© 2008 prentice hall, inc.4 – 1 outline – continued time-series forecasting decomposition of...
Post on 26-Dec-2015
224 Views
Preview:
TRANSCRIPT
© 2008 Prentice Hall, Inc. 4 – 1
Outline – ContinuedOutline – Continued Time-Series ForecastingTime-Series Forecasting
Decomposition of a Time SeriesDecomposition of a Time Series Naive ApproachNaive Approach Moving AveragesMoving Averages Exponential SmoothingExponential Smoothing Exponential Smoothing with Trend Exponential Smoothing with Trend
AdjustmentAdjustment Trend ProjectionsTrend Projections Seasonal Variations in DataSeasonal Variations in Data Cyclical Variations in DataCyclical Variations in Data
© 2008 Prentice Hall, Inc. 4 – 2
Outline – ContinuedOutline – Continued
Associative Forecasting Methods: Associative Forecasting Methods: Regression and Correlation Regression and Correlation AnalysisAnalysis Using Regression Analysis for Using Regression Analysis for
ForecastingForecasting
Standard Error of the EstimateStandard Error of the Estimate
Correlation Coefficients for Correlation Coefficients for Regression LinesRegression Lines
Multiple-Regression AnalysisMultiple-Regression Analysis
© 2008 Prentice Hall, Inc. 4 – 3
Outline – ContinuedOutline – Continued
Monitoring and Controlling Monitoring and Controlling ForecastsForecasts Adaptive SmoothingAdaptive Smoothing
Focus ForecastingFocus Forecasting
Forecasting In The Service SectorForecasting In The Service Sector
© 2008 Prentice Hall, Inc. 4 – 4
Forecasting at Disney WorldForecasting at Disney World
Global portfolio includes parks in Hong Global portfolio includes parks in Hong Kong, Paris, Tokyo, Orlando, and Kong, Paris, Tokyo, Orlando, and AnaheimAnaheim
Revenues are derived from people – how Revenues are derived from people – how many visitors and how they spend their many visitors and how they spend their moneymoney
Daily management report contains only Daily management report contains only the forecast and actual attendance at the forecast and actual attendance at each parkeach park
© 2008 Prentice Hall, Inc. 4 – 5
Forecasting at Disney WorldForecasting at Disney World
Disney generates daily, weekly, monthly, Disney generates daily, weekly, monthly, annual, and 5-year forecastsannual, and 5-year forecasts
Forecast used by labor management, Forecast used by labor management, maintenance, operations, finance, and maintenance, operations, finance, and park schedulingpark scheduling
Forecast used to adjust opening times, Forecast used to adjust opening times, rides, shows, staffing levels, and guests rides, shows, staffing levels, and guests admittedadmitted
© 2008 Prentice Hall, Inc. 4 – 6
Forecasting at Disney WorldForecasting at Disney World
20% of customers come from outside the 20% of customers come from outside the USAUSA
Economic model includes gross Economic model includes gross domestic product, cross-exchange rates, domestic product, cross-exchange rates, arrivals into the USAarrivals into the USA
A staff of 35 analysts and 70 field people A staff of 35 analysts and 70 field people survey 1 million park guests, employees, survey 1 million park guests, employees, and travel professionals each yearand travel professionals each year
© 2008 Prentice Hall, Inc. 4 – 7
Forecasting at Disney WorldForecasting at Disney World
Inputs to the forecasting model include Inputs to the forecasting model include airline specials, Federal Reserve airline specials, Federal Reserve policies, Wall Street trends, policies, Wall Street trends, vacation/holiday schedules for 3,000 vacation/holiday schedules for 3,000 school districts around the worldschool districts around the world
Average forecast error for the 5-year Average forecast error for the 5-year forecast is 5%forecast is 5%
Average forecast error for annual Average forecast error for annual forecasts is between 0% and 3%forecasts is between 0% and 3%
© 2008 Prentice Hall, Inc. 4 – 8
What is Forecasting?What is Forecasting?
Process of Process of predicting a future predicting a future eventevent
Underlying basis of Underlying basis of
all business all business decisionsdecisions ProductionProduction
InventoryInventory
PersonnelPersonnel
FacilitiesFacilities
??
© 2008 Prentice Hall, Inc. 4 – 9
Short-range forecastShort-range forecast Up to 1 year, generally less than 3 monthsUp to 1 year, generally less than 3 months Purchasing, job scheduling, workforce Purchasing, job scheduling, workforce
levels, job assignments, production levelslevels, job assignments, production levels
Medium-range forecastMedium-range forecast 3 months to 3 years3 months to 3 years Sales and production planning, budgetingSales and production planning, budgeting
Long-range forecastLong-range forecast 33++ years years New product planning, facility location, New product planning, facility location,
research and developmentresearch and development
Forecasting Time HorizonsForecasting Time Horizons
© 2008 Prentice Hall, Inc. 4 – 10
Seven Steps in ForecastingSeven Steps in Forecasting
Determine the use of the forecastDetermine the use of the forecast
Select the items to be forecastedSelect the items to be forecasted
Determine the time horizon of the Determine the time horizon of the forecastforecast
Select the forecasting model(s)Select the forecasting model(s)
Gather the dataGather the data
Make the forecastMake the forecast
Validate and implement resultsValidate and implement results
© 2008 Prentice Hall, Inc. 4 – 11
The Realities!The Realities!
