<< стр. 3(всего 15)СОДЕРЖАНИЕ >>
7 1.08 1.07 1.01
8 1.05 1.02 1.02
9 0.97 0.98 0.98
10 1.01 1.02 0.98
11 0.96 0.99 0.97
12 1.09 1.08 1.01
В·В·В· В·В·В· В·В·В· В·В·В·
152 1.03 1.02 1.01
153 1.01 0.98 1.03
154 1.01 1.02 0.99
155 0.95 0.99 0.96
156 1.05 1.08 0.98
157 0.85 0.82 1.04

Table 2.12 Demand forecasts pT (П„ ), T = 163, П„ = 1, . . . , 6, in the P&D problem.

T +П„ qT (П„ ) vT (П„ ) sT (П„ ) pT (П„ )
Month

Aug 99 164 873.19 1.04 1.02 933.84
Sep 99 165 874.62 1.04 0.98 895.28
Oct 99 166 876.05 1.04 1.02 932.52
Nov 99 167 877.48 1.04 0.99 898.46
Dec 99 168 878.92 1.03 1.08 976.03
Jan 00 169 880.35 1.03 0.82 743.76

2.5.1 Elementary technique
The forecast for the п¬Ѓrst time period ahead is simply given by
pT +1 = dT .

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FORECASTING LOGISTICS REQUIREMENTS 43
(qv)t

1200

1000

800

600

400

200

0
t
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Figure 2.5 Combined trend and cyclical effects (qv)t , t = 7, . . . , 157, in the P&D problem.

(qv)t

1200

1000

800

600

400

200

0
t
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Figure 2.6 Linear trend (in grey) qt , t = 7, . . . , 157, in the P&D problem.

The method is straightforward. The forecast time series reproduces the demand pattern
with one period delay. Consequently, it usually produces rather poor predictions.

Sarath is a Malaysia-based distributor of Korean appliances. The sales volume of
portable TV sets during the last 12 weeks in Kuala Lumpur is shown in Table 2.13.
The demand pattern is depicted in Figure 2.13. It can be seen that the trend is
constant. By using the elementary technique, we obtain
p13 = d12 = 1177.

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44 FORECASTING LOGISTICS REQUIREMENTS
vt

1.20

1.10

1.00

0.90

0.80

0.70

0.60
t
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Figure 2.7 Cyclical effect vt , t = 7, . . . , 157, in the P&D problem.

(sr)t

1.20

1.10

1.00

0.90

0.80

0.70

0.60
t
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Figure 2.8 Combined seasonal effect and random п¬‚uctuation (sr)t ,
t = 7, . . . , 157, in the P&D problem.

2.5.2 Moving average method
The moving average method uses the average of the r most recent demand entries as
the forecast for п¬Ѓrst period ahead (r 1):
rв€’1
dT в€’k
pT +1 = .
r
k=0

If r is chosen equal to 1, the moving average method reduces to the elementary
technique.

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st

1.20

1.10

1.00

0.90

0.80

0.70

0.60
t
1 2 3 4 5 6 7 8 9 10 11 12

Figure 2.9 Seasonal effect st = st , t = 1, . . . , 12, in the P&D problem.
ВЇ

rt

1.20

1.10

1.00

0.90

0.80

0.70

0.60
t
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Figure 2.10 Random variation rt , t = 7, . . . , 157, in the P&D problem.

Table 2.13 Number of portable TV sets sold by Sarath company in the last 12 weeks.

Time period Quantity Time period Quantity

1 1180 7 1162
2 1176 8 1163
3 1185 9 1180
4 1163 10 1170
5 1188 11 1161
6 1172 12 1177

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46 FORECASTING LOGISTICS REQUIREMENTS
vt_ (П„)

1.20

1.10

1.00

0.90

0.80

0.70

0.60
П„
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

ВЇ
Figure 2.11 Extrapolation of the cyclical effect (in grey) vtВЇ(П„ ), t = 157,
П„ = 1, . . . , 12, in the P&D problem.

dt (П„)

1200

1000

800

600

400

200

0
П„
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Figure 2.12 Demand forecasting (in grey) of electrosurgical equipment in
France for the subsequent six months.

