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。     

A joint dynamic pricing and advertising model of perishable products

Advertising and dynamic pricing play key roles in maximizing profit of a firm. In this paper a joint dynamic pricing and advertising problem for perishable products is investigated, where the time-varying demand rate is decreasing in sales price and increasing in goodwill. A dynamic optimization model is proposed to maximize total profit by setting a joint pricing and advertising policy under the constraint of a limited advertising capacity. By solving the dynamic optimization problem on the basis of Pontryagin’s maximum principle, the analytical solutions of the optimal joint dynamic pricing and advertising policy are obtained. Additionally, to highlight the advantage of the joint dynamic strategy, the case of the optimal advertising with static pricing policy is considered. Numerical examples are presented to illustrate the validness of the theoretical results, and some managerial implications for the pricing and advertising of the perishable products are provided.     

Inventory control with fixed cost and price optimization in continuous time

We continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper [8], we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see [11]), i.e., the (s, S, p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s, S, p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Se- lecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy.     

运单体积不确定对货运共享平台定价决策的影响研究

货运共享平台订单分析显示顾客对运输货物申报的体积往往比实际体积偏大,平台则会按照客户申报的体积进行收费。本文站在平台利润视角,构建了考虑运单体积不确定的装载与配送路径联合优化模型来研究运单体积不确定对定价决策的影响,研究发现可以提升平台利润达24%。进一步,为了实现平台、顾客与司机的收益共享,并提高平台市场占有率,平台考虑在收费体积存在偏差的情况下,采用降价的策略与平台实际定价策略进行对比。通过数值分析发现,在存在体积偏差率的情况下,价格下降18%以内时,平台的利润呈现增长的趋势,最高可以达到23.26%;当价格下降超过18%时,平台利润出现负增长,因此不适合采取降价的措施。在降价过程中,顾客需要支付的成本最多可减少19.17%,司机的收益最多可增加13.82%,结果表明,采用新型分段定价策略可以更好的实现三方共赢,对平台的实际运营有一定的借鉴意义。     

Pricing and lot-sizing for a deteriorating item in a periodic review inventory system with shortages

We consider a single product that is, subject to continuous decay, a multivariate demand function of price and time, shortages allowed and completely backlogged in a periodic review inventory system in which the selling price is allowed to adjust upward or downward periodically. The objective of this paper is to determine the periodic selling price and lot-size over multiperiod planning horizon so that the total discount profit is maximized. The proposed model can be used as an add-in optimizer like an advanced planning system in an enterprise resource planning system that coordinates distinct functions within a firm. Particular attention is placed on investigating the effect of periodic pricing jointly with shortages on the total discount profit. The problem is formulated as a bivariate optimization model solved by dynamic programming techniques coupled with an iterative search process. An intensive numerical study shows that the periodic pricing is superior to the fixed pricing in profit maximization. It also clarifies that shortages strategy can be an effective cost control mechanism for managing deterioration inventory.     

Optimal pricing,production, and inventory for deteriorating items under demand uncertainty: The finite horizon case

This paper aims to investigate the joint dynamic pricing and production decisions of deteriorating items with uncertain demand over a finite selling season, where the demand is price sensitive and the potential demand is characterized by a stochastic process. The stocks deteriorate physically at a constant fraction of the on-hand inventory. A joint dynamic pricing and production problem to maximize the total expected profit is modeled as a stochastic optimal control problem. We derive the closed-form solutions, which are in time-dependent linear feedback form of the inventory level when it is either positive or negative. It is shown that the manufacturer always benefits from a reduction in the volatility of potential market demand. In addition, to highlight the effectiveness of the joint dynamic strategy, we also consider the case of optimal production with a static price. A numerical example is presented to illustrate the validity of the optimal control policy, and sensitivity analysis on major parameters is performed to provide more managerial insights into deteriorating items.     

Utility-based indifference pricing in regime-switching models

In this paper, we study utility-based indifference pricing and hedging of a contingent claim in a continuous-time, Markov, regime-switching model. The market in this model is incomplete, so there is more than one price kernel. We specify the parametric form of price kernels so that both market risk and economic risk are taken into account. The pricing and hedging problem is formulated as a stochastic optimal control problem and is discussed using the dynamic programming approach. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution to the problem is given. An issuer’s price kernel is obtained from a solution of a system of linear programming problems and an optimal hedged portfolio is determined.     

