1bmoving average and exp smoothing
Moving Average Moving Average And And Exponential Exponential Smoothing Smoothing Prepared by Lee Revere and John Large for Management, 9e To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. Prepared by Lee Revere and John Large
5-1
Upper Saddle River, NJ 07458
Moving Averages
Moving Averages
Moving average methods consist of
computing an average of the most recent
n data values for the time series andusing this average for the forecast of the
next period.Simple moving average = demand in previous n periods
n
F =(X +X …+X )/n t+1 t t-1 t-n+1 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-2 Upper Saddle River, NJ 07458 To accompany Quantitative Analysis for Management, 9e 5-3 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Wallace Garden Supply’s
Three-Month Moving Average
Wallace Garden Supply’s
Three-Month Moving Average
Month ActualShed Sales Three-Month Moving Average
January
10 February
12 March
13 April
16 May
19 June
23 July
26 (10+12+13)/3 = 11 2 / 3 (12+13+16)/3 = 13 2 / 3 (13+16+19)/3 = 16
(16+19+23)/3 = 19 1 / 3
You’re manager of a museum store that sells historical replicas. You want to forecast sales (000) for 2003 using a 3 - period moving average. 1998
To accompany Quantitative Analysis for Management, 9e 5-4 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
4 1999
6 2000
5 2001
3 2002
7 © 1995 Corel Corp. Moving Average Moving Average Example Example
Moving Average Moving Average Solution Solution
Time Response Moving Moving
iY Total Average (n=3) (n=3) 1998
4 NA NA 1999
6 NA NA 2000
5 NA NA 2001 3 4+6+5=15 15/3 = 5 2002 7 2003 NA To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-5 Upper Saddle River, NJ 07458
Moving Average Moving Average Solution Solution
Time Response Moving Moving
iY Total Average (n=3) (n=3) 1998
4 NA NA 1999
6 NA NA 2000
5 NA NA 2001 3 4+6+5=15 15/3 = 5 2002 7 6+5+3=14 14/3=4 2/3 2003 NA To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-6 Upper Saddle River, NJ 07458
Moving Average Moving Average Solution Solution
Time Response Moving Moving
iY Total Average (n=3) (n=3) 1998
4 NA NA 1999
6 NA NA 2000
5 NA NA 2001 3 4+6+5=15 15/3=5.0 2002 7 6+5+3=14 14/3=4.7 2003 NA 5+3+7=15 15/3=5.0 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-7 Upper Saddle River, NJ 07458
Moving Average Moving Average Graph Graph
Sales
8 Actual
6 Forecast
4
2 95 96 97 98 99 00 Year To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-8 Upper Saddle River, NJ 07458
Weighted Moving
Weighted Moving
Averages
Averages
Weighted moving averages use weights
to put more emphasis on recent periods.
Weighted moving average = (weight for period n ) (demand in period n ) ∑ weights
F =(w t+1 1 t X +w 2 t-1 n t-(n-1) X …+w X )/(w +…+ w ) 1 2 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-9 Upper Saddle River, NJ 07458
Calculating Weighted Calculating Weighted Moving Averages Moving Averages
To accompany Quantitative Analysis for Management, 9e 5-10 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Weights Applied Period
3 Last month
2 Two months ago
1 Three months ago 3*Sales last month + 2*Sales two months ago +
1*Sales three months ago
Wallace Garden’s Weighted
Wallace Garden’s Weighted
Three-Month Moving Average
Three-Month Moving Average
Month Actual Three-Month Weighted Moving Average
Shed Sales
January
10 February
12 March
13 1 April
16 [3*13+2*12+1*10]/6 = 12 / 1 6 May
19 [3*16+2*13+1*12]/6 =14 / 3 June
23 [3*19+2*16+1*13]/6 = 17 1 July To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. 26 [3*23+2*19+1*16]/6 = 20 /
5-11
2 for Management, 9e Upper Saddle River, NJ 07458
Weighted Moving
Weighted Moving
Average Method
Average Method
- Used when trend is present
- Older data usually less important
- Weights based on intuition
- Often lay between 0 & 1, & sum to 1.0
- Equation
Σ n n ) Σ (Weight for period n (Weight for period ) (Demand in period n ) (Demand in period )
WMA = WMA =
Σ Weights Σ Weights To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-12 Upper Saddle River, NJ 07458 To accompany Quantitative Analysis for Management, 9e 5-13 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Actual Demand, Moving
Actual Demand, Moving
Average, Weighted Moving
Average, Weighted Moving
Average
Average
5
10
15
20
25
30
Actual sales Moving average
Weighted moving average
Disadvantages of Disadvantages of Moving Average Moving Average Methods Methods
- Increasing n makes forecast less sensitive to changes
- Do not forecast trend well
- Require much historical data © 1984-1994 T/Maker Co. To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-14 Upper Saddle River, NJ 07458
Exponential Smoothing
Exponential Smoothing
Exponential smoothing is a type of moving average technique that involves little record keeping of past data.
