Chapter 4 Forecasting. Ch. 4: What is covered? Moving AverageMoving Average Weighted Moving AverageWeighted Moving Average Exponential SmoothingExponential

Chapter 4Chapter 4

ForecastingForecasting

Ch. 4: What is covered?Ch. 4: What is covered?• Moving AverageMoving Average• Weighted Moving AverageWeighted Moving Average• Exponential SmoothingExponential Smoothing• Trend ProjectionsTrend Projections• Seasonal Index/Adjusted-ForecastSeasonal Index/Adjusted-Forecast• MAD, CE, Bias, MSE, MAPDMAD, CE, Bias, MSE, MAPD• Linear Regression, r, rLinear Regression, r, r22

Ch. 4: What is Ch. 4: What is notnot covered? covered?• Trend Adjusted Exponential SmoothingTrend Adjusted Exponential Smoothing• Tracking SignalTracking Signal• By-Hand computation of anything done by By-Hand computation of anything done by

ExcelExcel– a and b using Least Squaresa and b using Least Squares– MAD, CE, Bias, MSE, MAPD, r and rMAD, CE, Bias, MSE, MAPD, r and r22

• Multiple RegressionMultiple Regression

ForecastingForecastingPredicting future eventsPredicting future eventsUsually demand behavior Usually demand behavior

over a time frameover a time frameQualitative methodsQualitative methods

Based on subjective methodsBased on subjective methodsQuantitative methodsQuantitative methods

Based on mathematical formulasBased on mathematical formulas

Demand BehaviorDemand BehaviorTrendTrend

gradual, long-term up or down gradual, long-term up or down movementmovement

CycleCycle up & down movement repeating over up & down movement repeating over

long time framelong time frameSeasonal patternSeasonal pattern

periodic oscillation in demand which periodic oscillation in demand which repeatsrepeats

Random movements follow no patternRandom movements follow no pattern

Forms of Forecast MovementForms of Forecast Movement

TimeTime(a) Trend(a) Trend

TimeTime(d) Trend with seasonal pattern(d) Trend with seasonal pattern

TimeTime(c) Seasonal pattern(c) Seasonal pattern

TimeTime(b) Cycle(b) Cycle

Dem

and

Dem

and

Dem

and

Dem

and

Dem

and

Dem

and

Dem

and

Dem

and

Random Random movementmovement

Moving AverageMoving Average

MAMAnn = =

nn

ii = 1= 1 DDii

nnwherewhere

nn ==number of periods number of periods in the moving in the moving averageaverage

DDii ==demand in period demand in period ii

Average several Average several periods of dataperiods of data

Dampen, smooth out Dampen, smooth out changeschanges

Use when demand is Use when demand is stable with no trend stable with no trend or seasonal patternor seasonal pattern

JanJan 120120FebFeb 9090MarMar 100100AprApr 7575MayMay 110110JuneJune 5050JulyJuly 7575AugAug 130130SeptSept 110110OctOct 9090

ORDERSORDERSMONTHMONTH PER MONTHPER MONTH

MAMA33 = =

33

ii = 1= 1 DDii

33

== 90 + 110 + 13090 + 110 + 13033

= 110 orders for Nov= 110 orders for Nov

Simple Moving AverageSimple Moving Average

JanJan 120120 ––FebFeb 9090 – –MarMar 100100 – –AprApr 7575 103.3103.3MayMay 110110 88.388.3JuneJune 5050 95.095.0JulyJuly 7575 78.378.3AugAug 130130 78.378.3SeptSept 110110 85.085.0OctOct 9090 105.0105.0NovNov – – 110.0110.0

ORDERSORDERS THREE-MONTHTHREE-MONTHMONTHMONTH PER MONTHPER MONTH MOVING AVERAGEMOVING AVERAGE


JanJan 120120 ––FebFeb 9090 – –MarMar 100100 – –AprApr 7575 103.3103.3MayMay 110110 88.388.3JuneJune 5050 95.095.0JulyJuly 7575 78.378.3AugAug 130130 78.378.3SeptSept 110110 85.085.0OctOct 9090 105.0105.0NovNov – – 110.0110.0

