This appendix contains these topics:
Section A.4, "Method 2 - Calculated Percent Over Last Year,"
Section A.6, "Method 4 - Moving Average,"Section A.7, "Method 5 - Linear Approximation,"
Section A.14, "Method 12 - Exponential Smoothing with Trend and Seasonality,"
Twelve methods of calculating forecasts are available. Most of these methods provide for limited user control. For example, the weight placed on recent historical data or the date range of historical data used in the calculations might be specified. The following examples show the calculation procedure for each of the available forecasting methods, given an identical set of historical data.
The following examples use the same 2004 and 2005 sales data to produce a 2006 sales forecast. In addition to the forecast calculation, each example includes a simulated 2005 forecast for a three month holdout period (processing option 19 = '3') which is then used for percent of accuracy and mean absolute deviation calculations (actual sales compared to simulated forecast).
Depending on your selection of processing options and on the trends and patterns existing in the sales data, some forecasting methods will perform better than others for a given historical data set. A forecasting method that is appropriate for one product may not be appropriate for another product. It is also unlikely that a forecasting method that provides good results at one stage of a product's life cycle will remain appropriate throughout the entire life cycle.
You can choose between two methods to evaluate the current performance of the forecasting methods. These are Mean Absolute Deviation (MAD) and Percent of Accuracy (POA). Both of these performance evaluation methods require historical sales data for a user specified period of time. This period of time is called a holdout period or periods best fit (PBF). The data in this period is used as the basis for recommending which of the forecasting methods to use in making the next forecast projection. This recommendation is specific to each product, and may change from one forecast generation to the next. The two forecast performance evaluation methods are demonstrated in the pages following the examples of the twelve forecasting methods.
This method multiplies sales data from the previous year by a user specified factor; for example, 1.10 for a 10% increase, or 0.97 for a 3% decrease.
Required sales history: One year for calculating the forecast plus the user specified number of time periods for evaluating forecast performance (processing option 19).
Month | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 147 | |
February | 132 | 117 | 135 | |
March | 115 | 115 | 132 | |
April | 137 | 125 | 144 | |
May | 122 | 122 | 140 | |
June | 130 | 137 | 158 | |
July | 141 | 129 | 148 | |
August | 128 | 140 | 161 | |
September | 118 | 131 | 151 | |
October | 123 | 114 | 131 | 141.45 |
November | 139 | 119 | 137 | 159.85 |
December | 133 | 137 | 158 | 152.95 |
October, 2004 sales = 123 * 1.15 = 141.45
November, 2004 sales = 139 * 1.15 = 159.85
December, 2004 sales = 133 * 1.15 = 152.95
POA = (141.45 + 159.85 + 152.95) / (114 + 119 + 137) * 100 = 454.25 / 370 = 122.770
This method multiplies sales data from the previous year by a factor calculated by the system.
Required sales history: One year for calculating the forecast plus the user specified number of time periods for evaluating forecast performance (processing option 19).
Month | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 120 | |
February | 132 | 117 | 110 | |
March | 115 | 115 | 108 | |
April | 137 | 125 | 117 | |
May | 122 | 122 | 114 | |
June | 130 | 137 | 128 | |
July | 141 | 129 | 121 | |
August | 128 | 140 | 131 | |
September | 118 | 131 | 123 | |
October | 123 | 114 | 107 | 127.13178 |
November | 139 | 119 | 111 | 143.66925 |
December | 133 | 137 | 128 | 137.4677 |
Range of sales history to use in calculating growth factor (processing option 2a) = 3 in this example.
Sum the final three months of 2005: 114 + 119 + 137 = 370
Sum the same three months for the previous year: 123 + 139 + 133 = 395
The calculated factor = 370/395 = 0.9367
Calculate the forecasts:
January, 2005 sales = 128 * 0.9367 = 119.8036 or about 120
February, 2005 sales = 117 * 0.9367 = 109.5939 or about 110
March, 2005 sales = 115 * 0.9367 = 107.7205 or about 108
Sum the three months of 2005 prior to holdout period (July, Aug, Sept):
129 + 140 + 131 = 400
Sum the same three months for the previous year:
141 + 128 + 118 = 387
The calculated factor = 400/387 = 1.033591731
Calculate simulated forecast:
October, 2004 sales = 123 * 1.033591731 = 127.13178
November, 2004 sales = 139 * 1.033591731 = 143.66925
December, 2004 sales = 133 * 1.033591731 = 137.4677
POA = (127.13178 + 143.66925 + 137.4677) / (114 + 119 + 137) * 100 = 408.26873 / 370 * 100 = 110.3429
This method copies sales data from the previous year to the next year.
