广义均值保持变换的随机单调性及其在库存系统中的应用

本文提出一类非线性且均值可能不等的广义均值保持变换,研究实现其变换前后随机变量比较的充分条件或充分必要条件,并用此变换来定量刻画需求不确定性对库存系统决策和利润的影响。首先给出变换前后或不同参数下分布函数的关系及其满足一阶随机占优和割准则序的充分条件,特征刻画此变换与广义TTT变换之间的关系。进一步,用三类特殊的广义均值保持变换进行验证。最后,将此变换应用到报童模型中,得出该变换对包含最小化成本及最大化利润的一致化报童问题的随机单调性。     

Dynamic inventory system with pricing adjustment for price-comparison shoppers

With numerous price-comparison websites and applications, consumers today are frequently conducting price-comparison shopping. As a result, retailers face an increasing challenge in predicting consumer demand and determining the optimal product price and inventory level accordingly. To address this issue, this paper proposes an inventory model with joint decisions of price and inventory to optimize the retailer's long-run average profit under price-comparison consumer shopping. We first formulate the demand arrival process for a retailer under price-comparison shopping to be affected by not only its own price but also its competitors'. Based on this demand arrival process, we then formulate the retailer's long-run average profit and derive properties of its optimal solution. Our model focuses on capturing the impact of price-comparison consumers on a retailer's optimal price and inventory decisions. In particular, we allow competitors' prices to affect the retailer's demand via two key factors: the manufacturer's suggested price and the variability of the outside lowest price. According to our results, when the suggested price increases, the retailer should lower its price to obtain more price-comparison customers from competitors, whereas when the variability of outside lowest price increases, the retailer should raise its price to increase per unit profit from nonprice-comparison customers.     

Risk-averse order policies with random prices in complete market and retailers’ private information

We consider a retailer who orders products before the price for them becomes known. The price is an outcome of perfect competition in a complete market. Since the demand is price sensitive, the uncertainty in prices induces uncertain profits and associated risks. In this paper we show that if the retailer is risk averse and, as a result, selects a utility function of profit to maximize, then his subjective assessment of future prices is affected by the risk attitude. This, in turn, introduces a bias in retailer’s ordering policies. By considering coordinated pricing and ordering policies we derive a relationship between risk aversion, retailer’s subjective (private) assessment and the market implied, risk neutral forecast. This relationship and the induced bias are then illustrated for two typical operations management strategies which involve either inventory considerations or promotions avoiding accumulation of stocks.     

A single-period inventory placement problem for a supply system with the satisficing objective

Consider the inventory placement problem in an N-stage supply system facing a stochastic demand for a single planning period. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy demand. Stocking decisions are made before demand occurs. Unsatisfied demands are lost. The revenue, salvage value, ordering, transformation, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. The objective is to maximize the probability of achieving a given target profit level.     

需求依赖销售努力库存系统的随机比较

研究需求依赖销售努力库存系统中需求不确定性对系统最优订货量、利润和销售努力的影响.对一般需求模型给出期望利润关于订货量和努力水平为联合凹的充分条件,证明期望利润函数的超模性质.对加乘需求模型证明系统最优利润和最优努力水平都可由一类与需求分布有关的广义TTT变换来表示.通过引入定义在不同支撑分布集合上一阶、二阶和三阶随机占优,得到广义TTT变换之差与二阶和三阶随机占优之间的关系式,建立了比较库存系统最优利润或努力水平的理论基础.在一阶和二阶随机占优意义下对加乘需求模型得到比较系统最优利润和努力水平的充分条件或充分必要条件.进一步,证明存在一类需求分布当系统关键比(或市场价格)足够大时系统最优利润和努力水平随需求可变性的增加而增加.最后给出几个数值例子验证了研究结果.     

Dynamic pricing and inventory control: robust vs. stochastic uncertainty models—a computational study

In this paper, we consider a variety of models for dealing with demand uncertainty for a joint dynamic pricing and inventory control problem in a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting, where demand depends linearly on the price. Our goal is to address demand uncertainty using various robust and stochastic optimization approaches. For each of these approaches, we first introduce closed-loop formulations (adjustable robust and dynamic programming), where decisions for a given time period are made at the beginning of the time period, and uncertainty unfolds as time evolves. We then describe models in an open-loop setting, where decisions for the entire time horizon must be made at time zero. We conclude that the affine adjustable robust approach performs well (when compared to the other approaches such as dynamic programming, stochastic programming and robust open loop approaches) in terms of realized profits and protection against constraint violation while at the same time it is computationally tractable. Furthermore, we compare the complexity of these models and discuss some insights on a numerical example.     

