Chapter 5 Forecasting

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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)

• 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).

• 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

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

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

• 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 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

• 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

• 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

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 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

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