Some Optimum Weighing Designs

Suppose we are given N objects to be weighed in N weighings with a chemical balance having no bias. Let xᵢⱼ = 1 if the jth object is placed in the left pan in the ith weighing, $= -1$ if the jth object is placed in the right pan in the ith weighing, $= 0$ if the jth object is not weighed in the ith weighing. The Nth order matrix X = (xᵢⱼ) is known as the design matrix. Also let yᵢ be the result recorded in the ith weighing, εᵢ be the error in this result and wⱼ be the true weight of the jth object, so that we have the N equations xᵢ₁w₁ + xᵢ₂w₂ + ⋯ + xiNwN = yᵢ + εᵢ, i = 1, ⋯, N. We assume X to be a non-singular matrix. The method of Least Squares or theory of Linear Estimation gives the estimated weights ( wᵢ) by the equation w = (X'X)⁻¹X'Y, where Y is the column vector of the observations and w is the column vector of the estimated weights. If σ² is the variance of each weighing, then Var ( w) = (X'X)⁻¹σ² = (cᵢⱼ)σ², where (cᵢⱼ) is the inverse matrix of $(X'X)$. An expository article reviewing the work done in weighing designs is given by Banerjee [2]. Kishen [4] treats the reciprocal of the increase in variance resulting from the adoption of any design other than the most efficient design, with mean variance σ²/N, as the efficiency of the design. This efficiency can be measured by 1/∑Nᵢ₌₁ cᵢᵢ. Mood [5] gives an alternative definition for the best weighing design. In his view the best weighing design should give the smallest confidence region in the wᵢ(i = 1, ⋯, N) space for the estimates of the weights. Hence a design will be called best if the determinant of the matrix (cᵢⱼ) is minimised. In this paper we follow Kishen's definition in obtaining the best weighing designs. Hotelling [3] proved that for the best weighing design cᵢᵢ = 1/N and cᵢⱼ = 0 (i ≠ j). The weighing designs for which cᵢᵢ = 1/N and cᵢⱼ = 0 are best in the sense of both Kishen and Mood. Later Mood proved that the above property is satisfied by Hadamard matrices. Plackett and Burman [6] have constructed Hadamard matrices, HN, up to and including $N = 100$, excepting $N = 92$. It may be remarked here that a necessary condition for the existence of HN is N ≡ 0 ( 4), with the exception of $N = 2$. It is not known whether this condition is sufficient or not. In this paper, the best weighing designs are obtained in the cases (i) N is odd and (ii) N ≡ 2 ( 4) subject to the conditions: i) The variances of the estimated weights are equal; and ii) The estimated weights are equally correlated. The 2nd condition here is the same as that of Banerjee [1].