Dynamic Pricing to Minimize Maximum Regret

We consider a dynamic pricing problem that involves selling a given inventory of a single product over a short, two‐period selling season. There is insufficient time to replenish inventory during this season, hence sales are made entirely from inventory. The demand for the product is a stochastic, nonincreasing function of price. We assume interval uncertainty for demand, that is, knowledge of upper and lower bounds but not a probability distribution, with no correlation between the two periods. We minimize the maximum total regret over the two periods that results from the pricing decisions. We consider a dynamic model where the decision maker chooses the price for each period contingent on the remaining inventory at the beginning of the period, and a static model where the decision maker chooses the prices for both periods at the beginning of the first period. Both models can be solved by a polynomial time algorithm that solves systems of linear inequalities. Our computational study demonstrates that the prices generated by both our models are insensitive to errors in estimating the demand intervals. Our dynamic model outperforms our static model and two classical approaches that do not use demand probability distributions, when evaluated by maximum regret, average relative regret, variability, and risk measures. Further, our dynamic model generates a total expected revenue which closely approximates that of a maximum expected revenue approach which requires demand probability distributions.     

THE COST OF IGNORING LEAD TIME UNRELIABILITY IN INVENTORY THEORY*

A computerized cost-minimization model was used to study the importance of lead time unreliability (variability of lead time from mean lead time) in inventory management. Using different combinations of stockout cost, demand variability, mean lead time, and variability of lead time around the means, changes in optimum safety stock and in inventory costs were observed. Lead time unreliability was found to be of greater importance than either the mean lead time or the variability of demand in explaining inventory cost behavior. Managers and researchers involved in the development of inventory theory are urged to note the great financial damage that may result from ignoring lead time unreliability; a dollar measure of this penalty is included in the presentation.     

航空公司机上周转品多基地库存优化模型

在民航业日益激烈的市场竞争环境下,航空公司对成本控制的精细化管理显得尤为重要。本文基于航空公司的实际背景,针对为旅客提供的机上周转品,建立了一个多基地库存系统的库存优化模型。该模型以系统的总库存成本最小为目标,在考虑需求、回收、调运、安全库存等因素下,决策一个订货周期内的订货及调运方案。基于最优解的分析,在成本参数满足一定条件下,可以得到系统最优的期初订货量;在此基础上,可以将该模型转化为一个网络流模型,参考最小费用最大流算法,设计了一个多项式求解算法来求解该网络流模型,进而得到各基地之间的最优调运量,并证明该算法的最优性;此外,根据期初订货量可行解的范围,设计了一个求解原库存模型的启发式算法。通过实例分析,验证了该模型与两种算法的有效性。     

Assessing Service Level Targets in Production and Inventory Planning*

In production and stock planning, the relationship between customer service, defined as the ability to meet demand for finished goods from in-stock inventory, and expected profits or expected costs can be represented by a simple reliability curve. The shape of this curve depends upon the parameters of the demand process, specifically the expected level of demand, standard deviation and correlation structure, as well as upon the capacities and initial state of the production and inventory system. A model is presented which explicitly determines this trade-off curve for a firm. The model is intended both as an operational model to aid managers in setting revenue and service targets which are compatible with the capacities and resources of the firm, and as a tool for exploring relationships between the parameters of the demand process and the constraints of the physical production and inventory system. The results illustrate that the level of risk depends strongly on the variability of the demand process, the cost structure, the capacities and initial state of the system and, to a lesser extent, the correlation in demand between succeeding periods. Results suggest that establishing service level targets consistent with the firm's strategic orientation must be done in consideration of both the characteristics of the demand process and the capacities of the production and inventory system. The model provides a tool for estimating the premium above unit cost which must be paid to provide a designated service level.     

