Preview Extract

Chapter 02 – Forecasting
Forecasting
Solutions To Problems From Chapter 2
2.1
Trend
Seasonality
Cycles
Randomness
2.2
Cycles have repeating patterns that vary in length and magnitude.
2.3
a)
b)
c)
2.4
Marketing:
New sales and existing sales forecasts. Causal models relating advertising
to sales
Accounting
Interest rate forecasts; cost components, bad debts.
Finance:
Changes in stock market, forecast return on investment return from
specific projects.
Production:
Forecast product demand (aggregate and individual), availability of
resources, labor.
2.5
Time Series
Regression or Causal Model
Delphi Method
a) Aggregate forecasts deal with item groups or families.
b) Short term forecasts are generally next day or month; Long term forecasts may be for
many months or years into the future.
c) Causal models are based on relationship between predictor variables and other
variables. Naive models are based on the past history of series only
2.6
The Delphi Method is a technique for achieving convergence of group opinion. The
method has several potential advantages over the Jury of Executive Opinion method
depending upon how that method is implemented. If the executives are allowed to reach a
consensus as a group, strong personalities may dominate. If the executives are
interviewed, the biases of the interviewer could affect the results.
2-1
Chapter 02 – Forecasting
2.7
Some of the issues that a graduating senior might want to consider when choosing a
college to attend include: a) how well have graduates fared on the job market, b) what are
the student attrition rates, c) what will the costs of the college education be and d) what
part-time job opportunities might be available in the region. Sources of data might be
college catalogues, surveys on salaries of graduating seniors, surveys on numbers of
graduating seniors going on to graduate or professional schools, etc.
2.8
The manager should have been prepared for the consequences of forecast error.
2.9
It is unlikely that such long term forecasts are accurate.
2.10
This type of criteria would be closest to MAPE, since the errors measured are relative not
absolute. It makes more sense to use a relative measure of error in golf. For example,
an error of 10 yards for a 200 yard shot would be fine for most golfers, but a similar
error for a 20 yard shot would not.
2.11
a) (26)(.1) + (21)(.1) + (38)(.2) + (32)(.2) + (41)(.4) = 35.1
b) (23)(.1) + (28)(.1) + (33)(.2) + (26)(.2) + (21)(.4) = 25.3
2.12
a) and b)
Forecast
Period
Actual
80.5
73.5
77.5
107.5
98.5
87.5
100.0
78.5
79.5
95.0
3
4
5
6
7
8
9
10
11
12
72
83
132
65
110
90
67
92
98
73
(86 + 75)/2
(75 + 72)/2
etc
=
=
c)
(216)/10
(7175)/10
MAD
MSE
=
=
MAPE
=
=
=
21.6
717.5
100 ๏ฆ๏ง 1 ๏ฅ ei ๏ถ๏ท
๏จn
= 25.61
Di ๏ธ
2-2
et
+8.5
-9.5
-54.5
42.5
-11.5
-2.5
+33.0
-13.5
-18.5
+22.0
Chapter 02 – Forecasting
2.13
Fcst 1 Fcst 2
Demand
Err 1
Err 2
223
289
430
134
190
550
256
340
375
110
225
525
33
51
-55
-24
35
-25
46
20
-15
-2
75
35
๏ฝe1/D๏ฝ*100
๏ฝe2/D๏ฝ๏ช100
46
20
15
2
75
35
12.89062
15.0000
14.66667
21.81818
15.55556
4.761905
17.96875
5.88253
4.00000
1.81818
33.33333
6.66667
14.11549
(MAPE1)
11.61155
(MAPE2)
It means that E(ei) ๏น 0. This will show up by considering
n
๏ฅ ei
i =1
A bias is indicated when this sum deviates too far from zero.
2.15
Using the MAD: 1.25 MAD = (1.25)(21.6) = 27.0
(Using s, the sample standard deviation, one obtains 28.23)
2.16
MA (3) forecast:
MA (6) forecast:
MA (12) forecast:
258.33
249.33
205.33
2-3
Er2^2
1089
2116
2601
400
3025
225
576
4
1225
5625
625
1225
1523.5 1599.166
(MSE1
(MSE2)
๏ฝErr2๏ฝ
32.16666
(MAD2)
2.14
210
320
390
112
150
490
Er1^2
|Err1|
33
51
55
24
35
25
37.16666
(MAD1)
Chapter 02 – Forecasting
2.17, 2.18, and 2.19.
One-step-ahead
Two-step-ahead
Month
Forecast
Forecast
Demand
July
August
September
October
November
December
205.50
225.25
241.50
250.25
249.00
240.25
149.75
205.50
225.25
241.50
250.25
249.00
223
286
212
275
188
312
e1
MAD =
e2
-17.50
-60.75
29.50
-24.75
61.00
-71.75
-73.25
-80.50
13.25
-33.50
62.25
-63.00
44.2
54.3
The one step ahead forecasts gave better results (and should have according to the
theory).
2.20
Month
Demand
MA(3)
MA(6)
July
August
September
October
November
December
223
286
212
275
188
312
226.00
226.67
263.00
240.33
257.67
225.00
161.33
183.67
221.83
233.17
242.17
244.00
MA (6) Forecasts exhibit less variation from period to period.
2.21
An MA(1) forecast means that the forecast for next period is simply the current period’s
demand.
