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Chapter 15 - Forecasting 1
Chapter 15Forecasting
Introduction to Management Science8th Edition
byBernard W. Taylor III
Chapter 15 - Forecasting 2
Forecasting Components
Time Series Methods
Forecast Accuracy
Time Series Forecasting Using Excel
Time Series Forecasting Using QM for Windows
Regression Methods
Chapter Topics
Chapter 15 - Forecasting 3
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:
TrendRandom variationsCyclesSeasonal pattern
Forecasting Components
Chapter 15 - Forecasting 4
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)
Chapter 15 - Forecasting 5
Figure 15.1Forms of Forecast Movement: (a) Trend, (b) Cycle, (c) Seasonal
Pattern, (d) Trend with Seasonal Pattern
Forecasting ComponentsPatterns (2 of 2)
Chapter 15 - Forecasting 6
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.
Chapter 15 - Forecasting 7
Forecasting ComponentsQualitative Methods
Qualitative methods, the “jury of executive opinion,” is the most common type of forecasting method for long-term strategic planning.
Performed by individuals or groups within an organization, sometimes assisted by consultants and other experts, whose judgments and opinion 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.
Chapter 15 - Forecasting 8
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.
Chapter 15 - Forecasting 9
i periodin data average moving in the periods ofnumber
:where1
iDn
n
n
iiD
MAn
Time Series MethodsMoving Average (1 of 5)
Moving average uses values from the recent past to develop forecasts.
This dampens or smoothes out random increases and decreases.
Useful for forecasting relatively stable items that do not display any trend or seasonal pattern.
Formula for:
Chapter 15 - Forecasting 10
Example: Instant Paper Clip Supply Company forecast of orders for the next month.
Three-month moving average:
Five-month moving average:
orders 110313011090
3
3
13
iiD
MA
orders 915507513011090
5
5
15
iiD
MA
Time Series MethodsMoving Average (2 of 5)
Chapter 15 - Forecasting 11
Figure 15.2Three- and Five-Month Moving Averages
Time Series MethodsMoving Average (3 of 5)
Chapter 15 - Forecasting 12
Figure 15.2Three- and Five-Month Moving Averages
Time Series MethodsMoving Average (4 of 5)
Chapter 15 - Forecasting 13
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.
Chapter 15 - Forecasting 14
In a weighted moving average, weights are assigned to the most recent data.Formula:
orders 103.4 )130)(17(.)110)(33(.)90)(50(. 3
13
:Augustfor 17% September,for 33% October,for 50% ightscompany we clipPaper :Example
1.00
100% and 0%between i, periodfor weight the where
1
iiDiWWMA
iW
iW
n
iiDiWnWMA
Time Series MethodsWeighted Moving Average (1 of 2)
Chapter 15 - Forecasting 15
Determining precise weights and number of periods requires trial-and-error experimentation.
Time Series MethodsWeighted Moving Average (2 of 2)
Chapter 15 - Forecasting 16
Exponential smoothing weights recent past data more strongly than more distant data.
Two forms: simple exponential smoothing and adjusted exponential smoothing.
Simple exponential smoothing:
Ft + 1 = Dt + (1 - )Ft
where: Ft + 1 = the forecast for the next period Dt = actual demand in the present period Ft = the previously determined forecast for
the present period = a weighting factor (smoothing constant).
Time Series MethodsExponential Smoothing (1 of 11)
Chapter 15 - Forecasting 17
The most commonly used values of are between .10 and .50.
Determination of is usually judgmental and subjective and often based on trial-and -error experimentation.
Time Series MethodsExponential Smoothing (2 of 11)
Chapter 15 - Forecasting 18
Example: PM Computer Services (see Table 15.4).
Exponential smoothing forecasts using smoothing constant of .30.
Forecast for period 2 (February):
F2 = D1 + (1- )F1 = (.30)(.37) + (.70)(.37) = 37 units
Forecast for period 3 (March):
F3 = D2 + (1- )F2 = (.30)(.40) + (.70)(37) = 37.9 units
Time Series MethodsExponential Smoothing (3 of 11)
Chapter 15 - Forecasting 19
Table 15.4Exponential Smoothing Forecasts, = .30 and = .50
Time Series MethodsExponential Smoothing (4 of 11)
Chapter 15 - Forecasting 20
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)
Chapter 15 - Forecasting 21
Figure 15.3Exponential Smoothing Forecasts
Time Series MethodsExponential Smoothing (6 of 11)
Chapter 15 - Forecasting 22
Adjusted exponential smoothing: exponential smoothing with a trend adjustment factor added.
