A comparison of fixed and dynamic pricing policies in revenue management

We consider the problem of selling a fixed capacity or inventory of items over a finite selling period. Earlier research has shown that using a properly set fixed price during the selling period is asymptotically optimal as the demand potential and capacity grow large and that dynamic pricing has only a secondary effect on revenues. However, additional revenue improvements through dynamic pricing can be important in practice and need to be further explored. We suggest two simple dynamic heuristics that continuously update prices based on remaining inventory and time in the selling period. The first heuristic is based on approximating the optimal expected revenue function and the second heuristic is based on the solution of the deterministic version of the problem. We show through a numerical study that the revenue impact of using these dynamic pricing heuristics rather than fixed pricing may be substantial. In particular, the first heuristic has a consistent and remarkable performance leading to at most 0.2% gap compared to optimal dynamic pricing. We also show that the benefits of these dynamic pricing heuristics persist under a periodic setting. This is especially true for the first heuristic for which the performance is monotone in the frequency of price changes. We conclude that dynamic pricing should be considered as a more favorable option in practice.     

Optimal Policies for a Two‐Product Inventory System under a Flexible Substitution Scheme

We study a two‐product inventory model that allows substitution. Both products can be used to supply demand over a selling season of N periods, with a one‐time replenishment opportunity at the beginning of the season. A substitution could be offered even when the demanded product is available. The substitution rule is flexible in the sense that the seller can choose whether or not to offer substitution and at what price or discount level, and the customer may or may not accept the offer, with the acceptance probability being a decreasing function of the substitution price. The decisions are the replenishment quantities at the beginning of the season, and the dynamic substitution‐pricing policy in each period of the season. Using a stochastic dynamic programming approach, we present a complete solution to the problem. Furthermore, we show that the objective function is concave and submodular in the inventory levels—structural properties that facilitate the solution procedure and help identify threshold policies for the optimal substitution/pricing decisions. Finally, with a state transformation, we also show that the objective function is ‐concave, which allows us to derive similar structural properties of the optimal policy for multiple‐season problems.     

Combined Pricing and Portfolio Option Procurement

In this paper, we study a single‐product periodic‐review inventory system that faces random and price‐dependent demand. The firm can purchase the product either from option contracts or from the spot market. Different option contracts are offered by a set of suppliers with a two‐part fee structure: a unit reservation cost and a unit exercising cost. The spot market price is random and its realization may affect the subsequent option contract prices. The firm decides the reservation quantity from each supplier and the product selling price at the beginning of each period and the number of options to exercise (inventory replenishment) at the end of the period to maximize the total expected profit over its planning horizon. We show that the optimal inventory replenishment policy is order‐up‐to type with a sequence of decreasing thresholds. We also investigate the optimal option‐reservation policy and the optimal pricing strategy. The optimal reservation quantities and selling price are shown to be both decreasing in the starting inventory level when demand function is additive. Building upon the analytical results, we conduct a numerical study to unveil additional managerial insights. Among other things, we quantify the values of the option contracts and dynamic pricing to the firm and show that they are more significant when the market demand becomes more volatile.     

Dynamic Pricing in the Presence of Myopic and Strategic Consumers: Theory and Experiment

We investigate retailers’ dynamic pricing decisions in a stylized two‐period setting with possible supply constraints and demand from both myopic and strategic consumers. We present an analytical model and then test its predictions in a behavioral experiment in which human subjects played the role of pricing managers. We find that the fraction of strategic consumers in the market systematically moderates the optimal pricing structure. When this fraction exceeds a certain threshold, the retailer offers relatively small late season markdowns to discourage strategic consumers from waiting and to incentivize them to buy during the early season; otherwise, the retailer offers relatively large markdowns to divert all strategic consumers to the late season, where the majority of revenue is made. Our model analyses suggest that the latter policy is optimal under fairly broad conditions. Our experiment shows that after some significant learning, aggregate behavior is able to approximate the key qualitative predictions from our model analysis, with one notable deviation: in the presence of a mixture of myopic and strategic consumers, subjects act somewhat myopically – they underprice and oversell in the main selling season, which significantly limits their ability to generate revenue in the markdown season.     

