$D$-Optimum Weighing Designs

Abstract

For the problem of weighing $k$ objects in $n$ weighings $(n \geq k)$ on a chemical balance, and certain related problems, we obtain new results and list the designs which have been proved $D$-optimum up to this time. While some of these optimality results have been known for some time, others are fairly recent. In particular, in the most difficult case $n \equiv 3(\operatorname{mod} 4)$ we prove a result characterizing optimum designs when $n \geq 2k - 5$. In addition, by a combination of theoretical bounds and computer search we find previously unknown optimum designs in the cases $(k, n) = (9, 11), (11, 15)$, and (12, 15), and establish the optimality of Mitchell's (10, 11) design. In some cases the optimum $X'X$ is not unique. Thus, we find two optimum $X'X$'s for the (6, 7), (8, 11), (10, 11), and (10, 15) cases. As a consequence of these results and other constructions, $D$-optimum designs are now known in all cases $k \leq 12$ (for all $n \geq k$), and in many other cases. Essentially complete listings for all $n \geq k$ had been given previously only for $k \leq 5$.