Transcript Document
Introduction to Management Science 8th Edition by Bernard W. Taylor III Chapter 5
Forecasting
Chapter 5 - Forecasting 1
Chapter Topics
Forecasting Components Time Series Methods Forecast Accuracy Time Series Forecasting Using Excel Time Series Forecasting Using QM for Windows Regression Methods Chapter 5 - Forecasting 2
Forecasting Components
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 Chapter 5 - Forecasting 3
Forecasting Components Patterns (1 of 2)
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.
Chapter 5 - Forecasting 4
Forecasting Components Patterns (2 of 2)
trend-line
Figure 5.1
Forms of Forecast Movement: (a) Trend, (b) Cycle, (c) Seasonal Pattern, (d) Trend with Seasonal Pattern Chapter 5 - Forecasting 5
Forecasting Components Forecasting 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 5 - Forecasting 6
Forecasting Components Qualitative 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 5 - Forecasting 7
Time Series Methods Overview
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 5 - Forecasting 8
Time Series Methods Moving Average (1 of 5)
Moving average uses values from the recent past to develop forecasts.
This
dampens
decreases.
or
smoothes
out random increases and Useful for forecasting relatively stable items that do not display any trend or seasonal pattern.
Formula for:
MA n
1
n n
i
1
D i
where:
n
number of periods in the moving average
D i
data in period
i
9
Time Series Methods Moving Average (2 of 5)
Example: Instant Paper Clip Supply Company forecast of orders for the next month.
Three-month moving average:
MA
3 1 3 3
i
1
D i
90 110 3 130 110 orders Five-month moving average:
MA
3 1 5 5
i
1
D i
90 110 130 5 75 50 91 orders Chapter 5 - Forecasting 10
Time Series Methods Moving Average (3 of 5)
Chapter 5 - Forecasting
Figure 5.2
Three- and Five-Month Moving Averages 11
Time Series Methods Moving Average (4 of 5)
Chapter 5 - Forecasting
Figure 5.2
Three- and Five-Month Moving Averages 12
Time Series Methods Moving 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 5 - Forecasting 13
Time Series Methods Weighted Moving Average (1 of 2)
In a weighted moving average, weights are assigned to the most recent data.
Formula:
WMA n
i n
1
W i D i
where
W i
the weight for period i, between 0% and 100% 1.00
Example : Paper clip company we ights 50% for October, 33% for September, 17% for August :
WMA
3
i
3 1
W i D i
(.
50 )( 90 ) (.
33 )( 110 ) (.
17 )( 130 ) 103.4
orders Chapter 5 - Forecasting 14
Time Series Methods Weighted Moving Average (2 of 2)
Determining precise weights and number of periods requires trial-and-error experimentation.
Chapter 5 - Forecasting 15
Time Series Methods Exponential Smoothing (1 of 11)
Exponential smoothing weights recent past data more strongly than more distant data.
Two forms
: simple
exponential smoothing and
adjusted
exponential smoothing.
Simple exponential smoothing: F t + 1 = D t + (1 )F t where: F t + 1 D t = the forecast for the next period = actual demand in the present period F t = the previously determined forecast for the present period = a weighting factor (smoothing constant) use F 1 = D 1 .
Chapter 5 - Forecasting 16
Time Series Methods Exponential Smoothing (2 of 11)
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.
Chapter 5 - Forecasting 17
Time Series Methods Exponential Smoothing (3 of 11)
Example: PM Computer Services (see Table 5.4).
Exponential smoothing forecasts using smoothing constant of .30.
Forecast for period 2 (February): F 2 = D 1 + (1 )F 1 = ( .30
)(37) + ( .70
)(37) = 37 units Forecast for period 3 (March): F 3 = D 2 + (1 )F 2 = ( .30
)(40) + ( .70
)(37) = 37.9 units Chapter 5 - Forecasting 18
Time Series Methods Exponential Smoothing (4 of 11)
Using F 1 = D 1 Chapter 5 - Forecasting
Table 5.4
Exponential Smoothing Forecasts, = .30 and = .50
19
Time Series Methods Exponential Smoothing (5 of 11)
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.
Chapter 5 - Forecasting 20
Time Series Methods Exponential Smoothing (6 of 11)
Chapter 5 - Forecasting
Figure 5.3
Exponential Smoothing Forecasts 21
Time Series Methods Exponential Smoothing (7 of 11)
Adjusted exponential smoothing
: exponential smoothing with a trend adjustment factor added.
Formula: AF t + 1 = F t + 1 + T t+1 where: T t = an exponentially smoothed trend factor: T t + 1 T t = (F t + 1 - F t ) + (1 )T t = the last (previous) period’s trend factor = smoothing constant for trend ( a value between zero and one).
Reflects the weight given to the most recent trend data.
