How does dimensional analysis account for additive variables in equations?

[Mr Davis 97]

Under the standard form of dimensional analysis, I know that we relate a dependent variable to a function of the independent variable(s). However, what if there is some additive variable needed in the equation? How does this method, which expresses all of the independent variables as a product of the variables (with the exponents being any real number) times a constant, account for the needed addition of operations such as subtraction and addition? How can we derive correct equations if there is a missing sum or difference needed in the formula?

 

[mfb]

If you can construct a specific unit with multiple independent expressions, then dimensional analysis for this unit does not work.
A trivial example is a setup where you have two velocities, or two masses or something similar. There is no way to figure out which velocity/mass/... to use just by dimensional analysis.