1 1 Slide © 2005 Thomson/South-Western EMGT 501 HW Solutions Chapter 15 - SELF TEST 3 Chapter 15 -...

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

EMGT 501

HW Solutions

Chapter 15 - SELF TEST 3

Chapter 15 - SELF TEST 14

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15-3a.

Let x1 = number of units of product 1 producedx2 = number of units of product 2 produced

Min P1( d1 ) + P1( d1

) + P1( d2 ) + P1( d2

) + P2( d3 )

s.t. 1x1 + 1x2 - d1

+ d1 = 350 Goal 1

2x1 + 5x2 - d2 + d2

= 1000 Goal 2

4x1 + 2x2 - d3 + d3

= 1300 Goal 3

d1 d1

d2 d2

d3 d3

, , , , , , , 01x 2x

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b.

In the graphical solution, point A provides the optimal solution. Note that with x1 = 250 and x2 = 100, this solution achieves goals 1 and 2, but underachieves goal 3 (profit) by $100 since 4(250) + 2(100) = $1200.

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western0 100 200 300 400 500

100

200

300

400

500

600

700

Goal 1

Goal 2

A (250, 100)

B (281.25, 87.5)G

oal 3

x2

x1

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c. Max 4x1 + 2x2 s.t.

1x1 + 1x2 350 Dept. A 2x1 + 5x2 1000 Dept. B x1, x2 0

The graphical solution indicates that there are four extreme points. The profit corresponding to each extreme point is as follows:

Extreme Point Profit 1 4(0) + 2(0) = 0 2 4(350) + 2(0) = 1400 3 4(250) + 2(100) = 1200 4 4(0) + 2(200) = 400

Thus, the optimal product mix is x1 = 350 and x2 = 0 with a profit of $1400.

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. x2

x10 100 200 300 400 500

100

200

300

400

1

3

4

2

(250,100)

(0,250)

(0,0)

(350,0)

Department B

Feasible Region

Department A

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d. The solution to part (a) achieves both labor goals, whereas the solution to part (b) results in using only 2(350) + 5(0) = 700 hours of labor in department B. Although (c) results in a $100 increase in profit, the problems associated with underachieving the original department labor goal by 300 hours may be more significant in terms of long-term considerations.

e. Refer to the graphical solution in part (b). The solution to the revised problem is point B, with x1 = 281.25 and x2 = 87.5. Although this solution achieves the original department B labor goal and the profit goal, this solution uses 1(281.25) + 1(87.5) = 368.75 hours of labor in department A, which is 18.75 hours more than the original goal.

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15-14a.123456789

1011

A B C D ECriteria 220 Bowrider 230 Overnighter 240 SundancerCost 40 25 15Overnight Capability 6 18 27Kitchen/Bath Facilities 2 8 14Appearance 35 35 30Engine/Speed 30 40 20Towing/Handling 32 20 8Maintenance 28 20 12Resale Value 21 15 18

Score 194 181 144

123456789

1011

A B C D ECriteria 220 Bowrider 230 Overnighter 240 SundancerCost 21 18 15Overnight Capability 5 30 40Kitchen/Bath Facilities 5 15 35Appearance 20 28 28Engine/Speed 8 10 6Towing/Handling 16 12 4Maintenance 6 5 4Resale Value 10 12 12

Score 91 130 144

b.

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Home WorkHome Work

16-9 and 16-3516-9 and 16-35

Due Day: Nov 28, Due Day: Nov 28, 20052005

No Class on Nov 22, No Class on Nov 22, 20052005

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Chapter 16Chapter 16ForecastingForecasting

Quantitative Approaches to ForecastingQuantitative Approaches to Forecasting The Components of a Time SeriesThe Components of a Time Series Measures of Forecast AccuracyMeasures of Forecast Accuracy Using Smoothing Methods in Forecasting Using Smoothing Methods in Forecasting Using Trend Projection in Forecasting Using Trend Projection in Forecasting Using Trend and Seasonal Components Using Trend and Seasonal Components

in Forecastingin Forecasting Using Regression Analysis in ForecastingUsing Regression Analysis in Forecasting Qualitative Approaches to ForecastingQualitative Approaches to Forecasting

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Quantitative Approaches to ForecastingQuantitative Approaches to Forecasting

Quantitative methodsQuantitative methods are based on an analysis of are based on an analysis of historical data concerning one or more time historical data concerning one or more time series.series.

A A time seriestime series is a set of observations measured at is a set of observations measured at successive points in time or over successive successive points in time or over successive periods of time.periods of time.

If the historical data used are restricted to past If the historical data used are restricted to past values of the series that we are trying to forecast, values of the series that we are trying to forecast, the procedure is called a the procedure is called a time series methodtime series method..

If the historical data used involve other time If the historical data used involve other time series that are believed to be related to the time series that are believed to be related to the time series that we are trying to forecast, the series that we are trying to forecast, the procedure is called a procedure is called a causal methodcausal method. .

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Time Series MethodsTime Series Methods

Three time series methods are: Three time series methods are:

•smoothingsmoothing

•trend projectiontrend projection

•trend projection adjusted for trend projection adjusted for seasonal influenceseasonal influence

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Components of a Time SeriesComponents of a Time Series

The The trend componenttrend component accounts for accounts for the gradual shifting of the time the gradual shifting of the time series over a long period of time.series over a long period of time.

Any regular pattern of sequences of Any regular pattern of sequences of values above and below the trend values above and below the trend line is attributable to the line is attributable to the cyclical cyclical componentcomponent of the series. of the series.

