Online Assortment of Reusable Resources with Exogenous Replenishment

Abstract

We introduce and study a variant of online assortment planning where the products are for rental---in contrast to more classic models where products are for sale---and their inventories replenish over time. This problem is motivated in part by online platforms such as Thumbtack or Catchafire for allocating rental services, where additional workforce or volunteers can recharge the inventory for a particular service or product over time. The goal of the online platform is to sequentially pick assortments to maximize the total rental reward of allocated products (where rewards can be interpreted as fees or any other type of scores). For the online (adversarial) setting where the consumers and replenishment amounts are unknown upfront, we design a new family of algorithms which we call batched inventory balancing (BIB). By extending the framework for the non-reusable online assortment problem, we develop a novel randomized primal-dual method to bound the competitive ratio of our BIB algorithm against any feasible policy, for the objective of maximizing total collected rental rewards. We show with the proper choice of a certain parameter, this competitive ratio is asymptotically optimal and converges to (1-1/e), as initial inventories converge to infinity. We then show how to characterize BIB's competitive ratio in its general form. We use a refined analysis that reduces the dual construction to a combinatorial problem referred as the interval assignment problem. Our solution to this problem is algorithmic and might be of independent interest.