SP

© 2006 Prentice Hall, Inc. 4 – 1
Operations
Management
Chapter 4 -
Forecasting
© 2006 Prentice Hall, Inc.
PowerPoint presentation to accompanyPowerPoint presentation to accompany
Heizer/RenderHeizer/Render
Principles of Operations Management, 6ePrinciples of Operations Management, 6e
Operations Management, 8eOperations Management, 8e

OutlineOutline
 Global Company Profile:Global Company Profile:
Tupperware CorporationTupperware Corporation
 What Is Forecasting?What Is Forecasting?
 Forecasting Time HorizonsForecasting Time Horizons
 The Influence of Product Life CycleThe Influence of Product Life Cycle
 Types Of ForecastsTypes Of Forecasts

Outline – ContinuedOutline – Continued
 The Strategic Importance OfThe Strategic Importance Of
ForecastingForecasting
 Human ResourcesHuman Resources
 CapacityCapacity
 Supply-Chain ManagementSupply-Chain Management
 Seven Steps In The ForecastingSeven Steps In The Forecasting
SystemSystem

 Forecasting ApproachesForecasting Approaches
 Overview of Qualitative MethodsOverview of Qualitative Methods
 Overview of Quantitative MethodsOverview of Quantitative Methods

 Time-series ForecastingTime-series Forecasting
 Decomposition of a Time SeriesDecomposition of a Time Series
 Naïve ApproachNaïve Approach
 Moving AveragesMoving Averages
 Exponential SmoothingExponential Smoothing
 Exponential Smoothing with TrendExponential Smoothing with Trend
AdjustmentAdjustment
 Trend ProjectionsTrend Projections
 Seasonal Variations in DataSeasonal Variations in Data
 Cyclical Variations in DataCyclical Variations in Data

 Associative Forecasting Methods:Associative Forecasting Methods:
Regression And CorrelationRegression And Correlation
AnalysisAnalysis
 Using Regression Analysis toUsing Regression Analysis to
ForecastForecast
 Standard Error of the EstimateStandard Error of the Estimate
 Correlation Coefficients forCorrelation Coefficients for
Regression LinesRegression Lines
 Multiple-Regression AnalysisMultiple-Regression Analysis

 Monitoring And ControllingMonitoring And Controlling
ForecastsForecasts
 Adaptive SmoothingAdaptive Smoothing
 Focus ForecastingFocus Forecasting
 Forecasting In The Service SectorForecasting In The Service Sector

Learning ObjectivesLearning Objectives
When you complete this chapter, youWhen you complete this chapter, you
should be able to :should be able to :
Identify or Define:Identify or Define:
 ForecastingForecasting
 Types of forecastsTypes of forecasts
 Time horizonsTime horizons
 Approaches to forecastsApproaches to forecasts

Learning ObjectivesLearning Objectives
When you complete this chapter, youWhen you complete this chapter, you
should be able to :should be able to :
Describe or Explain:Describe or Explain:
 Moving averagesMoving averages
 Exponential smoothingExponential smoothing
 Trend projectionsTrend projections
 Regression and correlation analysisRegression and correlation analysis
 Measures of forecast accuracyMeasures of forecast accuracy

Forecasting at TupperwareForecasting at Tupperware
 Each of 50 profit centers around theEach of 50 profit centers around the
world is responsible forworld is responsible for
computerized monthly, quarterly,computerized monthly, quarterly,
and 12-month sales projectionsand 12-month sales projections
 These projections are aggregated byThese projections are aggregated by
region, then globally, atregion, then globally, at
Tupperware’s World HeadquartersTupperware’s World Headquarters
 Tupperware uses all techniquesTupperware uses all techniques
discussed in textdiscussed in text

Tupperware’sTupperware’s
ProcessProcess

Three Key Factors forThree Key Factors for
TupperwareTupperware
 The number of registeredThe number of registered
“consultants” or sales“consultants” or sales
representativesrepresentatives
 The percentage of currently “active”The percentage of currently “active”
dealers (this number changes eachdealers (this number changes each
week and month)week and month)
 Sales per active dealer, on a weeklySales per active dealer, on a weekly
basisbasis

Forecast by ConsensusForecast by Consensus
 Although inputs come from sales,Although inputs come from sales,
marketing, finance, and production,marketing, finance, and production,
final forecasts are the consensus offinal forecasts are the consensus of
all participating managersall participating managers
 The final step is Tupperware’sThe final step is Tupperware’s
version of the “jury of executiveversion of the “jury of executive
opinion”opinion”

What is Forecasting?What is Forecasting?
 Process ofProcess of
predicting a futurepredicting a future
eventevent
 Underlying basis ofUnderlying basis of
all businessall business
decisionsdecisions
 ProductionProduction
 InventoryInventory
 PersonnelPersonnel
 FacilitiesFacilities
??

