Abstract The optimum allocation for stratified sampling is studied under the assumption that prior information concerning the unknown stratum means is available. The prior information is assumed to be expressed in the form of a multivariate normal prior distribution for the unknown stratum means. There are ri variates of interest in the ith stratum, so that each stratum mean is a vector. The cost of sampling is the sum of the costs of sampling the various strata, and the cost of sampling the ith stratum is taken to be a constant ci times the size of the sample taken from that stratum. Not all strata need be sampled. Several allocation problems are formulated and shown to be solvable by the standard algorithms of nonlinear programming.