New forecast
= previous forecast + (previous actual –previous
forecast)Mathematically this is expressed as: F = F - F ) t t-1 t-1 t-1 + (Y
F = new forecast t F = previous forecast t-1 = smoothing constant Y = previous period actual t-1 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-15 Upper Saddle River, NJ 07458
Equations
Exponential Smoothing
Equations
Exponential Smoothing
- F
- - F
- + (A
Use for computing forecast
- F
- + (1- )
t-1
t-1 )
t = A t - 1
t - 2
- + (1-)A
- + (1- )
- + (1- )
- F
- A
3 A t - 4 + ...
t-1 ·A
t = Forecast value
t = Actual value
= Smoothing constant
t = F t-1
To accompany Quantitative Analysis for Management, 9e 5-16 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
2 ·A t - 3
Exponential Smoothing
Exponential Smoothing
Method
Method
- Form of weighted moving average
- Weights decline exponentially
- Most recent data weighted most
Requires smoothing constant
()- Ranges from 0 to 1
- Subjectively chosen
Involves little record keeping
of past data To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-17 Upper Saddle River, NJ 07458
Forecast Effects of Forecast Effects of Smoothing Constant Smoothing Constant
2 F = A (1- )A (1- ) A + ... + +
t t - 1 t - 2 t - 3
Weights
= Prior 2 periods 3 periods Period ago ago 2
(1 - ) (1 - ) = 0.10 10%
= 0.90 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-18 Upper Saddle River, NJ 07458
Forecast Effects of
Forecast Effects of
Smoothing Constant Smoothing Constant
2 F = A (1- ) A (1- ) A + ... + + t t - 1 t - 2 t - 3
Weights
= Prior 2 periods 3 periods Period ago ago 2
(1 - ) (1 - ) = 0.10
10% 9%
= 0.90 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-19 Upper Saddle River, NJ 07458
Forecast Effects of
Forecast Effects of
Smoothing Constant Smoothing Constant
- + 2 = A (1- )A (1- ) A + ...
- + F
t t - 1 t - 2 t - 3
Weights
= Prior 2 periods 3 periods Period ago ago 2
(1 - ) (1 - ) = 0.10 8.1%
10% 9%
= 0.90 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-20 Upper Saddle River, NJ 07458
Forecast Effects of
Forecast Effects of
Smoothing Constant Smoothing Constant
- + 2 = A (1- )A (1- ) A + ...
