I'm a bit at a loss here. I know that this seems a bit like a "here's my problem, now go solve it" thing, and I'm sorry about that, but I just want a nudge in the right direction since I've never dealt with this type of problem (in more than two variables). See, I have 5 variables $A, P, H, T,$ and $M$ and I would like to find a type of relation between them they satisfy with certain constraints. Of course all this is pretty vague so let me go into the specifics.
I have:
$0 \leq A \leq 100, A \in \mathbb{R}$
$0 \leq H \leq 10, H \in \mathbb{N}$
$0 \leq T \leq 20, T \in \mathbb{N}$
$0 \leq P \leq 30, P \in \mathbb{N}$
$-14 \leq M \leq 14, M \in \mathbb{R}$
And I need a relation of the form $A = f(H,P,M,T)$. In addition, I have the following conditions the relation must satisfy:
- If $H, P$, or $|M|$ increases, then A increases.
- If T increases, then A decreases.
- $P = 0 \implies A = 0$ regardless of the other variables. This can be written as $f(H, 0, M, T) = 0$.
- The function satisfies $f(10, 30, \pm 14, 0) = 100$, so when H, P and |M| are maximized and T is minimized, the function achieves its maximum.
Till now I've thought of something like: $A = |M|\frac{PH}{T}$, but this fails to meet the conditions since T is not allowed to be 0 here and A is not always in the range $[0,100]$. From here, I can add constants to tweak the exact values I'd like the function to take on, but I need a general form of a relation I can work with.
As requested, here is my comment in answer form:
Well we can almost take $$A=cP(H+P+|M|−T)$$ for any positive constant c. Only problem is that the term in the parentheses can go negative in the region (which is problematic for condition 1 in P) but the smallest it can be is -20 so take $$A=cP(H+P+|M|−T+21)$$ This satisfies the first three constraints, and the fourth lets you solve for c