Lecture.chap15

8/3/2019 Lecture.chap15

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Chapter 15 Ex panded Notes

Slide 1

1 2003 Thomson/South-Western Slide

Chapter 15Forecasting

Qualitative Approaches to Forecasting

Quantitative Approaches to Forecasting

The Components of a Time Series

Using Smoothing Methods in Forecasting

Measures of Forecast Accuracy

Using Trend Projection in Forecasting

We will cover the above-mentioned forecasting methods, covering sections 1 through 3 andsection 6. Read over section 5 (Regression analysis which you must have covered in statisticsclass)

Slide 2

Defining forecasting

2 2003 Thomson/South-Western Slide Heizer Render 2001 by Prentice

Hall, Inc.,

4-2

What is Forecasting?

Process of predicting a

future event

Underlying basis of

all business decisions Production

Inventory

Personnel

Facilities

Sales will be$200 Million!


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Slide 3

3 2003 Thomson/South-Western Slide Heizer Render,2001 by

Prentice Hall,

4-3

Short-range forecast

Up to 1 year; usually less than 3 months

Job scheduling, worker assignments

Medium-range forecast

3 months to 3 years

Sales & production planning, budgeting

Long-range forecast

3+ years

New product planning, facility location

Types of Forecasts by Time Horizon

Slide 4

4 2003 Thomson/South-Western Slide Heizer Render, 2001 by Prentice

Hall, Inc.,

4-4

Forecasting Approaches

Used when situation is

stable & historical data

exist Existing products

Current technology

Involves mathematical

techniques

e.g., forecasting sales of

color televisions

Quantitative Methods Used when situation is

vague & littl e data exist New product s

New technol ogy

Involves intuition,

experience

e.g., forecasting sales on

Internet

Qualitative Methods

Forecasting methods fall under two main umbrellas: quantitative based and qualitative based.

The slide provides the main difference between the two approaches and their applicability.


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Slide 5



Delphi Approach

A panel of experts, each of whom is physicallyseparated from the others and is anonymous, isasked to respond to a sequential series ofquestionnaires.

After each questionnaire, the responses are tabulatedand the information and opinions of the entire groupare made known to each of the other panel membersso that they may revise their previous forecastresponse.

The process continues until some degree ofconsensus is achieved.

Qualitative approaches to forecasting can take on a variety of different methods. In the DelphiApproach a panel of experts is used to determine what the forecast will be. A good example isthe forecasted price of stocks. Stock price forecasts are based on expert opinions from variousfund managers and industry experts.

Slide 6



Scenario WritingScenario writing consists of developing a conceptual

scenario of the future based on a well defined set ofassumptions.

After several different scenarios have beendeveloped, the decision maker determines which ismost likely to occur in the future and makesdecisions accordingly.

Scenario writing is another approach to qualitative forecasting. Here a variety of scenarios arewritten and the decision maker determines what he / she feels is most likely. A businessmanager might use this to gather input from each of the different business areas. For example,production might write a scenario, sales might write another scenario, and finance might write ascenario. The decision maker would then evaluate each of the potential scenarios anddetermine the forecast.


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Slide 7



Subjective or Interactive Approaches

These techniques are often used by committees orpanels seeking to develop new ideas or solvecomplex problems.

They often involve "brainstorming sessions".

It is important in such sessions that any ideas oropinions be permitted to be presented withoutregard to its relevancy and without fear of criticism.

Subjective approaches typically involve brainstorming to determine what the committee or groupviews as the likely forecast. A great example is when the sales manager asks each employee toestimate their likely sales. The group then discusses their potential as a whole and from thisdata the sales manager will forecast his sales.

Slide 8


Quantitative Approaches to Forecasting

Quantitative methods are based on an analysis ofhistorical data concerning one or more time series.

A time series is a set of observations measured atsuccessive points in time or over successive periodsof time.

If the historical data used are restricted to past valuesof the series that we are trying to forecast, theprocedure is called a time series method.

If the historical data used involve other time seriesthat are believed to be related to the time series thatwe are trying to forecast, the procedure is called acausal method.

