DECISION MODELING WITH
MICROSOFT EXCEL
Chapter 13
Part 2
Copyright 2001
Prentice Hall Publishers and
Ardith E. Baker
__________forecasting models produce forecasts
by extrapolating the __________behavior of the
values of a particular single variable of interest.
For example, we would forecast the ______of an
item or fluctuation of a particular market price
with________.
Figuratively, the series is being lifted into the
future “by its own______________.”
Time-series data are historical data in
_______________order, with only one value per
time period.
Extrapolating Historical Behavior
To provide examples of _____________methods,
suppose we have the daily closing prices of a
March cocoa futures contract for the past 12 days
(including today) from the Wall Street Journal .
Now, we wish to __________tomorrow's closing
price from this past stream of data. Consider the
following possibilities:
1. If it is felt that all historical values are
equally important, then the _________of
the past 12 values could be used as the
best _________for tomorrow.
2. If it is felt that today’s value (the 12th) is
the most________, then this value might
be the best prediction for tomorrow.
3. If, in the current “____________market,”
the first six values are too antiquated, but
the most recent six are equally important,
then the ________of the most recent six
values may be the best estimate for
tomorrow.
This method is called a _____________
average.
In this case, it is a simple _________
moving average since we are averaging
the previous 6 periods together to obtain
an estimate of the____________.
4. Perhaps ___past values contain useful
information, but today’s (the 12th obs.) is
the most important of all, and in
succession, the 11th, 10th, 9th, etc.,
observations have _________importance.
In this case, a _______________of all 12
observations (with increasing weights
assigned to each value in order 1 through
12 and with the 12 weights summing to
______) could be used.
This method is called ___________
smoothing.
5. We might actually plot the 12 values as a
function of _____and then draw a linear
“_________” near the values. This line
could then be used to predict tomorrow’s
value.
This is a ____________method.
Note that values for a particular (single) variable
of interest, which can be plotted against time, are
often termed a______________.
Any method used to analyze and _________such
a series into the future falls within the general
category of time-series____________.
For time-series models, we will use the error
measures of ______(mean absolute deviation)
and ______(mean absolute percent error).
CURVE FITTING
The main difference between curve fitting in
_________models and _________models is that
in time-series models, the independent variable
is________.
The historical observations of the _________
variable are plotted against_______, and a curve
is then fitted to these data.
As in causal models, this ______is then extended
into the ________to yield a forecast.
Sales
Forecast for Period t + 2
Historical data
Forecast for
Period t + 1
t
Time
In the context of time-series, extending the curve
simply means evaluating the derived _______for
larger values of t, the_______.
One of the assumptions with curve fitting is that
all the data are equally__________.
This method produces a very ______forecast that
is fairly __________to slight changes in the data.
Although the mathematical techniques for fitting
the curves are identical, the ___________behind
the two models is basically different.
To understand this difference, think of the values
of___, the variable of interest, as being produced
by a particular underlying ________or system.
The ______model assumes that as the underlying
system ________to produce different values of y,
it will also produce corresponding differences in
the ___________variables and thus, by knowing
the independent variables, a good forecast of y
can be deduced.
The ___________model assumes that the system
that produces y is essentially ____________(or
stable) and will continue to act in the future as it
has in the________.
Future patterns of y will closely resemble past
___________.
If the system that produces y significantly
changes, then the assumption of a _________
process is invalid and the forecast will be in____.
Just as for causal models, it is possible to use
___________functions (such as higher-order
polynomials in t) to _____________a series of
observations. For example:
yt = b0 + b1t + b2t2 + … + bktk
As before, appropriate values for the _________
b0 , b1 , …, bk must be mathematically derived
from the values of previous______________.
However, just as with causal models, these
perfect historical fits have little or no _________
power.
MOVING AVERAGES:
FORECASTING STECO’S STRUT SALES
The assumption behind models of this type is that
the ________performance over the recent past is
a good ________of the future.
The emphasis on recent data produces a forecast
that is much more _____________than a curve
fitting model.
This new type of model will be sensitive to
increases or decreases in_____, or other changes
in the data.
Let’s return to an ________control problem from
Chapter 7 (STECO’s stainless steel struts) and use
different forecasting models on __________data.