Required sales history: One year for calculating the forecast plus the number of time periods specified for evaluating forecast performance (processing option 19).
Month | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 128 | |
February | 132 | 117 | 117 | |
March | 115 | 115 | 115 | |
April | 137 | 125 | 125 | |
May | 122 | 122 | 122 | |
June | 130 | 137 | 137 | |
July | 141 | 129 | 129 | |
August | 128 | 140 | 140 | |
September | 118 | 131 | 131 | |
October | 123 | 114 | 114 | 123 |
November | 139 | 119 | 119 | 139 |
December | 133 | 137 | 137 | 133 |
January 2005 sales = January 2006 forecast = 128
February 2005 sales = February 2006 forecast = 117
March 2005 sales = March 2006 forecast = 115
October 2004 sales = 123
November 2004 sales = 139
December 2004 sales = 133
POA = (123 + 139 + 133) / (114 + 119 + 137) * 100 = 395/370 * 100 = 106.7567
This method averages a user specified number of months (processing option 4a) to project the next months demand.
Required sales history: Twice the number of periods to be included in the average (processing option 4a), plus number of time periods for evaluating forecast performance (processing option 19).
Month | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 123 | |
February | 132 | 117 | 126 | |
March | 115 | 115 | 129 | |
April | 137 | 125 | 126 | |
May | 122 | 122 | 127 | |
June | 130 | 137 | 127 | |
July | 141 | 129 | 127 | |
August | 128 | 140 | 127 | |
September | 118 | 131 | 127 | |
October | 123 | 114 | 127 | 133.3333 |
November | 139 | 119 | 127 | 128.3333 |
December | 133 | 137 | 127 | 121.3333 |
Number of periods to be included in the average (processing option 4a) = 3 in this example
For each month of the forecast, average the previous three month's data.
January forecast: 114 + 119 + 137 = 370, 370 / 3 = 123.333 or 123
February forecast: 119 + 137 + 123 = 379, 379 / 3 = 126.333 or 126
March forecast: 137 + 123 + 126 = 379, 386 / 3 = 128.667 or 129
October 2005 sales = (129 + 140 + 131)/3 = 133.3333
November 2005 sales = (140 + 131 + 114)/3 = 128.3333
December 2005 sales = (131 + 114 + 119)/3 = 121.3333
POA = (133.3333 + 128.3333 + 121.3333) / (114 + 119 + 137) * 100 = 103.513
Linear Approximation calculates a trend based upon two sales history data points.
Those two points define a straight trend line that is projected into the future. Use this method with caution, as long range forecasts are leveraged by small changes in just two data points.
Required sales history: The number of periods to include in regression (processing option 5a), plus 1 plus the number of time periods for evaluating forecast performance (processing option 19).
Month | 2004 Sales | 2005 Sales | 2006 Sales | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 149 | |
February | 132 | 117 | 160 | |
March | 115 | 115 | 172 | |
April | 137 | 125 | 183 | |
May | 122 | 122 | 195 | |
June | 130 | 137 | 206 | |
July | 141 | 129 | 218 | |
August | 128 | 140 | 229 | |
September | 118 | 131 | 241 | |
October | 123 | 114 | 252 | 132 |
November | 139 | 119 | 264 | 101 |
December | 133 | 137 | 275 | 113 |
Number of periods to include in regression (processing option 5a) = 3 in this example
For each month of the forecast, add the increase or decrease during the specified periods prior to holdout period the previous period.
January forecast: (137 - 114)/2 + 137 = 148.5 or 149
February forecast: (137 - 114)/2 * 2 + 137 = 160
March forecast: (137 - 114)/2 * 3 + 137 = 171.5 or 172
October 2004 sales = (131 - 129) / 2 + 131 = 132
November 2004 sales = (114 - 140) / 2 + 114 = 101
December 2004 sales = (119 - 131) /2 + 119 = 113
Linear Regression or Least Squares Regression (LSR) is the most popular method for identifying a linear trend in historical sales data. The method calculates the values for "a" and "b" to be used in the formula: Y = a + bX. The equation describes a straight line where Y represents sales, and X represents time. Linear regression is slow to recognize turning points and step function shifts in demand. Linear regression fits a straight line to the data, even when the data is seasonal or would better be described by a curve. When the sales history data follows a curve or has a strong seasonal pattern, forecast bias and systematic errors occur.
Forecast specifications: n = identifies the periods of sales history that will be used in calculating the values for a and b. For example, specify n = 3 to use the history from October through December, 2005 as the basis for the calculations. When data is available a larger n (such as n = 24) would ordinarily be used. LSR will define a line for as few as two data points. For this example, a small value for n (n = 3) was chosen to reduce the manual calculations required to verify the results.
