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To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna

5-1 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Chapter 5Chapter 5

ForecastingForecasting

Prepared by Lee Revere and John LargePrepared by Lee Revere and John Large



Learning ObjectivesLearning ObjectivesStudents will be able to:

1. Understand and know when to use various families of forecasting models.

2. Compare moving averages, exponential smoothing, and trend time-series models.

3. Seasonally adjust data.4. Understand Delphi and other

qualitative decision-making approaches.

5. Compute a variety of error measures.



Chapter OutlineChapter Outline5.1 Introduction5.2 Types of Forecasts5.3 Scatter Diagrams and Time

Series5.4 Measures of Forecast Accuracy5.5 Time-Series Forecasting Models5.6 Monitoring and Controlling

Forecasts5.7 Using the Computer to Forecast



IntroductionIntroduction

Eight steps to forecasting:1. Determine the use of the forecast.2. Select the items or quantities to be

forecasted.3. Determine the time horizon of the

forecast.4. Select the forecasting model or

models.5. Gather the data needed to make the

forecast.6. Validate the forecasting model.7. Make the forecast.8. Implement the results.

These steps provide a systematic way of initiating, designing, and implementing a forecasting system.



Types of ForecastsTypes of Forecasts

MovingAverage

Exponential Smoothing

Trend Projections

Time-Series Methods: include historical data over a time interval

Forecasting Techniques

No single method is superior

Delphi Methods

Jury of Executive Opinion

Sales ForceComposite

Consumer Market Survey

Qualitative Models: attempt to include subjective factors

Causal Methods: include a variety of factors

Regression Analysis

Multiple Regression

Decomposition



Qualitative MethodsQualitative Methods Delphi Method

interactive group process consisting of obtaining information from a group of respondents through questionnaires and surveys

Jury of Executive Opinionobtains opinions of a small group of high-level managers in combination with statistical models

Sales Force Compositeallows each sales person to estimate the sales for his/her region and then compiles the data at a district or national level

Consumer Market Surveysolicits input from customers or potential customers regarding their future purchasing plans



Scatter DiagramsScatter Diagrams

050

100150200250300350400450

0 2 4 6 8 10 12

Time (Years)

Annu

al S

ales

Radios

Televisions

Compact Discs

Scatter diagrams are helpful when forecasting time-series data because they depict the relationship between variables.



Measures of Forecast Measures of Forecast AccuracyAccuracy

Forecast errors allow one to see how well the forecast model works andcompare that model with other forecast models.

Forecast error = actual value – forecast value



Measures of Forecast Measures of Forecast Accuracy Accuracy (continued)(continued)

Measures of forecast accuracy include: Mean Absolute Deviation (MAD)

Mean Squared Error (MSE)

Mean Absolute Percent Error (MAPE)

= |forecast errors| n

= (errors) n

= actual n

100%

error

2



Hospital Days – Forecast Hospital Days – Forecast Error ExampleError Example

Ms. Smith forecastedtotal hospitalinpatient days lastyear. Now that theactual data areknown, she isreevaluatingher forecastingmodel. Compute theMAD, MSE, andMAPE for herforecast.

Month Forecast ActualJAN 250 243

FEB 320 315

MAR 275 286

APR 260 256

MAY 250 241

JUN 275 298

JUL 300 292

AUG 325 333

SEP 320 326

OCT 350 378

NOV 365 382

DEC 380 396



Hospital Days – Hospital Days – Forecast Error ExampleForecast Error Example

Forecast Actual |error| error^2 |error/actual|JAN 250 243 7 49 0.03FEB 320 315 5 25 0.02MAR 275 286 11 121 0.04APR 260 256 4 16 0.02MAY 250 241 9 81 0.04JUN 275 298 23 529 0.08JUL 300 292 8 64 0.03AUG 325 333 8 64 0.02SEP 320 326 6 36 0.02OCT 350 378 28 784 0.07NOV 365 382 17 289 0.04DEC 380 396 16 256 0.04AVERAGE

11.83 192.83 3.68MAD = MSE = MAPE = .0368*100

=



Decomposition of a Time-Decomposition of a Time-SeriesSeries

Time series can be decomposed into: Trend (T): gradual up or down

movement over time Seasonality (S): pattern of

fluctuations above or below trend line that occurs every year

Cycles(C): patterns in data that occur every several years

Random variations (R): “blips”in the data caused by chance and unusual situations



Components of Components of DecompositionDecomposition

-150-5050

150250350450550650

0 1 2 3 4 5

Time (Years)

Dem

and

Trend

Actual Data

Cyclic

Random



Decomposition of Time-Decomposition of Time-Series: Two ModelsSeries: Two Models

Multiplicative model assumes demandis the product of the four components.

demand = T * S * C * R

Additive model assumes demand is thesummation of the four components.

demand = T + S + C + R



Moving AveragesMoving Averages

n

Simple moving average = demand in previous n periods

Moving average methods consist of computing an average of the most recent n data values for the time series and using this average for the forecast of the next period.



