How to ( and How Not to ) Estimate the Salvage Value in the Newsvendor Model

Despite the importance of the Newsvendor model, little attention has been paid to its robustness with respect to the estimation of one of its inputs, the salvage value. This oversight is inappropriate because we show that the performance of the Newsvendor model is quite sensitive to the particular method chosen to estimate the salvage value. We highlight two reasons for this sensitivity: small errors in the estimate of the salvage value can lead to a signi.cant pro.t loss and, more interestingly, there exists a dependence between the estimation method used and the resulting data obtained. To explain, the ex ante salvage value inputted into the model determines the order quantity which then results in some ex post observed salvage value. An estimation method should be evaluated in terms of its equilibrium performance, i.e., the inputted salvage value equals the expected observed salvage value. We evaluate several intuitive estimation methods and . nd that they perform poorly in equilibrium. We also identify an estimation method that yields the optimal pro.t. We conclude that the Newsvendor model is robust to errors in the estimation of the salvage value as long as the proper estimation method is utilized. ¤The authors would like to thank Awi Federgruen, Marshall Fisher, Ananth Iyer, Anton Kleywegt, Martin Lariviere, Erica Plambeck, Sridhar Seshadri, Jim Smith, Matthew Sobel, Jing-Sheng Song, Christian Terwiesch, Sridhar Tayur, Jan Van Mieghem, Garrett Van Ryzin, Eitan Zemel and Paul Zipkin for their helpful comments, as well as the seminar participants at Carnegie Mellon and New York University. This paper is available via the authors’ webpages. The previous version of this paper was titled “Heuristic Equilibrium and the Estimation of the Salvage Value in the Newsvendor Model with Clearance Pricing”. The newsvendor model is certainly among the most important models in operations management. It is applied in a wide variety of areas: centralized and decentralized supply chain inventory management (e.g., Shang and Song 2003, Cachon 2003), retail assortment planning (e.g., van Ryzin and Mahajan, 1999), international operations (e.g., Kouvelis and Gutierrez 1997), horizontal competition among . rms facing stochastic demand (e.g., Lippman and McCardle, 1995), lead time competition (e.g., Li 1992), outsourcing and subcontracting decisions (e.g., Van Mieghem 1999), product and process redesign (Fisher and Raman 1996 and Lee 1996), and spot markets and inventory control (e.g., Lee and Whang 2002) to name a few. It is taught in most introductory courses in operations management and it is described in detail in most operations management text books. The newsvendor model is not complicated: the only decision is an order quantity, the purchase cost per unit is ! " units are sold during a selling season for a . xed price, #" demand is stochastic during the selling season with a known distribution, sales are bounded by the order quantity and left over inventory is salvaged at the end of the season for a . xed salvage value, $% While there is little uncertainty associated with the cost and price parameters (! and #)" and there is a substantial literature on how to choose the demand distribution, there has been surprisingly little research conducted on the remaining input to the model, the salvage value. In many applications of the model the salvage value is not a . xed amount received from left over inventory, but rather, it is the output of a clearance pricing decision: if there is little inventory left at the end of the season, the salvage value is likely to be high (a small discount is needed with a popular product in short supply) whereas if there is plenty of inventory at the end of the season, the salvage value is likely to be low (a deep discount is needed with an unpopular product in ample supply). As a result, it is not clear how historical data could be used to estimate the salvage value, nor whether the procedure for estimating the salvage value matters in terms of pro.ts. This paper studies how a manager should use historical data to estimate the salvage value when applying the newsvendor model to an actual decision. Besides the stature of this model in the operations literature, we o¤er three reasons why this is an important research topic. First, a small error in the salvage value estimate can translate into a signi.cant pro.t loss, especially if the salvage value is overestimated. This point is illustrated by the data reported in Table 1 from a simple numerical example: e.g., if demand is gamma distributed,