Forecasting error distribution

The parameters that the proposed methodology requires from a forecasting module are the mean forecast (6) and the forecast error distribution for each uncertain parameter that is considered in the predictive model. 

The forecast errors distributions depend on previous errors (e ) and on how many periods ahead the forecast is being done. If the correct forecasting model has been chosen, and if the statistical procedure used to estimate parameters in the model yields unbiased estimates, then the expected forecast error will be zero. When the forecasting error (e) is assumed to be normally distributed (7V(0, forecasting error for the period ahead can be calculated by means of Eq. (7.57). This approximation holds even for non-normally distributed errors. For more details about estimation of forecast errors, please refer to Montgomery et al. (1990). 

According to Armstrong and Fildes (1995), the objective of a forecast accuracy measure is to provide an informative and clear understanding of the error distribution. Theoretically, when the forecast errors are randomly structured, the form of the forecasts is independent of the selected accuracy measure. Otherwise, it is generally accepted that there is no single best accuracy measure, and deciding on the assessment method is essentially subjective. In this study, a simple form of relative error (E) is selected as the forecast accuracy measure, since it offers a number of desirable properties ... 

For clarity, this example first calculated the forecast error. It then took the absolute value of that forecast error, and, finally, the average of the absolute value was calculated. The MAD gives us a measure of the distribution of the forecast error, so we can estimate a minimum estimate and a maximum estimate for demand. For example, if our forecast model forecast 130 for period 6, we would know that that if the actual demand was within 1 MAD from the point estimate of demand (i.e., 130) it could be as high as 140 and as low as 120. Statisticians have calculated that 1 MAD = 0.80 standard deviations, so that 3.75 MADs is equivalent to 3.00 standard deviations (see Melnyk and Denzler, 1996, for more information). Because 3.0 standard deviations includes 99.87% of the population, when forecasters provide a forecast and the value of one MAD, they allow the user to quickly determine the accuracy of the forecast. This is illustrated in Figure 3.2. In this example, 3.0 standard deviations means that there is a 99.87% chance that the actual demand will be between 39 of the forecast of 130. Or, the actual demand has a 99.87% chance of being between 91 and 169. This is calculated as 130 (3.75) (10.4). 

The terms and Ef stand for the forecast errors of the retailer s and the supplier s estimates of their lead-time demand. We are particularly interested in the lagged forecast errors i.e., that made by the supplier during a particular period, and that made by the retailer r periods later. Proposition 15 tells us that the cost performance of the supply chain is uniquely determined by the safety stock parameters 7 and 7, and by the characteristics of the joint distribution of the lagged forecasts E] and which is described in the next theorem. 

Lawrence [68] uses a forecasting based approach for estimating flow times. In particular, he focuses on approximating the flow time estimation error distribution (which is assumed to be stationary, and denoted by G) by using the method of moments [111]. He uses six methods for forecasting flow times. NUL sets flow times to zero, such that G becomes an estimator for the flow time. ESF uses exponential smoothing, such that the flow time estimate after the completion of k jobs is fk = otFk + (1 — oc)fk-i- The other four rules... 

When the difference data approximation obeys normal distribution, we can use Matlab statistics function parameters for the forecast error, such as average value, variance and significant level d of confidence interval estimation. Meanwhile, it is examined whether the difference between unknown average error parameter is equal to the estimation of mean value. 

We then estimate that the mean of the random component is 0, and the standard deviation of the random component of demand is a. MAD is a better measure of error than MSE if the forecast error does not have a symmetric distribution. Even when the error distribution is symmetric, MAD is an appropriate choice when selecting forecasting methods if the cost of a forecast error is proportional to the size of the error. 

When a forecast method stops reflecting the underlying demand pattern (for instance, if demand drops considerably as it did for the automotive industry in 2(X)8-2(X)9), the forecast errors are unlikely to be randomly distributed around 0. In general, one needs a method to track and control the forecasting method. One approach is to use the sum of forecast errors to evaluate the bias, where the following holds ... 