ORDERSORDERS THREE-MONTHTHREE-MONTHMONTHMONTH PER MONTHPER MONTH MOVING AVERAGEMOVING AVERAGE

MAMA55 = =

55

ii = 1= 1 DDii

55

== 90 + 110 + 130 + 75 + 5090 + 110 + 130 + 75 + 5055

= 91 orders for Nov= 91 orders for Nov



JanJan 120120 –– – –FebFeb 9090 – – – –MarMar 100100 – – – –AprApr 7575 103.3103.3 – –MayMay 110110 88.388.3 – –JuneJune 5050 95.095.0 99.099.0JulyJuly 7575 78.378.3 85.085.0AugAug 130130 78.378.3 82.082.0SeptSept 110110 85.085.0 88.088.0OctOct 9090 105.0105.0 95.095.0NovNov – – 110.0110.0 91.091.0

ORDERSORDERS THREE-MONTHTHREE-MONTH FIVE-MONTHFIVE-MONTHMONTHMONTH PER MONTHPER MONTH MOVING AVERAGEMOVING AVERAGE MOVING AVERAGEMOVING AVERAGE

Smoothing EffectsSmoothing Effects150 150 –

125 125 –

100 100 –

75 75 –

50 50 –

25 25 –

0 0 –| | | | | | | | | | |

JanJan FebFeb MarMar AprApr MayMay JuneJune JulyJuly AugAug SeptSept OctOct NovNov

5-month5-month

3-month3-month

ActualActual

Ord

ers

Ord

ers

MonthMonth

Weighted Moving AverageWeighted Moving Average

WMAWMAnn = = ii = 1 = 1 WWii D Dii

wherewhere

WWii = the weight for period = the weight for period ii, , between 0 and 100 between 0 and 100 percentpercent

WWii = 1.00= 1.00

Adjusts Adjusts moving moving average average method to method to more closely more closely reflect data reflect data fluctuationsfluctuations

Weighted Moving Weighted Moving Average ExampleAverage Example

MONTH MONTH WEIGHT WEIGHT DATADATAAugustAugust 17%17% 130130SeptemberSeptember 33%33% 110110OctoberOctober 50%50% 9090

Weighted Moving Weighted Moving Average ExampleAverage Example

MONTH MONTH WEIGHT WEIGHT DATADATAAugustAugust 17%17% 130130SeptemberSeptember 33%33% 110110OctoberOctober 50%50% 9090

November forecastNovember forecast WMAWMA33 = = 33

ii = 1 = 1 WWii D Dii

= (0.50)(90) + (0.33)(110) + (0.17)(130)= (0.50)(90) + (0.33)(110) + (0.17)(130)

= 103.4 orders= 103.4 orders

FFt t +1 +1 = = DDtt + (1 - + (1 - ))FFtt

wherewhereFFt t +1+1 = =forecast for next forecast for next periodperiodDDtt ==actual demand actual demand for present periodfor present periodFFtt ==previously previously determined forecast determined forecast for present periodfor present period

== weighting factor, weighting factor, smoothing constantsmoothing constant

Averaging method Averaging method Weights most Weights most

recent data more recent data more stronglystrongly

Reacts more to Reacts more to recent changesrecent changes

Widely used, Widely used, accurate methodaccurate method

Exponential SmoothingExponential Smoothing

Effect of Smoothing Effect of Smoothing ConstantConstant0.0 0.0 1.0 1.0

If If = 0.20, then = 0.20, then FFt t +1 +1 = 0.20= 0.20DDtt + 0.80 + 0.80 FFtt

If If = 0, then = 0, then FFtt +1 +1 = 0= 0DDtt + 1 + 1 FFtt 0 = 0 = FFtt

Forecast does not reflect recent dataForecast does not reflect recent data

If If = 1, then = 1, then FFt t +1 +1 = 1= 1DDtt + 0 + 0 FFtt ==DDtt Forecast based only on most recent dataForecast based only on most recent data

