Using Data From Problem 31 On Page 128 Of Your Book
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 corresponding to a 12-month moving average, and 3) Winter's method using Centered MA12, including calculations of MAD for each method.
Paper For Above instruction
The assignment requires developing three distinct forecasting methods for January 2015 based on data from Problem 3.1 on page 128 of the referenced textbook. These methods include a simple moving average, exponential smoothing, and Winters’ seasonal method. Each approach utilizes different techniques to capture the data's underlying trend and seasonal components, providing comprehensive forecast insights for the specified period.
Introduction
Forecasting is a critical element of supply chain management and business planning, allowing organizations to anticipate future demand and adjust their operations accordingly. In this context, three forecasting methods—simple moving average, exponential smoothing, and Winters' method—offer distinct approaches suitable for different data characteristics, such as trend and seasonality. This paper develops forecasts for January 2015 utilizing these methods, grounded in the data from Problem 3.1, and evaluates their accuracy via MAD (Mean Absolute Deviation).
Data Overview and Assumptions
Since the specific data from Problem 3.1 is not provided here, it is assumed to include monthly observations spanning multiple years, exhibiting seasonal patterns with a period of 12 months. The data's historical context is essential for choosing appropriate smoothing parameters and models, especially considering the seasonal nature and potential trend components.
1. 12-Month Moving Average
The simplest approach involves calculating a 12-month moving average to smooth out seasonal fluctuations and identify underlying trends. The forecast for January 2015 is obtained by averaging months December 2014 through January 2014, depending on the available data.
Calculation Steps:
- Sum the observed values over the last 12 months.
- Divide the sum by 12 to obtain the average.
- Use this average as the forecast for January 2015.
This method assumes that the recent 12 months' average provides a stable estimate of the next period's demand, effectively smoothing out irregular variations.
2. Exponential Smoothing with Corresponding Alpha
Exponential smoothing assigns exponentially decreasing weights to older observations, making it more responsive to recent changes. To align the smoothing factor alpha with a 12-month moving average, the value of alpha is derived from formula 3.26, which approximates alpha as:
\[
\alpha = 1 - e^{-\frac{1}{n}}
\]
where \( n = 12 \). Substituting,
\[
\alpha = 1 - e^{-\frac{1}{12}} \approx 1 - e^{-0.0833} \approx 1 - 0.9200 = 0.08
\]
Initialization:
The initial level for exponential smoothing is the average of the first 11 months, computed as:
\[
L_0 = \frac{1}{11} \sum_{i=1}^{11} y_i
\]
Subsequently, the recursive formula is:
\[
L_t = \alpha y_t + (1 - \alpha) L_{t-1}
\]
Forecast for January 2015 is then:
\[
\hat{y}_{2015\,Jan} = L_{t}
\]
which is the last smoothed level after all historical data is processed.
3. Winters’ Method Using Centered MA12
Winter's method is suitable for data with trend and seasonal patterns. It involves estimating seasonal components and trend, combined with a smoothing model.
Centered MA12 (moving average):
Since the data exhibits seasonality with period 12 months, the centered MA12 is calculated by averaging two consecutive 12-month moving averages:
\[
\text{Centered MA}_t = \frac{MA_{t-6} + MA_{t-5}}{2}
\]
because the middle of 12 months is between months 6 and 7, specifically at 6.5.
Calculating MA2 for Centering:
The MA2 component helps to estimate the trend:
\[
MA2_t = \frac{(y_{t-1} + y_t)}{2}
\]
Forecasting:
Using formulas 3.34 and 3.35, the smoothing parameters for Holt-Winters’ additive model are computed:
\[
\alpha_{HW} = \text{as per formula 3.34} \quad \text{and} \quad \beta_{HW} = \text{as per formula 3.35}
\]
with \(\gamma_{HW} = 0.15\) (given). The level, trend, and seasonal components are updated iteratively based on the historical data, and the forecast for January 2015 is derived from the sum of the level, trend, and seasonal components.
Calculation of MAD:
Mean Absolute Deviation (MAD) is used to evaluate forecast accuracy:
\[
MAD = \frac{1}{n} \sum_{i=1}^{n} | y_i - \hat{y}_i |
\]
Calculated separately for each method by comparing the forecasted values with actual known data points (from the historical series), MAD guides the assessment of each forecasting approach's precision.
Conclusion
By applying these three methods, the best model for predicting January 2015 can be identified based on MAD values. Moving averages offer robust smoothing for stable data, exponential smoothing provides adaptability to recent changes, and Winters’ method captures seasonal variations. Each method's suitability depends on the data's underlying characteristics, making this comprehensive approach valuable for effective seasonal forecasting.
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