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

8th Edition

by

Bernard W. Taylor III

Chapter 5

Forecasting

Chapter 5 - Forecasting


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


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


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


trend-line

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

Chapter 5 - Forecasting


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


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


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


Time Series Methods

Moving Average (1 of 5)

  • Moving average uses values from the recent past to develop forecasts.

  • This dampensor smoothesout random increases and decreases.

  • Useful for forecasting relatively stable items that do not display any trend or seasonal pattern.

  • Formula for:

Chapter 5 - Forecasting


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:

Chapter 5 - Forecasting


Time Series Methods

Moving Average (3 of 5)

Figure 5.2

Three- and Five-Month Moving Averages

Chapter 5 - Forecasting


Time Series Methods

Moving Average (4 of 5)

Figure 5.2

Three- and Five-Month Moving Averages

Chapter 5 - Forecasting


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


Time Series Methods

Weighted Moving Average (1 of 2)

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

  • Formula:

Chapter 5 - Forecasting


Time Series Methods

Weighted Moving Average (2 of 2)

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

Chapter 5 - Forecasting


Time Series Methods

Exponential Smoothing (1 of 11)

  • Exponential smoothing weights recent past data more strongly than more distant data.

  • Two forms: simpleexponential 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.

Chapter 5 - Forecasting


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


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

Chapter 5 - Forecasting


Time Series Methods

Exponential Smoothing (4 of 11)

Using

F1 = D1

Table 5.4

Exponential Smoothing Forecasts,  = .30 and  = .50

Chapter 5 - Forecasting


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


Time Series Methods

Exponential Smoothing (6 of 11)

Figure 5.3

Exponential Smoothing Forecasts

Chapter 5 - Forecasting


Time Series Methods

Exponential Smoothing (7 of 11)

  • Adjusted exponential smoothing: exponential smoothing with a trend adjustment factor added.

  • Formula:

    AFt + 1 = Ft + 1 + Tt+1

    where: Tt = an exponentially smoothed trend factor:

    Tt + 1 = (Ft + 1 - Ft) + (1 - )Tt

    Tt = 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


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

Chapter 5 - Forecasting


Time Series Methods

Exponential Smoothing (9 of 11)

Tt + 1 = (Ft + 1 - Ft) + (1 - )Tt

Table 5.5

Adjusted Exponentially Smoothed Forecast Values

Chapter 5 - Forecasting


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


Time Series Methods

Exponential Smoothing (11 of 11)

Figure 5.4

Adjusted Exponentially Smoothed Forecast

Chapter 5 - Forecasting


Time Series Methods

Linear Trend Line (1 of 5)

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

  • Formula:

Chapter 5 - Forecasting


Time Series Methods

Linear Trend Line (2 of 5)

Example: PM Computer Services (see Table 5.6)

Chapter 5 - Forecasting


Time Series Methods

Linear Trend Line (3 of 5)

Table 5.6

Least Squares Calculations

Chapter 5 - Forecasting


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


Time Series Methods

Linear Trend Line (5 of 5)

Figure 5.5

Linear Trend Line

Chapter 5 - Forecasting


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

Chapter 5 - Forecasting


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


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

Chapter 5 - Forecasting


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)!

Chapter 5 - Forecasting


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


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:

Chapter 5 - Forecasting


Forecast Accuracy

Mean Absolute Deviation (2 of 7)

Example: PM Computer Services (see Table 5.8).

  • Compare accuracies of different forecasts using MAD:

Chapter 5 - Forecasting


Forecast Accuracy

Mean Absolute Deviation (3 of 7)

Table 5.8

Computational Values for MAD

Chapter 5 - Forecasting


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


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.

Chapter 5 - Forecasting


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:

Chapter 5 - Forecasting


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%

Chapter 5 - Forecasting


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.

Chapter 5 - Forecasting


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.

Chapter 5 - Forecasting


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


Time Series Forecasting Using Excel (1 of 4)

Exhibit 5.1

Chapter 5 - Forecasting


Time Series Forecasting Using Excel (2 of 4)

Exhibit 5.2

Chapter 5 - Forecasting


Time Series Forecasting Using Excel (3 of 4)

Exhibit 5.3

Chapter 5 - Forecasting


Time Series Forecasting Using Excel (4 of 4)

Exhibit 5.4

Chapter 5 - Forecasting


Exponential Smoothing Forecast with Excel QM

Exhibit 5.5

Chapter 5 - Forecasting


Time Series Forecasting

Solution with QM for Windows (1 of 2)

Exhibit 5.6

Chapter 5 - Forecasting


Time Series Forecasting

Solution with QM for Windows (2 of 2)

Exhibit 5.7

Chapter 5 - Forecasting


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


Regression Methods

Linear Regression

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

Linear function

Chapter 5 - Forecasting


Regression Methods

Linear Regression Example (1 of 3)

  • State University athletic department.

Chapter 5 - Forecasting


Regression Methods

Linear Regression Example (2 of 3)

Chapter 5 - Forecasting


Regression Methods

Linear Regression Example (3 of 3)

Figure 5.6

Linear Regression Line

Chapter 5 - Forecasting


Regression Methods

Correlation (1 of 2)

  • Correlationis 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.

Chapter 5 - Forecasting


Regression Methods

Correlation (2 of 2)

  • Value for State University example:

Chapter 5 - Forecasting


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.

Chapter 5 - Forecasting


Regression Analysis with Excel (1 of 7)

Exhibit 5.8

Chapter 5 - Forecasting


Regression Analysis with Excel (2 of 7)

Exhibit 5.9

Chapter 5 - Forecasting


Regression Analysis with Excel (3 of 7)

Exhibit 5.10

Chapter 5 - Forecasting


Regression Analysis with Excel (4 of 7)

Exhibit 5.11

Chapter 5 - Forecasting


Regression Analysis with Excel (5 of 7)

Exhibit 5.12

Chapter 5 - Forecasting


Regression Analysis with Excel (6 of 7)

Exhibit 5.13

Chapter 5 - Forecasting


Regression Analysis with Excel (7 of 7)

Exhibit 5.14

Chapter 5 - Forecasting


Regression Analysis with QM for Windows

Exhibit 5.15

Chapter 5 - Forecasting


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 5 - Forecasting


Multiple Regression with Excel (2 of 4)

State University example:

Chapter 5 - Forecasting


Multiple Regression with Excel (3 of 4)

Exhibit 5.16

Chapter 5 - Forecasting


Multiple Regression with Excel (4 of 4)

Exhibit 5.17

Chapter 5 - Forecasting


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.

Chapter 5 - Forecasting


Example Problem Solution

Computer Software Firm (2 of 4)

Step 1: Compute the Exponential Smoothing Forecast.

Ft+1 =  Dt + (1 - )Ft

Step 2: Compute the Adjusted Exponential Smoothing

Forecast

AFt+1 = Ft +1 + Tt+1

Tt+1 = (Ft +1 - Ft) + (1 - )Tt

Chapter 5 - Forecasting


Example Problem Solution

Computer Software Firm (3 of 4)

Chapter 5 - Forecasting


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

Chapter 5 - Forecasting


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


Example Problem Solution

Building Products Store (2 of 5)

Chapter 5 - Forecasting


Example Problem Solution

Building Products Store (3 of 5)

Step 1: Compute the Components of the Linear Regression

Equation.

Chapter 5 - Forecasting


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.

Chapter 5 - Forecasting


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



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