Bundle pricing of inventories with stochastic demand

We consider a retailer selling a fixed inventory of two perishable products over a finite horizon. Assuming Poisson arrivals and a bivariate reservation price distribution, we determine the optimal product and bundle prices that maximize the expected revenue. Our results indicate that the performances of mixed bundling, pure bundling and unbundled sales strategies heavily depend on the parameters of the demand process and the initial inventory levels. Bundling appears to be most effective with negatively correlated reservation prices and high starting inventory levels. When the starting inventory levels are equal and in excess of average demand, most of the benefits of bundling can be achieved through pure bundling. However, the mixed bundling strategy dominates the other two when the starting inventory levels are not equal. We also observe that an incorrect modeling of the reservation prices may lead to significant losses. The model is extended to allow for price changes during the selling horizon. It is shown that offering price bundles mid-season may be more effective than changing individual product prices.     

Speculative production and anticipative reservation of reactive capacity by a multi-product newsvendor

In this paper the optimal sourcing decisions of a multi-product newsvendor prior to the selling season of the products are studied. To satisfy the uncertain demands, the newsvendor can either utilize speculative production, or anticipatively reserve capacity. During the selling season when demand has become known, the newsvendor can utilize its reserved capacity and reactively satisfy demand uncovered by its speculative production. For the case where capacity for speculative production may be limited, but potential reservation of reactive capacity is unlimited two capacity reservation settings are analyzed and compared. In the first one capacity for each product has to be reserved separately, while in the second setting one joint capacity reservation for all products is permitted which can then be allocated to the different products optimally during the selling season. For the case of separate individual reservations the optimal strategies are analytically derived and structural insights concerning their existence are presented. As the model allowing for joint reservation can not be tackled analytically in general an approximation based on an LP formulation is used. Through a numerical example insights on the value of the increased flexibility induced by joint reservation, the cost-premium acceptable for joint reservation and the relative levels of capacity reservation in the two settings are given.     

Two-period pricing and decision strategies in a two-echelon supply chain under price-dependent demand

This paper presents a two-period supply chain model which is comprised of one manufacturer and one retailer who are involved in trading a single product. The demand rate in each period is dependent on the selling prices of the current period and the previous period. We assume that the manufacturer acts as the Stackelberg leader and declares wholesale price(s) to the retailer who follows the manufacturer’s decision and sets his selling prices for two consecutive periods. The manufacturer adopts one of the two pricing options: (1) setting the same wholesale price to both the selling periods (2) setting different wholesale prices to two different selling periods. Based on these pricing options, we develop four decision strategies of the manufacturer and the retailer and compare them. For a numerical example, we study the effects of these decision strategies on the optimal results of the supply chain. Further, we graphically analyze under what circumstances a particular decision strategy plays a dominant role.     

考虑保质期的非即时易腐品最优订购与定价决策

针对有保质期约束的非即时易腐品,零售商可以在产品非腐败阶段和腐败阶段进行差异化定价来调整市场需求。即零售商可以调整产品非腐败阶段的定价策略,使得非即时易腐品在进入腐败阶段之前全部售出;或者在产品腐败阶段降低零售价格以刺激市场需求。因此,在需求依赖价格的假设下,本文研究了零售商关于有保质期约束的非即时易腐品的最优订购与定价决策。通过分析零售商单位时间利润函数的理论性质,得到了零售商的最优订购与定价策略。基于线性和指数型两种需求函数形式进行数值算例,可以发现针对保质期较长或者非腐败阶段较长的易腐品,零售商会延长订货周期来增加销售收入。特别是在指数型需求函数情境下,当市场需求与零售价格高度相关时,零售商更有意愿降价来刺激销量,从而使得易腐品在非腐败阶段内销售完毕。     

Double optimal stopping times and dynamic pricing problem: description of the mathematical model

