Consider a category of product variants distinguished by some attribute such as color or flavor. A retailer must construct an assortment for the category, i.e., select a subset variants to stock and determine purchase quantities for each offered variant. We analyze this problem using a multinomial logit model to describe the consumer choice process and a newsboy model to represent the retailer's inventory cost. We show that the optimal assortment has a simple structure and provide insights on how various factors affect the optimal level of assortment variety. We also develop a formal definition of the level of fashion in a category using the theory of majorization and examine its implications for category profits.variety, inventory, retailing, consumer choice, assortment, optimization, newsboy, fashion, majorization, multinomial logit

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.

We analyze a model of inventory competition among n firms that provide competing, substitutable goods. Each firm chooses initial inventory levels for their good in a single period (newsboy-like) inventory model. Customers choose dynamically based on current availability, so the inventory levels at one firm affect the demand of all competing firms. This creates a strategic interaction among the firms' inventory decisions. Our work extends earlier work on variations of this problem by Karjalainen (1992), Lippman and McCardle (1997) and Parlar (1988). Specifically, we model demand in a more realistic way as a stochastic sequence of heterogeneous consumers who choose dynamically from among the available goods (or choose not to purchase) based on a utility maximization criterion. We also use a sample path analysis, so minimal assumptions are imposed on this demand process. We characterize the Nash equilibrium of the resulting stocking game and prove it is unique in the symmetric case. We show there is a bias toward overstocking caused by competition; specifically, reducing the quantity stocked at any equilibrium of the game increases total system profits, and at any joint-optimal set of stocking levels, each firm has an individual incentive to increase its own stock. For the symmetric case, we show that as the number of competing firms increases, the overstocking becomes so severe that total system (and individual firm) profits approach zero. Finally, we propose a stochastic gradient algorithm for computing equilibria and provide several numerical examples.

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