Render/Stair/Hanna Chapter 5

Report
Chapter 5
Forecasting
To accompany
Quantitative Analysis for Management, Tenth Edition,
by Render, Stair, and Hanna
Power Point slides created by Jeff Heyl
© 2008 Prentice-Hall, Inc.
© 2009 Prentice-Hall, Inc.
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.
5–2
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 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
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5–3
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
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5–4
Forecasting Models
Forecasting
Techniques
Qualitative
Models
Time-Series
Methods
Causal
Methods
Delphi
Methods
Moving
Average
Regression
Analysis
Jury of Executive
Opinion
Exponential
Smoothing
Multiple
Regression
Sales Force
Composite
Trend
Projections
Figure 5.1
Consumer
Market Survey
Decomposition
© 2009 Prentice-Hall, Inc.
5–5
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.
5–6
Causal Models
 Causal 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.
5–7
Qualitative Models
 Qualitative 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.
5–8
Qualitative Models
 Delphi Method – an iterative group process where
(possibly geographically dispersed) respondents
provide input to decision makers
 Jury of Executive Opinion – collects opinions of a
small group of high-level managers, possibly
using statistical models for analysis
 Sales Force Composite – individual salespersons
estimate the sales in their region and the data is
compiled at a district or national level
 Consumer Market Survey – input is solicited from
customers or potential customers regarding their
purchasing plans
© 2009 Prentice-Hall, Inc.
5–9
Scatter Diagrams
Annual Sales
Scatter diagrams are helpful when forecasting time-series
data because they depict the relationship between variables.
450
400
350
300
250
200
150
100
50
0
Televisions
0
2
4
6
8
10
12
Time (Years)
© 2009 Prentice-Hall, Inc.
5 – 10
Scatter Diagrams
 Wacker Distributors wants to forecast sales for
three different products
YEAR
TELEVISION SETS
RADIOS
COMPACT DISC PLAYERS
1
2
3
4
5
6
7
8
9
10
250
250
250
250
250
250
250
250
250
250
300
310
320
330
340
350
360
370
380
390
110
100
120
140
170
150
160
190
200
190
Table 5.1
© 2009 Prentice-Hall, Inc.
5 – 11
Scatter Diagrams
Annual Sales of Televisions
(a)
 Sales appear to be
330 –
250 –          
200 –
150 –
100 –
constant over time
Sales = 250
 A good estimate of
sales in year 11 is
250 televisions
50 –
|
|
|
|
|
|
|
|
|
|
0 1 2 3 4 5 6 7 8 9 10
Time (Years)
Figure 5.2
© 2009 Prentice-Hall, Inc.
5 – 12
Scatter Diagrams
(b)
Annual Sales of Radios
420 –
 Sales appear to be
400 –
380 –
360 –
340 –
320 –
300 –  
280 –
|
|

|

|

|

|

|

|

|

|
0 1 2 3 4 5 6 7 8 9 10
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
Time (Years)
Figure 5.2
© 2009 Prentice-Hall, Inc.
5 – 13
Scatter Diagrams
(c)
Annual Sales of CD Players
 This trend line may
200 –

180 –

160 –
140 –
120 –
100 –



|
 



|
|
|
|
|
|
|
|
|
0 1 2 3 4 5 6 7 8 9 10
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
Time (Years)
Figure 5.2
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5 – 14
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 absolute
deviation (MAD)
forecast error

