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2 FORECASTING Two main reasons why an inventory control system needs to order items some time before customers demand them. lead-time order in batches instead of unit for unit. This means that we need to look ahead and forecast the future demand. A demand forecast is an estimated
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Two main reasons why an inventory control system needs
to order items some time before customers demand them.
This means that we need to look ahead and forecast the
future demand. A demand forecast is an estimated
average of the demand size over some future period. We
also need to determine how uncertain the forecast is. If
the forecast is more uncertain, a larger safety stock is
required. Consequently, it is also necessary to estimate
the forecast error. In general, demand forecast is a
Two Types of Approaches
1.statistical methods for analysis of time series 2.easy to apply and use in computerized
inventory control systems to regularly update
forecasts for thousands of items
1.Forecast the demand for final products and demands for the components are then obtained directly from the production plan; used in Material Requirements Planning (MRP)
2.Other factors – sales campaign
3.Demand of ice cream can be based on the weather forecast
4.Dependencies between the demand for the spare part and previous sales of the machines
5.Dependence of refrigerator demand to forecast of housing constructions
xt = demand in period t,
a = average demand per period (assumed to vary slowly),
t = independent random deviation with mean zero.
a = average demand in period 0,
b = trend, that is the systematic increase or decrease per period (assumed to vary slowly).
Ft = seasonal index in period t (assumed to vary slowly).
If there are T periods in one year, we must require that for any T consecutive periods,
By setting b = 0 in (2.3) we obtain a constant-seasonal model.
The best forecast for t is always zero.
In constant model (2.1), the best forecast is simply our best estimate of a. In (2.2) the best forecast for the demand in period t is similarly our best estimate of a + bt. In (2.3) our best forecast is the estimate of (a + bt)Ft.
A more general demand model covers a wider class of demands, but, on the other hand, we need to estimate more parameters.
A more general model should be avoided unless there is some evidence that the generality will give certain advantages.
= estimate of a after observing the demand in period t,
= forecast for period > t after observing the demand
in period t.
The forecasted demand is the same for any value of > t.
If a is varying more slowly and the stochastic deviations
are larger, we should use a larger value of N.
If we use one month as our period length and set
N = 12, the forecast is the average over the preceding
year. This may be an advantage if we want to prevent
seasonal variations from affecting the forecast.
To update the forecast in period t we use a
linear combination of the previous forecast and
the most recent demand xt,
where > t and
= smoothing constant (0 < < 1).
Compare exponential smoothing to a moving average
the forecast is based on the demands in periods
t, t - 1, ..., t – (N- 1).
The ages of these data are respectively 0, 1, ..., and
N - 1 periods. The weights are all equal to 1/N. The
average age is therefore (N - 1)/2 periods.
A value of “corresponding” to N = 12 is according to (2.9)
obtained as = 2/(12 + 1) = 2/13 0.15.
Note: S’()=1/ 2.
practice to use a smoothing constant between 0.1
Table 2.1 Weights for demand data in exponential smoothing
emphasis on old values of demand. The same
is accomplished by a smaller .
we use = 0.3. On the other hand, stochastic
deviation will influence the demand forecast more compared to when = 0.1. Have to compromise when choosing .
each week, a smaller should be used.
to change N to 52. The “corresponding” value of
is obtained from (2.9) as = 2/(52 + 1) 0.04
by exponential smoothing gives essentially an average
of more recent demands. The forecast cannot predict the independent stochastic deviations.
recent demand as in exponential smoothing.
influence of seasonal variations on the forecast.
where and are smoothing constants between
0 and 1.
The forecast for a future period, t + k is obtained as
the smoothing constants and will mean that the forecasting system reacts faster to changes but will also make the forecasts more sensitive to stochastic deviations.
one and five periods ahead respectively. The smoothing constants are = 0.2 and = 0.1. At the end of period 2,
Our forecast for period 4 is then 94.4 - 0.56 = 93.84
94. At the end of period 4 we obtain the real
demand 170. Applying (2.10) and (2.11) again we get
Note that in (2.3), a + bt represents the development
of demand if we disregard the seasonal variations.
When we record the demand xt in period t we can
similarly interpret xt/ as the demand without seasonal
for i=1, 2,..., T-1, (2.16)
where 0 < < 1 is another smoothing constant.
T consecutive seasonal indices is equal to T.
