I have a problem, which has $n$ variables. Let's call these variables as $a_1$, $a_2$, $a_3$ ... $a_n$. Each of these variables have their own range (i.e, $a_{imin} \leq a_i \leq a_{imax}$ for all $i$). And most importantly, each $a_n$ is ranked in terms of importance. Without a loss of generality one can assume that that $a_1$ is more important than $a_2$, which in turn is more important than $a_3$, and so on. This means that as long as the constraint is satisfied, it is better to have a smaller $a_1$ compare to have a smaller $a_2$, and a smaller $a_3$ is better compare to a smaller $a_4$
These variables are required to fulfill one ( yes, just one!!) constraint function
$$F_{min} \leq f(a_1,a_2,a_3, .... a_n) \leq F_{max}$$
The question now is, how to construct a minimizing function that satisfy the above condition? If I have two variables ($a_1$ and $a_2$), and both of them are between or equal to 0 and 9, then I can easily write the minimizing function as
$$L=10a_1 + a_2$$
given that $a_1$ is more important than $a_2$.
But for complicated case, I am a bit confused.
Edit: Maybe I should clarify further. My point is, for the above case, if $a_1=3, a_2=4$ and $a_1=4, a_2=3$ both satisfy the constraint equation, $a_1=3, a_2=4$ should be selected. So how to construct $L$ in this case to reflect the situation?
Edit 2 : Referring to the answer by Rahul, it seems that it is not possible to formulate such a minimizing function. Is there other methods of attack I can use?
Edit 3: alext87 answer is a good start, it contains general guidelines on how to proceed. But I want a hard, quantitative rule that tells me how to select for $lambda_i$ for all $i$.