Using the moving average method for solving the Sarath problem above, we obtain
the forecasts,
d12 + d11
p13 = = 1169,
2
d12 + d11 + d10
p13 = = 1169.33,
3
with r = 2 and r = 3, respectively.

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dt

1600

1400

1200

1000

800

600

400

200

0
t
1 2 3 4 5 6 7 8 9 10 11 12

Figure 2.13 Demand pattern of portable TV sets sold by
the Sarath company in the last 12 weeks.

When using the moving average method, one should wait for the п¬Ѓrst r demand data
to be available before producing a forecast. To overcome this drawback, the forecasts
for time periods T < r are obtained by using the average of the data available in the
п¬Ѓrst T periods, i.e.
T в€’1
dT в€’k
pT +1 = , T < r.
r
k=0

For example, if T = 1, then p2 = d1 , whereas if T = 2, then p3 = (d1 + d2 )/2.
A key aspect of the moving average method is the choice of parameter r. A small
value of r allows a rapid adjustment of the forecast to demand п¬‚uctuations but, at
the same time, increases the inп¬‚uence of random perturbations. In contrast, a high
value of r effectively п¬Ѓlters the random effect, but produces a slow adaptation to
demand variations. This phenomenon can be explained as follows. Let d1 , . . . , dT be
independent random variables with expected value Вµ and standard deviation Пѓ . The
random variable pT +1 will have an expected value ВµpT +1 deп¬Ѓned as
rв€’1 rв€’1
ВµdT в€’k Вµ
ВµpT +1 = = =Вµ
r r
k=0 k=0

(i.e. the same expected value as d1 , . . . , dT ), and a standard deviation
1
ПѓpT +1 = в€љ Пѓ.
r
It follows that the dispersion of pT +1 is less than that of d1 , . . . , dT and decreases
as r increases.

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48 FORECASTING LOGISTICS REQUIREMENTS

2.5.3 Exponential smoothing method
The exponential smoothing method (also known as the Brown method) can be seen
as an evolution over the moving average technique. The demand forecast is obtained
by taking into account all historical data and assigning lower weights to older data.
The demand forecast for the п¬Ѓrst period ahead is given by

pT +1 = О±dT + (1 в€’ О±)pT , (2.5)

where О± в€€ (0, 1) is a smoothing constant. Here, pT represents the demand forecast
for period T made at time period T в€’ 1.

Suppose that at time period T we have dT = 1177 and pT = 1182. Then the
forecast pT +1 obtained by means of the Brown method is given by (О± = 0.2)

pT +1 = 0.2 Г— 1177 + (1 в€’ 0.2) Г— 1182 = 1181.

Rewriting Equation (2.5) as

pT +1 = pT + О±(dT в€’ pT ) = pT + О±eT ,

we obtain the following interpretation: the demand forecast at time period T + 1
corresponds to the sum of the demand value estimated at time period T в€’ 1 and a
fraction of the forecasting error at time period T . This means that if the value of pT is
overestimated with respect to dT , the forecasting value pT +1 is lower than pT . Vice
versa, if pT is an underestimate of dT , then pT +1 is increased.
The demand history is embedded into pT , and hence does not appear explicitly
in the previous formula. Applying Equation (2.5) recursively, all the demand history
appears explicitly:
pT = О±dT в€’1 + (1 в€’ О±)pT в€’1 .
From Equation (2.5), we obtain

pT +1 = О±dT + (1 в€’ О±)[О±dT в€’1 + (1 в€’ О±)pT в€’1 ].

Iterating these substitutions (and taking into account that p2 can be assumed equal
to d1 ), the following relation is obtained:
T в€’2
(1 в€’ О±)k dT в€’k + (1 в€’ О±)T в€’1 d1 .
pT +1 = О±
k=0

In this relation past demand entries are multiplied by exponentially decreasing weights
(this is where the name of the method comes from). Finally, we observe that the sum
of all weights О± T в€’2 (1 в€’ О±)k + (1 в€’ О±)T в€’1 is equal to 1.
k=0

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Table 2.14 Demand forecasts of portable TV sets in the Sarath problem.

t pt t pt

2 1180.00 8 1172.51
3 1178.80 9 1169.65
4 1180.66 10 1172.76
5 1175.36 11 1171.93
6 1179.15 12 1168.65
7 1177.01 13 1171.16

If we use the exponential smoothing method (with О± = 0.3) for solving the fore-
casting problem of Sarath company, we get
p2 = d1 = 1180.