PERIODPERIOD MONTHMONTH DEMANDDEMAND

11 JanJan 373722 FebFeb 404033 MarMar 414144 AprApr 373755 May May 454566 JunJun 505077 Jul Jul 434388 Aug Aug 474799 Sep Sep 5656

1010 OctOct 52521111 NovNov 55551212 Dec Dec 5454

FF22 = = DD11 + (1 - + (1 - ))FF11

= (0.30)(37) + (0.70)(37)= (0.30)(37) + (0.70)(37)= 37= 37

FF33 = = DD22 + (1 - + (1 - ))FF22

= (0.30)(40) + (0.70)(37)= (0.30)(40) + (0.70)(37)= 37.9= 37.9

FF1313 = = DD1212 + (1 - + (1 - ))FF1212

= (0.30)(54) + (0.70)(50.84)= (0.30)(54) + (0.70)(50.84)= 51.79= 51.79


FORECAST, FORECAST, FFtt + 1 + 1

PERIODPERIOD MONTHMONTH DEMANDDEMAND (( = 0.3) = 0.3)

11 JanJan 3737 ––22 FebFeb 4040 37.0037.0033 MarMar 4141 37.9037.9044 AprApr 3737 38.8338.8355 May May 4545 38.2838.2866 JunJun 5050 40.2940.2977 Jul Jul 4343 43.2043.2088 Aug Aug 4747 43.1443.1499 Sep Sep 5656 44.3044.30

1010 OctOct 5252 47.8147.811111 NovNov 5555 49.0649.061212 Dec Dec 5454 50.8450.841313 JanJan –– 51.7951.79


FORECAST, FORECAST, FFtt + 1 + 1

PERIODPERIOD MONTHMONTH DEMANDDEMAND (( = 0.3) = 0.3) (( = 0.5) = 0.5)

11 JanJan 3737 –– ––22 FebFeb 4040 37.0037.00 37.0037.0033 MarMar 4141 37.9037.90 38.5038.5044 AprApr 3737 38.8338.83 39.7539.7555 May May 4545 38.2838.28 38.3738.3766 JunJun 5050 40.2940.29 41.6841.6877 Jul Jul 4343 43.2043.20 45.8445.8488 Aug Aug 4747 43.1443.14 44.4244.4299 Sep Sep 5656 44.3044.30 45.7145.71

1010 OctOct 5252 47.8147.81 50.8550.851111 NovNov 5555 49.0649.06 51.4251.421212 Dec Dec 5454 50.8450.84 53.2153.211313 JanJan –– 51.7951.79 53.6153.61


70 70 –

60 60 –

50 50 –

40 40 –

30 30 –

20 20 –

1010 –

0 0 –| | | | | | | | | | | | |11 22 33 44 55 66 77 88 99 1010 1111 1212 1313

ActualActual

Ord

ers

Ord

ers

MonthMonth

Exponential Smoothing Exponential Smoothing ForecastsForecasts

70 70 –

60 60 –

50 50 –

40 40 –

30 30 –

20 20 –

1010 –

0 0 –| | | | | | | | | | | | |11 22 33 44 55 66 77 88 99 1010 1111 1212 1313

ActualActual

Ord

ers

Ord

ers

MonthMonth

= 0.30= 0.30


70 70 –

60 60 –

50 50 –

40 40 –

30 30 –

20 20 –

1010 –

0 0 –| | | | | | | | | | | | |11 22 33 44 55 66 77 88 99 1010 1111 1212 1313

= 0.50= 0.50ActualActual

Ord

ers

Ord

ers

MonthMonth

= 0.30= 0.30


Mean Absolute Mean Absolute Deviation (MAD)Deviation (MAD)

wherewhere tt = the period number= the period number

DDtt = demand in period = demand in period tt FFtt = the forecast for period = the forecast for period tt

nn = the total number of periods= the total number of periods= the absolute value= the absolute value

