Transcript 15.060 Data, Models and Decisions

15.Math-Review
Review 1
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Algebra
 Example: After “careful” study our marketing team has
estimated that the demand for knobs is related to the
price as: q = 400 -10p . And, considering all the
different producers of knobs the supply is estimated as:
q = 150 + 15p .
 Find the market’s equilibrium to estimate what should be
the market price of knobs and the volume of sales.
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Algebra
 Example: After a “more careful” study our marketing
team has refined the estimates for the demand and supply
to the following non-linear relations:
demand: q = e 9.1 p -0.10
supply: q = e 2.3 p 1.5.
 Find the market’s equilibrium to estimate what should be
the market price of knobs and the volume of sales.
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Differentiation
 To differentiate is a trade….
f(x) log 5 (( x 1)( x 1))
f(x)  3x 2 
f(x) 
1
2
e
2x

1
x
 e 2 x


f ( x)   x use f ' ( x) to get an expression for  ix i
i
i 0
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i 1
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Stationary Points
 Example:
Consider the function defined over all x>0, f(x) = x - ln(x).
Find any local or global minimum or maximum points. What
type are they?
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Optimization
 Example: Due to the interaction of supply and
demand, we are able to affect p the price of door
knobs with the quantity q of door knobs produced
according to the following linear model:
p = 100 - 0.1q
 Consider now variable operative costs = 20q
 Maximize profit, with the consideration that the
production level has to be at least 450 units due to
contracts with clients.
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LP
 Example: Write the constraints associated with the
solution space shown:
5
3
-1
1
2
5
-1
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LP
 Example: Graphically solve the following LP. Repeat
replacing x = 5 by x  5.
Max 5 x  2 y
s.t. x  y  10
x5
x, y  0
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LP
 Example: Our company now produces two types of
knobs. We can produce at most 300 knobs. The market
limits daily sales of the first and second types to 150 and
200 knobs. Assume that the profit per knob is $8 for type
1 and $5 for type 2. Try to maximize your profit.
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LP
 Example: Our company can advertise our knobs by using local
radio and TV stations. Our budget limits the advertisement
expenditures to $1000 a month. Each minute of radio advertisement
costs $5 and each minute of TV advertisement costs $100. Our
company would like to use the radio at least twice as much as the TV.
Past experience show that each minute of TV advertisement will
usually generate 25 times as many sales as each minute of radio
advertisement. Determine the optimum allocation of the monthly
budget to radio and TV advertisements in order to maximize the
estimated generation of sales.
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Equality Constrained
Optimization
 Example: Suppose we have the following model to
explain q, the quantity of knobs produced: q=L0.3K0.9,
where:
L: Labor, and has a cost of $1 per unit of labor.
K: Capital, and has a cost of $2 per unit of capital.
 Interpret the model. Is it reasonable? (not the units,
please)
 Find the mix of labor and capital that will produce
q=100 at minimum cost.
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Probability
 Example: Suppose that of 100 MBA students in the firstyear class, 20 of them have two years of work experience,
30 have three years, 15 have four years, and 35 have five
years or more. Suppose that we select one of these 100
students at random.
What is the probability that this student has at least four years of
work experience?
Suppose that you are told that this student has at least three years
of work experience. What is the (conditional) probability that
this student has at least four years of work experience?
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Probability
 Example: It is a relatively rare event that a new
television show becomes a long-term success. A new
television show that is introduced during the regular
season has a 10% chance of becoming a success. A new
television show that is introduced as a mid-season
replacement has only a 5% chance of becoming a
success. Approximately 60% of all new television shows
are introduced during the regular season. What is the
probability that a randomly selected new television show
will become a success.
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Tough examples to kill time
 Application of derivative: L’Hopital rule.
If lim g ( x)  , and lim h( x)  
x
x
then we have that
g ( x)
g '( x)
lim f ( x)  lim
 lim
x
x h( x)
x h'( x)
 Use this rule to find a limit for f(x)=g(x)/h(x):
If g ( x)  ln( x), and h( x)  x.0001
If g ( x)  4 x 2 -1, and h( x)  e x
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Tough examples to kill time
 Example:

( x 2 ) 2
2
Let us consider the function
f ( x)  e
Obtain a sketch of this function using all the information
about stationary points you can obtain.
Sketch the function
f ( x)  ( x 2 3)e x
Hint: for this we will need to know that the ex ‘beats’ any
polynomial for very large and very small x.
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