Month
Demand
MA(4)
MA(1)
Error
Month
Demand
MA(4)
MA(1)
Error
July
August
September
October
November
December
223
286
212
275
188
312
205.50
225.25
241.50
250.25
249.00
240.25
280
223
286
212
275
188
57
-63
74
-63
87
-124
MAD
=
78.0
(Much worse than MA(4))
2-4
Chapter 02 – Forecasting
Ft = ๏กDt-1 + (1-๏ก)Ft-1
2.22
a)
FFeb
= (.15)(23.3) + (.85)(25) = 24.745
FMarch = (.15)(72.3) + (.85)(24.745) = 31.88
FApr = (.15)(30.3) + (.85)(31.88) = 31.64
FMay = (.15)(15.5) + (.85)(31.63) = 29.22
b)
FFeb = (.40)(23.3) + (.60)(25) = 24.32
FMarch = 43.47
FApr = 38.20
FMay = 29.12
c)
Compute MSE for February through April:
Month
Error (a)
(๏ก = .15)
Error (b)
(๏ก = .40)
Feb
Mar
Apr
47.45
1.56
16.13
47.88
13.17
22.70
838.04
993.74
MSE
๏ก
2.23
=
= .15 gave a better
forecast
Small ๏ก implies little weight is given to the current forecast and virtually all weight is
given to past history of demand. This means that the forecast will be stable but not
responsive.
Large ๏ก implies that a great deal of weight is applied to current observation of demand.
This means that the forecast will adjust quickly to changes in the demand pattern but will
vary considerably from period to period.
2-5
Chapter 02 – Forecasting
2.24
a)
Week
MA(3) Forecast
4
5
6
7
8
17.67
20.33
28.67
22.67
21.67
b) and c
Week
ES(.15)
Demand
MA(3)
|err|
|err|
4
5
6
7
8
17.67
18.32
20.67
19.37
19.32
22
34
12
19
23
17.67
20.33
28.67
22.67
21.67
4.33
15.68
8.67
0.37
3.68
4.33
13.67
16.67
3.67
1.33
6.547540
MAD-ES
7.934
MAD-MA
Based on these results, ES(.15) had a lower MAD over the five weeks
d) It is the same as the exponential smoothing forecast made in week 6 for the demand
in week 7, which is 19.37 from part c).
2.25
2
2
๏ต ๏ก = = .286
N+1
7
a)
๏ก =
b)
N=
c)
๏ณ2 2
From Appendix 2-A ๏ณ e =
=1.1๏ณ2
2 โ๏ก
2 โ๏ก
๏ก
2 โ.05
= 39
.05
2
Hence
2.26
๏ตN=
2
= 1.1 Solving gives
2 โ๏ก
๏ก = .1818
It is the same as the one step ahead forecast made at the end of March which is
31.64.
2-6
Chapter 02 – Forecasting
2.27
The average demand from Jan to June is 161.33. Assume this is the forecast for
July.
a)
b)
Month
Forecast
Aug
Sept
Oct
Nov
Dec
173.7
196.2
199.4
214.5
209.2
[.2(223) + (.8)(161.33)]
etc.
Month
Demand
ES(.2)
(Error)
Aug
Sept
Oct
Nov
Dec
286
212
275
188
312
173.7
196.2
199.4
214.5
209.2
112.3
15.8
75.6
26.5
102.8
MAD
66.6
MA(6)
183.7
221.8
233.2
242.2
244
(Error)
102.3
9.8
41.8
54.2
68.0
55.2
MA(6) gave more accurate forecasts.
c) For ๏ก = .2 the consistent value of N is (2-๏ก)/๏ก = 9. Hence MA(6) will be somewhat
more responsive. Also the ES method may suffer from not being able to flush out
“bad” data in the past.
3000
2000
1000
500
1
Jan
2
Feb
3
Mar
4
Apr
Month
2-7
5
May
6
Jun
Chapter 02 – Forecasting
a)
โEyeballโ estimates: slope = 2750/6 = 458.33, intercept = -500.
b) Regression solution obtained is
Sxy = (6)(28,594) – (21)(5667) = 52,557
Sxx = (6)(91) – (21)2 = 105
b =
Sxy
S xx
=
52, 577
= 500.54
105
a = D โ b (n + 1) / 2 = -.807.4
c)
Regression equation
๏ค = -807.4 + (500.54)t
๏ D
t
Month
Forecasted Usage
July (t = 7)
Aug (t = 8)
Sept (t = 9)
Oct (t = 10)
Nov (t = 11)
Dec (t = 12)
2696
3197
3698
4198
4699
5199
d) One would think that peak usage would be in the summer months and the increasing
trend would not continue indefinitely.
2.29
a)
Month
Forecast
Month
Forecast
Jan
Feb
Mar
Apr
May
June
5700
6200
6700
7201
7702
8202
July
Aug
Sept
Oct
Nov
Dec
8703
9203
9704
10,204
10,705
11,206
(note that these are obtained from the regression equation
๏ค t = 807.4 + 500.54 t with t = 13, 14,. . . .)
D
The total usage is obtained by summing forecasted monthly usage.
Total forecasted usage for 1994 = 101,431
2-8
Chapter 02 – Forecasting
b) Moving average forecast made in June = 944.5/mo.
