Resume.

Der beskrives en metode til forudsigelse af fremtidige salg, som
disse ma forventes at blive, dersom de fundne salgsbeveegelser
lortsaetter i fremtiden, d. v. s. et forecastingprogram.

Den praktiske udnyttelse af metoden er muliggjort ved kodning
af et regneprogram til cifferregnemaskinen DASK.

Efter regneprogrammet antages salget at vaere deleligt i en langtidskomponent
(trend), en saesonkomponent og en tilfa:ldig variabel.

Scesonkomponenten findes ved kvotientmetoden, der er beskrevet detailleret, med en tiirig salgsperiode som beregningsgrundlag. Ved multiplikation med salget i en basis-maned forvandles kvotienterne til en salgsseeson i absolutte tal.

Selv om denne beregningsmetode forudssetter multiplikative sxsonudsving, tillader multiplikationen med den senest kendte basismaned, at langtidskomponenten findes ved subtraktion af saesonen fra totalsalget i de seneste 12 maneder.

Differensen indeholder langtidskomponent plus tilfasldig komponent.
Langtidskomponenten tilnaermes ved antagelse af linearitet
og beregning efter mindste kvadraters metode.

Prognosetallene fremkommer ved extrapolation af trendlinier
og pafolgende addition af saesonkomponenter.

I vurderingen af regneprogrammet og resultaterne tages hensyn til spredningen om trendlinierne, der angiver den tilfaeldige komponents spredning, og en beregning af denne spredning muliggor en fastsasttelse a* laengden pa perioden, der bor anvendes til be-

---

*) Chemical engineer, Carlsberg Bieweries, Copenhagen.

In modern industrial developments there is an increasing need for
reliable methods of estimating future sales.

In the inventory planning it is necessary to have an approximate knowledge of the demand for raw materials in the production process, and if possible the oscillation in the demand too. For this purpose a sales prediction should be very useful.

As a basis for a sales prognosis it is possible to employ a statistical
forecasting method.

Such forecasting includes periodical recalculations as knowledge is gained of new sales figures, whence the use of an electronic computer is imperative to make the latest corrections of the expected sales figures available for the planning departments without undue delay.

On the other hand it is not advisable to use the unscrutinized forecasting
results in the inventory planning.

The word forecasting means only the projection of the past into the future, whereas the influence of future sales efforts, competition, expanding markets and other independent variables are not taken into consideration in a sales forecasting programme. Accordingly the statistical computations will not be ready for use before further adjustments have been carried out in the sales and marketing departments, where the effect of planned advertising and other sales efforts is estimated.

As any sales prediction may be subject to more or less grave errors
it is the objective of the sales and advertising departments to keep the
sales as close to the stipulated figures as possible to insure optimum

regning af sæsonen. Desuden vises fremgangsmåden ved valg af antallet af måneder, der skal bruges til beregning af langtidskomponenten, og det anbefales at give salgsprognosen på basis af såvel en relativ kort salgshistorie som på basis af en noget længere. Herved opnås dels en hurtig reaktion på ændringer i salgstendensen ved den kortperiodiske prognose, dels en prognose, der ikke reagerer på pludselige tilfældige salgsudsving.

Der beskrives en metode til beregning af kontrolgrænser for en kurve over den akkumulerede differens mellem salgs- og forecast værdier. Kontrolgrænserne bruges som indikator for skift mellem anvendelsen af flere eller færre måneders salgshistorie i trendberegningen.

Regneprogrammets reaktion på visse typiske ændringer i salget
er beregnet.

Endelig indeholder artiklen en beskrivelse af regneprogrammets
praktiske udnyttelse på DASK, samt en omtale af resultaterne ved
anvendelse på ølsalget på Carlsberg Bryggerierne.

economy of the investment plans which have been put into operation.
This means that the above mentioned departments have to agree upon
the results arrived at before any further use.

In this paper only the statistical part of the sales forecasting is described as it is used to-day at the Garlsberg Breweries. As the described method only deals with the sales forecasting in the near future, that is, not more than six months to a year ahead, a short cut has been made in the analysis of the sales history.

The forecasting of the sales figures months ahead is based on a very simple mathematical model that splits beersales (Yt) into three components defined as the trend (It), the seasonal component (St), and the random deviation (Rt) from the seasonal component. (Figure 1).

(1)

I. Estimation of seasonal component, (St).

The sales season is obtained by means of the so-called link relative
method.

This method is based on a calculation of the relative sales figures (qt) where each month's sales (Yt) are divided by the sales of the preceding month Yt-i. The possible effect of a long time trend will be eliminated in the relative sales figures thus obtained. (Table 1).