MAD 
n
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5 – 15
Measures of Forecast Accuracy
 Using a naïve forecasting model
YEAR
ACTUAL
SALES OF CD
PLAYERS
FORECAST
SALES
ABSOLUTE VALUE OF
ERRORS (DEVIATION),
(ACTUAL – FORECAST)
1
110
—
—
2
100
110
|100 – 110| = 10
3
120
100
|120 – 110| = 20
4
140
120
|140 – 120| = 20
5
170
140
|170 – 140| = 30
6
150
170
|150 – 170| = 20
7
160
150
|160 – 150| = 10
8
190
160
|190 – 160| = 30
9
200
190
|200 – 190| = 10
10
190
200
|190 – 200| = 10
11
—
190
—
Sum of |errors| = 160
MAD = 160/9 = 17.8
Table 5.2
© 2009 Prentice-Hall, Inc.
5 – 16
Measures of Forecast Accuracy
 Using a naïve forecasting model
YEAR
ACTUAL
SALES OF CD
PLAYERS
FORECAST
SALES
ABSOLUTE VALUE OF
ERRORS (DEVIATION),
(ACTUAL – FORECAST)
1
110
—
—
2
100
110
|100 – 110| = 10
3
120
100
|120 – 110| = 20
4
140
120
|140 – 120| = 20
5
170
140
|170 – 140| = 30
6
150
7
160
150
|160 – 150| = 10
8
190
160
|190 – 160| = 30
9
200
190
|200 – 190| = 10
10
190
200
|190 – 200| = 10
11
—
190
—
forecast error 160

MAD 

 17.8
n
170
9
|150 – 170| = 20
Sum of |errors| = 160
MAD = 160/9 = 17.8
Table 5.2
© 2009 Prentice-Hall, Inc.
5 – 17
Measures of Forecast Accuracy
 There are other popular measures of forecast
accuracy
 The mean squared error
2
(
error)

MSE 
n
 The mean absolute percent error
MAPE 

error
actual
100%
n
 And bias 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 – 18
Measures of Forecast Accuracy
Year
Actual CD Sales
Forecast Sales
|Actual -Forecast|
1
110
2
100
110
10
3
120
100
20
4
140
120
20
5
170
140
30
6
150
170
20
7
160
150
10
8
190
160
30
9
200
190
10
10
190
200
10
11
190
Sum of |errors|
MAD
160
17.8
© 2009 Prentice-Hall, Inc.
5 – 19
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
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Forecast
Actual
250
320
275
260
250
275
300
325
320
350
365
380
243
315
286
256
241
298
292
333
326
378
382
396
© 2009 Prentice-Hall, Inc.
5 – 20
Hospital Days Forecast Error
Example
Actual
243
|error|
error2
|error/actual|
JAN
Forecast
250
7
49
0.03
FEB
320
315
5
25
0.02
MAR
275
286
11
121
0.04
APR
260
256
4
16
0.02
MAY
250
241
9
81
0.04
JUN
275
298
23
529
0.08
JUL
300
292
8
64
0.03
AUG
325
333
8
64
0.02
SEP
320
326
6
36
0.02
OCT
350
378
28
784
0.07
NOV
365
382
17
289
0.04
DEC
380
396
16
256
0.04
AVERAGE
MAD=
11.83
MSE=
192.83
MAPE=
.0381*100 =
3.81
© 2009 Prentice-Hall, Inc.
5 – 21
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
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5 – 22
Decomposition of a Time-Series
 A time series typically has four components
1. Trend (T) is the gradual upward or
downward movement of the data over time
2. Seasonality (S) is a pattern of demand
fluctuations above or below trend line that
repeats at regular intervals
3. Cycles (C) are patterns in annual data that
occur every several years
4. Random variations (R) are “blips” in the
data caused by chance and unusual
situations
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5 – 23
Demand for Product or Service
Decomposition of a Time-Series
Figure 5.3
Trend
Component
Seasonal Peaks
Actual
Demand
Line
Average Demand
over 4 Years
|
|
|
|
Year
1
Year
2
Year
3
Year
4
Time
© 2009 Prentice-Hall, Inc.
5 – 24
Decomposition of a Time-Series
 There are two general forms of time-series
models
 The multiplicative model
Demand = T x S x C x R
 The additive model
Demand = T + S + C + R
 Models may be combinations of these two
forms
 Forecasters often assume errors are
normally distributed with a mean of zero
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5 – 25
Moving Averages
 Moving 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 dropped
This methods tends to smooth out short-term
irregularities in the data series
Moving average forecast 
Sum of demands in previous n periods
n
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5 – 26
Moving Averages
 Mathematically
Ft 1 
Yt  Yt 1  ...  Yt  n1
n
where
Ft 1 = forecast for time period t + 1
Yt = actual value in time period t
n = number of periods to average
© 2009 Prentice-Hall, Inc.
5 – 27
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)
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5 – 28
Wallace Garden Supply Example
MONTH
ACTUAL SHED SALES
THREE-MONTH MOVING AVERAGE
January
10
February
12
March
13
April
16
(10 + 12 + 13)/3 = 11.67
May
19
(12 + 13 + 16)/3 = 13.67
June
23
(13 + 16 + 19)/3 = 16.00
July
26
(16 + 19 + 23)/3 = 19.33
August
30
(19 + 23 + 26)/3 = 22.67
September
28
(23 + 26 + 30)/3 = 26.33
October
18
(26 + 30 + 28)/3 = 28.00
November
16
(30 + 28 + 18)/3 = 25.33
December
14
(28 + 18 + 16)/3 = 20.67
January
—
(18 + 16 + 14)/3 = 16.00
Table 5.3
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5 – 29
Weighted Moving Averages
 Weighted moving averages use weights to put
more emphasis on recent periods
 Often used when a trend or other pattern is
emerging
Ft 1
( Weight in period i )( Actual value in period)