DDtt - - FFtt nnMAD =MAD =

MAD ExampleMAD Example11 3737 37.0037.0022 4040 37.0037.0033 4141 37.9037.9044 3737 38.8338.8355 4545 38.2838.2866 5050 40.2940.2977 4343 43.2043.2088 4747 43.1443.1499 5656 44.3044.30

1010 5252 47.8147.811111 5555 49.0649.061212 5454 50.8450.84

557557

PERIODPERIOD DEMAND, DEMAND, DDtt FFtt ( ( =0.3) =0.3)

MAD ExampleMAD Example11 3737 37.0037.00 –– ––22 4040 37.0037.00 3.003.00 3.003.0033 4141 37.9037.90 3.103.10 3.103.1044 3737 38.8338.83 -1.83-1.83 1.831.8355 4545 38.2838.28 6.726.72 6.726.7266 5050 40.2940.29 9.699.69 9.699.6977 4343 43.2043.20 -0.20-0.20 0.200.2088 4747 43.1443.14 3.863.86 3.863.8699 5656 44.3044.30 11.7011.70 11.7011.70

1010 5252 47.8147.81 4.194.19 4.194.191111 5555 49.0649.06 5.945.94 5.945.941212 5454 50.8450.84 3.153.15 3.153.15

557557 49.3149.31 53.3953.39

PERIODPERIOD DEMAND, DEMAND, DDtt FFtt ( ( =0.3) =0.3) ((DDtt - - FFtt)) | |DDtt - - FFtt||

MAD ExampleMAD Example11 3737 37.0037.00 –– ––22 4040 37.0037.00 3.003.00 3.003.0033 4141 37.9037.90 3.103.10 3.103.1044 3737 38.8338.83 -1.83-1.83 1.831.8355 4545 38.2838.28 6.726.72 6.726.7266 5050 40.2940.29 9.699.69 9.699.6977 4343 43.2043.20 -0.20-0.20 0.200.2088 4747 43.1443.14 3.863.86 3.863.8699 5656 44.3044.30 11.7011.70 11.7011.70

1010 5252 47.8147.81 4.194.19 4.194.191111 5555 49.0649.06 5.945.94 5.945.941212 5454 50.8450.84 3.153.15 3.153.15

557557 49.3149.31 53.3953.39

PERIODPERIOD DEMAND, DEMAND, DDtt FFtt ( ( =0.3) =0.3) ((DDtt - - FFtt)) | |DDtt - - FFtt||

Dt - Ft nMAD =

=

= 4.45

53.3912

Other Accuracy MeasuresOther Accuracy Measures

Average error = biasAverage error = bias

E =E =eett

nnMean squared error =Mean squared error =

E =E =ee22

tt

nn

yy = = aa + + bxbx

wherewhereaa == intercept (at period 0)intercept (at period 0)bb == slope of the lineslope of the linexx == the time periodthe time periodyy == forecast for demand for period forecast for demand for period xx

Linear Trend LineLinear Trend Line

yy = = aa + + bxbx

wherewhereaa == intercept (at period 0)intercept (at period 0)bb == slope of the lineslope of the linexx == the time periodthe time periodyy == forecast for demand for period forecast for demand for period xx

b =

a = y - b x

wheren = number of periods

x = = mean of the x values

y = = mean of the y values

xy - nxy

x2 - nx2

xn

yn

Linear Trend LineLinear Trend Line

xx(PERIOD)(PERIOD) yy(DEMAND)(DEMAND)

11 373722 404033 414144 373755 454566 505077 434388 474799 5656

1010 52521111 55551212 5454

7878 557557

Least Squares ExampleLeast Squares Example

xx(PERIOD)(PERIOD) yy(DEMAND)(DEMAND) xyxy xx22

11 3737 3737 1122 4040 8080 4433 4141 123123 9944 3737 148148 161655 4545 225225 252566 5050 300300 363677 4343 301301 494988 4747 376376 646499 5656 504504 8181

