Forecasting-2
Forecasting -2
Exponential Smoothing
Ardavan Asef-Vaziri
Based on
Operations management: Stevenson
Chapter 7
Operations Management: Jacobs, Chase, and Aquilano
Demand Forecasting
Supply Chain Management: Chopra and Meindl
in a Supply Chain
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 1
Forecasting-2
Time Series Methods
Moving Average
 Discard old records
 Assign same weight for recent records
Assign different weights
 Weighted moving average
Ft  0.4 At 1  0.3 At 2  0.2 Att33  0.1AAtt44
Exponential Smoothing
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 2
Forecasting-2
Exponential Smoothing
Ft 1  Ft  α( At  Ft )
Ft 1  Ft  αAt  αFt
Ft 1  (1  α) Ft  αAt
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 3
Forecasting-2
Exponential Smoothing
t
At
Ft
1
100
100
2
150
100
α=0.2
3
110
Since I have no information for F2, I just enter A1 which is 100. Alternatively we may
assume the average of all available data as our forecast for period 2.
A1  F2
F3 =(1-α)F2 + α A2
F3 =0.8(100) + 0.2(150)
F3 =80 + 30 = 110
F3 =(1-α)F2 + α A2
A1  F2
Ardavan Asef-Vaziri
F2 & A2  F3
A1 & A2  F3
6/4/2009
Exponential Smoothing 4
Forecasting-2
Exponential Smoothing
α=0.2
t
At
Ft
1
100
100
3
2
150 120
100 110
4
112
F4 =(1-α)F3 + α A3
F4 =0.8(110) + 0.2(120)
F4 =88 + 24 = 112
F4 =(1-α)F3 + α A3
A3 & F3  F4
A1 & A2  F3
A1& A2 & A3  F4
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 5
Forecasting-2
Example: Forecast for week 9 using a = 0.1
Week
Demand
1
200
2
250
3
175
4
186
5
225
6
285
7
305
8
190
Forecast
200
F3  1  a F2  aA2  0.9 * 200  0.1* 250  205
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 6
Forecasting-2
Week 4
Week
Demand
Forecast
1
200
2
250
200
3
175
205
4
186
5
225
6
285
7
305
8
190
F4  1  a F3  aA3  0.9 * 205  0.1*175  202
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 7
Forecasting-2
Exponential Smoothing
Week
Demand
1
200
2
250
200
3
175
205
4
186
202
5
225
200
6
285
203
7
305
211
8
190
220
Ardavan Asef-Vaziri
6/4/2009
Forecast
Exponential Smoothing 8
Forecasting-2
Two important questions
How to choose a? Large a or Small a

When does it work?

When does it not?
What is better exponential smoothing
OR moving average?
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 9
Forecasting-2
The Same Example: a = 0.4
Week
Demand
1
200
2
250
200
3
175
220
4
186
202
5
225
196
6
285
207
7
305
238
8
190
265
Ardavan Asef-Vaziri
6/4/2009
Forecast
Exponential Smoothing 10
Forecasting-2
Comparison
350
300
250
Demand
200
alpha = 0.1
150
alpha = 0.4
100
50
0
1
2
3
4
5
6
7
8
week
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 11
Forecasting-2
Comparison
As a becomes larger, the predicted values exhibit
more variation, because they are more responsive
to the demand in the previous period.
 A large a seems to track the series better.
 Value of stability
This parallels our observation regarding MA:
there is a trade-off between responsiveness and
smoothing out demand fluctuations.
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 12
Forecasting-2
Comparison
Forecast for
0.1 alpha
AD
Forecast for 0.4
alpha
AD
Week
Demand
1
200
2
250
200.00
50.00
200.00
50.00
3
175
205.00
30.00
220.00
45.00
4
186
202.00
16.00
202.00
16.00
5
225
200.40
24.60
195.60
29.40
6
285
202.86
82.14
207.36
77.64
7
305
211.07
93.93
238.42
66.58
8
190
220.47
30.47
265.05
75.05
46.73
51.38
Choose the forecast with lower MAD.
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 13
Forecasting-2
Which a to choose?
In general want to calculate MAD for many
different values of a and choose the one with the
lowest MAD.
Same idea to determine if Exponential Smoothing
or Moving Average is preferred.
Note that one advantage of exponential
smoothing requires less data storage to
implement.