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Components of a Time SeriesComponents of a Time Series

The The seasonal componentseasonal component of the of the series accounts for regular patterns series accounts for regular patterns of variability within certain time of variability within certain time periods, such as over a year.periods, such as over a year.

The The irregular componentirregular component of the of the series is caused by short-term, series is caused by short-term, unanticipated and non-recurring unanticipated and non-recurring factors that affect the values of the factors that affect the values of the time series. One cannot attempt to time series. One cannot attempt to predict its impact on the time series predict its impact on the time series in advance.in advance.

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Measures of Forecast AccuracyMeasures of Forecast Accuracy

Mean Squared ErrorMean Squared Error

The average of the squared forecast errors The average of the squared forecast errors for the historical data is calculated. The for the historical data is calculated. The forecasting method or parameter(s) which forecasting method or parameter(s) which minimize this mean squared error is then selected.minimize this mean squared error is then selected.

Mean Absolute DeviationMean Absolute Deviation

The mean of the absolute values of all The mean of the absolute values of all forecast errors is calculated, and the forecasting forecast errors is calculated, and the forecasting method or parameter(s) which minimize this method or parameter(s) which minimize this measure is selected. The mean absolute deviation measure is selected. The mean absolute deviation measure is less sensitive to individual large measure is less sensitive to individual large forecast errors than the mean squared error forecast errors than the mean squared error measure.measure.

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Smoothing MethodsSmoothing Methods

In cases in which the time series is In cases in which the time series is fairly stable and has no significant fairly stable and has no significant trend, seasonal, or cyclical effects, one trend, seasonal, or cyclical effects, one can use can use smoothing methodssmoothing methods to average to average out the irregular components of the out the irregular components of the time series. time series.

Four common smoothing methods are:Four common smoothing methods are:

•Moving averagesMoving averages

•Centered moving averagesCentered moving averages

•Weighted moving averagesWeighted moving averages

•Exponential smoothingExponential smoothing

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Moving Average MethodMoving Average Method

The The moving average methodmoving average method consists of computing an average of consists of computing an average of the most recent the most recent nn data values for data values for the series and using this average for the series and using this average for forecasting the value of the time forecasting the value of the time series for the next period.series for the next period.

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Time Series

A time series is a series of observations over time of some quantity of interest (a random variable).

Thus, if is the random variable of interest at time i, and if observations are taken at times i = 1, 2, …., t, then the observed values

are a time series. tt xXxXxX ,,, 2211

iX

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Several typical time series patterns:

Constant level

Linear trend Seasonal effect

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Constant level

Example

,,2,1for, ieAX ii

: the random variable observed at time I

: the constant level of the model

: the random error occurring at time i.

iX

ieA

forecast of the values of the time series at time t + 1, given the observed values,

1tF

tt xXxXxX ,,, 2211

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Forecasting Methods

for a Constant-Level Model

(1) Last-Value Forecasting Method

(2) Averaging Forecasting Method

(3) Moving-Average Forecasting Method

(4) Exponential Smoothing Forecasting Method

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(1) Last-Value Forecasting Method

.1 tt xF

By interpreting t as the current time, the last-value forecasting procedure uses the value of the time series observed at time , as the forecast at time t + 1.

The last-value forecasting method sometimes is called the naive method, because statisticians consider it naïve to use just a sample size of one when additional relevant data are available.

)( txt

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(2) Averaging Forecasting Method

This method uses all the data points in the time series and simply averages these points.

.1

1

t

i

tt t

xF

This estimate is an excellent one if the process is entirely stable.

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(3) Moving-Average Forecasting Method

This method averages the data for only the last n periods as the forecast for the next period.

.1

1

t

nti

it n

xF

The moving-average estimator combines the advantages of the last value and averaging estimators.

A disadvantage of this method is that it places as much weight on as on .1 ntX tX

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(4) Exponential Smoothing Forecasting Method

,)1(1 ttt FxF

Where is called the smoothing constant.

Thus, the forecast is just a weighted sum of the last observation and the preceding forecast for the period just ended.

)10(

tx tF

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Because of this recursive relationship between and , alternatively can be expressed as

.)1()1( 22

11 tttt xxxF

tF1tF 1tF

Another alternative form for the exponential smoothing technique is given by

),(1 tttt FxFF

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Seasonal Factor

It is fairly common for a time series to have a seasonal pattern with higher values at certain times of the year than others.

.average overall

period for the averagefactor Seasonal

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QuarterThree-Year

AverageSeasonal

Factor

Example

18.1529,7

880,8

99.0529,7

434,7

90.0529,7

784,6

93.0529,7

019,7

529,74

117,30Average30,117, Total

019,7

784,6

434,7

880,8

1

2

3

4

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QuarterActual

VolumeSeasonal

Factor

8724

7784

7064

7257

8266

6569

6465

6809

4

3

2

1

4

3

2

1

18.1

99.0

90.0

93.0

18.1

99.0

90.0

93.0

Seasonally Adjusted VolumeYear

2

2

2

2

1

1

1

1

7393

7863

7849

7803

7005

6635

7183

7322

.factor seasonal

valueactual valueadjusted Seasonally

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An Exponential Smoothing Method for a Linear Trend Model

Linear trend

Suppose that the generating process of the observed

time series can be represented by a linear trend

superimposed with random fluctuations.

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The model is represented by

,,2,1for, ieBiAX ii

Where is the random variable that is observed at time i, A is a constraint.