Optimal Control of Replenishment and Substitution in an Inventory System with Nonstationary Batch Demand

We study an inventory system in which a supplier supplies demand using two mutually substitutable products over a selling season of T periods, with a single replenishment opportunity at the beginning of the season. As the season starts, customer orders arrive in each period, for either type of products, following a nonstationary Poisson process with random batch sizes. The substitution model we consider combines the usual supplier‐driven and customer‐driven schemes, in that the supplier may choose to offer substitution, at a discount price, or may choose not to; whereas the customer may or may not accept the substitution when it is offered. The supplier's decisions are the supply and substitution rules in each period throughout the season, and the replenishment quantities for both products at the beginning of the season. With a stochastic dynamic programming formulation, we first prove the concavity of the value function, which facilitates the solution to the optimal replenishment quantities. We then show that the optimal substitution follows a threshold rule, and establish the monotonicity of the thresholds over time and with respect to key cost parameters. We also propose a heuristic exhaustive policy, and illustrate its performance through numerical examples.     

Recent Developments in Dynamic Pricing Research: Multiple Products,Competition, and Limited Demand Information

Dynamic pricing enables a firm to increase revenue by better matching supply with demand, responding to shifting demand patterns, and achieving customer segmentation. In the last 20 years, numerous success stories of dynamic pricing applications have motivated a rapidly growing research interest in a variety of dynamic pricing problems in the academic literature. A large class of problems that arise in various revenue management applications involve selling a given amount of inventory over a finite time horizon without inventory replenishment. In this study, we identify most recent trends in dynamic pricing research involving such problems. We review existing research on three new classes of problems that have attracted a rapidly growing interest in the last several years, namely, problems with multiple products, problems with competition, and problems with limited demand information. We also identify a number of possible directions for future research.     

Revenue Management with End‐of‐Period Discounts in the Presence of Customer Learning

Consider a firm that sells identical products over a series of selling periods (e.g., weekly all‐inclusive vacations at the same resort). To stimulate demand and enhance revenue, in some periods, the firm may choose to offer a part of its available inventory at a discount. As customers learn to expect such discounts, a fraction may wait rather than purchase at a regular price. A problem the firm faces is how to incorporate this waiting and learning into its revenue management decisions. To address this problem we summarize two types of learning behaviors and propose a general model that allows for both stochastic consumer demand and stochastic waiting. For the case with two customer classes, we develop a novel solution approach to the resulting dynamic program. We then examine two simplified models, where either the demand or the waiting behavior are deterministic, and present the solution in a closed form. We extend the model to incorporate three customer classes and discuss the effects of overselling the capacity and bumping customers. Through numerical simulations we study the value of offering end‐of‐period deals optimally and analyze how this value changes under different consumer behavior and demand scenarios.     

Competitive markdown timing for perishable and substitutable products

We model as a duopoly two firms selling their fixed stocks of two substitutable items over a selling season. Each firm starts with an initial price, and has the option to decrease the price once. The problem for each firm is to determine when to mark its price down in to maximize its revenue. We show that the existence and characterization of a pure-strategy equilibrium depend on the magnitude of the increase in the revenue rate of a firm when its competitor runs out of stock. When the increase is smaller than the change in the revenue rate of the price leader when both firms are in stock for all of the three possible scenarios, neither firm has the incentive to force its rival to run out of stock and if a firm marks its price down after the season starts, its inventory runs out precisely at the end of the season. When the increase is larger than the change of the price leader׳s revenue rate in one particular scenario, waiting until its rival runs out of inventory may be an equilibrium strategy for the larger firm even though this may lead to leftover inventory for itself. In other cases, there may be no pure-strategy equilibrium in the game. In certain regions of the parameter space, a firm׳s revenue may be decreasing in its starting inventory which shows that a firm may be better off if it can credibly salvage a portion of its inventory prior to the game. While most of our analysis is for open-loop strategies, in the final part of the paper, we show that the open-loop equilibrium survives as an equilibrium when we consider closed-loop strategies for an important subset of the parameter space.     

