describing the two figures.
  • Figure 5: Caption describing the two figures.

Theorems & Definitions (8)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3