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A Computational Method for Solving the Stochastic Joint Replenishment Problem in High Dimensions

Barış Ata, Wouter van Eekelen, Yuan Zhong

TL;DR

A novel, simulation-based computational method that relies on deep neural networks to solve the impulse control problem and proposes an implementable inventory control policy for the original (discrete-time) stochastic joint replenishment problem.

Abstract

We consider a discrete-time formulation for a class of high-dimensional stochastic joint replenishment problems. First, we approximate the problem by a continuous-time impulse control problem. Exploiting connections among the impulse control problem, backward stochastic differential equations (BSDEs) with jumps, and the stochastic target problem, we develop a novel, simulation-based computational method that relies on deep neural networks to solve the impulse control problem. Based on that solution, we propose an implementable inventory control policy for the original (discrete-time) stochastic joint replenishment problem, and test it against the best available benchmarks in a series of test problems. For the problems studied thus far, our method matches or beats the best benchmark we could find, and it is computationally feasible up to at least 50 dimensions -- that is, 50 stock-keeping units (SKUs).

A Computational Method for Solving the Stochastic Joint Replenishment Problem in High Dimensions

TL;DR

A novel, simulation-based computational method that relies on deep neural networks to solve the impulse control problem and proposes an implementable inventory control policy for the original (discrete-time) stochastic joint replenishment problem.

Abstract

We consider a discrete-time formulation for a class of high-dimensional stochastic joint replenishment problems. First, we approximate the problem by a continuous-time impulse control problem. Exploiting connections among the impulse control problem, backward stochastic differential equations (BSDEs) with jumps, and the stochastic target problem, we develop a novel, simulation-based computational method that relies on deep neural networks to solve the impulse control problem. Based on that solution, we propose an implementable inventory control policy for the original (discrete-time) stochastic joint replenishment problem, and test it against the best available benchmarks in a series of test problems. For the problems studied thus far, our method matches or beats the best benchmark we could find, and it is computationally feasible up to at least 50 dimensions -- that is, 50 stock-keeping units (SKUs).

Paper Structure

This paper contains 19 sections, 5 theorems, 61 equations, 3 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model
  4. The HJB Equation and an Equivalent Stochastic Identity
  5. Computational Method
  6. Test Problems and Benchmark Policies
  7. Computational Results
  8. Results in dimension 12.
  9. Low-dimensional results.
  10. Results in dimension 50.
  11. Concluding Remarks
  12. Formal Derivation of Stochastic Identity \ref{['eq:keyidentityprop']}
  13. Implementation Details
  14. Materials and Method Modifications
  15. Hyperparameter Configurations
  16. ...and 4 more sections

Key Result

Lemma 1

Let $x \in \mathbb{R}^d$. Suppose eq:align_ito holds a.s. Then, the stochastic identity eq:keyidentityprop holds if and only if the following is true:

Figures (3)

  • Figure 1: MDP and neural network policies (base case $d=2$).
  • Figure 2: Comparison of MDP and neural network policies across the remaining six two-dimensional test problems.
  • Figure 3: Value function and gradient computed using the expressions in sulem1986solvable, and the neural network approximations

Theorems & Definitions (10)

  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma:derivation']}
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof