Research Papers: Design Automation

A New Hybrid Algorithm for Multi-Objective Robust Optimization With Interval Uncertainty

[+] Author and Article Information
Shuo Cheng, Jianhua Zhou

University of Michigan-Shanghai Jiao Tong
University Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China

Mian Li

University of Michigan-Shanghai Jiao Tong
University Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: mianli@sjtu.edu.cn

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 2, 2014; final manuscript received October 27, 2014; published online November 26, 2014. Assoc. Editor: Christopher Mattson.

J. Mech. Des 137(2), 021401 (Feb 01, 2015) (9 pages) Paper No: MD-14-1100; doi: 10.1115/1.4029026 History: Received February 02, 2014; Revised October 27, 2014; Online November 26, 2014

Uncertainty is a very critical but inevitable issue in design optimization. Compared to single-objective optimization problems, the situation becomes more difficult for multi-objective engineering optimization problems under uncertainty. Multi-objective robust optimization (MORO) approaches have been developed to find Pareto robust solutions. While the literature reports on many techniques in MORO, few papers focus on using multi-objective differential evolution (MODE) for robust optimization (RO) and performance improvement of its solutions. In this article, MODE is first modified and developed for RO problems with interval uncertainty, formulating a new MODE-RO algorithm. To improve the solutions’ quality of MODE-RO, a new hybrid (MODE-sequential quadratic programming (SQP)-RO) algorithm is proposed further, where SQP is incorporated into the procedure to enhance the local search. The proposed hybrid approach takes the advantage of MODE for its capability of handling not-well behaved robust constraint functions and SQP for its fast local convergence. Two numerical and one engineering examples, with two or three objective functions, are tested to demonstrate the applicability and performance of the proposed algorithms. The results show that MODE-RO is effective in solving MORO problems while, on the average, MODE-SQP-RO improves the quality of robust solutions obtained by MODE-RO with comparable numbers of function evaluations.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Ben-Tal, A., and Nemirovski, A., 2002, “Robust Optimization–Methodology and Applications,” Math. Prog., 92(3), pp. 453–480. [CrossRef]
Park, G. J., Lee, T. H., Kwon, H. L., and Hwang, K. H., 2006, “Robust Design: An Overview,” AIAA J., 44(1), pp. 181–191. [CrossRef]
Hajimiragha, A. H., Canizares, C. A., Fowler, M. W., Moazeni, S., and Elkamel, A., 2010, “A Robust Optimization Approach for Planning the Transition to Plug-In Hybrid Electric Vehicles,” IEEE Trans. Power Syst., 26(4), pp. 2264–2274. [CrossRef]
Ferreira, R. S., Barroso, L. A., and Carvalho, M. M., 2012, “Demand Response Models With Correlated Price Data: A Robust Optimization Approach,” Appl. Energy, 96, pp. 133–149. [CrossRef]
Taguchi, G., 1978, “Performance Analysis Design,” Int. J. Prod. Res., 16(6), pp. 521–530. [CrossRef]
Riley, M. E., and Grandhi, R. V., 2011, “Quantification of Model-Form and Predictive Uncertainty for Multi-Physics Simulation,” Comput. Struct., 89(11), pp. 1206–1213. [CrossRef]
Parkinson, A., Sorensen, C., and Pourhassan, N., 1993, “A General Approach for Robust Optimal Design,” ASME J. Mech. Des., 115(1), pp. 74–80. [CrossRef]
Hughes, E., 2001, “Evolutionary Multi-Objective Ranking With Uncertainty and Noise,” Proceedings of the First International Conference on Evolutionary Multi-Criterion Optimization, Switzerland, Mar. 7–9, Springer-Verlag, London, UK, pp. 329–343. [CrossRef]
Tsutsui, S., and Ghosh, A., 1997, “Genetic Algorithms With a Robust Solution Searching Scheme,” IEEE Trans. Evol. Comput., 1(3), pp. 201–208. [CrossRef]
Youn, B. D., Choi, K. K., and Park, Y. H., 2003, “Hybrid Analysis Method for Reliability-Based Design Optimization,” ASME J. Mech. Des., 125(2), pp. 221–232. [CrossRef]
