Decision makers tend to define their optimization problems as multi-objective optimization problems. Generating the whole nondominated set is often unrealistic due to its size but also because most of these points are irrelevant to the decision maker. Another approach consists in obtaining preference information and integrating it a priori, in order to reduce the size of the nondominated set and have a gain in computation time. In this work we focus on a partial preference relation based on requirement and tolerance thresholds that translate the Pareto dominance cone. After introducing this preference relation, we present adaptations to use it in existing discrete multi-objective optimization algorithms. Numerical experiments on multi-objective combinatorial optimization problems show the applicabiliy of our approach.