A revenue management model for products with two capacity dimensions

Many perishable products and services have multiple capacity attributes. Shipping capacity of container liners, for example, is measured by both volume and weight. Containers with different size consume various capacities in the two dimensions. Restaurant revenue management aims to maximize the revenue per available seat-hour that captures both the number of dining tables and service manpower. Similar issues arise in the air cargo, trucking and health care industries.     

Dynamic pricing and warranty policies for products with fixed lifetime

We consider a repairable product with known market entry and departure times. A warranty policy is offered with product purchase, under which a customer can have a failed item repaired free of charge in the warranty period. It is assumed that customers are heterogeneous in their risk attitudes toward uncertain repair costs incurred after the warranty expires. The objective is to determine a joint dynamic pricing and warranty policy for the lifetime of the product, which maximizes the manufacturer’s expected profit. In the first part of the analysis, we consider a linearly decreasing price function and a constant warranty length. We first study customers’ purchase patterns under several different pricing strategies by the manufacturer and then discuss the optimal pricing and warranty strategy. In the second part, we assume that the warranty length can be altered once during the product lifetime in developing a joint pricing and warranty policy. Numerical studies show that a dynamic warranty policy can significantly outperform a fixed-length warranty policy.     

Dynamic pricing with uncertain production cost: An alternating-move approach

This article studies a two-firm dynamic pricing model with random production costs. The firms produce the same perishable products over an infinite time horizon when production (or operation) costs are random. In each period, each firm determines its price and production levels based on its current production cost and its opponent’s previous price level. We use an alternating-move game to model this problem and show that there exists a unique subgame perfect Nash equilibrium in production and pricing decisions. We provide a closed-form solution for the firm’s pricing policy. Finally, we study the game in the case of incomplete information, when both or one of the firms do not have access to the current prices charged by their opponents.     

Dynamic pricing,product and process innovation

The question of simultaneous dynamic pricing, product and process investment policies is crucial for manufacturing and high-tech industries. This paper models these policies in an optimal control setting. On the supply side, the firm sets prices, product and process investment levels over time. On the demand side, current demand depends on price and quality. Under an additive separable demand function, dynamic pricing increases with quality and cost. Therefore, both product innovation and process innovation impact the pricing policy. Under a multiplicative separable demand function, dynamic pricing policy follows the dynamic of production cost and is independent of the evolution of product quality. Thus, process innovation is the main determinant of a firm’s pricing policy over time and product innovation has no impact.     

Dynamic pricing model and algorithm for perishable products with fuzzy demand

This paper studies the dynamic pricing problem of selling fixed stock of perishable items over a finite horizon, where the decision maker does not have the necessary historic data to estimate the distribution of uncertain demand, but has imprecise information about the quantity demand. We model this uncertainty using fuzzy variables. The dynamic pricing problem based on credibility theory is formulated using three fuzzy programming models, viz.: the fuzzy expected revenue maximization model, α‐optimistic revenue maximization model, and credibility maximization model. Fuzzy simulations for functions with fuzzy parameters are given and embedded into a genetic algorithm to design a hybrid intelligent algorithm to solve these three models. Finally, a real‐world example is presented to highlight the effectiveness of the developed model and algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Revenue management for a supply chain with two streams of customers

In this paper, we consider revenue management for a service supply chain with one supplier and one retailer. The supplier has a limited capacity of a perishable product and both the supplier and the retailer face customers. Each customer may choose to buy a product from either the supplier or the retailer by considering prices and the cost associated with switching. For the centralized model, the supplier determines the selling prices for both herself and the retailer, and the retailer simply collects a commission fee for each product sold. We derive monotone properties for the revenue functions and pricing strategies. Further, we show that the commission fee increases the retailer’s price while decreasing the supplier’s and leads to efficiency loss of the chain. For the decentralized decision-making model, the supplier and the retailer compete in price over time. Two models are considered. In the first, the retailer buys products from the supplier before the selling season and in the second the retailer shares products with the supplier in retailing. For both models, we discuss the existence of the equilibrium and characterize the optimal decisions. Numerical results are presented to illustrate properties of the models and to compare the supply chain performance between the centralized and the decentralized models.     

Optimal dynamic pricing and inventory control with stock deterioration and partial backordering

This paper studies the optimal dynamic pricing and inventory control policies in a periodic-review inventory system with fixed ordering cost and additive demand. The inventory may deteriorate over time and the unmet demand may be partially backlogged. We identify two sufficient conditions under which (s,S,p) policies are optimal.     

