Re: Solver problem

[Sergei Steshenko]

--- On Thu, 8/30/12, Damien Bancal <address@hidden> wrote:

From: Damien Bancal <address@hidden>
Subject: Solver problem
To: "address@hidden" <address@hidden>
Date: Thursday, August 30, 2012, 7:36 AM

Hello
 I’m trying to use the solver to solve a problem at work.
My problem : I have product with 3 years lifespan. I know how much have been 
sold each year to a list of clients and how much they have now and I’m trying 
to find the yearly average survival rate of my product (percentage of the 
products
 sold in year XXXX which are still working). Once translated in an equation 
with 3 clients (named A to C).
X the number of item sold (XA12 : number of item sold in 2012 to client A) => 
known value.
S the survival rate (S12 the survival rate for the item sold in 2012, S11 for 
whose sold in 2011 and so on) =>
 unknow value.
P the population (PA Product population client A, PB for client B and so on) => 
known values.
XA12*S12+XA11*S11+XA10*S10=PAXB12*S12+XB11*S11+XB10*S10=PBXC12*S12+XC11*S11+XC10*S10=PC
I want to put limit to the potential value of them. For example, I know that 
the survival rate for the item sold in 2012 is limited like
 this : 0,9 < S12 < 1. Is it possible to create this kind of limits ? (and if 
the answer is yes, how ?)
Thanks for your help
Regards
Arkhos94 




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"I know that the survival rate for the item sold in 2012 is limited like
 this : 0,9 < S12 < 1. Is it possible to create this kind of limits ? (and if 
the answer is yes, how ?)"

I don't know whether it is possible to do it inside the solver of your choice, 
but _mathematically_ it is _always_ possible.

I mean the following. Suppose you want to find a fit of F(p, x) function, i.e. 
to find the 'p' set of parameters doing the best approximation of Y(x) measured 
data - this is what solvers do.

Suppose the solver does not limit 'x'.

The idea is then to rewrite F in the form:

modified_F(p, x) = F(p, x_limiter(x))

and tell the solver to work on modified_F(p, x). The answer of interest will 
then be x_limiter(x) and not plain 'x'.

The practical question is how to choose x_limiter.

Here are a couple of examples:

1) x_limiter(x) = (x ^ 2) / (1 + x ^ 2) # the limits are 0 .. 1
2) x_limiter(x) = abs(x) / (1 + abs(x))
2) x_limiter(x) = atan(x) # the limits are -pi/2 .. pi/2, probably slower than 
the two above
.

It is trivial to have any desired x_min .. _x_max limits through linear 
transformation (x_limiter -> k * x_limiter + b) of x_limiter(x) output - I hope 
you'll be able to figure this out yourself.

Regards,
  Sergei.