Publication: IIE Transactions
Publication Date: 01-SEP-07
Delivery: Immediate Online Access
Author: Hoberg, Kai ; Thonemann, Ulrich W. ; Bradley, James R.
Article Excerpt
1. Introduction
How should a supply chain react to a change in demand? According to Lee (2004) the top performing supply chains are agile, i.e., they react quickly to sudden changes in demand. Despite this rather intuitive statement, implementing an agile or responsive supply chain has many obstacles. Besides reducing lead times, accelerating information flows, and establishing trust in the supply chain, the performance of the inventory control policy is crucial.
When analyzing the performance of an inventory policy, the variability induced in the supply chain by the inventory policy has become a popular research topic in recent years (Chen, Drezner, Ryan, and Simchi-Levi, 2000; Dejonckheere et al., 2003; Hoberg et al., 2007). Among the root causes of increasing order and inventory variability as one moves up the supply chain, Lee et al. (1997) identify demand forecasting. Exponential smoothing is a forecasting technique that is commonly applied in practice, and typically implemented in supply chain software. This forecasting approach bases the new demand forecast on the previous demand forecast and the current demand observation. As the smoothing factor [alpha] increases, the current forecast puts more weight on the current demand observation and less weight on historic demand. Chen, Ryan, and Simchi-Levi (2000) and Dejonckheere et al. (2003) show that the variability induced into the supply chain when demand is stationary (Fig. 1(a)) can be reduced by decreasing the smoothing factor [alpha]. Therefore, setting the smoothing factor as low as possible seems appropriate to reduce the bullwhip effect, which drives supply chain cost. However, if demand is nonstationary, (e.g., trends or shocks occur) and a small smoothing factor is used, then the forecast would take a very long time to recognize the change in demand. Backorders and lost sales or large inventories would result.
[FIGURE 1 OMITTED]
In today's business environment nonstationary demand can be observed with increasing frequency. For example, the introduction of alternative products, competitors' price promotions and technological progress affect demand. For this reason, research increasingly addresses the case of nonstationary demand (Graves, 1999; Zhang, 2004; Gilbert, 2005). To study the performance of an inventory policy under nonstationary demand, ARIMA demand models (Box et al., 1994) are most commonly applied. An example of the resulting autocorrelation can be observed in Fig. 1(b). However, in many real-world settings demand will follow a very different pattern that cannot be described effectively by ARIMA models. For example, Fig. 1(c) conceptualizes the demand for a stock keeping unit as observed by a major European retailer. While demand is fairly stationary initially, a significant step change in demand occurs before it resumes a fairly stationary pattern albeit at a higher level. The step change in demand could be triggered by various events such as price reductions or shortages at competitors.
However, even if demand does not exhibit a step change, such a change in demand is often used as test signal to deduce a system's response to a variety of nonstationary inputs. For example, Sterman (1989) applied a step increase in demand in the beer game and it should be noted that the well-known Supply-Chain Operations Reference (SCOR) model measures supply chain performance in response to a step change in demand (Anon, 2004). Specifically, SCOR suggests that companies measure production upside flexibility, which is the time required to achieve an unplanned sustainable 20% increase in production, and supply chain response time, which measures how quickly a supply chain can return to normal operations after a significant change in demand.
Similarly, in linear control theory, the Heaviside step function is a common test signal for analyzing the responsiveness of a system. A system with a high responsiveness to a step change in the input typically has a good overall responsiveness to any input signal (demand in our case). For this reason, we apply linear control theory in this paper to analyze the responsiveness of a supply chain. In particular, we analyze the effect of the smoothing factor and the choice of the inventory policy on the responsiveness of the supply chain to nonstationary demand whilst also considering the stationary performance.
Transfer function analysis has been widely applied to inventory and supply chain management. Simon (1952) and Vassian (1955) applied linear control theory to a single-echelon inventory system. Meyer and Groover (1972) and Burns and Sivazlian (1978) were the first to analyze multiechelon problems. John et al. (1994) introduced the APIOPBCS inventory policy that bases orders on a demand forecast, the error between the target and actual inventories, and the error between target and actual work-in-progress levels. Dejonckheere et al. (2003) investigate the order amplification induced by order-up-to policies and the APIOPBCS policy. Dejonckheere et al. (2004) extend that work by analyzing the effect of information sharing on order variability. In addition to order variability, Disney and Towill (2003) study the inventory variability induced by an APIOPBCS-related inventory policy. Disney and Grubbstrom (2004) investigate the economic performance of a base stock policy in a single-echelon production system with an autoregressive demand process. Disney et al. (2006) demonstrate how inventory service levels can be incorporated into a linear control theoretic model.
The contribution of this paper is threefold. First, we present closed-form formulas for the transient behavior of the orders and the inventory in response to the Heaviside step change in demand. Applying this test signal we are able to obtain general results for the responsiveness of the inventory policies if demand is nonstationary. Second, we select appropriate performance measures for analyzing the stationary and nonstationary performance and illustrate the tradeoff between these performance measures based on the exponential smoothing factor. Third, we show the superiority of the echelon stock policy over the installation stock policy in a two-echelon setting. We also show that two performance measures that are commonly used in linear control theory are inappropriate to analyze the nonstationary performance in an inventory management context and show several properties of the relevant performance measures.
The remainder of this paper is organized as follows. In Section 2, we analyze a base stock policy in a single-echelon inventory system that forecasts customer demand with exponential smoothing. We analyze its stationary and nonstationary performance by applying a Heaviside step change in demand as a test signal and present closed-form formulas for the transient responses of the orders and the inventory. In Section 3, we analyze the nonstationary performance of two-echelon supply chains that operate under installation stock and echelon stock policies. We distinguish between two typical responses of the inventory policies based on their transient behavior and show the superiority of the echelon stock policy over the installation stock policy. We conclude in Section 4.
2. Single-echelon inventory system
In this section we analyze a single-echelon, periodic-review inventory system to gain basic insights into the dynamic behavior of a base stock policy, analyze its performance in response to nonstationary demand, and to introduce a building block for the subsequent two-echelon model. The inventory model has two parameters: the lead time L and the cover time C. While the lead time comprehends all delays between placing an order and receiving its shipment, the inventory cover time is a parameter that determines the amount of inventory that is held to hedge against demand uncertainty (Segerstedt, 1995). By applying the cover time concept rather than calculating the inventory based on demand variability (see Zipkin (2000) for the traditional approach used in inventory control literature) the analysis is largely simplified while the results are basically identical in many situations. In addition, many companies apply the cover time concept in practice, e.g., inventory planners at Gillette set the target inventory as 35 days demand (Duffy, 2004) and Dell chooses an inventory target of 10 days of demand (Kapuscinski et al., 2004).
2.1. Base stock policy
To study the dynamic behavior in the single-echelon case, we analyze the commonly applied base stock policy, which bases the order decisions on the installation stock or inventory position, i.e., the sum of the on-hand inventory and the open orders. In our inventory model the set of basic equations and the sequence of events are as...
NOTE: All illustrations and photos have been removed from this article.
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