Determining Optimal Selling Price and Lot Size When the Supplier Offers All-Unit Quantity Discounts*

This paper is concerned with finding the optimal price and lot size for a retailer purchasing a product for which the supplier offers all-unit quantity discounts. Demand for the product is assumed to be a decreasing function of price, and a procedure is developed for finding the optimal price and lot size for a class of demand functions. The procedure then is applied to two common demand functions: (1) the constant price-elasticity function, and (2) the linear demand function.     

A newsboy problem with an emergency order under a general backorder rate function

In this paper, a generalization is presented of the newsboy problem where an emergency lot can be ordered to provide for a certain fraction of shortage. This fraction is described by a general backorder rate function which is non-increasing with respect to the unsatisfied demand. An exponential distribution for the demand during the selling season is assumed. An expression is obtained in a closed form for the optimal lot size and the maximum expected profit. A general sensitivity analysis of the optimal policy with respect to the backorder rate function and the parameters of the inventory system is developed. When the backorder rate function is described by some particular functions, its behavior is analyzed with respect to changes in the parameters. To illustrate the theoretical results, some numerical examples are also given.     

运输成本对批量敏感时的供应链批量协调策略比较研究

以一个多周期、随机客户需求和单位产品运输成本对批量敏感的供应链为研究对象,建立了由生产商负责产品运输时,供应链分散决策情形下的最佳批量模型,并提出了基于批量折扣和改由批发商负责产品运输的供应链批量协调策略.研究结果表明:当由生产商负责产品运输时,单位产品运输成本对批量越不敏感,生产商的最佳生产批量越小,与批发商要求的短周期、小批量订货越接近;如果改由批发商负责产品运输,批发商的最佳订货批量更接近于生产商的最佳生产批量.最后,通过数例分析发现,对生产商来讲,当单位产品运输成本对批量不太敏感时,批量折扣政策优于改由批发商负责产品运输的策略,但当单位产品运输成本对批量较敏感时,后者优于前者.     

Joint pricing and ordering policy for a deteriorating inventory with partial backlogging

In this paper, a deterministic inventory model for deteriorating items with time-dependent backlogging rate is developed. The demand and deterioration rate are known, continuous, and differentiable function of price and time, respectively. Under these general assumptions, we first prove that the optimal replenishment schedule not only exists but is unique, for any given selling price. Next, we show that the total profit 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 numerical example to illustrate the algorithm.     

Ordering,Pricing, and Lead‐Time Quotation Under Lead‐Time and Demand Uncertainty

In this article, we study the newsvendor problem with endogenous setting of price and quoted lead‐time. This problem can be observed in situations where a firm orders semi‐finished product prior to the selling season and customizes the product in response to customer orders during the selling season. The total demand during the selling season and the lead‐time required for customization are uncertain. The demand for the product depends not only on the selling price but also on the quoted lead‐time. To set the quoted lead‐time, the firm has to carefully balance the benefit of increasing demand as the quoted lead‐time is reduced against the cost of increased tardiness. Our model enables the firm to determine the optimal selling price, quoted lead‐time, and order quantity simultaneously, and provides a new set of insights to managers.     

A comparison of fixed and dynamic pricing policies in revenue management

We consider the problem of selling a fixed capacity or inventory of items over a finite selling period. Earlier research has shown that using a properly set fixed price during the selling period is asymptotically optimal as the demand potential and capacity grow large and that dynamic pricing has only a secondary effect on revenues. However, additional revenue improvements through dynamic pricing can be important in practice and need to be further explored. We suggest two simple dynamic heuristics that continuously update prices based on remaining inventory and time in the selling period. The first heuristic is based on approximating the optimal expected revenue function and the second heuristic is based on the solution of the deterministic version of the problem. We show through a numerical study that the revenue impact of using these dynamic pricing heuristics rather than fixed pricing may be substantial. In particular, the first heuristic has a consistent and remarkable performance leading to at most 0.2% gap compared to optimal dynamic pricing. We also show that the benefits of these dynamic pricing heuristics persist under a periodic setting. This is especially true for the first heuristic for which the performance is monotone in the frequency of price changes. We conclude that dynamic pricing should be considered as a more favorable option in practice.     

考虑随机需求与收入共享的风险规避型V2G备用决策模型

在CVaR风险度量准则下,构建了考虑随机需求与收入共享的风险规避型V2G备用决策模型,推导了集中和分散两种决策下渠道成员最优决策行为的解析解,并进一步比较分析了随机需求变量服从均匀分布时的均衡策略。研究发现,集中决策下的最优V2G备用预留因子与渠道整体的风险规避度正相关,而均衡时的V2G备用销售价格与渠道整体的风险规避度的相关性不确定,且受到随机需求变量的分布函数影响;分散决策下的最优V2G备用预留因子仅与电网公司的风险规避度有关,而均衡时的V2G备用销售价格受到电网公司的风险规避度、购电价格以及电动汽车用户收入共享系数等的共同影响;电动汽车用户的最优V2G备用收入共享系数与其风险规避度正相关,而与电网公司的风险规避度负相关。数值仿真结果表明,在绝大多数情形下收入共享合约并不能完美协调此类V2G备用渠道的分散决策行为。     