 ( Weights)
 Mathematically
w1Yt  w2Yt 1  ...  wnYt  n1
Ft 1 
w1  w2  ...  wn
where
wi = weight for the ith observation
© 2009 Prentice-Hall, Inc.
5 – 30
Weighted Moving Averages
 Both simple and weighted averages are
effective in smoothing out fluctuations in
the demand pattern in order to provide
stable estimates
 Problems
Increasing the size of n smoothes out
fluctuations better, but makes the method
less sensitive to real changes in the data
Moving 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 – 31
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
PERIOD
3
2
1
Last month
Two months ago
Three months ago
3 x Sales last month + 2 x Sales two months ago + 1 X Sales three months ago
6
Sum of the weights
© 2009 Prentice-Hall, Inc.
5 – 32
Wallace Garden Supply Example
THREE-MONTH WEIGHTED
MOVING AVERAGE
MONTH
ACTUAL SHED SALES
January
10
February
12
March
13
April
16
[(3 X 13) + (2 X 12) + (10)]/6 = 12.17
May
19
[(3 X 16) + (2 X 13) + (12)]/6 = 14.33
June
23
[(3 X 19) + (2 X 16) + (13)]/6 = 17.00
July
26
[(3 X 23) + (2 X 19) + (16)]/6 = 20.50
August
30
[(3 X 26) + (2 X 23) + (19)]/6 = 23.83
September
28
[(3 X 30) + (2 X 26) + (23)]/6 = 27.50
October
18
[(3 X 28) + (2 X 30) + (26)]/6 = 28.33
November
16
[(3 X 18) + (2 X 28) + (30)]/6 = 23.33
December
14
[(3 X 16) + (2 X 18) + (28)]/6 = 18.67
January
—
[(3 X 14) + (2 X 16) + (18)]/6 = 15.33
Table 5.4
© 2009 Prentice-Hall, Inc.
5 – 33
Wallace Garden Supply Example
Program 5.1A
© 2009 Prentice-Hall, Inc.
5 – 34
Wallace Garden Supply Example
Program 5.1B
© 2009 Prentice-Hall, Inc.
5 – 35
Exponential Smoothing
 Exponential 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 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 – 36
Exponential Smoothing
 Mathematically
Ft 1  Ft   (Yt  Ft )
where
Ft+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 – 37
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 – 38
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 – 39
Port of Baltimore Example
 Exponential smoothing forecast for two values of
QUARTER
ACTUAL
TONNAGE
UNLOADED
1
180
175
175
2
168
175.5 = 175.00 + 0.10(180 – 175)
177.5
3
159
174.75 = 175.50 + 0.10(168 – 175.50)
172.75
4
175
173.18 = 174.75 + 0.10(159 – 174.75)
165.88
5
190
173.36 = 173.18 + 0.10(175 – 173.18)
170.44
6
205
175.02 = 173.36 + 0.10(190 – 173.36)
180.22
7
180
178.02 = 175.02 + 0.10(205 – 175.02)
192.61
8
182
178.22 = 178.02 + 0.10(180 – 178.02)
186.30
9
?
178.60 = 178.22 + 0.10(182 – 178.22)
184.15
FORECAST
USING  =0.10