For t = 3,
p3 = О±d2 + (1 в€’ О±)p2 = 1178.80.
Proceeding recursively up to t = 12, we obtain the results reported in Table 2.14.

2.5.4 Choice of the smoothing constant
The choice of a value for О± plays an important role in the exponential smoothing
method. High values of О± give a larger weight to the most recent historical data and
therefore allow us to follow rapidly the demand variations. On the other hand, lower
values of О± yield a forecasting method less dependent on the random п¬‚uctuation but,
at the same time, cannot take quickly into account the most recent variations of the
demand.
In practice, the value of О± is frequently chosen between 0.01 and 0.3. However, a
larger value may be preferable if rapid demand changes are anticipated.
In order to estimate the best value of О±, it is worth evaluating a posteriori the errors
that would have been made in the past if the Brown method had been applied with
different values of О± (e.g. the values between 0.1 and 0.5, with a step length equal to
0.05). A more detailed treatment of this topic will be given in Section 2.9.

2.5.5 The demand forecasts for the subsequent time periods
The methods just illustrated can be used to forecast demand one period ahead. In
order to predict demand for the subsequent time periods, it is sufп¬Ѓcient to recall that
the trend is assumed to be constant. Consequently,
pT (П„ ) = pT +1 , П„ = 2, 3, . . . , П„ ,
ВЇ
where the forecasting value pT +1 is obtained by using any technique described above
ВЇ
and П„ represents the duration of the forecasting time horizon. In such a context, the

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50 FORECASTING LOGISTICS REQUIREMENTS

forecasting time horizon is said to be rolling, because, once a new demand value
becomes available, the time horizon shifts one time period ahead.

The four-weeks-ahead forecasts are needed for the Sarath problem. By using the
moving average method with r = 2, the predictions are
d12 + d11
p13 [= p12 (1)] = p12 (2) = p12 (3) = = 1169.
2
At time period t = 13, the rolling forecasting horizon would cover the time periods
t = 14, 15, 16. For example, if d13 = 1173, then
d13 + d12
p13 (1) = p13 (2) = p13 (3) = = 1175.
2
Thus, for t = 14 a new updated value is available (p13 (1) = 1175), which substi-
tutes the previous one (p12 (2) = 1169). Similar considerations are valid for t = 15.

2.6 Further Time Series Extrapolation Methods:
the Linear Trend Case
If the trend is linear and no cyclical or seasonal effect is displayed, the forecasting
methods are based on the following computational scheme:
pT (П„ ) = aT + bT П„, П„ = 1, 2, . . . .
For estimating aT and bT , we can use the techniques illustrated below.

2.6.1 Elementary technique
This is the simplest technique:
aT = dT bT = dT в€’ dT в€’1 .
and

Sarath company also distributes satellite receivers. The items sold to the stores
located in Kuala Lumpur district during the last 12 weeks are reported in Table 2.15.
As shown in Figure 2.14, the trend is linear. By using the elementary technique to
forecast the demand in П„ th periods ahead, we get
p12 (П„ ) = a12 + b12 П„ = d12 + (d12 в€’ d11 )П„ = 1230 + 100П„, П„ = 1, 2, . . . .
In particular, for the п¬Ѓrst time period ahead (П„ = 1):
p13 = 1330.

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Table 2.15 Number of kits for satellite equipment furnished by
Sarath company in the last 12 weeks.

Time period Quantity Time period Quantity

1 630 7 895
2 730 8 1010
3 880 9 1030
4 850 10 1150
5 910 11 1130
6 890 12 1230

dt

1400

1200

1000

800

600

400

200

0
t
1 2 3 4 5 6 7 8 9 10 11 12

Figure 2.14 Demand pattern of kit for satellite equipment furnished by
Sarath company in the last 12 weeks.

