Enumeration of Strength Three Orthogonal Arrays and Their Implementation in Parameter Design


This paper describes the construction and enumeration of mixed orthogonal arrays (MOA) to produce optimal experimental designs. A MOA is a multiset whose rows are the different combinations of factor levels, discrete values of the variable under study, having very well defined features such as symmetry and strength three (all main interactions are taken in consideration). The applied methodology blends the fields of combinatorics and group theory by applying the ideas of orbits, stabilizers and isomorphisms to array generation and enumeration. Integer linear programming was used in order to exploit the symmetry property of the arrays under study. The backtrack search algorithm was used to find suitable arrays in the underlying space of possible solutions. To test the performance of the MOAs, an engineered system was used as a case study within the stage of parameter design. The analysis showed how the MOAs were capable of meeting the fundamental engineering design axioms and principles, creating optimal experimental designs within the desired context.

Share and Cite:

Romero, J. and Murray, S. (2015) Enumeration of Strength Three Orthogonal Arrays and Their Implementation in Parameter Design. Journal of Applied Mathematics and Physics, 3, 38-45. doi: 10.4236/jamp.2015.31006.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Fisher, R.A. (1954) The Design of Experiments. Hafner Publishing Company.
[2] Cochran, W.G. and Cox, G.M. (1950) Experimental designs. Wiley Classics Library, Wiley.
[3] Kempthorne, O. (1967) The Design and Analysis of Experiments. Wiley Publications in Statistics. R. E. Krieger Pub. Co.
[4] Scheffe, H. (1959) The Analysis of Variance. A Wiley Publication in Mathematical Statistics, Wiley.
[5] Ross, P.J. (2005) Taguchi Techniques for Quality Engineering. McGraw-Hill Education (India) Pvt Limited.
[6] Hedayat, A.S., Sloane, N.J.A. and Stufken, J. (1999) Orthogonal Arrays: Theory and Applications. Springer Series in Statistics. Springer-Verlag. http://dx.doi.org/10.1007/978-1-4612-1478-6
[7] Diamond, W.J. (2001) Practical Experiment Designs: For Engineers and Scientists. Industrial Engineering/Quality Management. Wiley.
[8] Montgomery, D.C. (2012) Design and Analysis of Experiments. Design and Analysis of Experiments. Wiley.
[9] Nguyen, M.M.M. (2005) Computer-Algebraic Methods for the Construction of Designs of Experiments. Technische Universiteit Eindhoven. Eindhoven University Press.
[10] DeVor, R.E., Chang, T. and Sutherland, J.W. (2007) Statistical Quality Design and Control: Contemporary Concepts and Methods. Pearson/Prentice Hall.
[11] Schoen, E.D. and Nguyen, M.V.M. (2007) Enumeration and Classification of Orthogonal Arrays. Journal of Algebraic Statistics.
[12] Schoen, E.D., Eendebak, P.T. and Nguyen, M.V.M. (2009) Complete Enumeration of Pure Level and Mixed-Level Orthogonal Arrays. University of Antwerp.
[13] Nguyen, M.V.M. and Murray, S.H. (2007) Enumeration of Strength Three Mixed Orthogonal Arrays. School of Mathematics and Statistics, University of Sydney.
[14] Brouwer, A.E. (2009) A c Program Constructs 3b2a Orthogonal Arrays of Strength Three Using Depth First Search. Technische Universiteit Eindhoven.
[15] Butler, G. (1982) Computing in Permutation and Matrix Groups II: Backtrack Algorithm. American Mathematical Socie-ty.
[16] Tang, B.X. and Zhou, J. (2013) D-Optimal Two-Level Orthogonal Arrays for Estimating Main Effects and Some Specified Two-Factor Interactions. Metrika.