Forecasts are seldom perfectForecasts are seldom perfect
Most techniques assume an Most techniques assume an underlying stability in the systemunderlying stability in the system
Product family and aggregated Product family and aggregated forecasts are more accurate than forecasts are more accurate than individual product forecastsindividual product forecasts
© 2008 Prentice Hall, Inc. 4 – 12
Forecasting ApproachesForecasting Approaches
Used when situation is ‘stable’ and Used when situation is ‘stable’ and historical data existhistorical data exist Existing productsExisting products
Current technologyCurrent technology
Involves mathematical techniquesInvolves mathematical techniques e.g., forecasting sales of color e.g., forecasting sales of color
televisionstelevisions
Quantitative MethodsQuantitative Methods
© 2008 Prentice Hall, Inc. 4 – 13
Overview of Quantitative Overview of Quantitative ApproachesApproaches
1.1. Naive approachNaive approach
2.2. Moving averagesMoving averages
3.3. Exponential Exponential smoothingsmoothing
4.4. Trend projectionTrend projection
5.5. Linear regressionLinear regression
Time-Series Time-Series ModelsModels
Associative Associative ModelModel
© 2008 Prentice Hall, Inc. 4 – 14
Set of evenly spaced numerical dataSet of evenly spaced numerical data Obtained by observing response Obtained by observing response
variable at regular time periodsvariable at regular time periods
Forecast based only on past values, Forecast based only on past values, no other variables importantno other variables important Assumes that factors influencing Assumes that factors influencing
past and present will continue past and present will continue influence in futureinfluence in future
Time Series ForecastingTime Series Forecasting
© 2008 Prentice Hall, Inc. 4 – 15
Trend
Seasonal
Cyclical
Random
Time Series ComponentsTime Series Components
© 2008 Prentice Hall, Inc. 4 – 16
Components of DemandComponents of DemandD
eman
d f
or
pro
du
ct o
r se
rvic
e
| | | |1 2 3 4
Year
Average demand over four years
Seasonal peaks
Trend component
Actual demand
Random variation
Figure 4.1Figure 4.1
© 2008 Prentice Hall, Inc. 4 – 17
Persistent, overall upward or Persistent, overall upward or downward patterndownward pattern
Changes due to population, Changes due to population, technology, age, culture, etc.technology, age, culture, etc.
Typically several years Typically several years duration duration
Trend ComponentTrend Component
© 2008 Prentice Hall, Inc. 4 – 18
Regular pattern of up and Regular pattern of up and down fluctuationsdown fluctuations
Due to weather, customs, etc.Due to weather, customs, etc.
Occurs within a single year Occurs within a single year
Seasonal ComponentSeasonal Component
Number ofPeriod Length Seasons
Week Day 7Month Week 4-4.5Month Day 28-31Year Quarter 4Year Month 12Year Week 52
© 2008 Prentice Hall, Inc. 4 – 19
Repeating up and down movementsRepeating up and down movements
Affected by business cycle, political, Affected by business cycle, political, and economic factorsand economic factors
Multiple years durationMultiple years duration
Often causal or Often causal or associative associative relationshipsrelationships
Cyclical ComponentCyclical Component
00 55 1010 1515 2020
© 2008 Prentice Hall, Inc. 4 – 20
Erratic, unsystematic, ‘residual’ Erratic, unsystematic, ‘residual’ fluctuationsfluctuations
Due to random variation or Due to random variation or unforeseen eventsunforeseen events
Short duration and Short duration and nonrepeating nonrepeating
Random ComponentRandom Component
MM TT WW TT FF
© 2008 Prentice Hall, Inc. 4 – 21
Naive ApproachNaive Approach
Assumes demand in next Assumes demand in next period is the same as period is the same as demand in most recent perioddemand in most recent period e.g., If January sales were 68, then e.g., If January sales were 68, then
February sales will be 68February sales will be 68
Sometimes cost effective and Sometimes cost effective and efficientefficient
Can be good starting pointCan be good starting point
© 2008 Prentice Hall, Inc. 4 – 22
MA is a series of arithmetic means MA is a series of arithmetic means