Unraveling quantity discounts

We consider the situation in which a buying organization deals with a discrete quantity discount schedule offered by a selling organization. Furthermore, the buying organization can negotiate with the selling organization about the lot size and purchase price, but does not know the underlying function that was used by the selling organization to determine the quantity discount schedule. In this paper, we provide an analytical and empirical basis for one general quantity discount function (QDF) that can be used to describe the underlying function of almost all different quantity discount types. We first develop such a QDF analytically. Among other things, this QDF enables buying organizations to calculate detailed prices for a large number of quantities. We subsequently show that the QDF fits very well with 66 discount schedules found in practice. We discuss that the QDF and related indicators can be a useful tool in supplier selection and negotiation processes. It can also be used for competitive analyses, multiple sourcing decisions, and allocating savings for purchasing groups. Additionally, the QDF can be included in research models incorporating quantity discounts. We conclude the paper with an outlook on further QDF research regarding the characterization of commodity markets from a demand elasticity point of view.     

Evaluating manufacturer's buyback policies in a single-period two-echelon framework under price-dependent stochastic demand

This paper attempts to model the profitability of a secondary market, in a newsvendor setting, to a profit-maximizing manufacturer, who is offering to the retailer a buyback policy for the unsold merchandise left at the end of the selling season. With a buyback agreement, the manufacturer shares the risks of demand uncertainty, thus inducing the buyer to place larger orders. The manufacturer's risk is mitigated to some extent by the availability of an extra market to dispose off the unsold merchandise. Both parties are risk-neutral profit-maximizers and both have the same information about the final demand for the product and its uncertainty. The manufacturer's decision is to arrive at an optimal wholesale price and the buyback price. Based on this offer, the retailer in turn sets the optimal amount of merchandise to purchase, as well as the unit selling price to meet a price-dependent uncertain demand for the merchandise in question. Due to the difficulty of obtaining analytical results, we have undertaken a numerical analysis to (i) compare the optimal policies across demand functions and error structures for three situations namely the no-incentive case and the buyback policies with and without a secondary market; (ii) indicate the conditions whereby the trade incentive is beneficial to both parties; (iii) assess the efficacy of the policies using two other performance indices (probability of achieving a target profit, and pass-through ratios) alternate to profit maximization; and (iv) conjecture the conditions for successful buyback policies and the nature of the benefits from the secondary market.     

供应链融资结构视角下的零售商订购策略研究

本文引入供应链融资结构因素,在传统库存管理模型基础上,研究零售商在面临随机市场需求情形下考虑股权和债权等融资方式的订购策略,建立允许零售商延期支付的最优订购模型,利用最优化理论与方法得到了该模型的最优解,证明了最优解的存在性和唯一性,并分析了债权融资比例、价格成本比例系数、销售价格、订货周期、回购价格、进货价格、缺货成本、库存成本、以及融资利率等因素对最优解的影响。进一步,通过数值示例深入探讨具体市场环境下零售商的最优订购量,并进行了敏感性分析。研究发现,债权融资比例、价格成本比例系数、进货价格、订购周期、库存成本与零售商订购量存在负相关关系;缺货成本、回购价格、销售价格与零售商订购量正相关。相关研究结论可以为零售商订购行为提供科学指导,并为其库存管理提供重要决策依据。     

EFFECTS OF INTEGRATING ORDER/BACKORDER QUANTITY AND PRICING DECISIONS

We consider a monopolistic situation where the retailer aims to find the profit-maximizing selling price, and the order and backorder quantity of an item. The price and demand are assumed to be inversely related. We develop a simple optimal approach, the proportion-balancing algorithm (PBA), which utilizes the proportions of cost components to sales revenue. We demonstrate that the PBA can handle three commonly used demand functions. However, applicability of the PBA depends on how the price-demand relationship is specified. A sensitivity analysis shows that the effects of changes in cost parameters on the order and backorder quantities can be very different from those of the classical economic order quantity model with backorders.     

Combined Pricing and Portfolio Option Procurement

In this paper, we study a single‐product periodic‐review inventory system that faces random and price‐dependent demand. The firm can purchase the product either from option contracts or from the spot market. Different option contracts are offered by a set of suppliers with a two‐part fee structure: a unit reservation cost and a unit exercising cost. The spot market price is random and its realization may affect the subsequent option contract prices. The firm decides the reservation quantity from each supplier and the product selling price at the beginning of each period and the number of options to exercise (inventory replenishment) at the end of the period to maximize the total expected profit over its planning horizon. We show that the optimal inventory replenishment policy is order‐up‐to type with a sequence of decreasing thresholds. We also investigate the optimal option‐reservation policy and the optimal pricing strategy. The optimal reservation quantities and selling price are shown to be both decreasing in the starting inventory level when demand function is additive. Building upon the analytical results, we conduct a numerical study to unveil additional managerial insights. Among other things, we quantify the values of the option contracts and dynamic pricing to the firm and show that they are more significant when the market demand becomes more volatile.     

