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# EMGT 501 HW Solutions Chapter 15 - SELF TEST 3 Chapter 15 - SELF TEST 14 - PowerPoint PPT Presentation

EMGT 501 HW Solutions Chapter 15 - SELF TEST 3 Chapter 15 - SELF TEST 14. 15-3 a. Let x 1 = number of units of product 1 produced x 2 = number of units of product 2 produced. , , , , , , , ³ 0. b.

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HW Solutions

Chapter 15 - SELF TEST 3

Chapter 15 - SELF TEST 14

a.

Let x1 = number of units of product 1 produced

x2 = number of units of product 2 produced

, , , , , , , ³ 0

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.

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

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

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.

a.

b.

16-9 and 16-35

Due Day: Nov 28, 2005

No Class on Nov 22, 2005

Chapter 16Forecasting

• Quantitative Approaches to Forecasting

• The Components of a Time Series

• Measures of Forecast Accuracy

• Using Smoothing Methods in Forecasting

• Using Trend Projection in Forecasting

• Using Trend and Seasonal Components in Forecasting

• Using Regression Analysis in Forecasting

• Qualitative Approaches to Forecasting

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

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

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

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

• Three time series methods are:

• smoothing

• trend projection

• trend projection adjusted for seasonal influence

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

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

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

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

• Mean Squared Error

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

• Mean Absolute Deviation

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

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

• Four common smoothing methods are:

• Moving averages

• Centered moving averages

• Weighted moving averages

• Exponential smoothing

• Moving Average Method

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

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.

Constant level

Seasonal effect

Linear trend

Constant level

: the random variable observed at time I

: the constant level of the model

: the random error occurring at time i.

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

for a Constant-Level Model

(1) Last-Value Forecasting Method

(2) Averaging Forecasting Method

(3) Moving-Average Forecasting Method

(4) Exponential Smoothing Forecasting Method

By interpreting t as thecurrent 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 thenaive method, because statisticians consider it naïve to use just asample size of onewhen additional relevant data are available.

This method uses all the data points in the time series and simplyaveragesthese points.

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

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

The moving-averageestimator combines the advantages of thelast value and averagingestimators.

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

Where is called thesmoothing constant.

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

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

Another alternative form for the exponential smoothing technique is given by

Seasonal Factor , alternatively can be expressed as

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

Example , alternatively can be expressed as

Three-Year

Average

Seasonal

Factor

Quarter

Seasonally Adjusted Volume , alternatively can be expressed as

Actual

Volume

Seasonal

Factor

Year

Quarter

An Exponential Smoothing Method for a Linear Trend Model , alternatively can be expressed as

Linear trend

Suppose that the generating process of the observed time series can be represented by a linear trend superimposed with random fluctuations.

The model is represented by , alternatively can be expressed as

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.

Adapting Exponential Smoothing to this Model , alternatively can be expressed as

Let

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

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

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

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

The exponential smoothing formula used for is

Then is calculated as estimating the current parameters.

where is thetrend smoothing constantwhich must be between 0 and 1.

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

initial estimate of theexpected valueof the time series

initial estimate of thetrendof the time series

Forecasting Errors two initial estimates.

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.

The two initial estimates.forecasting errorfor 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,

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

Mean Square Error (MSE)

(a) its ease of calculation

(b) its straightforward interpretation

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

forecasting error while almost ignoring

inconsequentially small forecasting errors.

Causal Forecasting with Linear Regression two initial estimates.

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.

Linear Regression two initial estimates.

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.

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.

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

In some cases, there exists a variable B is denoted by Y, then the random variables X and Y exhibit a functional relationship between two variables that may be linked linearly.

The previous example is

It follows that

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

With variable B is denoted by Y, then the random variables X and Y exhibit a 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

Method of Least Squares variable B is denoted by Y, then the random variables X and Y exhibit a

The usual method for identifying the “best” fitted line is the method of least squares.

Regression Line

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 computingthe sum of squaresof the vertical deviations of the actual points from the fitting line.

This method chooses that line , is drawn through the data.a + bx that makes Q a minimum.

and , is drawn through the data.

where

and

Re-write (1) , is drawn through the data.

Re-write (2) , is drawn through the data.

From (1)’

(1)’ in (2)

Example: Rosco Drugs , is drawn through the data.

Sales of Comfort brand headache medicine for

the past ten weeks at Rosco Drugs

are shown on the next slide. If

Rosco Drugs uses a 3-period

moving average to forecast sales,

what is the forecast for Week 11?

Example: Rosco Drugs , is drawn through the data.

• Past Sales

WeekSalesWeekSales

1 110 6 120

2 115 7 130

3 125 8 115

4 120 9 110

5 125 10 130

Example: Rosco Drugs , is drawn through the data.