Therefore, we need to normalize all indices
for i = 0, 1, ... , T-1. (2.17)
for i = 0, 1,..., T-1, and k = 1, 2,....(2.18)
The forecast for period t + k is obtained as
Manually setting seasonal indices is another
shall go through a complete updating of all parameters. Assume that we are dealing with monthly updates, i.e., that T = 12. The smoothing constants are = 0.2, = 0.05, and = 0.2. Assume that the last update took place in period 23 and that this update resulted in the following parameters: ,
0.4 and, . Note that the sum of the seasonal indices equals 12. At this stage according to (2.18).
Applying (2.13) - (2.15) we get
We obtain . By applying (2.17) we get the
updated normalized indices for periods 13 - 24 as
, , =0.403 ,
, and .
The forecast for period 26 is obtained from (2.19) as
where we apply according to (2.18).
Correlated stochastic deviations
ARMA (AutoRegressive Moving Average) model
Croston (1972) has suggested a simple technique to
handle such a situation. The forecast is only updated in
periods with positive demand. In case of a positive
demand two averages are updated by exponential
smoothing: the size of the positive demand, and the time
between two periods with positive demand.
Mean m = E(X).
2 is denoted the variance
Mean Absolute Deviation (MAD)
A common assumption is that the forecast errors are
normally distributed. In that case it is easy to show that
At the end of period t - 1 we obtained from the forecasting
system a forecast for period t, . At this stage we
could regard this as a “mean” for the stochastic demand in
period t, xt.
After period t we know xt and the corresponding absolute
deviation from the “mean”, . It is, in general,
assumed that these absolute variations can be seen as
independent random deviations from a mean which varies
relatively slowly, i.e., that they follow a constant model
according to (2.1).
where 0 < < 1 is a smoothing constant (not necessarily
the same as in (2.5)).
demand data as in Example 2.1 and 2.2.
Table 2.4 shows the updated values of MADt when
using = 0.1 and the initial value MAD2 = 20.
Table 2.4 Updated values of MADt with = 0.1 and initial value MAD2 = 20. The forecasts are obtained by exponential smoothing with trend. The smoothing constants are = 0.2 and = 0.1, and the initial forecast , , see Example 2.2.
was . In period 3 we then obtain from
and similarly in period 4
Note that when updating MAD we use forecasts that
are not rounded, e.g., 94.4 - 0.56 = 93.84 instead of
94, see Example 2.2. The corresponding standard
time periods is
Example 2.5 Assume that MADt is updated each
month and that the most recent value is MADt = 40.
From (2.25) we obtain t 1.25. 40 = 50. In case of
independence over time the standard deviation over
two months is obtained as (2) = 50.21/2 71, and
over 0.5 month as (0.5) = 50. 0.51/2 35.
where 0.5 c 1.
It is usually suitable to let the forecasting system itself
perform certain automatic tests to check whether an
item should go through a detailed manual examination.
These tests are similar to techniques used in connection
with statistical quality control.
Assume that that in period t-1 we obtained the forecast
and MADt-1. If can be regarded as the mean
and the deviations of the demand from the forecast are
normally distributed, we can determine the probability that
the next forecast error is within k standard deviations as,
with k1 = 4 to check whether xt is “reasonable”.
(When checking xt it is appropriate to use MADt-1
instead of MADt, which has been affected by the
demand that we are checking.)
By applying (2.28) with k = , we can see
that the probability that the test, under normal
conditions, should be satisfied is approximately 99.8
If (2.29) is not satisfied there is either some error in the
new demand or in the forecast or, alternatively, an
event with a very low probability has occurred.
It is also common to update the average error in
a similar way. Let
zt = estimate of the average error in period t.
If the forecast works as a correct mean, positive and
negative forecast errors should in the long run be of
about the same size, and zt can be expected to be
relatively close to zero. When updating zt ,it is natural
to use zero as the initial value and to have a relatively
small smoothing constant like = 0.1. A common test
approximately equal to MADt/N1/2. If we compare
(2.31) to (2.29) we can therefore say that it is
reasonable to choose k2 = k1/N1/2. With k1 = 4 and
N = 16, for example, we get k2 = 1.
Systematic errors that are detected by (2.31) can have
different explanations. One possibility is that the forecasting
method is inadequate. For example, if demand has a trend
and we are using simple exponential smoothing, there will
always be a systematic error. Another common reason is
that a large change in the average demand has occurred.
It will then take a long time for the forecast to approach the
new demand level. If such a situation is detected by (2.31),
we can improve the forecasts by restarting the system with
a new, more accurate initial forecast.
Examples of situations when manual forecasts could
be considered are:
A special problem with manual forecasts is that they
sometimes have systematic errors because of optimistic or
pessimistic attitudes by the forecaster.