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15-1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Forecasting
Chapter 15
15-2
■ Forecasting Components
■ Time Series Methods
■ Forecast Accuracy
■ Time Series Forecasting Using Excel
■ Time Series Forecasting Using QM for Windows
■ Regression Methods
Chapter Topics
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■ A variety of forecasting methods are available for use depending on the time frame of the forecast and the existence of patterns.
■ Time Frames:
Short-range (one to two months) Medium-range (two months to one or two years) Long-range (more than one or two years)
■ Patterns:
Trend Random variations Cycles Seasonal pattern
Forecasting Components
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Trend - A long-term movement of the item being forecast.
Random variations - movements that are not predictable and follow no pattern.
Cycle - A movement, up or down, that repeats itself over a lengthy time span.
Seasonal pattern - Oscillating movement in demand that occurs periodically in the short run and is repetitive.
Forecasting ComponentsPatterns (1 of 2)
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Figure 15.1 (a) Trend; (b) Cycle; (c) Seasonal; (d) Trend w/Season
Forecasting ComponentsPatterns (2 of 2)
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Forecasting ComponentsForecasting Methods
Times Series - Statistical techniques that use historical data to predict future behavior.
Regression Methods - Regression (or causal ) methods that attempt to develop a mathematical relationship between the item being forecast and factors that cause it to behave the way it does.
Qualitative Methods - Methods using judgment, expertise and opinion to make forecasts.
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Forecasting ComponentsQualitative Methods
“Jury of executive opinion,” a qualitative technique, is the most common type of forecast for long-term strategic planning.
Performed by individuals or groups within an organization, sometimes assisted by consultants and other experts, whose judgments and opinions are considered valid for the forecasting issue.
Usually includes specialty functions such as marketing, engineering, purchasing, etc. in which individuals have experience and knowledge of the forecasted item.
Supporting techniques include the Delphi Method, market research, surveys, etc.
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Time Series MethodsOverview
Statistical techniques that make use of historical data collected over a long period of time.
Methods assume that what has occurred in the past will continue to occur in the future.
Forecasts based on only one factor - time.
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i periodin data average moving in the periods ofnumber
:where
1
==
∑==
iDn
n
ni iD
nMA
Time Series MethodsMoving Average (1 of 5)
Moving average uses values from the recent past to develop forecasts.
This dampens or smoothes random increases and decreases.
Useful for forecasting relatively stable items that do not display any trend or seasonal pattern.
Formula for:
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Example: Instant Paper Clip Supply Company forecast of orders for the month of November.
Three-month moving average:
Five-month moving average:
orders 110313011090
3
3
13
=++=∑== i iD
MA
orders 915507513011090
5
5
15
=++++=∑== i iD
MA
Time Series MethodsMoving Average (2 of 5)
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Table 15.2 Three- and Five-Month Moving Averages
Time Series MethodsMoving Average (3 of 5)
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Figure 15.2 Three- and Five-Month Moving Averages
Time Series MethodsMoving Average (4 of 5)
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Time Series MethodsMoving Average (5 of 5)
Longer-period moving averages react more slowly to changes in demand than do shorter-period moving averages.
The appropriate number of periods to use often requires trial-and-error experimentation.
Moving average does not react well to changes (trends, seasonal effects, etc.) but is easy to use and inexpensive.
Good for short-term forecasting.
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In a weighted moving average, weights are assigned to the most recent data.
Determining precise weights and number of periods requires trial-and-error experimentation.
1
where the weight for period i, between 0% and 100%
1.00
Example: Paper clip company weights 50% for October, 33%for September, 17% for August:
3 (.50)(90) (.33)(110)13
nWMA W Dn i ii
Wi
Wi
WMA W Di ii
= ∑=
=
=∑
= = + +∑=
(.17)(130) 103.4 orders=
Time Series MethodsWeighted Moving Average
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Exponential smoothing weights recent past data more strongly than more distant data.
Two forms: simple exponential smoothing and adjustedexponential smoothing.
Simple exponential smoothing:
Ft + 1 = αDt + (1 - α)Ft
where: Ft + 1 = the forecast for the next periodDt = actual demand in the present periodFt = the previously determined forecast for the
present periodα = a weighting factor (smoothing constant).