 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, workforcePurchasing, 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++
yearsyears
 New product planning, facility location,New product planning, facility location,
research and developmentresearch and development
Forecasting Time HorizonsForecasting Time Horizons

Distinguishing DifferencesDistinguishing Differences
Medium/long rangeMedium/long range forecasts deal withforecasts deal with
more comprehensive issues and supportmore comprehensive issues and support
management decisions regardingmanagement decisions regarding
planning and products, plants andplanning and products, plants and
processesprocesses
Short-termShort-term forecasting usually employsforecasting usually employs
different methodologies than longer-termdifferent methodologies than longer-term
forecastingforecasting
Short-termShort-term forecasts tend to be moreforecasts tend to be more
accurate than longer-term forecastsaccurate than longer-term forecasts

Influence of Product LifeInfluence of Product Life
CycleCycle
 Introduction and growth require longerIntroduction and growth require longer
forecasts than maturity and declineforecasts than maturity and decline
 As product passes through life cycle,As product passes through life cycle,
forecasts are useful in projectingforecasts are useful in projecting
 Staffing levelsStaffing levels
 Inventory levelsInventory levels
 Factory capacityFactory capacity
Introduction – Growth – Maturity – Decline

Product Life CycleProduct Life Cycle
Best period toBest period to
increase marketincrease market
shareshare
R&D engineering isR&D engineering is
criticalcritical
Practical to changePractical to change
price or qualityprice or quality
imageimage
Strengthen nicheStrengthen niche
Poor time toPoor time to
change image,change image,
price, or qualityprice, or quality
Competitive costsCompetitive costs
become criticalbecome critical
Defend marketDefend market
positionposition
Cost controlCost control
criticalcritical
Introduction Growth Maturity Decline
CompanyStrategy/IssuesCompanyStrategy/Issues
InternetInternet
Flat-screenFlat-screen
monitorsmonitors
SalesSales
DVDDVD
CD-ROMCD-ROM
Drive-throughDrive-through
restaurantsrestaurants
Fax machinesFax machines
3 1/2”3 1/2”
FloppyFloppy
disksdisks
Color printersColor printers
Figure 2.5Figure 2.5

Product Life CycleProduct Life Cycle
Product designProduct design
andand
developmentdevelopment
criticalcritical
FrequentFrequent
product andproduct and
process designprocess design
changeschanges
Short productionShort production
runsruns
High productionHigh production
costscosts
Limited modelsLimited models
Attention toAttention to
qualityquality
Introduction Growth Maturity Decline
OMStrategy/IssuesOMStrategy/Issues
criticalcritical
Product andProduct and
processprocess
reliabilityreliability
CompetitiveCompetitive
productproduct
improvementsimprovements
and optionsand options
Increase capacityIncrease capacity
Shift towardShift toward
product focusproduct focus
EnhanceEnhance
distributiondistribution
StandardizationStandardization
Less rapidLess rapid
product changesproduct changes
– more minor– more minor
changeschanges
OptimumOptimum
capacitycapacity
IncreasingIncreasing
stability ofstability of
processprocess
Long productionLong production
runsruns
ProductProduct
improvement andimprovement and
cost cuttingcost cutting
Little productLittle product
differentiationdifferentiation
CostCost
minimizationminimization
OvercapacityOvercapacity
in thein the
industryindustry
Prune line toPrune line to
eliminateeliminate
items notitems not
returningreturning
good margingood margin
ReduceReduce
capacitycapacity

Types of ForecastsTypes of Forecasts
 Economic forecastsEconomic forecasts
 Address business cycle – inflation rate,Address business cycle – inflation rate,
money supply, housing starts, etc.money 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

Strategic Importance ofStrategic Importance of
 Human Resources – Hiring, training,Human Resources – Hiring, training,
laying off workerslaying off workers
 Capacity – Capacity shortages canCapacity – Capacity shortages can
result in undependable delivery, lossresult in undependable delivery, loss
of customers, loss of market shareof customers, loss of market share
 Supply-Chain Management – GoodSupply-Chain Management – Good
supplier relations and price advancesupplier relations and price advance

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 theDetermine 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

The Realities!The Realities!
 Forecasts are seldom perfectForecasts are seldom perfect
 Most techniques assume anMost techniques assume an
underlying stability in the systemunderlying stability in the system
 Product family and aggregatedProduct family and aggregated
forecasts are more accurate thanforecasts are more accurate than
individual product forecastsindividual product forecasts

Forecasting ApproachesForecasting Approaches
 Used when situation is vagueUsed when situation is vague
and little data existand little data exist
 New productsNew products
 New technologyNew technology
 Involves intuition, experienceInvolves intuition, experience
 e.g., forecasting sales on Internete.g., forecasting sales on Internet
Qualitative MethodsQualitative Methods

Forecasting ApproachesForecasting Approaches
 Used when situation is ‘stable’ andUsed 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 colore.g., forecasting sales of color
televisionstelevisions
Quantitative MethodsQuantitative Methods

Overview of QualitativeOverview of Qualitative
MethodsMethods
 Jury of executive opinionJury of executive opinion
 Pool opinions of high-levelPool opinions of high-level
executives, sometimes augment byexecutives, sometimes augment by
statistical modelsstatistical models
 Delphi methodDelphi method
 Panel of experts, queried iterativelyPanel of experts, queried iteratively

Overview of QualitativeOverview of Qualitative
MethodsMethods
 Sales force compositeSales force composite
 Estimates from individualEstimates from individual
salespersons are reviewed forsalespersons are reviewed for
reasonableness, then aggregatedreasonableness, then aggregated
 Consumer Market SurveyConsumer Market Survey
 Ask the customerAsk the customer