2.6.2 Linear regression method

In order to estimate aT and bT , this method determines the regression line which best
interpolates the r most recent demand entries (i.e. dT в€’r+1 , . . . , dT в€’1 , dT ):
rв€’1 rв€’1
в€’ 2 (r в€’ 1) k=0 dT в€’k +
1
k=0 kdT в€’k
bT = ,
в€’ 1)2 в€’ 1 r(r в€’ 1)(2r в€’ 1)
1
4 r(r 6
rв€’1
+ 2 bT r(r в€’ 1)
1
k=0 dT в€’k
aT = .
r

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The satellite receiver demand of Sarath company can be predicted by means of the
linear regression method. If r = 4, then
rв€’1
dT в€’k = 4540,
k=0
rв€’1
kdT в€’k = 6520,
k=0

and
b12 = 58,
a12 = 1222.

Therefore,
p12 (П„ ) = a12 + b12 П„ = 1222 + 58П„, П„ = 1, 2, . . . .
In particular, for the п¬Ѓrst time period ahead (П„ = 1):
p13 = 1280.

2.6.3 Double moving average method
The method is an extension of the moving average method illustrated above. Let r
(> 1) be a double moving average parameter. We get

aT = 2ОіT в€’ О·T ,
2
bT = (ОіT в€’ О·T ),
r в€’1
where ОіT is the average of the r most recent demand entries,
rв€’1
dT в€’k
ОіT = ,
r
k=0

and О·T represents the average of the r most recent average demand entries, i.e.
rв€’1
ОіT в€’k
О·T = .
r
k=0

Whenever r past demand data are not available (T < r), the computation of ОіT and
О·T can be executed along the guidelines illustrated for the moving average method.

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We will now solve the satellite receiver problem of Sarath company by means of
the double moving average method (with r = 3). We get
d12 + d11 + d10
Оі12 = = 1170,
3
Оі12 + Оі11 + Оі10
О·12 = ,
3
where
d11 + d10 + d9
Оі11 = = 1103.33,
3
d10 + d9 + d8
Оі10 = = 1063.33.
3
Therefore,
О·12 = 1112.22,
from which

a12 = 2Оі12 в€’ О·12 = 1227.78,
b12 = Оі12 в€’ О·12 = 57.78.

Therefore,
p12 (П„ ) = a12 + b12 П„ = 1227.78 + 57.78П„, П„ = 1, 2, . . . .

In particular, for the п¬Ѓrst period ahead (П„ = 1):

p13 = 1285.56.

2.6.4 The Holt method
The exponential smoothing method, introduced in Section 2.5.3, is unable to deal
with a linear trend. The Holt method is a modiп¬Ѓcation of the exponential smoothing
method and is based on the following two relations:

aT = О±dT + (1 в€’ О±)(aT в€’1 + bT в€’1 ), (2.6)
bT = ОІ(aT в€’ aT в€’1 ) + (1 в€’ ОІ)bT в€’1 . (2.7)

Applying recursively Equations (2.6) and (2.7), it is possible to express aT and bT
as a function of the past demand entries d1 , . . . , dT .

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Table 2.16 Computation of at and bt , t = 1, . . . , 12, in
the Sarath satellite receiver forecasting problem.

t at bt

1 630.00 0.00
2 660.00 9.00
3 732.30 27.99
4 787.20 36.06
5 849.29 43.87
6 892.21 43.59
7 923.56 39.91
8 977.43 44.10
9 1024.07 44.86
10 1093.26 52.16
11 1140.79 50.77
12 1203.09 54.23

In order to start the procedure, a1 and b1 must be speciп¬Ѓed. They can be chosen in
the following way:

a1 = d1 b1 = 0.
and

In this way, we have p2 = p1 (1) = a1 + b1 = d1 , as in the exponential smoothing
method. The choice of parameters О± and ОІ is conducted according to the same criteria
illustrated for the exponential smoothing method.

Applying the Holt method (with О± = ОІ = 0.3) to the Sarath satellite receiver fore-
casting problem, the values of at and bt , t = 1, . . . , 12, in Table 2.16 are generated.
As a result,
p12 (П„ ) = a12 + b12 П„ = 1203.09 + 54.23П„, П„ = 1, 2, . . . .
In particular, for the п¬Ѓrst period ahead (П„ = 1):
p13 = 1257.32.

2.7 Further Time Series Extrapolation Methods:
the Seasonal Effect Case
This section describes the main forecasting method when the demand pattern displays
a constant or linear trend, and a seasonal effect.

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Table 2.17 Number of air conditioners sold in the last 24 months by Sarath company.