Beyer, H.-G., and Sendhoff, B., 2007, “Robust Optimization—A Comprehensive Survey,” Comput. Methods Appl. Mech. Eng., 196(33–34), pp. 3190–3218. [CrossRef]
Mourelatos, Z. P., and Zhou, J., 2005, “A Design Optimization Method Using Evidence Theory,” ASME J. Mech. Design, 128(4), pp. 901–908. [CrossRef]
Du, L., Choi, K. K., and Lee, I., 2007, “Robust Design Concept in Possibility Theory and Optimization for System With Both Random and Fuzzy Input Variables,” ASME Paper No. DETC2007-35106. [CrossRef]
Deb, K., and Gupta, H., 2006, “Introducing Robustness in Multi-Objective Optimization,” Evol. Comput., 14(4), pp. 463–494. [CrossRef] [PubMed]
Gaspar-Cunha, A., and Covas, J. A., 2007, “Robustness in Multi-Objective Optimization Using Evolutionary Algorithms,” Comput. Optim. Appl., 39(1), pp. 75–96. [CrossRef]
Gunawan, S., and Azarm, S., 2005, “Multi-Objective Robust Optimization Using a Sensitivity Region Concept,” Struct. Multidiscip. Optim., 29(1), pp. 50–60. [CrossRef]
Li, M., Azarm, S., and Boyars, A., 2006, “A New Deterministic Approach Using Sensitivity Region Measures for Multi-Objective Robust and Feasibility Robust Design Optimization,” ASME J. Mech. Des., 128(4), pp. 874–883. [CrossRef]
Hu, W., Li, M., Azarm, S., and Almansoori, A., 2011, “Multi-Objective Robust Optimization Under Interval Uncertainty Using Online Approximation and Constraint Cuts,” ASME J. Mech. Des., 133(6), p. 061002. [CrossRef]
Deb, K., Gupta, S., Daum, D., and Branke, J., 2009, “Reliability-Based Optimization Using Evolutionary Algorithm,” IEEE Trans. Evol. Comput., 13(5), pp. 1054–1074. [CrossRef]
Dudy, L., Yaochu, J., Yew-Soon, O., and Sendhoff, B., 2010, “Generalizing Surrogate-Assisted Evolutionary Computation,” IEEE Trans. Evol. Comput., 134(3), pp. 329–355. [CrossRef]
Daum, D. A., Deb, K., and Branke, J., 2007, “Reliability-Based Optimization for Multiple Constraints With Evolutionary Algorithms,” IEEE Congress on Evolutionary Computation, Singapore, Sept. 25–28, pp. 911–918. [CrossRef]
Yew-Soon, O., Nair, P. B., and Lum, K. Y., 2006, “Max–Min Surrogate-Assisted Evolutionary Algorithm for Robust Design,” IEEE Trans. Evol. Comput., 10(4), pp. 392–404. [CrossRef]
Saha, A., Ray, T., and Smith, W., 2011, “Towards Practical Evolutionary Robust Multi-Objective Optimization,” IEEE Congress on Evolutionary Computation, New Orleans, LA, June 5–8, pp. 2123–2130. [CrossRef]
Kukkonen, S., and Lampinen, J., 2004, “An Extension of Generalized Differential Evolution for Multi-Objective Optimization,” 8th International Conference, Birmingham, UK, Sept. 18–22, pp. 752–761. [CrossRef]
Kukkonen, S., and Lampinen, J., 2005, “GDE3: The Third Evolution Step of Generalized Differential Evolution,” IEEE Congress on Evolutionary Computation, Edinburgh, Scotland, Sept. 5, Vol. 1, pp. 443–450. [CrossRef]
Kumar, A., Sharma, D., and Deb, K., 2007, “A Hybrid Multi-Objective Optimization Procedure Using PCX Based NSGA-II and Sequential Quadratic Programming,” Proceedings of the Congress on Evolutionary Computation (CEC-2007), Singapore, Sept. 25–28, pp. 3011–3018. [CrossRef]
Gao, X., Chen, B., He, X., Qiu, T., Li, J., Wang, C., and Zhang, L., 2008, “Multi-Objective Optimization for the Periodic Operation of the Naphtha Pyrolysis Process Using a New Parallel Hybrid Algorithm Combining NSGA-II With SQP,” Comput. Chem. Eng., 32(11), pp. 2801–2811. [CrossRef]
Mansoornejad, B., Mostoufi, N., and Jalali-Farahani, F., 2008, “A Hybrid GA–SQP Optimization Technique for Determination of Kinetic Parameters of Hydrogenation Reactions,” Comput. Chem. Eng., 32(7), pp. 1447–1455. [CrossRef]
Hu, X., Huang, Z., and Wang, Z., 2003, “Hybridization of the Multi-Objective Evolutionary Algorithms and the Gradient-Based Algorithms,” 2003 Congress on Evolutionary Computation, CEC’03, Canberra, Australia, Dec. 8–12, Vol. 2, pp. 870–877. [CrossRef]
Knowles, J., and Corne, D., 2005, “Memetic Algorithms for Multiobjective Optimization: Issues, Methods and Prospects,” Recent Advances Memetic Algorithms, Springer, Berlin, Germany, pp. 313–352. [CrossRef]
