IEOR 165 Discussion 10
April 24, 2015
Question 1. Assume that the demand is:
Year 1
1
2
3
4
Demand
10
20
26
17
Year 2
1
2
3
4
Demand
12
23
30
22
a. Using the data, determine initial values of the intercept, slope and seasonal factors for
multiplicative Winter’s method.
b. Assume that the observed demand for the first quarter of year 3 was 16. Using α = 0.2,
β = 0.1 and γ = 0.1, update the estimates of the series, the slope and the seasonal
factors.
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Solution 2.
a. Seasonal factor initialization
Dt
Vi − [(N + 1)/2 − j]G0
where i = 1 for the first season and i = 2 for the second season and j is the period of
the season.
Initialization procedure:
ct =
V1
1
=
N
V2 =
G0 =
S0 =
c−7 =
c−6 =
c−5 =
c−4 =
c−3 =
c−2 =
c−1 =
c0 =
c−3
c−2
c−1
c0
sum:
1
N
−N
∑
1
Dj = (10 + 20 + 26 + 17) = 18.25
4
j=−2N +1
0
∑
1
Dj = (12 + 23 + 30 + 22) = 21.75
4
j=−N +1
V2 − V1
21.75 − 18.25
=
= 0.875
N
4
3
N −1
= 21.75 + (0.875) = 23.06
V2 + G0
2
2
10
= 0.5904
18.25 − [5/2 − 1]0.875
20
= 1.123
18.25 − [5/2 − 2]0.875
26
= 1.391
18.25 − [5/2 − 3]0.875
17
= 0.869
18.25 − [5/2 − 4]0.875
12
= 0.5872
21.75 − [5/2 − 1]0.875
23
= 1.079
21.75 − [5/2 − 2]0.875
30
= 1.352
21.75 − [5/2 − 3]0.875
22
= 0.9539
21.75 − [5/2 − 4]0.875
avg seasonal factors
(c−7 + c−3 )/2 = 0.5888
(c−6 + c−2 )/2 = 1.1010
(c−5 + c−1 )/2 = 1.372
(c−4 + c0 )/2 = 0.9115
3.9733
2
normalized seasonal factors
0.59
1.11
1.38
0.92
4
Suppose we wish to forecast the following year’s demand at t = 0. The forecasting
equation is:
Ft,t+τ = (St + τ Gt )ct+τ −N
which results in
F0,1
F0,2
F0,3
F0,4
=
=
=
=
(S0 + 1G0 )c−3 = (23.06 + 0.875)0.59 = 14.12
(S0 + 2G0 )c−2 = [23.06 + (2)(0.875)]1.11 = 27.54
(S0 + 3G0 )c−1 = [23.06 + (3)(0.875)]1.38 = 35.44
(S0 + 4G0 )c0 = [23.06 + (4)(0.875)]0.92 = 24.38
b.
α = 0.2 β = 0.1 γ = 0.1 D1 = 18
S1 = α(D1 /c−3 ) + (1 − α)(S0 + G0 ) = 0.2(16/0.59) + 0.8(23.06 + 0.875) = 24.57
G1 = β(S1 − S0 ) + (1 − β)G0 = 0.1(24.57 − 23.06) + 0.9(0.875) = 0.9385
c1 = γ(D1 /S1 ) + (1 − γ)c−3 = 0.1(16/24.57) + 0.9(0.59) = 0.5961
At this point it is recommended to renorm c−2 , c−1 , c0 and the new value of c1 to add
to 4.
c1
c−2
c−1
c0
sum:
avg seasonal factors
0.5961
1.11
1.38
0.92
4.0061
normalized seasonal factors
0.59
1.11
1.38
0.92
4
Forecasts for 2nd , 3rd and 4th quarters of 1993
F1,2 = [S1 + G1 ]c−2 = (24.57 + 0.9385)1.11 = 28.3144
F1,3 = [S1 + 2G1 ]c−1 = [24.57 + 2(0.9385)]1.38 = 36.4969
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