D-optimal design of the additive mixture model with multi-response

Zheng Gong; Xiaoyuan Zhu; Chongqi Zhang; Zheng Gong; Xiaoyuan Zhu; Chongqi Zhang

doi:10.3934/mbe.2022221

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 5: 4737-4748. doi: 10.3934/mbe.2022221

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D-optimal design of the additive mixture model with multi-response

1.
School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China
2.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Academic Editor：Daniel Rossit

Received: 26 December 2021 Revised: 29 January 2022 Accepted: 23 February 2022 Published: 10 March 2022

This paper proposes the D-optimal design for the additive mixture model with two-response, which is linear model with no interaction terms. The optimality was validated by using the general equivalence theorem, and the corresponding weights are found under which additive model satisfies D-optimality. In addition, relevant statistics and graphics are given to illustrate our results.
- mixture experiment,
- D-optimal design,
- additive model,
- multi-response
Citation: Zheng Gong, Xiaoyuan Zhu, Chongqi Zhang. D-optimal design of the additive mixture model with multi-response[J]. Mathematical Biosciences and Engineering, 2022, 19(5): 4737-4748. doi: 10.3934/mbe.2022221

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Abstract

This paper proposes the D-optimal design for the additive mixture model with two-response, which is linear model with no interaction terms. The optimality was validated by using the general equivalence theorem, and the corresponding weights are found under which additive model satisfies D-optimality. In addition, relevant statistics and graphics are given to illustrate our results.

