Underage Costs Research Articles

Managing Inventories: The Newsvendor Model

This note uses the traditional newsvendor model to present strategies for inventory management when demand is uncertain. Methods for determining the critical ratio of overage and underage costs and the optimal stock level are discussed. Excerpt UVA-OM-1456 Rev. Oct. 9, 2013 Managing inventories: The Newsvendor model Managing inventories is important in every company and under many circumstances, but it is especially critical for firms facing uncertain demand in only one selling season. This problem is commonly referred to as the newsvendor problem or newsvendor model, derived from the problem faced by a newsvendor who must decide on the number of newspapers to purchase at the beginning of the day before demand can be observed. This model, however, can be applied to any situation where the selling season is short enough that ordering decisions must be made before the season starts, such as for fashion goods, influenza vaccines and other health care products, short-life-cycle technology products (consumer electronics and telecommunications equipment), and perishable goods. Determining the Optimal Inventory Level When determining how much inventory to hold before the selling season starts, a firm needs to balance two possible outcomes. If the firm orders too much, inventory remaining at the end of the selling season will have to be sold at a discount or thrown away. But if the firm doesn't order enough, it may lose potential sales and possibly also cause customers to obtain the product from a competitor. To take those two possible outcomes into account, we define two types of costs: the overstocking cost, Co, which is the loss for the firm for each unit of leftover, and the understocking cost, Cu, which is the loss of margin for the firm for each unit of lost sales. . . .

The risk-averse newsvendor problem under spectral risk measures: A classification with extensions

We study the risk-averse newsvendor problem by defining the objective function as a spectral risk measure. We analyze the problem under different types of return formulations, focusing on the impact of risk aversion and cost parameters on the optimal ordering decision. We show that the monotonicity of the return function with respect to random demand determines the structural properties of the problem. When the return function is monotone in demand realization, optimal order quantity does not depend on the return margin but only on the overage and underage costs, and it has a monotone relation to risk aversion. However, if return is non-monotone in demand impact of risk aversion depends on the specific setting and it can also be non-monotone. Additionally, it is non-increasing in the margin which leads to varying impact of selling price under distinct settings.

Contract Preferences and Performance for the Loss-Averse Supplier: Buyback vs. Revenue Sharing

Prior theory claims that buyback and revenue-sharing contracts achieve equivalent channel-coordinating solutions when applied in a dyadic supplier–retailer setting. This suggests that a supplier should be indifferent between the two contracts. However, the sequence and magnitude of costs and revenues (i.e., losses and gains) vary significantly between the contracts, suggesting the supplier’s preference of contract type, and associated contract parameter values, may vary with the level of loss aversion. We investigate this phenomenon through two studies. The first is a preliminary study investigating whether human suppliers are indeed indifferent between these two contracts. Using a controlled laboratory experiment, with human subjects taking on the role of the supplier having to choose between contracts, we find that contract preferences change with the ratio of overage and underage costs for the channel (i.e., the newsvendor critical ratio). In particular, a buyback contract is preferred for products with low critical ratio, whereas revenue sharing is preferred for products with high critical ratio. We show these results are consistent with the behavioral tendency of loss aversion and are more significant for subjects who exhibit higher loss aversion tendencies in an out of context task. In the second (main) study, we examine differences in the performance of buyback and revenue-sharing contracts when suppliers have the authority to set contract parameters. We find that the contract frame influences the way parameters are set and the critical ratio again plays an important role. More specifically, revenue-sharing contracts are more profitable for the supplier than buyback contracts in a high critical ratio environment when accounting for the supplier’s parameter-specification behavior. Also, there is little difference in performance between the two contracts in a low critical ratio environment. These results can help inform supply managers on what types of contracts to use in different critical ratio settings. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2182 . This paper was accepted by Martin Lariviere, operations management.

