Loss Functions for Inventory Control

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.