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 14
Forecasting-2
Pieces of Data and Age of Data in Exponential Smoothing
Ft = a At–1 + (1 – a) Ft–1
Ft–1 = a At–2 + (1 – a) Ft–2, etc
Ft = aAt–1+(1–a)aAt–2+(1–a)2Ft–2
= aAt–1+(1–a)aAt–2+(1–a)2aAt–3 +(1–a)3aAt–4
+(1–a)4aAt–5+(1–a)5aAt–6 +(1–a)6aAt–7+…
 A large number of data are taken into account– All data
are taken into account in ES.
 “Age” of data is about 1/a periods .
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 15
Forecasting-2
What is better? Exponential Smoothing or Moving
Average
If we set a = 2/(N+1), then MA and ES are
approximately equivalent.
What does it mean that the two systems are
equivalent?
 The variances of the errors are identical.
 Does it mean that the two systems have the same
forecasts?
Exponential smoothing requires less data
storage to implement.
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 16
Forecasting-2
Compute MAD & TS
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Ardavan Asef-Vaziri
6/4/2009
At
13400
14100
14700
15100
13400
16000
12700
15400
13000
16200
16100
13500
14900
15200
15200
15800
16100
16400
15300
15900
16300
15500
15800
16000
Alpha =
0.50
Ft
13912
13656
13878
14289
14695
14047
15024
13862
14631
13815
15008
15554
14527
14713
14957
15078
15439
15770
16085
15692
15796
16048
15774
15787
Dev
-512
444
822
811
-1295
1953
-2324
1538
-1631
2385
1092
-2054
373
487
243
722
661
630
-785
208
504
-548
26
213
AD
512
444
822
811
1295
1953
2324
1538
1631
2385
1092
2054
373
487
243
722
661
630
785
208
504
548
26
213
MAD =
927
MAD
512
478
593
647
777
973
1166
1212
1259
1371
1346
1405
1326
1266
1198
1168
1138
1110
1093
1048
1022
1001
959
927
Sum Dev
-512
-68
754
1565
271
2223
-100
1438
-193
2191
3284
1230
1603
2089
2333
3054
3715
4346
3561
3768
4272
3724
3750
3963
TS
-1.000
-0.142
1.272
2.418
0.348
2.286
-0.086
1.186
-0.153
1.598
2.440
0.875
1.209
1.651
1.948
2.616
3.265
3.916
3.259
3.594
4.178
3.721
3.912
4.273
Exponential Smoothing 17
Forecasting-2
Data Table Excel
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
At
13400
14100
14700
15100
13400
16000
12700
15400
13000
16200
16100
13500
14900
15200
15200
15800
16100
16400
15300
15900
16300
15500
15800
16000
Alpha =
0.50
Ft
13912
13656
13878
14289
14695
14047
15024
13862
14631
13815
15008
15554
14527
14713
14957
15078
15439
15770
16085
15692
15796
16048
15774
15787
Dev
-512
444
822
811
-1295
1953
-2324
1538
-1631
2385
1092
-2054
373
487
243
722
661
630
-785
208
504
-548
26
213
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
AD
512
444
822
811
1295
1953
2324
1538
1631
2385
1092
2054
373
487
243
722
661
630
785
208
504
548
26
213
927
1017
897
886
901
927
960
997
1036
1078
1130
Ardavan Asef-Vaziri
MAD =
927
MAD
512
478
593
647
777
973
1166
1212
1259
1371
1346
1405
1326
1266
1198
1168
1138
1110
1093
1048
1022
1001
959
927
Sum Dev
-512
-68
754
1565
271
2223
-100
1438
-193
2191
3284
1230
1603
2089
2333
3054
3715
4346
3561
3768
4272
3724
3750
3963
TS
-1.000
-0.142
1.272
2.418
0.348
2.286
-0.086
1.186
-0.153
1.598
2.440
0.875
1.209
1.651
1.948
2.616
3.265
3.916
3.259
3.594
4.178
3.721
3.912
4.273
927
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data, what if, Data table
This is a one variable Data Table
Min, conditional formatting
6/4/2009
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
927
1017
897
886
901
927
960
997
1036
1078
1130
886
Exponential Smoothing 18
Forecasting-2
Office Buttton
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 19
Forecasting-2
Add-Inns
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 20
Forecasting-2
Not OK, but GO, then Check Mark Solver
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 21
Forecasting-2
Data Tab/ Solver
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 22
Forecasting-2
Target Cell/Changing Cells
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 23
Forecasting-2
Optimal a Minimal MAD
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 24
Forecasting-2
Associative (Causal) Forecasting
The primary method for associative forecasting is Regression
Analysis.
