Increasing the precision and accuracy of top-loading balances:  application of experimental design

The traditional method of estimating the weight of multiple objects is to obtain the weight of each object individually. We demonstrate that the precision and accuracy of these estimates can be improved by using a weighing scheme in which multiple objects are simultaneously on the balance. The resulting system of linear equations is solved to yield the weight estimates for the objects. Precision and accuracy improvements can be made by using a weighing scheme without requiring any more weighings than the number of objects when a total of at least six objects are to be weighed. It is also necessary that multiple objects can be weighed with about the same precision as that obtained with a single object, and the scale bias remains relatively constant over the set of weighings. Simulated and empirical examples are given for a system of eight objects in which up to five objects can be weighed simultaneously. A modified Plackett-Burman weighing scheme yields a 25% improvement in precision over the traditional method and implicitly removes the scale bias from seven of the eight objects. Applications of this novel use of experimental design techniques are shown to have potential commercial importance for quality control methods that rely on the mass change rate of an object.