05-forecasting.ppt

© 2008 Prentice-Hall, Inc.

Chapter 5

Forecasting

© 2009 Prentice-Hall, Inc.

© 2009 Prentice-Hall, Inc. 5 – 2

Introduction

Managers are always trying to reduce uncertainty and make better estimates of what will happen in the future

This is the main purpose of forecasting Some firms use subjective methods

Seat-of-the pants methods, intuition, experience

There are also several quantitative techniques Moving averages, exponential smoothing,

trend projections, least squares regression analysis


Introduction Eight steps to forecasting :

1. Determine the use of the forecast—what objective are we trying to obtain?

2. Select the items or quantities that are to be forecasted

3. Determine the time horizon of the forecast4. Select the forecasting model or models5. Gather the data needed to make the

forecast6. Validate the forecasting model7. Make the forecast8. Implement the results


Introduction These steps are a systematic way of initiating,

designing, and implementing a forecasting system

When used regularly over time, data is collected routinely and calculations performed automatically

There is seldom one superior forecasting system

Different organizations may use different techniques

Whatever tool works best for a firm is the one they should use


Regression Analysis

Multiple Regression

MovingAverage

Exponential Smoothing

Trend Projections

Decomposition

Delphi Methods

Jury of Executive Opinion

Sales ForceComposite

Consumer Market Survey

Time-Series Methods

Qualitative Models

Causal Methods

Forecasting ModelsForecasting Techniques

Figure 5.1


Time-Series Models

Time-series models attempt to predict the future based on the past

Common time-series models are Moving average Exponential smoothing Trend projections Decomposition

Regression analysis is used in trend projections and one type of decomposition model


Causal Models

Causal modelsCausal models use variables or factors that might influence the quantity being forecasted

The objective is to build a model with the best statistical relationship between the variable being forecast and the independent variables

Regression analysis is the most common technique used in causal modeling


Qualitative Models

Qualitative modelsQualitative models incorporate judgmental or subjective factors

Useful when subjective factors are thought to be important or when accurate quantitative data is difficult to obtain

Common qualitative techniques are Delphi method Jury of executive opinion Sales force composite Consumer market surveys


Qualitative Models

Delphi MethodDelphi Method – an iterative group process where (possibly geographically dispersed) respondentsrespondents provide input to decision makersdecision makers

Jury of Executive OpinionJury of Executive Opinion – collects opinions of a small group of high-level managers, possibly using statistical models for analysis

Sales Force Composite Sales Force Composite – individual salespersons estimate the sales in their region and the data is compiled at a district or national level

Consumer Market SurveyConsumer Market Survey – input is solicited from customers or potential customers regarding their purchasing plans


Scatter 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.


Scatter Diagrams Wacker Distributors wants to forecast sales for

three different products

YEAR TELEVISION SETS RADIOS COMPACT DISC PLAYERS1 250 300 1102 250 310 1003 250 320 1204 250 330 1405 250 340 1706 250 350 1507 250 360 1608 250 370 1909 250 380 20010 250 390 190

Table 5.1


Scatter Diagrams

Figure 5.2

330 –

250 –

200 –

150 –

100 –

50 –| | | | | | | | | |

0 1 2 3 4 5 6 7 8 9 10Time (Years)

Ann

ual S

ales

of T

elev

isio

ns

(a) Sales appear to be

constant over timeSales = 250

A good estimate of sales in year 11 is 250 televisions


Scatter Diagrams

Sales appear to be increasing at a constant rate of 10 radios per year

Sales = 290 + 10(Year) A reasonable

estimate of sales in year 11 is 400 televisions

420 –400 –380 –360 –340 –320 –300 –280 –

| | | | | | | | | |

0 1 2 3 4 5 6 7 8 9 10Time (Years)

Ann

ual S

ales

of R

adio

s

(b)

Figure 5.2


Scatter Diagrams

This trend line may not be perfectly accurate because of variation from year to year