考虑码头内外堆场竞争的集装箱堆存定价模型

针对由于集装箱吞吐量增加造成翻箱成本增加与码头堆场拥堵,考虑内外堆场竞争,构建集装箱堆存定价模型,研究内外堆场竞争下的堆存定价决策和货主移箱决策,分析码头堆场操作成本、场外堆场运输成本、货主需求参数和码头堆场收益的内在关系。算例结果表明:一旦过了免费期立即移箱到场外堆场,货主成本最小。随着场外堆场运输成本的增加,码头堆场的堆存定价和箱量增加,提高码头堆场收益。随着码头堆场操作成本降低,码头堆场堆存价格随之下降,但码头堆存箱量增加,提高码头堆场收益。此外降低堆场定价对货主需求的影响参数以及提高运输成本对货主需求的影响系数,有利于提高码头堆场收益。     

一种电子商务环境下耐用品的动态定价模型与策略

针对电子商务环境下消费者对价格歧视的抗拒问题,以及耐用品生命周期长、产品需求依赖于时间、价格等特点,提出了一种动态定价模型与策略。该模型通过构造转移概率矩阵,推导出在线消费者浏览到耐用品的不同价格状态下的概率,接着根据消费者多阶段效用函数分析消费者的购买决策行为,进而给出零售商利润达到最大化时的最优定价策略集合。为了验证模型与策略的有效性,通过数值模拟实验,分析模型主要参数变化对最优定价策略的影响。研究发现当效用折扣因子越高,零售商应该降低促销频率和高价格并且提高低价格,从而诱导高端消费者在高价格购买产品。折扣效用因子大小还决定了网上零售商是否要隐藏自己的促销概率。     

存在空箱调运的非对称运输市场定价策略研究

考虑空箱调运成本,本文对垄断和双寡头市场分别研究运输企业在两条相向路径上的定价问题。对于垄断市场,建立了运输企业最优定价策略,并刻画出无空箱调运的潜在需求不平衡区间。对于双寡头市场,考虑同一路径上不同企业潜在运输需求不等的现实情境,求解了非对称企业的伯川德纳什均衡,给出最优定价策略。研究发现,无空箱调运并不意味着较高利润,运输企业没有必要刻意消除空箱调运现象。另外,增加单位载货运输成本和竞争强度会降低企业利润,而提升单位空箱重置成本、价格敏感度和市场不对称程度都会增大企业利润。     

Time-dependent and independent control rules for coordinated production and pricing under demand uncertainty and finite planning horizons

We address the effect of uncertainty on a manufacturer’s dynamic production and pricing decisions over a finite planning horizon. The demand for products, which depends on their price, is characterized by two stochastic processes: potential demand and customer price sensitivity. An optimal policy for coordinating production and pricing is a time-dependent feedback rule with respect to the state of the manufacturer’s inventories. We show that when the volatility of customer sensitivity to the product price is negligible, the optimal policy can be obtained analytically. Moreover, our simulations demonstrate that the volatility of stochastic customer price sensitivity does not have a strong effect on the manufacturer’s expected profit. Therefore, the solution derived for the case of customer price sensitivity with zero volatility can serve as a good approximation heuristic for the optimal policy if the true volatility of customer price sensitivity is within 40 % of its mean and the volatility of potential demand is within 25 % of its mean. Moreover, under these conditions, a simplified, time-independent control rule deteriorates expected profits by only 1.5 %.     

部分延迟订购的易变质品联合定价与生产策略

构建了一个需求同时依赖于销售价格和库存水平,生产率和变质率均为常数,允许缺货且缺货量部分延迟订购的易变质品联合定价与生产控制模型。首先证明了在销售价格给定的情况下,系统的总利润函数是关于生产计划的严格凹函数,平均利润函数是严格的伪凹函数,即存在唯一的最优解,并给出其充分条件。接着给出问题的一个数值求解算法。最后通过算例,展示了模型及相关算法的应用,并对相关参数进行了灵敏度分析,结果显示:当产品的生产成本、缺货成本和机会成本增加时,系统的平均利润将下降;生产成本和延迟订购阻力系数对最优定价和生产策略以及平均利润的影响较大。     

Optimal Contract Design of Supplier-Led Outsourcing Based on Pontryagin Maximum Principle

This paper provides a supplier-led outsourcing model to maximize the supplier’s profits based on a principal-agent framework with both asymmetric cost information and uncertain market demand information described by continuous random variables. The salvage value of the unsold product is processing-cost dependent. By converting the proposed model, which is a dynamic optimization problem, to an optimal control problem, we obtain the analytical form of the optimal supplier outsourcing contract composed of the wholesale price and the transfer payment by applying Pontryagin’s maximum principle. It is shown that the optimal contract is directly related to the supplier’s beliefs about the manufacturer’s unit cost and the salvage value function. The Pontryagin’s maximum principle-based solution method serves as a powerful tool to support the decision making for the best sourcing strategy, and it provides analytical insights for outsourcing management. Finally, numerical examples are presented to illustrate the validness of the theoretical results.     