Wallace Garden Supply’s Wallace Garden Supply’s Three-Month Moving AverageThree-Month Moving Average

Month Actual Shed Sales

Three-Month Moving Average

January 10

February 12

March 13

April 16

May 19

June 23

July 26

(10+12+13)/3 = 11 2/3

(12+13+16)/3 = 13 2/3

(13+16+19)/3 = 16

(16+19+23)/3 = 19 1/3



Weighted Moving Weighted Moving AveragesAverages

Weighted moving averages use weights to put more emphasis on recent periods.

(weight for period n) (demand in period n) ∑ weights

Weighted moving average =



Calculating Weighted Calculating Weighted Moving AveragesMoving AveragesWeights Applied Period

3 Last month

2 Two months ago

1 Three months ago

3*Sales last month +

2*Sales two months ago +1*Sales three months ago

6 Sum of weights



Wallace Garden’s Weighted Wallace Garden’s Weighted Three-Month Moving AverageThree-Month Moving Average

Month ActualShedSales

Three-Month Weighted Moving Average

10

12

13

16

19

23

January

February

March

April

May

June

July 26

[3*13+2*12+1*10]/6 = 12 1/6

[3*16+2*13+1*12]/6 =14 1/3

[3*19+2*16+1*13]/6 = 17

[3*23+2*19+1*16]/6 = 20 1/2



Exponential SmoothingExponential Smoothing

Exponential smoothing is a type ofmoving average technique that involveslittle record keeping of past data.

New forecast = previous forecast + (previous actual –previous

forecast)

Mathematically this is expressed as:

Ft = Ft-1 + (Yt-1 - Ft-1)

Ft-1 = previous forecast = smoothing constant

Ft = new forecast

Yt-1 = previous period actual



Port of Baltimore Port of Baltimore Exponential Smoothing Exponential Smoothing

ExampleExample

Qtr Actual Tonnage Unloaded

Rounded Forecast using =0.10

1 180 175

2 168 176= 175.00+0.10(180-175)

3 159 175 =175.50+0.10(168-175.50)

4 175 173 =174.75+0.10(159-174.75)

5 190 173 =173.18+0.10(175-173.18)

6 205 175 =173.36+0.10(190-173.36)

7 180 178 =175.02+0.10(205-175.02)

8 182 178 =178.02+0.10(180-178.02)

9 ? 179= 178.22+0.10(182-178.22)



Port of Baltimore Port of Baltimore Exponential Smoothing Exponential Smoothing

ExampleExample

Qtr Actual Tonnage Unloaded

Rounded Forecast using =0.50

1 180 175

2 168 178 =175.00+0.50(180-175)

3 159 173 =177.50+0.50(168-177.50)

4 175 166 =172.75+0.50(159-172.75)

5 190 170 =165.88+0.50(175-165.88)

6 205 180 =170.44+0.50(190-170.44)

7 180 193 =180.22+0.50(205-180.22)

8 182 186 =192.61+0.50(180-192.61)

9 ? 184 =186.30+0.50(182-186.30)



Selecting a Smoothing Selecting a Smoothing ConstantConstant

ActualForecast with

a = 0.10AbsoluteDeviations

Forecast with a = 0.50

AbsoluteDeviations

180 175 5 175 5

168 176 8 178 10

159 175 16 173 14

175 173 2 166 9

190 173 17 170 20

205 175 30 180 25

180 178 2 193 13

182 178 4 186 4

MAD 10.0 12

To select the best smoothing constant, evaluate the accuracy of each forecasting model.

The lowest MAD results from = 0.10



PM Computer: Moving PM Computer: Moving Average ExampleAverage Example

PM Computer assembles customized personal computers from generic parts.

The owners purchase generic computer parts in volume at a discount from a variety of sources whenever they see a good deal.

It is important that they develop a good forecast of demand for their computers so they can purchase component parts efficiently.



