forecasting time series methods time series … average uses values from the recent past ......

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15-1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Forecasting

Chapter 15


■ Forecasting Components

■ Time Series Methods

■ Forecast Accuracy

■ Time Series Forecasting Using Excel

■ Time Series Forecasting Using QM for Windows

■ Regression Methods

Chapter Topics


■ 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

Forecasting Components


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.


Patterns (1 of 2)


Figure 15.1 (a) Trend; (b) Cycle; (c) Seasonal; (d) Trend with Season


Patterns (2 of 2)



Forecasting Methods

1. Times Series - Statistical techniques that use historical

data to predict future behavior.

2. 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.

3. Qualitative Methods - Methods using judgment, expertise

and opinion to make forecasts.

2



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 Method, market research, surveys, and technological 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.


1

where: number of periods in the moving average

data in period i

nD

iiMA nn

n

Di

Time Series Methods

Moving Average (1 of 6)

Moving average uses values from the recent past to develop

forecasts.

This dampens random increases and decreases.

Useful for forecasting relatively stable items that do not display

any trend or seasonal pattern.

Formula for moving average (MA):


Example: Instant Paper Clip Supply Company wants to forecast

orders for the month of November. Develop three-month and

five-month moving averages using the data.

Time Series Methods


Table 15.1 Orders for 10-month period


Example: Instant Paper Clip Supply Company wants to forecast

orders for the month of November.

Three-month moving average:

Five-month moving average:

3

90 110 1301 110 orders3 33

Di

iMA

5

90 110 130 75 501 91 orders5 55

Di

iMA

Time Series Methods



Table 15.1 Three- and 5-month moving averages

Time Series Methods


3


Figure 15.2 Three- and 5-month moving averages

Time Series Methods



Time Series Methods


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.

A moving average does not react well to changes (trends,

seasonal effects, etc.) but is easy to use and inexpensive.

Good for short-term forecasting.


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

Determining precise weights and the number of periods requires

trial-and-error experimentation.

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)

13

nWMA W D

n i ii

Wi

Wi

WMA W Di i

i

(.17)(130) 103.4 orders

Time Series Methods

Weighted Moving Average


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).

Time Series Methods

Exponential Smoothing (1 of 11)


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.

Time Series Methods



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) + (1-.30)(.37) = 37 units

Forecast for period 3 (March):

F3 = D2 + (1- )F2 = (.30)(.40) + (1-.30)(37) = 37.9 units

Time Series Methods


4


Table 15.4 Exponential smoothing forecasts, = .30 and = .50

Time Series Methods



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.

Time Series Methods



Figure 15.3 Exponential smoothing forecasts

Time Series Methods



■ Adjusted exponential smoothing: exponential smoothing with a

trend adjustment factor added.

Formula AFt + 1 = Ft + 1 + Tt+1

where:

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.

■ Determined subjectively.

■

Time Series Methods



Example: PM Computer Services exponentially smoothed

forecasts with = .50 and = .30 (see Table 15.5 next slide).

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

Time Series Methods



Table 15.5 Adjusted exponentially smoothed forecast values

Time Series Methods


5


■ The adjusted forecast is consistently higher than the simple

exponentially smoothed forecast.

■ It is more reflective of the generally increasing trend of the

data.

Time Series Methods



Figure 15.4 Adjusted exponentially smoothed forecast

Time Series Methods



where: intercept (at period 0)

slope of the line

the time period

forecast for demand

for period x

y a bx

a

b

x

y

2

where: number of periods

x

xy nxyb

x nx

a y bx

nx

ny

y n

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

■ Formula:

Time Series Methods

Linear Trend Line (1 of 5)


Example: PM Computer Services (see Table 15.6)

2 22

78 5576.5 46.4212 12

3,867 (12)(6.5)(46.42) 1.72650 12 6.5

46.42 (1.72)(6.5) 35.2

35.2 1.72 linear trend line

for period 13, x 13, 35.2 1.72(13) 57.56

x y

xy nxyb

x nx

a y bx

y x

y

Time Series Methods


15-29 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Table 15.6 Least squares calculations

Time Series Methods



■ 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 the trend will

not change.

