Today, we are basically going to discuss the following concepts, that is, forecasting of
demand pattern where there is a trend either upward or downwards. Then we will be
discussing something related to forecast errors and associated forecasting accuracy, then
we will be discussing the role of tracking signals mainly to detect whether the
forecasting model is stable and the results that we are getting are consistent or not.
And, then we will take up one example where there is some evidence of seasonal
demand pattern; ok. We will find out the seasonal factors and then deployed at in
forecasting demand pattern for the next planning horizon.
(Refer Slide Time: 02:32)
See, the basic forecasting models for trends whatever we had discussed earlier they
compensate for the lagging that would otherwise occur because in the last week we had
discussed about simple moving average, weighted moving average, we have discussed
about simple exponential smoothing models.
Now, what we have basically discussed that these kind of models they take care of
smoothing the random fluctuations around the average level of demand. But, these
models they cannot take care of the trend in the demand pattern. In fact, single
exponential smoothing model they lag behind this trend by a factor of alpha by 1 minus
alpha, where alpha is the smoothing constant.
So, in order to take care of this trend in a demand pattern particularly for time series
related data, lots of techniques have been devised. One of the techniques is double
exponential smoothing model, there is Holt’s model; so many models are there. So, that
these basic forecasting models for particularly trend they compensate for the lagging that
would have occurred if we had deployed only single exponential smoothing models.
One such particular model which is basically known as trend adjusted exponential
smoothing uses the following 3 steps. In the first step what we do is that we smooth the
average level of the series it something what we had discussed in the last week. The
smoothed value of the level at the t-th period, it is given by S t.
S t represent the smoothed value of the level at the t-th period, that is basically equal to
alpha which is a smoothing constant into the actual demand in period t which is given by
A t plus 1 minus alpha into the smoothed value of the level 1 period ago which is
denoted by S t minus 1. Plus the trend factor which was computed 1 period ago and
denoted by t subscript T minus 1 T t minus 1.
Once again in the first step we compute the smoothed value of the average level which is
given by S t equals alpha which is a smoothing constant times the actual demand in the t-
th period denoted by A t plus 1 minus alpha times S t minus 1 that is the smoothed value
of the level computed 1 period before plus the smoothed value of trend which is denoted
by T t minus 1 which is computed 1 period ahead of the period t. Is that thing will be
cleared when we take up one example?
And, in this particular model which is trend adjusted exponential smoothing we need to
introduce another smoothing constant which is denoted by beta and this smoothing
constant is basically used for smoothing the trend. So, the smoothed value of the trend
factor at the period t is given by t subscript t which is nothing, but beta the new
smoothing constant multiplied by S t minus S t minus 1, where S t is given by this
formula plus 1 minus beta into the trend factor computed 1 period ago and denoted by T
subscript t minus 1.
So, once we compute the smoothed value of the level and the trend at the period t then,
the forecast value including the trend for the period t plus 1 that is 1 period ahead
standing at the end of period t if given by FI FTT t plus 1 which is nothing, but the sum
of S t plus the trend smoothed value of the trend T t computed at the end of period t this
is the third step.
Now, let us look at one example which will make things absolutely clear.
(Refer Slide Time: 09:59)
Here is one problem a company uses exponential smoothing techniques with trend to
forecast usage of it is lawn care products. At the end of July the company wishes to
forecast sales for the month of August. The demand for the month of July was 62 and
they have observed that the trend through June has been 15 additional gallons of product
sold per month. So, there is an upward trend and they had already computed that the
average sales have been 57 gallons per month that is the average level of demand.
And for this particular problem the company uses smoothing constant alpha as 0.2 and
beta as 0.10. The problem is to compute the forecast for the month of August. So, what
we have discussed just now that, at the first step we have to smooth the level of the
So, standing at the end of July the smoothed value of the level denoted by S July is alpha
times actual demand for July plus 1 minus alpha into the smoothed value which was
computed 1 period ahead that is the average value of sales till the month of June plus the
rate of growth or the upward trend that has been observed in the month of June.
So, what we have done? We have taken alpha equals 0.2, but in the problem it is very
clearly given the average value of sales had been for the month of July was 62. So, we
have put in that plus 1 minus betas. So, beta is actually in this problem it is not 1 minus
beta would have been 0.9 sorry, 1 minus alpha. So, 1 minus 0.2 that is 0.8 correct, plus
57 average sales have been 57 gallons per month ok. So, 57 plus the trend factor 15. So,
this total becomes 70.
Once again let us see 0.2 is alpha value and the actual demand for the month of July was
62 we have put in that plus 1 minus alpha that is 0.8 because 1 minus 0.2 is 0.8
multiplied by S t minus 1 which is the average sales is 57, plus the trend through June
has been 15 additional gallons of products so, 15; so, this is 70. So, this is the first step
smoothed value of the level at the end of July.
Second step is to smooth the trend the smoothed value for the trend factor end of July
denoted by T July equals beta multiplied by the smoothed value of the average level in
the month of July minus the same value computed 1 period ahead plus 1 minus beta into
the trend factor 1 period ahead; that means June.
