> It seems to me that it would be better to consider minimization case,
> because it is more obvious.

That is,

   minimize z = max(x1, x2) + max(x3, x4) + ...

The case 

   minimize z = max(x1, x2) + max(x3, x4) - min(x5, x6) - ...

can be reduced to the previous one by substituting 

   -min(x5, x6) = max(-x5, -x6)

And the case of maximization

   maximize z = min(x1, x2) + min(x3, x4) - max(x5, x6) - ...

can be reduced to minimization by changing the sign of the objective.


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