EMGT 501
HW Solutions
Chapter 14 - SELF TEST 3
Chapter 14 - SELF TEST 14
© 2005 Thomson/South-Western
Slide 1
14-3
a.
Let x1 = number of units of product 1 produced
x2 = number of units of product 2 produced
Min
P1( d 1 ) + P1( d 1 ) +
P1( d 2 ) +
P1( d 2 ) + P2( d 3 )
s.t.
1x1 +
2x1 +
1x2 5x2 -
d 1
+
d 1
=
350 Goal 1
d 2
+
d 2
=
1000 Goal 2
4x1 +
2x2 -
d 3
+
d 3
=
1300 Goal 3
x1, x,2 d,1 d, 1 d, 2 d, 2 d, 3 d3 0
© 2005 Thomson/South-Western
Slide 2
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.
© 2005 Thomson/South-Western
Slide 3
x2
700
600
500
al 3
Go
400
300
200
Go
al
1
Go a
l2
B (281.25, 87.5)
100
A (250, 100)
© 2005 Thomson/South-Western
0
100
200
300
400
500
x1
Slide 4
c.
Max
s.t.
4x1
+
2x2
1x1
2x1
+
+
x1,
1x2
5x2
x2



350
1000
0
Dept. A
Dept. B
The graphical solution indicates that there are four
extreme points. The profit corresponding to each
extreme point is as follows:
Extreme Point
1
2
3
4
Profit
4(0) + 2(0) = 0
4(350) + 2(0) = 1400
4(250) + 2(100) = 1200
4(0) + 2(200) = 400
Thus, the optimal product mix is x1 = 350 and x2 = 0
a profit of $1400.
© with
2005 Thomson/South-Western
Slide 5
x2
400
rtm
pa
De
300
tA
en
4 (0,250)
De p
artm
en
200
tB
100
3 (250,100)
Feasible Region
(0,0) 1
0
100
200
© 2005 Thomson/South-Western
300
2
x1
400
(350,0)
500
Slide 6
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.
© 2005
Thomson/South-Western
Slide 7
14-14
a.
A
Criteria
Cost
Overnight Capability
Kitchen/Bath Facilities
Appearance
Engine/Speed
Towing/Handling
M aintenance
Resale Value
B
S core
b.
A
Criteria
Cost
Overnight Capability
Kitchen/Bath Facilities
Appearance
Engine/Speed
Towing/Handling
M aintenance
Resale Value
B
S core © 2005 Thomson/South-Western
C
D
E
194
181
144
C
D
E
220 Bowrider
40
6
2
35
30
32
28
21
220 Bowrider
21
5
5
20
8
16
6
10
91
230 Overnighter 240 S undancer
25
15
18
27
8
14
35
30
40
20
20
8
20
12
15
18
230 Overnighter 240 S undancer
18
15
30
40
15
35
28
28
10
6
12
4
5
4
12
12
130
144
Slide 8
Home Work
15-9 and 15-35
Due Day: Dec 2, 2007
No Class on Nov 25, 2007
© 2005 Thomson/South-Western
Slide 9
Chapter 15
Forecasting








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
© 2005 Thomson/South-Western
Slide 10
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.
© 2005 Thomson/South-Western
Slide 11
Time Series Methods

Three time series methods are:
•smoothing
•trend projection
•trend projection adjusted for seasonal
influence
© 2005 Thomson/South-Western
Slide 12
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.

© 2005 Thomson/South-Western
Slide 13
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.

© 2005 Thomson/South-Western
Slide 14
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.
© 2005 Thomson/South-Western
Slide 15
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
© 2005 Thomson/South-Western
Slide 16
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.
© 2005 Thomson/South-Western
Slide 17
Time Series
A time series is a series of observations over time
of some quantity of interest (a random variable).
Thus, if X i is the random variable of interest at
time i, and if observations are taken at times
i = 1, 2, …., t, then the observed values
X1  x1 , X 2  x2 ,, X t  xt  are a time series.
© 2005 Thomson/South-Western
Slide 18
Several typical time series patterns:
Constant level
Linear trend
© 2005 Thomson/South-Western
Seasonal effect
Slide 19
Example
Constant level
X i  A  ei ,
for i  1, 2,,
X i : the random variable observed at time I
A : the constant level of the model
ei : the random error occurring at time i.
Ft 1  forecast of the values of the time series at
time t + 1, given the observed values,
X

