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1. Dynamic Pricing and Inventory Control with Fixed Ordering Cost and Incomplete Demand Information

   https://doi.org/10.1287/mnsc.2021.4171  

   Chen, Boxiao; Simchi-Levi, David; Wang, Yining; Zhou, Yuan (December 2021, Management Science)

   We consider the periodic review dynamic pricing and inventory control problem with fixed ordering cost. Demand is random and price dependent, and unsatisfied demand is backlogged. With complete demand information, the celebrated [Formula: see text] policy is proved to be optimal, where s and S are the reorder point and order-up-to level for ordering strategy, and [Formula: see text], a function of on-hand inventory level, characterizes the pricing strategy. In this paper, we consider incomplete demand information and develop online learning algorithms whose average profit approaches that of the optimal [Formula: see text] with a tight [Formula: see text] regret rate. A number of salient features differentiate our work from the existing online learning researches in the operations management (OM) literature. First, computing the optimal [Formula: see text] policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in OM that only require solving single-period optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profit-to-go function. The necessity of analyzing action-dependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult than existing bandit learning algorithms. Second, the pricing function [Formula: see text] is of infinite dimension, and approaching it is much more challenging than approaching a finite number of parameters as seen in existing researches. The demand-price relationship is estimated based on upper confidence bound, but the confidence interval cannot be explicitly calculated due to the complexity of the DP recursion. Finally, because of the multiperiod nature of [Formula: see text] policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy [Formula: see text], which is unknown to the learner a priori. In this paper, the demand randomness is approximated by an empirical distribution constructed using dependent samples, and a novel Wasserstein metric-based argument is employed to prove convergence of the empirical distribution. This paper was accepted by J. George Shanthikumar, big data analytics. 

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2. The Power of Opaque Products in Pricing

   https://doi.org/10.1287/mnsc.2020.3750  

   Elmachtoub, Adam N.; Hamilton, Michael L. (August 2021, Management Science)

   null (Ed.)

   We study the power of selling opaque products, that is, products where a feature (such as color) is hidden from the customer until after purchase. Opaque products, which are sold with a price discount, have emerged as a powerful vehicle to increase revenue for many online retailers and service providers that offer horizontally differentiated items. In the opaque selling models we consider, all of the items are sold at a single common price alongside opaque products that may correspond to various subsets of the items. We consider two types of customers, risk-neutral ones, who assume they will receive a truly random item of the opaque product, and pessimistic ones, who assume they will receive their least favorite item of the opaque product. We benchmark opaque selling against two common selling strategies: discriminatory pricing, where one explicitly charges different prices for each item, and single pricing, where a single price is charged for all the items. We give a sharp characterization of when opaque selling outperforms discriminatory pricing; namely, this result holds for situations where all customers are pessimistic or the item valuations are supported on two points. In the latter case, we also show that opaque selling with just one opaque product guarantees at least 71.9% of the revenue from discriminatory pricing. We then provide upper bounds on the potential revenue increase from opaque selling strategies over single pricing and describe cases where the increase can be significantly more than that of discriminatory pricing. Finally, we provide pricing algorithms and conduct an extensive numerical study to assess the power of opaque selling for a variety valuation distributions and model extensions. This paper was accepted by Gabriel Weintraub, revenue management and market analytics. 

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3. On Time-Dependent Trip Distance Distribution with For-Hire Vehicle Trips in Chicago

   https://doi.org/10.1177/03611981211021552  

   Martínez, Irene; Jin, Wen-Long (November 2021, Transportation Research Record: Journal of the Transportation Research Board)

