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

··· ··· ··· ···

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 („ ), T = 163, „ = 1, . . . , 6, in the P&D problem.

T +„ qT („ ) vT („ ) sT („ ) pT („ )

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 ¬rst 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 ¬‚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 ¬rst period ahead (r 1):

r’1

dT ’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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FORECASTING LOGISTICS REQUIREMENTS 45

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.

¯

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_ („)

1.20

1.10

1.00

0.90

0.80

0.70

0.60

„

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

¯

Figure 2.11 Extrapolation of the cyclical effect (in grey) vt¯(„ ), t = 157,

„ = 1, . . . , 12, in the P&D problem.

dt („)

1200

1000

800

600

400

200

0

„

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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FORECASTING LOGISTICS REQUIREMENTS 47

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 ¬rst 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

¬rst T periods, i.e.

T ’1

dT ’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 ¬‚uctuations but, at

the same time, increases the in¬‚uence of random perturbations. In contrast, a high

value of r effectively ¬lters 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 µ and standard deviation σ . The

random variable pT +1 will have an expected value µpT +1 de¬ned as

r’1 r’1

µdT ’k µ

µpT +1 = = =µ

r r

k=0 k=0

(i.e. the same expected value as d1 , . . . , dT ), and a standard deviation

1

σpT +1 = √ σ.

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 ¬rst period ahead is given by

pT +1 = ±dT + (1 ’ ±)pT , (2.5)

where ± ∈ (0, 1) is a smoothing constant. Here, pT represents the demand forecast

for period T made at time period T ’ 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 (± = 0.2)

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

Rewriting Equation (2.5) as

pT +1 = pT + ±(dT ’ pT ) = pT + ±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 ’ 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 = ±dT ’1 + (1 ’ ±)pT ’1 .

From Equation (2.5), we obtain

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

Iterating these substitutions (and taking into account that p2 can be assumed equal

to d1 ), the following relation is obtained:

T ’2

(1 ’ ±)k dT ’k + (1 ’ ±)T ’1 d1 .

pT +1 = ±

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 ± T ’2 (1 ’ ±)k + (1 ’ ±)T ’1 is equal to 1.

k=0

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FORECASTING LOGISTICS REQUIREMENTS 49

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 ± = 0.3) for solving the fore-

casting problem of Sarath company, we get

p2 = d1 = 1180.

For t = 3,

p3 = ±d2 + (1 ’ ±)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 ± plays an important role in the exponential smoothing

method. High values of ± 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 ± yield a forecasting method less dependent on the random ¬‚uctuation but,

at the same time, cannot take quickly into account the most recent variations of the

demand.

In practice, the value of ± 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 ±, 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 ± (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 suf¬cient to recall that

the trend is assumed to be constant. Consequently,

pT („ ) = pT +1 , „ = 2, 3, . . . , „ ,

¯

where the forecasting value pT +1 is obtained by using any technique described above

¯

and „ 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 („ ) = aT + bT „, „ = 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 ’ dT ’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 „ th periods ahead, we get

p12 („ ) = a12 + b12 „ = d12 + (d12 ’ d11 )„ = 1230 + 100„, „ = 1, 2, . . . .

In particular, for the ¬rst time period ahead („ = 1):

p13 = 1330.

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FORECASTING LOGISTICS REQUIREMENTS 51

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 ’r+1 , . . . , dT ’1 , dT ):

r’1 r’1

’ 2 (r ’ 1) k=0 dT ’k +

1

k=0 kdT ’k

bT = ,

’ 1)2 ’ 1 r(r ’ 1)(2r ’ 1)

1

4 r(r 6

r’1

+ 2 bT r(r ’ 1)

1

k=0 dT ’k

aT = .

r

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52 FORECASTING LOGISTICS REQUIREMENTS

The satellite receiver demand of Sarath company can be predicted by means of the

linear regression method. If r = 4, then

r’1

dT ’k = 4540,

k=0

r’1

kdT ’k = 6520,

k=0

and

b12 = 58,

a12 = 1222.

Therefore,

p12 („ ) = a12 + b12 „ = 1222 + 58„, „ = 1, 2, . . . .

In particular, for the ¬rst time period ahead („ = 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 = 2γT ’ ·T ,

2

bT = (γT ’ ·T ),

r ’1

where γT is the average of the r most recent demand entries,

r’1

dT ’k

γT = ,

r

k=0

and ·T represents the average of the r most recent average demand entries, i.e.

r’1

γT ’k

·T = .

r

k=0

Whenever r past demand data are not available (T < r), the computation of γT and

·T can be executed along the guidelines illustrated for the moving average method.

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FORECASTING LOGISTICS REQUIREMENTS 53

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

γ12 = = 1170,

3

γ12 + γ11 + γ10

·12 = ,

3

where

d11 + d10 + d9

γ11 = = 1103.33,

3

d10 + d9 + d8

γ10 = = 1063.33.

3

Therefore,

·12 = 1112.22,

from which

a12 = 2γ12 ’ ·12 = 1227.78,

b12 = γ12 ’ ·12 = 57.78.

Therefore,

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

In particular, for the ¬rst period ahead („ = 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 modi¬cation of the exponential smoothing

method and is based on the following two relations:

aT = ±dT + (1 ’ ±)(aT ’1 + bT ’1 ), (2.6)

bT = β(aT ’ aT ’1 ) + (1 ’ β)bT ’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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54 FORECASTING LOGISTICS REQUIREMENTS

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 speci¬ed. 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 ± and β is conducted according to the same criteria

illustrated for the exponential smoothing method.

Applying the Holt method (with ± = β = 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 („ ) = a12 + b12 „ = 1203.09 + 54.23„, „ = 1, 2, . . . .