Sales appear to be increasing

A forecast would probably be a larger figure each year

200 –

180 –

160 –

140 –

120 –

100 –

| | | | | | | | | |

0 1 2 3 4 5 6 7 8 9 10Time (Years)

Ann

ual S

ales

of C

D P

laye

rs

(c)

Figure 5.2


Measures of Forecast Accuracy

We compare forecasted values with actual values to see how well one model works or to compare models

Forecast error = Actual value – Forecast value

One measure of accuracy is the mean absolutemean absolute deviationdeviation (MADMAD)

n

errorforecast MAD


Measures of Forecast Accuracy Using a naïvenaïve forecasting model

YEAR

ACTUAL SALES OF CD

PLAYERS FORECAST SALES

ABSOLUTE VALUE OF ERRORS (DEVIATION), (ACTUAL – FORECAST)

1 110 — —2 100 110 |100 – 110| = 103 120 100 |120 – 110| = 204 140 120 |140 – 120| = 205 170 140 |170 – 140| = 306 150 170 |150 – 170| = 207 160 150 |160 – 150| = 108 190 160 |190 – 160| = 309 200 190 |200 – 190| = 10

10 190 200 |190 – 200| = 1011 — 190 —

Sum of |errors| = 160MAD = 160/9 = 17.8

Table 5.2


Measures of Forecast Accuracy Using a naïvenaïve forecasting model

YEAR

ACTUAL SALES OF CD

PLAYERS FORECAST SALES

ABSOLUTE VALUE OF ERRORS (DEVIATION), (ACTUAL – FORECAST)

1 110 — —2 100 110 |100 – 110| = 103 120 100 |120 – 110| = 204 140 120 |140 – 120| = 205 170 140 |170 – 140| = 306 150 170 |150 – 170| = 207 160 150 |160 – 150| = 108 190 160 |190 – 160| = 309 200 190 |200 – 190| = 10

10 190 200 |190 – 200| = 1011 — 190 —

Sum of |errors| = 160MAD = 160/9 = 17.8

Table 5.2

8179

160errorforecast .MAD

n



There are other popular measures of forecast accuracy

The mean squared errormean squared error

n

2error)(MSE

The mean absolute percent errormean absolute percent error

%MAPE 100actualerror

n

And biasbias is the average error and tells whether the forecast tends to be too high or too low and by how much. Thus, it can be negative or positive.



Year Actual CD Sales Forecast Sales |Actual -Forecast|

1 1102 100 110 103 120 100 204 140 120 205 170 140 306 150 170 207 160 150 108 190 160 309 200 190 1010 190 200 10

11 190

Sum of |errors| 160MAD 17.8


Hospital Days Forecast Error Example

Ms. Smith forecasted total hospital inpatient days last year. Now that the actual data are known, she is reevaluating her forecasting model. Compute the MAD, MSE, and MAPE for her forecast.

Month Forecast ActualJAN 250 243FEB 320 315MAR 275 286APR 260 256MAY 250 241JUN 275 298JUL 300 292AUG 325 333SEP 320 326OCT 350 378NOV 365 382DEC 380 396


Hospital Days Forecast Error Example

Forecast Actual |error| error2 |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 0.04

AVERAGEMAD= 11.83

MSE=192.83

MAPE=.0381*100 = 3.81


Time-Series Forecasting Models

A time series is a sequence of evenly spaced events (weekly, monthly, quarterly, etc.)