Supply Chain Contracts that Prevent Information Leakage

This paper determines two groups of contracts -- downside-protection and upside-protection contracts -- that facilitate vertical information sharing in a supply chain while precluding horizontal information leakage among competing newsvendors. In particular, we consider a supply chain in which retailers replenish inventory from a common supplier to satisfy uncertain demand and are engaged in newsvendor competition. Each retailer has imperfect demand information. Yet, one of the retailers (the incumbent) has a more accurate demand forecast than the other (the entrant). Substantial anecdotal evidence and academic research have shown that information leakage among such competing retailers precludes vertical information sharing and is a reason for retailers to abandon collaborative forecast-sharing initiatives, leading to sub-optimized supply chains. We show that whether a contract can prevent information leakage is due only to how the supply chain's expected supply-demand mismatch costs are distributed among the supplier and retailers, but does not depend on each party's expected revenue. In addition, we show that wholesale-price contracts and two-part tariff contracts, which are extensively used in practice, cannot prevent information leakage. By using the wholesale-price contract as a benchmark contract, we determine that downside-protection contracts and upside-protection contracts can prevent information leakage. We define a downside-protection contract as one that effectively reduces retailers' cost of excess inventory by shifting some of their overage cost to the supplier. Examples of such contracts include buy-back and revenue-sharing contracts. We define an upside-protection contract as one that effectively increases retailers' cost of inventory shortage by shifting some of the supplier's underage cost to retailers. Examples of such contracts include penalty and rebate contracts. We show that these two groups of contracts can prevent information leakage. We identify necessary and sufficient nonleakage conditions and mechanisms for each group to prevent information leakage in the supply chain.

Implementation of gradient methods for optimization of underage costs in aviation industry

Underage costs are not easily quantifiable in spare parts management. These costs occur when a spare part is required and none are available in inventory. This paper provides another approach to underage cost optimization for subassemblies and assemblies in aviation industry. The quantity of spare parts is determined by using a method for airplane spare parts forecasting based on Rayleigh's model. Based on that, the underage cost per unit is determined by using the Newsvendor model. Then, by implementing a transformed accelerated double-step size gradient method, the underage costs for spare sub-assemblies and assemblies in airline industry are optimized.

Managing supply disruption through procurement strategy and price competition

When facing supply disruptions, the emergency procurement strategy and the optimal allocation procurement strategy are widely used strategies to manage supply risks. In this paper, buyers use these types of procurement strategies under the threat of supply disruption and engage in price competition. The structural properties of the procurement strategies are characterised by their reliability thresholds. We find that reliability thresholds play a critical role in buyer procurement strategy choices, which are related to the sales price, underage cost and differentials in unit procurement cost. A solution procedure is proposed to determine the equilibrium strategy profile. The effects of reliability levels and costs on the equilibrium prices, expected profits and equilibrium strategy profiles are explored. We extend the basic model to investigate the case of symmetric competition where buyers can freely choose their procurement strategy. The results show that in most cases, the competing buyers will choose the same strategy, whether an optimal allocation strategy with single sourcing or an emergency procurement strategy with dual sourcing. In a special parameter setting, the buyers will choose either strategy because they yield identical expected profits; this leads to multiple equilibria. We also find the equilibrium to be Pareto efficient.

Per-unit cost minimisation of the newsvendor problem

Due to the advancements in technology, the newsvendor-type items become very common; this led to great interest in newsvendor model in recent years. In this paper, we model the newsvendor problem with the objective of minimising the per-unit cost where the holding cost, setup cost, underage cost as well as overage cost are included in the model. We establish the convexity of the cost function and we then devise easy solution methods to solve the proposed model. It turns out that the proposed solution methods are based on functions similar to the standard normal loss function that is well-known in the classical stochastic inventory model. These functions are tabulated so that the optimal solution can be found without difficulty. Moreover, numerical analysis is conducted on key parameters of the model.

A Reliability-Based Approach to Nonrepairable Spare Part Forecasting in Aircraft Maintenance System

In recent times, spare parts inventory systems have been extensively researched, but most inventory models were not fully adequate due to the stochastic nature of inventory environment. This paper proposes an approach that supports decision making process in planning and controlling of spare parts in aircraft maintenance systems. Reliability characteristics of aircraft consumable parts were analyzed in order to substantiate this approach. Moreover, the proposed reliability model was used to evaluate characteristics of subassemblies and/or assemblies these parts belong to. Finally, an innovative approach for determining the total amount of parts required in inventory and underage costs, based on observing the total unit time as a stochastic process, is presented herein.

