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Forcasting Methods
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© 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
© 2009 Prentice-Hall, Inc. 5 – 3
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
© 2009 Prentice-Hall, Inc. 5 – 4
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
© 2009 Prentice-Hall, Inc. 5 – 5
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
© 2009 Prentice-Hall, Inc. 5 – 6
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
© 2009 Prentice-Hall, Inc. 5 – 7
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
© 2009 Prentice-Hall, Inc. 5 – 8
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
© 2009 Prentice-Hall, Inc. 5 – 9
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
© 2009 Prentice-Hall, Inc. 5 – 10
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.
© 2009 Prentice-Hall, Inc. 5 – 11
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
© 2009 Prentice-Hall, Inc. 5 – 12
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
© 2009 Prentice-Hall, Inc. 5 – 13
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
© 2009 Prentice-Hall, Inc. 5 – 14
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
© 2009 Prentice-Hall, Inc. 5 – 15
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
© 2009 Prentice-Hall, Inc. 5 – 16
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
© 2009 Prentice-Hall, Inc. 5 – 17
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
© 2009 Prentice-Hall, Inc. 5 – 18
Measures of Forecast Accuracy
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.
© 2009 Prentice-Hall, Inc. 5 – 19
Measures of Forecast Accuracy
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
© 2009 Prentice-Hall, Inc. 5 – 20
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
© 2009 Prentice-Hall, Inc. 5 – 21
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
© 2009 Prentice-Hall, Inc. 5 – 22
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
© 2009 Prentice-Hall, Inc. 5 – 23
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
© 2009 Prentice-Hall, Inc. 5 – 24
Decomposition of a Time-Series
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
© 2009 Prentice-Hall, Inc. 5 – 25
Decomposition of a Time-Series
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
© 2009 Prentice-Hall, Inc. 5 – 26
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
© 2009 Prentice-Hall, Inc. 5 – 27
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
© 2009 Prentice-Hall, Inc. 5 – 28
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)
© 2009 Prentice-Hall, Inc. 5 – 29
Wallace Garden Supply Example
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
© 2009 Prentice-Hall, Inc. 5 – 30
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
© 2009 Prentice-Hall, Inc. 5 – 31
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
© 2009 Prentice-Hall, Inc. 5 – 32
Wallace Garden Supply Example
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
© 2009 Prentice-Hall, Inc. 5 – 33
Wallace Garden Supply Example
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
© 2009 Prentice-Hall, Inc. 5 – 34
Wallace Garden Supply Example
Program 5.1A
© 2009 Prentice-Hall, Inc. 5 – 35
Wallace Garden Supply Example
Program 5.1B
© 2009 Prentice-Hall, Inc. 5 – 36
Exponential Smoothing
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
© 2009 Prentice-Hall, Inc. 5 – 37
Exponential Smoothing
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
© 2009 Prentice-Hall, Inc. 5 – 38
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
© 2009 Prentice-Hall, Inc. 5 – 39
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
© 2009 Prentice-Hall, Inc. 5 – 40
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
© 2009 Prentice-Hall, Inc. 5 – 41
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
© 2009 Prentice-Hall, Inc. 5 – 42
Port of Baltimore Example
Program 5.2A
© 2009 Prentice-Hall, Inc. 5 – 43
Port of Baltimore Example
Program 5.2B
© 2009 Prentice-Hall, Inc. 5 – 44
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.
© 2009 Prentice-Hall, Inc. 5 – 45
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?
© 2009 Prentice-Hall, Inc. 5 – 46
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.
© 2009 Prentice-Hall, Inc. 5 – 47
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)
© 2009 Prentice-Hall, Inc. 5 – 48
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
© 2009 Prentice-Hall, Inc. 5 – 49
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
© 2009 Prentice-Hall, Inc. 5 – 50
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
© 2009 Prentice-Hall, Inc. 5 – 51
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.
© 2009 Prentice-Hall, Inc. 5 – 52
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
© 2009 Prentice-Hall, Inc. 5 – 53
Trend Projection
Valu
e of
Dep
ende
nt V
aria
ble
Time
*
*
*
**
*
*Dist2
Dist4
Dist6
Dist1
Dist3
Dist5
Dist7
Figure 5.4
© 2009 Prentice-Hall, Inc. 5 – 54
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
© 2009 Prentice-Hall, Inc. 5 – 55
Midwestern Manufacturing Company Example
Program 5.3A
Notice code instead of
actual years
© 2009 Prentice-Hall, Inc. 5 – 56
Midwestern Manufacturing Company Example
Program 5.3B
r2 says model predicts about 80% of the
variability in demand
Significance level for F-test indicates a
definite relationship
© 2009 Prentice-Hall, Inc. 5 – 57
Midwestern Manufacturing Company Example
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
© 2009 Prentice-Hall, Inc. 5 – 58
Midwestern Manufacturing Company Example
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
© 2009 Prentice-Hall, Inc. 5 – 59
Midwestern Manufacturing Company Example
Program 5.4A
© 2009 Prentice-Hall, Inc. 5 – 60
Midwestern Manufacturing Company Example
Program 5.4B
© 2009 Prentice-Hall, Inc. 5 – 61
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
© 2009 Prentice-Hall, Inc. 5 – 62
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
© 2009 Prentice-Hall, Inc. 5 – 63
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
© 2009 Prentice-Hall, Inc. 5 – 64
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
.,
© 2009 Prentice-Hall, Inc. 5 – 65
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
© 2009 Prentice-Hall, Inc. 5 – 66
Regression with Trend and Seasonal Components
Program 5.6A
© 2009 Prentice-Hall, Inc. 5 – 67
Regression with Trend and Seasonal Components
Program 5.6B (partial)
© 2009 Prentice-Hall, Inc. 5 – 68
Regression with Trend and Seasonal Components
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
© 2009 Prentice-Hall, Inc. 5 – 69
Regression with Trend and Seasonal Components
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
© 2009 Prentice-Hall, Inc. 5 – 70
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
© 2009 Prentice-Hall, Inc. 5 – 71
Monitoring and Controlling Forecasts
Acceptable Range
Signal Tripped
Upper Control Limit
Lower Control Limit
0 MADs
+
–
Time
Figure 5.7
Tracking Signal
© 2009 Prentice-Hall, Inc. 5 – 72
Monitoring and Controlling Forecasts
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
© 2009 Prentice-Hall, Inc. 5 – 73
Regression with Trend and Seasonal Components
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
© 2009 Prentice-Hall, Inc. 5 – 74
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
© 2009 Prentice-Hall, Inc. 5 – 75
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
© 2009 Prentice-Hall, Inc. 5 – 76
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