Time Series MethodsExponential Smoothing (1 of 11)
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The most commonly used values of α are between 0.10 and 0.50.
Determination of α is usually judgmental and subjective and often based on trial-and -error experimentation.
Time Series MethodsExponential Smoothing (2 of 11)
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Example: PM Computer Services (see Table 15.4).
Exponential smoothing forecasts using smoothing constant of .30.
Forecast for period 2 (February):
F2 = α D1 + (1- α)F1 = (.30)(.37) + (.70)(.37) = 37 units
Forecast for period 3 (March):
F3 = α D2 + (1- α)F2 = (.30)(.40) + (.70)(37) = 37.9 units
Time Series MethodsExponential Smoothing (3 of 11)
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Table 15.4 Exponential Smoothing Forecasts, α = .30 and α = .50
Time Series MethodsExponential Smoothing (4 of 11)
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The forecast that uses the higher smoothing constant (.50) reacts more strongly to changes in demand than does the forecast with the lower constant (.30).
Both forecasts lag behind actual demand.
Both forecasts tend to be consistently lower than actual demand.
Low smoothing constants are appropriate for stable data without trend; higher constants appropriate for data with trends.
Time Series MethodsExponential Smoothing (5 of 11)
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Figure 15.3 Exponential Smoothing Forecasts
Time Series MethodsExponential Smoothing (6 of 11)
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■ Adjusted exponential smoothing: exponential smoothing with a trend adjustment factor added.
Formula:
AFt + 1 = Ft + 1 + Tt+1
where: T = an exponentially smoothed trend factorTt + 1 + β(Ft + 1 - Ft) + (1 - β)TtTt = the last period trend factorβ = smoothing constant for trend ( a value
between zero and one).
■ Reflects the weight given to the most recent trend data.
■ Determined subjectively.
Time Series MethodsExponential Smoothing (7 of 11)
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Example: PM Computer Services exponential smoothedforecasts with α = .50 and β = .30 (see Table 15.5).
Adjusted forecast for period 3:
T3 = β(F3 - F2) + (1 - β)T2
= (.30)(38.5 - 37.0) + (.70)(0) = 0.45
AF3 = F3 + T3 = 38.5 + 0.45 = 38.95
Time Series MethodsExponential Smoothing (8 of 11)
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Table 15.5 Adjusted Exponentially Smoothed Forecast Values
Time Series MethodsExponential Smoothing (9 of 11)
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■ Adjusted forecast is consistently higher than the simple exponentially smoothed forecast.
■ It is more reflective of the generally increasing trend of the data.
Time Series MethodsExponential Smoothing (10 of 11)
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Figure 15.4 Adjusted Exponentially Smoothed Forecast
Time Series MethodsExponential Smoothing (11 of 11)
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xperiod for demand for forecast
period time the line the of slope
0) period (at intercept :where
====
+=
yxba
bxay
nyy
nx
n
xbyaxnxyxnxyb
∑=
∑==
−=−∑
−∑=
x
periods of number :where
2
■ When demand displays an obvious trend over time, a least squares regression line , or linear trend line, can be used to forecast.
■ Formula:
Time Series MethodsLinear Trend Line (1 of 5)
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Example: PM Computer Services (see Table 15.6)
565713721235 13, x 13, period for
line trend linear 721235
235567214246
72125612650424656128673
22
424612557 5612
78
.)(..
..
.).)(.(.
.).(
).)(.)((,
..
=+==
+=
=−=−=
=−
−=−∑
−∑=
====
y
xy
xbya
xnxyxnxyb
yx
Time Series MethodsLinear Trend Line (2 of 5)
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Table 15.6Least Squares Calculations
Time Series MethodsLinear Trend Line (3 of 5)
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■ A trend line does not adjust to a change in the trend as does the exponential smoothing method.
■ This limits its use to shorter time frames in which trend will not change.
Time Series MethodsLinear Trend Line (4 of 5)
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Figure 15.5 Linear Trend Line
Time Series MethodsLinear Trend (5 of 5)
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■ A seasonal pattern is a repetitive up-and-down movement in demand.
■ Seasonal patterns can occur on a quarterly, monthly, weekly, or daily basis.
■ A seasonally adjusted forecast can be developed by multiplying the normal forecast by a seasonal factor.
