基于多级折扣价格的易逝品订货策略研究

基于价格折扣条件下的报童模型,研究了分销商面向折拐的需求价格曲线.采用呈等差数列下降的多级折扣价格销售产品,以期望利润最大为目标,给出了求解最优折扣次数的算法,进而得到相应的分销商最优订货量.与需求价格曲线的斜率b不变时相比较,b增大对最优订货量无显著影响;b减小对最优订货量有显著影响.     

A Note on Determining Optimal Selling Price and Lot Size Under All-Unit Quantity Discounts

A recent study conducted by Abad [1] described a method of determining the optimal price and lot size when the supplier offers all-unit quantity discounts. The author developed a procedure using centralized and decentralized approaches. According to the author's suggestion, further study is necessary to determine if the decentralized approach always provides an optimal solution. In the present study, an attempt is made to investigate the uniformity between the centralized and decentralized approaches and to verify whether the decentralized approach always yields optimal solutions.     

在弹性需求和物品易变质条件下数量折扣定价模型

研究在弹性需求和易变质物品条件下,当供应商和零售商独自决策时,供应商如何确定最优数量折扣问题。基于Stackelberg博奕建立了数学模型,证明了零售商的出售价格随供应商出售价格的降低而降低,从而需求量增大。当供应商给予价格折扣时,零售商的利润是增加的。在此基础上,给出了最优数量折扣的计算方法。对供应商和零售商单独决策时,供应商利用数量折扣对供应链进行协调产生的系统利润与供应商和零售商联合决策时的系统利润做了数值分析和比较。结果表明,供应商采用数量折扣的方法使供应链协调是有效的;价格折扣随价格敏感系数的增大而增大,随变质率的增大而减小。     

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.     

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.     

网络购物平台最优价格促销策略：价格折扣还是现金券？

平台价格促销已成为平台营销中的重要问题，不同于传统模式下商家决定商品的价格促销，平台价格促销是以平台作为价格促销的决策者。本文建立了平台价格折扣和现金券两种由平台作为决策者的价格促销模型，探讨平台的最优价格促销策略，并在原有模型的基础上考虑商家广告投资决策并构建了扩展模型。研究结果表明，(1)只有当商品的基本需求、消费者价格敏感度和商品价格满足一定条件时，平台价格促销才能提升平台利润。(2)在两种价格促销都能提升平台利润情况下，平台的交易费率较高且商品价格较低则平台实施现金券策略的利润高于价格折扣策略；其他情况下平台实施价格折扣策略的利润高于现金券策略。(3)在平台交易费率较低的情况下，价格折扣策略下消费者能得到更多优惠；否则现金券策略下消费者得到更多优惠。(4)增加商家广告投资能够提高平台促销活动的利润，但是平台的价格优惠降低，且平台实施价格促销活动的条件区间减小。     

Discount Pricing Policies and the Coordination of Decentralized Distribution Systems*

A challenge of supply chain management is to align the objectives, and hence coordinate the activities, of independent supply chain members. In this study, we approach this problem in a simple way by extending traditional quantity discounts that are based solely on buyers' individual order size to discount policies that are based on both buyers' individual order size and their annual volume. We show that discount policies are able to achieve nearly optimal system profit and, hence, provide effective coordination, for a decentralized two‐echelon distribution system, whereby a supplier sells a product to a group of heterogeneous and independent retailers each facing a downward‐sloping demand curve of its retail price. When buyers are heterogeneous, a critical issue of coordination is to motivate different customers to increase their demand and lot size according to their potential so as to improve profits. We show that market heterogeneity presents an effective discriminating factor for the supplier to segment customers in the design of a coordination mechanism.     

产品召回供应链中线性保险合约与订货决策

考虑一个由单供应商和单制造商组成的二级供应链,在不确定市场需求情形下考虑产品召回,站在制造商角度研究供应链线性保险合约下的最优订货决策以及最优线性保险合约决策。研究表明,最优的订货决策与产品召回概率以及线性保险系数相关,存在唯一的最优订货量来最大化制造商的利润。对于密度函数倒数的导数较大的一般需求分布函数,最优的线性保险系数与线性保险合约的固定费用以及产品召回概率有关。当产品召回概率和线性保险合约的固定费用均较小时或者产品召回概率较大但线性保险合约的固定费用适中时,制造商通过选择线性保险合约可以改善自己的利润。     

An Inventory Model with Planned Shortages and Price Dependent Demand

Previous research has yielded a procedure for a retailer to determine the optimal lot size and selling price when a supplier offers all-unit quantity discounts and demand is a decreasing function of price. In this paper, we extend that research by allowing for shortages. An algorithm is presented that determines the optimal lot size, order level, and selling price for a class of demand functions, including the constant price-elasticity and linear demand functions.     