Ishibuchi, H., Yoshida, T., and Murata, T., 2003, “Balance Between Genetic Search and Local Search in Memetic Algorithms for Multiobjective Permutation Flowshop Scheduling,” IEEE Trans. Evol. Comput., 7(2), pp. 204–223. [CrossRef]
Chiang, T.-C., and Fu, L.-C., 2010, “An Improved Multiobjective Memetic Algorithm for Permutation Flow Shop Scheduling,” IEEE Congress on Evolutionary Computation, Barcelona, July 18–23, pp. 1–8. [CrossRef]
Ono, S., and Nakayama, S., 2009, “Multi-Objective Particle Swarm Optimization for Robust Optimization and Its Hybridization With Gradient Search,” IEEE Congress on Evolutionary Computation, CEC'09, Trondheim, May 18–21, pp. 1629–1636. [CrossRef]
Li, M., 2007, “Robust Optimization and Sensitivity Analysis With Multi-Objective Genetic Algorithms: Single- and Multi-Disciplinary Applications,” Ph.D. thesis, University of Maryland, College Park, MD.
Tušar, T., and Filipič, B., 2007, “Differential Evolution Versus Genetic Algorithms in Multiobjective Optimization,” Evolutionary Multi-Criterion Optimization, S.Obayashi, K.Deb, C.Poloni, T.Hiroyasu, and T.Murata, eds., Springer, Berlin, Germany, pp. 257–271.
Storn, R., and Price, K., 1997, “Differential Evolution—A Simple and Efficient Heuristic for Global Optimization Over Continuous Spaces,” J. Global Optim., 11(4), pp. 341–359. [CrossRef]
Zhou, J., Cheng, S., and Li, M., 2012, “Sequential Quadratic Programming for Robust Optimization With Interval Uncertainty,” ASME J. Mech. Des., 134(10), p. 100913. [CrossRef]
Deb, K., 2001, Multiobjective Optimization Using Evolutionary Algorithms, Wiley, NY.
Arora, J. S., 2004, Introduction to Optimum Design, Elsevier, NY.
Sarker, R., and Abbass, H. A., 2004, “Differential Evolution for Solving Multi-Objective Optimization Problems,” Asia-Pacific J. Oper. Res., 21(2), pp. 225–240. [CrossRef]
Santana-Quintero, L. V., and Coello Coello, C. A., 2005, “An Algorithm Based on Differential Evolution for Multi-Objective Problems,” Int. J. Comput. Intell. Res., 1(2), pp. 151–169. [CrossRef]
Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T., 2002, “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II,” IEEE Trans. Evol. Comput., 6(2), pp. 182–197. [CrossRef]
Mezura-Montes, E., Coello Coello, C., and Tun-Morales, E., 2004, “Simple Feasibility Rules and Differential Evolution for Constrained Optimization,” Proceedings of the 3rd Mexican International Conference on Artificial Intelligence, Mexico, City, Mexico, Apr. 26–30, Springer-Verlag, Berlin, Heidelberg, pp. 707–716. [CrossRef]
Haimes, Y., Lasdon, L., and Wismer, D., 1971, “On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization,” IEEE Trans. Syst., Man Cybern., 1(3), pp. 296–297. [CrossRef]
Azarm, S., and Wu, J., 2001, “Metrics for Quality Assessment of a Multiobjective Design Optimization Solution Set,” ASME J. Mech. Des., 123(1), pp. 18–25. [CrossRef]
Deb, K., Thiele, L., Laumanns, M., and Zitzler, E., 2001, “Scalable Test Problems for Evolutionary Multi-Objective Optimization,” TIK-Technical Report No. 112.
Narayanan, S., and Azarm, S., 1999, “On Improving Multiobjective Genetic Algorithms for Design Optimization,” Struct. Optim., 18(2), pp. 146–155. [CrossRef]


Grahic Jump Location
Fig. 3

Flowchart of MODE-RO

Grahic Jump Location
Fig. 4

Flowchart of MODE-SQP-RO

Grahic Jump Location
Fig. 5

Average hyper area (AHA)

Grahic Jump Location
Fig. 10

Robust optimal solutions for three-objective example

Grahic Jump Location
Fig. 11

Robust optimal solutions for vibrating platform

Grahic Jump Location
Fig. 12

(a) Objective robustness and (b) constraint robustness verification for R1

Grahic Jump Location
Fig. 6

Robust optimal solutions for OSY

Grahic Jump Location
Fig. 13

(a) Objective robustness and (b) constraint robustness verification for R2

Grahic Jump Location
Fig. 7

(a) Objective robustness and (b) constraint robustness verification for R1

Grahic Jump Location
Fig. 8

(a) Objective robustness and (b) constraint robustness verification for R2

Grahic Jump Location
Fig. 9

Overall spread (OS)




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In