Today, we will be talking about adaptive forecasting, moving average techniques and
simple exponential smoothing. Unlike the static method of forecasting whatever we
discussed in the previous module, in adaptive forecasting the estimates of level, trend,
and seasonality are updated after each demand observation.
This estimates incorporate all new data that will be observed and we will be discussed
being about a framework for a mixed model, where the demand data contains a level, a
trend, and the seasonal factor.
(Refer Slide Time: 11:46)
We will be taking the same example, the quarterly demand for a rock salt, where there is
seasonality along with trend as we had discussed earlier. You see, we start from the 2nd
quarter of the 1st year, where the demand is 8000. Then, you see you come over here for
the 2nd quarter of the 2nd year, the demand is 10,000 and then, you come to the 2nd
quarter of the 3rd year a 12,000.
So, you will see these points the demand is the lowest and again, there is peak at a
respective period. So, the periodicity is 4. We have 3 years of data and we want to
discuss what is the forecast for the 4th year; 2nd quarter of the 4th year and things like
that.
Because the 1st quarter already we know, 4th of the 4th year. So, we will be computing
the forecast for the 2nd quarter of the 4th year. So, we have data for 3 successive years.
(Refer Slide Time: 03:17)
So, in here, we have a set of historical data for n periods. The demand is seasonal with
periodicity p. We have quarterly data, wherein the pattern repeats itself every year. So,
the periodicity p equals 4.
(Refer Slide Time: 03:41)
So, in adaptive forecasting, we will be utilizing this particular expression, where the
forecast of demand for the period t plus l is given by the expression L t which is the
estimate of level at the end of period t plus l times T into S t plus l. Here it will be
actually T t, the estimate of trend at the end of period t is T t which is L into t.
(Refer Slide Time: 04:37)
So, the forecast for period t plus l, we are interested in and we are making this period at
this forecast standing at the point in time at period t. So, we will be using the estimate L t
and T t which we had estimated at the point or at period t and then, we will substitute in
this expression to compute the forecast for the period t plus l.
(Refer Slide Time: 05:21)
So, the steps involved in adaptive forecasting are first one is initialize. In the initialized
phase which is the first step, we will computing the initial estimates of level and trend
given by L 0 and T 0 and the initial seasonal factors S 1 to S p from the given data. And,
this will be done in the same manner as in the case of static forecasting method, which
has been discussed in the previous module and once, we compute L 0 that will be equal
to L and T 0 will be T.
(Refer Slide Time: 06:25)
Then, given the estimates in period t, the forecast demand for period t plus 1 or say t plus
l uses the equation F t plus l if L t plus l into T into S t plus l. This is actually l; t plus l.
Since our first forecast is for period 1 made with the estimates of level, trend and
seasonal factor at period 0.
(Refer Slide Time: 07:12)
We will be computing the error for the first period which is the difference between the
forecast that we compute for the first period and the difference between that forecast and
the demand that we observe in the period t plus 1. After doing that we will be modifying
the estimates of level, trend and seasonal factors. Once we get the error, the entire thing
will be clear when we take an example.
(Refer Slide Time: 07:55)
Among these adaptive forecasting methods, the most simple is the moving average
method. This moving average method is used when demand has no observable trend or
seasonality. Herein, the systematic component of demand is just the average level and
the level in period t is the average demand over the last N periods that is the expression
for L t is this divided by N. Once we compute that, then the forecast for the next period t
plus 1 is the estimate of L t.
So, F t plus 1 is L t and that estimate will remain same because you know there is no
growth, there is no seasonality. So, F t plus N will also be L t. After observing the
demand for period t plus 1, we can revise the estimates and in that case L t plus 1 will be
given by this expression and having computed L t plus 1, we forecast for the period t
plus 2 is the estimate of level computed at the period t plus 1.
(Refer Slide Time: 10:08)
We will take an example. See do not get afraid with all these expressions and all.
Moving average technique is very simple. If there is a series of data that is given and
suppose, you are computing 3 months moving average.
Then, for the first 3 periods 1 2 3, you just compute the average. That will be the
estimate of the level, that estimate will become the forecast for the fourth period or
fourth month. Then, having observed the demand for the fourth month, again you
recompute the estimate of level. How will you do that?
You discard the value corresponding to the first month’s demand and then, compute the
average considering the period second month, third month and fourth month because in
this case you know the actual demand for the fourth month. So, the average value of
demand for the second, third and fourth period will be the estimate of level that you
compute at this point in time and that will be the forecast for the fifth month.
Again, you will be observing the demand for the fifth month. In that case, while
recomputing the estimate of level, you will discard the demand data corresponding to
second month and compute the average considering the demand for third, fourth and fifth
month and that estimate will be the forecast for the sixth month. That is rolling period.
So, in this example, a supermarket has experienced weekly demand of milk. For the first
period, it is 120; for the second period it is 127; for the third period, it is 114 and for the
fourth period, it is 122 gallon a weekly demand. Problem is to forecast the demand for
period 5 using a four-period moving average and the next portion is what is the forecast
error if demand in period 5 turns out to be 125 gallons? It is very simple.