 Involves small group of high-levelInvolves small group of high-level
managersmanagers
 Group estimates demand by workingGroup estimates demand by working
togethertogether
 Combines managerial experience withCombines managerial experience with
statistical modelsstatistical models
 Relatively quickRelatively quick
 ‘‘Group-think’Group-think’
disadvantagedisadvantage
Jury of Executive OpinionJury of Executive Opinion

Sales Force CompositeSales Force Composite
 Each salesperson projects his orEach salesperson projects his or
her salesher sales
 Combined at district and nationalCombined at district and national
levelslevels
 Sales reps know customers’ wantsSales reps know customers’ wants
 Tends to be overly optimisticTends to be overly optimistic

Delphi MethodDelphi Method
 Iterative groupIterative group
process,process,
continues untilcontinues until
consensus isconsensus is
reachedreached
 3 types of3 types of
participantsparticipants
 Decision makersDecision makers
 StaffStaff
 RespondentsRespondents
Staff
(Administering
survey)
Decision Makers
(Evaluate
responses and
make decisions)
Respondents
(People who can
make valuable
judgments)

Consumer Market SurveyConsumer Market Survey
 Ask customers about purchasingAsk customers about purchasing
plansplans
 What consumers say, and whatWhat consumers say, and what
they actually do are often differentthey actually do are often different
 Sometimes difficult to answerSometimes difficult to answer

Overview of QuantitativeOverview of Quantitative
ApproachesApproaches
1.1. Naive approachNaive approach
2.2. Moving averagesMoving averages
3.3. ExponentialExponential
smoothingsmoothing
4.4. Trend projectionTrend projection
5.5. Linear regressionLinear regression
Time-SeriesTime-Series
ModelsModels
AssociativeAssociative
ModelModel

 Set of evenly spaced numericalSet of evenly spaced numerical
datadata
 Obtained by observing responseObtained by observing response
variable at regular time periodsvariable at regular time periods
 Forecast based only on pastForecast based only on past
valuesvalues
 Assumes that factors influencingAssumes that factors influencing
past and present will continuepast and present will continue
influence in futureinfluence in future
Time Series ForecastingTime Series Forecasting

Trend
Seasonal
Cyclical
Random
Time Series ComponentsTime Series Components

Components of DemandComponents of DemandDemandforproductorservice
| | | |
1 2 3 4
Year
Average
demand over
four years
Seasonal peaks
Trend
component
Actual
demand
Random
variation

 Persistent, overall upward orPersistent, overall upward or
downward patterndownward pattern
 Changes due to population,Changes due to population,
technology, age, culture, etc.technology, age, culture, etc.
 Typically several yearsTypically several years
durationduration
Trend ComponentTrend Component

 Regular pattern of up andRegular pattern of up and
down fluctuationsdown fluctuations
 Due to weather, customs, etc.Due to weather, customs, etc.
 Occurs within a single yearOccurs within a single year
Seasonal ComponentSeasonal Component
Number of
Period Length Seasons
Week Day 7
Month Week 4-4.5
Month Day 28-31
Year Quarter 4
Year Month 12
Year Week 52

 Repeating up and down movementsRepeating up and down movements
 Affected by business cycle,Affected by business cycle,
political, and economic factorspolitical, and economic factors
 Multiple years durationMultiple years duration
 Often causal orOften causal or
associativeassociative
relationshipsrelationships
Cyclical ComponentCyclical Component
00 55 1010 1515 2020

 Erratic, unsystematic, ‘residual’Erratic, unsystematic, ‘residual’
fluctuationsfluctuations
 Due to random variation orDue to random variation or
unforeseen eventsunforeseen events
 Short duration andShort duration and
nonrepeatingnonrepeating
Random ComponentRandom Component
MM TT WW TT FF

Naive ApproachNaive Approach
 Assumes demand in next period isAssumes demand in next period is
the same as demand in mostthe same as demand in most
recent periodrecent period
 e.g., If May sales were 48, then Junee.g., If May sales were 48, then June
sales will be 48sales will be 48
 Sometimes cost effective andSometimes cost effective and
efficientefficient

 MA is a series of arithmetic meansMA is a series of arithmetic means
 Used if little or no trendUsed if little or no trend
 Used often for smoothingUsed often for smoothing
 Provides overall impression of dataProvides 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

JanuaryJanuary 1010
FebruaryFebruary 1212
MarchMarch 1313
AprilApril 1616
MayMay 1919
JuneJune 2323
JulyJuly 2626
ActualActual 3-Month3-Month
MonthMonth 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
1010
1212
1313
((1010 ++ 1212 ++ 1313)/3 = 11)/3 = 11 22
//33

Graph of Moving AverageGraph of Moving Average
| | | | | | | | | | | |
JJ FF MM AA MM JJ JJ AA SS OO NN DD
ShedSalesShedSales
3030 –
2828 –
2626 –
2424 –
2222 –
2020 –
1818 –
1616 –
1414 –
1212 –
1010 –
ActualActual
SalesSales
MovingMoving
AverageAverage
ForecastForecast

 Used when trend is presentUsed when trend is present
 Older data usually less importantOlder data usually less important
 Weights based on experience andWeights based on experience and
intuitionintuition
Weighted Moving AverageWeighted Moving Average
WeightedWeighted
moving averagemoving average ==
∑∑ ((weight for period nweight for period n))
xx ((demand in period ndemand in period n))
∑∑ weightsweights

