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Stochastic Production Planning with Regime Switching: Numerical and Sensitivity Analysis, Optimal Control, and Python Implementation

Dragos-Patru Covei

TL;DR

This paper develops a stochastic production planning framework with regime switching to capture economic cycles, modeling inventory dynamics under regime-dependent volatility and discounting. The authors formulate a quadratic cost objective and derive a regime-dependent pair of HJB equations, which are transformed into a solvable elliptic PDE system for transformed variables $u_1,u_2$. A constructive monotone-iterative numerical scheme is proposed to compute the value functions $z_1,z_2$ and the associated optimal production policies $p_1^*,p_2^*$, with a detailed procedure to determine auxiliary parameters $K_1,K_2$ and to simulate inventory trajectories. Sensitivity analysis and model comparisons reveal conservative behavior of regime-switching models relative to static ones, and the work offers practical guidance for production planning under fluctuating economic conditions, along with clear pathways for extensions such as correlated noise and alternative loss functions. The study bridges theory and practice by providing an implementable Python pipeline for solving the PDEs and evaluating policy performance across regimes.

Abstract

This study investigates a stochastic production planning problem with regime-switching parameters, inspired by economic cycles impacting production and inventory costs. The model considers types of goods and employs a Markov chain to capture probabilistic regime transitions, coupled with a multidimensional Brownian motion representing stochastic demand dynamics. The production and inventory cost optimization problem is formulated as a quadratic cost functional, with the solution characterized by a regime-dependent system of elliptic partial differential equations (PDEs). Numerical solutions to the PDE system are computed using a monotone iteration algorithm, enabling quantitative analysis. Sensitivity analysis and model risk evaluation illustrate the effects of regime-dependent volatility, holding costs, and discount factors, revealing the conservative bias of regime-switching models when compared to static alternatives. Practical implications include optimizing production strategies under fluctuating economic conditions and exploring future extensions such as correlated Brownian dynamics, non-quadratic cost functions, and geometric inventory frameworks. This research bridges the gap between theoretical modeling and practical applications, offering a robust framework for dynamic production planning.

Stochastic Production Planning with Regime Switching: Numerical and Sensitivity Analysis, Optimal Control, and Python Implementation

TL;DR

This paper develops a stochastic production planning framework with regime switching to capture economic cycles, modeling inventory dynamics under regime-dependent volatility and discounting. The authors formulate a quadratic cost objective and derive a regime-dependent pair of HJB equations, which are transformed into a solvable elliptic PDE system for transformed variables . A constructive monotone-iterative numerical scheme is proposed to compute the value functions and the associated optimal production policies , with a detailed procedure to determine auxiliary parameters and to simulate inventory trajectories. Sensitivity analysis and model comparisons reveal conservative behavior of regime-switching models relative to static ones, and the work offers practical guidance for production planning under fluctuating economic conditions, along with clear pathways for extensions such as correlated noise and alternative loss functions. The study bridges theory and practice by providing an implementable Python pipeline for solving the PDEs and evaluating policy performance across regimes.

Abstract

This study investigates a stochastic production planning problem with regime-switching parameters, inspired by economic cycles impacting production and inventory costs. The model considers types of goods and employs a Markov chain to capture probabilistic regime transitions, coupled with a multidimensional Brownian motion representing stochastic demand dynamics. The production and inventory cost optimization problem is formulated as a quadratic cost functional, with the solution characterized by a regime-dependent system of elliptic partial differential equations (PDEs). Numerical solutions to the PDE system are computed using a monotone iteration algorithm, enabling quantitative analysis. Sensitivity analysis and model risk evaluation illustrate the effects of regime-dependent volatility, holding costs, and discount factors, revealing the conservative bias of regime-switching models when compared to static alternatives. Practical implications include optimizing production strategies under fluctuating economic conditions and exploring future extensions such as correlated Brownian dynamics, non-quadratic cost functions, and geometric inventory frameworks. This research bridges the gap between theoretical modeling and practical applications, offering a robust framework for dynamic production planning.
Paper Structure (56 sections, 7 theorems, 103 equations, 4 figures)

This paper contains 56 sections, 7 theorems, 103 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Regime Switching and Dynamics
  4. Inventory Dynamics and State Variables
  5. Objective Function
  6. Optimization Problem
  7. Optimization Objective
  8. Hamilton-Jacobi-Bellman Equations
  9. Assumptions
  10. Transformation and Simplification
  11. Existence and Uniqueness of Solutions
  12. Optimal Control
  13. Optimal Production Policy
  14. Verification of Optimality
  15. Numerical Computation
  16. ...and 41 more sections

Key Result

Lemma 3.1

For any $a_{1},\alpha _{1},a_{2},\alpha _{2},\sigma _{1},\sigma _{2},M_{1},M_{2},R\in (0,\infty )$ and $N\in \mathbb{N}^{\ast }$, there exist and are unique $K_{1},K_{2}\in (-\infty ,0)$ such that:

Figures (4)

  • 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.

Theorems & Definitions (11)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 6.1
  • Theorem 6.2
  • Theorem 6.3
  • Theorem 6.4
  • ...and 1 more