\magnification=\magstep1
\centerline{MONT 101N -- Analyzing Environmental Data}
\centerline{Problem Set 1 -- {\bf due}: Friday, February 2}
\bigskip
\noindent
A.   On page 33 of {\it Thinking in Systems}, Meadows
gives the {\it rule of thumb} that ``the time it takes 
for an exponentially growing stock to double in size, the 
``doubling time,'' equals approximately 70 divided by the 
growth rate (expressed
as a percentage)'' (this means to use the growth rate
per unit time).
\item{(1)} Do Exercise 7 in Chapter 5 of our
text {\it Elementary Mathematical Modeling and Data 
Analysis with Environmental Applications}.  
\item{(2)} Then check the accuracy of the {\it rule
of thumb} above if $Q(t)$ is growing at $5\%$ per
unit time and $10\%$ per unit time.
\item{(3)}  Why does the {\it rule of thumb} work
as well as it does?
\bigskip
\noindent
B.  Download and examine the spreadsheet {\tt Figure32.xlsx}
from the course homepage.  This gives a difference equation
model duplicating the behavior described in the text leading
to the oscillations shown Figure 32 in {\it Thinking in Systems}.  
\item{$\bullet$} Column C gives the inventory on hand
\item{$\bullet$} Column D gives the actual sales, with the 
$10\%$ increase from 20 cars to 22 cars starting in Day 6.
\item{$\bullet$} Column G gives the ``perceived sales'' -- 
the sales, {\it averaged over the 5 most recent days}.  
\item{$\bullet$} Column I gives the Desired Inventory (always
$10 \cdot $Perceived Sales)
\item{$\bullet$}  Column J (the ``Discrepancy'') 
is the difference between the Desired Inventory and the Inventory
on the lot from Column C.  
\item{$\bullet$}  Column E gives the orders to the factory, 
computed by adding the perceived sales and $1/3$ of the Discrepancy.
(The numerator of $3$ represents the response delay described
in the text -- each day the manager orders an additional number
of cars from the factory representing $1/3$ of the Discrepancy).  
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\noindent
Using this information, and modifying the spreadsheet as needed,
Answer these questions:
\medskip
\item{(1)}  Why is the Discrepancy value negative on some days?
What does that mean in real-world terms?
\item{(2)}  Describe how the inventory on hand changes (relative
to the original plot) if the perception
delay is increased to 10 days instead of 5.  (You'll need to start the 
averaging in day 11 rather than day 6).  
\item{(3)}  Describe how the inventory on hand changes (relative
to the original plot) if the delivery
delay is reduced to 3 days instead of 5 days.  (Note: You'll
see where the delivery delay is included if you look 
at the entries in Column F, starting at Row 10.)
\item{(4)}  Describe how the inventory on hand changes (relative
to the original plot) if the 
response delay is increased from 3 days to 5 days.  (Note: the
response delay is the denominator of 3 in formulas in Column
E, starting in row 10).  Roughly speaking this means that
the manager is going to be slower in making up discrepancies
between the desired inventory and the actual inventory.
\item{(5)} Describe a strategy for reducing the severity
of the oscillations in the inventory on the lot (making
the amplitude smaller, and/or the period longer).  Note:
The oscillations cannot be removed entirely if these delays
are taken into account(!)
\bye
 
