A method for the rational choice of control, design and error variables in optimization problems is devised, based on the reduction of the maximum matching of the problem graph to the Dulmage-Mendelsohn canonical form. The method allows the designer to find with minimum effort appropriate sets of control, design and error variables that lead to an ultimate decomposition in design optimization problems of any dimension. In design automation, this procedure is useful as a rationale to plan manual interventions, where designers guide the process according to domain-specific knowledge. The proposed technique is rigorous and intuitive, thanks to the application of sound graph-theoretic concepts. A real-life example of mechanical engineering design shows the applicability of the method.