Linear Programming Based Online Policies for Real-Time Assortment of Reusable Resources

Abstract

Motivated by applications of rental services in e-commerce, we consider real-time assortment of reusable products. In our model, arriving consumers with heterogeneous types choose rental products from the offered assortment, pay the rental fees, and return the product to the platform after a rental time. Consumers' types specify their choice models, rental fees and rental time distributions for various products. Our goal is to design competitive online policies against an appropriate benchmark for both the prior-free setting, in which types are arbitrary (or adversarial), and the Bayesian setting, in which types are drawn independently from known distributions.

Our contribution is threefold. We first introduce offline linear programming benchmarks in both settings, that use time-varying inventory constraints to capture feasibility of a policy under rentals, and are required to satisfy these constraints only in expectation. Second, in the prior-free setting, we develop a randomized primal-dual framework based on our introduced LP to settle that inventory balancing policies of Golrezaei et al. (2014) obtain same (asymptotically optimal) competitive ratios as with non-reusable resources when product rental times are fixed over time. As a corollary, we obtain the optimal competitive ratio of (1 − 1/e) when inventories are large. We also show this family of policies are constant competitive under i.i.d. (over time) stochastic rental times.

Third, we change gear to the Bayesian setting by introducing simulation-based policies that use the expected LP solution as guidance. By using primal-dual analysis, we obtain a (1/2)-competitive simple and static simulation-based policy against the expected LP for general type-varying rental time distributions and rental fees. We further show optimal (1 − 1/ (c_min+3)^0.5)-competitive adaptive policies against the same benchmark when rental times are infinite, where c_min is the smallest product inventory. Our analysis extends tools in the literature on prophet inequalities to design discarding threshold rules that maintain feasibility of a simulation-based policy in the Bayesian real-time assortment. We further justify the revenue performance of our proposed policies using numerical simulations.

Keywords: real-time assortment; reusable resources; revenue management; online algorithms