Forecast Formulas in detail

Moving Average

Forecast calculation with formula Moving Average projects the values in the forecast period based upon the average demand (sales) over a specific number of preceding periods, that is, weeks or months. The number of preceding periods used in the forecast calculation is referred to as n. The number of periods used determines how quickly the forecast will react to changes in actual trends and how sensitive it is to random variations. The more periods included will result more stable method for random variations but will also react slowly to real trends. Moving Average is calculated based on:

In M3 BE, a various number of hybrids of Moving Average exist. In M3 DMP, one formula replaces these due to a more dynamic and flexible set up feature. This is solved by introducing the concept Offset periods and Offset years. You can determine the first historical period to use in the forecast calculation when calculating the Moving Average based upon n number of calculation periods. For example, if you calculate the next period forecast as the average actual demand of the preceding, same, and following period from the preceding year, the Offset period is set to 11 and Number of calculation periods to 3. The use of Offset period in combination with various values of n provides endless possibilities to calculate the moving average.

Centered Weighted Moving Averages (CWMA)

A weighted average is any average that has multiplying factors to give different weights to different data points.

DMP uses a table of predefined smoothing constants depending on the selected weight function. It includes the most used variations including the weight functions of Spencer and Henderson.

These Weighted Moving Average Functions options are available:

• 3x3 MA
• 5x5 MA
• Spencer 15 MA
• Spencer 21 MA
• Henderson 5 MA
• Henderson 9 MA
• Henderson 13 MA
• Henderson 23 MA

This table shows the commonly used weights in the weighted moving average, where S=Spencer’s weighted moving average and H=Henderson’s weighted moving average:

Centered Moving Averages (CMA)

The Centered Moving Average is a specific form of Centered Weighted Moving Average, where the smoothing constants are the same, except for the start and end if the length is an equal number.

The weight function for CMA is defined by specifying the number of calculation periods (>=2).

Exponential Smoothing

Exponential Smoothing calculates the forecast based on the previous demand and forecast of the period. To achieve this, Offset periods is set to 1. You can also elaborate and use different Offset periods depending on purpose. The Smoothing constant - Alpha, α, is a value between zero (0) and one (1). The higher the α, the faster the method reacts to changes in demand. This model is good for non-seasonal data that is fairly level, not trend.

Exponential Smoothing of two period values

The formula weights the average demand of the last 25 percent of the total number of calculation periods with the average demand for n number of periods. The formula uses the smoothing constant α which determines how strongly the forecast responds to changes in demand.

The values of 0.2 and 0.3 are reasonable smoothing constants. These values indicate that the current forecast should be adjusted to 20 to 30 percent for the error in the prior forecast.

For example, Exponential Smoothing with n = 4 and α = 0.3 and latest number of calculation periods = 1, which is used to calculate M.

Adaptive Exponential Smoothing

Adaptive Exponential Smoothing is similar to basic exponential smoothing where the latest base demand value is weighted with a smoothing constant. In adaptive exponential smoothing, the smoothing constant is calculated every time a new forecast is made.

The smoothing constant is recalculated using this equation:

In order to control and follow up forecast precision, two types of measurements are used, Mean Absolute Deviation (MAD) and Mean Error (Forecast error, AVER). These measure the deviation between forecast and actual demand. You can calculate the Mean Error and MAD in three ways:

Method 1: Exponential Smoothing

Method 2: Mean F/C deviation

Method 3: Mean Demand deviation

Factor	Description
	Mean absolute deviation for period (i)
α	Smoothing constant for exponential smoothing in period (i)
	The absolute amount of a difference (without minus sign)
	Base demand during period (i)
	Base forecast for period (i)
	Average demand for (n) periods
i	Period number
n	Number of periods included in calculating the mean

Adaptive-response-rate SES (Trigg & Leach)

The Adaptive-response-rate single exponential smoothing, also known as Trigg & Leach’s method, is also a hybrid of exponential smoothing but where α is calculated based upon the smoothed forecast error and smoothed absolute forecast error. Parameter Beta is the smoothing constant applied to the trend in forecast versus demand.

Forecast as demand from preceding period

The forecast for the next period is equal to actual demand n periods back in time. You can use Offset Periods, Offset Years, or both.

For example, if Offset Periods is equal to zero (0), the forecast for the preceding period is equal to the demand for the previous period. If the Offset Periods is equal to eleven (11), the forecast preceding period is equal to the demand for the same period last year. On the last example, you can set Offset Years = 1 and Offset Periods = 0.

Croston’s Method

Exponential smoothing is often used to forecast demand for inventory in stock management. However, if the demand is sporadic or intermittent, the exponent smoothing usually produces stock levels that are too low. Croston’s Method improves the exponential smoothing by estimating the average size of demand (Z) and the average interval between demands (X). Croston’s Method is useful when forecasting spare parts or equipment where demand is in batches for replenishment of inventories, when the demand appears random or with irregular patterns, and when there are several periods with zero demand.

Initialization

Iteration over periods i and using q as the interval between last two periods of demand.

Forecast, which is estimated by Z considering average interval X, is calculated either as constant (average) or sporadic.

• Sporadic: Forecast is distributed according to the mean interval X

  

• Constant (average): Forecast is distributed according to the mean interval X

  

Croston’s – Modified Method

Studies show that in some cases you can estimate the forecast better by using a modified version of Croston’s Method that considers the effects of the last two periods of demands, especially if the interval between demands is changing.

Initialization same as Croston’s Method

Iteration over periods i and using q as the interval between last three periods of demand

Forecast is calculated either as constant (average) or sporadic.

• Sporadic: Forecast is distributed according to the mean interval X

  

• Constant (average): Forecast is distributed according to the mean interval X

  

Period Relation

The Period Relation forecast algorithm calculates the forecast equal to the relation between previous x number of periods and the same x number of periods last year multiplied with the average demand for y number of periods last starting with the current period previous year.

For example, = p 12 (number of months per year) = A 2 = n 4

Holt’s Linear Method

Holt’s Linear Method is an extension of exponential smoothing to take into account a possible linear trend. This model is good for non-seasonal data with a trend. There are two smoothing constants α and β. It follows the equations:

Holt-Winters trend and seasonality method

This method takes both trend and seasonality into consideration. A seasonal index is added to the equation.

Where s is the seasonality period length.

Extrapolation of Holt-Winter

For extrapolation, the last season range is repeated and Trending (B in period 24) is added.

At the End Calculation Point (P=25), the forecast is calculated with m=1.

• Forecast26 = (L24+B24*2)*S22
• Forecast27 = (L24+B24*3)*S23
• Forecast28 = (L24+B24*4)*S24