Although qualitative methods have their purpose, quantitative methods are rooted in data andshould provide a better forecast. It is important to use good, historical data that depicts what hashappened and can likely predict what will happen. We will study a number of time seriesmethods for quantitative forecasting and we will use regression analysis to develop a causalforecast.


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Slide 9


Quantitative Forecasting Methods

Quantitative

Forecasting

Linear

Regression

Associative

Models

ExponentialSmoothing

MovingAverage

Time Series

Models

TrendProjection

Heizer Render, 2001 by Prentice Hall,

Inc

Quantitative methods in forecasting fall in turn into 2 main streams: time series based and

Associative or Causal Model. Linear regression is the most popular causal method used (read

over section 5). We will focus on times series models.


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Slide 10


Components of a Time Series

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

Any regular pattern of sequences of values above andbelow the trend line is attributable to the cyclicalcomponent of the series.

The seasonal component of the series accounts forregular patterns of variability within certain timeperiods, such as over a year.

The irregular component of the series is caused byshort-term, unanticipated and non-recurring factorsthat affect the values of the time series. One cannotattempt to predict its impact on the time series in

advance.

Time series implies that the historical data have been gathered over a defined period of regulartime intervals. This could be weeks, months, days, etc. Plotting the data is always the best step.A simple line chart will show if there is any pattern to the data. Patterns might be the result ofseasonal increases in sales or regular, cyclical occurring influencers. I always think about myhusbands company who purchases a significant number of cars for their employees. The carpurchasing is cyclical every three years new cars are bought. If you are the lucky dealershipfrom whom these cars are purchased you would show a cyclical (every 3 years) pattern for thiscustomer. Car dealerships might also show seasonal patterns, for example near the end of the

year when they have significant sales in an effort to reduce their inventory (they do this for taxpurposes, so I am told ~ not really because of the volume of incoming new models!)


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Slide 11

11 2003 Thomson/South-Western Slide Heizer Render, 2001 by Prentice

Hall, Inc.,

4-11

Trend

Seasonal

Cyclical

Random

Time Series Components

Slide summarizes again the main time series components


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Slide 12

12 2003 Thomson/South-Western Slide4-12

Persistent, overall upward or downward pattern

Due to population, technology etc.

Several years duration

Mo., Qtr., Yr.

Response

1984-1994T/Maker Co.

Trend Component

Heizer Render, 2001 by Prentice

Hall, Inc.,

Slide 13


Regular pattern of up & down fluctuations Due to weather, customs etc.

Occurs within 1 year

Mo., Qtr.

Response

Summer

1984-1994T/Maker Co.

Seasonal Component


Hall, Inc.,


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Slide 14


Repeating up & down movements

Due to interactions of factors influencing economy

Usually 2-10 years duration

Mo., Qtr., Yr.

Response

Cycle

Cyclical Component


Hall, Inc.,

Slide 15


Erratic, unsystematic, residual fluctuations Due to random variation or unforeseen events

Union strike

Tornado

Short duration &nonrepeating

1984-1994 T/Maker Co.

Random Component


Hall, Inc.,


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Slide 16


Time Series Methods

Three time series methods are:

smoothing

trend projection

trend projection adjusted for seasonal influence

These are the three time series methods that we will review.


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Slide 17


Smoothing Methods

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

Three common smoothing methods are:

Moving averages

Weighted moving averages

Exponential smoothing

Our goal in forecasting is to smooth out the irregular patterns and plan for the known seasonalor cyclical patterns. Three common smoothing methods are: Moving averages, weighted movingaverages, and exponential smoothing. Each of these methods has a different approach tosmoothing out the patterns in the data.


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Slide 18


Smoothing Methods

Moving Average Method

The moving average method consists of computingan average of the most recent n data values for theseries and using this average for forecasting the value ofthe time series for the next period.

The moving average method is the least complex. Basically you use the average of some periodof recent data to predict future data. Lets take a look through an example.

Slide 19

19 2003 Thomson

/South-Western Slide

During the past ten weeks, sales of cases ofComfort brand headache medicine at Robert's Drugshave been as follows:

Week Sales Week Sales1 110 6 1202 115 7 1303 125 8 1154 120 9 1105 125 10 130

If Robert's uses a 3-period moving average toforecast sales, what is the forecast for Week 11?

