Integration of pricing and capacity allocation for perishable products

For years pricing and capacity allocation decisions in most revenue management models have been carried out independently. This article presents a comprehensive model to integrate these two decisions for perishable products. We assume that the supplier sells the same products to different micro-markets at distinct prices. Throughout the sales season, the supplier faces decisions as to which micro-markets or customer classes should be served and at what prices. We show that (i) at any time, a customer class is active (being served) if and only if the price offered is over a threshold level, but the optimal price may not be the highest one of the supplier’s choice; (ii) when the price decision is made in conjunction with inventory, it is similar to the procedure shown in pure pricing models, i.e., the optimal price comes from a subset of prices that forms a maximum increasing concave envelope; (iii) because of dynamic changes in the optimal prices, the nested-price structure does not necessarily hold in general and needs to be redefined; and (iv) the optimal pricing and capacity control policy is based on a sequence of threshold points that incorporate inventory, price and demand intensity. Numerical examples are provided.     

Production planning and pricing policy in a make-to-stock system with uncertain demand subject to machine breakdowns

We consider a make-to-stock system served by an unreliable machine that produces one type of product, which is sold to customers at one of two possible prices depending on the inventory level at the time when a customer arrives (i.e., the decision point). The system manager must determine the production level and selling price at each decision point. We first show that the optimal production and pricing policy is a threshold control, which is characterized by three threshold parameters under both the long-run discounted profit and long-run average profit criteria. We then establish the structural relationships among the three threshold parameters that production is off when inventory is above the threshold, and that the optimal selling price should be low when inventory is above the threshold under the scenario where the machine is down or up. Finally we provide some numerical examples to illustrate the analytical results and gain additional insights.     

Optimal ordering policies in inventory systems with random demand and random deal offerings

In this paper, we determine the optimal order policies for a firm facing random demand and random deal offerings. In a periodic review setting, a firm may first place an order at the regular price. Later in the period, if a price promotion is offered by the supplier (with a certain probability), the firm may decide to place another order. We consider two models in the paper. In the first model, the firm does not share the cost savings (due to the promotion offered by the supplier) with its own customers, i.e. its demand distribution remains fixed. In the second model, the cost savings are shared with the final customers. As a result, the demand distribution shifts to the right. For both the models, in a dynamic finite-horizon problem, the order policy structure is divided into three regions and is as follows. If the initial inventory level for the firm exceeds a certain threshold level, it is optimal not to order anything. If it is in the medium range, it is optimal to wait for the promotion and order only if it is offered. The order quantity when the promotion is offered has an ‘order up to’ policy structure. Finally, if the inventory level is below another threshold, it is optimal to place an order at the regular price, and to place a second order if the promotion is offered. The low initial inventory level makes it risky to just wait for the promotion to be offered. The sum of the order quantities in this case has an ‘order up to’ structure. Finally, we model the supplier's problem as a Stackelberg game and discuss the motivation for the supplier to offer a promotion for the case of uniform demand distribution for the firm. In the first model (when the firm does not share the cost savings with its customers), we show that it is rarely optimal for the supplier to offer a promotion. In the second model, the supplier may offer a promotion depending on the price elasticity of the product.     

Dynamic pricing of airline tickets with competition

Competition has a huge influence on customer buying behaviour and will impact on the optimal price that companies should charge for goods or services. To date, many dynamic pricing models have not modelled competition explicitly. In this paper, we introduce pricing strategies that maximize revenue when selling an inventory of identical items by a fixed time and where there is a competing seller. The model used incorporates a probabilistic formulation of customer demand, which is influenced by the prices offered by the company and the competitor, and the time remaining until the end of the selling period. Calculus of variations is used to solve the problem and simple conditions are given that ensure the uniqueness of a solution. Illustrative examples are included. A practical implementation that uses dynamic updating is proposed and tested using simulated data, showing the effectiveness of the method.     

