Contracting in a Supply Chain with Stochastic Demand and Substitute Products

Retailers often stock competing products from multiple manufacturers. When the retailer stocks out of a particular item, customers who prefer the item are likely, with some probability, to switch to a substitute product from another manufacturer at the same store. In such an event, a “lost sale” for the manufacturer is not a “lost sale” for the retailer. This exacerbates differences in manufacturer's and retailer's stockout costs for the item. Such differences in stockout cost influence the optimal contract between the manufacturer and the retailer and also impose agency costs on the channel. Such contracts, in turn, determine equilibrium inventory levels and fill rates. We study these issues in a single‐period supply chain, consisting of a manufacturer and a retailer, under three different scenarios (when the two firms are integrated into a single entity, when the retailer makes stocking decisions, and when the manufacturer makes stocking decisions). We compare, and present a methodology for comparing, stocking quantities, manufacturer efforts, and supply chain profits across different scenarios. We find that VMI performs better when manufacturer effort is a substantial driver of consumer demand and when consumers are unlikely to substitute to another brand in case of a stockout. On the other hand, if non‐contractible manufacturer effort is unimportant, or when substitution is significant, VMI can exacerbate, rather than mitigate, channel inefficiencies, and can perform worse than traditional Retailer Managed Inventory.     

Stockout risk of production-inventory systems with compound Poisson demands

Production-inventory systems with continuous production or continuous manufacturing have been implemented in a variety of manufacturing contexts. Most recently, the Commissioner of the FDA has called on drug and biological product manufacturers to begin switching from batch manufacturing processes to continuous production. Motivated by prevailing applications and the emerging and promising landscape in the healthcare and pharmaceutical industries, this paper studies a continuous-review production-inventory system with a constant production rate and compound Poisson demands, in which the cost of the system is assessed via inventory holding, stockout penalty and production costs. For any initial inventory, we derive a closed-form expression for the expected discounted cost function until stockout occurrence. We systemically quantify the stockout risk on four different dimensions (i.e., time, volume, frequency and percentage) and derive explicit expressions for each type of risk metric. The objective is to derive the production rate that minimizes the expected discounted system cost subject to a given risk tolerance level on stockouts. With the aid of the derived explicit forms of stockout risk and the cost function, we develop a computationally-efficient algorithm for the optimal solution. Extensive numerical studies are conducted to illustrate our results with rich insights. Numerically, we show that it is outrageously costly to reduce stockout risk, especially when this risk is relatively low; the value of risk is more sensitive to the stockout risk level if the demand distribution has a higher volatility.     

Two Backorder Compensation Mechanisms in Inventory Systems with Impatient Customers

We study a compensation mechanism design problem with customer‐choice behavior in a continuous review setting where the production and demand processes are stochastic. When a stockout occurs, the firm controls backorders on the basis of certain compensation policies. Customers make decisions to maximize their utility, which is decreasing in the price, the waiting time, and the customer's impatience factor. We assume that the impatience factor is private information held by the customer only. Two compensation mechanisms are designed to control backorders, namely uniform compensation and priority auction with an admission price. Under uniform compensation, the firm offers the same discount to all customers, whereas under auction compensation, priority is granted according to the customers' bid prices. We obtain the optimal stockout price and base stock level under each mechanism, and analyze the properties of the respective optimal policies. Assuming linear waiting costs with uniformly distributed impatience factor, we find that the auction mechanism (1) maintains a lower base stock level and results in greater profit and (2) benefits customers with relatively lower or higher impatience factors, but customers with a medium impatience factor may be rendered worse off. We further show that both compensation mechanisms are suitable for products with a high unit profit, a high lost sales penalty cost, and a high holding cost.     