Example: Roberts Drugs

Roberts Drugs shows their sales data for the last 10 weeks. They would like to use a 3-periodmoving average to forecast week 11. To do this, the forecaster will use the historical data forweeks 8, 9, and 10 and determine the average. This value is the forecast for week 11.


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Slide 20


Example: Rosco Drugs

Excel Spreadsheet Showing Input Data

A B C

1 Robert's Drugs

2

3 Week (t) Salest Forect+14 1 110

5 2 115

6 3 125

7 4 120

8 5 125

9 6 120

10 7 130

11 8 115

12 9 110

13 10 130

Slide 21


Example: Rosco Drugs

Steps to Moving Average Using Excel

Step 1: Select the Tools pull-down menu.Step 2: Select the Data Analysis option.

Step 3: When the Data Analysis Tools dialogappears, choose Moving Average.

Step 4: When the Moving Average dialog boxappears:

Enter B4:B13 in the Input Range box.

Enter 3 in the Interval box.

Enter C4 in the Output Range box.

Select OK.

Excel has a function that easily computes the moving average. See instructions in slide above.For Excell 2007, need to upload the ToolPack Analysis tool. Will show in class and will uploadinstructions on shell.


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Slide 22



Spreadsheet Showing Results Using n = 3

A B C

1 Robert's Drugs

2

3 Week (t) Salest Forect+14 1 110 #N/A

5 2 115 #N/A

6 3 125 116.7

7 4 120 120.0

8 5 125 123.3

9 6 120 121.7

10 7 130 125.0

11 8 115 121.7

12 9 110 118.313 10 130 118.3

Excel has a function that easily computes the moving average. In the above slide you can see

that excel has computed the forecasted 3-period moving average for weeks 4-14. It is helpful to

have a forecast for historical data ~ that is, we did not need to forecast week 4, because we

already knew the sales figures (Remember from the previous slide we had data through week

10). But by doing this, we can determine the accuracy of our forecast methodology. We will look

at measures of forecast accuracy later in this lecture.

Take a minute and notice how Excel outputs the data. The forecast that is horizontal with week

3 is 116.7. This is actually the forecast for week 4 so, if we wanted to see how well our model

would have predicted the forecast for week 4, we can compare the forecast values for week 4

(116.7) with the actual values for week 4 (120).


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Slide 23


Smoothing Methods

Weighted Moving Average Method

In the weighted moving average method forcomputing the average of the most recent n periods,the more recent observations are typically given moreweight than older observations. For convenience, theweights usually sum to 1.

The weighted moving average is similar to the moving average except we assign weights toeach of the time periods. Remember our goal: to smooth out variations in the data that areirregular. We might find that our sales show an increasing pattern. In this case, using a movingaverage of 3-periods will not account for the increasing pattern. Instead, we might assign a highweight to the most recent period and a lower weight to the less recent periods. Lets look atan example.


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Slide 24


Example: Augers Plumbing Service

Forecast for December (Month 10) using a three-period (n = 3) weighted moving average with weightsof .6, .3, and .1.

Month Jobs

March 353

April 387

May 342

June 374

July 396

Aug 409

Sept 399

Oct 412

Nov 408

Lets use a 3-period weighted moving average to forecast Augers plumbing service. Theweights are given above.

Slide 25


Example: Augers Plumbing Service (B)

Three-Month Weighted Moving Average

The forecast for December will be the weightedaverage of the preceding three months: September,

October, and November.F10 = .1YSep. + .3YOct. + .6YNov.

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

= 408.3

The forecast for period 10 (December) uses the previous 3 months of data (Sept, Oct, Nov).November is the most recent month so it is assigned a weight of .6. October is assigned aweight of .3 and September is assigned a weight of .1. We will multiple each of the previousmonths actual data times the weight that has been assigned to that month. The resulting value,408.3, is the forecast for December.

Although this is not difficult to do in Excel, you must manually set up the equations. Excel doesnot have a function that computes the weighted moving average.


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Slide 26


Smoothing Methods

Exponential Smoothing

Using exponential smoothing, the forecast for thenext period is equal to the forecast for the currentperiod plus a proportion () of the forecast errorin the current period.