Since this moving average is used for both one-step-ahead and multiple-step-ahead
forecasts, the total forecast for 1994 is (944.5)(12) = 11,334.)
c )
1200
Jan
Feb Mar Apr May Jun
Jul
Aug
Sep Oct Nov Dec
The monthly average is about 1200 based on a usage graph of this shape. This graph
assumes peak usage in summer months. The yearly usage is (1200)(12) = 14,400
which is much closer to (b), since the moving average method does not project trend
indefinitely.
2-9
Chapter 02 – Forecasting
2.30
From the solution of problem 24,
a)
slope = 500.54
value of regression in June = -807.4 + (500.54)(6) = 2196
S0 = 2196
G0 = 500.54
๏ก = .15
๏ข = .10
S1 = ๏กD1 + (1-๏ก)(S0 + G0)
= (.15)(2150) + (.85)(2196 + 500.54)
= 2615
G1 = (.1)[2615 – 2196] + (.9)(500.54) = 492.4
S2 = (.15)(2660) + (.85)(2615 + 492.4) = 3040
G2 = .1 [3040 – 2615] + (.9) (492.4) = 485.7
b) One-step-ahead forecast made in Aug. for Sept. is
S2 + G2 = 3525.7
Two-step-ahead forecast made in Aug for Oct is
S2 + G2 = 3040 + 2(485.7) = 4011.4
c) S1 + 5(G1) = 2615 + 5(492.4) = 5077.
2.31
This observation would lower future forecasts. Since it is probably an “outlier” (nonrepresentative observation) one should not include it in forecast calculations.
2.32
Both regression and Holt’s method are based on the assumption of constant linear trend.
It is likely in many cases that the trend will not continue indefinitely or that the observed
trend is just part of a cycle. If that were the case, significant forecast errors could result.
2.33
Month
Yr
1
2
3
4
5
6
7
8
9
10
12
18
36
53
79
134
112
90
66
45
1
Yr
16
14
46
48
88
160
130
83
52
49
2
Dem1/Mean
Dem2/Mean
0.20
0.31
0.61
0.90
1.34
2.27
1.90
1.53
1.12
0.76
0.27
0.24
0.78
0.81
1.49
2.71
2.20
1.41
0.88
0.83
2-10
Avg (factor)”
0.24
0.27
0.70
0.86
1.42
2.49
2.05
1.47
1.00
0.80
Chapter 02 – Forecasting
11
12
23
21
14
26
Totals
689
726
0.39
0.36
0.24
0.44
0.31
0.40
12
We used the Quick and Dirty Method here. The average demand for the two years
was (689 + 726)/2 = 707.5.
2.34
a)
(1)
MA
Centered MA
(2)
Centered MA
on periods
Ratio
(1)/(2)
42.440
42.440
0.2828
0.5891
41.750
1.8204
43.125
1.2058
43.375
0.3689
44.000
0.7272
45.000
1.5778
46.375
1.3369
49.625
0.2821
49.375
0.9114
Quarter
Demand
1
2
12
25
3
76
4
52
41.25
5
16
42.25
6
32
44.00
7
71
42.75
8
62
45.25
9
14
44.75
10
45
48.00
11
84
51.25
49.500
1.6970
12
47
47.50
49.500
0.9494
41.25
42.25
44.00
42.75
45.25
44.75
48.00
51.25
47.50
The four seasonal factors are obtained by averaging the appropriate quarters (1, 5, 9 for
first; 2, 6, 10 for the second, etc.)
One obtains the following seasonal factors
0.3112
0.7458
1.6984
1.1641
The sum is 3.9163. Norming the factors by multiplying each by
4
= 1.0214
3, 9163
2-11
Chapter 02 – Forecasting
we finally obtain the factors:
0.318
0.758
1.735
1.189
b)
2.35
Quarter
Demand
Factor
Deseasonalized
Series
1
2
3
4
5
6
7
8
9
10
11
12
12
25
76
52
16
32
71
62
14
45
84
47
0.318
0.758
1.735
1.189
0.318
0.758
1.735
1.189
0.318
0.758
1.735
1.189
37.74
32.98
43.80
43.73
50.31
42.22
40.92
52.14
44.03
59.37
48.41
39.53
c)
47.40
d)
Must “re-seasonalize” the forecast from part (c)
(47.40)(.318) = 15.07
a)
V1 = (16 + 32 + 71 + 62)/4 = 45.25
V2 = (14 + 45 + 84 + 47)/4 = 47.5
1. G0 = (V2 – V1)/N = 0.5625
2. S0 = V2 + G0 (N-1/2) = 47.5 + (0.5625)(3/2) = 48.34
3. ct =
Dt
Vi ๏N + 1/ 2 โ j ๏G0
-2N+1 = ๏ฃ t ๏ฃ 0
c-7 =
16
= 0.36
45.25 โ (5/ 2 โ 1)(..56)
c-6 =
32
= 0.71
45.25 โ (5/ 2 โ 2)(.56)
2-12
Chapter 02 – Forecasting
c-5 =
71
= 1.56
43.25 โ (5/ 2 โ 3)(.56)
