The Assortment Packing Problem: Multiperiod Assortment Planning for Short-Lived Products

Motivated by retailers ’ frequent introduction of new items to refresh product lines and maintain their market shares, we present the assortment packing problem in which a firm must decide, in advance, the release date of each product in a given collection over a selling season. Our formulation models the trade-offs among profit margins, preference weights, and limited life cycles. A key aspect of the problem is that each product is short-lived in the sense that, once introduced, its attractiveness lasts only a few periods and vanishes over time. The objective is to determine when to introduce each product to maximize the total profit over the selling season. Even for two periods, the corresponding optimization problem is shown to be NP-complete. As a result, we study a continuous relaxation of the problem that approximates the problem well, when the number of products is large. When margins are identical and product preferences decay exponentially, its solution can be characterized: it is optimal to introduce products with slower decays earlier. The structural properties of the relaxation also help us to develop several heuristics, for which we establish performance guarantees. We test our heuristics with data on sales and release dates of woman handbags from an accessories retailer. The numerical experiments show that the heuristics perform very well and can yield significant improvements in profitability. 1

Learning Consumer Tastes Through Dynamic Assortments

How should a firm modify its product assortment over time when learning about consumer tastes? In this paper, we study dynamic assortment decisions in a horizontally differentiated product category for which consumers' diverse tastes can be represented as locations on a Hotelling line. We presume that the firm knows all possible consumer locations, comprising a finite set, but does not know their probability distribution. We model this problem as a discrete-time dynamic program; each period, the firm chooses an assortment and sets prices to maximize the total expected profit over a finite horizon, given its subjective beliefs over consumer tastes. The consumers then choose a product from the assortment that maximizes their own utility. The firm observes sales, which provide censored information on consumer tastes, and it updates beliefs in a Bayesian fashion. There is a recurring trade-off between the immediate profits from sales in the current period (exploitation) and the informational gains to be exploited in all future periods (exploration). We show that one can (partially) order assortments based on their information content and that in any given period the optimal assortment cannot be less informative than the myopically optimal assortment. This result is akin to the well-known "stock more" result in censored newsvendor problems with the newsvendor learning about demand through sales when lost sales are not observable. We demonstrate that it can be optimal for the firm to alternate between exploration and exploitation, and even offer assortments that lead to losses in the current period in order to gain information on consumer tastes. We also develop a Bayesian conjugate model that reduces the state space of the dynamic program and study value of learning using this conjugate model

Mass Customization versus Mass Production: Variety and Price Competition.

We study competition between two multi-product firms with distinct production technologies in a market where customers have heterogeneous preferences on a single taste attribute. The mass customizer (MC) has a perfectly flexible production technology, thus can o.er any variety within a product space, represented by Hotelling’s (1929) linear city. The mass producer (MP) has a more focused production technology, and therefore, it offers a finite set of products in the same space. MP can invest in more flexible technology, which reduces its cost of variety and hence allows it to offer a larger set of products; in the extreme, MP can emulate MC’s technology and offer infinite variety. The firms simultaneously decide whether to enter the market, and MP chooses its degree of product-mix flexibility upon entry. Next, MP designs its product line, i.e., the number and position of its products; MC’s perfectly flexible technology makes this unnecessary. Finally, both firms simultaneously set prices. We analyze the sub-game perfect Nash equilibrium in this three-stage game, allowing firm-specific fixed and variable costs that together characterize their production technology. We find that an MP facing competition from an MC offers lower product variety compared to an MP monopolist, in order to reduce the intensity of price competition. We also find that MP can survive this competition even if it has higher fixed cost of production technology or higher marginal cost of production or both

Learning Consumer Tastes Through Dynamic Assortments

How should a firm modify its product assortment over time when learning about consumer tastes? In this paper, we study dynamic assortment decisions in a horizontally differentiated product category for which consumers' diverse tastes can be represented as locations on a Hotelling line. We presume that the firm knows all possible consumer locations, comprising a finite set, but does not know their probability distribution. We model this problem as a discrete-time dynamic program; each period, the firm chooses an assortment and sets prices to maximize the total expected profit over a finite horizon, given its subjective beliefs over consumer tastes. The consumers then choose a product from the assortment that maximizes their own utility. The firm observes sales, which provide censored information on consumer tastes, and it updates beliefs in a Bayesian fashion. There is a recurring trade-off between the immediate profits from sales in the current period (exploitation) and the informational gains to be exploited in all future periods (exploration). We show that one can (partially) order assortments based on their information content and that in any given period the optimal assortment cannot be less informative than the myopically optimal assortment. This result is akin to the well-known "stock more" result in censored newsvendor problems with the newsvendor learning about demand through sales when lost sales are not observable. We demonstrate that it can be optimal for the firm to alternate between exploration and exploitation, and even offer assortments that lead to losses in the current period in order to gain information on consumer tastes. We also develop a Bayesian conjugate model that reduces the state space of the dynamic program and study value of learning using this conjugate model

Assortment Planning and Inventory Decisions Under a Locational Choice Model

We consider a single-period assortment planning and inventory management problem for a retailer, using a locational choice model to represent consumer demand. We first determine the optimal variety, product location, and inventory decisions under static substitution, and show that the optimal assortment consists of products equally spaced out such that there is no substitution among them regardless of the distribution of consumer preferences. The optimal solution can be such that some customers prefer not to buy any product in the assortment, and such that the most popular product is not offered. We then obtain bounds on profit when customers dynamically substitute, using the static substitution for the lower bound, and a retailer-controlled substitution for the upper bound. We thus define two heuristics to solve the problem under dynamic substitution and numerically evaluate their performance. This analysis shows the value of modeling dynamic substitution and identifies conditions in which the static substitution solution serves as a good approximation.product variety, inventory management, consumer choice models, assortment planning, retail operations