Using exponential smoothing, the forecast iscalculated by:

[the actual value for the current period] +

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

where the smoothing constant, , is a numberbetween 0 and 1. OR

F = Y + Ft t-1 (1- )t-1

The third smoothing method that we will consider is exponential smoothing. This method is veryuseful for companies with little historical data. Exponential smoothing only requires one monthof historical data. The formula is shown above in a yellow box. The forecast for period t is equalto the value of the smoothing constant times the actual sales for the previous month plus 1minus the smoothing constant times the forecast for the previous month.

Exponential smoothing uses a smoothing constant and the accuracy of the previous monthsforecast to determine the forecast for the next period. The smoothing constant is determined by

the decision maker. It is a value between 0 and 1. The closer the value is to 1 the more theemphasis is on the previous months forecast (and more recent periods). Lets look at anexample.


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Slide 27



During the past ten weeks, sales of cases ofComfort brand headache medicine at Robert's Drugshave been as follows:

Week Sales Week Sales1 110 6 1202 115 7 1303 125 8 1154 120 9 1105 125 10 130

If Robert's uses exponential smoothing toforecast sales, which value for the smoothingconstant , = .1 or = .8, gives better forecasts?

Roberts Drugs is not convinced that his moving average forecast is good. He would like to useexponential smoothing to forecast his sales.Robert is going to use an exponential smoothing constant of .1 and .8. He plans to compare theaccuracy of each forecast and determine the best one.


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Slide 28



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

F1 = 110F2 = .1Y1 + .9F1 = .1(110) + .9(110) = 110

F3 = .1Y2 + .9F2 = .1(115) + .9(110) = 110.5

F4 = .1Y3 + .9F3 = .1(125) + .9(110.5) = 111.95

F5 = .1Y4 + .9F4 = .1(120) + .9(111.95) = 112.76

F6 = .1Y5 + .9F5 = .1(125) + .9(112.76) = 113.98

F7 = .1Y6 + .9F6 = .1(120) + .9(113.98) = 114.58

F8 = .1Y7 + .9F7 = .1(130) + .9(114.58) = 116.12

F9 = .1Y8 + .9F8 = .1(115) + .9(116.12) = 116.01

F10= .1Y9 + .9F9 = .1(110) + .9(116.01) = 115.41

The forecast for period 1 is 110. This is because the forecast for the first period is always setequal to the actual for that period (unless otherwise given in the problem). This does force theforecast for period 2 to be equal to the first months actual data, but after the first 2 periods theforecast works fine. So to forecast period 2, Robert multiplies the smoothing constant of .1 timesthe actual sales for period 1 (110). He adds to this the multiplication of .9 (1 - .1) to the forecastfor period 1. This process continues Lets focus in on period 10s forecast. The forecast forperiod 10 is .1 times the actual for period 9 (110) plus .9 times the forecast for period 9 (116.01).

If you look closely you can see that a smoothing constant close to 1 will put a lot of emphasis onthe actual value for the previous month. So the smoothing constant that you choose isdependent upon how much you want to rely on the previous months data versus the previouslyforecasted data.


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Slide 29



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

F1 = 110

F2 = .8(110) + .2(110) = 110

F3 = .8(115) + .2(110) = 114

F4 = .8(125) + .2(114) = 122.80

F5 = .8(120) + .2(122.80) = 120.56

F6 = .8(125) + .2(120.56) = 124.11

F7 = .8(120) + .2(124.11) = 120.82

F8 = .8(130) + .2(120.82) = 128.16

F9 = .8(115) + .2(128.16) = 117.63F10= .8(110) + .2(117.63) = 111.53

Using an exponential smoothing constant of .8, we can determine another forecast for RobertsDrugs. In this example, the smoothing constant is close to 1, so a lot of emphasis is beingplaced on the actual sales data for the previous month.

Exponential smoothing can be done easily in Excel. The next slides show how to set up in Exceland an excel poutput for the example.


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Slide 30


Example: Rosco Drugs (B)


A B C

1 Robert's Drugs

2

3 Week Sales

4 1 110

5 2 115

6 3 125

7 4 120

8 5 125

9 6 120

10 7 13011 8 115

12 9 110

13 10 130

Slide 31



Steps to Exponential Smoothing Using Excel

Step 1: Select the Tools pull-down menu.