c-4 =
62
= 1.35
45.25 โ (5/ 2 โ 4)(.56)
c-3 =
14
= 0.30
47.5 โ (5/ 2 โ 1)(.56)
c-2 =
45
= 0.95
47.5 โ (5/ 2 โ 2)(.56)
c-1 =
84
= 1.76
47.5 โ (5/ 2 โ 3)(.56)
c0 =
47
= 0.97
47.5 โ (5/ 2 โ 4)(.56)
(c7 + c3)/2 = .33
(c6 + c2)/2 = .83
(c5 + c1)/2 = 1.66
(c4 + c0)/2 = 1.16
Sum =
3.98
Norming factor = 4/3.9 = 1.01
Hence the initial seasonal factors are:
b)
c-3 = .33
c-1 = 1.67
c-2 = .83
c-0 = 1.17
๏ก = 0.2, ๏ข = 0.15, ๏ง = 0.1, D1 = 18
S1 = ๏ก(D1/c-3) + (1-๏ก)(S0 + G0) = 0.2(18/0.33)
+ 0.8(48.34 + 0.56) = 50.03
G1 = ๏ง(S1 – S0) + (1 – ๏ง) = G0 = 0.1(50.03 – 48.34)
+ 0.9(0.56) = 0.70
c1 = ๏ข(D1/S1) + (1-๏ข)c3 = 0.15(18/50.03) + 0.85(0.33)
2-13
Chapter 02 – Forecasting
= .3345
c)
Forecasts for 2nd, 3rd and 4th quarters of 1993
F1,2 = [S1 + G1]c2 = (50 + .70)0.83 = 42.08
F1,3 = [S1 + 2G1]c3 = (50 + 2(.70))1.67 = 85.84
F1,4 = [S1 + 3G1]c4 = (50 + 3(.70))1.17 = 60.96
2.36
Period
Dt
Forecast
Forecast
from
from
30(d) ๏ฝet ๏ฝ 31(c) ๏ฝ et ๏ฝ
1
2
3
4
51
86
66
35.8
82.4
56.5
15.2
3.6
9.5
42.08
85.84
60.96
8.92
0.16
5.04
MAD = 9.43 MAD = 4.71
MSE = 111.42 MSE = 35.00
Hence we conclude that Winter’s method is more accurate.
2.37
S1 = 50.03
G1 = 0.67
๏ก = 0.2
๏ข = 0.15
๏ง = 0.1
D1 = 18
D2 = 51
D3 = 85
D4 = 66
S2 = 0.2(51/0.83) + 0.8(50.03 + 0.70) = 52.87
G2 = 0.1(52.87 – 50.03) + 0.9(0.70) = 0.914
S3 = 0.2(86/1.67) + 0.8(52.87 + 0.914) = 53.33
G3 = 0.1(53.33 – 52.85) + 0.9(0.885) = 0.8445
S4 = 0.2(66/1.17) + 0.8(53.33 + 0.8445) = 54.62
G4 = 0.1(54.62 – 53.33) + 0.9(0.8445) = 0.8891
c1 = (.15)[18/50] + (0.85)(.33) = .3345 ๏ป .34
c2 = (.15)[51/52.85] + 0.85(0.83) = .8502 ๏ป .85
c3 = (.15)(86/53.29) + 0.85(1.67) = 1.6616 ๏ป 1.66
c4 = (.15)(66/54.59) + 0.85(1.17) = 1.1758 ๏ป 1.18
The sum of the factors is 4.02. Norming each of the factors by multiplying by
4/4.02 = .995 gives the final factors as:
c1 = .34
2-14
Chapter 02 – Forecasting
c2 = .84
c3 = 1.65
c4 = 1.17
The forecasts for all of 1995 made at the end of 1993 are:
F4,9 = [S4 + 5G4]c1 = [54.62 + 5(0.89)]0.34 = 20.08
F4,10 = [S4 + 6G4]c2 = [54.62 + 6(0.89)]0.84 = 50.37
F4,11 = [S4 + 7G4]c3 = [54.62 + 7(0.89)]1.65 = 100.40
F4,12 = [S4 + 8G4]c4 = [54.62 + 8(0.89)]1.17 = 72.24
2.42. ARIMA(2,1,1) means 2 autoregressive terms, one level of differencing, and 1 moving
average term. The model may be written ut = a0 + a1ut โ1 + a2ut โ2 + ๏ฅ t โ b1๏ฅ t โ1
where ut = Dt โ Dt โ1 . Since ut = (1 โ B) Dt , we have
a) (1 โ B) Dt = a0 + (a1 B + a2 B 2 )(1 โ B) Dt + (1 โ b1B)๏ฅ t
b) ๏Dt = a0 + (a1B + a2 B 2 )๏Dt + (1 โ b1B)๏ฅ t
c) Dt โ Dt โ1 = a0 + a1 ( Dt โ1 โ Dt โ2 ) + a2 ( Dt โ2 โ Dt โ3 ) + ๏ฅ t โ b1๏ฅ t โ1 or
Dt = a0 + (1 + a1 ) Dt โ1 โ a1Dt โ2 + a2 ( Dt โ2 โ Dt โ3 ) + ๏ฅ t โ b1๏ฅ t โ1
2.43. ARIMA(0,2,2) means no autoregressive terms, 2 levels of differencing, and 2 moving
average terms. The model may be written as
wt = b0 + ๏ฅ t โ b1๏ฅ t โ1 โ b2๏ฅ t โ2
Where wt = ut โ ut โ1 and ut = Dt โ Dt โ1 . Using backshift notation, we may also write
wt = (1 โ B)2 Dt , so that we have for part a)
a) (1 โ B) 2 Dt = b0 + (1 โ b1B โ b2 B 2 )๏ฅ t
b) ๏ 2 Dt = b0 + (1 โ b1 B โ b2 B 2 )๏ฅ t
c) Dt โ 2Dt โ1 + Dt โ2 = b0 + ๏ฅ t โ b1๏ฅ t โ1 โ b2๏ฅ t โ2 or
Dt = 2Dt โ1 โ Dt โ2 + b0 + ๏ฅ t โ b1๏ฅ t โ1 โ b2๏ฅ t โ2
2.44. The ARMA(1,1) model may be written Dt = a0 + a1Dt โ1 โ b1๏ฅ t โ1 + ๏ฅ t . If we substitute for
Dt โ1 , Dt โ2 ,… one can easily see this reduces to a polynomial in (๏ฅ t , ๏ฅ t โ1 ,…) and if we substitute for
๏ฅ t , ๏ฅ t โ1 ,… we see that this reduces to a polynomial in Dt โ1 , Dt โ2 ,… . .
