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4 – 1
Operations ManagementOperations ManagementChapter 4 - ForecastingChapter 4 - Forecasting
© 2006 Prentice Hall, Inc.
PowerPoint presentation to accompanyPowerPoint presentation to accompany Heizer/Render Heizer/Render Principles of Operations Management, 6ePrinciples of Operations Management, 6eOperations Management, 8e Operations Management, 8e
4 – 2
OutlineOutline
What Is Forecasting?What Is Forecasting? Forecasting Time HorizonsForecasting Time Horizons
Types Of ForecastsTypes Of Forecasts
The Strategic Importance Of The Strategic Importance Of ForecastingForecasting
Seven Steps In The Forecasting Seven Steps In The Forecasting SystemSystem
4 – 3
Outline – ContinuedOutline – Continued
Forecasting ApproachesForecasting Approaches Overview of Qualitative MethodsOverview of Qualitative Methods
Overview of Quantitative MethodsOverview of Quantitative Methods
Time-series ForecastingTime-series Forecasting
Forecasting In The Service SectorForecasting In The Service Sector
4 – 4
What is Forecasting?What is Forecasting?
Process of trying to Process of trying to predict a future eventpredict a future event
Underlying basis of Underlying basis of all business all business decisionsdecisions ProductionProduction
InventoryInventory
PersonnelPersonnel
FacilitiesFacilities
??
4 – 5
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 levels, Purchasing, job scheduling, workforce levels,
job assignments, production levelsjob 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, research New product planning, facility location, research
and developmentand development
Forecasting Time HorizonsForecasting Time Horizons
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Distinguishing DifferencesDistinguishing Differences
Medium/long rangeMedium/long range forecasts deal with more forecasts deal with more comprehensive issues and support comprehensive issues and support management decisions regarding planning management decisions regarding planning and products, plants and processesand products, plants and processes
Short-termShort-term forecasting usually employs forecasting usually employs different methodologies than longer-term different methodologies than longer-term forecastingforecasting
Short-termShort-term forecasts tend to be more forecasts tend to be more accurate than longer-term forecastsaccurate than longer-term forecasts
4 – 7
Types of ForecastsTypes of Forecasts
Economic forecastsEconomic forecasts Address business cycle – inflation rate, money Address business cycle – inflation rate, money
supply, housing starts, etc.supply, housing starts, etc.
Technological forecastsTechnological forecasts Predict rate of technological progressPredict rate of technological progress
Impacts development of new productsImpacts development of new products
Demand forecastsDemand forecasts Predict sales of existing productPredict sales of existing product
4 – 8
Strategic Importance of Strategic Importance of ForecastingForecasting
Human Resources – Hiring, training, Human Resources – Hiring, training, laying off workerslaying off workers
Capacity – Capacity shortages can result Capacity – Capacity shortages can result in undependable delivery, loss of in undependable delivery, loss of customers, loss of market sharecustomers, loss of market share
Supply-Chain Management – Good Supply-Chain Management – Good supplier relations and price advancesupplier relations and price advance
4 – 9
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
4 – 10
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 generally more accurate forecasts are generally more accurate than individual product forecaststhan individual product forecasts
4 – 11
Forecasting ApproachesForecasting Approaches
Used when situation is vague and Used when situation is vague and little data existlittle data exist New productsNew products
New technologyNew technology
Involves intuition, experienceInvolves intuition, experience e.g., forecasting demand for a new e.g., forecasting demand for a new
productproduct
Qualitative MethodsQualitative Methods
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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
4 – 13
Overview of Qualitative MethodsOverview of Qualitative Methods
Jury of executive opinionJury of executive opinion Pool opinions of high-level executives, Pool opinions of high-level executives,