服务水平约束下动态定价与库存管理

本文研究服务水平约束下的动态定价与库存管理问题。企业在有限期内销售某种产品,产品的需求为随机需求,且期望需求依赖于产品价格。在每一期期初,企业需要在满足服务水平约束的条件下同时决定订货量和产品价格。本文首先构建了动态定价和订购联合决策的随机动态规划模型,并证明了最优解的存在性。进一步,通过对最优解的结构进行刻画,将原问题的求解转化为若干子问题的求解,降低了问题求解的难度。通过对最优解的分析发现,当期初库存增大时,产品最优价格降低。通过分析目标服务水平对利润的影响,证明了服务水平与利润之间存在权衡,实现高的服务水平需要承受利润损失。数值模拟表明,相对于传统的静态定价策略,采用动态定价策略可以降低追求服务水平所带来的利润损失,验证了动态定价策略的有效性。     

Revenue management on an on-demand service platform

I consider a worker’s optimal per-unit-time pricing problem on an on-demand service platform. I establish that if the customer arrival process is Poisson, then any price discrimination must result only from differences in the customers’ willingness-to-pay distributions, and not from differences in their arrival or service-time characteristics. For multiple customer classes that differ in their willingness-to-pay distributions, I present a simple procedure to compute the optimal prices. Finally, I analyze price competition across workers in the presence of reputation effects.     

Dynamic pricing of inventory/capacity with infrequent price changes

We consider a problem of dynamically pricing a single product sold by a monopolist over a short time period. If demand characteristics change throughout the period, it becomes attractive for the company to adjust price continuously to respond to such changes (i.e., price-discriminate intertemporally). However, in practice there is typically a limit on the number of times the price can be adjusted due to the high costs associated with frequent price changes. If that is the case, instead of a continuous pricing rule the company might want to establish a piece-wise constant pricing policy in order to limit the number of price adjustments. Such a pricing policy, which involves optimal choice of prices and timing of price changes, is the focus of this paper.We analyze the pricing problem with a limited number of price changes in a dynamic, deterministic environment in which demand depends on the current price and time, and there is a capacity/inventory constraint that may be set optimally ahead of the selling season. The arrival rate can evolve in time arbitrarily, allowing us to model situations in which prices decrease, increase, or neither. We consider several plausible scenarios where pricing and/or timing of price changes are endogenized. Various notions of complementarity (single-crossing property, supermodularity and total positivity) are explored to derive structural results: conditions sufficient for the uniqueness of the solution and the monotonicity of prices throughout the sales period. Furthermore, we characterize the impact of the capacity constraint on the optimal prices and the timing of price changes and provide several other comparative statics results. Additional insights are obtained directly from the solutions of various special cases.     

不同价格模式下考虑需求波动性和差异化服务的网约车平台定价决策

针对单个平台两种品牌网约车的最优定价问题,考虑平台服务质量的差异化和市场需求波动性,分别建立动态价格、差异化价格和静态价格模式下的网约车动态服务模型,运用多元函数和泛函的条件极值求得两种品牌网约车的最优定价策略。研究发现,平台最优动态价格和差异化价格均随需求波动时长单调变化,而最优静态价格并非单调。此外,平台提高差异化服务时,两种品牌网约车的最优价格均提高,但高服务质量的网约车会有更高的提价幅度;固定佣金报酬率增大时,平台最优价格均提高,但边际损失成本较大的网约车会有更高的提价幅度。最后,通过数值仿真对不同价格模式下的平台利润进行比较和灵敏度分析,并发现平台利润在市场需求稳定时差异不大。     

Integrating inventory control and a price change in the presence of reference price effects: a two-period model

Demand and procurement planning for consumer electronics products must cope with short life cycles, limited replenishment opportunities and a willingness to pay that is influenced by past prices and decreases over time. We therefore propose the use of an integrated pricing and inventory control model with a two-period linear demand model, in which demand also depends on the difference between a price-history-based reference price and the current price. For this model we prove that the optimal joint pricing/inventory policy for the replenishment opportunity after the first period is a base-stock list-price policy. That is, stock is either replenished up to a base-stock level and a list-price is charged, or it is not replenished and a discount is given that increases with the stock-level. Furthermore, we use real-world cell phone data to study the differences between an integrated policy and traditional sequential optimization, where prices are initially optimized based on the expected demand and ordering cost, and the resulting demand distribution is used to determine an optimal inventory policy. Finally, we discuss possible extensions of the model.     

Revenue management model for on-demand IT services

This paper presents a model for applying revenue management to on-demand IT services. The multinomial logit model is used to describe customer choice over multiple classes with different service-level agreements (SLAs). A nonlinear programming model is provided to determine the optimal price or service level for each class. Through a numerical analysis, we examine the impacts of system capacity and customer waiting incentives on the service provider’s profit and pricing strategies.     

