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A hybrid elitist pareto-based coordinate exchange algorithm for constructing multi-criteria optimal experimental designs


This paper presents a new Pareto-based coordinate exchange algorithm for populating or approximating the true Pareto front for multi-criteria optimal experimental design problems that arise naturally in a range of industrial applications. This heuristic combines an elitist-like operator inspired by evolutionary multi-objective optimization algorithms with a coordinate exchange operator that is commonly used to construct optimal designs. Benchmarking results from both a two-dimensional and three-dimensional example demonstrate that the proposed hybrid algorithm can generate highly reliable Pareto fronts with less computational effort than existing procedures in the statistics literature. The proposed algorithm also utilizes a multi-start operator, which makes it readily parallelizable for high performance computing infrastructures.

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  1. Ajith, A., Crina, G., Hisao, I.: Hybrid Evolutionary Algorithms. Springer, Berlin (2007)

    MATH  Google Scholar 

  2. Borkowski, J.J.: Using a genetic algorithm to generate small exact response surface designs. J. Probab. Stat. Sci. 1(1), 65–88 (2003)

    Google Scholar 

  3. Bursztyn, D., Steinberg, D.M.: Comparison of designs for computer experiments. J. Stat. Plan. Inference 136, 1103–1119 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  4. Cao, Y., Smucker, B.J., Robinson, T.J.: On using the hypervolume indicator to compare pareto fronts: applications to multi-criteria optimal experimental design. J. Stat. Plan. Inference 160, 60–74 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  5. Cook, D.R., Nachtsheim, C.J.: A comparison of algorithms for constructing exact d-optimal designs. Technometrics 22(3), 315–324 (1980)

    Article  MATH  Google Scholar 

  6. Das, I., Dennis, J.E.: A closer look at drawbacks of minimizing weighted sums of objectives for pareto set generation in multicriteria optimization problems. Struct. Optim. 14, 63–69 (1997)

    Article  Google Scholar 

  7. Elhossini, A., Areibi, S., Dony, R.: Strength pareto particle swarm optimization and hybrid ea-pso for multi-objective optimization. Evo. Comput. 18, 127–156 (2010)

    Article  Google Scholar 

  8. Farrell, R.H., Kiefer, J., Walbran, A.: Optimum multivariate designs. In LM LeCam and J. Neyman (eds.), Proceedings of the fifth Berkeley symposium on mathematical statistics and probability. University of California Press 1, 113–138 (1967)

  9. Fedorov, V.V.: Theory of Optimal Experiments. Elsevier Science, Philadelphia (1972)

    Google Scholar 

  10. Gert van, V., Tommi, T.: “hit and run” and “shake and bake” for sampling uniformly from convex shapes. R Package (2015)

  11. Jones, B., Nachtsheim, C.J.: Efficient designs with minimal aliasing. Technometrics 53, 62–71 (2011)

    MathSciNet  Article  Google Scholar 

  12. Kiefer, J., Wolfowitz, J.: Optimum designs in regression problems. Ann. Math. Stat. 30, 271–294 (1959)

    MathSciNet  Article  MATH  Google Scholar 

  13. Kiefer, J.: Optimum experimental designs. J. R. Stat. Soc. Ser. B (Methodological) 21, 272–319 (1959)

    MathSciNet  MATH  Google Scholar 

  14. Kiefer, J.: Optimum designs in regression problems, ii. Ann. Math. Stat. 32, 298–325 (1961)

    MathSciNet  Article  MATH  Google Scholar 

  15. Limmun, W., Borkowski, J.J., Chomtee, B.: Using a genetic algorithm to generate d-optimal designs for mixture experiments. Qual. Reliab. Eng. Int. 29(7), 1099–1638 (2013)

    Article  Google Scholar 

  16. Lu, L., Anderson-Cook, C.M., Robinson, T.J.: Optimization of designed experiments based on multiple criteria utilizing a pareto frontier. Technometrics 53(4), 353–365 (2011)

    MathSciNet  Article  Google Scholar 

  17. Mashwani, W.K.: Hybrid multi-objective evolutionary algorithms: A survey of the state-of-the-art. Int. J. Comput. Sci. Issues 8, 374–392 (2011)

    Google Scholar 

  18. Meyer, R.K., Nachtsheim, C.J.: The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 37(1), 60–69 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  19. Park, Y.J.: Multi-optimal designs for second-order response surface models. Commun. Korean Stat. Soc. 16(1), 195–208 (2009)

    Google Scholar 

  20. Sambo, F., Borrotti, M., Mylona, K.: A coordinate-exchange two-phase local search algorithm for the d- and i-optimal designs of split-plot experiments. Comput. Stat. Data Anal. 71, 1193–1207 (2014)

  21. Scrucca, L.: R package ‘ga’ (2014)

  22. Sexton, C.J., Anthony, D.K., Lewis, S.M., Please, C.P., Keane, A.J.: Design of experiment algorithms for assembled products. J. Qual. Technol. 38, 298–308 (2006)

    Google Scholar 

  23. Sindhya, K., Miettinen, K., Deb, K.: A hybrid framework for evolutionary multi-objective optimization. Evol. Comput. IEEE Trans. 17, 495–511 (2013)

    Article  MATH  Google Scholar 

  24. Smith, N.A., Tromble, R.W.: Sampling uniformly from the unit simplex. Johns Hopkins University, Technical Report, pp. 1–6 (2004)

  25. Wang, L., Wu, H., Tang, F., Zheng, D.Z.: A hybrid quantum-inspired genetic algorithm for flow shop scheduling. In: Huang, D.S., Zhang, X.P., Huang, G.B. (eds.) Advances in Intelligent Computing, pp. 636–644. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  26. Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. Evol. Comput. IEEE Trans. 1, 67–82 (1997)

    Article  Google Scholar 

  27. Yang, D., Jiao, L., Gong, M.: Adaptive multi-objective optimization based on nondominated solutions. Comput. Intell. 25, 84–108 (2009)

    MathSciNet  Article  Google Scholar 

  28. Zhou, A., Qu, B.Y., Li, H., Zhao, S.Z., Suganthan, P.N., Zhang, Q.: Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol. Comput. 1, 32–49 (2011)

    Article  Google Scholar 

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The authors would like to thank Drs. Christine Anderson-Cook, Lu Lu and You-jin Park for sharing their examples. We also wish to express gratitude to the reviewers and editor who reviewed this work and allowed us the opportunity to improve the paper. The first author is also grateful to Dr. Nancy Flournoy for her valuable suggestions and comments.

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Correspondence to Yongtao Cao.

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Cao, Y., Smucker, B.J. & Robinson, T.J. A hybrid elitist pareto-based coordinate exchange algorithm for constructing multi-criteria optimal experimental designs. Stat Comput 27, 423–437 (2017).

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  • Multi-criteria optimal experimental design
  • Pareto front
  • Hybrid algorithm