Chapter 4.07: Optimization

2, 3: MTH/MTHT

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42 WEP for physics people
43 WEP for electrical-engineering people
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QA: Consider Example 6, a store selling an item and looking
for an optimal price. Suppose a new government regulation is
imposed that costs the store $5 per item sold. Should they
pass that entire price increase on to their customers by
increasing the item price by $5? Explain.

QB: (based on a real problem I've worked on with a student;
numbers have been changed) An army base in a hot location
uses 80 MREs (prepackaged meals) per day. Getting a shipment
of MREs costs $1000 no matter how many are delivered. Storing
an MRE overnight costs $0.25. Decide how many MREs should be
in each shipment to minimize the total cost of shipping plus
storage.

QC: (from Winston, pg 648)
 
Each morning during rush hour, 10,000 people want to travel from
New Jersey to New York City.  If a person takes the subway, the
trip lasts 40 minutes.  If x thousand people per morning drive to
NYC, it takes 20+5x minutes to make the trip.  This problem illustrates
a basic fact of life: if people are left to their own devices, they will
cause more congestion than need actually occur!
a) Show that if people are left to their own devices, an average
of 4,000 people will travel by road from NJ to NYC.  Here you
should assume that people will divide up between subway and
road in a way that makes the average travel time by road
equal to the average travel time by subway.  When this
"equilibrium" occurs, nobody has an incentive to switch
from road to subway or vice versa.  This is known as the
"user optimal" scenario.
b) Show that the average travel time per person is minimized
if 2,000 people travel by road.  This is known as the "system optimal"
scenario.
c) What percent worse than system optimal is the user-optimal scenario?

QD: (WEP) Do Bees Know Calculus? Read this:
http://courses.ncssm.edu/math/POW/POW07_08/Calculus%20Problem%206%20SOLUTION.pdf