On a problem of counting by weighing

Suppose it is desired to obtain a large number N s of items for which individual counting is impractical, but one can demand a batch to weigh at least w units so that the number of items N in the batch may be close to the desired number N s . If the items have mean weight ωTH, it is reasonable to have w equal to ωTH N s when ωTH is known. When ωTH is unknown, one can take a sample of size n , not bigger than N s , estimate ωTH by a good estimator ω n , and set w equal to ω n N s . Let R n = K p 2 N 2 s / n + K s n be a measure of loss, where K e and K s are the coefficients representing the cost of the error in estimation and the cost of the sampling respectively, and p is the coefficient of variation for the weight of the items. If one determines the sample size to be the integer closest to p CN s when p is known, where C is ( K e / K s ) 1/2 , then R n will be minimized. If p is unknown, a simple sequential procedure is proposed for which the average sample number is shown to be asymptotically equal to the optimal fixed sample size. When the weights are assumed to have a gamma distribution given ω and ω has a prior inverted gamma distribution, the optimal sample size can be found to be the nonnegative integer closest to p CN s + p 2 A (p C – 1), where A is a known constant given in the prior distribution.