Let there be variables $A$, $B$, $C$, $D$, and $E$ such that a total of $N$ points is allocated among the variables: $A$+$B$+$C$+$D$+$E$=$N$, $N$∈$ℝ$.
Let the corresponding point values returned by each function be as follows:
$S_A$=$A$
$S_B$=2$B$-3
$S_C$=0.01$e^C$
$S_D$=$D^2$
$S_E$=$e^{1.1E}$-2$e^E$
S=$S_A$+$S_B$+$S_C$+$S_D$+$S_E$
What allocation of points to $A$, $B$, $C$, $D$, and $E$ maximizes $S$ for $A$+$B$+$C$+$D$+$E$=$N$?
The functions above are chosen arbitrarily and used solely as an example. I am curious as to whether a solution exists for $n$ generalized functions, each with a corresponding variable.
What if $S$=$S_A$⋅$S_B$⋅$S_C$⋅$S_D$⋅$S_E$?
What if $A$, $B$, $C$, $D$, and $E$ are limited to sets such as $ℤ$, $ℚ$, or $ℕ^+$?
I am not even sure which area of mathematics would focus upon a solution to this problem, but my teacher recommended that I investigate stochastic matrices and Markov chains. However, I have studied neither formally since I am a high school student.
First of all, you should use constraint to exclude any of the variables (express it using N and other variables).
For example, I will express $A=N-B-C-D-E$. Now I have got a four argument function: $$S=(N-B-C-D-E)+(2B-3)+(0.01e^C)+(D^2)+(e^{1,1E}-2e^{E})$$ (I've put parentheses where wach of the $S_i$ used to be)
Now this is a multivariate function and there are methods to look for maxima and minima. Generally you should look for points where partial derivatives are zero and check if that point is a saddle or a real extremum. The directions you'd want to investigate more would probably be Hessian matrices.
In you example you have exponents which are strictly increasing/decreasing therefore you will have no extrema.
If you want to solve this numerically, Solver Add-In in Microsoft Excel does exactly this.