Near-Optimal Bisection Search for Nonparametric Dynamic Pricing with Inventory Constraint PDF Download
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Author: Yanzhe (Murray) Lei
Publisher:
ISBN:
Category :
Languages : en
Pages : 38
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Book Description
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 novel 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 relative revenue loss compared to the optimal dynamic pricing solution 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.
Author: Yanzhe (Murray) Lei
Publisher:
ISBN:
Category :
Languages : en
Pages : 38
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Book Description
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 novel 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 relative revenue loss compared to the optimal dynamic pricing solution 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.
Author: Xi Chen
Publisher: Springer Nature
ISBN: 3031019261
Category : Business & Economics
Languages : en
Pages : 444
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Book Description
This book examines recent developments in Operations Management, and focuses on four major application areas: dynamic pricing, assortment optimization, supply chain and inventory management, and healthcare operations. Data-driven optimization in which real-time input of data is being used to simultaneously learn the (true) underlying model of a system and optimize its performance, is becoming increasingly important in the last few years, especially with the rise of Big Data.
Author: Boxiao Chen
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
We study the fundamental model in joint pricing and inventory replenishment control under the learning-while-doing framework, with T consecutive review periods and the firm not knowing the demand curve a priori. At the beginning of each period, the retailer makes both a price decision and an inventory order-up-to level decision, and collects revenues from consumers' realized demands while suffering costs from either holding unsold inventory items, or lost sales from unsatisfied customer demands. We make the following contributions to this fundamental problem as follows:1. We propose a novel inversion method based on empirical measures to consistently estimate the difference of the instantaneous reward functions at two prices, directly tackling the fundamental challenge brought by censored demands, without raising the order-up-to levels to unnaturally high levels to collect more demand information. Based on this technical innovation, we design bisection and trisection search methods that attain an O(T^{1/2}) regret, assuming the reward function is concave and only twice continuously differentiable.2. In the more general case of non-concave reward functions, we design an active tournament elimination method that attains O(T^{3/5}) regret, based also on the technical innovation of consistent estimates of reward differences at two prices.3. We complement the O(T^{3/5}) regret upper bound with a matching Omega(T^{3/5}) regret lower bound. The lower bound is established by a novel information-theoretical argument based on generalized squared Hellinger distance, which is significantly different from conventional arguments that are based on Kullback-Leibler divergence. This lower bound shows that no learning-while-doing algorithm could achieve O(T^{1/2}) regret without assuming the reward function is concave, even if the sales revenue as a function of demand rate or price is concave.Both the upper bound technique based on the "difference estimator" and the lower bound technique based on generalized Hellinger distance are new in the literature, and can be potentially applied to solve other inventory or censored demand type problems that involve learning.
Author: Ningyuan Chen
Publisher:
ISBN:
Category :
Languages : en
Pages : 41
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Book Description
We consider the problem of a firm seeking to use personalized pricing to sell an exogenously given stock of a product over a finite selling horizon to different consumer types. We assume that the type of an arriving consumer can be observed but the demand function associated with each type is initially unknown. The firm sets personalized prices dynamically for each type and attempts to maximize the revenue over the season. We provide a learning algorithm that is near-optimal when the demand and capacity scale in proportion. The algorithm utilizes the primal-dual formulation of the problem and learns the dual optimal solution explicitly. It allows the algorithm to overcome the curse of dimensionality (the rate of regret is independent of the number of types) and sheds light on novel algorithmic designs for learning problems with resource constraints.
Author: Maxime C. Cohen
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Following the increasing popularity of personalized pricing, there is a growing concern from customers and policy makers regarding fairness considerations. This paper studies the problem of dynamic pricing with unknown demand under two types of fairness constraints: price fairness and demand fairness. For price fairness, the retailer is required to (i) set similar prices for different customer groups (called group fairness) and (ii) ensure that the prices over time for each customer group are relatively stable (called time fairness). We propose an algorithm based on an infrequently-changed upper-confidence-bound (UCB) method, which is proved to yield a near-optimal regret performance. In particular, we show that imposing group fairness does not affect the demand learning problem, in contrast to imposing time fairness. On the flip side, we show that imposing time fairness does not impact the clairvoyant optimal revenue, in contrast to imposing group fairness. For demand fairness, the retailer is required to satisfy that the resulting demand from different customer groups is relatively similar (e.g., the retailer offers a lower price to students to increase their demand to a similar level as non-students). In this case, we design an algorithm adapted from a primal-dual learning framework and prove that our algorithm also achieves a near-optimal performance.
