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Author: Boxiao Chen
Publisher:
ISBN:
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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: 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: Wen Chen
Publisher:
ISBN:
Category :
Languages : en
Pages : 33
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Book Description
We study an average-cost stochastic inventory control problem in which the firm can replenish inventory and adjust price at anytime. We establish the optimality to change the price from low to high in each replenishment cycle as inventory is depleted. With costly price adjustment, scale economies of inventory replenishment are reflected in the cycle time instead of lot size -- An increased fixed ordering cost leads to an extended replenishment cycle but does not necessarily increase the order quantity. A reduced marginal cost of ordering calls for an increased order quantity, as well as speeding up product selling within a cycle. We derive useful properties of the profit function that allows for reducing computational complexity of the problem. For systems requiring short replenishment cycles, the optimal solution can be easily computed by applying these properties. For systems requiring long replenishment cycles, we further consider a relaxed problem that is computational tractable. Under this relaxation, the sum of fixed ordering cost and price adjustment cost is equal to (greater than, less than) the total inventory holding cost within a replenishment cycle when the inventory holding cost is linear (convex, concave) in the stock level. Moreover, under the optimal solution, the time-average profit is the same across all price segments when the inventory holding cost is accounted properly. Through a numerical study, we demonstrate that inventory-based dynamic pricing can lead to significant profit improvement compared with static pricing and limited price adjustment can yield a benefit that is close to unlimited price adjustment. To be able to enjoy the benefit of dynamic pricing, however, it is important to appropriately choose inventory levels at which the price is revised.
Author: Boxiao Chen
Publisher:
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Languages : en
Pages : 0
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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: Omar Besbes
Publisher:
ISBN:
Category :
Languages : en
Pages : 33
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Book Description
Many factors introduce the prospect of changes in the demand environment that a firm faces, with the specifics of such changes not necessarily known in advance. If and when realized, such changes affect the delicate balance between demand and supply and thus current prices should account for these future possibilities. We study the dynamic pricing problem of a retailer facing the prospect of a change in the demand function during a finite selling season with no inventory replenishment opportunity. In particular, the time of the change and the postchange demand function are unknown upfront, and we focus on the fundamental trade-off between collecting revenues from current demand and doing so for postchange demand, with the capacity constraint introducing the main tension. We develop a formulation that allows for isolating the role of dynamic pricing in balancing inventory consumption throughout the horizon. We establish that, in many settings, optimal pricing policies follow a monotone path up to the change in demand. We show how one may compare upfront the attractiveness of pre- and postchange demand conditions and how such a comparison depends on the problem primitives. We further analyze the impact of the model inputs on the optimal policy and its structure, ranging from the impact of model parameter changes to the impact of different representations of uncertainty about future demand.
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: Oben Ceryan
Publisher:
ISBN:
Category :
Languages : en
Pages :
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We study a joint implementation of price- and availability-based product substitution to better match demand and constrained supply across vertically differentiated products. Our study is motivated by firms that utilize dynamic pricing as well as customer upgrades, as ex-ante and ex-post mechanisms, respectively, to mitigate inventory mismatches. To gain insight into how offering product upgrades impacts optimal price selection, we formulate a multiple period, nested two-stage model where the firm first sets prices and replenishment levels for each product while the demand is still uncertain, and after observing the demand, decides how many (if any) of the customers to upgrade to a higher quality product. We characterize the structure of the optimal upgrade, pricing and replenishment policies and find that firms having greater flexibility to offer product upgrades can restrain their reliance on dynamic pricing, enabling them to better protect the price differentiation between the products. We also show how the quality differential between the products or changes in the replenishment cost structures influence the optimal policy. Using insights gained from the optimal policy structure, we construct a heuristic policy and find that it performs well across various parameter values. Finally, we consider an extension in which the firm dynamically sets upgrade fees in each period. Our results overall help further our understanding of the intricate relationship among a firm's decisions on pricing, replenishment, and product upgrades in an effort to better match demand and constrained supply.
Author: Parthasarathy, S.
Publisher: IGI Global
ISBN: 1615206264
Category : Computers
Languages : en
Pages : 504
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Book Description
"This book aims at identifying potential research problems and issues in the EIS such as Enterprise Resource Planning (ERP), Supply Chain Management (SCM), and Customer Relationship Management (CRM)"--Provided by publisher.
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: Shiyu Liu
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
In recent years, omni-channel retailing has become immensely popular among both retailers and consumers. In this approach, retailers often leverage their brick-and-mortar stores to fulfill online orders, leading to the need for simultaneous decision-making on replenishment and inventory rationing. This inventory strategy presents significant complexities in traditional dynamic pricing and inventory management problems, particularly in unpredictable market environments. Therefore, we have developed a dynamic pricing, replenishment, and rationing model for omni-channel retailers using a two-level partially observed Markov decision process to visualize the dynamic process. We design a deep reinforcement learning algorithm, called Maskable LSTM-Proximal Policy Optimization (ML-PPO), which integrates the current observations and future predictions as input to the agent and uses the invalid action mask to guarantee the allowable actions. Our simulation experiments have demonstrated the ML-PPO's efficiency in maximizing retailer profit and service level, along with its generalized ability to tackle dynamic pricing and inventory management problems.
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.