Optimal pricing strategies under co-existence of price-takers and bargainers in a supply chain

We investigate how the co-existence of two types of customers, price-takers, and bargainers, influences the pricing decisions in a supply chain. We consider a stylized supply chain that includes one manufacturer and one retailer, and we characterize the optimal prices of the retailer and the manufacturer. We further discuss the effects of the fraction of the bargainers in the customer population and the relative bargaining power of the bargainers on these optimal prices. Our results show that, given the wholesale price, the lowest price at which the retailer is willing to sell (ie, cut-off price) increases with the relative bargaining power of the bargainers. Both posted and cut-off prices increase in the fraction of the bargainers in the customer population. Moreover, depending on the type of negotiation cost, the variations of both prices will vary. In equilibrium, both posted and cut-off prices do not monotonically increase with the fraction of the bargainers in the customer population. When the maximum reservation price of the customers is low, and/or the negotiation costs are high, and/or the relationship between the bargainer's negotiation cost and reservation price is high, the retailer may reduce both posted and cut-off prices as the fraction of the bargainers increases.     

基于战略顾客行为的最优定价与容量选择模型

研究了当市场中同时存在战略顾客和短视顾客时零售商的最优定价与容量选择问题。零售商在正常销售阶段和出清销售阶段制定不同的销售价格,同时通过容量选择影响战略顾客的购买行为,而战略顾客则根据零售商的定价和容量选择确定最优购买时机。分别分析了零售商在无限容量时的定价决策、固定价格时的容量选择、固定容量时的定价决策以及有限容量下的定价与容量选择四种情形。研究结果表明,零售商在无容量限制时的最优定价决策是制定两阶段定价策略,在固定价格时的最优容量选择依赖于模型的参数,而当零售商的容量固定时,部分满足出清销售阶段的顾客需求始终优于完全满足出清销售阶段的顾客需求。     

Dynamic pricing with real-time demand learning

In many service industries, the firm adjusts the product price dynamically by taking into account the current product inventory and the future demand distribution. Because the firm can easily monitor the product inventory, the success of dynamic pricing relies on an accurate demand forecast. In this paper, we consider a situation where the firm does not have an accurate demand forecast, but can only roughly estimate the customer arrival rate before the sale begins. As the sale moves forward, the firm uses real-time sales data to fine-tune this arrival rate estimation. We show how the firm can first use this modified arrival rate estimation to forecast the future demand distribution with better precision, and then use the new information to dynamically adjust the product price in order to maximize the expected total revenue. Numerical study shows that this strategy not only is nearly optimal, but also is robust when the true customer arrival rate is much different from the original forecast. Finally, we extend the results to four situations commonly encountered in practice: unobservable lost customers, time dependent arrival rate, batch demand, and discrete set of allowable prices.     

Optimal policies for inventory systems with two types of product sharing common hardware platforms: Single period and finite horizon

In this paper, we consider an inventory system whose products share a common hardware platform but are differentiated by two types of software. Choice of different software results in different installation cost and different selling price of the whole product. Product with different software also faces different customer demand. We investigate the optimal proportion of an order to be installed with software 1 or 2, that maximizes expected profit in the single and multiple period scenarios, respectively. The optimal policy is analytically obtained and proved to be an order-up-to policy in each scenario. Our investigation reveals that whether to replenish, and how much to replenish each product depend not only on its own initial inventory level, and system parameters, but also the initial inventory level of the other product. We perform numerical experiments using the optimal policies we have derived in the paper.     

The effect of market segmentation with demand leakage between market segments on a firm’s price and inventory decisions

In this paper, we address the simultaneous determination of price and inventory replenishment in a newsvendor setting when the firm faces demand from two or more market segments in which the firm can set different prices. We allow for demand leakage from higher-priced segments to lower-priced segments and assume that unsatisfied demand can be backlogged. We examine the case where the demands occur concurrently without priority and are met from a single inventory. We consider customer’s buy-down behavior explicitly by modeling demand leakage as a function of segment price differentiation, and characterize the structure of optimal inventory and pricing policies.     