Required sales history: The number of periods to include in regression (processing option 6a) plus the number of time periods for evaluating forecast performance (processing option 19).
Month | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 146 | |
February | 132 | 117 | 158 | |
March | 115 | 115 | 169 | |
April | 137 | 125 | 181 | |
May | 122 | 122 | 192 | |
June | 130 | 137 | 204 | |
July | 141 | 129 | 215 | |
August | 128 | 140 | 227 | |
September | 118 | 131 | 238 | |
October | 123 | 114 | 250 | 135.333 |
November | 139 | 119 | 261 | 102.333 |
December | 133 | 137 | 273 | 109.333 |
Number of periods to include in regression (processing option 6a) = 3 in this example
For each month of the forecast, add the increase or decrease during the specified periods prior to holdout period the previous period.
January forecast:
Average of the previous three months = (114 + 119 + 137)/3 = 123.3333
Summary of the previous three months with weight considered
= (114 * 1) + (119 * 2) + (137 * 3) = 763
Difference between the values
= 763 - 123.3333 * (1 + 2 + 3) = 23
Ratio = (1^2 + 2^2 + 3^2) - {(1 + 2 + 3)/3}^2 * 3 = 14 - 12 = 2
Value1 = Difference/Ratio = 23/2 = 11.5
Value2 = Average - value1 * ratio = 123.3333 - 11.5 * 2 = 100.3333
Forecast = (1 + n) * value1 + value2 = 4 * 11.5 + 100.3333 = 146.333 or 146
February forecast:
Forecast = 5 * 11.5 + 100.3333 = 157.8333 or 158
March forecast:
Forecast = 6 * 11.5 + 100.3333 = 169.3333 or 169
October 2004 sales:
Average of the previous three months
= (129 + 140 + 131)/3 = 133.3333
Summary of the previous three months with weight considered
= (129 * 1) + (140 * 2) + (131 * 3) = 802
Difference between the values
= 802 - 133.3333 * (1 + 2 + 3) = 2
Ratio = (1^2 + 2^2 + 3^2) - {(1 + 2 + 3)/3}^2 * 3 = 14 - 12 = 2
Value1 = Difference/Ratio = 2/2 = 1
Value2 = Average - value1 * ratio = 133.3333 - 1 * 2 = 131.3333
Forecast = (1 + n) * value1 + value2 = 4 * 1 + 131.3333 = 135.3333
November 2004 sales
Average of the previous three months
= (140 + 131 + 114)/3 = 128.3333
Summary of the previous three months with weight considered
= (140 * 1) + (131 * 2) + (114 * 3) = 744
Difference between the values = 744 - 128.3333 * (1 + 2 + 3) = -25.9999
Value1 = Difference/Ratio = -25.9999/2 = -12.9999
Value2 = Average - value1 * ratio = 128.3333 - (-12.9999) * 2 = 154.3333
Forecast = 4 * -12.9999 + 154.3333 = 102.3333
December 2004 sales
Average of the previous three months
= (131 + 114 + 119)/3 = 121.3333
Summary of the previous three months with weight considered
= (131 * 1) + (114 * 2) + (119 * 3) = 716
Difference between the values
= 716 - 121.3333 * (1 + 2 + 3) = -11.9999
Value1 = Difference/Ratio = -11.9999/2 = -5.9999
Value2 = Average - value1 * ratio = 121.3333 - (-5.9999) * 2 = 133.3333
Forecast = 4 * (-5.9999) + 133.3333 = 109.3333
POA = (135.33 + 102.33 + 109.33) / (114 + 119 + 137) * 100 = 93.78
Linear Regression determines values for a and b in the forecast formula Y = a + bX with the objective of fitting a straight line to the sales history data. Second Degree Approximation is similar. However, this method determines values for a, b, and c in the forecast formula Y = a + bX + cX2 with the objective of fitting a curve to the sales history data. This method may be useful when a product is in the transition between stages of a life cycle. For example, when a new product moves from introduction to growth stages, the sales trend may accelerate. Because of the second order term, the forecast can quickly approach infinity or drop to zero (depending on whether coefficient c is positive or negative). Therefore, this method is useful only in the short term.
Forecast specifications: The formulae finds a, b, and c to fit a curve to exactly three points. You specify n in the processing option 7a, the number of time periods of data to accumulate into each of the three points. In this example n = 3. Therefore, actual sales data for April through June are combined into the first point, Q1. July through September are added together to create Q2, and October through December sum to Q3. The curve will be fitted to the three values Q1, Q2, and Q3.