Step 2: Select the Data Analysis option.

Step 3: When the Data Analysis Tools dialogappears, choose Exponential Smoothing.

Step 4: When the Exponential Smoothing dialog boxappears:


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

Enter C4 in the Output Range box.

Select OK.


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Slide 32



Spreadsheet Showing Results Using = 0.1

A B C

1 Robert's Drugs

2 = 0.1

3 Week (t) Salest Forect+1

4 1 110 #N/A

5 2 115 110.0

6 3 125 110.5

7 4 120 112.0

8 5 125 112.8

9 6 120 114.0

10 7 130 114.611 8 115 116.1

12 9 110 116.0

13 10 130 115.4

Slide 33



Repeating the Process for = 0.8

Step 4: When the Exponential Smoothing dialog boxappears:


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

Enter D4 in the Output Range box.

Select OK.


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Slide 34



Spreadsheet Results for = 0.1 and = 0.8

A B C D

1 Robert's Drugs

2 = 0.1 = 0.8

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

4 1 110 #N/A #N/A

5 2 115 110.0 110.0

6 3 125 110.5 114.0

7 4 120 112.0 122.8

8 5 125 112.8 120.6

9 6 120 114.0 124.1

10 7 130 114.6 120.8

11 8 115 116.1 128.2

12 9 110 116.0 117.613 10 130 115.4 111.5

Slide 35

35 2003 Thomson/South-Western Slide Heizer Render, 2001 by Prentice Hall, Inc.4-35

Seven Steps in Forecasting

Select the items to be forecast Determine the time horizon of the forecast

Collect Data

Plot and analyze

Select the forecasting model(s)

Make the forecast

Validate and implement results (monitoring)

Accuracy (magnitude of error) and bias

This slides puts forecasting process in perspective


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Slide 36


Measures of Forecast Accuracy

Mean Squared Error

The average of the squared forecast errors for thehistorical data is calculated. The forecasting method orparameter(s) which minimize this mean squared erroris then selected.

Mean Absolute Deviation

The mean of the absolute values of all forecasterrors is calculated, and the forecasting method orparameter(s) which minimize this measure is selected.The mean absolute deviation measure is less sensitive

to individual large forecast errors than the meansquared error measure.

Now that Robert has two different forecasts using exponential smoothing, we need to evaluatewhich forecast is more accurate. There are two approaches to forecast accuracy. If we subtractthe forecast value from the actual value we have an indication of how well the forecastperformed for each period. But lets get an overall measure of accuracy. One

method is the

mean square errors. Here we square the forecast errors for each period and then take theiraverage. The goal is to minimize your MSE. The issue with MSE is that your MSE is in squaredterms, so it is not always easy to interpret.

One other approach is the mean absolute deviation. To compute the MAD, we take the absolutevalue of the forecast errors for each period and average these values. The MAD is easier tointerpret than the MAD because the units are not squared. The MAD provides an averageamount of deviation (over or under) that is given with the forecast. Again, the goal is to minimizethe MAD.


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Slide 37



Mean Squared Error

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

[(Y2-F2)2 + (Y3-F3)

2 + (Y4-F4)2 + . . . + (Y10-F10)

2]/9

Using Roberts Drugs we can determine the MSE for both the smoothing constant of .1 and .8

Slide 38



= .1 = .8

Week Yt Ft (Yt - Ft)2 Ft (Yt - Ft)

2

1 1102 115 110.00 25.00 110.00 25.003 125 110.50 210.25 114.00 121.00

4 120 111.95 64.80 122.80 7.845 125 112.76 149.94 120.56 19.716 120 113.98 36.25 124.11 16.917 130 114.58 237.73 120.82 84.238 115 116.12 1.26 128.16 173.309 110 116.01 36.12 117.63 58.26

10 130 115.41 212.87 111.53 341.27

Sum 974.22 Sum 847.52MSE Sum/9 Sum/9108.25 94.17

The above table shows that the smoothing constant of .8 provides a lower MSE, thus it ispreferred to the smoothing constant of .1.


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Slide 39


Trend Projection

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

Least squares, also used in regression analysis,determines the unique trend line forecast whichminimizes the mean square error between the trendline forecasts and the actual observed values for thetime series.

