提前期、构建成本和短缺量拖后率均可控的EOQ模型

传统库存模型通常将提前期和构建成本视为不可控制。事实上可以通过追加投资缩短提前期和降低构建成本。缺货期间,为减少订单丢失量和补偿顾客的损失,供应商会给予一定的价格折扣。现实库存系统中,容易得到需求的期望值和标准差,但较难得到其分布规律。基于此,考虑短缺量拖后率与价格折扣和缺货期间库存水平相关,提出了一种需求为任意分布且提前期和构建成本均可控的EOQ模型,证明了模型存在唯一最优解,给出了一种寻优算法。数值仿真分析表明,一般情况下,压缩提前期和降低构建成本能降低订购批量和安全库存,降低库存总成本;短缺量拖后系数和缺货概率对库存总成本影响较大,企业应尽量降低缺货概率,尤其在短缺量拖后系数较小时。     

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.     

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.     

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。     

促进低碳产品需求的供应链减排成本分担模型

减排成本的存在,导致低碳产品的零售价格偏高,市场需求偏低,从而制约低碳产品的推广。以此为背景,针对单一制造商和单一零售商构成的双边垄断供应链,建立了基于减排成本分担契约的Stackelberg模型,分析比较了有、无减排成本分担契约两种情况下订货量、供应链各成员利润及供应链整体利润变化。研究发现,引入减排成本分担契约后,低碳产品最优订货量增加,零售价格降低,且在一定条件下制造商和零售商的利润得到帕累托改进,并推导出了制造商和零售商的参与约束和最优分担率。在此基础上,进一步分析了消费者对低碳产品的认知与零售商的产品订货量、供应链整体利润呈正相关关系,与制造商承担的减排成本分担率呈负相关关系。最后结合数值分析验证了成本分担契约的有效性。     

Decision Making and the Price Setting Newsvendor: Experimental Evidence

We present an experimental study of the price‐setting newsvendor problem, which extends the traditional framework by allowing the decision maker to determine both the selling price and the order quantity of a given item. We compare behavior under this model with two benchmark conditions where subjects have a single decision to make (price or quantity). We observe that subjects deviate from the theoretical benchmarks when they are tasked with a single decision. They also exhibit anchoring behavior, where their anchor is the expected demand when quantity is the decision variable and is the initial inventory level when price is the decision variable. When decision makers set quantity and price concurrently, we observe no significant difference between the normative (i.e., expected profit‐maximizing) prices and the decision makers’ price choices. Quantity decisions move further from the normative benchmarks (compared to when subjects have a single decision to make) when the ratio of cost to price is less than half. When this ratio is reversed, there is no significant difference between order levels in single‐ and multi‐task settings. In the multidecision framework, we also observe a tendency to match orders and expected demand levels, which subjects can control using prices.     

Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost

In this paper, an economic order quantity inventory model is analyzed, considering that the unit cumulative holding cost has two significant components: a fixed cost which represents the cost of accommodating the item in the warehouse and a variable cost given by a potential function of the length of time over which the item is held in stock. Shortages are allowed and, during the stockout period, only a fraction of demand is partially backordered. The backordering cost includes a fixed cost and a cost linearly dependent on the length of time for which backorder exists. A solution procedure is developed for determining the optimal inventory policy. Moreover, to illustrate the effects of some parameters on the optimal policy and the minimum total inventory cost, a numerical study is developed.     

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。     

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.     

Optimal Quantity Discount Design with Limited Information Sharing*

In this article we address the optimal quantity discount design problem of a supplier in a two‐stage supply chain where the supplier and the buyer share annual demand information only. The supply chain faces a constant deterministic demand that is not price sensitive and operates with fixed setup costs in both stages. We show that the supplier can actually moderate a cost‐minimizing buyer to order in quantities different than the buyer's optimal order quantity in the traditional setting and develop a multi‐breakpoint quantity discount scheme that maximizes supplier's expected net savings. The proposed multi‐breakpoint discount scheme can be easily computed from the available information and, while also maximizing the supplier's net savings, is very effective in achieving high levels of supply chain coordination efficiency in the presence of limited information.     

Quantity Flexibility Contracts: Optimal Decisions with Information Updates*

We study single and multiperiod quantity flexibility contracts involving one demand forecast update in each period and a spot market. We obtain the optimal order quantity at the beginning of a period and order quantities on contract and from the spot market at the then prevailing price after the forecast revision and before the demand materialization. The amount that can be purchased on contract is bounded by a given flexibility limit. We discuss the impact of the forecast quality and the level of flexibility on the optimal decisions and managerial insights behind the results.     

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.     

