Weak Pareto optimality

As mentioned in the introduction, we here assume that a DM is able to participate in the solution process. (S)he is expected to know the problem domain and be able to specify preference information related to the objectives and/or different solutions. We assume that less is preferred to more in each objective for him/her. (In other words, all the objective functions are to be minimized.) If the problem is correctly formulated, the final solution of a rational DM is always Pareto optimal. Thus, we can restrict our consideration to Pareto optimal solutions. For this reason, it is important that the multi-objective optimization method used is able to find any Pareto op>-timal solution and produce only Pareto optimal solutions. However, weakly Pareto optimal solutions are sometimes used because they may be easier to generate than Pareto optimal ones. A decision vector x G S (and the corresponding objective vector) is weakly Pareto optimal if there does not exist another x G S such that /i(x) < /i(x ) for alH = 1,..., A . Note that Pareto optimality implies weak Pareto optimality but not vice versa. 

The solution of problem (6.3) is always weakly Pareto optimal and Pareto optimal if it is unique. On the other hand, x G S is Pareto optimal if and only if it solves (6.3) for every I = 1,..., k, where Sj = for j = 1,..., k, j 1. Thus, ensuring Pareto optimality means either solving k problems or obtaining a unique solution (which is not necessarily easy to verify in practice). What is positive when compared to the weighting method is that the e-constraint method can find any Pareto optimal solution even for nonconvex problems. 

We can say that interactive methods overcome weaknesses of a priori and a posteriori methods the DM does not need a global preference structure and only interesting Pareto optimal solutions need to be considered. The latter means both savings in computational cost, which in many computationally complicated real problems is a significant advantage, and avoids setting cognitive overload on the DM, which the comparison of many solutions typically implies. 

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