PM Computers: DataPM Computers: DataPeriod month actual demand1 Jan 372 Feb 403 Mar 414 Apr 375 May 456 June 507 July 438 Aug 479 Sept 56

Compute a 2-month moving average Compute a 3-month weighted average using weights

of 4,2,1 for the past three months of data Compute an exponential smoothing forecast using

= 0.7 Using MAD, what forecast is most accurate?



PM Computers: Moving PM Computers: Moving Average SolutionAverage Solution

2 month MA Abs. Dev 3 month WMA Abs. Dev Exp.Sm. Abs. Dev

37.00

37.00 3.00

38.50 2.50 39.10 1.90

40.50 3.50 40.14 3.14 40.43 3.43

39.00 6.00 38.57 6.43 38.03 6.97

41.00 9.00 42.14 7.86 42.91 7.09

47.50 4.50 46.71 3.71 47.87 4.87

46.50 0.50 45.29 1.71 44.46 2.54

45.00 11.00 46.29 9.71 46.24 9.76

51.50 51.57 53.075.29 5.43 4.95MAD

Exponential smoothing resulted in the lowest MAD.



Exponential Smoothing Exponential Smoothing with Trend Adjustmentwith Trend Adjustment

Simple exponential smoothing - first-order smoothing

Trend adjusted smoothing - second-order smoothing

Low gives less weight to more recent trends, while high gives higher weight to more recent trends.

Simple exponential smoothing fails to respond to trends, so a more complex model is necessary with trend adjustment.



Exponential Smoothing Exponential Smoothing with Trend Adjustmentwith Trend Adjustment

Forecast including trend (FITt+1) = new forecast (Ft) + trend correction(Tt)

where

Tt = (1 - )Tt-1 + (Ft – Ft-1)

Ti = smoothed trend for period 1

Ti-1 = smoothed trend for the preceding period

= trend smoothing constant

Ft = simple exponential smoothed forecast for period t

Ft-1 = forecast for period t-1



Trend ProjectionTrend ProjectionTrend projections are used to forecasttime-series data that exhibit a lineartrend. Least squares may be used to determine

a trend projection for future forecasts. Least squares determines the trend line

forecast by minimizing the mean squared error between the trend line forecasts and the actual observed values.

The independent variable is the time period and the dependent variable is the actual observed value in the time series.



Trend Projection Trend Projection (continued)(continued)

The formula for the trendprojection is:

Y = b + b X

where: Y = predicted value b1 = slope of the trend line b0 = intercept X = time period (1,2,3…n)

0 1



Midwestern Midwestern Manufacturing Trend Manufacturing Trend Projection ExampleProjection Example

Midwestern Manufacturing Company’s demand for electrical generators over the period of 1996 – 2000 is given below.

Year Time Sales

1996 1 74

1997 2 79

1998 3 80

1999 4 90

2000 5 105

2001 6 142

2002 7 122



Midwestern Midwestern Manufacturing Company Manufacturing Company

Trend SolutionTrend Solution

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.89R Square 0.80Adjusted R Square 0.76Standard Error 12.43Observations 7.00

ANOVAdf SS MS F Sign. F

Regression 1.00 3108.04 3108.04 20.11 0.01Residual 5.00 772.82 154.56Total 6.00 3880.86

Coefficients

Standard Error t Stat P-value

Lower 95%

Intercept 56.71 10.51 5.40 0.00 29.70Time 10.54 2.35 4.48 0.01 4.50

Sales = 56.71 + 10.54 (time)



Midwestern Midwestern Manufacturing’s Trend Manufacturing’s Trend

60708090

100110120130140150160

1993 1994 1995 1996 1997 1998 1999 2000 2001

Forecast points

Trend Line

Actual demand line

Xy 54.1070.56



Seasonal VariationsSeasonal Variations

Seasonal indices can be used to make adjustments in the forecast for seasonality.

A seasonal index indicates how a particular season compares with an average season.

The seasonal index can be found by dividing the average value for a particular season by the average of all the data.