In many industries, managers face the problem of selling a given stock of items by a deadline. We investigate the problem of dynamically pricing such inventories when demand is price sensitive and stochastic and the firm’s objective is to maximize expected revenues. Examples that fit this framework include retailers selling fashion and seasonal goods and the travel and leisure industry, which markets space such as seats on airline flights, cabins on vacation cruises, hotels renting rooms before midnight and theaters selling seats before curtain time that become worthless if not sold by a specific time. Given a fixed number of seats, rooms, or coats, the objective for these industries is to maximize revenues in excess of salvage value. When demand is price sensitive and stochastic, pricing is an effective tool to maximize revenues. In this paper, we address the problem of deciding the optimal timing of a double price changes from a given initial price to given lower or higher prices. Under mild conditions, it is shown that it is optimal to decrease the initial price as soon as the time-to-go falls below a time threshold and increase the price if time-to-go is longer than adequate time threshold. These thresholds depend on the number of yet unsold items.      

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。     

The ability to compensate for suboptimal capacity decisions by optimal pricing decisions

Certain companies have high capacity cost and rather moderate production cost. These companies usually assume that deciding about their capacity is quite critical. Frequently, however, they are able to adjust the demand for their products to the available capacity by setting appropriate prices, that is higher (lower) than current prices in the presence of under-capacity (over-capacity). We argue that appropriate prices can reduce the adverse effects of non-optimal capacities. We analyze the sensitivity of profit in such a situation for a company in a monopolistic market, selling a non-storable product and facing fluctuating but interdependent demand across two time periods which allows to profitably differentiate prices. Therefore, we state optimality conditions for prices in situations of variable and given capacities and describe a procedure to determine them. The main suggestion of this analysis is that, within the bounds of the normative models and specific parameters examined, optimal prices can substantially reduce the adverse effects of capacity deviating from its optimum. In this way, profit is rather insensitive to deviations of capacity from its optimum. The implications of this finding are discussed for a number of situations.     

Dynamic switching times from season to single tickets in sports and entertainment

Revenue management can be used in many industries where there is a limited, perishable capacity and the market can be segmented. In this paper we focus on the sales of event tickets in the Sports and Entertainment industries, where tickets are sold exclusively as season tickets initially or as single events later in the selling horizon. We specifically study the optimal time to switch between these market segments dynamically as a function of the state of the system. Under Poisson demand processes, we find the optimal switching time is a set of time thresholds that depends on the remaining inventory and time left in the horizon. We use numerical experiments to show that significant profit improvements can be obtained by dynamically deciding the optimal switch time over the case when the date is announced in advance. We also study an extension where ??early switch to a low-demand event?? is allowed.     

Designing coalition-based fair and stable pricing mechanisms under private information on consumers’ reservation prices

We model the relation between an aggregator and consumers joining a coalition to reduce the risk resulting from the unpredictability of their base load demand, as a Stackelberg game formulated as a mathematical bilevel program with private information on the consumers’ reservation prices. At the upper-level of the Stackelberg game, the aggregator optimizes his daily price profile so as to reach a net targeted profit which is the maximum value guaranteeing that no consumer will leave the coalition - to contract with a conventional retailer considered here as a fixed alternative - while meeting fairness criterion imposed by the cost-sharing mechanism. At the lower-level, the consumers are asked to provide in day ahead an estimate of their base load hourly demand profile and to schedule their shiftable loads depending on the price signal sent by the aggregator. We provide algorithms that determine the unique price profile and consumer shiftable load schedules as functions of the reservation price estimates. The Stackelberg game between the aggregator and the consumers being repeated for a period of time, the aggregator has the possibility to update his estimates of the reservation prices relying on a feedback function which depends on the percentage of activated loads. A randomized algorithm for consumers’ reservation price learning based on regret minimization is provided. For four cost-sharing mechanisms such as uniform allocation, stand-alone cost, Shapley value, separable and non-separable costs, we determine the closed form of the aggregator’s optimal net targeted profit guaranteeing the stability of the coalition. We also determine conditions guaranteeing the core non-emptiness and prove that for a profit-maximizing aggregator, the stand-alone cost is always preferable to the Shapley value, which coincides with the uniform allocation. Furthermore, the optimal size of the coalition - in terms of the aggregator’s profit - can be determined analytically when the Shapley value is implemented as cost-sharing mechanism. The results are illustrated on a case study where we show that there exists an optimal net targeted profit below which the consumers energy bill is lower when joining the aggregator than with the conventional retailer. Coalition dynamics is also analyzed numerically depending on the consumer inertia in their energy supplier choice process, for each cost-sharing mechanism.     