供需不确定条件下基于CVaR的零售商库存鲁棒优化模型

在经典报童模型下考虑供应和需求不确定性,研究了具有风险厌恶的零售商库存优化问题。采用条件风险值(CVaR)对库存绩效进行度量,构建了基于CVaR的零售商库存运作模型;在此基础上,考虑上游供应商供货能力和下游市场需求不确定性,并采用一系列未知概率的离散情景进行描述,给出了供需不确定条件下基于CVaR的零售商库存鲁棒优化模型。进一步,采用区间不确定集对未知情景概率进行建模,给出了基于最大最小准则的鲁棒对应模型。针对同时考虑供需不确定性导致的模型非凸性,采用标准对偶理论将其转化为易于求解的数学规划问题。最后,通过数值计算分析了不同风险厌恶程度和不确定性程度对零售商库存决策以及库存绩效的影响。结果表明,供需不确定性的存在虽然会导致零售商库存绩效损失,但损失值较小。特别地,依据文中模型得到的鲁棒库存策略在多数情况下能够保证零售商获得更优的库存绩效。此外,不确定性和风险厌恶程度的增加虽然会影响零售商库存决策和运作绩效,但在同等风险厌恶态度下,随着不确定性程度的增加,基于文中方法得到的鲁棒库存策略仍能确保零售商获得理想的库存绩效,表明文中所建模型在应对供需不确定性方面具有良好的鲁棒性。     

The impact of distributions of uncertain lumpy demand on inventories

Most studies on inventory management under demand uncertainty analyse the detrimental effects of demand variability (i.e. its coefficient of variation). In contrast, the consequences of the shape of demand distribution (e.g. its multi-modality and asymmetry) have received only minor attention. The aim of the present paper is to assess the impact of the shape of demand distribution on inventories. In particular, it considers contexts in which demand distributions have more than one 'mode' and are asymmetric, thus making requirements lumpy. The analysis is based on a simulation model which, in order to ensure comparable conditions, keeps all the other factors constant (i.e. ceteris paribus), and compares different demand distributions with the same mean and coefficient of variation. Results show that that demand shape is a key driver of inventory investment. Specifically: (i) the magnitude of the impact of multi-modality and asymmetry is comparable to that of variability; (ii) multi-modality and asymmetry can have either a negative or positive effect depending on the target service level sought. The managerial implications of these findings are addressed in the conclusions.     

On the Benefit of Inventory‐Based Dynamic Pricing Strategies

We study the optimal pricing and replenishment decisions in an inventory system with a price‐sensitive demand, focusing on the benefit of the inventory‐based dynamic pricing strategy. We find that demand variability impacts the benefit of dynamic pricing not only through the magnitude of the variability but also through its functional form (e.g., whether it is additive, multiplicative, or others). We provide an approach to quantify the profit improvement of dynamic pricing over static pricing without having to solve the dynamic pricing problem. We also demonstrate that dynamic pricing is most effective when it is jointly optimized with inventory replenishment decisions, and that its advantage can be mostly realized by using one or two price changes over a replenishment cycle.     

Buffer stock under the effect of fluctuating demand

This paper deals with an inventory situation where the rate of demand for units stocked is known, but fluctuating over time. The focus of attention is on reserve stock. In an ideal world, where demand is known well in advance and where suppliers maintain scheduled shipping dates, there would be little need to hold any form of inventory other than a limited in-process stock. In practice however, supply interruptions are likely to occur. In this case an inventory acts as buffer to withstand variations between supply and demand. A mathematical model to determine the optimum reserve stock needed to minimize the expected total inventory cost of carrying the reserve inventory and cost of stockouts is developed.     

周期性需求下基于延迟订货的库存控制

本文在假设产品需求为周期性需求,订货周期为需求周期的整数倍,不能满足的需求在一定的时间间隔内允许缺货的前提下,提出一种确定性库存模型,并在此基础上,以满足单位时间内利润最大化为目标,研究并确定了优化订货批量、订货周期的时间间隔长度和缺货的时间长度的算法,最后给出算例,并验证了算法的可行性.     

短生命周期产品的三种库存模型的比较

讨论短生命周期产品库存模型,考虑无形变质因素,并假定无形变质率与需求率成反比关系.在线性需求和常数生产率的条件下,假设,若需求率为1,则视作产品的市场生命周期结束.并且,在产品生命周期的前期不存在缺货的情况下,给出了短生命周期产品分别在理想状态、允许缺货以及价格折扣导致需求率变化等3种状态下的库存模型,最后在相同参数条件下对这3种模型进行比较,找出了单位时间内平均总成本最小的最优库存模型,厂商可以根据自身的情况选择合适的库存策略.     