Style goods pricing with demand learning

For many industries (e.g., apparel retailing) managing demand through price adjustments is often the only tool left to companies once the replenishment decisions are made. A significant amount of uncertainty about the magnitude and price sensitivity of demand can be resolved using the early sales information. In this study, a Bayesian model is developed to summarize sales information and pricing history in an efficient way. This model is incorporated into a periodic pricing model to optimize revenues for a given stock of items over a finite horizon. A computational study is carried out in order to find out the circumstances under which learning is most beneficial. The model is extended to allow for replenishments within the season, in order to understand global sourcing decisions made by apparel retailers. Some of the findings are empirically validated using data from U.S. apparel industry.     

A continuous-time dynamic pricing model knowing the competitor’s pricing strategy

In this paper we consider a dynamic pricing model for a firm knowing that a competitor adopts a static pricing strategy. We establish a continuous time model to analyze the effect of dynamic pricing on the improvement in expected revenue in the duopoly. We assume that customers arrive to purchase tickets in accordance with a geometric Brownian motion. We derive an explicit closed-form expression for an optimal pricing policy to maximize the expected revenue. It is shown that when the competitor adopts a static pricing policy, dynamic pricing is not always effective in terms of maximizing expected revenue compared to a fixed pricing strategy. Moreover, we show that the size of the reduction in the expected revenue depends on the competitor’s pricing strategy. Numerical results are presented to illustrate the dynamic pricing policy.     

Ruin probability in the dual risk model with two revenue streams

The dual risk model describes the surplus of a company with fixed expense rate and occasional random income inflows, called gains. Consider the dual risk model with two streams of gains. Type I gains arrive according to a Poisson process, and type II gains arrive according to a general renewal process. We show that the survival probability of the company can be expressed in terms of the survival probability in a dual risk process with renewal arrivals with initial reserve 0, and the survival probability in the dual risk process with Poisson arrivals in finite time.     

Strategic and operational decisions in restaurant revenue management

The paper addresses restaurant revenue management from both a strategic and an operational point of view. Strategic decisions in restaurants are mainly related to defining the most profitable combination of tables that will constitute the restaurant. We propose new formulations of the so-called “Tables Mix Problem” by taking into account several features of the real setting. We compare the proposed models in a computational study showing that restaurants, with the capacity of managing tables as renewable resources and of combining different-sized tables, can improve expected revenue performances. Operational decisions are mainly concerned with the more profitable assignment of tables to customers. Indeed, the “Parties Mix Problem” consists of deciding on accepting or denying a booking request from different groups of customers, with the aim of maximizing the total expected revenue. A dynamic formulation of the “Parties Mix Problem” is presented together with a linear programming approximation, whose solutions can be used to define capacity control policies based on booking limits and bid prices. Computational results compare the proposed policies and show that they lead to higher revenues than the traditional strategies used to support decision makers.     

考虑盗版容忍度的软件产品动态定价模型

基于收益管理理论,对产品瞬态扩散模型进行拓展,选择盗版容忍度、价格和创新改进作为控制变量,扩散速度和创新水平作为状态变量,构建收益最大化的动态定价模型,依据最优化控制理论,建立哈密尔顿方程进行求解,探讨了三个控制变量的联合优化策略和相关特性,通过算例仿真验证其结论。结论显示:应采取撇脂定价策略;当由盗版引起边际收益增加的价值量与损失价值相等时,盗版容忍度达到最优;与产品生命周期的成长阶段相比较,在成熟阶段,应逐渐增加盗版容忍度,同时,减缓创新改进的速度;当损失结构降低时,应提高盗版容忍度,并且可采用创新与保护监管相互互补的策略。动态建模能够拓展静态模型中关于网络外部性的概念,分析连续状态下的变量,与自适应学习机制相一致,对实践具有更好的借鉴作用。     

Dynamic cruise ship revenue management

Recently, it has been recognized that revenue management of cruise ships is different from that of airlines or hotels. Among the main differences is the presence of multiple capacity constraints in cruise ships, i.e., the number of cabins in different categories and the number of lifeboat seats, versus a single constraint in airlines and hotels (i.e., number of seats or rooms). We develop a discrete-time dynamic capacity control model for a cruise ship characterized by multiple constraints on cabin and lifeboat capacities. Customers (families) arrive sequentially according to a stochastic process and request one cabin of a certain category and one or more lifeboat seats. The cruise ship revenue manager decides which requests to accept based on the remaining cabin and lifeboat capacities at the time of an arrival as well as the type of the arrival. We show that the opportunity cost of accepting a customer is not always monotone in the reservation levels or time. This non-monotone behavior implies that “conventional” booking limits or critical time periods capacity control policies are not optimal. We provide analysis and insights justifying the non-monotone behavior in our cruise ship context. In the absence of monotonicity, and with the optimal solution requiring heavy storage for “large” (industry-size) problems, we develop several heuristics and thoroughly test their performance, via simulation, against the optimal solution, well-crafted upper bounds, and a first-come first-served lower bound. Our heuristics are based on rolling-up the multi-dimensional state space into one or two dimensions and solving the resulting dynamic program (DP). This is a strength of our approach since our DP-based heuristics are easy to understand, solve and analyze. We find that single-dimensional heuristics based on decoupling the cabins and lifeboat problems perform quite well in most cases.     

