The stochastic ordering of mean-preserving transformations and its applications

•A unified theory that enables variability ordering for any mean preserving transformation.•Application to affine transformation and correction of an error in the literature.•Application to truncation and inventory models in supply chain management.•Application to capping and procurement risk management.

The stochastic variability measures the degree of uncertainty for random demand and/or price in various operations problems. Its ordering property under mean-preserving transformation allows us to study the impact of demand/price uncertainty on the optimal decisions and the associated objective values. Based on Chebyshev’s algebraic inequality, we provide a general framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and apply it to a broad class of specific transformations, including the widely used mean-preserving affine transformation, truncation, and capping. The application to mean-preserving affine transformation rectifies an incorrect proof of an important result in the inventory literature, which has gone unnoticed for more than two decades. The application to mean-preserving truncation addresses inventory strategies in decentralized supply chains, and the application to mean-preserving capping sheds light on using option contracts for procurement risk management.