Impacts of demand uncertainty on retailer's dominance and manufacturer-retailer supply chain cooperation

Recent studies in marketing and distribution channels have shown that the balance of power between manufacturers and retailers is shifting. Based on this observation, we investigate a two-echelon supply chain with a manufacturer and a retailer in this paper. We first develop retailer-dominant non-cooperative game models by introducing a sensitivity of retailer's order quantity to manufacturer's wholesale price; then we analyze two cooperative scenarios, in which the Nash bargaining model is utilized to implement profit sharing between the manufacturer and the retailer. Under the assumption that the manufacturer and the retailer are risk-neutral, we find that the manufacturer and the retailer can bargain to cooperate at any level of retail-market demand uncertainty with exogenous retail price. However, the cooperation is conditional on retail-market demand uncertainty with endogenous retail price: it can be implemented if the fluctuation of retail-market demand is relatively small, and the measure of retail-market demand uncertainty does not exceed an upper bound. Theoretical and numerical analyses show that the retailer's dominance over the manufacturer increases with the increase in the sensitivity of retailer's order quantity to manufacturer's wholesale price under a limitation of retail-market demand uncertainty. Numerical analyses also show that the retailer's dominance decreases with the increase in retail-market demand uncertainty.     

An EOQ model with deteriorating items in response to a temporary sale price

In economic order quantity models, it is often assumed that the unit purchase cost is constant. Such an assumption is usually not fulfilled in many practical situations. In practice, it is observed that suppliers sometimes offer temporary price discounts to stimulate demand, boost market share or decrease inventories of certain items. In this paper, a deteriorating inventory model with a temporary sale price has been developed. We shall be concerned with finding the optimal total cost saving for deteriorating items during the special replenishment period. Numerical examples are presented to illustrate the proposed model.     

信息不对称下价格随机的应急数量折扣契约研究

以市场需求和市场价格随机波动的二级供应链为研究对象,寻找其在信息不对称条件下绩效优化的路径。通过显示原理分别构建生产成本和销售成本信息不对称时应急供应链的数量折扣契约模型,得到最优订货量与最佳批发价策略。通过具体的算例加以验证,分析了信息不对称程度对供应链上成员及整个供应链绩效的影响。研究结果表明：当突发事件造成市场价格随机波动和市场需求变化时,供应链上的跟随者能从隐瞒私人信息中获利。但当市场需求增大时,整个供应链绩效受损,反之则反是。这与以往研究在市场需求随机、市场价格固定和信息不对称条件下的结论迥异。     

多类顾客环境下报童模型中库存分配策略研究

考虑一个报童模型中多类顾客的库存分配问题,将顾客按照他们愿意支付价格的高低划分为不同级别。零售商在销售期初决定产品订货量,并在销售期内决定接受或者拒绝不同顾客的需求,以最大化销售期内的期望总利润。将销售期分成大量足够小的时间单位,通过建立一个反向Bellman动态规划方程,以优化每个时间单位内的库存分配策略,并得到了零售商最优的期初订货量。通过与没有库存分配策略下零售商的期望利润进行比较,算例分析得出库存分配策略可以大幅提高零售商的利润。这主要是因为通过库存分配可以使得零售商从高端顾客中获取更多利润,同时能够减小期初的订货量,以节约采购成本和库存持有成本。     

考虑保鲜努力与数量/质量弹性的农产品三级供应链协调优化

以供应具有季节性的农产品为研究对象,研究了由农户-第三方物流供应商（TPLP）-零售商组成的三级供应链协调问题。将农产品的数量和质量损耗均视为TPLP保鲜努力的内生变量,分别建立了分散与集中决策模式下的动态博弈模型。分析表明分散决策模式下供应链各成员的决策会导致产品销售量和销售价格扭曲,严重影响供应链绩效。针对三级供应链中的两个交易过程,并基于对系统中相关决策变量的变化及影响因素分析,分别设计了成本分担和收益共享协调契约,并论证了在一定条件下该组合协调机制可以显著扩大农产品的销售量,实现供应链各成员收益的帕累托改进。最后,通过算例验证了文章结论,并分析了努力水平对不同农产品质量和数量损耗的改善程度（数量/质量弹性）对农产品三级供应链决策及协调效果的影响。     

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.     

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.     

Inventory‐Based Dynamic Pricing with Costly Price Adjustment

We study an average‐cost stochastic inventory control problem in which the firm can replenish inventory and adjust the price at anytime. We establish the optimality to change the price from low to high in each replenishment cycle as inventory is depleted. With costly price adjustment, scale economies of inventory replenishment are reflected in the cycle time instead of lot size—An increased fixed ordering cost leads to an extended replenishment cycle but does not necessarily increase the order quantity. A reduced marginal cost of ordering calls for an increased order quantity, as well as speeding up product selling within a cycle. We derive useful properties of the profit function that allows for reducing computational complexity of the problem. For systems requiring short replenishment cycles, the optimal solution can be easily computed by applying these properties. For systems requiring long replenishment cycles, we further consider a relaxed problem that is computational tractable. Under this relaxation, the sum of fixed ordering cost and price adjustment cost is equal to (greater than, less than) the total inventory holding cost within a replenishment cycle when the inventory holding cost is linear (convex, concave) in the stock level. Moreover, under the optimal solution, the time‐average profit is the same across all price segments when the inventory holding cost is accounted properly. Through a numerical study, we demonstrate that inventory‐based dynamic pricing can lead to significant profit improvement compared with static pricing and limited price adjustment can yield a benefit that is close to unlimited price adjustment. To be able to enjoy the benefit of dynamic pricing, however, it is important to appropriately choose inventory levels at which the price is revised.