Stochastic Production Planning: Optimal Control and Analytical Insights

TL;DR

This paper formulates and solves a stochastic production planning problem using optimal control theory, deriving explicit solutions and properties for minimizing expected costs with inventory constraints.

Contribution

It introduces a novel stochastic control framework for production planning with explicit analytical solutions and insights into cost and inventory dynamics.

Findings

Optimal feedback control derived from HJB equation

Solution exhibits monotonicity and convexity properties

Practical example confirms theoretical results

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.