Chapter 11 What Is Forecasting?

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2006 - Shuguang Liu

Chapter 11

Forecasting Demand for Services

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What Is Forecasting?Predicting the future

How many people will buy dell dimension 8200 on December?

Estimating/measuring the reliability of the prediction

Forecasts are always wrong… except by accident

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Learning ObjectivesUnderstand sources of demand variability.Able to pick the appropriate forecasting model.Exponential smoothing and moving average.Linear regression and time series.Seasonality.

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How Famous People Feel About Forecasts?"Prediction is very difficult, especially if it's

about the future."--Nils Bohr, Nobel laureate in physics.

"If you have to forecast, forecast often. "--A useful survival tactic for a consultant.

“I think there is a world market for maybe five computers.”

--Thomas Watson (1874-1956), chairman of IBM, 1943.IBM alone produces over 1 million computers per year.

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We Classify ForecastingMethods Into Three Types

JudgmentalUses subjective inputs

Time seriesAssumes past is best predictor of future

AssociativeUses explanatory variables to predict the future

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Judgmental ForecastsJudgmental forecasts are subjective, based on

Executive opinionsSales force compositeConsumer surveysOutside opinionOpinions of managers and staff

Delphi method

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Time Series ForecastsTime series forecasts are often prepared by decomposing the dataTrend

Long-term movement in dataSeasonality

Short-term regular variations in dataIrregular variations

Caused by unusual circumstancesRandom variations

Caused by chance 2006 - Shuguang Liu

Associative ForecastsAssociative forecasts assume a correlation with predictor variablesPredictor variables

Used to predict values of variable interestRegression

Technique for fitting a line to a setLeast squares line

Minimizes sum of squared deviations around the line

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Judgmental ForecastsPros

Easy to prepareCan be used when no appropriate data exist for other methodsRelatively easy to start with new items

ConsAccuracy is poorMore often represent the forecasters’ hopes rather than probable outcomes

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Next, Associative ForecastsPros

Some methods (e.G. Linear regression) are straight forwardUse causal relationshipsUseful for sensitivityProduce measures of correlation

ConsData intensiveComputation intensiveNon-linear methods may not be easyLinear regression may not be feasibleDifficult to start with new items

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Lastly, Time Series ForecastsPros

Methods range from easy to complexSome methods are data and computation efficientDoes not require causal variable identification

ConsSome methods are data and/or computation intensiveCannot incorporate causal/co-relational relationshipsVery difficult to start with new items

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Demand Patterns

Qua

ntity

Time

(a) Horizontal: Data cluster about a horizontal line.

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Demand Patterns

Qua

ntity

Time

(b) Trend: Data consistently increase or decrease.2006 - Shuguang Liu

Demand Patterns

Qua

ntity

| | | | | | | | | | | |J F M A M J J A S O N D

Months

Year 1

Year 2

(c) Seasonal: Data consistently show peaks and valleys.

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Demand Patterns

Qua

ntity

| | | | | |1 2 3 4 5 6

Years(d) Cyclical: Data reveal gradual increases and decreases

over extended periods.

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Demand Patterns

(E) seasonal multiplicative pattern

Period

Dem

and

| | | | | | | | | | | | | | | |0 2 4 5 8 10 12 14 16

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Demand Patterns

(F) seasonal additive patternPeriod

| | | | | | | | | | | | | | | |0 2 4 5 8 10 12 14 16

Dem

and

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Linear RegressionRegression line

Y = a + bxLeast-squares regression line

Predictionr2

Causation and association

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PredictionChoose the explanatory and response variablesPlot the dataLook for a straight-line relationshipFit the data for regression lineSubstitute the value of x and calculate y

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Causal Methods Linear Regression

Dep

ende

nt v

aria

ble

Independent variableX

Y

Actualvalueof Y

Estimate ofY fromregression

equation

Value of X usedto estimate Y

Deviation,or error

{

Regressionequation:Y = a + bX

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Regression LineSummarizes the relationship between two variables.The variables must be explanatory and response variables.Describes how y changes as x takes different values.Use a regression line to predict the value of y for a given value of x.

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Regression Line: EquationDifferent people can draw different regression lines for a given graph

A regression line that is the closest to the points in the vertical direction (y). A way to find an equation for the line.

