Optimal software pricing in the presence of piracy and word-of-mouth effect

We develop an analytical model that embeds empirical findings on software diffusion to examine optimal pricing strategies for a spreadsheet software product under coalescing effects of piracy and word-of-mouth through its entire life cycle. We find that the demand of the innovators has the most significant impact on the firm's pricing decision. Our research recommends market skimming pricing strategy if innovators' demand is high and the market penetration pricing strategy is preferred otherwise. Surprisingly, the increase of conversion rate of imitators to buyers never significantly alters the pricing strategy pre-determined by the demand of innovators. Most interestingly, the optimal profit from instituting a two prices policy for a software product with five years lifespan outperforms that from a one price policy by no more than 4%, a finding that corroborates the common one price policy observed in reality.     

Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts

Retailer promotional activity has become prevalent in the business world. Promotional efforts impact the replenishment policy and the sale price of goods. In this paper, the problem of replenishment policy and pricing for non-instantaneous deteriorating items subject to promotional effort is considered. We adopt a price dependent stochastic demand function in which shortages are allowed and partially backlogged. The major objective is to simultaneously determine the optimal selling price, the optimal replenishment schedule, and the optimal order quantity to maximize the total profit. First, we prove that a unique optimal replenishment schedule exists for any given selling price. Second, we prove that the total profit is a concave function of price. Third, we present an algorithm to obtain the optimal solution and solve a numerical example. Last, we extend the numerical example by performing a sensitivity analysis of the model parameters and discuss specific managerial insights.     

需求依赖于产品价格和消费者时间偏好的定价与订货决策

研究产品的市场需求依赖于价格和消费者时间偏好情形下的供应链及其成员的最优定价与订货问题. 利用时间偏好因子刻画消费者的时间偏好, 利用产品市场需求关于零售价格的弹性指数反映消费者的价格敏感程度, 建立存在强势零售商、强势供应商和供需双方势力均衡3 种渠道权力结构下的定价与订货模型. 数值结果表明, 在3 种渠道权力结构下, 消费者的时间偏好和产品市场需求关于零售价格的弹性指数是供应链主导者最优定价、订货量和期望利润的单调减函数, 将其纳入定价与订货决策的影响因素予以考量能够增加供应链主导者的最大期望利润.

     

库存水平影响需求下变质品订购、定价和保鲜技术投资的联合决策

研究变质品的订购、定价和保鲜技术投资联合决策问题.假设产品需求同时受价格和库存水平的影响,系统不允许缺货并放松期末库存水平为零的约束,零售商拥有有限的货架空间或存储空间,同时考虑零售商可以通过投资保鲜技术减低产品的变质率,以平均利润最大化为目标构建库存水平影响需求下变质品的订购、定价和保鲜技术投资联合决策模型.首先证明最优策略的存在性和唯一性,并给出零售商建立期末库存的条件;然后利用最优解的相关性质设计一个求解模型的多阶段迭代算法;最后通过具体算例验证展示模型和算法的可行性和实用性,并完成相关参数的敏感性分析,获得一定的管理启示.     

Inventory and pricing strategies for deteriorating items with shortages: A discounted cash flow approach

In this article, we consider an infinite horizon, single product economic order quantity where demand and deterioration rate are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and completely backlogged. The objective is to find the optimal inventory and pricing strategies maximizing the net present value of total profit over the infinite horizon. For any given selling price, we first prove that the optimal replenishment schedule not only exists but is unique. Next, we show that the total profit per unit time is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use a couple of numerical examples to illustrate the algorithm.     

基于消费者时间偏好和决策者风险偏好的定价与订货策略*

在供应链管理过程中, 消费者时间偏好和决策者风险偏好是影响产品定价与订货决策的两个重要因素。本文以累积前景理论为框架, 将消费者时间偏好与价格依赖等影响产品市场需求的因素和决策者风险偏好与参考依赖等影响决策的因素共同纳入模型考量范围, 建立了基于消费者的时间偏好和决策者风险偏好的定价与订货模型。研究表明：随着消费者时间偏好的增大, 产品的最优定价和订货量都将减少;随着决策者心理参考点的增大, 产品的最优定价降低,最优订货量增加;随着决策者损失规避程度的增大, 产品最优定价增加, 而最优订货量减少。考虑消费者时间偏好和决策者的风险偏好的定价与订货模型能够提高供应链中决策者的最大累积前景效用。     

动态定价策略下的精确库存成本建模与优化

提出一种更接近实际的需求率公式,在式中同时考虑了价格和出厂时间对客户需求率的影响.基于新的需求率公式,建立了动态定价策略下的精确库存持有成本模型和库存商品的利润函数.注意到利润函数的复杂性,使用遗传算法分析了利润函数的性质,得出最优定价时间、定价价格和最大利润的关系,并分析了库存持有成本变化和消费者购买欲望变化对各定价参数的影响.     