References

[1]	C. L. Atwood, Optimal and efficient designs of experiments, Ann. Math. Stat., 40 (1969), 1570–1602.
[2]	J. A. Cornell, Experiments with Mixtures (Designs, Models, and the Analysis of Mixture Data), Wiley, New York, 2002.
[3]	R. Zaitri, M. Bederina, T. Bouziani, Z. Makhloufi, M. Hadjoudja, Development of high performances concrete based on the addition of grinded dune sand and limestone rock using the mixture design modelling approach, Constr. Build. Mater., 60 (2014), 8–16. https://doi.org/10.1016/j.conbuildmat.2014.02.062 doi: 10.1016/j.conbuildmat.2014.02.062
[4]	C. Chandan, R. K. Maheshwari, Mixed solvency concept in reducing surfactant concentration of selfemulsifying drug delivery systems of candesartan cilexetil using d-optimal mixture design, Asian J. Pharm. Sci., 7 (2014), 83–91. http://dx.doi.org/10.22377/ajp.v7i2.31 doi: 10.22377/ajp.v7i2.31
[5]	S. Campos-Barreiro, J. López-Fidalgo, KL-optimal experimental design for discriminating between two growth models applied to a beef farm, Math. Biosci. Eng., 13 (2013), 67–82. https://doi.org/10.3934/mbe.2016.13.67 doi: 10.3934/mbe.2016.13.67
[6]	G. F. Piepel, S. K. Cooley, B. Jones, Construction of a 21-component layered mixture experiment design using a new mixture coordinate-exchange algorithm, Qual. Eng., 17 (2005), 579–594.
[7]	W. J. Welch, ACED: algorithms for the construction of experimental designs, Am. Stat., 39 (1985), 146.
[8]	W. K. Wong, R. B. Chen, C. C. Huang, W. Wang, A modified Particle Swarm Optimization technique for finding optimal designs for mixture models, PLOS ONE, 10 (2015), 1–23. https://doi.org/10.1371/journal.pone.0124720 doi: 10.1371/journal.pone.0124720
[9]	R. Coetzer, L. M. Haines, The construction of D- and I-optimal designs for mixture experiments with linear constraints on the components, Chemom. Intell. Lab. Syst., 171 (2017), 112–124. https://doi.org/10.1016/j.chemolab.2017.10.007 doi: 10.1016/j.chemolab.2017.10.007
[10]	U. Syafitri, B. Sartono, P. Goos, I-optimal design of mixture experiments in the presence of ingredient availability constraints, J. Qual. Technol., 47 (2015), 220–234. https://doi.org/10.1080/00224065.2015.11918129 doi: 10.1080/00224065.2015.11918129
[11]	L. Brown, A. N. Donev, A. C. Bissett, General blending models for data from mixture experiments, Technometrics, 57 (2015), 449–456. https://doi.org/10.1080/00401706.2014.947003 doi: 10.1080/00401706.2014.947003
[12]	B. P. M. Duarte, A. C. Atkinson, J. F. O. Granjo, N. M. C. Oliveira, Optimal design of mixture experiments for general blending models, Chemom. Intell. Lab. Syst., 217 (2021), 104–400. https://doi.org/10.1016/j.chemolab.2021.104400 doi: 10.1016/j.chemolab.2021.104400
[13]	A. Atkinson, A. Donev, R. Tobias, Optimum Experimental Designs, with SAS, Oxford University Press, 2007.
[14]	J. A. Cornell, Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, John Wiley & Sons, 2011.
[15]	P. Goos, U. Syafitri, B. Sartono, A. R. Vazquez, A nonlinear multidimensional knapsack problem in the optimal design of mixture experiments, Eur. J. Oper. Res., 281 (2020), 201–221. https://doi.org/10.1016/j.ejor.2019.08.020 doi: 10.1016/j.ejor.2019.08.020
[16]	B. K. Sinha, N. K. Mandal, M. Pal, P. Das, Optimal Mixture Experiments, Springer, 2014.
[17]	V. V. Fedorov, Theory of Optimal Experiments, Academic Press, 1972.
[18]	N. R. Draper, W. G. Hunter, Design of experiments for parameter estimation in multiresponse situations, Biometrika, 53 (1966), 525–533. https://doi.org/10.2307/2333656 doi: 10.2307/2333656
[19]	L. Imhof, Optimal designs for a multiresponse regression model, J. Multivar. Anal., 72 (2000), 120–131. https://doi.org/10.1006/jmva.1999.1841 doi: 10.1006/jmva.1999.1841
[20]	C. E. Rolz, Statistical design and analysis of experiments, Computer and Information Science Applications in Bioprocess Engineering, Springer, 1996.
[21]	X. Liu, R. X. Yue, W. K. Wong, D-optimal designs for multi-response linear mixed models, Metrika, 82 (2019), 87–98. https://doi.org/10.1007/s00184-018-0679-7 doi: 10.1007/s00184-018-0679-7
[22]	H. Dette, L. Hoyden, S. Kuhnt, K. Schorning, Optimal designs for multi-response generalized linear models with applications in thermal spraying, preprint, arXiv: 1312.4472.
[23]	L. Y. Chan, Optimal design for experiment with mixtures: a survey, Commun. Stat. Theory Methods, 29 (2000), 342–373. https://doi.org/10.1080/03610920008832607 doi: 10.1080/03610920008832607
[24]	J. N. Darroch, J. Waller, Additivity and interaction in three-component experiments with mixtures, Biometrika, 72 (1985), 153–163. https://doi.org/10.1093/biomet/72.1.153 doi: 10.1093/biomet/72.1.153
[25]	L. Y. Chan, Y. N. Guan, C. Q. Zhang, A-optimal designs for an additive quadratic mixture model, Stat. Sin., 8 (1998), 979–990.
[26]	H. Zhao, Y. Guan, D. Han, R-optimal designs for an additive quadratic mixture model, Stat. Sin., 22 (2001), 979–990.
[27]	C. Zhang, Y. Guan, Generalized additive mixture model and its D-optimal designs, Northeast Univ., 1992 (1992).
[28]	C. Q. Zhang, L. Y. Chan, Y. N. Guan, K. H. Li, T. S. Lau, Optimal designs for an additive quadratic mixture model involving the amount of mixture, Stat. Sin., 15 (2005), 165–176.
[29]	J. Kiefer, General equivalence theory for optimal designs (approximate theory), Ann. Stat., 2 (1974), 849–879.

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