Loss Functions for Inventory Control

TL;DR

This paper derives analytic expressions for various loss functions related to inventory control, facilitating their implementation in optimization models and filling a gap in existing literature.

Contribution

It provides the first comprehensive set of analytic formulas for loss functions across multiple probability distributions, aiding researchers and practitioners.

Findings

Analytic expressions for first-order, complementary, and second-order loss functions.

Enhanced ability to incorporate diverse probability distributions in inventory models.

Facilitates implementation of loss functions using standard probability functions.

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.