Used if little or no trendUsed if little or no trend
Used often for smoothingUsed often for smoothingProvides overall impression of data Provides overall impression of data
over timeover time
Moving Average MethodMoving Average Method
Moving average =Moving average =∑∑ demand in previous n periodsdemand in previous n periods
nn
© 2008 Prentice Hall, Inc. 4 – 23
JanuaryJanuary 1010FebruaryFebruary 1212MarchMarch 1313AprilApril 1616MayMay 1919JuneJune 2323JulyJuly 2626
ActualActual 3-Month3-MonthMonthMonth Shed SalesShed Sales Moving AverageMoving Average
(12 + 13 + 16)/3 = 13 (12 + 13 + 16)/3 = 13 22//33
(13 + 16 + 19)/3 = 16(13 + 16 + 19)/3 = 16(16 + 19 + 23)/3 = 19 (16 + 19 + 23)/3 = 19 11//33
Moving Average ExampleMoving Average Example
101012121313
((1010 + + 1212 + + 1313)/3 = 11 )/3 = 11 22//33
© 2008 Prentice Hall, Inc. 4 – 24
Graph of Moving AverageGraph of Moving Average
| | | | | | | | | | | |
JJ FF MM AA MM JJ JJ AA SS OO NN DD
Sh
ed S
ales
Sh
ed S
ales
30 30 –28 28 –26 26 –24 24 –22 22 –20 20 –18 18 –16 16 –14 14 –12 12 –10 10 –
Actual Actual SalesSales
Moving Moving Average Average ForecastForecast
© 2008 Prentice Hall, Inc. 4 – 25
Increasing n smooths the forecast Increasing n smooths the forecast but makes it less sensitive to but makes it less sensitive to changeschanges
Do not forecast trends wellDo not forecast trends well
Require extensive historical dataRequire extensive historical data
Potential Problems WithPotential Problems With Moving Average Moving Average
© 2008 Prentice Hall, Inc. 4 – 26
Form of weighted moving averageForm of weighted moving average Weights decline exponentiallyWeights decline exponentially
Most recent data weighted mostMost recent data weighted most
Requires smoothing constant Requires smoothing constant (()) Ranges from 0 to 1Ranges from 0 to 1
Subjectively chosenSubjectively chosen
Involves little record keeping of past Involves little record keeping of past datadata
Exponential SmoothingExponential Smoothing
© 2008 Prentice Hall, Inc. 4 – 27
Exponential SmoothingExponential Smoothing
New forecast =New forecast = Last period’s forecastLast period’s forecast+ + ((Last period’s actual demand Last period’s actual demand
– – Last period’s forecastLast period’s forecast))
FFtt = F = Ft t – 1– 1 + + ((AAt t – 1– 1 - - F Ft t – 1– 1))
wherewhere FFtt == new forecastnew forecast
FFt t – 1– 1 == previous forecastprevious forecast
== smoothing (or weighting) smoothing (or weighting) constant constant (0 (0 ≤≤ ≤≤ 1) 1)
© 2008 Prentice Hall, Inc. 4 – 28
Exponential Smoothing Exponential Smoothing ExampleExample
Predicted demand Predicted demand = 142= 142 Ford Mustangs Ford MustangsActual demand Actual demand = 153= 153Smoothing constant Smoothing constant = .20 = .20
© 2008 Prentice Hall, Inc. 4 – 29
Exponential Smoothing Exponential Smoothing ExampleExample
Predicted demand Predicted demand = 142= 142 Ford Mustangs Ford MustangsActual demand Actual demand = 153= 153Smoothing constant Smoothing constant = .20 = .20
New forecastNew forecast = 142 + .2(153 – 142)= 142 + .2(153 – 142)
© 2008 Prentice Hall, Inc. 4 – 30
Exponential Smoothing Exponential Smoothing ExampleExample
Predicted demand Predicted demand = 142= 142 Ford Mustangs Ford MustangsActual demand Actual demand = 153= 153Smoothing constant Smoothing constant = .20 = .20
New forecastNew forecast = 142 + .2(153 – 142)= 142 + .2(153 – 142)
= 142 + 2.2= 142 + 2.2
= 144.2 ≈ 144 cars= 144.2 ≈ 144 cars
© 2008 Prentice Hall, Inc. 4 – 31
Effect ofEffect of Smoothing Constants Smoothing Constants
Weight Assigned toWeight Assigned to
MostMost 2nd Most2nd Most 3rd Most3rd Most 4th Most4th Most 5th Most5th MostRecentRecent RecentRecent RecentRecent RecentRecent RecentRecent
SmoothingSmoothing PeriodPeriod PeriodPeriod PeriodPeriod PeriodPeriod PeriodPeriodConstantConstant (()) (1 - (1 - )) (1 - (1 - ))22 (1 - (1 - ))33 (1 - (1 - ))44
= .1= .1 .1.1 .09.09 .081.081 .073.073 .066.066
= .5= .5 .5.5 .25.25 .125.125 .063.063 .031.031
© 2008 Prentice Hall, Inc. 4 – 32
Impact of Different Impact of Different
225 225 –
200 200 –
175 175 –
150 150 –| | | | | | | | |
