Loss Functions for Inventory Control
Steven R. Pauly
TL;DR
The paper addresses the need for consistent analytic expressions of three loss functions in inventory control and provides closed-form formulas for eight common distributions (four discrete, four continuous). It presents derivations, relationships among L1, Lc, and L2, and demonstrates how these expressions improve numerical accuracy and computational efficiency in quantitative models. The main contributions are a complete reference of analytic loss-function expressions and a discussion of their practical impact for policy optimization and software implementation. The work enables more accurate modeling of lead-time demand and supports broader adoption of diverse distributions in inventory optimization.
Abstract
In this paper, we provide analytic expressions for the first-order loss function, the complementary loss function and the second-order loss function for several probability distributions. These loss functions are important functions in inventory optimization and other quantitative fields. For several reasons, which will become apparent throughout this paper, the implementation of these loss functions prefers the use of an analytic expression, only using standard probability functions. However, complete and consistent references of analytic expressions for these loss functions are lacking in literature. This paper aims to close this gap and can serve as a reference for researchers, software engineers and practitioners that are concerned with the optimization of a quantitative system. This should lead directly to easily using different probability distributions in quantitive models which is at the core of optimization. Also, this paper serves as a broad introduction to loss functions and their use in inventory control.