[ReSolved]Limits, both Lower and Upper...

[Something Else]

Here's the problem. Lets say I have a series of variables, F1 thru Fn.
Each one of these, individually, are >= B1. They also are individually <= T1. { For any Fn, B1 <= Fa <= T1 }
Also, If you sum two of them together, any combination of two are >= B2, and <= T2.

{ For any 2 Fn, B2 <= Fa + Fb <= T2 }
So, more explixitely, if you had F1 + F2, B2<= (F1+F2) <= T2.
And More Generically, if you sum together N elements of {F1, F2, F3, ...}, then
These N elements are <= Tn, and >= Bn.

{ For any K Fn, Bk <= Fa + Fb + ... <= Tk }

Now, Some of these F's combine together in such a way to produce other F's.
Given these equations, I would like to produce as many minima and maxima as possible.

For example, lets say I had the equations as seen in the attached picture.




I can build the limits as seen by hand thru visual inspection.

Question #1) Can anyone add or improve upon the illustrated limits.
Please feel free to build some upper limits.

Question #2) Can anyone help me with how I should approach creating a progie that,
given equations, variables, and elemental limits {All The Bn's and Tn's}
produces the optimal Lower and Upper limits of the control variables?

-Thanks,
-Lou