11 22 33 44 55 66 77 88 99
QuarterQuarter
De
ma
nd
De
ma
nd
= .1= .1
Actual Actual demanddemand
= .5= .5
© 2008 Prentice Hall, Inc. 4 – 33
Impact of Different Impact of Different
225 225 –
200 200 –
175 175 –
150 150 –| | | | | | | | |
11 22 33 44 55 66 77 88 99
QuarterQuarter
De
ma
nd
De
ma
nd
= .1= .1
Actual Actual demanddemand
= .5= .5Chose high values of Chose high values of when underlying average when underlying average is likely to changeis likely to change
Choose low values of Choose low values of when underlying average when underlying average is stableis stable
© 2008 Prentice Hall, Inc. 4 – 34
Choosing Choosing
The objective is to obtain the most The objective is to obtain the most accurate forecast no matter the accurate forecast no matter the techniquetechnique
We generally do this by selecting the We generally do this by selecting the model that gives us the lowest forecast model that gives us the lowest forecast errorerror
Forecast errorForecast error = Actual demand - Forecast value= Actual demand - Forecast value
= A= Att - F - Ftt
© 2008 Prentice Hall, Inc. 4 – 35
Common Measures of ErrorCommon Measures of Error
Mean Absolute Deviation Mean Absolute Deviation ((MADMAD))
MAD =MAD =∑∑ |Actual - Forecast||Actual - Forecast|
nn
Mean Squared Error Mean Squared Error ((MSEMSE))
MSE =MSE =∑∑ ((Forecast ErrorsForecast Errors))22
nn
© 2008 Prentice Hall, Inc. 4 – 36
Common Measures of ErrorCommon Measures of Error
Mean Absolute Percent Error Mean Absolute Percent Error ((MAPEMAPE))
MAPE =MAPE =∑∑100100|Actual|Actualii - Forecast - Forecastii|/Actual|/Actualii
nn
nn
i i = 1= 1
© 2008 Prentice Hall, Inc. 4 – 37
Comparison of Forecast Comparison of Forecast Error Error
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviation
TonnageTonnage withwith forfor withwith forforQuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50
11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30
82.4582.45 98.6298.62
© 2008 Prentice Hall, Inc. 4 – 38
Comparison of Forecast Comparison of Forecast Error Error
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviation
TonnageTonnage withwith forfor withwith forforQuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50
11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30
82.4582.45 98.6298.62
MAD =∑ |deviations|
n
= 82.45/8 = 10.31
For = .10
= 98.62/8 = 12.33
For = .50
© 2008 Prentice Hall, Inc. 4 – 39
Comparison of Forecast Comparison of Forecast Error Error
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviation
TonnageTonnage withwith forfor withwith forforQuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50
11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30
82.4582.45 98.6298.62MADMAD 10.3110.31 12.3312.33
= 1,526.54/8 = 190.82
For = .10
= 1,561.91/8 = 195.24
For = .50
MSE =∑ (forecast errors)2
n
© 2008 Prentice Hall, Inc. 4 – 40
Comparison of Forecast Comparison of Forecast Error Error
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviation
TonnageTonnage withwith forfor withwith forforQuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50
11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30
82.4582.45 98.6298.62MADMAD 10.3110.31 12.3312.33MSEMSE 190.82190.82 195.24195.24
= 44.75/8 = 5.59%
For = .10
= 54.05/8 = 6.76%
For = .50
MAPE =∑100|deviationi|/actuali
n
n
i = 1
© 2008 Prentice Hall, Inc. 4 – 41
Comparison of Forecast Comparison of Forecast Error Error
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviation
TonnageTonnage withwith forfor withwith forforQuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50
11 180180 175175 5.005.00 175175 5.005.0022 168168 175.5175.5 7.507.50 177.50177.50 9.509.5033 159159 174.75174.75 15.7515.75 172.75172.75 13.7513.7544 175175 173.18173.18 1.821.82 165.88165.88 9.129.1255 190190 173.36173.36 16.6416.64 170.44170.44 19.5619.5666 205205 175.02175.02 29.9829.98 180.22180.22 24.7824.7877 180180 178.02178.02 1.981.98 192.61192.61 12.6112.6188 182182 178.22178.22 3.783.78 186.30186.30 4.304.30
82.4582.45 98.6298.62MADMAD 10.3110.31 12.3312.33MSEMSE 190.82190.82 195.24195.24MAPEMAPE 5.59%5.59% 6.76%6.76%
© 2008 Prentice Hall, Inc. 4 – 42
