I-Optimal Axial Designs for Four Ingredient Concrete Experiment

Njoroge Elizabeth Wambui; Koske Joseph; Mutiso John

doi:doi:10.11648/j.ajtas.20211001.15

| Peer-Reviewed

I-Optimal Axial Designs for Four Ingredient Concrete Experiment

Njoroge Elizabeth Wambui, Koske Joseph, Mutiso John

Published in American Journal of Theoretical and Applied Statistics (Volume 10, Issue 1)

Received: 13 November 2020 Accepted: 1 December 2020 Published: 28 January 2021

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Stakeholders in the construction industry work towards obtaining optimal concrete mixes with an aim of producing structures with the best compressive strength. In many instances, Kenya has witnessed collapse of buildings leading to death and huge financial loses, which has been associated largely to poor concrete mixes. This paper aims at evaluating the I-optimal designs for a concrete mixture experiment for both Equally Weighted Simplex Centroid Axial Design and Unequally Weighted Simplex Centroid Axial Design, based on the second-degree Kronecker model. Optimality tests are performed to locate the optimum values of a design. In various studies, I-optimality has been shown to be among the best criteria in obtaining the most optimal outcomes. In this study, Response Surface Methodology is applied in evaluating I-optimal designs, which are known to minimize average or integrated prediction variance over the experimental region. I-optimality equivalence conditions for the inscribed tetrahedral design and for the concrete experiment model are identical with the boundary points, mid-face points and the centroid, denoted by η₂, η₃ and η₄ respectively. Equally, Weighted Simplex Centroid Axial Design proved to be a more I-efficient design than the Unequally Weighted Simplex Centroid Axial Design for both the tetrahedral design and the concrete model, with 87.85% and 79.54% respectively. The optimal response surface occurred in the region of the I-optimal designs. The Kronecker model derived from the concrete mixture experiment proved effective and efficient in describing the observed results.

Published in	American Journal of Theoretical and Applied Statistics (Volume 10, Issue 1)
DOI	10.11648/j.ajtas.20211001.15
Page(s)	32-37
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

I-Optimality, Tetrahedral, Efficiency, Equivalence, Average Prediction

References

[1]	Atkinson A. C., Donev A. N., Tobias R. D (2007). Optimum Experimental Designs, with SAS. Oxford University Press, Oxford.
[2]	Cornell, J. A. (2002). Experiments with Mixtures, Third Edition. John Wiley & Sons Inc, New York.
[3]	Draper N. and John R. (1977). Designs in three and four components for mixture models with inverse terms. Technometrices, 19, 17-30.
[4]	Draper, N. R. and Pukelsheim, F. (1998). Mixture models based on homogeneous polynomials. Journal of statistical planning and inference, 71, pp. 303-311.
[5]	Fedorov, V. (1972). Theory of optimal experiments. Academic Press, New York.
[6]	Jones B., and Goos P., (2012). I-optimal Versus D-optimal Slit-plot response surface designs. Journal of Quality Technology vol 44 (2).
[7]	Kerich G., Koske J., Rutto M., Korir B., Ronoh B., Kinyanjui J., Kungu P., (2014). D-Optimal Designs for Third-Kronecker Model Mixture Experiments with an Application to Artificial Sweetener Experiment. IOSR Journal of Mathematics vol 10 (6). pp 32-41.
[8]	Montgomery D. C., (2001). Design and Analysis of Experiments 5^th Ed. John Wiley and Sons, Inc.
[9]	Njoroge E. W., Koske, J., Mutiso J. (2020). D- and G- Optimal Axial Slope Designs for Four Ingredient Mixture. Journal of Applied Mathematics and Physics vol 8 (1). pp 20-25.
[10]	Njoroge E. W., Koske, J., Mutiso J. (2020). Optimization of Plinth Concrete Mix Using Quad-Axial Weighted Simplex Centroid Designs And Second Degree Kronecker Model. Not published.
[11]	Peter G., Bradely J. and Utami S. (2013) I- optimal mixture designs.www.repec.org, Research Papers in Economics D/2013/1169/033.
[12]	Rady, E. A., Abd EL-Monsef, M. M. E., Seyam, M. M., (2009). Relationship among several optimality criteria. Interstat Journal volume 15 (6). Pp 1-11.
[13]	Sinha B. K., Das P., Mandal N. K. and Pal M., (2010). Parameter Estimation of Linear and Quadratic Mixture Models. A review. Pak. Statist, vol 26 (1), 77-96.
[14]	Sinha B. K., Das P., Mandal N. K. and Pal M., (2014). Optimal Mixture Experiments. http://www.Springer.com/978-81-322-1785-5.
[15]	Ugbe T. A. and Akpan S. S., (2013). On the comparision of the boundary and interior support points of a response surface under optimality criteria. International journal of Mathematics and Statistics studies. Vol 1 (4), pp 48-58.

