Normal Constraint Method with Guarantee of Even Representation of Complete Pareto Frontier

Abstract

Multiobjective optimization is rapidly becoming an invaluable tool in engineering design. A particular class of solutions to the multiobjective optimization problem is said to belong to the Pareto frontier. A Pareto solution, the set of which comprises the Pareto frontier, is optimal in the sense that any improvement in one design objective can only occur with the worsening of at least one other. Accordingly, the Pareto frontier plays an important role in engineering design -it characterizes the tradeoffs between conflicting design objectives. Some optimization methods can be used to automatically generate a set of Pareto solutions from which a final design is subjectively chosen by the designer. For this approach to be successful, the generated Pareto set must be truly representative of the complete optimal design space (Pareto frontier). In other words, the set must not over represent one region of the design space, or neglect others. Some commonly used methods comply with this requirement, while others do not. This paper offers a new phase in the development of the Normal Constraint method, which is a simple approach for generating Pareto solutions that are evenly distributed in the design space of an arbitrary number of objectives. The even distribution of the generated Pareto solutions can facilitate the process of developing an analytical expression for the Pareto frontier in n-dimension. An even distribution of Pareto solutions also facilitates the task of choosing the most desirable (final) design from among the set of Pareto solutions. The Normal Constraint method bears some similarities to the Normal Boundary Intersection and ε-Constraint methods. Importantly, the developments presented in this paper define its critical distinction: Namely, the ability to generate a set of evenly distributed Pareto solutions over the complete Pareto frontier. Examples are provided that show the Normal Constraint method to perform favorably under the new developments when compared with the Normal Boundary Intersection method, as well as with the original Normal Constraint method.

A novel approach for traffic accidents sanitary resource allocation based on multi-objective genetic algorithms

“…In the existing literature, we can find different analytical and numeric algorithms to approximate the Pareto front, such as Normal Boundary Intersection (NBI) (Indraneel and Dennis, 1998), Normal Constraint (NC) (Messac et al, 2003;Messac and Mattson, 2004), and Directed Search Domain (DSD) (Erfani and Utyuzhnikov, 2010). However, these methods present several shortcomings due to the significant number of redundant solutions that they can generate (Utyuzhnikov et al, 2009), and the problems that arise when the Pareto frontier is not continuous (Shukla and Deb, 2007).…”

Computational system for multi disciplinary optimization at conceptual design stage

“…The DHCBI method shares conceptual similarities with the NBI and NC methods [6][7][8][9][10] for which an optimization is performed for each of the Pareto points sought. However, as it will be described below, the DHCBI is designed to improve the effectiveness of the search.…”

Performance assessment of the hybrid Archive-based Micro Genetic Algorithm (AMGA) on the CEC09 test problems

“…This scenario is not desirable, since it would not speed-up the convergence towards the Pareto-optimal frontier. It should be noted that there exist several other scalarization schemes such as Epsilonconstraint method [18] and Normal-constraint method [18]. These scalarization schemes are designed to obtain a uniform distribution of points on the Pareto-optimal front.…”

“…16 Update the archive (using the off-spring population). 17 until (termination) 18 Report desired number of solutions from the archive.…”