Time Series Methods


6


Figure 15.5 Linear trend line

Time Series Methods

Linear Trend (5 of 5)


■ A seasonal pattern is a repetitive up-and-down movement in

demand.

■ Seasonal patterns can occur on a quarterly, 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

Time Series Methods

Seasonal Adjustments (1 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.

Time Series Methods



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

Table 15.7 Demand for turkeys at Wishbone Farms

Example: Wishbone Farms

Time Series Methods



Multiply forecasted demand for an entire year by seasonal factors to

determine the quarterly demand.

Forecast for entire year (trend line for data in Table 15.7):

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

Time Series Methods



Forecasts will always deviate from actual values.

Difference between forecasts and actual values are referred to as

forecast error.

We would like forecast error to be as small as possible.

If forecast error is large, either the technique being used is the wrong one, or the 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)

Forecast Accuracy

Overview

7


MAD is the average absolute difference between the forecast

and actual demand.

The most popular and simplest-to-use measures of forecast error.

Formula:

where:

t the period number

D demand in period tt

F the forecast for period tt

n the total number of periods

D FttMAD n

Forecast Accuracy

Mean Absolute Deviation (1 of 7)


Example: PM Computer Services (see Table 15.8).

Compare accuracies of different forecasts using MAD:

53.41 4.8511

D Ft tMAD n

Forecast Accuracy



Table 15.8 Computational values for MAD and error

Forecast Accuracy



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.

MAD must be considered in light of magnitude of the data.

Forecast Accuracy



Can be used to compare the 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

The linear trend line has the lowest MAD; increasing from .30 to

.50 improved the smoothed forecast.

Forecast Accuracy



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

Measures the absolute error as a percentage of demand rather

than per period.

Eliminates the problem of interpreting the measure of accuracy

relative to the magnitude of the demand and forecast values.

Formula:

53.41 .103 or 10.3%520

D Ft tMAPDD

t

Forecast Accuracy


8


MAPD for other three forecasts:

Exponential smoothing ( = .50): MAPD = 8.5%

Adjusted exponential smoothing ( = .50, = .30):

MAPD = 8.1%

Linear trend: MAPD = 4.9%

Forecast Accuracy



Cumulative error is the sum of the forecast errors (E =et).

A relatively large positive value indicates the forecast is biased low, a large negative value indicates the forecast is biased high.

If the preponderance of errors are positive, the forecast is

consistently low; and vice versa.

The cumulative error for a trend line is always almost zero, and

is therefore not a good measure for this method.

The cumulative error for PM Computer Services can be read

directly from Table 15.8.

E = et = 49.31, indicating the forecasts are frequently below

actual demand.

Forecast Accuracy

Cumulative Error (1 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 the 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.

Forecast Accuracy

Cumulative Error (2 of 2)

49.31 4.4811

etE n


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 Computer Services

Forecast Accuracy

Example Forecasts by Different Measures


Exhibit 15.1

Time Series Forecasting Using Excel (1 of 4)

=G21/11

=B3*B8+(1-B3)*C8

=C9+D9

=B9-E9

=ABS(B9-E9)

=SUM(F9:F20)


Exhibit 15.2


To access this window, click

on “Data” on the toolbar

ribbon and then the “Data

Analysis” add-in

9


Exhibit 15.3


Demand values

= 0.5

Cells in which the

forecasted values

will be placed


Exhibit 15.4



Exhibit 15.5

Exponential Smoothing Forecast with Excel QM

Click “Add-Ins” to access

forecasting macros

Input problem data in

cells B7 and B10:B21


Time Series Forecasting

Solution with QM for Windows (1 of 2)

Exhibit 15.6

15-53 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 15.7

Time Series Forecasting

Solution with QM for Windows (2 of 2)


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.

The simplest form of regression is linear regression.

