A DECISION THEORETIC APPROACH TO PRICING

The problem of price determination and revision is considered as a case of decision making under uncertainty in which profit is to be maximized. Typically price is a simple function of cost which in turn determines the quantity which will be demanded. This paper proposes that maximum profits could be realized in the long run, if a quantity corresponding to the lowest cost per unit of product under the attendant circumstances was fixed and a price established at which demand would exactly equal the fixed quantity. It further suggests a theoretical approach to the determination of this price based on decision theory. The decision theoretic approach considers the set of possible price levels at which demand will equal the fixed quantity of product as the state of nature. The set of acts consist of the establishment of the product price at each of the possible levels. After an initial price is established, empirical information can then be utilized according to some optimal decision rule for subsequent price revisions.     

Using Prices More Realistically as Decision Variables in Perishable-Asset Revenue Management Problems

This paper addresses a new slant to a problem which is general to many of the transportation industries—perishable asset revenue management. Traditional approaches have assumed that prices are fixed and solved for the optimal allocation quantities. Our approach recognizes that prices of the different classes affect demand and should therefore be included as decision variables to be optimized. We solve three different types of problems: (a) up to n price classes, distinct asset control mechanism, and no diversion, (b) up to 3 price classes, serial nested asset control mechanism, and no diversion, (c) up to 3 price classes, serial nested asset control mechanism, and diversion. Analytical results are provided in most cases and examples illustrate the results as well as the time required to solve these complex problems. Finally we look at the tradeoff involved between computational time and expected contribution when using heuristic decisions obtained from less realistic assumptions relative to the true optimal decisions. On average, the suboptimality ranged from 3.19% to 4.88% with a corresponding decrease in computing time required on the order of several minutes. Some trends are presented to help determine a priori which type of problems would tend to benefit most from the more accurate formulation. This should help managers decide when it is worth the extra computing time to come up with the true optimal solution.     

关键词拍卖中基于广义第一价格的投标价格决策分析

在关键词拍卖中,由于广告商不能保证对关键词的估价是最优投标价格,因此如何选取最优的关键词投标价格一直是广告商急于解决的问题。基于与Friedman投标模型假设条件等价的关键词拍卖投标假设条件,计算了各个关键词广告位置的胜标概率,并在广义第一价格拍卖机制下建立了关键词投标价格决策模型。该模型可以使广告商选取最优的关键词投标价格。最后通过仿真验证了该模型的有效性。     

招标与投标竞争系统决策模型及其应用

招投标理论所研究的首要问题是投标商的投标策略问题,其中最优报价尤为重要。本文利用两种方法-解析法和利润法,解决了独立投标中的最优报价问题。     

Distributed Decision‐Making in Supply Chains and Private E‐Marketplaces

This paper develops a distributed decision‐making framework for the players in a supply chain or a private e‐marketplace to collaboratively arrive at a global Pareto‐optimal solution. In this model, no player has complete knowledge about all the costs and constraints of the other players. The decision‐making framework employs an iterative procedure, based on the Integer L‐shaped method, in which a master problem is solved to propose global solutions, and each player uses his local problems to construct feasibility and optimality cuts on the master problem. The master problem is modeled as a mixed‐integer program, and the players' local problems are formulated as linear programs. Collaborative planning scenarios in private e‐marketplaces and in supply chains were formulated and solved for test data. The results show that this distributed model is able to achieve near‐optimal solutions considerably faster than the traditional centralized approach.     

UNINSURED COUNTERCYCLICAL RISK: AN AGGREGATION RESULT AND APPLICATION TO OPTIMAL MONETARY POLICY

We consider an incomplete markets economy with capital accumulation and endogenous labor supply. Individuals face countercyclical idiosyncratic labor and asset risk. We derive conditions under which the aggregate allocations and price system can be found by solving a representative agent problem. This result is applied to analyze the properties of an optimal monetary policy in a New Keynesian economy with uninsured countercyclical individual risk. The optimal monetary policy that emerges from our incomplete markets economy is the same as the optimal monetary policy in a representative agent model with preference shocks. When price rigidity is the only friction the optimal monetary policy calls for stabilizing the inflation rate at zero.     