具有定价和开机成本的生产-库存系统的最优(s,d,S)策略

考虑一个具有有限容量和开机成本的连续盘点生产-库存系统, 其控制策略为(s,d,S)策略. 未被满足的需求都会丢失. 当机器处于关闭状态时,库存产品可以两个不同的价格进行销售. 当机器处于开机状态时,库存只能以较高的价格进行销售. 研究了如何发现该系统下的最优(s,d,S)策略,并开发了用于计算最优控制参数的有效算法.     

权力结构影响下考虑预售的双渠道供应链决策研究

针对存在预售且通过网络与传统渠道销售的现实状况,基于消费者剩余理论和博弈论,构建不同权力结构下的双渠道供应链博弈模型:制造商主导的Stackelberg、权力对等的Vertical Nash和零售商主导的Stackelberg。比较三种权力结构下各成员最优策略及绩效,分析关键因素的敏感性,检验模型的鲁棒性。研究发现:三种博弈下各权力主体的最优策略及绩效均受渠道替代程度、单位生产成本等关键因素影响。渠道替代程度越高,制定的双渠道价格越高;消费者对价格更敏感,预售市场需求呈现向现售市场转移的趋势。     

An EOQ model with stock and price sensitive demand

Many business practices show that the presence of a larger quantity of goods displayed may attract more customers than that with a smaller quantity of goods. This phenomenon implies that the demand may have a positive correlative with stock level. Under such a circumstance, a firm should seriously consider its pricing and ordering strategy since the demand for their goods may be affected by their selling prices and inventory level. This paper aims to develop a continuous inventory model for finding the strategy for a firm that sells a seasonal item over a finite planning time. The purpose of this firm is to maximize its expected profit by determining the optimal ordering quantity and price setting/changing strategy. Some sufficient conditions are found for finding the optimal decision rules.     

Coordination mechanism for the supply chain with leadtime consideration and price-dependent demand

We study a coordination contract for a supplier–retailer channel producing and selling a fashionable product exhibiting a stochastic price-dependent demand. The product’s selling season is short, and the supply chain faces great demand uncertainty. We consider a scenario where the supplier reserves production capacity for the retailer in advance, and permits the retailer to place an order not exceeding the reserved capacity after a demand information update during a leadtime. We formulate a two-stage optimization problem in which the supplier decides the amount of capacity reservation in the first stage, and the retailer determines the order quantity and the retail price after observing the demand information in the second stage. We propose a three-parameter risk and profit sharing contract that coordinates the supply chain. The proposed contract permits any agreed-upon division of the supply-chain profit between the channel members.     

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.     

Competing trade mechanisms and monotone mechanism choice

We investigate the choice between posted prices and auctions of competing sellers with private valuations. Assuming that buyers face higher hassle costs in auctions, we show the existence of monotone pure strategy equilibria where sellers offer posted prices rather than auctions if and only if they have a sufficiently high reservation value. Posted prices sell with lower probability but yield a larger revenue in case of trade. Using an empirical strategy to compare revenues of posted prices and auctions that takes selling probabilities explicitly into account, we find our theoretical predictions supported by data from eBay auctions on ticket sales for the EURO 2008 European Football Championship.     

