Combinatorial (?) assignment problem

[Shrisha Rao]

Greetz.

I am looking at a problem, probably a simple enough one, for which I don't know
a solution. Here's how it goes. Any help appreciated.

Let s_i(a) and r_i(a) be two groups of functions, that each take one of two
values, 0 or \sigma_i in the case of s_i, and 0 or \rho_i in the case of r_i,
for any a. Let us further say that there is an \alpha_i that is the sum of
these two, i.e., \sigma_i(a) = s_i(a) + r_i(a), for all a. By the constraints
of the model I am developing, the situation s_i(a) = \sigma_i and r_i(a) = 0
never occurs, so \alpha_i(a) always has one of three values, 0, \rho_i, or
\sigma_i + \rho_i.

Each of the \alpha_i needs to be scaled by some factor c_i before being
aggregated, i.e., define a \beta(a) = \sum^i c_i\alpha_i(a).

Now, the problem is to make an assignment of values of \sigma_i, \rho_i, and
c_i, such that given any \beta(a) value, we can uniquely fix which of the
\alpha_i have non-zero values, and what those values are. (I'm thinking of
positive integers/reals as values for all.)

It seems obvious that \sigma_i \neq \sigma_j and \rho_i \neq \rho_j for all i,j
is a must, as else we cannot know, given the value of \beta(a), which of the
\alpha_i(a) has a non-zero value (in fact, a more general constraint exists,
that no sum of certain \sigma_i for some values i should equal a sigma_j).
However, simple ideas like letting \sigma_i = \rho_i = 2^i do not seem to work.

Any tips/suggestions appreciated.

Regards,

Shrisha Rao
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