In particular, for the ¬rst period ahead („ = 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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FORECASTING LOGISTICS REQUIREMENTS 55

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 („ ) = dT +„ ’M , „ = 1, . . . , M. (2.8)

On the basis of Equation (2.8), the forecast related to the time period T + „ 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 + „ ) = dT +„ ’M , „ = 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 („ ) = d24+„ ’12 , „ = 1, . . . , 12.

In particular,

p25 = d25’12 = d13 = 815,

p24 (2) = d26’12 = d14 = 1015.

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56 FORECASTING LOGISTICS REQUIREMENTS

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 („ ) = aT sT +„ , „ = 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 +„ is the seasonal index (see

Section 2.4.1) for period T + „ . More generally, for a time horizon whose duration

is greater than one cycle time, we get

pT (kM + „ ) = aT sT +„ , „ = 1, . . . , M, k = 1, 2, . . . .

Assuming, without loss of generality, that the available historical data are suf¬cient

to cover an integer number K = T /M of cycles, the parameters aT and sT +„ , „ =

1, . . . , M, can be computed by the following relations,

dT

aT = ± + (1 ’ ±)aT ’1 , (2.10)

sT

d(K’1)M+„

sT +„ = skM+„ = β + (1 ’ β)s(K’1)M+„ , „ = 1, . . . , M, (2.11)

a(K’1)M+„

where ± and β are smoothing constants (0 ±, β 1). Equation (2.10) expresses

aT as the weighted sum of two components: the ¬rst, 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 ’ 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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FORECASTING LOGISTICS REQUIREMENTS 57

way that all demand entries d1 , . . . , dT appear explicitly. To start the procedure we

set a0 equal to the average demand in the ¬rst time cycle, i.e.

d1 + · · · + dM

¯

a0 = d(1) = ,

M

whereas we can select the following initial estimate of st , t = 1, . . . , M:

¯ ¯ ¯

dt /d(1) + dt+M /d(2) + · · · + dt+(K’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

¯ ¯

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 ¬rst cycle is equal to 1. However, this con-

dition cannot be satis¬ed 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 ’ 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 ± = β = 0.3. To this end, we compute

the mean value of the demand during the two time cycles,

d1 + · · · + d12

¯

d(1) = = 1031.67,

12

d13 + · · · + d24

¯

d(2) = = 1056.67,

12

from which we obtain

¯

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=

¯ 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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58 FORECASTING LOGISTICS REQUIREMENTS

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 ¬rst 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 + „ ) = [aT + bT (kM + „ )]sT +„ , „ = 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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FORECASTING LOGISTICS REQUIREMENTS 59

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 („)

1800

1600

1400

1200

1000

800

600

400

200

0

„

10 20 30

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

sT +„ , „ = 1, . . . , M:

dT

aT = ± + (1 ’ ±)(aT ’1 + bT ’1 ), (2.14)

sT

bT = ·(aT ’ aT ’1 ) + (1 ’ ·)bT ’1 , (2.15)

d(K’1)M+„

sT +„ = sKM+„ = β + (1 ’ β)s(K’1)M+„ , „ = 1, . . . , M,

a(K’1)M+„

(2.16)

where ±, · and β are smoothing constants chosen in the interval (0, 1). In order to

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60 FORECASTING LOGISTICS REQUIREMENTS

dt („)

16000

14000

12000

10000

8000

6000

4000

2000

0

„

10 20 30

Figure 2.17 Demand pattern and demand forecasting (in grey) of

microwave ovens in the Sarath problem.

use Equations (2.14)“(2.16) recursively, we need an estimate of the values a0 , b0 and

st , t = 1, . . . , M. We can use

¯ ¯

d(K) ’ d1

b0 = . (2.17)

T ’M

To explain this formula, we observe that the numerator represents the variation of the

mean value of the demand between the ¬rst and the last period. In addition, we note

that the mean value of the demand in the ¬rst cycle corresponds to time period

t = 2 (M + 1),

1

which is the ˜centre™ of the ¬rst 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 ’ 2 (M ’ 1).

1

Hence, there are T ’ M time periods between the centres of the ¬rst and the last

period. The parameter a0 can be determined as follows:

¯

a0 = d(1) ’ 2 (M + 1)b0 .

1

(2.18)

An estimate of the seasonal indices during the ¬rst period is given by

dt

st = t = 1, . . . , M,

, (2.19)

a0 + b0 t

subject to a normalization, if necessary.

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FORECASTING LOGISTICS REQUIREMENTS 61

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 + · · · + d12

¯

d(1) = = 2089.58,

12

d13 + · · · + d24

¯

d(2) = = 3674.50.

12

Then, we determine the value b0 through Equation (2.17),

¯ ¯

d(1) + d(2)

b0 = = 132.08,

12

and the value a0 through Equation (2.18),

¯

a0 = d(1) ’ 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)“(2.16) with ± = · = β = 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 ¬rst 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 ¬rst ¬ve approaches can be classi¬ed 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.

Input“output models. Input“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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62 FORECASTING LOGISTICS REQUIREMENTS

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 ˜S curve™ 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 clients™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 ¬nished 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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FORECASTING LOGISTICS REQUIREMENTS 63

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

Box“Jenkins method. The Box“Jenkins method is made up of three steps (iden-

ti¬cation, parameter evaluation and diagnostic check). In the ¬rst 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 coef¬cients 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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64 FORECASTING LOGISTICS REQUIREMENTS

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

„ p24 („ ) „ p24 („ )

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 de¬ned:

t

k=2 |ek |

MADt = , (2.20)

t ’1

t

k=2 |ek |/dk

MAPDt = 100 , (2.21)

t ’1

t 2

k=2 ek

MSEt = , (2.22)

t ’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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FORECASTING LOGISTICS REQUIREMENTS 65

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.

± MAD12

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