考虑策略型消费者的在线零售商库存信息披露及联合定价、库存决策模型

在两期销售环境下,研究了面向策略型消费者的在线零售商库存信息披露及联合定价、库存决策问题。考虑在线零售商库存信息共享和隐藏两种库存披露策略,构建了在线零售商两期销售利润模型。通过分析消费者的购买行为,讨论了在具有策略型消费者的市场中,在线零售商应如何选择库存信息披露策略,以及如何制定最优定价和库存决策。在此基础上,分析了相关参数对在线零售商决策及利润的影响。研究结果表明,在线零售商的最优定价及库存决策受消费者估值折扣系数和第二期定价策略的影响。特别地,当在线零售商选择第一期缺货时,更倾向于共享其库存水平信息;当在线零售商选择第二期缺货或者两期均不缺货时,消费者估值折扣系数存在一个阈值,当低于该阈值时,在线零售商会选择隐藏库存水平信息,当高于该阈值时,在线零售商会选择共享库存水平信息。     

Managing Clearance Sales in the Presence of Strategic Customers

We study the effect of strategic customer behavior on pricing and rationing decisions of a firm selling a single product over two periods. The seller may limit the availability of the product (that is, ration) in the second (clearance) period. Some customers are strategic and respond to the firm's decisions by timing their purchases. When capacity is nonconstraining and the seller has pricing flexibility, we show that rationing in the clearance period cannot improve revenue. However, when prices are fixed in advance, rationing can improve revenue. In the latter case, we conduct a detailed analysis for linear and exponential demand curves and derive explicit expressions for optimal rationing levels. We find that the policy of doing the better of not restricting availability at the clearance price or not offering the product at the clearance price is typically near optimal. Our analysis also suggests that rationing—although sometimes offering considerable benefit over allowing unrestricted availability in the clearance period—may allow the seller to obtain only a small fraction of the optimal revenue when the prices are chosen optimally without rationing. We extend the analysis to cases where the capacity is constraining and obtain similar results.     

Dynamic Pricing,Production, and Channel Coordination with Stochastic Learning

We consider a decentralized two‐period supply chain in which a manufacturer produces a product with benefits of cost learning, and sells it through a retailer facing a price‐dependent demand. The manufacturer's second‐period production cost declines linearly in the first‐period production, but with a random learning rate. The manufacturer may or may not have the inventory carryover option. We formulate the resulting problems as two‐period Stackelberg games and obtain their feedback equilibrium solutions explicitly. We then examine the impact of mean learning rate and learning rate variability on the pricing strategies of the channel members, on the manufacturer's production decisions, and on the retailer's procurement decisions. We show that as the mean learning rate or the learning rate variability increases, the traditional double marginalization problem becomes more severe, leading to greater efficiency loss in the channel. We obtain revenue sharing contracts that can coordinate the dynamic supply chain. In particular, when the manufacturer may hold inventory, we identify two major drivers for inventory carryover: market growth and learning rate variability. Finally, we demonstrate the robustness of our results by examining a model in which cost learning takes place continuously.     

引入策略型消费者遗憾心理的时尚产品定价策略

通过对时尚策略型消费行为的研究,把消费者心理动机归结为两种,其一是因冲动购买而错过降价产生的“高价遗憾”;其二是因等待降价并延迟购买而遭遇缺货产生的“缺货遗憾”,并在此基础上,在一个两周期销售中分析其对企业的降低标价(Markdown, MD)与抬高标价(Markup, MU)两种定价模式的影响机理。首先把两种遗憾心理因素纳入消费者的效用函数,建模分析了需求随机下降低标价(抬高标价)两阶段销售模式中供应商如何通过考虑消费者行为设置不同阶段价格和限量配给水平的决策。并在此基础上通过不同定价模式下供应商收益参数的对比,重点分析了两种遗憾心理因素效应对销售阶段分割的综合影响。研究发现：有遗憾心理的消费者将削弱延迟购买倾向,供应商可通过制造适当的配给风险诱导顾客高价期购买;其次两种定价类型下,当满足一定条件时,面对遗憾心理的消费群体,企业均能实现比统一定价更高的最优收益,并且抬高标价比降低标价的优化效果更明显。     