x
,
X

x
,

,
X

x
1
1
2
2
t
t
© 2005 Thomson/South-Western
Slide 20
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
© 2005 Thomson/South-Western
Slide 21
(1) Last-Value Forecasting Method
Ft 1  xt .
By interpreting t as the current time, the lastvalue forecasting procedure uses the value of the
time series observed at time t ( xt ) , 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.
© 2005 Thomson/South-Western
Slide 22
(2) Averaging Forecasting Method
This method uses all the data points in the
time series and simply averages these
points.
t
xt
Ft 1   .
i 1 t
This estimate is an excellent one if the
process is entirely stable.
© 2005 Thomson/South-Western
Slide 23
(3) Moving-Average Forecasting Method
This method averages the data for only the last n
periods as the forecast for the next period.
t
xi
Ft 1  
.
i  t  n 1 n
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 X t n1 as on X t .
© 2005 Thomson/South-Western
Slide 24
(4) Exponential Smoothing Forecasting Method
Ft 1  xt  (1   ) Ft ,
Where  (0    1) is called the smoothing
constant.
Thus, the forecast is just a weighted sum of the last
observation xt and the preceding forecast Ft
for the period just ended.
© 2005 Thomson/South-Western
Slide 25
Because of this recursive relationship
between Ft 1 and Ft , alternatively Ft 1can
be expressed as
Ft 1  xt   (1   ) xt 1   (1   ) 2 xt  2  .
Another alternative form for the exponential
smoothing technique is given by
Ft 1  Ft   ( xt  Ft ),
© 2005 Thomson/South-Western
Slide 26
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 for the period
Seasonal factor 
.
overall average
© 2005 Thomson/South-Western
Slide 27
Example
Quarter
Three-Year
Average
1
7,019
2
6,784
3
7,434
4
8,880
Total  30,117, Average
© 2005 Thomson/South-Western
Seasonal
Factor
7,019
 0.93
7,529
6,784
 0.90
7,529
7,434
 0.99
7,529
8,880
 1.18
7,529
30,117

 7,529
Slide 28
4
Seasonally
Actual Seasonal Adjusted
Year Quarter Volume Factor
Volume
1
1
6809
0.93
7322
1
1
2
3
6465
6569
0.90
0.99
7183
6635
1
4
8266
1.18
7005
2
2
1
2
7257
0.93
7803
2
3
7064
7784
0.90
0.99
7849
7863
7393
1.18
actual value
Seasonally adjusted value 
. Slide 29
© 2005 Thomson/South-Western
seasonal factor
2
4
8724
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.
© 2005 Thomson/South-Western
Slide 30
The model is represented by
X i  A  Bi  ei ,
for i  1, 2,,
Where X i is the random variable that is observed
at time i, A is a constraint.
B is the trend factor, and ei is the random error
occurring at time i.
© 2005 Thomson/South-Western
Slide 31
Adapting Exponential Smoothing to this Model
Let
Tt 1  Exponential smoothing estimate of the
trend factor B at time t + 1, given the
observed values, X1  x1 , X 2  x2 ,
, X t  xt .
Given Tt 1 , the forecast of the value of the time
series at time t + 1( Ft 1 ) is obtained simply by
adding Tt 1 to the formula for Ft 1 .
Ft 1  xt  (1   ) Ft  Ft 1.
© 2005 Thomson/South-Western
Slide 32
The most recent observations are the most
reliable ones for estimating the current
parameters.
Lt 1  latest trend at time t + 1 based on the last
two values ( xt and xt 1 ) and the last two
forecasts ( Ft and Ft 1 ).
The exponential smoothing formula used for Lt 1 is
Lt 1   ( xt  xt 1 )  (1   )( Ft  Ft 1 ).
© 2005 Thomson/South-Western
Slide 33
Then Tt 1 is calculated as
Tt 1  Lt 1  (1   )Tt ,
where  is the trend smoothing constant
which must be between 0 and 1.
© 2005 Thomson/South-Western
Slide 34
Getting started with this forecasting method
requires making two initial estimates.
x0  initial estimate of the expected value of
the time series
T1  initial estimate of the trend of the time
series
© 2005 Thomson/South-Western
Slide 35
The resulting forecasts for the first two periods are
F1  x0  T1 ,
L2   ( x1  x0 )  (1   )( F1  x0 ),
T2  L2  (1   )T1 ,
F2  x1  (1   ) F1  T2 .
© 2005 Thomson/South-Western
Slide 36
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.
© 2005 Thomson/South-Western
Slide 37
The forecasting error for any period t is the
absolute value of the deviation of the forecast
for period t ( Ft ) from what then turns out to be
the observed value of the time series for period
t ( xt ) .
Thus, letting Et denote this error,
Et | xt  Ft | .
© 2005 Thomson/South-Western
Slide 38
Given the forecasting errors for n time periods
(t =1, 2, …, n), two popular measures of
performance are available.
Mean Absolute Deviation (MAD)
n
MAD 
E
t
t 1
.
n
Mean Square Error (MSE)
n
MSE 
© 2005 Thomson/South-Western
E
t 1
n
2
t
.
Slide 39
n
n
MAD 
E
t 1
n
t
.
MSE 
E
t 1
2
t
n
.
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.
© 2005 Thomson/South-Western
Slide 40
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.
© 2005 Thomson/South-Western
Slide 41
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.
© 2005 Thomson/South-Western
Slide 42
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.
E[Y | X  x]  A  Bx.
This relationship between X and Y is referred to
as a degree of association model.
© 2005 Thomson/South-Western
Slide 43
In some cases, there exists a functional
relationship between two variables that may be
linked linearly.
The previous example is
X i  A  Bi  ei ,
It follows that
E[ X t ] A  Bt .
Both the degree of association model and the
exact functional relationship model lead to the
same
linear relationship.
© 2005 Thomson/South-Western
Slide 44
E[ X t ] A  Bt .
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
Yt  A  Bt  et .
© 2005 Thomson/South-Western
Slide 45
Method of Least Squares
The usual method for identifying the “best” fitted
line is the method of least squares.
Regression Line
© 2005 Thomson/South-Western
Slide 46
Suppose that an arbitrary line, given by the
y  a  bx , is drawn through the data.
expression ~
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.
2
2
2
~
~
~
Q  ( y1  y1 )  ( y2  y2 )    ( yn  yn )
n
2
~
  ( yi  yi ) .
i 1
© 2005 Thomson/South-Western
Slide 47
This method chooses that line a + bx that
makes Q a minimum.
n
b
 ( x  x )( y
i
i 1
i
 y)
n
 (x  x)
i 1
2
i
n
 n