FORECAST
USING  =0.50
Table 5.5
© 2009 Prentice-Hall, Inc.
5 – 40
Selecting the Best Value of 
QUARTER
ACTUAL
TONNAGE
UNLOADED
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..
FORECAST
WITH  = 0.10
ABSOLUTE
DEVIATIONS
FOR  = 0.10
Sum of absolute deviations
MAD =
Table 5.6
FORECAST
WITH  = 0.50
ABSOLUTE
DEVIATIONS
FOR  = 0.50
82.45
Σ|deviations|
n
=
10.31
98.63
MAD =
12.33
Best choice
© 2009 Prentice-Hall, Inc.
5 – 41
Port of Baltimore Example
Program 5.2A
© 2009 Prentice-Hall, Inc.
5 – 42
Port of Baltimore Example
Program 5.2B
© 2009 Prentice-Hall, Inc.
5 – 43
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 – 44
PM Computers: Data
Period
Month
Actual Demand
1
Jan
37
2
Feb
40
3
Mar
41
4
Apr
37
5
May
45
6
June
50
7
July
43
8
Aug
47
9
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 – 45
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
39.10
1.90
38.50
2.50
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
MAD
51.57
5.29
53.07
5.43
4.95
Exponential smoothing resulted in the lowest MAD.
© 2009 Prentice-Hall, Inc.
5 – 46
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 – 47
Exponential Smoothing with
Trend Adjustment
 The equation for the trend correction uses a
new smoothing constant 
 Tt is computed by
Tt 1  (1   )Tt   ( Ft 1  Ft )
where
Tt+1 =
Tt =
=
Ft+1 =
smoothed trend for period t + 1
smoothed trend for preceding period
trend smooth constant that we select
simple exponential smoothed forecast for
period t + 1
Ft = forecast for pervious period
© 2009 Prentice-Hall, Inc.
5 – 48
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-anderror approach based on the value of the MAD for
different values of 
Simple exponential smoothing is often referred to
as first-order smoothing
Trend-adjusted smoothing is called second-order,
double smoothing, or Holt’s method
© 2009 Prentice-Hall, Inc.
5 – 49
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 – 50
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.
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Trend Projection
 The mathematical form is
Yˆ  b0  b1 X
where
Yˆ = predicted value
b0 = intercept
b1 = slope of the line
X = time period (i.e., X = 1, 2, 3, …, n)
© 2009 Prentice-Hall, Inc.
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Trend Projection
Value of Dependent Variable
Dist7
Dist5
*
*
Dist3
*
*
Dist6
Dist4
Dist1
*
*
Dist2
Time
*
Figure 5.4
© 2009 Prentice-Hall, Inc.
5 – 53
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
2002
2003
2004
2005
2006
2007
74
79
80
90
105
142
122
Table 5.7
© 2009 Prentice-Hall, Inc.
5 – 54
Midwestern Manufacturing
Company Example
Notice code
instead of
actual years
Program 5.3A
© 2009 Prentice-Hall, Inc.
5 – 55
Midwestern Manufacturing
Company Example
r2 says model predicts
about 80% of the
variability in demand
Significance level for
F-test indicates a
definite relationship
Program 5.3B
© 2009 Prentice-Hall, Inc.
5 – 56
Midwestern Manufacturing
Company Example
 The forecast equation is
Yˆ  56.71 10.54 X
 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 – 57
Midwestern Manufacturing
Company Example
160 –