Time period Quantity Time period Quantity

1 915 13 815
2 815 14 1015
3 1015 15 915
4 1115 16 1315
5 1415 17 1215
6 1615 18 1615
7 1515 19 1315
8 1415 20 1115
9 815 21 1115
10 615 22 915
11 315 23 715
12 815 24 615

2.7.1 Elementary technique

If the trend is constant, then

pT (П„ ) = dT +П„ в€’M , П„ = 1, . . . , M. (2.8)

On the basis of Equation (2.8), the forecast related to the time period T + П„ cor-
responds to the demand value M time periods back. More generally, for a temporal
horizon whose length is superior to one cycle, we have

pT (kM + П„ ) = dT +П„ в€’M , П„ = 1, . . . , M, k = 1, 2, . . . .

The air conditioners supplied by Sarath company during the last 24 months are
reported in Table 2.17. The demand pattern (Figure 2.15) displays a linear trend and
a seasonal effect with a cycle duration M = 12. By using the elementary technique,
we get
p24 (П„ ) = d24+П„ в€’12 , П„ = 1, . . . , 12.
In particular,

p25 = d25в€’12 = d13 = 815,

p24 (2) = d26в€’12 = d14 = 1015.

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dt

1800

1600

1400

1200

1000

800

600

400

200

0
t
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figure 2.15 Demand pattern of Sarath air conditioners for the last 24 months.

2.7.2 Revised exponential smoothing method
This method can be used whenever the trend is constant. It is based on the following
computational scheme,
pT (П„ ) = aT sT +П„ , П„ = 1, . . . , M, (2.9)
where aT takes into account the constant trend (and can be interpreted as the fore-
casted demand without the seasonal effect), whereas sT +П„ is the seasonal index (see
Section 2.4.1) for period T + П„ . More generally, for a time horizon whose duration
is greater than one cycle time, we get
pT (kM + П„ ) = aT sT +П„ , П„ = 1, . . . , M, k = 1, 2, . . . .
Assuming, without loss of generality, that the available historical data are sufп¬Ѓcient
to cover an integer number K = T /M of cycles, the parameters aT and sT +П„ , П„ =
1, . . . , M, can be computed by the following relations,
dT
aT = О± + (1 в€’ О±)aT в€’1 , (2.10)
sT
d(Kв€’1)M+П„
sT +П„ = skM+П„ = ОІ + (1 в€’ ОІ)s(Kв€’1)M+П„ , П„ = 1, . . . , M, (2.11)
a(Kв€’1)M+П„
where О± and ОІ are smoothing constants (0 О±, ОІ 1). Equation (2.10) expresses
aT as the weighted sum of two components: the п¬Ѓrst, dT /sT , represents the value of
the demand at time period T without the seasonal effect, while the second represents
the forecast, without the seasonal effect, at time period T в€’ 1. A similar interpretation
can be given to Equation (2.11). However, in this case, it is necessary to take into
account the periodicity of the seasonal effect. Using the reasoning of Sections 2.5.3
and 2.6.4, it is possible to develop recursively Equations (2.10) and (2.11) in such a

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way that all demand entries d1 , . . . , dT appear explicitly. To start the procedure we
set a0 equal to the average demand in the п¬Ѓrst time cycle, i.e.
d1 + В· В· В· + dM
ВЇ
a0 = d(1) = ,
M
whereas we can select the following initial estimate of st , t = 1, . . . , M:
ВЇ ВЇ ВЇ
dt /d(1) + dt+M /d(2) + В· В· В· + dt+(Kв€’1)M /d(K)
st = . (2.12)
K
It is worth noting that the numerator is the sum of the demand entries of the tth
time period for each cycle (dt , dt+M , . . . ) divided by the average demand of the
ВЇ ВЇ
corresponding cycles (d(1) , d(2) , . . . ). Equation (2.12) implies
M
T
st = = M,
K
t=1

i.e. the average seasonal index for the п¬Ѓrst cycle is equal to 1. However, this con-
dition cannot be satisп¬Ѓed for the subsequent cycles, and for this reason, in order to
respect Equation (2.2), it is necessary to normalize the indices st , t = (k в€’ 1)M +
1, . . . , kM, k = 2, 3, . . . .