Asymmetric Forecast Information and the Value of Demand Observation in Repeated Procurement

In many supply chain relationships that continue over multiple periods, information about the hidden properties of the supply chain partners can be revealed throughout the course of the relationship. This article examines how the availability of such information affects the contracting scheme between a supplier and his manufacturer in a relationship that persists over several selling seasons. At the beginning of the first selling season, the manufacturer observes private information about the demand distribution, whereas the supplier who is less familiar with the market is endowed only with a prior belief about the market condition. When the supplier cannot observe the demand realization during the first selling period, she offers a contract that induces the manufacturer to reveal the market condition in the first selling season. However, the opportunity for the supplier to observe demand realization can result in the supplier offering the manufacturer a simple contract that does not induce the manufacturer to reveal his private information during the first selling season. In the latter case, the supplier observes the demand realization and designs the second period contract based on this information. We show that when the supplier chooses to offer such a contract, the manufacturer becomes worse‐off, and it has an ambiguous effect on the performance of the supply chain.     

On the Benefit of Inventory‐Based Dynamic Pricing Strategies

We study the optimal pricing and replenishment decisions in an inventory system with a price‐sensitive demand, focusing on the benefit of the inventory‐based dynamic pricing strategy. We find that demand variability impacts the benefit of dynamic pricing not only through the magnitude of the variability but also through its functional form (e.g., whether it is additive, multiplicative, or others). We provide an approach to quantify the profit improvement of dynamic pricing over static pricing without having to solve the dynamic pricing problem. We also demonstrate that dynamic pricing is most effective when it is jointly optimized with inventory replenishment decisions, and that its advantage can be mostly realized by using one or two price changes over a replenishment cycle.     

Dynamic Pricing and Inventory Management with Regular and Expedited Supplies

We consider a periodic‐review inventory system with regular and expedited supply modes. The expedited supply is faster than the regular supply but incurs a higher cost. Demand for the product in each period is random and sensitive to its selling price. The firm determines its order quantity from each supply in each period as well as its selling price to maximize the expected total discounted profit over a finite or an infinite planning horizon. We show that, in each period if it is optimal to order from both supplies, the optimal inventory policy is determined by two state‐independent thresholds, one for each supply mode, and a list price is set for the product; if only the regular supply is used, the optimal policy is a state‐dependent base‐stock policy, that is, the optimal base‐stock level depends on the starting inventory level, and the optimal selling price is a markdown price that decreases with the starting inventory level. We further study the operational impact of such supply diversification and show that it increases the firm's expected profit, reduces the optimal safety‐stock levels, and lowers the optimal selling price. Thus that diversification is beneficial to both the firm and its customers. Building upon these results, we conduct a numerical study to assess and compare the respective benefit of dynamic pricing and supply diversification.     

Optimal ordering and pricing decisions for a target oriented newsvendor

This paper is the first to study pricing and target oriented decision making together in the newsvendor model. Specifically, this paper studies a newsvendor who decides on order quantity and selling price to maximize the probability of achieving both profit and revenue targets simultaneously. First, it is shown that the probability of a newsvendor achieving both targets depends critically on the relative magnitudes of the profit margin and the ratio between the profit target and the revenue target. Second, the closed-form expressions of the optimal order quantity, the optimal selling price, and the maximal profit and revenue probability are obtained. It is shown that if the product has greater price elasticity, the best strategy is always to price lower and order more.     

