I have a data set containing both positive and negative numbers. Say, for example the following $10$ values:
11,
45,
68,
-18,
29,
22,
-8,
-12,
54,
33
I have to assign proportionate weights to each of these values such that the highest value ($68$) gets maximum weight and the lowest one ($-18$) gets the least weight. However, the weights should be POSITIVE.
If it had been a list of only positive numbers, I would simply take the sum of all $10$ numbers, and divide each number by that sum to get proportionate positive weight for each.
However, this data set contains negative numbers also, so a few of the weights will go negative if the above method is followed.
What would be the best way to "adjust" the weights so as to arrive only at positive weight values? Is there any method to adjust each of the core data values themselves ($10$ in this case)?
There are multiple applications for this: allocating budgets to states which are running surplus while others running deficit, or making proportionate investments in projects which have positive and negative NPVs, etc.
I am open to use any mathematical function as suitable, like log, exponent, etc.