The time series decomposition method is based on the assumption that the demand pattern of a product (or a service) can be decomposed into the following four effects:
trend,cyclical variation,seasonal variationandresidual variation.
Trend. The trend is the long-term modification of demand over time; it may depend on changes in population and on the product (or service) life cycle (see Figure 2.3).
Cyclical variation. Cyclical variation is caused by the so-called business cycle, which depends on macro-economic issues. It is quite irregular, but its pattern is roughly periodic.
Seasonal variation. Seasonal variation is caused by the periodicity of several hu- man activities. Typical examples are the ups and downs in the demand of some items over the year. This type of effect can also be observed on a weekly basis (e.g. some product sales are higher on weekends than on working days).
Residual variation. Residual variation is the portion of demand that cannot be interpreted as trend, cyclical or seasonal variation. It is often the result of numerous causes, each of which has a small impact. If there are no other predictable variations in the demand, the residual effect is a random variable with unit expected value (assuming that demand is modelled as the product of the four effects).
In the sequel we assume that the way the four components are combined together is multiplicative,
dt =qtvtstrt, t =1, . . . , T ,
FORECASTING LOGISTICS REQUIREMENTS 35 whereqt represents the trend at time period t (expressed in the same units as the demand),vt is the cyclical effect at time periodt,st is the seasonal variation at time periodt, andrt is the residual variation at time periodt. It is worth noting that all factors are greater than or equal to 0. Also note that ifMis the periodicity of the seasonal variation, then the average of the seasonal effects overMconsecutive time periods is equal to 1:
j+M
t=j+1st
M =1, j =0,1, . . . , T −M. (2.2)
In Figure 2.4 a typical demand pattern is reported. The decomposition method is made up of three steps: in the first phase, the demand time seriesdt, t =1, . . . , T, is decomposed into the four componentsqt,vt,st,rt, t =1, . . . , T; in the second phase, the time series ofq,vandsare projected into one or more future time periods (it is worth noting that the residual variation cannot be predicted); finally, in the third phase the projected values are combined,
pT(τ )=qT(τ )vT(τ )sT(τ ), τ =1,2, . . . , (2.3) to obtain the required demand forecasts. The decomposition phase is carried out as follows.
Evaluation of the product(qv)t
The product(qv)t is obtained by removing from the time seriesdt, t =1, . . . , T, the seasonal effect and the random fluctuation. This can be done by observing that the average value of the demand overMconsecutive time periods is not affected by the seasonal fluctuations. Furthermore, by so doing we also remarkably reduce the influence of the random fluctuations, especially ifMis relatively high (see also the next section). Therefore, the computation of the following quantities,
d1+ ã ã ã +dM
M ,
d2+ ã ã ã +dM+1
M ,
...
dT−M+1+ ã ã ã +dT
M ,
allows us to determine a series of demand entries without the seasonal and residual effects.
36 FORECASTING LOGISTICS REQUIREMENTS IfM is odd, each average can be associated with the central period of the corre- sponding time interval. Thus,
(qv)M/2=d1+ ã ã ã +dM
M ,
(qv)M/2+1=d2+ ã ã ã +dM+1
M ,
(qv)T−M/2+1=dT−M+1+ ã ã ã +dT
M .
Hence the required time series is(qv)t, t= 12M, . . . , T − 12M +1.
IfMis even, one can use a weighted average ofM+1 demand entries in which the first and the last ones have a weight of 12and all other values have a unit weight.
Then, the time series(qv)t, t = 12M+1, . . . , T −12Mis given by (qv)t =
1
2dt−M/2+dt−M/2+1+ ã ã ã +12dt+M/2
M , t = 12M+1, . . . , T−12M.
Evaluation ofqtandvt
In most cases, it can be assumed that the trend is described by a simple functional relation, such as a linear or quadratic function. Then the trend is obtained by applying a simple regression to the time series(qv)t, t=1, . . . , T (for the sake of simplicity, we assume that(qv)t spanst =1, . . . , T, although, as we have seen previously, it is defined over a shorter time interval). Onceqt, t = 1, . . . , T, is determined, the cyclical effectvt, t=1, . . . , T can be computed for eacht =1, . . . , T as follows:
vt = (qv)t qt . Evaluation ofstandrt
The time series(sr)t, t =1, . . . , T, which includes both the seasonal variation and the random fluctuation, can be computed for eacht =1, . . . , T as follows:
(sr)t = dt (qv)t.
The seasonal effect can then be expressed by means ofM indices s¯1, . . . ,s¯M, defined as
skM+t = ¯st, t=1, . . . , M, k =0,1, . . . .
Each index s¯t, t = 1, . . . , M, represents the average of the values (sr)t, t = 1, . . . , T, associated with homologous time periods (i.e.s¯t, t=1, . . . , M, is the aver- age of(sr)t, (sr)M+t, (sr)2M+t, . . .). This procedure is correct because, as explained previously, the average calculation reduces greatly the random fluctuation. Finally, we observe that, on the basis of the definition of seasonal index, we have
M
t=1s¯t
M =1. (2.4)
FORECASTING LOGISTICS REQUIREMENTS 37
0 200 400 600 800 1000 1200 dt
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 t
Figure 2.4 Demand pattern of electrosurgical equipment in France.
