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 and

using 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 Actual

  Shed 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

i

  Y 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

i

  Y 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

i

  Y 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%

= 0.90

  90% 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

7 180

  175.02 + .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

7 180

  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 ?

  

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.