To solve the air conditioner forecasting problem of Sarath company, we use the
revised exponential smoothing method with О± = ОІ = 0.3. To this end, we compute
the mean value of the demand during the two time cycles,
d1 + В· В· В· + d12
ВЇ
d(1) = = 1031.67,
12
d13 + В· В· В· + d24
ВЇ
d(2) = = 1056.67,
12
from which we obtain
ВЇ
a0 = d(1) = 1031.67.
Then we compute the seasonal indices st , t = 1, . . . , 12 (see Table 2.18), using
Equation (2.12). We observe that
12
1
s=
ВЇ st = 1.
12
t=1

Using Equations (2.10) and (2.11), we obtain the results reported in Tables 2.19
and 2.20, where the values of st , t = 13, . . . , 36, are already normalized. From
Equation (2.9), we get
p25 = p24 (1) = 959.20,
p24 (2) = 1016.66,

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Table 2.18 Time series of st , t = 1, . . . , 12, in
the Sarath air conditioner forecasting problem.

t st t st

1 0.83 7 1.36
2 0.88 8 1.21
3 0.92 9 0.92
4 1.16 10 0.73
5 1.26 11 0.49
6 1.55 12 0.69

Table 2.19 Time series of at , t = 1, . . . , 24, in
the Sarath air conditioner forecasting problem.

t at t at

1 1053.25 13 969.16
2 1016.61 14 1035.30
3 1040.86 15 1016.74
4 1016.31 16 1056.87
5 1048.13 17 1022.92
6 1046.90 18 1029.51
7 1067.89 19 1007.53
8 1097.37 20 975.86
9 1033.17 21 1062.15
10 975.60 22 1135.18
11 875.39 23 1269.56
12 969.19 24 1140.58

and so on. Figure 2.16 shows both the demand pattern during the п¬Ѓrst 24 months and
the forecasting of the subsequent 12 months.

2.7.3 The Winters method
The Winters method can be used whenever there is a linear trend and a seasonal effect:

pT (kM + П„ ) = [aT + bT (kM + П„ )]sT +П„ , П„ = 1, . . . , M, k = 1, 2, . . . . (2.13)

As in the revised exponential smoothing method, we assume that the historical data
available are enough to have an integer number K = T /M of cycles. The Winters
method is based on the following relationships for the computation of aT , bT and

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Table 2.20 Time series of st , t = 13, . . . , 36, in
the Sarath air conditioner forecasting problem.

t st t st

13 0.84 25 0.84
14 0.85 26 0.89
15 0.94 27 0.93
16 1.14 28 1.17
17 1.29 29 1.26
18 1.55 30 1.55
19 1.38 31 1.35
20 1.24 32 1.21
21 0.88 33 0.93
22 0.70 34 0.73
23 0.45 35 0.49
24 0.73 36 0.67

dt (П„)
1800

1600

1400

1200

1000

800

600

400

200

0
П„
10 20 30

Figure 2.16 Demand forecasting (in grey) of air conditioners in the Sarath problem.

sT +П„ , П„ = 1, . . . , M:
dT
aT = О± + (1 в€’ О±)(aT в€’1 + bT в€’1 ), (2.14)
sT
bT = О·(aT в€’ aT в€’1 ) + (1 в€’ О·)bT в€’1 , (2.15)
d(Kв€’1)M+П„
sT +П„ = sKM+П„ = ОІ + (1 в€’ ОІ)s(Kв€’1)M+П„ , П„ = 1, . . . , M,
a(Kв€’1)M+П„
(2.16)
where О±, О· and ОІ are smoothing constants chosen in the interval (0, 1). In order to

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60 FORECASTING LOGISTICS REQUIREMENTS
dt (П„)
16000

14000

12000

10000

8000

6000

4000

2000

0
П„
10 20 30

Figure 2.17 Demand pattern and demand forecasting (in grey) of
microwave ovens in the Sarath problem.

use Equations (2.14)вЂ“(2.16) recursively, we need an estimate of the values a0 , b0 and
st , t = 1, . . . , M. We can use
ВЇ ВЇ
d(K) в€’ d1
b0 = . (2.17)
T в€’M
To explain this formula, we observe that the numerator represents the variation of the
mean value of the demand between the п¬Ѓrst and the last period. In addition, we note
that the mean value of the demand in the п¬Ѓrst cycle corresponds to time period

t = 2 (M + 1),
1

which is the вЂ˜centreвЂ™ of the п¬Ѓrst M time periods (see Section 2.4.1). Similarly, the
mean value of the demand of the last time period is assumed to correspond to the time
period
t = T в€’ 2 (M в€’ 1).
1