The cost of using stationary inventory policies when demand is non-stationary

Non-stationary stochastic demands are very common in industrial settings with seasonal patterns, trends, business cycles, and limited-life items. In such cases, the optimal inventory control policies are also non-stationary. However, due to high computational complexity, non-stationary inventory policies are not usually preferred in real-life applications. In this paper, we investigate the cost of using a stationary policy as an approximation to the optimal non-stationary one. Our numerical study points to two important results: (i) Using stationary policies can be very expensive depending on the magnitude of demand variability. (ii) Stationary policies may be efficient approximations to optimal non-stationary policies when demand information contains high uncertainty, setup costs are high and penalty costs are low.     

一种需求和采购价均为时变的EOQ模型

本文提出了一种需求和采购价均为时变的EOQ模型,证明了该模型的总库存成本目标函数在给定条件下为凸函数,给出了寻求最佳采购次数及服务水平的算法,并对该模型进行了数值仿真和灵敏度分析。     

Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates

This paper considers an inventory system with non-instantaneous deteriorating item in which demand rate is a function of advertisement of an item and selling price. This paper aids the retailer in maximizing the total profit by determining optimal inventory and marketing parameters. In contrast to previous inventory models, an arbitrary holding cost rate and arbitrary deterioration rate have been incorporated to provide general framework to the model. First, a mathematical model is formulated and then some useful theoretical results have been framed to characterize the optimal solutions. The necessary and sufficient conditions for the existence and uniqueness of the optimal solutions are also derived. An algorithm is designed to find the optimum solutions of the proposed model. Numerical examples are included to illustrate the algorithmic procedure and the effects of key parameters are studied to analyze the behavior of the model.     

A hybrid inventory management system responding to regular demand and surge demand

This paper proposes a hybrid policy for a stochastic inventory system facing regular demand and surge demand. The combination of two different demand patterns can be observed in many areas, such as healthcare inventory and humanitarian supply chain management. The surge demand has a lower arrival rate but higher demand volume per arrival. The solution approach proposed in this paper incorporates the level crossing method and mixed integer programming technique to optimize the hybrid inventory policy with both regular orders and emergency orders. The level crossing method is applied to obtain the equilibrium distributions of inventory levels under a given policy. The model is further transformed into a mixed integer program to identify an optimal hybrid policy. A sensitivity analysis is conducted to investigate the impact of parameters on the optimal inventory policy and minimum cost. Numerical results clearly show the benefit of using the proposed hybrid inventory model. The model and solution approach could help healthcare providers or humanitarian logistics providers in managing their emergency supplies in responding to surge demands.     

基于混合差分进化算法的联合补货-配送优化模型

本文综合考虑联合补货与配送决策,研究了随机需求、允许缺货环境下多企业多产品联合补货与配送集成优化模型,设计了混合差分进化算法(Hybrid Differential Evolution, HDE)对该模型进行求解,同时通过算例与遗传算法、标准的DE算法进行了比较,证实HDE算法高效且稳定;另外,设计了一个先补货再配送的两阶段优化模型,对比优化结果发现采用供应链协同时补货成本较高,配送成本较低,且总成本较低。最后,对相关参数进行了敏感性分析,发现需求率和库存维持成本的变动对总成本的影响远远大过次要订货成本对总成本的影响。     

Supply Performance in Multi‐Echelon Inventory Systems with Intermediate Product Demand: A Perspective on Allocation

In this article, we study the performance of multi‐echelon inventory systems with intermediate, external product demand in one or more upper echelons. This type of problem is of general interest in inventory theory and of particular importance in supply chain systems with both end‐product demand and spare parts (subassemblies) demand. The multi‐echelon inventory system considered here is a combination of assembly and serial stages with direct demand from more than one node. The aspect of multiple sources of demands leads to interesting inventory allocation problems. The demand and capacity at each node are considered stochastic in nature. A fixed supply and manufacturing lead time is used between the stages. We develop mathematical models for these multi‐echelon systems, which describe the inventory dynamics and allow simulation of the system. A simulation‐based inventory optimization approach is developed to search for the best base‐stock levels for these systems. The gradient estimation technique of perturbation analysis is used to derive sample‐path estimators. We consider four allocation schemes: lexicographic with priority to intermediate demand, lexiographic with priority to downstream demand, predetermined proportional allocation, and proportional allocation. Based on the numerical results we find that no single allocation policy is appropriate under all conditions. Depending on the combinations of variability and utilization we identify conditions under which use of certain allocation polices across the supply chain result in lower costs. Further, we determine how selection of an inappropriate allocation policy in the presence of scarce on‐hand inventory could result in downstream nodes facing acute shortages. Consequently we provide insight on why good allocation policies work well under differing sets of operating conditions.     