Required sales history: 3 * n periods for calculating the forecast plus the number of time periods required for evaluating the forecast performance (PBF).
Month | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 98 | |
February | 132 | 117 | 98 | |
March | 115 | 115 | 98 | |
April | 137 | 125 | 57 | |
May | 122 | 122 | 57 | |
June | 130 | 137 | 57 | |
July | 141 | 129 | 1 | |
August | 128 | 140 | 1 | |
September | 118 | 131 | 1 | |
October | 123 | 114 | 136 | |
November | 139 | 119 | 136 | |
December | 133 | 137 | 136 |
Number of periods to include (processing option 7a) = 3 in this example
Use the previous (3 * n) months in three-month blocks:
Q1(Apr - Jun) = 125 + 122 + 137 = 384
Q2(Jul - Sep) = 129 + 140 + 131 = 400
Q3(Oct - Dec) = 114 + 119 + 137 = 370
The next step involves calculating the three coefficients a, b, and c to be used in the forecasting formula Y = a + bX + cX^2
(1) Q1 = a + bX + cX^2 (where X = 1) = a + b + c
(2) Q2 = a + bX + cX^2 (where X = 2) = a + 2b + 4c
(3) Q3 = a + bX + cX^2 (where X = 3) = a + 3b + 9c
Solve the three equations simultaneously to find b, a, and c:
Subtract equation (1) from equation (2) and solve for b
(2) - (1) = Q2 - Q1 = b + 3c
b = (Q2 - Q1) - 3c
Substitute this equation for b into equation (3)
(3) Q3 = a + 3[(Q2 - Q1) - 3c] + c
a = Q3 - 3(Q2 - Q1)
Finally, substitute these equations for a and b into equation (1)
[Q3 - 3(Q2 - Q1)] + [(q2 - Q1) - 3c] + c = Q1
c = [(Q3 - Q2) + (Q1 - Q2)]/2
The Second Degree Approximation method calculates a, b, and c as follows:
a = Q3 - 3(Q2 - Q1) = 370 - 3(400 - 384) = 322
c = [(Q3 - Q2) + (Q1 - Q2)]/2 = [(370 - 400) + (384 - 400)]/2 = -23
b = (Q2 - Q1) - 3c = (400 - 384) - (3 * -23) = 85
Y = a + bX + cX^2 = 322 + 85*X + (-23)X^2
January thru March forecast (X=4):
(322 + 340 - 368)/3 = 294/3 = 98 per period
April thru June forecast (X=5):
(322 + 425 - 575)/3 = 57.333 or 57 per period
July thru September forecast (X=6):
(322 + 510 - 828)/3 = 1.33 or 1 per period
October thru December (X=7)
(322 + 595 - 1127/3 = -70
October, November and December, 2004 sales:
Q1(Jan - Mar) = 360
Q2(Apr - Jun) = 384
Q3(Jul - Sep) = 400
a = 400 - 3(384 - 360) = 328
c = [(400 - 384) + (360 - 384)]/2 = -4
b = (384 - 360) - 3 * (-4) = 36
[328 + 36 * 4 + (-4) * 16]/3 = 136
The Flexible Method (Percent Over n Months Prior) is similar to Method 1, Percent Over Last Year. Both methods multiply sales data from a previous time period by a user specified factor, then project that result into the future. In the Percent Over Last Year method, the projection is based on data from the same time period in the previous year. The Flexible Method adds the capability to specify a time period other than the same period last year to use as the basis for the calculations.
Forecast specifications:
Multiplication factor. For example, specify 1.15 in the processing option 8b to increase the previous sales history data by 15%.
Base period. For example, n = 3 will cause the first forecast to be based upon sales data in October, 2005.
Minimum sales history: The user specified number of periods back to the base period, plus the number of time periods required for evaluating the forecast performance (PBF).
