The weighted-Tchebycheff approach to design optimization

Bi-objective optimization is a useful mathematical method for the investigation of real-world problems with conflicting two objectives that arise in decision making in order to incorporate a tradeoff between the two objectives. In most of such studies, an assumption of convexity on the problems has been made. However, the method to obtain a non-convex Pareto frontier has not yet been fully explored. In many practical industrial situations, at least one objective function can be a higher-order than the second. Consequently, the associated Pareto frontier can be nonconvex. In this situation, it is known that obtaining efficient solutions using the conventional weighted-sum method is unlikely because this method do not yield solutions that lie in nonconvex regions of the feasible design space. In practice, this means that weighted-sum methods could miss potentially preferable solutions. For this reason, we develop a weighted-Tchebycheff-based bi-objective optimization model to obtain all efficient solutions and a nonconvex Pareto frontier. The weighted-Tchebycheff method is far more effective than any other method when a function has higher-order terms or is neither convex nor concave. Our numerical example clearly shows the advantages of the proposed model.