B is the trend factor, and is the random error occurring at time i.

iX

ie

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Adapting Exponential Smoothing to this Model

Let1tT

,, 2211 xXxX

Exponential smoothing estimate of the trend factor B at time t + 1, given the observed values,

., tt xX

Given , the forecast of the value of the time series at time t + 1( ) is obtained simply by adding to the formula for .

1tT1tF

1tT 1tF

.)1( 11 tttt FFxF

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The most recent observations are the most reliable ones for estimating the current parameters.

1tL latest trend at time t + 1 based on the last two values ( and ) and the last two forecasts ( and ).tF 1tF

1txtx

The exponential smoothing formula used for is1tL

).)(1()( 111 ttttt FFxxL

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Then is calculated as1tT

,)1(11 ttt TLT

where is the trend smoothing constant

which must be between 0 and 1.

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Getting started with this forecasting method requires making two initial estimates.

initial estimate of the expected value of the time series

initial estimate of the trend of the time series

0x

1T

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.)1(

,)1(

),)(1()(

,

2112

122

01012

101

TFxF

TLT

xFxxL

TxF

The resulting forecasts for the first two periods are

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Forecasting Errors

The goal of several forecasting methods is to

generate forecasts that are as accurate as

possible, so it is natural to base a measure of

performance on the forecasting errors.

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The forecasting error for any period t is the

absolute value of the deviation of the forecast

for period t ( ) from what then turns out to be

the observed value of the time series for period

.

Thus, letting denote this error,

tF

)( txt

tE

.|| ttt FxE

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Given the forecasting errors for n time periods (t =1, 2, …, n), two popular measures of performance are available.

Mean Absolute Deviation (MAD)

.MAD 1

n

En

tt

.MSE 1

2

n

En

tt

Mean Square Error (MSE)

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.MAD 1

n

En

tt

.MSE 1

2

n

En

tt

The advantages of MAD

(a) its ease of calculation

(b) its straightforward interpretation

The advantages of MSE

(c) it imposes a relatively large penalty for a large

forecasting error while almost ignoring

inconsequentially small forecasting errors.

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Causal Forecasting with Linear Regression

In the preceding sections, we have focused on time series forecasting methods.

We now turn to another type of approach to forecasting.

Causal forecasting:

Causal forecasting obtains a forecast of the quantity of interest by relating it directly to one ore more other quantities that drive the quantity of interest.

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Linear Regression

We will focus on the type of causal forecasting

where the mathematical relationship between the

dependent variable and the independent

variable(s) is assumed to be a linear one.

The analysis in this case is referred to as linear

regression.

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The number of variable A is denoted by X and the number of variable B is denoted by Y, then the random variables X and Y exhibit a degree of association.

For any given number of variable A, there is a range of possible variable B, and vice versa.

.]|[ BxAxXYE

This relationship between X and Y is referred to as a degree of association model.

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In some cases, there exists a functional relationship between two variables that may be linked linearly.

The previous example is

,ii eBiAX

.][ BtAXE t It follows that

Both the degree of association model and the exact functional relationship model lead to the same linear relationship.

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With t taking on integer values starting with 1, leads to certain simplified expressions.

In the standard notation of regression analysis, X represents the independent variable and Y represents the dependent variable of interest.

Consequently, the notational expression for this special time series model becomes

.tt eBtAY

.][ BtAXE t

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Method of Least Squares

The usual method for identifying the “best” fitted

line is the method of least squares.

Regression Line

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Suppose that an arbitrary line, given by the expression , is drawn through the data.

A measure of how well this line fits the data can be obtained by computing the sum of squares of the vertical deviations of the actual points from the fitting line.

bxay ~

n

iii

nn

yy

yyyyyyQ

1

2

2222

211

.)~(

)~()~()~(

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This method chooses that line a + bx that makes Q a minimum.

n

i

n

iii

n

i

n

ii

n

iiii

n

ii

n

iii

nxx

nyxyx

xx

yyxxb

1

2

1

2

1 11

1

2

1

)(

))((

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Slide


and

where

and

,xbya

n

i

i

n

xx

1

.1

n

i

i

n

yy

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)2(0)(2

)1(0)(2

)(

1

1

2

1

1

2

1

2

1

2

t

n

ttt

n

tt

n

ttt

n

tt

n

ttt

n

tt

xbxaybb

L

bxayaa

L

bxayL

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Slide


)'1( xy

0

0)(

11

11

11

1111

ba

n

xb

n

ya

xbyna

xbnay

bxaybxay

n

tt

n

tt

n

tt

n

tt

n

tt

n

tt

n

tt

n

t

n

tt

n

ttt

Re-write (1)

y x

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Re-write (2)

0

0

0

0)(

1

2

1

11

1

11

1

2

11

1

2

11

1

n

tt

n

tt

n

tt

n

ttn

ttt

n

tt

n

tt

n

tt

n

tt

n

ttt

n

tt

n

tt

n

ttt

t

n

ttt

xbxn

xb

n

yxy

n

xb

n

yxbya

xbxaxy

bxaxxy

xbxay

From (1)’

(1)’ in (2)

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(2)'

0

2

1

1

2

11

1

2

1

1

211

1

1

2

2

111

1

n

x

x

n

xy

xyb

n

x

xbn

xy

xy

xbn

x

bn

xy

xy

n

ttn

tt

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ttn

ttt

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Sales of Comfort brand headache Sales of Comfort brand headache medicine formedicine for

the past ten weeks at Rosco Drugsthe past ten weeks at Rosco Drugs

are shown on the next slide. If are shown on the next slide. If

Rosco Drugs uses a 3-periodRosco Drugs uses a 3-period

moving average to forecast sales,moving average to forecast sales,

what is the forecast for Week 11?what is the forecast for Week 11?