The independent variable is the time period and thedependent variable is the actual observed value inthe time series.

Trend projection is a time series method that uses the least squares to determine a trend line.Least squares method is also used in regression. Rather than have you labor throughcomputing a trend line by hand, I prefer to use regression to demonstrate both trend projectionand causal method forecasting.

For regression, an independent variable is used to predict a dependent variable. In forecasting

the dependent variable is what we are trying to forecast (i.e. sales, jobs, etc.). The independentvariable is what we are using to predict the forecast. For trend projection the independentvariable is always time; period 1, 2, 3, etc.

For causal regression the sky is our limit with respect to independent variables. For example,we can predict season ticket sales for the Astros based on their previous seasons record. Orwe can predict our grade on final exam based on our grades on tests 1 and 2. As you can see,in casual regression, we can use any other variable to help us forecast. A great example is theinterest rate and new home sales! Lets look further.


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Slide 40


Trend Projection

Using the method of least squares, the formula for thetrend projection is: Tt = b0 + b1t.

where: Tt = trend forecast for time period t

b1 = slope of the trend line

b0 = trend line projection for time 0

b1 = ntYt - t Yt

nt 2 - (t )2

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

period t

= average of the observed values for Yt

= average time period for the n observations

0 1b Y b t=

Y

t

I have provided you the equations that are used to determine the least sum of squares trendprojection line. As you can see there are a lot of opportunities for error, so I suggest we useExcel.


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Slide 41



The number of plumbing repair jobs performed byAuger's Plumbing Service in each of the last ninemonths are listed below.

Month Jobs Month Jobs Month Jobs

March 353 June 374 September 399

April 387 July 396 October 412

May 342 August 409 November 408

Forecast the number of repair jobs Auger's will

perform in December using the least squares method.

We will use Augers Plumbing service to demonstrate trend projections. We will see how to useExcel to compute Trend projection.

Slide 42



Trend Projection (continued)

= 45/9 = 5 = 3480/9 = 386.667

ntYt - t Yt (9)(17844) - (45)(3480)b1 = = = 7.4

nt 2 - (t)2 (9)(285) - (45)2

= 386.667 - 7.4(5) = 349.667

T10 = 349.667 + (7.4)(10) = 423.667

0 1b Y b t=

Yt


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Slide 43




A B C

1 Auger's Plumbing Service

2

3 Month Calls

4 1 353

5 2 387

6 3 342

7 4 374

8 5 396

9 6 409

10 7 399

11 8 412

12 9 408

13

Slide 44

Slide shows how to set up in Excel.



Steps to Trend Projection Using Excel

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

Step 2: Select the Insert pull-down menu.

Step 3: Choose the Function option.

Step 4: When the Paste Function dialog box appears:

Choose Statistical in Function Categorybox.

Choose Forecast in the Function Name box.

Select OK.

more . . . . . . .


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Slide 45

Continue



Steps to Trend Projecting Using Excel (continued)

Step 5: When the Forecast dialog box appears:

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

Enter B4:B12 in the Known ys box.

Enter A4:A12 in the Known xs box.

Select OK.

Slide 46



Spreadsheet with Trend Projection for Month 10

A B C

1 Auger's Plumbing Service

2

3 Month Cal ls

4 1 353

5 2 387

6 3 342

7 4 374

8 5 396

9 6 409

10 7 399

11 8 412

12 9 408

13 10 423.667 Projected


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Slide 47



Forecast for December (Month 10) using a

three-period (n = 3) weighted moving average with

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

Then, compare this Month 10 weighted moving

average forecast with the Month 10 trend projection

forecast.

Slide 48



Three-Month Weighted Moving Average

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

F10 = .1YSep. + .3YOct. + .6YNov.= .1(399) + .3(412) + .6(408)

=

Trend Projection

F10 = 423.7 (from earlier slide)

408.3


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Slide 49



Conclusion

Due to the positive trend component in the time

series, the trend projection produced a forecast that is

more in tune with the trend that exists. The weighted

moving average, even with heavy (.6) placed on the

current period, produced a forecast that is lagging

behind the changing data.

Slide 50


End of Chapter 15

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