竞争环境下风险规避的航班动态定价

以同一航线上的多个竞争航班为研究对象,在假设各竞争航班之间具有完全信息的基础上,利用马尔可夫决策过程和博弈论,建立了竞争环境下风险规避的航班动态定价的数学模型,证明了均衡价格的存在性。在此基础上,进一步讨论了信息不完全情况下风险规避的竞争航班的动态定价问题。数值实验表明:在竞争环境下,各风险规避航班的均衡价格随自身剩余座位数量和风险规避系数的增加而下降,随其他竞争航班的剩余座位数量和风险规避系数的增加而提高。     

Periodic pricing and replenishment policy for continuously decaying inventory with multivariate demand

This paper deals with the joint decisions on pricing and replenishment schedule for a periodic review inventory system in which a replenishment order may be placed at the beginning of some or all of the periods. We consider a single product which is subject to continuous decay and a demand which is a function of price and time, without backlogging over a finite planning horizon. The proposed scheme may adjust periodically the selling price upward or downward that makes the pricing policy more responsive to structure changes in supply or demand. The problem is formulated as a dynamic programming model and solved by numerical search techniques. An extensive numerical study is conducted to attend qualitative insights into the structures of the proposed policy and its sensitivity with respect to major parameters. The numerical result shows that the solution generated by the periodic policy outperforms that by the fixed pricing policy in maximizing discount profit.     

Differential game approach to pricing and advertising decisions

This study proposes a model to make concurrent decisions on dynamic pricing and advertising to maximise firms' profitability over an infinite time horizon in a duopoly market. To this end, the Nerlove-Arrow pricing and advertising model is designed in the presence of shifting costs in a dynamic duopolistic competition as a differential game. The Nash equilibrium solution is defined based upon a set of Hamilton–Jacobi–Bellman. Four scenarios are applied for economic interpretations and the efficacy of the model.     

Optimal ordering and pricing policy for an inventory system with trial periods

It is a business practice that home shopping companies offer a free trial period for their products with a goal of increasing sales. Under this policy, if for any reason customers are not satisfied with the purchase, they can return the product for a refund within the trial period. To develop inventory strategies in such environment, home shopping companies should take the return phenomenon into account so as to increase their profit. This paper considers this phenomenon and develops a seasonal inventory model to deal with the problem. Two scenarios are analyzed. In the first scenario, demand is assumed to be linearly price-dependent while in the second one, it is assumed to be exponentially price-dependent. The purpose of this research is to maximize the total profit over a given planning period by determining the optimal ordering quantity and price. The analytical results demonstrate that the optimal ordering quantity and prices are obtained using closed-form formulas.     

Risk-sensitive dynamic pricing for a single perishable product

We show that the monotone structures of dynamic pricing for a single perishable product under risk-neutrality are preserved under risk-sensitivity with the additive general utility and atemporal exponential utility functions. We also show that the optimal price is decreasing over the degree of risk-sensitivity under the exponential class of both additive and atemporal utility functions.     

考虑消费者从众特性的动态定价研究

面对掌握大量信息的消费者,厂商需考虑多方因素制定价格策略。研究运用经典的博弈模型,讨论不同消费者特性情况下厂商的定价机制。模型按从众特性和等待特性将消费者分为六类,并引入一个价格偏差变量,分析各类型消费者比例变化对厂商定价和利润的影响。结果显示,一般情况下降价幅度、消费者期望购买总数量和期望销售利润都随着反从众消费者比例的增大而增加。最后扩展分析了效益贴现率,反从众消费者比例和短视型消费者比例三因素对厂商总利润的影响,研究认为效益贴现率和从众消费者比例较高时,厂商利润会随着短视型消费者比例的降低而有所提高。因此,为使收益最大化,厂商在制定定价策略时需同时考虑贴现因素和消费者行为特性。     

Priority option pricing in an M/M/m queue

We study a system where the service provider offers priority options. We identify the optimal option pricing policy, by deriving the optimal number a customer would buy and the customer’s exercise policy as a function of system congestion, options remaining, time to expiration and possibility of balking.