Time-series forecasts predict the future based solely of the past values of the variable

Other variables, no matter how potentially valuable, are ignored


Decomposition of a Time-Series

A time series typically has four components1.1. TrendTrend (TT) is the gradual upward or

downward movement of the data over time2.2. SeasonalitySeasonality (SS) is a pattern of demand

fluctuations above or below trend line that repeats at regular intervals

3.3. CyclesCycles (CC) are patterns in annual data that occur every several years

4.4. Random variationsRandom variations (RR) are “blips” in the data caused by chance and unusual situations



Average Demand over 4 Years

Trend Component

Actual Demand

Line

Time

Dem

and

for P

rodu

ct o

r Ser

vice

| | | |

Year Year Year Year1 2 3 4

Seasonal Peaks

Figure 5.3



There are two general forms of time-series models

The multiplicative modelDemand = T x S x C x R

The additive modelDemand = T + S + C + R

Models may be combinations of these two forms

Forecasters often assume errors are normally distributed with a mean of zero


Moving Averages

Moving averagesMoving averages can be used when demand is relatively steady over time

The next forecast is the average of the most recent n data values from the time series

The most recent period of data is added and the oldest is droppedThis methods tends to smooth out short-term irregularities in the data series

nnperiods previous in demands of Sumforecast average Moving


Moving Averages

Mathematically

nYYYF nttt

t11

1

...

where= forecast for time period t + 1= actual value in time period tn= number of periods to average

tY1tF


Wallace Garden Supply Example

Wallace Garden Supply wants to forecast demand for its Storage Shed

They have collected data for the past year

They are using a three-month moving average to forecast demand (n = 3)



Table 5.3

MONTH ACTUAL SHED SALES THREE-MONTH MOVING AVERAGEJanuary 10February 12March 13April 16May 19June 23July 26August 30September 28October 18November 16December 14January —

(12 + 13 + 16)/3 = 13.67(13 + 16 + 19)/3 = 16.00(16 + 19 + 23)/3 = 19.33(19 + 23 + 26)/3 = 22.67(23 + 26 + 30)/3 = 26.33(26 + 30 + 28)/3 = 28.00(30 + 28 + 18)/3 = 25.33(28 + 18 + 16)/3 = 20.67(18 + 16 + 14)/3 = 16.00

(10 + 12 + 13)/3 = 11.67


Weighted Moving Averages Weighted moving averagesWeighted moving averages use weights to put

more emphasis on recent periods Often used when a trend or other pattern is

emerging

)(

))((Weights

period in value Actual period inWeight 1

iFt

Mathematically

n

ntnttt www

YwYwYwF

...

...21

11211

wherewi= weight for the ith observation


Weighted Moving Averages Both simple and weighted averages are

effective in smoothing out fluctuations in the demand pattern in order to provide stable estimates

ProblemsIncreasing the size of n smoothes out fluctuations better, but makes the method less sensitive to real changes in the dataMoving averages can not pick up trends very well – they will always stay within past levels and not predict a change to a higher or lower level



Wallace Garden Supply decides to try a weighted moving average model to forecast demand for its Storage Shed

They decide on the following weighting scheme

WEIGHTS APPLIED PERIOD3 Last month2 Two months ago1 Three months ago

6

3 x Sales last month + 2 x Sales two months ago + 1 X Sales three months ago

Sum of the weights



Table 5.4

MONTH ACTUAL SHED SALESTHREE-MONTH WEIGHTED

MOVING AVERAGEJanuary 10February 12March 13April 16May 19June 23July 26August 30September 28October 18November 16December 14January —

[(3 X 13) + (2 X 12) + (10)]/6 = 12.17[(3 X 16) + (2 X 13) + (12)]/6 = 14.33[(3 X 19) + (2 X 16) + (13)]/6 = 17.00[(3 X 23) + (2 X 19) + (16)]/6 = 20.50[(3 X 26) + (2 X 23) + (19)]/6 = 23.83[(3 X 30) + (2 X 26) + (23)]/6 = 27.50[(3 X 28) + (2 X 30) + (26)]/6 = 28.33[(3 X 18) + (2 X 28) + (30)]/6 = 23.33[(3 X 16) + (2 X 18) + (28)]/6 = 18.67[(3 X 14) + (2 X 16) + (18)]/6 = 15.33



Program 5.1A



Program 5.1B



Exponential smoothingExponential smoothing is easy to use and requires little record keeping of data