Example: Rosco DrugsExample: Rosco Drugs

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Past SalesPast Sales

WeekWeek SalesSales WeekWeek SalesSales 1 110 6 1201 110 6 120 2 115 7 1302 115 7 130 3 125 8 1153 125 8 115 4 120 9 1104 120 9 110 5 125 10 1305 125 10 130


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Excel Spreadsheet Showing Input DataExcel Spreadsheet Showing Input DataA B C

1 Robert's Drugs2

3 Week (t ) Salest Forect+1

4 1 1105 2 1156 3 1257 4 1208 5 1259 6 120

10 7 13011 8 11512 9 11013 10 130

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Steps to Moving Average Using ExcelSteps to Moving Average Using Excel

Step 1:Step 1: Select the Select the ToolsTools pull-down menu. pull-down menu.

Step 2:Step 2: Select the Select the Data AnalysisData Analysis option. option.

Step 3:Step 3: When the Data Analysis Tools dialog When the Data Analysis Tools dialog appears, choose Mappears, choose Moving Averageoving Average..

Step 4:Step 4: When the Moving Average dialog box When the Moving Average dialog box appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input Input RangeRange box. box.

Enter 3 in the Enter 3 in the IntervalInterval box. box.

Enter C4 in the Enter C4 in the Output RangeOutput Range box.box.

Select Select OKOK..

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Spreadsheet Showing Results Using Spreadsheet Showing Results Using nn = 3 = 3A B C

1 Robert's Drugs2

3 Week (t ) Salest Forect+1

4 1 110 #N/A5 2 115 #N/A6 3 125 116.77 4 120 120.08 5 125 123.39 6 120 121.7

10 7 130 125.011 8 115 121.712 9 110 118.313 10 130 118.3

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Centered Moving Average MethodCentered Moving Average Method

The The centered moving average methodcentered moving average method consists of computing an average of consists of computing an average of n n periods' periods' data and associating it with the midpoint of the data and associating it with the midpoint of the periods. For example, the average for periods 5, periods. For example, the average for periods 5, 6, and 7 is associated with period 6. This 6, and 7 is associated with period 6. This methodology is useful in the process of methodology is useful in the process of computing season indexes.computing season indexes.

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Weighted Moving Average MethodWeighted Moving Average Method

In the In the weighted moving average methodweighted moving average method for computing the average of the most recent for computing the average of the most recent n n periods, the more recent observations are periods, the more recent observations are typically given more weight than older typically given more weight than older observations. For convenience, the weights observations. For convenience, the weights usually sum to 1.usually sum to 1.

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Exponential SmoothingExponential Smoothing

• Using Using exponential smoothingexponential smoothing, the forecast , the forecast for the next period is equal to the forecast for the next period is equal to the forecast for the current period plus a proportion (for the current period plus a proportion () ) of the forecast error in the current period.of the forecast error in the current period.

• Using exponential smoothing, the forecast is Using exponential smoothing, the forecast is calculated by: calculated by:

[the actual value for the current [the actual value for the current period] +period] +

(1- (1- )[the forecasted value for the current )[the forecasted value for the current period], period],

where the smoothing constant, where the smoothing constant, , is a , is a number between 0 and 1.number between 0 and 1.

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Trend ProjectionTrend Projection

If a time series exhibits a linear trend, the If a time series exhibits a linear trend, the method of method of least squaresleast squares may be used to may be used to determine a trend line (projection) for future determine a trend line (projection) for future forecasts. forecasts.

Least squares, also used in regression analysis, Least squares, also used in regression analysis, determines the unique determines the unique trend line forecasttrend line forecast which which minimizes the mean square error between the minimizes the mean square error between the trend line forecasts and the actual observed trend line forecasts and the actual observed values for the time series.values for the time series.

The independent variable is the time period and The independent variable is the time period and the dependent variable is the actual observed the dependent variable is the actual observed value in the time series.value in the time series.

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Using the method of least squares, the formula for the Using the method of least squares, the formula for the trend projection is: trend projection is: TTtt = = bb00 + + bb11tt. .

where: where: TTtt = trend forecast for time period = trend forecast for time period tt

bb1 1 = slope of the trend line= slope of the trend line

bb00 = trend line projection for time 0 = trend line projection for time 0

bb11 = = nntYtYtt - - t t YYtt

nnt t 22 - ( - (t t ))22

where: where: YYtt = observed value of the time series at time = observed value of the time series at time

period period tt

= average of the observed values for = average of the observed values for YYtt

= average time period for the = average time period for the nn observationsobservations

0 1b Y b t 0 1b Y b t

YYtt

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If Rosco Drugs uses exponentialIf Rosco Drugs uses exponential

smoothing to forecast sales, which value for smoothing to forecast sales, which value for thethe

smoothing constant smoothing constant , .1 or .8, gives better , .1 or .8, gives better forecasts?forecasts?

WeekWeek SalesSales WeekWeek SalesSales 1 110 6 1201 110 6 120 2 115 7 1302 115 7 130 3 125 8 1153 125 8 115 4 120 9 1104 120 9 110 5 125 10 1305 125 10 130

Example: Rosco Drugs (B)Example: Rosco Drugs (B)

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Exponential SmoothingExponential Smoothing

To evaluate the two smoothing constants, To evaluate the two smoothing constants, determine how the forecasted values would determine how the forecasted values would compare with the actual historical values in compare with the actual historical values in each case. each case.