sometimes augment by statistical sometimes augment by statistical modelsmodels
Delphi methodDelphi method Panel of experts, queried iterativelyPanel of experts, queried iteratively
4 – 14
Overview of Qualitative MethodsOverview of Qualitative Methods
Sales force compositeSales force composite Estimates from individual salespersons Estimates from individual salespersons
are reviewed for reasonableness, then are reviewed for reasonableness, then aggregated aggregated
Consumer Market SurveyConsumer Market Survey Ask the customerAsk the customer
4 – 15
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
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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 valuesForecast based only on past values Assumes that factors influencing past Assumes that factors influencing past
and present will continue influence in and present will continue influence in futurefuture
Relates the forecast to only one Relates the forecast to only one factor – time factor – time
Time Series ForecastingTime Series Forecasting
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Trend
Seasonal
Cyclical
Random
Time Series ComponentsTime Series Components
4 – 18
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
4 – 19
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 duration Typically several years duration
Trend ComponentTrend Component
4 – 20
Regular pattern of up and down Regular pattern of up and down fluctuationsfluctuations
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
4 – 21
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
4 – 22
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
4 – 23
Naive ApproachNaive Approach
Assumes demand in next period is Assumes demand in next period is the same as demand in most recent the same as demand in most recent periodperiod e.g., If May sales were 48, then June e.g., If May sales were 48, then June
sales will be 48sales will be 48
Sometimes cost effective and Sometimes cost effective and efficientefficient
4 – 24
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
4 – 25
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
4 – 26
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
4 – 27
Used when trend is present Used when trend is present Older data usually less importantOlder data usually less important
Weights based on experience and Weights based on experience and intuitionintuition
Weighted Moving AverageWeighted Moving Average
WeightedWeightedmoving averagemoving average ==
∑∑ ((weight for period nweight for period n)) x x ((demand in period ndemand in period n))
∑∑ weightsweights
4 – 28
JanuaryJanuary 1010FebruaryFebruary 1212MarchMarch 1313AprilApril 1616MayMay 1919JuneJune 2323JulyJuly 2626
ActualActual 3-Month Weighted3-Month WeightedMonthMonth Shed SalesShed Sales Moving AverageMoving Average
[(3 x 16) + (2 x 13) + (12)]/6 = 14[(3 x 16) + (2 x 13) + (12)]/6 = 1411//33
[(3 x 19) + (2 x 16) + (13)]/6 = 17[(3 x 19) + (2 x 16) + (13)]/6 = 17[(3 x 23) + (2 x 19) + (16)]/6 = 20[(3 x 23) + (2 x 19) + (16)]/6 = 2011//22
Weighted Moving AverageWeighted Moving Average
101012121313
[(3 x [(3 x 1313) + (2 x ) + (2 x 1212) + () + (1010)]/6 = 12)]/6 = 1211//66
Weights Applied Period
3 Last month2 Two months ago1 Three months ago6 Sum of weights
4 – 29
Increasing Increasing nn smooths the forecast but smooths the forecast but makes it less sensitive to changesmakes it less sensitive to changes
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
4 – 30
Moving Average And Moving Average And Weighted Moving AverageWeighted Moving Average
30 30 –
25 25 –
20 20 –
15 15 –
10 10 –
5 5 –
Sa
les
de
man
dS
ale
s d
em
and
| | | | | | | | | | | |
JJ FF MM AA MM JJ JJ AA SS OO NN DD
Actual Actual salessales
Moving Moving averageaverage
Weighted Weighted moving moving averageaverage
Figure 4.2Figure 4.2
4 – 31
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
4 – 32
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)
4 – 33
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
4 – 34
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)
4 – 35
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
4 – 36
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
4 – 37
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
4 – 38
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
4 – 39
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
4 – 40
Common Measures of ErrorCommon Measures of Error
Mean Absolute Percent Error Mean Absolute Percent Error ((MAPEMAPE))
