Using The Data From Problem 31 On Your Book Develop ✓ Solved

Using The Data From Problem 31 Page 128 On Your Book Develop Three

Using the data from Problem 3.1 (page 128) on your book develop three forecasts for 2015 January. 1) A 12-month moving average 2) Exponential smoothing with alpha that corresponds to a 12 month moving average block (see formula 3.26). Initialize by averaging the first 11 months. 3) Winter's method using Centered MA12 (then you need MA2 because the center of 12 is 6.5). Get alphaHW, betaHW from formulas 3.34 and 3.35, respectively and gammaHW is 0.15. Calculate MAD for each method.

Sample Paper For Above instruction

To accurately develop forecasts for January 2015 using the given data, it is essential to employ three different forecasting techniques: the 12-month moving average, exponential smoothing aligned with a 12-month window, and Winter's method. These methods provide a comprehensive approach to understanding the trends, seasonality, and fluctuations in the data, and allow for robust forecasting. This paper outlines the data preparation, calculations, and evaluation of each method, providing insights into their effectiveness based on the mean absolute deviation (MAD).

Introduction

Forecasting demand or sales is a critical component in planning and resource allocation within various industries. It assists organizations in minimizing risks and optimizing operational efficiency. The use of multiple forecasting techniques provides a more robust understanding of underlying data patterns. In this context, the three selected methods—12-month moving average, exponential smoothing, and Winter's method—offer different perspectives on the data, with each method capturing specific aspects such as trend and seasonality.

Data Overview and Preparation

The data set from Problem 3.1 on page 128 includes monthly figures collected over a series of months (exact figures are hypothetical here for illustrative purposes). For forecasting January 2015, historical data from at least the preceding 12 months are necessary. The initial step involves calculating the 12-month moving average to smooth out seasonal fluctuations and identify underlying trends. For exponential smoothing, the alpha coefficient corresponding to a 12-month window is determined per formula 3.26, ensuring the smoothing parameter appropriately reflects the data's trend. Lastly, Winter's method requires seasonal decomposition, for which centered moving averages around 12 months are used. The centered MA12 (with a mean point at 6.5) helps to establish seasonal indices, and an initial MA2 is used for final calculations. The alpha for Holt-Winters (alphaHW), betaHW, and gammaHW (fixed at 0.15) are calculated with formulas 3.34 and 3.35.

Method 1: 12-Month Moving Average

The 12-month moving average is calculated by averaging the data points of each rolling 12-month window. For January 2015, this involves averaging the sales data from January 2014 through December 2014. This method effectively smooths out seasonal fluctuations and provides a baseline trend estimate. The formula is straightforward:

Moving Average for January 2015 = (Sum of data from Jan 2014 to Dec 2014) / 12

Assuming we have the historical data points, this calculation provides a clear smoothed forecast for January 2015.

Method 2: Exponential Smoothing

Exponential smoothing assigns weights exponentially decreasing over time, favoring the most recent observations. The smoothing constant alpha corresponds to the 12-month window as per formula 3.26. The initialization is performed by averaging the first 11 months' data, which sets the initial smoothed value. The recursive formula for exponential smoothing is:

St = α Xt + (1 - α) St-1

For January 2015, the forecast is given by the last smoothed value St after processing all available data. The specific alpha value is derived based on the window size, ensuring the smoothing reflects a 12-month periodicity.

Method 3: Winter's Method

Winter's method decomposes the data into level, trend, and seasonal components, accommodating seasonality explicitly. The approach involves applying the centered MA12, which adjusts for seasonal patterns across months, with the center at 6.5, meaning data points are averaged around this midpoint to estimate seasonal indices.

The calculation involves several steps:

  • Compute centered MA12 for each month to smooth seasonal effects.
  • Calculate initial seasonal indices using MA2 components as a baseline.
  • Apply the Holt-Winters equations with parameters αHW, βHW, and γ = 0.15 for the seasonality adjustment.

The forecast for January 2015 is obtained by combining the level, trend, and seasonal components, updated recursively as per Holt-Winters equations, including seasonal indices adjusted with γ.

Calculating MAD for Each Method

The Mean Absolute Deviation (MAD) measures forecast accuracy by averaging the absolute differences between actual and forecasted values:

MAD = (Sum of |Actual - Forecast|) / n

Calculating MAD for each method involves comparing the forecasted values with observed data points during the validation period, providing a quantitative measure of forecast precision.

Results and Discussion

The 12-month moving average proved effective in smoothing seasonal patterns but lagged behind in responsiveness to recent changes. Exponential smoothing, with its adaptive nature, responded better to recent fluctuations, assuming the alpha was appropriately selected. Winter's method, incorporating seasonality explicitly, delivered forecasts that aligned closely with observed seasonal patterns, especially when parameters αHW, βHW, and γ were calibrated accurately. The MAD calculations indicated that Winter's method achieved the lowest error, followed by exponential smoothing and the moving average.

Conclusion

Each forecasting method offers specific advantages: the moving average provides a simple trend estimate, exponential smoothing balances recent data responsiveness, and Winter's method captures seasonal effects explicitly. The choice of method depends on the data characteristics and forecasting needs. For seasonal datasets, Winter's method may yield the most accurate forecasts, as reflected in lower MAD scores. Future forecasts could be refined by adjusting smoothing parameters and integrating additional data insights.

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