Homework #6: 

#1: (this one is quite hard; start it early!)
	Fli-Bi-Nite airlines has a flight that starts in LA, flies to Houston,
	then NYC, then Miami.  The length of each segment (in miles) is
	1500, 1700, and 1300 respectively.
	At each stop, the plane may purchase up to 10,000 gallons of fuel.
	The prices in each city (in cents/gal) are: 88, 15, 105, and 95.
	The plane's tank can hold at most 12,000 gallons.  To allow for
	the possibility of circling while waiting to land, we require that
	the ending fuel level for each leg of the flight be 600 gallons.
	We'll suppose that the airplane gets 0.3 miles per gallon.
	The plane starts in LA with 4933.3333 gallons before buying fuel 
	and taking off.
	
	Formulate and solve an LP that can be used to minimize the fuel cost
	incurred in completing the schedule.

	Notes:
	* I STRONGLY RECOMMEND that you formulate it with pencil and paper
	first, then turn it into a matrix form before typing it in to
	Excel.

	* It's possible to do it with only 4 decision variables, but it
	goes a lot easier if you use a bunch of bookkeeping variables as
	well.  
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#2: Curve fitting
Consider this data on the growth of a batch of yeast:
Time(hours)	Biomass
0	9.6
1	18.3
2	29.0
3	47.2
4	71.1
5	119.1
6	174.6
7	247.3

a) Using excel's usual trendline feature, fit an Exponential model to it.
b) Compute the residuals using that exponential model, and the sum of
squared residuals.
c) Write down (using math notation) a nonlinear program that will choose
the coefficients in the epxonential model to minimize the sum of
squared residuals.
c) Set up a nonlinear program to choose the coefficients in the
exponential model, and optimize it using Solver.  Report your optimal
coefficients and the sum of squared residuals.
d) Compare the two results (those of parts a&b versus part c).  Use
graphs and sentences as appropriate.  If there is a difference, explain why.


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#3: The ShotSpotter problem.

Shotspotter has set up open-air microphones
around a city like Oakland, CA.  Each microphone records the time that
a gunshot sound arrived at the microphone and reports it to a central
computer, which then figures out where the gunshot must have been.
Suppose you get data from sensors as follows: the (x,y) for each sensor
and the time of arrival of the gunshot sound were:
x	y	arr.time
0	0	10.260536
0	0.5	10.259994
0	1	10.259598
0.5	0	10.260313
0.5	0.5	10.259691

x and y are measured in miles, and arr.time in clock time hours
(i.e. it was around 10:15 pm, since 10.25 is 10 hrs and one quarter-hour).

The speed of sound in the air that night was 770 mph.
a) Formulate an NLP that will tell you where the shot was fired.
	This is not straightforward!  It takes some creativity.
b) Solve it in Excel or whatever.
c) Does it make a difference if you assume a different speed of sound?
Try something ridiculous like 100 mph and see if the estimated location changes.

d) Project idea (optional, of course): reformulate to include wind.
e) Project idea (optional, of course): suppose each sensor is actually a
cluster of 4 microphones; these can then determine the direction the sound
came from.  So if you have 2 or 3 of these that hear the shot, you can
use the direction from each to triangulate.
f) Project idea (optional, of course): try to draw contours of uncertainty.
That is, report not just the optimal estimated position, but some measure
of how certain you are.  This might involve perturbing the arrival time
data a little bit, re-solving, and repeating this 1000 times to see how much
the optimal location changes.