A performance measure approach for risk optimization

  • André Jacomel ToriiEmail author
  • Rafael Holdorf Lopez
  • André Teófilo Beck
  • Leandro Fleck Fadel Miguel
Research Paper


In recent years, several approaches have been proposed for solving reliability-based design optimization (RBDO), where the probability of failure is a design constraint. The same cannot be said about risk optimization (RO), where probabilities of failure are part of the objective function. In this work, we propose a performance measure approach (PMA) for RO problems. We first demonstrate that RO problems can be solved as a sequence of RBDO sub-problems. The main idea is to take target reliability indexes (i.e., probabilities of failure) as design variables. This allows the use of existing RBDO algorithms to solve RO problems. The idea also extends the literature concerning RBDO to the context of RO. Here, we solve the resulting RBDO sub-problems using the PMA. Sensitivity expressions required by the proposed approach are also presented. The proposed approach is compared to an algorithm that employs the first-order reliability method (FORM) for evaluation of the probabilities of failure. The numerical examples show that the proposed approach is efficient and more stable than direct employment of FORM. This behavior has also been observed in the context of RBDO, and was the main reason for the development of PMA. Consequently, the proposed approach can be seen as an extension of PMA approaches to RO, which result in more stable optimization algorithms.


Risk optimization Reliability-based design optimization Performance measure approach Reliability index Probability of failure 


Funding information

This research was financially supported by CNPq-Brazil.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Replication of results

The results obtained with the proposed approach are presented in Section 5.


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Copyright information

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

Authors and Affiliations

  • André Jacomel Torii
    • 1
    Email author
  • Rafael Holdorf Lopez
    • 2
  • André Teófilo Beck
    • 3
  • Leandro Fleck Fadel Miguel
    • 2
  1. 1.Latin American Institute of Technology, Infrastructure and Territory (ILATIT)Federal University of Latin American Integration (UNILA)Foz do IguaçuBrazil
  2. 2.Center for Optimization and Reliability in Engineering (CORE), Department of Civil EngineeringFederal University of Santa Catarina (UFSC)FlorianópolisBrazil
  3. 3.Structural Engineering DepartmentUniversity of São PauloSão CarlosBrazil

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