Solver Not Working

Can I assume you are using the default "GRG non-linear" algorithm? Usually when I see this, it is because of something "not optimal" for the default algorithm in the objective function (H6). For example, your COUNTIF() functions in D6 and D7 that are part of H6 are "step" functions. Meaning that small changes in the "by changing" cell have zero effect on these functions. Of course, "No change in H6 for a small change in D4" is the convergence ("Solver has found a solution") criterion for a Newton-Raphson type algorithm.

If you have not yet tried it, you might try an evolutionary algorithm (assuming you don't need this to be backwards compatible to pre 2010 versions of Excel). EA algorithms are computationally more intense, but they are better suited for "step" functions like this. If you need this to use the NR type algorithm so it will be backwards compatible with older versions, you will need to re-think your calculations steps between D4 and H6 to eliminate the "step" functions.