MAPE =MAPE =100 100 ∑∑ |actual |actualii - forecast - forecastii|/actual|/actualii
nn
nn
i i = 1= 1
4 – 41
Comparison of Forecast Error Comparison of Forecast 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 55 175175 5522 168168 176176 88 178178 101033 159159 175175 1616 173173 141444 175175 173173 22 166166 9955 190190 173173 1717 170170 202066 205205 175175 3030 180180 252577 180180 178178 22 193193 131388 182182 178178 44 186186 44
8484 100100
4 – 42
Comparison of Forecast Error Comparison of Forecast Error
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviationTonageTonage withwith forfor withwith forfor
QuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50
11 180180 175175 55 175175 5522 168168 176176 88 178178 101033 159159 175175 1616 173173 141444 175175 173173 22 166166 9955 190190 173173 1717 170170 202066 205205 175175 3030 180180 252577 180180 178178 22 193193 131388 182182 178178 44 186186 44
8484 100100
MAD =∑ |deviations|
n
= 84/8 = 10.50
For = .10
= 100/8 = 12.50
For = .50
4 – 43
Comparison of Forecast Error Comparison of Forecast Error
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviationTonageTonage withwith forfor withwith forfor
QuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50
11 180180 175175 55 175175 5522 168168 176176 88 178178 101033 159159 175175 1616 173173 141444 175175 173173 22 166166 9955 190190 173173 1717 170170 202066 205205 175175 3030 180180 252577 180180 178178 22 193193 131388 182182 178178 44 186186 44
8484 100100MADMAD 10.5010.50 12.5012.50
= 1,558/8 = 194.75
For = .10
= 1,612/8 = 201.50
For = .50
MSE =∑ (forecast errors)2
n
4 – 44
Comparison of Forecast Error Comparison of Forecast Error
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsoluteActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviationTonageTonage withwith forfor withwith forfor
QuarterQuarter UnloadedUnloaded = .10 = .10 = .10 = .10 = .50 = .50 = .50 = .50
11 180180 175175 55 175175 5522 168168 176176 88 178178 101033 159159 175175 1616 173173 141444 175175 173173 22 166166 9955 190190 173173 1717 170170 202066 205205 175175 3030 180180 252577 180180 178178 22 193193 131388 182182 178178 44 186186 44
8484 100100MADMAD 10.5010.50 12.5012.50MSEMSE 194.75194.75 201.50201.50
= 45.62/8 = 5.70%
For = .10
= 54.8/8 = 6.85%
For = .50
MAPE =100 ∑ |deviationi|/actuali
n
n
i = 1
4 – 45
Comparison of Forecast Error Comparison of Forecast 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 55 175175 5522 168168 176176 88 178178 101033 159159 175175 1616 173173 141444 175175 173173 22 166166 9955 190190 173173 1717 170170 202066 205205 175175 3030 180180 252577 180180 178178 22 193193 131388 182182 178178 44 186186 44
8484 100100MADMAD 10.5010.50 12.5012.50MSEMSE 194.75194.75 201.50201.50
MAPEMAPE 5.70%5.70% 6.85%6.85%
4 – 46
Trend ProjectionsTrend Projections
Fitting a trend line to historical data points to Fitting a trend line to historical data points to project into the medium-to-long-rangeproject 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 where yy= 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
^̂
4 – 47
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^̂
4 – 48
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)
4 – 49
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
4 – 50
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
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
4 – 51
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^
4 – 52
Least Squares ExampleLeast Squares Example
| | | | | | | | |19991999 20002000 20012001 20022002 20032003 20042004 20052005 20062006 20072007
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^̂
4 – 54
Forecasting in the Service Forecasting in the Service SectorSector
Presents unusual challengesPresents unusual challenges Special need for short term recordsSpecial need for short term records
Needs differ greatly as function of Needs differ greatly as function of industry and productindustry and product
Holidays and other calendar eventsHolidays and other calendar events
Unusual eventsUnusual events
4 – 55
Fast Food Restaurant Fast Food Restaurant ForecastForecast
20% 20% –
15% 15% –
10% 10% –
5% 5% –
11-1211-12 1-21-2 3-43-4 5-65-6 7-87-8 9-109-1012-112-1 2-32-3 4-54-5 6-76-7 8-98-9 10-1110-11
(Lunchtime)(Lunchtime) (Dinnertime)(Dinnertime)
Hour of dayHour of day
Per
cen
tag
e o
f sa
les
Per
cen
tag
e o
f sa
les
Figure 4.12Figure 4.12