A-optimal chemical balance weighing design with nonhomogeneity of variances of errors

The problem studied in the paper concerns the estimation of individual weights of p   objects according to the model of an A-optimal chemical balance weighing design with a positive definite diagonal variance matrix of errors under the restriction p1+p2=q⩽pp1+p2=q⩽p, where p1p1 and p2p2 represent the numbers of objects placed on the left and on the right pan, respectively, in each of the measurement operations. The lower bound of tr(X′G-1X)-1tr(X′G-1X)-1 is obtained and the necessary and sufficient condition for this lower bound to be attained is given. To construct the A-optimal chemical balance weighing design the incidence matrices of the balanced bipartite weighing designs are used.