150 –

Generator Demand
140 –
Trend Line
Yˆ  56.71 10.54 X
130 –

120 –
110 –

100 –
90 –
80 –
70 –




|
|
Actual Demand Line
60 –
50 –
|
Figure 5.5
|
|
|
|
|
|
2001 2002 2003 2004 2005 2006 2007 2008 2009
Year
© 2009 Prentice-Hall, Inc.
5 – 58
Midwestern Manufacturing
Company Example
Program 5.4A
© 2009 Prentice-Hall, Inc.
5 – 59
Midwestern Manufacturing
Company Example
Program 5.4B
© 2009 Prentice-Hall, Inc.
5 – 60
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 – 61
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 – 62
Seasonal Variations
SALES DEMAND
MONTH
YEAR 1
YEAR 2
AVERAGE TWOYEAR DEMAND
MONTHLY
DEMAND
AVERAGE
SEASONAL
INDEX
January
80
100
90
94
0.957
February
85
75
80
94
0.851
March
80
90
85
94
0.904
April
110
90
100
94
1.064
May
115
131
123
94
1.309
June
120
110
115
94
1.223
July
100
110
105
94
1.117
August
110
90
100
94
1.064
September
85
95
90
94
0.957
October
75
85
80
94
0.851
November
85
75
80
94
0.851
December
80
80
80
94
0.851
Total average demand = 1,128
Average monthly demand =
Table 5.8
1,128
= 94
12 months
Average two-year demand
Seasonal index = Average monthly demand
© 2009 Prentice-Hall, Inc.
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Seasonal Variations
 The calculations for the seasonal indices are
Jan.
1,200
 0.957  96
12
July
1,200
 1.117  112
12
Feb.
1,200
 0.851  85
12
Aug.
1,200
 1.064  106
12
Mar.
1,200
 0.904  90
12
Sept.
1,200
 0.957  96
12
Apr.
1,200
 1.064  106
12
Oct.
1,200
 0.851  85
12
May
1,200
 1.309  131
12
Nov.
1,200
 0.851  85
12
June
1,200
 1.223  122
12
Dec.
1,200
 0.851  85
12
© 2009 Prentice-Hall, Inc.
5 – 64
Regression with Trend and
Seasonal Components
 Multiple 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
Yˆ  a  b1 X 1  b2 X 2  b3 X 3  b4 X 4
where
X1
X2
X3
X4
= time period
= 1 if quarter 2, 0 otherwise
= 1 if quarter 3, 0 otherwise
= 1 if quarter 4, 0 otherwise
© 2009 Prentice-Hall, Inc.
5 – 65
Regression with Trend and
Seasonal Components
Program 5.6A
© 2009 Prentice-Hall, Inc.
5 – 66
Regression with Trend and
Seasonal Components
Program 5.6B (partial)
© 2009 Prentice-Hall, Inc.
5 – 67
Regression with Trend and
Seasonal Components
 The resulting regression equation is
Yˆ  104.1 2.3 X 1  15.7 X 2  38.7 X 3  30.1X 4
 Using the model to forecast sales for the first two
quarters of next year
Yˆ  104.1 2.3(13)  15.7(0)  38.7(0)  30.1(0)  134
Yˆ  104.1 2.3(14)  15.7(1)  38.7(0)  30.1(0)  152
 These are different from the results obtained
using the multiplicative decomposition method
 Use MAD and MSE to determine the best model
© 2009 Prentice-Hall, Inc.
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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 – 69
Monitoring and Controlling Forecasts
 Tracking signals can be used to monitor
the performance of a forecast
 Tacking signals are computed using the
following equation
RSFE
Tracking signal 
MAD
where
forecast error

MAD 
n
© 2009 Prentice-Hall, Inc.
5 – 70
Monitoring and Controlling Forecasts
Signal Tripped
Upper Control Limit
+
Tracking Signal
Acceptable
Range
0 MADs
–
Lower Control Limit
Time
Figure 5.7
© 2009 Prentice-Hall, Inc.
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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 – 72
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.
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Kimball’s Bakery Example
 Tracking signal for quarterly sales of croissants
TIME
PERIOD
FORECAST
DEMAND
ACTUAL
DEMAND
CUMULATIVE
ERROR
MAD
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
ERROR
RSFE
|FORECAST |
| ERROR |
TRACKING
SIGNAL
forecast error 85

MAD 

 14.2
n
6
RSFE 35
Tracking signal 

 2.5MAD s
MAD 14.2
© 2009 Prentice-Hall, Inc.
5 – 74
Forecasting at Disney
 The Disney chairman receives a daily
report from his main theme parks that
contains only two numbers – the forecast
of yesterday’s attendance at the parks and
the actual attendance
 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 – 75
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
© 2009 Prentice-Hall, Inc.
5 – 76

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