## Garrett van Ryzin

*Stocking retail assortments under dynamic consumer substitution*

Coauthor(s): Siddharth Mahajan.

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**Abstract:**

We analyze a single-period, stochastic inventory model (newsboy-like model) in which a sequence of heterogeneous customers dynamically substitute among product variants within a retail assortment when inventory is depleted. The customer choice decisions are based on a natural and classical utility maximization criterion. Faced with such substitution behavior, the retailer must choose initial inventory levels for the assortment to maximize expected profits.

Using a sample path analysis, we analyze structural properties of the expected profit function. We show that, under very general assumptions on the demand process, total sales of each product are concave in their own inventory levels and possess the so-called *decreasing differences* property, meaning that the marginal value of an additional unit of the given product is decreasing in the inventory levels of all other products. For a continuous relaxation of the problem, we then show, via counterexamples, that the expected profit function is in general not even quasiconcave. Thus, global optimization may be difficult. However, we propose and analyze a stochastic gradient algorithm for the problem, and prove that it converges to a stationary point of the expected profit function under mild conditions. Finally, we apply the algorithm to a set of numerical examples and compare the resulting inventory decisions to those of some simpler, naive heuristics. The examples show that substitution effects can have a significant impact on an assortment's gross profits. The examples also illustrate some systematic distortions in inventory decisions if substitution effects are ignored. In particular, under substitution one should stock relatively more of popular variants and relatively less of unpopular variants than a traditional newsboy analysis indicates.

**Source:** *Operations Research*

**Exact Citation:**

Mahajan, Siddharth, and Garrett van Ryzin. "Stocking retail assortments under dynamic consumer substitution." *Operations Research* 49, no. 3 (2001): 334-351.

**Volume:** 49

**Number:** 3

**Pages:** 334-351

**Date:**
2001