# How to split $100$% to various components with different priorities?

For the following scenario :

$E=100$ (energy available = $100$%)

Components = $N$ (can be $1,2,3,\dots,\infty$)

Now I want to split $E=100$ for each component, however, components are prioritised :

Example, component $1$ has first priority, component $2$ has second highest priority, and the rest have low priorities.

Application : If $E=100$

• $C_1=1$ (component $1$ has priority 1 )
• $C_2=2$
• $C_3=3,C_4=3,C_5=3$

Then $C_1$ must get the maximum value of $E$, while $C_2$ gets the second highest value ... $C_{345}$ must get the same equal lowest value of $E$

What mathematical model achieves this ?

Splitting $100/5$ will only give me equal values for all components, but im not sure how to achieve this with priorities.

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Intriguing selection of tags. – joriki Nov 6 '12 at 15:19
@joriki thank you – NLed Nov 6 '12 at 16:01
This is some kind of optimization problem, yet you do not specify how to compare two distributions. – Alexei Averchenko Nov 6 '12 at 16:17
@AlexeiAverchenko to be honest, I have no idea what to call this .. I am not familiar with the mathematical terms or titles. I just thought they have something to do with models and division ... Can you elaborate what you mean by specifying how to compare two distributions ? -- Also, did you mark my question down ? – NLed Nov 6 '12 at 16:19
Suppose you have two distributions $(a_1, a_2, \ldots, a_n)$ and $(b_1, b_2, \ldots, b_n)$. How do you determine which one is better for you purposes? – Alexei Averchenko Nov 6 '12 at 16:21

Unless you have a model of how benefit changes with money spent, the solution will simply always be 100/0/0/0/0 split, since you are maximising the highest priority.

If you had a utility function specifying that say, Roads were top priority up to 5%, then lowest priority afterwards then you would get a 95/5/0/0/0 split etc.

If you just want a simple weighting, then just add up the weights , eg 5+2+1+1+1=10 then assign 100/10*weight percentage points to each category.

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Im interested in the simple weighting you mentioned ... Can you please elaborate on that ? – NLed Nov 6 '12 at 15:53
can someone please explain what @Nick mentioned as 'If you just want a simple weighting, then just add up the weights , eg 5+2+1+1+1=10 then assign 100/10*weight percentage points to each category.' ?? Can you please explain this and use another example, I dont get it – NLed Nov 6 '12 at 23:13

C1 gets 90, C2 gets 7, C3-C5 get 1. For a mathematically unique answer you need to define priorities more completely. There is an infinite collection of solutions to your example as stated.

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