If Equation (2.4) is not satisfied, the following normalized indicess˜t, t=1, . . . , M, are used:
˜
st = Ms¯t M
t=1s¯t, t=1, . . . , M.
It is easy to show that the indicess˜t, t =1, . . . , M, verify the relation, M
t=1s˜t
M =1.
The random valuesrt, t =1, . . . , T, can be obtained by dividing each term of the time series(sr)t, t = 1, . . . , T, by the correspondent seasonal indexst, t = 1, . . . , T, that is
rt =(sr)t st .
If the decomposition has been executed correctly, the time seriesrt, t =1, . . . , T, has an expected value close to 1.
The second phase of the decomposition method amounts to projecting the effects q,vandspreviously determined over one or more future time periods; this is very easy to accomplish for the trend and the seasonal effect. However, the cyclical trend is much harder to extrapolate in a quantitative fashion. As a result, it is often estimated qualitatively, on the basis of the macro-economic forecasts. If no such information is available, it can be assumed that
vT(τ )=vT, τ =1,2, . . . .
Finally, in the third phase, a forecast is generated by combining the projections obtained in step two, according to Equation (2.3).
38 FORECASTING LOGISTICS REQUIREMENTS Table 2.5 Demand history (in thousands of euros) of electrosurgical
equipment in France (Part I).
Year Month Period Demand Year Month Period Demand
1986 Jan 1 511.70 1989 May 41 848.40
1986 Feb 2 468.30 1989 Jun 42 820.40
1986 Mar 3 571.90 1989 Jul 43 795.90
1986 Apr 4 648.20 1989 Aug 44 774.90
1986 May 5 705.60 1989 Sep 45 750.40
1986 Jun 6 709.10 1989 Oct 46 759.50
1986 Jul 7 676.90 1989 Nov 47 740.60
1986 Aug 8 661.50 1989 Dec 48 809.90
1986 Sep 9 611.80 1990 Jan 49 603.40
1986 Oct 10 640.50 1990 Feb 50 558.60
1986 Nov 11 611.10 1990 Mar 51 711.20
1986 Dec 12 697.20 1990 Apr 52 760.90
1987 Jan 13 548.80 1990 May 53 840.00
1987 Feb 14 492.10 1990 Jun 54 835.80
1987 Mar 15 613.20 1990 Jul 55 777.00
1987 Apr 16 692.30 1990 Aug 56 727.30
1987 May 17 721.70 1990 Sep 57 714.00
1987 Jun 18 672.00 1990 Oct 58 744.80
1987 Jul 19 670.60 1990 Nov 59 723.10
1987 Aug 20 635.60 1990 Dec 60 770.70
1987 Sep 21 611.80 1991 Jan 61 581.00
1987 Oct 22 686.00 1991 Feb 62 555.80
1987 Nov 23 630.70 1991 Mar 63 665.70
1987 Dec 24 750.40 1991 Apr 64 770.70
1988 Jan 25 515.20 1991 May 65 836.50
1988 Feb 26 498.40 1991 Jun 66 779.10
1988 Mar 27 627.20 1991 Jul 67 745.50
1988 Apr 28 741.30 1991 Aug 68 739.20
1988 May 29 760.90 1991 Sep 69 676.20
1988 Jun 30 754.60 1991 Oct 70 710.50
1988 Jul 31 733.60 1991 Nov 71 711.90
1988 Aug 32 704.90 1991 Dec 72 731.50
1988 Sep 33 709.80 1992 Jan 73 598.50
1988 Oct 34 733.60 1992 Feb 74 578.90
1988 Nov 35 714.70 1992 Mar 75 675.50
1988 Dec 36 831.60 1992 Apr 76 756.00
1989 Jan 37 586.60 1992 May 77 865.20
1989 Feb 38 536.90 1992 Jun 78 819.00
1989 Mar 39 654.50 1992 Jul 79 800.80
1989 Apr 40 767.90 1992 Aug 80 758.10
FORECASTING LOGISTICS REQUIREMENTS 39 Table 2.6 Demand history (in thousands of euros) of electrosurgical
equipment in France (Part II).