合理制定铁路客票价格的优化模型及算法

在充分考虑出行者和铁路客运部门两方面的利益情况下 ,提出一个双层规划模型以得到在多种交通方式竞争条件下的铁路客票价格制定的最优策略 .既保障了出行者使自己的广义出行费用最小 ,又能使铁路客运部门在运输市场竞争中取得的经济效益最大 .并且给出了求解该模型的 SAB算法。最后用一个简单的算例说明了模型及算法的应用     

An optimal credit policy to increase supplier's profits with price-dependent demand functions

This paper deals with the problem of determining an optimal length of credit period from the perspective of supplier. We assume that a retailer jointly determines the unit retail price and order size to maximize profit when he/she purchases a product for which the supplier offers a trade credit. Two widely used demand functions are adopted for the study in which demands are decreasing functions of the retail price. A procedure is presented which shows how to achieve an optimal length of credit period for suppliers. The effects of credit period on the behaviour of retailers are also investigated using an example.     

Determining Order Quantity and Selling Price by Geometric Programming: Optimal Solution,Bounds, and Sensitivity*

This paper presents a geometric programming (GP) approach to finding a profit-maximizing selling price and order quantity for a retailer. Demand is treated as a nonlinear function of price with a constant elasticity. The proposed GP approach finds optimal solutions for both no-quantity discounts and continuous quantity discounts cases. This approach is superior to the traditional approaches of solving a system of nonlinear equations. Since the profit function is not concave, the traditional approaches may require an exhaustive search, especially for the continuous discounts schedule case. By applying readily available theories in GP, we easily can find global optimal solutions for both cases. More importantly, the GP approach provides lower and upper bounds on the optimal profit level and sensitivity results which are unavailable from the traditional approaches. These bounding and sensitivity results are further utilized to provide additional important managerial implications on pricing and lot-sizing policies.     

价格下界随时间递增的占线采购问题

本文研究的是价格不确定且其下界随时间递增的原材料采购问题。在实际的原材料采购问题中,原材料的价格随时间的变动往往是不可预测的。之前的学者在研究价格不确定的占线采购问题时,假设价格在一个统一的常数上下界内,这没有考虑到经过时间的变化,价格的上下界可能也是变化的。本文提出并研究价格下界随时间递增的原材料占线采购问题。构建了相应数学模型,给出了相应的竞争采购策略并证明了竞争比,同时通过证明问题的匹配竞争比下界,说明给出的竞争采购策略是最优的,最后利用数值分析进一步说明竞争策略具有较好的竞争性能。     

A UNIFIED APPROACH TO THE PRICE-BREAK ECONOMIC ORDER QUANTITY (EOQ) PROBLEM*

In many developing countries where supply is limited, a premium rather than a discounted price is paid for buying in larger amounts. In order for management science/operations research solutions to be relevant to developing economies, such realities need to be explicitly recognized and any decisions evaluated from that perspective. To illustrate this point, a typical inventory-purchasing problem is solved. In the process, formulae are developed that introduce simultaneous price and quantity sensitivity into the inventory equation. A systematic procedure is suggested to locate the optimal order quantity for a given schedule of discounts or premiums.     

基于两阶段决策的封闭式基金价格控制问题研究

在给定相关假设的基础上,考虑了在某一交易时间段的初始时刻,“庄家”型封闭式基金投资者初始持有资金额度有限的状况,将“庄家”型基金投资者的基金价格控制决策分为两个阶段：第一阶段为在确定各时刻期望收益率及其收益离差的条件下求得该投资者在对应时刻的最优投资值,第二阶段以各时刻的最优投资值为约束来获得最优基金价格控制序列。对于第一阶段可建立使“庄家”型基金投资者期望收益最大、收益平均绝对离差最小（风险最小）的双目标模型,变换为一个单目标模型后利用库恩-塔克条件进行求解,从而获得各时刻的最优投资值;对于第二阶段则以第一阶段得到的各时刻最优投资值为约束,以“庄家”型基金投资者在该交易时间段内的现金支付为极小值（现金收益的极大值）建立目标函数,而其为一个带约束的非线性规划问题,对此采用了一种改进的遗传算法进行求解,最终获得基金的最优价格控制序列。     

