Introduction to management science 10e by bernard taylor chapter 16

■ Forecasting Components■ Time Series Methods ■ Forecast Accuracy ■ Time Series Forecasting Using Excel ■ Time Series Forecasting Using QM for Windows ■ Regression Methods Chapter Topic

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■ Forecasting Components

■ Time Series Methods

■ Forecast Accuracy

■ Time Series Forecasting Using Excel

■ Time Series Forecasting Using QM for

Windows

■ Regression Methods

Chapter Topics

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■ 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)

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 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 Components

Patterns (1 of 2)

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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

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Qualitative Methods

 “Jury of executive opinion,” a qualitative

technique, is the most common type of forecast for long-term strategic planning

 Performed by individuals or groups within an

organization, sometimes assisted by consultants and other experts, whose judgments and

opinions 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

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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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i period

in data

averagemoving

in theperiods

ofnumber

:where

n

i D i n

 Useful for forecasting relatively stable items that

do not display any trend or seasonal pattern

 Formula for:

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Example: Instant Paper Clip Supply Company

forecast of orders for the month of November

 Three-month moving average:

 Five-month moving average:

31





i D i MA

orders

91

130110

905

51

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Figure 15.2 Three- and Five-Month Moving Time Series Methods

Moving Average (4 of 5)

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15-Time Series Methods

Moving Average (5 of 5)

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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 In a weighted moving average, weights are

assigned to the most recent data

 Determining precise weights and number of

periods requires trial-and-error

1 where the weight for period i, between 0% and 100%

1.00

Example: Paper clip company weights 50% for October, 33%

for September, 17% for August:

3

(.50)(90) (.33)(110) 1

3

n

i Wi Wi

Weighted Moving Average

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15- Exponential smoothing weights recent past data

 Two forms: simple exponential smoothing and

 Simple exponential smoothing:

Ft + 1 = Dt + (1 - )Ftwhere: 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)

Exponential Smoothing (1 of 11)

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 The most commonly used values of  are

between 0.10 and 0.50.

 Determination of  is usually judgmental and subjective and often based on trial-and -error experimentation

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15-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

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Table 15.4 Exponential Smoothing Forecasts, 

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15- 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

stable data without trend; higher constants

appropriate for data with trends

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Figure 15.3 Exponential Smoothing ForecastsTime Series Methods

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21

smoothing with a trend adjustment factor added

Formula:

AFt + 1 = Ft + 1 + Tt+1where: 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

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Example: PM Computer Services exponential

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■ 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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xperiodfor

demandfor

forecast

periodtime

the

linethe

ofslope

0)period(at

bx a

y

n y y

n x n

x b y a

x n x

y x n xy b

periodsof

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:

Linear Trend Line (1 of 5)

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56 57

13 72

1 2 35 13,

x 13, period

for

line trend

linear

72 1

2 35

2 35 5

6 72 1

42 46

72

1 2

5 6 12 650

42 46

5 6 12 867

3 2

2

42

46 12

557

5

6

12 78

)

(

.

)

)(

(

)

(

)

)(

( ,

.

x b y a

x n x

y x n xy b

y x

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Table 15.6 Least Squares Calculations

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■ This limits its use to shorter time frames in which trend will not change

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Figure 15.5 Linear Trend LineTime Series Methods

Linear Trend (5 of 5)

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15-■ A seasonal pattern is a repetitive up-and-down

movement in demand

■ Seasonal patterns can occur on a quarterly,

monthly, weekly, or daily basis

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:

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■ 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

Seasonal Adjustments (2 of 4)

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Example: Wishbone Farms

Seasonal Adjustments (3 of 4)

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 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):

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15- 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

 Measures of forecast errors:

 Mean Absolute deviation (MAD)

 Mean absolute percentage deviation (MAPD)

 Cumulative error (E bar)

 Average error, or bias (E)

Forecast Accuracy

Overview

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 MAD is the average absolute difference

between the forecast and actual demand

 Most popular and simplest-to-use measures of

forecast error

 Formula:

periodsof

number total

the

n

tperiodfor

forecast the

tF

tperiod

in demandt

D

numberperiod

the

t

:where

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Compare accuracies of different forecasts using MAD:

85

411

D MAD

Mean Absolute Deviation (2 of 7)

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Table 15.8 Computational Values for Forecast Accuracy

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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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 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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15- A variation on MAD is the mean absolute percent deviation (MAPD).

 Measures absolute error as a percentage of

 Eliminates problem of interpreting the measure of accuracy relative to the magnitude of the demand and forecast values

 Formula:

10.3%

or 103

D MAPD

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MAPD for other three forecasts:

Exponential smoothing ( = 50): MAPD =

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15- 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

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

Cumulative Error (1 of 2)

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 Cumulative error for other forecasts:

Exponential smoothing ( = 50): E = 33.21Adjusted 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

Cumulative Error (2 of 2)

48

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Results consistent for all forecasts:

 Larger value of alpha is preferable

 Adjusted forecast is more accurate than exponential smoothing

 Linear trend is more accurate than all the others

Table 15.9 Comparison of Forecasts for PM

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Exhibit 15.1Time Series Forecasting Using Excel (1 of 4)

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Exhibit 15.3Time Series Forecasting Using Excel (3 of 4)

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Exhibit 15.5Exponential Smoothing Forecast

with Excel QM

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15-Time Series Forecasting

Solution with QM for Windows (1 of 2)

Exhibit 15.6

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Time Series Forecasting

Solution with QM for Windows (2 of 2)

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15- 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

 Simplest form of regression is linear regression

Regression Methods

Overview

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y theof

mean

data

x theof

mean

:where

22

n x x

x n x

y x n xy b

x b y a

bx a

y

variable ) to an independent variable

Linear Regression

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x (wins)

145.2 240.6 247.2 424.0 264.0 319.2 195.0 332.5 2,167.7

Linear Regression Example (1 of 3)

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or88467

06446

Attendance

0644618Therefore,

4618125

640634

43

06

42

1256

8311

3443125

6870

167

22

49

,

)(

.)

.)(

(

.)

.)(

()(

)

)(

.)(

(

,(

x b y a

x n x

y x n xy b

y x

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15-Figure 15.6 Linear Regression Line

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 Correlation is a measure of the strength of the

variables

Formula:

 Value lies between +1 and -1

 Value of zero indicates little or no relationship

2

x n

y x xy

n r

Correlation (1 of 2)

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15-948

.2)7.346(

)7.224,

15)(

8()49)(

49()311)(

8(

)7.346)(

49()7.167,2)(

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 The Coefficient of determination is the

percentage of the variation in the dependent variable that results from the independent

 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

Coefficient of Determination

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Regression Analysis with Excel (2 of 6)

Exhibit 15.9

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Exhibit 15.11

Regression Analysis with Excel (4 of 6)

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Exhibit 15.13Regression Analysis with Excel (6 of

6)

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15-Multiple Regression with Excel (1 of 4)

General form:

y = 0 +  1x1 +  2x2 + +  kxk where  0 = the intercept

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36,300 40,100 41,200 53,000 44,000 45.600 39,000 47,500

Multiple Regression with Excel (2 of 4)

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Exhibit 15.15Multiple Regression with Excel (4 of 4)

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 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 Solution

Computer Software Firm (1 of 4)

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Step 1: Compute the Exponential Smoothing Forecast.

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 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

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Step 3: Compute the MAD Values

Step 4: Compute the Cumulative Error.

E(Ft) = 35.97

E(AFt) = 30.60

34

5

37)

(

99

5

41)

D t

AF MAD

n F t t

D t

F MAD

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