Eichler Supplies: Eichler Supplies: Seasonal Index ExampleSeasonal Index Example

Month Sales Demand

AverageTwo-YearDemand

AverageMonthlyDemand

SeasonalIndex

Year 1

Year 2

80 100 90 94 0.957

75 85 80 94 0.851

80 90 85 94 0.904

90 110 100 94 1.064

Jan

Feb

Mar

Apr

May 115 131 123 94 1.309

… … … … … …

Total Average Demand 1,128Seasonal Index: = Average 2 -year demand/Average monthly demand



Seasonal Variations Seasonal Variations with Trendwith Trend

Steps of Multiplicative Time-Series Model

1. Compute the CMA for each observation.2. Compute seasonal ratio

(observation/CMA).3. Average seasonal ratios to get seasonal

indices.4. If seasonal indices do not add to the

number of seasons, multiply each index by (number of seasons)/(sum of the indices).

Centered Moving Average (CMA) is an approach that prevents a variation due to trend from being incorrectly interpreted as a variation due to the season.



Turner IndustriesTurner Industries Seasonal Seasonal Variations with TrendVariations with Trend

Turner Industries’ sales figures are shown below with the CMA and seasonal ratio.Year Quarter Sales CMA

Seasonal Ratio

1 1 1082 1253 150 132 1.1364 141 134.125 1.051

2 1 116 136.375 0.8512 34 138.875 0.9653 159 141.125 1.1274 152 143 1.063

3 1 123 145.125 0.8482 142 147.187 0.963 1684 165

CMA (qtr 3 / yr 1 ) = .5(108) + 125 + 150 + 141+ .5(116)4

Seasonal Ratio = Sales Qtr 3 = 150 CMA 132



Decomposition Method Decomposition Method with Trend and with Trend and

Seasonal ComponentsSeasonal ComponentsDecomposition is the process ofisolating linear trend and seasonalfactors to develop more accurateforecasts.

There are five steps to decomposition:

1. Compute the seasonal index for each season.2. Deseasonalize the data by dividing

each number by its seasonal index. 3. Compute a trend line with the

deseasonalized data.4. Use the trend line to forecast.5. Multiply the forecasts by the seasonal

index.



Turner Industries: Turner Industries: Decomposition MethodDecomposition Method

Turner Industries has noticed a trend in quarterly sales figures. There is also a seasonal component. Below is the seasonal index and deseasonalized sales data.

Yr Qtr Sales Seasonal

IndexDesasonalized

Sales1 1 108 0.85 127.059

2 125 0.96 130.2083 150 1.13 132.7434 141 1.06 133.019

2 1 116 0.85 136.4712 134 0.96 139.5833 159 1.13 140.7084 152 1.06 143.396

3 1 123 0.85 144.7062 142 0.96 147.9173 168 1.13 148.6734 165 1.06 155.660

*

•This value is derived by averaging the season rations for each quarter. Refer to slide 5-37.

Seasonal Index for Qtr 1 = 0.851+0.848 = 0.85 2

108 0.85=




Using the deseasonalized data, the following trend line was computed:

Sales = 124.78 + 2.34X

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.98981389R Square 0.97973154Adjusted R Square 0.9777047Standard Error 1.26913895Observations 12

ANOVAdf SS MS F Significance F

Regression 1 778.5826246 778.5826246 483.3774 8.49E-10Residual 10 16.10713664 1.610713664Total 11 794.6897613

Coefficients Standard Error t Stat P-value Lower 95%Intercept 124.78 0.781101025 159.8320597 2.26E-18 123.1046Time 2.34 0.10613073 21.98584604 8.49E-10 2.0969




Using the trend line, the following forecast was computed:

Sales = 124.78 + 2.34X

For period 13 (quarter 1/ year 4):

Sales = 124.78 + 2.34 (13) = 155.2 (before seasonality adjustment)

After seasonality adjustment:

Sales = 155.2 (0.85) = 131.92

Seasonal index for quarter 1



Multiple Regression Multiple Regression with Trend and with Trend and

Seasonal ComponentsSeasonal ComponentsMultiple regression can be used todevelop an additive decompositionmodel.

One independent variable is time. Seasons are represented by dummy

independent variables.

Y = a + b X + b X + b X + b X

Where X = time period X = 1 if quarter 2 = 0 otherwise X = 1 if quarter 3 = 0 otherwise X = 1 if quarter 4 = 0 otherwise

1 1 2 2 3 3 4 4

1

2

3

4



Monitoring and Monitoring and Controlling ForecastsControlling Forecasts

n|errorsforecast |Σ MAD

MADi) periodin demandforecast i periodin demand Σ(actual

MADRSFEgnalTrackingSi

where

Tracking signals measure how well predictions fit actual data.



Monitoring and Monitoring and Controlling ForecastsControlling Forecasts

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