1bmoving average and exp smoothing

(1)

Moving Average

And

Exponential

Smoothing

(2)

Moving Averages

n

Simple moving average =



demand in previous

n

periods

Moving average

methods consist of

computing an average of the most recent

n

data values for the time series and

using this average for the forecast of the

next period.

(3)

Wallace Garden Supply’s

Three-Month Moving Average

Month

Actual

Shed

Sales

Three-Month

Moving Average

January

10

(4)

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

4 1999

6 2000

5 2001

3 2002

7 Moving Average

Moving Average

Example

(5)

Moving Average

Solution

Time Response

Y

i

Moving

Total

(n=3)

Moving

Average

(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)

Moving Average

Solution

Time Response

Y

i

Moving

Total

(n=3)

Moving

Average

(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

(7)

Moving Average

Solution

Time Response

Y

i

Moving

Total

(n=3)

Moving

Average

(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

(8)

95 96 97 98 99 00

Year

Sales

2

4

6

8 _Actual

Forecast

Moving Average

Graph

(9)

Weighted Moving

Averages

Weighted moving averages

use weights

to put more emphasis on recent periods.



_{(weight for period}

_n

_{) (demand in period}

_n

₎

∑ weights

Weighted moving average =

(10)

Calculating Weighted

Moving Averages

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

(11)

Wallace Garden’s Weighted

Three-Month Moving Average

Month

Actual

Shed

Sales

Three-Month Weighted

Moving Average

10 [319+216+1*13]/6 = 17

(12)

• _{Used when trend is present}

• _{Older data usually less}

important

• _{Weights based on intuition}

• _{Often lay between 0 & 1, & sum}

to 1.0

• _Equation

WMA =

Σ

(Weight for period n

(Weight for period

n

) (Demand in period n

) (Demand in period

n

)

Σ

Weights

Weighted Moving

Average Method

(13)

Actual Demand, Moving

Average, Weighted Moving

Average

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

(14)

• _Increasing

_n

_makes

forecast less sensitive to

changes

• _{Do not forecast trend}

well

• _{Require much historical}

data

Disadvantages of

Moving Average

Methods

(15)

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

_t

= F

_t-1

+

_

(Y

_t-1

- F

_t-1

)

F

_t-1

= previous forecast



= smoothing constant

F

_t

= new forecast

(16)

• _F

_t

₌

_F

_t

_-1

₊

_

₍

_A

_t

_-1

_-

_F

_t

_-1

₎

• _{Use for computing forecast}

• _F

_t

₌

_

_A

_t

_{- 1}

₊

_

_(1-

_

₎

_A

_t

_{- 2}

₊



(1-



)

2 ·A

_t

_{- 3}

+

_

(1-

_

)

3 A

_t

_{- 4}

+ ...

+

_

(1-

_

)

t-

1 ·A

₀

• F

t

= Forecast value

• A

_t

= Actual value



_

_{= Smoothing constant}

Exponential Smoothing

Equations

(17)

• 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

Exponential Smoothing

Method

(18)

F

_t

=



A

_t

_{- 1}

+



(1-



)

A

_t

_{- 2}

+



(1-



)

2 A

_t

_{- 3}

+ ...

Forecast Effects of

Smoothing Constant

(19)

F

_t

=



A

_t

_{- 1}

+



(1-



)

A

_t

_{- 2}

+



(1-



)

2 A

_t

_{- 3}

+ ...

Forecast Effects of

Smoothing Constant

(20)

F

_t

=



A

_t

_{- 1}

+



(1-



)

A

_t

_{- 2}

+



(1-



)

2 A

_t

_{- 3}

+ ...

Forecast Effects of

Smoothing Constant

(21)

F

_t

=



A

_t

_{- 1}

+



(1-



)

A

_t

_{- 2}

+



(1-



)

2 A

_t

_{- 3}

+ ...

Forecast Effects of

Smoothing Constant

(22)

F

_t

=



A

_t

_{- 1}

+



(1-



)

A

_t

_{- 2}

+



(1-



)

2 A

_t

_{- 3}

+ ...

