Modified Algorithm to Compute Pareto-Optimal Vectors

Bartering integer commodities with exogenous prices

“…Let us define g(α) = u k (x + αS kh ij ), differentiable with respect to α. It will be shown that for all α in the interval (26), and 0 < τ ∈ R, g ′ (α) < 0 implies g ′ (α+τ ) < 0, which is a sufficient condition for the unimodality of g(α). By the chain rule, and using ( 24) and ( 25), the derivative of g(α) can be written as…”

“…Proposition 2. Under the definition of u k and S kh ij , for every feasible point x ∈ R mn , u k (x + αS kh ij ) is a unimodal function with respect to α in the interval defined by (26).…”

“…and let[α down , α up ] be the interval of feasible step lengths defined in (26). Then, due to Proposition 2, the set V * of Pareto efficient solutions of an ERP can be obtained as follows:…”

“…. , c n and let again Γ be the set of integer points in the interval [a down , a up ], where a down = min{argmax α αc k S kh ij , argmax α αc h S kh ij } and a up = max{argmax α αc k S kh ij , argmax α αc h S kh ij }, and let [α down , α up ] be the interval of feasible step lengths defined in (26). It might be easily seen that either…”

Robust design strategy applied to a vehicle suspension system with high camber angle tyres

“…The robust optimization of the most influential parameters of order 1, based on simplified models and objective criteria related to the driving situation is performed from a multiobjective genetic algorithm so as to converge on the Pareto front which represents the tradeoff between the different criteria for each driving situation (Sastry, 1999). The algorithm does not converge to a unique solution but to a frontier (Pareto front).…”