Optimal dynamic pricing and replenishment policy for perishable items with inventory-level-dependent demand

An inventory system for perishable items with limited replenishment capacity is introduced in this paper. The demand rate depends on the stock quantity displayed in the store as well as the sales price. With the goal to realise profit maximisation, an optimisation problem is addressed to seek for the optimal joint dynamic pricing and replenishment policy which is obtained by solving the optimisation problem with Pontryagin’s maximum principle. A joint mixed policy, in which the sales price is a static decision variable and the replenishment rate remains to be a dynamic decision variable, is presented to compare with the joint dynamic policy. Numerical results demonstrate the advantages of the joint dynamic one, and further show the effects of different system parameters on the optimal joint dynamic policy and the maximal total profit.     

Dynamic joint pricing and production policy for perishable products

A joint dynamic pricing and production problem for perishable products without shortages is considered. The demand rate is price‐dependent and time‐varying. This paper constructs an optimal control model to maximize the total profit under a general nonlinear production cost function. The feature of the optimal joint dynamic pricing and production policy is analyzed by solving the corresponding optimal control problem on the basis of improved Pontryagin's maximum principle. Then, an effective algorithm is designed to obtain the optimal joint policy. The case of the joint static optimal policy is also investigated and compared with the dynamic one. Finally, numerical examples are presented to illustrate the effectiveness of the proposed methods, and some managerial implications are provided for the management of perishable items.     

损失厌恶下考虑参照利润效应的供应链决策模型

针对由一个风险中性供应商和一个损失厌恶零售商构成的二级供应链,研究随机需求下考虑零售商参照利润效应的供应链决策问题.在回购政策下,建立以供应商为主方、零售商为从方的Stackelberg主从博弈模型.结合参照依赖偏好模型分别得到集中和分散供应链决策,分析供应链最优决策与损失厌恶程度、参照利润强度和零售商乐观水平之间的关系,并进一步设计能够实现供应链完美协调的回购契约机制.研究结果表明,在集中和分散供应链决策下,零售商订货量均随着损失厌恶和乐观程度的增加而减少.而当零售商损失厌恶程度较低时,订货量随参照利润强度的增加而增加;反之,亦成立.对于批发价格决策,则存在一个阈值,当高于该阈值时,批发价格随着零售商损失厌恶、乐观程度和参照利润强度的增大而增加;低于该阈值时,批发价格随着损失厌恶、乐观程度和参照利润强度的增大而降低.     

Optimal dynamic pricing for deteriorating items with reference-price effects

In this paper, a dynamic pricing problem for deteriorating items with the consumers’ reference-price effect is studied. An optimal control model is established to maximise the total profit, where the demand not only depends on the current price, but also is sensitive to the historical price. The continuous-time dynamic optimal pricing strategy with reference-price effect is obtained through solving the optimal control model on the basis of Pontryagin's maximum principle. In addition, numerical simulations and sensitivity analysis are carried out. Finally, some managerial suggestions that firm may adopt to formulate its pricing policy are proposed.     

Pricing and inventory policies for Hi-tech products under replacement warranty

Companies, especially in the Hi-tech (high-technology) industry (such as computer, communication and consumer electronic products), often provide a replacement warranty period for purchased items. In reality, simultaneously determining the price and inventory decisions under warranty policy is an important issue. The objective of this paper is to develop a joint pricing and inventory model for Hi-tech products under replacement warranty policy. In the first model, we consider a Hi-tech product feature in which the selling price is declining in a trend. We determine the optimal inventory level for each period and retail price for the first period while maximising the total profit. In the second model, we further determine the optimal retail price and inventory level for each period in the dynamic demand market. This study develops solution approaches to solve the problems described above. Numerical analysis discusses the influence of system parameters on the company's decisions and behaviours. The results of this study could serve as a reference for business managers or administrators.     

