Mika Sumida

We consider revenue management problems with heterogenous resources, each with unit capacity. An arriving
customer makes a booking request for a particular interval of days in the future. We offer an assortment
of resources in response to each booking request. The customer makes a choice within the assortment to
use the chosen resource for her desired interval of days. The goal is to find a policy that determines an
assortment of resources to offer to each customer to maximize the total expected revenue over a finite selling
horizon. The problem has two useful features. First, each resource is unique with unit capacity. Second, each
customer uses the chosen resource for a number of consecutive days. We consider static policies that offer
each assortment of resources with a fixed probability. We show that we can efficiently perform rollout on
any static policy, allowing us to build on any static policy and construct an even better policy. Next, we
develop two static policies, each of which is derived from linear and polynomial approximations of the value
functions. We give performance guarantees for both policies, so the rollout policies based on these static
policies inherit the same guarantee. Lastly, we develop an approach for computing an upper bound on the
optimal total expected revenue. Our results for efficient rollout, static policies, and upper bounds all exploit
the aforementioned two useful features of our problem. We use our model to manage hotel bookings based on
a dataset from a real-world boutique hotel, demonstrating that our rollout approach can provide remarkably
good policies and our upper bounds can significantly improve those provided by existing techniques.

We examine the revenue–utility assortment optimization problem with the goal of finding an assortment that maximizes a linear combination of the expected revenue of the firm and the expected utility of the customer. This criterion captures the trade-off between the firm-centric objective of maximizing the expected revenue and the customer-centric objective of maximizing the expected utility. The customers choose according to the multinomial logit model, and there is a constraint on the offered assortments characterized by a totally unimodular matrix. We show that we can solve the revenue–utility assortment optimization problem by finding the assortment that maximizes only the expected revenue after adjusting the revenue of each product by the same constant. Finding the appropriate revenue adjustment requires solving a nonconvex optimization problem. We give a parametric linear program to generate a collection of candidate assortments that is guaranteed to include an optimal solution to the revenue–utility assortment optimization problem. This collection of candidate assortments also allows us to construct an efficient frontier that shows the optimal expected revenue–utility pairs as we vary the weights in the objective function. Moreover, we develop an approximation scheme that limits the number of candidate assortments while ensuring a prespecified solution quality. Finally, we discuss practical assortment optimization problems that involve totally unimodular constraints. In our computational experiments, we demonstrate that we can obtain significant improvements in the expected utility without incurring a significant loss in the expected revenue.

We consider dynamic assortment problems with reusable products, in which each arriving customer chooses a product within an offered assortment, uses the product for a random duration of time, and returns the product back to the firm to be used by other customers. The goal is to find a policy for deciding on the assortment to offer to each customer so that the total expected revenue over a finite selling horizon is maximized. The dynamic-programming formulation of this problem requires a high-dimensional state variable that keeps track of the on-hand product inventories, as well as the products that are currently in use. We present a tractable approach to compute a policy that is guaranteed to obtain at least 50% of the optimal total expected revenue. This policy is based on constructing linear approximations to the optimal value functions. When the usage duration is infinite or follows a negative binomial distribution, we also show how to efficiently perform rollout on a simple static policy. Performing rollout corresponds to using separable and nonlinear value function approximations. The resulting policy is also guaranteed to obtain at least 50% of the optimal total expected revenue. The special case of our model with infinite usage durations captures the case where the customers purchase the products outright without returning them at all. Under infinite usage durations, we give a variant of our rollout approach whose total expected revenue differs from the optimal by a factor that approaches 1 with rate cubic-root of Cmin, where Cmin is the smallest inventory of a product. We provide computational experiments based on simulated data for dynamic assortment management, as well as real parking transaction data for the city of Seattle. Our computational experiments demonstrate that the practical performance of our policies is substantially better than their performance guarantees and that performing rollout yields noticeable improvements. This paper was accepted by Yinyu Ye, optimization.