We consider a single-product revenue management problem with an inventory constraint and unknown, noisy, demand function. The objective of the firm is to dynamically adjust the prices to maximize total expected revenue. We restrict our scope to the nonparametric approach where we only assume some common regularity conditions on the demand function instead of a specific functional form. We propose a family of pricing heuristics that successfully balance the tradeoff between exploration and exploitation. The idea is to generalize the classic bisection search method to a problem that is affected both by stochastic noise and an inventory constraint. Our algorithm extends the bisection method to produce a sequence of pricing intervals that converge to the optimal static price with high probability. Using regret (the revenue loss compared to the deterministic pricing problem for a clairvoyant) as the performance metric, we show that one of our heuristics exactly matches the theoretical asymptotic lower bound that has been previously shown to hold for any feasible pricing heuristic. Although the results are presented in the context of revenue management problems, our analysis of the bisection technique for stochastic optimization with learning can be potentially applied to other application areas.3 revenue loss? It turns out that it is possible: If we use Stochastic Approximation algorithms (i.e., Kiefer-Wolfowitz and Robbins-Monro, see Broadie et al. [10]) during the exploitation phase instead of another bisection search, then the resulting revenue loss is exactly Θ( √ θ). Thus, we have provided an "optimal" nonparametric pricing heuristic for the setting of a single-product problem with inventory constraint. (In the case where the firms know the functional form of the demand function, i.e., parametric model, the Ω( √ θ) lower bound has been repeatedly shown to be tight. For example, in the setting without inventory constraints, Keskin and Zeevi [27], den Boer and Zwart [19], and Broder and Rusmevichientong [11], each proposes a parametric pricing heuristic that guarantees a revenue loss of the order of O( √ θ) . As for the setting with inventory constraints, recently Chen et al. [13] propose a heuristic that exactly matches this lower bound. Their result holds for a general parametric model with an arbitrary set of inventory constraints. Thus, they have resolved the parametric dynamic pricing problem with inventory constraints.)

We consider an e-commerce retailer (e-tailer) who sells a catalog of products to customers from different regions during a finite selling season and fulfills orders through multiple fulfillment centers. The e-tailer faces a Joint Pricing and Fulfillment (JPF) problem: At the beginning of each period, she needs to jointly decide the price for each product and how to fulfill an incoming order. The objective is to maximize the total expected profits defined as total expected revenues minus total expected shipping costs (all other costs are fixed in this problem). The exact optimal policy for JPF is difficult to solve; so, we propose two heuristics that have provably good performance compared to reasonable benchmarks. Our first heuristic directly uses the solution of a deterministic approximation of JPF as its control parameters whereas our second heuristic improves the first heuristic by adaptively adjusting the original control parameters at the beginning of every period. An important feature of the second heuristic is that it decouples the pricing and fulfillment decisions, making it easy to implement. We show theoretically and numerically that the second heuristic significantly outperforms the first heuristic and is very close to a benchmark that jointly re-optimizes the full deterministic problem at every period.

We consider an e-commerce retailer (e-tailer) who sells a catalog of products to customers from different regions during a finite selling season and fulfills orders through multiple fulfillment centers. The e-tailer faces a joint pricing and fulfillment (JPF) optimization problem: at the beginning of each period, the e-tailer needs to jointly decide the price for each product and how to fulfill an incoming order (i.e., from which warehouse to ship the order). The objective of the e-tailer is to maximize its total expected profits defined as total expected revenues minus total expected shipping costs. (All other costs are fixed in this problem.) The exact optimal policy for JPF is difficult to solve; so, we propose two heuristic controls that have provably good performance compared to reasonable benchmarks. Our first heuristic control directly uses the solution of a deterministic approximation of JPF as its control parameters. Our second heuristic control improves the first one by adaptively adjusting the original control parameters according to the realized demand. An important feature of the second heuristic control is that it decouples the real-time pricing and fulfillment decisions, making it easy to implement. We show theoretically and numerically that the second heuristic control significantly outperforms the first heuristic control and is very close to a benchmark that jointly reoptimizes the full deterministic problem at the beginning of every period. The online appendix is available at https://doi.org/10.1287/msom.2017.0641 .

In “Real-Time Dynamic Pricing for Revenue Management with Reusable Resources, Advance Reservation, and Deterministic Service Time Requirements,” Lei and Jasin consider a fundamental dynamic pricing problem when resources are reusable. In this problem, demand arrives according to a price-sensitive nonstationary rate, requesting a service that uses a combination of different types of resources for a deterministic duration of time. The resources are reusable in the sense that they can be immediately used to serve a new customer on the completion of the previous service. Moreover, different customers may have different service time requirement and may book the service in advance. The objective is to construct a dynamic pricing control that maximizes expected total revenues. They develop real-time heuristic controls based on the solution of the deterministic relaxation of the original stochastic problem and show that the proposed controls are near optimal in the regime of large demand and large resource capacity.