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218 José-Oscar H. Sendín, Antonio A. Alonso, and Julio R. Banga3025λ = 150λ = 250302510 FMINCON / K = 15FMINCON / K = 20FMINCON / K = 308SRES / K = 15620204J 2(ISE)15J 2(ISE)15201000 1100 1200 1300 1400 1500 1600 170010105501000 1200 1400 1600 1800 2000 2200 2400 2600J 1(Cost)01000 1200 1400 1600 1800 2000 2200 2400 2600J 1(Cost)Figure 4. Pareto-optimal sets obtained with theNBI-based ES.Figure 5. Pareto-optimal sets obtained with thestandard NBI.6. ConclusionsIn this work, we have presented an alternative method for the multi-objective optimization ofnonlinear dynamic systems. The NBI-based Evolution Strategy has been specially designedto produce an even spread of Pareto-optimal solutions in one single optimization run, insteadof solving repeatedly a set of NLPs. This novel approach has been successfully applied to theintegrated design and control of a wastewater treatment plant. A very good approximation ofthe optimal trade-offs between cost and controllability has been generated with less computationaleffort than the standard NBI (using also SRES as solver). It should be noted that thisproblem could not be solved satisfactorily with traditional gradient-based methods.In the near future, we will consider selected case studies where the NBI-based ES will becompared with other techniques. Also, new ideas will be suggested to increase its efficiency.AcknowledgmentsAuthor José-Oscar H. Sendín acknowledges a pre-doctoral grant from the I3P programme ofthe Spanish Council for Scientific Research (CSIC).References[1] Banga, J.R., Moles, C.G. and Alonso, A.A. (2003). “Global Optimization of Bioprocesses using Stochastic andHybrid Methods,” In Frontiers in Global Optimization, (Floudas, C.A. and Pardalos, P.M., eds.). Kluwer AcademicPublishers, Dordrecht.[2] Das, I and Dennis, J.E. (1998). “Normal-Boundary Intersection: A New Method for Generating the ParetoSurface in Nonlinear Multicriteria Optimization Problems,” SIAM J. Optimization, Vol 8, 631–657.[3] Deb, K. (2001). Multi-Objective Optimization using Evolutionary Algorithms. Wiley, Chichester.[4] Miettinen, K. (1999). Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Dordrecht.[5] Moles, C.G., Gutierrez, G., Alonso, A.A. and Banga, J.R. (2003). “Integrated design and control via globaloptimization: A wastewater treatment plant case study,” Chemical Engineering Research & Design, 81, 507–517.[6] Runarsson T.P. and Yao X. (2000), “Stochastic Ranking for Constrained Evolutionary Optimization,” IEEETransac. Evol. Comp., Vol. 4, 284–294.[7] Schweiger C.A. and Floudas A. (1997), “Interaction of Design and Control: Optimization with Dyamic Models,”In Optimal Control Theory, Algorithms and Applications, (Hager, W.W. and Pardalos, P.M., eds.), KluwerAcademic Publishers, Dordrecht.