Trend ProjectionsTrend Projections
Fitting a trend line to historical data points Fitting a trend line to historical data points to project into the medium to long-rangeto project into the medium to long-range
Linear trends can be found using the least Linear trends can be found using the least squares techniquesquares technique
y y = = a a + + bxbx^̂
where ywhere y= computed value of the = computed value of the variable to be predicted (dependent variable to be predicted (dependent variable)variable)aa= y-axis intercept= y-axis interceptbb= slope of the regression line= slope of the regression linexx= the independent variable= the independent variable
^̂
© 2008 Prentice Hall, Inc. 4 – 43
Least Squares MethodLeast Squares Method
Time periodTime period
Va
lue
s o
f D
ep
end
en
t V
ari
able
Figure 4.4Figure 4.4
DeviationDeviation11
(error)(error)
DeviationDeviation55
DeviationDeviation77
DeviationDeviation22
DeviationDeviation66
DeviationDeviation44
DeviationDeviation33
Actual observation Actual observation (y value)(y value)
Trend line, y = a + bxTrend line, y = a + bx^̂
© 2008 Prentice Hall, Inc. 4 – 44
Least Squares MethodLeast Squares Method
Time periodTime period
Va
lue
s o
f D
ep
end
en
t V
ari
able
Figure 4.4Figure 4.4
DeviationDeviation11
DeviationDeviation55
DeviationDeviation77
DeviationDeviation22
DeviationDeviation66
DeviationDeviation44
DeviationDeviation33
Actual observation Actual observation (y value)(y value)
Trend line, y = a + bxTrend line, y = a + bx^̂
Least squares method minimizes the sum of the
squared errors (deviations)
© 2008 Prentice Hall, Inc. 4 – 45
Least Squares MethodLeast Squares Method
Equations to calculate the regression variablesEquations to calculate the regression variables
b =b =xy - nxyxy - nxy
xx22 - nx - nx22
y y = = a a + + bxbx^̂
a = y - bxa = y - bx
© 2008 Prentice Hall, Inc. 4 – 46
Least Squares ExampleLeast Squares Example
b b = = = 10.54= = = 10.54∑∑xy - nxyxy - nxy
∑∑xx22 - nx - nx22
3,063 - (7)(4)(98.86)3,063 - (7)(4)(98.86)
140 - (7)(4140 - (7)(422))
aa = = yy - - bxbx = 98.86 - 10.54(4) = 56.70 = 98.86 - 10.54(4) = 56.70
TimeTime Electrical Power Electrical Power YearYear Period (x)Period (x) DemandDemand xx22 xyxy
20012001 11 7474 11 747420022002 22 7979 44 15815820032003 33 8080 99 24024020042004 44 9090 1616 36036020052005 55 105105 2525 52552520052005 66 142142 3636 85285220072007 77 122122 4949 854854
∑∑xx = 28 = 28 ∑∑yy = 692 = 692 ∑∑xx22 = 140 = 140 ∑∑xyxy = 3,063 = 3,063xx = 4 = 4 yy = 98.86 = 98.86
© 2008 Prentice Hall, Inc. 4 – 47
Least Squares ExampleLeast Squares Example
b b = = = 10.54= = = 10.54xy - nxyxy - nxy
xx22 - nx - nx22
3,063 - (7)(4)(98.86)3,063 - (7)(4)(98.86)
140 - (7)(4140 - (7)(422))
aa = = yy - - bxbx = 98.86 - 10.54(4) = 56.70 = 98.86 - 10.54(4) = 56.70
TimeTime Electrical Power Electrical Power YearYear Period (x)Period (x) DemandDemand xx22 xyxy
19991999 11 7474 11 747420002000 22 7979 44 15815820012001 33 8080 99 24024020022002 44 9090 1616 36036020032003 55 105105 2525 52552520042004 66 142142 3636 85285220052005 77 122122 4949 854854
xx = 28 = 28 yy = 692 = 692 xx22 = 140 = 140 xyxy = 3,063 = 3,063xx = 4 = 4 yy = 98.86 = 98.86
The trend line is
y = 56.70 + 10.54x^
© 2008 Prentice Hall, Inc. 4 – 48
Least Squares ExampleLeast Squares Example
| | | | | | | | |20012001 20022002 20032003 20042004 20052005 20062006 20072007 20082008 20092009
160 160 –
150 150 –
140 140 –
130 130 –
120 120 –
110 110 –
100 100 –
90 90 –
80 80 –
70 70 –
60 60 –
50 50 –
YearYear
Po
wer
dem
and
Po
wer
dem
and
Trend line,Trend line,y y = 56.70 + 10.54x= 56.70 + 10.54x^̂
© 2008 Prentice Hall, Inc. 4 – 50
Seasonal Variations In DataSeasonal Variations In Data
The multiplicative The multiplicative seasonal model seasonal model can adjust trend can adjust trend data for seasonal data for seasonal variations in variations in demanddemand
© 2008 Prentice Hall, Inc. 4 – 51
Seasonal Variations In DataSeasonal Variations In Data
1.1. Find average historical demand for each Find average historical demand for each season season
2.2. Compute the average demand over all Compute the average demand over all seasons seasons
