$A$-Optimal Weighing Designs when $N \equiv 3 (\operatorname{mod} 4)$

Abstract

In this paper we consider the problem of $A$-optimal weighing designs for $n$ objects in $N$ weighings on a chemical balance when $N \equiv 3(\operatorname{mod} 4)$. Let $D(N, n)$ denote the class of $N \times n$ design matrices $X_d$ whose elements are $+1$ and $-1$. It is shown that if $X_d$ is such that $X'_dX_d$ is a block matrix having a specified block structure, then $X_d$ is $A$-optimal in $D(N, n)$. It is found that in some cases the $A$-optimal design in $D(N, n)$ is not unique. A larger class of chemical balance weighing designs is $D^0(N, n)$, where $X_d$ may have some elements equal to zero. It is observed that the designs which are $A$-optimal in $D(N, n)$ are not necessarily $A$-optimal in $D^0(N, n)$.