© 2008 prentice hall, inc.4 – 1 operations management chapter 4 – forecasting delivered by:...
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© 2008 Prentice Hall, Inc. 4 – 1
Operations ManagementOperations ManagementChapter 4 – Chapter 4 – ForecastingForecasting
Delivered by:Delivered by:
Eng.Mosab I. TabashEng.Mosab I. Tabash
© 2008 Prentice Hall, Inc. 4 – 2
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
Sales will be $200 Million!
© 2008 Prentice Hall, Inc. 4 – 3
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 – 4
Distinguishing DifferencesDistinguishing Differences
Medium/long rangeMedium/long range forecasts deal with forecasts deal with more comprehensive issues and support more comprehensive issues and support management decisions regarding management decisions regarding planning and products, plants and planning and products, plants and processesprocesses
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
© 2008 Prentice Hall, Inc. 4 – 5
Influence of Product Life Influence of Product Life CycleCycle
Introduction and growth require longer Introduction 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
© 2008 Prentice Hall, Inc. 4 – 6
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 products and Predict sales of existing products and
servicesservices
© 2008 Prentice Hall, Inc. 4 – 7
Strategic Importance of Strategic Importance of ForecastingForecasting
Human Resources – Hiring, training, Human Resources – Hiring, training, laying off workerslaying off workers
Capacity – Capacity shortages can Capacity – Capacity shortages can result in undependable delivery, loss result in undependable delivery, loss of customers, loss of market shareof customers, loss of market share
Supply Chain Management – Good Supply Chain Management – Good supplier relations and price supplier relations and price advantagesadvantages
© 2008 Prentice Hall, Inc. 4 – 8
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 – 9
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 – 10
Forecasting ApproachesForecasting Approaches
Used when situation is vague Used 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
© 2008 Prentice Hall, Inc. 4 – 11
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 – 12
Overview of Qualitative Overview of Qualitative MethodsMethods
Jury of executive opinionJury of executive opinion Pool opinions of high-level experts, Pool opinions of high-level experts,
sometimes augment by statistical sometimes augment by statistical modelsmodels
Delphi methodDelphi method Panel of experts, queried iterativelyPanel of experts, queried iteratively
© 2008 Prentice Hall, Inc. 4 – 13
Overview of Qualitative Overview of Qualitative MethodsMethods
Sales force compositeSales force composite Estimates from individual Estimates from individual
salespersons are reviewed for salespersons are reviewed for reasonableness, then aggregated reasonableness, then aggregated
Consumer Market SurveyConsumer Market Survey Ask the customerAsk the customer
© 2008 Prentice Hall, Inc. 4 – 14
Involves small group of high-level experts Involves small group of high-level experts and managersand managers
Group estimates demand by working Group estimates demand by working togethertogether
Combines managerial experience with Combines managerial experience with statistical modelsstatistical models
Relatively quickRelatively quick
‘‘Group-think’Group-think’disadvantagedisadvantage
Jury of Executive OpinionJury of Executive Opinion
© 2008 Prentice Hall, Inc. 4 – 15
Sales Force CompositeSales Force Composite
Each salesperson projects his or Each salesperson projects his or her salesher sales
Combined at district and national Combined at district and national levelslevels
Sales reps know customers’ wantsSales reps know customers’ wants
Tends to be overly optimisticTends to be overly optimistic
© 2008 Prentice Hall, Inc. 4 – 16
Delphi MethodDelphi Method
Iterative group Iterative group process, process, continues until continues until consensus is consensus is reachedreached
3 types of 3 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)
© 2008 Prentice Hall, Inc. 4 – 17
Consumer Market SurveyConsumer Market Survey
Ask customers about purchasing Ask customers about purchasing plansplans
What consumers say,What consumers say,
and what they actually and what they actually
do are often differentdo are often different
Sometimes difficult to answerSometimes difficult to answer
How many hours will you use the
Internet next week?
How many hours will you use the
Internet next week?
© 1995 Corel Corp.
© 2008 Prentice Hall, Inc. 4 – 18
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 – 19
Set of evenly spaced numerical dataSet of evenly spaced numerical data
Obtained by observing response variable at Obtained by observing response variable at regular time periodsregular time periods
Forecast based only on past values, no other Forecast based only on past values, no other variables importantvariables important
Assumes that factors influencing past and Assumes that factors influencing past and present will continue influence in futurepresent will continue influence in future
Time Series ForecastingTime Series Forecasting
ExampleYear: 1993 1994 1995 1996 1997Sales:78.7 63.5 89.7 93.2 92.1
© 2008 Prentice Hall, Inc. 4 – 20
Trend
Seasonal
Cyclical
Random
Time Series ComponentsTime Series Components
© 2008 Prentice Hall, Inc. 4 – 21
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 – 22
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
Mo., Qtr., Yr.
Response
© 1984-1994 T/Maker Co.
© 2008 Prentice Hall, Inc. 4 – 23
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
Mo., Qtr.
Response
Summer
© 1984-1994 T/Maker Co.