3.3. Compute a seasonal index for each season Compute a seasonal index for each season
4.4. Estimate next year’s total demandEstimate next year’s total demand
5.5. Divide this estimate of total demand by the Divide this estimate of total demand by the number of seasons, then multiply it by the number of seasons, then multiply it by the seasonal index for that seasonseasonal index for that season
Steps in the process:Steps in the process:
© 2008 Prentice Hall, Inc. 4 – 52
Seasonal Index ExampleSeasonal Index Example
JanJan 8080 8585 105105 9090 9494
FebFeb 7070 8585 8585 8080 9494
MarMar 8080 9393 8282 8585 9494
AprApr 9090 9595 115115 100100 9494
MayMay 113113 125125 131131 123123 9494
JunJun 110110 115115 120120 115115 9494
JulJul 100100 102102 113113 105105 9494
AugAug 8888 102102 110110 100100 9494
SeptSept 8585 9090 9595 9090 9494
OctOct 7777 7878 8585 8080 9494
NovNov 7575 7272 8383 8080 9494
DecDec 8282 7878 8080 8080 9494
DemandDemand AverageAverage AverageAverage Seasonal Seasonal MonthMonth 20052005 20062006 20072007 2005-20072005-2007 MonthlyMonthly IndexIndex
© 2008 Prentice Hall, Inc. 4 – 53
Seasonal Index ExampleSeasonal Index Example
JanJan 8080 8585 105105 9090 9494
FebFeb 7070 8585 8585 8080 9494
MarMar 8080 9393 8282 8585 9494
AprApr 9090 9595 115115 100100 9494
MayMay 113113 125125 131131 123123 9494
JunJun 110110 115115 120120 115115 9494
JulJul 100100 102102 113113 105105 9494
AugAug 8888 102102 110110 100100 9494
SeptSept 8585 9090 9595 9090 9494
OctOct 7777 7878 8585 8080 9494
NovNov 7575 7272 8383 8080 9494
DecDec 8282 7878 8080 8080 9494
DemandDemand AverageAverage AverageAverage Seasonal Seasonal MonthMonth 20052005 20062006 20072007 2005-20072005-2007 MonthlyMonthly IndexIndex
0.9570.957
Seasonal index = average 2005-2007 monthly demand
average monthly demand
= 90/94 = .957
© 2008 Prentice Hall, Inc. 4 – 54
Seasonal Index ExampleSeasonal Index Example
JanJan 8080 8585 105105 9090 9494 0.9570.957
FebFeb 7070 8585 8585 8080 9494 0.8510.851
MarMar 8080 9393 8282 8585 9494 0.9040.904
AprApr 9090 9595 115115 100100 9494 1.0641.064
MayMay 113113 125125 131131 123123 9494 1.3091.309
JunJun 110110 115115 120120 115115 9494 1.2231.223
JulJul 100100 102102 113113 105105 9494 1.1171.117
AugAug 8888 102102 110110 100100 9494 1.0641.064
SeptSept 8585 9090 9595 9090 9494 0.9570.957
OctOct 7777 7878 8585 8080 9494 0.8510.851
NovNov 7575 7272 8383 8080 9494 0.8510.851
DecDec 8282 7878 8080 8080 9494 0.8510.851
DemandDemand AverageAverage AverageAverage Seasonal Seasonal MonthMonth 20052005 20062006 20072007 2005-20072005-2007 MonthlyMonthly IndexIndex
© 2008 Prentice Hall, Inc. 4 – 55
Seasonal Index ExampleSeasonal Index Example
JanJan 8080 8585 105105 9090 9494 0.9570.957
FebFeb 7070 8585 8585 8080 9494 0.8510.851
MarMar 8080 9393 8282 8585 9494 0.9040.904
AprApr 9090 9595 115115 100100 9494 1.0641.064
MayMay 113113 125125 131131 123123 9494 1.3091.309
JunJun 110110 115115 120120 115115 9494 1.2231.223
JulJul 100100 102102 113113 105105 9494 1.1171.117
AugAug 8888 102102 110110 100100 9494 1.0641.064
SeptSept 8585 9090 9595 9090 9494 0.9570.957
OctOct 7777 7878 8585 8080 9494 0.8510.851
NovNov 7575 7272 8383 8080 9494 0.8510.851
DecDec 8282 7878 8080 8080 9494 0.8510.851
DemandDemand AverageAverage AverageAverage Seasonal Seasonal MonthMonth 20052005 20062006 20072007 2005-20072005-2007 MonthlyMonthly IndexIndex
Expected annual demand = 1,200
Jan x .957 = 961,200
12
Feb x .851 = 851,200
12
Forecast for 2008
© 2008 Prentice Hall, Inc. 4 – 56
Seasonal Index ExampleSeasonal Index Example
140 140 –
130 130 –
120 120 –
110 110 –
100 100 –
90 90 –
80 80 –
70 70 –| | | | | | | | | | | |
JJ FF MM AA MM JJ JJ AA SS OO NN DD
TimeTime
Dem
and
Dem
and
2008 Forecast2008 Forecast
2007 Demand 2007 Demand
2006 Demand2006 Demand
2005 Demand2005 Demand
© 2008 Prentice Hall, Inc. 4 – 57
Associative ForecastingAssociative Forecasting
Used when changes in one or more Used when changes in one or more independent variables can be used to predict independent variables can be used to predict
the changes in the dependent variablethe changes in the dependent variable
