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Causal models and the statistical tools of regression and correlation analysis can be usedto augment the managerial insight and judgment needed for making these forecasts. Suchmodels relate the firms business to indicators that are more easily forecast or are availableas general information.

LINEAR REGRESSION OR LEAST SQUARES METHOD:

Linear regression refers to the special class of regressionwhere the relationship between variables forms a straight line.

The linear regression line is of the form Y_a_bX, where Yis the value of the dependentvariable that we are solving for, a is the Y-intercept, b is the slope, andXis the independentvariable. (In time series analysis,Xis units of time.)Linear regression is useful for long-term forecasting of major occurrences and aggregateplanning. For example, linear regression would be very useful to forecast demandsfor product families. Even though demand for individual products within a family mayvary widely during a time period, demand for the total product family is surprisinglysmooth.

The major restriction in using linear regression forecasting is, as the name implies,that past data and future projections are assumed to fall about a straight line. Although thisdoes limit its application, sometimes, if we use a shorter period of time, linear regressionanalysis can still be used. For example, there may be short segments of the longer periodthat are approximately linear.

DECOMPOSITION OF TIME SERIES:

In practice, it is relatively easy to identify the trend (even without mathematicalanalysis, it is usually easy to plot and see the direction of movement) and the seasonalcomponent(by comparing the same period year to year). It is considerably more difficult toidentify the cycles (these may be many months or years long), autocorrelation, and randomcomponents. (The forecaster usually calls random anything left over that cannot beidentifiedas another component.)

When demand contains both seasonal and trend effects at the same time, the question ishow they relate to each other. In this description, we examine two types of seasonalvariation:additive and multiplicative.

Additive Seasonal VariationAdditive seasonal variation simply assumes that the seasonal amount is a constant nomatterwhat the trend or average amount is.Forecast including trend and seasonal _Trend _ SeasonalFigure 3.5A shows an example of increasing trend with constant seasonal amounts.

Multiplicative Seasonal Variation

In multiplicative seasonal variation, the trend is multiplied by the seasonal factors.Forecast including trend and seasonal _Trend _ Seasonal factorFigure 3.5B shows the seasonal variation increasing as the trend increases because its sizedepends on the trend.

The multiplicative seasonal variation is the usual experience. Essentially, this says thatthe larger the basic amount projected, the larger the variation around this that we can

expect.

Seasonal Factor (or Index)

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A seasonal factor is the amount of correction needed in a time series to adjust for theseasonof the year.We usually associate seasonal with a period of the year characterized by some particularactivity. We use the word cyclical to indicate other than annual recurrent periods ofrepetitive activity.

The following examples show how seasonal indexes are determined and used to forecast(1) a simple calculation based on past seasonal data and (2) the trend and seasonal indexJanuaryAmountA. Additive seasonal

JulyJanuaryJulyJanuaryJulyJanuaryJulyJanuary JanuaryAmountB. Multiplicative seasonal

JulyJanuaryJulyJanuaryJulyJanuaryJulyJanuary

FIGURE 3.5 Additive and Multiplicative Seasonal Variation Superimposed on ChangingTrend62 Chapter 3 Forecasting

from a hand-fit regression line. We follow this with a more formal procedure for thedecompositionof data and forecasting using least squares regression.

Short-Term Forecasting TechniquesIn this section,well introduce two very common short-termforecasting techniques:movingaverages and exponential smoothing. We choose these procedures since they arecommonlyavailable in commercial software and meet the criteria of low cost and little managementinvolvement. The techniques are simple mathematical means for converting pastinformation into forecasts.

Moving-Average ForecastingMoving-average and exponential smoothing forecasting are both concerned with averagingpast demand to project a forecast for future demand. This implies that the underlyingdemand pattern, at least for the next few days or weeks, is constant with random

fluctuations about the average.

Since were interested in averaged pastdata to project into the future, we could even use an average of all past demand dataavailablefor forecasting purposes. There are several reasons, however, why this may not be adesirableway of smoothing. In the first place, there may be so many periods of past data that

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storing them all is an issue. Second, often the most recent history is most relevant inforecastingshort-term demand in the near future. Recent data may reveal current conditionsbetter than data several months or years old. For these reasons, the moving averageprocedureuses only a few of the most recent demand observations.

Whenever a forecast is needed, the most recent past history of demand is used to do theaveraging. Youll note the moving-average model does smooth the historical data, but itdoes so withan equal weight on each piece of historical information

Exponential Smoothing ForecastingThe exponential smoothing model for forecasting doesnt eliminate anypast information,but so adjusts the weights given to past data that older data get increasingly less weight(hence the name exponential smoothing).

The proportion of the error that will be incorporated into the forecast is calledtheexponential smoothing constant and is identified as_.

result shows larger values of_give more weight to recent demands and utilize olderdemand data less than is the case for smaller values of_; that is, larger values of_providemore responsive forecasts, and smaller values produce more stable forecasts. The sameargumentcan be made for the number of periods in an MAF model. This is the basic trade-offin determining what smoothing constant (or length of moving average) to use in aforecastingprocedure. The higher the smoothing constant or the shorter the moving average, themore responsive forecasts are to changes in underlying demand, but the more nervousthey are in the presence of randomness. Similarly, smaller smoothing constants or longermoving averages provide stability in the face of randomness but slower reactions to changesin the underlying demand. Ultimately, however, the trade-off between stability andresponsiveness

is reflected in the quality of the forecasts, a subject to which we now turn.