Dynamic channel control and pricing of a single perishable product on multiple distribution channels

This paper studies dynamic channel control and pricing of a single perishable product distributed through multiple channels with the objective of maximizing the total expected profit over a finite horizon. We consider two types of commissions, namely proportional and fixed commissions, on the third-party channels and utilize stylized linear functions to characterize dependent demand flows from different channels. We show that, the magnitude of the opportunity cost of capacity uniquely determines the optimal channel control, at any given inventory level and periods to go. Consequently, we are able to derive the optimal price offered on each channel as a function of the opportunity cost of capacity in closed form. This significantly reduces the computational complexity of the stochastic dynamic program when parameters are constant with time. When channels are independent, we provide a necessary and sufficient condition for the optimality of a nested channel control policy by commission rates. The same condition is also sufficient for the optimality of the nested channel control policy in a distribution system with two dependent channels. We then characterize the structural properties of the optimal pricing and channel control policies. Finally, we explore the impact of the substitution effect on the channel control through numerical studies and gain managerial insights.     

Integrated inventory model with quantity discount and price-sensitive demand

Quantity discount has been a subject of study for a long time; however, little is known about its effect on integrated inventory models when price-sensitive demand is placed. The objective of this study is to find the optimal pricing and ordering strategies for an integrated inventory system when a quantity discount policy is applied. The pricing strategy discussed here is one in which the vendor offers a quantity discount to the buyer. Then, the buyer will adjust his retail price based on the purchasing cost, which will influence the customer demand as a result. Consequently, an integrated inventory model is established to find the optimal solutions for order quantity, retail price, and the number of shipments from vendor to buyer in one production run, so that the joint total profit incurred has the maximum value. Also, numerical examples and a sensitivity analysis are given to illustrate the results of the model.     

Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit

This study models a finite horizon inventory problem for deteriorating and fashion goods under trade credit and partial backlogging conditions. Demand may vary with price or time. The supplier can extend credit to the retailer. As a result, the retailer does not have to pay for goods immediately upon acquiring them, and can instead earn interest on the retail price of the goods between the time they are sold and the end of the credit period. The proposed model considers two-phase pricing and inventory decisions. In other words, it determines both the optimal prices and the lengths of the in-stock and stock-out period. This paper is the first to consider different price decisions for in-stock and stock-out periods under trade credit. We develop an algorithm to determine the optimal pricing and replenishment strategy while still maximizing the total profit. Further, this study shows that the proposed two-phase pricing strategy is superior to a one-phase pricing strategy in terms of profit maximization. Computational analysis illustrates the solution procedures and the impacts of the related parameters on decisions and profits. The results of this study can serve as references for business managers or administrators.     

考虑横向整合战略的库存路径动态区域定价

库存路径和定价是供货商管理库存(Vendor Management Inventory, VMI)中三个互相制约和影响的决策问题,是降低供货商成本,提高其利润的关键。针对VMI拉式供应链中多供货商、多商品和多区域的库存路径定价问题,提出了对不同区域客户、在不同时段进行商品差异化定价策略,并设计一种共同配送车辆司机成本和燃油成本分摊方案,据此构建基于横向整合战略的库存路径动态区域定价模型。算例结果显示,在横向整合战略下,供货商商品定价会有所降低,配送车辆行驶距离显著缩短,各时段配货量更为均衡,期末库存数量显著降低。研究表明,无论供货商之间供货规模比例差异多大,开展库存路径动态区域定价,供货商联盟成员的利润均能得到显著提高,实现合作共赢的目标。     

Optimal admission and pricing control problems in service industries with multiple servers and sideline profit

This paper deals with the problem of selecting profitable customer orders sequentially arriving at a company operating in service industries with multiple servers in which two classes of services are provided. The first class of service is designed to meet the particular needs of customers; and the company (1) makes a decision on whether to accept or to reject the order for this service (admission control) and (2) decides a price of the order and offers it to an arriving customer (pricing control). The second class of service is provided as a sideline, which prevents servers from being idle when the number of customer orders for the first class is less than the number of servers. This yields the sideline profit. A cost is paid to search for customer orders, which is called the search cost. In the context of search cost, the company has an option whether to conduct the search or not. In this paper, we discuss both admission control and pricing control problems within an identical framework as well as examine the structure of the optimal policies to maximize the total expected net profit gained over an infinite planning horizon. We show that when the sideline profit is large, the optimal policies may not be monotone in the number of customer orders in the system. Copyright © 2008 John Wiley & Sons, Ltd.