An integrated supplier selection and inventory problem with multi-sourcing and lateral transshipments

In this research, we consider the supplier selection problem of a firm offering a single product via multiple warehouses. The warehouses face stationary, stochastic demand and replenish their inventory via multiple suppliers, to be determined from a set of candidates, with varying price, capacity, quality, and disruption characteristics. Additionally, the warehouses may simultaneously replenish their inventory from other warehouses proactively. With these characteristics, the problem is a multi-sourcing, supplier selection, and inventory problem with lateral transshipments. Even though the benefits of multi-sourcing and lateral transshipments have been presented in the literature individually to mitigate risks associated with uncertain demand and disrupted supply, the intertwined sourcing and inventory decisions under these settings have not been investigated from a quantitative perspective. We develop a decomposition based heuristic algorithm, powered with simulation. While the decomposition based heuristic determines a solution with supplier selection and inventory decisions, the simulation model evaluates the objective function value corresponding to each generated solution. Experimental results show, contrary to the existing literature, inferior decisions may result when considering the selection of suppliers solely on unit and/or contractual costs. We also evaluate the impact of multi-sourcing with rare but long disruptions compared to frequent but short ones.     

Robust Stochastic Lot‐Sizing by Means of Histograms

Traditional approaches in inventory control first estimate the demand distribution among a predefined family of distributions based on data fitting of historical demand observations, and then optimize the inventory control using the estimated distributions. These approaches often lead to fragile solutions whenever the preselected family of distributions was inadequate. In this article, we propose a minimax robust model that integrates data fitting and inventory optimization for the single‐item multi‐period periodic review stochastic lot‐sizing problem. In contrast with the standard assumption of given distributions, we assume that histograms are part of the input. The robust model generalizes the Bayesian model, and it can be interpreted as minimizing history‐dependent risk measures. We prove that the optimal inventory control policies of the robust model share the same structure as the traditional stochastic dynamic programming counterpart. In particular, we analyze the robust model based on the chi‐square goodness‐of‐fit test. If demand samples are obtained from a known distribution, the robust model converges to the stochastic model with true distribution under generous conditions. Its effectiveness is also validated by numerical experiments.     

Real options valuation principle in the multi-period base-stock problem

This paper analyzes the multi-period base-stock problem where there is a financial risk associated with a stochastic demand. For the single-period problem, it is known that the optimal inventory policy can be obtained with the Black and Scholes option pricing formula. This paper pushes the analysis further by applying the options valuation framework to the multi-period problem and presenting an algorithm for finding the optimal inventory policy. A computational study indicates that the effect of systematic risk is typically negligible (as for the single-period problem). Therefore, it can be concluded that systematic risk in demand is of little importance for optimal inventory control.     

区分仓库与货架库存需求影响效应的零售商库存决策

既往有关库存水平影响需求条件下的库存问题研究中,通常对终端库存水平是否存在货架与零售商仓库库存水平的区别未作深入探讨。本文的研究认为,现实中许多零售商拥有仓库,其现有库存水平包括仓库库存和货架库存两部分,而影响需求的仅为与货架展示能力相关的库存,因此有必要对二者的需求影响效应进行区分。在明确这一区别的前提下,本文首先建立了供应商管理库存情况下库存水平影响需求问题的一般库存模型,给出零售商的最优订货策略;并考虑货架的容量限制,给出零售商启用仓库的判断条件。由于仓库库存仅在能够影响货架展示能力的条件下才能够影响消费需求,本文还进一步讨论了在零售商拥有仓库时,区分货架与仓库的库存水平影响需求条件下的最优库存与订货决策。这对于经营不同特征商品的零售商在进行是否需要拥有仓库,以及拥有仓库条件下的库存决策具有很好的参考价值。