Combinatorial Patterns of D-Optimal Weighing Designs Using a Spring Balance

Abstract

Given a spring balance that reports the true total weight of items plus a white noise of an unknown variance, which n subsets of n items will you weigh in order to estimate the true weights of each item with the highest possible precision? For n ≤ 6, we classify all D-optimal weighing designs according to the combinatorial patterns they exhibit (modulo permutation), we count the D-optimal designs exhibiting each pattern, and we explain how a D-optimal design for n items may arise out of a D-optimal design for (n − 1) items. For n = 7, 11 we exhibit D-optimal designs obtained from balanced incomplete block designs (BIBDs). We discuss some strategies to construct D-optimal designs of larger sizes, and pose some unsolved problems.