Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions

Nyakundi Omwando Cornelious; Evans Mbuthi Kilonzo

doi:doi:10.11648/j.ijssam.20210602.11

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Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions

Nyakundi Omwando Cornelious, Evans Mbuthi Kilonzo

Published in International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 2)

Received: 6 April 2021 Accepted: 19 April 2021 Published: 29 April 2021

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Abstract

Response surface methodology (RSM) is a statistical technique used to evaluate the relationship between multiple input variables and one or more response variables with the aim of optimizing the response variables. Sequential experiments are very economical and useful in practice. Therefore, rotatable designs such as the third order rotatable design (TORD) may be run sequentially in three stages with three or four blocks depending on the model adequacy. Normally, the first section consisting of first order is run and the response function is approximated using a first order model. If the first order model is found to be adequate, as the representation of the unknown function by noting the evidence of the goodness of fit, the experiment may be stopped at this stage. However, if the first model is found to be unfit, the trials of the second order are run and ultimately, proceed to fit a third order if a second order model is also found to be inadequate. In this paper, two sets of second order rotatable designs are combined to form sequential third order rotatable designs (TORD) in three, four and five dimensions. The TORDs are then evaluated on their alphabetic optimality criteria with the aim of reducing the costs of experimentation. The classical optimality criteria includes; D-criterion, A-criterion, T-criterion and E- criterion.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 2)
DOI	10.11648/j.ijssam.20210602.11
Page(s)	35-39
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Response Surface Methodology, Third Order Rotatable Designs, Optimality Criteria

References

[1]	Box, G., & Hunter, J. (1957). Multifactor Experimental Designs for Exploring Response Surfaces. The Annals of Mathematical Statistics, 28 (1), 195-241.
[2]	cornelious, N. (2019). A Hypothetical sequential third order rotatable design in five dimensions. Eoropean International Journal of Science and Technology, 8 (4), 40-50.
[3]	Cornelious, N. (2019). A sequential third order rotatable design in four dimensions. current journal of applied science and technology, 33 (6), 1-7.
[4]	Cornelious, N. (2019). construction of thirty nine points second order rotatable design in three dimensions with a practical hypothetical example. European International Journal of Science and Technology, 8 (4), 51-57.
[5]	cornelious, N. (2019). construction of thirty six points second order rotatable design in three dimensions with a practical hypothetical case study. International Journal of Advances in Scientific Reseach and Engineering, 5 (6).
[6]	cornelious, N., & Cruiff, M. (2019). A sequential third order rotatable design of eighty points in four dimensions with an hypothetical case study. Asian Journal of Probability and Statistics, 4 (4), 1-9.
[7]	Draper, N. (1960). Third order rotatable designs in three dimensions. The Annals of Mathematical statistics, 31 (4), 865-874.
[8]	Gardiner, D., Grandage, A., & Hader. R. J. (1959). Third order rotatable designs for exploring response surfaces. Annals of Mathematical Statistics, 30 (4), 1082-1096.
[9]	Herzberg, A. (1967). A method for the construction of second order rotatable designs in K dimensions. The Annals of Mathematical Statistics, 38 (1), 177-180.
[10]	Kiefer, J. (1960). Optimum experimental designs V, with applications to systematic and rotatable designs. In Proc. 4th Berkeley Symp, 1 (1), 381-405.
[11]	Kosgei, M. (2002). optimality criteria for the specific second order rotatable designs. M. Phil. Thesis, Moi Univesrity, Eldoret Kenya.
[12]	Mutai, C., Koske, J., & Mutiso., J. (2012). construction of four dimensional third order rotatable design through balanced incomplete block designs. Advances and Applications in Statistics, 27 (1).
[13]	Mutai, N. H. (2020). Modified non-sequential third order rotatable designs constructed using pairwise balanced design. Statistical Theory and Related Fields, 6 (4), 1-5.
[14]	Nyakundi, C. O. (2016). Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions. M. Phil. Thesis, Moi University, Eldoret, Kenya.
[15]	Rotich, J., Kosgei, M., & Kerich., G. (2017). Optimal Third Order Rotatable Designs constructed from balanced incomplete block designs (BIBD). Current journal of applied science and technology, 22 (3), 1-5.

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Nyakundi Omwando Cornelious, Evans Mbuthi Kilonzo. (2021). Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions. International Journal of Systems Science and Applied Mathematics, 6(2), 35-39. https://doi.org/10.11648/j.ijssam.20210602.11

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ACS Style

Nyakundi Omwando Cornelious; Evans Mbuthi Kilonzo. Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions. Int. J. Syst. Sci. Appl. Math. 2021, 6(2), 35-39. doi: 10.11648/j.ijssam.20210602.11

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AMA Style

Nyakundi Omwando Cornelious, Evans Mbuthi Kilonzo. Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions. Int J Syst Sci Appl Math. 2021;6(2):35-39. doi: 10.11648/j.ijssam.20210602.11

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@article{10.11648/j.ijssam.20210602.11,
  author = {Nyakundi Omwando Cornelious and Evans Mbuthi Kilonzo},
  title = {Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {2},
  pages = {35-39},
  doi = {10.11648/j.ijssam.20210602.11},
  url = {https://doi.org/10.11648/j.ijssam.20210602.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210602.11},
  abstract = {Response surface methodology (RSM) is a statistical technique used to evaluate the relationship between multiple input variables and one or more response variables with the aim of optimizing the response variables. Sequential experiments are very economical and useful in practice. Therefore, rotatable designs such as the third order rotatable design (TORD) may be run sequentially in three stages with three or four blocks depending on the model adequacy. Normally, the first section consisting of first order is run and the response function is approximated using a first order model. If the first order model is found to be adequate, as the representation of the unknown function by noting the evidence of the goodness of fit, the experiment may be stopped at this stage. However, if the first model is found to be unfit, the trials of the second order are run and ultimately, proceed to fit a third order if a second order model is also found to be inadequate. In this paper, two sets of second order rotatable designs are combined to form sequential third order rotatable designs (TORD) in three, four and five dimensions. The TORDs are then evaluated on their alphabetic optimality criteria with the aim of reducing the costs of experimentation. The classical optimality criteria includes; D-criterion, A-criterion, T-criterion and E- criterion.},
 year = {2021}
}

