Nonscalarized multiobjective global optimization

Sensitivity of Pareto Solutions in Multiobjective Optimization

“…In this case, k = 1; thus, l 1 = 1 and by (13) we have v * 1 = 1. Dropping the subscript, we get from (20)…”

Interactive balance space approach for solving multi-level multi-objective programming problems

“…In this way, we can present satisfactoriness and the preferred solution from the perspective of singular-level multi-objective decision-making problems. We now introduce several theorems [7,9,12] with the help of the quality of balance space approach to provide a theoretical basis for the first-level, second-level, and third-level multi-objective decision-making problems.…”

“…. ; a m Þ be the vector of partial global solutions, and let X 0 i ðgÞ ¼ fx 2 X jf i ðxÞ À a i 6 gg be the g-suboptimal solution sets, then we have the following definitions [9,12]:…”

“…In [9,12] the balance space approach was introduced along with the concepts of balance number, balance points, and apportioned balance number. A comparison between Pareto analysis and the balance space approach was presented in [10] to demonstrate both the interrelations and differences between the two methods.…”

“…It is based on monotonic set contraction (expansion), see [6][7][8][9][10][11][12][13], which solves multi-objective programming problems in their original form and provides a new instrument, called a balance set [9]. It is represented by a collection of equations in the function value space that allows DMs to choose in advance the quality of a particular optimal solution from a continuum (a set) of similar ones.…”