Formula:
AFt + 1 = Ft + 1 + Tt+1
where: T = an exponentially smoothed trend factor Tt + 1 + (Ft + 1 - Ft) + (1 - )Tt
Tt = the last period trend factor = 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)
Chapter 15 - Forecasting 23
Example: PM Computer Services exponential smoothedforecasts with = .50 and = .30 (see Table 15.5).
Adjusted forecast for period 3:
T3 = (F3 - F2) + (1 - )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)
Chapter 15 - Forecasting 24
Table 15.5Adjusted Exponentially Smoothed Forecast Values
Time Series MethodsExponential Smoothing (9 of 11)
Chapter 15 - Forecasting 25
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)
Chapter 15 - Forecasting 26
Figure 15.4Adjusted Exponentially Smoothed Forecast
Time Series MethodsExponential Smoothing (11 of 11)
Chapter 15 - Forecasting 27
xperiod for demand for forecast
period time the line the of slope
0) period (at intercept :where
yxba
bxay
nyynx
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)
Chapter 15 - Forecasting 28
Example: PM Computer Services (see Table 15.6)
565713721235 13, x 13, period for
line trend linear 721235
235567214246
72125612650
42465612867322
424612557 56
1278
.)(..
..
.).)(.(.
.).(
).)(.)((,
..
y
xy
xbya
xnx
yxnxyb
yx
Time Series MethodsLinear Trend Line (2 of 5)
Chapter 15 - Forecasting 29
Table 15.6Least Squares Calculations
Time Series MethodsLinear Trend Line (3 of 5)
Chapter 15 - Forecasting 30
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)
Chapter 15 - Forecasting 31
Figure 15.5Linear Trend Line
Time Series MethodsLinear Trend Line (5 of 5)
Chapter 15 - Forecasting 32
A seasonal pattern is a repetitive up-and-down movement in demand.
Seasonal patterns can occur on a 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)
Chapter 15 - Forecasting 33
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)
Chapter 15 - Forecasting 34
S1 = D1/ D = 42.0/148.7 = 0.28 S2 = D2/ D = 29.5/148.7 = 0.20 S3 = D3/ D = 21.9/148.7 = 0.15 S4 = D4/ D = 55.3/148.7 = 0.37
Table 15.7Demand for Turkeys at Wishbone Farms
Example: Wishbone Farms
Time Series MethodsSeasonal Adjustments (3 of 4)
Chapter 15 - Forecasting 35
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)
Chapter 15 - Forecasting 36
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
Chapter 15 - Forecasting 37
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)
Chapter 15 - Forecasting 38
Example: PM Computer Services (see Table 15.8).
Compare accuracies of different forecasts using MAD:
85.41153.41
ntFtDMAD
Forecast AccuracyMean Absolute Deviation (2 of 7)
Chapter 15 - Forecasting 39
Table 15.8Computational Values for MAD
Forecast AccuracyMean Absolute Deviation (3 of 7)
Chapter 15 - Forecasting 40
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)
Chapter 15 - Forecasting 41
Can be used to compare accuracy of different forecasting techniques working on the same set of demand data (PM Computer Services):
Exponential smoothing ( = .50): MAD = 4.04
Adjusted exponential smoothing ( = .50, = .30): MAD = 3.81
Linear trend line: MAD = 2.29
Linear trend line has lowest MAD; increasing from .30 to .50 improved smoothed forecast.
Forecast AccuracyMean Absolute Deviation (5 of 7)
Chapter 15 - Forecasting 42
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
tDtFtDMAPD
Forecast AccuracyMean Absolute Deviation (6 of 7)
Chapter 15 - Forecasting 43
MAPD for other three forecasts:
Exponential smoothing ( = .50): MAPD = 8.5%
Adjusted exponential smoothing ( = .50, = .30): MAPD = 8.1%
Linear trend: MAPD = 4.9%
Forecast AccuracyMean Absolute Deviation (7 of 7)
Chapter 15 - Forecasting 44
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)
Chapter 15 - Forecasting 45
Cumulative error for other forecasts:
Exponential smoothing ( = .50): E = 33.21
Adjusted exponential smoothing ( = .50, =.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)
48411
3149 .. netE
Chapter 15 - Forecasting 46
Results consistent for all forecasts:Larger value of alpha is preferable.Adjusted forecast is more accurate than exponential
smoothing forecasts.Linear trend is more accurate than all the others.