xi yi    xi  yi  n

i 1
i 1
i 1



2
n
n


2
xi    xi  n

© 2005 Thomson/South-Western
i 1
 i 1 
n
Slide 48
and
a  y  bx ,
where
n
xi
x
i 1 n
and
n
y
i 1
© 2005 Thomson/South-Western
yi
.
n
Slide 49
n
n
t 1
t 1
L    t2   ( yt  a  bxt ) 2
n
L

a
 
t 1
2
t
a
n
 2 ( yt  a  bxt )  0
(1)
t 1
n
L

b
   t2
t 1
b
n
 2 ( yt  a  bxt )xt  0 (2)
t 1
© 2005 Thomson/South-Western
Slide 50
Re-write (1)
n
(y
t 1
t
n
n
n
t 1
t 1
t 1
 a  bxt )   yt   a   bxt  0
n
n
t 1
t 1
  yt  na  b xt  0
n
n
t 1
t 1
 na   yt  b xt
n
a
y
t 1
n
t
n
y

a  y  bx
© 2005 Thomson/South-Western
b
x
t 1
t
n
x
(1)'
Slide 51
Re-write (2)
n
(y
t 1

 a  bxt )xt  0
t
n
n
 y x   ax   bx
t t
t 1

n
n
y x
t t
t 1
t
t 1
n
n
t 1
t 1
y
t 1
n
y x
t 1
t t

y
t 1
© 2005 Thomson/South-Western
n
2
n
t
b
n
n
0
 a  xt  b xt  0
a  y  bx 
(1)’ in (2)
t
t 1
n
From (1)’
2
x
t 1
t
n
n
t
b
x
t 1
n
t
n
x
t 1
t
n
 b xt  0
t 1
2
Slide 52
2





  yt   xt 
  xt 
n
n
2
t 1
t 1
t 1






  yt xt 
b
 b xt  0
n
n
t 1
t 1
n
n
n



  yt   xt 
n
  yt xt   t 1  t 1   b
n
t 1
n

© 2005
b
n


  xt 
n
2
 t 1 
x


t
n
t 1
 n  n 
  yt   xt 
n
 t 1  t 1 
y
x


t t
n
t 1
 n 
  xt 
n
2
 t 1 
x


t
n
t 1
Thomson/South-Western
2
n
2
(2)'
Slide 53
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?
© 2005 Thomson/South-Western
Slide 54
Example: Rosco Drugs

Past Sales
Week
1
2
3
4
5
Sales
110
115
125
120
125
© 2005 Thomson/South-Western
Week
6
7
8
9
10
Sales
120
130
115
110
130
Slide 55
Example: Rosco Drugs