Determined subjectively.
Chapter 5 - Forecasting 22
Time Series Methods Exponential Smoothing (8 of 11)
Example: PM Computer Services exponential smoothed forecasts with = .50
and = .30
(see Table 5.5).
Start with T 2 = 0.00
Adjusted forecast for period 3: T 3 = (F 3 - F 2 ) + (1 )T 2 = ( .30
)(38.5 - 37.0) + ( .70
)(0) = 0.45
AF 3 = F 3 + T 3 = 38.5 + 0.45 = 38.95
Chapter 5 - Forecasting 23
Time Series Methods Exponential Smoothing (9 of 11)
T t + 1 = (F t + 1 + (1 - F t ) )T t
Table 5.5
Adjusted Exponentially Smoothed Forecast Values Chapter 5 - Forecasting 24
Time Series Methods Exponential Smoothing (10 of 11)
Adjusted forecast is
consistently higher
than the simple exponentially smoothed forecast.
It is more reflective of the generally increasing trend of the data.
Chapter 5 - Forecasting 25
Time Series Methods Exponential Smoothing (11 of 11)
Chapter 5 - Forecasting
Figure 5.4
Adjusted Exponentially Smoothed Forecast 26
Time Series Methods Linear Trend Line (1 of 5)
When demand displays an obvious trend over time, a least squares regression line , or forecast.
linear trend line
, can be used to Formula:
y
a
bx
where:
a
intercept (at period 0)
b
slope of the line
x
the time period
y
forecast for demand for period
x b
x
2
nx a
y
bx
where:
n xy
number of periods x 1
n y
1
n nxy
x y
Chapter 5 - Forecasting 27
Time Series Methods Linear Trend Line (2 of 5)
Example: PM Computer Services (see Table 5.6)
x
78 12 6.5
y
557 12 46.42
b
xy nxy x
2
nx
2 3,867 (12)(6.5)(46.42) 650 12(6.5) 2 1.72
a
y
bx
46.42
(1.72)(6.5) 35.2
y
35.2
1.72
x
linear trend line for period 13,
x
13,
y
35.2
1.72(13) 57.56
Chapter 5 - Forecasting 28
Time Series Methods Linear Trend Line (3 of 5) Table 5.6
Least Squares Calculations Chapter 5 - Forecasting 29
Time Series Methods Linear Trend Line (4 of 5)
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.
Chapter 5 - Forecasting 30
Time Series Methods Linear Trend Line (5 of 5)
Chapter 5 - Forecasting
Figure 5.5
Linear Trend Line 31
Time Series Methods Seasonal Adjustments (1 of 4)
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: S i =D i / D Chapter 5 - Forecasting 32
Time Series Methods Seasonal Adjustments (2 of 4)
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.
Chapter 5 - Forecasting 33
Time Series Methods Seasonal Adjustments (3 of 4)
Example: Wishbone Farms Chapter 5 - Forecasting
Table 5.7
Demand for Turkeys at Wishbone Farms S 1 S 2 S 3 S 4 = D 1 / D = 42.0/148.7 = 0.28
= D 2 / D = 29.5/148.7 = 0.20
= D 3/ D = 21.9/148.7 = 0.15
= D 4 / D = 55.3/148.7 = 0.37
34
Time Series Methods Seasonal Adjustments (4 of 4)
Multiply forecasted demand for entire year by seasonal factors to determine quarterly demand.
Forecast for entire year (
trend line
for data in Table 5.7): y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17
Seasonally adjusted forecasts: SF 1 = (S 1 )(F 5 ) = (.28)(58.17) = 16.28
SF 2 = (S 2 )(F 5 ) = (.20)(58.17) = 11.63
SF 3 = (S 3 )(F 5 ) = (.15)(58.17) = 8.73
SF 4 = (S 4 )(F 5 ) = (.37)(58.17) = 21.53
Note the potential for confusion: the S i whereas the SF i are “seasonal factors” (fractions), are “seasonally adjusted forecasts” (commodity values)!
Chapter 5 - Forecasting 35
Forecast Accuracy Overview
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) Chapter 5 - Forecasting 36
Forecast Accuracy Mean Absolute Deviation (1 of 7)
MAD is the average absolute difference between the forecast and actual demand.
Most popular and simplest-to-use measures of forecast error.
Formula:
MAD
1
n
D t
F t
where:
t
the period number
D t
demand in period
t F t
the forecast for period
t n
the total number of periods 37
Forecast Accuracy Mean Absolute Deviation (2 of 7)
Example: PM Computer Services (see Table 5.8).
Compare accuracies of different forecasts using MAD:
MAD
1
n
D t
F t
53.41
11 4.85
Chapter 5 - Forecasting 38
Forecast Accuracy Mean Absolute Deviation (3 of 7)
Chapter 5 - Forecasting
Table 5.8
Computational Values for MAD 39
Forecast Accuracy Mean Absolute Deviation (4 of 7)
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.
