### A Property Preserving Method for Extending a Single-Objective Problem Instance to Multiple Objectives with Specific Correlation

Abstract

A method is proposed to generate multi-objective optimization problem instances from a corresponding
single-objective instance. The user of the method can specify the correlations between the generated
the objectives. Different from existing instance generation methods the new method allows to keep
certain properties of the original single-objective instance. In particular, we consider optimization
problems where the objective is defined by a matrix, e.g., a distance matrix for the Traveling
Salesperson problem (TSP) or a flow matrix for the Quadratic Assignment problem. It is shown that the
method creates new distance matrices with specific correlations between each other and also have the
same average distance and variance of distances as the distance matrix of the original instance. This
property is important, e.g., when the influence of correlations between the objectives on the behavior
of metaheuristics for the multi-objective TSP are investigated. Some properties of the new method are
shown theoretically. In an empirical analysis the new method is compared with instance generation
methods from the literature.