decision analysis

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EMGT 501 HW Solutions Chapter 14 - SELF TEST
3 Chapter 14 - SELF TEST 14
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14-3 a.
Let x1 number of units of product 1
produced x2 number of units of product 2
produced
, , , , , , ,
³ 0
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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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c.
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.
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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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14-14 a.
b.
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Home Work 15-9 and 15-35 Due Day Dec 2,
2007 No Class on Nov 25, 2007
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Chapter 15Forecasting

• 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

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Quantitative 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.

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Time Series Methods

• Three time series methods are
• smoothing
• trend projection
• trend projection adjusted for seasonal influence

13
Components of a Time Series

• 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.

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Components of a Time 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.

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Measures of Forecast Accuracy

• 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.

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Smoothing Methods

• 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

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Smoothing Methods

• 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.

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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.
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Several typical time series patterns
Constant level
Seasonal effect
Linear trend
20
Example
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,
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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
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.
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(2) Averaging Forecasting Method This method uses
all the data points in the time series and simply
averages these points.
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.
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
.
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(4) Exponential Smoothing Forecasting Method
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.
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Because of this recursive relationship between
and , alternatively can be
expressed as
Another alternative form for the exponential
smoothing technique is given by
27
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.
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Example
Three-Year Average
Seasonal Factor
Quarter
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Seasonally Adjusted Volume
Actual Volume
Seasonal Factor
Quarter
Year
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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
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.
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Adapting Exponential Smoothing to this Model 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
.
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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
34
Then is calculated as
where is the trend smoothing constant which
must be between 0 and 1.
35
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
36
The resulting forecasts for the first two periods
are
37
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.
38
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,
39
Given the forecasting errors for n time periods
(t 1, 2, , n), two popular measures of
performance are available. Mean Absolute
Deviation (MAD)
Mean Square Error (MSE)
40
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.
43
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.
44
In some cases, there exists 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.
45
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
46
Method of Least Squares The usual method for
identifying the best fitted line is the method
of least squares.
Regression Line
47
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.
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This method chooses that line a bx that makes Q
a minimum.
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and
where
and
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Re-write (1)
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Re-write (2)
From (1)
(1) in (2)
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Example Rosco Drugs
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?
55
Example Rosco Drugs

• Past Sales
• Week Sales Week
  Sales
• 1 110
  6 120
• 2 115
  7 130
• 3 125
  8 115
• 4 120
  9 110
• 5 125
  10 130

56
Example Rosco Drugs

• Excel Spreadsheet Showing Input Data

57
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 dialog
  appears, choose Moving Average.
• Step 4 When the Moving Average dialog box
  appears
• Enter B4B13 in the Input Range box.
• Enter 3 in the Interval box.
• Enter C4 in the Output Range box.
• Select OK.

58
Example Rosco Drugs

• Spreadsheet Showing Results Using n 3

59
Smoothing Methods

• 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.

60
Smoothing Methods

• 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.

61
Smoothing Methods

• 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.

62
Trend Projection

• 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.

63
Trend Projection

• 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

64
Example Rosco Drugs (B)

• If Rosco Drugs uses exponential
• smoothing to forecast sales, which value for the
• smoothing constant ?, .1 or .8, gives better
  forecasts?
• Week Sales Week Sales
• 1 110
  6 120
• 2 115
  7 130
• 3 125
  8 115
• 4 120
  9 110
• 5 125
  10 130

65
Example Rosco Drugs (B)

• 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, Ft1 .1Yt .9Ft

66
Example Rosco Drugs (B)

• 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

67
Example Rosco Drugs (B)

• 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

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Example Rosco Drugs (B)

• 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

69
Example Rosco Drugs (B)

• ? .1 ? .8
• Week Yt Ft (Yt -
  Ft)2 Ft (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
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Example Rosco Drugs (B)

• Excel Spreadsheet Showing Input Data

71
Example Rosco Drugs (B)

• 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 B4B13 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.

72
Example Rosco Drugs (B)

• Spreadsheet Showing Results Using a 0.1

73
Example Rosco Drugs (B)

• Repeating the Process for a 0.8
• Step 4 When the Exponential Smoothing dialog
  box appears
• Enter B4B13 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.

74
Example Rosco Drugs (B)

• Spreadsheet Results for a 0.1 and a 0.8

75
Example Augers Plumbing Service

• 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.

76
Example Augers Plumbing Service
Month Jobs Month Jobs
Month Jobs March 353
June 374 September 399
April 387 July 396
October 412 May 342
August 409 November 408
77
Example Augers Plumbing Service

• Trend Projection
• (month) t Yt
  tYt t 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

78
Example Augers Plumbing Service

• 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
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Example Augers Plumbing Service

• Excel Spreadsheet Showing Input Data

80
Example Augers Plumbing Service

• 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 . . . . . . .

81
Example Augers Plumbing Service

• 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 B4B12 in the Known ys box.
• Enter A4A12 in the Known xs box.
• Select OK.

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Example Augers Plumbing Service

• Spreadsheet Showing Trend Projection for Month 10

83
Example Augers Plumbing Service (B)

• 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.

84
Example Augers Plumbing Service (B)

• 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
85
Example Augers Plumbing Service (B)

• 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.

86
Forecasting with Trendand 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.

87
Example Terrys Tie Shop

• 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.

88
Example Terrys Tie Shop

• Past Sales ()
• Year
• Season 1 2
  3 4
• 1 1856 1995
  2241 2280
• 2 2012 2168
  2306 2408
• 3 985 1072
  1105 1120

89
Example Terrys Tie Shop

• Dollar Moving
  Scaled
• Year Season Sales (Yt) Average StIt
  St Yt/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

90
Example Terrys Tie Shop

• 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.

91
Example Terrys Tie Shop

• 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.

92
Example Terrys Tie Shop

• 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 )

93
Example Terrys Tie Shop

• 4. Determine the average seasonal factors.
• Averaging all St It values 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

94
Example Terrys Tie Shop

• 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

95
Example Terrys Tie Shop

• 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

96
Example Terrys Tie Shop

• 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

97
Example Terrys Tie Shop

• 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
98
Qualitative Approaches to Forecasting

• 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.

99
Qualitative Approaches to Forecasting

• 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.

100
Qualitative Approaches to Forecasting

• 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.