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 share, 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 relaxation also helps us to develop several heuristics, for which we establish performance guarantees. Numerical experiments show that these heuristics perform very well, yielding profits within 1% of the optimal in most cases.