■ A seasonal factor can be determined by dividing the actual demand for each seasonal period by total annual demand:
Si =Di/∑D
Time Series MethodsSeasonal Adjustments (1 of 4)
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■ Seasonal factors lie between zero and one and represent the portion of total annual demand assigned to each season.
■ Seasonal factors are multiplied by annual demand to provide adjusted forecasts for each period.
Time Series MethodsSeasonal Adjustments (2 of 4)
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S1 = D1/ ∑D = 42.0/148.7 = 0.28S2 = D2/ ∑D = 29.5/148.7 = 0.20S3 = D3/ ∑D = 21.9/148.7 = 0.15S4 = D4/ ∑D = 55.3/148.7 = 0.37
Table 15.7 Demand for Turkeys at Wishbone Farms
Example: Wishbone Farms
Time Series MethodsSeasonal Adjustments (3 of 4)
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Multiply forecasted demand for entire year by seasonal factors to determine quarterly demand.
Forecast for entire year (trend line for data in Table 15.7):
y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17
Seasonally adjusted forecasts:
SF1 = (S1)(F5) = (.28)(58.17) = 16.28
SF2 = (S2)(F5) = (.20)(58.17) = 11.63
SF3 = (S3)(F5) = (.15)(58.17) = 8.73
SF4 = (S4)(F5) = (.37)(58.17) = 21.53
Time Series MethodsSeasonal Adjustments (4 of 4)
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Forecasts will always deviate from actual values.
Difference between forecasts and actual values referred to as forecast error.
Would like forecast error to be as small as possible.
If error is large, either technique being used is the wrong one, or parameters need adjusting.
Measures of forecast errors:
Mean Absolute deviation (MAD) Mean absolute percentage deviation (MAPD) Cumulative error (E bar) Average error, or bias (E)
Forecast AccuracyOverview
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MAD is the average absolute difference between the forecast and actual demand.
Most popular and simplest-to-use measures of forecast error.
Formula:
periods ofnumber total then tperiodfor forecast the tF
tperiodin demand tD number period the t
:where
====
∑ −= n
tFtDMAD
Forecast AccuracyMean Absolute Deviation (1 of 7)
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Example: PM Computer Services (see Table 15.8).
Compare accuracies of different forecasts using MAD:
85.41153.41 ==∑ −
= ntFtD
MAD
Forecast AccuracyMean Absolute Deviation (2 of 7)
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Table 15.8 Computational Values for MAD and error
Forecast AccuracyMean Absolute Deviation (3 of 7)
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The lower the value of MAD relative to the magnitude of the data, the more accurate the forecast.
When viewed alone, MAD is difficult to assess.
Must be considered in light of magnitude of the data.
Forecast AccuracyMean Absolute Deviation (4 of 7)
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Can be used to compare accuracy of different forecasting techniques working on the same set of demand data (PM Computer Services):
Exponential smoothing (α = .50): MAD = 4.04
Adjusted exponential smoothing (α = .50, β = .30): MAD = 3.81
Linear trend line: MAD = 2.29
Linear trend line has lowest MAD; increasing α from .30 to .50 improved smoothed forecast.
Forecast AccuracyMean Absolute Deviation (5 of 7)
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A variation on MAD is the mean absolute percent deviation (MAPD).
Measures absolute error as a percentage of demand rather than per period.
Eliminates problem of interpreting the measure of accuracy relative to the magnitude of the demand and forecast values.
Formula:
10.3%or 103.52041.53 ==
∑∑ −
=tD
tFtDMAPD
Forecast AccuracyMean Absolute Deviation (6 of 7)
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MAPD for other three forecasts:
Exponential smoothing (α = .50): MAPD = 8.5%
Adjusted exponential smoothing (α = .50, β = .30): MAPD = 8.1%
Linear trend: MAPD = 4.9%
Forecast AccuracyMean Absolute Deviation (7 of 7)
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Cumulative error is the sum of the forecast errors (E =∑et). A relatively large positive value indicates forecast is biased low,
a large negative value indicates forecast is biased high. If preponderance of errors are positive, forecast is consistently low;
and vice versa. Cumulative error for trend line is always almost zero, and is
therefore not a good measure for this method. Cumulative error for PM Computer Services can be read directly
from Table 15.8. E = ∑ et = 49.31 indicating forecasts are frequently below actual
demand.