- + F
t t - 1 t - 2 t - 3
Weights
= Prior 2 periods 3 periods Period ago ago 2
(1 - ) (1 - ) = 0.10 8.1%
10% 9%
= 0.9090% To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-21 Upper Saddle River, NJ 07458
- + (1- ) A
- + (1- )
To accompany Quantitative Analysis for Management, 9e 5-22 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
F t
= A t - 1
t - 2
2 A t - 3 + ...
Forecast Effects of
Forecast Effects of
Smoothing Constant Smoothing Constant
Weights Prior Period
2 periods ago
(1 - ) 3 periods ago
(1 - ) 2 =
= 0.10 = 0.90
10% 9%
8.1%
90% 9%
- + (1- ) A
- + (1- )
To accompany Quantitative Analysis for Management, 9e 5-23 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
F t
= A t - 1
t - 2
2 A t - 3 + ...
Forecast Effects of
Forecast Effects of
Smoothing Constant Smoothing Constant
Weights Prior Period
2 periods ago
(1 - ) 3 periods ago
(1 - ) 2 =
= 0.10 = 0.90
10% 9%
8.1%90% 9% 0.9% To accompany Quantitative Analysis for Management, 9e 5-24 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Port of Baltimore
Port of Baltimore
Exponential Smoothing
Exponential Smoothing
Example
Example
Qtr Actual Tonnage Unloaded Rounded Forecast using =0.10
1 180 175 2 168 176= 175.00+0.10(180-175) 3 159 175 =175.50+0.10(168-175.50) 4 175 173 =174.75+0.10(159-174.75) 5 190 173 =173.18+0.10(175-173.18) 6 205 175 =173.36+0.10(190-173.36) 7 180 178 =175.02+0.10(205-175.02) 8 182 178 =178.02+0.10(180-178.02) 9 ? 179= 178.22+0.10(182-178.22) To accompany Quantitative Analysis for Management, 9e 5-25 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Port of Baltimore
Port of Baltimore
Exponential Smoothing
Exponential Smoothing
Example
Example
Qtr Actual Tonnage Unloaded Rounded Forecast using =0.50
1 180 175 2 168 178 =175.00+0.50(180-175) 3 159 173 =177.50+0.50(168-177.50) 4 175 166 =172.75+0.50(159-172.75) 5 190 170 =165.88+0.50(175-165.88) 6 205 180 =170.44+0.50(190-170.44) 7 180 193 =180.22+0.50(205-180.22) 8 182 186 =192.61+0.50(180-192.61) 9 ? 184 =186.30+0.50(182-186.30)
Selecting a Smoothing Selecting a Smoothing Constant Constant Actual Forecast with a = 0.10 Absolute Deviations Forecast with a = 0.50 Absolute Deviations
To accompany Quantitative Analysis for Management, 9e 5-26 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
180 175 5 175
5 168 176 8 178
10 159 175 16 173
14 175 173 2 166
9 190 173 17 170
20 205 175 30 180
25 180 178 2 193
13 182 178 4 186
4 MAD
10.0
12 To select the best smoothing constant,
evaluate the accuracy of each forecasting
model.The lowest MAD results from = 0.10 To accompany Quantitative Analysis for Management, 9e 5-27 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
During the past 8 quarters, the Port of Baltimore
has unloaded large quantities of grain. ( = .10 ).The first quarter forecast was 175. .
Quarter Actual 1 180 2 168 3 159 4 175 5 190 6 205 7 180 8 182 9 ?