Chapter 5 - Forecasting 40
Forecast Accuracy Mean Absolute Deviation (5 of 7)
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, MAD = 3.81
= .30): Linear trend line: MAD = 2.29
Linear trend line has lowest MAD; increasing from .30 to .50 improved smoothed forecast.
Chapter 5 - Forecasting 41
Forecast Accuracy Mean Absolute Deviation (6 of 7)
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:
MAPD
D t F t
53 .
41 520 .
103 or 10.3% Chapter 5 - Forecasting 42
Forecast Accuracy Mean Absolute Deviation (7 of 7)
MAPD for other three forecasts: Exponential smoothing ( = .50): MAPD = 8.5% Adjusted exponential smoothing ( MAPD = 8.1% = .50, = .30): Linear trend: MAPD = 4.9% Chapter 5 - Forecasting 43
Forecast Accuracy Cumulative Error (1 of 2)
Cumulative error
is the sum of the forecast errors (E = e t ).
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 5.8.
E = e t = 49.31 indicating forecasts are frequently below actual demand.
Chapter 5 - Forecasting 44
Forecast Accuracy Cumulative Error (2 of 2)
Cumulative error for other forecasts: Exponential smoothing ( = .50): E = 33.21
Adjusted exponential smoothing ( = .50, E = 21.14
=.30):
Average error (bias
) is the per period average of cumulative error.
Average error for exponential smoothing forecast:
E
n e t
49.31
11 4.48
A large positive value of average error indicates a forecast high.
Chapter 5 - Forecasting 45
Forecast Accuracy Example Forecasts by Different Measures Table 5.9
Comparison of Forecasts for PM Computer Services 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.
Chapter 5 - Forecasting 46
Time Series Forecasting Using Excel (1 of 4)
Chapter 5 - Forecasting
Exhibit 5.1
47
Time Series Forecasting Using Excel (2 of 4)
Chapter 5 - Forecasting
Exhibit 5.2
48
Time Series Forecasting Using Excel (3 of 4)
Chapter 5 - Forecasting
Exhibit 5.3
49
Time Series Forecasting Using Excel (4 of 4)
Chapter 5 - Forecasting
Exhibit 5.4
50
Exponential Smoothing Forecast with Excel QM
Chapter 5 - Forecasting
Exhibit 5.5
51
Time Series Forecasting Solution with QM for Windows (1 of 2)
Chapter 5 - Forecasting
Exhibit 5.6
52
Time Series Forecasting Solution with QM for Windows (2 of 2)
Chapter 5 - Forecasting
Exhibit 5.7
53
Regression Methods Overview
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.
Chapter 5 - Forecasting 54
Regression Methods Linear Regression
Linear regression
relates demand (dependent variable ) to an independent variable.
y
a
bx Linear function a
y
bx b
xy
nxy
x
2
nx
2 where:
x
n x
mean of the x data
y
n y
mean of the y data 55
Regression Methods Linear Regression Example (1 of 3)
State University athletic department.
Wins Attendance 4 36,300 6 6 40,100 41,200 8 6 7 5 7 53,000 44,000 45,600 39,000 47,500 x (wins) 4 6 6 8 6 7 5 7 49 y (attendance, 1,000s) 36.3 40.1 41.2 53.0 44.0 45.6 39.0 47.5 346.7 xy 145.2 240.6 247.2 424.0 264.0 319.2 195.0 332.5 2,167.7 x 2 16 36 36 64 36 49 25 49 311
Chapter 5 - Forecasting 56
Regression Methods Linear Regression Example (2 of 3)
x
49 8 6.125
y
346.9
8 43.34
b
xy nxy
x
2
nx
2 (2,167.70
(8)(6.125)(43.34) (311) (8)(6.125) 2 4.06
a
y
bx
43.34
(.406)(6.125) 18.46
Therefore,
y
18.46
4.06
x
Attendance forecast for
x
7 wins is
y
18.46
4.06(7) 46.88 or 46,880 57
Regression Methods Linear Regression Example (3 of 3) Figure 5.6
Linear Regression Line Chapter 5 - Forecasting 58
Regression Methods Correlation (1 of 2)
Correlation
is a measure of the strength of the relationship between independent and dependent variables.
Formula:
r
n
x
2
x
n y y
2 2 Value lies between +1 and -1.
variables.
Values near 1.00 and -1.00 indicate strong linear relationship:
correlation
and
anti-correlation
.