不确定性对混合CVaR约束库存系统的影响

运用应用概率中的随机占优研究需求不确定性对混合CVaR约束库存系统最优订购量和最优利润的影响。引入刻画决策者风险态度的“风险偏好系数”,得到系统最优订购量和最优利润关于风险偏好系数的单调性。研究表明随机大需求总会导致系统较高的最优订购量和最优利润;在割准则序意义下,最优订购量可能随需求可变性的增加而增加也可能随需求可变性的增加而减少;在二阶随机占优且风险偏好系数大于等于1的情况下系统最优利润具有随机单调性,然而当风险偏好系数小于1时最优利润在二阶随机占优意义下的结论不一定成立,我们通过一个数值例子来说明。     

Coordinating pricing and inventory control in a fluctuating environment

This paper addresses the simultaneous determination of pricing and inventory replenishment strate- gies under a fluctuating environment. Specifically, we analyze the single item, periodic review model. The demand consists of two parts： the deterministic component, which is influenced by the price, and the stochastic component （perturbation）. The distribution of the stochastic component is determined by the current state of an exogenous Markov chain. The price that is charged in any given period can be specified dynamically. A replenishment order may be placed at the beginning of some or all of the periods, and stockouts are fully backlogged. Ordering costs that are lower semicontinuous, and inventory/backlog （or surplus） costs that are continuous with polynomial growth. Finite-horizon and infinite-horizon problems are addressed. Existence of optimal policies is established. Furthermore, optimality of （s,S,p）-type policies is proved when the ordering cost consists of fixed and proportional cost components and the surplus cost （these costs are all state-dependent） is convex.     

考虑即时采购价格的物流服务供应链优化

物流服务市场的不确定性会影响物流服务供应链的服务水平和收益。在考虑随机即时采购价格以及基础物流提供商在正常情况和应急情况下具有不同物流能力投资成本的情形下,引入期权机制研究不确定市场环境下物流服务供应链的优化决策,以提高物流服务供应链柔性和降低市场不确定性带来的风险。构建物流服务供应链的期权契约模型,采用Stackelberg博弈理论和优化算法分析和求得物流服务集成商的最优期权采购和即时采购策略,以及基础物流提供商的最优物流能力投资策略。结果表明即时采购价格将影响基础物流提供商和物流服务集成商的决策。最后通过数值分析研究即时采购价格的不确定程度对物流服务集成商和提供商的优化策略和利润的影响。     

Coordinating ordering and pricing decisions in a two-stage distribution system with price-sensitive demand through short-term discounting

We consider a short-term discounting model in which the distributor offers a discounted price for the retailers’ orders placed at the beginning of its replenishment cycle, in a non-cooperative distribution system with one distributor and multiple retailers, each facing price-sensitive demand. We examine the value of the price discount strategy as a mechanism for the distributor to coordinate the retailers’ ordering and pricing decisions under two common types of demand, linear demand in price and constant elasticity demand in price. Our numerical study reveals that, in the presence of homogeneous retailers (namely, retailers with identical demand rates), the distributor’s profit improvement due to coordination generally decreases as the number of retailers or the inventory holding cost rate increases, but increases as price elasticity increases. Although an increase in the inventory holding cost rate has a negative effect on the distributor’s profit, it may have a positive effect on the retailers’ profits. We further find that with heterogeneous retailers (namely, retailers with different demand rates), offering a discounted price under linear demand benefits the distributor when both the inventory holding cost rate and the variation in demand are either small or large. This cross effect, however, is absent under constant elasticity demand.     

Selling to the “Newsvendor” with a forecast update: Analysis of a dual purchase contract

We consider a supply chain in which a manufacturer sells to a procure-to-stock retailer facing a newsvendor problem with a forecast update. Under a wholesale price contract, the retailer waits as long as she can and optimally places her order after observing the forecast update. We show that the retailer’s wait-and-decide strategy, induced by the wholesale price contract, hinders the manufacturer’s ability to (1) set the wholesale price and maximize his profit, (2) hedge against excess inventory risk, and (3) reduce his profit uncertainty. To mitigate the adverse effect of wholesale price contract, we propose the dual purchase contract, through which the manufacturer provides a discount for orders placed before the forecast update. We characterize how and when a dual purchase contract creates strict Pareto improvement over a wholesale price contract. To do so, we establish the retailer’s optimal ordering policy and the manufacturer’s optimal pricing and production policies. We show how the dual purchase contract reduces profit variability and how it can be used as a risk hedging tool for a risk averse manufacturer. Through a numerical study, we provide additional managerial insights and show, for example, that market uncertainty is a key factor that defines when the dual purchase contract provides strict Pareto improvement over the wholesale price contract.     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.     

Dynamic pricing inventory control under fixed cost and lost sales

This paper studies a periodic review pricing and inventory replenishment problem which encounters stochastic demands in multiple periods. In many inventory control problems, the unsatisfied demand is traditionally assumed to be backlogged but in this paper is assumed to be lost. In many practical problems, a consumer who could not buy what he/she wants in one store is not willing to wait until that store restocks it but tries to buy alternatives in other stores. Also, in this paper, the random variable for the demand function is assumed to be general, which means that any probability function for the random variable can be applied to our result. Cost terms consist of the holding cost by the leftover, the shortage cost by lost sales, and the strictly positive fixed ordering cost. The objective of this paper is to dynamically and simultaneously decide the optimal selling price and replenishment in each period by maximizing the expected profit over the finite selling horizon. We show that, under the general assumption on the random variable for the demand, the objective function is K$K$-concave, an (s,S)$(s\text{,}S)$ policy is optimal for the replenishment and the optimal price is determined based on the inventory level after the replenishment in each period.     