QUANTIFYING THE IMPACT OF INVENTORY HOLDING COST AND REACTIVE CAPACITY ON AN APPAREL MANUFACTURER'S PROFITABILITY

This paper was motivated by the operational problems faced by Northco, a school uniform manufacturer in the Northeastern United States. Northco was facing high working capital costs while also incurring high stockout and markdown costs. This paper models the impact of inventory holding cost and reactive capacity on Northco's targeted understocking and overstocking cost and offers a solution methodology for such problems. We quantify the impact of varying inventory carrying costs (and hence, high working capital costs) on stockout costs and the value of additional capacity. Our results illustrate that apparel manufacturers with high working capital costs, and hence high inventory carrying costs, should target higher stockout costs and achieve lower capacity utilization. The results presented have application beyond Northco because high working capital cost is endemic to many supply chains.     

Retailer Transshipment versus Central Depot Allocation for Supply Network Design

We consider a supply chain structure with shipments from an external warehouse directly to retailers and compare two enhancement options: costly transshipment among retailers after demand has been realized vs. cost‐free allocation to the retailers from the development of a centralized depot. Stochastic programming models are developed for both the transshipment and allocation structures. We study the impact of cost parameters and demand coefficient of variation on both system structures. Our results show an increasing convex relationship between average costs and demand coefficient of variation, and furthermore that this increase is more pronounced for the allocation structure. We employ simulation and nonlinear search techniques to computationally compare the cost performance of allocation and transshipment structures under a wide range of system parameters such as demand uncertainty and correlation; lead times from the external warehouse to retailers, from warehouse to central depot, and from depot to retailers; and transshipment, holding, and penalty costs. The transshipment approach is found to outperform allocation for a broad range of parameter inputs including many situations for which transshipment is not an economically sound decision for a single period. The insights provided enable the manager to choose whether to invest in reducing lead times or demand uncertainty and assist in the selection of investments across identical and nonidentical retailers.     

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.     

Contracting Under Vendor Managed Inventory Systems Using Holding Cost Subsidies

Vendor managed inventory systems are becoming increasingly popular. An important issue in implementing a vendor managed inventory scheme is the contracting terms that dictate the ownership of the inventory and the responsibility of inventory replenishment decisions. Thus the performance of a vendor managed system crucially depends on these terms and on how inventory‐related costs are shared in a supply chain. We consider a system where a manufacturer supplies a single product to a retailer who faces random demand in a competitive market. The retailer incurs a fixed cost per order, inventory holding cost, and a penalty cost for a stockout (unsatisfied demand is back‐ordered). Further, the manufacturer incurs a penalty cost when there is a stockout at the retailer and a fixed replenishment cost. We assume that the players are rational and act noncooperatively. We compare the performance of retailer managed inventory systems, where the retailer places orders and makes replenishment decisions, with vendor managed inventory systems, wherein the vendor or manufacturer makes inventory and replenishment decisions. Specifically, in the vendor managed inventory system, we propose and evaluate holding cost subsidy‐type contracts on inventories offered by the retailer to improve system performance. We evaluate this contract in the context of three widely used inventory systems—deterministic economic order quantity, continuous review (Q, r) policies, and periodic review policies—and show when such contracts may improve channel performance.     

Mean–variance analysis of the newsvendor model with stockout cost

We study the risk-averse newsvendor model with a mean–variance objective function. We show that stockout cost has a significant impact on the newsvendor's optimal ordering decisions. In particular, with stockout cost, the risk-averse newsvendor does not necessarily order less than the risk-neutral newsvendor. We illustrate this finding analytically for the case where the demand follows the power distribution.     

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.     

RELIABILITY-CONSTRAINED PRICING,CAPACITY, AND QUALITY: AN ILLUSTRATION OF ECONOMIC CONCEPTS TO POSTAL SERVICES

A model is developed from which welfare-optimal prices, capacities, and reliabilities for a service provider are simultaneously determined. Solutions are determined under conditions of stochastic demand subject to a reliability constraint on service quality. Both quality of service provided, as well as price, impact on demand for services rendered. Results indicate that (i) optimal prices are equated to the reliability-constrained marginal costs, (ii) optimal reliabilities require that the marginal benefits of increasing reliability are equated to the marginal costs of doing so, and (iii) optimal capacity allocation involves minimizing the system's expected costs subject to meeting the prespecified reliability constraint for service quality. The model is applied to postal delivery services in light of the growing competition that has emerged in this industry.     

