Finding a constant

[alexisellis]

Hi All,
My goal is to create a set of values with a predetermined average. The problem is that each value has a condition such that they can not be set to a % that would create this average value. I will explain the issue verbally and then mathematically.

Verbally,

I have a set of values.
Each value has another value associated with it and the total cannot be greater than 100%
I wish to make the average of all values say 50%
Right now some values are 50%, others are less (because of the conditional values)
If I add (1 - (1 - <the current value>) * <a constant> to any values that can be increased,
I can make the final average 50%.
I can determine the constant value through trial and error, but would like to use an equation to figure out the constant.

Mathematically,

i = each value
C = The current i
T = The highest number that this i could be based on the condition
x = The constant <= 1
N = The total of all values
Y = The current average value
Z = The target average value
min = The lower of the values

Find x such that N=N*Z
In order to do so;
N = ∑(min(T[i], 1-1-C[i]*x)

This is a GIS problem, so it is kind of hard to explain. Any help I can get
would be truely appreciated!!
Thanks
Alexis

[alexisellis]

Dr. Math,
I hope I can explain myself well enough for you to change your mind.
First let me describe the problem.

C = The current value of each square
T = The highest number that this square could be based on the condition
x = The constant <= 1
M = The total area of all squares
N = The desired total
Z = The target average value
Let me better explain the issue.

I have a raster dataset (think of it like a piece of graph paper where each square represents a value). Each square represent a piece of land. I want say, 50% of the entire page to be covered with trees or N=M*Z. Easy enough, but there is a condition for each square. Some squares have too much other stuff to fill them half full with trees. So, some of the squares will have to be more than 50% covered and some with < 50% tree cover (due to the condition), such that the total tree cover is 50%.

I have so far set up my dataset so that each square is set as close to 50% as possible based on the condition. I have figure out that in order to determine how much to add to these cells I can run the following equation;
1-1-C*x and when this value is less than T (the portion of the square available to be tree cover) I add this value to the current value for this square.
By trial and error, I can get a constant (x) to make the total coverage = 50%.
I would like to somehow calculate this value on the fly???

If you could help that would be great!
Thanks
Alexis