In particular, let’s use last year’s _________sales
data to see how well these models work. This is
called a ___________study.
The following notation will be used. Let
yt-1 = __________sales of struts in month t-1
y^ = ________of sales for struts in period t
t
Sales will be forecast only one month (_______)
ahead.
That is, the known historical values yt , …, yt-1
(demand in months 1 through t-1) will be used to
^ the forecast for demand in month t.
produce y
t
This will result in a sequence of ^
yt values.
By comparing these values with the _________yt
values, an indication of how the forecasting
model would have worked can be obtained.
Simple n-Period Moving Average This is the
________model in the moving average category.
In this model, the average of a _______number
(say, n) of the most recent observations is used
as an __________of the next value of y.
For example, if n = 4, then after the value of y in
period 15 was observed, the estimate for period
16 would be:
y15 + y14 + y13 + y12
^
y16 =
4
In general,
^
(yt + yt-1 + … + yt-n+1)
yt+1 = 1
n
Here are 3-period and 4-period moving averages
for STECO’s strut sales data.
The 3-month moving average forecast for sales in
April is the average of January, February, and
March sales:
(20 + 24 + 27)/3 = 23.67
________(i.e., after the forecast) actual sales in
April were 31. Thus, in this case, the forecasted
sales differed from _________sales by
31 – 23.67 = 7.33
We could __________compare each of the actual
sales to the corresponding forecasted sales,
however, it is more useful to have a _________
measure of how well the two methods performed.
We will use the _____________deviation (MAD)
MAD =
|actual sales – forecast sales|
S
all forecasts
number of forecasts
and the mean absolute ______________(MAPE).
|actual sales – forecast sales|
MAPE =
S
all forecasts
actual sales
number of forecasts
*100%
This spreadsheet calculates the 3-month and 4month moving averages and the associated
MAD’s
The 3-month moving average MAD is 12.67.
The 4-month moving average MAD is 15.59
The smallest MAD indicates a more accurate
forecast.
Historically, in this example, including more data
harms rather than helps the forecasting_______.
A simple moving average will always ________
rising data and ____________declining data.
Thus, if there are broad rises and falls, simple
moving averages will not______________.
They are best suited to data with ______erratic
ups and downs, providing some stability in the
face of the random_________________.
The philosophical problem with simple moving
averages is that in calculating a forecast, the
most __________observation receives no more
weight or importance than an _____observation.
This is because each of the last n observations is
assigned the weight_________.
The ____________shortcoming of simple moving
average models is that if __observations are to be
included in the moving average, then (__) pieces
of past data must be brought forward to be
combined with the _______(the nth) observation.
All this data must be _________in some way, in
order to calculate the forecast.
This may become a problem when a company
needs to forecast the demand for thousands of
individual products on an ______________basis.
Weighted n-Period Moving Average The notion
that recent data are more important than old data
can be implemented with a weighted _______
moving average.
In this more general form, taking n = 3 as a
specific example, we would set
^
y7 = a0y6 + a1y5 + a2y4
where the a’s (______) are nonnegative numbers
and the weights __________as the data become
older. Also, all the weights sum to___. For
example,
^
2 y + 1y
y7 = 3
y
+
6 6
6 5
6 4
or
^
5y + 3y + 2 y
y7 = 10
6
10 5 10 4
Now, apply a 3-month weighted moving average
with initial weights of 3/6, 2/6, 1/6 to the
historical stainless strut data.
Now, compare this new MAD with the MADs from
the _______________average models:
The 3-month weighted moving average MAD is
______and confirms the suggestion that ______
sales results are a better indicator of future sales.
However, how do we know which _________to
use that will give us the best _____and ultimately
the best forecast?
Solver can choose the optimal weights for us.
First, click on the Tools menu and choose Solver.
Specify the following
parameters in the
resulting dialog:
Click Solve to
perform the
analysis.
Here are the results of the Solver analysis:
The resulting MAD is 7.56.
Although the weighted moving average places
more weight on ______data, it does not solve the
operational problems of data_______, since (n-1)
pieces of historical sales data must still be stored.
The next weighting scheme addresses this
problem.
EXPONENTIAL SMOOTHING:
THE BASIC MODEL
___________________is a scheme that weights
________data more heavily than ____data, with
weights summing to___, but it avoids the
previously discussed operational problems.