Table 15.9Comparison of Forecasts for PM Computer Services
Forecast AccuracyExample Forecasts by Different Measures
Chapter 15 - Forecasting 47Exhibit 15.1
Time Series Forecasting Using Excel (1 of 4)
Chapter 15 - Forecasting 48
Exhibit 15.2
Time Series Forecasting Using Excel (2 of 4)
Chapter 15 - Forecasting 49
Exhibit 15.3
Time Series Forecasting Using Excel (3 of 4)
Chapter 15 - Forecasting 50
Exhibit 15.4
Time Series Forecasting Using Excel (4 of 4)
Chapter 15 - Forecasting 51
Exhibit 15.5
Exponential Smoothing Forecast with Excel QM
Chapter 15 - Forecasting 52
Time Series ForecastingSolution with QM for Windows (1 of 2)
Exhibit 15.6
Chapter 15 - Forecasting 53
Exhibit 15.7
Time Series ForecastingSolution with QM for Windows (2 of 2)
Chapter 15 - Forecasting 54
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
Chapter 15 - Forecasting 55
data y the of mean
data x the of mean :where
22
nyy
nxx
xnx
yxnxyb
xbya
bxay
Linear regression relates demand (dependent variable ) to an independent variable.
Regression MethodsLinear Regression
Chapter 15 - Forecasting 56
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)
Chapter 15 - Forecasting 57
88046 or 884670644618is wins7 x for forecast Attendance
0644618 Therefore,
461812564063443
064212568311
34431256870167222
34438
9346
1256849
,.)(..
..
.).)((..
.).)(()(
).)(.)((.,(
..
.
y
xy
xbya
xnx
yxnxyb
y
x
Regression MethodsLinear Regression Example (2 of 3)
Chapter 15 - Forecasting 58
Figure 15.6Linear Regression Line
Regression MethodsLinear Regression Example (3 of 3)
Chapter 15 - Forecasting 59
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)
Chapter 15 - Forecasting 60
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)
Chapter 15 - Forecasting 61
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 = .899This 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
Chapter 15 - Forecasting 62
Exhibit 15.8
Regression Analysis with Excel (1 of 7)
Chapter 15 - Forecasting 63
Exhibit 15.9
Regression Analysis with Excel (2 of 7)
Chapter 15 - Forecasting 64
Exhibit 15.10
Regression Analysis with Excel (3 of 7)
Chapter 15 - Forecasting 65
Exhibit 15.11
Regression Analysis with Excel (4 of 7)
Chapter 15 - Forecasting 66
Exhibit 15.12
Regression Analysis with Excel (5 of 7)
Chapter 15 - Forecasting 67
Exhibit 15.13
Regression Analysis with Excel (6 of 7)
Chapter 15 - Forecasting 68
Exhibit 15.14
Regression Analysis with Excel (7 of 7)
Chapter 15 - Forecasting 69
Exhibit 15.15
Regression Analysis with QM for Windows
Chapter 15 - Forecasting 70
Multiple Regression with Excel (1 of 4)
Multiple regression relates demand to two or more independent variables.
General form:
y = 0 + 1x1 + 2x2 + . . . + kxk
where 0 = the intercept
1 . . . k = parameters representing contributions of the independent
variables
x1 . . . xk = independent variables
Chapter 15 - Forecasting 71
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)
Chapter 15 - Forecasting 72
Exhibit 15.16
Multiple Regression with Excel (3 of 4)
Chapter 15 - Forecasting 73
Exhibit 15.17
Multiple Regression with Excel (4 of 4)
Chapter 15 - Forecasting 74
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 = .40, and an adjusted exponential smoothing forecast using = .40 and = .20.
• Compare the accuracy of the forecasts using MAD and cumulative error.
Example Problem SolutionComputer Software Firm (1 of 4)
Chapter 15 - Forecasting 75
Step 1: Compute the Exponential Smoothing Forecast.
Ft+1 = Dt + (1 - )Ft
Step 2: Compute the Adjusted Exponential SmoothingForecast
AFt+1 = Ft +1 + Tt+1
Tt+1 = (Ft +1 - Ft) + (1 - )Tt
Example Problem SolutionComputer Software Firm (2 of 4)
Chapter 15 - Forecasting 76
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)
Chapter 15 - Forecasting 77
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)
Chapter 15 - Forecasting 78
Problem Statement:
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)
Chapter 15 - Forecasting 79
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)
Chapter 15 - Forecasting 80
Step 1: Compute the Components of the Linear RegressionEquation.
361292518612
25122910932
861229103290122
861210
6128
921092
.).)(.(.
.).)(()(
).)(.)(().,(
..
xbya
xnx
yxnxyb
y
x
Example Problem SolutionBuilding Products Store (3 of 5)
Chapter 15 - Forecasting 81
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)
Chapter 15 - Forecasting 82
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