Optimal ordering and pricing decisions for a target oriented newsvendor

This paper is the first to study pricing and target oriented decision making together in the newsvendor model. Specifically, this paper studies a newsvendor who decides on order quantity and selling price to maximize the probability of achieving both profit and revenue targets simultaneously. First, it is shown that the probability of a newsvendor achieving both targets depends critically on the relative magnitudes of the profit margin and the ratio between the profit target and the revenue target. Second, the closed-form expressions of the optimal order quantity, the optimal selling price, and the maximal profit and revenue probability are obtained. It is shown that if the product has greater price elasticity, the best strategy is always to price lower and order more.     

ECONOMIC ORDER QUANTITIES WITH QUANTITY DISCOUNTS: GRANDMA DOES IT BEST

We examine a new algorithm developed by Kuzdrall and Britney [5] for locating the optimal order quantity in the presence of quantity discounts. Their algorithm, based on a model for the supplier's formulation of the price schedule, involves a regression analysis to identify the supplier's variable cost per unit and the fixed cost that the supplier seeks to recover, followed by an iterative search for the optimum. The authors describe this method as a “convenient alternative to the aimless searching of traditional approaches” [5, p. 101]. We examine the allegation of superiority of their total setup lot-sizing model over the classical method and dispute their claim of superiority.     

Optimal price,warranty length and production rate for free replacement policy in the static demand market

In the present paper, a decision model is developed for producers in the static demand market to determine the optimal price, warranty length and production rate of a product to maximize profit based on the pre-determined life cycle. The free renewal warranty policy is considered under which failed products are renewed before the end of warranty length at no cost to consumers. The expected number of renewals based on warranty length is derived for Weibull life distributed products. The objective function includes both demand and cost functions, where production cost, warranty cost and inventory cost are involved. A solution approach using the maximum principle is described, and is applied to two specific cases of markets. The first case of market considers positive discount rate, and the second case of market considers zero discount rate. The economic sensitivity analysis is conducted to evaluate the effect of model parameters on the optimal solution. Some conclusions are drawn based on the sensitivity analysis.     

A continuous approximation method for dynamic pricing problem under costly price modifications

This paper presents a heuristic method to solve a dynamic pricing problem under costly price modifications. This is an extremely difficult nonlinear problem that has been solved only for a few special instances. Here we provide a new approach that involves an approximate reformulation of the problem, which can subsequently be solved in closed-form using elementary calculus techniques. Numerical results show that the approach is quite accurate; approximating the optimal revenue with errors usually much less than 1%. Moreover, the accuracy rapidly improves as the optimal number of price changes increases, which are precisely the cases conventional approaches would fail.     

随机需求环境下联产品系统价格与产量决策研究

基于固定比例生产技术和多产品随机需求的情形,研究了联产品制造商的两阶段产量和价格联合优化模型。通过反向倒推的优化求解,得到了联产品制造商的最优产量和价格决策以及变化规律。同时研究了需求服从均匀分布时,需求波动对均衡的影响。研究结果表明,在订货成本较低和一种产品的需求波动性较大时,当另外一种产品的波动增大,则该产品的最优订货量增大,同时价格下降。借助数值仿真,分析了价格敏感度和产出比例对最优决策和利润的影响。结果表明,在给定一种产品的价格敏感度时,另外一种的价格敏感度越大,那么该产品的价格就越低,制造商的订货量就下降。当一种产品的产出比例固定时,另一种产品的产出比例上升时,则该产品的价格下降,制造商的订货量下降,总利润上升。     

不确定需求下的冷链分销商最优决策

针对新鲜产品供应链 (冷链) 中分销商, 引入保鲜努力刻画分销商投入的保鲜工作, 考虑新鲜产品到达市场的数量与质量, 分析冷链分销商的最优决策问题.与前人研究相比, 本文使用加法形式需求函数刻画市场需求, 并使用相关企业数据验证加法形式需求函数合理性, 完善了冷链分销商决策问题的相关研究.研究发现, 当保鲜努力外生时, 分销商的最优订货数量和最优零售价格只与保鲜努力有关;当零售价格外生时, 分销商的最优保鲜努力与产品实际单位成本关于保鲜努力函数的一阶最优条件有关;当订货数量外生时, 分销商制定的零售价格随保鲜努力增加而增大, 所获得的利润也随保鲜努力递增, 但边际利润随保鲜努力递减.最后, 将所得结论应用于中国500强民营企业卓尔控股有限公司旗下的小雪冷链 (武汉) 物流有限公司, 为该公司提供相应的保鲜努力投入策略.     

