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Keywords: multiobjective optimization, multidimensional Pareto frontier, normal boundary intersection method, normal constraint method.

Summary

Advances in computational and numerical capabilities allow for more efficient engineering design through the solution of more realistic optimization problems. For instance, considering that the true goal of a design is the improvement of a set of objectives, leading to the so-called multiobjective optimization problem (MOP).

The Pareto optimality concept [1] is, in general, adopted in the multiobjective optimization problem and several approaches have been proposed in the literature to obtain Pareto points distribution, such as: the weighting sum method (WS), the min-max method, the normal boundary intersection method (NBI) and the normal constraint method (NC). Among those the NBI and NC methods are very efficient approaches to obtain good Pareto distributions for two objective optimization problems. However, when more than two objective functions are considered, all standard techniques referred above may fail in covering the whole Pareto frontier. To overcome such problem a novel modified procedure to the standard NBI and NC methods for more than two objective functions is presented in this work. The purpose is to obtain an even distribution of points over the whole Pareto frontier without significant additional computational cost. Other modification of standard methods, like that proposed by Messac and Mattson [2] to the NC method, is also examined.

In this paper, the MOP and the concept of Pareto optimality are briefly revisited. Then, the NBI method is detailed, and the difficulties of the standard existing methods to handle problems involving more than two objectives are illustrated, especially for the NBI and NC methods. A detailed description of the proposed modification is presented and the scheme is tested and compared to other existing schemes by the solution of several model problems, including three and four objective applications. For the applications analyzed the results obtained validate the proposed scheme and demonstrate its superiority, or at least equivalent performance, when compared against the other tested schemes. The improvements are both in terms of quality and computational efficiency.

References

1: C.L. Hwang, S.R. Paidy, K. Yoon, A.S.M. Masud, "Mathematical Programming with Multiple Objectives: A Tutorial", Comput. and Ops. Res., 7, 5-31, 1980. doi:10.1016/0305-0548(80)90011-8
2: A. Messac, C.A. Mattson, "Normal constraint method with guarantee of even representation of complete Pareto frontier", 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Material Conference, Palm Springs, CA, 2004. doi:10.2514/1.8977

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 96 PROCEEDINGS OF THE THIRTEENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping and Y. Tsompanakis Paper 143 The Solution of N-Multiobjective Optimization Problems using Modified Normal Boundary Intersection and Normal Constraint Methods R.S. Motta¹, S.M.B.A. da Silva¹ and P.R.M. Lyra² ¹Civil Engineering Department, ²Mechanics Engineering Department, Federal university of Pernambuco (UFPE), Recife PE, Brazil doi:10.4203/ccp.96.143 purchase the full-text of this paper Full Bibliographic Reference for this paper R.S. Motta, S.M.B.A. da Silva, P.R.M. Lyra, "The Solution of N-Multiobjective Optimization Problems using Modified Normal Boundary Intersection and Normal Constraint Methods", in B.H.V. Topping, Y. Tsompanakis, (Editors), "Proceedings of the Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 143, 2011. doi:10.4203/ccp.96.143 Keywords: multiobjective optimization, multidimensional Pareto frontier, normal boundary intersection method, normal constraint method. Summary Advances in computational and numerical capabilities allow for more efficient engineering design through the solution of more realistic optimization problems. For instance, considering that the true goal of a design is the improvement of a set of objectives, leading to the so-called multiobjective optimization problem (MOP). The Pareto optimality concept [1] is, in general, adopted in the multiobjective optimization problem and several approaches have been proposed in the literature to obtain Pareto points distribution, such as: the weighting sum method (WS), the min-max method, the normal boundary intersection method (NBI) and the normal constraint method (NC). Among those the NBI and NC methods are very efficient approaches to obtain good Pareto distributions for two objective optimization problems. However, when more than two objective functions are considered, all standard techniques referred above may fail in covering the whole Pareto frontier. To overcome such problem a novel modified procedure to the standard NBI and NC methods for more than two objective functions is presented in this work. The purpose is to obtain an even distribution of points over the whole Pareto frontier without significant additional computational cost. Other modification of standard methods, like that proposed by Messac and Mattson [2] to the NC method, is also examined. In this paper, the MOP and the concept of Pareto optimality are briefly revisited. Then, the NBI method is detailed, and the difficulties of the standard existing methods to handle problems involving more than two objectives are illustrated, especially for the NBI and NC methods. A detailed description of the proposed modification is presented and the scheme is tested and compared to other existing schemes by the solution of several model problems, including three and four objective applications. For the applications analyzed the results obtained validate the proposed scheme and demonstrate its superiority, or at least equivalent performance, when compared against the other tested schemes. The improvements are both in terms of quality and computational efficiency. References 1 C.L. Hwang, S.R. Paidy, K. Yoon, A.S.M. Masud, "Mathematical Programming with Multiple Objectives: A Tutorial", Comput. and Ops. Res., 7, 5-31, 1980. doi:10.1016/0305-0548(80)90011-8 2 A. Messac, C.A. Mattson, "Normal constraint method with guarantee of even representation of complete Pareto frontier", 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Material Conference, Palm Springs, CA, 2004. doi:10.2514/1.8977 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £130 +P&P)
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