Hence, there are T в€’ M time periods between the centres of the п¬Ѓrst and the last
period. The parameter a0 can be determined as follows:
ВЇ
a0 = d(1) в€’ 2 (M + 1)b0 .
1
(2.18)

An estimate of the seasonal indices during the п¬Ѓrst period is given by
dt
st = t = 1, . . . , M,
, (2.19)
a0 + b0 t
subject to a normalization, if necessary.

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Table 2.21 reports the number of microwave ovens sold by Sarath company in the
last 24 months in Southern Malaysia. The demand pattern (see Figure 2.17) has both
a linear trend and a seasonal effect (M = 12). In order to forecast the demand for
the subsequent 12 months, we use the Winters method. First, we compute the mean
demand value in the K = 2 time cycles, that is
d1 + В· В· В· + d12
ВЇ
d(1) = = 2089.58,
12
d13 + В· В· В· + d24
ВЇ
d(2) = = 3674.50.
12
Then, we determine the value b0 through Equation (2.17),
ВЇ ВЇ
d(1) + d(2)
b0 = = 132.08,
12
and the value a0 through Equation (2.18),
ВЇ
a0 = d(1) в€’ 6.5b0 = 1231.09.
The seasonal indices st , t = 1, . . . , M (already normalized), determined by Equa-
tion (2.19), are reported in Table 2.22.
Using Equations (2.14)вЂ“(2.16) with О± = О· = ОІ = 0.1, we obtain the results
reported in Tables 2.23 and 2.24.
The st values, t = 13, . . . , 36, are already in normalized form. From Equa-
tion (2.13) we obtain the forecasting values for the subsequent 12 months, reported
in Table 2.25. Figure 2.17 shows both the demand pattern in the п¬Ѓrst 24 months and
the forecasts in the subsequent 12 months.

2.8 Advanced Forecasting Methods
For the sake of completeness, six advanced forecasting techniques are outlined in this
section. As stated in Section 2.2, the п¬Ѓrst п¬Ѓve approaches can be classiп¬Ѓed as casual
methods, whereas the sixth one is a complex time series extrapolation technique.

Econometric models. Econometric models consist of several interrelated regres-
sion equations linking the demand to be forecasted and its main determinants. The
parameters of such relations have to be estimated simultaneously.

InputвЂ“output models. InputвЂ“output analysis, introduced by Wassily Leontief, is
concerned with the interdependence among the various industries and sectors of the
economy. Once such a dependence has been established, the variation in demand of a
commodity can be forecasted given the forecasts for the other commodities (obtained,
for example, through econometric models).

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Table 2.21 Number of microwave ovens produced in
the last 24 months in the Sarath problem.

Time period Quantity Time period Quantity

1 682 13 416
2 416 14 1746
3 1613 15 2411
4 1613 16 2544
5 1746 17 4140
6 2677 18 4539
7 4672 19 7997
8 5603 20 8263
9 3741 21 7465
10 1480 22 3209
11 682 23 1081
12 150 24 283

Table 2.22 Time series of st , t = 1, . . . , 12, in
the Sarath microwave ovens forecasting problem.

t st t st

1 0.50 7 2.17
2 0.28 8 2.45
3 0.99 9 1.55
4 0.92 10 0.58
5 0.92 11 0.25
6 1.32 12 0.05

Life-cycle analysis. Most items (and services) pass through the usual stages of
introduction, growth, maturity and decline, as shown by the вЂ˜S curveвЂ™ of Figure 2.3.
In each stage the product (or service) is demanded by a particular subset of potential
customers. Life-cycle analysis attempts to predict clientsвЂ™demand through an analysis
of their behaviour.

Computer simulation models. Computer simulation can be used to estimate the
impact of changes of policy (e.g. inventory policies, production schedules) on the
demand for п¬Ѓnished goods.

Neural networks. Neural networks are made up of a set of elementary nonlinear
systems reproducing the behaviour of biological neurons. If properly trained by means
of the past demand entries, they can be used to make a forecast.