For any t > 1, the forecast for period t+1 is a
weighted sum of the ______sales in period t and
the ________for period t.
^
yt+1 = ayt + (1-a)^
yt
a is a user-specified _________constant such
that 0 < a < 1.
If a is close to 1, then then almost all of the
weight is placed on the _____demand in period t.
Exponential smoothing has some important
computational advantages:
^ , only y^ need be stored along
To compute y
t+1
t
with the value of___.
By saving a and the last__________, all the
previous forecasts are being stored_______.
As soon as the actual yt is_________, we
compute ^
yt+1 = ayt + (1-a)^
yt
Note that when t = 1, the expression used to
define ^
y2 is
^
y2 = ay1 + (1-a)^
y1
^ is an “__________” at the
In this expression, y
1
value for y in period 1 and y1 is the __________
value in period 1.
Several options are available to obtain this “initial
guess”
^ = y (this assumes a “_______”
1. We let y
1
1
forecast, but we don’t count this error of
zero in the MAD calculation).
2. Look ahead at the _________data and let
y^1 = y (the _______of all available data).
^ = the average of just the first
3. Let y
1
couple of months.
^ = y and a = .5.
For this example, we will let y
1
1
Now, compare this exponential smoothing MAD
with the MADs from the previous models:
The exponential smoothing model with a = .5
yields a ___________MAD.
In exponential smoothing models, the forecasts
depend on the values selected for the ________
^ y .
constant a and the “____________”
1
Solver can be used to select the _________value
for a (one that ___________the MAD) by setting
up a __________optimization model.
Click on Tools – Solver to open the Solver dialog.
Specify the
Target Cell,
Minimize the
objective, and
specify the
Changing Cells,
and the
Constraints.
Click Solve to perform the analysis.
Solver chose a value of 1 for a. The resulting
MAD is 6.82.
As before, the more weight that is put on the
most ______observation, the _____the forecast.
Because of the importance of the exponential
smoothing model, we will now examine some it
its______________.
Note that if t > 2, then it is possible to ________
t-1 for t to obtain
^
yt = ayt-1 + (1-a)^
yt-1
^ back into the
Substituting this relationship for y
t
original expression for ^
yt+1 yields for t > 2 ,
^
^
yt+1 = ayt + a(1-a)yt-1 + (1-a)2y
t-1
By ________performing similar substitutions,
one is led to the following general expression for
^
yt+1:
^
yt+1 = ayt + a(1-a)yt-1 + a(1-a)2yt-2 +… + a(1-a)t-1y1 + (1-a)t^
y1
Since 0 < a < 1, it follows that 0 < 1-a < 1. Thus,
a > a(1-a) > a(1-a)2
This illustrates the general property of an
exponential smoothing model – that the
_________of the y’s decrease as the data
become_________.
It can also be seen that the _____of all of the
coefficients is 1.
Remember that in the exponential smoothing
^ was a “______” at y .
formula, the value of y
1
1
Observe now that as __increases, the influence of
^
^ decreases and in time becomes
y1 on y
t+1
_____________.
Obviously, the value of_, affects the performance
of the_________.
The _______the value of a, the more strongly the
model will react to the _____observation (this is
called a ____________forecast).
When a  0.0, this means almost complete _____
in the last forecast and almost completely
________the most recent observation. This
would be an extremely _________forecast.
Consider the following table which shows values
for the weights when a = 0.1, 0.3, and 0.5.
You can see that for ________values of a more
relative ________is assigned to the more recent
___________, and the influence of older data is
more rapidly______________.
To illustrate further the effect of choosing various
value for a, consider the following three cases:
Case 1 (Response to a Sudden_______) Suppose
that at a certain point in time the underlying
system experiences a rapid and ________change.
How does the choice of a ___________the way in
which the exponential smoothing model will
react?
Consider the following __________case in which
yt = 0 for t = 1, 2, …, 99
yt = 1 for t = 100, 101, …
yt
1
0
95
96
97
98
99
100
101
102
t
^ = 0, then ^
Note that in this case if y
y100 = 0 for
1
any value of a, since we are taking the weighted
sum of a series of________.
Thus, at time 99, our best estimate of y100 is 0,
whereas the _________value will be 1.