Month | 2004 Sales | 2005 Sales | 2006 Sales | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 131 | |
February | 132 | 117 | 137 | |
March | 115 | 115 | 158 | |
April | 137 | 125 | 151 | |
May | 122 | 122 | 157 | |
June | 130 | 137 | 181 | |
July | 141 | 129 | 173 | |
August | 128 | 140 | 181 | |
September | 118 | 131 | 208 | |
October | 123 | 114 | 199 | 148.35 |
November | 139 | 119 | 208 | 161 |
December | 133 | 137 | 240 | 150.65 |
Number of periods prior (processing option 8a) = 3, and the percent over the previous period (processing option 8b) is 1.15 in this example
For each month of the forecast, multiple the sales history n periods prior by the specified percent
January forecast: (114 * 1.15) = 131.1 or 131
February forecast: (119 * 1.15) = 136.85 or 137
March forecast: (137 * 1.15) = 157.55 or 158
October 2004 sales = 129 * 1.15 = 148.35
November 2004 sales = 140 * 1.15 = 161
December 2004 sales = 131 * 1.15 = 150.65
The Weighted Moving Average (WMA) method is similar to Method 4, Moving Average (MA). However, with the Weighted Moving Average you can assign unequal weights to the historical data. The method calculates a weighted average of recent sales history to arrive at a projection for the short term. More recent data is usually assigned a greater weight than older data, so this makes WMA more responsive to shifts in the level of sales. However, forecast bias and systematic errors still do occur when the product sales history exhibits strong trend or seasonal patterns. This method works better for short range forecasts of mature products rather than for products in the growth or obsolescence stages of the life cycle.
Forecast specifications:
n = the number of periods of sales history to use in the forecast calculation. For example, specify n = 3 in the processing option 9a to use the most recent three periods as the basis for the projection into the next time period. A large value for n (such as 12) requires more sales history. It results in a stable forecast, but will be slow to recognize shifts in the level of sales. On the other hand, a small value for n (such as 3) will respond quicker to shifts in the level of sales, but the forecast may fluctuate so widely that production can not respond to the variations.
The weight assigned to each of the historical data periods. The assigned weights must total to 1.00. For example, when n = 3, assign weights of 0.6, 0.3, and 0.1, with the most recent data receiving the greatest weight.
Minimum required sales history: n plus the number of time periods required for evaluating the forecast performance (PBF).
Month | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | 129 | |
February | 132 | 117 | 131 | |
March | 115 | 115 | 131 | |
April | 137 | 125 | 131 | |
May | 122 | 122 | 131 | |
June | 130 | 137 | 131 | |
July | 141 | 129 | 131 | |
August | 128 | 140 | 131 | |
September | 118 | 131 | 131 | |
October | 123 | 114 | 131 | 133.5 |
November | 139 | 119 | 131 | 121.7 |
December | 133 | 137 | 131 | 118.7 |
Number of periods prior (processing option 9a) = 3, and the weight for one, two and three periods prior (processing option 9b, 9c and 9d) are 0.6, 03, and 0.1 in this example.
January forecast: 137 * 0.6 + 119 * 0.3 + 114 * 0.1 = 129.3 or 129
February forecast: 129.3 * 0.6 + 137 * 0.3 + 119 * 0.1 = 130.58 or 131
March forecast: 131 * 0.6 + 129 * 0.3 + 137 * 0.1 = 130.748 or 131
October 2004 sales = 129 * 0.1 + 140 * 0.3 * 131 * 0.6 = 133.5
November 2004 sales = 140 * 0.1 + 131 * 0.3 + 114 * 0.6 = 121.7
December 2004 sales = 131 * 0.1 + 114 * 0.3 + 119 * 0.6 = 118.7
POA = (133.5 + 121.7 + 118.7) / (114 + 119 + 137) * 100 = 101.05
This method is similar to Method 9, Weighted Moving Average (WMA). However, instead of arbitrarily assigning weights to the historical data, a formula is used to assign weights that decline linearly and sum to 1.00. The method then calculates a weighted average of recent sales history to arrive at a projection for the short term.
As is true of all linear moving average forecasting techniques, forecast bias and systematic errors occur when the product sales history exhibits strong trend or seasonal patterns. This method works better for short range forecasts of mature products rather than for products in the growth or obsolescence stages of the life cycle.
Forecast specifications:
n = the number of periods of sales history to use in the forecast calculation. This is specified in the processing option 10a. For example, specify n = 3 in the processing option 10b to use the most recent three periods as the basis for the projection into the next time period. The system will automatically assign the weights to the historical data that decline linearly and sum to 1.00. For example, when n = 3, the system will assign weights of 0.5, 0.3333, and 0.1, with the most recent data receiving the greatest weight.
Minimum required sales history: n plus the number of time periods required for evaluating the forecast performance (PBF).
Month | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|
January | 125 | 128 | ||
February | 132 | 117 | 127 | |
March | 115 | 115 | 129 | |
April | 137 | 125 | 130 | |
May | 122 | 122 | 129 | |
June | 130 | 137 | 129 | |
July | 141 | 129 | 129 | |
August | 128 | 140 | 129 | |
September | 118 | 131 | 129 | |
October | 123 | 114 | 129 | 133.6666 |
November | 139 | 119 | 129 | 124 |
December | 133 | 137 | 129 | 119.3333 |
Number of periods to include in smoothing average (processing option 10a) = 3 in this example
Ratio for one period prior = 3/(n^2 + n)/2 = 3/(3^2 + 3)/2 = 3/6 = 0.5
Ratio for two periods prior = 2/(n^2 + n)/2 = 2/(3^2 + 3)/2 = 2/6 = 0.3333..
