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Author: Xiaocheng Li
Publisher:
ISBN:
Category :
Languages : en
Pages : 41
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Book Description
A merchant sells a product over a selling season of T time periods in presence of a limited inventory. The merchant observes new external information at the beginning of each time period and then sets a price for that time period. Initially, the merchant does not know the distribution of the external information and the demand function, i.e., how the external information and price jointly impact the demand distribution in a single time period. The seller's decision, setting price dynamically, serves dual roles to learn the unknown demand function and to balance inventory, with an ultimate goal to maximize the expected revenue over the selling season. We characterize and prove a full spectrum of relations between the optimal revenue achieved in three decision-making regimes: the merchant's online decision-making regime, a clairvoyant regime with complete knowledge about the demand function and all the external information in advance, and a deterministic regime in which the demand function and all the uncertainties are revealed at the beginning. In the analyses, we derive an unconstrained representation of the optimality gap for generic constrained online learning problems, which renders tractable lower and upper bounds for the expected revenue achieved by dynamic pricing algorithms between different regimes. This analytical framework also inspires the design of two dual-based history-dependent dynamic pricing algorithms for the clairvoyant regime and the online regime. Numerical experiments are conducted to demonstrate the performances of our algorithms.
Author: Xiaocheng Li
Publisher:
ISBN:
Category :
Languages : en
Pages : 41
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Book Description
A merchant sells a product over a selling season of T time periods in presence of a limited inventory. The merchant observes new external information at the beginning of each time period and then sets a price for that time period. Initially, the merchant does not know the distribution of the external information and the demand function, i.e., how the external information and price jointly impact the demand distribution in a single time period. The seller's decision, setting price dynamically, serves dual roles to learn the unknown demand function and to balance inventory, with an ultimate goal to maximize the expected revenue over the selling season. We characterize and prove a full spectrum of relations between the optimal revenue achieved in three decision-making regimes: the merchant's online decision-making regime, a clairvoyant regime with complete knowledge about the demand function and all the external information in advance, and a deterministic regime in which the demand function and all the uncertainties are revealed at the beginning. In the analyses, we derive an unconstrained representation of the optimality gap for generic constrained online learning problems, which renders tractable lower and upper bounds for the expected revenue achieved by dynamic pricing algorithms between different regimes. This analytical framework also inspires the design of two dual-based history-dependent dynamic pricing algorithms for the clairvoyant regime and the online regime. Numerical experiments are conducted to demonstrate the performances of our algorithms.
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: Boxiao Chen
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
We consider a joint pricing and inventory control problem where pricing can be adjusted more frequently, such as every period, than inventory ordering decisions, which are made every epoch that consists of multiple periods. This is motivated by many examples, especially for online retailers, where price is indeed much easier to change than inventory level, because changing the latter is subject to logistic and capacity constraints. In this setting, the retailer determines the inventory level at the beginning of each epoch and solves a dynamic pricing problem within each epoch with no further replenishment opportunities. The optimal pricing and inventory control policy is characterized by an intricate dynamic programming (DP) solution. We consider the situation where the demand-price function and the distribution of random demand noise are both unknown to the retailer, and the retailer needs to develop an online learning algorithm to learn those information and at the same time maximize total profit. We propose a learning algorithm based on least squares estimation and construction of an empirical noise distribution under a UCB framework and prove that the algorithm converges through the DP recursions to approach the optimal pricing and inventory control policy under complete demand information. The theoretical lower bound for convergence rate of a learning algorithm is proved based on the multivariate Van Trees inequality coupled with some structural DP analyses, and we show that the upper bound of our algorithm's convergence rate matches the theoretical lower bound.
Author: Elodie Adida
Publisher: VDM Publishing
ISBN: 9783836421430
Category : Business & Economics
Languages : en
Pages : 288
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Book Description
(cont.) We introduce and study a solution method that enables to compute the optimal solution on a finite time horizon in a monopoly setting. Our results illustrate the role of capacity and the effects of the dynamic nature of demand. We then introduce an additive model of demand uncertainty. We use a robust optimization approach to protect the solution against data uncertainty in a tractable manner, and without imposing stringent assumptions on available information. We show that the robust formulation is of the same order of complexity as the deterministic problem and demonstrate how to adapt solution method. Finally, we consider a duopoly setting and use a more general model of additive and multiplicative demand uncertainty. We formulate the robust problem as a coupled constraint differential game. Using a quasi-variational inequality reformulation, we prove the existence of Nash equilibria in continuous time and study issues of uniqueness. Finally, we introduce a relaxation-type algorithm and prove its convergence to a particular Nash equilibrium (normalized Nash equilibrium) in discrete time.
