A forecast formula is a mathematical relationship that can be used to automatically calculate forecasts of demand. This is done using demand history data.

Forecast formulas are used to calculate fresh base forecast from actual demand adjusted for seasonal and period length variations. A formula is specified for each forecast method. The method also contains specified parameters and limits which regulate the calculation performed using the formula.

**Formulas**

There are four forecast formulas, as follows:

**Moving average**

This forecast formula calculates the base forecast for the next period as the average of historic base demand for a specified number of periods. This is denoted in the following equation:

F(i + 1) = (D(i) + D(i - 1) + .... + D(i - (n - 1))) / n

The number of periods used determines how quickly the averaging will react to changes in actual trends and how sensitive it will be to random variations. The more periods included will make the calculation method more stable from random variations, but it will also react more slowly to changes resulting from real trends.

**Forecast Formulas: Two-period weighted average**

This forecast formula weighs the average demand from the latest quarter (of periods included in the forecast) with the average demand for all historic periods. The weight factor is the smoothing constant for exponential smoothing, (, and 1 - (, respectively. This is denoted in the following equation:

F(i + 1) = ((i) * M + (1 - ((i)) * L

**Exponential smoothing**

This forecast formula weighs latest base demand value with the smoothing constant (, while the previous base forecast value is weighted with 1 - (. This is denoted in the following equation:

F(i + 1) = ((i) * D(i) + (1 - ((i)) * F(i)

The value of smoothing constant ( determines how quickly the forecast will react to changes in actual trends and how sensitive it will be to random variations. The lower the value, the more stable the calculation is from random variations, but it will also react more slowly to changes resulting from real trends. Smoothing constant ( must be between 0 and 1.

**Forecast Formulas: Adaptive exponential smoothing**

This forecast formula is similar to basic exponential smoothing in that the latest base demand value is weighted with smoothing constant (, while the previous base forecast value is weighted with 1 - (. However, in adaptive exponential smoothing, the smoothing constant is recalculated every time a new forecast is made. This is denoted in the following equation:

F(i + 1) = ((i) * D(i) + (1 - ((i)) * F(i)

The smoothing constant is recalculated using the following equation:

((i) = ((min.) + ((max.) * (ABS(ME(i)) / MAD(i))

This forecast formula uses ( values that are adjusted for the current systematic forecast error. A larger mean forecast error results in a higher ( value. This results in a quicker corrections to the forecast towards reflecting actual demand.

**Key:**

((i) | = | Allowable smoothing constant for smoothing in period (i) |

((min.) | = | Minimum allowable smoothing constant |

((max.) | = | Maximum allowable smoothing constant |

D(i) | = | Base demand for period (i) |

F(i) | = | Base forecast for period (i) |

A(n) | = | Average demand for (n) periods |

i | = | Period number |

n | = | Number of periods included in calculating the average |

L | = | Average demand for the latest (n) periods |

M | = | Average demand for the latest 25 % periods out of the total of (n) periods |

MAD(i) | = | Forecast MAD for period (i) |

ME(i) | = | Mean forecast error for period (i) |

ABS( ) | = | Absolute difference, the difference without minus sign |

Base demand is the demand for a period adjusted for seasonal variations and, if applicable, the effect of a varying number of workdays per period. Non-representative demand is not included either. Base forecast is the forecast calculated from base demand adjusted for seasonal and period length variations.

When a manual forecast has been entered (as when an erroneous forecast is corrected) it is used as the base forecast for the previous period when calculating subsequent forecasts with exponential smoothing. This is after it is adjusted for seasonal and period length variations and any other manual changes.

The examples below describe using each of the formulas based on the following data.

Aug. | Sep. | Oct. | Nov. | |

Base demand | 120 | 145 | 138 | 129 |

Base forecast for Nov. | 136 | |||

(-factor used | 0.3 | |||

((min.) | 0.2 | |||

((max.) | 0.5 | |||

MAD(Nov.) | 10 | |||

ME(Nov.) | -2 |

The following forecast values will be calculated for December using the four methods as listed:

**Moving average**

F(Dec.) = (D(Aug.) + D(Sep.) + D(Oct.) + D(Nov.)) / 4 = (120 + 145 + 138 + 129) / 4 = 133

**Two-period weighted average**

F(Dec.) = 0.3 * 129 / 1+ 0.7 * (120 + 145 + 138 + 129) / 4 = 0.3 * 129 + 0.7 * 133 = 131.8

**Exponential smoothing**

F(Dec.) = 0.3 * D(Nov.) + 0.7 * F(Nov.) = 0.3 * 129 + 0.7 * 136 = 133.9

**Adaptive exponential smoothing**

((Nov.) = ((min.) + ((max.) * (ABS(ME(i)) / MAD(i)) = 0.2 + 0.5 * ABS(-2) / 10 = 0.2 + 0.5 * 0.2 = 0.21

F(Dec.) = ((Nov.) * D(Nov.) + (1 - ((Nov.)) * F(Nov.) = 0.21 * 129 + 0.79 * 136 = 134.5