Supervised Learning for the (s,S) Inventory Model with General Interarrival Demands and General Lead Times

TL;DR

The paper tackles the analytical intractability of non-Markovian continuous-review $(s,S)$ inventories with general interarrival and lead times under lost sales. It develops a supervised learning framework that uses PH-based input sampling, moment-encoded representations, and offline discrete-event simulation labels to train three neural networks that predict the stationary distribution, average cycle time, and loss probability with high accuracy and near-instantaneous inference. Key contributions include quantifying the usefulness of the first five moments, providing an end-to-end data-generation and inference pipeline, and releasing an open-source package for fast steady-state analysis across a broad class of inventory models. The approach offers a practical, model-agnostic tool for real-time policy evaluation and optimization in complex stochastic supply chains, significantly reducing computational burden versus repeated simulations.

Abstract

The continuous-review (s,S) inventory model is a cornerstone of stochastic inventory theory, yet its analysis becomes analytically intractable when dealing with non-Markovian systems. In such systems, evaluating long-run performance measures typically relies on costly simulation. This paper proposes a supervised learning framework via a neural network model for approximating stationary performance measures of (s,S) inventory systems with general distributions for the interarrival time between demands and lead times under lost sales. Simulations are first used to generate training labels, after which the neural network is trained. After training, the neural network provides almost instantaneous predictions of various metrics of the system, such as the stationary distribution of inventory levels, the expected cycle time, and the probability of lost sales. We find that using a small number of low-order moments of the distributions as input is sufficient to train the neural networks and to accurately capture the steady-state distribution. Extensive numerical experiments demonstrate high accuracy over a wide range of system parameters. As such, it effectively replaces repeated and costly simulation runs. Our framework is easily extendable to other inventory models, offering an efficient and fast alternative for analyzing complex stochastic systems.

Figures (14)

• Figure 1: Work-flow diagram of our learning procedure.
• Figure 2: Work-flow diagram for inference.
• Figure 3: SCV, Skewness, and Kurtosis scatter.
• Figure 4: Learning scheme.
• Figure 5: NN 1 architecture for approximating $\textbf{P}$.