A Choice‐Based Dynamic Programming Approach for Setting Opaque Prices

Opaque pricing is a form of pricing where certain characteristics of the product or service are hidden from the consumer until after purchase. In essence, opaque selling transforms a differentiated good into a commodity. Opaque pricing has become popular in service pricing as it allows firms to sell their differentiated product at higher prices to regular brand loyal customers while simultaneously selling to non‐brand loyal customers at discounted prices. We use a nested logit model in combination with logistic regression and dynamic programming to illustrate how a service firm can optimally set prices on an opaque sales channel. The choice model allows the characterization of consumer trade‐offs when purchasing opaque products while the dynamic programming approach allows the characterization of the optimal pricing policy as a function of inventory and time remaining. We compare optimal prices and expected revenues when dynamic pricing is restricted to daily price changes. We provide an illustrative example using data from an opaque selling mechanism ( Hotwire.com ) and a Washington DC‐based hotel.     

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 and Inventory Management with Dual Suppliers of Different Lead Times and Disruption Risks

It is common for a firm to make use of multiple suppliers of different delivery lead times, reliabilities, and costs. In this study, we are concerned with the joint pricing and inventory control problem for such a firm that has a quick‐response supplier and a regular supplier that both suffer random disruptions, and faces price‐sensitive random demands. We aim at characterizing the optimal ordering and pricing policies in each period over a planning horizon, and analyzing the impacts of supply source diversification. We show that, when both suppliers are unreliable, the optimal inventory policy in each period is a reorder point policy and the optimal price is decreasing in the starting inventory level in that period. In addition, we show that having supply source diversification or higher supplier reliability increases the firm's optimal profit and lowers the optimal selling price. We also demonstrate that, with the selling price as a decision, a supplier may receive even more orders from the firm after an additional supplier is introduced. For the special case where the quick‐response supplier is perfectly reliable, we further show that the optimal inventory policy is of a base‐stock type and the optimal pricing policy is a list‐price policy with markdowns.     

Advance Selling by a Newsvendor Retailer

Retailers often face a newsvendor problem. Advance selling helps retailers to reduce demand uncertainty. Consumers, however, may prefer not to purchase in advance unless given a discount because they are uncertain about their valuation for the product in advance. It is then unclear whether or when advance selling to pass some uncertainty risk to consumers is optimal for the retailer. This paper examines the advance selling price and inventory decisions in a two‐period setting, where the first period is the advance selling period and the second is the selling (and consumption) period. We find that an advance selling strategy is not always optimal, but is contingent on parameters of the market (e.g., market potential and uncertainty) and the consumers (e.g., valuation, risk aversion, and heterogeneity). For example, we find that retailers should sell in advance if the consumers' expected valuation exceeds consumers' expected surplus when not buying early by a certain threshold. This threshold increases with the degree of risk aversion but decreases with stock out risk. If the degree of risk aversion varies across consumers, then a retailer should sell in advance if the probability for a consumer to spot buy is less than a critical fractile.     

Pricing Decisions During Inter‐Generational Product Transition

How should companies price products during an inter‐generational transition? High uncertainty in a new product introduction often leads to extreme cases of demand and supply mismatches. Pricing is an effective tool to either prevent or alleviate these problems. We study the optimal pricing decisions in the context of a product transition in which a new‐generation product replaces an old one. We formulate the dynamic pricing problem and derive the optimal prices for both the old and new products. Our analysis sheds light on the pattern of the optimal prices for the two products during the transition and on how product replacement, along with several other dynamics including substitution, external competition, scarcity, and inventory, affect the optimal prices. We also determine the optimal initial inventory for each product and discuss a heuristic method.     

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.     

Allocation of Returned Products among Different Recovery Options through an Opportunity Cost–Based Dynamic Approach*

In a make‐to‐order product recovery environment, we consider the allocation decision for returned products decision under stochastic demand of a firm with three options: refurbishing to resell, parts harvesting, and recycling. We formulate the problem as a multiperiod Markov decision process (MDP) and present a linear programming (LP) approximation that provides an upper bound on the optimal objective function value of the MDP model. We then present two solution approaches to the MDP using the LP solution: a static approach that uses the LP solution directly and a dynamic approach that adopts a revenue management perspective and employs bid‐price controls technique where the LP is resolved after each demand arrival. We calculate the bid prices based on the shadow price interpretation of the dual variables for the inventory constraints and accept a demand if the marginal value is higher than the bid price. Since the need for solving the LP at each demand arrival requires a very efficient solution procedure, we present a transportation problem formulation of the LP via variable redefinitions and develop a one‐pass optimal solution procedure for it. We carry out an extensive numerical analysis to compare the two approaches and find that the dynamic approach provides better performance in all of the tested scenarios. Furthermore, the solutions obtained are within 2% of the upper bound on the optimal objective function value of the MDP model.     

