An Adaptive Augmented Weighted Tchebycheff Method to solve Discrete, Integer-valued Bicriteria Optimization Problems

The augmented weighted Tchebycheff norm was introduced in the context of multicriteria optimization by Steuer and Choo (1983) in order to avoid the generation of weakly nondominated points. It augments a weighted l∞-norm with an l1-term, multiplied by a “small” parameter ρ > 0. However, the appropriate selection of the parameter ρ remained an open question: A too small value of ρ may cause numerical difficulties while a too large value of ρ may lead to the oversight of some nondominated points. For discrete bicriteria optimization problems, we derive a method for a problem dependent determination of all parameters of the augmented weighted Tchebycheff norm such that all nondominated points can be found and ρ is as large as possible. In a computational study based on randomly generated instances of a bicriteria knapsack problem, the resulting adaptive augmented weighted Tchebycheff method is compared with the lexicographic weighted Tchebycheff method and with the augmented weighted Tchebycheff method with preset parameter values.