Forecast AccuracyCumulative Error (1 of 2)
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Cumulative error for other forecasts:
Exponential smoothing (α = .50): E = 33.21
Adjusted exponential smoothing (α = .50, β =.30):E = 21.14
Average error (bias) is the per period average of cumulative error.
Average error for exponential smoothing forecast:
A large positive value of average error indicates a forecast is biased low; a large negative error indicates it is biased high.
Forecast AccuracyCumulative Error (2 of 2)
48.41131.49 ==
∑= n
teE
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Results consistent for all forecasts:
Larger value of alpha is preferable.
Adjusted forecast is more accurate than exponential smoothing.
Linear trend is more accurate than all the others.
Table 15.9 Comparison of Forecasts for PM Computer Services
Forecast AccuracyExample Forecasts by Different Measures
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Exhibit 15.1
Time Series Forecasting Using Excel (1 of 4)
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Exhibit 15.2
Time Series Forecasting Using Excel (2 of 4)
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Exhibit 15.3
Time Series Forecasting Using Excel (3 of 4)
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Exhibit 15.4
Time Series Forecasting Using Excel (4 of 4)
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15-50Exhibit 15.5
Exponential Smoothing Forecast with Excel QM
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Time Series ForecastingSolution with QM for Windows (1 of 2)
Exhibit 15.6Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
15-52Exhibit 15.7
Time Series ForecastingSolution with QM for Windows (2 of 2)
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Time series techniques relate a single variable being forecast to time.
Regression is a forecasting technique that measures the relationship of one variable to one or more other variables.
Simplest form of regression is linear regression.
Regression MethodsOverview
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data y the of mean
data x the of mean :where
22
=∑=
=∑=
∑ −∑ −=
−=
+=
nyy
nxx
xnxyxnxyb
xbya
bxay
Linear regression relates demand (dependent variable ) to an independent variable.
Regression MethodsLinear Regression
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State University Athletic Department.
Wins Attendance 4 6 6 8 6 7 5 7
36,300 40,100 41,200 53,000 44,000 45,600 39,000 47,500
x (wins)
y (attendance, 1,000s)
xy
x2
4 6 6 8 6 7 5 7
49
36.3 40.1 41.2 53.0 44.0 45.6 39.0 47.5 346.7
145.2 240.6 247.2 424.0 264.0 319.2 195.0 332.5
2,167.7
16 36 36 64 36 49 25 49
311
Regression MethodsLinear Regression Example (1 of 3)
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88046 or 884670644618is wins7 x for forecast Attendance
0644618 Therefore,
461812564063443
064212568311344312568701672
22
344389346
1256849
,.)(..
..
.).)((..
.).)(()(
).)(.)((.,(
..
.
=+==
+=
=−=−=
=−−=
∑ −
−∑=
==
==
y
xy
xbya
xnxyxnxyb
y
x
Regression MethodsLinear Regression Example (2 of 3)
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Figure 15.6Linear Regression Line
Regression MethodsLinear Regression Example (3 of 3)
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Correlation is a measure of the strength of the relationship between independent and dependent variables.
Formula:
Value lies between +1 and -1. Value of zero indicates little or no relationship between
variables. Values near 1.00 and -1.00 indicate strong linear relationship.
( )
∑ ∑−∑ ∑−
∑ ∑∑−=2222 yynxxn
yxxynr
Regression MethodsCorrelation (1 of 2)
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948.2)7.346()7.224,15)(8()49)(49()311)(8(
)7.346)(49()7.167,2)(8( =−−
−=
r
Value for State University example:
Regression MethodsCorrelation (2 of 2)
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The Coefficient of determination is the percentage of the variation in the dependent variable that results from the independent variable.
Computed by squaring the correlation coefficient, r.
For State University example:r = .948, r2 = .899
This value indicates that 89.9% of the amount of variation in attendance can be attributed to the number of wins by the team, with the remaining 10.1% due to other, unexplained, factors.
