Maximum Load Assortment Optimization: Approximation Algorithms and Adaptivity Gaps

Motivated by modern-day applications such as Attended Home Delivery and Preference-based Group Scheduling, where decision makers wish to steer a large number of customers toward choosing the exact same alternative, we introduce a novel class of assortment optimization problems, referred to as Maximum Load Assortment Optimization. In such settings, given a universe of substitutable products, we are facing a stream of customers, each choosing between either selecting a product out of an offered assortment or opting to leave without making a selection. Assuming that these decisions are governed by the Multinomial Logit choice model, we define the random load of any underlying product as the total number of customers who select it. Our objective is to offer an assortment of products to each customer so that the expected maximum load across all products is maximized. We consider both static and dynamic formulations. In the static setting, a single offer set is carried throughout the entire process of customer arrivals, whereas in the dynamic setting, the decision maker offers a personalized assortment to each customer, based on the entire information available at that time. The main contribution of this paper resides in proposing efficient algorithmic approaches for computing near-optimal static and dynamic assortment policies. In particular, we develop a polynomial-time approximation scheme (PTAS) for the static formulation. Additionally, we demonstrate that an elegant policy utilizing weight-ordered assortments yields a 1/2- approximation. Concurrently, we prove that such policies are sufficiently strong to provide a 1/4-approximation with respect to the dynamic formulation, establishing a constant-factor bound on its adaptivity gap. Finally, we design an adaptive policy whose expected maximum load is within factor 1-\eps of optimal, admitting a quasi-polynomial time implementation

Aplicación del método de entropía cruzada al problema de optimización dinámica de surtido

This work considers an assortment optimization problem, under capacity constraint and unknown demand, where a retailer offers an assortment and observes the sale of one of the products according to a multinomial logit choice model. In this problem, named as the dynamic assortment optimization problem (DAOP), the retailer must offer different assortments in each period to learn the customer preferences. Therefore, the trade-off between exploration of new assortments and the exploitation of the best known assortment must be balanced. Similarities between sampling and exploration are established in order to apply the cross-entropy method as a policy for the solution of the DAOP. The cross-entropy method finds a probability distribution that samples an optimal solution by minimizing the cross-entropy between a target probability distribution and an arbitrarily selected probability distribution. This requires the DAOP to be formulated as a knapsack problem with a penalty for offering assortments that exceed capacity. The results are compared with adaptive exploration algorithms and, experimentally, the cross-entropy method shows competitive results. These results suggest that the cross-entropy method can be used to solve other sequential decision-making problems.Este trabajo considera un problema de optimización de surtido, bajo restricción de capacidad y demanda desconocida, donde un vendedor ofrece un surtido y observa la venta de un producto según un modelo de elección logit multinomial. En este problema, llamado como el problema de optimización dinámica de surtido (PODS), el vendedor debe ofrecer diferentes surtidos en cada per´ıodo para aprender las preferencias del consumidor. Por lo tanto, el trade-off entre la exploraci´on de nuevos surtidos y la explotación del mejor surtido conocido debe ser equilibrado. Se estableció similitudes entre el muestreo y la exploración con el fin de aplicar el método de entrop´ıa cruzada como política para la solución del PODS. El método de entropía cruzada encuentra una distribución de probabilidad que muestrea una solución óptima al minimizar la entropía cruzada entre una distribución de probabilidad objetivo y una distribución de probabilidad seleccionada arbitrariamente. Esto requiere que el PODS se formule como un problema de la mochila con una penalización por ofrecer surtidos que superan la capacidad. Los resultados se comparan con algoritmos de exploración adaptativa y, experimentalmente, el método de entropía cruzada muestra resultados competitivos. Estos resultados sugieren que el método de entropía cruzada se puede utilizar para resolver otros problemas de toma de decisiones secuenciales

Robust Dynamic Assortment Optimization in the Presence of Outlier Customers

We consider the dynamic assortment optimization problem under the multinomial logit model (MNL) with unknown utility parameters. The main question investigated in this paper is model mis-specification under the $\varepsilon$-contamination model, which is a fundamental model in robust statistics and machine learning. In particular, throughout a selling horizon of length $T$, we assume that customers make purchases according to a well specified underlying multinomial logit choice model in a ($1-\varepsilon$)-fraction of the time periods, and make arbitrary purchasing decisions instead in the remaining $\varepsilon$-fraction of the time periods. In this model, we develop a new robust online assortment optimization policy via an active elimination strategy. We establish both upper and lower bounds on the regret, and show that our policy is optimal up to logarithmic factor in T when the assortment capacity is constant. Furthermore, we develop a fully adaptive policy that does not require any prior knowledge of the contamination parameter $\varepsilon$. Our simulation study shows that our policy outperforms the existing policies based on upper confidence bounds (UCB) and Thompson sampling.Comment: 27 pages, 1 figur

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