Dominance-Based Rough Set Approach to Interactive Multiobjective Optimization

In this chapter, we present a new method for interactive multiobjective optimization, which is based on application of a logical preference model built using the Dominance-based Rough Set Approach (DRSA). The method is composed of two main stages that alternate in an interactive procedure. In the first stage, a sample of solutions from the Pareto optimal set (or from its approximation) is generated. In the second stage, the Decision Maker (DM) indicates relatively good solutions in the generated sample. From this information, a preference model expressed in terms of "if ..., then ... " decision rules is induced using DRSA. These rules define some new constraints which can be added to original constraints of the problem, cutting-off non-interesting solutions from the currently considered Pareto optimal set. A new sample of solutions is generated in the next iteration from the reduced Pareto optimal set. The interaction continues until the DM finds a satisfactory solution in the generated sample. This procedure permits a progressive exploration of the Pareto optimal set in zones which are interesting from the point of view of DM's preferences. The "driving model" of this exploration is a set of user-friendly decision rules, such as "if the value of objective i 1 is not smaller than $\alpha_{i_1}$ and the value of objective i 2 is not smaller than $\alpha_{i_2}$, then the solution is good". The sampling of the reduced Pareto optimal set becomes finer with the advancement of the procedure and, moreover, a return to previously abandoned zones is possible. Another feature of the method is the possibility of learning about relationships between values of objective functions in the currently considered zone of the Pareto optimal set. These relationships are expressed by DRSA association rules, such as "if objective j 1 is not greater than $\alpha_{j_1}$ and objective j 2 is not greater than $\alpha_{j_2}$, then objective j 3 is not smaller than $\beta_{j_3}$ and objective j 4 is not smaller than $\beta_{j_4}$".

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