Balancing learning and economies of scale for adaptive clinical trials

Prior to the start of an adaptive clinical trial, demand for an investigational drug can be highly uncertain. Treatment length, recommended dosages, and forecasted patient recruitment can fluctuate in response to early trial results. While initial demand forecasts can be very wrong, the factors influencing future demand can be learned during the trial. To take advantage of this learning, intra-trial production and/or packaging can be leveraged, but this is done at the expense of scale economies. In this paper, we study the balance between learning and economies of scale for adaptive clinical trials. We characterize the optimal production (or packaging) decisions and through analytical and numerical studies, we develop insights on the impact of fixed costs, learning rates in terms of forecasting future demand, inventory overage costs, and inventory underage costs on the value of having intra-trial flexibility and on decisions regarding quantity and timing of drug supply.

Task Decomposition and Newsvendor Decision Making

We separate newsvendor order decisions into point forecasts, uncertainty judgments, and service level decisions in a behavioral laboratory experiment. Structuring order decisions in such a way can lead to performance improvements, and enable us to more closely examine these components. Point forecasts are influenced by the underlying overage and underage cost parameters, and suffer from demand chasing. Uncertainty judgments are subject to overconfidence, and service level decisions suffer from a cognitive dissonance bias where service levels are rarely set below 50 percent. In a set of additional experiments, we show that these biases can be partially mitigated by properly structuring and framing these lower level judgments and decisions.

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Anatomy of a Newsvendor Decision: Observations from a Verbal Protocol Analysis

An exploratory analysis of verbal protocols from a think‐aloud newsvendor experiment provided deeper insights into the decision‐making process, enabling us to formulate a number of questions that are worth answering in future research. In a think‐aloud experiment, subjects verbalize their cognitions while performing a task; responses are then recorded, transcribed, and analyzed. A majority of the subjects struggled with the abstractness of the business setting and were keen to know information on the product type, industry setting, decisions taken in the past, competitor's situation, etc. A large portion of the participants correctly identified the overage and underage costs, but failed to convert that information into the optimal order quantity. Finally, the bias in the order quantity was significantly influenced by the specific type of risk (overage or underage) that was identified closer to the decision, alluding to the presence of a recency effect. As a first application of verbal protocol analysis to inventory decision making, this study gives us an opportunity to highlight the strengths and weaknesses of this research methodology.

Role of transfer prices in global supply chains with random demands

Transfer prices are an effective strategy for improving theafter-tax profits of global supply chains with differential taxrates. Rigorous evidence of their effectiveness has beenestablished by many researchers for deterministic settings. Tothe best of our knowledge, there has been no research studying theimpact of transfer prices in stochastic supply chain settings.We attempt to fill this void by studying a two-stage supply chainin which the end-customer demand is random. This forces theretailer to behave like a newsvendor and balance overage costswith underage costs. Using a combination of analytical andcomputational techniques, we show that randomness in a supplychain magnifies the impact of transfer prices. We analyze possiblereasons behind this behavior and also summarize the impact ofvarious supply chain parameters (customer base, price elasticity,overage and underage costs, etc.) on the magnitude of profitimprovement.

Supply chain coordination with downstream operating costs: Coordination and investment to improve downstream operating efficiency

We endow the downstream firm in a supply chain with operating costs in addition to the traditional overage and underage costs. We reanalyze Lariviere and Porteus's “Selling to a Newsvendor” model (2001), allowing for non-linear production costs, and provide comparative statics. We then explore investment in reducing downstream operating costs. To overcome the fact that investment is lower in a decentralized chain than in an integrated one, we propose several coordination mechanisms––buybacks, revenue sharing and operating subsidy with a license fee.