Measuring and Achieving Equity in Multiperiod Emergency Material Allocation

Emergency material allocation is an important part of postdisaster emergency logistics that is significant for improving rescue effectiveness and reducing disaster losses. However, the traditional single‐period allocation model often causes local surpluses or shortages and high cost, and prevents the system from achieving an equitable or optimal multiperiod allocation. To achieve equitable allocation of emergency materials in the case of serious shortages relative to the demand by victims, this article introduces a multiperiod model for allocation of emergency materials to multiple affected locations (using an exponential utility function to reflect the disutility loss due to material shortfalls), and illustrates the relationship between equity of allocations and the cost of emergency response. Finally, numerical examples are presented to demonstrate both the feasibility and the usefulness of the proposed model for achieving multiperiod equitable allocation of emergency material among multiple disaster locations. The results indicate that the introduction of a nonlinear utility function to reflect the disutility of large shortfalls can make the material allocation fairer, and minimize large losses due to shortfalls. We found that achieving equity has a significant but not unreasonable impact on emergency costs. We also illustrate that using differing utility functions for different types of materials adds an important dimension of flexibility.     

Optimal responses to stockouts: Lateral transshipment versus emergency order policies

In the retail industry, stockouts have a significant effect on a firm׳s profitability. When a stockout takes place, retailers often apply one of two strategies to resolve the issue – placing an emergency order with their supplier or arranging a lateral transshipment with a nearby partner store. Choosing the optimal response to a stockout is complicated by customers׳ spontaneous reactions. Customers who find that a product is out of stock may choose to give up on the purchase, to wait for delivery (through emergency order or lateral transshipment), or go to a partner store to search for the product on their own. In this study, under a single-period setting with two retail stores, we investigate the optimal inventory decisions under each strategy, and conduct a comparison between lateral transshipment and emergency order options. We also analyze the effects of the customer requesting rate and switching rate on the optimal inventory decision. Through numerical analysis, the two strategies are compared in terms of inventory levels and profitability. The results suggest that in addition to the cost associated with each of these strategies, the customers׳ behavior in response to a stockout has a significant effect on the optimal decision. The emergency order strategy is a better option when more customers request deliveries or when more customers switch to another store. Extending this analysis, we also examine the combined strategy when an emergency order is placed after a transshipment fails to fulfill unmet demands, and explore the circumstances under which this strategy provides the highest additional profit for the stores. Finally, we also find that a higher requesting rate does not necessarily increase profits, particularly when there is a high customer switching rate, because requesting emergency order or transshipment reduces switching demand.     

A Simple Algorithm for the Mathematical Programming Approach to Joint Cost Allocation for Decision Making (Demand-Adjusted-Sales-Value Algorithm)*

This paper investigates the mathematical programming (MP) approach for decision making and cost allocation in the context of joint products. Through analysis of the MP approach to marginal-value-based cost allocations, an algorithm is developed to show that the allocations contain two elements: an element that is specific to individual joint products due to differences in demand elasticities, and an element based on relative adjusted sales values. If demand elasticities are identical for all joint products, then the traditional sales value method will generate the same cost allocation as the MP method. The new algorithm, the demand-adjusted-sales-value algorithm (DASV), is especially helpful for solving the MP model with linear demand functions. The DASV algorithm provides an easy way to generate costs that are useful for optimal decision making.     

Cross‐Training with Imperfect Training Schemes

Cross‐training workers is one of the most efficient ways of achieving flexibility in manufacturing and service systems for increasing responsiveness to demand variability. However, it is generally the case that cross‐trained employees are not as productive on a specific task as employees who were originally trained for that task. Also, the productivity of the cross‐trained workers depends on when they are cross‐trained. In this work, we consider a two‐stage model to analyze the effects of variations in productivity levels on cross‐training policies. We define a new metric called achievable capacity and show that it plays a key role in determining the structure of the problem. If cross‐training can be done in a consistent manner, the achievable capacity is not affected when the training is done, which implies that the cross‐training decisions are independent of the opportunity cost of lost demand and are based on a trade‐off between cross‐training costs at different times. When the productivities of workers trained at different times differ, there is a three‐way trade‐off between cross‐training costs at different times and the opportunity cost of lost demand due to lost achievable capacity. We analyze the effects of variability and show that if the productivity levels of workers trained at different times are consistent, the decision maker is inclined to defer the cross‐training decisions as the variability of demand or productivity levels increases. However, when the productivities of workers trained at different times differ, an increase in the variability may make investing more in cross‐training earlier more preferable.     