Combined pricing and inventory control with multiple unreliable suppliers

We study a general finite horizon, periodic review combined inventory and pricing model with N suppliers and T periods, where both the demands and the supply mechanisms are random. The random supply mechanisms are of a general type that includes most structures encountered in practice. Demands are price dependent according to general, stochastic demand functions. We characterize the optimal combined pricing and ordering policies to all N suppliers. The general results pertain to general independent supply mechanisms. Under random capacities—one of the special random supply mechanisms—they also extend to suppliers that are positively dependent on each other.     

不确定性需求下易逝品零售商横向调货策略研究

本文通过建立易逝品零售商间横向调货模型,研究了不确定性需求与不确定性需求更新零售商之间协调调货行为下最优调货价格和调货量。研究结果表明,在不确定性需求条件下,零售商间横向调货模型中调货价格和调货量的存在纯策略纳什均衡解;在不确定性需求更新条件时,零售商间横向调货价格和调货量在任意时刻均存在纯策略纳什均衡解。数值实验也证实了理论部分的研究结论。     

Inventory systems with stochastic demand and supply: Properties and approximations

We model a retailer whose supplier is subject to complete supply disruptions. We combine discrete-event uncertainty (disruptions) and continuous sources of uncertainty (stochastic demand or supply yield), which have different impacts on optimal inventory settings. This prevents optimal solutions from being found in closed form. We develop a closed-form approximate solution by focusing on a single stochastic period of demand or yield. We show how the familiar newsboy fractile is a critical trade-off in these systems, since the optimal base-stock policies balance inventory holding costs with the risk of shortage costs generated by a disruption.     

Manufacturer’s revenue-sharing contract and retail competition

This paper investigates a revenue-sharing contract for coordinating a supply chain comprising one manufacturer and two competing retailers. The manufacturer, as a Stackelberg leader, offers a revenue-sharing contract to two competing retailers who face stochastic demand before the selling season. Under the offered contract terms, the competing retailers are to determine the quantities to be ordered from the manufacturer, prior to the season, and the retail price at which to sell the items during the season. The process of pricing and ordering is expected to result in an equilibrium as in the Bayesian Nash game. On the basis of anticipated responses and actions of the retailers, the manufacturer designs the revenue-sharing contract. Adopting the classic newsvendor problem model framework and using numerical methods, the study finds that the provision of revenue-sharing in the contract can obtain better performance than a price-only contract. However, the benefits earned under the revenue-sharing contract by different supply chain partners differ because of the impact of demand variability and price-sensitivity factors. The paper also analyses the impact of demand variability on decisions about optimal retail price, order quantity and profit sharing between the manufacturer and the retailers. Lastly, it investigates how the competition (between retailers) factor influences the decision-making of supply chain members in response to uncertain demand and profit variability.     

基于损耗率和需求不确定情况下的订购批量

库存管理模型在现实生活中有着广泛的运用。然而,在实际生活中,由于种种不确定性原因的影响,使得经典的确定型的EOQ模型的运用越来越不符合现实的需要;本文将需求和损耗率看成模糊数的同时,将物品的销售价格分成两部分来进行处理,即:没有损耗的产品以一种较高价格出售,对于有部分损耗的产品则按较低的价格出售;采用概率论置信区间估计的方法构建模糊变量的波动区间,构建使得总的利润达到最大的模糊库存模型,并利用三角模糊数、符号距离的方法以及最优化理论进行处理,得出满足条件的最优订购批量。最后,给出了模型分析和算例分析;通过分析,我们发现模糊库存模型的优点在于它自生所具有的不确定性;从数据上看,模糊库存模型比经典的库存模型更能反映出实际情况。     

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。     

Inventory rationing via drop-shipping in Internet retailing: A sensitivity analysis

In this paper, we study a threshold level inventory rationing policy that is of interest to e-tailers, operating in a business to consumer (B2C) environment and selling non-perishable, made-to-stock items such as books, CDs, consumer electronics, and body and bath products. A Monte Carlo simulation model is developed to examine this policy when the demand process is stochastic, lead-time is stochastic, and the e-tailer uses ‘drop-shipping’ as an order fulfillment option. The methodology presented, which includes computer simulation and a full factorial experimental design, permits understanding of the complexity of the decision-making environment and implications of different sources of uncertainty (e.g. demand variability and lead-time variability) on a profit-maximizing threshold level of inventory, a stock level below which low margin orders are drop-shipped directly from the e-tailer’s supplier rather than fulfilled from internal stock.