Exponential Smoothing
Exponential Smoothing Example Example Find the forecast for the 9 th quarter.- + 0.1(A
- - F
5
3
3
159
159
4
4
175
175
5
168
190
190
6
6
205
205
175.00 +
175.00 +
168
2
To accompany Quantitative Analysis for Management, 9e 5-28 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
(
F t
= F t-1
t-1
t-1 )
Quarter
Quarter
Actual
Actual Forecast, F t
α
2
α
=
=
.10
.10
)
)
1
1 180 175.00 (Given)
Exponential Smoothing Exponential Smoothing Solution
Solution
Exponential Smoothing Exponential Smoothing Solution
Solution
F t= F t-1
- + 0.1(A
- - F
Actual Forecast, F t
190
159
159
4
4
175
175
5
5
190
3
6
6
205
205
Actua
Quarter
t-1
t-1 )
3
(
(
1
α
α
=
=
.10
.10
)
)
1
175.00 + 175.00 + .10 .10 (
180
180
175.00 (Given)
175.00 (Given)
2
2
168
To accompany Quantitative Analysis for Management, 9e 5-29 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Quarter
168
- + 0.1(A
- - F
- - 3
5
(180 -
3
159
159
4
4
175
175
5
190
175.00 + 175.00 + .10 .10
190
6
6
205
205
Exponential Smoothing Exponential Smoothing
Solution
Solution
F t= F t-1
t-1
(180
168
To accompany Quantitative Analysis for Management, 9e 5-30 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Quarter
.10
Quarter
Actual
Actual
Forecast,
Forecast, F F t t
(
( α α
=
=
.10
168
)
)
1
1
180
180
175.00 (Given)
175.00 (Given)
2
2
t-1 )
- + 0.1(A
- - F
5
3
159
159
4
4
175
175
5
190
(180 - 175.00 - 175.00) )
190
6
6
205
205
Exponential Smoothing Exponential Smoothing
Solution
Solution
F t= F t-1
t-1
3
(180
To accompany Quantitative Analysis for Management, 9e 5-31 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Quarter
)
Quarter
Actual
Actual Forecast, F t
( α α
=
=
.10
.10
)
1
175.00 + 175.00 + .10
.10
1
180
180
175.00 (Given)
175.00 (Given)
2
2
168
168
t-1 )
- + 0.1(A
- - F
5
= 175.50
3
3
159
159
4
4
175
175
5
(180
190
190
6
6
205
205
Exponential Smoothing Exponential Smoothing
Solution
Solution
F t= F t-1
t-1
(180 - 175.00 - 175.00) ) = 175.50
175.00 + 175.00 + .10 .10
To accompany Quantitative Analysis for Management, 9e 5-32 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Quarter
=
Quarter
Actual
Actual
Forecast,
Forecast, F F t t
(
( α α
=
.10
168
.10
)
)
1
1 180
180 175.00 (Given)
175.00 (Given)
2
2
168
t-1 )
F t
Exponential Smoothing
Solution
Exponential Smoothing
= F t-1
Solution
- + 0.1(A
- - F
Quarter
5
(168 -
(168 - 175.50 175.50) ) = 174.75
= 174.75
4
4
175
175
5
159
190
190
6
6
205
205
t-1 )
t-1
175.50 175.50 + + .10 .10
159
Quarter
.10
Actual
Actual Forecast, F t
(
α
α
=
=
.10
3
)
)
1 180 175.00 (Given)
2
2 168
168 175.00 + .10(180 - 175.00) = 175.50
To accompany Quantitative Analysis for Management, 9e 5-33 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
3
175.00 + .10(180 - 175.00) = 175.50
Exponential Smoothing
Exponential Smoothing
Solution Solution
F = F + 0.1(A - F ) t t-1 t-1 t-1
Forecast, F t
QuarterActual ( α = .10 )
1995 180 175.00 (Given) 1996 168175.00 + .10(180 - 175.00) = 175.50 1997 159 175.50 + .10(168 - 175.50) = 174.75 1998 175 174.75 .10 (159 - 174.75 ) + = 173.18 1999 190 2000 205 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-34 Upper Saddle River, NJ 07458
- + 0.1(A
- - F
To accompany Quantitative Analysis for Management, 9e 5-35 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
F t
= F t-1
t-1
t-1 ) Quarter Actual Forecast, F t
( α = .10 ) 1 180 175.00 (Given) 2 168175.00 + .10(180 - 175.00) = 175.50 3 159175.50 + .10(168 - 175.50) = 174.75
4 175 174.75 + .10(159 - 174.75) = 173.18 5 190173.18 + .10 (175 - 173.18 ) = 173.36 6 205