At time 100, the _______has changed. Thus, the
question is: How quickly will the forecasting
system _________as time passes and the
information that the system has changed
becomes___________?
^ for a = 0.1 and a = 0.5.
To answer this, plot y
t+1
^
1.2
yt+1
When a = 0.5
1.1
^
y106 = 0.984
1
0.9
a = 0.5
0.8
0.7
When a = 0.1
^
0.6
y106 = 0.468
a = 0.1
0.5
0.4
0.3
0.2
0.1
0
100
105
110
115
120
t
125
A forecast system with a = 0.5 responds more
________to changes in the data than the system
with a = 0.1.
Thus, a larger a would be ________if the system
is characterized by a low level of _______
behavior, but is subject to occasional _______
shocks.
However, suppose that the data are characterized
by ______random error but a stable_______.
In this case, if a is large, a large random _____in
^ , way off.
yt will throw the forecast value, y
t+1
Hence, for this type of process, a ________value
of a would be preferred.
Case 2 (Response to a _______Change) Suppose
now that a system experiences a steady change
in the value of____.
For example, the following graph displays
steadily increasing values of yt (a________).
yt
t
How will the _________________model respond
and will this response be affected by a?
In this case, recall that
^
yt+1 = ayt + a(1-a)yt-1 + …
Since all the previous y’s (y1, …, yt-1) are _____
than yt and since the weights sum to 1, it can be
shown that, for any a between_________,
^
yt+1 < yt
Also, since yt+1 > yt , we see that
^
yt+1 < yt < yt+1
Thus, the forecast will _______be too small.
Finally, since smaller values of a put more weight
on _______data, the smaller the value of a, the
_______the forecast becomes.
Even with a very close to 1, the forecast is not
very good if the ________is steep.
Exponential smoothing (or any ________moving
average), without an appropriate___________, is
not a good forecasting tool in a rapidly growing
market or a ___________market.
The model can be adjusted to include the______.
This is called ________model or exponential
smoothing with trend.
Case 3 (Response to a _______Change) Suppose
that a system experiences a regular seasonal
pattern in___.
How then will the exponential smoothing model
respond, and how will this response be affected
by the choice of a?
Consider the following seasonal pattern:
Demand
7
8
9 10 11
t
Suppose it is desired to ______________several
periods forward.
For example, suppose we wish to forecast
demand in periods 8 through 11 based only on
data through period 7. Then
^
^
y8 = ay7 + (1-a)y7
Now to obtain ^
y9, since we have data only
through period 7, we assume that y8=^
y8. Then
^
^
y9 = ay8 + (1-a)y
8
= a^
y8 + (1-a)^
y8
^
=y
8
Now, we know that
^
yt+1 = ayt + a(1-a)yt-1 + (1-a)2yt-2 + …
Suppose that a ______value of a is chosen.
The coefficients
for the most
recent terms
change relatively
slowly. Thus ^
yt+1
will resemble a
______________
average.
In this case, the future predictions will all be
somewhere near the ___________of the past
observations.
Even if a ______value of a is chosen, the forecast
will be close to the ______observations.
The forecast thus essentially ___________the
seasonal pattern.
The exponential smoothing model is intended for
situations in which the _________of the variable
of interest is essentially______, in the sense that
deviations over time have nothing to do with
______, per se, but are caused by __________
that do not follow a regular pattern.
This is referred to as the __________assumption
HOLT’S MODEL
(EXPONENTIAL SMOOTHING WITH TREND)
Simple ___________________models don’t
perform well on models that have obvious up or
down trend in the data (and no_____________).
To correct this, ________developed the following
model:
^
yt+k = Lt + kTt
where
Lt = ayt + (1-a)(Lt-1 + Tt-1)
Tt = b(Lt - Lt-1) + (1-b)Tt-1
Holt’s model allows us to forecast up to ___time
periods ahead.
In this model, we now have two _________
parameters, a and b, both of which must be
between________.
The Lt term indicates the _________level or base
value for the time-series data.
The Tt term indicates the _________increase or
decrease per period (i.e., the________).
Consider the following data of quarterly earnings
of Startup Airlines.
There is obviously an ___________trend.
Now, apply Holt’s trend model to the data in
order to generate the forecast of _________per
share (EPS) for the next (13th) quarter.