   For transportation system analysis in a new space dimension with respect to individual trips’ remaining distances, vehicle trips demand has two main components: the departure time and the trip distance. In particular, the trip distance distribution (TDD) is a direct input to the bathtub model in the new space dimension, and is a very important variable to consider in many applications, such as the development of distance-based congestion pricing strategies or mileage tax. For a good understanding of the demand pattern, both the distribution of trip initiation and trip distance should be calibrated from real data. In this paper, it is assumed that the demand pattern can be described by the joint distribution of trip distance and departure time. In other words, TDD is assumed to be time-dependent, and a calibration and validation methodology of the joint probability is proposed, based on log-likelihood maximization and the Kolmogorov–Smirnov test. The calibration method is applied to empirical for-hire vehicle trips in Chicago, and it is concluded that TDD varies more within a day than across weekdays. The hypothesis that TDD follows a negative exponential, log-normal, or Gamma distribution is rejected. However, the best fit is systematically observed for the time-dependent log-normal probability density function. In the future, other trip distributions should be considered and also non-parametric probability density estimation should be explored for a better understanding of the demand pattern. 

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4. Dynamic Pricing of Wireless Internet Based on Usage and Stochastically Changing Capacity

   Batur, Demet; Ryan, Jennifer K.; Zhao, Zhongyuan; Vuran, Mehmet C. (January 2019, Manufacturing & service operations management)

   Problem definition: Inspired by new developments in dynamic spectrum access, we study the dynamic pricing of wireless Internet access when demand and capacity (bandwidth) are stochastic. Academic/practical relevance: The demand for wireless Internet access has increased enormously. However, the spectrum available to wireless service providers is limited. The industry has, thus, altered conventional license-based spectrum access policies through unlicensed spectrum operations. The additional spectrum obtained through these operations has stochastic capacity. Thus, the pricing of this service by the service provider has novel challenges. The problem considered in this paper is, therefore, of high practical relevance and new to the academic literature. Methodology: We study this pricing problem using a Markov decision process model in which customers are posted dynamic prices based on their bandwidth requirement and the available capacity. Results: We characterize the structure of the optimal pricing policy as a function of the system state and of the input parameters. Because it is impossible to solve this problem for practically large state spaces, we propose a heuristic dynamic pricing policy that performs very well, particularly when the ratio of capacity to demand rate is low. Managerial implications: We demonstrate the value of using a dynamic heuristic pricing policy compared with the myopic and optimal static policies. The previous literature has studied similar systems with fixed capacity and has characterized conditions under which myopic policies perform well. In contrast, our setting has dynamic (stochastic) capacity, and we find that identifying good state-dependent heuristic pricing policies is of greater importance. Our heuristic policy is computationally more tractable and easier to implement than the optimal dynamic and static pricing policies. It also provides a significant performance improvement relative to the myopic and optimal static policies when capacity is scarce, a condition that holds for the practical setting that motivated this research. 

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5. Offline Feature-Based Pricing Under Censored Demand: A Causal Inference Approach

   https://doi.org/10.1287/msom.2024.1061  

   Tang, Jingwen; Qi, Zhengling; Fang, Ethan; Shi, Cong (March 2025, Manufacturing & Service Operations Management)

   Problem definition: We study a feature-based pricing problem with demand censoring in an offline, data-driven setting. In this problem, a firm is endowed with a finite amount of inventory and faces a random demand that is dependent on the offered price and the features (from products, customers, or both). Any unsatisfied demand that exceeds the inventory level is lost and unobservable. The firm does not know the demand function but has access to an offline data set consisting of quadruplets of historical features, inventory, price, and potentially censored sales quantity. Our objective is to use the offline data set to find the optimal feature-based pricing rule so as to maximize the expected profit. Methodology/results: Through the lens of causal inference, we propose a novel data-driven algorithm that is motivated by survival analysis and doubly robust estimation. We derive a finite sample regret bound to justify the proposed offline learning algorithm and prove its robustness. Numerical experiments demonstrate the robust performance of our proposed algorithm in accurately estimating optimal prices on both training and testing data. Managerial implications: The work provides practitioners with an innovative modeling and algorithmic framework for the feature-based pricing problem with demand censoring through the lens of causal inference. Our numerical experiments underscore the value of considering demand censoring in the context of feature-based pricing. Funding: The research of E. Fang is partially supported by the National Science Foundation [Grants NSF DMS-2346292, NSF DMS-2434666] and the Whitehead Scholarship. The research of C. Shi is partially supported by the Amazon Research Award. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2024.1061 . 

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