Forecasting

1
Introduction to Management Science 8th
Edition by Bernard W. Taylor III
Chapter 5 Forecasting
2
Chapter Topics

• Forecasting Components
• Time Series Methods
• Forecast Accuracy
• Time Series Forecasting Using Excel
• Time Series Forecasting Using QM for Windows
• Regression Methods

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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

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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.

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Forecasting Components Patterns (2 of 2)
Figure 5.1 Forms of Forecast Movement (a)
Trend, (b) Cycle, (c) Seasonal Pattern, (d) Trend
with Seasonal Pattern
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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.

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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.

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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.

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Time Series Methods Moving 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

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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
• Five-month moving average

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Time Series Methods Moving Average (3 of 5)
Figure 5.2 Three- and Five-Month Moving Averages
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Time Series Methods Moving Average (4 of 5)
Figure 5.2 Three- and Five-Month Moving Averages
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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.

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Time Series Methods Weighted Moving Average (1 of
2)

• In a weighted moving average, weights are
  assigned to the most recent data.
• Formula

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Time Series Methods Weighted Moving Average (2 of
2)

• Determining precise weights and number of periods
  requires trial-and-error experimentation.

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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
• 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)
• use F1 D1.

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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.

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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)
• 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

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Time Series Methods Exponential Smoothing (4 of
11)
Using F1 D1
Table 5.4 Exponential Smoothing Forecasts, ?
.30 and ? .50
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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.

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Time Series Methods Exponential Smoothing (6 of
11)
Figure 5.3 Exponential Smoothing Forecasts
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Time Series Methods Exponential Smoothing (7 of
11)

• Adjusted exponential smoothing exponential
  smoothing with a trend adjustment factor added.
• Formula
• AFt 1 Ft 1 Tt1
• where Tt an exponentially smoothed trend
  factor
• Tt 1 ?(Ft 1 - Ft) (1 - ?)Tt
• Tt the last (previous) periods trend
  factor
• ? smoothing constant for trend ( a value
  between zero and one).
• Reflects the weight given to the most recent
  trend data.
• Determined subjectively.

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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 T2 0.00
• 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

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Time Series Methods Exponential Smoothing (9 of
11)
Tt 1 ?(Ft 1 - Ft) (1 - ?)Tt
Table 5.5 Adjusted Exponentially Smoothed
Forecast Values
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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.

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Time Series Methods Exponential Smoothing (11 of
11)
Figure 5.4 Adjusted Exponentially Smoothed
Forecast
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Time Series Methods Linear Trend Line (1 of 5)

• When demand displays an obvious trend over time,
  a least squares regression line , or linear trend
  line, can be used to forecast.
• Formula

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Time Series Methods Linear Trend Line (2 of 5)

Example PM Computer Services (see Table 5.6)
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Time Series Methods Linear Trend Line (3 of 5)

Table 5.6 Least Squares Calculations
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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.

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Time Series Methods Linear Trend Line (5 of 5)
Figure 5.5 Linear Trend Line
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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
• Si Di/?D

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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.

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Time Series Methods Seasonal Adjustments (3 of 4)

• Example Wishbone Farms

Table 5.7 Demand for Turkeys at Wishbone Farms
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
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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
• 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

Note the potential for confusion the Si are
seasonal factors (fractions), whereas the SFi
are seasonally adjusted forecasts (commodity
values)!
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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)

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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

38
Forecast Accuracy Mean Absolute Deviation (2 of 7)

• Example PM Computer Services (see Table 5.8).
• Compare accuracies of different forecasts using
  MAD

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Forecast Accuracy Mean Absolute Deviation (3 of 7)
Table 5.8 Computational Values for MAD
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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.

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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, ?
  .30) MAD 3.81
• Linear trend line MAD 2.29
• Linear trend line has lowest MAD increasing ?
  from .30 to .50 improved smoothed forecast.

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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

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Forecast Accuracy Mean Absolute Deviation (7 of 7)
MAPD for other three forecasts Exponential
smoothing (? .50) MAPD 8.5 Adjusted
exponential smoothing (? .50, ? .30) MAPD
8.1 Linear trend MAPD 4.9
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Forecast Accuracy Cumulative Error (1 of 2)

• 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 5.8.
• E ? et 49.31 indicating forecasts are
  frequently below actual demand.

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Forecast Accuracy Cumulative Error (2 of 2)

• 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.

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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.

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Time Series Forecasting Using Excel (1 of 4)
Exhibit 5.1
48
Time Series Forecasting Using Excel (2 of 4)
Exhibit 5.2
49
Time Series Forecasting Using Excel (3 of 4)
Exhibit 5.3
50
Time Series Forecasting Using Excel (4 of 4)
Exhibit 5.4
51
Exponential Smoothing Forecast with Excel QM
Exhibit 5.5
52
Time Series Forecasting Solution with QM for
Windows (1 of 2)
Exhibit 5.6
53
Time Series Forecasting Solution with QM for
Windows (2 of 2)
Exhibit 5.7
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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.

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Regression Methods Linear Regression

• Linear regression relates demand (dependent
  variable ) to an independent variable.

Linear function
56
Regression Methods Linear Regression Example (1
of 3)

• State University athletic department.

57
Regression Methods Linear Regression Example (2
of 3)

58
Regression Methods Linear Regression Example (3
of 3)
Figure 5.6 Linear Regression Line
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Regression Methods Correlation (1 of 2)

• 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 correlation and anti-correlation.

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Regression Methods Correlation (2 of 2)

• Value for State University example

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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, 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.

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Regression Analysis with Excel (1 of 7)
Exhibit 5.8
63
Regression Analysis with Excel (2 of 7)
Exhibit 5.9
64
Regression Analysis with Excel (3 of 7)
Exhibit 5.10
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Regression Analysis with Excel (4 of 7)
Exhibit 5.11
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Regression Analysis with Excel (5 of 7)
Exhibit 5.12
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Regression Analysis with Excel (6 of 7)
Exhibit 5.13
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Regression Analysis with Excel (7 of 7)
Exhibit 5.14
69
Regression Analysis with QM for Windows
Exhibit 5.15
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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

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Multiple Regression with Excel (2 of 4)
State University example
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Multiple Regression with Excel (3 of 4)
Exhibit 5.16
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Multiple Regression with Excel (4 of 4)
Exhibit 5.17
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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.

75
Example Problem Solution Computer Software Firm
(2 of 4)
Step 1 Compute the Exponential Smoothing
Forecast. Ft1 ?
Dt (1 - ?)Ft Step 2 Compute the Adjusted
Exponential Smoothing Forecast
AFt1 Ft 1 Tt1
Tt1 ?(Ft 1 - Ft) (1 - ?)Tt

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Example Problem Solution Computer Software Firm
(3 of 4)
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Example Problem Solution Computer Software Firm
(4 of 4)
Step 3 Compute the MAD Values Step 4
Compute the Cumulative Error.
E(Ft) 35.97 E(AFt)
30.60
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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.

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Example Problem Solution Building Products Store
(2 of 5)
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Example Problem Solution Building Products Store
(3 of 5)
Step 1 Compute the Components of the Linear
Regression Equation.

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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.
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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

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