Most common technique is linear Most common technique is linear regression analysisregression analysis
We apply this technique just as we did We apply this technique just as we did in the time series examplein the time series example
© 2008 Prentice Hall, Inc. 4 – 58
Associative ForecastingAssociative Forecasting
Forecasting an outcome based on predictor Forecasting an outcome based on predictor variables using the least squares techniquevariables using the least squares technique
y y = = a a + + bxbx^̂
where ywhere y= computed value of the = computed value of the variable to be predicted (dependent variable to be predicted (dependent variable)variable)aa= y-axis intercept= y-axis interceptbb= slope of the regression line= slope of the regression linexx= the independent variable though = the independent variable though to predict the value of the to predict the value of the dependent variabledependent variable
^̂
© 2008 Prentice Hall, Inc. 4 – 59
Associative Forecasting Associative Forecasting ExampleExample
SalesSales Local PayrollLocal Payroll($ millions), y($ millions), y ($ billions), x($ billions), x
2.02.0 113.03.0 332.52.5 442.02.0 222.02.0 113.53.5 77
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |0 1 2 3 4 5 6 7
Sal
es
Area payroll
© 2008 Prentice Hall, Inc. 4 – 60
Associative Forecasting Associative Forecasting ExampleExample
Sales, y Payroll, x x2 xy
2.0 1 1 2.03.0 3 9 9.02.5 4 16 10.02.0 2 4 4.02.0 1 1 2.03.5 7 49 24.5
∑y = 15.0 ∑x = 18 ∑x2 = 80 ∑xy = 51.5
xx = = ∑∑xx/6 = 18/6 = 3/6 = 18/6 = 3
yy = = ∑∑yy/6 = 15/6 = 2.5/6 = 15/6 = 2.5
bb = = = .25 = = = .25∑∑xy - nxyxy - nxy
∑∑xx22 - nx - nx22
51.5 - (6)(3)(2.5)51.5 - (6)(3)(2.5)
80 - (6)(380 - (6)(322))
aa = = yy - - bbx = 2.5 - (.25)(3) = 1.75x = 2.5 - (.25)(3) = 1.75
© 2008 Prentice Hall, Inc. 4 – 61
Associative Forecasting Associative Forecasting ExampleExample
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |0 1 2 3 4 5 6 7
Sal
es
Area payroll
y y = 1.75 + .25= 1.75 + .25xx^̂ Sales Sales = 1.75 + .25(= 1.75 + .25(payrollpayroll))
If payroll next year If payroll next year is estimated to be is estimated to be $6$6 billion, then: billion, then:
Sales Sales = 1.75 + .25(6)= 1.75 + .25(6)SalesSales = $3,250,000 = $3,250,000
3.25
© 2008 Prentice Hall, Inc. 4 – 62
Standard Error of the Standard Error of the EstimateEstimate
A forecast is just a point estimate of a A forecast is just a point estimate of a future valuefuture value
This point is This point is actually the actually the mean of a mean of a probability probability distributiondistribution
Figure 4.9Figure 4.9
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |0 1 2 3 4 5 6 7
Sal
es
Area payroll
3.25
© 2008 Prentice Hall, Inc. 4 – 63
Standard Error of the Standard Error of the EstimateEstimate
wherewhere yy == y-value of each data y-value of each data pointpoint
yycc == computed value of the computed value of the dependent variable, from the dependent variable, from the regression equationregression equation
nn == number of data pointsnumber of data points
SSy,xy,x = =∑∑((y - yy - ycc))22
n n - 2- 2
© 2008 Prentice Hall, Inc. 4 – 64
Standard Error of the Standard Error of the EstimateEstimate
Computationally, this equation is Computationally, this equation is considerably easier to useconsiderably easier to use
We use the standard error to set up We use the standard error to set up prediction intervals around the prediction intervals around the
point estimatepoint estimate
SSy,xy,x = =∑∑yy22 - a - a∑∑y - by - b∑∑xyxy
n n - 2- 2
© 2008 Prentice Hall, Inc. 4 – 65
Standard Error of the Standard Error of the EstimateEstimate
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |0 1 2 3 4 5 6 7
Sal
es
Area payroll
3.25
SSy,xy,x = = = =∑∑yy22 - a - a∑∑y - by - b∑∑xyxyn n - 2- 2
39.5 - 1.75(15) - .25(51.5)39.5 - 1.75(15) - .25(51.5)6 - 26 - 2
SSy,xy,x = = .306.306
The standard error The standard error of the estimate is of the estimate is $306,000$306,000 in sales in sales
© 2008 Prentice Hall, Inc. 4 – 66
How strong is the linear How strong is the linear relationship between the relationship between the variables?variables?