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TY  - JOUR
T1  - Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions
AU  - Nyakundi Omwando Cornelious
AU  - Evans Mbuthi Kilonzo
Y1  - 2021/04/29
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210602.11
DO  - 10.11648/j.ijssam.20210602.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 35
EP  - 39
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210602.11
AB  - Response surface methodology (RSM) is a statistical technique used to evaluate the relationship between multiple input variables and one or more response variables with the aim of optimizing the response variables. Sequential experiments are very economical and useful in practice. Therefore, rotatable designs such as the third order rotatable design (TORD) may be run sequentially in three stages with three or four blocks depending on the model adequacy. Normally, the first section consisting of first order is run and the response function is approximated using a first order model. If the first order model is found to be adequate, as the representation of the unknown function by noting the evidence of the goodness of fit, the experiment may be stopped at this stage. However, if the first model is found to be unfit, the trials of the second order are run and ultimately, proceed to fit a third order if a second order model is also found to be inadequate. In this paper, two sets of second order rotatable designs are combined to form sequential third order rotatable designs (TORD) in three, four and five dimensions. The TORDs are then evaluated on their alphabetic optimality criteria with the aim of reducing the costs of experimentation. The classical optimality criteria includes; D-criterion, A-criterion, T-criterion and E- criterion.
VL  - 6
IS  - 2
ER  -

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Author Information

Nyakundi Omwando Cornelious

Department of Mathematics, Physics, and Computing, Moi University, Eldoret, Kenya
Evans Mbuthi Kilonzo

Department of Mathematics, Physics, and Computing, Moi University, Eldoret, Kenya

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APA Style

Nyakundi Omwando Cornelious, Evans Mbuthi Kilonzo. (2021). Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions. International Journal of Systems Science and Applied Mathematics, 6(2), 35-39. https://doi.org/10.11648/j.ijssam.20210602.11

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ACS Style

Nyakundi Omwando Cornelious; Evans Mbuthi Kilonzo. Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions. Int. J. Syst. Sci. Appl. Math. 2021, 6(2), 35-39. doi: 10.11648/j.ijssam.20210602.11

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AMA Style

Nyakundi Omwando Cornelious, Evans Mbuthi Kilonzo. Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions. Int J Syst Sci Appl Math. 2021;6(2):35-39. doi: 10.11648/j.ijssam.20210602.11

Copy | Download

@article{10.11648/j.ijssam.20210602.11,
  author = {Nyakundi Omwando Cornelious and Evans Mbuthi Kilonzo},
  title = {Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {2},
  pages = {35-39},
  doi = {10.11648/j.ijssam.20210602.11},
  url = {https://doi.org/10.11648/j.ijssam.20210602.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210602.11},
  abstract = {Response surface methodology (RSM) is a statistical technique used to evaluate the relationship between multiple input variables and one or more response variables with the aim of optimizing the response variables. Sequential experiments are very economical and useful in practice. Therefore, rotatable designs such as the third order rotatable design (TORD) may be run sequentially in three stages with three or four blocks depending on the model adequacy. Normally, the first section consisting of first order is run and the response function is approximated using a first order model. If the first order model is found to be adequate, as the representation of the unknown function by noting the evidence of the goodness of fit, the experiment may be stopped at this stage. However, if the first model is found to be unfit, the trials of the second order are run and ultimately, proceed to fit a third order if a second order model is also found to be inadequate. In this paper, two sets of second order rotatable designs are combined to form sequential third order rotatable designs (TORD) in three, four and five dimensions. The TORDs are then evaluated on their alphabetic optimality criteria with the aim of reducing the costs of experimentation. The classical optimality criteria includes; D-criterion, A-criterion, T-criterion and E- criterion.},
 year = {2021}
}

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TY  - JOUR
T1  - Optimal Sequential Third Order Rotatable Designs in Three, Four and Five Dimensions
AU  - Nyakundi Omwando Cornelious
AU  - Evans Mbuthi Kilonzo
Y1  - 2021/04/29
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210602.11
DO  - 10.11648/j.ijssam.20210602.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 35
EP  - 39
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210602.11
AB  - Response surface methodology (RSM) is a statistical technique used to evaluate the relationship between multiple input variables and one or more response variables with the aim of optimizing the response variables. Sequential experiments are very economical and useful in practice. Therefore, rotatable designs such as the third order rotatable design (TORD) may be run sequentially in three stages with three or four blocks depending on the model adequacy. Normally, the first section consisting of first order is run and the response function is approximated using a first order model. If the first order model is found to be adequate, as the representation of the unknown function by noting the evidence of the goodness of fit, the experiment may be stopped at this stage. However, if the first model is found to be unfit, the trials of the second order are run and ultimately, proceed to fit a third order if a second order model is also found to be inadequate. In this paper, two sets of second order rotatable designs are combined to form sequential third order rotatable designs (TORD) in three, four and five dimensions. The TORDs are then evaluated on their alphabetic optimality criteria with the aim of reducing the costs of experimentation. The classical optimality criteria includes; D-criterion, A-criterion, T-criterion and E- criterion.
VL  - 6
IS  - 2
ER  -

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