MBMUDs: a combinatorial extension of BIBDs showing good optimality behaviour

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 including 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;\lambda,\lambda+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.