Approximation algorithms for dynamic assortment optimization models

We consider the single-period joint assortment and inventory planning problem with stochastic demand and dynamic substitution across products, motivated by applications in highly differentiated markets, such as online retailing and airlines. This class of problems is known to be notoriously hard to deal with from a computational standpoint. In fact, prior to the present paper, only a handful of modeling approaches were shown to admit provably good algorithms, at the cost of strong restrictions on customers’ choice outcomes. Our main contribution is to provide the first efficient algorithms with provable performance guarantees for a broad class of dynamic assortment optimization models. Under general rank-based choice models, our approximation algorithm is best possible with respect to the price parameters, up to lower-order terms. In particular, we obtain a constant-factor approximation under horizontal differentiation, where product prices are uniform. In more structured settings, where the customers’ ranking behavior is motivated by price and quality cues, we derive improved guarantees through tailor-made algorithms. In extensive computational experiments, our approach dominates existing heuristics in terms of revenue performance, as well as in terms of speed, given the myopic nature of our methods. From a technical perspective, we introduce a number of novel algorithmic ideas of independent interest, and unravel hidden relations to submodular maximization

Dynamic Assortment Optimization with Changing Contextual Information

In this paper, we study the dynamic assortment optimization problem under a finite selling season of length $T$. At each time period, the seller offers an arriving customer an assortment of substitutable products under a cardinality constraint, and the customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contexutal information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and makes the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\widetilde O(d\sqrt{T})$, where $d$ is the dimension of the feature and $\widetilde O$ suppresses logarithmic dependence. We further established the lower bound $\Omega(d\sqrt{T}/K)$ where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.Comment: 4 pages, 4 figures. Minor revision and polishing of presentatio

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

Assortment optimisation under a general discrete choice model: A tight analysis of revenue-ordered assortments

The assortment problem in revenue management is the problem of deciding which subset of products to offer to consumers in order to maximise revenue. A simple and natural strategy is to select the best assortment out of all those that are constructed by fixing a threshold revenue $\pi$ and then choosing all products with revenue at least $\pi$. This is known as the revenue-ordered assortments strategy. In this paper we study the approximation guarantees provided by revenue-ordered assortments when customers are rational in the following sense: the probability of selecting a specific product from the set being offered cannot increase if the set is enlarged. This rationality assumption, known as regularity, is satisfied by almost all discrete choice models considered in the revenue management and choice theory literature, and in particular by random utility models. The bounds we obtain are tight and improve on recent results in that direction, such as for the Mixed Multinomial Logit model by Rusmevichientong et al. (2014). An appealing feature of our analysis is its simplicity, as it relies only on the regularity condition. We also draw a connection between assortment optimisation and two pricing problems called unit demand envy-free pricing and Stackelberg minimum spanning tree: These problems can be restated as assortment problems under discrete choice models satisfying the regularity condition, and moreover revenue-ordered assortments correspond then to the well-studied uniform pricing heuristic. When specialised to that setting, the general bounds we establish for revenue-ordered assortments match and unify the best known results on uniform pricing.Comment: Minor changes following referees' comment