JanuaryJanuary 1010
FebruaryFebruary 1212
MarchMarch 1313
AprilApril 1616
MayMay 1919
JuneJune 2323
JulyJuly 2626
ActualActual 3-Month Weighted3-Month Weighted
MonthMonth 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
1010
1212
1313
[(3 x[(3 x 1313) + (2 x) + (2 x 1212) + () + (1010)]/6 = 12)]/6 = 1211
//66
Weights Applied Period
3 Last month
2 Two months ago
1 Three months ago
6 Sum of weights

 Increasing n smooths the forecastIncreasing n smooths the forecast
but makes it less sensitive tobut 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 AverageMoving Average

Moving Average AndMoving Average And
3030 –
2525 –
2020 –
1515 –
1010 –
55 –
SalesdemandSalesdemand
| | | | | | | | | | | |
ActualActual
salessales
MovingMoving
averageaverage
WeightedWeighted
movingmoving
averageaverage

 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 constantRequires smoothing constant ((αα))
 Ranges from 0 to 1Ranges from 0 to 1
 Subjectively chosenSubjectively chosen
 Involves little record keeping of pastInvolves little record keeping of past
datadata
Exponential SmoothingExponential Smoothing

t =t = last period’s forecastlast period’s forecast
++ αα ((last period’s actual demandlast period’s actual demand
–– last period’s forecastlast period’s forecast))
FFtt = F= Ftt – 1– 1 ++ αα((AAtt – 1– 1 -- FFtt – 1– 1))
wherewhere FFtt == new forecastnew forecast
FFtt – 1– 1 == previous forecastprevious forecast
αα == smoothing (or weighting)smoothing (or weighting)
constantconstant (0(0 ≤≤ αα ≥≥ 1)1)

ExampleExample
Predicted demandPredicted demand = 142= 142 Ford MustangsFord Mustangs
Actual demandActual demand = 153= 153
Smoothing constantSmoothing constant αα = .20= .20

ExampleExample
New forecastNew forecast = 142 + .2(153 – 142)= 142 + .2(153 – 142)

ExampleExample
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

Effect ofEffect of
Smoothing ConstantsSmoothing Constants
Weight Assigned toWeight Assigned to
MostMost 2nd Most2nd Most 3rd Most3rd Most 4th Most4th Most 5th Most5th Most
RecentRecent RecentRecent RecentRecent RecentRecent RecentRecent
SmoothingSmoothing PeriodPeriod PeriodPeriod PeriodPeriod PeriodPeriod PeriodPeriod
ConstantConstant ((αα)) αα(1 -(1 - αα)) αα(1 -(1 - αα))22
αα(1 -(1 - αα))33
αα(1 -(1 - αα))44
αα = .1= .1 .1.1 .09.09 .081.081 .073.073 .066.066
αα = .5= .5 .5.5 .25.25 .125.125 .063.063 .031.031

Impact of DifferentImpact of Different αα
225225 –
200200 –
175175 –
150150 –
| | | | | | | | |
11 22 33 44 55 66 77 88 99
QuarterQuarter
DemandDemand
αα = .1= .1
ActualActual
demanddemand
αα = .5= .5

ChoosingChoosing αα
The objective is to obtain the mostThe objective is to obtain the most
accurate forecast no matter theaccurate forecast no matter the
techniquetechnique
We generally do this by selecting theWe generally do this by selecting the
model that gives us the lowest forecastmodel that gives us the lowest forecast
errorerror
Forecast errorForecast error = Actual demand - Forecast value= Actual demand - Forecast value
= A= Att - F- Ftt

Common Measures of ErrorCommon Measures of Error
Mean Absolute DeviationMean Absolute Deviation ((MADMAD))
MAD =MAD =
∑∑ |actual - forecast||actual - forecast|
nn
Mean Squared ErrorMean Squared Error ((MSEMSE))
MSE =MSE =
∑∑ ((forecast errorsforecast errors))22
nn

Common Measures of ErrorCommon Measures of Error
Mean Absolute Percent ErrorMean Absolute Percent Error ((MAPEMAPE))
MAPE =MAPE =
100100 ∑∑ |actual|actualii - forecast- forecastii|/actual|/actualii
nn
nn
ii = 1= 1

Comparison of ForecastComparison of Forecast
ErrorError
RoundedRounded AbsoluteAbsolute RoundedRounded AbsoluteAbsolute
ActualActual ForecastForecast DeviationDeviation ForecastForecast DeviationDeviation
TonnageTonnage withwith forfor withwith forfor
QuarterQuarter UnloadedUnloaded αα = .10= .10 αα = .10= .10 αα = .50= .50 αα = .50= .50
11 180180 175175 55 175175 55
22 168168 176176 88 178178 1010
33 159159 175175 1616 173173 1414
44 175175 173173 22 166166 99
55 190190 173173 1717 170170 2020
66 205205 175175 3030 180180 2525
77 180180 178178 22 193193 1313
88 182182 178178 44 186186 44
8484 100100

ErrorError
TonageTonage withwith forfor withwith forfor
11 180180 175175 55 175175 55
22 168168 176176 88 178178 1010
33 159159 175175 1616 173173 1414
44 175175 173173 22 166166 99
55 190190 173173 1717 170170 2020
66 205205 175175 3030 180180 2525
77 180180 178178 22 193193 1313
88 182182 178178 44 186186 44
8484 100100
MAD =
∑ |deviations|
n
= 84/8 = 10.50
For α = .10
= 100/8 = 12.50
For α = .50