• Excel Spreadsheet Showing Input Data

Example: Rosco Drugs , is drawn through the data.

• 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 dialog appears, choose Moving Average.

Step 4: When the Moving Average dialog box appears:

Enter B4:B13 in the Input Range box.

Enter 3 in the Interval box.

Enter C4 in the Output Range box.

Select OK.

Example: Rosco Drugs , is drawn through the data.

• Spreadsheet Showing Results Using n = 3

Smoothing Methods , is drawn through the data.

• Centered Moving Average Method

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

Smoothing Methods , is drawn through the data.

• Weighted Moving Average Method

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

Smoothing Methods , is drawn through the data.

• Exponential Smoothing

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

• Using exponential smoothing, the forecast is calculated by:

[the actual value for the current period] +

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

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

Trend Projection , is drawn through the data.

• If a time series exhibits a linear trend, the method of least 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 which minimizes the mean square error between the trend line forecasts and the actual observed values for the time series.

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

Trend Projection , is drawn through the data.

• Using the method of least squares, the formula for the trend 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 = ntYt - t Yt

nt 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

Example: Rosco Drugs (B) , is drawn through the data.

If Rosco Drugs uses exponential

smoothing to forecast sales, which value for the

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

WeekSalesWeekSales

1 110 6 120

2 115 7 130

3 125 8 115

4 120 9 110

5 125 10 130

Example: Rosco Drugs (B) , is drawn through the data.

• Exponential Smoothing

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

Let: Yt = actual sales in week t

Ft = forecasted sales in week t

F1 = Y1 = 110

For other weeks, Ft+1 = .1Yt + .9Ft

Example: Rosco Drugs (B) , is drawn through the data.

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

F1 = 110

F2 = .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

Example: Rosco Drugs (B) , is drawn through the data.

• 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) = 117.63

F10= .8(110) + .2(117.63) = 111.53

Example: Rosco Drugs (B) , is drawn through the data.

• Mean Squared Error

In order to determine which smoothing constant gives the better performance, calculate, for each, the mean 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

Example: Rosco Drugs (B) , is drawn through the data.

 = .1  = .8

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

1 110

2 115 110.00 25.00 110.00 25.00

3 125 110.50 210.25 114.00 121.00

4 120 111.95 64.80 122.80 7.84

5 125 112.76 149.94 120.56 19.71

6 120 113.98 36.25 124.11 16.91

7 130 114.58 237.73 120.82 84.23

8 115 116.12 1.26 128.16 173.30

9 110 116.01 36.12 117.63 58.26

10 130 115.41 212.87 111.53 341.27

Sum 974.22 Sum 847.52

MSE Sum/9 Sum/9

108.25

94.17

Example: Rosco Drugs (B) , is drawn through the data.

• Excel Spreadsheet Showing Input Data

Example: Rosco Drugs (B) , is drawn through the data.

• 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 dialog appears, choose Exponential Smoothing.

Step 4: When the Exponential Smoothing dialog box appears:

Enter B4:B13 in the Input Range box.

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

Enter C4 in the Output Range box.

Select OK.

Example: Rosco Drugs (B) , is drawn through the data.

• Spreadsheet Showing Results Using a = 0.1

Example: Rosco Drugs (B) , is drawn through the data.

• Repeating the Process for a = 0.8

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

Enter B4:B13 in the Input Range box.

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

Enter D4 in the Output Range box.

Select OK.

Example: Rosco Drugs (B) , is drawn through the data.

• Spreadsheet Results for a = 0.1 and a = 0.8

Example: Auger’s Plumbing Service , is drawn through the data.

The number of plumbing repair jobs performed by

Auger's Plumbing Service in each of the last nine

months is listed on the next slide. Forecast

the number of repair jobs Auger's will

perform in December using the least

squares method.

Example: Auger’s Plumbing Service , is drawn through the data.

MonthJobsMonthJobsMonthJobs

March 353 June 374 September 399

April 387 July 396 October 412

May 342 August 409 November 408

Example: Auger’s Plumbing Service , is drawn through the data.

• Trend Projection

(month) tYttYtt 2

(Mar.) 1 353 353 1

(Apr.) 2 387 774 4

(May) 3 342 1026 9

(June) 4 374 1496 16

(July) 5 396 1980 25

(Aug.) 6 409 2454 36

(Sep.) 7 399 2793 49

(Oct.) 8 412 3296 64

(Nov.) 9 408 3672 81

Sum 45 3480 17844 285

Example: Auger’s Plumbing Service , is drawn through the data.

• Trend Projection (continued)

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

ntYt - t Yt (9)(17844) - (45)(3480)

b1 = = = 7.4

nt 2 - (t)2 (9)(285) - (45)2

= 386.667 - 7.4(5) = 349.667

T10 = 349.667 + (7.4)(10) =

423.667

Example: Auger’s Plumbing Service , is drawn through the data.

