© 2006 prentice hall, inc.4 – 1 short-range forecast up to 1 year, generally less than 3 months...
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© 2006 Prentice Hall, Inc. 4 – 1
Short-range forecast Up to 1 year, generally less than 3 months Purchasing, job scheduling, workforce
levels, job assignments, production levels Medium-range forecast
3 months to 3 years Sales and production planning, budgeting
Long-range forecast 3+ years New product planning, facility location,
research and development
Forecasting Time Horizons
© 2006 Prentice Hall, Inc. 4 – 3
Components of DemandD
eman
d f
or
pro
du
ct o
r se
rvic
e
| | | |1 2 3 4
Year
Average demand over four years
Seasonal peaks
Trend component
Actual demand
Random variation
Figure 4.1
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Graph of Moving Average
| | | | | | | | | | | |
J F M A M J J A S O N D
Sh
ed S
ales
30 –28 –26 –24 –22 –20 –18 –16 –14 –12 –10 –
Actual Sales
Moving Average Forecast
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Impact of Different
225 –
200 –
175 –
150 –| | | | | | | | |
1 2 3 4 5 6 7 8 9
Quarter
De
ma
nd
a = .1
Actual demand
a = .5
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Least Squares Method
Time period
Va
lue
s o
f D
ep
end
en
t V
ari
able
Figure 4.4
Deviation1
Deviation5
Deviation7
Deviation2
Deviation6
Deviation4
Deviation3
Actual observation (y value)
Trend line, y = a + bx^
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Least Squares Method
Time period
Va
lue
s o
f D
ep
end
en
t V
ari
able
Figure 4.4
Deviation1
Deviation5
Deviation7
Deviation2
Deviation6
Deviation4
Deviation3
Actual observation (y value)
Trend line, y = a + bx^
Least squares method minimizes the sum of the
squared errors (deviations)
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Least Squares Example
b = = = 10.54∑xy - nxy
∑x2 - nx2
3,063 - (7)(4)(98.86)
140 - (7)(42)
a = y - bx = 98.86 - 10.54(4) = 56.70
Time Electrical Power Year Period (x) Demand x2 xy
1999 1 74 1 742000 2 79 4 1582001 3 80 9 2402002 4 90 16 3602003 5 105 25 5252004 6 142 36 8522005 7 122 49 854
∑x = 28 ∑y = 692 ∑x2 = 140 ∑xy = 3,063x = 4 y = 98.86
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Least Squares Example
b = = = 10.54Sxy - nxy
Sx2 - nx2
3,063 - (7)(4)(98.86)
140 - (7)(42)
a = y - bx = 98.86 - 10.54(4) = 56.70
Time Electrical Power Year Period (x) Demand x2 xy
1999 1 74 1 742000 2 79 4 1582001 3 80 9 2402002 4 90 16 3602003 5 105 25 5252004 6 142 36 8522005 7 122 49 854
Sx = 28 Sy = 692 Sx2 = 140 Sxy = 3,063x = 4 y = 98.86
The trend line is
y = 56.70 + 10.54x^
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Least Squares Example
| | | | | | | | |1999 2000 2001 2002 2003 2004 2005 2006 2007
160 –150 –140 –130 –120 –110 –100 –90 –80 –70 –60 –50 –
Year
Po
wer
dem
and
Trend line,y = 56.70 + 10.54x^
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Associative Forecasting
Forecasting an outcome based on predictor variables using the least squares technique
y = a + bx^
where y = computed value of the variable to be predicted (dependent variable)a = y-axis interceptb = slope of the regression linex = the independent variable though to predict the value of the dependent variable
^
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Associative Forecasting Example
Sales Local Payroll($000,000), y ($000,000,000), x
2.0 13.0 32.5 42.0 22.0 13.5 7
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |0 1 2 3 4 5 6 7
Sal
es
Area payroll
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Associative Forecasting Example
Sales, y Payroll, x x2 xy
2.0 1 1 2.03.0 3 9 9.02.5 4 16 10.02.0 2 4 4.02.0 1 1 2.03.5 7 49 24.5
∑y = 15.0 ∑x = 18 ∑x2 = 80 ∑xy = 51.5
x = ∑x/6 = 18/6 = 3
y = ∑y/6 = 15/6 = 2.5
b = = = .25∑xy - nxy
∑x2 - nx2
51.5 - (6)(3)(2.5)
80 - (6)(32)
a = y - bx = 2.5 - (.25)(3) = 1.75
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Associative Forecasting Example
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |0 1 2 3 4 5 6 7
Sal
es
Area payroll
y = 1.75 + .25x^ Sales = 1.75 + .25(payroll)
If payroll next year is estimated to be $600 million, then:
Sales = 1.75 + .25(6)Sales = $325,000
3.25
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Standard Error of the Estimate