© 2008 Prentice Hall, Inc. 4 – 24
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 – 25
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 – 26
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 – 27
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 – 28
JanuaryJanuary 1010FebruaryFebruary 1212MarchMarch 1313AprilApril 1616MayMay 1919JuneJune 2323JulyJuly 2626
ActualActual 3-Month3-MonthMonthMonth car Salescar 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 – 29
Graph of Moving AverageGraph of Moving Average
| | | | | | | | | | | |
JJ FF MM AA MM JJ JJ AA SS OO NN DD
Car
s S
ales
Car
s 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 – 30
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
© 2008 Prentice Hall, Inc. 4 – 31
JanuaryJanuary 1010FebruaryFebruary 1212MarchMarch 1313AprilApril 1616MayMay 1919JuneJune 2323JulyJuly 2626
ActualActual 3-Month Weighted3-Month WeightedMonthMonth car Salescar 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 ago
6 Sum of weights
© 2008 Prentice Hall, Inc. 4 – 32
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 – 33
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
© 2008 Prentice Hall, Inc. 4 – 34
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 – 35
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 – 36
You’re organizing a planning meeting. You’re organizing a planning meeting. You want to forecast attendance for You want to forecast attendance for 2000 using exponential smoothing 2000 using exponential smoothing ((αα= .10). The1995 forecast was 175.= .10). The1995 forecast was 175.
19951995 1801801996 1996 16816819971997 15915919961996 17517519991999 190190
© 1995 Corel Corp.
Exponential Smoothing Exponential Smoothing ExampleExample
© 2008 Prentice Hall, Inc. 4 – 37
Ft = Ft-1 + ·(At-1 - Ft-1)
TimeTime ActualForecast, F t
(αα = = .10.10))
19951995 180 175.00 (Given)
19961996 168168
19971997 159159
19981998 175175
19991999 190190
20002000 NANA
175.00 +175.00 +
Exponential Smoothing Exponential Smoothing SolutionSolution
© 2008 Prentice Hall, Inc. 4 – 38
Ft = Ft-1 + ·(At-1 - Ft-1)
TimeTime ActualActualForecast, Forecast, FFtt
((αα= = .10.10))
19951995 180180 175.00 (Given)175.00 (Given)
19961996 168168 175.00 +175.00 + .10.10(180 (180 - 175.00- 175.00)) = 175.50 = 175.50
19971997 159159
19981998 175175
19991999 190190
20002000 NANA
Exponential Smoothing Exponential Smoothing SolutionSolution
© 2008 Prentice Hall, Inc. 4 – 39
Ft = Ft-1 + ·(At-1 - Ft-1)
Time ActualForecast, F t
(α = .10)
19951995 180180 175.00 (Given)175.00 (Given)
19961996 168168 175.00 + .10(180 - 175.00) = 175.50175.00 + .10(180 - 175.00) = 175.50
19971997 159159 175.50 + .10(168 - 175.50) = 174.75175.50 + .10(168 - 175.50) = 174.75
19981998 175175 174.75 + .10(159 - 174.75) = 173.18174.75 + .10(159 - 174.75) = 173.18
19991999 190190 173.18 + .10(175 - 173.18) = 173.36173.18 + .10(175 - 173.18) = 173.36
20002000 NANA 173.36173.36 + + .10.10(190(190 - 173.36- 173.36) = 175.02) = 175.02
Exponential Smoothing Exponential Smoothing SolutionSolution
© 2008 Prentice Hall, Inc. 4 – 40
Year
Sales
140150160170180190
93 94 95 96 97 98
Actual
Forecast
Exponential Smoothing Exponential Smoothing GraphGraph
© 2008 Prentice Hall, Inc. 4 – 41
Forecast errorForecast error
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 – 42
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 – 43
Exponential Smoothing with Exponential Smoothing with Trend AdjustmentTrend Adjustment
When a trend is present, exponential When a trend is present, exponential smoothing must be modifiedsmoothing must be modified
Forecast Forecast including including ((FITFITtt)) = = trendtrend
ExponentiallyExponentially ExponentiallyExponentiallysmoothed smoothed ((FFtt)) + + ((TTtt)) smoothedsmoothedforecastforecast trendtrend
© 2008 Prentice Hall, Inc. 4 – 44
Exponential Smoothing with Exponential Smoothing with Trend AdjustmentTrend Adjustment
FFtt = = ((AAtt - 1 - 1) + (1 - ) + (1 - )()(FFtt - 1 - 1 + + TTtt - 1 - 1))
TTtt = = ((FFtt - - FFtt - 1 - 1) + (1 - ) + (1 - ))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 == F Ftt + + TTtt
© 2008 Prentice Hall, Inc. 4 – 45
Exponential Smoothing with Exponential Smoothing with Trend Adjustment ExampleTrend Adjustment Example
ForecastForecastActualActual SmoothedSmoothed SmoothedSmoothed IncludingIncluding
MonthMonth((tt)) Demand Demand ((AAtt)) Forecast, FForecast, Ftt Trend, TTrend, Ttt Trend, FITTrend, FITtt
11 1212 1111 22 13.0013.0022 171733 202044 191955 242466 212177 313188 282899 3636
1010
Table 4.1Table 4.1
© 2008 Prentice Hall, Inc. 4 – 46
Exponential Smoothing with Exponential Smoothing with Trend Adjustment ExampleTrend Adjustment Example
ForecastForecastActualActual SmoothedSmoothed SmoothedSmoothed IncludingIncluding
MonthMonth((tt)) Demand Demand ((AAtt)) Forecast, FForecast, Ftt Trend, TTrend, Ttt Trend, FITTrend, FITtt
11 1212 1111 22 13.0013.0022 171733 202044 191955 242466 212177 313188 282899 3636
