Heuristic optimization of Balanced Incomplete Block Designs under practical constraints

An approach to the construction of Balanced Incomplete Block Designs (BIBD) is described. The exact pairwise balance of treatments within blocks (second-order balancing condition) is required by standard BIBD. This requirement is attainable when $$\lambda $$ is an integer, namely when the relationship among the number of treatments t, blocks b, and block size k allows for an equal number of co-occurrences for every treatment pair. This work presents an algorithm for generating quasi-BIBD with particular attention to settings where a second blocking variable is taken into account (Youden squares) and the blocks are assigned to s groups (sessions) where all treatments are equally represented inside each one. Unlike classical resolvable or nested BIBDs, which require an exact combinatorial solution and therefore an integer value of $$\lambda $$ , the designs developed here explicitly allow $$\lambda $$ to be non-integer and do not rely on the existence of an exact BIBD. Two real-world applications are presented. The first refers to an exploratory experiment with 8 two-level factors leading to t=20 trials, assessed by k=4 evaluators through b=20 blocks divided into s=4 balanced sessions. The second example refers to a robust design experiment based on a Central Composite Design involving 4 technological and 2 environmental factors leading to t=30 trials, evaluated in k=5 environmental conditions within b=30 blocks divided into s=5 balanced sessions. In these experimental settings, the second-order balancing condition is not attainable since $$\lambda $$ is not an integer. The proposed algorithm has been designed to approximate this condition as closely as possible, maintaining the first-order balancing condition, highest D-optimality and connectedness.