Let: Let: YYtt = actual sales in week = actual sales in week tt

FFt t = forecasted sales in week = forecasted sales in week tt

FF11 = = YY11 = 110 = 110

For other weeks, For other weeks, FFtt+1+1 = .1 = .1YYtt + .9 + .9FFtt

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Exponential Smoothing (Exponential Smoothing ( = .1, 1 - = .1, 1 - = .9) = .9)

FF11 = 110 = 110

FF2 2 = .1= .1YY11 + .9 + .9FF11 = .1(110) + .9(110) = 110 = .1(110) + .9(110) = 110

FF33 = .1 = .1YY22 + .9 + .9FF22 = .1(115) + .9(110) = 110.5 = .1(115) + .9(110) = 110.5

FF44 = .1 = .1YY33 + .9 + .9FF33 = .1(125) + .9(110.5) = 111.95 = .1(125) + .9(110.5) = 111.95

FF55 = .1 = .1YY44 + .9 + .9FF44 = .1(120) + .9(111.95) = 112.76 = .1(120) + .9(111.95) = 112.76

FF66 = .1 = .1YY55 + .9 + .9FF55 = .1(125) + .9(112.76) = 113.98 = .1(125) + .9(112.76) = 113.98

FF77 = .1 = .1YY66 + .9 + .9FF66 = .1(120) + .9(113.98) = 114.58 = .1(120) + .9(113.98) = 114.58

FF88 = .1 = .1YY77 + .9 + .9FF77 = .1(130) + .9(114.58) = 116.12 = .1(130) + .9(114.58) = 116.12

FF99 = .1 = .1YY88 + .9 + .9FF88 = .1(115) + .9(116.12) = 116.01 = .1(115) + .9(116.12) = 116.01

FF1010= .1= .1YY99 + .9 + .9FF99 = .1(110) + .9(116.01) = 115.41 = .1(110) + .9(116.01) = 115.41

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Exponential Smoothing (Exponential Smoothing ( = .8, 1 - = .8, 1 - = .2) = .2)

FF11 = 110 = 110

FF22 = .8(110) + .2(110) = 110 = .8(110) + .2(110) = 110

FF33 = .8(115) + .2(110) = 114 = .8(115) + .2(110) = 114

FF44 = .8(125) + .2(114) = 122.80 = .8(125) + .2(114) = 122.80

FF55 = .8(120) + .2(122.80) = 120.56 = .8(120) + .2(122.80) = 120.56

FF66 = .8(125) + .2(120.56) = 124.11 = .8(125) + .2(120.56) = 124.11

FF77 = .8(120) + .2(124.11) = 120.82 = .8(120) + .2(124.11) = 120.82

FF88 = .8(130) + .2(120.82) = 128.16 = .8(130) + .2(120.82) = 128.16

FF99 = .8(115) + .2(128.16) = 117.63 = .8(115) + .2(128.16) = 117.63

FF1010= .8(110) + .2(117.63) = 111.53= .8(110) + .2(117.63) = 111.53

68 68 Slide

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Mean Squared ErrorMean Squared Error

In order to determine which smoothing In order to determine which smoothing constant gives the better performance, constant gives the better performance, calculate, for each, the mean squared error for calculate, for each, the mean squared error for the nine weeks of forecasts, weeks 2 through 10 the nine weeks of forecasts, weeks 2 through 10 by:by:

[([(YY22--FF22))22 + ( + (YY33--FF33))22 + ( + (YY44--FF44))22 + . . . + ( + . . . + (YY1010--FF1010))22]/9]/9

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= .1 = .1 = .8 = .8

Week Week YYtt FFtt ( (YYtt - - FFtt))22 FFt t ((YYtt - - FFtt))22

1 110 1 110 2 115 110.00 25.00 110.00 25.002 115 110.00 25.00 110.00 25.00 3 125 110.50 210.25 114.00 121.003 125 110.50 210.25 114.00 121.00 4 120 111.95 64.80 122.80 7.844 120 111.95 64.80 122.80 7.84 5 125 112.76 149.94 120.56 19.715 125 112.76 149.94 120.56 19.71 6 120 113.98 36.25 124.11 16.916 120 113.98 36.25 124.11 16.91 7 130 114.58 237.73 120.82 84.237 130 114.58 237.73 120.82 84.23 8 115 116.12 1.26 128.16 173.308 115 116.12 1.26 128.16 173.30 9 110 116.01 36.12 117.63 58.269 110 116.01 36.12 117.63 58.26 10 130 115.41 212.87 111.53 341.2710 130 115.41 212.87 111.53 341.27

Sum 974.22 Sum 847.52Sum 974.22 Sum 847.52 MSE Sum/9 Sum/9MSE Sum/9 Sum/9

108.25108.25108.25108.25 94.1794.1794.1794.17

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1 Robert's Drugs23 Week Sales4 1 1105 2 1156 3 1257 4 1208 5 1259 6 120

10 7 13011 8 11512 9 11013 10 130

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Steps to Exponential Smoothing Using ExcelSteps to Exponential Smoothing Using Excel

Step 1:Step 1: Select the Select the ToolsTools pull-down menu. pull-down menu.

Step 2:Step 2: Select the Select the Data AnalysisData Analysis option. option.