• Excel Spreadsheet Showing Input Data

Example: Auger’s Plumbing Service , is drawn through the data.

• 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 Category box.

Choose Forecast in the Function Name box.

Select OK.

more . . . . . . .

Example: Auger’s Plumbing Service , is drawn through the data.

• 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 y’s box.

Enter A4:A12 in the Known x’s box.

Select OK.

Example: Auger’s Plumbing Service , is drawn through the data.

• Spreadsheet Showing Trend Projection for Month 10

Example: Auger’s Plumbing Service (B) , is drawn through the data.

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.

Example: Auger’s Plumbing Service (B) , is drawn through the data.

• Three-Month Weighted Moving Average

The forecast for December will be the weighted average 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

Example: Auger’s Plumbing Service (B) , is drawn through the data.

• 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.

Forecasting with Trend , is drawn through the data.and Seasonal Components

• Steps of Multiplicative Time Series Model

1. Calculate the centered moving averages (CMAs).

2. Center the CMAs on integer-valued periods.

3. Determine the seasonal and irregular factors (StIt ).

4. Determine the average seasonal factors.

5. Scale the seasonal factors (St ).

6. Determine the deseasonalized data.

7. Determine a trend line of the deseasonalized data.

8. Determine the deseasonalized predictions.

9. Take into account the seasonality.

Example: Terry’s Tie Shop , is drawn through the data.

Business at Terry's Tie Shop can be viewed as

falling into three distinct seasons:

(1) Christmas (November-December);

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

and (3) all other times. Average weekly

sales (\$) during each of the three seasons

during the past four years are shown on

the next slide.

Determine a forecast for the average weekly sales

in year 5 for each of the three seasons.

Example: Terry’s Tie Shop , is drawn through the data.

• Past Sales (\$)

Year

Season1234

1 1856 1995 2241 2280

2 2012 2168 2306 2408

3 985 1072 1105 1120

Example: Terry’s Tie Shop , is drawn through the data.

Dollar Moving Scaled

Year Season Sales (Yt) Average StItStYt/St

1 1 1856 1.178 1576

2 2012 1617.67 1.244 1.236 1628

3 985 1664.00 .592 .586 1681

2 1 1995 1716.00 1.163 1.178 1694

2 2168 1745.00 1.242 1.236 1754

3 1072 1827.00 .587 .586 1829

3 1 2241 1873.00 1.196 1.178 1902

2 2306 1884.00 1.224 1.236 1866

3 1105 1897.00 .582 .586 1886

4 1 2280 1931.00 1.181 1.178 1935

2 2408 1936.00 1.244 1.236 1948

3 1120 .586 1911

Example: Terry’s Tie Shop , is drawn through the data.

• 1. Calculate the centered moving averages.

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

1st MA = (1856 + 2012 + 985)/3 = 1617.67

2nd MA = (2012 + 985 + 1995)/3 = 1664.00

etc.

Example: Terry’s Tie Shop , is drawn through the data.

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

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

Example: Terry’s Tie Shop , is drawn through the data.

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

St It =Yt /(Moving Average for period t )

Example: Terry’s Tie Shop , is drawn through the data.

• 4. Determine the average seasonal factors.

Averaging all St Itvalues corresponding to that season:

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

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

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

Example: Terry’s Tie Shop , is drawn through the data.

• 5. Scale the seasonal factors (St ).

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

Season 1: 1.180/1.002 = 1.178

Season 2: 1.238/1.002 = 1.236

Season 3: .587/1.002 = .586

Total = 3.000

Example: Terry’s Tie Shop , is drawn through the data.

• 6. Determine the deseasonalized data.

Divide the data point values, Yt , by St .

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

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

Tt = 1580.11 + 33.96t

Example: Terry’s Tie Shop , is drawn through the data.

• 8. Determine the deseasonalized predictions.

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

T13 = 1580.11 + (33.96)(13) = 2022

T14 = 1580.11 + (33.96)(14) = 2056

T15 = 1580.11 + (33.96)(15) = 2090

Example: Terry’s Tie Shop , is drawn through the data.

• 9. Take into account the seasonality.

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

Season 1: (1.178)(2022) =

Season 2: (1.236)(2056) =

Season 3: ( .586)(2090) =

2382

2541

1225

Qualitative Approaches to Forecasting , is drawn through the data.

• Delphi Approach

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

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

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

Qualitative Approaches to Forecasting , is drawn through the data.

• Scenario Writing

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

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

Qualitative Approaches to Forecasting , is drawn through the data.

• Subjective or Interactive Approaches

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

• They often involve "brainstorming sessions".

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

End of Chapter 16 , is drawn through the data.