Integer programming formulations applied to optimal allocation in stratified sampling

The problem of optimal allocation of samples in surveys using a stratified sampling plan was first discussed by Neyman in 1934. Since then, many researchers have studied the problem of the sample allocation in multivariate surveys and several methods have been proposed. Basically, these methods are divided into two classes: The first class comprises methods that seek an allocation which minimizes survey costs while keeping the coefficients of variation of estimators of totals below specified thresholds for all survey variables of interest. The second aims to minimize a weighted average of the relative variances of the estimators of totals given a maximum overall sample size or a maximum cost. This paper proposes a new optimization approach for the sample allocation problem in multivariate surveys. This approach is based on a binary integer programming formulation. Several numerical experiments showed that the proposed approach provides efficient solutions to this problem, which improve upon a `textbook algorithm' and can be more efficient than the algorithm by Bethel (1985, 1989).