考虑策略性消费者的生鲜农产品定价和库存决策

消费者的策略性行为使零售商的生鲜农产品的定价和库存决策面临更大挑战。本文基于报童模型,综合考虑消费者的策略性行为,对生鲜农产品价值下降进行离散化处理。刻画策略性消费者的决策行为,构建零售商的单阶段和两阶段定价及库存决策模型,分析了产品价值剩余率对消费者行为、零售商最优定价、最优库存水平以及零售商利润的影响机理。研究发现,在单阶段模型中零售商最优价格和最优库存水平均随产品价值剩余率的递增而递增;而在两阶段模型中,第二阶段最优价格随价值剩余率的变化趋势可能存在阈值。     

固定价格与英式拍卖同时使用时顾客行为分析

很多网站采用在英式拍卖中加入固定价格选项的方式销售多件同类商品.我们证明了,在此规则下存在一个弱占优投标策略.若顾客对商品的估价低于固定价格,那么,当拍卖价格低于他的估价时,该顾客参与拍卖,且以自己的估价作为报价是他的弱占优投标策略;否则,当拍卖价格超过他的估价时,他退出拍卖.若顾客对商品的估价高于或等于固定价格,他会在参与拍卖和接受固定价格之间进行选择;若他参与拍卖,将固定价格作为报价是他的弱占优投标策略.我们发现,该顾客是选择参与拍卖、还是选择以固定价格购买主要依赖于顾客到达拍卖时距离拍卖结束的剩余时间.最后,通过算例,我们对影响顾客参与策略的因素进行了数值分析并得到了一些结论.     

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。     

Optimal Reserve Prices in Name‐Your‐Own‐Price Auctions with Bidding and Channel Options

Few papers have explored the optimal reserve prices in the name‐your‐own‐price (NYOP) channel with bidding options in a multiple channel environment. In this paper, we investigate a double‐bid business model in which the consumers can bid twice in the NYOP channel, and compare it with the single‐bid case. We also study the impact of adding a retailer‐own list‐price channel on the optimal reserve prices. This paper focuses on achieving some basic understanding on the potential gain of adding a second bid option to a single‐bid system and on the potential benefits of adding a list‐price channel by the NYOP retailer. We show that a double‐bid scenario can outperform a single‐bid scenario in both single‐channel and dual‐channel situations. The optimal reserve price in the double‐bid scenario is no less than that in the single‐bid case. Furthermore, the addition of a retailer‐own list‐price channel could push up the reserve prices in both single‐bid and double‐bid scenarios.     

Inventory‐Based Dynamic Pricing with Costly Price Adjustment

We study an average‐cost stochastic inventory control problem in which the firm can replenish inventory and adjust the 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.     

考虑消费者细分的网购拼单机制研究

本文提供理论模型分析网购拼单机制分别给销售者和消费者带来的影响，为销售者（消费者）的抉择提供参考。研究结果显示，当拼单价格和正常价格均很低时，销售者（消费者）无动机实施（参与）拼单机制。但随着正常价格的提高，拼单机制的营利能力（消费者剩余）会不断增大，进而激励销售者（消费者）实施（参与）拼单机制，并有会将低估价类型消费者驱逐出市场。这一结果从另一个方面反映了拼单机制的最终目标不是为了迎合低估价类型的消费者，而是为了吸引高估价类型的消费者参与交易。进一步地，在拼单机制有效实施的情形下，最优拼单数量、最优期望收益均与两种类型消费者估值比例密切相关。所以，在实施拼单机制过程中，应该有意识地对消费者进行细分，并挖掘不同类型消费者的私人信息，从而制定有效的拼单策略。     

Pre‐Orders for New To‐be‐Released Products Considering Consumer Loss Aversion

Advance selling through pre‐orders is a strategy to transfer inventory risk from a retailer to consumers. A newsvendor retailer can have three strategies to choose from: no advance selling allowed (NAS), moderate advance selling with a moderate discount for pre‐orders (MAS), and deep advance selling with a deep discount for pre‐orders (DAS). This research studies how a retailer could design an advance selling strategy to maximize her own profits. We find some interesting results. For example, there exist two thresholds for the selling season profit margin and two thresholds for consumer's expected valuation. For products with higher profit margin than the high threshold on profit margin, a retailer should always use DAS. For products with medium profit margin within the two thresholds, a retailer should adopt MAS if consumer's expected valuation is lower than the high threshold and use DAS otherwise. For products with lower profit margin than the low threshold, a retailer should use NAS, DAS, or MAS, respectively, if consumer's expected valuation is lower than the low threshold, higher than the high threshold, or between the two thresholds, respectively. Through sensitivity analyses, we also show the effects of multiple consumer characteristics on a retailer's optimal advance selling strategy.     