2-15
Chapter 02 – Forecasting
2.45
a) 1400 – 1200 = 200
200/5 = 40 Change = -40 (He should decrease the forecast by 40.)
b) (0.2)(0.8)4 = 0.08192
200(0.08192) = 16.384
16.384)
2.46
Change = -16.384 (He should decrease the forecast by
From Example 2.2 we have the following:
Forecast
(ES(.1))
Observed
Error (et)
Quarter
Failures
2
3
4
5
6
7
250
175
186
225
285
305
200
205
202
201
203
211
-50
+30
+16
-24
-82
-94
8
190
220
+30
Using MADt = ๏ก |et| + (1 -๏ก)MADt-1, we would obtain the following values:
MAD1 = 50 (given)
MAD2 = (.1)(50) + (.9)(50) = 50.0
MAD3 = (.1)(30) + (.9)(50) = 48.0
MAD4 = (.1)(16) + (.9)(48) = 44.8
MAD5 = (.1)(24) + (.9)(44.8) = 42.7
MAD6 = (.1)(82) + (.9)(42.7) = 46.6
MAD7 = (.1)(94) + (.9)(46.6) = 51.3
MAD8 = (.1)(30) + (.9)(51.3) = 49.2
The MAD obtained from direct computation is 46.6, so this method gives a pretty
good approximation after eight periods. It has the important advantage of not
requiring the user to save past error values in computing the MAD.
2.47
c1 = 0.7
c2 = 0.8
c3 = 1.0
c4 = 1.5
2-16
Chapter 02 – Forecasting
2.48
Dept
yr 1
yr 2
yr 3
ratio 1
ratio 2
ratio 3
Management
Marketing
Accounting
Production
Finance
Economics
835
620
440
695
380
1220
956
540
490
680
425
1040
774
575
525
624
410
1312
1.20
0.89
0.63
1.00
0.55
1.75
1.37
0.78
0.70
0.98
0.61
1.49
1.11
0.83
0.75
0.90
0.59
1.88
average
1.23
0.83
0.70
0.96
0.58
1.71
6
Mean pages over all fields and years = 696.72.
The multiplicative factors in the final column give the percentages above or below the
grand mean when multiplied by 100.
2.49 a) and b)
Month
Sales
MA(3
Error
Abs Err
Sq Err
Per Err
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
238
220
195
245
345
380
270
220
280
120
110
85
135
145
185
219
240
420
520
410
380
320
290
240
217.67
220.00
261.67
323.33
331.67
290.00
256.67
206.67
170.00
105.00
110.00
121.67
155.00
183.00
214.67
293.00
393.33
450.00
436.67
370.00
330.00
-27.33
-125.00
-118.33
53.33
111.67
10.00
136.67
96.67
85.00
-30.00
-35.00
-63.33
-64.00
-57.00
-205.33
-227.00
-16.67
70.00
116.67
80.00
90.00
27.33
125.00
118.33
53.33
111.67
10.00
136.67
96.67
85.00
30.00
35.00
63.33
64.00
57.00
205.33
227.00
16.67
70.00
116.67
80.00
90.00
747.11
15625.00
14002.78
2844.44
12469.44
100.00
18677.78
9344.44
7225.00
900.00
1225.00
4011.11
4096.00
3249.00
42161.78
51529.00
277.78
4900.00
13611.11
6400.00
8100.00
11.16
36.23
31.14
19.75
50.76
3.57
113.89
87.88
100.00
22.22
24.14
34.23
29.22
23.75
48.89
43.65
4.07
18.42
36.46
27.59
37.50
86.62
MAD
10547.47
MSE
38.31
MAPE
2-17
Chapter 02 – Forecasting
2.49
c)
Month
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Sales
238
220
195
245
345
380
270
220
280
120
110
85
135
145
185
219
240
420
520
410
380
320
290
240
MA(6
Error
270.50
0.50
275.83
55.83
275.83
-4.17
290.00
170.00
269.17
159.17
230.00
145.00
180.83
45.83
158.33
13.33
145.83
-39.17
130.00
-89.00
146.50
-93.50
168.17 -251.83
224.00 -296.00
288.17 -121.83
332.33
-47.67
364.83
44.83
381.67
91.67
390.00
150.00
Abs Err
Sq Err
Per Err
0.50
55.83
4.17
170.00
159.17
145.00
45.83
13.33
39.17
89.00
93.50
251.83
296.00
121.83
47.67
44.83
91.67
150.00
0.25
3117.36
17.36
28900.00
25334.03
21025.00
2100.69
177.78
1534.03
7921.00
8742.25
63420.03
87616.00
14843.36
2272.11
2010.03
8402.78
22500.00
0.19
25.38
1.49
141.67
144.70
170.59
33.95
9.20
21.17
40.64
38.96
59.96
56.92
29.72
12.54
14.01
31.61
62.50
86.63
MAD
14282.57
MSE
42.63
MAPE
MA(6) has about the same MAD and higher MSE and MAPE.