As discussed in Chapter 7, demand has a systematic as well as a random component. The random component is a measure of demand uncertainty. The goal of forecasting is to predict the systematic component and estimate the random component. The random component is usually estimated as the standard deviation of forecast error. We illustrate our ideas using uncertain demand for a smartphone at B M Office Supplies as the context. We assume that periodic demand for the phone at B M is normally distributed with the following inputs ... 

Consider a buyer at Bloomingdale s who is responsible for purchasing dinnerware with Christmas patterns. The dinnerware sells only during the Christmas season, and the buyer places an order for delivery in early November. Each dinnerware set costs c = 100 and sells for a retail price of p = 250. Any sets unsold by Christmas are heavily discounted in the post-Christmas sales and are sold for a salvage value of i = 80. The buyer has estimated that demand is normally distributed, with a mean of p = 350. Historically, forecast errors have had a standard deviation of a- = 150. The buyer has decided to conduct additional market research to get a better forecast. Evaluate the impact of improved forecast accuracy on profitability and inventories as the buyer reduces o- from 150 to 0 in increments of 30. 

Application of our model requires a method to estimate demand probability distributions. This is particularly challenging as there was no sales history for any of the new dresses. However, we were able to calculate forecast errors defined as the difference between buyer forecast and actual sales for similar products appearing in the same catalog from the past 2 years. We used this information to conclude that the distribution of forecast errors were normally distributed with a large degree of confidence (x test holds at a = 0.01 level). We assumed that forecast errors would follow a similar distribution in past and future seasons. This seemed reasonable as the same individuals who forecast product demand in the past seasons were also forecasting current season demand. 

Since the demand for any given product is equal to its forecast plus the associated forecast error, this implies that the demand distribution for U for a given product during the entire season is normally distributed. While probability distributions for retail products seem to have long tails, these result from plotting actual demand for products that seem indistinguishable (or at least similar) ex ante. However, in contrast, U represents the demand distribution for a given product. 

To estimate normal parameters p and o of this distribution, we implemented the procedure developed by Fisher and Raman (1996). In this method, the members of a committee (comprised of four buyers in our case), independently provide a forecast of sales for each product. The mean p is set to the average of these forecasts. The standard deviation of demand a is set to 0cTc, where Oo is the standard deviation of the individual committee member forecasts and the factor 0 is chosen so that the average standard deviation of historical forecast errors equals the average standard deviation assigned to new products. In our application, we found 0 to be 1.4. 

The classical, frequentist approach in statistics requires the concept of the sampling distribution of an estimator. In classical statistics, a data set is commonly treated as a random sample from a population. Of course, in some situations the data actually have been collected according to a probability-sampling scheme. Whether that is the case or not, processes generating the data will be snbject to stochastic-ity and variation, which is a sonrce of uncertainty in nse of the data. Therefore, sampling concepts may be invoked in order to provide a model that accounts for the random processes, and that will lead to confidence intervals or standard errors. The population may or may not be conceived as a finite set of individnals. In some situations, such as when forecasting a fnture value, a continuous probability distribution plays the role of the popnlation. 

Multiple forecasts for the same line of business within the organization are common (if any planning is even done). The gap between what is planned and what actually happens represents lost profits and lost opportunities. The new paradigm for retail supply chain management begins with an accurate view of customer demand. That demand drives planning for inventory, production, and distribution within some understood error parameters. Consumers will never be completely predictable. At the same time, prediction is bounded by limitations in our statistical and modeling sciences. We can... 

The empirical assessment of experts relative error of estimates revealed that over 45% of errors were close to one (expert estimate true value). Additionally, lognormal was identified as one of the best fitted distributions, considering the selection of relative error as the forecast accuracy measure. The study also showed 285% average improvements in experts estimates with 77% of estimates improved, applying the likelihood function developed by relative errors for homogenous and nonhomogenous cases. 

Remember, no forecast will be correct. If there is just random error, then about 50% of the time the forecasts will be higher than the actual demand and about 50% of the time the forecasts will be lower than actual demand. If the MFE is close to zero, then the forecast model is not biased. If the MFE is negative then the forecasts are biased on the high side of the distribution (i.e., the forecast is too large). If the MFE is positive then the forecasts are biased on the low side of the distribution. In Table 8.3, the MFE is 6.4. This means that if there is any bias, it is that that the forecast is consistently low. 

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