Supply and Demand Functions in Inventory Models

The central issue in supply chain management is to match supply with demand, and the heart of a planning model is the modeling of supply and demand functions. To allow for analytical tractability, the existing literature often assumes almost surely linear supply and demand functions, which greatly limits the applicability of the models. The goal of this paper is to provide a unified approach to analyze general random supply and demand functions. By transforming the problem into one defined on a higher dimension, we show that many of the seemingly highly nonlinear supply and demand functions (in the almost sure sense) are linear in the stochastic sense. With this new notion of linearity, called stochastic linearity in mid-point, our ability to analyze supply chain problems is much enhanced. We are able to prove the concavity of the profit function in the transformed supply and demand decisions for a general class of supply and demand functions that include, but are not restricted to, the ones studied in the existing literature. Thus, many of the challenging problems now become tractable. Moreover, we characterize a set of easy-to-verify conditions for the stochastic linearity in mid-point that allows us to test functions with known forms. For functions with unknown forms, we develop a nonparametric approach to allow for empirical estimation and verification of the needed stochastic properties directly from the data