Omnichannel Assortment Optimization Under the Multinomial Logit Model with a Features Tree

Problem definition: We consider the assortment optimization problem of a retailer that operates a physical store and an online store. The products that can be offered are described by their features. Customers purchase among the products that are offered in their preferred store. However, customers who purchase from the online store can first test out products offered in the physical store. These customers revise their preferences for online products based on the features that are shared with the in-store products. The full assortment is offered online, and the goal is to select an assortment for the physical store to maximize the retailer’s total expected revenue. Academic/practical relevance: The physical store’s assortment affects preferences for online products. Unlike traditional assortment optimization, the physical store’s assortment influences revenue from both stores. Methodology: We introduce a features tree to organize products by features. The nonleaf vertices on the tree correspond to features, and the leaf vertices correspond to products. The ancestors of a leaf correspond to features of the product. Customers choose among the products within their store’s assortment according to the multinomial logit model. We consider two settings; either all customers purchase online after viewing products in the physical store, or we have a mix of customers purchasing from each store. Results: When all customers purchase online, we give an efficient algorithm to find the optimal assortment to display in the physical store. With a mix of customers, the problem becomes NP-hard, and we give a fully polynomial-time approximation scheme. We numerically demonstrate that we can closely approximate the case where products have arbitrary combinations of features without a tree structure and that our fully polynomial-time approximation scheme performs remarkably well. Managerial implications: We characterize conditions under which it is optimal to display expensive products with underrated features and expose inexpensive products with overrated features.

Approximating Geometric Knapsack via L-packings

We study the two-dimensional geometric knapsack problem, in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is 2+ε [Jansen and Zhang, SODA 2004]. In this article we present a polynomial-time 17/9+ε < 1.89-approximation, which improves to 558/325+ε < 1.72 in the cardinality case. Prior results pack items into a constant number of rectangular containers that are filled via greedy strategies. We deviate from this setting and show that there exists a large profit solution where items are packed into a constant number of containers plus one L-shaped region at the boundary of the knapsack containing narrow-high items and thin-wide items. These items may interact in complex manners at the corner of the L. The best-known approximation ratio for the subproblem in the L-shaped region is 2+ε (via a trivial reduction to one-dimensional knapsack); hence, as a second major result we present a PTAS for this case that we believe might be of broader utility. We also consider the variant with rotations, where items can be rotated by 90 degrees. Again, the best-known polynomial-time approximation factor (even for the cardinality case) is 2+ε [Jansen and Zhang, SODA 2004]. We present a polynomial-time (3/2+ε)-approximation for this setting, which improves to 4/3+ε in the cardinality case.

Approximating the product knapsack problem

We consider the product knapsack problem, which is the variant of the classical 0-1 knapsack problem where the objective consists of maximizing the product of the profits of the selected items. These profits are allowed to be positive or negative. We present the first fully polynomial-time approximation scheme for the product knapsack problem, which is known to be weakly -hard. Moreover, we analyze the approximation quality achieved by a natural extension of the classical knapsack greedy procedure to the product knapsack problem.

A Polynomial-Time Approximation Scheme for Sequential Batch Testing of Series Systems

The paper studies a recently introduced generalization of the classic sequential testing problem for series systems, consisting of multiple stochastic components. The conventional assumption in such settings is that the overall system state can be expressed as an AND function, defined with respect to the states of individual components. However, unlike the classic setting, rather than testing components separately, one after the other, we allow aggregating multiple tests to be conducted simultaneously, while incurring an additional set-up cost. This feature is present in numerous practical applications, where decision makers are incentivized to exploit economy of scale by testing subsets of components in batches. The main contribution of this paper is to devise a polynomial-time approximation scheme for the sequential batch-testing problem, by leveraging a number of techniques in approximate dynamic programming, based on a synthesis of ideas related to efficient enumeration methods, state-space collapse, and charging schemes.