It is a type of moving average

New forecast = Last period’s forecast+ (Last period’s actual demand – Last period’s forecast)

Where is a weight (or smoothing constantsmoothing constant) with a value between 0 and 1 inclusive

A larger gives more importance to recent data while a smaller value gives more importance to past data



Mathematically

)( tttt FYFF 1

whereFt+1= new forecast (for time period t + 1)Ft= pervious forecast (for time period t)= smoothing constant (0 ≤ ≤ 1)Yt= pervious period’s actual demand

The idea is simple – the new estimate is the old estimate plus some fraction of the error in the last period


Exponential Smoothing Example

In January, February’s demand for a certain car model was predicted to be 142

Actual February demand was 153 autos Using a smoothing constant of = 0.20, what

is the forecast for March?

New forecast (for March demand) = 142 + 0.2(153 – 142)= 144.2 or 144 autos

If actual demand in March was 136 autos, the April forecast would be

New forecast (for April demand) = 144.2 + 0.2(136 – 144.2)= 142.6 or 143 autos


Selecting the Smoothing Constant

Selecting the appropriate value for is key to obtaining a good forecast

The objective is always to generate an accurate forecast

The general approach is to develop trial forecasts with different values of and select the that results in the lowest MAD


Port of Baltimore Example

QUARTER

ACTUAL TONNAGE

UNLOADEDFORECAST

USING =0.10FORECAST

USING =0.501 180 175 1752 168 175.5 = 175.00 + 0.10(180 – 175) 177.53 159 174.75 = 175.50 + 0.10(168 – 175.50) 172.754 175 173.18 = 174.75 + 0.10(159 – 174.75) 165.885 190 173.36 = 173.18 + 0.10(175 – 173.18) 170.446 205 175.02 = 173.36 + 0.10(190 – 173.36) 180.227 180 178.02 = 175.02 + 0.10(205 – 175.02) 192.618 182 178.22 = 178.02 + 0.10(180 – 178.02) 186.309 ? 178.60 = 178.22 + 0.10(182 – 178.22) 184.15

Table 5.5

Exponential smoothing forecast for two values of


Selecting the Best Value of

QUARTER

ACTUAL TONNAGE

UNLOADED

FORECAST WITH =

0.10

ABSOLUTEDEVIATIONS FOR = 0.10

FORECAST WITH = 0.50

ABSOLUTEDEVIATIONS FOR = 0.50

1 180 175 5….. 175 5….

2 168 175.5 7.5.. 177.5 9.5..

3 159 174.75 15.75 172.75 13.75

4 175 173.18 1.82 165.88 9.12

5 190 173.36 16.64 170.44 19.56

6 205 175.02 29.98 180.22 24.78

7 180 178.02 1.98 192.61 12.61

8 182 178.22 3.78 186.30 4.3..Sum of absolute deviations 82.45 98.63

MAD =Σ|deviations|

= 10.31 MAD = 12.33n

Table 5.6Best choiceBest choice



Program 5.2A



Program 5.2B


PM Computer: Moving Average 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: 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, previous forecast of 40 Using MAD, what forecast is most accurate?


PM Computers: Moving Average 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 with Trend Adjustment

Like all averaging techniques, exponential smoothing does not respond to trends

A more complex model can be used that adjusts for trends

The basic approach is to develop an exponential smoothing forecast then adjust it for the trend

Forecast including trend (FITt) = New forecast (Ft)+ Trend correction (Tt)


Exponential Smoothing with Trend Adjustment

The equation for the trend correction uses a new smoothing constant

Tt is computed by

)()1( 11 tttt FFTT

whereTt+1 =smoothed trend for period t + 1Tt =smoothed trend for preceding period =trend smooth constant that we selectFt+1 =simple exponential smoothed forecast for period t + 1Ft =forecast for pervious period


Selecting a Smoothing Constant

As with exponential smoothing, a high value of makes the forecast more responsive to changes in trend