REFINEMENTS IN THE ISMAIL-LOUDERBACK STOCHASTIC CVP MODEL*

This paper presents results concerning the Ismail-Louderback stochastic cost-volume-profit model. For the case where demand is normally distributed, an analytic expression for the output that maximizes the probability of achieving specified profits is obtained and a new procedure for determining the output that maximizes the profits, given a target level of probability of their being achieved, is developed. If opportunity costs of lost sales are zero, then analytic expressions for both the above outputs are obtained for any demand distribution.     

需求不确定条件下的制造商订单分配模型

制造商的订单分配作为供应链模型微观层面的重要组成部分,对提升整个供应链效率有很大影响,但需求层面的不确定因素加大了订单分配的难度。以按比例分配为原则,讨论在需求不确定条件下完全信息与不完全信息两类多供应商-单制造商的订单分配模型。重点研究完全信息条件下各方的分散决策和集中决策,由于后者能避免各参与方对其他决策方的边际影响,所以能够实现供应链总利润的最大化;其次又将不完全信息引入模型,讨论制造商如何通过折算因子结合已有信息对供应商的私人信息进行估计,进而做出决策。最后以需求服从正态分布为例对两类模型进行验证。     

多受灾点应急物资动态多阶段分配模型研究

针对传统的单阶段物资分配模型可能导致应急物资分配的局部冗余或短缺、高成本、系统无法达到全局最优等现实情况,通过在指数效用函数中引入灾民物资需求的比例短缺测度公平,以物资短缺的延迟损失最小化与物资分配的总成本最小化为目标构建考虑多集散点、多配送中心和多受灾点的三级配送网络的应急物资动态多阶段分配模型,设计了目标转化与线性近似相结合的模型求解方法,并通过算例对所提出模型的有效性和可行性进行了验证。结果表明：所提出的多阶段模型能够兼顾物资分配的效率与公平,最大程度地降低物资短缺的延迟损失以及物资分配的总成本;运用灾民物资需求的比例短缺量化公平,避免了由于各受灾点的需求量差异而对公平分配产生的影响,可以使各受灾点即使在应急救援初期物资有限、中期物资持续供不应求等情况下,仍然能够在每阶段获得一定比率的所需物资,进而避免较大的物资短缺损失,确保多受灾点之间多阶段应急物资分配的公平性,更符合灾害救援实际,可为现实大规模灾害应急救援物资多阶段分配提供决策支持。     

OPTIMALITY OF STATE-DEPENDENT (s,S) POLICIES IN INVENTORY MODELS WITH MARKOV-MODULATED DEMAND AND LOST SALES

Markov-modulated processes have been used in stochastic inventory models with setup costs for modeling demand under the influence of uncertain environmental factors, such as fluctuating economic and market conditions. The analyses of these models have been carried out in the literature only under the assumption that unsatisfied demand is fully backlogged. The lost sales situation occurs in many retail establishments such as department stores and supermarkets. We use the analysis of the Markovian demand model with backlogging to analyze the lost sales case; in particular, we establish the optimality of an (s, S)-type policy under fairly general conditions.     

The Economic Impacts of a Terrorist Attack on the U.S. Commercial Aviation System

We use data on air passenger travel expenditures per passenger as well as statistical analysis of the air traffic lost for the two-year aftermath of the September 11, 2001, attacks to estimate direct demand losses for air transportation services. These are used along with a national input-output model to assess the full costs of these losses. Depending on assumptions made, the full losses to the U.S. economy were between $214.3 and $420.5 billion. These estimates are similar to those from other studies of such an event, and suggest that the high costs of effective countermeasures may be justified.