First, choose the ____________for both L and T.
We could choose to
1. Let L1 = actual ____for quarter 1 and
T1 = 0.
2. Let L1 = actual EPS for all 12 quarters
and T1 = ______trend for all 12 quarters,
and many other variations in between.
Let’s choose the first option for this example and
initial ________of .5 for both a and b.
Also, let’s use the mean absolute percent error
(_______) to determine forecast__________.
Notice that the MAPE is 43.3% which is fairly
high.
Try putting in a b of 0 (as if there were no trend)
to see if we can gain anything by this new model.
The MAPE is now 78.1% which is much worse.
Let’s use Solver to help find the optimal values
for a and b. Open the Solver dialog by clicking on
Tools – Solver.
Specify the
Target Cell,
Minimize the
objective, and
specify the
Changing Cells,
and the
Constraints.
Click Solve to perform the analysis.
Solver found the optimal values for a and b.
The MAPE has been reduced to 38%.
SEASONALITY
_______comprises movements up and down in a
pattern of _______length that repeats itself.
For example, monthly sales of ice cream would
expectedly be higher in the warmer months (say
June to August in the Northern Hemisphere) than
in the winter months, year after year.
This seasonal pattern would be 12 months long.
Optionally, we could use weekly data with the
seasonal pattern repeating every 52 periods.
The number of _____________in a season
pattern depends on how often the ___________
are collected.
The approach for treating such seasonal patterns
consists of four steps:
1. Look at the _______data that exhibit a
seasonal pattern and _____________an
m-period seasonal pattern.
2. Using the numerical approach,
________________the data.
3. Using the best forecasting method
available, make a _________in
deseasonalized terms.
4. ___________the forecast to account for
the seasonal pattern.
This concept will be illustrated with data on U.S.
coal receipts by the commercial/residential
sectors over a nine-year period (measured in
thousands of tons). The data are graphed below:
3,000
2,500
Coal (000 Tons)
2,000
1,500
1,000
500
0
1- 1- 1- 1- 2- 2- 2- 2- 3- 3- 3- 3- 4- 4- 4- 4- 5- 5- 5- 5- 6- 6- 6- 6- 7- 7- 7- 7- 8- 8- 8- 8- 9- 9- 9- 91 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Tim e (Year and Quarter)
Notice that the data ______in the 1st and 4th
quarters and _____in the 2nd and 3rd quarters.
The Gillette Coal Mine would like to forecast
________in the upcoming two quarters.
Deasonalizing The procedure to deseasonalize
data is simply to ________out all variations that
occur within one_______.
Thus, for quarterly data, an average of four
periods is used to eliminate ___________
seasonality.
In order to deseasonalize a whole time series, the
first step is to _________a series of m-period
moving averages, where __is the ________of the
seasonal pattern.
Note that two new columns have been added to
the Excel spreadsheet:
If m is_______, as here, then the task is more
complicated, using an addition step to get the
moving averages___________.
Here is a graph of the original data along with
their centered moving averages.
Coal (000 Tons)
3000
2500
Actual data
2000
1500
Cntrd Moving
Average
1000
500
11
21
31
41
51
61
71
81
91
0
Time (Year & Qtr)
Notice that coal receipts are above the centered
moving average in the 1st and 4th quarters and
below average in the 2nd and 3rd quarters.
The averaging process _________the quarter-toquarter movement.
The third step is to _______the actual data at a
given point in the series by the ________moving
average corresponding to the same point.
This calculation cannot be done for all possible
________, since at the beginning and end of the
series, we are unable to compute a _______
moving average.
These _______represent the degree to which a
particular observation is below or above the
__________level.
These ratios form the basis for developing a
seasonal_________.
To develop the seasonal index, first group the
ratios according to____________.
Then _______all of the ratios to moving averages
quarter by quarter.
This is a _______________for each quarter and
represents what that particular season’s data
look line on __________compared to the average
of the entire_________.
A seasonal index >__ means that season is higher
than the ________for the year; likewise an index
< 1 means that season is _____than the average
for the year.
The last step of deseasonalization is to take the
actual data and divide it by the appropriate
seasonal index.
