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214 José-Oscar H. Sendín, Antonio A. Alonso, and Julio R. BangaOn the other hand, an increasing number of evolutionary, population-based, algorithmshave been developed in the last decade for handling multi-objective optimization problems.Since a set of candidate solutions is used in each iteration, these methods are capable of findingmultiple Pareto-optimal solutions in one single optimization run.In this contribution, we present an alternative multi-objective optimization approach whichis ultimately based on an extension of a recent Evolution Strategy (ES) for single-objectiveNLPs. The proposed approach makes use of a well known multicriteria optimization methodto generate an even spread of points in the Pareto front. The usefulness and efficiency of thisnovel strategy are illustrated by solving a wastewater treatment plant case study.2. Problem StatementGiven a process dynamic model, the multi-objective optimization problem can be mathematicallystated, without loss of generality, as follows:Find v to minimize simultaneouslyJ(ẋ, x, v) = [ J 1 (ẋ, x, v), J 2 (ẋ, x, v), . . . , J m (ẋ, x, v) ] T(1)subject to:f(ẋ, x, v) = 0 (2)ẋ(t 0 ) = x 0 (3)h(x, v) = 0 (4)g(x, v) ≤ 0 (5)v L ≤ v ≤ v U (6)where J is the vector of objective functions, v is the vector of decision variables, f is theset of differential and algebraic equality constraints describing the system dynamics (i.e. thenonlinear process model), x is the vector of state variables, h and g are possible equality andinequality path and point constraints which express additional requirements for the processperformance, and v L and v U are the lower and upper bounds for the decision variables.The solution to this problem is a set of points known as Pareto-optimal. A feasible solutionv ∗ is said to be Pareto-optimal solution if there is no v such that J i (v) ≤ J i (v ∗ ), for all i =1, ..., m, with at least one strict inequality. The vector J(v ∗ ) is said to be non-dominated.3. Multi-Objective Optimization MethodsComputing the Pareto-optimal set can be a very challenging task due to the highly constrainedand non-linear nature of most biochemical systems. In this regard, the ability of evolutionary,population-based, algorithms to deal with problems involving non-convex Pareto frontsmakes them attractive to solve highly nonlinear multi-objective problems. As a drawback,a very large population size is usually required, which is translated into a large number ofPareto-optimal solutions. Besides the associated rise in computational effort, such a large setcan be very difficult to handle, especially as the number of objectives increases. In order tofacilitate the selection of a suitable compromise, from a practical point of view it would be desirableto generate only a small number of optimal solutions capturing the complete trade-offamong the objectives.3.1 NBI-based Evolution StrategyHere we propose a new multi-objective evolutionary algorithm which combines the recentSRES (Stochastic Ranking Evolution Strategy) by Runarsson and Yao [6] with the NormalBoundary Intersection (NBI) method developed by Das & Dennis [2].