搜寻成本下可替代产品的定价策略与库存决策

为探寻存在搜寻成本情况下消费者购买可替代产品时的定价与库存问题,从消费者效用出发,对厂商收益构建了基于马尔可夫决策过程的优化模型。在消费者方面,分析了其购买与继续搜寻的条件,并分别在搜寻成本不变和搜寻成本边际递减的情况下研究了消费者保留效用的变化情况以及购买相应产品的概率。此外,与很多相关文献不同的是,由于搜寻成本的存在,该情形下的消费者并不一定会在完成购买之前搜寻完所有的产品。在厂商方面,根据实际情况构建不同搜寻顺序下的收益模型并求解出最优定价策略与库存策略,并将定价模型与库存策略扩展到了动态的环境,为厂商制定价格及库存方案提供相应的决策支持。     

A capacitated firm’s pricing strategies for strategic consumers with different search costs

This paper studies pricing strategies of a seller with budget constraints facing two types of strategic consumers with different search costs, and proposes three pricing strategies to motivate all consumers to visit his shop. These are the basic price strategy, differentiated compensation strategy and an improved differentiated compensation strategy. Based on the rational expectations paradigm, we characterize the rational expectations equilibrium in the game and propose a basic pricing strategy. In order to address the interplay between price and demand, we further propose a differentiated compensation strategy to improve the basic model. We then compare the differentiated compensation strategy to the basic pricing strategy when both are feasible. We find that selection of the optimal strategy is independent of composition of consumers but is dependent on the seller’s budget level and the difference between the two search costs. If the budget is large enough and the difference between the search costs is small enough, a differentiated compensation strategy can further improve the seller’s profitability. In addition to these findings, we first propose an improved differentiated compensation strategy to further enhance the firm’s profit. We find that the optimal strategy is to implement the improved differentiated compensation strategy when all three strategies are feasible. Interestingly, the firm may benefit from paying a high compensation to the consumers.     

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.     

考虑云安全风险的个人云存储服务最优定价策略

针对现有个人云存储服务纷纷关闭的现象,如何根据不同市场时期和云用户特征,制定合理的定价策略已成为个人云存储服务商面临的挑战性问题。本文在考虑云安全风险,云用户感知价值及弹性成本对个人云存储服务定价的影响,建立了基于云安全风险的两阶段定价模型,给出了免费试用时长确定下的最优决策及免费试用时长不确定下的最优免费试用时长及最优价格。借助数值分析,进一步研究了云安全风险系数,云用户感知价值,云安全运营成本系数对最优免费时长,最优利润的影响,结果表明,当云安全风险系数适中时,个人云存储供应商(PCSP)采取两阶段定价策略。另外,最优利润随着云安全风险系数,免费试用时长,单位安全运营成本的增加先增加后减少。     

Optimal Portfolio Selection Under Concave Price Impact

In this paper we study an optimal portfolio selection problem under instantaneous price impact. Based on some empirical analysis in the literature, we model such impact as a concave function of the trading size when the trading size is small. The price impact can be thought of as either a liquidity cost or a transaction cost, but the concavity nature of the cost leads to some fundamental difference from those in the existing literature. We show that the problem can be reduced to an impulse control problem, but without fixed cost, and that the value function is a viscosity solution to a special type of Quasi-Variational Inequality (QVI). We also prove directly (without using the solution to the QVI) that the optimal strategy exists and more importantly, despite the absence of a fixed cost, it is still in a “piecewise constant” form, reflecting a more practical perspective.     

Optimal pricing and inventory policies with reliable and random-yield suppliers: characterization and comparison

In this paper, we study the joint pricing and inventory replenishment problem for a periodic-review inventory system with random demand and dual suppliers, one of the suppliers is reliable but more expensive, the other supplier is less expensive but is unreliable with random yield. We characterize the firm’s optimal policies that simultaneously determine the optimal ordering and pricing decisions in each period over a finite planning horizon, and investigate the impacts of supply source diversification and supplier reliability on the firm and on its customers. We show that having source diversification or higher reliability of suppliers not only increases the firm’s expected profit, but also results in a lower optimal selling price, thus they benefit both the firm and its customers.     

基于贝叶斯信息更新的动态库存与定价研究

考虑价格对需求量的扰动并利用贝叶斯公式对需求分布函数中的未知参数进行不断学习更新,研究缺货部分比例延迟交货情形下的动态库存与动态定价问题,刻画了最优利润函数的性质并证明了"基准库存列表价格"是最优的库存价格水平,并由此得到了最优的补货策略和定价策略。