Operations Management Forecasting Chapter 4  Part 2. Forecasting a Trend. Trend is increasing or decreasing pattern. First, plot data to verify trend. If trend exists, then moving averages and exponential smoothing will always lag. 20. Actual. 16. 12. 8. 4. 1. 4. 5. 2. 3. 6.
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MA
Error
MA
Period
Sales
1
8
2
11
3
13
4.33
4
15
10.67
6.00
13.00
5
19
15.67
?
6
MA = 3 period Moving Average
MA
Error
ES
Error
ES
MA
Period
Sales
1
8
11
2
11
11
3
13
4.33
12
4
15
10.67
3.0
6.00
13.5
13.00
5
19
5.5
15.67
?
6
16.25
?
MA = 3 period Moving Average
ES = Exponential Smoothing with =0.5 (F2=11)
Actual observation
Deviation
Deviation
Deviation
Deviation
Values of Dependent Variable (Y)
Deviation
Point on regression line
Deviation
Deviation
Time (x)
(x)
Sales
(y)
x2
xy
1
1
8
8
4
2
11
22
39
9
3
13
60
16
4
15
25
5
19
95
x=3
xy=224
x2=55
y=13.2
Linear Trend Projection Example(x)
TP
Err.
Sales
(y)
ES
Err.
MA
Err.
ES
MA
TP
1
8
11
2
11
11
3
13
12
4
15
10.67
15.8
3.0
4.33
0.8
13.5
18.4
13.00
5
19
0.6
5.5
6.00
15.67
6
16.25
21.0
Linear Trend Projection ExampleTP = Trend Projection: Y = 5.4 + 2.6x
Small errors!
Y
a
+
b
X
=
i
i
Independent variable (number of ads).
Dependent variable (sales).
Actual observation
Deviation
Deviation
Deviation
Deviation
Values of Dependent Variable (Y)
Deviation
Point on regression line
Deviation
Deviation
Values of Independent Variable (x)
Forecast is for negative sales!
Sales
0
Number of TV ads per month
You’re a marketing analyst for Hasbro Toys. You’ve forecast sales with a linear regression model & exponential smoothing. Which model do you use?
Linear Regression Exponential
Actual Model Smoothing
YearSales Forecast Forecast (.9)
1 1 0.6 1.00 2 1 1.3 1.00 3 2 2.0 1.00 4 2 2.7 1.90 5 4 3.4 1.99
F’cast
Error
Error2
Error
Year
i
1
1
0.6
0.4
0.16
0.4
2
1
1.3
0.3
0.09
0.3
3
2
2.0
0.0
0.00
0.0
4
2
2.7
0.7
0.49
0.7
5
4
3.4
0.6
0.36
0.6
Total
0.0
2.0
Linear Regression Model1.10
MSE = Σ Error2 / n = 1.10 / 5 = 0.220
MAD = Σ Error / n = 2.0 / 5 = 0.400
MAPE = Σ[Error/Actual]/n = 1.2/5 = 0.24 = 24%
Year
F’cast
Error
Error2
Error
i
1
1
1.00
0.0
0.00
0.0
2
1
1.00
0.0
0.00
0.0
3
2
1.00
1.0
1.00
1.0
4
2
1.90
0.1
0.01
0.1
5
4
2.01
4.04
2.01
Total
0.3
5.05
3.11
Exponential Smoothing Model1.99
MSE = Σ Error2 / n = 5.05 / 5 = 1.01
MAD = Σ Error / n = 3.11 / 5 = 0.622
MAPE = Σ[Error/Actual]/n = 1.0525/5 = 0.2105 = 21%
Linear Regression Model:
MSE = Σ Error2 / n = 1.10 / 5 = 0.220
MAD = Σ Error / n = 2.0 / 5 = 0.400
MAPE = Σ[Error/Actual]/n = 1.2/5 = 0.24 = 24%
Exponential Smoothing Model:
MSE = Σ Error2 / n = 5.05 / 5 = 1.01
MAD = Σ Error / n = 3.11 / 5 = 0.622
MAPE = Σ[Error/Actual]/n = 1.0525/5 = 0.2105 = 21%
Signal exceeded limit
Tracking signal
Upper control limit
+
0
MAD
Acceptable range

Lower control limit
Time
Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
1
100
90
Error = Actual  Forecast = 90  100 = 10
RSFE = Errors = 10
Tracking Signal  Month 1Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
MAD = Errors/n = 10/1 = 10
Tracking Signal  Month 1Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
1
TS = RSFE/MAD = 10/10 = 1
Tracking Signal  Month 1Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
1
5
2
99
94
Error = Actual  Forecast = 94  99 = 5
Tracking Signal  Month 2Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
1
15
5
2
99
94
RSFE = Errors = (10) + (5) = 15
Tracking Signal  Month 2Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
1
15
15
5
2
99
94
Cum Error = Errors = 10 + 5 = 15
Tracking Signal  Month 2Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
1
15
15
5
2
99
94
7.5
MAD = Errors/n = 15/2 = 7.5
Tracking Signal  Month 2Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
1
15
15
5
2
99
94
7.5
2
TS = RSFE/MAD = 15/7.5 = 2
Tracking Signal  Month 2Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
1
15
15
5
2
99
94
7.5
2
30
10
0
15
0
3
98
113
Tracking Signal  Month 3Mo
F’cst
Act
RSFE
MAD
TS
Error
Error
10
10
10
1
100
90
10.0
1
15
15
5
2
99
94
7.5
2
30
10
0
15
0
3
98
113
10
40
10
10
1
4
105
95
5
15
55
11
.45
5
104
119
14.2
35
30
85
2.47
6
110
140
Tracking Signal  Months 46You’re a marketing analyst for Hasbro Toys. You’ve forecast sales with a linear regression model & exponential smoothing. Which model do you use?
Linear Regression Exponential
Actual Model Smoothing
YearSales Forecast Forecast (.9)
1 1 0.6 1.00 2 1 1.3 1.00 3 2 2.0 1.00 4 2 2.7 1.90 5 4 3.4 1.99
Y
F’cast
Error
MAD
Year
i
1
1
0.6
0.4
0.4
1.0
2
1
1.3
0.3
0.35
0.29
3
2
2.0
0.0
0.233
0.43
1.71
4
2
2.7
0.7
0.35
5
4
3.4
0.6
0.40
0.0
Linear Regression Model Tracking SignalYear
F’cast
Error
TS
MAD
i
1
1
1.00
0.0
0.0
0.0
2
1
1.00
0.0
0.0
0.0
3
2
1.00
1.0
0.33
3.0
4
2
1.90
0.1
0.275
4.0
5
4
2.01
0.622
5.0
1.99
Exponential Smoothing Model Tracking Signal