Least-squares regression lineMinimizes the sum of squares of the vertical distances of the data pointsY = a + bx

A is the y-interceptB is the slope of the line

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Finding the LS EquationSales Advertising

Month (000 units) (000 $)1 264 2.52 116 1.33 165 1.44 101 1.05 209 2.0

a = Y – bX b = ΣXY – nXYΣX 2 – nX 2

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Finding the LS Equation

a = Y – bX b = 1560.8 – 5(1.64)(171)

14.90 – 5(1.64)2

Sales, Y Advertising, XMonth (000 units) (000 $) XY X 2 Y 2

1 264 2.5 660.0 6.25 69,6962 116 1.3 150.8 1.69 13,4563 165 1.4 231.0 1.96 27,2254 101 1.0 101.0 1.00 10,2015 209 2.0 418.0 4.00 43,681

Total 855 8.2 1560.8 14.90 164,259Y = 171 X = 1.64

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Finding the LS Equation

a = – 8.136 b = 109.229


1 264 2.5 660.0 6.25 69,6962 116 1.3 150.8 1.69 13,4563 165 1.4 231.0 1.96 27,2254 101 1.0 101.0 1.00 10,2015 209 2.0 418.0 4.00 43,681

Total 855 8.2 1560.8 14.90 164,259Y = 171 X = 1.64

Y = – 8.136 + 109.229(X)2006 - Shuguang Liu

Forecasting/predictionY = – 8.136 + 109.229(x)

Forecast for month 6:Advertising expenditure = $1750Y = - 8.136 + 109.229(1.75) = 183.015

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Prediction: MoreGiven the equation of the regression line

Y = – 8.136 + 109.229(x)What is the value for the intercept?What does the intercept mean here?What is the value for the slope?What does the slope mean in this example?

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Understanding PredictionPrediction works best when the model fits the data closely.Prediction outside the range of data of available data is risky (extrapolation).Check for outliers

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What is r2

The square of the correlation is the fraction of the variation in the values of y that is explained by xThe variation that occurs in y can be attributed to the changing x values

variation in predicted y as x pulls it along the line

total variation in observed values of yr2 =

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Using r2

Correlation tells how close the points are to the liner2 is the measure of how successful the regression was in explaining the response

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Finding r and r2


1 264 2.5 660.0 6.25 69,6962 116 1.3 150.8 1.69 13,4563 165 1.4 231.0 1.96 27,2254 101 1.0 101.0 1.00 10,2015 209 2.0 418.0 4.00 43,681

Total 855 8.2 1560.8 14.90 164,259Y = 171 X = 1.64

nΣXY – ΣX ΣY

[nΣX 2 – (ΣX) 2][nΣY 2 – (ΣY) 2]r =

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Finding r and r2


1 264 2.5 660.0 6.25 69,6962 116 1.3 150.8 1.69 13,4563 165 1.4 231.0 1.96 27,2254 101 1.0 101.0 1.00 10,2015 209 2.0 418.0 4.00 43,681

Total 855 8.2 1560.8 14.90 164,259Y = 171 X = 1.64

r = 0.98 r 2 = 0.96 σYX = 15.61

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r2: ExampleGiven the numerical value of the correlation for the sales/advertising data

r = 0.98What percent of variation in sales is explained by how much the advertising expenditure is?

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The Question of CausationA strong relationship between two variables does not always mean that changes in one variable cause changes in the other. The relationship between two variables is often influenced by other variables in the background.The best evidence for causation comes form randomized comparative experiments.

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Association and CausationA high association can mean one of the following scenarios:

CausationCommon responseConfounding

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Now We’re Going to Focus on Time Series Methods

Quantitative forecasting methodsForecasts are not based on predictive variablesInherent assumption: the past is the best predictor of the future

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Time Series ForecastsTime series forecasts can address all of these components

TrendSeasonalityCyclesRandom variations

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Time Series ForecastsMost time series methods involve some form of averagingMoving average (MA)Weighted moving average (WMA)Exponential smoothing (ES)Trend-adjusted ES

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Basic NotationsD1 ,D2 , Dt: the demand of values observed during periods 1,2,...,t.Ft : the forecast made for period t in period t-1 before Dt is observed

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Naive Forecast MethodWhat is the forecast for month 5?

