MBMUDs: a combinatorial extension of BIBDs showing good optimality behaviour |
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| Institution: | 1. Departament d''Arquitectura de Computadors (UPC), Campus Nord- Mòdul-D6, c/ Gran Capità s/n, Barcelona 08071, Spain;2. Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Parc Tecnològic de Barcelona - Edifici U, c/ Llorens i Artigas 4-6, Barcelona 08028, Spain;1. Moulay Ismaïl University, Faculty of Science and Technology, Department of Mathematics, Algebra and Applications Group, P.O. Box 509, Boutalamine, Errachidia, Morocco;2. Ibn Zohr University, Faculty of Sciences of Agadir, Department of Mathematics, P.O. Box 8106, Agadir, Morocco;3. Moulay Ismaïl University, Faculty of Science and Technology, Computer Science Department, Errachidia, Morocco;1. Department of Mathematics and Computer Science, University of Northern British Columbia, Prince George, B.C. Canada V2N 4Z9;2. Department of Computer Science and Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0115, USA;1. Laboratory of Experimental and Computational Biology, National Cancer Institute, National Institute of Health, 12 South Drive, Bldg 12B, Rm B116, Bethesda, MD 20892-5677, USA;2. Analytical Biostatistics Section, Mathematical and Statistical Computing Laboratory, CIT, National Institute of Health, 12 South Drive, Bldg 12A, Rm 2039, Bethesda, MD 20892-5626, USA;1. Institute for Studies in Theoretical Physics and Mathematics, IPM, P.O. Box 19395-5746, Tehran, Iran;2. Department of Mathematics & Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alta, Canada T1K 3M4;3. Department of Mathematics, University of Tehran and Institute for Studies in Theoretical Physics and Mathematics, IPM, P.O. Box 19395, Tehran, Iran |
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| Abstract: | The construction of a balanced incomplete block design (BIBD) is formulated in terms of combinatorial optimization by defining a cost function that reaches its lower bound on all and only those configurations corresponding to a BIBD. This cost function is a linear combination of distribution measures for each of the properties of a block design (number of plots, uniformity of rows, uniformity of columns, and balance). The approach generalizes naturally to a super-class BIBDs, which we call maximally balanced maximally uniform designs (MBMUDs), that allow two consecutive values for their design parameters r,r+1;k,k+1;λ,λ+1]. In terms of combinatorial balance, MBMUDs are the closest possible approximation to BIBDs for all experimental settings where no set of admissible parameters exists. Thus, other design classes previously proposed with the same approximation aim—such as RDGs, SRDGs and NBIBDs of type I—can be viewed as particular cases of MBMUDs. Interestingly, experimental results show that the proposed combinatorial cost function has a monotonic relation with A- and D-statistical optimality in the space of designs with uniform rows and columns, while its computational cost is much lower. |
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