考虑参照价格效应的易逝品定价与订购策略

在考虑消费者参照价格效应的基础上,构建一个易逝品的定价与订购联合决策模型,其中产品的需求不仅依赖于销售价格还与该产品在消费者心目中的参照价格相关,变质率为常数,系统不允许缺货.分别讨论了对称参照价格效应和非对称参照价格效应两种情况下零售商的最优定价与订购决策问题,证明并得到关于模型结构的一些性质,进而设计了问题的求解算法.通过数值方法分析了参照价格效应参数和变质率对系统最优解的影响,以及两种情况下最优解之间的关系.结果显示:当面对具有参照价格依赖的消费者时,采用适当的营销策略来提高消费者的参照价格对零售商总是有利的;对高变质率产品而言,零售商可保持一个较稳定的订购策略,更多地关注产品的定价策略;面对损失厌恶型消费者,随着消费者参照价格的逐渐提高,零售商的定价与订购策略均应缓慢地改变,而不宜急剧变化.     

Multi-period optimization with loss-averse customer behavior: Joint pricing and inventory decisions with stochastic demand

To maximize a firm's profit over a finite planning horizon, we develop a dynamic optimization model by considering loss aversion when making pricing and inventory decisions. We estimate customer demand through a choice model, which incorporates reference price, utility function and customer loss aversion. Our model forms the core of the expert system for decision support. Through a sequence of Bellman equations, we find that the firm's profit is a concave function of price and inventory, and we solve the model optimally. The profit is positively correlated with the reference price, and the price and inventory decisions are non-monotonic functions of loss aversion intensity. Our results shed new light on pricing and inventory management with customer behavior in a multi-period system. Through various theorem developments, we are able to identify the optimal inventory level and the corresponding price. Numerical examples are provided to illustrate and validate the model and to derive managerial insights. To show the potential significance, we demonstrate how a dynamic programming model yields good results with customer loss aversion under realistic customer behavior assumptions. Our system can improve the efficiency of decision making and provide better customer service.     

The value of modeling with reference effects in stochastic inventory and pricing problems

We analyze a periodic review inventory system in which the random demand is contingent on the current price and the reference price. The reference price captures the price history and acts as a benchmark against which the current price is compared. The randomness is due to additive and multiplicative random terms. The objective is to maximize the discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We study three key issues using numerical computation and simulation. First, we study the effects of reference price mechanism on the total expected profit. It is shown that high dependence on a good history increases the profit. Second, we investigate the value of dynamic programming and show that the firm that ignores the dynamic structure suffers from the revenue. Third, we analyze the value of estimating the correct demand model with reference effects. We observe that this value is significant when the inventory related costs are low.     

A myopic policy for the gradual obsolescence problem with price-dependent demand

The purpose of this paper is to develop a retailer's profit-maximizing myopic inventory policy for an item recognized as subject to gradual obsolescence. Demand is assumed to be a decreasing function of both the retailer's sale price and of time, up to a certain stochastic time point when obsolescence occurs and, as a result, the demand suddenly drops to zero. For each ordering cycle, the decision variables are the retailer's selling price and the order size. A stop-ordering rule is developed on the basis of finding the time point beyond which it is profitable to stop ordering, even if there is still some demand for the item. In addition, the sudden obsolescence problem is shown to be a limiting and non-trivial case of its gradual counterpart. The numerical example illustrates the main features of the model, including the importance of the vendor dropping the price charged to retailers, so as to provide the needed incentives for the retailers to drop the price charged to their own customers and thereby palliate as much as possible to negative effects of obsolescence.Scope and purposeThis paper develops a myopic policy to evaluate a retailer's decision process, when an item is recognized as subject to gradual obsolescence. The model considers demand to be a decreasing function of both the retailer's sale price and of time, up to a certain stochastic time point when obsolescence occurs and, as a result, the demand suddenly drops to zero. The retailer's profit-maximizing policy consists of an optimal selling price and an order size for each ordering cycle, as well as the time point beyond which it is profitable to stop ordering, even if there is still some demand for the item. This is in contrast to alternate formulations, where the stopping rule is based upon minimizing the cost of obsolescence, rather than evaluating the profitability of the item in question. Finally, the numerical example illustrates the need for vendor/retailer collaboration in the development of the pricing policies. Otherwise, the retailer has no incentive to keep prices low and thus counteract the normal decreases in demand that occur as time passes by and the probability of obsolescence increases.     

Optimal pricing policies for services with consideration of facility maintenance costs

For survival and success, pricing is an essential issue for service firms. This article deals with the pricing strategies for services with substantial facility maintenance costs. For this purpose, a mathematical framework that incorporates service demand and facility deterioration is proposed to address the problem. The facility and customers constitute a service system driven by Poisson arrivals and exponential service times. A service demand with increasing price elasticity and a facility lifetime with strictly increasing failure rate are also adopted in modelling. By examining the bidirectional relationship between customer demand and facility deterioration in the profit model, the pricing policies of the service are investigated. Then analytical conditions of customer demand and facility lifetime are derived to achieve a unique optimal pricing policy. The comparative statics properties of the optimal policy are also explored. Finally, numerical examples are presented to illustrate the effects of parameter variations on the optimal pricing policy.     