ErrorError
11 180180 175175 55 175175 55
22 168168 176176 88 178178 1010
33 159159 175175 1616 173173 1414
44 175175 173173 22 166166 99
55 190190 173173 1717 170170 2020
66 205205 175175 3030 180180 2525
77 180180 178178 22 193193 1313
88 182182 178178 44 186186 44
8484 100100
MADMAD 10.5010.50 12.5012.50
= 1,558/8 = 194.75
For α = .10
= 1,612/8 = 201.50
For α = .50
MSE =
∑ (forecast errors)2
n

ErrorError
11 180180 175175 55 175175 55
22 168168 176176 88 178178 1010
33 159159 175175 1616 173173 1414
44 175175 173173 22 166166 99
55 190190 173173 1717 170170 2020
66 205205 175175 3030 180180 2525
77 180180 178178 22 193193 1313
88 182182 178178 44 186186 44
8484 100100
MADMAD 10.5010.50 12.5012.50
MSEMSE 194.75194.75 201.50201.50
= 45.62/8 = 5.70%
For α = .10
= 54.8/8 = 6.85%
For α = .50
MAPE =
100 ∑ |deviationi|/actuali
n
n
i = 1

ErrorError
TonnageTonnage withwith forfor withwith forfor
11 180180 175175 55 175175 55
22 168168 176176 88 178178 1010
33 159159 175175 1616 173173 1414
44 175175 173173 22 166166 99
55 190190 173173 1717 170170 2020
66 205205 175175 3030 180180 2525
77 180180 178178 22 193193 1313
88 182182 178178 44 186186 44
8484 100100
MADMAD 10.5010.50 12.5012.50
MSEMSE 194.75194.75 201.50201.50
MAPEMAPE 5.70%5.70% 6.85%6.85%

Exponential Smoothing withExponential Smoothing with
Trend AdjustmentTrend Adjustment
When a trend is present, exponentialWhen a trend is present, exponential
smoothing must be modifiedsmoothing must be modified
ForecastForecast
includingincluding ((FITFITtt)) ==
trendtrend
exponentiallyexponentially exponentiallyexponentially
smoothedsmoothed ((FFtt)) ++ ((TTtt)) smoothedsmoothed
forecastforecast trendtrend

Trend AdjustmentTrend Adjustment
FFtt == αα((AAtt - 1- 1) + (1 -) + (1 - αα)()(FFtt - 1- 1 ++ TTtt - 1- 1))
TTtt == ββ((FFtt -- FFtt - 1- 1) + (1 -) + (1 - ββ))TTtt - 1- 1
Step 1: Compute FStep 1: Compute Ftt
Step 2: Compute TStep 2: Compute Ttt
Step 3: Calculate the forecast FITStep 3: Calculate the forecast FITtt == FFtt ++ TTtt

Trend Adjustment ExampleTrend Adjustment Example
ForecastForecast
ActualActual SmoothedSmoothed SmoothedSmoothed IncludingIncluding
MonthMonth((tt)) DemandDemand ((AAtt)) Forecast, FForecast, Ftt Trend, TTrend, Ttt Trend, FITTrend, FITtt
11 1212 1111 22 13.0013.00
22 1717
33 2020
44 1919
55 2424
66 2121
77 3131
88 2828
99 3636
1010
Table 4.1Table 4.1

ForecastForecast
11 1212 1111 22 13.0013.00
22 1717
33 2020
44 1919
55 2424
66 2121
77 3131
88 2828
99 3636
1010
Table 4.1Table 4.1
F2 = αA1 + (1 - α)(F1 + T1)
F2 = (.2)(12) + (1 - .2)(11 + 2)
= 2.4 + 10.4 = 12.8 units
Step 1: Forecast for Month 2

ForecastForecast
11 1212 1111 22 13.0013.00
22 1717 12.8012.80
33 2020
44 1919
55 2424
66 2121
77 3131
88 2828
99 3636
1010
Table 4.1Table 4.1
T2 = β(F2 - F1) + (1 - β)T1
T2 = (.4)(12.8 - 11) + (1 - .4)(2)
= .72 + 1.2 = 1.92 units
Step 2: Trend for Month 2

ForecastForecast
11 1212 1111 22 13.0013.00
22 1717 12.8012.80 1.921.92
33 2020
44 1919
55 2424
66 2121
77 3131
88 2828
99 3636
1010
Table 4.1Table 4.1
FIT2 = F2 + T1
FIT2 = 12.8 + 1.92
= 14.72 units
Step 3: Calculate FIT for Month 2

ForecastForecast
11 1212 1111 22 13.0013.00
22 1717 12.8012.80 1.921.92 14.7214.72
33 2020
44 1919
55 2424
66 2121
77 3131
88 2828
99 3636
1010
Table 4.1Table 4.1
15.1815.18 2.102.10 17.2817.28
17.8217.82 2.322.32 20.1420.14
19.9119.91 2.232.23 22.1422.14
22.5122.51 2.382.38 24.8924.89
24.1124.11 2.072.07 26.1826.18
27.1427.14 2.452.45 29.5929.59
29.2829.28 2.322.32 31.6031.60
32.4832.48 2.682.68 35.1635.16

| | | | | | | | |
11 22 33 44 55 66 77 88 99
Time (month)Time (month)
ProductdemandProductdemand
3535 –
3030 –
2525 –
2020 –
1515 –
1010 –
55 –
00 –
Actual demandActual demand ((AAtt))
Forecast including trendForecast including trend ((FITFITtt))