2.50
Month
1
2
3
4
5
6
7
8
9
10
11
12
13
Sales
238
220
195
245
345
380
270
220
280
120
110
85
135
ES(.1)
225
226.30
225.67
222.60
224.84
236.86
251.17
253.06
249.75
252.77
239.50
226.55
212.39
Error
-13.00
6.30
30.67
-22.40
-120.16
-143.14
-18.83
33.06
-30.25
132.77
129.50
141.55
77.39
Abs Err
13.00
6.30
30.67
22.40
120.16
143.14
18.83
33.06
30.25
132.77
129.50
141.55
77.39
2-18
Sq Err
169.00
39.69
940.65
501.63
14437.78
20489.51
354.47
1092.65
915.07
17629.15
16769.56
20035.72
5989.65
Per Err
5.46
2.86
15.73
9.14
34.83
37.67
6.97
15.03
10.80
110.65
117.72
166.53
57.33
Alpha
0.1
Chapter 02 – Forecasting
14
15
16
17
18
19
20
21
22
23
24
145
185
219
240
420
520
410
380
320
290
240
204.65
198.69
197.32
199.49
203.54
225.18
254.67
270.20
281.18
285.06
285.56
59.65
13.69
-21.68
-40.51
-216.46
-294.82
-155.33
-109.80
-38.82
-4.94
45.56
59.65
13.69
21.68
40.51
216.46
294.82
155.33
109.80
38.82
4.94
45.56
3558.55
187.37
470.05
1641.27
46855.50
86915.99
24128.54
12056.10
1507.01
24.39
2075.31
41.14
7.40
9.90
16.88
51.54
56.70
37.89
28.89
12.13
1.70
18.98
79.18
MAD
11616.03
MSE
36.41
MAPE
The error turns out to be a declining function of ๏ก for this data. Hence, ๏ก = 1 gives the
lowest error.
2.51
a)
Year
1
2
3
4
5
6
7
8
(Yi)
Sales
($100,000)
(Xi)
Births
Preceding Year
6.4
8.3
8.8
5.1
9.2
7.3
12.5
2.9
3.4
3.5
3.1
3.8
2.8
4.2
Obtain ๏ฅ Xi – 23.7, ๏ฅ Yi = 57.6, ๏ฅ XiYi = 201.29
๏ฅ Xi = 81.75, ๏ฅ Yi = 507.48
2
2
Sxx = 10.56
Sxy = 43.91
b = SXY = 4.158
SXX
a = ๏ y – b๏ x = -5.8
Hence Yt = – 5.8 + 4.158Xt-1 is the resulting regression equation.
b)
Y10 = -5.8 + (4.158)(3.3) = 7.9214 (that is, $792,140)
2-19
Chapter 02 – Forecasting
c)
Year
1
2
3
4
5
6
7
8
9
10
US Births
(in 1,000,000)
(Xi)
Forecasted
Births
Using ES(.15)
2.9
3.4
3.5
3.1
3.8
2.8
4.2
3.7
3.2
3.3
3.2
3.4
3.4
3.4
Hence, forecasted births for years 9 and 10 is 3.4 million.
d)
Yt = -5.8 + 4.158 Xt-1
Xt-1 = 3.4 million in years 8 and 9.
Substituting gives Yt = -5.8 + (4.158)(3.4) = 8.3372 for sales in each of years 9 and
10. Hence the forecast of total aggregate sales in these years is (8.3372)(2) = 16.6744 or
$1,667,440.
2.52
Month
1
2
3
4
5
6
Ice cream
Sales
325
335
172
645
770
950
Xi
Yi
Month
Ice Cream Sales
a)
Sum
Avg
=
Park
Attendees
880
976
440
1823
1885
2436
XiYi
1
2
3
4
5
6
325
335
172
645
770
950
325
670
516
2580
3850
5700
= 21
3.5
3197.
532.8
13641
Sxx = 105
Sxy = 14709
2-20
Chapter 02 – Forecasting
b = Sxy/Sxx = 140.1
a = ๏ Y – b๏ X = 42.5
Y30 = 42.5 + (30)(140) = $4245.1
We would not be very confident about this answer since it assumes the trend
observed over the first six months continues into month 30 which is very unlikely.
Xi
Park
attendees
b)
Sum
Avg
=
=
Yi
Ice Cream
Sales
XiYi
880
976
440
1823
1885
2436
325
335
172
645
770
950
286000
326960
75680
1175835
1451450
2314200
8440
1406.666
3197
532.8333
5630125
Sxx = 17,153,756
Sxy = 6,798,070
b = Sxy/Sxx = 0.396302
a = ๏ Y -b๏ X = 24.6316
Hence the resulting regression equation is:
Yi = -24.63 + 0.4Xi
2-21
Chapter 02 – Forecasting
c)
6000
5000
Attendees
4000
3000
2000
1000
2
4
6
8
10
12
14
16
18
20
Months
Readng the values from the curve:
X12 ๏ป 5100
X13 ๏ป 5350
X14 ๏ป 5600
X15 ๏ป 5800
X16 ๏ป 5900
X17 ๏ป 5950
X18 ๏ป 5980
Using the regression equation Yi = -24.63 + 0.4Xi derived in part (b) we obtain
the ice cream sales predictions below.
