Linear Programming - How to maximise the maximum I want to do the following:
max: greatest(a1+b1+c1, a2+b2+c2, a3+b3+c3);
... constraints involving a1,a2,a3,b1,b2,b3,c1,c2,c3...

Since there is no greatest() function, I restructured it like this:
max: greatest_val;
greatest_val >= a1+b1+c1;
greatest_val >= a2+b2+c2;
greatest_val >= a3+b3+c3;
... constraints involving a1,a2,a3,b1,b2,b3,c1,c2,c3...

But this leads to a boundless problem as greatest_val can go to infinity.
How do I structure this problem so that it has upper and lower bounds??
Thanks in advance.
 A: It may seem strange, but you should try to minimise greatest_val. This will give you the smallest of the possible maxima. It is sometimes called a minimax problem.
A: I like Tomi's suggestion for finding the smallest "greatest_val" that is at least as big as all of the individual sums.  As an alternative, if you have access to routines that solve mixed integer linear programs, you might consider the following reformulation of your problem:
Define a new set of indicator variables $k_i$ and redefine your objective function as $\sum_{i=1}^{N}k_i(a_i+b_i+c_i)$ where $N$ represents the number of $(a,b,c)$ triples that you have.  Define each $k_i$ to be either $0$ or $1$ (binary) and add the constraint that $\sum_{i=1}^{N}k_i=1$.  Along with the binary constraints this additional constraint says that exactly one value of $k_i$ can be equal to $1$ and all the rest will be $0$.
You would keep all of the other constraints that you already have.  Maximizing this new objective function will then seek to find the optimal $k_i$ and thus the largest value that $a_i+b_i+c_i$ can take on, subject to all the other constraints.  If the problem is feasible then this should work for you.   