Trend ProjectionsTrend Projections
Fitting a trend line to historical data pointsFitting 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 leastLinear trends can be found using the least
squares techniquesquares technique
yy == aa ++ bxbx^^
where ywhere y = computed value of= computed value of
the variable to be predictedthe variable to be predicted
(dependent variable)(dependent variable)
aa = y-axis intercept= y-axis intercept
bb = slope of the regression line= slope of the regression line
xx = the independent variable= the independent variable
^^

Least Squares MethodLeast Squares Method
Time periodTime period
ValuesofDependentVariable
DeviationDeviation11
Actual observationActual observation
(y value)(y value)
Trend line, y = a + bxTrend line, y = a + bx^^

Time periodTime period
ValuesofDependentVariable
Actual observationActual 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)

Equations to calculate the regression variablesEquations to calculate the regression variables
b =b =
ΣΣxy - nxyxy - nxy
ΣΣxx22
- nx- nx22
yy == aa ++ bxbx^^
a = y - bxa = y - bx

Least Squares ExampleLeast Squares Example
bb = = = 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 PowerElectrical Power
YearYear Period (x)Period (x) DemandDemand xx22
xyxy
19991999 11 7474 11 7474
20002000 22 7979 44 158158
20012001 33 8080 99 240240
20022002 44 9090 1616 360360
20032003 55 105105 2525 525525
20042004 66 142142 3636 852852
20052005 77 122122 4949 854854
∑∑xx = 28= 28 ∑∑yy = 692= 692 ∑∑xx22
= 140= 140 ∑∑xyxy = 3,063= 3,063
xx = 4= 4 yy = 98.86= 98.86

bb = = = 10.54= = = 10.54
ΣΣxy - nxyxy - nxy
ΣΣxx22
- 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 PowerElectrical Power
YearYear Period (x)Period (x) DemandDemand xx22
xyxy
19991999 11 7474 11 7474
20002000 22 7979 44 158158
20012001 33 8080 99 240240
20022002 44 9090 1616 360360
20032003 55 105105 2525 525525
20042004 66 142142 3636 852852
20052005 77 122122 4949 854854
ΣΣxx = 28= 28 ΣΣyy = 692= 692 ΣΣxx22
= 140= 140 ΣΣxyxy = 3,063= 3,063
xx = 4= 4 yy = 98.86= 98.86
The trend line is
y = 56.70 + 10.54x^

| | | | | | | | |
19991999 20002000 20012001 20022002 20032003 20042004 20052005 20062006 20072007
160160 –
150150 –
140140 –
130130 –
120120 –
110110 –
100100 –
9090 –
8080 –
7070 –
6060 –
5050 –
YearYear
PowerdemandPowerdemand
Trend line,Trend line,
yy = 56.70 + 10.54x= 56.70 + 10.54x^^

Seasonal Variations In DataSeasonal Variations In Data
The multiplicative seasonal model canThe multiplicative seasonal model can
modify trend data to accommodatemodify trend data to accommodate
seasonal variations in demandseasonal variations in demand
1.1. Find average historical demand for each seasonFind average historical demand for each season
2.2. Compute the average demand over all seasonsCompute the average demand over all seasons
3.3. Compute a seasonal index for each seasonCompute 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 theDivide this estimate of total demand by the
number of seasons, then multiply it by thenumber of seasons, then multiply it by the
seasonal index for that seasonseasonal index for that season

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 SeasonalSeasonal
MonthMonth 20032003 20042004 20052005 2003-20052003-2005 MonthlyMonthly IndexIndex

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
0.9570.957
Seasonal index =
average 2003-2005 monthly demand
average monthly demand
= 90/94 = .957

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

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
Expected annual demand = 1,200
Jan x .957 = 96
1,200
12
Feb x .851 = 85
1,200
12
Forecast for 2006

140140 –
130130 –
120120 –
110110 –
100100 –
9090 –
8080 –
7070 –
| | | | | | | | | | | |
TimeTime
DemandDemand
2006 Forecast2006 Forecast
2005 Demand2005 Demand

San Diego HospitalSan Diego Hospital
10,20010,200 –
10,00010,000 –
9,8009,800 –
9,6009,600 –
9,4009,400 –
9,2009,200 –
9,0009,000 –
| | | | | | | | | | | |
JanJan FebFeb MarMar AprApr MayMay JuneJune JulyJuly AugAug SeptSept OctOct NovNov DecDec
6767 6868 6969 7070 7171 7272 7373 7474 7575 7676 7777 7878
MonthMonth
InpatientDaysInpatientDays
95309530
95519551
95739573
95949594
96169616
96379637
96599659
96809680
97029702
97239723
97459745
97669766
Trend DataTrend Data

1.061.06 –
1.041.04 –
1.021.02 –
1.001.00 –
0.980.98 –
0.960.96 –
0.940.94 –
0.92 –
| | | | | | | | | | | |
6767 6868 6969 7070 7171 7272 7373 7474 7575 7676 7777 7878
MonthMonth
IndexforInpatientDaysIndexforInpatientDays
1.041.04
1.021.02
1.011.01
0.990.99
1.031.03
1.041.04
1.001.00
0.980.98
0.970.97
0.990.99
0.970.97
0.960.96
Seasonal IndicesSeasonal Indices