Excel Spreadsheet Showing Input Data
1
2
3
4
5
6
7
8
9
10
11
12
13
A
B
Robert's Drugs
Week (t )
1
2
3
4
5
6
7
8
9
10
© 2005 Thomson/South-Western
Salest
110
115
125
120
125
120
130
115
110
130
C
Forect+1
Slide 56
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 B4:B13 in the Input Range box.
Enter 3 in the Interval box.
Enter C4 in the Output Range box.
Select OK.
© 2005 Thomson/South-Western
Slide 57
Example: Rosco Drugs

Spreadsheet Showing Results Using n = 3
1
2
3
4
5
6
7
8
9
10
11
12
13
A
B
Robert's Drugs
Week (t )
1
2
3
4
5
6
7
8
9
10
© 2005 Thomson/South-Western
Salest
110
115
125
120
125
120
130
115
110
130
C
Forect+1
#N/A
#N/A
116.7
120.0
123.3
121.7
125.0
121.7
118.3
118.3
Slide 58
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.
© 2005 Thomson/South-Western
Slide 59
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.
© 2005 Thomson/South-Western
Slide 60
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.
© 2005 Thomson/South-Western
Slide 61
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.
© 2005 Thomson/South-Western
Slide 62
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
b0  Y  b1 t
nt 2 - (t )2
where: Yt = observed value of the time series at time
period t
Y = average of the observed values for Yt
t = average time period for the n observations
© 2005 Thomson/South-Western
Slide 63
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
1
2
3
4
5
Sales
110
115
125
120
125
© 2005 Thomson/South-Western
Week
6
7
8
9
10
Sales
120
130
115
110
130
Slide 64
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, Ft+1 = .1Yt + .9Ft
© 2005 Thomson/South-Western
Slide 65
Example: Rosco Drugs (B)

Exponential Smoothing ( = .1, 1 -  = .9)
F1
F2 = .1Y1 + .9F1 = .1(110) + .9(110)
F3 = .1Y2 + .9F2 = .1(115) + .9(110)
F4 = .1Y3 + .9F3 = .1(125) + .9(110.5)
F5 = .1Y4 + .9F4 = .1(120) + .9(111.95)
F6 = .1Y5 + .9F5 = .1(125) + .9(112.76)
F7 = .1Y6 + .9F6 = .1(120) + .9(113.98)
F8 = .1Y7 + .9F7 = .1(130) + .9(114.58)
F9 = .1Y8 + .9F8 = .1(115) + .9(116.12)
F10= .1Y9 + .9F9 = .1(110) + .9(116.01)
© 2005 Thomson/South-Western
= 110
= 110
= 110.5
= 111.95
= 112.76
= 113.98
= 114.58
= 116.12
= 116.01
= 115.41
Slide 66
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
© 2005 Thomson/South-Western
Slide 67
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
© 2005 Thomson/South-Western
Slide 68
Example: Rosco Drugs (B)
 = .1
Ft
(Yt - Ft)2
 = .8
Ft
(Yt - Ft)2
Week
Yt
1
2
3
4
5
6
7
8
9
10
110
115
125
120
125
120
130
115
110
130
110.00
25.00
110.50 210.25
111.95
64.80
112.76 149.94
113.98 36.25
114.58 237.73
116.12
1.26
116.01
36.12
115.41 212.87
110.00
25.00
114.00 121.00
122.80
7.84
120.56
19.71
124.11 16.91
120.82
84.23
128.16 173.30
117.63
58.26
111.53 341.27
MSE
Sum
974.22
Sum/9 108.25
Sum
847.52
Sum/9 94.17
© 2005 Thomson/South-Western
Slide 69
Example: Rosco Drugs (B)

Excel Spreadsheet Showing Input Data
1
2
3
4
5
6
7
8
9
10
11
12
13
A
B
Robert's Drugs
Week
1
2
3
4
5
6
7
8
9
10
© 2005 Thomson/South-Western
C
Sales
110
115
125
120
125
120
130
115
110
130
Slide 70
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 B4:B13 in the Input Range box.
Enter 0.9 (for  = 0.1) in Damping Factor box.
Enter C4 in the Output Range box.
Select OK.
© 2005 Thomson/South-Western
Slide 71
Example: Rosco Drugs (B)

Spreadsheet Showing Results Using  = 0.1
1
2
A
B
Robert's Drugs
C
 = 0.1
3
Week (t )
Sales t
Forec t +1
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
110
115
125
120
125
120
130
115
110
130
#N/A
110.0
110.5
112.0
112.8
114.0
114.6
116.1
116.0
115.4
© 2005 Thomson/South-Western
Slide 72
Example: Rosco Drugs (B)