1010
Table 4.1Table 4.1
F2 = A1 + (1 - )(F1 + T1)
F2 = (.2)(12) + (1 - .2)(11 + 2)
= 2.4 + 10.4 = 12.8 units
Step 1: Forecast for Month 2
© 2008 Prentice Hall, Inc. 4 – 47
Exponential Smoothing with Exponential Smoothing with Trend Adjustment ExampleTrend Adjustment Example
ForecastForecastActualActual SmoothedSmoothed SmoothedSmoothed IncludingIncluding
MonthMonth((tt)) Demand Demand ((AAtt)) Forecast, FForecast, Ftt Trend, TTrend, Ttt Trend, FITTrend, FITtt
11 1212 1111 22 13.0013.0022 1717 12.8012.8033 202044 191955 242466 212177 313188 282899 3636
1010
Table 4.1Table 4.1
T2 = (F2 - F1) + (1 - )T1
T2 = (.4)(12.8 - 11) + (1 - .4)(2)
= .72 + 1.2 = 1.92 units
Step 2: Trend for Month 2
© 2008 Prentice Hall, Inc. 4 – 48
Exponential Smoothing with Exponential Smoothing with Trend Adjustment ExampleTrend Adjustment Example
ForecastForecastActualActual SmoothedSmoothed SmoothedSmoothed IncludingIncluding
MonthMonth((tt)) Demand Demand ((AAtt)) Forecast, FForecast, Ftt Trend, TTrend, Ttt Trend, FITTrend, FITtt
11 1212 1111 22 13.0013.0022 1717 12.8012.80 1.921.9233 202044 191955 242466 212177 313188 282899 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
© 2008 Prentice Hall, Inc. 4 – 49
Exponential Smoothing with Exponential Smoothing with Trend Adjustment ExampleTrend Adjustment Example
ForecastForecastActualActual SmoothedSmoothed SmoothedSmoothed IncludingIncluding
MonthMonth((tt)) Demand Demand ((AAtt)) Forecast, FForecast, Ftt Trend, TTrend, Ttt Trend, FITTrend, FITtt
11 1212 1111 22 13.0013.0022 1717 12.8012.80 1.921.92 14.7214.7233 202044 191955 242466 212177 313188 282899 3636
1010
Table 4.1Table 4.1
15.1815.18 2.102.10 17.2817.2817.8217.82 2.322.32 20.1420.1419.9119.91 2.232.23 22.1422.1422.5122.51 2.382.38 24.8924.8924.1124.11 2.072.07 26.1826.1827.1427.14 2.452.45 29.5929.5929.2829.28 2.322.32 31.6031.6032.4832.48 2.682.68 35.1635.16
© 2008 Prentice Hall, Inc. 4 – 50
Exponential Smoothing with Exponential Smoothing with Trend Adjustment ExampleTrend Adjustment Example
Figure 4.3Figure 4.3
| | | | | | | | |
11 22 33 44 55 66 77 88 99
Time (month)Time (month)
Pro
du
ct d
eman
dP
rod
uct
dem
and
35 35 –
30 30 –
25 25 –
20 20 –
15 15 –
10 10 –
5 5 –
0 0 –
Actual demand Actual demand ((AAtt))
Forecast including trend Forecast including trend ((FITFITtt))
withwith = .2 = .2 andand = .4 = .4
© 2008 Prentice Hall, Inc. 4 – 51
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 – 52
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 – 53
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 – 54
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 – 55
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 – 56
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 – 57
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 – 59
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 – 60
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 – 61
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 – 62
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
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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 – 64
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
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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
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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 – 67
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
^̂
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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
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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
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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
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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
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Correlation CoefficientCorrelation Coefficient
r = r = nnxyxy - - xxy y
[[nnxx22 - ( - (xx))22][][nnyy22 - ( - (yy))22]]
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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
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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 – 75
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 – 76
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 – 77
Adaptive ForecastingAdaptive Forecasting
It’s possible to use the computer to It’s possible to use the computer to continually monitor forecast error and continually monitor forecast error and adjust the values of the adjust the values of the and and coefficients used in exponential coefficients used in exponential smoothing to continually minimize smoothing to continually minimize forecast errorforecast error
This technique is called adaptive This technique is called adaptive smoothingsmoothing