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Science Forum Index » Statistics - Math Forum » Optimal Allocation Under Failure...
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 Posted: Mon Jun 09, 2008 3:55 am |
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Guest
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Let i=1... m to denote m different fridges. For each fridge i you know
Pe_i, probability of event "The egg x in this fridge is corrupted",
and Pc_i, probability of event "The fridge is broken". When a fridge
is broken, all eggs inside get corrupted.
In order to cook a simple dish, you need at least K eggs not
corrupted. If you have initially N eggs, which is the optimal
allocation mix (x_1, x_2, .. x_m) of eggs in the fridges that maximize
the probability of having at least K eggs not corrupted? (Under Sum_j
x_j = N condition)
Thank you for reading this strange request... |
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| Richard Ulrich... |
| Posted: Tue Jun 10, 2008 8:50 pm |
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On Mon, 9 Jun 2008 06:55:19 -0700 (PDT), ameba.spugnosa at (no spam) gmail.com
wrote:
Quote: Let i=1... m to denote m different fridges. For each fridge i you know
Pe_i, probability of event "The egg x in this fridge is corrupted",
and Pc_i, probability of event "The fridge is broken". When a fridge
is broken, all eggs inside get corrupted.
In order to cook a simple dish, you need at least K eggs not
corrupted. If you have initially N eggs, which is the optimal
allocation mix (x_1, x_2, .. x_m) of eggs in the fridges that maximize
the probability of having at least K eggs not corrupted? (Under Sum_j
x_j = N condition)
Thank you for reading this strange request...
It is strange, and it is also hard to grasp as concrete.
I translated it into "people in bomb shelters" before
I came up with --
- If you only need 1 survivor, put an equal number
in each shelter (or refrigerator).
- If you need almost *all* as surviving, put them all
in the same shelter.
You are not asking for the mean, but for something
about the thickness of the tail... for a problem with
integers. I think you would need a tabulation to compare
the results of different specifications. It might be a
linear-programming problem, with integers.
--
Rich Ulrich
http://www.pitt.edu/~wpilib/index.html |
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| Aniko... |
| Posted: Wed Jun 11, 2008 3:22 am |
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On Jun 9, 8:55 am, ameba.spugn... at (no spam) gmail.com wrote:
Quote: Let i=1... m to denote m different fridges. For each fridge i you know
Pe_i, probability of event "The egg x in this fridge is corrupted",
and Pc_i, probability of event "The fridge is broken". When a fridge
is broken, all eggs inside get corrupted.
In order to cook a simple dish, you need at least K eggs not
corrupted. If you have initially N eggs, which is the optimal
allocation mix (x_1, x_2, .. x_m) of eggs in the fridges that maximize
the probability of having at least K eggs not corrupted? (Under Sum_j
x_j = N condition)
Thank you for reading this strange request...
I have not been able to find an explicit solution, but here is a
possible approach.
Yij = 1 if egg j in fridge i is spoiled, 0 if not spoiled
P(Yij=1)=P(i-th fridge breaks) + P(i-th fridge does not break)*P(egg
spoils) = Pc_i + (1-Pc_i)*Pe_i=p_i
Yij ~ Bernoulli(p_i)
Yi=sum_j Yij - number of spoiled eggs in fridge i ~ Binomial(x_i,p_i)
~~ N(x_ip*_i, x_i*p_i*(1-p_i))
Y=sum_i Yi - total number of spoiled eggs ~~ N(sum_i x_i*p_i, sum
x_i*p_i*(1-p_i))
The normal approximations work if x_i are not too small, and p_i is
not too close to 0 or 1. Otherwise you have a convolution
of the binomials which can be difficult to handle even numerically
(depending on m). Even if the normal approximation does not work
though, the mean and variance of the resulting distribution are
correct, so the following calculations will certainly be reasonable
approximations.
P(at least K eggs not spoiled) = P(Y<N-K) = Phi((sum_i x_i*p_i-K)/
sqrt(sum x_i*p_i*(1-p_i)).
To maximize this, you need to maximize the argument of Phi under the
constraint of sum_i x_i=N, x_i>=0 [that's why the exact nature of Phi
is not that important]. As Rich Ulrich mentions, that's an integer
linear programming problem. Ignoring the integer constraints, you can
use the Lagrange multiplier method to get a straightforward
optimization problem. I calculated the derivatives, but could not see
an explicit solution - it probably has to be done numerically.
Hope this helps,
Aniko |
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| Posted: Wed Jun 11, 2008 10:23 am |
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Guest
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On Jun 11, 3:22 pm, Aniko <aniko123... at (no spam) yahoo.com> wrote:
Quote: Hope this helps,
Aniko
Thank you, yours replies are really helpful! |
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