Combinatorial Assortment Optimization

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a model of consumer choice where the relative value of different bundles is described by a valuation function, while individual customers may differ in their absolute willingness to pay, and study the complexity of the resulting optimization problem. We show that any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even for succinctly-representable submodular valuations. On the positive side, we show how to obtain constant approximations under a "well-priced" condition, where each product's price is sufficiently high. We also provide an exact algorithm for $k$-additive valuations, and show how to extend our results to a learning setting where the seller must infer the customers' preferences from their purchasing behavior

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

Guiding the formation of single-handed enantiomeric porphyrin domains using kinked and chiral stepped surfaces

The self-assembly of nickel tetraphenyl porphyrin (NiTPP) on achiral Au(111), which includes randomly located kinked steps, and chiral Au(1036 1070 1035) and (1036 1035 1070) surfaces has been studied using UHV scanning tunneling microscopy. The clean surfaces of the achiral and chiral gold crystals were characterized with STM. Subsequently, NiTPP molecules were deposited on each surface. On achiral Au(111), the porphyrins assemble into racemic mixtures of enantiomerically resolved domains. It is concluded that, on large flat terraces, intermolecular interactions are the dominant factor in the chiral assembly. Moreover, it is found that the chirality of the molecular array can be guided using the handedness of locally kinked step edges. Preliminary work has begun on the chiral crystal surfaces. Initial findings suggest that the chirality of the kinked step edges induces formation of a single-handed domain of molecules across a terrace