Author: Qi Feng
Publisher:
ISBN:
Category :
Languages : en
Pages : 43
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Book Description
Inventory-based pricing under lost sales is an important, yet notoriously challenging problem in the operations management literature. The vast existing literature on this problem focuses on identifying optimality conditions for a simple management policy, while restricting to special classes of demand functions and to the special case of single-period or long-term stationary settings. In view of the existing developments, it seems unlikely to find general, easy-to-verify conditions for a tractable optimal policy in a possibly nonstationary environment. Instead, we take a different approach to tackle this problem. Specifically, we refine our analysis to a class of intuitively appealing policies, under which the price is decreasing in the post-order inventory level. Using properties of stochastic functions, we show that, under very general conditions on the stochastic demand function, the objective function is concave along such price paths, leading to a simple base stock list price policy. We identify the upper and lower boundaries for a candidate set of decreasing price paths and show that any decreasing path outside of this set is always dominated by some inside the set in terms of profit performance. The boundary policies can be computed efficiently through a single-dimensional search. An extensive numerical analysis suggests that choosing boundary policies yields close-to-optimal profit--in most instances, one of the boundary policies indeed generates the optimal profit, even when they are not, the profit loss is very marginal.
Author: Boxiao Chen
Publisher:
ISBN:
Category :
Languages : en
Pages : 61
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Book Description
A firm makes pricing and inventory replenishment decisions for a product over T periods to maximize its expected total profit. Demand is random and price sensitive, and unsatisfied demands are lost and unobservable (censored demand). The firm knows the demand process up to some parameters and needs to learn them through pricing and inventory experimentation. However, due to business constraints the firm is prevented from making frequent price changes, leading to correlated and dependent sales data. We develop data-driven algorithms by actively experimenting inventory and pricing decisions and construct maximum likelihood estimator with censored and correlated samples for parameter estimation. We analyze the algorithms using the T-period regret, defined as the profit loss of the algorithms over T periods compared with the clairvoyant optimal policy that knew the parameters a priori. For a so-called well-separated case, we show that the regret of our algorithm is O(T^{1/(m+1)}) when the number of price changes is limited by m >= 1, and is O( log T) when limited by beta log T for some positive constant beta>0; while for a more general case, the regret is O(T^{1/2}) when the underlying demand is bounded and O(T^{1/2} log T) when the underlying demand is unbounded. We further prove that our algorithm for each case is the best possible in the sense that its regret rate matches with the theoretical lower bound.
Author: Boxiao Chen
Publisher:
ISBN:
Category :
Languages : en
Pages : 42
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Book Description
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 (s,S,p) policy is proved to be optimal, where s and S are the reorder point and order-up-to level for ordering strategy, and p, 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 (s,S,p) with a tight O ̃(√T) regret rate. A number of salient features differentiate our work from the existing online learning researches in the OM literature. First, computing the optimal (s,S,p) policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in operations management 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 p 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, due to the multi-period nature of (s,S,p) policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy p, 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.
Author: Jing-Sheng J. Song
Publisher: Edward Elgar Publishing
ISBN: 180037710X
Category : Technology & Engineering
Languages : en
Pages : 565
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Book Description
This comprehensive Handbook provides an overview of state-of-the-art research on quantitative models for inventory management. Despite over half a century’s progress, inventory management remains a challenge, as evidenced by the recent Covid-19 pandemic. With an expanse of world-renowned inventory scholars from major international research universities, this Handbook explores key areas including mathematical modelling, the interplay of inventory decisions and other business decisions and the unique challenges posed to multiple industries.
Author: JUBI R
Publisher: NEHAS PUBLICATIONS
ISBN: 819681142X
Category : Business & Economics
Languages : en
Pages : 367
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Book Description
In today's dynamic and data-driven business landscape, the art and science of Business Analytics have emerged as critical tools for exploration, introspection, and informed decision-making. "Business Analytics," the book at hand, delves into the practices and competencies essential for unraveling the complexities of business performance, facilitating purposeful, intuitive, and expedient decision-making processes. The essence of Business Analytics lies in the extensive exploration of business data, aiming to extract meaningful information usable by managers across various organizational levels. This book positions Business Analytics as a catalyst for fact-based decision-making, elevating accountability in the decision-making process. It defines Business Analytics as a methodical process that involves scrutinizing and summarizing data with the explicit purpose of uncovering hidden predictive insights. This book places a particular emphasis on the science and artistry of business analytics, with a special focus on financial analytics. It not only explores the practical aspects but also lays the theoretical foundations, providing a comprehensive context for various elements of business analytics within specific business situations. A distinctive feature of this book is its commitment to showcasing the implementation of analytics by illustrating how leading companies leverage this power to enhance their investments. Acknowledging that scientific knowledge alone may not suffice for sound decision-making, the book underscores the importance of combining scientific expertise with a deep understanding of the business context and the best available information. Addressing a notable gap in existing literature, this book goes beyond traditional academic texts that predominantly concentrate on quantitative methods. Instead, it extends its reach to cover analytics for non-quantitative managers. In doing so, the book aims to equip a broader audience with the knowledge and tools necessary to harness the benefits of Business Analytics in diverse business scenarios. As you embark on this journey through the pages of "Business Analytics," you will gain insights into the transformative power of analytics in decision-making, and how it has become an indispensable asset for businesses navigating the intricacies of the contemporary corporate landscape.
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.