So, what we have done? We have substituted beta equals 0.1 because beta has been given
as 0.1 multiplied by the smoothed value of the average level which is 70 which was
computed in the first step minus the smoothed value of the average level till 1 period
back that is up to June which is given as 57 gallons per month plus 1 minus beta that is
0.9 multiplied by the trend factor through June which is 15. So, this comes out to be
So, forecast including the trend for the month of August standing at end of July equals
the sum of these 2 components S July and T July which is nothing, but 70 plus 14.8 is
84.8 gallons. I think it is now very clear.
(Refer Slide Time: 17:19)
Next we had discussed that the forecast error for the period t is the difference between
the actual demand that had observed; that had been observed in the t-th period is A t
minus whatever was the forecast demand for the t-th period standing at the end of t
minus 1th period which is F t. So, forecast error is given by E t equals A t minus F t.
We had also mentioned that forecasts are never perfect. There is always an error; a
forecast is a forecast. So, we need to know how much we should rely on a chosen
forecasting method. For this we need to keep track of the errors that get accumulated
over several periods of forecast. So, we measure forecast error with this expression of E t
equals A t minus F t.
Now, if this value E t comes out to be negative; that means, it is a negative error then it
implies that we have over forecasts and if E t comes out to be positive then the forecasts
is basically an under forecast.
(Refer Slide Time: 19:10)
There are several indicators for measuring the accuracy of forecast. The most simple of
them is the Mean Absolute Deviation popularly known as MAD. MAD measures the
total error in a forecast without regard to sign.
That is why we basically take the absolute value of the difference between the actual
demand and the forecast and we sum it over n periods that is why this summation symbol
and divided by n, the number of periods for which these errors have been considered.
So, MAD stands for mean average Mean Absolute Deviation. We are taking the absolute
deviation between the actual demand and the forecast demand, and then trying to take the
average value of these absolute demand over n periods. So, it is basically sum over the
absolute value of these differences between actual and forecast for period n divided by
the number of periods n. So, this is one indicator or one measure of forecast error.
The second one is Cumulative Forecast Error; it measures any bias in the forecast, bias
means error. So, cumulative forecast error is summation of the actual demand and the
forecast demand over n periods. Here we are not taking absolute value. Whatever is the
actual forecast error we are summing it up over n periods and that is why, we are saying
that it is a cumulative forecast error.
Then Mean Square Error here we are first taking the difference between actual demand
and forecast demand squaring it up and then we are summing this over n period and then
divided by n; it is something like variance calculation. And we define tracking signal as
TS which is the ratio of the cumulative forecast error divided by the mean absolute
(Refer Slide Time: 22:03)
So, tracking signal is a measure that indicates whether a method of forecasting is
accurately predicting the actual changes in demand; that means, whether the forecasting
system is stable and consistent or not. So, the tracking signal has been defined as the
ratio of the cumulative forecasting error divided by the mean absolute deviation.
Each period we compute the cumulative forecast error and the Mean Absolute Deviation
MAD to reflect current error; that means we have to update the values of CFE and MAD
for every period and then we compare the value of tracking signal with some
(Refer Slide Time: 23:13)
Also, while updating the value of mean absolute deviation we use this particular
expression which is nothing but the mean absolute deviation for the period t; denoted by
MAD t equals the smoothing constant alpha multiplied by the absolute value of error
computed at the period t plus 1 minus alpha into the same value of mad computed 1
period ahead before that is MAD with a subscript t minus 1.
And, if we assume that the forecast errors are normally distributed with a mean of 0, the
relationship between the standard deviation of forecast errors and Mean Absolute
Deviation is very simple is given by this expression sigma equals root over pi by 2 into
mad which is equal to 1.25 into MAD. Therefore, mad is nothing but 0.7978 sigma from
this relationship equals almost 0.8 times the standard deviation of the distribution of
error and with this expression sometimes we can also compute the mean absolute
(Refer Slide Time: 24:58)
And, then we can draw control charts and plot the forecast errors. And, if any of these
values fall outside the control limits then we can basically infer that there is some error
that has crypt in, the forecasting system is not stable and we have to really find out what
are the assignable causes for that whether the forecasting system needs any division or
correction something like that.
(Refer Slide Time: 25:39)
Let us look at this example. A company is comparing the accuracy of 2 forecasting
methods. Forecast using both methods are shown in the next slide along with the actual
values for January through May. The company also uses a tracking signal with plus
minus 4 limits to decide when a forecast should be reviewed. Which forecasting method
So, here you see the control limits value are plus minus 4.
(Refer Slide Time: 26:17)
So, we will use the error.
(Refer Slide Time: 26:24)
So, basically the values are given here the demand values and the forecast values and
you see the errors have been computed for every period. The cumulative forecast error is
minus 15 and then for every period we have to compute the square of the errors. So, we
compute the square of the errors. We compute the absolute value of error for every
period and we also compute the percentage of this absolute error using this expression.
And, the sum of the error square is this 5275, the sum of the absolute values of error is
this, and the absolute percentage error is this much from this table. So, once we know
these values whatever has been asked for in this problem becomes very simple to
(Refer Slide Time: 27:37)
And, as you have seen from the last table that the cumulative forecast error which is also
the mean bias is this much, average forecast error is this one and the mean squared error
comes out to be this much.