Coordinating pricing and production decisions for multiple products

We consider a dynamic problem of joint pricing and production decisions for a profit-maximizing firm that produces multiple products. We model the problem as a mixed integer nonlinear program, incorporating capacity constraints, setup costs, and dynamic demand. We assume demand functions to be convex, continuous, differentiable, and strictly decreasing in price. We present a solution approach which is more general than previous approaches that require the assumption of a specific demand function. Using real-world data from a manufacturer, we study problem instances for different demand scenarios and capacities and solve for optimal prices and production plans. We present analytical results that provide managerial insights on how the optimal prices change for different production plans and capacities. We extend some of the earlier works that consider single product problems to the case of multiple products and time variant production capacities. We also benchmark performance of proposed algorithm with a commercial solver and show that it outperforms the solver both in terms of solution quality and computational times.     

考虑资金收益情况下的供应链回购契约

在制造商的资金收益率大于零售商资金收益率的情况下,以报童模型为基础,研究了资金收益对回购契约的影响.系统预期利润变为与批发价相关,利用拉格朗日方程求解了满足零售商参与约束、同时使系统和制造商预期利润取得最大值的契约参数.与不考虑资金收益情况下的供应链协调不同的是:订货批量大于相应批发价下不考虑零售商参与约束的最优订货批量;若零售商的保留利润一定,则回购价增大;回购价可以大于批发价,批发价可以小于制造商的边际生产成本.     

需求受销售价格与变质时间影响的时滞变质品供应链协调研究

在由单个生产商和单个销售商组成的两级时滞变质品供应链中，销售商面临的市场需求受销售价格和物品变质时间影响。分别建立分散决策与集中决策下的时滞变质品供应链定价与补货模型，得到两种决策模式下的销售价格与订货批量均衡解。通过比较两种决策模式发现，集中决策下的销售价格更低且订货批量和预期利润更高。引入数量折扣契约对时滞变质品供应链进行协调，并借助常相对风险规避型效用函数和纳什讨价还价理论来分配协调后的预期利润增量。通过数值算例演示了决策变量求解过程，敏感性分析给出了时滞变质品特性参数变动时对决策变量产生的影响。     

允许延期支付条件下退化性商品的销售与定购策略

针对退化性商品,允许延期支付的条件下,分别从供应商和零售商的角度出发,利用斯坦博格博弈模型给出了其如何制定最优的销售和定购策略,以实现其利润的最大化。通过算例给出了这一问题的求解算法,并且分析了利率对于信用交易期限、供应商利润、零售商利润、销售价格、订购周期及订货批量的影响。为供应商信用交易策略和零售商库存决策的制定提供依据。     

The price-setting newsvendor with service and loss constraints

Current performance measurement systems consider not only financial measures, like costs and profits, but also non-financial indicators with respect to customer service, quality and flexibility. Using the newsvendor model, we analyse the respective influence of these possibly conflicting performance measures on important operations and marketing decisions, for instance the order quantity and the selling price of a product. As in the classical newsvendor model for price-independent as well as price-dependent demand distribution, the objective of the firm is to maximise expected profit. In this paper, we also consider a service constraint—a lower bound for the level of product availability—and a loss constraint—an upper bound for the probability of a loss occurring. For both models, we provide conditions for the existence of solutions. We then analyse the influence of demand variability using a set of conditions specifying the quantiles of the predetermined performance measures: a higher variability of demand implies a smaller admissible region of the decision variables. In the price-independent case, the optimal solution has a control-limit structure: the optimal order quantity is thus given either by the classical newsvendor solution or by the control-limits corresponding to the constraints. In the price-setting model with multiplicative demand, this structure is used to check whether small admissible prices are determined by the service constraint or by the loss constraint. Using these structural results, a procedure is developed to more easily enable the computation of the optimal values of the order quantity, selling price and expected profit.