Metamodel-assisted Kalai-Smorodinsky equilibria to solve a multicriteria optimization problem

shape optimization of an aerosol can
  • Benki Aalae
  • Habbal Abderrahmane
  • Mathis Gael
Industrial Application


The main idea of solving a multicriteria optimization problem (MOP) is to find non-dominated solutions belonging to the Pareto frontier (e.g., a compromise between the criteria). If we consider our criteria as players, we can remark that the obtained solutions are the result of a cooperation of the players to increase their profits which is the same principle of solving a bargaining problem (BP). The Kalai-Smorodinsky (KS) model suggests an attractive solution for the BP called KS equilibria that can be also a MOP solution without having to calculate the Pareto Frontier known to be computationally so expensive. In this paper, we propose an algorithm aimed to rapidly finding the KS solution. The idea is to make a coupling between the KS algorithm and a Radial Basis Function (RBF) metamodel called KS-RBF. In fact, the KS algorithm transforms the MOP into a succession of single objective problems (SOP); for our proposed algorithm, the objective function of each SOP will be replaced by an approximate one using a reliable RBF metamodel (SOP-RBF). The performance of the proposed approach is firstly validated by some well-known mathematical multicriteria problems (Tanaka, Poloni, and ConstMIN problems) by finding a KS solution belonging to the Pareto Frontier; then, we used it to solve a realistic industrial case, namely, shape optimization of the bottom of an aerosol can undergoing non-linear elasto-plastic deformation in order to simultaneously minimize the dome growth (DG) (e.g., displacement of can base) at a proof pressure and maximize the dome reversal pressure (DRP), a critical pressure at which the aerosol can’s bottom loses stability (e.g., initiates buckling). The KS solutions are compared with Pareto Frontier results previously suggested in other papers. The comparison leads us to confirm the performance of the KS-RBF coupling.


Kalai-Smorodinsky equilibria Radial basis function metamodel Multicriteria optimization problem Aerosol can shape optimization 


Funding information

The present work was achieved within the framework of the partnership between the INRIA Sophia Antipolis research center and ArcelorMittal France Leader Company in steel manufacturing, which funded this work.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Benki Aalae
    • 1
  • Habbal Abderrahmane
    • 1
  • Mathis Gael
    • 1
  1. 1.ToulouseFrance

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