A low value of gives less weight to the recent trend and tends to smooth out the trend

Values are generally selected using a trial-and-error approach based on the value of the MAD for different values of

Simple exponential smoothing is often referred to as first-order smoothingfirst-order smoothing

Trend-adjusted smoothing is called second-second-orderorder, double smoothingdouble smoothing, or Holt’s methodHolt’s method


Trend Projection

Trend projection fits a trend line to a series of historical data points

The line is projected into the future for medium- to long-range forecasts

Several trend equations can be developed based on exponential or quadratic models

The simplest is a linear model developed using regression analysis


Trend Projection Trend projections are used to forecast time-

series data that exhibit a linear trend. A trend line is simply a linear regression

equation in which the independent variable (X) is the time period

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

The mathematical form is

XbbY 10 ˆ

where= predicted valueb0= interceptb1= slope of the lineX= time period (i.e., X = 1, 2, 3, …, n)

Y


Trend Projection

Valu

e of

Dep

ende

nt V

aria

ble

Time

*

*

*

**

*

*Dist2

Dist4

Dist6

Dist1

Dist3

Dist5

Dist7

Figure 5.4


Midwestern Manufacturing Company Example

Midwestern Manufacturing Company has experienced the following demand for it’s electrical generators over the period of 2001 – 2007

YEAR ELECTRICAL GENERATORS SOLD

2001 742002 792003 802004 902005 1052006 1422007 122

Table 5.7



Program 5.3A

Notice code instead of

actual years



Program 5.3B

r2 says model predicts about 80% of the

variability in demand

Significance level for F-test indicates a

definite relationship



The forecast equation is

XY 54107156 ..ˆ

To project demand for 2008, we use the coding system to define X = 8

(sales in 2008) = 56.71 + 10.54(8)= 141.03, or 141 generators

Likewise for X = 9

(sales in 2009) = 56.71 + 10.54(9)= 151.57, or 152 generators



Gen

erat

or D

eman

d

Year

160 –150 –140 –130 –120 –110 –100 –

90 –80 –70 –60 –50 –

| | | | | | | | |

2001 2002 2003 2004 2005 2006 2007 2008 2009

Actual Demand Line

Trend LineXY 54107156 ..ˆ

Figure 5.5



Program 5.4A



Program 5.4B


Seasonal Variations Recurring variations over time may

indicate the need for seasonal adjustments in the trend line

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

When no trend is present, the seasonal index can be found by dividing the average value for a particular season by the average of all the data


Seasonal Variations

Eichler Supplies sells telephone answering machines

Data has been collected for the past two years sales of one particular model

They want to create a forecast that includes seasonality


Seasonal Variations

MONTH

SALES DEMAND

AVERAGE TWO- YEAR DEMAND

MONTHLY DEMAND

AVERAGE SEASONAL

INDEXYEAR 1 YEAR 2January 80 100

9094 0.957

February 85 7580

94 0.851

March 80 9085

94 0.904

April 110 90100

94 1.064

May 115 131123

94 1.309

June 120 110115

94 1.223

July 100 110105

94 1.117

August 110 90100

94 1.064

September 85 9590

94 0.957

October 75 8580

94 0.851

November 85 7580

94 0.851

December 80 8080

94 0.851

Total average demand = 1,128

Seasonal index =Average two-year demandAverage monthly demandAverage monthly demand = = 941,128

12 monthsTable 5.8


Seasonal Variations The calculations for the seasonal indices are

Jan. July969570122001

., 1121171122001

.,

Feb. Aug.858510122001

., 1060641122001

.,

Mar. Sept.909040122001

., 969570122001

.,

Apr. Oct.1060641122001

., 858510122001

.,

May Nov.1313091122001

., 858510122001

.,

June Dec.1222231122001

., 858510122001

.,


Regression with Trend and Seasonal Components

Multiple regressionMultiple regression can be used to forecast both trend and seasonal components in a time series One independent variable is time Dummy independent variables are used to represent

the seasons The model is an additive decomposition model

where X1 = time periodX2 = 1 if quarter 2, 0 otherwiseX3 = 1 if quarter 3, 0 otherwiseX4 = 1 if quarter 4, 0 otherwise