(2)

To arrive at a reasonably well founded expression for a seasonal factor it is necessary to use figures from a period of some length as for instance ten years monthly sales figures, each of which is transformed into a monthly link relative by a division as described above. The choice of the length of the preceding period used for the calculation of the seasonal component will be discussed later in the paper.

Using a total of 121 consecutive months, January will be estimated by
ten link relatives, February also by ten and so on.

As the ten relatives are varying due to the random component of the
season, a mean must be found.

Besides the random variation a certain number of sales anomalies may occur, causing abnormal deviations of the relatives during a ten year period. In order to obtain a mean relative which is not influenced by such anomalies, a so-called positional mean is calculated for each month.

In the programme for the electronic computer the two highest and
the two lowest link relatives are excluded from the estimated mean
link relative, qt (table 2).

Until now the calculations have resulted in one link relative (qt) for
each of the twelve months of the year.

In the further calculations it is necessary to get a sales season where
the individual month is represented by a real, not by a relative figure.

To reach such a season a so-called basic month is chosen. In the present
instance the month of November has been used for this purpose.

The typical link relative for November, the basic month, is fixed at
Q'o = 1.00.

The typical relative for December is thus 1.00 multiplied by the link
relative for December (Table 3). Q'i = Q'o ■ qi.

The typical relative for January is found by a multiplication of the
link relative for January by the newly found typical relative for December.
(Table 3).

In this way the typical relative season is found through a series of
consecutive multiplications. (Table 3).

(3)

In the results thus obtained the fixed November relative (Q'o = 1,00) will rarely coincide with the November relative found as the last result in the series of successive multiplications, (1,02 in table 3). This lack of coincidence is due to the resulting trend component for the used ten years, mostly different from zero. Because of this difference from 1.00, 1 fi2 of the difference is subtracted from the December typical relative, 2 fi2 from the January typical relative and so forth, concluding in the subtraction of 12\\2 from the last November typical relative, transforming it to 1.00. Qt = Q'_t - (Q'i2 - Q'o) t/12. (Table 3).

The sales season is now easily obtained by a multiplication of the
newly found typical relative season by the figure from the latest known
November sales. (Table 4).

(4)

Even if the link relative method fundamentally is based on the assumption that the seasonal variations are proportional to the sales, the multiplication with the November sales will make it possible to consider the seasonal component additive or multiplicative in the further use. That is, a quite satisfactory estimate for the seasonal component has been reached.

The practical adaption, however, will be much too time-consuming
for routine calculation unless electronic computers are available.

II. Estimation of trend, (T1).

Complying with the hypothesis of the possibility to split the sales into seasonal, trend and random components, it is now possible to find the trend plus the random component by means of a simple subtraction of the seasonal component from the total sales. (Fig. 1).

In the computer programme the subtraction is carried out for the last known twelve months only, as the hypothesis of an additive season will be less correct for a longer period of time. The twelve figures obtained will contain the trend plus the random deviations of the last twelve-months' season from the calculated typical season. (Fig. 1).

If there is a systematic deviation from the calculated season during
the last twelve months, this will cause an abnormal distribution of the
resulting figures, illustrated in fig. 1 as trend + random points.

If there is no reason to assume that the recent season has been abnormal,
a trend curve may be fitted to the resulting points for the last
twelve months.

The fitting of a polynomial expression to the points will be possible,
but the values found by extrapolations will be less correct the more
complicated the used polynomium is.

To avoid this complication the trend is considered linear, and estimated
as the regression line corresponding to the points using a trivial least
squares method. (Table 5).

Table 5

Least squares method:

Equation of the regression line.

(5)

where

Complying with

III. Estimation of random component, (Rt).

It should now be possible to make calculations of the random deviations
from the season during the period treated.

The trend is considered linear in each of the twelve-months' periods
used for the calculation of the sales season.

When the sales season is subtracted from the actual sales figures throughout the said period, the difference will consist of the trend plus the random deviation from the season, as pointed out in the description of the estimation of the trend component.

For each twelve-months' period a calculation of the regression line will
give the trend component, considered linear, and the further subtraction
of the trend

will give the values of the random component. Using separate twelvemonths' periods, the use of an additive seasonal component in the computation can still be justified. The standard deviation about the regression line is thus the standard deviation of the random seasonal component.

(6)

IV. Length of period for calculation of seasonal component, (Si).

As previously mentioned the length of the period used for the calculation need some thought. Given a value for the standard deviation of the random seasonal component for each year used for the calculation of the season, it is possible to detect systematic alterations of the sales season from year to year.

If the figures from the earlier part of the period are considerably
higher than those from the later part, this earlier period may be excluded
in the estimation of the sales season.

A statistical test on the standard deviation found to detect a "jump"
in the figures will hardly be necessary.

V. The sales forecasting.

Roughly spoken the sales forecasting when looked upon as a geometric problem will result from an extrapolation of the trend curve followed by an addition of the seasonal components to the extrapolated future trend curve.