Cite This Article

Plain Text BibTeX RIS

APA Style

Njoroge Elizabeth Wambui, Koske Joseph, Mutiso John. (2021). I-Optimal Axial Designs for Four Ingredient Concrete Experiment. American Journal of Theoretical and Applied Statistics, 10(1), 32-37. https://doi.org/10.11648/j.ajtas.20211001.15

Copy | Download

ACS Style

Njoroge Elizabeth Wambui; Koske Joseph; Mutiso John. I-Optimal Axial Designs for Four Ingredient Concrete Experiment. Am. J. Theor. Appl. Stat. 2021, 10(1), 32-37. doi: 10.11648/j.ajtas.20211001.15

Copy | Download

AMA Style

Njoroge Elizabeth Wambui, Koske Joseph, Mutiso John. I-Optimal Axial Designs for Four Ingredient Concrete Experiment. Am J Theor Appl Stat. 2021;10(1):32-37. doi: 10.11648/j.ajtas.20211001.15

Copy | Download

@article{10.11648/j.ajtas.20211001.15,
  author = {Njoroge Elizabeth Wambui and Koske Joseph and Mutiso John},
  title = {I-Optimal Axial Designs for Four Ingredient Concrete Experiment},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {1},
  pages = {32-37},
  doi = {10.11648/j.ajtas.20211001.15},
  url = {https://doi.org/10.11648/j.ajtas.20211001.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211001.15},
  abstract = {Stakeholders in the construction industry work towards obtaining optimal concrete mixes with an aim of producing structures with the best compressive strength. In many instances, Kenya has witnessed collapse of buildings leading to death and huge financial loses, which has been associated largely to poor concrete mixes. This paper aims at evaluating the I-optimal designs for a concrete mixture experiment for both Equally Weighted Simplex Centroid Axial Design and Unequally Weighted Simplex Centroid Axial Design, based on the second-degree Kronecker model. Optimality tests are performed to locate the optimum values of a design. In various studies, I-optimality has been shown to be among the best criteria in obtaining the most optimal outcomes. In this study, Response Surface Methodology is applied in evaluating I-optimal designs, which are known to minimize average or integrated prediction variance over the experimental region. I-optimality equivalence conditions for the inscribed tetrahedral design and for the concrete experiment model are identical with the boundary points, mid-face points and the centroid, denoted by η2, η3 and η4 respectively. Equally, Weighted Simplex Centroid Axial Design proved to be a more I-efficient design than the Unequally Weighted Simplex Centroid Axial Design for both the tetrahedral design and the concrete model, with 87.85% and 79.54% respectively. The optimal response surface occurred in the region of the I-optimal designs. The Kronecker model derived from the concrete mixture experiment proved effective and efficient in describing the observed results.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - I-Optimal Axial Designs for Four Ingredient Concrete Experiment
AU  - Njoroge Elizabeth Wambui
AU  - Koske Joseph
AU  - Mutiso John
Y1  - 2021/01/28
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211001.15
DO  - 10.11648/j.ajtas.20211001.15
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 32
EP  - 37
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211001.15
AB  - Stakeholders in the construction industry work towards obtaining optimal concrete mixes with an aim of producing structures with the best compressive strength. In many instances, Kenya has witnessed collapse of buildings leading to death and huge financial loses, which has been associated largely to poor concrete mixes. This paper aims at evaluating the I-optimal designs for a concrete mixture experiment for both Equally Weighted Simplex Centroid Axial Design and Unequally Weighted Simplex Centroid Axial Design, based on the second-degree Kronecker model. Optimality tests are performed to locate the optimum values of a design. In various studies, I-optimality has been shown to be among the best criteria in obtaining the most optimal outcomes. In this study, Response Surface Methodology is applied in evaluating I-optimal designs, which are known to minimize average or integrated prediction variance over the experimental region. I-optimality equivalence conditions for the inscribed tetrahedral design and for the concrete experiment model are identical with the boundary points, mid-face points and the centroid, denoted by η2, η3 and η4 respectively. Equally, Weighted Simplex Centroid Axial Design proved to be a more I-efficient design than the Unequally Weighted Simplex Centroid Axial Design for both the tetrahedral design and the concrete model, with 87.85% and 79.54% respectively. The optimal response surface occurred in the region of the I-optimal designs. The Kronecker model derived from the concrete mixture experiment proved effective and efficient in describing the observed results.
VL  - 10
IS  - 1
ER  -