可加效用下基于消费者惰性的易逝品动态定价

本文在考虑销售商的风险规避态度和消费者惰性的情况下，研究了易逝品的动态定价策略。首先给出风险规避的销售商实施动态定价的马尔可夫决策过程，并运用MNL随机效用模型来刻画惰性消费者的购买决策，在此基础上，以风险规避销售商的期望效用最大化为目标，利用动态规划方法建立了可加效用下的易逝品动态定价模型，并探讨了消费者的惰性行为和销售商的风险规避态度对最优价格的影响。结论表明：在销售商的风险态度为风险规避的情况下，考虑消费者惰性行为的最优价格随库存水平的增加而降低，随剩余销售时间的增加而提高，消费者的惰性行为和销售商的风险规避态度对最优价格都存在负向影响，即最优价格随惰性深度的增加而降低，随惰性宽度的增加而降低，同时，销售商的风险规避态度越强烈，其制定的产品价格越低。     

考虑残值变化和顾客异质性的零售商定价与库存决策

本文研究策略型消费者对零售商的定价和库存决策的影响。与以往的研究假设残值固定不变,且消费者均为策略型消费者不同的是,本文考虑了顾客的异质性以及残值由清仓期库存决定这一特点。本文的研究表明:在消费者均为策略型消费者,且清楚零售商将采取残值定价策略的情况下,零售商可以通过数量保证策略来提高自身的期望利润,降低库存订货量。在考虑顾客异质性的条件下,当零售商采取估值定价策略时,零售商的最大期望利润与短视型消费者在市场上的比例成正相关。并且,存在一个唯一的阈值点,当短视型消费者在市场上的比例高于这一阈值点时,估值定价策略优于保留价格定价策略,反之,则保留价格定价策略更优。     

The Impact of Capacity Costs on Product Differentiation in Delivery Time,Delivery Reliability,and Price

We develop an analytical framework for studying the role capacity costs play in shaping the optimal differentiation strategy in terms of prices, delivery times, and delivery reliabilities of a profit‐maximizing firm selling two variants (express and regular) of a product in a capacitated environment. We first investigate three special cases. The first is an existing model of price and delivery time differentiation with exogenous reliabilities, which we only review. The second focuses on time‐based (i.e., length and reliability) differentiation with exogenous prices. The third deals with deciding on all features for an express variant when a regular product already exists in the marketplace. We subsequently address the integrative framework of time‐and‐price‐based differentiation for both products in a numerical study. Our results shed light on the role that customer preferences towards delivery times, reliabilities and prices, and the capacity costs (absolute and relative) have on the firm's optimal product positioning policy.     

Pricing and Replenishment of Competing Perishable Product Variants under Dynamic Demand Substitution

I consider pricing and ordering decisions faced by a retailer selling a perishable product with a two‐period shelf life over an infinite horizon. In the first period, the product is “new”; in the next, it becomes “old.” The new product is perceived by customers to have a higher quality than the old product. Every period, the retailer makes three decisions: prices for the new and old products and how much new product to order. I first show, with some simple cases, that demand uncertainty can make the sale of the old product profitable. I then consider a more realistic case with dynamic demand substitution among customers. I recognize that the retailer's decisions may be constant or may vary across different periods, under different contexts. For instance, varying the price of the new product can sometimes be difficult due to the negative impact it generates among customers. I find that (i) the benefit obtained from selling the old product with constant decisions is much higher than the benefit from allowing all the decisions to vary; (ii) the former benefit increases with a higher procurement cost, a higher quality of the new product, and higher demand volatility; however, the latter benefit is non‐monotone in these parameters; (iii) most of the latter benefit can be obtained by just changing the order quantity; and (iv) as the inventory of the old product increases, when all the decisions vary, the optimal price of the new product may increase or decrease.     

A Dynamic Disaggregation Approach to Approximate Linear Programs for Network Revenue Management

The linear programming approach to approximate dynamic programming has received considerable attention in the recent network revenue management (RM) literature. A major challenge of the approach lies in solving the resulting approximate linear programs (ALPs), which often have a huge number of constraints and/or variables. Starting from a recently developed compact affine ALP for network RM, we develop a novel dynamic disaggregation algorithm to solve the problem, which combines column and constraint generation and exploits the structure of the underlying problem. We show that the formulation can be further tightened by considering structural properties satisfied by an optimal solution. We prove that the sum of dynamic bid‐prices across resources is concave over time. We also give a counterexample to demonstrate that the dynamic bid‐prices of individual resources are not concave in general. Numerical experiments demonstrate that dynamic disaggregation is often orders of magnitude faster than existing algorithms in the literature for problem instances with and without choice. In addition, adding the concavity constraints can further speed up the algorithm, often by an order of magnitude, for problem instances with choice.