A forecast is just a point estimate of a future value
This point is actually the mean of a probability distribution
Figure 4.9
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |0 1 2 3 4 5 6 7
Sal
es
Area payroll
3.25
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Standard Error of the Estimate
where y = y-value of each data point
yc = computed value of the dependent variable, from the regression equation
n = number of data points
Sy,x =∑(y - yc)2
n - 2
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Standard Error of the Estimate
Computationally, this equation is considerably easier to use
We use the standard error to set up prediction intervals around the
point estimate
Sy,x =∑y2 - a∑y - b∑xy
n - 2
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Standard Error of the Estimate
4.0 –
3.0 –
2.0 –
1.0 –
| | | | | | |0 1 2 3 4 5 6 7
Sal
es
Area payroll
3.25
Sy,x = =∑y2 - a∑y - b∑xyn - 2
39.5 - 1.75(15) - .25(51.5)6 - 2
Sy,x = .306
The standard error of the estimate is $30,600 in sales
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How strong is the linear relationship between the variables?
Correlation does not necessarily imply causality!
Coefficient of correlation, r, measures degree of associationValues range from -1 to +1
Correlation
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Correlation Coefficient
r = nSxy - SxSy
[nSx2 - (Sx)2][nSy2 - (Sy)2]
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Correlation Coefficient
r = n∑xy - ∑x∑y
[n∑x2 - (∑x)2][n∑y2 - (∑y)2]
y
x(a) Perfect positive correlation: r = +1
y
x(b) Positive correlation: 0 < r < 1
y
x(c) No correlation: r = 0
y
x(d) Perfect negative correlation: r = -1
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Coefficient of Determination, r2, measures the percent of change in y predicted by the change in xValues range from 0 to 1Easy to interpret
Correlation
For the Nodel Construction example:
r = .901
r2 = .81
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Multiple Regression Analysis
If more than one independent variable is to be used in the model, linear regression can be
extended to multiple regression to accommodate several independent variables
y = a + b1x1 + b2x2 …^
Computationally, this is quite complex and generally done on the
computer
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Multiple Regression Analysis
y = 1.80 + .30x1 - 5.0x2^
In the Nodel example, including interest rates in the model gives the new equation:
An improved correlation coefficient of r = .96 means this model does a better job of predicting the change in construction sales
Sales = 1.80 + .30(6) - 5.0(.12) = 3.00Sales = $300,000
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Measures how well the forecast is predicting actual values
Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD) Good tracking signal has low values If forecasts are continually high or low, the
forecast has a bias error
Monitoring and Controlling Forecasts
Tracking Signal
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Monitoring and Controlling Forecasts
Tracking signal
RSFEMAD
=
Tracking signal =
∑(actual demand in period i -
forecast demand in period i)
(∑|actual - forecast|/n)
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Tracking Signal
Tracking signal
+
0 MADs
–
Upper control limit
Lower control limit
Time
Signal exceeding limit
Acceptable range
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Tracking Signal ExampleCumulative
Absolute AbsoluteActual Forecast Forecast Forecast
Qtr Demand Demand Error RSFE Error Error MAD
1 90 100 -10 -10 10 10 10.02 95 100 -5 -15 5 15 7.53 115 100 +15 0 15 30 10.04 100 110 -10 -10 10 40 10.05 125 110 +15 +5 15 55 11.06 140 110 +30 +35 30 85 14.2
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CumulativeAbsolute Absolute
Actual Forecast Forecast ForecastQtr Demand Demand Error RSFE Error Error MAD
1 90 100 -10 -10 10 10 10.02 95 100 -5 -15 5 15 7.53 115 100 +15 0 15 30 10.04 100 110 -10 -10 10 40 10.05 125 110 +15 +5 15 55 11.06 140 110 +30 +35 30 85 14.2
Tracking Signal ExampleTracking
Signal(RSFE/MAD)
-10/10 = -1-15/7.5 = -2
0/10 = 0-10/10 = -1
+5/11 = +0.5+35/14.2 = +2.5
The variation of the tracking signal between -2.0 and +2.5 is within acceptable limits