10,20010,200 –
10,00010,000 –
9,8009,800 –
9,6009,600 –
9,4009,400 –
9,2009,200 –
9,0009,000 –
| | | | | | | | | | | |
6767 6868 6969 7070 7171 7272 7373 7474 7575 7676 7777 7878
MonthMonth
InpatientDaysInpatientDays
99119911
92659265
97649764
95209520
96919691
94119411
99499949
97249724
95429542
93559355
1006810068
95729572
Combined Trend and Seasonal ForecastCombined Trend and Seasonal Forecast

Associative ForecastingAssociative Forecasting
Used when changes in one or moreUsed when changes in one or more
independent variables can be used to predictindependent variables can be used to predict
the changes in the dependent variablethe changes in the dependent variable
Most common technique is linearMost common technique is linear
regression analysisregression analysis
We apply this technique just as we didWe apply this technique just as we did
in the time series examplein the time series example

Forecasting an outcome based on predictorForecasting an outcome based on predictor
variables using the least squares techniquevariables using the least squares technique
yy == aa ++ bxbx^^
where ywhere y = computed value of= computed value of
the variable to be predictedthe variable to be predicted
(dependent variable)(dependent variable)
aa = y-axis intercept= y-axis intercept
bb = slope of the regression line= slope of the regression line
xx = the independent variable= the independent variable
though to predict the value of thethough to predict the value of the
dependent variabledependent variable
^^

ExampleExample
SalesSales Local PayrollLocal Payroll
($000,000), y($000,000), y ($000,000,000), x($000,000,000), x
2.02.0 11
3.03.0 33
2.52.5 44
2.02.0 22
2.02.0 11
3.53.5 77
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Sales
Area payroll

ExampleExample
Sales, y Payroll, x x2
xy
2.0 1 1 2.0
3.0 3 9 9.0
2.5 4 16 10.0
2.0 2 4 4.0
2.0 1 1 2.0
3.5 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

ExampleExample
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Sales
Area payroll
yy = 1.75 + .25= 1.75 + .25xx^^ SalesSales = 1.75 + .25(= 1.75 + .25(payrollpayroll))
If payroll next yearIf payroll next year
is estimated to beis estimated to be
$600$600 million, then:million, then:
SalesSales = 1.75 + .25(6)= 1.75 + .25(6)
SalesSales = $325,000= $325,000
3.25

Standard Error of theStandard Error of the
EstimateEstimate
 A forecast is just a point estimate of aA forecast is just a point estimate of a
future valuefuture value
 This point isThis point is
actually theactually the
mean of amean of a
probabilityprobability
distributiondistribution
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Sales
Area payroll
3.25

EstimateEstimate
wherewhere yy == y-value of each datay-value of each data
pointpoint
yycc == computed value ofcomputed value of
the dependent variable, from thethe dependent variable, from the
regression equationregression equation
nn == number of datanumber of data
pointspoints
SSy,xy,x == ∑∑((y - yy - ycc))22
nn - 2- 2

EstimateEstimate
Computationally, this equation isComputationally, this equation is
considerably easier to useconsiderably easier to use
We use the standard error to set upWe use the standard error to set up
prediction intervals around theprediction intervals around the
point estimatepoint estimate
SSy,xy,x ==
∑∑yy22
- a- a∑∑y - by - b∑∑xyxy
nn - 2- 2

EstimateEstimate
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Sales
Area payroll
3.25
SSy,xy,x = == =∑∑yy22
- a- a∑∑y - by - b∑∑xyxy
nn - 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 errorThe standard error
of the estimate isof the estimate is
$30,600$30,600 in salesin sales

 How strong is the linearHow strong is the linear
relationship between therelationship between the
variables?variables?
 Correlation does not necessarilyCorrelation does not necessarily
imply causality!imply causality!
 Coefficient of correlation, r,Coefficient of correlation, r,
measures degree of associationmeasures degree of association
 Values range fromValues range from -1-1 toto +1+1
CorrelationCorrelation

Correlation CoefficientCorrelation Coefficient
r =r =
nnΣΣxyxy -- ΣΣxxΣΣyy
[[nnΣΣxx22
- (- (ΣΣxx))22
][][nnΣΣyy22
- (- (ΣΣyy))22
]]

Correlation CoefficientCorrelation Coefficient
r =r =
nn∑∑xyxy -- ∑∑xx∑∑yy
[[nn∑∑xx22
- (- (∑∑xx))22
][][nn∑∑yy22
- (- (∑∑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

 Coefficient of Determination, rCoefficient of Determination, r22
,,
measures the percent of change inmeasures the percent of change in
y predicted by the change in xy predicted by the change in x
 Values range fromValues range from 00 toto 11
 Easy to interpretEasy to interpret
CorrelationCorrelation
For the Nodel Construction example:For the Nodel Construction example:
rr = .901= .901
rr22
= .81= .81

Multiple RegressionMultiple Regression
AnalysisAnalysis
If more than one independent variable is to beIf more than one independent variable is to be
used in the model, linear regression can beused in the model, linear regression can be
extended to multiple regression toextended to multiple regression to
accommodate several independent variablesaccommodate several independent variables
yy == aa ++ bb11xx11 + b+ b22xx22 ……^^
Computationally, this is quiteComputationally, this is quite
complex and generally done on thecomplex and generally done on the
computercomputer