1010 5252 520520 1001001111 5555 605605 1211211212 5454 648648 144144

7878 557557 38673867 650650

Least Squares ExampleLeast Squares Example

Least Squares ExampleLeast Squares Examplexx(PERIOD)(PERIOD) yy(DEMAND)(DEMAND) xyxy xx22

11 3737 3737 1122 4040 8080 4433 4141 123123 9944 3737 148148 161655 4545 225225 252566 5050 300300 363677 4343 301301 494988 4747 376376 646499 5656 504504 8181

1010 5252 520520 1001001111 5555 605605 1211211212 5454 648648 144144

7878 557557 38673867 650650

x = = 6.5

y = = 46.42

b =

=

= 1.72a = y - bx

= 46.42 - (1.72)(6.5)= 35.2

3867 - (12)(6.5)(46.42)650 - 12(6.5)2

xy - nxyx2 - nx2

781255712


11 7373 3737 1122 4040 8080 4433 4141 123123 9944 3737 148148 161655 4545 225225 252566 5050 300300 363677 4343 301301 494988 4747 376376 646499 5656 504504 8181

1010 5252 520520 1001001111 5555 605605 1211211212 5454 648648 144144

7878 557557 38673867 650650

x = = 6.5

y = = 46.42

b =

=

= 1.72a = y - bx

= 46.42 - (1.72)(6.5)= 35.2

3867 - (12)(6.5)(46.42)650 - 12(6.5)2

xy - nxyx2 - nx2

781255712

Linear trend liney = 35.2 + 1.72x


11 7373 3737 1122 4040 8080 4433 4141 123123 9944 3737 148148 161655 4545 225225 252566 5050 300300 363677 4343 301301 494988 4747 376376 646499 5656 504504 8181

1010 5252 520520 1001001111 5555 605605 1211211212 5454 648648 144144

7878 557557 38673867 650650

Linear trend liney = 35.2 + 1.72x

Forecast for period 13y = 35.2 + 1.72(13)

y = 57.56 units

Linear Trend LineLinear Trend Line70 70 –

60 60 –

50 50 –

40 40 –

30 30 –

20 20 –

1010 –

0 0 –| | | | | | | | | | | | |11 22 33 44 55 66 77 88 99 1010 1111 1212 1313

ActualActual

Dem

and

Dem

and

PeriodPeriod

Linear Trend LineLinear Trend Line70 70 –

60 60 –

50 50 –

40 40 –

30 30 –

20 20 –

1010 –

0 0 –| | | | | | | | | | | | |11 22 33 44 55 66 77 88 99 1010 1111 1212 1313

ActualActual

Dem

and

Dem

and

PeriodPeriod

Linear trend lineLinear trend line

Seasonal AdjustmentsSeasonal Adjustments

Repetitive increase/ Repetitive increase/ decrease in demanddecrease in demand

Use seasonal factor Use seasonal factor to adjust forecastto adjust forecast

Seasonal factor = Seasonal factor = SSii = =DDii

DD

Seasonal AdjustmentSeasonal Adjustment

1999 12.61999 12.6 8.68.6 6.36.3 17.517.5 45.045.02000 14.12000 14.1 10.310.3 7.57.5 18.218.2 50.150.12001 15.32001 15.3 10.610.6 8.18.1 19.619.6 53.653.6Total 42.0Total 42.0 29.529.5 21.921.9 55.355.3 148.7148.7

DEMAND (1000’S PER QUARTER)DEMAND (1000’S PER QUARTER)YEARYEAR 11 22 33 44 TotalTotal


1999 12.61999 12.6 8.68.6 6.36.3 17.517.5 45.045.02000 14.12000 14.1 10.310.3 7.57.5 18.218.2 50.150.12001 15.32001 15.3 10.610.6 8.18.1 19.619.6 53.653.6Total 42.0Total 42.0 29.529.5 21.921.9 55.355.3 148.7148.7


