In Search of the Most Balanced Sampling Design
Balanced sampling aims to select random samples in which the estimated totals of the auxiliary variables, weighted by the inverse of the inclusion probabilities, correspond as closely as possible to the known population totals. While several methods, such as rejective sampling, rerandomization, and the cube method, have been proposed to improve balance, identifying the most balanced sampling design under fixed inclusion probabilities remains a challenging combinatorial problem. This problem can be formulated as a linear program defined over the set of all possible samples, but the number of samples grows exponentially with population size, making exact optimization infeasible except for very small populations. To address this issue, we propose a heuristic approach based on a genetic algorithm that iteratively improves the balance of sampling designs by combining minimum support designs with highly balanced candidate samples. Although optimality cannot be guaranteed, the proposed method can substantially improve balance relative to standard procedures such as the cube method. The approach is applicable to both survey sampling and experimental design.