1 800

2 740

3 810

Month Customer arrivals

4 790

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Simple Moving AveragesThe procedure

Dt: actual demand in period tn: number of periods in the average

Ft+1 =Dt + Dt−1 + ... + Dt−n +1

n

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Moving AveragesThe main idea behind a MA is to smooth random variations

Uses a number of the most recent data valuesEach data value has equal weight

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Simple Moving Averages

Week

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Actual patientarrivals

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rriv

als

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PatientWeek Arrivals

1 4002 3803 411

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rriv

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Week


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1 4002 3803 411

F4 = 411 + 380 + 4003

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1 4002 3803 411

F4 = 397.0

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Week

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2 3803 4114 415

F5 = 415 + 411 + 3803

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2 3803 4114 415

F5 = 402.0

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Implications of Increasing Ndifferences in responsiveness

Week

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3-week MAforecast

6-week MAforecast

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Simple Moving Average

1 800

2 740

3 810


4 790

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Weighted Moving Average

Wi1

n∑ = 1

11211 ... +−−+ +++= ntnttt DWDWDWF

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Week

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3-week MAforecast

6-week MAforecastWeighted Moving Average

Assigned weightst 0.70

t-1 0.20t-2 0.10

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Weighted Moving Average450 —

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3-week MAforecast



t-1 0.20t-2 0.10

F4 = 0.70(411) + 0.20(380) +0.10(400)

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430 —

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3-week MAforecast



t-1 0.20t-2 0.10

F4 = 403.7

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430 —

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3-week MAforecast



t-1 0.20t-2 0.10

F4 = 404 F5 = 410.7

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1 800

2 740

3 810


4 790

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Moving Averages (MA & WMA)

Let’s review what we know about moving averages (MA & WMA)

We need n data values for an n-period moving averageWe need n weighting factors for an n-period weighted moving averageThe responsiveness of the forecast is determined by n

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Exponential Smoothing (ES)Now let’s look at exponential smoothing (ES)Requires only three parameters

Previous forecastCurrent data valueSmoothing factor (often referred to as a smoothing constant)

Smoothing controls responsiveness2006 - Shuguang Liu

Exponential Smoothing

1 and 0between valueawith parameter smoothing a :)(1

αα tttt FDFF −+=+

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Exponential SmoothingIs just a weighted average between the old forecast (old ES average) and the new data valueEfficiently incorporates more distant data values through the old forecast

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Exponential Smoothing450 —

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Exponential Smoothingα = 0.10

Ft +1 = Ft + α (Dt – Ft )

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430 —

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F4 = 0.10(411) + 0.90(390)

F3 = (400 + 380)/2D3 = 411

Ft +1 = Ft + α (Dt – Ft )

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F4 = 392.1


F3 = (400 + 380)/2D3 = 411

Ft +1 = Ft + α (Dt – Ft )

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Week

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F4 = 392.1D4 = 415


F4 = 392.1 F5 = 394.4

Ft +1 = Ft + α (Dt – Ft )

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Exponential smoothing

α = 0.10

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430 —

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3-week MAforecast

6-week MAforecast

Exponential smoothing

α = 0.10

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1 800

2 740

3 810


4 790

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Exponential SmoothingAdvantages.

Simplicity.Minimal data requirements compared to moving average.Inexpensive.

Disadvantages.Lags behind changes in underlying average.Does not account for any factors other than the series past performance.

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Exponential SmoothingRequire only three parameters.Are controlled by the smoothing factor on interval [0,1].

Values close to 0 (large n) provide very stable (unresponsive) forecasts.Values close to 1 (small n) provide very aggressive (responsive) forecasts.

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Moving Average and ESWe can approximate moving average with ES!

12+

≈n

α

αα−

≈2n

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Trend-adjusted ESSmooth estimates for the series average as well as the trend

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Trend-adjusted ES

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Actual blood test requests

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Trend-adjusted ES

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Trend-adjusted forecast


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Trend-adjusted ES

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Number of time periods 15.00Demand smoothing coefficient (α ) 0.20Initial demand value 28.00Trend-smoothing coefficient (β ) 0.20Estimate of trend 3.00

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Trend-adjusted ES

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Smoothed Trend ForecastWeek Arrivals Average Average Forecast Error

0 28 28.00 3.00 0.00 0.001 27 30.20 2.84 31.00 –4.002 44 35.23 3.27 33.04 10.963 37 38.20 3.21 38.51 –1.514 35 40.14 2.96 41.42 –6.425 53 45.08 3.35 43.10 9.896 38 46.35 2.93 48.43 –10.437 57 50.83 3.24 49.29 7.718 61 55.46 3.52 54.08 6.929 39 54.99 2.72 58.98 –19.98

10 55 57.17 2.61 57.71 –2.7111 54 58.63 2.38 59.78 –5.7812 52 59.21 2.02 61.01 –9.0113 60 60.99 1.97 61.23 –1.2314 60 62.37 1.85 62.96 –2.9615 75 66.38 2.28 64.22 10.77