Month
Attendees
Predicted
Ice Cream Sales
12
13
14
15
16
17
18
5100
5350
5600
5800
5900
5950
5980
2015.37
2115.37
2215.37
2295.37
2335.37
2355.37
2367.37
2-22
Chapter 02 – Forecasting
2.53
The method assumes that the “best” ๏ก based on a past sequence of specific demands will
be the “best” ๏ก for future demands, which may not be true. Furthermore, the best value
of the smoothing constant based on a retrospective fit of the data may be either larger or
smaller than is desirable on the basis of stability and responsiveness of forecasts.
2.54
Year Demand S sub t
0
1981
0.2
6.44
1982
4.3
12.16
1983
8.8
17.33
1984 18.6
23.08
1985 34.5
30.68
1986 68.2
43.65
1987 85.0
58.37
1988 58.0
65.81
G sub t
8
7.69
7.29
6.87
6.64
6.84
8.06
9.39
9.00
Forecast
alpha
0.2
8.00
14.13
19.46
24.19
29.72
37.51
51.71
67.77
beta
0.2
|error| error^2
7.80
9.83
10.66
5.59
4.78
30.69
33.29
9.77
60.84
96.59
113.58
31.30
22.85
941.74
1108.00
95.37
14.05
MAD
308.78
MSE
The forecast error appears to decrease with decreasing values of ๏ก and ๏ข. That is, ๏ก =
๏ข = 0 appears to give the lowest value of the forecast error.
2.55
a) We are given in problem 22 that the forecast for January was 25.
Hence e1 = 25-23.3 = 1.7 = E1 and M1 = |e1 | = 1.7 as well. Hence ๏ก1 = 1.
FFeb = (1)(23.3) + (0)(25) = 23.3
e2 = 23.3 – 72.2 = -48.9
E2 = (.1)(-48.9)(.9)(1.7) = -3.36
M2 = (.1)(48.9) + (.9)(1.7) = 6.42
๏ก2 = 3.36/6.42 = .5234
FMarch = (.5234)(72.2) + (.4766)(23.3) = 48.73
e3 = 48.73 – 30.3 = 18.43
E3 = (.1)(18.43) + (.9)(-3.36) = -3.024
M3 = (.1)(18.43) + (.9)(6.42) = 7.621
๏ก3 = 3.024/7.621 = .396 ~ .40
FApr = (.40)(30.3) + (.60)(48.73) = 41.358
2-23
Chapter 02 – Forecasting
Comparison of Methods
Month
Demand
ES(.15)
|Error|
Trigg-Leach
|Error|
Feb
March
April
72.2
30.3
15.5
24.745
31.87
31.63
47.5
1.6
16.1
23.3
48.7
41.4
48.9
18.4
25.9
Obviously Trigg-Leach performed much worse for this 3-month period than did
ES(.12). (The respective MAD’s are 21.7 for ES and 31.1 for Trigg-Leach.)
b) Consider only the period July to December as in problem 36. As in part (a) ๏ก7 = 1.
Use E6 = 567.1 – 480 = 87.
F7 = 480
e7 = 480 – 500 = -20
E7 = (.2)(-20) + (.8)(87) = 65.6
M7 = (.2)(20) + (.8)(87) = 73.6
๏ก7 = 65.6/73.6 = .89
F8 = (.89)(500) + (.11)(480) = 498
e8 = 498 – 950 = -452
E8 = (.2)(-452) + (.8)(65.6) = -37.9
M8 = (.2)(452) + (.8)(73.6) = 149.3
๏ก8 = 37.9/149.3 = .25
F9 = (.25)(950) + (.75)(498) = 611
e9 = 611 – 350 = 261
E9 = (.2)(261) + (.8)(-37.9) = 21.9
M9 = (.2)(261) + (.8)(149.3) = 171.6
๏ก9 = 21.9/171.6 = .13
F10 = (.13)(350) + (.87)(620) = 584.9
e10 = 584.9 – 600 = -15.1
E10 = (.2)(-15.1) + (.8)(21.9) = 14.5
M10 = (.2)(17.8) + (.8)(171.6) = 140.8
๏ก10 = 14.5/140.8 = .10
F11= (.10)(600) + (.90)(584.9) = 586.4
e11 = 586.4 – 870 = -283.6
E11 = (.2)(-283.6) + (.8)(14.5) = -45.1
M11 =(.2)(283.6) + (.8)(140.8) = 169.4
2-24
Chapter 02 – Forecasting
๏ก11 = 45.1/169.4 = .27
F12 = (.27)(870) + (.73)(586.4) = 663.0
Performance Comparison
Month
Demand
7
8
9
10
11
12
500
950
350
600
870
740
MAD
=
Trigg-Leach
Forecast
480
498
611
585
586
663
|Error|
20
452
261
15
284
77
185
The MAD for ES(.2) from problem 36 was 194.5.
Hence Trigg-Leach was slightly better for this problem.
c) Trigg-Leach will out-perform simple exponential smoothing when there is a trend in
the data or a sudden shift in the series to a new level, since ๏ก will be adjusted upward
in these cases and the forecast will be more responsive. However, if the changes are
due to random fluctuations, as in part (a), Trigg-Leach will give poor performance as
the forecast tries to “chase” the series.