The relationship between a dependent variable and one or more
independent variables.
The independent variables are also referred to as predictor variables.
We only discuss linear regression between two variables.
We consider the relationship between the dependent variable
(demand) and the independent variable (time).
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 25
Forecasting-2
Regression Method
Ft  b0  b1t
Computed
relationship
50
40
30
20
10
0
0
5
10
15
20
25
Least Squares Line
minimizes sum of squared deviations around the line
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 26
Forecasting-2
Regression: Chart the Data
Period
1
2
3
4
5
6
7
8
9
10
Demand
117
126
210
222
262
310
278
338
379
388
Demand
450
400
350
300
250
Demand
200
150
100
50
0
0
Ardavan Asef-Vaziri
6/4/2009
2
4
6
8
10
12
Exponential Smoothing 27
Forecasting-2
Regression: The Same as Solver but This Time Data Analysis
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 28
Forecasting-2
Data/Data Analysis/ Regression
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 29
Forecasting-2
Regression: Tools / Data Analysis / Regression
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 30
Forecasting-2
Regression Output
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.98
R Square
0.95
Adjusted R Square
0.95
Standard Error
22.21
Observations
10
ANOVA
Regression
Residual
Total
Intercept
X Variable 1
df
1
8
9
SS
77771
3945
81716
MS
77771
493
F
158
Coefficients Standard Error t Stat P-value
94.13
15.17
6.21 0.000258
30.70
2.44
12.56 0.000002
Significance F
1.51524E-06
Lower 95%
59.15
25.07
Upper 95%
129.12
36.34
Ft = 94.13 +30.71t
Forecast for the next period.
F11 = 94.13 +30.71(11) = 431.7
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 31
Assignment . Problem 1.
Forecasting-2
Due at the beginning of the next class
Based on the data below forecast the demand for September
using the listed techniques:
Month
Sales (1000)
Feb
19
Mar
18
Apr
15
May
20
Jun
18
Jul
22
Aug
20
Ardavan Asef-Vaziri
6/4/2009
•
•
•
•
•
Linear regression
5 period moving average
Exponential smoothing.
α=.2
March forecast=19
Naive method
Compute MAD for naive method
and exponential smoothing. Which
one is preferred? NM or ES?
Exponential Smoothing 32
Assignment Problem 2
Forecasting-2
Due at the beginning of the next class
(a) Exponential smoothing is being used to forecast demand. The
previous forecast of 66 turned out to be 5 units larger than actual
demand. The next forecast is 65. Compute a?
(b) The 5-period moving average in month 6 was 150 units. Actual
demand in month 7 is 180 units. What is the 6 period moving
average in month 7?
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 33
Forecasting-2
Practice
The president of State University wants to forecast student
enrollments for this academic year based on the following
historical data:
5 years ago
4 years ago
3 years ago
2 years ago
Last year
15,000
16,000
18,000
20,000
21,000
What is the forecast for this year using exponential
smoothing with α = 0.4, if the forecast for two years ago
was 16,000?
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 34
Forecasting-2
Practice
t
1
At 15000
2
16000
3
18000
Ft
4
20000
16000
5
21000
17600
Forecast for last year
F5 = (1-α)F4+ α(A4)
F5 = .6(16000)+.4(20000)=17600
Forecast for this year
F6 = (1-α)F5+ α(A5)
F6 = .6(17600)+.4(21000)=18960
Ardavan Asef-Vaziri
6/4/2009
Exponential Smoothing 35
Forecasting-2
Practice ……… For your own practice
Based on the data below forecast the total number of new
customers in year 9. Use the listed techniques:
Year
1
2
3
4
5
6
7
8
Ardavan Asef-Vaziri
Customers
35
43
41
46
48
63
67
79
6/4/2009
•
•
•
•
•
Linear regression
(show equation)
4 period moving average
Exponential Smoothing.
α=.3
Year 3 forecast=43
Naive method
Compute MAD for naive
method and exponential
smoothing. Which one is
preferred? NM or ES?
Exponential Smoothing 36