D-Optimal Design for Mixture Process Variable with Split-Plot Approach Using a Genetic Algorithm

Authors

  • Dea Handayani Juniar Department of Statistics, IPB University, Bogor, 16680, Indonesia
  • Aji Hamim Wigena Department of Statistics, IPB University, Bogor, 16680, Indonesia
  • Utami Dyah Syafitri Department of Statistics, IPB University, Bogor, 16680, Indonesia

Keywords:

D-Optimality criterion, Genetic Algorithm, Mixture Experiment, Process Variable, Split-plot

Abstract

Mixture-process variables (MPV) is an experiment that responds not only to the proportions of the components but also to the conditions of the process. The impact of MPV is a large number of experimental runs. A large number of experimental runs has a consequence on cost, time, and resource constraints. However, choosing an optimal design with limited runs will ensure efficiency in such cases. In practice, the composition of the mixture experiment will run for each level of the process variables. Therefore, it causes a limitation in randomization. A split-plot approach can be an option to solve the problem, where the whole-plot is the process variables, and the sub-plot is the mixture component. In this research, the authors developed a genetic algorithm (GA) to find the optimal design. The genetic algorithm maintains a population of candidate solutions to a problem. It then selects the candidate point that has the most suitable criteria for solving the problem. The selection criterion used is the D-optimality criterion which is focused on parameter optimization. The case study is an experiment consisting of three ingredients and a process variable with three levels. The result concluded that GA provided an excellent design compared to the coordinate exchange algorithm with the value of D-efficiency for   is 1.195,  is  1.082, and  is 1.078.

References

[1] Cornell, J. (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (3rd ed.). Wiley.
[2] Goos, P., Jones, B. 2011. Optimal Design of Experiment: A Case Study Approach. Wiley.
[3] John J. Borkowski. 2003. Using a Genetic Algorithm to Generate Small Exact Response Surface Designs. Journal of Probability and Statistical Science.1(1), 65-88.
[4] Limmun, W., Borkowski, J. J., & Chomtee, B. (2012). Using a Genetic Algorithm to Generate D-optimal Designs for Mixture Experiments. Quality and Reliability Engineering International, 29(7), 1055–1068. https://doi.org/10.1002/qre.1457
[5] Pradubsri, W., Chomtee, B., & Borkowski, J. J. (2019). Using a genetic algorithm to generate D?optimal designs for mixture?process variable experiments. Quality and Reliability Engineering International, 35(8), 2657–2676. https://doi.org/10.1002/qre.2549
[6] Cornell, J. A. (1988). Analyzing Data from Mixture Experiments Containing Process Variables: A Split-Plot Approach. Journal of Quality Technology, 20(1), 2–23. https://doi.org/10.1080/00224065.1988.11979079
[7] Goos, P., & Donev, A. N. (2007). Tailor-Made Split-Plot Designs for Mixture and Process Variables. Journal of Quality Technology, 39(4), 326–339. https://doi.org/10.1080/00224065.2007.11917699
[8] Kiefer, J., & Wolfowitz, J. (1960). The Equivalence of Two Extremum Problems. Canadian Journal of Mathematics, 12, 363–366. https://doi.org/10.4153/cjm-1960-030-4
[9] Goos, P., & Vandebroek, M. (2001). Optimal Split-Plot Designs. Journal of Quality Technology, 33(4), 436–450. https://doi.org/10.1080/00224065.2001.11980103
[10] Goos, P., & Vanderbroek, M. (2003). D-Optimal Split-Plot Designs With Given Numbers and Sizes of Whole Plots. Technometrics, 45(3), 235–245. https://doi.org/10.1198/004017003000000050
[11] Atkinson, Donev, Tobias. 2007. Optimum Experimental Designs with SAS. Oxford: Oxford University Press.
[12] Holland, J. H.(1975). Adaption in Natural and Artificial Systems, Universitas of Michigan Press, Ann Arbor, MI.
[13] Davis, L. (1991). Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York.
[14] Golberg, D. E.(1991). Real-coded Genetic Algorithms, Virtual Alphabets, and Blocking, Complex System, 5, 139-168.
[15] Michalewicz, Z. (1992). Genetic algorithm + Data Structures = Evolution Programs, Springer-Verlag, New York.
[16] Heredia-Langner, A., Carlyle, W. M., Montgomery, D. C., Borror, C. M., & Runger, G. C. (2003). Genetic Algorithms for the Construction of D-optimal Designs. Journal of Quality Technology, 35(1), 28–46. https://doi.org/10.1080/00224065.2003.11980189

Downloads

Published

2021-12-08

How to Cite

Dea Handayani Juniar, Wigena, A. H., & Utami Dyah Syafitri. (2021). D-Optimal Design for Mixture Process Variable with Split-Plot Approach Using a Genetic Algorithm . International Journal of Sciences: Basic and Applied Research (IJSBAR), 60(4), 360–375. Retrieved from https://gssrr.org/index.php/JournalOfBasicAndApplied/article/view/13562

Issue

Vol. 60 No. 4 (2021)

Section

Articles

License

Authors who submit papers with this journal agree to the following terms.