Forecast Effects of

Smoothing Constant

(23)

F

_t

=



A

_t

_{- 1}

+



(1-



)

A

_t

_{- 2}

+



(1-



)

2 A

_t

_{- 3}

+ ...

Forecast Effects of

Smoothing Constant

(24)

Port of Baltimore

Exponential Smoothing

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)

(25)

Port of Baltimore

Exponential Smoothing

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)

(26)

Selecting a Smoothing

Constant

Actual

Forecast with

a = 0.10

Absolute

Deviations

Forecast with

a = 0.50

To select the best smoothing constant,

evaluate the accuracy of each forecasting

model.

(27)

During the past 8 quarters, the Port of Baltimore

has unloaded large quantities of grain.

(

_

= .10

).

The first quarter forecast was

175. .

Quarter

Actual

Exponential Smoothing

Example

(28)

F

_t

=

F

_t

_-1

+ 0.1(

A

_t

_-1

-

F

_t

_-1

)

Quarter

Actual

Forecast,

F

t

(

α

=

.10

)

1

180 175.00 (Given)

2 Exponential Smoothing

Exponential Smoothing

Solution

(29)

Quarter

Actual

Actua

Forecast,

F

t

(

α

=

.10

)

1

180

180 175.00 (Given)

175.00 (Given)

2 Exponential Smoothing

Exponential Smoothing

Solution

(30)

Quarter

Actual

Forecast,

F

t

(

α

=

.10

)

1

180

180 175.00 (Given)

175.00 (Given)

2 Exponential Smoothing

Exponential Smoothing

Solution

(31)

Quarter

Actual

Forecast,

F

t

(

α

=

.10

)

1

180

180 175.00 (Given)

175.00 (Given)

2 Exponential Smoothing

Exponential Smoothing

Solution

(32)

Quarter

Actual

Forecast,

F

t

(

α

=



=



.10

)

1

180

180 175.00 (Given)

175.00 (Given)

2 Exponential Smoothing

Exponential Smoothing

Solution

(33)

F

_t

=

F

_t

_-1

+ 0.1(

A

_t

_-1

-

F

_t

_-1

)

Quarter

Actual

Forecast,

F

t

(

α

=

.10

)

1

180 175.00 (Given)

2 Exponential Smoothing

Exponential Smoothing

Solution

(34)

F

_t

=

F

_t

_-1

+ 0.1(

A

_t

_-1

-

F

_t

_-1

)

QuarterActual

Forecast,

F

t

(

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

1999 190

2000 205

174.75 +

.10

(159

- 174.75

)

= 173.18

Exponential Smoothing

Solution

(35)

F

_t

=

F

_t

_-1

+ 0.1(

A

_t

_-1

-

F

_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

(36)

F

_t

=

F

_t

_-1

+ 0.1(

A

_t

_-1

-

F

_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

Solution

(37)

F

_t

=

F

_t

_-1

+ 0.1(

A

_t

_-1

-

F

_t

_-1

)

Time Actual

Forecast,

F

t

(

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

Exponential Smoothing

Solution

7

180

8 175.02 +

.10

(205

- 175.02

) = 178.02

(38)

F

_t

=

F

_t

_-1

+ 0.1(

A

_t

_-1

-

F

_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

Solution

7

180

8 175.02 + .10(205 - 175.02) = 178.02

9 178.22 +

.10

(182

- 178.22) =

178.58

182 178.02 + .10(180 - 178.02) =

178.22

(39)

Impact of

(40)

Choosing

_

Seek to minimize the Mean Absolute Deviation (MAD)

If:

Forecast error = demand - forecast

Then:

n

errors

-forecast

(41)

PM Computer: Moving

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.

(42)

PM Computers: Data

Period month actual demand

1 Jan

37

2 Feb

40

3 Mar

41

4 Apr

37

5 May

45

6 June

50

7 July

43

8 Aug

47

9 Sept

56 

Compute a 2-month moving average



Compute a 3-month weighted average using weights

of 4,2,1 for the past three months of data



Compute an exponential smoothing forecast using



= 0.7

(43)

PM Computers: Moving

Average Solution

2 month

MA

Abs. Dev

3 month WMA

Abs. Dev

Exp.Sm.

Abs. Dev

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