Inventory Pooling Under Heavy-Tailed Demand

Abstract

Risk pooling has been studied extensively in the operations management literature as the basic driver behind strategies such as transshipment, manufacturing flexibility, component commonality, and drop-shipping. This paper explores the benefit of risk pooling in the context of inventory management using the canonical model first studied in Eppen (1979). Specifically, we consider a single-period multi-location newsvendor model, where n different locations face independent and identically distributed demands and linear holding and backorder costs. We show that Eppen's celebrated result, i.e., that the expected cost savings from centralized inventory management scale with the square root of the number of locations, depends critically on the "light-tailed" nature of the demand uncertainty. In particular, we establish that the benefit from pooling relative to the decentralized case, in terms of both expected cost and safety stock, is equal to n α−1 α for a class of heavy-tailed demand distributions (stable distributions), whose power-law asymptotic decay rate is determined by the parameter α ∈ (1, 2). Thus, the benefit from pooling under heavy-tailed demand uncertainty can be significantly lower than √ n, which is predicted for normally distributed demands. We discuss the implications of this result on the performance of periodic-review policies in multi-period inventory management, as well as for the profits associated with drop-shipping fulfilment strategies. Corroborated by an extensive simulation analysis with heavy-tailed distributions that arise frequently in practice, such as power-law and log-normal, our findings highlight the importance of taking into account the shape of the tail of demand uncertainty when considering a risk pooling initiative.

On the Heavy-Tail Behavior of the Distributionally Robust Newsvendor

“…In another study, Gaffeo, Scorcu and Vici [20] analyzed the demand of books in Italy and found that for the three categories -local novels, foreign novels and non-fiction books, a power law distribution where the exponent is typically lesser than 2 is a good fit to the right tail of the demand distribution. Bimpkis and Markakis [8] used the ratings of movies on Netflix as an approximation to the demand of a movie and estimated a power law distribution with an exponent of around 1.04 for the number of movies per number of distinct ratings. Using data from a North American retailer over a one year period with 626 products, their statistical tests showed that the exponential and normal distributions were a poor fit to the data while the power law provided a reasonable approximation to the dataset.…”

On the Learning Benefits of Resource Flexibility

“…Stable distributions represent a broad family that subsumes many distributions frequently used by academics and practitioners, such as Gaussian, Cauchy, and Lévy, which can be tailored in terms of their mean, scale, skewness, and tail asymptotics. Importantly, the stable class includes both light-tailed and heavy-tailed distributions, capturing fundamentally different demand variability regimes; for further discussion and empirical evidence of heavy-tailed demand uncertainty see Bimpikis and Markakis [2016]. We restrict attention to symmetric stable distributions that have a well defined mean, which we also assume to be high enough so that the probability of negative demand is negligible, i.e., F i (0) ≈ 0, i = 1, 2.…”

Dual sourcing under heavy-tailed demand: an extreme value theory approach

“…They revealed that the profit gain by inventory pooling is strongly affected by the tail dependence between products. Bimpikis and Markakis (2014) studied the inventory pooling problem under heavy-tailed demand. They showed that the benefit from pooling decreases as the tail of demand becomes heavier.…”