Effective generation of the Pareto frontier using the Normal Constraint method

In engineering optimization, Pareto solutions are those where improvement in one design objective can take place if at least one other design objective worsens. Accordingly Pareto solutions play an important role in engineering design. Some engineering design optimization methods automatically generate a Pareto solution as a possible final solution, while other methods generate a set of Pareto solutions from which one is chosen as final. For the latter approach to be successful, the generated Pareto set must be truly representative of the Pareto solutions in the design space. In other words, the set must not overly represent one region of the design space, while neglecting other regions. Many commonly used methods fail to comply with this requirement, while others do. This paper offers a new and simple method to generate an evenly spaced set of Pareto solutions in the design space. This method bears some similarities to the Normal Boundary Intersection and to the e-Constraint methods, and it works for an arbitrary number of objectives. The even spread of the generated Pareto solutions makes it possible to effectively develop an analytical expression for the Pareto frontier (complete set of Pareto solutions) in ^-dimension. This even spread also facilitates decision making in choosing the most desirable (final) Pareto solution.