Step 3:Step 3: When the Data Analysis Tools dialog When the Data Analysis Tools dialog appears, choose appears, choose Exponential SmoothingExponential Smoothing..

Step 4:Step 4: When the Exponential Smoothing dialog box When the Exponential Smoothing dialog box appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input RangeInput Range box.box.

Enter 0.9 (for Enter 0.9 (for = 0.1) in = 0.1) in Damping Damping FactorFactor box. box.

Enter C4 in the Enter C4 in the Output RangeOutput Range box. box.


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Spreadsheet Showing Results Using Spreadsheet Showing Results Using = 0.1 = 0.1

A B C1 Robert's Drugs2 = 0.1

3 Week (t ) Salest Forect +1

4 1 110 #N/A5 2 115 110.06 3 125 110.57 4 120 112.08 5 125 112.89 6 120 114.0

10 7 130 114.611 8 115 116.112 9 110 116.013 10 130 115.4

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Repeating the Process for Repeating the Process for = 0.8 = 0.8

• Step 4: When the Exponential Smoothing Step 4: When the Exponential Smoothing dialog box dialog box appears:appears:

Enter B4:B13 in the Enter B4:B13 in the Input Input RangeRange box. box.

Enter 0.2 (for Enter 0.2 (for = 0.8) in = 0.8) in Damping FactorDamping Factor box. box.

Enter D4 in the Enter D4 in the Output RangeOutput Range box.box.


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Spreadsheet Results for Spreadsheet Results for = 0.1 and = 0.1 and = 0.8 = 0.8

A B C D1 Robert's Drugs2 = 0.1 = 0.8

3 Week (t ) Salest Forect +1 Forect +1

4 1 110 #N/A #N/A5 2 115 110.0 110.06 3 125 110.5 114.07 4 120 112.0 122.88 5 125 112.8 120.69 6 120 114.0 124.1

10 7 130 114.6 120.811 8 115 116.1 128.212 9 110 116.0 117.613 10 130 115.4 111.5

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The number of plumbing repair jobs The number of plumbing repair jobs performed byperformed by

Auger's Plumbing Service in each of the last nineAuger's Plumbing Service in each of the last nine

months is listed on the next slide. Forecastmonths is listed on the next slide. Forecast

the number of repair jobs Auger's willthe number of repair jobs Auger's will

perform in December using the leastperform in December using the least

squares method. squares method.

Example: Auger’s Plumbing ServiceExample: Auger’s Plumbing Service

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MonthMonth JobsJobs MonthMonth JobsJobs MonthMonth JobsJobs

March 353 June 374 March 353 June 374 September 399September 399

April 387 July 396 October April 387 July 396 October 412 412

May 342 August 409 May 342 August 409 November 408November 408


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(month) (month) tt YYtt tYtYtt t t 22

(Mar.) 1 353 353 1 (Mar.) 1 353 353 1 (Apr.) 2 387 774 4(Apr.) 2 387 774 4 (May) 3 342 1026 9(May) 3 342 1026 9 (June) 4 374 1496 16(June) 4 374 1496 16 (July) 5 396 1980 25(July) 5 396 1980 25 (Aug.) 6 409 2454 36(Aug.) 6 409 2454 36 (Sep.) 7 399 2793 49(Sep.) 7 399 2793 49 (Oct.) 8 412 3296 64(Oct.) 8 412 3296 64 (Nov.) 9 408 3672 81(Nov.) 9 408 3672 81

Sum 45 3480 17844 285Sum 45 3480 17844 285

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Trend Projection (continued)Trend Projection (continued)

= 45/9 = 5 = 3480/9 = = 45/9 = 5 = 3480/9 = 386.667386.667

nntYtYtt - - t t YYtt (9)(17844) - (45) (9)(17844) - (45)(3480)(3480)

bb11 = = = = = = 7.47.4

nnt t 22 - ( - (tt))22 (9)(285) - (45) (9)(285) - (45)22

= 386.667 - 7.4(5) = 349.667= 386.667 - 7.4(5) = 349.667

TT1010 = 349.667 + (7.4)(10) = = 349.667 + (7.4)(10) =

423.667423.667423.667423.667

0 1b Y b t 0 1b Y b t

YYtt

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1 Auger's Plumbing Service23 Month Calls4 1 3535 2 3876 3 3427 4 3748 5 3969 6 409

10 7 39911 8 41212 9 40813

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Steps to Trend Projection Using ExcelSteps to Trend Projection Using Excel

Step 1:Step 1: Select an empty cell (B13) in the worksheet. Select an empty cell (B13) in the worksheet.

Step 2:Step 2: Select the Select the InsertInsert pull-down menu. pull-down menu.

Step 3:Step 3: Choose the Choose the FunctionFunction option. option.

Step 4:Step 4: When the Paste Function dialog box When the Paste Function dialog box appears:appears:

Choose Choose StatisticalStatistical in Function Category in Function Category box.box.

Choose Choose ForecastForecast in the Function Name in the Function Name box.box.


more . . . . . . .

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Steps to Trend Projecting Using Excel Steps to Trend Projecting Using Excel (continued)(continued)

Step 5:Step 5: When the Forecast dialog box appears: When the Forecast dialog box appears:

Enter 10 in the Enter 10 in the xx box (for box (for month 10).month 10).

Enter B4:B12 in the Enter B4:B12 in the Known y’sKnown y’s box.box.

Enter A4:A12 in the Enter A4:A12 in the Known x’sKnown x’s box.box.