The deseasonalized data are graphed below:
Deseasonalized
3,000.0
Coal (000 Tons)
2,500.0
2,000.0
1,500.0
Deseasonalized
1,000.0
500.0
1- 1- 1- 1- 2- 2- 2- 2- 3- 3- 3- 3- 4- 4- 4- 4- 5- 5- 5- 5- 6- 6- 6- 6- 7- 7- 7- 7- 8- 8- 8- 8- 9- 9- 9- 91 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Tim e (Year & Qtr)
Notice that the data seem to “jump around” a lot
less than the original data.
Forecasting Now a deseasonalized forecast can
be made based on an appropriate ___________
that accounts for the pattern in the
_____________data (e.g., if there is trend in the
data, use a _______________model).
In this example, let’s forecast coal receipts for
the 1st quarter of the 10th year by simple
exponential smoothing.
Using this method, it turns out that the _______
smoothing constant a*= .653, which yields an
______of 47,920.
The deseasonalized forecast would be 1726
thousand tons for the 1st quarter of next year.
This would also be the deseasonalized forecast
amount for the 2nd quarter given the current data.
Reseasonalizing The last step in the process is to
___________the forecast of 1726.
To do this, _________1726 by the seasonal index
for the 1st quarter:
1726(1.108) = 1912
A seasonalized forecast for the 2nd quarter would
be:
1726(.784) = 1353
THE RANDOM WALK
Recently, more ___________time-series analysis
methods (based on developments by G.E.P. Box
and G.M. Jenkins) have become available.
These time-series forecasting techniques are
based on the ___________that the true values of
the variable of interest, yt , are generated by a
__________(i.e., probabilistic) model.
A very important and simple process, called a
___________will be used to illustrate a
stochastic model.
Here, the variable yt is assumed to be produced
by the_____________
yt = yt-1 + e
The value of e is determined by a_____________.
Now, consider a man standing at a street corner
on a north-south street. He flips a fair______.
If it lands with a _____showing (H), he walks one
block_____. If it lands with a tail showing (T), he
walks one block_____. When he arrives at the
next corner, he _________the process.
This is the _________example of a random walk.
To put this example in the form of the model,
label the original corner ______(y1).
Starting at this point, label ___________corners
going north +1, +2, … and successive corners
going ______–1, –2, … (these labels describe the
location of the random walker and are the yt’s).
In the model, yt = yt-1 + e, where (assuming a fair
coin) e = 1 with ____________½ and e = -1 with
probability ½.
If the walker observes the sequence H, H, H, T, T,
H, T, T, T, he will follow the ______shown below.
Forecasts Based on Conditional Expected Value
Now, continuing with the random walk, we have
ten _____________(starting with corner 0 and 9
coin tosses) and would like to _______where the
walker will be after another move.
This is the typical forecasting problem in the
_____________context.
We have observed y1, y2, … y10, and we need a
good forecast ^
y11 of the forthcoming value y11.
^ is the
In this case, the best value for y
11
_______________value of the random quantity
y11.
In other words, the best forecast is the _______
value of y11 given that we know y1, y2, …, y10.
From the model we know that
y11 = (y10 + 1) with probability ½
and
y11 = (y10 - 1) with probability ½
Thus, the __________expected value of y11 given
y1, y2, …, y10, is calculated as follows
E(y11|y1, …, y10) = (y10 + 1)½ + (y10 - 1)½ = y10
Thus, we see that for this model, the data
y1, …, y9 are_______, and the best forecast of the
random walker’s position one move from now is
his _________position.
The best forecast for any future value of y1, given
this particular model, is its_____________.
Seeing What Isn’t There There is a great deal of
________that supports the idea that stock prices
and foreign _________exchange rates behave
like a random walk and that the best _______of a
future stock price or of an exchange rate is its
current value.
However, this conclusion is not warmly ________
by research directors and technical chartist who
make their living forecasting stock prices or
exchange rates.
Once reason for ___________to the random walk
hypothesis is the almost universal human
tendency when looking at a set of data to observe
certain patterns or_________, no matter how the
data are produced.
Consider the time-series data plotted below.
It appears
that the
data follow
a _______
pattern.
However,
the data
were in fact
generated
by the
random walk model.
Any attempt to _________future values by
____________the sinusoidal pattern would have
no more ___________than flipping a coin.
End of Part 2
Please continue to Part 3