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Table 2.23 Time series of at and bt , t = 1, . . . , 24, in
the Sarath microwave ovens forecasting problem.

t at bt t at bt

1 1362.96 132.06 13 2733.31 110.74
2 1494.82 132.04 14 3186.32 144.96
3 1626.67 132.02 15 3241.05 135.94
4 1758.50 132.00 16 3316.38 129.88
5 1890.32 131.98 17 3549.47 140.20
6 2022.13 131.96 18 3663.31 137.57
7 2153.93 131.95 19 3789.22 136.40
8 2285.72 131.93 20 3869.94 130.83
9 2417.51 131.92 21 4082.83 139.04
10 2549.30 131.90 22 4352.16 152.07
11 2681.08 131.89 23 4478.58 149.50
12 2812.87 131.88 24 4695.76 156.27

Table 2.24 Time series of st , t = 13, . . . , 36, in
the Sarath microwave ovens forecasting problem.

t st t st

13 0.50 25 0.47
14 0.28 26 0.31
15 0.99 27 0.97
16 0.92 28 0.91
17 0.92 29 0.95
18 1.32 30 1.32
19 2.17 31 2.17
20 2.45 32 2.43
21 1.55 33 1.58
22 0.58 34 0.60
23 0.25 35 0.25
24 0.05 36 0.05

BoxвЂ“Jenkins method. The BoxвЂ“Jenkins method is made up of three steps (iden-
tiп¬Ѓcation, parameter evaluation and diagnostic check). In the п¬Ѓrst phase, the most
appropriate forecasting method is selected from a set of techniques. To this end, the
past demand entries are used to generate a set of autocorrelation functions, which are
then compared. In the second phase, the coefп¬Ѓcients of the forecasting method are
selected so as to minimize the mean squared error. Finally, an autocorrelation func-
tion of the error is determined to verify the adequacy of the method chosen and its
corresponding parameters. In case of negative result, the entire procedure is executed

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Table 2.25 Demand forecast of Sarath microwave ovens in the subsequent 12 months.

П„ p24 (П„ ) П„ p24 (П„ )

1 2266.79 7 12 560.23
2 1533.73 8 14 427.88
3 5009.31 9 9 640.73
4 4815.66 10 3 741.53
5 5207.48 11 1 627.72
6 7431.86 12 355.89

again by discarding the forecasting method previously chosen. Of course, when new
demand entries become available, the whole procedure is run again.

2.9 Selection and Control of Forecasting Methods
Forecasting methods can be evaluated through accuracy measures calculated on the
basis of errors made in the past. Such measures can be employed to select the most
precise approach. Moreover, in the case of periodic predictions (like those required
by inventory management), forecasting errors should be monitored in order to adjust
parameters if needed. For the sake of brevity, we examine these issues for the case
where a one-period-ahead forecast has to be generated.

2.9.1 Accuracy measures
To evaluate the accuracy of a forecasting method, the errors made in the past have
to be computed. Then a number of indices (the mean absolute deviation (MAD), the
mean absolute percentage deviation (MAPD) and the mean squared error (MSE)) at
time period t can be deп¬Ѓned:
t
k=2 |ek |
MADt = , (2.20)
t в€’1
t
k=2 |ek |/dk
MAPDt = 100 , (2.21)
t в€’1
t 2
k=2 ek
MSEt = , (2.22)
t в€’2
where 1 < t T for Equations (2.20) and (2.21), and 2 < t T for Equation (2.22).
These three accuracy measures can be used at time period t = T to establish a
comparison between different forecasting methods. In particular, MAPDT can be
used to evaluate the quality of a forecasting method (see Table 2.26).

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Table 2.26 Evaluation of the forecasting accuracy through MAPDT .

MAPDT Quality of forecast

10% Very good
>10%, 20% Good
>20%, 30% Moderate
>30% Poor

Table 2.27 Mean absolute deviation in the Sarath microwave ovens forecasting problem.

0.05 9.30
0.10 9.27
0.15 9.28
0.20 9.33
0.25 9.40
0.30 9.50
0.35 9.63
0.40 9.79
0.45 9.96

The accuracy of the exponential smoothing method will be evaluated for different
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