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 overrepresent one region of the design space, or neglect others. Some commonly used methods comply with this requirement, whereas 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 � -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.

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

Optimization is one of the most important and challenging parts of any engineering design. In real-world design, multiobjective optimization with constraints has to be considered. The optimal solution in this case is not unique because the objectives can contradict each other. Therefore, a set of optimal solutions, which forms the Pareto frontier, should be considered. There are many algorithms to generate a Pareto set. However, only a few of them are potentially capable of providing an evenly distributed set of solutions. This property is especially important in real-life design because a decision maker is usually able to analyse only a very limited number of solutions. The main objective of this article is to develop and give detailed description of an algorithm that is able to generate an evenly distributed Pareto set in a general formulation. The approach is based on shrinking a search domain to generate a Pareto optimal solution in a selected area on the Pareto frontier. The effectiveness of the algo...

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Directed search domain: a method for even generation of the Pareto frontier in multiobjective optimization

Robust parameter design (RPD) has recently been applied in modern industries in a large deal of processes. This technique is occasionally employed as a multiobjective optimization approach using weighted sums as a trade-off strategy; in such cases, however, a considerable number of gaps have arisen. In this paper, the use of normal boundary intersection (NBI) method coupled with mean-squared error (MSE) functions is proposed. This approach is capable of generating equispaced Pareto frontiers for a bi-objective robust design model, independent of the relative scales of the objective functions. To verify the adequacy of this proposal, a central composite design (CCD) is developed with combined arrays for the AISI 1045 steel end milling process. In this case study, a CCD with three noise factors and four control factors are used to create the mean and variance equations for MSE of two quality characteristics. The numerical results indicate the NBI-MSE approach is capable of generating a convex and equispaced Pareto frontier to MSE functions of surface roughness, thus nullifying the drawbacks of weighted sums. Moreover, the results show that the achieved optimum lessens the sensitivity of the end milling process to the variability transmitted by the noise factors.

https://www.pedro.unifei.edu.br/Artigos%20Publicados/2014PrecisionNBI.pdf

A normal boundary intersection approach to multiresponse robust optimization of the surface roughness in end milling process with combined arrays

This article proposes a novel predictor–corrector (PC) method for the numerical treatment of multi-objective optimization problems (MOPs). The algorithm, Pareto Tracer (PT), is capable of performing a continuation along the set of (local) solutions of a given MOP with k objectives, and can cope with equality and box constraints. Additionally, the first steps towards a method that manages general inequality constraints are also introduced. The properties of PT are first discussed theoretically and later numerically on several examples.

Pareto Tracer: a predictor–corrector method for multi-objective optimization problems

On ε-Constraint Based Methods for the Generation of Pareto Frontiers

In multidisciplinary optimization the designer needs to find solution to optimization problems which include a number of usually contradicting criteria. Such a problem is mathematically related to the field of nonlinear vector optimization where there are many numerical methods capable of providing a solution. However, only a few of those are suitable for real multidisciplinary design in industry because an iteration design circle usually is very time-consuming. This is due to the time scales and computational resources associated with each iterative design cycle. The recently suggested Physical Programming Method appears to match many requirements raised in industrial applications. The method is modified to make its realization easier and more efficient, the main focus being the even generation of the complete Pareto set. The method is used to find the Pareto surface for different test cases.

A Method for Generating a Well-Distributed Pareto Set in Nonlinear Multiobjective Optimization

https://personalpages.manchester.ac.uk/staff/S.Utyuzhnikov/Papers/WebsitePapers/UFG2009.pdf

A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

In the design process of complex systems, the designer is solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the space of the objective functions. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution in order to explore its sensitivity. In this paper, a method for obtaining linear and quadratic local approximations of the Pareto surface is derived. The concept of a local quick Pareto analyser is proposed. This concept is based on a local sensitivity analysis, which provides the relation between variations of the different objective functions under constraints. A few examples are considered.

Local Approximation of Pareto Surface.

In the design process of complex systems, the designer is solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously Mathematically this problem can be reduced to a vector optimization problem The solution of this problem is not unique and is represented by a Pareto surface in the space of the objective functions Once a Pareto solution is obtained, it may be very useful for the decision‐maker to be able to perform a quick local approximation in the vicinity of this Pareto solution in order to explore its sensitivity In this paper, a method for obtaining linear and quadratic local approximations of the Pareto surface is derived A few examples are considered

S. V. Utyuzhnikov

Papers

A Method for Generating a Well-Distributed Pareto Set in Nonlinear Multiobjective Optimization

A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

Local Approximation of Pareto Surface.

Local Pareto Approximation