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Trend-adjusted ES

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Smoothed Trend ForecastWeek Arrivals Average Average Forecast Error

0 28 28.00 3.00 0.00 0.001 27 30.20 2.84 31.00 –4.002 44 35.23 3.27 33.04 10.963 37 38.20 3.21 38.51 –1.514 35 40.14 2.96 41.42 –6.425 53 45.08 3.35 43.10 9.896 38 46.35 2.93 48.43 –10.437 57 50.83 3.24 49.29 7.718 61 55.46 3.52 54.08 6.929 39 54.99 2.72 58.98 –19.98

10 55 57.17 2.61 57.71 –2.7111 54 58.63 2.38 59.78 –5.7812 52 59.21 2.02 61.01 –9.0113 60 60.99 1.97 61.23 –1.2314 60 62.37 1.85 62.96 –2.9615 75 66.38 2.28 64.22 10.77

SUMMARY

Average demand 49.80Mean square error 76.13Mean absolute deviation 7.35Forecast for week 16 68.66Forecast for week 17 70.95Forecast for week 18 73.24

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Trend-adjusted ESThe forecaster for Canine Gourmet dog breath fresheners estimated (in March) the series average to be 300,000 cases sold per month and the trend to be +8,000 per month. The actual sales for April were 330,000 cases. What are the forecasts for May and July, assuming alpha 0.2 and beta 0.1?

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Trend-adjusted ES

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Seasonal Patterns

Period

Dem

and

(a) Multiplicative pattern

| | | | | | | | | | | | | | | |0 2 4 6 8 10 12 14 16

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Seasonal Patterns

Period

| | | | | | | | | | | | | | | |0 2 4 6 8 10 12 14 16

Dem

and

(b) Additive pattern

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Seasonal InfluencesMultiplicative seasonal method

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Quarter Year 1 Year 2 Year 3 Year 41 45 70 100 1002 335 370 585 7253 520 590 830 11604 100 170 285 215

Total 1000 1200 1800 2200Average 250 300 450 550

Seasonal Influences

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Seasonal Patterns

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Forecast ErrorForecast error is the difference between the actual demand and forecast.

Et = Dt – Ft

Cumulative sum of forecast errors (CFE).Useful for bias measurement.Used in tracking signals.

Average forecast error equals CFE divided by the number of data points.

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Forecast ErrorMean absolute deviation (MAD)

Measures dispersion of forecast errors Easier for managers to understand than standard deviation

Standard deviation (σ)Measures dispersion of forecast errors Want to have small values of σMAD = 0.8 σσ = 1.25MAD

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Forecast ErrorMean squared error (MSE).

Measures dispersion of forecast errors.Places greater weight than MAD on large errors.

Mean absolute percent error (MAPE) Relates forecast error to level of demand Puts the size of a forecast error in proper perspective

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Forecast Error

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Absolute Error Absolute Percent

Month, Demand, Forecast, Error, Squared, Error, Error, t Dt Ft Et Et

2 |Et| (|Et|/Dt)(100)1 200 225 -25 625 25 12.5% 2 240 220 20 400 20 8.3 3 300 285 15 225 15 5.0 4 270 290 –20 400 20 7.4 5 230 250 –20 400 20 8.7 6 260 240 20 400 20 7.7 7 210 250 –40 1600 40 19.0 8 275 240 35 1225 35 12.7

Total –15 5275 195 81.3%

Forecast Error

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Forecast ErrorAbsolute

Error Absolute Percent Month, Demand, Forecast, Error, Squared, Error, Error,

t Dt Ft Et Et2

|Et| (|Et|/Dt)(100)1 200 225 –25 625 25 12.5% 2 240 220 20 400 20 8.3 3 300 285 15 225 15 5.0 4 270 290 –20 400 20 7.4 5 230 250 –20 400 20 8.7 6 260 240 20 400 20 7.7 7 210 250 –40 1600 40 19.0 8 275 240 35 1225 35 12.7

Total –15 5275 195 81.3%

MSE = = 659.452758

CFE = – 15

Measures of Error

MAD = = 24.41958

MAPE = = 10.2%81.3%8

E = = – 1.875– 15 8

σ = 27.4

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Criteria for Selecting Time Series Methods

Statistical criteria Minimize bias. Minimize MAD or MSE. Meet managerial expectations of changes.Minimize the forecast error last period. Use a holdout set (data from more recent periods) as a final test, after constructing model from data from earlier periods.

Chapter 11 What Is Forecasting?

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Transcript of Chapter 11 What Is Forecasting?