(Refer Slide Time: 13:35)
What do you do? First, you compute the estimates of level for the fourth period that is L
4 because it is a four period, four-month, four week moving average. So, you compute L
4 as 120.75. This estimate will be the forecast for the period 5, 5th period. So, forecast
for the 5th period standing at the end of fourth period is 120.75 gallons and the error can
be only noted after observing the demand for the period 5.
So, error in forecast is the difference between the forecast value and the demand which is
equal to minus 4.25. Then, you basically revise that estimate. So, in that case, you will be
neglecting or discarding the demand for the first period and then, you the revised
estimate value will be 122 which will be the forecast for the sixth period on a rolling
basis as if the average is moving as simple as that.
(Refer Slide Time: 15:18)
Now, we will talk about single exponential smoothing model. This model is used when
there is no observable trend or seasonality. So, in this case, the systematic component of
demand is only the level, estimate of level. So, initial estimate of level which we denote
by L 0 is assumed to be the average of all historical data and that assumption is based on
the fact that the demand has been assumed to have no observable trend or seasonality.
So, whatever demand data you have got, you just take the average demand considering
all the periods over which the demand data is given that will be the initial estimate of
level which we denote by L 0.
(Refer Slide Time: 16:51)
And that initial estimate of level computed at period 0 will be the forecast for the period
F 1, the first period. So, thus F 1 equals L 0. So, the generalized expression for the
current forecast is F t plus l is L t.
(Refer Slide Time: 17:35)
After observing the demand for the period t plus 1, we revise the estimate of the level
and the revised estimate of the level for the period t plus 1 is given by the expression L t
plus 1 is alpha times the demand for the period t plus 1 plus 1 minus alpha times the
estimate that was computed initially that is standing at for the period t which is L 0.
So, in this case, we compute the revised estimate L 1 which becomes the basis for the
forecast, for the period for the second period that is F 2 equals L 1.
(Refer Slide Time: 18:54)
So, this is the general expression, if you note this particular expression.
(Refer Slide Time: 19:09)
Then, what you find that the current estimate of the level is nothing but a weighted
average of all the past observations of demand, with recent observations weighted higher
than the older observations. So, the significance of which it means or implies that a
higher value of this moving constant alpha corresponds to a forecast that is more
responsive to recent observations; whereas, a lower value of the smoothing constant
represents a more stable forecast which is less responsive to recent observations.
(Refer Slide Time: 20:23)
So, again here do not get you know swayed away with all these expressions and all.
Simple exponential smoothing or single exponential smoothing whatever you might call
it, is extremely simple; suppose, you are trying to make a forecast for the same month of
October.
And you are standing at the end of September, standing at the end of September, if you
want to make a forecast for the demand of that item in the month of October, it is very
simple. Forecast for the month of October is nothing but whatever you had forecasted for
the month of September plus the smoothing constant times the difference between the
forecast and the actual demand that was observed in the month of September.
As simple as that; forecast for the month of October is whatever we had forecasted for
the month of September plus smoothing constant times the difference between the
forecast and the actual demand that was observed in the month of September.
That is this. And if we give a very high value of alpha that is the smoothing constant, we
are giving more importance or weightage to the recent observations compared to
whatever has happened in the past. But if the manager or the decision maker feels that
ok, due to some particular reason which may not hold good in future.
The demand for this recent observation has become very high, but actually whatever we
have observed in the past that is the actual thing or pattern. Then, you give a low value to
the smoothing constant. Let us take this example. Same example what we had discussed
in case of this moving average method, same supermarket example.
A supermarket has experienced weekly demand of milk and the data is given. We have
to forecast the demand for period 5, using a simple exponential smoothing method and
what is the forecast error if demand in period 5 turns out to be 125 gallon. So, what we
will do? We will take first the average of all these 4 weeks to compute the initial level of
the estimate that is L 0.
(Refer Slide Time: 25:28)
So, L 0 is nothing but the sum of the past 4 weeks of demand which if you look at which
is D 1 plus D 2 plus D 3 plus D 4 divided by 4 which is 120.75 and that initial level
initial estimate of the level that is L 0, we will serve as the forecast for the first period.
So, with this method the forecast for the first period F 1 is L 0 is 120.75 and the error
corresponding to the forecast that was made for the first period is E 1 given by the
difference between F 1 and D 1 which is in this case 120.75 minus the demand for the
first period over 120. So, what you see? This is 0.75.
This error you can only compute, when you observe the actual demand for the first
period. Then, at the end of the first period, you revise the estimate for the level and that
is given by L 1. This revised estimate for the level standing at the end of period 1 is
given by the expression L 1 equals alpha times D 1 plus 1 minus alpha into the estimate
of level computed at the initial starting period which we have denoted by L 0.
The smoothing constant varies between 0 to 1 and in here, we have give chosen a
smoothing constant value of 0.1, alpha equals 0.1; that means, we are not giving much
more importance to the recent periods data because there is no growth, there is no
seasonality, it is a stable demand pattern.
So, there is no reason why should we give more weight or importance to the demand data
that we have observed in the first period. Having done that, we compute L 1. In this case,
it turns out to be 120.68. Now, this estimate L 1 will serve as the forecast for the second
period because F 2 is nothing but L 1.
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