Regression Methods

Overview

10


22

where:

mean of the x data

mean of the y data

y a bx

a y bx

xy nxyb

x nx

xx n

yy n

Linear regression relates demand (dependent variable ) to an

independent variable.

Regression Methods

Linear Regression


State University Athletic Department.

Wins Attendance

4 6 6 8 6 7 5 7

36,300 40,100 41,200 53,000 44,000 45,600 39,000 47,500

x (wins)

y (attendance, 1,000s)

xy

x

2

4 6 6 8 6 7 5 7

49

36.3 40.1 41.2 53.0 44.0 45.6 39.0 47.5 346.7

145.2 240.6 247.2 424.0 264.0 319.2 195.0 332.5

2,167.7

16 36 36 64 36 49 25 49

311

Regression Methods

Linear Regression Example (1 of 3)


49 6.1258

346.9 43.348

(2,167.70 (8)(6.125)(43.34) 4.062 22 (311) (8)(6.125)

43.34 (.406)(6.125) 18.46

Therefore, 18.46 4.06

Attendance forecast for x 7 wins is18.46 4.06(7)

x

y

xy nxyb

x nx

a y bx

y x

y

46.88 or 46,880

Regression Methods



Figure 15.6

Linear regression line

Regression Methods



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 a strong linear relationship.

2 22 2

n xy x yr

n x x n y y

Regression Methods

Correlation (1 of 2)


2 2

(8)(2,167.7) (49)(346.7) .948(8)(311) (49) (8)(15,224.7) (346.7)

r

Value for State University example:

Since the value is close to one, we have evidence of a

strong linear relationship.

Regression Methods

Correlation (2 of 2)

11


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 the 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.

Regression Methods

Coefficient of Determination


Regression Analysis with Excel (1 of 6)

Exhibit 15.8

=INTERCEPT(B5:B12,A5:A12)

=CORREL(B5:B12,A5:A12)



Exhibit 15.9

Click on “Insert” to access “Charts” Click on “Scatter”


Exhibit 15.10



Exhibit 15.11



Exhibit 15.12


12


Exhibit 15.13

Regression Analysis with QM for Windows (6 of 6)


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


State University example revisited; does the addition of

promotional and advertising expenditures to wins improve the

prediction of attendance?

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


15-70 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 15.14


r2, the coefficient of determination

Regression equation

coefficients for x1 and x2


Exhibit 15.15


Includes x1 and x2

columns


Period Units

1 2 3 4 5 6 7 8

56 61 55 70 66 65 72 75

Problem Statement:

For the 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)

13


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




Period Dt Ft AFt Dt - Ft Dt - AFt

1 2 3 4 5 6 7 8 9

56 61 55 70 66 65 72 75

56.00 58.00 56.80 62.08 63.65 64.18 67.31 70.39

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




Step 3: Compute the MAD Values

Step 4: Compute the Cumulative Error.

E(Ft) = 35.97

E(AFt) = 30.60

41.97( ) 5.997

37.39( ) 5.347

D Ft tMAD F nt

D AFt tMAD AF nt




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.


Building Products Store (1 of 5)


Quarter

Building Permits, x

Lumber Sales (1,000s of board ft), y

1 2 3 4 5 6 7 8 9

10

8 12 7 9 15 6 5 8 10 12

12.6 16.3 9.3 11.5 18.1 7.6 6.2 14.2 15.0 17.8




Step 1: Compute the Components of the Linear Regression Equation.

92 9.210

128.6 12.8610

(1,290.3) (10)(9.2)(12.86)2 22 (932) (10)(9.2)

1.25

12.86 (1.25)(9.2)

1.36

x

y

xy nxyb

x nx

a y bx



14


Step 2: Develop the Linear regression equation.

y = a + bx, y = 1.36 + 1.25x

Step 3: Compute the Correlation Coefficient.

2 22 2

n xy x yr

n x x n y y

22

(10)(1,290.3) (92)(128.6)

(10)(932) (92) (10)(1,810.48) (128.6)

.925

r






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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forecasting time series methods time series … average uses values from the recent past ......

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