Application of the Kalai-Smorodinsky approach in multi-objective optimization of metal forming processes

The problem of multi-objective optimization (MOP) is approached from the theoretical background of the Game Theory, which consists in finding a compromise between two rational players of a bargaining problem. In particular, the Kalai and Smorodinsky (K-S) model offers a balanced and attractive solution resulting from cooperative players. This approach allows avoiding the computationally expensive and uncertain reconstruction of the full Pareto Frontier usually required by MOPs. The search for the K-S solution can be implemented into methodologies with useful applications in engineering MOPs where two or more functions must be minimized. This paper presents an optimization algorithm aimed at rapidly finding the K-S solution where the MOP is transformed into a succession of single objective problems (SOP). Each SOP is solved by meta-model assisted evolution strategies used in interaction with an FEM simulation software for metal forming applications. The proposed method is first tested and demonstrated with known mathematical multi-objective problems, showing its ability to find a solution lying on the Pareto Frontier, even with a largely incomplete knowledge of it. The algorithm is then applied to the FEM optimization problem of wire drawing process with one and two passes, in order to simultaneously minimize the pulling force and the material damage. The K-S solutions are compared to results previously suggested in literature using more conventional methodologies and engineering expertise. The paper shows that K-S solutions are very promising for finding quite satisfactory engineering compromises, in a very efficient manner, in metal forming applications.