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Description
1. Introduction
Heath and Jackson (1994) use a simulation model for the manner in which sales forecasts evolve over time in order to estimate the need for safety stock in a production-distribution system. The model is fitted to historical forecast data. This approach has the advantage that it does not require a specification of the actual forecasting techniques employed by the sales department. It also captures correlations in the forecast history. These correlations are important for correctly assessing the effectiveness of any safety stock policy.
In this paper, we extend this approach to handle the situation in which the forecasting system provides forecasts at different aggregation levels over different time intervals. From a history of these aggregate-disaggregate forecasts, we seek to build a model for how these forecasts will evolve in the future. The primary use of such a model would be as input to a production and inventory planning model.
It is frequently the case in manufacturing that sales forecasts are available at the detailed product level for only a relatively short time horizon. For the rest of the forecast horizon, only aggregate sales forecasts at the product family level are available. The problem addressed in this paper is how to fit a forecast simulation model to a history of these aggregate and disaggregate forecasts.
Our approach to develop such a model is to combine a forecast update model with a forecast disaggregation model. The parameters of the two models must be estimated from the forecast history data. It is this statistical parameter estimation problem that occupies the major part of our investigation. With one exception, to our knowledge the specific statistical estimators that we use do not appear in the literature. However, they are applications of two of the most commonly used methods for finding statistical estimators, the method of moments and least squares estimation (see, for example, Shao (2003)).
The forecast update model, called the Martingale Model of Forecast Evolution (MMFE) was developed by Graves et al. (1986), Heath and Jackson (1994) and Graves et al. (1998). It is a general probabilistic model for modeling the evolution of demand forecasts over time. It can be traced to early work by Hausman (1969) which suggests modeling the time series of ratios of successive forecasts as a quasi-Markovian system. Analysis of inventory systems using the MMFE as the underlying model can be found in Gullu (1996), Chen et al. (1999), Aviv (2001), Iida and Zipkin (2001), Toktay and Wein (2001), Chod and Rudi (2003), Dong and Lee (2003), and Lu et al. (2003). A survey of many of these papers can be found in Toktay and Wein (2001).
We use a ratio-splitting model to capture the forecast disaggregation process. We are aware of very little literature for this particular problem. Gross and Sohl (1990) propose 21 different ratio-splitting disaggregation methods and recommend three methods (called methods A, F, and I) which are the most accurate in their application. Method F is identical to our method of moments estimation I. Method I is based on time series methodology and is not applicable in our Martingale setting. In the context of this paper, their method A is not competitive with our methods in either accuracy or stability over the range of different scenarios we consider.
The paper is organized as follows. In Section 2, we review the original MMFE model. In Section 3 we combine the MMFE with a forecast disaggregation model to model the forecasting system of interest. Following Heath and Jackson (1994), we provide both an additive (Section 3.1) and a multiplicative (Section 3.2) version of the model. The additive model is simpler for exposition and analysis, and in certain settings leads to more tractable inventory optimization. In particular, there exists research with theoretical results on inventory management for the additive MMFE model that have not been established, and might not hold, for the multiplicative MMFE (see, for example, Iida and Zipkin (2001) and Lu et al. (2003)). However, for realistic choices of parameters, there is a significant probability that the additive MMFE will give negative demand values; the multiplicative MMFE never does so. Secondly, in our experience, industry forecasts tend to be updated in a relative sense (as done by the multiplicative MMFE) rather than an absolute sense (as done by the additive MMFE). In Sections 4 and 5, we assume that a history of forecasts is available and we focus on the parameter estimation problem. We propose three parameter estimation techniques based on different views of the approximation process. We have no a priori basis to select one approach so, in Section 6, we present the results of numerical tests designed to measure the effectiveness of these techniques. One technique, a method of moments approach, dominates the others in effectiveness. We continue the numerical study, focusing on the method of moments technique, and draw conclusions on the sensitivity of the effectiveness of the technique to changes in the underlying parameters. In Section 7, we conclude and suggest future research.
2. The MMFE
We now describe the single-item additive MMFE, a general probabilistic model for modeling the evolution of demand forecasts. At the end of period s, demand forecasts, denoted as [F.sub.s,t], are generated for s [less than or equal to] t [less than or equal to] s + l, where l is the forecast horizon and [F.sub.s,s] is the actual demand in period s, since the forecast is made after the true demand is revealed. These forecasts form a forecast vector [F.sub.s] = ([F.sub.s,s], [F.sub.s,s+1], ..., [F.sub.s,s+l]). No forecast beyond period s + l is available, apart from a vector of estimated demand [[lambda].sub.t] for t > s + l, that is not subject to change. The [[lambda].sub.t] values reflect the long-term demand trend. When time advances to the end of the next period, period s + 1, additional information becomes available and a new demand forecast vector, [F.sub.s+1], is generated. Let [X.sub.s,t] = [F.sub.s,t] - [F.sub.s-1,t] be the forecast update made in period s for period t, s [less than or equal to] t. Then [X.sub.s] = ([X.sub.s,s], [X.sub.s,s+1], ..., [X.sub.s,s+l]) is called the forecast update vector. The MMFE is a descriptive model that characterizes the resulting sequence of forecast update vectors.
The MMFE characterizes two key properties of the forecast update vectors. The first property is that the expected updates are identically zero, i.e., E([X.sub.s]) = 0. This corresponds to the assumption that all forecasts, given the currently available information, are unbiased. The second key property is that the updates from different periods are uncorrelated, i.e., E([X.sub.s]'[X.sub.t]) = if s [not equal to] t. In Heath and Jackson (1994), this property can be justified from more basic assumptions, but we take it as an essential property that forecast updates arise from new information in each period and are therefore uncorrelated. Thus, the successive forecasts of future demands, {[F.sub.s]}, form a Martingale process, which leads to the name of this methodology. The flexibility of the MMFE comes from noting that whereas forecast update vectors from different periods and are uncorrelated, the elements in a particular forecast update vectors, [X.sub.s], can be highly correlated. For example, good (bad) news, in the sense of marketing intelligence, in a period can contribute to an increase (decrease) in demand forecasts in all future periods, which leads to positive correlation between entries. On the other hand, when the total sales are fixed due to long-term sales contracts, negative correlation tends to happen since an increase in sales forecasts in the current period should cause forecasts of sales in future periods to fall.
Note that MMFE is not a forecasting technique. Rather, it is a model of the behavior of forecasting systems and how forecast changes evolve over time. It is robust in that it does not specify a particular forecasting technique. It needs only historical forecast data as input and can be used to model forecast evolution without the specification of the particular forecasting techniques used by a forecaster.
Based on some relatively weak assumptions, Heath and Jackson (1994) show that the series of forecast update vectors {[X.sub.s]} can be modeled as a sequence of independent, identically distributed, multivariate normal random vectors with a... |
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