Repeating the Process for  = 0.8
• Step 4: When the Exponential Smoothing dialog box
appears:
Enter B4:B13 in the Input Range box.
Enter 0.2 (for  = 0.8) in Damping Factor box.
Enter D4 in the Output Range box.
Select OK.
© 2005 Thomson/South-Western
Slide 73
Example: Rosco Drugs (B)

Spreadsheet Results for  = 0.1 and  = 0.8
1
2
A
B
Robert's Drugs
C
D
 = 0.1
 = 0.8
3
Week (t )
Sales t
Forec t +1
Forec t +1
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
110
115
125
120
125
120
130
115
110
130
#N/A
110.0
110.5
112.0
112.8
114.0
114.6
116.1
116.0
115.4
#N/A
110.0
114.0
122.8
120.6
124.1
120.8
128.2
117.6
111.5
© 2005 Thomson/South-Western
Slide 74
Example: Auger’s 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.
© 2005 Thomson/South-Western
Slide 75
Example: Auger’s Plumbing Service
Month Jobs
March 353
April
387
May
342
Month
June
July
August
© 2005 Thomson/South-Western
Jobs
374
396
409
Month
Jobs
September 399
October
412
November 408
Slide 76
Example: Auger’s Plumbing Service

Trend Projection
(month) t
(Mar.) 1
(Apr.) 2
(May) 3
(June) 4
(July) 5
(Aug.) 6
(Sep.) 7
(Oct.) 8
(Nov.) 9
Sum 45
Yt
tYt
t2
353
353
1
387
774
4
342 1026
9
374 1496 16
396 1980
25
409 2454
36
399 2793 49
412 3296 64
408 3672
81
3480 17844 285
© 2005 Thomson/South-Western
Slide 77
Example: Auger’s Plumbing Service

Trend Projection (continued)
t = 45/9 = 5
b1 =
ntYt - t Yt
nt 2 - (t)2
Y = 3480/9 = 386.667
=
(9)(17844) - (45)(3480)
(9)(285) - (45)2
= 7.4
b0  Y  b1 t = 386.667 - 7.4(5) = 349.667
T10 = 349.667 + (7.4)(10) = 423.667
© 2005 Thomson/South-Western
Slide 78
Example: Auger’s Plumbing Service

Excel Spreadsheet Showing Input Data
1
2
3
4
5
6
7
8
9
10
11
12
13
A
B
C
Auger's Plumbing Service
Month
1
2
3
4
5
6
7
8
9
© 2005 Thomson/South-Western
Calls
353
387
342
374
396
409
399
412
408
Slide 79
Example: Auger’s 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 . . . . . . .
© 2005 Thomson/South-Western
Slide 80
Example: Auger’s 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 B4:B12 in the Known y’s box.
Enter A4:A12 in the Known x’s box.
Select OK.
© 2005 Thomson/South-Western
Slide 81
Example: Auger’s Plumbing Service

Spreadsheet Showing Trend Projection for Month 10
1
2
3
4
5
6
7
8
9
10
11
12
13
A
B
C
Auger's Plumbing Service
Month
1
2
3
4
5
6
7
8
9
10
© 2005 Thomson/South-Western
Calls
353
387
342
374
396
409
399
412
408
423.667
Projected
Slide 82
Example: Auger’s 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.
© 2005 Thomson/South-Western
Slide 83
Example: Auger’s 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)
= 408.3

Trend Projection
F10 = 423.7 (from earlier slide)
© 2005 Thomson/South-Western
Slide 84
Example: Auger’s 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.
© 2005 Thomson/South-Western
Slide 85
Forecasting with Trend
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.
© 2005 Thomson/South-Western
Slide 86
Example: Terry’s 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.
© 2005 Thomson/South-Western
Slide 87
Example: Terry’s 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
© 2005 Thomson/South-Western
Slide 88
Example: Terry’s 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
© 2005 Thomson/South-Western
Slide 89
Example: Terry’s 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.
© 2005 Thomson/South-Western
Slide 90
Example: Terry’s 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.
© 2005 Thomson/South-Western
Slide 91
Example: Terry’s 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 )
© 2005 Thomson/South-Western
Slide 92
Example: Terry’s 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
© 2005 Thomson/South-Western
Slide 93
Example: Terry’s 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
© 2005 Thomson/South-Western
Slide 94
Example: Terry’s 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
© 2005 Thomson/South-Western
Slide 95
Example: Terry’s 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
© 2005 Thomson/South-Western
Slide 96
Example: Terry’s 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) =
© 2005 Thomson/South-Western
2382
2541
1225
Slide 97
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.
© 2005 Thomson/South-Western
Slide 98
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.
© 2005 Thomson/South-Western
Slide 99
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.
© 2005 Thomson/South-Western
Slide 100