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Spreadsheet Showing Trend Projection for Spreadsheet Showing Trend Projection for Month 10Month 10 A B C

1 Auger's Plumbing Service23 Month Calls4 1 3535 2 3876 3 3427 4 3748 5 3969 6 409

10 7 39911 8 41212 9 40813 10 423.667 Projected

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Example: Auger’s Plumbing Service (B)Example: Auger’s Plumbing Service (B)

Forecast for December (Month 10) using aForecast for December (Month 10) using a

three-period (three-period (nn = 3) weighted moving average = 3) weighted moving average withwith

weights of .6, .3, and .1. weights of .6, .3, and .1.

Then, compare this Month 10 weighted Then, compare this Month 10 weighted movingmoving

average forecast with the Month 10 trend average forecast with the Month 10 trend projectionprojection

forecast.forecast.

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Three-Month Weighted Moving AverageThree-Month Weighted Moving Average

The forecast for December will be the The forecast for December will be the weighted average of the preceding three weighted average of the preceding three months: September, October, and November.months: September, October, and November.

FF1010 = .1 = .1YYSep.Sep. + .3 + .3YYOct.Oct. + .6 + .6YYNov.Nov.

= .1(399) + .3(412) + .6(408) = .1(399) + .3(412) + .6(408)

= =


FF1010 = 423.7 (from earlier slide) = 423.7 (from earlier slide)

408.3408.3408.3408.3

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ConclusionConclusion

Due to the positive trend component in Due to the positive trend component in the time series, the trend projection produced a the time series, the trend projection produced a forecast that is more in tune with the trend that forecast that is more in tune with the trend that exists. The weighted moving average, even exists. The weighted moving average, even with heavy (.6) placed on the current period, with heavy (.6) placed on the current period, produced a forecast that is lagging behind the produced a forecast that is lagging behind the changing data. changing data.

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Forecasting with TrendForecasting with Trendand Seasonal Componentsand Seasonal Components

Steps of Multiplicative Time Series ModelSteps of Multiplicative Time Series Model

1.1. Calculate the centered moving averages Calculate the centered moving averages (CMAs).(CMAs).

2.2. Center the CMAs on integer-valued periods. Center the CMAs on integer-valued periods.

3.3. Determine the seasonal and irregular factors Determine the seasonal and irregular factors ((SSttIIt t ).).

4.4. Determine the average seasonal factors. Determine the average seasonal factors.

5.5. Scale the seasonal factors ( Scale the seasonal factors (SSt t ).).

6.6. Determine the deseasonalized data. Determine the deseasonalized data.

7.7. Determine a trend line of the deseasonalized Determine a trend line of the deseasonalized data.data.

8.8. Determine the deseasonalized predictions. Determine the deseasonalized predictions.

9.9. Take into account the seasonality. Take into account the seasonality.

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Example: Terry’s Tie ShopExample: Terry’s Tie Shop

Business at Terry's Tie Shop can be viewed Business at Terry's Tie Shop can be viewed asas

falling into three distinct seasons:falling into three distinct seasons:

(1) Christmas (November-December);(1) Christmas (November-December);

(2) Father's Day (late May - mid-June);(2) Father's Day (late May - mid-June);

and (3) all other times. Average weeklyand (3) all other times. Average weekly

sales ($) during each of the three seasonssales ($) during each of the three seasons

during the past four years are shown onduring the past four years are shown on

the next slide.the next slide.

Determine a forecast for the average Determine a forecast for the average weekly salesweekly sales

in year 5 for each of the three seasons.in year 5 for each of the three seasons.

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Past Sales ($)Past Sales ($)

YearYear

SeasonSeason 11 22 33 44 1 1856 1995 2241 22801 1856 1995 2241 2280 2 2012 2168 2306 24082 2012 2168 2306 2408 3 985 1072 1105 11203 985 1072 1105 1120

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Dollar Moving Scaled Dollar Moving Scaled Year Season Sales (Year Season Sales (YYtt) Average ) Average SSttIItt SStt YYtt//SStt

1 1 1856 1.178 15761 1 1856 1.178 1576 2 2012 1617.67 1.244 1.236 16282 2012 1617.67 1.244 1.236 1628 3 985 1664.00 .592 .586 16813 985 1664.00 .592 .586 1681 2 1 1995 1716.00 1.163 1.178 16942 1 1995 1716.00 1.163 1.178 1694 2 2168 1745.00 1.242 1.236 17542 2168 1745.00 1.242 1.236 1754 3 1072 1827.00 .587 .586 18293 1072 1827.00 .587 .586 1829 3 1 2241 1873.00 1.196 1.178 19023 1 2241 1873.00 1.196 1.178 1902 2 2306 1884.00 1.224 1.236 18662 2306 1884.00 1.224 1.236 1866 3 1105 1897.00 .582 .586 18863 1105 1897.00 .582 .586 1886 4 1 2280 1931.00 1.181 1.178 19354 1 2280 1931.00 1.181 1.178 1935 2 2408 1936.00 1.244 1.236 19482 2408 1936.00 1.244 1.236 1948 3 1120 .586 19113 1120 .586 1911

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1. Calculate the centered moving averages.1. Calculate the centered moving averages.

There are three distinct seasons in each There are three distinct seasons in each year. Hence, take a three-season moving year. Hence, take a three-season moving average to eliminate seasonal and irregular average to eliminate seasonal and irregular factors. For example:factors. For example:

11stst MA = (1856 + 2012 + 985)/3 = MA = (1856 + 2012 + 985)/3 = 1617.671617.67

22ndnd MA = (2012 + 985 + 1995)/3 = MA = (2012 + 985 + 1995)/3 = 1664.001664.00

etc.etc.