More detailed the arithmetic treatment by the computer programme
will split the 121 known sales figures in a typical seasonal component,
and a trend plus a random component for the last twelve months.

Furthermore the programme will make extrapolations on the straight
trend lines resulting from the trend plus random components for the
last 2, 3, 4, - 12 months, that is, a total of 11 trend lines. (Fig. 2). The

extrapolations are made right up to twelve months after the last known, and the values of the typical season are added resulting in a total of 11 • 12 = 132 forecast figures. Furthermore the typical season and an estimate of the standard deviation in the figures used for computation of the link relatives are typed of the computer.

VI. Practical use of the computer programme.

Every month the newly realized month's sales are used as input to the computer programme together with the 120 sales figures from the preceding ten years. The computer result will be a tabular arrangement as shown in table 6.

A. Choice of number of months retrograde. (Table 6).

In the computer output it is possible to choose between eleven different
results for every month to come, depending on how many months retrograde
are taken into consideration.

A smooth nearly linear development of the trend curve will make the
use of many trend points reasonable, while an oscillating trend curve
only allows a forecasting based on a few trend points retrograde.

In order to make the best choice of the number of months retrograde
it is necessary to treat the forecasting results as "a dry running" of the
results from the last one or two years known.

In the dry running the eleven possible forecast results for each
month are compared with the actual sales in the same month, and thus
it is found which result is closer to the actual sales.

In many cases the comparison will give two numbers of months retrograde giving a partial minimum difference between sales and forecast values, one corresponding to only a few months retrograde, another corresponding to some 9-12 months retrograde.

It is recommended to use both possibilities, as a small number of months retrograde will give a "nervous" curve rapidly tracking new developments in the sales trend, and therefore also reacting on random variations in a marked way.

A greater number will give a more stable curve where random variations
will have only smaller effects, and on the other hand new sales
developments will be tracked more slowly.

The two most frequent numbers occurring will consequently serve as
the starting points of the forecasting.

B. Control diagram.

The control diagram for a forecasting sequence is obtained as follows:

The forecast is compared with the new sales result when known, and
we start the drawing of a graph of the cumulated difference between
the forecast and the actual sales plotted as a time series.

(7)

The cumulated diagram will serve as a control chart for the choice of
the number of months retrograde to be used in the computation of the
trend.

The control limits are an angle placed with the top point on a horizontalline

zontallinethrough the last point plotted on the cumulated curve, and
at a predestinated distance (a) after this point.

The characteristics of the angle can be set up as empirical values
found suitable in the dry running or estimated through a simple bit of
reasoning.

In fig. 3 and 4 the control angle is shown.

The letters A, B, C illustrate three accumulated differences between the actual sales and the forecast values when the values A and B fall just on the borderline of the control limits. (Fig. 4). L is the distance chosen corresponding to the time interval between the successive forecast

Given the values of A, B, C and L - the values of a and v can be found.

The maximum allowable difference that should not give rise to
reactions is set to Dtmax = 2 • o, where o is the standard deviation corresponding
to the random seasonal component.

The solution of the following two equations will give the values for
tan v and a

,
(8

(9)

The value of Dt + Dt-\ = (Yt - Ft) + (Yt-i - F_t-i), the sum of the differences between actual sales and forecast figures for the last two sales periods (months) should not exceed 2a]f2 if we consider the random component normally distributed with standard deviation a and mean equal to zero.

As the forecasting programme, however, reacts on every deviation from
the forecast value, a certain amount of quantitative knowledge of the
reaction of the programme is required.

Let us assume that the trend line used for the extrapolation in the
forecasting is based on n months retrograde.

The equation

(10)

will then describe the trend line when the time unit on the abscissa
equals 1, 7V being the (forecast) value of the trend component.

(11)

if no difference between sales and forecast has occurred in the r'th
month.

(In the following reasoning the disappearance of the oldest point on
the trend line, when new points are added, is assumed to have no influence
on the calculations).

If the new sales have exceeded the forecast figure with the amount
«o in the r'th month,

(12)

This reaction of the programme on deviations of sales from forecast
values means that a theoretical maximum value of Dt + Dt+i = 2 o ]/ 2
4 aa
will be reduced with when the deviation of the first of the two
n
months considered has been aa = Dt~\.

The maximum allowed Dt + Dt-i value is thus reduced from the
"ideal" value 2 a Y% to

(13)

As Dt-i =aa and Dtmux =2a

we have also

(14)

From equations (13) and (14) a is found to

(15)

and by substituting (15) in (13) or (14)

(16)

Entering the expressions for (Dt + Dt-\)max and (Dt)««* in (8) and (9)
we have:

from which

(17)

and

(]ft

The control angle is now fully described when a, L and n are known.

