Lecture 9:
                Multi-Objective 
                Optimization
      Suggested reading: K. Deb, Multi-Objective Optimization using Evolutionary 
              Algorithms, John Wiley & Sons, Inc., 2001 
       Multi-Objective Optimization 
       Problems (MOOP)
       „ Involve more than one objective function that 
         are to be minimized or maximized
       „ Answer is set of solutions that define the best 
         tradeoff between competing objectives
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            General Form of MOOP
            „ Mathematically
                      min/max    fm(x),         m=1, 2,L,M
                      subject to   gj(x)≥0,   j =1, 2,L,J
                                         h (x) = 0,   k =1, 2,L,K
                                            k
                                         x(L) ≤ x ≤ x(U),   i =1, 2,L,n
                                            i          i       i
                                          lower             upper
                                          bound             bound
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       Dominance
       „ In the single-objective optimization problem, 
         the superiority of a solution over other 
         solutions is easily determined by comparing 
         their objective function values
       „ In multi-objective optimization problem, the 
         goodness of a solution is determined by the 
         dominance
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