Optimal Policy for a Reseller When the Supplier Offers a Temporary Reduction in Price*

A common practice in product distribution is the case in which the supplier offers a temporary reduction in price. It is suggested in the literature that in such situations, the reseller may engage in forward buying (i.e., purchasing additional stock at the reduced price offered by the supplier for later sale at the regular selling price). In this paper, a model is formulated of the reseller's response when the supplier offers a temporary reduction in price. It is assumed that the market demand for the product is elastic with respect to the selling price the reseller sets. A procedure for determining the optimal response of the reseller is developed. The model presented in this paper can easily be adapted to the case in which the reseller faces a permanent increase in the price charged by the supplier.     

考虑消费者细分的网购拼单机制研究

本文提供理论模型分析网购拼单机制分别给销售者和消费者带来的影响，为销售者（消费者）的抉择提供参考。研究结果显示，当拼单价格和正常价格均很低时，销售者（消费者）无动机实施（参与）拼单机制。但随着正常价格的提高，拼单机制的营利能力（消费者剩余）会不断增大，进而激励销售者（消费者）实施（参与）拼单机制，并有会将低估价类型消费者驱逐出市场。这一结果从另一个方面反映了拼单机制的最终目标不是为了迎合低估价类型的消费者，而是为了吸引高估价类型的消费者参与交易。进一步地，在拼单机制有效实施的情形下，最优拼单数量、最优期望收益均与两种类型消费者估值比例密切相关。所以，在实施拼单机制过程中，应该有意识地对消费者进行细分，并挖掘不同类型消费者的私人信息，从而制定有效的拼单策略。     

双边不确定性条件下制造商质量投资与零售商销售努力决策

已有研究较少同时考察供应链中制造商产品质量信号传递和零售商销售努力调整的问题。本文考虑零售商通过努力水平改变市场需求,且制造商分摊一定比例销售努力成本的情形,制造商在决定进行质量投资后,双方对制造商能否成为高质量类型各有不同的信念,零售商只能通过观察制造商提供的契约来修正关于制造商类型的信念,并确定产品销售价格与销售努力水平。构建博弈模型分析了双方的质量投资决策、销售努力决策、定价决策以及成本分摊比例。结果表明,在不考虑质量投资成本的情形下,制造商的质量投资能够达到帕累托最优。为了激励零售商付出最优促销努力水平,销售契约中规定销售成本分摊比例与惩罚措施,由于讨价还价的存在,实际的销售成本分摊比例小于对制造商最有利的成本分摊比例。对零售商而言,制造商承担的销售成本分摊比例并非越高越好,这意味着如果超过一定范围,制造商的成本分摊反而会损害零售商的利益。     

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.     

Dynamic Pricing and Inventory Management with Regular and Expedited Supplies

We consider a periodic‐review inventory system with regular and expedited supply modes. The expedited supply is faster than the regular supply but incurs a higher cost. Demand for the product in each period is random and sensitive to its selling price. The firm determines its order quantity from each supply in each period as well as its selling price to maximize the expected total discounted profit over a finite or an infinite planning horizon. We show that, in each period if it is optimal to order from both supplies, the optimal inventory policy is determined by two state‐independent thresholds, one for each supply mode, and a list price is set for the product; if only the regular supply is used, the optimal policy is a state‐dependent base‐stock policy, that is, the optimal base‐stock level depends on the starting inventory level, and the optimal selling price is a markdown price that decreases with the starting inventory level. We further study the operational impact of such supply diversification and show that it increases the firm's expected profit, reduces the optimal safety‐stock levels, and lowers the optimal selling price. Thus that diversification is beneficial to both the firm and its customers. Building upon these results, we conduct a numerical study to assess and compare the respective benefit of dynamic pricing and supply diversification.     

Price-setting newsvendor with strategic consumers

We consider a newsvendor who sells a single product over a single season with the objective of determining both the selling price and stock quantity to maximize the expected profit. The customers are strategic and we consider two demand cases: additive and multiplicative. For each case, we derive the newsvendor׳s optimal decisions and demonstrate that neglecting the price-sensitivity of demand leads the newsvendor to make sub-optimal decisions. Moreover, we show that under certain conditions, strategic consumer behavior may positively affect the newsvendor׳s optimal expected profit in the additive demand case.     

基于补偿合约的供应链定价与能力设计的协调问题研究

研究和分析了在需求不确定并且受价格影响的情况下,供应链中制造商与供应商的产品定价与能力设计的协调问题。制造商从供应商处采购用于产品生产的关键零部件,供应链面临的潜在需求服从随机分布,有效需求则受产品定价的影响。当供应商的生产能力出现约束时,制造商可以从外部其他渠道获取关键零部件,但是需要付出一个更高的采购价格。制造商确定产品的销售价格,供应商确定生产能力。分析比较了在集成供应链与独立决策的供应链中的定价与能力计划策略,提出了一种能够有效协调制造商和供应商的决策行为的补偿合约。最后,进行了数值分析,证明补偿合约的有效性。     

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.