[17] Brouerius, V. (2006) Parallel Construction of Orthogonal Arrays. Technische Universiteit Eindhoven.
[18] He, Y. and Tang, B. (2013) Strong Orthogonal Arrays and Associated Latin Hypercubes for Computer Experiments. Biometrika. http://dx.doi.org/10.1093/biomet/ass065
[19] Angelopoulos, P., Evangelaras, H., Koukouvinos, C. and Lappas, E. (2006) An Effective Step-Down Algorithm for the Construction and the Identification of Non-Isomorphic Orthogonal Arrays. Springer Verlag.
[20] Nguyen, M.V.M. (2007) Some New Constructions of Strength 3 Mixed Orthogonal Arrays. Journal of Statistical Planning and Inference.
[21] Glynn, D.G. and Byatt, D. (2012) Graphs for Orthogonal Arrays and Projective Planes of Even Order. SIAM Journal on Discrete Mathematics.
[22] Tsai, K., Ye, K.Q. and Li, W. (2006) A Complete Catalog of Geometrically Non-Isomorphic 18-Run Orthogonal Arrays. Operations and Management Science, University of Minnesota.
[23] Sloane, N.J.A. and Stufken, J. (1996) A Linear Programming Bound for Orthogonal Arrays with Mixed Levels. Journal of Statistical Planning and Inference. http://dx.doi.org/10.1016/S0378-3758(96)00025-0
[24] Nguyen, M.V.M. and Murray, S.H. (2010) Enumeration of Orthogonal Arrays Using Group Theory and Integer Linear Programming. Journal of Algebraic Statistics.
[25] Huynh, T.V. (2010) Orthogonal Array Experiment in Systems Engineering and Architecting. Department of Systems Engineering.
[26] Chatzopoulos, S., Kolyva-Machera, F. and Chatterjee, K. (2011) Optimality Results on Orthogonal Arrays Plus p Runs for s Factorial Experiments. Metrika, 73, 385-394.
[27] Evangelaras, H., Kolaiti, E. and Koukouvinos, C. (2006) Robust Parameter Design: Optimization of Combined Array Approach with Orthogonal Arrays. Journal of Statistical Planning and Inference, 136, 3698-3709.
[28] Tsai, J.-T. (2010) Robust Optimal-Parameter Design Approach for Tolerance Design Problems. Engineering Optimization, 42, 1079-1093.
[29] Lekivetz, R. and Tang, B. (2012) Multi-Level Orthogonal Arrays for Estimating Main Effects and Specified Interactions. Journal of Statistical Planning and Inference.
[30] Lekivetz, R. (2011) Optimal Factorial Designs with Robust Properties. Simon Fraser University.
[31] Byth, K. and Gebski, V. (2005) Factorial Designs: A Graphical Aid for Choosing Study Designs Accounting for Interaction. SCT Journal, 1, 315-325.
[32] Suh, N.P. (1990) The Principles of Design. Oxford Series on Advanced Manufacturing. Oxford University Press on Demand.
[33] Blanchard, B.S. and Fabrycky, W.W.J. (2011) Systems Engineering and Analysis. Prentice-Hall International Series in Industrial and Systems Engineering. Prentice Hall PTR.
[34] Boylestad, R. and Nashelsky, L. (2009) Electronic Devices and Circuit Theory. Pearson-/Prentice Hall.
[35] Sedra, A.S. and Smith, K.C. (1998) Circuitos Microelectronicos. The Oxford Series in Electrical and Computer Engineering. Oxford University Press, Incorporated.
[36] Hulpke, A. (2010) Notes on Computational Group Theory. Department of Mathematics. Colorado State University.
[37] Holt, D.F., Eick, B. and O’Brien, E.A. (2005) Handbook of Computational Group Theory. Discrete Mathematics and Its Applications. Taylor & Francis. http://dx.doi.org/10.1201/9781420035216
[38] Fraleigh, J.B. (2004) A First Course in Abstract Algebra. Addison Wesley.
[39] Artin, M. (2011) Algebra. Pearson Prentice Hall.
[40] Hu, T.C. and Shing, M.T. (2002) Combinatorial Algorithms. In: Hu, T.C. and Shing, M.T., Eds., Dover Books on Computer Science Series, Dover Publ.

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.