DOI QR코드

DOI QR Code

Development of a Multiobjective Optimization Algorithm Using Data Distribution Characteristics

데이터 분포특성을 이용한 다목적함수 최적화 알고리즘 개발

  • Received : 2009.12.29
  • Accepted : 2010.10.14
  • Published : 2010.12.01

Abstract

The weighting method and goal programming require weighting factors or target values to obtain a Pareto optimal solution. However, it is difficult to define these parameters, and a Pareto solution is not guaranteed when the choice of the parameters is incorrect. Recently, the Mahalanobis Taguchi System (MTS) has been introduced to minimize the Mahalanobis distance (MD). However, the MTS method cannot obtain a Pareto optimal solution. We propose a function called the skewed Mahalanobis distance (SMD) to obtain a Pareto optimal solution while retaining the advantages of the MD. The SMD is a new distance scale that multiplies the skewed value of a design point by the MD. The weighting factors are automatically reflected when the SMD is calculated. The SMD always gives a unique Pareto optimal solution. To verify the efficiency of the SMD, we present two numerical examples and show that the SMD can obtain a unique Pareto optimal solution without any additional information.

가중치법이나 목표계획법을 이용하여 다목적함수 최적화를 수행할 때 설계자는 각 함수에 적절한 가중치나 목표값을 설정해 주어야 한다. 하지만 파라미터를 잘못 설정하게 되면 파레토 최적해를 얻지못하기 때문에 이는 설계자에게 큰 부담이 된다. 최근에 데이터의 분포특성만을 이용하여 데이터의 평균과 함수 사이의 거리를 표현하는 마하라노비스 거리(MD)를 최소화하는 MTS기법이 개발되었다. 이 방법은 파라미터를 설정하지 않아도 되는 장점이 있지만 최적해가 참고데이터의 평균으로 수렴하는 단점이 있다. 따라서 본 연구에서는 방향성이 없는 기존의 MD에 방향성을 부여한 새로운 거리 척도인 SMD를 제안하였다. 그리고 SMD법이 계산과정에서 각 함수의 가중치를 자동으로 반영하고 평균에서 가장 멀리 위치한 한 점을 항상 파레토 최적해로 제공한다는 것을 2개의 단순예제를 통해 검증하였다.

Keywords

References

  1. Marler, R.T. and Arora, J.S., 2004, “Survey of Multi- Objective Optimization Method for Engineering,” Struct Multidisc Optim 26, pp. 369-395. https://doi.org/10.1007/s00158-003-0368-6
  2. Taguchi, G. and Jugulum, R., 2002, “The Mahalanobis-Taguchi Strategy : A Pattern Technology ystem,” John Wiley & Sons.
  3. Maesschalck, R.D., Jouan, R. and Massart, D.L., 2002, “Tutorial: The Mahalanobis distance,” Chemometrics and Intelligent Laboratory Systems 50, pp. 1-18. https://doi.org/10.1016/S0169-7439(99)00047-7
  4. Kim, I.Y. and de Weck, O.L., 2005, “Adaptive Weighted-Sum Method for Bi-Objective Optimization: Pareto Front Generation,” Struct Multidisc Optim 29, pp. 149-158. https://doi.org/10.1007/s00158-004-0465-1
  5. Athan, T.W. and Papalambros, P.Y., 1996, “A Note on Weighted Criteria Methods for Compromise Solutions in Multi-Objective Optimization,” Engineering Optimization. 27, pp. 155-176. https://doi.org/10.1080/03052159608941404
  6. Osyczka, A., 1989, “Multicriterion Decision Making with Min-Max Approach,” Optimization Methods in Structural Design, Edited by H.Eschenauer, N. Olhoff, Euromech-Colloquium 164, Wissenschaftsverlag.
  7. Nakayama, H. and Furukawa, K., 1985, “Satisficing Trade-Off Method with an Application to Multiobjective Structure Design,” Large Scale Systems, Vol. 8, pp. 47-57.
  8. Miettinen, K. and Makela, M.M., 2002, “On Scalarizing Functions in Multiobjective Optimization,” OR Spectrum 24, pp. 193-213. https://doi.org/10.1007/s00291-001-0092-9
  9. Goel, P.S. and Singh, N., 1998, “A Multi-Objective Design-Optimization Model with Total Life Cycle Consideration,” Society of Automotive Engineer, 982167.
  10. Yoon, K.P. and Hwang, C.L., “Multiple Attribute Decision Making,” An Introduction. London: Sage Publications.
  11. Saaty, T.L., 1997, “A Scaling Method for Priorities in Hierarchies, Multiple Objectives and Fuzzy Sets,” J.Math. Psych.15, pp. 234-281. https://doi.org/10.1016/0022-2496(77)90033-5
  12. Charnes, A. and Cooper, W.W., 1961, “Management Models and Industrial Applications of Linear Programming,” Vols. 1 and 2. John Wiley, New York.
  13. Charnes, A. and Cooper, W.W., 1977, “Goal Programming and Multiple Objective Optimization,” Part 1. Eur. J. Oper. Res. 1, pp. 39-54. https://doi.org/10.1016/S0377-2217(77)81007-2
  14. Hwang, C.L. and Md.Masud, A.S., 1979, “Multiple Objective Decision Making, Methods and Applications: A State-of-the-Art Survey,” In: Beckmann, M. and Kunzi, H.P.(Eds.) Lecture Notes in Economics and mathematical Systems, No.164.
  15. Hwang, I.J., Hyun, Y.J. and Park, J.H., 2007, “The Study for the Improvement of On-Center Feel with MTS Technique,” Society of Automotive Engineers, 2007-01-0090.
  16. Kenney, J.F. and Keeping, E.S., 1962, “Mathematics of Statistics,” Pt.1,3rd ed. Princeton, NJ: Van Nostrand.
  17. Kim, W.C., Kim, J.J., Park, B.W., Park, S.H., Song, M.S., Lee, S.Y., Lee, Y.C., Chon, C.W. and Cho, S.S, 1998, “Modern Statistics, 4th Edition,” Young-ji MunhaSa.
  18. FRAMAX Inc., 2007, “PIAnO(Process Integration, Automation and Optimization) User’s manual”, ver 2.1.