Ratio for three periods prior = 1/(n^2 + n)/2 = 1/(3^2 + 3)/2 = 1/6 = 0.1666..
January forecast: 137 * 0.5 + 119 * 1/3 + 114 * 1/6 = 127.16 or 127
February forecast: 127 * 0.5 + 137 * 1/3 * 119 * 1/6 = 129
March forecast: 129 * 0.5 + 127 * 1/3 * 137 * 1/6 = 129.666 or 130
October 2004 sales = 129 * 1/6 + 140 * 2/6 * 131 * 3/6 = 133.6666
November 2004 sales = 140 * 1/6 + 131 * 2/6 + 114 * 3/6 = 124
December 2004 sales = 131 * 1/6 + 114 * 2/6 + 119 * 3/6 = 119.3333
POA = (133.6666 + 124 + 119.3333) / (114 + 119 + 137) * 100 = 101.891
This method is similar to Method 10, Linear Smoothing. In Linear Smoothing the system assigns weights to the historical data that decline linearly. In exponential smoothing, the system assigns weights that exponentially decay. The exponential smoothing forecasting equation is:
Forecast =a(Previous Actual Sales) + (1 -a) Previous Forecast
The forecast is a weighted average of the actual sales from the previous period and the forecast from the previous period. a is the weight applied to the actual sales for the previous period. (1 -a) is the weight applied to the forecast for the previous period. Valid values for a range from 0 to 1, and usually fall between 0.1 and 0.4. The sum of the weights is 1.00. a+ (1 -a) = 1
You should assign a value for the smoothing constant, a. If you do not assign values for the smoothing constant, the system calculates an assumed value based upon the number of periods of sales history specified in the processing option 11a.
Forecast specifications:
a = the smoothing constant used in calculating the smoothed average for the general level or magnitude of sales. Valid values for a range from 0 to 1.
n = the range of sales history data to include in the calculations. Generally one year of sales history data is sufficient to estimate the general level of sales. For this example, a small value for n (n = 3) was chosen in order to reduce the manual calculations required to verify the results. Exponential smoothing can generate a forecast based on as little as one historical data point.
Minimum required sales history: n plus the number of time periods required for evaluating the forecast performance (PBF).
Month | 2004 Sales | 2005 Sales | 2006 Sales | Simulated 2005 Forecasts |
---|---|---|---|---|
January | 125 | 128 | 127 | |
February | 132 | 117 | 127 | |
March | 115 | 115 | 127 | |
April | 137 | 125 | 127 | |
May | 122 | 122 | 127 | |
June | 130 | 137 | 127 | |
July | 141 | 129 | 127 | |
August | 128 | 140 | 127 | |
September | 118 | 131 | 127 | |
October | 123 | 114 | 127 | 133.6666 |
November | 139 | 119 | 127 | 124 |
December | 133 | 137 | 127 | 119.3333 |
Number of periods to include in smoothing average (processing option 11a) = 3, and alpha factor (processing option 11b) = blank in this example
a factor for the oldest sales data = 2/(1+1), or 1 when alpha is specified
a factor for the 2nd oldest sales data = 2/(1+2), or alpha when alpha is specified
a factor for the 3rd oldest sales data = 2/(1+3), or alpha when alpha is specified
a factor for the most recent sales data = 2/(1+n), or alpha when alpha is specified
November Sm. Avg. =a(October Actual) + (1 - a)October Sm. Avg. = 1 * 114 + 0 * 0 = 114
December Sm. Avg. =a(November Actual) + (1 - a)November Sm. Avg. = 2/3 * 119 + 1/3 * 114 = 117.3333
January Forecast =a(December Actual) + (1 - a)December Sm. Avg. = 2/4 * 137 + 2/4 * 117.3333 = 127.16665 or 127
February Forecast = January Forecast = 127
March Forecast = January Forecast = 127
July, 2004 Sm. Avg. = 2/2 * 129 = 129
August Sm. Avg. = 2/3 * 140 + 1/3 * 129 = 136.3333
September Sm. Avg. = 2/4 * 131 + 2/4 * 136.3333 = 133.6666
October, 2004 sales = Sep Sm. Avg. = 133.6666
August, 2004 Sm. Avg. = 2/2 * 140 = 140
September Sm. Avg. = 2/3 * 131 + 1/3 * 140 = 134
October Sm. Avg. = 2/4 * 114 + 2/4 * 134 = 124
November, 2004 sales = Sep Sm. Avg. = 124
September 2004 Sm. Avg. = 2/2 * 131 = 131
October Sm. Avg. = 2/3 * 114 + 1/3 * 131 = 119.6666
November Sm. Avg. = 2/4 * 119 + 2/4 * 119.6666 = 119.3333
December 2004 sales = Sep Sm. Avg. = 119.3333
POA = (133.6666 + 124 + 119.3333) / (114 + 119 + 137) * 100 = 101.891
This method is similar to Method 11, Exponential Smoothing in that a smoothed average is calculated. However, Method 12 also includes a term in the forecasting equation to calculate a smoothed trend. The forecast is composed of a smoothed averaged adjusted for a linear trend. When specified in the processing option, the forecast is also adjusted for seasonality.