Demand uncertainty and supplier's returns policies for a multi-store style-good retailer

As demand uncertainty grows in the marketplace, a critical issue today in most purchase contract negotiations between an independent retailer of a style-good and its supplier is the provision of a returns policy, i.e., a commitment by the supplier to buy back unsold inventory of the good at the end of its selling season. Management science research on the strategic role and optimal design of returns policies has grown in recent years but so far offers little treatment of how exactly the retailer's optimal order quantity decisions are affected by demand uncertainty and how a supplier's returns policy can influence these decisions. Employing the traditional “newsboy problem” modeling framework, the authors investigate these issues considering a supplier who faces a retailer with two or more store outlets with normally distributed and possibly correlated demands. To facilitate their analyses, the authors employ a methodology based on special error function representations of the highly nonlinear objective functions of the retailer and supplier. Utilizing this approach, the authors are able to provide explicit insights into how: (a) the buyer's total order quantity decision is affected by the variability in demand; (b) buyback prices in combination with wholesale prices can influence the buyer's order quantity response to demand uncertainty; (c) demand uncertainty moderates the effects of the buyback and wholesale prices; (d) supplier's optimal combination of actions are affected by demand variability; (e) retailer's and supplier's expected profits behave in response to changes in the supplier's actions under different levels of demand variability.     

The newsvendor problem with an in-season price adjustment

Consider a retailer stocking a seasonal item facing a stochastic demand where information about the demand becomes more accurate as the selling season progresses. The retailer places orders before the start of the season and in-season reorders are not possible. This article extends the classical newsvendor model by allowing the retailer to make an in-season price adjustment after conducting a review and using the realized demand to obtain an accurate estimate of the remaining demand. Our results include answers to the following questions. What price should the retailer choose? How much should the retailer have ordered at the start of the season given the option of adjusting prices in-season? This model was motivated by a problem in car rental revenue management and has applications in perishable assets revenue management (PARM), where price adjustments are needed towards the end of the selling season.     

A multi-item newsvendor problem with preseason production and capacitated reactive production

Given items with short life cycles or seasonal demands, one can potentially improve profits by producing during the selling season, especially when its production capacity is substantial. We develop a two-stage, multi-item model incorporating reactive production that employs a firm’s internal capacity. Production occurs in an uncapacitated preseason stage and a capacitated reactive stage. Demands occur in the reactive stage. Reactive capacities are pre-allocated to each item in the preseason stage and cannot be changed during the reactive stage. Reactive production occurs during the selling season with full knowledge of demands. The objective is expected profit maximization. Unsatisfied demand is lost. The revenue, salvage value, and production and lost sales costs are proportional. Assuming no fixed costs, we present a simple algorithm for computing optimal policies. For a model with fixed costs for allocating preseason stage production and reactive stage capacity to product families, we characterize optimal policies and develop optimal and heuristic algorithms.     

Production under periodic demand update prior to a single selling season: A decomposition approach

This paper focuses on dynamic, continuous-time production control problems in the fashion industry. Similar to the classical news-vendor problem, we consider a single product-type and the cumulative demand for items is not known until the end of the production horizon and therefore must be forecasted. Since there are periodic updates before a single selling season, actual demand during a period of time can only be determined by the end of the period. If the overall demand is overestimated, excessive inventory holding and production costs are paid and surpluses are sold at low prices at the end of the production horizon. If it is under-estimated, then sales are lost. The objective is to dynamically determine production orders which minimize overall expected costs. Since the optimal feedback for such a problem is characterized by thresholds evolving with time and system states, there is a significant computational burden in determining them. With the aid of the variational analysis and a decomposition, we derive a closed-form solution for the thresholds. A numerical study carried out to compare the decomposition and straightforward simulation-based solutions indicates the high accuracy of the suggested approach while the computational burden is dramatically reduced.     

Inventory policy for products with price and time-dependent demands

This paper investigates the problem of jointly determining the order size and optimal prices for a perishable inventory system under the condition that demand is time and price dependent. It is assumed that a decision-maker has the opportunity to adjust prices before the end of the sales season to influence demand and to improve revenues. A mathematical model is developed to find the optimal number of prices, the optimal prices and the order quantity. Analytical results show that a stationary solution to the Kuhn–Tucker necessary conditions can be found and it is shown to be the optimal solution. The analytical results lead us to derive a solution procedure for determining the optimal order size and prices.