Up Then Down: Bid‐Price Trends in Revenue Management

In the classic revenue management (RM) problem of selling a fixed quantity of perishable inventories to price‐sensitive non‐strategic consumers over a finite horizon, the optimal pricing decision at any time depends on two important factors: consumer valuation and bid price. The former is determined exogenously by the demand side, while the latter is determined jointly by the inventory level on the supply side and the consumer valuations in the time remaining within the selling horizon. Because of the importance of bid prices in theory and practice of RM, this study aims to enhance the understanding of the intertemporal behavior of bid prices in dynamic RM environments. We provide a probabilistic characterization of the optimal policies from the perspective of bid‐price processes. We show that an optimal bid‐price process has an upward trend over time before the inventory level falls to one and then has a downward trend. This intertemporal up‐then‐down pattern of bid‐price processes is related to two fundamental static properties of the optimal bid prices: (i) At any given time, a lower inventory level yields a higher optimal bid price, which is referred to as the resource scarcity effect; (ii) Given any inventory level, the optimal bid price decreases with time; that is referred to as the resource perishability effect. The demonstrated upward trend implies that the optimal bid‐price process is mainly driven by the resource scarcity effect, while the downward trend implies that the bid‐price process is mainly driven by the resource perishability effect. We also demonstrate how optimal bid price and consumer valuation, as two competing forces, interact over time to drive the optimal‐price process. The results are also extended to the network RM problems.     

Grocery Perishables Management

In this article, we study the joint pricing and inventory control problem for perishables when a retailer does not sell new and old inventory at the same time. At the beginning of a period, the retailer makes replenishment and pricing decisions, and at the end of a period, the retailer decides whether to dispose of ending inventory or carry it forward to the next period. The objective of the retailer is to maximize the long‐run average profit. Assuming zero lead time, we propose an efficient solution approach to the problem, which is also generalized to solve three extensions to the basic model. A feature of the present study is that we consider explicitly the influence of perishability on the demand. Among the insights gathered from the numerical analysis, we find that dynamic pricing aids extending shelf life and when disposal incurs a lower cost, or even a positive salvage value, the retailer is induced to dispose earlier since the benefit of selling new inventory offsets the loss due to disposal. We also observe that the faster the perceived rate of deterioration, the lower the threshold of the ending inventory for disposal. Perhaps a bit counter‐intuitive, maximizing profits does not mean eliminating disposals or expirations.     

Committed Versus Contingent Pricing Under Competition

Should capacitated firms set prices responsively to uncertain market conditions in a competitive environment? We study a duopoly selling differentiated substitutable products with fixed capacities under demand uncertainty, where firms can either commit to a fixed price ex ante, or elect to price contingently ex post, e.g., to charge high prices in booming markets, and low prices in slack markets. Interestingly, we analytically show that even for completely symmetric model primitives, asymmetric equilibria of strategic pricing decisions may arise, in which one firm commits statically and the other firm prices contingently; in this case, there also exists a unique mixed strategy equilibrium. Such equilibrium behavior tends to emerge, when capacity is ampler, and products are less differentiated or demand uncertainty is lower. With asymmetric fixed capacities, if demand uncertainty is low, a unique asymmetric equilibrium emerges, in which the firm with more capacity chooses committed pricing and the firm with less capacity chooses contingent pricing. We identify two countervailing profit effects of contingent pricing under competition: gains from responsively charging high price under high demand, and losses from intensified price competition under low demand. It is the latter detrimental effect that may prevent both firms from choosing a contingent pricing strategy in equilibrium. We show that the insights remain valid when capacity decisions are endogenized. We caution that responsive price changes under aggressive competition of less differentiated products can result in profit‐killing discounting.     

考虑战略顾客行为时的供应链性能分析与协调

动态定价策略的广泛应用使得越来越多的顾客具有了战略性,战略顾客会根据产品在销售期内的价格路径确定最优购买时机,零售商则根据顾客的购买行为确定订货数量和销售价格.研究了双方静态博弈时的理性预期均衡解和零售商进行数量承诺时的情形.研究表明:理性预期均衡时的最优销售价格、最优存货数量和最优期望利润分别小于标准报童模型的情形;数量承诺时的最优存货数量小于理性预期均衡时的最优存货数量;最优期望利润则大于理性预期均衡时的最优期望利润,并且在一定条件下可能会大于标准报童模型的最优期望利润,战略顾客行为的存在对零售商可能有利.最后分析了在分散式供应链中如何利用收入分享契约和数量折扣契约实现供应链协调.