Chapter 5 - Forecasting 59
Regression Methods Correlation (2 of 2)
Value for State University example:
r
(8)(2,167.7) (49)(346.7) (8)(311) (49)(49) (8)(15,224.7) (346.7) 2 .948
Chapter 5 - Forecasting 60
Regression Methods Coefficient of Determination
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, r 2 = .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.
Chapter 5 - Forecasting 61
Regression Analysis with Excel (1 of 7)
Chapter 5 - Forecasting
Exhibit 5.8
62
Regression Analysis with Excel (2 of 7)
Chapter 5 - Forecasting
Exhibit 5.9
63
Regression Analysis with Excel (3 of 7)
Chapter 5 - Forecasting
Exhibit 5.10
64
Regression Analysis with Excel (4 of 7)
Chapter 5 - Forecasting
Exhibit 5.11
65
Regression Analysis with Excel (5 of 7)
Chapter 5 - Forecasting
Exhibit 5.12
66
Regression Analysis with Excel (6 of 7)
Chapter 5 - Forecasting
Exhibit 5.13
67
Regression Analysis with Excel (7 of 7)
Chapter 5 - Forecasting
Exhibit 5.14
68
Regression Analysis with QM for Windows
Chapter 5 - Forecasting
Exhibit 5.15
69
Multiple Regression with Excel (1 of 4)
Multiple regression
relates demand to two or more independent variables.
General form: y = 0 + where 0 1 . . . k 1 x 1 + 2 x 2 + . . . + = the intercept k x k = parameters representing contributions of the independent variables x 1 . . . x k = independent variables Chapter 5 - Forecasting 70
Multiple Regression with Excel (2 of 4)
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
Chapter 5 - Forecasting 71
Multiple Regression with Excel (3 of 4)
Chapter 5 - Forecasting
Exhibit 5.16
72
Multiple Regression with Excel (4 of 4)
Chapter 5 - Forecasting
Exhibit 5.17
73
Example Problem Solution Computer Software Firm (1 of 4)
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.
Period Units 1 2 3 4 5 6 7 8 56 61 55 70 66 65 72 75
Chapter 5 - Forecasting 74
Example Problem Solution Computer Software Firm (2 of 4)
Step 1: Compute the Exponential Smoothing Forecast.
F t+1 = D t + (1 )F t Step 2: Compute the Adjusted Exponential Smoothing Forecast AF t+1 T t+1 = F t +1 = (F t +1 + T t+1 - F t ) + (1 )T t Chapter 5 - Forecasting 75
Example Problem Solution Computer Software Firm (3 of 4) Period 1 2 3 4 5 6 7 8 9 D t 56 61 55 70 66 65 72 75
F t
56.00 58.00 56.80 62.08 63.65 64.18 67.31 70.39 AF t
56.00 58.40 56.88 63.20 64.86 65.26 68.80 72.19 D t - F t
5.00 -3.00 13.20 3.92 1.35 7.81 7.68
35.97 D t - AF t
5.00 -3.40 13.12 2.80 0.14 6.73 6.20
30.60
Chapter 5 - Forecasting 76
Example Problem Solution Computer Software Firm (4 of 4)
Step 3: Compute the MAD Values
MAD
(
F t
) 1
n
D t
F t
41.97
5.99
7
MAD
(
AF t
) 1
n
D t
AF t
37.39
5.34
7 Step 4: Compute the Cumulative Error.
E(F t ) = 35.97
E(AF t ) = 30.60
Chapter 5 - Forecasting 77
Example Problem Solution Building Products Store (1 of 5)
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.
Chapter 5 - Forecasting 78
Example Problem Solution Building Products Store (2 of 5) Quarter 1 2 3 4 5 6 7 8 9 10 Building Permits x 8 12 7 9 15 6 5 8 10 12 Lumber Sales (1,000s of bd ft) y 12.6 16.3 9.3 11.5 18.1 7.6 6.2 14.2 15.0 17.8
Chapter 5 - Forecasting 79
Example Problem Solution Building Products Store (3 of 5)
Step 1: Compute the Components of the Linear Regression Equation.
x
92 10 92
y
128.6
10 12.86
b
xy
n x y
x
2
n x
2 (1,290.3) (10)(9.2)(12.86) (932) (10)(9.2) 2 1.25
a
y
bx
12.86
(1.25)(9.2) 1.36
Chapter 5 - Forecasting 80
Example Problem Solution Building Products Store (4 of 5)
Step 2: Develop the Linear regression equation.
y = a + bx, y = 1.36 + 1.25x
Step 3: Compute the Correlation Coefficient.
r
n
x
2
n xy
x
n y y
2 2
r
(10)(932) (10)(1,170.3) (92)(92) (92)(128.6) (10)(1,810.48) (128.6) 2 .925
Chapter 5 - Forecasting 81
Example Problem Solution Building 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 Chapter 5 - Forecasting 82
Chapter 5 - Forecasting 83