General idea: Lets say there's a finite supply (S) of resources which should be allocated to two tasks based on their demands, D1 and D2.

If Supply is larger than total demand (D), each task simply gets whatever it demands for:

If S=1000 and D=400 (D1=100, D2=300)

Then 100 units are assigned to task one (let's call it utilization 1, so U1=100) and 300 units to task two (U2=300) and the remaining 600 units are unused.

However, S is often less than D. In that case, each task receives a proportion of S, such as:

S=500 and D=1000 (D1=200, D2=800)

Then: U1=D1/D*S and U2=D2/D*S

To make sure that the tasks do not receive more than what they demand for:

U1=min(D1, D1/D2*S) and U2=min(D2, D2/D*S)

And let's call this as the base case.

Problem: I'd like to compare this simple allocation to a a more general allocation formulation that changes the priority of allocation based on α (a continuum that can vary between 0 and 1).

α=0 represents full priority to task one (so only resources remaining after meeting the demand for task one is spent on task two) and α=1 represents full priority to task two, and the base case results are obtained for α=0.5. To be more specific, here is an example:

D1=100, D2=900, and S=200

Three scenarios:

α=0: Full priority to task one

So, U1=100, then S-U1 goes to task 2, so U2=100

α=0.5: Base case

So, U1=20 and U2=180

α=1: Full priority to task two

So, U2=200 and nothing is left for task one, so U1=0

I'd like to formulate this (calculation of U1 and U2) without IF THEN ELSE and lots of Min and Max, just with mathematical equation(s). I already have something which may be too complex so I would like to see what others think of this formulation.

Thanks!