垄断双边平台的价格策略和数量策略设计问题

考虑垂直差异化下双边参与者的选择行为,利用无穷维多类别用户均衡分析方法建立了垄断双边平台企业的价格策略和数量策略的 MPEC模型.其中,着重研究价格策略中双边平台市场均衡的多重性问题,利用供应曲线和需求曲线综合分析用户均衡多重解的几何特征及价格可行集合.针对非零用户均衡多重性的问题,本文还讨论了乐观和悲观两种用户均衡预期下的最优价格策略设计方法及平台最优数量策略,并发现了最优乐观价格策略和最优数量策略的对偶性.为更好结合平台实际运营,本文考虑平台具有一定的初始用户规模,研究了双边平台用户均衡定点问题的用户规模轨迹迭代方程的极限解,进而获得给定用户规模的平台最优价格策略,为平台企业提供定价决策参考.     

Up Then Down: Bid‐Price Trends in Revenue Management

In the classic revenue management (RM) problem of selling a fixed quantity of perishable inventories to price‐sensitive non‐strategic consumers over a finite horizon, the optimal pricing decision at any time depends on two important factors: consumer valuation and bid price. The former is determined exogenously by the demand side, while the latter is determined jointly by the inventory level on the supply side and the consumer valuations in the time remaining within the selling horizon. Because of the importance of bid prices in theory and practice of RM, this study aims to enhance the understanding of the intertemporal behavior of bid prices in dynamic RM environments. We provide a probabilistic characterization of the optimal policies from the perspective of bid‐price processes. We show that an optimal bid‐price process has an upward trend over time before the inventory level falls to one and then has a downward trend. This intertemporal up‐then‐down pattern of bid‐price processes is related to two fundamental static properties of the optimal bid prices: (i) At any given time, a lower inventory level yields a higher optimal bid price, which is referred to as the resource scarcity effect; (ii) Given any inventory level, the optimal bid price decreases with time; that is referred to as the resource perishability effect. The demonstrated upward trend implies that the optimal bid‐price process is mainly driven by the resource scarcity effect, while the downward trend implies that the bid‐price process is mainly driven by the resource perishability effect. We also demonstrate how optimal bid price and consumer valuation, as two competing forces, interact over time to drive the optimal‐price process. The results are also extended to the network RM problems.     

分布式供应链中基于准时制的原油采购计划方法

本文研究了分布式供应链多供应商/多炼油厂的原油采购计划问题,建立了在有限资源约束下的准时制多目标采购优化数学模型,以实现总采购费用和供应链循环时间最小。然后,本文将非线性规划模型转化为线性规划模型,并利用层次分析法(Analytic Hierarchy Process,AHP)与多目标规划相结合的方法求解模型得到采购计划方案。数值仿真结果表明本文所提出的原油采购方法的有效性和实用性。     

产品众筹中价格歧视策略的优化研究

产品众筹是一种常见的众筹形式,它要求企业为支持者提供创新型产品或服务作为回报。合理的定价策略是产品众筹的关键,本文分别从项目成功率和利润最大化角度研究价格歧视的优化问题。针对产品众筹全有或全无（All-or-Nothing,AoN）的机制下采用价格歧视策略时,价格差值和低价销售定额的最优决策问题。本文首先采用随机模型定义购买概率函数,分析支持者在购买过程中的影响因素;然后从项目支持者效用角度出发,应用两阶段模型讨论价格差值的相关影响因素,研究发现两级价格差值会受到级联效应、项目的特征以及支持者特征的影响。并且从支持者估值类型方向对基础模型进行了扩展,讨论了不同估值类型的支持者在不同价格差值范围内的选择行为;再则从项目的利润最大化角度出发讨论价格差值问题,研究发现价格差值受支持者数量和项目目标值的影响;最后通过计算相对利润函数解出最优低价销售定额的数量。     