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2. Center the CMAs on integer-valued periods.2. Center the CMAs on integer-valued periods.

The first moving average computed in The first moving average computed in step 1 (1617.67) will be centered on season 2 of step 1 (1617.67) will be centered on season 2 of year 1. Note that the moving averages from year 1. Note that the moving averages from step 1 center themselves on integer-valued step 1 center themselves on integer-valued periods because periods because nn is an odd number. is an odd number.

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3. Determine the seasonal & irregular factors 3. Determine the seasonal & irregular factors ((SSt t IIt t ).). Isolate the trend and cyclical Isolate the trend and cyclical components. For each period components. For each period tt, this is given , this is given by:by:

SSt t IIt t = = YYt t /(Moving Average for period /(Moving Average for period t t ))

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4. Determine the average seasonal factors.4. Determine the average seasonal factors.

Averaging all Averaging all SSt t IItt values corresponding to values corresponding to that season:that season:

Season 1: (1.163 + 1.196 + 1.181) /3 Season 1: (1.163 + 1.196 + 1.181) /3 = 1.180 = 1.180

Season 2: (1.244 + 1.242 + 1.224 + Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.2381.244) /4 = 1.238

Season 3: (.592 + .587 + .582) /3 Season 3: (.592 + .587 + .582) /3 = .587 = .587

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5. Scale the seasonal factors (5. Scale the seasonal factors (SSt t ).).

Average the seasonal factors = (1.180 + Average the seasonal factors = (1.180 + 1.238 + .587)/3 = 1.002. Then, divide each 1.238 + .587)/3 = 1.002. Then, divide each seasonal factor by the average of the seasonal seasonal factor by the average of the seasonal factors. factors.

Season 1: 1.180/1.002 = 1.178Season 1: 1.180/1.002 = 1.178

Season 2: 1.238/1.002 = 1.236Season 2: 1.238/1.002 = 1.236

Season 3: .587/1.002 = Season 3: .587/1.002 = .586 .586

Total = 3.000Total = 3.000

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6. Determine the deseasonalized data.6. Determine the deseasonalized data.

Divide the data point values, Divide the data point values, YYt t , by , by SSt t ..

7. Determine a trend line of the deseasonalized 7. Determine a trend line of the deseasonalized data.data.

Using the least squares method for Using the least squares method for tt = 1, = 1, 2, ..., 12, gives:2, ..., 12, gives:

TTtt = 1580.11 + 33.96 = 1580.11 + 33.96tt

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8. Determine the deseasonalized predictions.8. Determine the deseasonalized predictions.

Substitute Substitute tt = 13, 14, and 15 into the = 13, 14, and 15 into the least squares equation:least squares equation:

TT1313 = 1580.11 + (33.96)(13) = 2022 = 1580.11 + (33.96)(13) = 2022

TT1414 = 1580.11 + (33.96)(14) = 2056 = 1580.11 + (33.96)(14) = 2056

TT1515 = 1580.11 + (33.96)(15) = 2090 = 1580.11 + (33.96)(15) = 2090

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9. Take into account the seasonality.9. Take into account the seasonality.

Multiply each deseasonalized prediction Multiply each deseasonalized prediction by its seasonal factor to give the following by its seasonal factor to give the following forecasts for year 5:forecasts for year 5:

Season 1: (1.178)(2022) =Season 1: (1.178)(2022) =

Season 2: (1.236)(2056) =Season 2: (1.236)(2056) =

Season 3: ( .586)(2090) =Season 3: ( .586)(2090) =

2382238223822382

2541254125412541

1225122512251225

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Qualitative Approaches to ForecastingQualitative Approaches to Forecasting

Delphi ApproachDelphi Approach

• A panel of experts, each of whom is A panel of experts, each of whom is physically separated from the others and is physically separated from the others and is anonymous, is asked to respond to a anonymous, is asked to respond to a sequential series of questionnaires. sequential series of questionnaires.

• After each questionnaire, the responses are After each questionnaire, the responses are tabulated and the information and opinions of tabulated and the information and opinions of the entire group are made known to each of the entire group are made known to each of the other panel members so that they may the other panel members so that they may revise their previous forecast response. revise their previous forecast response.

• The process continues until some degree of The process continues until some degree of consensus is achieved.consensus is achieved.

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Scenario WritingScenario Writing

• Scenario writing consists of developing a Scenario writing consists of developing a conceptual scenario of the future based on a conceptual scenario of the future based on a well defined set of assumptions. well defined set of assumptions.

• After several different scenarios have been After several different scenarios have been developed, the decision maker determines developed, the decision maker determines which is most likely to occur in the future and which is most likely to occur in the future and makes decisions accordingly.makes decisions accordingly.

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Subjective or Interactive ApproachesSubjective or Interactive Approaches

• These techniques are often used by These techniques are often used by committees or panels seeking to develop new committees or panels seeking to develop new ideas or solve complex problems.ideas or solve complex problems.

• They often involve "brainstorming sessions". They often involve "brainstorming sessions".

• It is important in such sessions that any ideas It is important in such sessions that any ideas or opinions be permitted to be presented or opinions be permitted to be presented without regard to its relevancy and without without regard to its relevancy and without fear of criticism.fear of criticism.

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End of Chapter 16End of Chapter 16

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