Multiple RegressionMultiple Regression
AnalysisAnalysis
yy = 1.80 + .30= 1.80 + .30xx11 - 5.0- 5.0xx22
^^
In the Nodel example, including interest rates inIn the Nodel example, including interest rates in
the model gives the new equation:the model gives the new equation:
An improved correlation coefficient of rAn improved correlation coefficient of r = .96= .96
means this model does a better job of predictingmeans this model does a better job of predicting
the change in construction salesthe change in construction sales
SalesSales = 1.80 + .30(6) - 5.0(.12) = 3.00= 1.80 + .30(6) - 5.0(.12) = 3.00
SalesSales = $300,000= $300,000

 Measures how well the forecast isMeasures how well the forecast is
predicting actual valuespredicting actual values
 Ratio of running sum of forecast errorsRatio 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, theIf forecasts are continually high or low, the
forecast has a bias errorforecast has a bias error
Monitoring and ControllingMonitoring and Controlling
ForecastsForecasts
Tracking SignalTracking Signal

Monitoring and ControllingMonitoring and Controlling
ForecastsForecasts
TrackingTracking
signalsignal
RSFERSFE
MADMAD
==
TrackingTracking
signalsignal ==
∑∑(actual demand in(actual demand in
period i -period i -
forecast demandforecast demand
in period i)in period i)
((∑∑|actual - forecast|/n|actual - forecast|/n))

Tracking SignalTracking Signal
Tracking signalTracking signal
++
00 MADsMADs
––
Upper control limitUpper control limit
Lower control limitLower control limit
TimeTime
Signal exceeding limitSignal exceeding limit
AcceptableAcceptable
rangerange

Tracking Signal ExampleTracking Signal Example
CumulativeCumulative
AbsoluteAbsolute AbsoluteAbsolute
ActualActual ForecastForecast ForecastForecast ForecastForecast
QtrQtr DemandDemand DemandDemand ErrorError RSFERSFE ErrorError ErrorError MADMAD
11 9090 100100 -10-10 -10-10 1010 1010 10.010.0
22 9595 100100 -5-5 -15-15 55 1515 7.57.5
33 115115 100100 +15+15 00 1515 3030 10.010.0
44 100100 110110 -10-10 -10-10 1010 4040 10.010.0
55 125125 110110 +15+15 +5+5 1515 5555 11.011.0
66 140140 110110 +30+30 +35+35 3030 8585 14.214.2

CumulativeCumulative
AbsoluteAbsolute AbsoluteAbsolute
ActualActual ForecastForecast ForecastForecast ForecastForecast
QtrQtr DemandDemand DemandDemand ErrorError RSFERSFE ErrorError ErrorError MADMAD
11 9090 100100 -10-10 -10-10 1010 1010 10.010.0
22 9595 100100 -5-5 -15-15 55 1515 7.57.5
33 115115 100100 +15+15 00 1515 3030 10.010.0
44 100100 110110 -10-10 -10-10 1010 4040 10.010.0
55 125125 110110 +15+15 +5+5 1515 5555 11.011.0
66 140140 110110 +30+30 +35+35 3030 8585 14.214.2
Tracking Signal ExampleTracking Signal Example
Tracking
Signal
(RSFE/MAD)
-10/10 = -1
-15/7.5 = -2
0/10 = 0
-10/10 = -1
+5/11 = +0.5
+35/14.2 = +2.5
The variation of the tracking signalThe variation of the tracking signal
betweenbetween -2.0-2.0 andand +2.5+2.5 is within acceptableis within acceptable
limitslimits

Adaptive ForecastingAdaptive Forecasting
It’s possible to use the computer toIt’s possible to use the computer to
continually monitor forecast error andcontinually monitor forecast error and
adjust the values of theadjust the values of the αα andand ββ
coefficients used in exponentialcoefficients used in exponential
smoothing to continually minimizesmoothing to continually minimize
forecast errorforecast error
This technique is called adaptiveThis technique is called adaptive
smoothingsmoothing

Focus ForecastingFocus Forecasting
Developed at American Hardware Supply,Developed at American Hardware Supply,
focus forecasting is based on two principles:focus forecasting is based on two principles:
1.1. Sophisticated forecasting models are notSophisticated forecasting models are not
always better than simple modelsalways better than simple models
2.2. There is no single techniques that shouldThere is no single techniques that should
be used for all products or servicesbe used for all products or services
This approach uses historical data to testThis approach uses historical data to test
multiple forecasting models for individual itemsmultiple forecasting models for individual items
The forecasting model with the lowest error isThe forecasting model with the lowest error is
then used to forecast the next demandthen used to forecast the next demand

Forecasting in the ServiceForecasting 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 ofNeeds differ greatly as function of
industry and productindustry and product
 Holidays and other calendar eventsHolidays and other calendar events
 Unusual eventsUnusual events

Fast Food RestaurantFast 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 9-109-10
12-112-1 2-32-3 4-54-5 6-76-7 8-98-9 10-1110-11
(Lunchtime)(Lunchtime) (Dinnertime)(Dinnertime)
Hour of dayHour of day
PercentageofsalesPercentageofsales

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