3.8 Forecast Accuracy Measures
In this section, we will calculate forecast accuracy measures such as Mean Absolute Deviation (MAD), Mean Squared Error (MSE), and Mean Absolute Percentage Error (MAPE).
Mean Absolute Deviation (MAD)
The Mean Absolute Deviation (MAD) measures the average magnitude of the forecast errors without considering their direction (positive or negative). It is calculated as the arithmetic mean of the absolute differences between the forecasted and actual values:
[latex]MAD\;=\;\frac{\sum\left|\text{Actual Value}\;-\;\text{Forecasted Value}\right|}n[/latex]
Where n is the number of observations.
MAD is expressed in the same units as the data, making it easy to interpret and understand. A lower MAD value indicates better forecast accuracy.
To obtain the MAD of the given data, the following steps should be followed:
Step 1: Calculate forecast errors, i.e. find the difference between forecast and actual data.
Step 2: Take an absolute of forecast error, i.e. consider the positive of even those negative forecast error values.
Step 3: Find the average of these absolute values. (Singla, 2018)
Mean Square of Errors (MSE)
The Mean Squared Error (MSE) is a measure that squares the forecast errors before averaging them. This squaring process gives more weight to larger errors, making MSE more sensitive to outliers compared to MAD:
[latex]MSE\;=\;\frac{\sum\left(\text{Actual Value}\;-\;\text{Forecasted Value}\right)^2}n\\[/latex]
MSE is expressed in squared units of the data, which can make interpretation more difficult. However, it is a statistically valuable measure as it is used in many advanced forecasting models and optimization techniques.
To obtain the MSE of given data, the following steps should be followed:
Step 1: Calculate forecast errors, i.e. find the difference between forecast and actual data.
Step 2: Take a square of each forecast error value.
Step 3: Find the average of these squared values. (Singla, 2018)
Mean Absolute Percent Error (MAPE)
The Mean Absolute Percentage Error (MAPE) is a relative measure that expresses the forecast errors as a percentage of the actual values. It is particularly useful when working with data from different scales or units:
[latex]MAPE\;=\;\frac{\frac{\sum\left|\text{Actual Value}\;-\;\text{Forecasted Value}\right|}{\text{Actual Value}}\times100\%}n\\[/latex]
MAPE is expressed as a percentage, making it easy to interpret and compare across different data sets. However, it can be problematic when dealing with actual values close to zero, resulting in extremely high or undefined MAPE values.
To obtain MAPE of given data, the following steps should be followed:
Step 1: Calculate forecast errors, i.e. find the difference between forecast and actual data.
Step 2: Take an absolute of forecast error, i.e. consider the positive of even those negative forecast error values.
Step 3: Divide each absolute value by actual demand value and multiply it by 100 to get data in percentage terms.
Step 4: Find the average of these calculated percentage values. (Singla, 2018)
The computation of methods of forecast error has been illustrated in the following example.
Example
The following actual demand and forecast values are given for the past four periods. We want to calculate MAD, MSE and MAPE for this forecast to see how well it is doing.
Note that Abs (et) refers to the absolute value of the error in period t (et).
| Period | Actual Demand | Forecast | et | Abs (et) | et2 | [Abs (et) ÷ Dt] × 100% |
|---|---|---|---|---|---|---|
| 1 | 63 | 68 | ||||
| 2 | 59 | 65 | ||||
| 3 | 54 | 61 | ||||
| 4 | 65 | 59 |
Here are the steps:
Step 1: Calculate the error as et = Dt − Ft (the difference between the actual demand and the forecast) for any period t and enter the values in the table above.
Step 2: Calculate the absolute value of the errors calculated in step 1 [i.e., Abs (et)], and enter the values in the table above.
Step 3: Calculate the squared error (i.e., et2) for each period and enter the values in the table above.
Step 4: Calculate [Abs (et) ÷ Dt] x 100% for each period and enter the value under its column in the table above.
Solution
| Period | Actual Demand | Forecast | et | Abs (et) | et2 | [Abs (et) ÷ Dt] × 100% |
|---|---|---|---|---|---|---|
| 1 | 63 | 68 | -5 | 5 | 25 | 7.94% |
| 2 | 59 | 65 | -6 | 6 | 36 | 10.17% |
| 3 | 54 | 61 | -7 | 7 | 49 | 12.96% |
| 4 | 65 | 59 | 6 | 6 | 36 | 9.23% |
“3 Forecasting” from Introduction to Operations Management Copyright © by Hamid Faramarzi and Mary Drane is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted. —modifications: added additional explanations