A two-period newsvendor model with concave capacity acquisition costs

The classical newsvendor formulation does not consider multiperiod planning horizons and economies of scale in the acquisition of capacity. This paper extends the newsvendor formulation by incorporating a planning horizon consisting of more than one period and a concave capacity acquisition cost function. A solution cannot be easily obtained in the most general case. A solution procedure is provided for a special case of this problem, in which the planning horizon consists of two periods and demand in each period is independent and identically distributed as a uniform random variable. The analysis provides insights into the tradeoff between capacity acquisition costs on the one hand, and overage and underage costs on the other.

Good buy? Delaying end-of-life purchases

We study the effects of delaying an end-of-life buy. Manufacturers sometimes are required to place a final, end-of-life buy for a component that the supplier will no longer provide. The manufacturer performs a newsvendor analysis with the possible result of significant expected overage and underage costs. If the decision can be delayed, the expected overage and underage costs can be reduced. We model the effects of a delay of the final purchase under various scenarios of remaining demand. We contrast the manufacturing benefits with the costs incurred by the supplier and show that the supplier, who benefits greatly from the end-of-life buy, likely will require an incentive to enact a delay. Our results provide an insight to the observation of increasing numbers of end-of-life buys and provides a framework for analysis as manufacturers strive to cope with the issue.

Simple expressions for finding recovery system inventory control parameter values

In recent years considerable effort has been devoted to the development of inventory control models for joint manufacturing and remanufacturing. Optimality of control policies is analyzed and algorithms for the determination of parameter values have been developed. However, there is still a lack of formulae or algorithms that allow for an easy computation of optimal or near optimal policy parameter values. This paper addresses the problem of computing the produce-up-to level S and the remanufacture-up-to level M in a periodic review inventory control model. We provide simple formulae for the policy parameter values, which can easily be implemented within spreadsheet applications. The approach is to derive news-vendor-type formulae that are based on underage and overage cost considerations. We propose different formulae depending on whether lead times for production and remanufacturing are identical or not. A numerical study shows that the obtained solutions provide relatively small cost deviations compared to the optimal solution within the investigated class of inventory control policies.

Contract Assembly: Dealing with Combined Supply Lead Time and Demand Quantity Uncertainty

We consider a problem faced by a contract assembler that both assembles finished goods and procures the associated component parts for one of its major customers. Because of rapid changes in technology and ongoing engineering changes, all parts subject to obsolescence are purchased only for the current customer order. The procurement lead times of the components are random. Moreover, although the order for the finished product has a defined due date, the contract allows the customer to change the order quantity. Consequently, the assembler also faces a random demand. The assembler must determine how much to order and when to order each component part. The objective is to minimize the total expected cost, including the cost of holding components prior to their assembly, penalties for tardiness visa-vis the assembly due date, and overage and underage costs in satisfying the demand quantity. We present some structural results and discuss insights regarding optimal policies. We also present several simple heuristic policies and compare them to optimal policies. Computational results indicate that ignoring lead time variability can be costly, but relatively simple heuristics that consider lead time variability perform quite well.

MINIMAX POLICIES FOR THE PRODUCTION OF SEASONAL STYLE-GOODS

The problem considered in this note deals with the optimal minimax policies for the sequential production problem in which seasonal atyle­ goods are produced in anticipation of the actual demand in future, when the production cost and the anticipated demand in each period follow arith­ metic random walks. Also after the actual demand is known, terminal costs representing overage and underage costs are incurred at the terminal period.

Multiproduct Production Scheduling for Style Goods with Limited Capacity, Forecast Revisions and Terminal Delivery

A general mathematical model is formulated for the problem of scheduling production quantities for a group of products with seasonal stochastic demand through a common production facility. It is assumed that revised forecasts of total demand over the season for each product become available as the season progresses; delivery is not required until the end of the selling season. Limited production capacity requires that some production take place early in the season, when forecasts are less accurate. At the end of the season, there are overage costs and underage costs representing costs of producing excess quantities and opportunity costs of not producing enough units. Under certain assumptions concerning the data-generating process for forecast revisions, it is possible to formulate the entire problem as a dynamic programming problem; however, the formulation is not computationally feasible if two or more products are considered. Three heuristic approaches to the multi-product problem are presented, and their cost performance is evaluated in some numerical examples. In these particular examples, more frequent reforecasting and rescheduling produces substantial reduction in costs.