Correlation does not necessarily Correlation does not necessarily imply causality!imply causality!
Coefficient of correlation, r, Coefficient of correlation, r, measures degree of associationmeasures degree of association Values range from Values range from -1-1 to to +1+1
CorrelationCorrelation
© 2008 Prentice Hall, Inc. 4 – 67
Correlation CoefficientCorrelation Coefficient
r = r = nnxyxy - - xxy y
[[nnxx22 - ( - (xx))22][][nnyy22 - ( - (yy))22]]
© 2008 Prentice Hall, Inc. 4 – 68
Correlation CoefficientCorrelation Coefficient
r = r = nnxyxy - - xxy y
[[nnxx22 - ( - (xx))22][][nnyy22 - ( - (yy))22]]
y
x(a) Perfect positive correlation: r = +1
y
x(b) Positive correlation: 0 < r < 1
y
x(c) No correlation: r = 0
y
x(d) Perfect negative correlation: r = -1
© 2008 Prentice Hall, Inc. 4 – 69
Coefficient of Determination, rCoefficient of Determination, r22, , measures the percent of change in measures the percent of change in y predicted by the change in xy predicted by the change in x Values range from Values range from 00 to to 11
Easy to interpretEasy to interpret
CorrelationCorrelation
For the Nodel Construction example:For the Nodel Construction example:
r r = .901= .901
rr22 = .81 = .81
© 2008 Prentice Hall, Inc. 4 – 70
Multiple Regression Multiple Regression AnalysisAnalysis
If more than one independent variable is to be If more than one independent variable is to be used in the model, linear regression can be used in the model, linear regression can be
extended to multiple regression to extended to multiple regression to accommodate several independent variablesaccommodate several independent variables
y y = = a a + + bb11xx11 + b + b22xx22 … …^̂
Computationally, this is quite Computationally, this is quite complex and generally done on the complex and generally done on the
computercomputer
© 2008 Prentice Hall, Inc. 4 – 71
Multiple Regression Multiple Regression AnalysisAnalysis
y y = 1.80 + .30= 1.80 + .30xx11 - 5.0 - 5.0xx22^̂
In the Nodel example, including interest rates in In the Nodel example, including interest rates in the model gives the new equation:the model gives the new equation:
An improved correlation coefficient of r An improved correlation coefficient of r = .96= .96 means this model does a better job of predicting means this model does a better job of predicting the change in construction salesthe change in construction sales
Sales Sales = 1.80 + .30(6) - 5.0(.12) = 3.00= 1.80 + .30(6) - 5.0(.12) = 3.00Sales Sales = $3,000,000= $3,000,000
© 2008 Prentice Hall, Inc. 4 – 72
Measures how well the forecast is Measures how well the forecast is predicting actual valuespredicting actual values
Ratio of running sum of forecast errors Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)(RSFE) to mean absolute deviation (MAD) Good tracking signal has low valuesGood tracking signal has low values
If forecasts are continually high or low, the If forecasts are continually high or low, the forecast has a bias errorforecast has a bias error
Monitoring and Controlling Monitoring and Controlling ForecastsForecasts
Tracking SignalTracking Signal
© 2008 Prentice Hall, Inc. 4 – 73
Monitoring and Controlling Monitoring and Controlling ForecastsForecasts
Tracking Tracking signalsignal
RSFERSFEMADMAD==
Tracking Tracking signalsignal ==
∑∑(Actual demand in (Actual demand in period i - period i -
Forecast demand Forecast demand in period i)in period i)
∑∑|Actual - Forecast|/n|Actual - Forecast|/n))
© 2008 Prentice Hall, Inc. 4 – 74
Tracking SignalTracking Signal
Tracking signalTracking signal
++
00 MADs MADs
––
Upper control limitUpper control limit
Lower control limitLower control limit
TimeTime
Signal exceeding limitSignal exceeding limit
Acceptable Acceptable rangerange
top related