Copy | Download

Author Information

Njoroge Elizabeth Wambui

Department of Physical Sciences, Chuka University, Chuka, Kenya
Koske Joseph

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

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

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Njoroge Elizabeth Wambui, Koske Joseph, Mutiso John. (2021). I-Optimal Axial Designs for Four Ingredient Concrete Experiment. American Journal of Theoretical and Applied Statistics, 10(1), 32-37. https://doi.org/10.11648/j.ajtas.20211001.15

Copy | Download

ACS Style

Njoroge Elizabeth Wambui; Koske Joseph; Mutiso John. I-Optimal Axial Designs for Four Ingredient Concrete Experiment. Am. J. Theor. Appl. Stat. 2021, 10(1), 32-37. doi: 10.11648/j.ajtas.20211001.15

Copy | Download

AMA Style

Njoroge Elizabeth Wambui, Koske Joseph, Mutiso John. I-Optimal Axial Designs for Four Ingredient Concrete Experiment. Am J Theor Appl Stat. 2021;10(1):32-37. doi: 10.11648/j.ajtas.20211001.15

Copy | Download

@article{10.11648/j.ajtas.20211001.15,
  author = {Njoroge Elizabeth Wambui and Koske Joseph and Mutiso John},
  title = {I-Optimal Axial Designs for Four Ingredient Concrete Experiment},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {1},
  pages = {32-37},
  doi = {10.11648/j.ajtas.20211001.15},
  url = {https://doi.org/10.11648/j.ajtas.20211001.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211001.15},
  abstract = {Stakeholders in the construction industry work towards obtaining optimal concrete mixes with an aim of producing structures with the best compressive strength. In many instances, Kenya has witnessed collapse of buildings leading to death and huge financial loses, which has been associated largely to poor concrete mixes. This paper aims at evaluating the I-optimal designs for a concrete mixture experiment for both Equally Weighted Simplex Centroid Axial Design and Unequally Weighted Simplex Centroid Axial Design, based on the second-degree Kronecker model. Optimality tests are performed to locate the optimum values of a design. In various studies, I-optimality has been shown to be among the best criteria in obtaining the most optimal outcomes. In this study, Response Surface Methodology is applied in evaluating I-optimal designs, which are known to minimize average or integrated prediction variance over the experimental region. I-optimality equivalence conditions for the inscribed tetrahedral design and for the concrete experiment model are identical with the boundary points, mid-face points and the centroid, denoted by η2, η3 and η4 respectively. Equally, Weighted Simplex Centroid Axial Design proved to be a more I-efficient design than the Unequally Weighted Simplex Centroid Axial Design for both the tetrahedral design and the concrete model, with 87.85% and 79.54% respectively. The optimal response surface occurred in the region of the I-optimal designs. The Kronecker model derived from the concrete mixture experiment proved effective and efficient in describing the observed results.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - I-Optimal Axial Designs for Four Ingredient Concrete Experiment
AU  - Njoroge Elizabeth Wambui
AU  - Koske Joseph
AU  - Mutiso John
Y1  - 2021/01/28
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211001.15
DO  - 10.11648/j.ajtas.20211001.15
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 32
EP  - 37
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211001.15
AB  - Stakeholders in the construction industry work towards obtaining optimal concrete mixes with an aim of producing structures with the best compressive strength. In many instances, Kenya has witnessed collapse of buildings leading to death and huge financial loses, which has been associated largely to poor concrete mixes. This paper aims at evaluating the I-optimal designs for a concrete mixture experiment for both Equally Weighted Simplex Centroid Axial Design and Unequally Weighted Simplex Centroid Axial Design, based on the second-degree Kronecker model. Optimality tests are performed to locate the optimum values of a design. In various studies, I-optimality has been shown to be among the best criteria in obtaining the most optimal outcomes. In this study, Response Surface Methodology is applied in evaluating I-optimal designs, which are known to minimize average or integrated prediction variance over the experimental region. I-optimality equivalence conditions for the inscribed tetrahedral design and for the concrete experiment model are identical with the boundary points, mid-face points and the centroid, denoted by η2, η3 and η4 respectively. Equally, Weighted Simplex Centroid Axial Design proved to be a more I-efficient design than the Unequally Weighted Simplex Centroid Axial Design for both the tetrahedral design and the concrete model, with 87.85% and 79.54% respectively. The optimal response surface occurred in the region of the I-optimal designs. The Kronecker model derived from the concrete mixture experiment proved effective and efficient in describing the observed results.
VL  - 10
IS  - 1
ER  -

Copy | Download