A joint optimal pricing and order quantity model under parameter uncertainty and its practical implementation

We consider a robust optimization model of determining a joint optimal bundle of price and order quantity for a retailer in a two-stage supply chain under uncertainty of parameters in demand and purchase cost functions. Demand is modeled as a decreasing power function of product price, and unit purchase cost is modeled as a decreasing power function of order quantity and demand. While the general form of the power functions are given, it is assumed that parameters defining the two power functions involve a certain degree of uncertainty and their possible values can be characterized by ellipsoids. We show that the robust optimization problem can be transformed into an equivalent convex optimization which can be solved efficiently and effectively using interior-point methods. In addition, we propose a practical implementation of the model, where the stochastic characteristics of parameters are obtained from regression analysis on past sales and production data, and ellipsoidal representations of the parameter uncertainties are obtained based on a combined use of genetic algorithm and Monte Carlo simulation. An illustrative example is provided to demonstrate the model and its implementation.     

On Pricing and Composition of Bundles

This paper addresses the problem of how to determine the composition and price of a bundle so as to maximize the total expected profit. To motivate the problem, we use as a setting a high‐tech manufacturing company that operates in a competitive environment, is not a leader in the industry, and is constantly reacting to bundles introduced by the leader. Bundles are sets of components that must meet technical constraints. The company's objective is to build a bundle and offer it in a market where it will compete with other bundles. Consumers purchase the bundle that maximizes their utility after examining all available bundles. The company selection of the bundle's components and its price is made in light of the bundles against which it will be competing and the uncertainty in the consumer choice process. The optimal decision could be found by solving a nonlinear mixed integer program, which is difficult to solve. Instead, we propose an efficient solution procedure to determine the optimal composition of the bundle and the price at which it should be offered. The paper concludes with a brief discussion of extensions of the research to cases that consider multiple segments of customers and/or multiple bundles.     

EXPLORING MARKET PRICING STRATEGIES VIA DYNAMIC PROGRAMMING

Price determinants as well as strategies can be studies by use of simulation, particularly if cost and price relationships can be related to market activity [1] [9] [11]. But, through the use of dynamic programming, given the market conditions, one can extend the analysis to include an optimal strategy. This paper describes a dynamic programming approach to studying price strategy. A model is developed to show that in a market characterized by cost/volume and price/volume relationships, profitability can be extended beyond that resulting from a dominant market strategy to an optimal maximizing strategy. Extension of the model is suggested for studying (a) sensitivity of a strategy (solution) to price level and cost changes, (b) optimal timing of withdrawal, and (c) present value analysis.     

Coordinating Service‐Sensitive Demand and Capacity by Adaptive Decision Making: An Application of Case‐Based Decision Theory*

The subject of this article is the simultaneous choice of product price and manufacturing capacity if demand is stochastic and service‐level sensitive. In this setting, capacity as well as price have an impact on demand because several aspects of service level depend on capacity. For example, delivery time will be reduced if capacity is increased given a constant demand rate. We illustrate the relationship between service level, capacity, and demand reaction by a stylized application problem from the after‐sales services industry. The reaction of customers to variations in service level and price is represented by a kinked price‐demand‐rate function. We first derive the optimal price‐capacity combination for the resulting decision problem under full information. Subsequently, we focus on a decision maker (DM) who lacks complete knowledge of the demand function. Hence the DM is unable to anticipate the service level and consequently cannot identify the optimal solution. However, the DM will acquire additional information during the sales process and use it in subsequent revisions of the price‐capacity decision. Thus, this decision making is adaptive and based on experience. In contrast to the literature, which assumes certain repetitive procedures somewhat ad hoc, we develop an adaptive decision process based on case‐based decision theory (CBDT) for the price‐capacity problem. Finally, we show that a CBDT DM in our setting eventually finds the optimal solution, if the DM sets the price based on absorption costs and adequately adjusts the capacity with respect to the observed demand.