Forecast specifications:
a = the smoothing constant used in calculating the smoothed average for the general level or magnitude of sales. Valid values for alpha range from 0 to 1.
b = the smoothing constant used in calculating the smoothed average for the trend component of the forecast. Valid values for beta range from 0 to 1.
Whether a seasonal index is applied to the forecast
Note:
a and b are independent of each other. They do not have to add to 1.0.Minimum required sales history: two years plus the number of time periods required for evaluating the forecast performance (PBF).
Method 12 uses two exponential smoothing equations and one simple average to calculate a smoothed average, a smoothed trend, and a simple average seasonal factor.
A) An exponentially smoothed average
B) An exponentially smoothed trend
C) A simple average seasonal index
When a is not specified in the processing option, it is calculated.
When b is not specified in the processing option, it is calculated.
Note:
A "t" is considered 6 when it is 6 or greater.The forecast is then calculated using the results of the three equations:
D)
Where:
L is the length of seasonality (L=12 months or 52 weeks)
t is the current time period
m is the number of time periods into the future of the forecast
S is the multiplicative seasonal adjustment factor indexed to the appropriate time periods
Month | 2003 Sales | 2004 Sales | 2005 Sales | 2006 Forecast | Simulated 2005 Forecast |
---|---|---|---|---|---|
January | 115 | 115 | 116 | ||
February | 137 | 125 | 132 | ||
March | 122 | 122 | 123 | ||
April | 130 | 137 | 135 | ||
May | 141 | 129 | 137 | ||
June | 128 | 140 | 136 | ||
July | 118 | 131 | 127 | ||
August | 118 | 123 | 114 | 121 | 122.81 |
September | 121 | 139 | 119 | 132 | 133.14 |
October | 130 | 133 | 137 | 139 | 135.33 |
November | 1543 | 1514 | |||
December | |||||
Total |
Alpha, and beta factor (processing option 12a, and 12b) = blank, and seasonality (processing option 13c) is '1' in this example
Description | Calculation |
---|---|
January, 05 Seasonal Index, S1 | = (125 + 128)/(1543 + 1514) * 12
= 0.99313 |
January, 05 Smoothed Average, A1 | = Jan, 05 Actual/Jan, 05 Seasonal Index
= 128/0.99313 = 128.885 |
January, 05 Smoothed Trend, T1 | = 0 insufficient information to calculate first smoothed trend |
February Seasonal Index, S2 | =(132 + 117)/(1543 + 1514) * 12
= 0.97742 |
February Smoothed Average, A2 |
= 122.7519 |
February Smoothed Trend, T2 |
= 2/3 * (122.7519 - 128.885) + 1/3 * 0 = -4.0887333 |
March Seasonal Index, S3 | = (115 + 115)/3057 * 12 = 0.90284 |
March Smoothed Average, A3 | = 2/4 * 115/0.90284 + 2/4 * [122.7519 + (-4.088733)] = 123.01950 |
March Smoothed Trend, T3 | = 2/4 (123.01950 - 122.7519) + 2/4 (-4.0888733) = -1.91063665 |
(Continued through December '06) | |
December '06 Seasonal Index, S12 | = (133 + 137)/3057 * 12 = 1.05986 |
December Smoothed Average, A12 | = (2/13)137/1.05986 + (11/13)(A11 + T11) = 19.8865 + 107.47247 = 127.35897 |
Calculation of Linear and Seasonal Exponentially Smoothed Forecast
January '06 = (A12+T12)S1 = (127.35897 + 0.28814 * 1) * 0.99313 = 126.77 or 127
February '06 = (A12+T12)S2 = (127.35897 + 0.28814 * 2 ) * 0.9774 = 125.04 or 125
March '06 = (A12+T12)S3 = (127.35897 + 0.28814 * 3) * 0.902845 = 115.77 or 116
December '06 = (A12+T12)S12 = (127.35897 + 0.28814 * 12) * 1.059862 = 138.65 or 139
Description | Calculation |
---|---|
October, 04 Seasonal Index, S1 | = (118 + 123)/(3056) * 12
= 0.94633 |
October, 04 Smoothed Average, A1 | =a * Oct, 04 Actual/Oct, 04 Seasonal Index
= 123/0.94633 = 129.9758 |
October, 04 Smoothed Trend, T1 | = 0 insufficient information to calculate first smoothed trend |
(Continued through September '05) | |
September, 05 Seasonal Index, S12 | = (118 + 131)/(3056) * 12
= 0.97774869 |
September, 05 Smoothed Average, A12 | = 2/13*131/0.97774869 + 11/13*(A11 + T11)
= 129.1410 |
September, 05 Smoothed Trend, T12 | = 2/7 * (129.141025630 - A11) + 5/7 * T11 = 0.6343542 |
October 2005 sales = (A12 + T12*1)S1 = (129.1410 + 0.6343542 * 1) * 0.94633 = 129.775379872 * 0.94633 = 122.81