Exponential Smoothing
Exponential Smoothing
Solution
Solution
- + 0.1(A
- - F
To accompany Quantitative Analysis for Management, 9e 5-36 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
F t
= F t-1
t-1
t-1 ) Quarter Actual Forecast, F t
( α = .10 ) 1 180 175.00 (Given) 2 168175.00 + .10(180 - 175.00) = 175.50 3 159175.50 + .10(168 - 175.50) = 174.75 4 175174.75 + .10(159 - 174.75) = 173.18 5 190 173.18 + .10(175 - 173.18) = 173.36 6 205 173.36 + .10 (190 - 173.36 ) = 175.02
Exponential Smoothing
Exponential Smoothing
Solution
Solution
Exponential Smoothing
Exponential Smoothing
Solution Solution
F = F + 0.1(A - F ) t t-1 t-1 t-1
Forecast, F t
Time Actual ( α = .10 )
4 175174.75 + .10(159 - 174.75) = 173.18
5 190173.18 + .10(175 - 173.18) = 173.36
6 205 173.36 + .10(190 - 173.36) = 175.02
7 180175.02 + .10 (205 - 175.02 ) = 178.02
8
9 To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-37 Upper Saddle River, NJ 07458
- + 0.1(A
- - F
To accompany Quantitative Analysis for Management, 9e 5-38 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
F t
= F t-1
t-1
t-1 ) Time Actual Forecast, F t
( α = .10 )
4 175 174.75 + .10(159 - 174.75) = 173.18
5 190 173.18 + .10(175 - 173.18) = 173.36
6 205 173.36 + .10(190 - 173.36) = 175.02
Exponential Smoothing
Exponential Smoothing
Solution
Solution
7 1808 175.02 + .10(205 - 175.02) = 178.02
9 178.22 + .10 (182 - 178.22) = 178.58 182 178.02 + .10(180 - 178.02) = 178.22 ?
Impact of
Impact of
250 Forecast (0.5)
200 Forecast (0.1)
150
Actual n a g e l T o a tu A c 100
50
1
2
3
4
5
6
7
8
9 for Management, 9e
To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc.
5-39 Quarter Upper Saddle River, NJ 07458- - errors forecast MAD
To accompany Quantitative Analysis for Management, 9e 5-40 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Choosing
Choosing
Seek to minimize the Mean Absolute Deviation (MAD) If: Forecast error = demand - forecast Then: n
PM Computer: Moving PM Computer: Moving Average Example Average Example
PM Computer assembles customized
personal computers from generic parts. The owners purchase generic computer parts in volume at a discount from a variety of sources whenever they see a good deal.
It is important that they develop a good forecast of demand for their computers so they can purchase component parts efficiently. To accompany Quantitative Analysis © 2006 by Prentice Hall, Inc. for Management, 9e 5-41 Upper Saddle River, NJ 07458
PM Computers: Data
50
Using MAD, what forecast is most accurate?
Compute an exponential smoothing forecast using = 0.7
Compute a 3-month weighted average using weights of 4,2,1 for the past three months of data
Compute a 2-month moving average
56
9 Sept
47
8 Aug
43
7 July
To accompany Quantitative Analysis for Management, 9e 5-42 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
PM Computers: Data
45
5 May
37
4 Apr
41
3 Mar
40
2 Feb
37
1 Jan
Period month actual demand
6 June
PM Computers: Moving PM Computers: Moving Average Solution Average Solution
To accompany Quantitative Analysis for Management, 9e 5-43 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
MA Abs. Dev 3 month WMA Abs. Dev Exp.Sm. Abs. Dev 2 month
37.00 37.00 3.00
38.50 2.50 39.10 1.90 40.50 3.50 40.14 3.14 40.43 3.43
39.00 6.00 38.57 6.43 38.03 6.97 41.00 9.00 42.14 7.86 42.91 7.09
47.50 4.50 46.71 3.71 47.87 4.87 46.50 0.50 45.29 1.71 44.46 2.54
45.00 11.00 46.29 9.71 46.24 9.76 51.50 51.57 53.07 5.29 5.43 4.95 MAD
Exponential smoothing resulted in the lowest MAD.