Research Article Comparing Solutions under Uncertainty in Multiobjective Optimization Miha Mlakar, Tea Tušar, and Bogdan FilipiI Department of Intelligent Systems, Joˇ zef Stefan Institute and Joˇ zef Stefan International Postgraduate School, Jamova cesta 39, 1000 Ljubljana, Slovenia Correspondence should be addressed to Miha Mlakar; miha.mlakar@ijs.si Received 19 December 2013; Revised 14 April 2014; Accepted 17 April 2014; Published 18 May 2014 Academic Editor: Zhan Shu Copyright © 2014 Miha Mlakar et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Due to various reasons the solutions in real-world optimization problems cannot always be exactly evaluated but are sometimes represented with approximated values and confdence intervals. In order to address this issue, the comparison of solutions has to be done diferently than for exactly evaluated solutions. In this paper, we defne new relations under uncertainty between solutions in multiobjective optimization that are represented with approximated values and confdence intervals. Te new relations extend the Pareto dominance relations, can handle constraints, and can be used to compare solutions, both with and without the confdence interval. We also show that by including confdence intervals into the comparisons, the possibility of incorrect comparisons, due to inaccurate approximations, is reduced. Without considering confdence intervals, the comparison of inaccurately approximated solutions can result in the promising solutions being rejected and the worse ones preserved. Te efect of new relations in the comparison of solutions in a multiobjective optimization algorithm is also demonstrated. 1. Introduction Multiobjective optimization is the process of simultaneously optimizing two or more conficting objectives. Problems with multiple objectives can be found in various felds, from product design and process optimization to fnancial applications. Teir specifcity is that the result is not just one solution, but a set of solutions representing trade-ofs between objectives. Multiobjective evolutionary algorithms (MOEAs) are known for efciently solving these kind of problems [1]. However, MOEAs can also be used for solving optimization problems with uncertain objective values. Te reason for uncertainty can be noise, robustness, ftness approximations, or time-varying ftness functions. When solving uncertain optimization problems, it is better if the algorithm takes uncertainty into account. Uncertain solutions can be represented with approxi- mated values and variances of these approximations. From the variance, the confdence interval of the approximation can be calculated. Tis interval indicates the region in which the exactly evaluated solution is most likely to appear. Te confdence interval width indicates the certainty of the approximation. If the confdence interval is narrow, we can be more certain about the approximation and vice versa. Since the confdence intervals ofer additional information on the approximations, they can be efectively used to compare solutions and an algorithm using confdence intervals can perform better by exploiting this additional information [2]. During optimization that does not consider confdence intervals, an approximated solution may be incorrectly iden- tifed as the better of the two compared solutions. Ofen the solution that is incorrectly determined as worse is then discarded. Similarly, a promising solution can get discarded if a worse solution is incorrectly determined as being better. In both cases good solutions are lost due to the comparison of solutions which only considers approximated values. To prevent these unwanted efects, we propose new relations for comparing solutions under uncertainty, where, in addition to the approximated values of a solution, their confdence intervals are considered. Tese relations cover all possible combinations that can occur when comparing Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 817964, 10 pages http://dx.doi.org/10.1155/2014/817964