44332211 XbXbXbXbaY ˆ



Program 5.6A



Program 5.6B (partial)



The resulting regression equation is

4321 130738715321104 XXXXY .....ˆ

Using the model to forecast sales for the first two quarters of next year

These are different from the results obtained using the multiplicative decomposition method

Use MAD and MSE to determine the best model

13401300738071513321104 )(.)(.)(.)(..Y

15201300738171514321104 )(.)(.)(.)(..Y



American Airlines original spare parts inventory system used only time-series methods to forecast the demand for spare parts This method was slow to responds to even moderate changes in aircraft utilization let alone major fleet expansions

They developed a PC-based system named RAPS which uses linear regression to establish a relationship between monthly part removals and various functions of monthly flying hours The computation now takes only one hour instead of the days the old system needed Using RAPS provided a one time savings of $7 million and a recurring annual savings of nearly $1 million


Monitoring and Controlling Forecasts

Tracking signalsTracking signals can be used to monitor the performance of a forecast

Tacking signals are computed using the following equation

MADRSFE

signal Tracking

n

errorforecast MAD

where



Acceptable Range

Signal Tripped

Upper Control Limit

Lower Control Limit

0 MADs

+

–

Time

Figure 5.7

Tracking Signal



Positive tracking signals indicate demand is greater than forecast

Negative tracking signals indicate demand is less than forecast

Some variation is expected, but a good forecast will have about as much positive error as negative error

Problems are indicated when the signal trips either the upper or lower predetermined limits

This indicates there has been an unacceptable amount of variation

Limits should be reasonable and may vary from item to item



How do you decide on the upper and lower limits? Too small a value will trip the signal too often and

too large will cause a bad forecast Plossl & Wight – use maximums of ±4 MADs for

high volume stock items and ±8 MADs for lower volume items One MAD is equivalent to approximately 0.8

standard deviation so that ±4 MADs =3.2 s.d. For a forecast to be “in control”, 89% of the errors

are expected to fall within ±2 MADs, 98% with ±3 MADs or 99.9% within ±4 MADs whenever the errors are approximately normally distributed


Kimball’s Bakery Example Tracking signal for quarterly sales of croissants

TIME PERIOD

FORECAST DEMAND

ACTUAL DEMAND ERROR RSFE

|FORECAST || ERROR |

CUMULATIVE ERROR MAD

TRACKING SIGNAL

1 100 90 –10 –10 10 10 10.0 –1

2 100 95 –5 –15 5 15 7.5 –2

3 100 115 +15 0 15 30 10.0 0

4 110 100 –10 –10 10 40 10.0 –1

5 110 125 +15 +5 15 55 11.0 +0.5

6 110 140 +30 +35 30 85 14.2 +2.5

2146

85errorforecast .MAD

n

sMAD..MAD

RSFE 52214

35signal Tracking


Forecasting at Disney

The Disney chairman receives a daily The Disney chairman receives a daily report from his main theme parks that report from his main theme parks that contains only two numbers – the contains only two numbers – the forecastforecast of yesterday’s attendance at the parks and of yesterday’s attendance at the parks and the the actual actual attendanceattendance An error close to zero (using MAPE as the

measure) is expected The annual forecast of total volume

conducted in 1999 for the year 2000 resulted in a MAPE of 0


Using The Computer to Forecast

Spreadsheets can be used by small and medium-sized forecasting problems

More advanced programs (SAS, SPSS, Minitab) handle time-series and causal models

May automatically select best model parameters

Dedicated forecasting packages may be fully automatic

May be integrated with inventory planning and control

05-forecasting.ppt

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