Author: Hanzhang Qin (S. M.)
Publisher:
ISBN:
Category :
Languages : en
Pages : 96
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Book Description
The thesis studies the classical multi-period joint pricing and inventory control problem in a data-driven setting. In the problem, a retailer makes periodic decisions of the prices and inventory levels of an item that the retailer wishes to sell. The objective is to match the inventory level with a random demand that depends on the price in each period, while maximizing the expected profit over finite horizon. In reality, the demand functions or the distribution of the random noise are usually unavailable, whereas past demand data are relatively easy to collect. A novel data-driven nonparametric algorithm is proposed, which uses the past demand data to solve the joint pricing and inventory control problem, without assuming the parameters of the demand functions and the noise distributions are known. Explicit sample complexity bounds are given, on the number of data samples needed to guarantee a near-optimal profit. A simulation study suggests that the algorithm is efficient in practice.
Author: Georgia Perakis
Publisher:
ISBN:
Category :
Languages : en
Pages : 50
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Book Description
We consider the dynamic pricing problem of a retailer who does not have any information on the underlying demand for a product. The retailer aims to maximize cumulative revenue collected over a finite time horizon by balancing two objectives: textit{learning} demand and textit{maximizing} revenue. The retailer also seeks to reduce the amount of price experimentation because of the potential costs associated with price changes. Existing literature solves this problem in the case where the unknown demand is parametric. We consider the pricing problem when demand is non-parametric. We construct a pricing algorithm that uses piecewise linear approximations of the unknown demand function and establish when the proposed policy achieves near-optimal rate of regret, tilde{O}( sqrt{T}), while making O( log log T) price changes. Hence, we show considerable reduction in price changes from the previously known mathcal{O}( log T) rate of price change guarantee in the literature. We also perform extensive numerical experiments to show that the algorithm substantially improves over existing methods in terms of the total price changes, with comparable performance on the cumulative regret metric.
Author: Stephen P. Boyd
Publisher: Cambridge University Press
ISBN: 9780521833783
Category : Business & Economics
Languages : en
Pages : 744
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Book Description
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.
Author: David Edward Goldberg
Publisher: Addison-Wesley Professional
ISBN:
Category : Computers
Languages : en
Pages : 436
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Book Description
A gentle introduction to genetic algorithms. Genetic algorithms revisited: mathematical foundations. Computer implementation of a genetic algorithm. Some applications of genetic algorithms. Advanced operators and techniques in genetic search. Introduction to genetics-based machine learning. Applications of genetics-based machine learning. A look back, a glance ahead. A review of combinatorics and elementary probability. Pascal with random number generation for fortran, basic, and cobol programmers. A simple genetic algorithm (SGA) in pascal. A simple classifier system(SCS) in pascal. Partition coefficient transforms for problem-coding analysis.
Author: Alex Gershkov
Publisher: MIT Press
ISBN: 0262028409
Category : Business & Economics
Languages : en
Pages : 209
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Book Description
A new approach to dynamic allocation and pricing that blends dynamic paradigms from the operations research and management science literature with classical mechanism design methods. Dynamic allocation and pricing problems occur in numerous frameworks, including the pricing of seasonal goods in retail, the allocation of a fixed inventory in a given period of time, and the assignment of personnel to incoming tasks. Although most of these problems deal with issues treated in the mechanism design literature, the modern revenue management (RM) literature focuses instead on analyzing properties of restricted classes of allocation and pricing schemes. In this book, Alex Gershkov and Benny Moldovanu propose an approach to optimal allocations and prices based on the theory of mechanism design, adapted to dynamic settings. Drawing on their own recent work on the topic, the authors describe a modern theory of RM that blends the elegant dynamic models from the operations research (OR), management science, and computer science literatures with techniques from the classical mechanism design literature. Illustrating this blending of approaches, they start with well-known complete information, nonstrategic dynamic models that yield elegant explicit solutions. They then add strategic agents that are privately informed and then examine the consequences of these changes on the optimization problem of the designer. Their sequential modeling of both nonstrategic and strategic logic allows a clear picture of the delicate interplay between dynamic trade-offs and strategic incentives. Topics include the sequential assignment of heterogeneous objects, dynamic revenue optimization with heterogeneous objects, revenue maximization in the stochastic and dynamic knapsack model, the interaction between learning about demand and dynamic efficiency, and dynamic models with long-lived, strategic agents.
