© 2006 Prentice Hall, Inc.4 – 1 Short-range forecast Up to 1 year, generally less than 3 months Purchasing, job scheduling, workforce levels, job

© 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


Trend

Seasonal

Cyclical

Random

Time Series Components


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


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


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


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

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)


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



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^



| | | | | | | | |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^


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

^


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



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



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


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



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



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



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


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


Correlation Coefficient

r = nSxy - SxSy

[nSx2 - (Sx)2][nSy2 - (Sy)2]


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


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


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


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


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


Monitoring and Controlling Forecasts

Tracking signal

RSFEMAD

=

Tracking signal =

∑(actual demand in period i -

forecast demand in period i)

(∑|actual - forecast|/n)


Tracking Signal

Tracking signal

+

0 MADs

–

Upper control limit

Lower control limit

Time

Signal exceeding limit

Acceptable range


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


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

Documents

© 2006 Prentice Hall, Inc.4 – 1 Short-range forecast Up to 1 year, generally less than 3 months Purchasing, job scheduling, workforce levels, job