*Dominance*

It is required to check each solution with the rest on the dominance basis. Assuming that *X*1, *X*2 are two solutions, *m* is the number of objective functions, if *X*1 ≺ *X*2 (*X*1 dominates *X*2) is true, the Pareto dominance conditions must all be true, and they are:

$$1. \quad f\_{\vec{j}}(X\_1) \not\rightsquigarrow f\_{\vec{j}}(X\_2) \\ \forall j = \{1, \ldots, m\}.$$

$$\text{2.} \quad f\_{\rangle}(X\_1) \lhd f\_{\rangle}(X\_2) \\ \exists j = \{1, \ldots, m\}$$

Dominance is investigated for all of the solutions. The solution that is not dominated by any of the remaining solutions is a non-dominated solution and it is removed from the solutions matrix. This is repeated for all solutions and the resulted non-dominated solutions are considered *rank 1*. The same process is repeated for the remaining ranks.