Pricing and production lot-size/scheduling with finite capacity for a deteriorating item over a finite horizon

Although the lately evolved manufacturing technologies such as enterprise resource planning (ERP) provide a unified platform for managing and integrating core business processes within a firm, the decision-making between marketing and production planning still remains rather disjoint. It is due in large parts to the inherent weaknesses of ERP such as the fixed and static parameter settings and uncapacitated assumption. To rectify these drawbacks, we propose a decision model that solves optimally the production lot-size/scheduling problem taking into account the dynamic aspects of customer's demand as well as the restriction of finite capacity in a plant. More specifically, we consider a single product that is subject to continuous decay, faces a price-dependent and time-varying demand, and time-varying deteriorating rate, production rate, and variable production cost, with the objective of maximizing the profit stream over multi-period planning horizon. We propose both coordinated and decentralized decision-making policies that drive the solution of the multivariate maximization problem. Both policies are formulated as dynamic programming models and solved by numerical search techniques. In our numerical experiments, the solution procedure is demonstrated, comparative study is conducted, and sensitivity analysis is carried out with respect to major parameters. The numerical result shows that the solution generated by the coordinated policy outperforms that by the decentralized policy in maximizing net profit and many other quantifiable measures such as minimizing inventory investment and storage capacity.Scope and purposeWe consider a manufacturing firm who produces and sells a single product that is subjected to continuous decay over a lifetime, faces a price-dependent and time-varying demand function, shortages are allowed and a completely backlogged, and has the objective of determining price and production lot-size/scheduling so as to maximize the total profit stream over multi-period planning horizon. We develop a tactical-level decision model that solves the production scheduling problem taking into account the dynamic nature of customer's demand which is partially controllable through pricing schemes. As analogous to the sales and operations planning, the proposed scheme can be used as a coordination center of the APS system within a generic enterprise resource planning framework which integrates and coordinates distinct functions within a firm.This paper differs from the existing works in several ways. First, we propose a dynamic version of the joint pricing and lot-size/scheduling problem taking into account the capacitated constraint. Second, several key factors being considered in the model, such as the demand rate, deteriorating rate, production rate, and variable production cost are assumed time-varying that reflect the dynamic nature of the market and the learning effect of the production system. A third difference between the past research and ours is that the price can be adjusted upward or downward in our model, making the proposed pricing policy more responsive to the structural change in demand or supply.     

Optimal decision making for online referral marketing

Widely available web 2.0 technologies not only bring rich and interactive user experiences, but also easily help users advertise products or services on their own blogs and social network webpages. Online referral marketing, for example, is a business practice that rewards customers who successfully refer other customers to a website or upon completion of a sale usually via their own social contacts. The referral rewards come in different forms such as shopping vouchers, redeemable points, discounts, prizes, cash payments, etc. We develop an analytical model to evaluate the business potential of incorporating an online referral marketing program into the firm's product selling strategies. Under different demand dynamics, we investigate the optimal decision making including the pricing and referral strategies to maximize the seller's profitability. We find that, under simple decision making environment such as fixed product price and myopic strategy, different demand dynamics yield the same prediction of the referral payment, which turns out to be a static policy. However, under complex market situations, both the optimal product pricing and referral offering critically depend on the demand side dynamics. Under the nonlinear demand dynamics, the referral payment is an all-or-nothing decision throughout the product selling horizon. In contrast, under the linear demand assumption, the referral payment can be partially offered in initial phase of the product introduction. We further offer some managerial insights to guide practical implementation of the online referral marketing strategy.     

零售电商和回收政策对闭环供应链的影响研究

构建第三方回收下的双渠道闭环供应链模型,考虑了三种不同的决策情形来探讨零售电商参与和处理基金政策对供应链中决策、利润、需求和消费者剩余的影响。结果表明：零售电商参与使得传统零售商的产品价格和利润水平降低,制造商可以通过调整对零售电商的批发价格来提高其利润;第三方回收商的回收决策和最优利润不受到零售电商参与的影响,处理基金政策可以有效地提高产品回收数量,但也会使得正向供应链中企业的定价提高;零售电商会蚕食制造商原有的市场份额,但产品总需求量不发生变化。当进一步考虑处理基金政策时,市场总需求量出现下降的趋势;此外,零售电商参与和处理基金政策都会造成供应链中消费者总剩余的减少。