November 2005 sales = (A12 + T12*2)S2 = (129.1410 + 0.6343542 * 2) * 1.02094236 = 133.14
December 2005 sales = (A12 + T12*3)S3 = (129.1410 + 0.6343542 * 3) * 1.032722508 = 135.33
POA = (122.81 + 133.14 + 135.33) / (114 + 119 + 137) * 100 = 105.75
You can select forecasting methods to generate as many as twelve forecasts for each product. Each forecasting method will probably create a slightly different projection. When thousands of products are forecast, it is impractical to make a subjective decision regarding which of the forecasts to use in your plans for each of the products.
The system automatically evaluates performance for each of the forecasting methods that you select, and for each of the products forecast. You can choose between two performance criteria, Mean Absolute Deviation (MAD) and Percent of Accuracy (POA). MAD is a measure of forecast error. POA is a measure of forecast bias. Both of these performance evaluation techniques require actual sales history data for a user specified period of time. This period of recent history is called a "holdout period" or "periods best fit" (PBF).
To measure the performance of a forecasting method, use the forecast formulae to simulate a forecast for the historical holdout period. There will usually be differences between actual sales data and the simulated forecast for the holdout period.
When multiple forecast methods are selected, this same process occurs for each method. Multiple forecasts are calculated for the holdout period, and compared to the known sales history for that same period of time. The forecasting method producing the best match (best fit) between the forecast and the actual sales during the holdout period is recommended for use in your plans. This recommendation is specific to each product, and might change from one forecast generation to the next.
MAD is the mean (or average) of the absolute values (or magnitude) of the deviations (or errors) between actual and forecast data. MAD is a measure of the average magnitude of errors to expect, given a forecasting method and data history. Because absolute values are used in the calculation, positive errors do not cancel out negative errors. When comparing several forecasting methods, the one with the smallest MAD has shown to be the most reliable for that product for that holdout period. When the forecast is unbiased and errors are normally distributed, there is a simple mathematical relationship between MAD and two other common measures of distribution, standard deviation and Mean Squared Error:
Percent of Accuracy (POA) is a measure of forecast bias. When forecasts are consistently too high, inventories accumulate and inventory costs rise. When forecasts are consistently two low, inventories are consumed and customer service declines. A forecast that is 10 units too low, then 8 units too high, then 2 units too high, would be an unbiased forecast. The positive error of 10 is canceled by negative errors of 8 and 2.
Error = Actual - Forecast
When a product can be stored in inventory, and when the forecast is unbiased, a small amount of safety stock can be used to buffer the errors. In this situation, it is not so important to eliminate forecast errors as it is to generate unbiased forecasts. However in service industries, the above situation would be viewed as three errors. The service would be understaffed in the first period, then overstaffed for the next two periods. In services, the magnitude of forecast errors is usually more important than is forecast bias.
Note:
The summation over the holdout period allows positive errors to cancel negative errors. When the total of actual sales exceeds the total of forecast sales, the ratio is greater than 100%. Of course, it is impossible to be more than 100% accurate. When a forecast is unbiased, the POA ratio will be 100%. Therefore, it is more desirable to be 95% accurate than to be 110% accurate. The POA criteria select the forecasting method that has a POA ratio closest to 100%.