Abstract
Generation of orthogonal fractional factorial designs (OFFDs) is an important and extensively studied subject in applied statistics. In this paper we show how searching for an OFFD that satisfies a set of constraints, expressed in terms of orthogonality between simple and interaction effects, is, in many applications, equivalent to solving an integer linear programming problem. We use a recent methodology, based on polynomial counting functions and strata, that represents OFFDs as the positive integer solutions of a system of linear equations. We use this system to set up an optimization problem where the cost function to be minimized is the size of the OFFD and the constraints are represented by the system itself. Finally we search for a solution using standard integer programming techniques. Some applications are also presented in the computational results section. It is worth noting that the methodology does not put any restriction either on the number of levels of each factor or on the orthogonality constraints and so it can be applied to a very wide range of designs, including mixed orthogonal arrays.
Similar content being viewed by others
References
Bailey RA (2008) Design of comparative experiments. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511611483
Bailey RA, Cameron PJ, Connelly R (2008) Sudoku, gerechte designs, resolutions, affine space, spreads, Reguli, and hamming codes. Amer Math Monthly 115(5): 383–404
Berkelaar M, Eikland K, Notebaert P (2004) lpsolve, version 5.5. Available at http://lpsolve.sourceforge.net/5.5/
Bose RC (1947) Mathematical theory of the symmetrical factorial design. Sankhyā 8: 107–166
Chakravarti IM (1956) Fractional replication in asymmetrical factorial designs and partially balanced arrays. Sankhyā 17: 143–164
Collombier D (1996) Plans D’Expérience Factoriels. Construction et propriétés des fractions de plans. No. 21 in Mathématiques et applications. Springer, Paris
Dey A, Mukerjee R (1999) Fractional factorial plans. Wiley series in probability and statistics: probability and statistics. Wiley, New York. doi:10.1002/9780470316986, a Wiley-Interscience Publication
Fontana R, Pistone G (2010a) Algebraic generation of orthogonal fractional factorial designs. In: Proceedings of the 45th scientific meeting of the Italian statistical society, Padova, 16–18 June 2010. “http://homes.stat.unipd.it/mgri/SIS2010/Program/contributedpaper/654-1297-3-DR.pdf”
Fontana R, Pistone G (2010b) Algebraic strata for non symmetrical orthogonal fractional factorial designs and application. La matematica e le sue applicazioni 2010/1, Politecnico di Torino DIMAT
Fontana R, Rogantin MP (2010) Indicator function and sudoku designs. In: Algebraic and geometric methods in statistics. Cambridge University Press, Cambridge, pp 203–224
Fontana R, Pistone G, Rogantin MP (2000) Classification of two-level factorial fractions. J Stat Plan Inference 87(1): 149–172
Hedayat AS, Sloane NJA, Stufken J (1999) Orthogonal arrays. Theory and applications. Springer series in statistics. Springer, New York
Lang S (1965) Algebra. Addison Wesley, Reading
Mukerjee R, Wu C (2006) A modern theory of factorial designs. Springer series in statistics. Springer. URL http://books.google.com/books?id=p47Mv7QjDAQC
Pistone G, Rogantin M (2008) Indicator function and complex coding for mixed fractional factorial designs. J Stat Plann Inference 138(3): 787–802
Pistone G, Wynn HP (1996) Generalised confounding with Gröbner bases. Biometrika 83(3): 653–666
Plackett R, Burman J (1946) The design of optimum multifactorial experiments. Biometrika 33: 305–325
Raktoe BL, Hedayat A, Federer WT (1981) Factorial designs. Wiley series in probability and mathematical statistics. Wiley, New York
SAS Institute Inc (2010) SAS/QC 9.2 user’s guide, 2nd edn. SAS Institute Inc., Cary, NC
Schoen ED, Eendebak PT, Nguyen MVM (2010) Complete enumeration of pure-level and mixed-level orthogonal arrays. J Comb Des 18(2):123–140. doi:10.1002/jcd.20236
Wu CFJ, Hamada M (2000) Experiments. Planning, analysis, and parameter design optimization. A Wiley-interscience publication. Wiley, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fontana, R. Algebraic generation of minimum size orthogonal fractional factorial designs: an approach based on integer linear programming. Comput Stat 28, 241–253 (2013). https://doi.org/10.1007/s00180-011-0296-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-011-0296-7