SS11 = = = 0.28 = = = 0.28 DD11

DD42.042.0

148.7148.7

SS22 = = = 0.20 = = = 0.20 DD22

DD29.529.5

148.7148.7 SS44 = = = 0.37 = = = 0.37 DD44

DD55.355.3

148.7148.7

SS33 = = = 0.15 = = = 0.15 DD33

DD21.921.9

148.7148.7


1999 12.61999 12.6 8.68.6 6.36.3 17.517.5 45.045.02000 14.12000 14.1 10.310.3 7.57.5 18.218.2 50.150.12001 15.32001 15.3 10.610.6 8.18.1 19.619.6 53.653.6Total 42.0Total 42.0 29.529.5 21.921.9 55.355.3 148.7148.7


SSii 0.280.28 0.200.20 0.150.15 0.370.37


1999 12.61999 12.6 8.68.6 6.36.3 17.517.5 45.045.02000 14.12000 14.1 10.310.3 7.57.5 18.218.2 50.150.12001 15.32001 15.3 10.610.6 8.18.1 19.619.6 53.653.6Total 42.0Total 42.0 29.529.5 21.921.9 55.355.3 148.7148.7


SSii 0.280.28 0.200.20 0.150.15 0.370.37

yy = 40.97 + 4.30= 40.97 + 4.30xx= 40.97 + 4.30(4)= 40.97 + 4.30(4)= 58.17= 58.17

For 2002For 2002


SFSF11 = (= (SS11) () (FF44)) SFSF33 = (= (SS33) () (FF44) ) = (0.28)(58.17) = 16.28= (0.28)(58.17) = 16.28 = (0.15)(58.17) = 8.73= (0.15)(58.17) = 8.73

SFSF22 = (= (SS22) () (FF44)) SFSF44 = (= (SS44) () (FF44) ) = (0.20)(58.17) = 11.63= (0.20)(58.17) = 11.63 = (0.37)(58.17) = 21.53= (0.37)(58.17) = 21.53

1999 12.61999 12.6 8.68.6 6.36.3 17.517.5 45.045.02000 14.12000 14.1 10.310.3 7.57.5 18.218.2 50.150.12001 15.32001 15.3 10.610.6 8.18.1 19.619.6 53.653.6Total 42.0Total 42.0 29.529.5 21.921.9 55.355.3 148.7148.7


SSii 0.280.28 0.200.20 0.150.15 0.370.37

yy = 40.97 + 4.30= 40.97 + 4.30xx= 40.97 + 4.30(4)= 40.97 + 4.30(4)= 58.17= 58.17

For 2002For 2002

Causal Modeling with Causal Modeling with Linear RegressionLinear Regression

Study relationship between two Study relationship between two or more variablesor more variables

Dependent variable Dependent variable yy depends depends on independent variable on independent variable xx

yy = = aa + + bxbx

CorrelationCorrelationCorrelation, Correlation, rr

Measure of strength of relationshipMeasure of strength of relationship Varies between -1.00 and +1.00Varies between -1.00 and +1.00 1.00 => an increase in the 1.00 => an increase in the

independent variable results in a independent variable results in a linear increase in the dependentlinear increase in the dependent

-1.00 -1.00 => an increase in the independent => an increase in the independent variable results in a linear decrease in the variable results in a linear decrease in the dependentdependent

0.0 => there does not seem to be a linear 0.0 => there does not seem to be a linear relationship between therelationship between the

Coefficient of Coefficient of DeterminationDetermination

• Coefficient of determination, rCoefficient of determination, r22

– Percentage of variation in dependent Percentage of variation in dependent variable resulting from changes in the variable resulting from changes in the independent variableindependent variable

Documents

Chapter 4 Forecasting. Ch. 4: What is covered? Moving AverageMoving Average Weighted Moving AverageWeighted Moving Average Exponential SmoothingExponential