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216 José-Oscar H. Sendín, Antonio A. Alonso, and Julio R. Bangapenalty coefficient, and ¯γ k is:¯γ k =(1 − γ − γ )min.γ max − γ minFor a bi-objective optimization problem, the upper bound γ max is defined as the euclideandistance to the Utopia vector from the point in the CHIM given by w max = [0.5, 0.5] T (Fig. 2).Similarly, the lower bound γ min is the euclidean distance to the Nadir point, J Nadir (i.e., thevector of upper bounds on each objective in the entire Pareto set), multiplied by -1.In each iteration (generation), the basic procedure of the algorithm is as follows:1. For each individual j ∈ {1, . . . , λ} evaluate the objective vector and the constraints;2. For each weight vector k ∈ {1, . . . , K}:3. For each individual j ∈ {1, . . . , λ}:4. Evaluate the fitness function ϕ k (Eq. 9);5. Perform Stochastic Ranking;6. Select the best µ k individuals (µ k ≈ λ/(3K ));7. Update the mutation strength (non-isotropic self-adaptation);8. Mutate the (K · µ k ) parents to create the new population of size λ.The algorithm has been implemented in Matlab version 6.5. For the sake of comparison,we have made an implementation of the standard NBI in which the set of NLPs are solvedsequentially by means of SRES. Also, the original Matlab 5.0 implementation of NBI has beenrewritten for running on version 6.5 (with the FMINCON code, which is part of the MatlabOptimization Toolbox).4. Case Study: A Wastewater Treatment PlantThis case study, solved previously by Moles et al. [5] using a weighted sum approach, representsan alternative configuration of a real wastewater treatment plant located in Manresa(Spain). The process flowsheet (Fig. 3) consists of two aerated bioreactors, where thebiodegradable substrate is transformed by a microbial population (biomass), and two settlers,where the activated sludge is separated from the clean water and recycled to the correspondingreactor. The activated sludge in excess is eliminated via a purge stream (q p ). The aim ofthe control system is to keep the substrate concentration at the output (s 2 ) under a certain admissiblevalue. We consider a step disturbance to the input substrate concentration (s i ), takingthe flowrate of the sludge recirculation to the first tank (qr 1 ) as the manipulated variable.q i s i x iq 12sir 1fk 1s 1q q122c 1 xd sx 1 1 sir 2 fk 21sxb 21xir 2cxr 21xdx 22 xb 2xr 2q sals 2xir 1Figure 3. Process flowsheet of the wastewater treatment plant.qr 1qr 2q 3q 2qr 3qpqrsrxr