(Refer Slide Time: 28:01)
The standard deviation of error is again using the same expression and the data values
that we have already computed comes out to be 27.4, mean absolute deviation is 24.4
and mean absolute percentage error from this expression is 10.2 percent. So, all the
data’s are already computed.
(Refer Slide Time: 28:21)
So, basically what is the implications? A cumulative forecast error of minus 15 indicates
that the forecast has a slight bias to overestimate the demand. The mean squared error
sigma which is the standard deviation of the distribution of errors and the MAD statistics
provide measures of forecast error variability. MAD of 24.4 means that the average
forecast error was 24.4 units in absolute value.
Similarly, the standard deviation of the distribution of errors has been computed to be
27.4 and the map value is 10.2 percent implies that on an average the forecast error was
about 10 percent of the actual demand. Now, these measures become more reliable as the
number of periods of data increases so that is why we take beta over a larger horizon.
(Refer Slide Time: 29:47)
Then the next topic of our discussion is to compute the forecast when there is
seasonality. So, there are two methods the first one the multiplicative seasonal method is
more widely used. This is a method where the seasonal factors are multiplied by an
estimate of average demand to arrive at a seasonal forecast. So, the task is to compute the
Second method is almost same, only thing instead of multiplication the seasonal forecast
are generated by adding a constant to the estimate of average demand per season. So, we
will talk about only this multiplicative seasonal method because that is more widely
(Refer Slide Time: 30:54)
So, in the multiplicative seasonal method for each year first we calculate the average
demand for each season, say the season is by quarter. So, for each year we will calculate
the average demand for each season say each quarter by dividing the annual demand by
the number of seasons per year. So, if it is a; you know if this season extends over a
quarter, so, this value will be 4.
For each year, divide the actual demand for each season by the average demand per
season resulting in a seasonal factor for each season. Then, we will compute the average
seasonal factor for each season using the results from step 2, and then we will calculate
the each seasons forecast for the next year.
(Refer Slide Time: 32:08)
It will very simple if we look at this example. The manager of the Stanley Steemer carpet
cleaning company needs a quarterly forecast of the number of customers expected next
year. So, the carpet cleaning business is seasonal, with a peak in the third quarter and a
trough in the first quarter. The next 2 slides show the quarterly demand data from the
past 4 years.
The manager wants to forecast the customer demand for each quarter of year 5 based on
an estimate of total year 5, demand of 2,600 customers.
(Refer Slide Time: 32:57)
So, this is the first year, you see there are 4 quarters. There is a peak in the 3rd quarter
520 and a trough in the first quarter. The sum of the demand for the 4 quarters is 1000.
So, average quarterly demand comes out to be 1000 by 4 that is 250. Each quarters
demand is then divided by the average demand for the year.
So, the seasonal factors are computed for the first year and have been given as 0.18, 1.34,
2.08, 0.4. We repeat the same for the second year and we calculate the seasonal factors
for each quarter in the second year. Like this we compute it for the 3rd year and the 4th
(Refer Slide Time: 34:02)
So, for each year we compute the seasonal factors for every quarter and then what we
do? We take these seasonal factors for every year for each quarter and then compute the
average of that. That means for the year one for the quarter one we will take the seasonal
factor add it with the seasonal factor for the quarter 1, for the year 2. Then add to that the
seasonal factor for the quarter 1, for the year 3 that is 0.22 plus say 0.18 computed for the
year 4 and then divide it by 4 to get the average value.
(Refer Slide Time: 35:01)
So, the average seasonal factors for all the quarters is computed for the quarter 1 it is
0.2043, for the quarter 2 is 1.2979, for quarter 3 it is 2.0001 and for the 4th quarter it is
this. So, you see from the seasonal factors; you see that there is a peak demand in the
quarter 3 and for quarter 1, there is a trough.
Now, for the next planning horizon; that means, for the 5th year it has been given that
they expect that the total sales will be 2600 units. So, we divide it by 4 to find out the
average level that is 650 units, then for each quarter we multiply it by the corresponding
average seasonal factor for that quarter to arrive at the estimated demand for that
particular quarter considering the seasonality in the demand pattern.
So, we do that and we find that the forecast for quarter 1 is 139.795 units for the 5th year
and look at the demand estimated demand for the 3rd quarter for the 5th year is 1300.06.
So, you see there is a peak here and there is a trough here and you will see that the sum
of all these will lead to the total estimated demand for the fifth year which is 2600 units.
(Refer Slide Time: 37:19)
Similarly, another example. Suppose the multiplicative seasonal method is being used to
forecast customer demand. The actual demand and the seasonal indices are shown; ok.
(Refer Slide Time: 37:37)
So, if the projected demand for year 3 is 1320 units. The problem is what is the forecast
for each quarter of that year. So, we have found out this average index for each quarter
then multiply it by this 330 units to get the forecast for quarter 1, quarter 3 and then
quarter 4 the absolutely. This is easiest method, most simple way in handling seasonality
ok. There are other complicated models like winters method for handling seasonality.
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