C. Use of control diagram.

In the application of the sales forecasting programme the following
practice may be suggested.

n, the optimum number of months retrograde to be used, is found
from a two year period.

The following reasoning covers the "nervous* curve (n = 3, 4, 5, 6)
as well as the "stable" curve (n = 10, 11, 12).

° is found as the mean standard deviation about the regression lines

calculated for each twelve-month's period in the years used for calculationof
the sales season.

L is fixed as the time interval on the abscissa of the diagram illustrating
the accumulated difference between sales and forecast figures 2Dt.

The characteristics of the control angle, a and tan v, are found in
accordance with the values of n, a and L, from equations (17) and (18).

The control chart (showing the aforementioned accumulated difference
2Dt between sales and forecast values) is started with n months retrograde
as the basis for the forecasting.

The points on the (graph are tested in respect to the control limits. If a newly plotted point results in part of the accumulated curve falling outside the control limits this may be due to alterations in the trend, and the consequence will be a shift to re — 1 months retrograde as the basis for the forecasting.

Only the part of the curve corresponding to the last n months will
have to be taken into consideration in the use of the control angle.

The shift, however, should be made only when the diagram goes out of control because of increasing or decreasing -SDi-values during twothree months. If the lack of control follows a shift from positive to negative elevation of the accumulated curve, i. e. abnormal high sales followed by abnormal low sales et vice versa, the matter must be given further consideration.

It should be noted that the programme reacts on an increase aa in
the value of ZDt in the way that the next forecast figure will be
4 aa
increased with as calculated in equation (5).
n

A decrease AD in value of 2Dt in this situation therefore should be
4 aa
corrected to AD , which value should be considered in relation
n
to the control diagram.

As the reaction of the programme on alterations in the elevation of
the trend curve is very quick, there should be no need to stay on n — 1
months retrograde as the forecasting basis for more than a few months.

If in the forecasting a constant number of months retrograde has been used for some time it should be possible to consider the trend so stable that a shift to n + 1 months could give just as good a basis for the forecasting.

Besides the control diagram a check on the "best" choice of the number of months retrograde to be used is made in the form of a calculation repeated after every twelve months for those in each case foregoing two years to insure that the most frequent numbers of months used are still unaltered.

The numbers of months are then corrected corresponding to the result
of the computation.

In the application of the programme to beersales it has been found that a correction should be made only to the extent that the number of months retrograde used is not allowed to differ more than one from the result found in the computation.

The shift to n + 1 months because of an apparently stable trend has
been made when n months retrograde has been used for n months.

If a month shows a marked low or high value compared with the forecast value it must be noted that this abnormal value will influence the future forecasting especially when the said month appears as earliest of the months retrograde used in the determination of the trend line.

When such a situation occurs it is advisable either to use the mean of the two surrounding forecast values corresponding to n — 1 and n + 1 months retrograde, or to exclude the value in future computations. However, it should be noted that this practice should not be carried too far. Only when it is evident that something really abnormal has taken place, such correction may be allowed.

To test the programme the influence of theoretical alterations in the
input figures i. e. sales figures have been examined.

1) a "jump" J occurring in one month only (month 0) will cause the
following oscillations.

Increase of forecast value:

etc. the disturbing influence of the jump disappearing after n months.

For* = 6.he values are-^/, ±J. ±J, ±J. -L], -±J, 0
(fig. 5).

2) a "step" S shifting the niveau of the trend will be tracked as follows.

Increase of forecast value:

etc.

For n — 6 the values are

iis,ils,iLs,^s,iis,iis,.iis
15 15 15 15 15 15 15

(fig- 6)

3) a "ramp" R shifting the elevation of the trend will be tracked as
follows.

Increase of forecast value:

For n = 6 the values are

(fig- 7)

etc.

VIII. In some instances it has been possible to predict abnormal sales
developments due to strike-situations, higher beer taxes, etc.

This may cause a jump in the sales figures followed by a retraction
the month afterwards.

Under such peculiar circumstances it may be justified to straighten out the sales development in accordance with for instance stipulated storage figures for the retailers and the like, to avoid oscillations in the forecasting. Such precautions are only justified when first a clear relationship exists between sales figures and the said circumstances, and secondly such causal factors are not periodical, i. e. represents part of the seasonal component. If it is not possible to make a reasonable correction of the sales figure in this way, it should be excluded in the future calculations.

The programme may be used universally, not limited to beersales
only, but minor corrections may be necessary in more specialized applications.

IX. In conclusion of this paper it should be noted that the programme has been used on the Carlsberg Breweries for some 18 months giving a relative standard deviation of the sales figures about the forecast values of 6% taken over 21/z2^l/z years, and a cumulated difference between the sales and forecast values not exceeding 9 million bottles on a sales total of about 400 million bottles per year.