Regression MethodsCoefficient of Determination
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Regression Analysis with Excel (1 of 6)
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Exhibit 15.8
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Regression Analysis with Excel (2 of 6)
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Exhibit 15.9
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Exhibit 15.10
Regression Analysis with Excel (3 of 6)
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Exhibit 15.11
Regression Analysis with Excel (4 of 6)
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Exhibit 15.12
Regression Analysis with Excel (5 of 6)
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Exhibit 15.13
Regression Analysis with Excel (6 of 6)
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Multiple Regression with Excel (1 of 4)
Multiple regression relates demand to two or more independent variables.
General form:
y = β0 + β 1x1 + β 2x2 + . . . + β kxk
where β 0 = the intercept
β 1 . . . β k = parameters representing contributions of the independent variables
x1 . . . xk = independent variables
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State University example:
Wins Promotion ($) Attendance 4 6 6 8 6 7 5 7
29,500 55,700 71,300 87,000 75,000 72,000 55,300 81,600
36,300 40,100 41,200 53,000 44,000 45.600 39,000 47,500
Multiple Regression with Excel (2 of 4)
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15-69Exhibit 15.14
Multiple Regression with Excel (3 of 4)
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15-70Exhibit 15.15
Multiple Regression with Excel (4 of 4)
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Period Units 1 2 3 4 5 6 7 8
56 61 55 70 66 65 72 75
Problem Statement:
For data below, develop an exponential smoothing forecast using α = .40, and an adjusted exponential smoothing forecast using α = .40 and β= .20.
Compare the accuracy of the forecasts using MAD and cumulative error.
Example Problem SolutionComputer Software Firm (1 of 4)
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Step 1: Compute the Exponential Smoothing Forecast.
Ft+1 = α Dt + (1 - α)Ft
Step 2: Compute the Adjusted Exponential SmoothingForecast
AFt+1 = Ft +1 + Tt+1
Tt+1 = β(Ft +1 - Ft) + (1 - β)Tt
Example Problem SolutionComputer Software Firm (2 of 4)
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Period Dt Ft AFt Dt - Ft Dt - AFt 1 2 3 4 5 6 7 8 9
56 61 55 70 66 65 72 75
56.00 58.00 56.80 62.08 63.65 64.18 67.31 70.39
56.00 58.40 56.88 63.20 64.86 65.26 68.80 72.19
5.00
-3.00 13.20 3.92 1.35 7.81 7.68
35.97
5.00
-3.40 13.12 2.80 0.14 6.73 6.20
30.60
Example Problem SolutionComputer Software Firm (3 of 4)
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Step 3: Compute the MAD Values
Step 4: Compute the Cumulative Error.
E(Ft) = 35.97
E(AFt) = 30.60
34.5739.37)(
99.5797.41)(
==∑ −=
==∑ −=
ntAFtD
tAFMAD
ntFtD
tFMAD
Example Problem SolutionComputer Software Firm (4 of 4)
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For the following data:
Develop a linear regression model
Determine the strength of the linear relationship using correlation.
Determine a forecast for lumber given 10 building permits in the next quarter.
Example Problem SolutionBuilding Products Store (1 of 5)
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Quarter
Building Permits, x
Lumber Sales (1,000s of bd ft), y
1 2 3 4 5 6 7 8 9
10
8 12 7 9 15 6 5 8 10 12
12.6 16.3 9.3 11.5 18.1 7.6 6.2 14.2 15.0 17.8
Example Problem SolutionBuilding Products Store (2 of 5)
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Step 1: Compute the Components of the Linear Regression Equation.
361292518612
251229109328612291032901
22
8612106128
921092
.).)(.(.
.).)(()(
).)(.)(().,(
..
=−=−=
=−−=
−∑
∑ −=
==
==
xbya
xnxyxnxyb
y
x
Example Problem SolutionBuilding Products Store (3 of 5)
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Step 2: Develop the Linear regression equation.
y = a + bx, y = 1.36 + 1.25x
Step 3: Compute the Correlation Coefficient.
( )
∑ ∑−∑ ∑−
∑ ∑∑−=2222 yynxxn
yxxynr
9252612848810110929293210
6128923170110 .).().,)(())(())((
).)(().,)(( =−−
−=
r
Example Problem SolutionBuilding Products Store (4 of 5)
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Example Problem SolutionBuilding Products Store (5 of 5)
Step 4: Calculate the forecast for x = 10 permits.
Y = a + bx = 1.36 + 1.25(10) = 13.86 or 1,386 board ft
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