Year Month Period Demand Year Month Period Demand
1992 Sep 81 737.80 1996 Mar 123 721.00
1992 Oct 82 774.90 1996 Apr 124 877.10
1992 Nov 83 728.00 1996 May 125 959.70
1992 Dec 84 817.60 1996 Jun 126 916.30
1993 Jan 85 618.10 1996 Jul 127 870.80
1993 Feb 86 565.60 1996 Aug 128 832.30
1993 Mar 87 691.60 1996 Sep 129 760.20
1993 Apr 88 768.60 1996 Oct 130 833.70
1993 May 89 903.00 1996 Nov 131 827.40
1993 Jun 90 847.70 1996 Dec 132 864.50
1993 Jul 91 830.90 1997 Jan 133 705.60
1993 Aug 92 772.10 1997 Feb 134 619.50
1993 Sep 93 755.30 1997 Mar 135 723.10
1993 Oct 94 779.10 1997 Apr 136 847.70
1993 Nov 95 770.00 1997 May 137 942.90
1993 Dec 96 844.20 1997 Jun 138 917.00
1994 Jan 97 671.30 1997 Jul 139 897.40
1994 Feb 98 607.60 1997 Aug 140 859.60
1994 Mar 99 737.80 1997 Sep 141 821.80
1994 Apr 100 863.10 1997 Oct 142 872.20
1994 May 101 908.60 1997 Nov 143 795.90
1994 Jun 102 891.10 1997 Dec 144 824.60
1994 Jul 103 853.30 1998 Jan 145 669.90
1994 Aug 104 836.50 1998 Feb 146 618.10
1994 Sep 105 797.30 1998 Mar 147 756.00
1994 Oct 106 840.70 1998 Apr 148 901.60
1994 Nov 107 816.90 1998 May 149 968.80
1994 Dec 108 872.20 1998 Jun 150 968.80
1995 Jan 109 613.90 1998 Jul 151 921.20
1995 Feb 110 595.00 1998 Aug 152 891.10
1995 Mar 111 744.10 1998 Sep 153 882.00
1995 Apr 112 812.00 1998 Oct 154 887.60
1995 May 113 941.50 1998 Nov 155 840.00
1995 Jun 114 940.10 1998 Dec 156 935.90
1995 Jul 115 863.10 1999 Jan 157 763.70
1995 Aug 116 829.50 1999 Feb 158 700.00
1995 Sep 117 808.50 1999 Mar 159 844.20
1995 Oct 118 800.10 1999 Apr 160 989.10
1995 Nov 119 836.50 1999 May 161 1045.80
1995 Dec 120 870.80 1999 Jun 162 1012.90
1996 Jan 121 684.60 1999 Jul 163 970.90
1996 Feb 122 644.70
40 FORECASTING LOGISTICS REQUIREMENTS Table 2.7 Computation of combined trend and cyclical effects(qv)t,
t=1, . . . ,163, in the P&D problem.
t (qv)t t (qv)t
1 152 864.65
2 153 871.73
3 154 879.05
4 155 885.91
5 156 890.95
6 157 894.86
7 627.70 158
8 630.23 159
9 632.95 160
10 636.50 161
11 639.01 162
12 638.14 163
. . . . . .
P&D is a French consulting firm which was entrusted in July 1999 to estimate the future demand of electrosurgical equipment in France for the subsequent six months.
The sales over the past 13 years and seven months are available (Tables 2.5 and 2.6, Figure 2.4).
The durationM of the seasonal cycle was assumed to be equal to 12 and the trend was assumed to be linear. Then the decomposition method was applied. The intermediate and final results are summarized in Tables 2.7–2.12 and in Figures 2.5–
2.12. The trend equation isqt = 638.51+1.43t. The seasonal indicess¯1, . . . ,s¯12
(see Table 2.10 and Figure 2.9) satisfy Equation (2.4). As the expected value of the random variation is approximately 1, the demand decomposition can be deemed to be satisfactory. The demand forecasts from August 1999 to January 2000 (the first six months ahead) were obtained by combining the projections of the trend with the seasonal and cyclical effects. The latter was estimated (see Figure 2.11) by using a quadratic regression curve defined on the basis ofvt, t =150, . . . ,157. In particular, it was assumed that
vt =f (t−149)2+g(t−149)+h, t =150,151, . . . .
The values of the coefficientsf,g andh that best fit the cyclical effect for t = 150, . . . ,157 are
f = −0.0004, g=0.0094, h=0.9856.
FORECASTING LOGISTICS REQUIREMENTS 41 Table 2.8 Trendqt,t=7, . . . ,157, and cyclical effectvt,t=7, . . . ,157,
in the P&D problem.
t (qv)t qt vt
7 627.70 648.52 0.97 8 630.23 649.95 0.97 9 632.95 651.39 0.97 10 636.50 652.82 0.98 11 639.01 654.25 0.98 12 638.14 655.68 0.97
ã ã ã ã ã ã ã ã ã ã ã ã 152 864.65 856.02 1.01 153 871.73 857.45 1.02 154 879.05 858.88 1.02 155 885.91 860.31 1.03 156 890.95 861.74 1.03 157 894.86 863.17 1.04
Table 2.9 Evaluation of combined seasonal and residual effects(sr)t, t=1, . . . ,157, in the P&D problem.
t dt (qv)t (sr)t
7 676.90 627.70 1.08 8 661.50 630.23 1.05 9 611.80 632.95 0.97 10 640.50 636.50 1.01 11 611.10 639.01 0.96 12 697.20 638.14 1.09 . . . . . . . . . . . . 152 891.10 864.65 1.03 153 882.00 871.73 1.01 154 887.60 879.05 1.01 155 840.00 885.91 0.95 156 935.90 890.95 1.05 157 763.70 894.86 0.85