2.56
Given information:
๏ก = .2, ๏ข = 0.2, and ๏ง = 0.2
S10 = 120, G10 = 14
c10 = 1.2
c9 = 1.1
c8 = 0.8
c7 = 0.9
a)
F11 = (S10 + G10)c7 = (120 + 14)(0.9) = 120.6
b)
D11 = 128
S11 = ๏ก(D11/c7) + (1 – ๏ก)(S10 + G10) = 135.6
G11 = ๏ง(S11 – S10) + (1 – ๏ง)G10 = 14.3
c11 = ๏ข(D11/S11) + (1-๏ข)c7 = .909
2-25
Chapter 02 – Forecasting
11
๏ฅ C = 4.009
t
t=8
The factors are normed by multiplying each by 1/4.009 = .9978
They will not change appreciably.
F11,13 = (S11 + 2G11)C9 = (135.6 + (2)(14.3))1.1 = 180.6
Xi
Yi
XiYi
1
2
3
4
5
6
7
8
9
10
11
649.8
705.1
772.0
816.4
892.7
963.9
1015.5
1102.7
1212.8
1359.3
1472.8
649.8
1410.2
2316.0
3265.6
4463.5
5783.4
7108.5
8821.6
10915.2
13593.0
16200.8
66
6
10,963.0
996.64
74,527.6
2.57 a)
Sum =
Avg =
Sxy
= n ๏ฅ i Di โ
(n)(n + 1)
i=1
=
(11)(74,527.6) =
๏ฅD
i
2
(11)(12 )
2
(10,963.0) = 96,245.6
SXX = n (n +1)(2n + 1) n (n +1) (11 )(12 )(23) ((11) (12) ) = 1210
โ
=
โ
6
4
6
4
2
b =
2
Sxy
S xx
=
2
2
96, 245.6
= 79.54
1210
โ โ 10,963.0
66
a = Yโ b X =
= 519.4
โ (79.54)
11
11
Initialization for Holt’s Method
S0 = regression line in year 11 (1974)
= 519.4 + (11)(79.54) = 1394.34
2-26
2
2
Chapter 02 – Forecasting
Updating Equations
G0 = slope of regression line = 79.54
Si = ๏กDi + (1 -๏ก)(Si-1 + Gi+1)
Gi = ๏ข (Si – Si-1) + (1 -๏ข)Gi-1
GI
Si
|Error|
|Error|2
Obs
Yr
Di
1
1975
1598.4
1498.78
82.03 F0,1= S0+G0= 1473.88
2
1976
1782.8
1621.21
86.07 F1,2= S1+G1=
1580.81
201.99
40798.18
3
1977
1990.9
1764.01
91.74 F2,3= S2+G2=
1707.28
283.62
80439.38
4
1978
2249.7
1934.54
99.62 F3,4= S3+G3=
1855.75
393.95 155198.35
5
1979
2508.2
2128.97
109.10 F4,5= S4+G4=
2034.16
474.04 224714.16
6
1980
2732.0
2336.86
118.98 F5,6= S5+G5=
2238.07
493.93 243966.72
7
1981
3052.6
2575.20
130.92 F6,7= S6+G6=
2455.84
596.76 356126.04
8
1982
3166.0
2798.08
140.11 F7,8= S7 +G7= 2706.11
459.89 211502.67
9
1983
3401.6
3030.88
149.38 F8,9= S8 +G8= 2938.20
463.40 214740.75
10
1984
3774.7
3299.15
161.27 F9,10 =S9+G9= 3180.26
594.44 353357.64
124.52
Totals
MAD = 408.6,
15505.23
4086.54 1896349.11
MSE = 189,634.9
b)
Year
GNP
%
MA(6)
Forecast
GNP
|Error|
ES(.2)
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
0.2
1976
47.9
1977
45.2
649.8
705.1
772.0
816.4
892.7
963.9
1015.5
1102.7
1212.8
1359.3
1472.8
1598.4
8.51%
9.49%
5.75%
9.35%
7.98%
5.35%
8.59%
9.98%
12.08%
8.35%
8.53%
8.72%
1601.3
2.9
8.54%
1598.6
1782.8
11.54%
8.81%
1739.3
43.5
8.54%
1734.9
1990.9
11.67%
9.84%
1958.3
32.6
9.14%
1945.7
2-27
Forecast
GNP
|Error|
Chapter 02 – Forecasting
1978
66.8
1979
26.4
1980
40.8
1981
41.2
1982
208.0
1983
54.6
1984
73.1
2249.7
13.00%
10.36%
2197.1
52.6
9.65%
2182.9
2508.2
11.49%
10.86%
2494.0
14.2
10.32%
2481.8
2732.0
8.92%
10.76%
2778.2
46.2
10.55%
2772.8
3052.6
11.73%
10.86%
3028.6
24.0
10.23%
3011.4
3166.0
3.71%
10.98%
3387.9
221.9
10.53%
3374.0
3401.6
7.44%
10.30%
3492.0
90.4
9.16%
3456.2
3774.7
10.97%
9.71%
3731.9
42.8
8.82%
3701.6
=
57.1
*MAD
*MAD
=
60.4
The moving average and exponential smoothing forecasts based on percentage
increases are more accurate than Holt’s method.
c) One would expect that a causal model might be more accurate. Large-scale
econometric models for predicting GNP and other fundamental economic time series
are common.
2-28

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